Microscope generating a three-dimensional representation of an object and images generated by such a microscope

[0002] This invention relates to a microscope generating a three-dimensional representation of the observed object, operating on a principle derived from “image formation by scattered field inversion”, tomography and synthetic aperture systems.

[0003] 2.1. References

[0004] [Wolf]: Three-Dimensional Structure Determination of Semi-Transparent Object from Holographic Data,Emil Wolf, Optics Communications, Vol. 1, No. 4, p.153, October 1969.

[0005] [Dändliker]: Reconstruction of the Three-Dimensional-Refractive Index from Scattered Waves,R. Dändliker, K. Weiss, Optics Communications, Vol. 1, No. p.323, February 1970.

[0006] [Fercher]: Image Formation by Inversion of Scattered Field Data: Experiments and Computational Simulation.A. F. Fercher, H. Bartelt, H. Becker, E. Wiltschko, Applied Optics, Vol.18, No 14, p.2427, July 1979

[0007] [Kawata]: Optical Microscope Tomography. I. Support Constraint,S. Kawata, O. Nakamnura & S. Minami, Journal of the Optical Society of America A, Vol. 4, No.1, p.292, January 1987

[0008] [Noda]: Three-Dimensional Phase-Contrast Imaging by a Computed-Tomography Microscope,Tomoya Noda, Satoshi Kawata & Shigeo Minami, Applied Optics, Vol. 31, No. 5, p.670, Feb. 10, 1992

[0009] [Devaney]: The Coherent Optical Tomographic Microscope,A. J. Devaney and A. Schatzberg, SPIE, Vol.1767, p.62, 1992

[0010] [Wedberg]: Experimental Simulation of the Quantitative Imaging Properties of Optical Diffraction Tomography,Torolf A. Wedberg and Jacob J. Stamnes, Journal of the Optical Society of America A, Vol. 12, No 3, p.493, March 1995.

[0011] [Vishnyakov]: Interferometric Computed-Microtomography of3D Phase Objects,Gennady N. Vishnyakov & Gennady G. Levin, SPIE Proceedings, Vol.2984, p.64, 1997

[0012] [Ausherman]: Developments in Radar Imaging,D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, E. C. Poggio, IEEE Transactions on Aerospace and Electronic Systems, Vol. 20, No. 4, p.363, July 1984.

[0013] [Goodman]: Synthetic Aperture Optics,Progress in Optics, Vol. VIII, 1970, North Holland Publishing Company.

[0014] [Walker]: Range-Doppler Imaging of Rotating Objects,Jack L. Walker, IEEE transactions on Aerospace and Electronic Systems, Vol. 16, No. 1, p.23, Januasy 1980.

[0015] [Brown]: Walker model for Radar Sensing of Rigid Target Fields,William M. Brown, IEEE Transactions on Aerospace and Electronic Systems, Vol. 16, No 1, p.104, January 1980.

[0016] [Turpin 1]: U.S. Pat. No. 5,384,573

[0017] [Turpin 2]: Theory of the Synthetic Aperture Microscope,Terry Turpin, Leslie Gesell, Jeffrey Lapides, Craig Price, SPIE Proceedings, Vol. 2566, p.230, 1995

[0018] [Turpin 3]: The Synthetic Aperture Microscope, Experimental results,P. Woodford, T. Turpin, M. Rubin, J. Lapides, C. Price, SPIE Proceedings, Vol. 2751 p.230, 1996

[0019] [Lauer 1]: Patent WO 98/13715

[0020] 2.2. Description of the Prior Art

[0021] A three-dimensional object may be characterized optically by a certain number of local parameters, for example its index and its absorptivity at each point. Mathematically, this may be expressed by the data at each point of a complex number which is a function of the local parameters at the considered point. A three-dimensional spatial representation of the object can then be expressed in the form of a three-dimensional array of complex numbers.

[0022] By carrying out the three-dimensional Fourier transform of this three-dimensional spatial representation, a three-dimensional frequency representation of the object is obtained.

[0023] [Wolf] showed that a three-dimensional representation of a weakly diffracting object can be obtained from the acquisition of the wave diffracted by this object when it is illuminated successively by a series of plane waves of variable direction. [Wolf] also determined the maximum resolution thus obtainable, expressed as a function of the illuminating wavelength. This resolution corresponds to a maximum period of λ/2 for the sinusoidal components the object's representation, i.e. a sampling period in the Nyquist sense of λ/4 which is a resolution twice as fine as that of conventional microscopes. [Dändliker] improved [Wolf]'s formalism and provided a geometrical interpretation of it. From the wave diffracted by the object under a given illumination, part of the three-dimensional frequency representation of the object is obtained. This part is a sphere in a three-dimensional frequency space. By combining the spheres thus obtained for various illuminating waves, the frequency space can be filled, obtaining the three-dimensional frequency representation of the object. The latter can then be inverse transformed to obtain a spatial representation.

[0024] [Fercher] designed a microscope constituting the first practical application of the principles defined by [Wolf] and [Dandliker]. In that microscope, the wave diffracted by the object is picked up on a receiving surface on which it interferes with a reference wave not having passed through the object and the phase of which can be modified. From several interference figures differing from each other by the phase of the reference wave, [Fercher] obtains, at each point of the receiving surface, the amplitude and the phase of the wave diffracted by the object.

[0025] [Fercher] does not use several successive illumination waves but several illuminating waves generated simultaneously by means of a diffraction grating, thus limiting the number of possible illumination directions, even though the use of several successive illuminating waves does not present any particular technical difficulty. The reason for this choice is not clearly explained. However, it appears that this technique is adopted in order to obtain illuminating waves all having the same phase at a given point of the image. In fact, the equation (1) of the document [Wolf] assumes that each illuminating wave has a zero phase at the point of origin of the position vectors.

[0026] The method defined by [Wolf], [Dändliker] and [Fercher] is generally called “image formation by scattered field inversion”. Another conventional approach for obtaining three-dimensional images is tomography. Tomography, used for example in x-ray techniques, consists in reconstructing an image from a set of projections of this image along different directions. Each projection depends linearly upon a three-dimensional density function characterizing the object. From a sufficient number of projections it is possible to reconstitute the object by reversing this linear correspondence.

[0027] Tomography was adapted to optical microscopy by [Kawata]. In his tomographic microscope, a plane and non-coherent illuminating wave of variable direction is used. This illuminating wave passes through a sample and then a microscope objective focussed in the plane of the sample. It is received on a receiving surface placed in the plane in which the objective forms the image of the sample. Because the illumination is non-coherent, the intensities coming from each point of the object are added and the image intensity produced on the receiving surface consequently depends linearly on the three-dimensional density function characterizing the absorptivity of the object. From a sufficient number of images it is possible to reconstitute the image by reversing this linear correspondence. This microscope differs from usual tomographic systems in that the linear correspondence between the density function of the object and a given image is not a projection, but is characterized by a three-dimensional optical transfer function.

[0028] This microscope is not very suitable for obtaining images that take into account the index of the sample. [Noda] designed a modified microscope enabling this phase to be taken into account. The initial idea of that microscope [Noda] is to use phase contrast for obtaining an image which depends on the index of the sample, and to adapt to this configuration the linear correspondence reversal principle already implemented by [Kawata]. The use of the [Noda] microscope is however limited to the study of non-absorbing objects whose index variations are extremely small.

[0029] [Noda]'s text does not refer to holography nor to “image formation by scattered field inversion,” but the operating principle involved can be interpreted within this framework. In fact, the technique adopted by [Noda] is tantamount to using on the receiving surface a reference wave consisting of the illuminating wave alone. From the images received for a set of illuminating waves of variable direction, a three-dimensional frequency representation is obtained. The complex wave detected on the receiving surface is replaced here by a pure imaginary value obtained by multiplying by j the real value obtained using the single reference wave constituted by the illuminating wave shifted π/2 in phase. If the reference wave is sufficiently higher, at each point of the receiving surface, than the diffracted wave, then the quantity thus obtained is the imaginary part of the complex wave really received on the receiving surface, the phase reference being the phase of the illuminating wave. The object generating a pure imaginary wave equivalent to that detected by [Noda] on the receiving surface is made up of the superposition of the observed real object and a virtual object whose complex spatial representation is obtained from that of the real object by symmetry in relation to the plane of the object corresponding to the receiving surface, and by reversing the sign of the real part. Using the imaginary part thus detected in a manner similar to that used by [Fercher] for the detected complex wave, a function representing the superposition of the real object and the virtual object is generated in frequency representation. During each acquisition, the two-dimensional frequency representation obtained by performing the Fourier transform of the value detected on the receiving surface includes a part corresponding to the real object and a part corresponding to the virtual object, which coincide only at the point corresponding to the illumination frequency. It is thus possible to select only the part corresponding to the real object, so as to obtain a representation thereof. [Noda] in fact uses the superposition of the real object with the virtual object which he symmetrizes in relation to the plane of the object corresponding to the receiving surface, thus obtaining a pure imaginary representation corresponding to the imaginary part of the representation that would be obtained using [Wolf]'s method.

[0030] The theoretical explanations given in the document [Noda] are very different from those presented here and are perfectly valid. The principle consisting in reversing a filter by multiplication in the frequency domain, as applied by [Noda], is found to be equivalent to the explanations given above, although obtained by different reasoning. It may be considered that FIGS. 2 and 3 of the document [Noda] illustrate how the three-dimensional frequency representation of the object is generated from two-dimensional frequency representations.

[0031] [Devaney] proposed a tomographic microscope whose operating mode is derived essentially from the method of [Wolf]. In the [Devaney] microscope the reference wave coincides with the illuminating wave. Consequently, that microscope does not have means for varying the phase of the reference wave. As in the case of [Noda], the detected wave consequently corresponds to that which would be formed by the superposition of a real object and a virtual object. [Devaney] solves the problem by placing the receiving surface outside of the object, so that the real object and the virtual object do not overlap. When the direction of the illuminating wave varies, only one of the two objects is reconstituted. Two variants of the microscope are presented: a first in which the object is fixed and the direction of the illuminating wave is variable, and a second in which the object rotates around a fixed point, the illuminating wave then having a fixed direction in relation to the receiver. The first version of the [Devaney] microscope was built by [Wedberg].

[0032] Another approach allowing the adaptation of tomography to the production of phase images is that of [Vishnyakov]. [Vishnyakov] introduces a reference wave distinct from the illuminating wave and carries out a detection of the wave received on a receiving surface in accordance with a method similar to the one used by [Fercher]. He then generates a profile characteristic of the phase difference between the received wave and the illuminating wave. As this phase difference is considered to be the projection of the index along the direction of the illuminating wave, it regenerates the distribution of the index in the object according to the tomographic method used conventionally in x-ray techniques. This method may be compared to a method of the [Wolf] type, but in which the sphere portion acquired in the frequency space would be assimilated with a plane portion, which is largely unjustified in the case of an objective with a large aperture such as the one used here.

[0033] The technique of the synthetic aperture radar is an imagery method used in the field of radar waves and which was considered very early for application to the field of optics. [Ausherman] presents the history of this technique. The application of the synthetic aperture radar technique to the field of optical wavelengths would make it possible in principle to obtain images of an observed object. However, in order for the technique to be feasible, it is necessary, at all times, to have the position values, within a reference coordinate system linked to the object, of each element of the transmitter-receiver assembly. These values must be known to within a fraction of a wavelength. This is achievable in the area of radar frequencies, in which wavelengths are macroscopic and can be, for example, of a few tens of centimeters. In the optics field, where wavelengths are sub-micrometric, this is difficult to achieve. This problem is the basic reason for which the system is difficult to adapt to optics, as indicated in [Goodman], Pages 36 to 39.

[0034] [Walker] and [Brown] formalized the synthetic aperture radar method in a form similar to that already obtained by [Wolf] for optical systems. This formalism was originally used by [Walker] with a radar imagery method in which the transmitter-receiver assembly is fixed and in which the object rotates around a fixed point. This overcomes the problem of determining the position of the object at each instant.

[0035] [Turpin] recently described several microscopes constituting an adaptation of synthetic aperture radar principles to the field of optics.

[0036] In the microscope implemented by [Turpin 3], the material configuration used complies with the principle used in [Walker] to overcome the problem consisting in determining the position of the object at each instant, i.e. the transmitter and receiver are fixed and the object rotates around a fixed axis. This microscope is also similar to the second version of the [Devaney] microscope. Since the rotation axis, the transmitter and the receiver are fixed, the position of the transmitter-receiver assembly within a reference coordinate system linked to the object can be known with the required accuracy.

[0037] However, effective resolution calls not only for a mechanical system making it possible to determine the movement of the object, but also the taking of this movement into account in the definition of algorithms and/or appropriate adjustment of the system. In the absence of special precautions, the point of origin of the reference wave moves in relation to the object over a circle centered on the rotation axis of the object. If this movement is significant, this effect destroys the image. If this movement is small, resolution in the plane of this circle is affected in proportion to the amplitude of the movement.

[0038] To solve this problem in the absence of any specific compensation algorithm, the point of origin of the reference wave should be on the rotation axis of the object. This condition is in principle difficult to obtain. [Turpin] does not mention this problem and does not specify any means of appropriate adjustment.

[0039] This problem can however be solved when use is made, for example, of a flat specimen, by performing an adjustment intended to comply with the following conditions:

[0040] (i)—The image of the reference wave on the “image CCD array” of FIG. 1 in document [Turpin 3] must be a point image.

[0041] (ii)—When the object makes a rotation of 180 degrees, the image obtained must be symmetrized in relation to an axis passing through the point image of the reference wave.

[0042] The position of the “image CCD array” must be adjusted to comply with (i).

[0043] The position of the entire receiver must be adjusted to comply with (ii).

[0044] This solution is however not perfect, depending essentially on a visual interpretation. It can be used reasonably only for very simple objects.

[0045] The microscope described in [Turpin 3] is a particular case of the generalized system described in [Turpin 1] and [Turpin 2]. The generalized system specifies that the illuminating wave and/or the position of the receiver can vary. However, the material configurations proposed do not allow the solution of the problem consisting in determining, to within a fraction of a wavelength, the position of the transmitter and the receiver in relation to the object. In fact, the illuminating wave of variable direction is produced by mechanical devices which cannot be controlled with sub-micrometric accuracy.

[0046] The microscope described in [Lauer 1] allows the generation of the frequency representation of a wave coming from the object and the reconstitution of the object from several of these representations. The method used in [Lauer 1] is not directly related to the one described by [Wolf]. In fact, in the case in which it recombines several frequency representations to obtain the representation of the object:

[0047] it uses spatially incoherent lighting and not plane illuminating waves

[0048] it combines the frequency representations of the waves received by intensity summing in the spatial domain.

[0049] The image obtained by [Lauer 1] is affected by a residual granularity effect, does not allow a differentiation of the refractive index and of absorptivity, and does not yield the theoretical accuracy indicated by [Wolf].

[0050] 3.1. Problem to Be Solved by the Invention

[0051] Systems based on “image formation by scattered field inversion,” “tomographic” systems or “synthetic aperture” systems appear to be equivalent to each other, at least when examined from only the coherent optical domain. Two classes of systems may be distinguished depending on the reference wave generation mode:

[0052] the microscopes of [Noda] and [Devaney] use a reference wave coinciding with the illuminating wave.

[0053] the microscopes of [Fercher], [Vishnyakov] and [Turpin] use a reference wave distinct from the illuminating wave.

[0054] Microscopes of the first category exhibit limitations in terms of observed image size, characteristics imposed on the object, or displayable parameters. These limitations are due to the fact that, in the absence of a reference wave distinct from the illuminating wave, it is not possible to acquire under good conditions the complex value of the wave received on the receiving surface. It is thus necessary to use various expedients to eliminate spurious images and various disturbances generated by the drawbacks of the acquisition method.

[0055] Microscopes of the second group allow this problem to be solved. In principle, microscopes of the second group should allow the representation of the refractive index and the absorptivity of the object in three dimensions with quarter-wavelength accuracy, according to the theory devised by [Wolf] and [Dändliker]. Such performance is clearly higher than that of all existing optical microscopes including the confocal microscope, and these microscopes should in all logic have led to industrial applications. However, none of these microscopes has as yet made it possible to obtain quality images comparable to those produced, for example, by the [Noda] microscope, and thus these microscopes have not gone beyond the experimental stage. The reason for which these microscopes have never yielded high quality images, despite their theoretical possibilities, has never been clearly determined.

[0056] A first approach to the problem is contained implicitly in the equation (1) of the document [Wolf]: all the illuminating waves must have the same phase at the origin of the representation. However, in an optical system, the only values accessible to measurement are the phase differences between a reference wave and a wave to be analyzed. The fact that at the illuminating waves all have the same phase at a given point of the object, as indicated implicitly by the equation (1) in the document [Wolf], is thus not sufficient to ensure the proper functioning of the system: it is also necessary for the reference wave to meet appropriate conditions, so that the phase differences accessible to measurement lead to correct results.

[0057] A second approach to the problem is provided by [Goodman], Pages 36 to 39, in the terms of the numerical aperture radars: the position of the transmitter and receiver must be determined with an accuracy within the wavelength, and this is not feasible in the optical domain.

[0058] If the illuminating wave has a variable direction, these two approaches are similar: in fact, an indetermination on the position of the transmitter results, among other effects, in a phase shift of the illuminating wave at the origin of the representation. We shall confine ourselves here to these systems, i.e. to the microscopes of [ercher] and [Vishryakov], and to the versions of the microscope of [Turpin] which include a variable direction illuminating wave.

[0059] For example, in the microscopes of [Turpin], the illuminating wave is generated by a mechanical device. This device does not allow the illuminating wave phase to be controlled. When two successive three-dimensional images are taken, a given illuminating wave characterized by its direction will not have the same phase difference with the reference wave in each case. The wave detected on the receiving surface will consequently not have the same phase either, and hence finally that two successive three-dimensional images taken will not lead to the same result, even in the absence of any noise. This example highlights the basic problem which has up to now limited the performance of microscopes of the second group: lack of control of the phase difference between the illuminating wave and the reference wave leads to non-reproducibility of the results obtained and, in general, results which do not correspond to what is expected considering the theoretical approach of [Wolf].

[0060] 3.2. Solution of Problem According to the Invention

[0061] The entire three-dimensional frequency representation of an object can be multiplied by a complex number Ae^jα. It will then be said that the frequency representation of the object is affected by a phase shift α and by a multiplicative factor A. If an object is characterized by its three-dimensional frequency representation, its spatial representation can be obtained by taking the inverse Fourier transform of this frequency representation. If the frequency representation of the object is affected by a phase shift α and a multiplicative factor A, its spatial representation is affected by the same phase shift and the same multiplicative factor, corresponding to a modification of the function giving the complex number associated with a point as a function of the local parameters at this point.

[0062] It is also possible to multiply each point of the three-dimensional representation by a complex number Ae^jα depending on the point. Two different points of the three-dimensional frequency representation can then be affected by a phase shift and a multiplicative factor which are different. By performing the inverse Fourier transform of a frequency representation in which the phase shift and/or the multiplicative factor depend on the considered point, a modified representation of the object is obtained in which the complex number associated with a point depends not only on the local parameters at the considered point, but also on the local parameters at a set of other points. This modified representation of the object is a filtered spatial representation, the filter having a frequency representation consisting of the numbers Ae^jα defined at each point. Depending on the characteristics of this filter, more or less correct frequency representations will be obtained.

[0063] In the microscopes of [Turpin], the illuminating wave is generated by a mechanical device. With each change of direction of the illuminating wave, a random phase shift of this wave occurs, and hence a random phase shift in the corresponding part of the three-dimensional frequency representation.

[0064] In the document of [Vishnyakov], and for the same reasons, the phase of the illuminating wave varies randomly with each change of illumination direction. Equation (2) on Page 67 of the document [Vishnyakov] should be replaced by ψ(x,y)=Φ(x,y)+x sin α+φ, where φ is the phase of the illuminating wave at the origin of the three-dimensional representation obtained. When the direction of the illuminating wave varies, the value of φ varies. Non-determination of the correct value of φ results in the addition of a constant to each projection obtained, this constant varying randomly between two projections. This consequently makes inexact the assimilation of the phase profile obtained with the projection of the index. The method of [Vishnyakov] is roughly equivalent to the method of [Wolf] in which a portion of the sphere of the frequency space, obtained from a given illuminating wave, would have been assimilated to a plane. The non-determination of φ is equivalent to a random phase shift of the entire part of the two-dimensional frequency representation generated from a given illuminating wave. However, other sources of error are added to this effect, and in particular the fact that a conventional tomographic reconstruction is used.

[0065] In the document of [Fercher], owing to the use of a diffraction grating to generate the three illuminating waves simultaneously, there is no random phase shift in the illuminating waves. However, a detailed analysis of the system shows that, to obtain the same phase shift in the parts of the frequency representation obtained from each illuminating wave, the virtual image, in the object, of the point of focus of the reference wave, must coincide with a point in which the illuminating waves all have the same phase. This calls for very precise adjustment of the position of the reference wave origin. As the document of [Fercher] does not contain any mention of such an adjustment, it is likely that it has not been carried out. In any case, the solution adopted by [Fercher] limits significantly the number of illuminating waves that may be used and the aperture under which the wave coming from the object can be acquired.

[0066] In existing microscopes using variable direction illuminating waves and a reference wave distinct from the illuminating wave, the parts of the frequency representation obtained from different illuminating waves are thus affected by different phase shifts. Consequently, the inversion of the frequency representation obtained generates a filtered representation which is in general of rather poor quality and which is moreover not reproducible owing to the random nature of the phase shift.

[0067] The invention consists in providing a microscope in which the direction of the illuminating wave is variable, but comprising means for generating a three-dimensional representation of the object in which the distribution of the phase shift affecting each point obtained, in frequency representation, is concentrated around a constant value. Ideally, this phase shift should be constant, but the existence of disturbances such as Gaussian noise, a residual spherical aberration, or a small imprecision in controlling the phase difference between the reference wave and the illuminating wave, creates a certain spread of the distribution around the constant value.

[0068] In existing microscopes using variable direction illuminating waves and a reference wave distinct from the illuminating wave, and in the case in which a large number of distinct illuminating waves are used, such as in [Turpin], this phase shift tends to be of a random nature and its distribution is hence roughly homogeneous over the interval [0,2π] . Where a limited number of illuminating waves are used, as in [Fercher], the phase shift distribution exhibits peaks of comparable level centered on several distinct values corresponding to the phase shifts affecting the sub-representations obtained from each illuminating wave.

[0069] The fact that the phase shift affecting each point, in frequency representation, is roughly constant, constitutes a new functionality of the microscope which allows, for example, a better quality spatial representation. However, the generation of a three-dimensional representation of the observed object does not necessarily constitute the final purpose sought by the microscope user. For example, the microscope may be used to read three-dimensional optical memories. In this case, the data can be encoded before being stored in the optical memory. The microscope then allows a three-dimensional representation of this optical memory to be obtained from which the data can be decoded. The three-dimensional representation of the object is then a calculation intermediary finally enabling the decoded data to be obtained.

[0070] 3.3. Vocabulary Used and General Considerations

[0071] A three-dimensional object can be characterized optically by a certain number of local parameters. These parameters can, for example, be its refractive index and its absorptivity at each point, or its absorptivity and one of its indices in the case of a non-isotropic material. It is possible to define at each point a complex number which is a function of certain of these local parameters, this function generally being defined uniquely in the entire spatial representation and hence not depending on the considered point. It will then be possible to express a three-dimensional spatial representation of the object in the form of a three-dimensional array of complex numbers. The dependency between the complex number and the local parameters can be defined in various ways. For example, this complex number may reduce to a real number characterizing the index, as in the microscope of [Noda]. The definition that will be used most often will however be of the type given by [Wolf], but with a “complex index” representing both the refractive index and absorptivity. For example, the real part of the complex number may be proportional to absorptivity and its imaginary part to the refractive index. A complex number may be used which is obtained by rotating the preceding in the complex plane, corresponding to a phase shift. In every case, the three-dimensional spatial representation of the object is unique if the correspondence between the complex number and the local parameters has been defined and the central part of the representation has also been defined.

[0072] Performing the three-dimensional Fourier transform of this three-dimensional spatial representation, a three-dimensional frequency representation of the object is obtained.

[0073] The phase shift affecting a point of the three-dimensional frequency representation of the object is defined when the representation of the object has been defined uniquely, i.e. when the complex function of the local parameters and the point of origin characterizing the spatial representation have been defined.

[0074] When these parameters are not specified, it may be considered that the phase shift is defined in relation to the spatial representation which coincides at best with the obtained three-dimensional representation of the object.

