Optical lattice microscopy

[0001] This application is a continuation of International Patent Application No. PCT/US2005/042686, filed Nov. 23, 2005, entitled “OPTICAL LATTICE MICROSCOPY,” which itself claims priority to U.S. Patent Application No. 60/630,021, filed Nov. 23, 2004, entitled, OPTICAL LATTICE MICROSCOPY,” both of which are incorporated by reference herein in their entirety.

[0002] This disruption relates to optical lattices and related components, systems, and methods, and, in particular, to optical lattice microscopy.

[0003] Optical microscopy is useful for the study of living cells, because of its comparatively weak effect on the intracellular environment and because of the wealth of information that can be extracted from highly developed optical techniques, such as time resolved spectroscopy. Because living cells are three-dimensional (3D) systems that exhibit features on length scales from the atomic to macroscopic and dynamic evolution on time scales over many orders of magnitude, the refinement of optical microscopy to visualize these features and this evolution with ever increasing spatial and temporal resolution remains an area of active interest.

[0004] Referring to FIG. 1, a common approach in optical microscopy is to illuminate a sample 102 with light filling the rear pupil 100 of an objective lens 101, and then to collect the light resulting from the interaction of the focused illumination with the sample 102 using either the same lens 101 or a different lens. In widefield microscopy, as shown in FIG. 1, the excitation light 103 from a source 104 impinges on the plane of the rear pupil 100 from a multitude of angles, so that the entire field of view within the sample 102 is illuminated simultaneously. Then, a detector 106 images the collected light 105 representing the signal of interest. Because detection across the entire field of view usually occurs simultaneously, widefield imaging can be quite rapid.

[0005] Light 107 emitted near the focal plane 108 of the objective 101 produces a clear, focused image at the detector 106. However, a widefield system also collects light (e.g., 109 and 110) emitted from other planes (e.g., 111 and 112), which can create an out-of-focus background haze in thicker samples 102 and obscure the crisp image data from the focal plane 108.

[0006] As shown in FIG. 2, confocal microscopy addresses this issue by: (a) illuminating the objective lens 101 at its rear pupil 100 with a nearly parallel beam of light 200, which, for example, can be created by filtering the excitation light 103 through a pinhole mask 201 and subsequently collimating the light with a lens 205, thereby creating a single, concentrated spot of light 206 within the focal plane 207; and (b) centering a pinhole 208 in an opaque mask 210 at the confocal position in the image plane 207 relative to the focal spot 206. Light 209 emitted from the sample 210 near the focal spot 206 is largely concentrated at the pinhole 208 and is passed to the detector 211. However, light (e.g., 212 and 213) from other focal planes (e.g., 214 and 215) is not well concentrated at the confocal position in the image plane 207 and, therefore, is mostly excluded from passing through the pinhole 208. By scanning the sample 210 and the focal spot 206/detection pinhole 208 combination relative to one another on a point-by-point basis in one, two, or three dimensions, a composite image can be captured by detector 211.

[0007] As shown in FIG. 3, the resolution of images created by confocal microscopy techniques (e.g., as shown in FIGS. 3D, 3E, and 3F) are often much sharper than equivalent images created by widefield microscopy techniques (e.g., as shown in FIGS. 3A, 3B, and 3C) due to the significant rejection of out-of-focus background haze that confocal microscopy can achieve. However, confocal microscopy images generally take longer to acquire due to the sequential nature of the point-by-point imaging process.

[0008] As shown in FIG. 4, in both widefield microscopy (as shown in FIG. 4A) and confocal microscopy (as shown in FIG. 4B), the light illumination within a sample is not confined to the focal plane 400 or point of interest. Instead, the illumination light in each case extends throughout two solid cones 401 and 402 of incoming and outgoing illumination (relative to the focal plane), where the half-angle of each cone is dictated by the numerical aperture (NA) of the objective lens that focuses the light within the sample. The light within illumination cones 401 and 402 that does not contribute to the focused image increases the possibility of photo-induced damage (also known as “photobleaching”) within a sample.

[0009] Fluorescence microscopy can be used for intracellular imaging, and a wide variety of site-specific markers have been developed to yield high-contrast images of features of interest. Photobleaching can be problematic in fluorescence microscopy because many fluorescent marker molecules can undergo only a limited number of absorption/emission cycles before permanently photobleaching and becoming useless for creating a fluorescent image. Therefore, excitation of fluorescent marker molecules throughout the illumination cones 410 and 412 of widefield microscopy and confocal microscopy is wasteful, because only a small fraction of the photon budget for each molecule contributes to the desired signal.

[0010] Spatial resolution is usually somewhat superior in confocal microscopy, compared to widefield microscopy because the spatial resolution arises from a convolution of the effective excitation region with the effective detection region dictated by the pinhole 204. As shown in FIG. 5, an objective lens 101 can focus incoming monochromatic light 504 to a focal point 505. Because the light is monochromatic, rays of the focused light can have different wavevectors, k₁, k_z, and k₂, not because their wavelengths are different, but because their directions are different. The sharpness of either the effective excitation region or the effective detecting region created by the lens 101, and hence the resolution of the imaging system, is superior in the plane transverse to the objective axis, ê_z 500, as compared to any plane along ê_z because the highest spatial frequency in the plane transverse to the objective axis is:

[0011] $\begin{array}{cc}{\left(\Delta k_{\perp }\right)}_{\max }=\left(k_1-k_2\right)·{\hat{e}}_{\perp }=2k\frac{\mathrm{NA}}{n}& \left(1\right)\end{array}$

[0012] whereas the highest spatial frequency in the direction along the objective axis is:
(Δk_z)_max=|(k_z−k₁)·ê_z|=k(1−√{square root over (1−(NA/n)²)})  (2)
where, for monochromatic light, |k_m|=k for all values of m, k₁ 501 and k₂ 502 are wavevectors at opposite edges of the illumination cone of the objective lens 102, k_z=kê_z 503, and n is the refractive index in the sample and its surrounding medium. Increased confinement of the excitation and detection regions improves sensitivity as well as resolution of the imaging system, because less spurious signal from effects such as autofluorescence and Rayleigh scattering is generated outside the region of interest, and the remaining spurious signal is more efficiently rejected prior to detection.

[0013] Improving the performance of live cell imaging requires consideration of the demanding problem of time resolved three-dimensional imaging of P≧1 independent optical properties (e.g., polarization and/or wavelength at different three-dimensional positions and at different times). This requires binning the measured photons in hypervoxels of P+4 dimensions (where a “hypervoxel” is a multidimensional pixel), which vastly increases the detected flux necessary to generate statistically significant data, compared to static two-dimensional or even three-dimensional imaging. Increasing the resolution in any of the P+4 dimensions demands further subdivision, thereby only exacerbating the problem. Furthermore, the photons within each hypervoxel originate from a 3D volume that decreases rapidly in size with increasing spatial resolution, making it increasingly difficult to generate the requisite flux in a non-invasive manner. Thus, the issues of image resolution, signal intensity, imaging speed, and sensitivity of the sample to damage by the imaging light are interrelated.

[0014] In a first general aspect, a method of creating a periodic interference pattern of coherent waves in two or three dimensions, D, includes generating at least D+1 waves, each wave having substantially the same wavelength, λ, and each wave traveling substantially in a unique direction, k_n, and positioning said waves such that at least a portion of each wave intersects in a common excitation region to create the interference pattern. The directions of the waves, k_n, are selected such that the interference pattern within the excitation region forms a Bravais lattice having a symmetry, a shape to the primitive cell defined by the D-dimensional region closest to any chosen lattice point, and a periodicity relative to the wavelength, λ, that is larger than the smallest periodicity Bravais lattice having the same symmetry and identically defined primitive cell shape that can be created at the wavelength, λ. In addition, the directions of the waves, k_n, do not all lie on a D-dimensional set of mutually orthogonal axes.

[0015] In another general aspect, a method for creating a periodic interference pattern of coherent waves in two or three dimensions, D, includes generating at least D+2 waves, where each wave has substantially the same wavelength, λ, and travels substantially in a unique direction, k_n, and directing the waves such that at least a portion of each wave intersects in a common excitation region to create the interface pattern. The directions, k_n, of the waves are selected such that the interference pattern within the excitation region forms a Bravais lattice. At least one wave travels substantially in a direction, k_i, such that one or more symmetry operations characteristic of the Bravais lattice map the direction of the wave, k_i, onto a direction of a different wave, k_j, and the directions of the waves, k_n, do not all lie on a D-dimensional set of mutually orthogonal axes.

[0016] Implementations can include one or more of the following features. For example, the method can further include controlling an amplitude of at least one wave to enhance an amplitude of a field in the interference pattern at a predetermined point within each primitive cell of the lattice within the excitation region. The method can further include controlling a phase of at least one wave to enhance an amplitude of a field in the interference pattern at a predetermined point within each primitive cell of the lattice within the excitation region. The method can further include controlling a phase of at least two waves to enhance a degree of mutual constructive interference between the two waves at a predetermined point within each primitive cell of the lattice within the excitation region. A phase of substantially all of beams can be controlled to translate the interference pattern. The waves can be electromagnetic waves, and a property (e.g., a polarization, a phase, or an amplitude) of at least one wave can be controlled to enhance an intensity in the interference pattern at a predetermined point in each primitive cell within the excitation region, to enhance a contrast between an intensity at a predetermined point in each primitive cell within the excitation region and an intensity at regions adjacent to the predetermined point within each such primitive cell, or to enhance a predetermined component of the polarization of the electromagnetic interference pattern at a predetermined point in each primitive cell within the excitation region.

[0017] The method can also include controlling a complex electric field polarization, e_n(x,t), of at least one wave to enhance a real part of the wave's complex projection, Re{e_n*(x_d,t_d)′·e_d}, at a time, t_d, and at a relative point, x_d, in each primitive cell within the excitation region onto a complex electric field vector, e_d, describing the desired state of the total electric field in the interference pattern at the same time, t_d, and the same position, x_d. The electric field vector, e_d, can describe a state of linear polarization, or a state of circular polarization.