[0075] The term “three-dimensional representation of an object” will designate all the numerical data characterizing the spatial or frequency representation of the object, independent of the manner in which these data are combined or stored, for example in a computer memory. This representation may be expressed, for example:

[0076] in the spatial domain, in the form of a complex number dependent on spatial coordinates

[0077] in the frequency domain, in the form of a complex number dependent on the spatial frequency

[0078] in any other manner when the “three-dimensional representation of the object” allows the spatial or frequency representation of the object to be obtained by a known algorithm.

[0079] Part of the three-dimensional frequency representation of the object will be called frequency sub-representation of the object, and the term sub-representation will designate all the corresponding data, independent of how they are combined or stored.

[0080] A wave reaching a receiving surface is entirely characterized by its amplitude and its phase along each polarization direction and at every point of the receiving surface. A scalar representation of the wave may also be adopted by limiting oneself, for example, to a single polarization direction, the wave then being characterized by a single phase and a single intensity at every point of the receiving surface. From the wave measured on a receiving surface, a frequency sub-representation of the observed object can be generated. This sub-representation is two-dimensional in that it constitutes a sphere portion in the space of the frequencies, as indicated by [Dändliker]. A calculation intermediary may be constituted by a frequency representation of the wave, defined by the wave phase and intensity on each wave vector, a scalar representation being adopted.

[0081] The frequency representation of the wave is two-dimensional and can be projected on a plane without any loss of information. Such a projection yields a plane image that will be called “frequency plane image”. In a system such as that of [Turpin] or that of [Fercher], such a frequency plane image is obtained directly on the receiving surface. In other systems, such as the second embodiment of the present invention, such a frequency plane image is obtained by two-dimensional Fourier transformation of the scalar representation obtained directly on the receiving surface. A modified frequency plane image can also be obtained from several frequency plane images differing from each other in the polarization of the illuminating wave and the direction of analysis of the wave received on the receiving surface. The frequency plane image can constitute a calculation intermediary making it possible to obtain the frequency representation of the wave and then a corresponding sub-representation of the object.

[0082] The term “two-dimensional frequency representation” will designate either a two-dimensional frequency representation or a two-dimensional part of a three-dimensional frequency representation. In particular, it can designate either a frequency plane image, a frequency representation of a wave, or a two-dimensional sub-representation of the object.

[0083] The term “lens” will designate, throughout the text, either simple lenses, compound lenses, or achromats, generally designed to limit optical aberration.

[0084] In the rest of the text, five embodiments are described, referred to as Embodiments 1, 2, 3, 4 and 5.

[0085] 3.4. Obtaining the Three-Dimensional Frequency Representation Directly

[0086] For each direction of the illuminating wave, a frequency sub-representation of the object is obtained by direct application of the methods defined in [Fercher] and [Turpin]. In the systems of [Fercher] and [Turpin], the microscope is built so that the different sub-representations obtained are affected by different phase shifts. According to one embodiment of the invention, the microscope is built so that these phase shifts are constant. This variant of the invention implies:

[0087] (i)—that the microscope is built so that the phase difference between an illuminating wave and the reference wave with which it interferes is reproducible. This condition implies an appropriate construction of the microscope. This is shown, in the absence of vibrations, in Embodiments 3, 4 and 5. Embodiments 1 and 2 are affected by the same problems as the microscope of [Turpin]: the phase difference between reference wave and illuminating wave varies randomly owing to the overall mechanical design. Embodiments 1 and 2 consequently do not allow compliance with this first condition. Embodiments 3 to 5 enable this condition to be complied with because of the different design of the illuminating and reference wave generating system.

[0088] (ii)—that the microscope is built so that, through appropriate adjustment, the phase difference between the illuminating wave and the reference wave can be made constant. Embodiment 3 does not allow this condition to be met because there is no particular point at which all the illuminating waves have the same phase, which would be necessary in order to meet this condition since the reference wave used is spherical and constant. Embodiment 4, quite similar moreover to Embodiment 3, allows this condition to be met because, with an appropriate control of the beam deflection system, it is possible to generate illuminating waves whose phase at a given point is constant. Embodiment 5 also meets this condition.

[0089] (iii)—that the position of the optical elements is adjusted appropriately so that the phase difference between the illuminating wave and the reference wave is in fact constant. This adjustment is described in 8.6. for Embodiment 4 and in 9.20. for Embodiment 5.

[0090] Conditions (ii) and (iii) imply that the phase difference between the illuminating wave and the reference wave is constant. It is possible to define a virtual wave present in the object and such that its image, through the optical device modifying the wave coming from the object between the object and the receiving surface, is the reference wave. The phase difference between the illuminating wave and the reference wave means here the phase difference between the illuminating wave and the component of this virtual wave on the wave vector of the illuminating wave.

[0091] 3.5. Phase Correction Method

[0092] When the microscope is not built so that the phase shift affecting each frequency sub-representation is constant, the differences between the phase shifts affecting each sub-representation must be determined and compensated if necessary.

[0093] 3.5.1. General Phase Correction Method

[0094] Part of the three-dimensional representation of the object is considered:

[0095] consisting of a subset A of the three-dimensional representation.

[0096] characterized by a function a(f) defined on A and conventionally zero outside of A, where f is the spatial frequency vector and a(f) the value of the representation on this spatial frequency.

[0097] affected by a Gaussian noise, the standard deviation of the noise on a given frequency f being σ_a(f).

[0098] This sub-representation will be designated by the expression “sub-representation RA”. It will be said that A is the support of RA.

[0099] A second part of the three-dimensional representation of the object is considered, consisting of a subset B of the three-dimensional representation, characterized by a function b(f) defined on B, affected by a Gaussian noise σ_b(f), designated by the expression “sub-representation RB.”

[0100] These two parts of the representation are phase shifted in relation to each other and are assumed to have a non-zero intersection. From these two parts will be generated a sub-representation RC defined on a set C=A∪B (C is the union of A and B) and defined by a function c(f) affected by a Gaussian noise σ_c(f).

[0101] It is possible to proceed in two steps:

[0102] a complex ratio between the two representations can be obtained, for example, with the formula: $r=\frac{\sum _{f\in E}\frac{a\left(f\right)\stackrel{\_}{b\left(f\right)}}{{\sigma }_a^2\left(f\right)+{\sigma }_b^2\left(f\right)}}{\sum _{f\in E}\frac{{\left|b\left(f\right)\right|}^2}{{\sigma }_a^2\left(f\right)+{\sigma }_b^2\left(f\right)}}$

[0103] where the sums are over a set E of frequency vectors included in the intersection of the two sets A and B, or E⊂(A∩B) , the set E being if necessary limited to points for which the signal-to-noise ratio is sufficiently high. The phase difference between the two representations is the argument of r. The phase difference thus calculated is a good approximation of the most probable phase difference knowing the values of the representations RA and RB over the set E.

[0104] the representation RB may be “phase-corrected” in relation to RA by multiplying it by the ratio r:

[0105] b(f )←r.b(f) where the sign←means that b(f) is replaced by r.b(f).

[0106] the function c can be obtained, for example, by the formula: $c\left(f\right)=\frac{\frac{a\left(f\right)}{{\sigma }_a^2\left(f\right)}+\frac{b\left(f\right)}{{\sigma }_b^2\left(f\right)}}{\frac{1}{{\sigma }_a^2\left(f\right)}+\frac{1}{{\sigma }_b^2\left(f\right)}}$

[0107] The values thus assigned to the representation RC are the most probable values, knowing the phase-corrected representations RA and RB.

[0108] the function σ_c(f) can be obtained, for example, by the formula: $\frac{1}{{\sigma }_c^2\left(f\right)}=\frac{1}{{\sigma }_a^2\left(f\right)}+\frac{1}{{\sigma }_b^2\left(f\right)}$

[0109] Both of the preceding operations constitute the grouping of RA and RB.

[0110] More detailed explanations on the calculation of these functions in the form of arrays are given in Paragraph 7.17.1. The formulas indicated above for phase correction carry out an intensity normalization simultaneously, which however is not indispensable.

[0111] This method makes it possible, from the two sub-representations RA and RB whose supports A and B have a non-zero intersection, to obtain a sub-representation RC corresponding to the superposition of RA and RB.

[0112] If the three-dimensional frequency representation of the object must be reconstituted from many sub-representations whose phases are not known, the above method, applied iteratively, allows the grouping of all these sub-representations. For example, it is possible to begin with a given sub-representation, and group it with a second sub-representation. Then, it is possible to begin with the sub-representation generated by this grouping, and group it with a further sub-representation. By repeating this grouping operation until all the sub-representations have been integrated in a overall representation, one finally obtains the three-dimensional frequency representation of the object. The only conditions to be met in order for this method to succeed are:

[0113] that no sub-representation or group of sub-representations has an empty intersection with all the other sub-representations.

[0114] that the object does not have an excessively singular frequency representation, which would be for example zero on a set of points separating in two its three-dimensional frequency representation.

[0115] These conditions are easily met for all biological objects as soon as a large number of representations is acquired.

[0116] For example:

[0117] an embodiment may be derived from the system of [Fercher] in which the three simultaneous illuminating waves produced by the diffraction grating have been replaced by a single illuminating wave of variable direction. In this case, a sub-representation is constituted by a two-dimensional frequency representation obtained for a given illuminating wave.

[0118] an embodiment can be derived from the microscope implemented in [Turpin 3] in which, in addition to the rotation of the object, variations in the direction of the illuminating wave have been authorized. In this case, a sub-representation is constituted by all the two-dimensional frequency representations obtained for a given illuminating wave when the object is rotated.

[0119] in the case of Embodiment 5, Paragraph 9.19., a sub-representation is constituted by a two-dimensional frequency representation obtained for a given illuminating wave. A small number of sub-representations is first grouped in a basic representation having a non-zero intersection with all the other sub-representations obtained under similar conditions. All the representations are then phase-corrected in relation to the basic representation, and then an overall representation is generated.

[0120] in Embodiments 3, 4 and 5, four intermediate sub-representations are generated each time, as explained in 7.17.1.1. These four representations are grouped in a single representation by the application of this general method.

[0121] According to a variant of the invention, the microscope thus comprises means for:

[0122] determining, for each sub-representation RB, a coefficient characterizing the phase difference between this sub-representation and another sub-representation, part of a sub-representation or group of sub-representations RA, this coefficient being calculated from values of RA and RB over a set included in the intersection of the supports of RA and RB.

[0123] correct the phase of RB so as to obtain for RB the same phase reference as for RA.

[0124] The simplest way to correct the phase of RB is to carry out a multiplication by the coefficient r as indicated above. However, this correction can also be carried out by physical means, in which case the phase does not need to be corrected during the calculation stage. An example of such an embodiment is described in 7.18.7.

[0125] The phase difference affecting each sub-representation can be recalculated with each acquisition. This is necessary in Embodiments 1 and 2, for which these differences are not reproducible. In the case of Embodiments 3, 4 and 5, this phase difference is reproducible and can consequently be measured during a phase preparatory to acquisition. An example of such an embodiment is described in 7.18.1.

[0126] According to a variant of the invention, the microscope comprises means for:

[0127] determining, for a given sub-representation RB, the frequency representation RC resulting from the grouping of RB with another sub-representation RA.

[0128] determining a coefficient characteristic of the noise affecting RC defined over the entire support of RC, obtained from a coefficient characteristic of the noise affecting RA and defined on the support of RA, and from a coefficient characteristic of the noise affecting RB and defined on the support of RB.

[0129] The methods used can differ from the formalism set forth above. For example, in Embodiment 1, the quantity $\frac{1}{{\sigma }_c^2\left(f\right)}$

[0130] is assimilated with the number N of frequency representations reaching a given point.

[0131] The calculations can be grouped: after phase correction of each sub-representation, they can be grouped in an overall representation, without calculating each intermediate sub-representation. This is what is done in all the embodiments to group two-dimensional sub-representations into complete or partial three-dimensional sub-representations.

[0132] Representations of the object can be calculated without passing formally through its three-dimensional frequency representation. For example, in 7.17.3.3., a confocal representation of the object is generated using, for the final frequency representation, a value at each point which is the sum of the values obtained for each representation reaching this point. The representation thus obtained is not strictly speaking a frequency representation of the object, but nevertheless carries information on this object. It is also possible to generate real representations of the refractive index or absorptivity. These representations can be generated simply through the frequency representation of the object, but it is also possible to modify the algorithms so as not to formally use this procedure.

[0133] 3.5.2. Absolute Phase Correction

[0134] The method set forth in 3.5.1. makes it possible to obtain a three-dimensional representation of the object. However, the overall phase of this three-dimensional representation remains arbitrary.

[0135] In the three-dimensional spatial representation of the object, obtained from the three-dimensional frequency representation by taking the inverse Fourier transform, the complex number associated with each point characterizes the absorptivity and the refractive index of the considered point. If the overall phase of the three-dimensional representation is chosen appropriately, the real part of the said complex number characterizes the local absorptivity of the object, and the imaginary party of said complex number characterizes the local refractive index of the object. The overall phase is chosen appropriately when the origin of the three-dimensional frequency representation has a real value.

[0136] According to a variant of the invention, and in the case in which the origin of the three-dimensional frequency representation is one of the points that have been acquired, the microscope has means for dividing, by its value at the origin, the three-dimensional frequency representation obtained by the method pointed out in 3.5.1. This makes it possible to obtain a spatial representation in which the real part and the imaginary part of the complex numbers represent respectively the local absorptivity and the local refractive index. This also allows the entire representation to be normalized.

[0137] When the origin of the three-dimensional frequency representation is not part of the points that have been acquired, this operation is impossible. The operator who views an image sees, for example, the real part of the complex number, and must intuitively choose the total phase of the representation so as to obtain the most contrasted possible image.

[0138] 3.5.3. Phase Correction in Relation to the illuminating Wave

[0139] The general phase correction algorithms defined above exhibit the drawback of being relatively complex to implement. A simplified version can be obtained when the acquisition system allows the acquisition of the non-diffracted part of the illuminating wave. This corresponds to the origin of the three-dimensional frequency representation of the object. This point is common to all the two-dimensional frequency representations.

[0140] The phase correction described in 3.5.1. can then be carried out in relation to the sub-representation part constituted by this single point. This correction can be grouped with the absolute correction described in 3.5.2. The total of the two corrections then is equivalent to dividing the entire sub-representation of the object obtained from a given illuminating wave by its value at the origin. When a plane frequency image is generated as an intermediate calculation step, this is equivalent to dividing the entire plane frequency image by its value at the point corresponding to the non-diffracted part of the illuminating wave. According to a variant of the invention, the phase correction of the two-dimensional frequency representations is carried out by dividing each sub-representation of the object by its value at the origin of the three-dimensional frequency representation of the object. This method is used, for example, in Embodiments 1, 2, 3 and 4.

[0141] 3.5.4. Phase Correction in Relation to Pre-Recorded Phase Values

[0142] A three-dimensional frequency representation of the object is obtained from a series of two-dimensional frequency representations each corresponding to a different illuminating beam.

[0143] Each of these two-dimensional frequency representations can be phase corrected in relation to the sub-representation constituted by the origin alone, as indicated in 3.5.4. However, the high intensity of the corresponding point on each two-dimensional frequency representation makes difficult the simultaneous acquisition of the rest of the representation. According to a variant of the invention, the acquisition of plane frequency images takes place in two phases:

[0144] a preliminary phase during which are recorded the values obtained at the image point of the illuminating wave, for each illuminating wave;

[0145] an acquisition phase proper during which the direct beam can be obstructed and during which the values of the plane frequency images are recorded.

[0146] A two-dimensional frequency representation can then be obtained for each illuminating wave from these two recordings, the value at the image point of the illuminating wave being obtained from the first recording and the value at any other point being obtained from the second recording. The method described in 3.5.4. can then be applied to each two-dimensional frequency representation obtained.

[0147] When a series of two-dimensional frequency representations is obtained, for example to “film” the movement of cells, the preliminary phase must not be repeated. It only needs to be carried out once before the start of the acquisitions.

[0148] In order for this method to be functional, the phase difference between the illuminating beam and the reference beam, at the level of the receiving surface, must be reproducible. The illuminating beam generation systems used in Embodiments 3, 4 and 5 meet this condition. This phase correction phase is described for example in 7.18.1. and 9.18.2.

[0149] 3.6. Vibration Compensation

[0150] The method described in 3.5.4. presupposes the reproducibility of the illuminating beams. The method described in 3.5.3., in the case in which several objectives are used, presupposes a constant phase shift between the waves received on each of these objectives. However, in Embodiments 3, 4 and 5, the vibrations can make these methods ineffective or less robust. In order for the results to be reliable, these vibrations must be compensated.

[0151] For this purpose, the system can periodically acquire a reference image. The reference image consists, for example, of an image obtained on the receiving surface for a fixed illuminating wave, which is not modified when the illuminating wave used to obtain the “useful” plane frequency images varies. Each acquisition then corresponds to a reference image acquired at a near instant. A reference image acquired at an initial instant is chosen as absolute reference. By “useful” image is meant an image obtained on the receiving surface and for which will be calculated a two-dimensional frequency representation used to generate the representation of the object.

[0152] With the vector v going through the entire support of an image, we denote by m(v) a “useful” image obtained on this receiving surface, and h(v) the corresponding reference image. We denote as h_o(v) the reference image chosen as absolute reference, obtained on the receiving surface. v represents the plane frequency projection and is thus a two-dimensional vector varying over the entire receiving surface. By σ(v) is denoted the standard deviation of the Gaussian noise affecting the function h(v) at each point.

[0153] The phase variation of vibratory origin can be characterized, for example, by the coefficient $r=\frac{\sum _f\frac{h_0\left(v\right)\stackrel{\_}{h\left(v\right)}}{{\sigma }^2\left(v\right)}}{\sum _f\frac{{\left|h\left(v\right)\right|}^2}{{\sigma }^2\left(v\right)}}$

[0154] which represents the most probable phase difference between the reference images h(v) and h_o(v).

[0155] The image m(v) can then be phase corrected as follows:

[0156] m(v)←r.m(v) where the sign←designates the sense.

[0157] When this preliminary correction has been carried out the images thus phase corrected can be used in the algorithms defined in 3.5.3. and 3.5.4.

[0158] If the system is totally free from vibrations, this preliminary correction is not required.

[0159] If the vibrations are of the low-frequency type, the reference image can be acquired only at a low frequency, however higher than the frequency of the vibrations of the system.

[0160] If the vibrations are of the high-frequency type, it is possible to acquire a reference image at each useful image.

[0161] In the presence of vibrations,

[0162] this preliminary correction is essential for the application of the algorithm defined in 3.5.4.

[0163] this preliminary correction is not essential for the application of the algorithm defined in 3.5.3. in the case in which a single objective is used. However, if several objectives are used, it allows the fixing of the phase difference between the plane frequency images generated from each objective. In this case, it is thus also essential.

[0164] This technique is used in Embodiments 3 and 4 and described in 7.17. It is also used in Embodiment 5 when the phase correction is carried out in accordance with paragraph 9.18.

[0165] A variant of the invention thus consists in periodically acquiring reference images corresponding to a fixed illuminating direction, and using these images to compensate for phase differences of vibratory origin affecting the plane frequency images.

[0166] 3.7. Characterizing the Wave Vector of the Illuminating Wave

[0167] It is necessary to control the direction of the illuminating wave, i.e. its wave vector f_e, for instance by mechanical means as described for example in [Turpin]. However, these mechanical means must be very accurate and are costly to implement.

[0168] In the conditions defined in 3.5.3., i.e. if the acquisition system allows the acquisition of the non-diffracted part of the illuminating wave, the non-diffracted part of the illuminating wave corresponds to the maximum modulus value on the frequency representation of the wave coming from the object. According to a variant of the invention, the microscope has means for determining the coordinates of this maximum and for calculating, from these coordinates, the wave vector f_eof the illuminating wave. This method is used, for example, in Embodiments 1 and 2.

[0169] However, the presence of the object can slightly distort the value of the wave vector thus obtained. According to a variant of the invention, the wave vectors f_eof each illuminating wave are obtained in a preliminary phase in which the object is eliminated or replaced by a transparent plate. The wave vectors thus obtained are thus not distorted by the presence of the object. This method presupposes that the wave vectors are reproducible from one acquisition to the other. On the other hand, it obviates the need to calculate these wave vectors according to mechanical parameters. This method is used in Embodiments 3, 4 and 5.

[0170] 3.8. Characteristics of the Receiver

[0171] 3.8.1. Use of a Microscope Objective

[0172] According to a variant of the invention, the receiver comprises a large-aperture microscope objective that transforms the rays coming from the object under a large aperture into paraxial rays that can be directed towards a receiving surface. This configuration offers better performance than the configurations defined in [Fercher] (absence of objective) or [Turpin] (small-aperture objective). While [Vishnyakov] uses such a configuration without benefiting from it, owing to the use of poorly suited tomographic methods, the algorithms defined in the present invention make it possible to take full advantage of this configuration.

[0173] 3.8.2. Use of a Receiving Surface in a Frequency Plane

[0174] It is advantageous to directly acquire the image in the frequency domain, as in [Turpin 3]. This can be done, for example, by means of the receiver described in [Lauer 1] that allows improved performance compared to the receiver of [Turpin 3].

[0175] 3.8.3. Use of a Receiving Surface in a Spatial Plane

[0176] Besides the microscope objective itself, the receiving system defined in 3.8.2. presents a paraxial part allowing the modification of the optical signal captured by the objective, to obtain a frequency representation. The signal from the objective first passes through the plane in which the objective normally forms the image of the observed sample. This plane will be called the spatial plane. It is then transformed by a paraxial system so that, in the plane in which the receiving surface is placed, a plane wave coming from the object has a point image. This plane, in which the receiving surface is placed, will be called the frequency plane. The paraxial part of the optical system used can include intermediate spatial or frequency planes. The receiving surface can be placed in a spatial plane, in a frequency plane, or in an intermediate plane. However, to simplify calculations, it will always be placed either in a spatial plane or in a frequency plane. In order for the received image to be correct, the following conditions must moreover be complied with:

[0177] if the receiving surface is in a frequency plane, the reference wave must be centered virtually at a central point of the observed object.

[0178] if the receiving surface is in a spatial plane, the reference wave must be the image of a virtual wave which is plane on the crossing of the observed object.

[0179] Under these conditions, the signal detected on a receiving surface placed in a frequency plane is the optical Fourier transform of the signal that would be detected in a spatial plane. One variant of the invention constituting an alternative to the receiver defined in 3.8.2. is thus to use a receiving surface positioned in a spatial plane and a reference wave which is the image of a virtual wave which is plane on the crossing of the observed object. A numerical Fourier transform then replaces the optical Fourier transform.

[0180] Embodiments 1, 3 and 4 use sensor means in a frequency plane and Embodiments 2 and 5 use sensor means a spatial plane. A plane frequency image can thus be obtained either directly on a receiving surface placed in a frequency plane, or by the Fourier transform of an image received on a receiving surface placed in a spatial plane.

[0181] 3.9. Beam Attenuation

[0182] If the sensor is placed in a spatial plane, the direct illuminating wave has, as a representation on the sensor, a constant modulus value which is superimposed on the wave diffracted by the object. An excessively weak diffracted wave in relation to this constant basic level cannot be detected correctly. On the other hand, this basic level is not very high because the intensity of the reference beam is spread over the entire sensor, thus generally allowing good images to be obtained. If the sensor is placed in a frequency plane, the illuminating wave is concentrated at a point and, as in the previous case, an excessively weak diffracted wave in relation to this constant basic level cannot be correctly detected. As the wave is concentrated at a point, this basic level is high and this limitation is troublesome.

[0183] According to an advantageous variant of the invention, a device controlling the attenuation of the beam, having one or more attenuation levels, is introduced to solve this problem. The attenuation device makes it possible to obtain successively several recordings differing in the intensity of the illuminating wave. A less noisy value of the diffracted wave is then obtained by combining these recordings. The final value of the diffracted wave is calculated for example at each point from the recording for which the intensity of the wave received at the considered point is highest, but for which the sensor remains unsaturated at the considered point and in its immediate surroundings for all the interference patterns allowing said recording to be obtained.

[0184] Given the use of the beam attenuation device, the variant of the invention in which a plane wave has a point image allows the detection of weaker diffracted waves. In fact, the waves are not superimposed on the sensor with any other wave and very low levels can be detected when the illuminating beam intensity is high.

[0185] Such a device is used in Embodiments 1, 3 and 4.

[0186] 3.10.

[0187] Illuminating Beam Generation System

[0188] Whatever the embodiment, an illuminating beam generation method must be designed. The methods proposed by [Turpin] have the drawback of requiring significant mechanical movements and hence of greatly slowing down the system.

[0189] An optical system, for example the one described in 8.1.1., can transform a parallel beam having a given spatial extension and variable direction into a parallel beam whose spatial extension has been reduced and whose directional variations have been amplified. In general, small directional variations applied to a beam having a broad spatial extension can be amplified by an optical system through a reduction in the spatial extension of the beam. As the spatial extension of the illuminating beam required for a microscope is small, this principle can be used for the optical amplification of small mechanical movements.

[0190] A beam with a variable direction can be transformed by a lens into a beam with a variable position in the back (image) focal plane of this lens. A directional variation of the beam in a part of its optical path is thus equivalent to a position variation in another part of its optical path and vice-versa. In intermediate planes, the variation is a joint position and directional variation. There is thus no reason to differentiate between a system generating illuminating wave position variations and a system generating directional variations, as these systems are equivalent.

[0191] According to one variant of the invention, the illuminating beam generation system comprises:

[0192] a beam deflector generating variations in a paraxial beam.

[0193] a large-aperture optical element (for example, a microscope objective or a condenser) transforming said incoming paraxial beam variations into significant directional variations in the outgoing beam.

[0194] Said system can also comprise a lens system designed so that the beam is parallel at the exit of said large-aperture optical element.

[0195] The beam deflector may, for example, be a mirror mounted on a positioner allowing the control of its orientation. This solution is implemented in Embodiments 1 and 2. However, this solution has two drawbacks:

[0196] the movement of the mirror generates vibrations which disturb the system. After each movement of the mirror, it is necessary to wait for the absorption of vibrations before proceeding with the acquisition.

[0197] the phase difference between the reference beam and the illuminating beam is not reproducible, thus preventing the use of certain algorithms such as those defined in 3.5.4.

[0198] Each of the beam deflectors described in 3.10.1., 3.10.2 and 3.10.3. enables these two problems to be solved.

[0199] 3.10.1. Beam Deflector Based on a Series of Binary Deflectors

[0200] A system capable of sending the beam back in two directions can be built by means of a birefringent prism which transmits the ordinary beam and the extraordinary beam in two different directions. The laser beam used must then be polarized. A polarization rotator placed before the prism allows the orientation of its polarization in the ordinary direction or the extraordinary direction, which implies a different angle of deflection by the prism. However, available ferroelectric polarization rotators, which have the advantage of being fast, do not allow a rotation of 90 degrees but a rotation of about 80 degrees. This prevents the possibility of having both a beam polarized exactly in the ordinary direction, for one of the positions of the polarization rotator, and a beam polarized exactly in the extraordinary direction for the other position. Hence, in one of the positions, is created a spurious beam deflected in an undesired direction. To eliminate this spurious beam, it is necessary to use at the output of the birefringent prism a polarizer selecting only the desired beam. In order for this polarizer not to eliminate the beam in the other position of the rotator, a second rotator must be introduced between this polarizer and the prism, said second rotator being used to bring the electric field vector of the beam back to the passing direction of the polarizer when it is not there directly at the exit of the prism.

[0201] A system capable of returning a beam in many directions can be constituted by combining in series several of these elementary systems. By combining two of them, which produce a deflection of the same amplitude but in two different orthogonal directions, a doublet is formed. By combining N doublets in series, each doublet being characterized by birefringent prisms having characteristics such that the deflection angle of doublet number i is proportional to 2ⁱ, one obtains 2^Npossible deflection values in each direction. For example, with N=8, there is a total of 256×256 beam deflection directions.

[0202] According to a variant of the invention, the beam deflection system is constituted by the association of elementary deflectors in series, each of these elementary deflectors comprising a birefringent prism deflecting differently the ordinary beam and the extraordinary beam, preceded by an electronically controlled polarization rotator and allowing the orientation of the electric field vector of the beam along the ordinary axis or the extraordinary axis of said prism, and followed by a second rotator and by a polarizer allowing the elimination of spurious beams.

[0203] Such a device is used in Embodiment 3.

[0204] 3.10.2. Beam deflector based on spatial modulators

[0205] A spatial modulator is a two-dimensional matrix of pixels allowing the phase or intensity modulation of a wave in a plane. Most spatial modulators are based on liquid crystals. Common LCD screens constitute an example of a spatial intensity modulator.

[0206] A spatial plane, on the path of the illuminating beam, will be defined by a plane in which this beam is parallel and is centered on the optical axis. A frequency plane will be defined as a plane in which this beam has a point image.

[0207] A parallel beam reaching a spatial plane has, in this plane, a complex representation of the form exp{j2π(f_xx+f_yy)} in which (x,y) are coordinates of a point of the plane and in which (f_x,f_y) are the coordinates of the projection of the wave vector in this plane. If a phase modulation device is placed in this plane and if a phase shift of the form θ=2π(g_xx+g_yy+c) is applied by means of this device, the wave has, after passing through said device, a complex representation exp{j2π((f_x+g_x)x+(f_y+g_y)y+c)}. The spatial modulation device has thus modified the direction of the incident wave. Wave vectors that can generate such a spatial modulation device are included in a cone the aperture of which depends on the maximum values of g_xand g_ypermitted by the modulator. This cone will be called “deflection cone.”

[0208] If an intensity modulation device is used instead of the phase modulation device, it is possible to apply an attenuation function of the type cos{2π(g_xx+g_yy+c)}. After passing through the device, the wave then has a form of the type exp{j2π((f_x+g_x)x+(f_y+g_y)y+c)}+exp{j2π((f_x−g_x)x+(f_y−g_y)y−c)} which corresponds to the superposition of two plane waves whose wave vectors are symmetrical in relation to the axis oriented along the wave vector of the wave leaving the device in the absence of modulation. One of the two waves can be stopped by a diaphragm, in which case the device constitutes a beam deflector comparable to the preceding one.

[0209] Intermediate modulation devices performing joint phase and intensity modulation may also be used.

[0210] A variant of the invention thus consists in using, as beam deflector, a suitably controlled spatial modulator.

[0211] According to a variant of the invention, said modulator is a phase modulator controlled so as to generate a phase shift in a form as near as possible to θ=2π(g_xx+g_yy).

[0212] Prior-art modulation devices operate pixel by pixel. This discretization leads to the generation of spurious frequencies outside of the deflection cone. One variant of the invention consists in eliminating these spurious frequencies by means of a diaphragm placed in a frequency plane on the path of the wave coming from the phase modulator.

[0213] Modulation devices allowing fast modulation are binary, i.e. to a given pixel there correspond only two possible phase or intensity values. The use of a binary modulation device results in the presence of a spurious plane wave symmetrical with the wave to be obtained in relation to an axis constituted by the direction of an undeflected beam. In the case of binary modulators, this is true even in the case of a phase modulator, whereas in the case of modulators generating a continuous modulation, this problem can be avoided by using a phase modulator. According to a variant of the invention, the diaphragm filtering the spurious frequencies is dimensioned so as to filter not only the frequencies located outside of the deflection cone, but also part of the frequencies located within the deflection cone, so as to stop the spurious plane wave.

[0214] Binary modulation devices also have the drawback of generating spurious frequencies included within the deflection cone and constituting a frequency “noise.” According to a variant of the invention, these frequencies are stopped by an intensity modulator placed in a frequency plane along the path of the beam coming from the phase modulator, and controlled so as to allow only the sought frequency to pass.

[0215] Such a device is used in Embodiment 4.

[0216] 3.10.3. Beam deflector consisting of a mobile mirror designed so that vibrations are not troublesome

[0217] The beam deflectors described in 3.10.1. and 3.10.2. are based on the use of liquid crystal devices and polarizers. These devices are not available in the ultraviolet radiation range. To use ultraviolet rays, other means are required.

[0218] In prior-art devices, the entire system was placed on an optical table.

[0219] According to one variant of the invention, the beam deflection device is made up of a mirror placed outside the optical table, the separation between the illuminating beam and the reference beam being provided by a separator fixed on the optical table and positioned after said mirror on the path of the beam.

[0220] As the mirror is placed outside the optical table, it does not generate vibrations in this table. As the separation of the beams takes place after the mirror, its vibrations also do not generate phase shifts between illuminating beam and reference beam. This consequently solves the problem of vibrations.

[0221] On the other hand, the fact that beam separation takes place after the mobile mirror means that the reference beam varies at the same time as the illuminating beam. This variation must be taken into account in the design of the system and compensated. For example, if the receiving surface is placed in a spatial plane, variations in the direction of the reference wave result in translations of the plane frequency image. According to a variant of the invention, this effect is compensated by providing a translation in the opposite direction of the plane frequency images obtained.

[0222] This technique is used, for example, in Embodiment 5.

[0223] 3.11. Compensation of errors due to polarization

[0224] The image generation method described by [Wolf] is based on a scalar diffraction theory and assumes that the wave passing through the object is diffracted isotropically in all directions by each point of the object.

[0225] It is only on the basis of this assumption that the theoretical resolution of λ/4 can be obtained. Scalar diffraction theory is however not valid for high diffraction angles. The intensity diffracted by a point of the object depends on the direction of the diffracted wave, the direction of the illuminating wave, the polarization direction of the illuminating wave and the polarization direction of the diffracted wave.

[0226] 3.11.1. Compensation by real-coefficient multiplication

[0227] The wave diffracted by the object differs from the wave which would be diffracted if the diffraction were isotropic by a real multiplicative factor depending on the:

[0228] illuminating wave propagation direction

[0229] illuminating wave polarization

[0230] diffracted wave propagation direction

[0231] diffracted wave polarization

[0232] According to one variant of the invention, the microscope comprises means for determining this multiplicative factor and compensating it by multiplying the received waves by the inverse of said factor. Such a variant of the invention is described in 7.18.8.

[0233] 3.11.2. Compensation in the case of an anisotropic material

[0234] If the observed object is made up essentially of an anisotropic material, the effects of the diffraction differ from what they are in an isotropic material. The material is then characterized at each point by six crystalline parameters plus absorptivity.

[0235] In the particular case in which the observed object is a uniaxial crystal, the refractive index of the ordinary ray has a constant value. According to one variant of the invention, a three-dimensional representation in which the complex numbers obtained characterize the absorptivity and the ordinary index of refraction can be calculated. According to a variant of the invention, this representation is calculated from plane frequency images obtained for an illuminating wave polarized so that it constitutes an ordinary beam.

[0236] As the ordinary polarization direction varies with the wave's propagation direction, it is necessary to be able to modify the polarization direction of the illuminating wave. However, as the phenomena are linear, one need only record the wave received at every point for two polarization directions of the illuminating wave to be able to deduce the wave diffracted by the object for an illuminating wave of any direction. According to a variant of the invention, the microscope comprises means for generating two polarization directions of the illuminating wave. According to a further variant of the invention, the microscope also comprises means for analyzing the diffracted wave along two polarization directions, thus making it possible to distinguish the ordinary diffracted wave from the extraordinary diffracted wave. According to this variant of the invention, the plane frequency images are then obtained from four elementary images corresponding to each combination of two polarization directions and two analysis directions.

[0237] Such a variant of the invention is described in 7.18.9. In the description of 7.18.9. only the analysis direction corresponding to the ordinary ray is used, but it is also possible to take into account the analysis direction corresponding to the extraordinary ray. Two plane frequency images are then obtained for each illuminating wave propagation direction, one corresponding to the ordinary index and the other to the extraordinary index. The final frequency representation is obtained from this set of images, taking into account the variations in the extraordinary index at each point of the plane frequency images corresponding to the extraordinary index.

[0238] 3.11.3. Compensation by combining several polarization and analysis directions

[0239] In the case of the isotropic material, the method described in 3.11.1. has the drawback of causing the noise level to rise considerably. One way to avoid this problem is to acquire at least four plane frequency images corresponding to each combination of two distinct polarizations of the illuminating wave and two distinct polarization directions of the diffracted wave. An appropriate algorithm then makes it possible, from these four images, to calculate a single image corresponding to the scalar parameter sought. According to a variant of the invention, the microscope consequently comprises means for generating two distinct illuminating wave polarizations, and means for analyzing the diffracted wave along two distinct polarization directions. A variant of the invention is that the microscope comprises means for calculating, from the plane frequency images corresponding to each combination of two polarization directions and two analysis directions, a single plane frequency image representing a complex scalar quantity complying with the condition of uniform diffraction in all directions. This principle is used in Embodiments 3, 4 and 5. The principle for calculating said scalar quantity is described in detail in 7.12.1.

[0240] In the case of wavelengths in the visible, said means for varying the illuminating wave polarization can consist of a liquid crystal polarization rotator. Said means for varying the diffracted wave analysis direction can consist of a polarization rotator associated with a polarizer. In the case of ultraviolet, these devices are not available.

[0241] 3.11.4. Variation of illuminating wave polarization direction in the UV range

[0242] In the UV range, polarization rotators can be replaced by a quartz retardation plate rotating around an axis by mechanical means. However, these mechanical movements slow the system down considerably. For this reason, use is made of a system in which the only mechanical movements are those of shutters and in which the beam to be closed off has a spatial extension which is as small as possible so that the movement of the shutter is as small as possible. It is then possible to use a high-speed shutter or a shutter system with a rotating crownwheel.

[0243] According to one variant of the invention, such a system includes:

[0244] a beam separator which breaks the beam down into a beam A and a beam B.

[0245] lenses placed on each beam A and B and focussing these beams on points of focus at which the shutters are placed.

[0246] a device making it possible to again superimpose beams A and B having passed through their respective shutters.

[0247] a device placed on the place of one of the beams A or B, in the part of the path in which the two beams are distinct, and modifying the polarization of this beam.

[0248] Such a system can also comprise additional lenses designed to reform parallel beams after passage through the shutters. It can also comprise a second polarization modification device. The beam polarization modification device can be a retardation plate. The beam separation and beam superposition devices can be semi-transparent mirrors. Such a device is used in Embodiment 5.

[0249] 3.11.5. Variation of analysis direction in the UV range

[0250] The wave coming from the object can be broken down into a wave whose electric field vector is parallel to that of the reference wave and a wave whose electric field vector is orthogonal to that of the reference wave. The intensity received on the receiving surface is the sum of the intensity of the wave whose electric field is orthogonal to the reference wave and the intensity produced by the interference of the reference wave and the wave whose electric field is parallel to the reference wave. The first of these intensities does not depend on the phase of the reference wave and consequently does not modify the complex value of the wave coming from the object measured by the combination of interference patterns corresponding to different phases of the reference wave. It is thus only the wave whose electric field vector is parallel to that of the reference wave which is obtained on the receiving surface.

[0251] The analysis direction of a wave can consequently be modified simply by modifying the polarization direction of the reference wave or, symmetrically, by modifying the polarization direction of the wave coming from the object.

[0252] According to a variant of the invention, the analysis direction is modified by varying the polarization direction of the reference wave or of the wave coming from the object.

[0253] According to a variant of the invention, the polarization of the reference wave or of the wave coming from the object is modified by a device comprising:

[0254] a beam separator breaking the beam down into two beams A and B

[0255] a retardation plate LA placed on the path of beam A and a retardation plate LB placed on the path of beam B, the angle between the neutral axes of these two retardation plates being 45 degrees.

[0256] The polarization direction of the beam having passed through retardation plate LA is then deduced from the polarization direction of the beam having passed through retardation plate LB by a rotation whose angle is the angle between the neutral axes of retardation plates LA and LB. If this angle is 45 degrees, the beams coming from plates LA and LB will always have orthogonal polarizations whatever the polarization direction of the incident beam.

[0257] The two beams A and B can then be combined by a superposition device after having passed through shutters, as in the case of the device described in 3.11.4. This has the drawback of requiring the use of shutters at a point of the beam path in which the direction of the beam is variable and in which the beam can consequently not be focussed on a fixed point.

[0258] According to a variant of the invention, the two beams A and B are separately superimposed on the beam coming from the object (if they constitute the reference wave) or the reference wave (if they come from the object). The interference patterns corresponding to each polarization direction are then formed on two distinct receiving surfaces.

[0259] Whether the plane frequency images corresponding to each polarization are obtained on distinct receiving surfaces or on the same receiving surface, a phase difference is then created between these images and must be compensated. According to a variant of the invention, the retardation plates are positioned so that the reference and illuminating beams reaching the receiving surface have different polarization directions, preferably 45 degrees from each other. Then, the part of the wave coming from the object which comprises frequencies near the illuminating wave is detected on the two receiving surfaces and can be used to calculate said phase difference.

[0260] 3.12. System for eliminating direct illumination

[0261] The illuminating wave is generally much more intense than the diffracted wave. It can saturate the sensors or reduce considerably the signal-to-noise ratio of the system by requiring acquisition of high signal levels. The illumination of the non-diffracted part of the illuminating wave during the acquisition phase or during part of the acquisition phase clearly improves the performance of the system. In a frequency plane, the non-diffracted part of the illuminating wave has a point image and can be eliminated by placing an absorbing element on this point. According to one variant of the invention, the system accordingly comprises a device for eliminating the non-diffracted part of the illuminating wave, placed in a frequency plane, and absorbing the beam on a small zone around the point corresponding to this illuminating wave.

[0262] According to a variant of the invention, this device is made up of a spatial intensity modulator controlled to allow passage at all points except on a limited zone around the impact point of the non-diffracted part of the illuminating wave. This variant is implemented in Embodiment 4.

[0263] According to a further variant of the invention, this device comprises a glass which is mobile in translation within a frequency plane, an absorbing black spot placed on the glass being designed to stop the direct beam, the position of the glass being controlled so that this black spot coincides with the point of impact of the non-diffracted part of the illuminating wave. This variant of the invention is implemented in Embodiment 5.

[0264] 3.13. Spatial modulator utilization device

[0265] The spatial modulators used in 3.10.2. or in 3.12. may in particular be high-speed binary modulators operating by reflection. They are generally used by means of a polarizing semi-transparent mirror which is a device of cubic form sending back incident beams in two different directions depending on their polarization. This device, owing to its thickness, generates a slight aberration which widens the point corresponding to a given frequency, which is detrimental to the quality of the images obtained.

[0266] To avoid the use of this device, it is possible to use beams coming from the modulator under an oblique angle. The incident and reflected beams are then separated. However, this method deforms the distribution of the generated frequencies. To prevent this deformation, said oblique angle must be small.

[0267] According to one variant of the invention, this problem is solved by using a device consisting of a mirror with two orthogonal reflecting faces and a lens traversed in one direction by the beam directed towards the modulator and in the other direction by the beam reflected by the modulator. The incident beam is reflected by one side of the mirror, passes through the lens, is reflected by the modulator, passes again through the lens in the opposite direction and is reflected by the second side of the mirror, resuming its initial direction. The incident and reflected beams can be partially superimposed on the lens, but are separated on the two-sided mirror. In order to obtain this separation on the mirror, while maintaining as small an oblique angle as possible, the two-sided mirror must be positioned approximately in one focal plane of the lens and the modulator must be positioned in the other focal plane of the lens.

[0268] 3.14. Use of objectives traversed in both directions and/or of several objectives

[0269] The object can be illuminated on one side and the wave coming from the object can be picked up on the opposite side by an objective, thus allowing the reconstitution of part of the frequency representation of the object. However, other parts of the frequency representation of the object can be constituted only from the wave moving towards the side of the object from which the illuminating waves come. According to one variant of the invention, an objective is associated with an optical system allowing, on the one hand, the measurement on a sensor of the wave coming from the sample and having gone through the objective and, on the other, the formation of an illuminating wave which, after crossing the objective, becomes in the sample a plane illuminating wave of variable direction. This objective is then crossed in one direction by the illuminating wave moving towards the object, and in the other direction by the diffracted wave coming from the object. It plays both the role of objective receiving the wave coming from the object and the role of high-aperture system transforming the small directional variations of the wave coming from the beam splitter into large directional variations of the wave in the object. This can be achieved for example by means of a semi-transparent mirror placed on the path of the beam coming from the object and superimposing the beam coming from the object and directed in a given direction on the illuminating beam moving in the opposite direction. This variant of the invention is implemented in Embodiments 3, 4 and 5 which comprise several objectives. One arrangement in which only one objective is used is described in 7.18.10.

[0270] Possible limitations having to do with the direction of the illuminating wave, characterized by its frequency vector, influence the performance of the system. Maximum precision is obtained when all the possible directions are used. Similarly, it is desirable to record the wave diffracted by the object in all directions. When a single microscope objective is used, its aperture limits the directions in which it is possible to record the wave diffracted by the object. According to an advantageous variant of the invention, several objectives focussed on the sample are used, which makes it possible to record the wave coming from the sample along more directions. The objectives then cover almost the entire space around the sample, and the illuminating waves must necessarily pass through these objects to reach the sample. According to one variant of the invention, each objective is associated with an optical system allowing, on the one hand, the measurement on a sensor of the wave coming from the sample and having passed through the objective and, on the other, the formation of an illuminating wave which, after passing through the objective, becomes a variable direction plane illuminating wave in the sample. The illuminating waves can thus be generated in all the directions covered by the aperture of the objectives, and similarly the waves coming from the object can be measured in all these directions. Considering a variant of the invention, the acquisition and calculation system takes into account all the waves measured on all the sensors for all the illuminations used and, from these data, generates the three-dimensional frequency representation of the object. Each pair (illuminating wave direction-direction of wave from object) corresponds to a point of the frequency representation of the object, and the frequency representation thus generated consequently comprises all the points obtainable from the illuminating waves and the diffracted waves respectively produced and received by all the objectives.

[0271] A large number of objectives can be used in order to receive all the waves coming from the sample, or in order to increase the working distance by using low-aperture objectives. However, most of the samples observed in practice are flat and can be placed conveniently between two cover glasses. By virtue of a variant of the invention constituting the best compromise between utilization difficulty and performance, use is made of two large-aperture microscope objectives positioned opposite each other, the flat sample being introduced between these two objectives. This solution is used in Embodiments 3, 4 and 5. In Embodiments 1 and 2, which exhibit lower performance but are easier to fabricate, a single microscope objective is used.

[0272] 3.15. Generation of inverse beams

[0273] When two or more microscope objectives are used, a plane frequency image is generated from the wave received by each objective. Each point of a plane frequency image corresponds to a given wave vector of the diffracted wave. In order to calculate the three-dimensional frequency representation of the object, it is necessary to determine these wave vectors correctly, and this in a coordinate system common to the wave vectors received by each objective.

[0274] Knowing the K factor and the optical center, defined in [Lauer 1], allows the determination of the wave vectors corresponding to each point of a frequency image. However, the coordinate system used for two-dimensional representations RA (before translation of vector −f_e) reconstituted from the plane frequency image obtained from a given objective is different from that used for the representations RB obtained from the facing objective. To establish a correspondence between these two coordinate systems it is necessary to determine the coordinates of certain points in the coordinate system used for RA as well as in the coordinate system used for RB.

[0275] Each point PA of the representation RA is the image of a wave vector f_eof the illuminating wave which reaches this point in the absence of an object, and has the coordinates of this wave vector. To this vector there corresponds a wave vector −f_eof opposite direction whose image is a point PB of representation RB. The coordinates of point PB in a coordinate system used for RA are the opposite of the coordinates of point PA in this coordinate system.

[0276] The correspondence between the two coordinate systems can thus be established if the coordinates of point PB are also determined in coordinate system RB. This can be done by generating, by optical means, a wave vector beam opposite to the illuminating beam and by determining the coordinates of the image point of this beam in the coordinate system used for RB. If this correspondence is established at a sufficient number of points, the relationship between the coordinate systems used for RB and RA can be easily determined and these representations can be modified to use a common coordinate system.

[0277] In like manner, it is possible to obtain, by optical means, a direct correspondence between coordinate systems RB and RA. This requires the adjustment of a certain number of optical elements. To perform this adjustment, it is possible to check continuously the correspondence between the coordinates of point PB obtained in each of the coordinate systems used, and this for a certain number of points PB (three points in principle).

[0278] In both cases, it is necessary to generate a beam having the same characteristics as the illuminating beam, but propagating in the opposite direction. In general, given a beam used in the system, the term “opposite indicator beam” will designate a beam having the same characteristics but propagating in the opposite direction.

[0279] Considering one variant of the invention, the microscope consequently comprises, during the 35 adjustment phase, means for generating an indicator beam opposite to the illuminating wave. These means may be eliminated after the adjustment phase corresponding to the determination, by calculation means or optical means, of the correspondences between coordinate systems RB and RA.

[0280] According to a variant of the invention, the microscope also comprises means for generating, during an adjustment phase, an indicator beam opposite to the reference beam. This beam will also be used in certain adjustment phases. For example, if the receiving surface is in a frequency plane, the reference wave is centered virtually at a central point of the object. The indicator beam opposite to the reference wave makes it possible to adjust the position of the objectives so that these objectives are focussed on the same point.

[0281] According to a variant of the invention, when the receiving surface is a spatial plane, an additional beam centered on this spatial plane is also used during an adjustment phase, as is its opposite indicator beam. This beam makes it easy, for example, to adjust the position of the objectives in the absence of a reference wave centered on a point of the object.

[0282] In Embodiments 3, 4 and 5, each beam used has an opposite indicator beam, and the means of generating these opposite indicators are described as forming part of the microscope and are not eliminated after the adjustments have been completed: shutters are simply used to cut off these beams.