[0018] The method can further include controlling a property of at least one wave to enhance a gradient of a total electric field in the interference pattern, ∂e(x,t)/∂x_j, or the total magnetic field, ∂h(x,t)/∂x_j, in at least one particular direction, ê_j, at a predetermined point in each primitive cell within the excitation region.

[0019] The method can further include controlling a property of at least one wave to enhance a magnitude of a gradient of the total electric field in the interference pattern, |∇e(x,t)|, or the total magnetic field in the interference pattern, |∇h(x,t)|, at a predetermined point in each primitive cell within the excitation region.

[0020] The method can further include a property of at least one wave to reduce an intensity in the interference pattern at a predetermined point within each primitive cell of the lattice within the excitation region. A property of the at least one wave can be controlled to enhance an intensity in the interference pattern in at least a portion of a region surrounding the predetermined point in each primitive cell where the intensity has been reduced.

[0021] The method can further include varying the direction, k_n, of each wave such that ratios between the lattice constants of the lattice are changed. The change in the ratios can cause enhance an intensity at a predetermined point in each primitive cell within the excitation region, can enhance a contrast between an intensity at a predetermined point in each primitive cell within the excitation region and an intensity elsewhere within each such primitive cell, or can enhance a magnitude of at least one component of an electric field in the interference pattern at a predetermined point in each primitive cell within the excitation region is enhanced. By varying the direction of each wave an aspect ratio of the lattice can be increased in a direction parallel to an axis defined by the component of the electric field that is enhanced.

[0022] The method can further include rotating the direction, k_n, of substantially every wave by a substantially identical angle to enhance a magnitude of at least one component of an electric field of the interference pattern at a predetermined point in each primitive cell within the excitation region.

[0023] The waves can be acoustic waves. The waves can be composed of particles of matter (e.g., electrons or electrically neutral particles) having a deBroglie wavelength, λ. The waves can exist at an interface between two dissimilar media (e.g., water and air).

[0024] In one implementation, the direction, k_n, of each wave forms substantially the same angle, θ, with respect to a fixed axis, ê_2D. The method can further include varying the direction, k_n, of each wave such that the angle, θ, of each wave with the fixed axis, ê_2D, changes and thereby varies the periodicity of the lattice in the plane substantially perpendicular to the fixed axis, ê_2D. The waves can travel substantially within a single plane, and the symmetry of the lattice can be a square symmetry, a primitive rectangular symmetry, a centered rectangular symmetry, or a hexagonal symmetry.

[0025] In certain implementations, the interference pattern can be a three-dimensional pattern, and the symmetry of the lattice is a simple cubic symmetry, a hexagonal symmetry, a body-centered cubic symmetry, or a face-centered cubic symmetry. The lattice can be a member of the tetragonal crystal group, a member of the orthorhombic crystal group.

[0026] The method can further include generating more than D+1 waves, where at least one wave travels substantially in a direction, k_i, such that one or more symmetry operations characteristic of the Bravais lattice can map the direction of the wave, k_i, onto a direction, k_j, of a different wave. For substantially each wave traveling substantially in a unique direction, k_i, another wave can exist traveling substantially in a direction, k_j, that is determined by applying a combination of symmetry operations associated with the Bravais lattice to the direction of the wave, k_i.

[0027] The method can further include rotating the optical lattice and its constituent waves together with respect to a longitudinal axis, ê_z, of a lens to increase the number of waves traveling substantially in directions that fall within the solid angle defined by the numerical aperture of the lens.

[0028] The method can further include controlling the directions, k_n, in which the waves travel to control a ratio of the lattice constants of the lattice and thereby increase the number of waves traveling substantially in directions that fall within a solid angle defined by the numerical aperture of a lens.

[0029] In another general aspect, an optical system for generating an optical lattice in two or three dimensions, D, substantially confined to an excitation region, includes a source of electromagnetic radiation having a wavelength, λ, and one or more optical elements configured to divide the radiation into at least D+1 independent excitation beams. Also included are optical elements configured to direct each excitation beam substantially in a unique direction, k_n, such that substantially all the beams intersect to create the optical lattice within the excitation region, and a first lens, having a rear pupil side and an opposite focusing side facing the excitation region The optical elements are adapted and arranged for directing at least two of the excitation beams into the rear pupil of the lens at distinct beam-center positions, x_n″, chosen such that the beams exit from the focusing side of the lens in the unique directions, k_n, and the directions, k_n, of the beams that exit from the focusing side of the lens lie substantially within a solid angle defined by a numerical aperture of the lens.

[0030] Implementations may include one or more of the following features. For example, the optical elements can be configured such that the optical lattice includes a Bravais lattice within the excitation region having a symmetry, a shape to the primitive cell defined by the D-dimensional region closest to any chosen lattice point, and a periodicity relative to the wavelength, λ, that is larger than the smallest periodicity Bravais lattice having the same symmetry and identically defined primitive cell shape that can be created at the wavelength, λ, and wherein the directions, k_n, of all the beams do not lie on a D-dimensional set of mutually orthogonal axes. The optical elements can be configured such that the optical lattice is a Bravais lattice within the excitation region, and at least one beam travels substantially in a direction, k_i, such that one or more symmetry operations characteristic of the Bravais lattice can map the direction of the beam, k_i, onto a direction, k_j, of a different beam, and wherein the directions, k_n, of all beams do not lie on a D-dimensional set of mutually orthogonal axes.

[0031] The optical system can further include a first optical element (e.g., a substantially opaque mask having a substantially transmissive aperture, a diffractive optical element, or a spatial light modulator) configured to control an effective cross-section at the plane of the rear pupil of a beam. The optical elements can be adapted and arranged to control the cross-sectional extent of at least one beam along the radial axis, ê_ρ, in the plane of the rear pupil intersecting the beam-center position, x_n″, and the longitudinal axis, ê_z, of the lens, so that the extent of the focal waist of the beam after it exits from the focusing side of the lens is inversely controlled along the axis, ê_θ, perpendicular to its direction, k_n, and in the plane defined by k_n and ê_z. The optical elements can be are adapted and arranged to control the cross-sectional extent of at least one beam along the tangential axis, ê_φ, parallel to the plane of the rear pupil and perpendicular to the radial axis, ê_ρ, that lies in this plane and intersects the beam-center position, x_n″, and the longitudinal axis, ê_z, of the lens, so that the extent of the focal waist of the beam after it exits from the focusing side of the lens is inversely controlled in the same tangential direction, ê_φ. The optical elements can be adapted and arranged to control the cross-sectional extent of at least one beam such that the cross-sectional extent of the resulting convergent beam that exits from the focusing side of the lens is substantially similar to the desired cross-sectional extent of the excitation region in a desired plane within said region perpendicular to the direction, k_n, of the convergent beam. The optical system can further include a second optical element (e.g., an attenuating filter, a polarization sensitive optical element, a spatial light modulator) adapted and arranged for controlling the intensity of at least one of the beams entering the lens.

[0032] The optical system can further include an optical element (e.g., a reflector mounted on a linear translator, an optical phase modulator, a spatial light modulator) for controlling the phase at the plane of the rear pupil across at least one of the beams entering the lens. The optical element can be a means (e.g., a pivot-mounted reflector) for controlling the phase across the beam at the plane of the rear pupil by tilting the beam direction, (k_in)_n, with respect to the longitudinal axis, ê_z, of the lens. The optical elements can be adapted and arranged to control the tilt of at least one beam that enters the lens, said tilt being controlled in the plane defined by the longitudinal axis, ê_z, of the lens and the direction, k_n, of the resulting convergent beam that exits from the focusing side of the lens, such that the position of the focal waist of the convergent beam is controlled along the axis, ê_θ, in the same plane that is perpendicular to k_n. The optical elements can be adapted and arranged to control the tilt of at least one beam that enters the lens, said tilt being controlled perpendicular to the plane defined by the longitudinal axis, ê_z, of the lens and the direction, k_n, of the resulting convergent beam that exits from the focusing side of the lens, such that the position of the focal waist of the convergent beam is controlled along the axis, ê_φ, perpendicular to both the same plane and to k_n. The optical element can be adapted and arranged to control the wavefront curvature of at least one beam, such that the position of the focal waist of the resulting convergent beam that exits from the focusing side of the lens is controlled along its direction, k_n. The third optical element can be a second lens through which the beam passes prior to entering the first lens.

[0033] The system can further include an optical element (e.g., a phase retardation plate, a linear polarizer, or a polarization sensitive beam splitter) adapted and arranged for controlling a polarization of a beam entering the lens. The optical elements can be adapted and arranged to control the magnitude of the electric field of at least one beam along the radial axis, ê_ρ, in the plane of the rear pupil pointing from the center of the lens to the center of the beam, such that the magnitude of the electric field of the beam is controlled along an axis ê_θ as it exits from the lens, where ê_θ is defined as perpendicular to the direction k_n of the exiting beam and parallel to the plane defined by k_n and the longitudinal axis of the lens. The optical elements can be adapted and arranged to control the magnitude of the electric field of the beam along the tangential axis, ê_φ, perpendicular to plane defined by the longitudinal axis of the lens and the direction of the beam, k_n, as it exits from the lens, such that the magnitude of the electric field of the exiting beam is controlled along ê_φ.

[0034] The optical lattice can be created at least in part by the intersection of substantially all beams substantially traveling in unique directions, k_n, that reside within the solid angle defined by the numerical aperture of the lens and that can be found by applying any combination of the symmetry operations of the Bravais lattice to the direction vector k_m of any other constituent beam of the lattice. The optical lattice can be created exclusively by the intersection of beams that reside within the solid angle defined by the numerical aperture of the lens.

[0035] The optical system can further include a substrate, a sample at least partially overlapping with the excitation region and mounted to the substrate, and a fifth optical element (e.g., lens or a spatial light modulator) adapted for correcting for aberrations introduced in a beam that passes through the substrate before reaching the sample.