[0283] According to one variant of the invention, the device generating an opposite indicator beam from an original beam comprises:

[0284] a semi-transparent mirror separating the original beam into an unmodified beam and a secondary beam.

[0285] a lens focussing the secondary beam in a focussing plane.

[0286] a mirror placed at the point of focus, which reflects the opposite beam back to said lens.

[0287] The fact that the mirror is placed at the point of focus guarantees that the reflected beam has exactly the opposite direction of the incident beam. The reflected beam passes through the lens again in the opposite direction. The part of this beam which is then reflected again by the semi-transparent mirror has the same characteristics as the unmodified beam but is directed in the opposite direction.

[0288] 3.16. Determination of differences in the position of objectives

[0289] From waves coming from the object and passing through a given objective, it is possible to generate a three-dimensional representation of the observed object. In spatial representation, this representation is relative to a given origin, that will be called characteristic point of the objective. In general, characteristic points of the objectives used do not coincide. Consequently the part of the frequency representation generated from an objective is translated in relation to that obtained from another objective. This translation results in a frequency modulation, the points in the frequency space obtained from a given objective consequently being affected by a variable phase shift corresponding to this modulation. Considering a particular variant of the invention, the microscope comprises means for compensating this translation and generating a representation of the object in which the phase shift affecting each point of the representation is constant. To be able to superimpose the representations obtained from each objective, and in accordance with one variant of the invention, the microscope comprises means for determining the coordinates of the characteristic points of each microscope objective, in a common coordinate system. It is then possible to translate appropriately each representation before superimposing them. This translation in the spatial domain is equivalent to a demodulation in the frequency domain, which can be carried out directly on the plane frequency images.

[0290] 3.16.1. Determination of coordinates of characteristic points of each objective

[0291] According to a variant of the invention, this can be obtained by using the beam centered on a central point of the observed object and its opposite indicator beam. This beam is received on one sensor after having passed through the objectives, and its opposite indicator is received on another sensor. From the beam received on a sensor, the two-dimensional frequency representation of this beam can be obtained and the coordinates of its point of focus can be determined. The point of focus of the beam is the same as that of its opposite indicator beam. The difference between the coordinates of the point of focus of the beam obtained from one objective and those of its opposite indicator obtained from another objective is equal to the difference between the coordinates of the characteristic points of these objectives in a common coordinate system.

[0292] This method can be implemented with any number of objectives, provided that the configuration is such that no group of objectives is optically isolated, i.e. that a given objective group, if it does not include all the objectives used, can always be reached by a beam coming from an objective outside the group. For example, if six objectives are used, they must be grouped two by two: each objective must receive beams coming from the other two objectives.

[0293] This aspect of the invention is implemented in 7.9.1. and in 9.12.

[0294] 3.16.2. Determination of displacements of each objective

[0295] It is generally necessary to move the objectives in order to introduce the observed object. This operation modifies the coordinates obtained and calls for the repetition of the preceding operation. However, the presence of the object disturbs the beam which passes through it and prevents any precise result from being obtained. According to one variant of the invention, this problem is solved by using parallel beams of variable direction. A parallel beam has a point plane frequency image and the value obtained at this point is affected little by the local irregularities of the observed sample.

[0296] The difference between the phase of such a beam received on a sensor before the movement of the objectives and the phase of the same beam after the movement of the objectives depends on the vector characterizing the displacement of the characteristic point of the objective receiving the beam in relation to the characteristic point of the objective from which the beam comes. From these phase differences established for a sufficient number of beams, it is possible by means of an appropriate algorithm to determine this displacement. According to this variant of the invention, the phase of a set of parallel beams reaching a given sensor is consequently measured a first time in the absence of the object and a second time in the presence of the object. From the phase differences and possibly from the intensity ratios thus measured, an appropriate algorithm can recalculate the displacement of the characteristic points of each objective. These phase and intensity differences can be characterized, for each parallel beam, by a complex value obtained as the quotient of the value obtained in the presence of the object by the value obtained in the absence of the object at a corresponding point of the plane frequency image. This value will be called the phase and intensity ratio on a given parallel beam.

[0297] Knowing the initial coordinates of the origin of each representation and its displacement, its current coordinates can be deduced and the difference in position can be compensated. This method can be implemented with any number of objectives, with the same provision as in 3.16.1.

[0298] The first measurement in the absence of the object is carried out, for example, in 7.9.2. and in 9.13. In these two cases, the correction related to the position values determined as indicated in 3.16.1. is described in the same paragraph as that measurement.

[0299] The second measurement in the presence of the object and the calculation of the displacements are carried out for example in 7.11. and in 9.15. In both cases, the absolute positions are calculated directly without intermediate displacement determination, the correction related to the position values having already been performed as indicated above. This determination is coupled with the calculation of the average index of the object whose principle is given in the next paragraph.

[0300] 3.17. Determination of index and thickness of the object

[0301] 3.17.1. Principle

[0302] When two microscope objectives opposite to each other are used and when the observed object forms a layer between two flat cover glasses, the average index of the object, if it differs from the nominal index of the objective (index of optical liquid designed to be used with the objective), creates a spherical aberration which distorts the three-dimensional representations obtained. This spherical aberration is reflected in phase and intensity variations of the beams measured in 3.16.2., these variations depending on the index and thickness of the object. According to one variant of the invention, a program uses the values measured in 3.16.2. to determine simultaneously the displacements of the objectives, and the index and thickness of the object.

[0303] The calculation carried out in 7.11. or in 9.15. is an embodiment of this variant of the invention.

[0304] 3.17.2. Minimization algorithm

[0305] For given values of displacements and of the index and thickness of the object it is possible to calculate the phase and intensity ratios on each parallel beam. According to a particular variant of the invention, the calculation algorithm determines the values of the displacements and the index and thickness of the object which minimize the standard deviation between the theoretical values thus calculated and the values actually measured.

[0306] This algorithm must determine an absolute minimum on a set of five parameters of a noisy function, the standard deviation function exhibiting, even in the absence of noise, local minimum values other than the absolute maximum. This problem consequently lends itself poorly to conventional minimization methods.

[0307] When the values of the parameters are known approximately, the algorithm can maximize, in a first phase, the value at the origin obtained by compensating the phase differences due to these parameters. Maximization of a function being equivalent to a minimization of its opposite, the term maximization will be used only to define the algorithm, but it would also be possible to speak of minimization.

[0308] According to a variant of the invention, the algorithm comprises iterative phases during which it determines, for each iteration, an absolute maximum on a set of vales of the parameters varying discretely, the function to be maximized having previously been filtered to eliminate the frequencies that would cause aliasing when sampling at the variation intervals of the parameters. According to this variant of the invention, the interval is reduced with each iteration and the central point of the set on which the parameters vary during an iteration is the maximum determined in the preceding iteration.

[0309] Such an algorithm makes it possible in general to converge towards the solution despite the local maxima and the large number of parameters.

[0310] Such an algorithm is described in 7.8.2.

[0311] 3.18. Object position determination

[0312] The exact position of the object has no influence on the values measured in 3.16.2. and can consequently not be obtained from these values. On the other hand, it can modify the three-dimensional representations obtained: the spherical aberration affecting a given three-dimensional representation depends on the position of the object. If the index of the object differs from the nominal index of the objectives, this position must be determined.

[0313] The position of the object in relation to the objectives affects the proper superposition of the phase corrected two-dimensional representations. If it is not correctly evaluated and taken into account as a correction factor, abnormal phase differences appear between pairs of phase corrective two-dimensional representations, on the part of the frequency space that corresponds to the intersection of these representations.

[0314] According to a variant of the invention, the measurement of the position of the object in relation to the objectives comprises an acquisition phase during which a series of plane frequency images are acquired, corresponding to a series of illuminating beams of variable orientation. Knowing the position parameters, the index and the thickness of the object, previously calculated by the maximizing algorithm described in 3.17., and knowing the sought position, a program can determine the two-dimensional representations corresponding to each plane frequency image. According to this variant of the invention, the program for determining the position of the object in relation to the objectives is consequently a minimization program that determines the value of the position parameter that minimizes the abnormal phase differences. Such a program is described in detail in paragraph 17.15.

[0315] 3.19. Use of objectives exhibiting aberrations

[0316] The design of an objective without aberrations is difficult. Aberrations increase in proportion to the size of the optical elements used, so that it is difficult to obtain a great working distance. It is also difficult to obtain a high numerical aperture.

[0317] In the present microscope, the complex value of the wave received on a receiving surface is recorded. The frequency representation of part of the wave coming from the object can be reconstituted when this part of the wave coming from the object reaches the receiving surface. The frequency representation of the wave coming from the object is obtained, in every case, by a linear relationship from the wave reaching the receiving surface. The only “indispensable” property of the objective is consequently its ability to pick up a significant part of the wave coming from the object, and transform it into a paraxial beam reaching the receiving surface. An objective having this property can easily be designed and have a high numerical aperture and long working distance. According to one variant of the invention, an objective affected by aberrations greater than the limit imposed by the diffraction is used, and the calculation program reverses the linear relationship between the wave coming from the object and the wave picked up on the receiving surface so as to compensate these aberrations.

[0318] According to a variant of the invention, the microscope objective is designed to comply with the following property:

[0319] (1)—The aberration affecting the image formed in the image plane of the objective must be less than a fraction of the diameter of the observed part of this image.

[0320] The requirement (1) ensures that, for the major part of the object studied, the entire beam coming from a given point reaches the receiving surface. In fact, the spatial extent of the observed sample is limited, depending on the systems, by a diaphragm or by the size of the sensor used. In the presence of spherical aberration, when the distance between a point and the limit of the observed area is less than the characteristic distance of the spherical aberration, part of the waves coming from this point do not reach the receiving surface and the image of this point can consequently not be reconstituted with precision. If the area concerned remains small in size, this drawback is not very troublesome: this is what is guaranteed by compliance with condition (1). If the area concerned were too large, the precision would be affected on the entire image.

[0321] Requirement (1) is similar to the usual requirement concerning spherical aberration, but is considerably reduced. In fact, results of good quality can be obtained with a spherical aberration of the order of 10 or so wavelengths, whereas in a conventional microscope spherical aberration must be a fraction of a wavelength.

[0322] However, an objective not having certain additional properties may be difficult to use. In fact, the linear relation linking the wave received on the receiving surface to the frequency representation of the wave coming from the object may be relatively complex. The algorithmic compensation of aberrations may thus require high computation volumes. According to one variant of the invention, this problem is solved by using an objective having, in addition, the following property:

[0323] (2)—The image, in the back (image) focal plane of a beam which is parallel in the observed object, must be a point image.

[0324] The plane frequency image used in all the embodiments is equivalent to the image formed directly in the rear focal plane. The requirement (2) means that each point of a plane frequency image corresponds to a given frequency of the wave in the object. If requirement (2) were not complied with, it would be necessary to use a generalized algorithm consisting in obtaining the value associated with a given point as a linear combination of the values detected on a set of neighboring points. Requirement (2) is relatively simple to comply with. The use of an objective not complying with condition (2) would only be of limited interest, but would considerably complicate the calculations needed to obtain an image. An example of the use of an objective complying with conditions (1) and (2) is described in paragraph 7.21.

[0325] In the case of an objective complying only with conditions (1) and (2), the image reconstitution algorithm must take into account the relation between the coordinates of a point on the plane frequency image and the corresponding frequency of the wave in the object.

[0326] By B(α) is denoted the image point, in the back focal plane, of a parallel beam in the object and forming in the object an angle a with the optical axis.

[0327] According to a variant of the invention, the algorithms are simplified by using an objective complying in addition with the following condition:

[0328] (3)—The distance between point B(α) and point B(0) must be proportional to sin α.

[0329] Compliance with condition (3) means that the coordinates of a point on the plane frequency image must be directly proportional to the components along two corresponding axes of the wave vector of the wave in the object, which considerably simplifies the calculations. If an objective complies with conditions (1)(2)(3), spherical aberration results only in a phase shift of each point of a given frequency representation, a phase shift which can then be easily compensated. An example of the use of an objective complying with conditions (1)(2)(3) is described in 7.20. Condition (3) is roughly equivalent to the condition of Abbe's sines.

[0330] In an objective complying only with conditions (1)(2)(3), the image is disturbed in the vicinity of the diaphragm over a distance equivalent to the characteristic distance of the spherical aberration. This disturbance is eliminated by the use of a conventional objective rid of any spherical aberration, which thus makes it possible in principle to obtain the best images. However, the use of an objective complying only with (1)(2)(3) allows a significant reduction in the design requirements of the objective. This makes it possible to obtain an objective with a longer working distance, a less expensive objective, or an objective with a higher numerical aperture. This technical solution may thus be preferable in certain cases.

[0331] The use of such a microscope objective requires the algorithmic compensation of phase differences affecting the plane frequency image and which are the consequence of spherical aberration. This calls for the determination of the function characterizing the spherical aberration induced by the microscope. Because the objective exhibits a symmetry of revolution, the phase difference affecting a point of a plane frequency image depends only on the distance between this point and the optical center. The spherical aberration may thus be characterized by a one-dimensional function representing the phase difference as a function of this distance.

[0332] According to a variant of the invention, this function can be obtained by the optical calculation program used for the design of the objective. In fact, this program allows the tracing of beams and can easily be improved to also enable optical path calculations. As the phase differences are proportional to the optical path differences affecting the rays coming from a given point, they can be deduced from geometrical considerations of this type.

[0333] According to a further variant of the invention, this function can be measured by optical means. In accordance with this variant of the invention:

[0334] two identical microscope objectives facing each other are used.

[0335] use is made of an illuminating wave centered on a point of the plane at which a first objective normally forms the image of the observed sample.

[0336] the wave received in the plane at which the second objective forms the observed sample's image is detected.

[0337] In the absence of spherical aberration, the phase of the detected wave must be constant. In the presence of spherical aberration, the phase difference due to the aberration is twice the difference due to a single microscope objective. This makes it possible to obtain the sought function.

[0338] 3.20. Compensation for spherical aberration and position differences

[0339] Spherical aberration due to the refractive index of the object, spherical aberration due to the properties of the objective, and positioning errors relative to the objectives, all result in phase differences applied to the plane frequency image. These phase differences must be corrected in order to obtain a good quality image.

[0340] According to a particular variant of the invention, this corrected is carried out by multiplying each plane frequency image by a correction function taking into account the various parameters determined above. The calculation of such a function is described in 7.16. and the multiplication operation is carried out for example in step 2 of the algorithm described in 7.17. If an objective exhibiting spherical aberration is used, the calculation described in 7.16. must be modified as indicated in 7.20.

[0341] 3.21. Regular sampling

[0342] When two microscope objectives are used, the sampling interval on the plane frequency image generated from one of the receiving surfaces can be taken as the basis for the sampling interval of the three-dimensional representation of the object along two corresponding axes. If no precaution is taken:

[0343] The image points of the illuminating waves on this plane frequency image do not correspond to integer values of the coordinates in pixels.

[0344] If two objectives are used, the sampling interval and the axes on the plane frequency image generated from a receiving surface associated with the objective located opposite do not correspond to the sampling interval and to the axes of the three-dimensional representation of the object.

[0345] The result of this is that the sampling of the three-dimensional representation of the object is not regular. According to one variant of the invention, this sampling is made regular along two axes corresponding to the axes of the plane frequency representations. The quality of the three-dimensional representation of the object is then clearly improved.

[0346] The plane frequency image can be modified in particular, due to imperfections in the optical system, by rotation or by similitude transformation. To obtain a regular sampling, it is necessary to cancel or compensate for these imperfections.

[0347] Further, in Embodiment 4, there must be a point to point correspondence between the different SLMs used and the CCDs. To obtain these correspondences, adjustments of the same type are needed.

[0348] According to a variant of the invention, the microscope consequently comprises one or more optical devices allowing a rotation adjustment of the images generated in the frequency planes and/or one or more devices allowing adjustment by magnifying the images generated in the frequency planes.

[0349] 3.21.1. Adjustment of representation scale (homothetic transformation)

[0350] This adjustment is in fact a magnification adjustment. According to a variant of the invention, the magnification of an image is adjusted by means of an optical system with a variable focal length. Such a system may, for example, be composed of two lenses, a variation in the distance between said lens resulting in a variation in the overall focal length. Such a device is used in Embodiments 4 and 5 and is described in 8.1.4.1.

[0351] 3.21.2. Image rotation adjustment

[0352] According to a variant of the invention, said adjustment is carried out by means of a device consisting of a first group of fixed mirrors and a second group of mirrors, complying with the following conditions:

[0353] the first group of mirrors symmetrizes the wave vector of the incident beam in relation to a given axis.

[0354] the second group of mirrors symmetrizes the wave vector of the incident beam in relation to a second axis.

[0355] the second group of mirrors is mobile in rotation around an axis orthogonal to the plane of these two axes.

[0356] Both groups of mirrors then have the effect of imparting a rotation to the beam represented in a frequency plane, the angle of rotation being twice the angle between the two axes of symmetry. Such a device is used in Embodiments 4 and 5 and is described in 8.1.4.2.

[0357] 3.22. Phase shift system

[0358] The phase shift system used can be a piezoelectric mirror, constituting the most usual solution. However, used at high speed, such a mirror generates vibrations. According to an advantageous variant of the invention, the phase shift system used is a birefringent blade inducing a phase shift of 120 degrees between its neutral axes, preceded by a polarization rotator allowing the orientation of the electric field vector of the beam along one or the other of said neutral axes, and followed by a second polarization rotator for bringing the polarization direction of the beam at the output of the device back to its direction at the input of the device. As this system allows only a phase shift of 120 degrees, it is necessary to combine two in series to obtain a phase shift of −120, 0, or +120 degrees.

[0359] 3.23. Data processine method in the case of a limited RAM

[0360] Three-dimensional frequency representation calculations involve large quantities of data. As these data are normally accessed in a random order, they cannot be stored during calculations on a sequential-access medium such as a hard disk and must be stored on a random-access medium such as an internal computer memory (RAM).

[0361] According to an advantageous variant of the invention, adapted to the case in which the system does not have sufficient RAM to allow storage of all the data, the calculation algorithm is modified so as to process the data block by block, a block corresponding to a large amount of data which can then be stored sequentially on a sequential-access medium and loaded into central memory only during the processing time of said block. For this purpose:

[0362] The modified algorithm carries out, within a three-dimensional space, processing operations horizontal plane by horizontal plane, each horizontal plane being stored on the sequential-access medium in a single block.

[0363] In order to be able to also carry out processing along the vertical dimension, the algorithm incorporates axis exchange phases which make it possible to bring the vertical axis temporarily into a horizontal plane.

[0364] The axis exchange procedure works block by block, the blocks generally having equal or similar dimensions along the two axes to be exchanged and having as byte size the maximum size that can be stored in the central memory of the system (random-access memory—RAM).

[0365] This method is implemented in Embodiment 1 and described in paragraph 5.21.

[0366] 3.24. Images venerated by the microscope

[0367] The three-dimensional representations generated by the present microscope can be stored and transmitted in the form of a three-dimensional array of complex numbers. According to one variant of the invention, it is possible to generate two-dimensional projections or sections representing either the refractive index or absorptivity in the object.

[0368] In the case of a projection, one generates a projection of the three-dimensional image on a projection plane and along a projection direction orthogonal to the projection plane. Each of the projection plane is obtained from all the values of the three-dimensional spatial representation which is located on a line passing through this point and directed along the projection direction.

[0369] According to a variant of the invention, the value associated with each point of the projection plane is obtained by extracting the maximum value of the real or imaginary part or the module from the points of the three-dimensional spatial representation located on the corresponding line.

[0370] According to a variant of the invention, the value associated with each point of the projection plane is obtained by integrating the complex value of the points of the three-dimensional spatial representation located on the corresponding line. It is then possible to display either the real part or the imaginary part of the projection thus obtained. According to this variant of the invention, the projection can be obtained more rapidly as follows, in two steps:

[0371] step 1: extraction, in frequency representation, of a plane passing through the origin and orthogonal to the projection direction.

[0372] step 2: inverse Fourier transformation of this plane.

[0373] The two-dimensional array thus obtained constitutes a projection along the direction having served to extract the frequency plane.

[0374] 3.25. Optical element positioning system

[0375] The embodiments described require the use of many high-precision positioners. These positioners are costly elements ill suited to mass production and capable of losing their adjustment with time.

[0376] According to a variant of the invention, this problem is solved by using removable positioners during the manufacture of the microscope, each element being positioned and then fixed by a binder, for example an adhesive, and the positioner being removed after final solidification of the binder.

[0377] 3.26. Shocks, vibration and dust protection system

[0378] The microscopes described consist of a set of elements fixed to an optical table. During possible transport, shocks, even minor ones, can lead to the misadjustment of the system. During prolonged use, dust may deposit on the different optical elements.

[0379] According to one variant of the invention, the greater part of the optical device is included in a hermetically closed box which is itself included in a larger box, the link between the two boxes being provided by shock absorbers placed on each side of said hermetically closed box. This system protects the microscope from shocks and from dust while providing a good suspension for the optical table.

[0380] FIGS. 1 to 24 relate to a first embodiment. FIG. 1 is a general diagram of the microscope optical system. FIGS. 2 and 3 represent in detail the angular positioner (110) already represented in FIG. 1. FIG. 4 is a general diagram of the vertical mechanical and anti-vibration support of the microscope. FIG. 5 represents in detail a tensioner of FIG. 4. FIG. 6 represents a dimensioning example of the optical part. FIGS. 7 to 9 and 15 and 16 are graphic representations serving as a support for the explanation of the microscope's operating principle. FIG. 10 represents the algorithm of a program allowing the adjustment of piezoelectric control voltages (122). FIG. 11 represents the detailed algorithm of an imaging procedure used in the preceding program. FIG. 12 represents the algorithm of a program for adjusting the intensity attenuator consisting of the polarizer (105) and of the polarization rotator (104) and for obtaining its characteristics. FIG. 13 represents the detailed algorithm of a “low-level” imaging procedure used in the preceding program and in the 2D or 3D image acquisition programs. FIG. 14 represents the algorithm of a focussing program for obtaining a 2D image and focussing the microscope objective (113). FIG. 17 is a support for the explanations concerning the adjustment of the condenser. FIG. 18 represents the algorithm of the three-dimensional image acquisition program. FIG. 19 represents the algorithm of a calculation program generating, from the results of the acquisition, a three-dimensional representation of the object. FIG. 20 represents schematically an operation carried out by the first part of this program. FIG. 21 represents in detail the algorithm of this first part. FIG. 22 represents schematically an operation carried out by the second part of this program. FIG. 23 represents the algorithm of a third part of this program. FIG. 24 represents the algorithm of a last part of this program.

[0381] FIGS. 25 and 26 relate to a second embodiment. FIG. 25 is a general diagram of the optical part of the microscope. FIG. 26 represents the algorithm of the “low-level” image acquisition procedure used.

[0382] FIG. 71 illustrates a specific device used in the adjustment operations for Embodiments 3 to 5.

[0383] FIGS. 27 to 59 relate to a third embodiment. FIGS. 27 and 28 constitute a general diagram of the optical part of the microscope. FIG. 29 shows schematically the optical path of the rays between the objective and the sensor. FIG. 30 represents the beam attenuation device used. FIG. 31 is a schematic diagram illustrating its operation in the case in which there is actually an attenuation. FIG. 32 is a schematic diagram illustrating its operation in the case in which there is no attenuation. FIG. 33 represents the phase shift device used. FIG. 34 is a schematic diagram illustrating its operation. FIG. 35 continues FIG. 34 indicating the phase differences of the various vectors. FIG. 36 continues FIG. 34 indicating the modulus of the different vectors. FIG. 37 represents a basic unit of the beam deflection device used. FIG. 38 represents the beam deflection and switching device, formed by the association of these basic units. FIGS. 39 and 40 are schematic diagrams explaining the operation of a basic unit. FIG. 39 corresponds to one deflection direction and FIG. 40 to the other deflection direction possible. FIG. 41 illustrates the calculation of the deflection of the beam by a prism. FIGS. 42 to 44 illustrate steps in a phase rotator marking procedure. FIG. 42 illustrates the first step and FIGS. 43 and 44 illustrate a second step, in two different cases. FIG. 45 illustrates the calculation of the path difference produced on a parallel beam by an object with a given refractive index and thickness. FIG. 46 illustrates the calculation of the path difference produced on a parallel beam by the displacement of the origin of the reference wave in relation to which the optical path is calculated. FIGS. 47 to 50 and FIG. 60 illustrate an algorithm for calculating the refractive index and thickness values of the sample as well as the displacement of the origin of the reference wave. FIG. 47 corresponds to the highest level of this algorithm and FIG. 50 to the lowest level. FIG. 51 represents, in a three-dimensional space, different vectors used to evaluate the effect on the diffracted wave of the polarization of the illuminating beam. FIG. 52 represents, in the plane of a sensor, vectors deduced from the preceding. FIG. 53 represents an algorithm for obtaining beam deflector control indices based on the coordinates of a point of direct impact sought for the illuminating beam. FIG. 54 illustrates the calculation of the path difference of a wave coming from a point of the object in relation to a reference wave. FIG. 55 illustrates the calculation of the path difference between a wave coming from a point of the object and the reference waves used on the two sensors of the system. FIG. 56 illustrates the path, on one of the sensors, of the point of direct impact of the illuminating wave during an imaging procedure. FIG. 57 represents an algorithm determining the position of the object in relation to the objectives. FIG. 58 illustrates how the three-dimensional frequency representation of the object is obtained by the superposition of two-dimensional representations. FIG. 59 represents a section of the three-dimensional frequency representation of the object.

[0384] FIGS. 61 to 70 and 72 relate to a fourth embodiment of the invention, constituting the preferred embodiment. FIGS. 61, 62 and 63 constitute a general diagram of the optical part of the microscope. FIGS. 64 and 65 illustrate the operation of the beam deflector used in this embodiment. FIG. 66 illustrates the operation of a direct illuminating wave eliminating system. FIGS. 67 and 68 illustrate the principle used to control the beam paths. FIGS. 69 and 70 illustrate a system used to obtain controlled rotation of a beam. FIG. 72 illustrates the image to be obtained in one of the adjustment operations.

[0385] FIGS. 73 to 82 relate to a fifth embodiment. FIGS. 73 and 74 constitute a general diagram of the optical part of the system. FIGS. 75 and 76 illustrate images obtained during an intermediate calculation step. FIG. 77 illustrates an algorithm used to determine the control sequence of a mirror deflecting the beam and to determine the control sequence of a glass stopping the direct illuminating wave. FIGS. 78 and 79 represent images obtained during an adjustment step. FIG. 80 illustrates the principle used to vary the polarization of the illuminating wave and the analysis direction. FIG. 81 represents the points corresponding to different illuminating waves used in a phase correction algorithm. FIG. 82 represents a diaphragm used in this embodiment.

[0386] FIGS. 83 to 85 relate to a device for positioning and fixing in a definitive manner the optical elements. FIG. 83 and FIG. 84 represent parts of this device, and FIG. 85 represents the entire device.

[0387] FIGS. 86 to 88 relate to a device designed to protect the microscope from shocks and dust. FIG. 86 represents a schematic diagram of the device. FIGS. 87 and 88 represent a specific configuration suited to the case of Embodiment 4.

[0388] FIGS. 89, 90 and 91 relate to the use of non-stigmatic objectives. FIG. 89 represents an embodiment of such an objective. FIG. 90 is used as a support for the statement of conditions to be complied with by this objective. FIG. 91 shows a plane frequency image obtained by means of this objective.

[0389] This embodiment is the simplest and is not costly.

[0390] 5.1. Material characteristics

[0391] A laser beam with a 633-nm wavelength polarized in the direction orthogonal to the figure is emitted by the helium-neon laser (100) and passes through the filter (101) made up of zero, one or more stacked filters in tinted glass of the Schoff type. It is then separated into an illuminating beam Fe and a reference beam Fr by the semi-transparent mirror (102). The illuminating beam then passes through a filter (103) of the same type as (101), then through a polarization rotator (104) based on ferroelectric liquid crystals and marketed by the company Displaytech Inc., 2602 Clover Basin Dr., Longmont, Colo. 80503, United States. It then passes through a polarizer (105). The beam then goes through an achromat (106) then a diaphragm (107) and an achromat (108). It is reflected by a mirror (109) fixed on the angular positioner (110) controlled by two stepping motors. It then passes through an achromat (124) and then a condenser (111) and reaches the object (112). The condenser is, for example, a Nikon aplanatic/achromatic condenser of 1.4 aperture designed for a light source located at infinity. The object (112) is a sample placed between plate and cover glass, the absorptivity and refractive index variations of which are relatively small, and a three-dimensional image of which is to be obtained. The achromat (124) is placed as near as possible to the lowest lens of the condenser (111), possibly in the condenser body. The front focal point of the achromat (124) coincide with the rotation center of the mirror (109). Lenses (106) and (108) must be such that the laser beam focuses in the front focal plane of the assembly made up of the condenser (111) and the achromat (124), and have a sufficient aperture upon its arrival in this focal plane. The beam is thus again parallel when it reaches the object (112). The object plane must be horizontal so that the optical oil required to use the objective and the immersion condenser does not flow. The optical axis is thus vertical and the elements of FIG. 1 are fixed on a vertical plate (300) shown in FIG. 4.

[0392] The angular positioner (110) is shown in detail in FIGS. 2 and 3. FIG. 2 represents a plate on which the mirror (109) is glued, in a bottom view. FIG. 3 represents the entire device seen from the side. The plate (200) is placed on mobile contact pads (205) and (212) whose contact points with the plate are (203) and (204), and a fixed contact pad whose contact point is at (201), the plate (200) being slightly hollowed at this point. It is maintained by a spring (210) fixed to the plate at (202) and to a fixed pad (211). The mobile contact pad (205) moves vertically in translation and is incorporated in a single-axis actuator of conventional type: it is blocked in rotation by a flexible element (207) and is driven in translation by the rotation of the rod (206) coming out of the stepping motor (208) which is, for example, a 400 steps/rev. stepping motor. The drive is obtained, for example, by means of a screw thread. The contact pad (212) is likewise driven by the rotation of the motor (213). The fixed contact point (201) must be in the optical axis of the objective (113) and the center of the mirror must be on this contact point. The motors (208) and (213) are fixed on a plate (209) itself fixed on the support plate (300). The center of rotation of the mirror, which must be positioned on the optical axis, is a point of the reflecting surface of the mirror located on a line orthogonal to the plane of the mirror and passing through the center of (201).

[0393] The wave coming from the object (112) passes through the microscope objective(113). This objective is an aplanatic objective (giving a flat image of a plane), with a high aperture (for example 1.25), of the immersion type, and forming a magnified image of the object at a finite distance.

[0394] In the plane in which the objective normally forms the image of the object to be observed, a diaphragm (114) is inserted and allows spatial filtering of the image. Behind this plane is positioned an achromat (115) whose front focal plane must coincide with the rear focal plane of the objective (113). A second achromat (117) whose rear focal plane is in the plane of a CCD sensor (118) forms in the plane of this CCD the image of the rear focal plane of the objective (113). The CCD (118) is incorporated in a camera (119) delivering an analog video signal and a pixel clock.

[0395] The reference beam first passes through a filter (120) of the same type as (101) and is then reflected by a mirror (121) mounted on the mobile end of a piezoelectric translator (122). It then passes through a lens (123) which focuses the beam at a point. The diverging beam coming from this point is reflected partially by the semi-reflecting mirror (116), thus superimposing it on the beam coming from the object, enabling the recording of their interferences on the CCD (118). The point of focus of the beam coming from the lens (123) must have a virtual image after reflection on the semi-transparent mirror (116) at the center of the image of the diaphragm (114) through the achromat (115). The piezoelectric translator (122) permits the modulation of the reference beam phase.

[0396] The manual positioners used in the system are not represented in the figure. The microscope objective (113) is mounted on a focussing device. The laser (100) is mounted on a two-axis positioner permitting the direction to be adjusted. The fixed part of the piezoelectric stack (122) is mounted on a two-axis positioner allowing rotation in relation to an axis orthogonal to the plane of the figure and passing through the center of the mirror (121), and in relation to a second axis located in the plane of the mirror (121), orthogonal to the first axis and passing through the center of the mirror (121). The condenser is mounted on a three-axis translation positioner. The camera (119) is mounted on a three-axis translation positioner. The angular position of the semi-transparent mirrors (102) and (116) can be adjusted manually. The lens (106) can be translated along its axis. The object (112) is fixed on a two-dimension positioner enabling it to be moved in a horizontal plane. The diaphragm (114) can be moved in a horizontal plane.

[0397] The assembly is fixed on the support plate (300), on the side of the plate opposite the viewpoint of FIG. 4. This plate is fixed to two triangular plates (301) and (302) themselves fixed to a square base (303). The plate (300) is also fixed directly to the square base (303). Plates (300)(301)(302)(303) are in rigid aluminum alloy AU4G, for example 20 mm thick. The attachment of the plates is possible by means of screws and tapped holes, and must be done in a sufficient number of points to ensure perfect overall rigidity. This makes it possible to keep the system vertical while providing sufficient rigidity. The assembly is placed on an anti-vibration support consisting of a granite plate (304) 30 mm thick placed on a van inner tube inflated to a low pressure (305) which dampens the vibrations and which is itself placed on a rigid wooden table (306). A rigid wooden frame (311) is fixed on top by means of uprights (307) (308)(309)(310) to the table (306). The entire wooden construction is reinforced so as to be perfectly rigid. The top of the plate (300) is connected by tensioners (312)(313)(314)(315) to the corners of the rigid frame. Each tensioner, shown in detail in FIG. 5, consists of a set of rubber bands (316) tensioned between two rings (318)(317), these rings themselves being fixed to the plate (300) and to the frame (311) by means of cords (320)(319), the assembly being placed under tension. The inner tube (305) makes it possible to have, for the suspended AU4G assembly, small resonance frequencies for translation movements, of the order of 2 Hz. The tensioners (312) to (315) allow the swing of the assembly to be limited. The swinging frequency can be evaluated simply by imparting a light swinging movement to the assembly and measuring the time required to have, for example, ten swings. The swinging frequency is adjusted by modifying the number of rubber bands used for each tensioner. The higher the number, the higher the swinging frequency. It must be adjusted so that the swinging frequency is of the same order as the resonance frequency for translation movements, i.e. about 2 Hz.

[0398] The mounting of the different elements on the plate (300), and in particular the mirrors and the semi-transparent mirrors, must be carried out so as to ensure maximum overall rigidity. All usual precautions must be taken so as to limit vibrations.

[0399] The following description relates to a particular practical design example for the device. The distances mentioned are represented in FIG. 6. The dimensions given are approximate and some of them must be corrected in an adjustment phase. The lens (106) is an achromat with a focal length of 10 mm. The lens (108) is an achromat of 24-mm diameter and 120-mm focal length. The distance D7 between (106) and (108) is 250 mm. The distance D6 between the lens (108) and the center of the mirror (109) is about 100 mm. The distance D5 between the center of rotation of the mirror (106) and the achromat (124) is 120 mm. The achromat (124) is placed about 5 mm from the lowest lens of the condenser, in the body of the condenser. The condenser is a clear-bottom achromatic/aplanatic immersion type condenser with an aperture of 1.4, for example, the Nikon model. The microscope objective is an ×100 planachromatic objective with an aperture of 1.25, finite distance, forming the image 160 mm from the shoulder of the objective, with a focal length of about 1.8 mm, for example the Zeiss model. The distance D4 between the shoulder of the objective and the diaphragm (114) is 160 mm. The distance D3 between the diaphragm (114) and the achromat (115) is 20 mm. The achromat (115) has a focal length of 200 mm and a diameter of 30 mm and its most curved side is oriented towards the semi-transparent mirror (116). The achromat (117) has the same characteristics and its most curved side is also oriented towards the mirror (116). The distance D2 between the two achromats is 85 mm, allowing the insertion of a semi-transparent mirror (116) of sufficient dimensions. The distance between the achromat (117) and the CCD (118) is 200 mm. The lens (123) has a diameter of 4 mm and a focal length of 6 mm. The distance D9 between this lens and the optical axis is about 70 mm. The distance D8 between the achromat (115) and the center of the semi-transparent mirror (116), located on the optical axis, is about 45 mm. The laser (100) is a helium-neon laser with a wavelength in vacuum λ=633 nm, polarized in the directional orthogonal to the figure, with a power of about 0.5 mW, and a beam diameter of 0.5 mm. The CCD sensor is a square pixel sensor, the surface of the pixel being about 8.5×8.5 micrometers, and the useful surface in pixels having dimensions at least equal to 512×512 pixels. The camera delivers a CCIR video signal and a pixel clock, and its exposure time is equal to half the duration of a field, i.e. {fraction (1/50)} second for a non-interleaved CCIR camera whose fields last {fraction (1/25)} second. This gives a delay between the end of a field and the start of the next exposure period, a delay which can be used to modify the illumination conditions without any influence by the transition on the image. The piezoelectric positioner (122) is a prestressed piezoelectric ‘stack’ in the form of a cylinder whose body is fixed and whose end moves 15 micrometers for an applied voltage of 100 volts.

[0400] The calculation system is, for example, a ‘PC’ type computer equipped with appropriate acquisition and control cards and possibly with additional computation resources, operating for example under the Windows 95 operating system. The video signal acquisition card, operating in real time, samples the signal on 8 bits and acquires images of hpix×vpix size in which hpix and vpix are higher than 512 and multiples of 4. The pixels are sampled according to the pixel clock, hence corresponding exactly to the pixels of the CCD. The piezoelectric positioner is controlled directly by a digital/analog conversion card delivering a signal included, for example, between zero and U_max, for example with U_max=10 volts. A resistance is inserted between the output of the conversion card and the terminals of the piezoelectric actuator so as to limit the current. Its value is set so that the rise time of the voltage at the terminals of the actuator (122) is about 1 ms. The polarization rotator (104) is equivalent to a half-wave plate whose axis can turn and has two equilibrium positions separated by an angle of 45 degrees. If it is positioned so that the polarized beam is parallel to this axis in one of the equilibrium positions, it will cause the beam to turn 90 degrees in the other position. The polarization rotator is controlled by applying a bipolar voltage, −5 V, corresponding to one equilibrium position and +5 V to the other. Each terminal of the polarization rotator is connected to an 0/5 V output of a digital output card, and its two positions are controlled by applying in one case 0 V to one output and 5 V to the other, and reversing for the other position. The stepping motors (208) and (212) are also controlled by computer, via an appropriate control card and electronics. The computer is equipped with sufficient internal memory (at least 32 MB) and a hard disk of sufficient size (at least 4 GB).

[0401] 5.2. Miscellaneous conventions

[0402] The following conventions will be used in the rest of this description, as well as in the other embodiments:

[0403] The letter j represents sometimes an index, sometimes the pure imaginary complex number of modulus 1. If there should be any ambiguity, the complex number j will be denoted {tilde over (j)}.

[0404] The sign=symbolizes the assignment of a value right of the sign to a symbol left of the sign, or equality, depending on the situation.

[0405] The expression a+=b means a=a+b

[0406] a multiplied by b is written ab, a.b or a*b

[0407] The modulus of a complex number z is denoted |z| and its conjugate is denoted {overscore (z)}.

[0408] The expression a%b will mean <<a modulo b >>

[0409] If a is a boolean integer, {overscore (a)} is its complement, hence {overscore (0)}=1 and {overscore (1)}=0

[0410] The discrete Fourier transform, in its most usual form, transforms a Dirac located at the origin into a constant and transforms a constant into a point located at the origin. The discrete Fourier transform used throughout this patent transforms a constant into a Dirac located in the middle of the transform array and transforms a Dirac located at this point into a constant. This means that the ‘zero’ in frequency or in position is placed at the middle of the array and not at the origin of the array. This modified transform is obtained from the usual form by carrying out, before and after the transformation, a permutation of indices. An array E of dimension fdim is transformed as follows:

[0411] 1-first permutation: E[i]=E[(i+fdim/2)%fdim]

[0412] 2-usual Fourier transformation of Array E

[0413] 3-inverse permutation: E[i]=E[(i+fdim/2)%fdim]

[0414] where the sign % designates the modulo.

[0415] The two-dimensional Fourier transform of an array of lines and columns is obtained by performing the one-dimensional Fourier transformation defined above on each line of the array, thus generating an intermediate array, and then performing this transformation on each column of the intermediate array to obtain the transformed array.

[0416] Similarly, the three-dimensional Fourier transform consists in carrying out successively, on each axis, one-dimensional transforms extended to the entire array.

[0417] 5.3. Opera tiding principles

[0418] The value of the light wave coming from the object under a given illumination at a point of the CCD sensor is obtained from the recording of three interference patterns received on the sensor, the phase of the reference wave being shifted 120 degrees between each of these patterns.

[0419] If s is the light vibration coming from the object and r is the light vibration constituting the reference wave during the first recording, the total light vibrations reaching the sensor during the three successive recordings are: $s_0=s+r,s_1=s+{\mathrm{re}}^{j\frac{2\pi }{3}},s_2=s+{\mathrm{re}}^{-j\frac{2\pi }{3}}.$

[0420] The intensities recorded successively are thus:

[0421] |s₀|²=|s|²+|r|²+(s{overscore (r)}+{overscore (s)}r) ${\left|s_1\right|}^2={\left|s\right|}^2+{\left|r\right|}^2+\left(s\stackrel{\_}{r}{e}^{-j\frac{2\pi }{3}}+s\stackrel{\_}{r}{e}^{-j\frac{2\pi }{3}}\right)$ ${\left|s_2\right|}^2={\left|s\right|}^2+{\left|r\right|}^2+\left(s\stackrel{\_}{r}{e}^{-j\frac{2\pi }{3}}+s\stackrel{\_}{r}{e}^{-j\frac{2\pi }{3}}\right)$

[0422] It is possible to reverse these formulas, thus obtaining: $s\frac{\stackrel{\_}{r}}{\left|r\right|}=\frac{1}{6\left|r\right|}\left(2{\left|s_0\right|}^2-{\left|s_1\right|}^2-{\left|s_2\right|}^2\right)+j\frac{1}{2\sqrt{3}\left|r\right|}\left({\left|s_1\right|}^2-{\left|s_2\right|}^2\right)$

[0423] The above value is the light vibration coming from the object alone, the phase reference being conventionally equal to zero for a vibration in phase with the reference wave. This calculation consequently allows the reconstitution of the complex value of the wave from the intensity recordings.

[0424] By construction, each point of the CCD sensor corresponds to a pure frequency f_cof the wave coming from the sample. By optical center of the sensor is meant the point of the sensor illuminated by a ray entering the objective along a direction strictly parallel to its axis of symmetry, and by C_x,C_yis denoted its coordinates in pixels. As the objective complies with the sine condition, the deflection at the exit of the objective of a ray originating at a central point of the object is proportional to the sine of its angle of entry into the objective, which is equal to $\sin \varphi =\frac{\sqrt{f_x^2+f_y^2}}{\sqrt{f_x^2+f_y^2+f_z^2}},$

[0425] where (f_x,f_y,f_z) are the coordinates of the spatial frequency vector of the beam at the entry of the objective, whose norm is 1/λ where λ is the wavelength in the observed medium, and whose direction is that of the beam. As the rest of the optical system is paraxial, the deflection of the point of arrival of the beam on the sensor in relation to the optical center of the sensor is also proportional to this value, and hence the coordinates (i−C_x,j−C_y) of this point in relation to the optical center are proportional to (f_x,f_y). If the sensor has square pixels, the frequency vector at the entry of the objective of the ray which illuminates the point of the sensor having coordinates (i,j) is thus equal to: $f_c\left(i,j\right)=\frac{1}{K\lambda }\left(i-C_x,j-C_y,\sqrt{K^2-{\left(i-C_x\right)}^2-{\left(j-C_y\right)}^2}\right)$

[0426] where K is a constant to be determined. This frequency vector will be called “characteristic frequency” of the corresponding point.

[0427] Knowing the characteristic frequency of each point of the sensor and the light vibration coming from the object and received at this point, one thus obtains, for a given illumination of the sample, the frequency representation of the wave coming from this sample.

[0428] When a sufficiently thin sample which is not highly absorbing and which exhibits low index variations is traversed by a parallel laser beam, each point of the sample is subjected to a light vibration Ae^j2πf.rwhere f_eis the spatial frequency vector of the illuminating beam and r is the position vector at the considered point of the object, the origin being taken at the virtual origin of the reference wave. The absorptivity and index variations of the sample result in the appearance of a secondary wave which is superimposed on the illuminating wave. A small volume dV creates a secondary wave Ae^j2πf.ru(r)dV where u(r) is a complex coefficient whose real part depends on the local absorptivity of the sample and whose imaginary part depends on the index of the sample. The three-dimensional image of the sample generated by this microscope is the set of the values u(r) at each point of the sample.

[0429] When a point of the sample, of position vector r, emits a light vibration s(r) locally, the light vibration received on the sensor at a point of characteristic frequency f_cis then equal to s(r)e^−j2πf.r. A small volume dV of the object, of position vector r, illuminated by a plane wave having a spatial frequency f_e, thus creates, at the point of sensor having a characteristic frequency sensor f_c, a vibration Ae^{j2π(f−f).r}u(r)dV which is superimposed on the main vibration. Integrated over the entire object, the vibration received at a point of the sensor is hence equal to: s(P)=∫∫∫Ae^{j2π(f−f).r}u(r)dV. This vibration is thus an element of the Fourier transform of the function u(r), corresponding to the Fourier frequency f_t=f_e−f_c. The principle of this microscope is to record this vibration for a set of frequencies f_eand f_c, and then to reconstitute the frequency representation of u(r) and finally u(r) by inverting the Fourier transform.

[0430] The samples studied are not highly absorbent and exhibit small variations in the refractive index. Consequently, the illuminating wave remains very intense and illuminates a point of the CCD, which will be called the point of impact of the illuminating wave and which is the point of the CCD at which the received light vibration is highest. If this point has the coordinates (imax,jmax) then the frequency of the illuminating wave is: $f_e=\frac{1}{K\lambda }\left(\mathrm{imax}-C_x,\mathrm{jmax}-C_y,\sqrt{K^2-{\left(\mathrm{imax}-C_x\right)}^2-{\left(\mathrm{jmax}-C_y\right)}^2}\right)$

[0431] FIG. 7 shows the set (500) of all characteristic frequencies corresponding to the points of the sensor. This set is a sphere portion of radius 1/λ, limited by the aperture of the objective, centered on the optical axis (501) of the system. An example of illuminating frequency vector (502) is superimposed on this set. Finally, the set (503) of the corresponding frequencies f_t=f_c−f_eis deduced therefrom. When the illuminating frequencies (502) are made to vary along a circular arc (504) as indicated in FIG. 8, each position of the illuminating frequency vector corresponds to a set of frequencies f_tforming a portion of a sphere. FIG. 9 shows a section of a set of such sphere portions (505) (506) and others, generated when the illuminating frequency vector is moved on the circular arc (504). When the movement of the illuminating frequency vector on the circular arc (504) becomes continuous, a volume is generated whose sectional view is represented in FIG. 15. In a top view, the circular arc (504) is represented by the segment (1102) of FIG. 16. When the extremity of the illuminating frequency vector passes through several circular arcs so as to generate a path represented in a top view in FIG. 16, the generated volume is roughly that which would be generated by rotating the surface (1100) around the vertical axis (1101). One thus obtains a three-dimensional frequency representation (and no longer a two-dimensional representation as in the case in which a single recording is used). From this three-dimensional representation in the frequency space, it is possible to generate the function u(r) by taking the inverse Fourier transform.

[0432] To obtain the frequency representation of u(r) from the set of two-dimensional frequency representations, at each point will be calculated the average value of the two-dimensional frequency representations reaching this point.

[0433] The calculation of the frequency representation of u(r) and then finally of u(r) can be carried out for example in six steps:

[0434] Step 1: For each illuminating wave, the two-dimensional frequency representation of the received wave is determined, which is a sphere portion of a three-dimensional space.

[0435] From a point of the CCD sensor having the coordinates (i,j), one obtains a point of the representation of the wave coming from the object, this point having the frequency $f_c\left(i,j\right)=\frac{1}{K\lambda }\left(i-C_x,j-C_y,\sqrt{K^2-{\left(i-C_x\right)}^2-{\left(j-C_y\right)}^2}\right)$

[0436] and having as complex value the complex value obtained at the considered point of the CCD sensor by the formula $s\frac{\stackrel{\_}{r}}{\left|r\right|}=\frac{1}{6\left|r\right|}\left(2{\left|s_0\right|}^2-{\left|s_1\right|}^2-{\left|s_2\right|}^2\right)+j\frac{1}{2\sqrt{3\left|r\right|}}\left({\left|s_1\right|}^2-{\left|s_2\right|}^2\right).$

[0437] Step 2: For each illuminating wave, one determines the frequency f_eof the illuminating wave by determining the coordinates (imax,jmax) of the point of maximum intensity on the CCD sensor and applying the formula: $f_e=\frac{1}{K\lambda }\left(\mathrm{imax}-C_x,\mathrm{jmax}-C_y,\sqrt{K^2-{\left(\mathrm{imax}-C_x\right)}^2-{\left(\mathrm{jmax}-C_y\right)}^2}\right)$

[0438] Step 3: For each illuminating wave, one translates the representation obtained after Step 1, of a vector−f_ewhere f_eis the frequency of the illuminating wave obtained after Step 2.