[0036] The lens of the optical system can be a high numerical aperture microscope objective having a numerical aperture greater than 0.9. The microscope objective is adapted for collecting light arising from an interaction of the lattice with the sample. The optical system can further include one additional beam contributing to creation of the lattice that reaches the excitation zone from outside the objective. The lattice can be a three-dimensional lattice, oriented such that multiple lattice planes are oriented substantially parallel to a focal plane of the objective. The lattice can be a two-dimensional lattice, and each beam can be directed at substantially the same angle with respect to a longitudinal axis, ê_z, of the objective, and reside within a solid angle defined by the numerical aperture of the lens.

[0037] The optical system can further include a second lens having a rear pupil side and an opposite focusing side, where the directing optical elements are adapted and arranged for directing at least two of the excitation beams into the second lens. The focusing sides of the first and second lenses can oppose one another and their longitudinal axes can be substantially parallel. The two lenses can be high numerical aperture microscope objectives having numerical apertures greater than 0.9. The system can further include a sample at least partially overlapping with the excitation region, and wherein the microscope objectives are adapted for collecting light arising from an interaction of the lattice with the sample. The optical lattice can be created at least in part by the intersection of all beams substantially traveling in unique directions, k_n, residing within the total solid angle defined by the numerical apertures of the two objectives and that can be found by applying any combination of the symmetry operations of the Bravais lattice to the direction vector k_m of any other constituent beam of the lattice. At least one additional beam contributing to the creation of the lattice can reach the excitation zone by passing through the space between the two objectives. The lattice can be generated by substantially all beams substantially traveling in unique directions, k_n, that can be found by applying any combination of the symmetry operations of the Bravais lattice to the direction, k_m, of any other constituent beam of the lattice and that either reside within the total solid angle defined by the numerical apertures of the two objectives or that travel outside the objectives and that can reach the excitation zone without being substantially occluded by the objectives.

[0038] The system can further include phase-controlling optical elements adapted and arranged for introducing a controllable, independent overall phase to substantially every beam that contributes to the lattice. At least one of the phase-controlling optical elements can be a reflector mounted on a linear translator or an optical phase modulator or a spatial light modulator. The phase-controlling optical elements can be adapted for translating a basis associated with the lattice by controlling the phase of substantially every beam. The system can further include a sample at least partially overlapping with the excitation region, and a detector adapted for detecting deviations from a desired basis associated with the lattice, where the phase-controlling optical elements are adapted for independently controlling the overall phase of substantially every beam to reduce deviations in the basis by reducing deviations in the phases of the beams relative to one another at a predetermined point within both the sample and the excitation region.

[0039] In another general aspect, an optical system includes a source of electromagnetic radiation having a wavelength, λ and one or more dividing optical elements configured for dividing the radiation into at least D+1 substantially independent excitation beams. Directing optical elements are configured for directing each excitation beam in a unique direction, k_n, such that substantially all the beams overlap within an excitation region to create an optical lattice in two or three dimensions, D. The optical elements are arranged and adapted such that in the excitation region the phase of substantially every beam approximates the phase of a plane wave traveling in the same direction as the beam.

[0040] Implementations can include one or more of the following features. For example, the optical elements can be further configured to focus substantially every beam and to control the position or cross-sectional extent of substantially every focused beam waist to control the location and extent of the excitation region. The optical elements can include a lens that focuses exactly one of the beams. One of the optical elements can include a spatial light modulator that focuses at least one of the beams.

[0041] The optical lattice can include a Bravais lattice within the excitation region having a symmetry, a shape to the primitive cell defined by the D-dimensional region closest to any chosen lattice point, and a periodicity relative to the wavelength, λ, that is larger than the smallest periodicity Bravais lattice having the same symmetry and identically defined primitive cell shape that can be created at the wavelength, λ, and wherein the directions, k_n, of all the beams do not lie on a D-dimensional set of mutually orthogonal axes.

[0042] The optical lattice can include a Bravais lattice within the excitation region, and a least one beam can travel substantially in a direction, k_i, such that one or more symmetry operations characteristic of the Bravais lattice can map the direction of the beam, k_i, onto a direction, k_j, of a different beam, and wherein the directions, k_n, of all beams do not lie on a D-dimensional set of mutually orthogonal axes.

[0043] The system can further include second optical elements configured for controlling the intensity of substantially every beam, such that they are of substantially similar intensity at a predetermined point within the excitation region. At least some of the optical elements can be configured for controlling the shape and sharpness of the excitation region by directing multiple beams substantially in the same direction, k_m, as one or more beams that make up the lattice, and by controlling the positions of the focal waists of all beams traveling in substantially the same direction so that their foci are offset from one another. The optical elements can include a diffractive optical element configured to create two or more beams of offset foci traveling in substantially the same direction. The optical elements can include a spatial light modulator configured to create two or more beams of offset foci traveling in substantially the same direction.

[0044] In another general aspect, an optical system includes a source of electromagnetic radiation, having a first wavelength, λ₁, and dividing optical elements configured for dividing the radiation from the source into a first set of at least D+1 excitation beams. Directing optical elements are configured for directing each excitation beam in a unique direction, k_n, such that the beams intersect within an excitation region to create an interference pattern having a symmetry of a first Bravais lattice in two or three dimensions, D, and a first lattice periodicity. Control optical elements are configured for controlling an electromagnetic field basis associated with the first lattice to produce at least one excitation maximum in each primitive cell of the first lattice, wherein the excitation maximum is substantially confined to a region less than the wavelength, λ₁, in at least two dimensions. A detector having individual detector elements configured for detecting signal light resulting from an interaction of the lattice with a sample located in the excitation region. The lattice and the directions, k_n, of the excitation beams are selected such that the lattice periodicity is large enough for signals from multiple excitation maxima within the sample to be individually resolved and simultaneously measured by separate detector elements.

[0045] Implementations can include one or more of the following features. For example, the directing optical elements can be configured for directing the beams in directions, k_n, such that the optical Bravais lattice has a shape to the primitive cell defined by the D-dimensional region closest to any chosen lattice point, and a periodicity relative to the wavelength, λ₁, that is larger than the smallest periodicity Bravais lattice having the same symmetry and identically defined primitive cell shape that can be created at the wavelength, λ₁, and wherein the directions, k_n, of all beams do not lie on a D-dimensional set of mutually orthogonal axes.

[0046] The directing optical elements can be configured for directing at least one of the beams in a direction, k_i, such that one or more symmetry operations characteristic of the Bravais lattice can map the direction, k_i, onto a direction, k_j, of a different beam, and wherein the directions k_n of all beams do not lie on a D-dimensional set of mutually orthogonal axes.

[0047] The detector can be configured for detecting a fluorescence emission signal arising from the interaction of the electromagnetic radiation with the sample, or for performing time correlated single photon counting of the signal resulting from the interaction of the electromagnetic radiation with the sample.

[0048] The source and detector can be configured for generating and detecting a signal resulting from a nonlinear interaction of the electromagnetic radiation with the sample, for generating and detecting light resulting from multiphoton absorption of the electromagnetic radiation by the sample, for generating and detecting light from a multiphoton scattering process (e.g., including second harmonic generation) between the electromagnetic radiation and the sample, for generating and detecting light resulting from Raman scattering of the electromagnetic radiation by the sample.

[0049] The source can be configured for emitting pulsed beams of electromagnetic radiation.

[0050] The system can further include a source of electromagnetic radiation having a second wavelength, λ₂, optical elements configured for dividing the radiation into a second set of at least D+1 excitation beams, optical elements configured for directing each excitation beam of the second set in a unique direction, k_n, such that the beams intersect within the excitation region to create an interference pattern having a symmetry of a second Bravais lattice in D dimensions, and a second lattice periodicity, and optical elements configured for controlling an electromagnetic field basis associated with the second lattice to produce at least one excitation maximum in each primitive cell of the second lattice, where the excitation maxima are substantially confined to a region less than the wavelength, λ₂, in at least two dimensions, and where at least a portion of the signal light arises from an interaction of both the first lattice and the second lattice with the sample.

[0051] The second lattice can have the same symmetry and substantially the same lattice constants as the first lattice, and the period of the second lattice can be a rational multiple of a period of the first lattice. The first wavelength can be selected to provide pump radiation to the sample and the second wavelength is selected to provide probe radiation to the sample. The sources and the detector can be configured for generating and detecting light resulting from Coherent Anti-Stokes Raman Scattering within the sample.

[0052] The system can further include translation optical elements configured for translating the excitation maxima relative to the sample, a memory configured for recording the detected light from each observed excitation maximum at multiple relative positions of the maxima and sample, and a processor configured for generating an image from the recorded detected light. A display can be configured for displaying the image.

[0053] The lattice can be a three dimensional lattice, and the optical elements for translating the excitation maxima can be configured for translating the excitation maxima across a volume that includes a primitive cell of the lattice. The lattice can be a three dimensional lattice, and the optical elements can be configured for scanning the excitation maxima relative to the sample in a plane substantially parallel to a lattice plane of the lattice and for scanning each maximum at least over an area defined by an intersection of the lattice plane with a primitive cell of the lattice.

[0054] The processor can be configured for generating simultaneously multiple two-dimensional images from excitation maxima within parallel lattice planes.

[0055] The lattice can be two-dimensional, and the processor can be adapted for generating a two-dimensional image from signal light detected when the excitation maxima are scanned over an area that includes a primitive cell of the lattice.

[0056] A translation stage can translate an optical element that is configured to translate the excitation maxima relative to the sample. The optical elements configured for translating the excitation maxima can include at least one phase shifter for adding an independent overall phase to each excitation beam. The system can further include a translation stage configured for translating the sample relative to the excitation maxima of the Bravais lattice, a memory for recording the detected light from each observed excitation maximum at multiple relative positions of the maxima and sample, and a processor for generating an image from the recorded detected light.

[0057] The lattice can be a three-dimensional lattice, and the translation stage can be configured for translating the sample is configured for translating the excitation maxima across a volume that includes a primitive cell of the lattice. The translation stage for translating the sample can be configured for translating the sample in a plane substantially parallel to a lattice plane of the lattice, and for translating the sample at least over an area defined by an intersection of the lattice plane with a primitive cell of the lattice. The processor can be configured for generating simultaneously multiple two dimensional images from excitation maxima within parallel lattice planes.