[0439] Step 4: For each illuminating wave, the two-dimensional frequency representation obtained after Step 3 is divided by its value at the point of coordinates (0,0). This step constitutes the phase correction operation described in 3.5.3. and is indispensable so that the two-dimensional frequency representations can be superimposed in a coherent manner.

[0440] Step 5: All the two-dimensional frequency representations obtained after Step 4 are superimposed, obtaining the frequency representation of u(r). The value assigned to a point not reached is 0 and the value assigned to a point reached is the average of the values at this point of each two-dimensional frequency representation reaching this point.

[0441] Step 6: A three-dimensional inverse Fourier transformation of the frequency representation is carried out, finally obtaining the function u(r) in spatial representation.

[0442] In practice, the steps will be carried out in a different order and in a modified form in order to optimize the calculation time and to limit the required memory space. The method actually used includes two steps, equivalent to the preceding six steps:

[0443] Acquisition phase: For each illuminating wave, this phase determines the plane frequency image obtained on the CCD sensor by applying, at each point, the formula $s\frac{\stackrel{\_}{r}}{\left|r\right|}=\frac{1}{6\left|r\right|}\left(2{\left|s_0\right|}^2-{\left|s_1\right|}^2-{\left|s_2\right|}^2\right)+j\frac{1}{2\sqrt{3\left|r\right|}}\left({\left|s_1\right|}^2-{\left|s_2\right|}^2\right).$

[0444] Also determined are the coordinates (imax,jmax) of the point of maximum intensity on the CCD sensor, and the entire frequency plane image obtained on the CCD sensor is divided by its value at the point of coordinates (imax,jmax).

[0445] Three-dimensional calculation phase: From each illuminating wave characterized by the values (imax,jmax), and from each point of coordinates (i,j) of the sensor, is obtained a point of the three-dimensional frequency representation with the frequency $f_t=\frac{1}{K\lambda }\left(i-C_x,j-C_y,\sqrt{K^2-{\left(i-C_x\right)}^2-{\left(j-C_y\right)}^2}\right)-\frac{1}{K\lambda }\left(\mathrm{imax}-C_x,\mathrm{jmax}-C_y,\sqrt{K^2-{\left(\mathrm{imax}-C_x\right)}^2-{\left(\mathrm{jmax}-C_y\right)}^2}\right)$

[0446] When a point is not reached, a zero value is assigned. When it is reached several times, the average of the values obtained each time is assigned to it. When this operation has been carried out for all the illuminating waves and all the points of the sensor, the thee-dimensional inverse Fourier transform can be taken.

[0447] This method raises a practical problem, namely that the point illuminated directly by the beam passing through the sample is illuminated much more intensely than the points corresponding to the diffracted wave. The three-dimensional representation generated when imaging contains basically the frequencies near that of the illuminating beam, the other frequencies being smothered in the noise. To remedy this drawback, use is made of a device allowing controlled attenuation of the beam. The part of the frequency representation corresponding to the frequencies on which the intensity is high is obtained with a strong attenuation, and that corresponding to the other frequencies is obtained with a weak attenuation. The values obtained with a strong attenuation are then multiplied by a complex coefficient characteristic of the phase shift and of the ratio of the illuminating wave amplitudes between the two positions of the controlled attenuation device. This controlled attenuation device consists of the polarization rotator (104) and the polarizer (105).

[0448] The three-dimensional representation obtained from images of size hpix×vpix corresponds to large file sizes. In order to limit the size of files and the calculation time, the size of the images will be divided by two by averaging during the three-dimensional representation acquisition procedure. This is equivalent to grouping the pixels 4 by 4, a group of 4 pixels on the original image being equivalent to an actual pixel used for the calculation. The size of the observable object is of course reduced accordingly. The values of C_x, C_yand K are divided by 2 to take into account the new coordinate system. This lniitation in the size of the observed image is of course optional.

[0449] The use of the microscope proper comprises:

[0450] a focussing phase on the sample, described in paragraph 5.17.

[0451] a condenser position adjustment, described in paragraph 5.18.

[0452] a filter adjustment, described in paragraph 5.19.

[0453] the acquisition phase for the standard two-dimensional frequency representations, described above, and described in detail in paragraph 5.20.

[0454] the three-dimensional calculation phase, described above, and described in detail in paragraph 5.21.

[0455] a display phase described in paragraph 5.22.

[0456] Before being able to use this microscope, several adjustments must be made:

[0457] The adjustments of the manual positioners, described in paragraphs 5.6, 5.7, 5.8, 5.9, allow the correct adjustment of the beam path so that the image of a plane wave on the camera is actually a point image, and that the condenser indeed forms a parallel beam at its exit. In particular, the adjustment described in paragraph 5.8. allows a first adjustment of the condenser position.

[0458] The adjustment described in paragraph 5.10. allows the number of motor steps per pixel to be obtained, useful for controlling the beam deflection mirror (109).

[0459] The level of the reference wave is adjusted as indicated in paragraph 5.11.

[0460] The adjustment described in paragraph 5.12. makes it possible to obtain appropriate control voltages for the piezoelectric actuator.

[0461] The adjustment described in paragraph 5.13. allows the adjustment of the beam attenuator and makes it possible to obtain the attenuation and phase shift constants characterizing it.

[0462] The adjustment described in paragraph 5.14. enables the constant K to be obtained.

[0463] The aperture of the diaphragms and the position of the semi-transparent mirror (116) must be adjusted so as to obtain a centered image without aliasing. This adjustment is described partly in paragraph 5.14 and is completed in paragraph 5.15.

[0464] The reference wave must be recorded, as described in paragraph 5.16.

[0465] 5.4. Handling of filters

[0466] Adjustment operations require constant handling of filters to adjust the intensity received on the sensor. These handling operations are not systematically described in the rest of the text.

[0467] The filter at (120) determines the intensity of the reference wave. Its value is determined in a particular adjustment phase. Subsequently, when a reference wave is necessary, the filter thus determined is inserted. When the reference wave must be eliminated, an opaque element is inserted at (120).

[0468] The filter at (103) determines the intensity of the illuminating wave. Its value depends on the ongoing adjustment operations. For most of the operations, the filter is adjusted so that the intensity received on the sensor (118) is high, without reaching saturation. For certain operations, the sensor is saturated. For others, the illuminating wave must be eliminated, which is done by inserting an opaque element.

[0469] The filter at (101) is used only for adjustment operations requiring visual monitoring of the beam, identified by its diffusion disk on a piece of white paper. The filter is then adjusted so that the diffusion disk is visible without being dangerous to the eye.

[0470] If only the illuminating wave is present, the term relative intensity of the image received on the CCD sensor will refer to the ratio of the intensity received on the sensor to the intensity at the output of the filter (103). In an operation in which the aim is to maximize the relative intensity received on the sensor, it is necessary to change filters regularly in order to keep the intensity at a level measurable by the sensor.

[0471] 5.5. Programs currently used

[0472] Certain simple programs are frequently used during adjustment, without this being indicated each time:

[0473] Displacement of the mirror (109): as this mirror is motor-driven, a program is needed in order to modify its position. This program asks the user for a number of steps and an axis number corresponding either to the motor (213) or to the motor (208), and then has this motor carry out the number of steps requested.

[0474] Display of image received on the sensor: a program permits the direct display on the computer screen of the image received on the sensor (118).

[0475] Display of image and characteristics of the maximum: this program allows the direct display on the computer screen of the image received on the sensor (118). It also displays the maximum value detected by the sensor, the coordinates of the corresponding point, and the ratio between the intensity of this point and the sum of the intensities of its 8 neighbors. This program is used to check the appearance of an image, to check the non-saturation of the sensor (maximum value lower than 255), to determined the coordinates and the value of the maximum, to evaluate the point nature (punctuality) of this maximum by direct observation of the image and by the use of the displayed values: the (relative) intensity of the maximum must be as high as possible, as must also the ratio of its intensity to that of its neighbors.

[0476] 5.6. Adjustment of position of laser (100) and mirror (121)

[0477] In a first phase, the illuminating wave is eliminated and the position of the sensor (100) is set so as to actually aim at the center of the mirror (121), which is verified by following the path of the beam by means of a piece of paper enabling it to be displayed. The position of the mirror (121) is then adjusted so that the reference beam actually passes through the lens (123) and reaches the camera. The reference beam must be centered on the sensor (118).

[0478] 5.7. Translation adjustment of camera position and adjustment of mirror (102)

[0479] The position of the camera is adjusted in translation by sending a parallel beam directly on to the microscope objective. For this purpose, elements (106)(108)(111)(105)(104) are temporarily removed, the reference wave is eliminated, and the microscope objective is placed in a position roughly focussed on the object. The object (112) used is a transparent plate and optical oil is placed between (112) and (113). The angular position of (102) is then adjusted so that the beam reaches directly the center of the mirror (109). The position of the mirror (109) is adjusted so that the parallel beam enters directly into the objective (113) and is fine-adjusted so as to maximize the relative intensity of the signal received on the CCD sensor. The position of the CCD sensor is then adjusted in translation in the direction of the optical axis so that the image produced is a perfect point image, then adjusted in translation in the directions orthogonal to the optical axis so that this point is at the center of the useful area of the sensor. It is finally re-adjusted in translation in the direction of the optical axis.

[0480] 5.8. Condenser (111) position adjustment and obtaining the position of the optical center

[0481] The elements (106)(108)(111) are putback in place. Oil for the microscope is placed between (111) and (112) and between (112) and (113). A piece of white cardboard is placed on the mirror (109) to diffuse the light. The diaphragm (107) is open to the maximum position and the aperture of the diaphragm (114) is about 6 mm. The microscope objective is placed in a roughly focussed position. The position of the condenser (111) is adjusted so as to obtain on the CCD a slightly granular clear disc with the highest possible radius and with a roughly constant intensity over the entire disc. In FIG. 17, the useful area of the CCD (1200) is represented, and the clear disc (1201) stands out against this black background. A specific program is then used to determine the coordinates C_x, C_yof the optical center and the radius R of the disc. This program determines:

[0482] the lines (1206) and (1205) constituting the right and left limits of the disc (1201).

[0483] the line (1207) defined as the middle of the lines (1206) and (1205).

[0484] the lines (1202) and (1203) constituting the upper and lower limits of the disc (1201).

[0485] the line (1204) defined as the middle of the lines (1202) and (1203).

[0486] the optical center, which is the intersection of the lines (1204) and (1207)

[0487] the radius R of the disc, equal to half the distance between the lines (1205) and (1206).

[0488] 5.9. Adjustment of lens (106) position

[0489] The piece of white cardboard placed on the mirror (109) is removed. The position of the mirror is modified in order to bring the illuminated point on to the edge of the disc previously obtained. The position of the lens (106) is then adjusted along its axis so as to have the most perfect possible point image on the CCD sensor.

[0490] 5.10. Determining the number of steps per pixel

[0491] The piece of paper blocking the mirror is then removed. The motor is moved along an axis by a known number of steps. The position in pixels of the point of maximum intensity is noted before and after the movement, and the number of pixels covered is deduced therefrom. The ratio $\mathrm{steps\_per}\mathrm{\_pixel}=\frac{\mathrm{number}\mathrm{of}\mathrm{steps}}{\mathrm{number}\mathrm{of}\mathrm{pixels}}$

[0492] is then calculated. The same is done on the other axis and the smallest ratio obtained is used.

[0493] 5.11. Reference wave level adjustment

[0494] If we consider a reference wave and an illuminating wave having the same maximum intensity upon their arrival on the sensor, adding up in amplitude when they are in phase, the non-saturation requirement of the sensor is that the common amplitude of the two waves is half the amplitude saturating the sensor, or equivalently that the common intensity of the two waves is one-fourth the intensity saturating the sensor. To adjust the level of the reference wave at this value, the illuminating wave is eliminated and the value of the filter (120) is adjusted to obtain an image whose maximum level is about one-fourth the maximum level authorized by the acquisition card, i.e. in the case of a sampling on 8 bits of the video signal, a level of about 64. Before imaging, the maximum level of the illuminating wave must be adjusted in the same manner.

[0495] 5.12. Piezoelectric actuator control voltage adjustment

[0496] This calibration of the piezoelectric actuator can be performed outside the system using a known interferometric method. Three positions of the actuator are used. The movement of the mirror between each position must be $\frac{\lambda }{3\sqrt{2}}.$

[0497] The control voltages corresponding to each position must be determined, the actuator having a regular cycle to prevent hysteresis effects.

[0498] However, it is also possible to carry out this adjustment while the actuator is in place. This allows compensation for the inaccuracy on the orientation of the mirror and provides a simple calibration procedure not requiring any specific equipment.

[0499] This is done by using a program which varies the control voltages and which is described by the algorithm of FIG. 10. Before starting the program, the position of the mirror (109) must be adjusted so that the point produced on the CCD sensor using an object consisting of a completely transparent plate is at the center of the sensor. To utilize this program, the object used must be highly diffusing. Use may be made, for example, of a piece of white paper soaked in gelatin and then placed between the plate and the cover glass. The paper must be sufficiently thick to stop the direct beam and sufficiently thin to allow the passage of a diffused beam. The diaphragm (107) is at its maximum aperture and the diaphragm (114) is adjusted for an aperture of about 0.8 mm. The reference wave then interferes on the CCD sensor with the clear disc produced by the wave coming from the object. The intensity of the illuminating wave must be adjusted so that the sensor is near its saturation limit.

[0500] If the maximum voltage applied to the actuator is Umax, the voltages corresponding to the phase shifts of $-\frac{2\pi }{3},0,\frac{2\pi }{3}$

[0501] will be respectively Umax/2−diff_bas, Umax/2, Umax/2+diff_haut, where diff_haut and diff_bas are chosen to produce the indicated phase shifts. To prevent any hysteresis effect, with each acquisition the voltage is initialized to 0, the different images are acquired in the increasing order of voltages applied to the actuator, and a final voltage Umax is finally applied so that the same cycle is always used. This cycle must also be used in the normal operating phase. Imaging with the phase shifts indicated above allows the calculation of a frequency representation by applying to each pixel P the formula $S\left(P\right)=\left[\frac{1}{6}\left(2I\left(P,0\right)-I\left(P,\frac{2\pi }{3}\right)-I\left(P,-\frac{2\pi }{3}\right)\right)+j\frac{1}{2\sqrt{3}}\left(I\left(P,\frac{2\pi }{3}\right)-I\left(P,-\frac{2\pi }{3}\right)\right)\right]$

[0502] in which the reference wave has been replaced by a constant and the expression I(P,α) designates the intensity recorded at the point P for a phase shift α.

[0503] The adjustment of the actuator control voltages consists in evaluating diff_haut and diff_bas. The principle is to obtain two frequency representations shifted by π/3 in relation to each other. The second image can be phase corrected by multiplying it by e^−jπ/3and the average difference between the first image and this phase-corrected image can be calculated. This difference is minimum when the voltages diff_haut and diff_bas are correctly adjusted. The phase shifts allowing the acquisition of the second elementary frequency image are thus $-\frac{2\pi }{3}+\frac{\pi }{3},\frac{\pi }{3},\frac{2\pi }{3}+\frac{\pi }{3}$

[0504] and correspond respectively, as a first approximation, to the voltages Umax/2−dif_bas+diff_bas/2, Umax/2+(diff_bas+diff_haut)/2, Umax/2+diff_haut+diff_haut/2. The adjustment program calculates the average difference for series of values of diff_haut and diff_bas and chooses those which correspond to the minimum average difference. Its algorithm is shown in detail in FIG. 10.

[0505] The basic steps of this algorithm are:

[0506] (600): Acquisition of images. The acquisition procedure is depicted in FIG. 11. By “images” is meant here six interference patterns received consecutively by the CCD sensor. This procedure always carries out the same cycle beginning with the voltage 0 and ending with the voltage Umax. The waiting time after the application of a zero voltage to the piezoelectric enables the latter to become stabilized. The waiting time before the start of the acquisition avoids the acquisition of a frame which may have been exposed before the application of the desired exposure conditions. The end of the exposure of an image is indicated on the acquisition cards by the start of the transfer of the corresponding image. The use of an exposure time shorter than the image transfer time prevents the image from being affected by transient states. Giving the process maximum priority prevents the disturbance of acquisition by other system tasks under a multitask operating system. The six images are acquired successively and in real time (no image lost) by the acquisition card in order to minimize the time during which vibrations can affect the result. Each image has the horizontal dimension of hpix and vertical dimension of vpix. During acquisition, images are transferred automatically by the acquisition card into an array which is reserved for them in the computer's central memory. Following the acquisition procedure, there is a resulting array I[a,b,i,j], the index a corresponding to the phase difference, the index b corresponding to the image (phase-shifted or not phase-shifted), the indices i and j being the coordinates in pixels and varying respectively from 0 to hpix−1 and from 0 to vpix−1.

[0507] (601): Calculation of the two frequency representations. These are stored in the arrays of complexes S0[i,j] and S1[i,j], applying the formulas: $\mathrm{S0}\left[i,j\right]=\frac{1}{6}\left(2I\left[0,0,i,j\right]-I\left[1,0,i,j\right]-I\left[2,0,i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3}}\left(I\left[1,0,i,j\right]-I\left[2,0,i,j\right]\right)$ $\mathrm{S1}\left[i,j\right]=\frac{1}{6}\left(2I\left[0,1,i,j\right]-I\left[1,1,i,j\right]-I\left[2,1,i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3}}\left(I\left[1,1,i,j\right]-I\left[2,1,i,j\right]\right)$

[0508] (602): The program modifies the array S1 by multiplying each of its elements by e^−jπ/3.

[0509] (603): The program calculates the maximum value $\mathrm{mod\_max}=\underset{\underset{0\le j\le \mathrm{vpix}-1}{0\le j\le \mathrm{hpix}-1}}{\max }\left|\mathrm{S0}\left[i,j\right]\right|$

[0510] of the modulus on S0.

[0511] (604): The program calculates the standard difference between the two arrays as follows:

[0512] The program initializes ecart and nombre_valeurs to 0 and goes through all the points i,j by testing the condition (605): |S0[i,j]|≧0,5 mod_max. Each time this condition is met, it carries out the following operations (606):

[0513] ecart+=|S0[i,j]−S1[i,j]|

[0514] nombre_valeurs+=1

[0515] When the program has gone through the indices (i,j) it divides the ecart by nombre valeurs (607), giving it the average difference. This difference consequently integrates only values for which the illuminating wave is sufficiently strong, in order to prevent too noisy a result. It is stored in an array.

[0516] (608): The program goes on to the next value of the pair (diff_haut,diff_bas) and reiterates the operations until the series of differences has been calculated.

[0517] (609): The program applies a low-pass filter to the array obtained in order to limit the noise due to system vibrations. The low-pass filter used is represented in the frequency domain by a step going from a value of 1 at low frequencies to a value of 0 at high frequencies, and its bandwidth is determined empirically to have a good limitation of noise without excessively deforming the curve.

[0518] (610): The array is represented graphically in order to check its shape.

[0519] (611): The value of the pair (diff_bas, diff_haut) corresponding to the minimum is shown.

[0520] The series of values of the pair (diff_bas,diff_haut) are determined as follows:

[0521] In a first stage, the program varies the values of diff_haut and diff_bas, leaving them equal to each other. For example, they may vary between 0 and $\frac{\mathrm{Umax}}{4}$

[0522] in steps of $\frac{\mathrm{Umax}}{4096}$

[0523] if use is made of 12 conversion bits. The result is an array of 1024 elements (the differences calculated for each value of (diff_haut,diff_bas) ) that the program filters in order to eliminate the noise (609) and which it represents graphically (610). The index value of the array corresponding to the minimum difference then corresponds to the correct value of the pair (diff_bas,diff_haut) and this pair is displayed (611).

[0524] In a second stage, the program is restarted by setting diff_bas to the value previously obtained and by varying only diff_haut. A new minimum is thus obtained, which corresponds to a more precise value of diff_haut.

[0525] In a third stage, the program is restarted by setting diff_haut to the value previously obtained and by varying only diff_bas, obtaining a more precise value of diff_bas.

[0526] The operator can reiterate these steps by varying separately and alternately diff_bas and diff_haut, but the maximum accuracy on these values is obtained quite rapidly.

[0527] 5.13. Adjustment of polarizer (105) and of polarization rotator (104)

[0528] The following step consists in adjusting the position of the polarization rotator (104) and of the polarizer (105). This device is designed to provide a controlled attenuation of the illuminating beam by controlling the phase rotator, and the assembly will be called the “optical switch”. It has a closed position corresponding to a low intensity through it, and an open position corresponding to a higher intensity.

[0529] The object used is the same as in the preceding step, and the adjustment of the mirror (109) and of the diaphragms is also the same. The reference wave is first eliminated. In a first stage, the polarizer is put in place and adjusted to maximize the intensity going through it and received on the sensor (118). In a second stage, the polarization rotator is put in place. The voltage corresponding to a state arbitrarily defined as closed is applied to it and it is positioned in rotation to minimize the intensity going through the entire switch. In a third stage, the reference wave is re-established and a program is used which calculates the ratio of the intensities and the phase difference between the two positions of the polarization rotator, a ratio which will be needed during the image acquisition phases in order to combine the waves corresponding to the open and closed states of the switch. The algorithm of this program is shown in FIG. 12. The stages are the following:

[0530] (800): The program first acquires the images using the acquiert_images procedure described in FIG. 13. By “images” is meant here six interference patterns received consecutively by the CCD sensor. This procedure always performs the same cycle beginning with the voltage 0 and ending with the voltage Umax. The waiting time after the application of a zero voltage to the piezoelectric allows the latter to become stabilized. The waiting time before the start of the acquisition prevents the acquisition of a frame which may have been exposed before the application of the desired exposure conditions. The end of the exposure of an image is indicated on the acquisition cards by the start of the transfer of the corresponding image. The use of an exposure time shorter than the image transfer time prevents the image from being affected by transient conditions. Giving the process maximum priority prevents the disturbance of the acquisition by other system tasks under a multitask operating system. The six images are acquired successively and in real time (no image lost) by the acquisition card in order to minimize the time during which vibrations may affect the result. Each image has the horizontal dimension of hpix and vertical dimension of vpix. During the acquisition, the images are transferred automatically by the acquisition card into the array reserved for them in the computer's central memory.

[0531] (801): The program calculates the two frequency representations S0 and S1 which differ by the state of the switch. S0 corresponds to the open switch and S1 to the closed switch. $\mathrm{S0}\left[i,j\right]=\frac{1}{6}\left(2I\left[0,0,i,j\right]-I\left[1,0,i,j\right]-I\left[2,0,i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3}}\left(I\left[1,0,i,j\right]-I\left[2,0,i,j\right]\right)$ $\mathrm{S1}\left[i,j\right]=\frac{1}{6}\left(2I\left[0,1,i,j\right]-I\left[1,1,i,j\right]-I\left[2,1,i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3}}\left(I\left[1,1,i,j\right]-I\left[2,1,i,j\right]\right)$

[0532] (802): The program calculates the maximum value $\mathrm{mod\_max}=\underset{\underset{0\le j\le \mathrm{vpix}-1}{0\le j\le \mathrm{hpix}-1}}{\max }\left|\mathrm{S0}\left[i,j\right]\right|$

[0533] reached by the modulus of the elements of array S0.

[0534] (803): The program calculates the average ratio between the two frequency representations. It initializes rapport and nombre_valeurs to 0 then goes through all the indices (i,j) while testing the condition (804): |S0[i,j]|≧0,5 mod_max

[0535] When the condition has been checked, it carries out (805): $\mathrm{rapport}+=\frac{\mathrm{S0}\left[i,j\right]}{\mathrm{S1}\left[i,j\right]}$  nombre_valeurs+=1

[0536] When all the indices i,j have been gone through, the program divides rapport by nombre_valeurs (806) which gives the sought ratio.

[0537] (807): The program calculates the average rapport_moy of the values of rapport obtained since it started.

[0538] (808): The program displays the real and imaginary parts as well as the modulus of rapport and rapport_moy.

[0539] (809): The program reiterates this procedure continuously to allow continuous adjustment. The program is ended on an instruction from the operator.

[0540] The angular position of the polarization rotator must be adjusted so that the modulus of rapport is roughly equal to 30. The program is then stopped and restarted and, after a sufficient number of iterations, the average complex value rapport_moy is noted and will serve as the basis for the following operations.

[0541] 5.14. Obtaining the constant K and adjusting the diaghram (114) and the mirror (116)

[0542] K is the maximum value in pixels corresponding to the maximum spatial frequency of the wave where 1/λ_vwhere λ_vis the wavelength in the observed medium, assumed to have an index equal to the nominal index n_vof the objective. The nominal index of the objective is the index for which it was designed and for which it creates no spherical aberration. It is also the index of the optical oil to be used with the objective.