[0058] The lattice can be two-dimensional, and the processor can be adapted for generating a two dimensional image from signal light detected when the sample is translated over an area that includes a primitive cell of the lattice.

[0059] The detector can include a microscope objective configured for collecting at least a portion of the signal light. The detector can include a lens configured for imaging signal light beyond the rear pupil of the objective. Detector elements of the detector can be located substantially at an image plane of the lens that images the signal light.

[0060] The lattice can be is three-dimensional and oriented such that a family of lattice planes is substantially parallel to the focal plane of the objective, the excitation maxima within at least one lattice plane within the family can reside at least in part within the focal plane.

[0061] A numerical aperture of the objective can be selected relative to a separation distance between adjacent lattice planes of the family of lattice planes such that substantially only excitation maxima within a single lattice plane lie within the region of focus defined by the numerical aperture.

[0062] The detector can be charged-coupled device (“CCD”) camera. The detector can include a substantially opaque mask located at the image plane of the lens that images the signal light, and the mask can contains multiple substantially transmissive apertures located at positions corresponding to projected images onto the image plane of multiple excitation maxima. A lens can refocus a spatially filtered signal transmitted through each aperture onto a distinct element or group of elements in a detector that converts the light into an electrical signal. The numerical aperture of the objective and the sizes of the apertures can be selected such that the signal transmitted through the apertures originates primarily from excitation maxima within the region of focus defined by the numerical aperture of the objective.

[0063] The lattice can be two-dimensional and oriented such that the excitation varies primarily in a plane parallel to the focal plane of the objective. The objective can be configured for passing at least one of the excitation beams and can include an optical element for separating an excitation beam from signal light in the region beyond the rear pupil. A phase shifter can be used to control a phase across an aggregate signal beam emerging from the rear pupil of the objective. The phase shifter can include a phase shift mask, a spatial light modulator, or a deformable mirror.

[0064] The objective can be infinity corrected and can include a tube lens for creating a magnified image of an excitation maximum at an image plane of the tube lens.

[0065] The system can further include a source of electromagnetic radiation having a trapping wavelength, λ_trap, and optical elements configured for directing radiation beams of the trapping wavelength into the sample to create a single or multi-focus optical trap for holding the sample substantially at a desired point with respect to the objective. At least a portion of the optical trap can be created with one or more radiation beams of the trapping wavelength that pass through the signal collection of objective, and further comprising an optical element for isolating the signal light from radiation having the trapping wavelength.

[0066] The detector of the system can include first and second microscope objectives, with each objective having a rear pupil end, a distal focusing end and a longitudinal axis, where the focusing ends of the two objectives face each another and face the excitation region, and the longitudinal axes of the two objectives are substantially parallel. In such a detector the first lens can be configured for imaging signal light beyond the rear pupil end of the first objective, and the second lens can be configured for imaging signal light beyond the rear pupil end of the second objective. First individual detector elements can be configured for detecting signal light imaged by the first lens, and second individual detector elements configured for detecting signal light imaged by the second lens. At least some excitation maxima can be located within the sample, at least in part within a region of focus of the first or second objective defined by a focal plane and a numerical aperture of the first or second objective.

[0067] The lattice can be three-dimensional, with the lattice being oriented such that a lattice plane is substantially parallel to a focal plane of the first or second objective, and where the excitation maxima within at least one lattice plane are located at least in part within the region of focus of the first or second objective. At least one lattice plane can be in the region of focus of the first objective and in the region of focus of the second objective, and signal light from the at least one lattice plane can be detected by the first individual detector elements and by second individual detector elements. A numerical aperture of each objective can be selected relative to a separation distance between adjacent lattice planes, such that only a single lattice plane is within the region of focus of both objectives.

[0068] At least one lattice plane can be within the region of focus of the first objective but not within the region of focus of the second objective, and at least one additional lattice plane can be within the region of focus of the second objective but not within the region of focus of the first objective. A numerical aperture of each objective can be selected relative to a separation distance between adjacent lattice planes, such that signal light from excitation maxima within exactly one lattice plane is detected only by the first individual detector elements but not by the second individual detector elements, and signal light from excitation maxima within a second lattice plane is detected only by the second individual detector elements but not by the first individual detector elements.

[0069] When at least one excitation beam can be transmitted by each objective, the system can further include optical elements configured for separating excitation beams from the signal light in a region beyond the rear pupil of each objective.

[0070] The system can include first and second microscope objectives, with each objective having a rear pupil end, a distal focusing end, and a longitudinal axis, where the focusing ends of the two objectives face each another and face the excitation region, and the longitudinal axes of the two objectives are substantially parallel. The system can also include optical elements configured to coherently combine at least a portion of a first aggregate signal beam emerging from the rear pupil of the first objective with at least a portion of a second aggregate signal beam emerging from the rear pupil of the second objective, and a lens configured to focus the combined signal beam to create an image at an image plane, where signal in focus at the image plane arises from either a first focal plane of the first objective, or from a second focal plane of the second objective. The system can also include optical elements configured to manipulate the polarization and phase of the aggregate signal beams prior to their combination, such that the two orthogonal polarization components of the individual signal beams substantially interfere constructively after combination, and matching optical elements configured to symmetrically match the image produced by focusing the first aggregate signal beam through the lens with the image produced by focusing the second aggregate signal beam through the lens, such that signal from any transverse location in the first focal plane is focused at the same point on the image plane as the signal from the same transverse location on the second focal plane. In such a case, the detector can have individual detector elements substantially located at the image plane, and at least some of the excitation maxima can reside at least in part within both the first focal plane and the second focal plane.

[0071] The system can further include a phase shifter configured for controlling the phase across the aggregate signal beam emerging from the rear pupil of at least one of the objectives prior to its combination with the aggregate signal beam from the other objective, such that the first and second focal planes associated with the image plane substantially coincide within the sample. The phase shifter can include a phase shift mask, a movable reflective element, or a phase retardation plate configured to control the phase of one component of the polarization of one of the aggregate signal beams relative to an orthogonal polarization component of the same aggregate signal beam.

[0072] The matching optical elements can include a penta prism and a roof prism, where the image produced by focusing the first aggregate signal beam through the lens is symmetrically matched to the image produced by focusing the second aggregate signal beam through the lens by passing one of the aggregate signal beams through the penta prism, and the other aggregate signal beam through the roof prism.

[0073] The system can include optical elements configured for controlling a polarization of an excitation beam. The system can include a polarizer configured to substantially isolate at least one of the two orthogonal polarization states of the signal light that emerges from the rear pupil of the objective. The polarizer can be configured to divide the signal light into two polarized beams of orthogonal polarization traveling in substantially different directions, and then the system can further include a first lens configured to image one of the polarized beams at a first image plane, a second lens configured to image the other polarized beam at a second image plane, first individual detector elements for detecting signal light imaged by the first lens, and second individual detector elements for detecting signal light imaged by the second lens. The polarizer can be configured to divide the signal light into two polarized beams of orthogonal polarization traveling in directions tilted slightly with respect to one another, and the system can further include a lens configured to image each of the polarized beams at an image plane, individual detector elements configured for detecting signal light imaged by the lens, where the different detector elements are configured for detecting signal in each of the orthogonal polarization states generated at excitation maxima within the sample.

[0074] The system can further include a spatial filter configured to isolate at least one portion of the aggregate signal beam that emerges from the rear pupil of the objective. The system can further include optical elements configured to focus light spatially filtered from two or more distinct portions of the aggregate signal beam such that the light from each portion is focused onto its own distinct group of detection elements. The system can further include a lens and a detector associated with each filtered portion of the aggregate signal beam, where each lens is configured to focus light from its associated filtered portion of the signal beam onto its associated detector.

[0075] The system can further include at least one optical element configured to direct light from the distinct filtered portions of the aggregate signal beam in slightly different directions, and a lens configured to focus light from the different portions of the aggregate signal beam onto an image plane, where the detector is located at the image plane, and the light from each portion of the aggregate signal beam is focused onto a different set of elements within the detector at positions determined by the direction imparted on that portion of the aggregate signal beam.

[0076] The system can include a spatial light modulator configured to direct two or more portions of the aggregate signal beam in different directions. The spatial light modulator can be configured to isolate the signal from at least one annular portion of the aggregate signal beam. A spatial light modulator can be configured to spatially filter the signal.

[0077] The detector can include a spectral filter configured for isolating and spectrally resolving the signal created by at least one of the excitation maxima. The detector can include an optical fiber bundle, a spectrometer, and a lens. The fiber bundle can include a face located at an image plane of the lens and configured to image the signal light and an array of optical fibers having proximal ends located at positions corresponding to positions of imaged signal light from a plurality of excitation maxima within a region of focus defined by the focal plane and numerical aperture of the objective lens and having distal ends substantially arranged in a single column. The a spectrometer can include a two-dimensional multi-element detector, and the lens can be configured for focusing signal light emerging from the array of optical fibers onto an entrance slit of the spectrometer. An aperture mask can be configured for spatially filtering the signal light from the excitation maxima before coupling the signal light into individual optical fibers of the array.

[0078] In another general aspect, an optical system includes a substrate adapted for supporting a sample, a source of electromagnetic radiation, a detector, first dividing optical elements, first directing optical elements, and first control optical elements. The substrate has a refractive index, n_sub, larger than a refractive index of the sample, n_sample. The source of electromagnetic radiation has a wavelength, λ_o. The detector has multiple individual detector elements configured for detecting a signal resulting from an interaction of the electromagnetic radiation with the sample. The first dividing optical elements are configured for dividing the electromagnetic radiation from the source into at least three substantially independent excitation beams. The first directing optical elements are configured for directing each excitation beam in a unique direction, k_n, through the substrate and incident upon an interface between the substrate an the sample at angles relative to a normal axis to the interface, ê_z, such that each excitation beam is divided into a reflected beam and an evanescent interfacial beam within the sample that travels in the plane of the interface, where the evanescent beams at least partially intersect with each other within at least a portion of the sample to create an excitation region with an interference pattern having a symmetry of a two dimensional interfacial Bravais lattice. The first control optical elements are configured for controlling an electromagnetic field of each excitation beam such that a basis associated with the interfacial Bravais lattice includes a substantially isolated excitation maximum in each primitive cell of the lattice substantially confined to a size less than a substrate wavelength, λ_sub=λ_o/n_sub, in any direction parallel to the interface. The directions, k_n, of the excitation beams are selected such that the periodicity of the lattice in any direction parallel to the interface is large enough for signals from adjacent excitation maxima to be individually resolved and simultaneously measured by separate elements within the detector.

[0079] Implementations can include one or more of the following features. For example, the directing and control optical elements can be configured such that the interfacial Bravais lattice has a symmetry, a shape to the primitive cell defined by the region within the interface closest to any chosen lattice point, and a periodicity within the sample that is larger relative to the substrate wavelength, λ_sub, than the smallest such lattice of the same symmetry and identically defined primitive cell shape that can be created at that wavelength, and wherein the directions of the interfacial beams do not all lie on a pair of mutually orthogonal axes.

[0080] The directing and control optical elements can be configured such that at least one of the interfacial beams travels substantially in a direction, k_n^interface, such that one or more symmetry operations characteristic of the lattice can map the direction, k_n^interface, onto a different direction, k_m^interface, associated with a different interfacial beam, and wherein the directions of all the interfacial beams do not lie on a pair of mutually orthogonal axes.

[0081] The directing and control optical elements can be configured such that for substantially every interfacial beam traveling in a unique direction, k_n^interface, another interfacial beam exists traveling in a direction, k_m^interface, that is determined by applying any combination of symmetry operations associated with the interfacial Bravais lattice to the direction k_n^interface.

[0082] The directing and control optical elements can be configured such that the incident excitation beams and the reflected beams intersect within the substrate to form a three-dimensional Bravais lattice oriented such that a lattice plane of the three-dimensional lattice is parallel to the interface.

[0083] The directing and control optical elements can be configured such that each incident beam is polarized in its plane of incidence, so that an electric field associated with the interfacial Bravais lattice is substantially polarized perpendicular to the interface.

[0084] The substrate can include a substantially transparent material having a refractive index greater than about 2. The substrate can include gallium nitride, gallium phosphide, silicon carbide, aluminum nitride, or aluminum arsinide. The substrate can be substantially hemispherical in shape, where the sample is supported on the flat surface of the hemisphere, and where the excitation beams enter the substrate through the curved surface of the hemisphere.

[0085] The system can include a hemisphere of a material having a refractive index, n_sphere, where the substrate is mounted to the flat surface of the hemisphere, and the directing and control optical elements can be configured such that the excitation beams enter the hemisphere through its curved surface and exit the hemisphere through its flat surface. The refractive index of the hemisphere can be selected such that at least a portion of each excitation beam is transmitted through the interface between the hemisphere and the substrate, and the directing and control optical elements can be configured such that the excitation beams enter the hemisphere in directions such that they travel in the substrate in the directions, k_n, that generate the interfacial lattice.

[0086] The system can include translation optical elements configured for translating the excitation maxima relative to the sample in the plane of the interface, a memory configured for recording a signal from each excitation maximum at multiple relative positions of the maxima and sample, and a processor configured for generating an image of the sample from the recorded signals. A display can be configured for displaying the image. The processor can be further configured for generating a two-dimensional image from signals recorded when the sample and the excitation maxima are scanned relative to another over an area that includes at least one primitive cell of the interfacial lattice.

[0087] The detector can include a microscope objective adapted for collecting at least a portion of the signal, and a lens adapted for creating an image of the interface at an image plane beyond a rear pupil of the objective, where the individual detector elements are located substantially at an image plane of the lens, and where a plurality of excitation maxima within the sample are focused upon distinct detector elements. The detector can include a substantially opaque mask located at the image plane having multiple substantially transmissive apertures at positions corresponding to projected image positions on the image plane of a plurality of excitation maxima near the interface and a lens adapted for refocusing signal light transmitted through each aperture onto a distinct detector element or group of detector elements.

[0088] The system can include optical elements configured for adjusting the direction, k_n, of each excitation beam, so that the projection of each direction, k_n, onto the interface can be changed simultaneously by substantially the same amount for each excitation beam.

[0089] The processor can be adapted for generating a plurality of images recorded at different incident beam projections onto the interface, and can be adapted for generating a three-dimensional image of the sample from the plurality of images.

[0090] The system can include a second source of pulsed electromagnetic radiation, having a second wavelength, λ_depl. Second dividing optical elements can be configured for dividing the radiation from the second source into a second set of at least D_depl+1 depletion beams of wavelength, λ_depl, second directing optical elements configured for directing each depletion beam of the second set in a unique direction, k_n^depl, such that a pulse within each beam intersects with a pulse within every other beam of the second set within the excitation region at time t₀ create a depletion interference pattern having a symmetry of a Bravais lattice in D_depl dimensions, and a depletion lattice periodicity, second control optical elements configured for controlling an electromagnetic field basis associated with the depletion lattice, such that at least one substantially isolated intensity minimum of the depletion lattice at least partially surrounded by a region of higher intensity and substantially confined to a size of less than the wavelength, λ_depl, in at least two dimensions, exists in each primitive cell of the depletion lattice. An optical element can be configured for gating the detector to detect signal at a time, t>t₀, after the pulses that intersect at an initial time, t₀, have left the excitation region, where the first and second directing and control optical elements are configured such that the symmetry and periodicity of the first Bravais lattice and the depletion Bravais lattice are chosen such that a plurality of excitation maxima within the first Bravais lattice at least partially overlap with a plurality of intensity minima within the depletion lattice. The wavelength, λ, of the first Bravais lattice can be chosen such that the first Bravais lattice raises a selected species of fluorescent emitters within the sample to an excited energy state, and the wavelength, λ_depl, of the depletion lattice can be chosen such that the depletion Bravais lattice lower the fluorescent emitters from the excited state to a lower energy state by a stimulated emission process.

[0091] The system can further include a pulse controller for controlling an intensity and a duration of the pulse of depletion beams, such that a desired portion of the emitters near each excitation maximum of the first Bravais lattice is de-excited prior to opening of a gate to the detector, so that a recorded signal arises substantially from a lattice of regions, each significantly smaller than the excitation wavelength in at least two dimensions.

[0092] The first and second directing and control optical elements can be configured such that the depletion lattice has the same symmetry and identically defined primitive cell shape as the first Bravais lattice, and wherein the wavelengths, λ and λ_depl, of the two lattices as well as their wavelength-normalized periodicities are chosen so that their absolute periodicities are an integral ratio of one another.

[0093] In another general aspect, optical system includes a source of electromagnetic radiation, dividing optical elements, field control optical elements, and intensity control optical elements. The source of electromagnetic radiation has a wavelength, λ. The dividing optical elements are configured for dividing the radiation from the source into at least D+1 excitation beams. The directing optical elements are configured for directing each excitation beam in a unique direction, k_n, such that the beams intersect within an excitation region within a photosensitive sample to create an interference pattern in the sample having a symmetry of a Bravais lattice in two or three dimensions, D, and a lattice periodicity. The field control optical elements are configured for controlling an electromagnetic field basis associated with the lattice. The intensity control optical elements configured for controlling the intensity of the excitation beams. The optical elements are configured to produce at least one substantially isolated intensity maximum in each primitive cell of the lattice substantially confined in any direction within at least two dimensions to a size that is less than half the periodicity of the lattice in the same direction.

[0094] In another general aspect, an optical system includes, a first source of electromagnetic radiation, first dividing optical elements, first directing optical elements, and first control optical elements. The first source of electromagnetic radiation has a trapping wavelength, λ_trap. The first dividing optical elements are configured for dividing the radiation from the first source into a first set of at least D+1 trapping beams. The first directing optical elements are configured for directing each trapping beam in a unique direction, k_n, such that the beams intersect within an trapping region to create an interference pattern having a symmetry of a trapping Bravais lattice in two or three dimensions, D, and a lattice periodicity. The first control optical elements are configured for controlling an electromagnetic field basis associated with the lattice, such that at least one substantially isolated intensity maximum is produced in each primitive cell of the lattice and is substantially confined in any direction within at least two dimensions to a size that is less than half the periodicity of the trapping lattice in the same direction. The control optical elements are configured for controlling the absolute intensity at the maxima in each primitive cell is controlled to produce points of an intensity gradient sufficiently strong for confining matter.

[0095] In another general aspect, a method for inducing a photophysical change in a sample at a plurality of isolated regions simultaneously includes generating within the sample a periodic interference pattern of coherent waves from a plurality of mutually intersecting electromagnetic beams of substantially the sample wavelength, λ, wherein the pattern has characteristics of a Bravais lattice in two or three dimensions. The basis associated with the lattice is controlled to produce at least one substantially isolated intensity maximum in each primitive cell of the lattice, with the intensity maximum being substantially confined in at least two dimensions to less than half the periodicity of the lattice defined by the intensity maxima. The intensity and duration of the exposure of the sample to the lattice is controlled to limit the photoinduced physical change in the sample to a desired region centered at each maximum.

[0096] In another general aspect, a method of creating a plurality of isolated intensity maxima capable of optically trapping a sample of matter in at least two dimensions includes generating within the matter a periodic interference pattern of coherent waves from a plurality of mutually intersecting electromagnetic beams of substantially the sample wavelength, λ, where the pattern has the characteristics of a Bravais lattice in two or three dimensions. The basis associated with the lattice is controlled to produce at least one substantially isolated intensity maximum in each primitive cell of the lattice, the intensity maxima being substantially confined in at least two dimensions to less than half the periodicity of the lattice defined by the intensity maxima. An absolute intensity at the maxima is controlled to produce an intensity gradient sufficiently strong to trap the matter.

[0097] In another general aspect, a microscope includes a source of electromagnetic radiation having a wavelength, λ₁, dividing optical elements configured for dividing the radiation from the source into multiple excitation beams, directing optical elements, and a detector. The directing optical elements are configured for directing each excitation beam in unique directions, such that the beams intersect in an excitation region within a sample to create a two-dimensional or three-dimensional interference pattern of multiple excitation maxima within the sample. The detector has individual detector elements, and the detector elements are configured for detecting light resulting from an interaction of an individual excitation maximum and the sample.