[0543] There are K pixels between the frequencies 0 and 1/λ_v. The frequency step along an axis is thus $\frac{1}{K{\lambda }_v}.$

[0544] The frequencies vary in all from −1/λ_vto 1/λ_vin steps of $\frac{1}{K{\lambda }_v}.$

[0545] If N is the total number of pixels along each axis taken into account for the Fourier transform, N values of frequencies are taken into account, from $-\frac{N}{2K{\lambda }_v}$

[0546] to $\frac{N}{2K{\lambda }_v}.$

[0547] After transformation, N position values are obtained with a position step equal to half the inverse of the maximum frequency before transformation.

[0548] The position step is thus $\frac{1}{\frac{N}{2\frac{N}{2K{\lambda }_v}}}=\frac{K{\lambda }_v}{N}.$

[0549] If we consider two points between which the distance in pixels is D_pixand the real distance is D_reel, we have: $D_{\mathrm{reel}}=\frac{K{\lambda }_v}{N}{D}_{\mathrm{pix}}$

[0550] so that $K=\frac{N}{{\lambda }_v}\frac{D_{\mathrm{reel}}}{D_{\mathrm{pix}}}.$

[0551] The wavelength to be considered here is the wavelength in the material, assumed to have an index equal to the nominal index n_vof the objective or: ${\lambda }_v=\frac{\lambda }{n_v}.$

[0552] We finally have: $K=\frac{n_v}{\lambda }\frac{N}{D_{\mathrm{pix}}}{D}_{\mathrm{reel}}.$

[0553] To obtain the constant K, we produce the image of an objective micrometer for which the real distances are known, and then apply the above formula.

[0554] This is done using a focussing program which will be re-used later whenever focussing on a sample is necessary before three-dimensional imaging. The algorithm of this program is shown in FIG. 14. Its main steps are:

[0555] (1000): The program acquires an image using the acquiert_images procedure of FIG. 13. This yields an array of integers (unsigned char type for 8 bits) I[p,c,i,j] where the index p varying from 0 to 2 corresponds to the phase state, the index c varying from 0 to 1 corresponds to the state of the switch (0=open, 1=closed) and indices i and j varying from 0 to hpix−1 and from 0 to vpix−1 correspond to the coordinates of the pixel.

[0556] (1001): An array of Booleans H[i,j] is generated: it is initialized to 0 then, for each pixel, the maximum value reached by I[p,0,i,j] on the three images corresponding to the open position of the switch is calculated. If this value is equal to 255 (highest value of the digitizer), the images corresponding to the closed position of the switch must be used in calculating the frequency associated with the pixel and with its 8 immediate neighbors, and the array H[i,j] is set to 1 for these 9 pixels.

[0557] (1002): The frequency representation S[i,j] of complex numbers is generated: for each pixel, the value is generated according to the following equations: $S\left[i,j\right]=[\frac{1}{6}\left(2I\left[0,H\left[i,j\right],i,j\right]-I\left[1,H\left[i,j\right],i,j\right]-I\left[2,H\left[i,j\right],i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3}}\left(I\left[1,H\left[i,j\right],i,j\right]-I\left[2,H\left[i,j\right],i,j\right]\right))]\left(1+\left(\mathrm{rapport\_moy}-1\right)H\left[i,j\right]\right)$

[0558] If H[i,j] is equal to 1, the complex value thus obtained is consequently multiplied by the complex number rapport_moy obtained during the switch calibration operation in order to give the final value of the element of array S[i,j], in order to take into account the phase shift and absorption induced by the switch.

[0559] (1003): The program limits the array S to dimensions of 512×512. The program then carries out optionally one or the other, or none, of the following two operations:

[0560] averaging over a width of 2, which brings the array S to an array S′ of dimensions 256×256 with S′[i,j]=S[2i,2j]+S[2i+1,2j]+S[2i,2j+1]+S[2i+1,2j+1]. This averaging, coupled with a reduction in the aperture of the diaphragm (114), makes it possible to reduce the diameter of the observed area and to reduce the calculation time. It is equivalent to a low-pass filter followed by sub-sampling.

[0561] limitation of all the observed frequencies to a square of 256×256 pixels with S″[i,j]=S[128+i,128+j]. This allows a reduction in the calculation time at the cost of a reduction in resolution. However, in the present case, the program carries out neither of these two operations.

[0562] (1004): The program then performs the inverse Fourier transform of the array thus obtained.

[0563] (1005): It displays the results on the screen, extracting the modulus, the real part or the imaginary part. In the presence case, it will display the modulus. Whatever the variable displayed, the corresponding array of real numbers is first normalized either in relation to the average value or in relation to the maximum value. The program also writes on the disk the corresponding file of real values. When the real part or the imaginary part are represented, it is essential that the point of impact of the direct beam should be at (256,256) on the image of size 512×512, otherwise a modulation becomes visible. When the modulus is represented, the exact point of impact of the direct beam does not affect the result signficantly.

[0564] (1006): The program starts again during the acquisition of a new image, thus operating continuously. It stops on an instruction from the operator.

[0565] To obtain the image of the micrometer, the objective is first placed in a roughly focussed position, the micrometer having been introduced as an object. In a first step, the reference wave is eliminated, the diaphragms are at maximum aperture, and the program for the direct real-time display of the image received on the sensor is started, the filters at (103) and the polarization rotator being set to allow the passage of sufficient intensity to saturate significantly the sensor at the direct point of impact of the beam. The object is then moved in the horizontal plane by means of the corresponding positioner until the image made up of many aligned intense points characteristic of the micrometer appears. The micrometer is then correctly positioned under the objective.

[0566] The diaphragms at (107) and (114) are then set for an aperture of about 8 mm. The filters at (103) are then set so that the maximum intensity on the CCD is at a level of about a fourth of the maximum value of the digitizer, or 256/4=64. The reference wave is re-introduced. The focussing program is then started. The diaphragm (114) must be adjusted so that it is clearly visible on the displayed image, while being as open as possible. If the image is not correctly centered, it is possible to improve its centering either by modifying the orientation of the mirror (116), in which case it may be necessary to readjust the orientation of the mirror (121), or by modifying the position of the diaphragm (114). The diaphragm at (107) must be adjusted so that the observed area appears uniformly illuminated. The focussing program is then stopped, the reference wave is eliminated and the intensity of the beam is readjusted as previously. The reference wave is then re-introduced and the focussing program restarted.

[0567] The microscope objective is then moved by the focussing device so as to obtain a good image of the micrometer. To facilitate focusing, it is advisable to display the part of the micrometer where lines of different length are present. This limits “false focussing” due to interference phenomena ahead of the micrometer. Between two movements it is necessary to release the manual focussing device to obtain an image not disturbed by the vibrations. When a good image has been obtained, the program is stopped and the obtained image is used to obtain the distance in number of pixels between two lines, the metric distance between these lines being known. If the distance between two graduations separated by D_realmicrometers is the image thus obtained of D_pixpixels, if the nominal index of the objective is n_v(in general, n_v=1.5) and if the wavelength of the laser in vacuum is λ (λ=0.633 micrometers) and if the number of points of the Fourier transform is N (N=512) then we have: $K=\frac{n_v}{\lambda }\frac{N}{D_{\mathrm{pix}}}{D}_{\mathrm{real}},$

[0568] where of course D_realand λ are in the same unit.

[0569] 5.15. Diaphraom adjustment

[0570] The three-dimensional image to be calculated has a side measuring 256 pixels, which allows the file sizes and calculation times to be limited. The adjustment of the diaphragm consists in re-using the focussing program, this time with the intermediate averaging option, and adjusting the diaphragm (114) so that its image is clearly visible while being as large as possible. The diaphragm (107) is then adjusted so that it is slightly more open than the minimum allowing regular illumination of the observed part of the sample.

[0571] 5.16. Reference wave recording

[0572] Knowing the reference wave is essential for the precise calculation of the complex values of the wave reaching the sensor. It must be recorded independently of the constant average noise value that characterizes each pixel. A specific program is used for this purpose. In a first step, the illuminating and reference waves are eliminated and the program records the optically dark background which results on the CCD sensor. It averages the intensity obtained on 100 images to have a noise-free optically dark background. In a second step, the reference wave is re-established and the illuminating wave remains eliminated. The program records the resulting image, the average on 100 images for noise removal. The program then calculates the difference between the image of the reference wave alone and the background image, and records the resulting image in an array Iref[i,j] where i varies from 0 to hpix−1 and j varies from 0 to vpix−1.

[0573] 5.17. Focussing on the studied object

[0574] This stage must be reiterated for each sample for which an image is desired. The illuminating wave is reestablished. The sample to be studied is put in place. The mirror (109) is adjusted so that the direct impact point of the illuminating beam is at the center of the sensor. The filters at (103) are adjusted so that, in the absence of a reference wave, the maximum intensity received on the CCD sensor is about 64. The reference wave is then re-established. The focussing program is started with the intermediate averaging option and the position of the objective is adjusted by means of the focussing device to obtain a clear image of the region of interest of the sample.

[0575] 5.18. Adjustment of condenser position and setting of diaphragm (107)

[0576] After the focussing phase, the position of the condenser must be adjusted so that the image, in the sample, of the illuminated part of the mirror coincides with the observed part of the sample. The reference wave is removed, the illuminating beam attenuation device is placed in the open position, the filters located along the path of the illuminating beam are removed, and a wet piece of white paper is placed on the surface of the mirror (109) so as to form a diffusing surface of small thickness that may be considered assimilated with the surface of the mirror. A specific program for the display of the image received on the CCD is used. This program averages the image, for example over eight successive acquisitions, and represents, on the screen, the root of the value obtained at each point so as to make the image more visible even in the absence of a high brightness. The diaphragm (107) is first opened widely, and the position of the condenser is adjusted to obtain a homogeneous clear disc of long radius. It is then closed again gradually up to the image visibility limit, the position of the condenser being itself readjusted gradually. The final aperture of the diaphragm (107) must be clearly smaller than that determined in 5.15.

[0577] The filters are then re-introduced, the reference wave is also re-introduced, and the focussing program is started. The image of the diaphragm (114) should be seen with a central light spot corresponding to the out-of-focus (hence blurred) image of the diaphragm (107). The position of the condenser can then be adjusted in a plane orthogonal to the optical axis so that the central light spot is centered.

[0578] The diaphragm (107) is then adjusted so as to be slightly more open than the minimum allowing regular illumination of the observed area.

[0579] 5.19. Filter adjustment

[0580] The filters at (103) are adjusted so that, in the absence of a reference wave and in the open position of the beam attenuator, the maximum intensity measured on the CCD sensor is about 64. It can also be useful, just before acquisition, to again adjust the position of the polarization rotator (104) and of the polarizer (101). This adjustment can be carried out using the same program as in 5.13, but leaving the observed sample in place and without modifying the adjustment of the diaphragms and filters. This adjustment allows compensation for the time-wise drift of polarization rotator characteristics.

[0581] 5.20. Acquisition stare

[0582] This stage allows the acquisition of two-dimensional frequency representations from which the three-dimensional representation will be calculated. The point of direct impact of the illuminating wave is moved along a series of radii starting at the optical center and having a length Rb slightly smaller than R, for example Rb=R−6, the optical center and the radius R having been determined in stage 5.8. nbangles will be used to denote the total number of radii that will be traversed, for example nbangles=9. nbim is the number of two-dimensional frequency representations to be acquired and its value is $\mathrm{nbim}=\frac{\mathrm{Rb}}{2}\mathrm{nbangles}.$

[0583] At each stage, the program calculates and stores in a file fich_acquis a frequency representation which will be used in the three-dimensional calculation phase. The size of the image acquired from the camera is hpix×vpix, but this size is divided by two to obtain frequency representations of dimension hel×vel, according to the intermediate averaging principle already used in the focussing program.

[0584] The acquisition program is represented in detail by the algorithm of FIG. 18 whose stages are the following:

[0585] (1300): The basic data of the acquisition are stored in file fich_acquis:

[0586] nbim: number of two-dimensional frequency representations

[0587] hel=hpix/2: final number of points in the horizontal direction

[0588] vel=vpix/2: final number of points in the vertical direction

[0589] (1301): Waiting time allowing the absorption of vibrations created by the movement of the positioner (110). A waiting time of about 2 s can be suitable.

[0590] (1302): The program acquires the images through the acquiert_images procedure of FIG. 13. One thus obtains an array of integers (unsigned char type for 8 bits) I[p,c,i,j] where the index p varying from 0 to 2 corresponds to the phase condition, the index c varying from 0 to 1 corresponds to the condition of the switch (0=open, 1=closed) and the indices i and j varying respectively from 0 to hpix−1 and from 0 to vpix−1 correspond to the coordinates of the pixel.

[0591] (1303): An array of Booleans H[i,j] is generated: it is initialized to 0, then, for each pixel, the maximum value reached on the three images corresponding to the open position of the switch is calculated. If this value is equal to 255 (highest value of the digitizer), the images corresponding to the closed position of the switch must be used in the calculation of the frequency associated with the pixel and with its 8 immediate neighbors, and the array H is set to 1 for these 9 pixels.

[0592] (1304): Calculation of rapport_moy. The calculation of rapport_moy, already carried out during the switch calibration operation, must be performed again with each image so as to compensate for time drifts. The program first determines: $\mathrm{Vmax}=\underset{\left(i,j\right)\epsilon \mathrm{EI}}{\max }\left|\frac{1}{6\sqrt{\mathrm{Iref}\left[i,j\right]}}\left(2I\left[0,0,i,j\right]-I\left[1,0,i,j\right]-I\left[2,0,i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3\mathrm{Iref}\left[i,j\right]}}\left(I\left[1,0,i,j\right]-I\left[2,0,i,j\right]\right)\right|$

[0593] where E1 is the set of pairs (i,j) verifying H[i,j]=0

[0594] The program then calculates: $\mathrm{rapport\_moy}=\frac{1}{\mathrm{N2}}\sum _{\left(i,j\right)\in \mathrm{E2}}^{}\frac{\frac{1}{6}\left(2I\left[0,0,i,j\right]-I\left[1,0,i,j\right]-I\left[2,0,i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3}}\left(I\left[1,0,i,j\right]-I(2,0,i,j]\right)}{\frac{1}{6}\left(2I\left[0,1,i,j\right]-I\left[1,1,i,j\right]-I\left[2,1,i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3}}\left(I\left[I,1,i,j\right]-I\left(2,1,i,j\right]\right)}$

[0595] E2 is the set of pairs (i,j) verifying H[i,j]=0 and $\left|\frac{1}{6\sqrt{\mathrm{Iref}\left[i,j\right]}}\left(2I\left[0,0,i,j\right]-I\left[1,0,i,j\right]-I\left[2,0,i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3\mathrm{Iref}\left[i,j\right]}}\left(I\left[1,0,i,j\right]\right)\right|\ge \mathrm{coef}·\mathrm{Vmax}$

[0596] with, for example, coef=0.5. The coefficient coef can be adjusted so that E2 contains a sufficient number of points. N2 is the number of elements of E2.

[0597] (1305): The frequency representation S[i,j] of complex numbers is generated: for each pixel, the value is generated according to the following equations: $S\left[i,j\right]=\left[\frac{1}{6\sqrt{\mathrm{Iref}\left[i,j\right]}}\left(2I\left[0,H\left[i,j\right],i,j\right]-I\left[1,H\left[i,j\right],i,j\right]-I\left[2,H\left[i,j\right],i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3\mathrm{Iref}\left[i,j\right]}}\left(I\left[1,H\left[i,j\right],i,j\right]-I\left[2,H\left[i,j\right],i,j\right]\right)\right]\left(1+\left(\mathrm{rapport\_moy}-1\right)H\left[i,j\right]\right)$

[0598] If H[i,j] is equal to 1, the complex value thus obtained is multiplied by the complex number rapport_moy obtained during the switch calibration operation to give the final value of the element of array S[i,j], in order to take into account the phase shift and absorption induced by the switch.

[0599] (1306): The point corresponding to the modulus maximum value of array S is determined. Its coordinates (i_max,j_max) are recorded.

[0600] (1307): The “saught” position is calculated, the angle having the value k.pas: $\mathrm{i\_obj}=G_x+\left(k\%\left(\frac{\mathrm{Rb}}{2}\right)\right)·\cos \left(E\left(\frac{2k}{\mathrm{Rb}}\frac{2\pi }{\mathrm{nbangles}}\right)\right)$ $\mathrm{j\_obj}=G_x+\left(k\%\left(\frac{\mathrm{Rb}}{2}\right)\right)·\sin \left(E\left(\frac{2k}{\mathrm{Rb}}\frac{2\pi }{\mathrm{nbangles}}\right)\right)$

[0601] where i_obj and j_obj are coordinates of the objective position in pixels, and where the sign % designates the modulo, and where E(x) designates x rounded to the nearest integer.

[0602] The movement of the motors is calculated:

[0603] pas_—i=(i_obj−i_max).pas_par_pixel

[0604] pas_—j=(i_obj−j_max).pas_par_pixel

[0605] where pas_par_pixel is the number of steps of the motors per pixel of movement, determined experimentally during the adjustments.

[0606] (1308): One of the motors allows the movement of the direct impact point of the beam in the direction of the axis i. It must complete a number of steps pas_i in the direction corresponding to an index i increasing for the position of the direct impact point of the illuminating beam (for pas_i<0, it must complete a number of steps −pas_i in the opposite direction). The other motor allows the movement of the direct impact point of the beam in the direction of the axis j. Similarly, it must complete a number of steps pas_j.

[0607] (1309): An array of averaged frequencies is generated:

[0608] Each dimension of the initial array S is divided by two to give an array M_kwith $M_k\left[i,j\right]=\sum _{\begin{array}{c}0\le p\le 1\\ 0\le q\le 1\end{array}}S\left[2i+p,2j+q\right]$

[0609] (1310): The point corresponding to the modulus maximum value of array M_kis determined, and its coordinates imax_k, jmax_kand the value of this point max_moy=M_k[imax_k, jmax_k] are recorded.

[0610] (1311): The elements of array M_kare normalized by dividing them by max_moy: $M_k\left[i,j\right]=\frac{M_k\left[i,j\right]}{\mathrm{max\_moy}}$

[0611] (1312): imax_k, jmax_kand the frequency representation M_kare stored in file fich_acquis.

[0612] (1313): The algorithm ends when the angle is equal to 2π. The motor then returns to its initial position and the acquisition file fich_acquis is closed.

[0613] 5.21. Three-dimensional calculation

[0614] The acquisition procedure generated two-dimensional frequency representations M_k, where the index k represents the order number of each representation. These representations have been averaged over a width of 2, so that the values C_x, C_yand K must be divided by 2 to correspond to the new system of coordinates. These divided values are the ones used below. The set of two-dimensional representations may be considered to be a representation with three dimensions (i,j,k) in which the index k represents the image index and the indices i and j represent the Cartesian coordinates of each representation. In the k-th two-dimensional frequency representation:

[0615] The relation between the coordinates and the characteristic frequency is:

[0616] Kλf_c=[i−C_x,j−C_y,{square root over (K

[0617] The relation between the coordinates and the illuminating frequency is:

[0618] Kλf_e=[imax_k−C_x,jmax_k−C_y,{square root over (K

[0619] The relation between the coordinates and the total frequency is thus:

[0620] Kλf_i=[i−imax_k,j−max_k,{square root over (K²−(i−C_x)²−(j−C_y)²)}−{square root over (K²−(imax_k−C_x)²−(jmax_k−C_y)²])}

[0621] The three-dimensional calculation procedure consists in principle in generating during a first stage a three-dimensional representation in the form of an array F of dimensions fdim×fdim×fdim where fdim=512, then in taking the Fourier transform to obtain an array U of the same dimensions corresponding to the representation u(r), and in which the indices thus correspond to the position vector. The representation F is an array in which the indices represent the coordinates of Kλf_T, the zero being brought by translation to the point of coordinates (fdim/2,fdim/2,fdim/2). One thus obtains, from each point of coordinates i,j of a two-dimensional representation M_k, a point of the three-dimensional representation by:

[0622] F[ni,nj,nk]=M_k[i,j]

[0623] where: $\mathrm{ni}=i-{\mathrm{imax}}_k+\frac{\mathrm{fdim}}{2}$ $\mathrm{nj}=j-{\mathrm{jmax}}_k+\frac{\mathrm{fdim}}{2}$ $\mathrm{nk}=\sqrt{K^2-{\left(i-C_x\right)}^2-{\left(j-C_y\right)}^2}-\sqrt{K^2-{\left({\mathrm{imax}}_k-C_x\right)}^2-{\left({\mathrm{jmax}}_k-C_y\right)}^2}+\frac{\mathrm{fdim}}{2}$

[0624] When a point of array F, of coordinates (ni,nj,nk), is obtained successively from several distinct two-dimensional representations, the value of F taken at this point is the average of the values obtained from each of the two-dimensional representations. When this point is never obtained, a zero value is assigned.

[0625] When the array F has been generated, the array U can be obtained by means of the inverse three-dimensional Fourier transform.

[0626] This method can be applied directly if the program has a sufficient random-access memory. The program whose algorithm is described in FIG. 19 however makes it possible to carry out the calculations on a system whose RAM is limited, the files being stored on a sequential-access medium (hard disk).

[0627] For practical reasons, part of the Fourier transformation will be carried out as the array F is being generated, so that it is consequently never really generated.

[0628] The program works in five stages. Each stage uses an input file stored on the hard disk of the computer and generates an output file also stored on the hard disk, whose name is italicized in the figure. The program could theoretically carry out more directly the operations necessary for the generation of the three-dimensional image, but the size of the files involved is too large to enable them to be contained entirely in a computer memory. It is thus necessary to manage their storage on disk. As the reading/writing of file elements on disk takes place faster if the elements are stored contiguously, the program must be designed to read and write on the disk only blocks of a sufficient size. This is what the described algorithm renders possible, whereas a direct method would require reading/writing to non-contiguous addresses and would not be practicable owing to the time lost in disk accesses. The stages of the algorithm are the following:

[0629] (1400): Centering of two-dimensional frequency representations:

[0630] This procedure consists in translating the two-dimensional representations to go from a representation in the system of coordinates (i,j,k) to a representation in the system (ni,nj,k) where ni=i−imax+fdim/2,nj=j−jmax+fdim/2, the dimensions following ni and nj in the array thus generated being fdim×fdim. The algorithm of this part is represented in FIG. 21. The stages are the following:

[0631] (1600): The values of hel,vel,nbim are read in the file fich_acquis.

[0632] (1601): The frequency representation M_kcorresponding to the index k is transferred to central memory with the corresponding values imax_kand jmax_k.

[0633] (1602): A translated frequency representation T_kof dimensions fdim×fdim is generated with:

[0634] .T_k[ni,nj]=M_k[ni−fdim/2+imax_k,nj−fdim/2+jmax_k]

[0635] when

[0636] 0≦ni−fdim/2+imax_k≦hel−1 and 0≦nj−fdim/2+jmax_k≦hel−1.

[0637] .T_k[ni,nj]=0 in the other cases.

[0638] FIG. 20 shows an image of an origin with its point of coordinates (imax_k, jmax_k) and an arbitrary drawing around this point, and the new translated image.

[0639] (1603): The values of imax_k, jmax_k, and the representation T_kare stored in the file fich_centréin the following order:

[0640] imax_k,jmax_k,T_k[0,0],T_k[1,0], . . . T_k[fdim−1,0],

[0641] imax_k,jmax_k,T_k[0,1],T_k[1,1], . . . T_k[fdim−1,1], . . .

[0642] imax_k,jmax_k,T_k[0,fdim]T_k[1,fdim], . . . T_k[fdim−1,fdim−1],

[0643] Note that imax_kand jmck_{k are repeated on each line of Tk}in the file fich_centré. This makes it possible for these data, indispensable for the index change k→nk, to remain available after the axis exchange operation.

[0644] (1604): The process is repeated as long as k is less than nbim.

[0645] (1401): Exchange of “plane” and “line” axes

[0646] The change of indices i→ni and j→nj having been carried out, it remains to perform the change of index k→nk. To carry out this change of index in a reasonable time, it is necessary to be able to load into central memory, rapidly, a plane (ni,k). For this operation to be possible, it is previously necessary to exchange the axes k and nj. This is what this algorithm does.

[0647] The file fich_centré previously created is reread, the order of the data is modified and a file fich_ech1 is written in which the data are in the following order:

[0648] imax₀,jmax₀,T₀[0,0],T₀[1,0], . . . T₀[fdim−1,0],

[0649] imax₁,jmax₁,T₁[0,0],T₁[1,0], . . . T₁[fdim−1,0], . . .

[0650] imax_nbim−1,jmax_nbim−1,T_nbim−1[0,0],T_nbim−1[1,0], . . . T_nbim−1[fdim −1,0],

[0651] imax₀,jmax₀,T₀[0,1],T₀[1,1], . . . T₀[fdim−1,1],

[0652] imax₁,jmax₁,T₁[0,1],T₁[1,1], . . . T₁[fdim−1,1], . . .