[0098] Implementations can include one or more of the following features. For example, the microscope can include a translation stage configured for translating the sample and the interference patter relative to each other. The detector can be configured to detect simultaneously signals of light resulting from interactions of multiple individual excitation maxima and the sample. The microscope can further include a processor configured for generating an image of the sample from the multiple signals. A spacing of the maxima can be less than the wavelength, λ₁.

Overview

[0272] A sample can be imaged by creating a periodic array of light intensity maxima within the sample. The periodic array can be created though the superposition and interference of a finite set of plane waves, i.e., multiple beams of monochromatic light. Such a construct can be known as an optical lattice. Portions of the sample illuminated by the intensity maxima can be detected and imaged by separate detector elements. The periodic array then can be translated within the sample to illuminate and image different portions of the sample. By translating the array of intensity maxima throughout the sample and imaging the maxima during the translation, an image of the entire sample can be created. Because the array is periodic, the array only needs to be shifted over one unit cell of the array to create an image of the entire sample.

[0273] Before three-dimensional optical lattices are described, the simpler cases of one- and two-dimensional optical lattices are described. The concepts that are developed for the one- and two-dimensional cases then are extended to the three-dimensional case.

[0274] One-Dimensional Lattices

[0275] Referring to FIG. 6A, two electromagnetic (EM) plane waves traveling in the x-z plane of a three-dimensional coordinate system defined by an x-axis 602, a z-axis 604, and a y-axis 605 can be superimposed. Each plane wave can be parameterized by a wavevector k₀ and k₁, respectively, which defines the propagation direction of the wave and the wavelength of the wave. The magnitude of the wavevector is related to the wavelength of the wave, λ, by the relationship |k|=k=2π/λ. Because the wavevector is defined by a direction and a magnitude, it is a vector quantity. Vector quantities, as used herein, are written as bold lower-case symbols. Wavevectors k₀ and k₁ have different directions but identical magnitudes, |k₀|=|k₁|, so the two plane waves each have the same wavelength. In addition, the phases of the waves are controlled so that they have a constant phase difference. Because the waves are monochromatic and have a well-defined phase difference, their intersection results in a coherent superposition of the waves. As shown in FIG. 6B, the superposition of the two plane waves yields a one-dimensional pattern of intensity maxima in the combined EM fields of the waves. Such a pattern can be called a one-dimensional optical lattice and is commonly known as a standing wave. As shown in FIG. 6C, the EM field of the one-dimensional lattice varies sinusoidally along the x-axis 602 of the x-z plane 603. Along the z-axis 604 (defined by the line in the x-z plane 603 that bisects the wavevectors 600 and 601), as well as along the y-axis 605 that is perpendicular to both wavevectors 600 and 601, the EM field is uniform.

[0276] As shown in FIG. 6C, by altering the angle, θ, between the two wavevectors 600 and 601, the spacing Δx=λ/(2 sin(θ/2)) of the periodic intensity maxima along the x-axis can be increased by any desired amount from a minimum of λ/2. Although altering the angle, θ, to increase the distance between adjacent intensity maxima permits signals from adjacent maxima to be more readily resolved, it also reduces the spatial confinement of each high-intensity region. This is the case because the spatial confinement at a local maximum in the light intensity can be defined as the width of a region within which the intensity is at least half of its greatest magnitude. In other words, the spatial confinement of an intensity maximum can be defined as the full width at half magnitude (“FWHM”), such that FWHM_x=Δ/2=λ/(4 sin(θ/2)). Thus, for purposes of detecting local maxima (e.g., to create a microscopic image of a portion of a sample of maximum light intensity), increased resolvability of adjacent maxima generally comes at the price of reduced spatial confinement and resolution of individual high-intensity regions. Furthermore, when only two EM plane waves are superimposed, as shown in FIG. 6A, although the intensity maxima are confined along the x-axis, the excitation of local maxima is not confined along the y and z axes. Thus, the three-dimensional microscopy resolution that can be achieved by creating a one-dimensional lattice in a sample is determined primarily by the detection characteristics of the particular microscope in the z- and y-directions, as well as the ability to isolate the signal from a single planar intensity maximum in the x-direction.

[0277] Two-Dimensional Lattices

[0278] Referring to FIG. 7A, a two-dimensional optical lattice can be created by adding a third plane wave having a wavevector, k₂, to the superposition of plane waves with wavevectors, k₀ and k₁, where the three plane waves are monochromatic (i.e., each plane wave has the same wavelength). The optical lattice is located in the plane that is orthogonal to the z-axis 703 of the cone 704 on which all three wavevectors k₀, k₁, and k₂ lie. In particular, FIG. 7B is a three-dimensional plot of surfaces of light intensity having 50% of the maximum light intensity in the optical lattice formed by the plane waves shown in FIG. 7A. FIG. 7B shows the periodic nature of the intensity maxima in an exemplary two-dimensional optical lattice created by the superposition of three z-polarized plane waves, all propagating in the x-y plane. FIG. 7C depicts the intensity maxima of the lattice of FIG. 7B as lighter regions that are shown in relation to the wavevectors k₀, k₁, and k₂ of the constituent plane waves that make up the lattice.

[0279] The superposition of exactly three wavevectors k₀, k₁, and k₂ creates only one spatial frequency in each of the three directions 705, 706, and 707 defined by the intersection of the lattice plane with the plane bisecting any pair of wavevectors. Thus, as in the case of a one-dimensional lattice, the spatial confinement (i.e., the intensity full width at half maximum) of high-intensity regions of the lattice is equal to half of the distance between maxima (i.e., half the periodicity of the lattice), in each of the directions 705, 706, and 707. As a result, spatial confinement of individual maxima and periodicity of the lattice can only be adjusted by changing the angle of the cone, shown in FIG. 7A, upon which the three wavevectors k₀, k₁, and k₂ lie (as shown in FIGS. 8B and 8C).

[0280] When certain relationships between the constituent wavevectors of a lattice are satisfied, a lattice of special symmetry can be created that is particularly useful for microscopy, such as, for example, the lattice of hexagonal symmetry shown in FIG. 8A which is formed by the superposition of three wavevectors k₀, k₁, and k₂ that all lie in the x-y plane. FIGS. 8B and 8C show an example of changing the angle of the cone upon which the wavevectors lie. By shifting the wavevectors k₀, k₁, and k₂ that create the lattice in FIG. 8A out of the x-y plane to form an angle of sin⁻¹(⅔) with respect to the z-axis, as illustrated in FIG. 8B, an optical lattice having a periodicity of 1.5 times greater than the original lattice can be created. The resulting lattice having the higher periodicity is shown in FIG. 8C.

[0281] When such lattices are used in optical microscopy, the periodicity of the lattice intensity maxima determines the ease with which the maxima can be resolved at a detector. The periodicity also determines the number of maxima required to cover a given field of view. On the other hand, the spatial confinement of individual maxima determines the spatial resolution of an image of one or more maxima. Although the periodicity of the lattice and the spatial confinement of individual maxima may only be adjusted in tandem for two-dimensional lattices constructed from three plane waves, control of the lattice periodicity and the spatial confinement of individual maxima can be decoupled if more than three plane waves with wavevectors k₀, k₁ . . . k_N lying on a single cone are used to create the optical lattice. Such lattices formed of four or more plane waves having different wavevectors lying on a cone and having a wavelength x, are hereafter called composite two-dimensional lattices. FIG. 8D, for example, illustrates a composite two-dimensional lattice formed from 12 plane waves that has increased spatial confinement but identical periodicity relative to the exemplary lattice of FIG. 8A. The direction of the 12 constituent wavevectors of the plane waves is explained in more detail below.

[0282] Three-Dimensional Lattices

[0283] Analogously to the case of two-dimensional lattices described above, the superposition of four or more plane waves results in a three-dimensional lattice when all constituent wavevectors k_m do not lie on one cone, as illustrated in FIG. 9B. As shown in FIG. 9B, wavevectors k₀, k₁, and k₂ fall on the positive x-, y-, and z-axes, respectively, and the wavevector k₃ and falls on the negative z-axis. Intensity confinement occurs in all three directions, as shown in intensity maxima of FIG. 9 and as indicated in the intensity maxima in the two-dimensional slice planes 900 and 901 shown in FIG. 9A, which are plotted separately in FIGS. 9C and 9D, respectively. Analogously to two-dimensional optical lattices, when certain relationships between the constituent wavevectors, k_m, are satisfied, three-dimensional lattices of special symmetry can be created that are particularly useful for purposes of microscopy. For example, a body-centered cubic three-dimensional lattice (shown in FIG. 10A) can be created from four plane waves having a wavelength, λ, and wavevectors k₀, k₁, k₂, and k_z arranged as shown in FIG. 9B. Maxima along the two-dimensional planes shown in FIG. 10A are shown in FIGS. 10B and 10C. Also, although periodicity of the lattice and the spatial confinement of the maxima may only be adjusted in tandem for three-dimensional lattices constructed from four plane waves, control of the lattice periodicity and spatial confinement of the intensity maxima can be decoupled if more than four plane waves with wavevectors k₀, k₁ . . . k_N not all lying on a single cone are used to create the lattice. Such lattices formed of five or more planewaves having a wavelength, λ, and wavevectors that do not lie on a single cone are hereafter termed composite three-dimensional lattices. The periodicity of the lattice is then decoupled from the spatial intensity confinement of the intensity maxima in the four directions defined by the cone axis associated with every grouping of three wavevectors. For example, FIG. 10D illustrates a three-dimensional lattice, formed from 24 plane waves, that has increased three-dimensional spatial confinement but identical three-dimensional periodicity relative to the exemplary lattice of FIG. 10A. FIGS. 10E and 10F illustrate the intensity maxima of the lattice shown in FIG. 10D viewed in the x-y and x-z planes, respectively.

[0284] The Basis Associated with an Optical Lattice

[0285] For two-dimensional and three-dimensional lattices, the interrelationship between the wavevectors, k₀, k₁ . . . k_N, of the N+1 constituent plane waves completely determines the symmetry and periodicity of the lattice relative to the wavelength of the plane waves. However, it is the electric field vectors, e₀, e₁, . . . e_N, (and associated magnetic field vectors, h_m=n(k_m×e_m)/k) of these plane waves as well as their wavevectors, k₀, k₁ . . . k_N, that determine the EM field pattern that exists within each primitive cell (i.e., the smallest two- or three-dimensional region that, when repeated, will fill the entire lattice) of the lattice. This EM field pattern, which can be referred to as the basis, is analogous to the arrangement of atoms in each primitive cell of a crystalline solid, such as the arrangement of the sodium and chlorine atoms that exists in every primitive cell of the simple cubic lattice of table salt.

[0286] In general, the basis associated with any lattice can be quite complex, leading to an intensity pattern within each unit cell that is unsuitable for microscopy. However, by carefully selecting the relative amplitudes and phases of each of the components of the electric field, e_m, for every plane wave in the lattice, the basis can be manipulated so as to be suitably tailored for microscopy applications.

[0287] A common example of such a basis is a basis consisting of a single dominant, tightly-confined intensity maximum within the primitive cell. For example, the two-dimensional hexagonal lattice represented in FIG. 8D clearly achieves this dominant maximum basis having a single tightly-confined intensity maximum in each cell by using plane waves polarized perpendicularly to their mutual plane of propagation and phases adjusted to achieve maximal constructive interference at the maximum. However, as shown in FIG. 11A, when the same wavevectors used to form the lattice of FIG. 8D but having plane waves polarized to maximize the y-component of the polarization at the intensity maximum, a more complicated basis results in which the intensity is no longer significantly confined to one point within the primitive cell. Similarly, the three-dimensional body-centered cubic lattice depicted in FIG. 11B uses plane waves of the same wavevectors and polarizations as those used to form the lattice in FIG. 10D, but the phases of the plane waves are chosen randomly for the lattice of FIG. 11B instead of being selected to achieve optimal constructive interference at a single point within the primitive cell, as in FIG. 10D. As seen from the comparison of FIGS. 11B and 10D, and from the comparison of FIGS. 11A and 8D, control of the relative phase and polarization, in addition to the wavevectors, of the constituent plane wave of a lattice can be used to control the basis of the lattice and to tailor the basis for particular applications.

[0288] Image Formation in Optical Lattice Microscopy

[0289] When the basis is selected to achieve a tightly confined intensity maximum in each primitive cell, as exemplified by the two- and three-dimensional lattices shown in FIG. 8D and FIG. 10D, the lattice and basis together are well suited to function as an optical crystal lattice that can provide a multi-point excitation field to excite a sample to be imaged during a microscopy process. Such an excitation field can be exploited in the context of microscopy by conducting parallel scanning of a sample in one or more dimensions, as depicted in the imaging and detection examples shown in FIG. 12 and FIG. 13, respectively. However, in other microscopy applications, such as those in which a microscopy signal is generated from two or more overlapping optical lattices, the formation of different types of bases can be more appropriate.

[0290] FIG. 12 illustrates one manner in which microscopy can detect a signal from a multi-point excitation field using a two-dimensional lattice. Signal light 1200 emitted from the interaction of an optical lattice 1201 with a sample 1202 is collected by a lens 1203 and imaged onto a two-dimensional multi-element detector 1205, such as, for example, a charge-coupled device (CCD) array. Lens 1203 may be an infinity-corrected microscope objective and the light 1200 may be imaged with a corresponding tube lens 1204. Prior to detection, the signal light 1200 can be spatially filtered by, for instance, using a pinhole array 1206 of the same symmetry and magnified periodicity as the lattice. Such filtration can block signal light 1200 that is emitted from regions not within the two-dimensionally confined maximum of each primitive cell and can limit detection in the third dimension to regions near the focal plane 1207. Indeed, the pinhole diameters within array 1206 together with the numerical aperture (NA) of the objective lens 1203 and the magnification of the objective 1203/tube lens 1204 pair define a detection point spread function (“PSF”) 1208 that characterizes a three-dimensional region from which signal light is predominantly measured within each primitive cell.

[0291] FIG. 13 illustrates one manner in which microscopy can detect signal from a multi-point excitation field of a three-dimensional lattice. Signal light 1300 emitted from the interaction of a three-dimensional optical lattice 1314 with a sample 1315 is collected by a lens 1305 and imaged onto one or more two-dimensional detectors 1312, such as one or more CCD arrays films or other photosensitive medium. Lens 1305 may be an infinity-corrected microscope objective and the light 1300 may be imaged with a corresponding tube lens 1316. Detection of a signal from a three-dimensional lattice with a basis of isolated intensity maxima (e.g., as shown in FIG. 10D) is rendered more complicated by the fact that the signal 1300 emitted from each maximum (e.g., 1307, 1308, and 1309) must be individually discriminated after originating on multiple, lattice planes (e.g., 1301, 1302, and 1303), and being imaged onto one or more two-dimensional detectors 1312. Therefore, prior to detection, the signal light 1200 can be spatially filtered such that signal light from only the intensity maxima within the desired lattice planes 1302 is detected while signal light from other lattice planes 1301 and 1303 is blocked from reaching the detector 1312. For example, a pinhole array 1304 can act as a spatial filter to pass light emitted from the plane 1302 (i.e., at the focal plane of lens 1305) but to block light from planes 1301 and 1303 (i.e., out of the focal plane of the lens 1305).

[0292] The pinhole diameters within array 1304 together with the numerical aperture (“NA”) of the objective lens 1305 and the magnification of the objective 1305/tube lens 1316 pair define a detection point spread function (“PSF”) 1306 of axial extent that primarily encompasses signal from only one intensity maximum element (e.g., 1308) within a given column 1317 of maxima whose axial projections onto the image plane 1318 are not resolvable transversely to the optical axis 1310. The three-dimensional lattice can be tilted relative to the optical axis 1310 to define the number of lattice planes that fall within the axial bounds 1311 of the detection PSF 1306 and to insure that the intensity maxima within these planes are transversely separated by distances 1313 greater than the transverse extent of the detection PSF, making them individually resolvable. In particular, the lattice can be tilted to an orientation dictated by the Miller indices of the various lattice plane families if it is desirous for the intensity maxima to lie in planes transverse to the optical axis 1310.

[0293] FIG. 14A depicts a three-dimensional plot of intensity maxima for an exemplary three-dimensional optical lattice where each three-dimensional detection PSF encompasses signal emitted from a single intensity maximum element of the lattice. FIG. 14B represents the convolution of the intensity maxima with the detection PSF in the x-z plane shown in FIG. 14A, and demonstrates that the detected signal is indeed limited to individual maxima within three adjacent lattice planes. FIG. 14C is a representation of the projection of the lattice at the image plane, and thus indicates the pinhole pattern within the mask 1304 needed to preferentially capture the signal from all intensity maxima in the three desired lattice planes over a region of given transverse extent. FIG. 14D, FIG. 14E, and FIG. 14F are plots similar to FIG. 14A, FIG. 14B, and FIG. 14C, except with the lattice tilted so that the (121) lattice planes are transverse to the optical axis instead of the (111) lattice planes. The pinhole pattern in FIG. 14F is designed to collect signal from four of the six lattice planes having different x-y positions of intensity maxima therein. Many other detection scenarios are also possible, as discussed in more detail herein.

[0294] Imaging an Entire Sample

[0295] For either a two-dimensional or a three-dimensional lattice with a basis of isolated intensity maxima, an image of the entire two-dimensional or three-dimensional field of view can be generated by scanning the optical lattice and the detection system over the dimensions of a single primitive cell of the lattice relative to a sample, and (2) recording the signal from all maxima at each scan position. Where L lattice points of intensity maxima are detected within the field of view, the signal-limited imaging speed for optical lattice microscopy is L times faster than for single focus confocal microscopy. Also, where an optical lattice has a high degree of symmetry (e.g., a three-dimensional lattice of the cubic crystal group), the intensity maxima can be tightly confined in optical lattice microscopy, such that the lattice has spatial frequencies that approach the diffraction-limited maximum of (Δk·ê_j)_max=2k in all directions, ê_j. Thus, for example, the intensity maxima 1502 of a particular simple cubic lattice shown in FIG. 15A can be significantly smaller than the single maximum 1504 of a confocal system shown in the same figure. This improvement is further demonstrated by comparing the intensity plots in the xy, xz, and yz planes for the lattice in FIGS. 14E, 14F, and 14G with the corresponding plots for the confocal focus in FIGS. 14B, 14C, and 14D.

[0296] Optical lattice microscopy can also reduce the rate of photobleaching in a sample, as compared with widefield or confocal fluorescence microscopy, because optical lattice microscopy detects intensity maxima at numerous points throughout a field of view, creates constructive interference leading to strong intensity maxima at such points, and creates partial destructive interference at other points in each primitive cell, such that a greater proportion of the light interacting with a sample results in the generation of detected signal. Optical lattice microscopy can also enhance sensitivity to nanometric structures and processes, such as fluorescence resonance energy transfer (“FRET”) between a single pair of molecules, because it can reduce background radiation and improve excitation confinement.

Design and Construction of Ideal Optical Lattices

[0297] Introduction

[0298] The first step in applying optical lattices to microscopy is to develop methods of identifying the plane wave sets and their wavevectors, k_m, and properties of their electric fields, e_m, that give rise to lattices and bases of greatest use for a given imaging application. Toward this end, maximal flexibility and effectiveness can be achieved by:

• • [0299] a) determining wavevector sets, {k_m}, for lattices of any possible desired symmetry;
  • [0300] b) determining different wavevector sets, {k_m}, corresponding to lattices of the same symmetry but having different wavelength-normalized periodicities;
  • [0301] c) increasing the number of wavevectors used to create a lattice of given symmetry and periodicity in order to increase the spatial frequency content and confinement of the excitation; and
  • [0302] d) determining complete descriptions of the electric field properties and wavevectors {e_m, k_m} for plane wave sets that optimize one or more basis parameters for a given lattice, such as the intensity or polarization state at a given point in the primitive cell.