[0653] imax_nbim−1,jmax_nbim−1,T_nbim−1[0,1],T_nbim−1[1,1], . . . T_nbim−1[fdim−1,1], . . .

[0654] imax₀,jmax₀,T₀[0,fdim−1],T₀[1,fdim−1], . . . T₀[fdim−1,fdim−1],

[0655] imax₁,jmax₁,T₁[0,fdim−1],T₁[1,fdim−1], . . . T₁[fdim−1,fdim−1], . . .

[0656] imax_nbim−1,jmax_nbim−1,T_nbim−1[0,fdim−1],T_nbim−1[1,fdim−1], . . . T_nbim−1[fdim−1,fdim−1,]

[0657] In other words, the axes nj and k are exchanged. This axis exchange operation is carried out block by block. FIG. 22 represents symbolically (1700) the content of a three-dimensional file corresponding to indices i,j,k, arranged in memory horizontal plane by horizontal plane and each plane being arranged line by line, one line being in the direction of the depth in the drawing. The axes i,j,k and the origin 0 of the coordinate system are indicated in the drawing. The content of the file obtained by inverting the axes j and k is represented in (1702). The transfer of data from one file to another takes place block by block, block (1701) being copied into central memory and then transferred at (1703). The reading and writing of the blocks takes place horizontal plane by horizontal plane, a horizontal plane in the file read not corresponding to a horizontal plane in the file written. The size of the block is the maximum size that the internal memory of the computer can hold. The operation is repeated for all the blocks, the blocks located “on the edge” generally not having the same dimension as the others. This procedure enables the exchange of axes with a computer whose internal memory size is smaller than that of the files used.

[0658] The procedure in C language (Microsoft C/C++ under Windows 95) which permits this operation is the following:

[0659] void echange_axes(FILE* read_file,FILE* write_file,int ktot,int jtot,int itot,int memory_limit)

[0660] {

[0661] int knum,jnum,bknum,bjnum,keff,jeff,k,j,bk,bj;

[0662] char* buff;

[0663] knum=(int)sqrt(((double)memory_limit)/((double)itot));

[0664] jnum=knum;

[0665] buff(char*)malloc(itot*knum*jnum);

[0666] bknum=ktot/knum;

[0667] if knum*bknum!=ktot) bknum+=1;

[0668] bjnum=jtot/jnum;

[0669] if(jnum*bjnum!jtot) bjnum+=1;

[0670] for (bk=0;bk<=bknum−1;bk++)

[0671] for (bj=0;bj<=bjnum−1;bj++)

[0672] {

[0673] if (bk==(bknum−1)) keff=ktot−knum*(bknum−1); else keff−knum;

[0674] if (bj==(binum−1)) jeff=tot−jnum*(bjnum−1); else jeff=num;

[0675] for (k=0;k<=keff−1;k++)

[0676] for (j=0;j<=jeff−1;j++)

[0677] {

[0678] fseek(read_file,(bk*knum+k)*jtot*itot+(bj*jnum+j)*itot,SEEK_SET);

[0679] fread(buff+k*jeff*itot+j*itot, 1, itot, read_file);

[0680] }

[0681] for (j=0;j<=jeff−1;j++)

[0682] for (k=0;k<=keff−1;k++)

[0683] {

[0684] fseek(write_file,(bj*jnum+j)*ktot*itot+(bk*knum+k)*itot,SEEK_SET);

[0685] fwrite(buff+k*jeff*itot+j*itot, 1, itot, write_file);

[0686] }

[0687] }

[0688] free(buff);

[0689] }

[0690] The parameters to be passed to this procedure are:

[0691] read_file: pointer to filed fich_centré

[0692] write_file: pointer to file fich_ech1

[0693] ktot: total number of images nbim

[0694] jtot: total number of lines in the frequency representation: fdim

[0695] itot: size in bytes of a line: fdim*sizeof(complex)+2*sizeof(int), where sizeof(complex) designates the size in bytes of a complex number (T_k[ij]for example) and sizeof(int) the size in bytes of an integer number (imax_kfor example).

[0696] memory_limit: maximum size in bytes of the random-access memory (RAM) available to the procedure for the storage of blocks.

[0697] The files must be opened in “committed” mode, i.e. the read/write operations are carried out directly from or towards the hard disk, without intermediate buffering in central memory.

[0698] (1402): Calculation phase

[0699] The purpose of this calculation phase is to replace the “image” index k by the index nk given by the formula: $\mathrm{nk}=\frac{\mathrm{fdim}}{2}+\sqrt{K^2-{\left(i-C_x\right)}^2-{\left(j-C_y\right)}^2}-\sqrt{K^2-{\left({\mathrm{imax}}_k-C_x\right)}^2-{\left({\mathrm{jmax}}_k-C_y\right)}^2}$

[0700] where i,j are the coordinates in the original system of coordinates (before centering). In the centered system of coordinates we thus have: $\mathrm{nk}=\sqrt{\begin{array}{c}{K}^2-{\left(\mathrm{ni}-\mathrm{fdim}\text{/}2+{\mathrm{imax}}_k-C_x\right)}^2-\\ {\left(\mathrm{nj}-\mathrm{fdim}\text{/}2+{\mathrm{jmax}}_k-C_y\right)}^2\end{array}}-\sqrt{\begin{array}{c}{K}^2-{\left({\mathrm{imax}}_k-C_x\right)}^2-\\ {\left({\mathrm{jmax}}_k-C_y\right)}^2\end{array}}+\frac{\mathrm{fdim}}{2}$

[0701] When the same indices (i,j,k) are obtained several time by the replacement of the image index, the value taken for the corresponding element of the three-dimensional frequency representation is the average of the values for which the indices (i,j,k) are obtained.

[0702] A point of the three-dimensional frequency representation, with coordinates (ni,nj,nk), can only be obtained by this change of coordinates from a given plane (ni,k) corresponding to its index nj. The planes (ni,k) can consequently be processed independently of each other. When, in a plane (ni,k), the index k has been replaced by the index nk, it is possible to directly take the inverse two-dimensional Fourier transform of this plane before going on to the next plane. This is what is done by this part of the program, whose algorithm is represented in detail in FIG. 23. Its main stages are:

[0703] (1800): The following elements are read in the file fich_ech1 and transferred to internal memory:

[0704] imax₀,jmax₀,T₀[0,nj],T₀[1,nj], . . . T₀[fdim−1,nj],

[0705] imax₀,jmax₀,T₁[0,nj],T₁[1,nj], . . . T₁[fdim−1,nj], . . .

[0706] imax_nbim−1,jmax_nbim−1,T_nbim−1[0,nj],T_nbim−1[1,nj], . . . T_{nbim −1}[fdim−1,nj]

[0707] (1801): The arrays D_njand Poids, having dimensions fdim×fdim, are initialized to 0.

[0708] (1802): The following condition is tested: ${\left(\mathrm{ni}-\mathrm{fdim}\text{/}2+{\mathrm{imax}}_k-C_x\right)}^2+{\left(\mathrm{nj}-\mathrm{fdim}\text{/}2+{\mathrm{jmax}}_k-C_y\right)}^2<{\left(K\frac{o}{n}\right)}^2$

[0709] where o is the aperture of the microscope, and n the index of the optical oil and of the cover glass used, i.e. approximately: $\frac{o}{n}=\frac{1.25}{1.51}.$

[0710] If the condition is true, the point corresponds to a frequency vector which is not beyond the aperture of the objective and is thus within the observable area.

[0711] (1803): The value nk is calculated by the formula: $\mathrm{nk}=\sqrt{\begin{array}{c}{K}^2-{\left(\mathrm{ni}-\mathrm{fdim}\text{/}2+{\mathrm{imax}}_k-C_x\right)}^2-\\ {\left(\mathrm{nj}-\mathrm{fdim}\text{/}2+{\mathrm{jmax}}_k-C_y\right)}^2\end{array}}-\sqrt{\begin{array}{c}{K}^2-{\left({\mathrm{imax}}_k-C_x\right)}^2-\\ {\left({\mathrm{jmax}}_k-C_y\right)}^2\end{array}}+\mathrm{fdim}\text{/}2$

[0712] (1804): The corresponding frequency value is added to the array D_nj. The corresponding element of the array of weights, which will be used to calculate the average value, is incremented.

[0713] D_nj[ni,nk]+=T_k[ni,nj]

[0714] Poids[ni,nk]+=1

[0715] (1805): When all the indices ni,nk have thus been covered, the program goes through all the indices ni and nk while testing the condition Poids[ni,nk]≠0 and, each time this condition is met, it performs: $D_{\mathrm{nj}}\left[\mathrm{ni},\mathrm{nk}\right]=\frac{D_{\mathrm{nj}}\left[\mathrm{ni},\mathrm{nk}\right]}{\mathrm{Poids}\left[\mathrm{ni},\mathrm{nk}\right]}$

[0716] (1806): The program performs the inverse two-dimensional Fourier transform of the array D_nj.

[0717] (1807): It stores the transformed array in the output file fich_calc in the following order:

[0718] D_nj[0,0],D_nj[1,0], . . . D_nj[fdim−1,0],

[0719] D_nj[0,1],D_nj[1,1], . . . D_nj[fdim−1,1], . . .

[0720] D_nj[0,fdim−1],D_nj[1,fdim−1], . . . D_nj[fdim−1,fdim−1]

[0721] (1403): Second exchanges of axes

[0722] At this level, it remains to carry out an inverse Fourier transform with a single dimension along the axis nj. In order to be able to carry out this transformation in a reasonable time, it is necessary to previously exchange the axes nj and nk. The program will then be able to load into central memory complete planes (ni,nj) in order to process them.

[0723] In the file fich_calc the data are arranged in the following order:

[0724] D₀[0,0],D₀[1,0], . . . D₀[fdim−1,0],

[0725] D₀[0,1],D₀[1,1], . . . D₀[fdim−1,1], . . .

[0726] D₀[0,fdim−1],D₀[1,fdim−1], . . . D₀[fdim−1,fdim−1],

[0727] D₁[0,0],D₁[1,0], . . . D₁[fdim−1,0],

[0728] D₁[0,1],D₁[1,1], . . . D₁[fdim−1,1], . . .

[0729] D₁[0,fdim−11],D₁[1,fdim−1], . . . D₁[fdim−1,fdim−1], . . .

[0730] D_fdim−1[0,0],D_fdim−1[1,0], . . . D_fdim−1[fdim−1,0],

[0731] D_fdim−1[0,1],D_fdim−1[1,1], . . . D_fdim−1[fdim−1,1], . . .

[0732] D_fdim−1[0,fdim−1],D_fdim−1[1,fdim−1], . . . D_fdim−1[fdim−1,fdim−1]

[0733] This file is reread, and a file fich_ech2 is generated, in which the data are rewritten in the following order:

[0734] D₀[0,0],D₀[1,0], . . . D₀[fdim−1,0],

[0735] D₁[0,0],D₁[1,0], . . . D₁[fdim−1,0], . . .

[0736] D_fdim−1[0,0],D_fdim−1[1,0], . . . D_fdim−1[fdim−1,0],

[0737] D₀[0,1],D₀[1,1], . . . D₀[fdim−1,1],

[0738] D₁[0,0],D₁[1,1], . . . D₁[fdim−1,1], . . .

[0739] D_fdim−1[0,1],D_fdim−1[1,1], . . . D_fdim−1[fdim−1,1], . . .

[0740] D₀[0,fdim−1],D₀[1,fdim−1], . . . D₀[fdim−1,fdim−1],

[0741] D₁[0,fdim−1],D₁[1,fdim−1], . . . D₁[fdim−1,fdim−1], . . .

[0742] D_fdim−1[0,fdim−1],D_fdim−1[1,fdim−1], . . . D_fdim−1[fdim−1,fdim−1]

[0743] This exchange of axes nj and nk is carried out by blocks as in the previous exchange of axes. The same procedure is used, the parameters to be processed being:

[0744] read_file: pointer to file fich_calc

[0745] write_file: pointer to file fich_ech2

[0746] ktot: fdim

[0747] jtot: fdim

[0748] itot: Size in bytes of a line: fdim*sizeof(complex) where sizeof(complex) designates the size in bytes of a complex number.

[0749] memory_limit: as previously, the size in bytes of the available memory.

[0750] (1404): Last Fourier transformation

[0751] This procedure consists in performing the inverse Fourier transform along the axis nj. It is an iterative process on the index nk. The algorithm of this part of the program is represented in FIG. 24. Its basic stages are the following:

[0752] (1900): The program loads the following values into internal memory:

[0753] D₀[0,nk],D₀[1,nk], . . . D₀[fdim−1,nk],

[0754] D₀[0,nk],D₁[1,nk], . . . D₁[fdim−1,nk], . . .

[0755] D_fdim−1[0,nk],D_fdim−1[1,nk], . . . D_fdim−1[fdim−1,nk]

[0756] (1901): The program generates an array E_nk,niwith one dimension:

[0757] E_nk,ni[nj]=D_nj[ni,nk]

[0758] It performs the inverse Fourier transform of this array, generating the array {tilde over (E)}_nk,ni

[0759] (1902): It stores the results in the file fich_rep, in the following order:

[0760] {tilde over (E)}_nk,0[0],{tilde over (E)}_nk,1[0], . . . {tilde over (E)}_nk,fdim−1[0],

[0761] {tilde over (E)}_nk,0[1],{tilde over (E)}_nk,1[1], . . . {tilde over (E)}_nk,fdim−1[1], . . .

[0762] {tilde over (E)}_nk,0[fdim−1],{tilde over (E)}_nk,1[fdim−1], . . . {tilde over (E)}_nk,fdim−[fdim−1]

[0763] The file thus generated then contains the three-dimensional representation of the object in the format U[ni,nj,nk] in which the complex element U[ni,nj,nk] is arranged in the file fich__rep at the address (nk*fdim*fdim+nj*fdim+ni) counted from the start of the file, the addressing taking place by “complex number” type elements.

[0764] 5.22. Display

[0765] The array U having been generated, its content can be displayed.

[0766] The simplest display consists in extracting a section, one of the indices being fixed at a constant value. On this section, the real part, the imaginary part or the modulus can be displayed. For example, to extract a section in an imaginary part with nk constant, the program first generates the array V[ni,nj]=Re(U[ni,nj,nk]) in which nk has a fixed value, and in which Re(x) designates the real part of x. It then determines the minimum and maximum values of the array $V,i.e.V_{\min }=,\underset{\mathrm{ni},\mathrm{nj}}{\min }\left(V\left[\mathrm{ni},\mathrm{nj}\right]\right),$ $V_{\min }=\underset{\mathrm{ni},\mathrm{nj}}{\min }\left(V\left[\mathrm{ni},\mathrm{nj}\right]\right).$

[0767] It then generates an array of pixels to be displayed on the screen (bitmap), the gray level of a given pixel with coordinates ni,nj being: $\mathrm{val}\left[\mathrm{ni},\mathrm{nj}\right]=\frac{V\left[\mathrm{ni},\mathrm{nj}\right]-V_{\min }}{V_{\max }-V_{\min }}.$

[0768] Another display method consists in extracting projections therefrom. The displayed image depends to a great extent on the type of projection used.

[0769] For example, for a projection of the real part along the axis nk, by integration, the array V will be represented with $V\left[\mathrm{ni},\mathrm{nj}\right]=\sum _k\mathrm{Re}\left(U\left[\mathrm{ni},\mathrm{nj},\mathrm{nk}\right]\right),$

[0770] where Re(x) designates the real part of x, the sum over nk being taken between two “limit” planes depending on what is to be represented, and the array V being represented as previously.

[0771] For a projection of the real part along the axis nk, by extraction of the maximum in absolute value, the array V will be represented with: $V\left[\mathrm{ni},\mathrm{nj}\right]=\underset{\mathrm{nk}}{\max }\left(\left|\mathrm{Re}\left(U\left[\mathrm{ni},\mathrm{nj},\mathrm{nk}\right]\right)\right|\right)$

[0772] In these two cases, it is possible to represent the array V on the screen according to the method already used for the representation of a section, i.e. $\mathrm{val}\left[\mathrm{ni},\mathrm{nj}\right]=\frac{V\left[\mathrm{ni},\mathrm{nj}\right]-V_{\min }}{V_{\max }-V_{\min }}$

[0773] in which V_minand V_maxare respectively the minimum and maximum values of V.

[0774] Even though the example used was of sections with nk fixed and projections along nk, these sections and projections can be carried out along any direction, including oblique directions.

[0775] As a significant part of the frequency representation was not acquired, there are a certain number of resulting defects in the images obtained under spatial representation, which depend on the type of image represented. In general, the defects are less significant for a top view. In the particular case of the projection by integration, the produced image will remain of good quality up to a limit angle of the projection direction with respect to the vertical: the projection direction must form with the vertical an angle sufficiently smaller than $\mathrm{Arc}\sin \left(\frac{\mathrm{ouv}}{n_v}\right).$

[0776] In the case of the projection by extraction of the maximum, defects will be present whatever the angle of observation. On the other hand, projection by extraction of the maximum is less sensitive to gaussian noise than projection by integration.

[0777] In the case of the projection by integration, a fast method may be obtained for the projection. This method requires storage in memory of the frequency representation of the object. The latter may have been obtained, for example, as indicated in 5.21 but without taking the Fourier transforms described in (1806) and in (1901). The fast method comprises two stages:

[0778] Stage 1: extraction, frequency representation, of a plane passing through the origin and orthogonal to the projection direction.

[0779] Stage 2: inverse Fourier transformation of this plane.

[0780] The two-dimensional array thus obtained constitutes a projection along the direction used to extract the frequency plane.

[0781] It is possible to generate a stereoscopic view by extracting two projections along appropriate directions and using the anaglyph method, for example, to view them. In this case, the use of the fast method makes it possible, if the computer has a sufficient memory for storing the three-dimensional representation, to modify in “real time” the observation angle of an image precalculated in frequency representation form.

[0782] This embodiment is a simple variant of the first and is represented in FIG. 25.

[0783] 6.1. Principle

[0784] In the first embodiment, the sensor (118) is in the back (image) focal plane of the optical assembly made up of the objective (113) and lenses (115) and (117). The plane illuminating wave thus has a point image in this plane, and a spherical reference wave centered virtually on the object must be used to obtain homogeneous illumination on the sensor (118). In this second embodiment, the sensor (2018) is placed directly in the image plane of the objective. A plane illuminating wave consequently no longer has a point image. The reference wave must be the image through the objective of a virtual plane wave passing through the object.

[0785] An image in complex numbers is obtained on the CCD (2018) from three images differing in the phase of the reference wave, using, as in the first embodiment, the formula $s\frac{\stackrel{\_}{r}}{\left|r\right|}=\frac{1}{6\left|r\right|}\left(2{\left|s_0\right|}^2-{\left|s_1\right|}^2-{\left|s_2\right|}^2\right)+j\frac{1}{2\sqrt{3}\left|r\right|}\left({\left|s_1\right|}^2-{\left|s_2\right|}^2\right)$

[0786] The two-dimensional Fourier transform of this image gives an image in complex numbers equivalent to that which, in the first embodiment, was obtained directly in the plane of the CCD sensor. This image thus replaces the image obtained directly on the sensor in the first embodiment. For the rest, this embodiment uses the same principles as the first.

[0787] 6.2. Equipment description

[0788] The system is represented in FIG. 25. The elements of this figure, identical to those of FIG. 1, are numbered by replacing the first digit 1 of the elements of FIG. 1 by the digit 20. For example (116) becomes (2016). This system is similar to the one used in the first embodiment, except that:

[0789] The device for controlled attenuation of the beam, made up of the elements (104) and (105) is eliminated.

[0790] The CCD (2018) is placed in the plane in which the diaphragm (114) was previously located, and consequently the elements (114) (117) (115) are eliminated.

[0791] The virtual image, after reflection on the semi-transparent mirror (2016), of the focussing point of the beam coming from the lens (2023), must be located on the optical axis and in the rear focal plane of the objective (2013). The elements (2023)(2022)(2021)(2016) are thus moved so as to meet this condition.

[0792] On the CCD sensor CCD (2018), the size of the basic cell (distance between the central points of two neighboring pixels) must be less than $\frac{\lambda }{2}\frac{g}{\mathrm{ouv}},$

[0793] where ouv designates the numerical aperture of the objective, λ the wavelength in vacuum of the laser used, g the magnification. For example, for an ×100 objective with an aperture of 1.25 we find 25 micrometers. It is possible to use an objective modified to have an ×50 magnification, in order to obtain 12 micrometers, thus allowing the use of a current camera with 10-micrometer graduations in a more optimum manner than with an ×100 objective.

[0794] The lens (2023) is mounted on a positioner allowing translation along the axis of the beam entering this lens.

[0795] 6.3. Adjustments: overall view

[0796] In the preceding embodiment, the image received on the sensor was in the frequency domain. An image in the spatial domain could if necessary be obtained from that frequency domain image by two-dimensional inverse Fourier transform, which was done by the focussing program described in FIG. 14. In this second embodiment, the image received on the sensor is in the spatial domain and an image in the frequency domain can be obtained by Fourier transform.

[0797] Each time the two-dimensional frequency image received directly on the CCD sensor (118) was used, it is necessary to use the two-dimensional Fourier transform of the image received on the CCD sensor (2018), which constitutes a frequency image which can have a dimension of 256×256 square pixels. Conversely, the focussing program must be replaced by a program for direct display of the image received on the CCD sensor.

[0798] As the beam attenuation device is eliminated, a single image must be used instead of two in the stages in which this device was used.

[0799] The camera (2019) is fixed. The adjustment of this camera's position along the optical axis is replaced by an adjustment of the position of the lens (2023) along the axis of the beam entering this lens. The camera position adjustment in the plane orthogonal to the optical axis is replaced by an angular adjustment of the mirror (2016).

[0800] For the rest, the operating procedure is similar to that of the previous system. The adjustment stages and the programs used are given in detail below:

[0801] 6.4. Current usage program

[0802] In addition to the programs described in 5.5., use is made of a frequency image display program. This program generally replaces the direct display program used in the first embodiment, which allowed the observation of an image in the frequency domain. To use this program, a reference wave must be used, whereas in the direct display program used in the first embodiment it was not necessary. When this program is used, it is necessary to avoid vibrations and hence to release at least temporarily the focussing device if it is used, or wait for the absorption of the vibration before each image when stepping motors are used.

[0803] This program is similar to the focussing program described in 5.14 and whose algorithm is given in FIG. 14. It is modified as follows:

[0804] The image acquisition stage (1000), represented in detail in FIG. 13, is modified as indicated in FIG. 26, in order to take into account the absence of the beam attenuation device.

[0805] The stage (1001) is eliminated.

[0806] The stage (1002) is modified to take into account the absence of the array H. For each pixel, the value is generated according to the following equations: $S\left[i,j\right]=[\frac{1}{6}\left(2I\left[0,i,j\right]-I\left[1,i,j\right]-I\left[2,i,j\right]\right)+\tilde{j}\frac{1}{2\sqrt{3}}\left(I\left[1,i,j\right]-I\left[2,i,j\right]\right)]$

[0807] During the stage (1002) the program limits the array S to dimensions of hpix×hpix but does not perform any averaging.

[0808] The stage (1004) is replaced by a direct Fourier transform.

[0809] During the stage (1005), the program displays the intensity, corresponding to the square of the modulus of the elements of the transformed array S, as well as the maximum value of this intensity, the coordinates of the corresponding point and the ratio between the intensity of this point and the sum of the intensities of its eight neighbors.

[0810] This program makes it possible to determine the point nature (punctuality) and the general appearance of a frequency image. On the other hand, to determine non-saturation (which must be checked in almost all the stages, a fact which will no longer be mentioned), the programs described in 5.5. continue to be used, and the non-saturation must be verified directly on the CCD sensor CCD (2018).

[0811] 6.5. Adjustment of position of laser (2000) and mirror (2021)

[0812] This stage is similar to the stage described in 5.6.

[0813] 6.6. Adjustment of piezoelectric actuator control voltages

[0814] The process is the same as the one described in 5.12., except that there is no adjustment of the diaphragm (114) which is eliminated and the position of (2009) is adjusted so as to maximize the intensity received on the sensor. The fact that the image received directly by the sensor is in the spatial domain does not affect the result.

[0815] 6.7. Adjustment of reference wave level

[0816] This stage is identical to the one described in 5.11, the level of the reference wave being measured on the direct image.

[0817] 6.8. Position adjustment of (2023) (2002) (2016)

[0818] This adjustment is similar to the one described in 5.7. The direct display program is replaced by the frequency image display program described in 6.4, for which the presence of the reference wave is necessary. The adjustment of the camera position along the optical axis is replaced by an adjustment of the position of the lens (2023) along the axis of the beam entering this lens. The camera position adjustment in the plane orthogonal to the optical axis is replaced by an angular adjustment of the mirror (2016).