[0303] Each of the above tasks can be accomplished by considering a mathematical model of optical lattices. The total basis (i.e., the excitation electric field pattern), e(x,t), for a given optical lattice results from the coherent superposition of all of its constituent plane waves. Where the selected plane waves have wavevectors, k₀, k₁ . . . k_N and electric fields e₀, e₁ . . . e_N, the resulting basis is described by the equation:

[0304] $\begin{array}{cc}e\left(x,t\right)=\mathrm{Re}\left\{\sum _{n=0}^N{e}_n\exp \left[i\left(k_n·x-i\omega t\right)\right]\right\},& \left(3\right)\end{array}$
where, by the coherence condition,
|k_n|=k=2π/λ=ω/c=constant for all values of n  (4)
and λ, ω, and c are the wavelength, angular frequency, and speed of light in the medium, respectively. According to Maxwell's equations, each plane wave must also satisfy the equation:
e_n·k_n=0 for all values of n  (5)
The total excitation intensity is given by the square of the electric field function and can be expressed by the equation:

[0305] $\begin{array}{cc}I\left(x\right)\propto {e\left(x,t\right)}^2=\sum _{m=0}^N\sum _{n=0}^N{e}_m^{*}·e_n\exp \left[i\left(k_n-k_m\right)·x\right].& \left(6\right)\end{array}$

[0306] Multidimensional Lattices

[0307] The electric fields of the basis of an optical lattice have a spatial periodicity such that the fields repeat themselves in space. The spatial periodicity of the basis described in Equation (3), which can be expressed as e(x+a,t)=e(x,t), is dictated solely by the exp(ik_n·x) terms. As a result, as mentioned above, the symmetry and wavelength-normalized size of an optical lattice depend only upon the wavevectors, k_n, associated with the constituent plane waves. However, the complete electromagnetic field pattern, or basis, depends upon both the wavevectors, k_n, and the electric fields, e_n, of the constituent plane waves. With exactly two plane waves having wavevectors k₀ and k₁ (i.e., N=1), as illustrated in FIG. 6A, the basis is uniform along the y and z axes and (k₀−k₁)·ê_y=0 and (k₀−k₁)·ê_z=0, where ê_y=(k₀×k₁)/|k₀×k₁| and ê_z=(k₀+k_1)/|k0+k₁|. In light of Equation (6), the superimposition of two such plane waves yields a one-dimensional lattice described by I(x)=I(x) where ê_x=ê_y×ê_z.

[0308] Similarly, as illustrated in FIG. 7A, exactly three plane waves having wavevectors k₀, k₁, and k_Z (i.e., N=2) results in a uniform basis ∀n along the z-axis and k_n·ê_z=0 for all values where the z-axis is described in the equation:
ê_z=(k₀×k₁)+(k₁×k₂)+(k₂×k₀)/|(k₀×k₁)+(k₁×k₂)+(k₂×k₀)|.  (7)

[0309] The superimposition of three such plane waves yields a two-dimensional lattice described by I(x)=I(x,y) and lying in the x-y plane orthogonal to ê_z.

[0310] With four or more plane waves (i.e., N≧3), a three-dimensional lattice can be created, provided that there is no axis, ê_z, for which k_n·ê_z=0 for all values of n. Thus, a D dimensional lattice requires at least D+1 plane waves.

[0311] Finding Wavevector Sets for a Given Lattice

[0312] Just as a lattice in solid-state physics is a purely geometrical construct specifying the periodicities and symmetry of the crystal, and the corresponding basis is the physical entity that determines the actual location of the atoms within each primitive cell, an optical lattice is analogously a geometrical construct specifying the periodicities and symmetry of an optical crystal and the corresponding basis is given by the field, e(x,t), that determines the actual intensity pattern within each primitive cell. Crystallography dictates that it is possible to create exactly five two-dimensional lattices, and exactly fourteen three-dimensional lattices, of unique symmetry, which are collectively referred to as Bravais lattices. Bravais Lattices are discussed in K. I. Petsas, et. al, Phys. Rev. A 50, 5173-5189 (1994), which is incorporated herein by reference. Each D-dimensional Bravais lattice (where D=2 or 3) can be characterized by a D-dimensional set of D primitive vectors, a_n, that satisfy the necessary and sufficient conditions of both (1) connecting symmetrically identical points and (2) spanning the entire real space lattice (i.e., the direct lattice). FIG. 16B illustrates several valid primitive vector pairs for a particular oblique two-dimensional lattice, and FIG. 16A illustrates an exemplary primitive vector trio for a particular three-dimensional lattice. Where each set of primitive vectors, a_n, is expressed as a D×1 column vector, the associated lattice can be characterized by a D×D direct lattice matrix. For example, where D is equal to two or three, respectively, the corresponding direct lattice matrices are given by the equations:

[0313] $\begin{array}{cc}\begin{array}{cc}A\equiv \left[\begin{array}{cc}{a}_1& a_2\end{array}\right]=\left[\begin{array}{cc}{a}_{1x}& a_{2x}\\ a_{1y}& a_{2y}\end{array}\right]& D=2\end{array}& \left(8a\right)\\ \begin{array}{cc}A\equiv \left[\begin{array}{cc}{a}_1& \begin{array}{cc}{a}_2& a_3\end{array}\end{array}\right]=\left[\begin{array}{ccc}{a}_{1x}& a_{2x}& a_{3x}\\ a_{1y}& a_{2y}& a_{3y}\\ a_{1z}& a_{2z}& a_{3z}\end{array}\right]& D=3.\end{array}& \left(8b\right)\end{array}$
where matrix quantities, as used herein, are written as bold upper-case symbols. The direct lattice matrix, A, is not unique because the choice of primitive vectors, a_n, is not unique, which can be seen from the existence of multiple valid primitive vector pairs for the particular two-dimensional lattice illustrated in FIG. 16B.

[0314] Every direct (i.e., real space) Bravais lattice has an associated reciprocal (i.e. reciprocal space) lattice that, by definition, is given by the set of all vectors, κ, for which the quantity exp(iκ·x) exhibits the same periodicity as the direct lattice. A reciprocal lattice is also a Bravais lattice, and hence can also be described by an D-dimensional set of D primitive vectors, b_n, or by a reciprocal lattice matrix B that is defined similarly to the direct lattice matrix A in Equations (8a) and (8b). For a given set of direct lattice primitive vectors, {a_n}, it is possible find a set of reciprocal lattice primitive vectors, {b_n}, that satisfy the condition b_i·a_j=2πδ_ij, where i,j=1 . . . D, and where δ is the Dirac delta function and is equal to one when i=j and is equal to zero when i≠j. In matrix form, the reciprocal lattice matrix is described by the equation:
B′·A=2πI  (9a)
or, equivalently:
B=2π(A′)⁻¹  (9b)
where I is the D×D identity matrix.

[0315] Next Equation (6), which describes the light intensity pattern due to the combination of the interfering individual light beams, can be examined in view of these definitions. There are D(D+1) non-zero wavevector differences, k_n−k_m, that can be identified with specific points in a reciprocal lattice, and these wavevector differences can be expressed as integral combinations of N independent primitive vectors, b_n. In particular, for the minimum set of N=D+1 plane waves, we can choose:
b_n=k₀−k_n where n=1 . . . D;  (10)
that is, the primitive vectors, b_n, are given by the difference between a single wavevector, k₀, in the minimum set and every other wavevector, k_n, in the same set. Combining Equations (4) and (10) then yields:
|k_n|²=k²=|k₀−b_n|²=k²−2b_n′·k₀+b_n′·b_n  (11)
or, in matrix form:

[0316] $\begin{array}{cc}\begin{array}{cc}{B}^{\prime }·k_0=\left[\begin{array}{c}{b}_1^{†}·b_1\\ ⋮\\ b_D^{†}·b_D\end{array}\right]/2\equiv \beta /2& D=2\mathrm{or}3\end{array}& \left(12\right)\end{array}$
Lastly, Equations (9b) and (12) together yield:
k₀=A·β/4π  (13)

[0317] Using the mathematical framework provided by Equations (8), (9b), (10), (12) and (13), a minimum set of wavevectors, {k_n}, can be selected that generates a two-dimensional or three-dimensional optical lattice of any desired Bravais symmetry, as characterized by a particular direct lattice matrix, A. First, the corresponding reciprocal lattice, B, can be determined using Equation (9b). Second, the column vectors b_n′ of B can be used to determine β in light of Equation (12). Third, A and β together can yield the first unknown wavevector, k₀, according to Equation (13). Finally, the remaining D wavevectors can be determined using Equation (10).

[0318] Direct lattice primitive vectors, a_n, are traditionally expressed as vector functions of the lattice constants a, b, and c that describe the lengths of the three sides (for a three-dimensional lattice) of the corresponding conventional unit cell. Such a unit cell has aspect ratios of η≡b/a and ξ≡c/a, and inclination angles γ and δ between its sides a to b and a to c, respectively (c, ξ, and δ being relevant only for the three-dimensional case). The direct lattice matrix, A, and, as a result, a, b, c, γ, and δ can be related to the wavelength λ of the excitation waves by squaring both sides of Equation (13) and applying Equation (4) (assuming, for simplicity, that the lattice is not of triclinic symmetry, and hence the c-axis is orthogonal to the ab plane). The result is the equation:
λ=8π²/√{square root over (β′·A′·A·β)}≡F(a,b,c,γ,δ)  (14)
where F(a, b, c, γ, δ) is a function of the lattice constants and inclination angles. Thus, any optical lattice of a given Bravais symmetry, with aspect ratios, η and ξ, and inclination angles, γ and δ, can only exist in discrete sizes relative to the wavelength of light used to create the lattice. These sizes are dictated by the valid choices of A (assuming, for D=2, that the wavevectors, k_n, are restricted to a single plane). Furthermore, for any optical lattice, a minimum possible lattice size will always exist, because the lattice basis e(x,t) can have no spatial frequencies greater than 2k.

[0319] Fundamental Lattices

[0320] A D-dimensional lattice of a given Bravais type that consists of D+1 plane waves and that has the minimum possible size relative to the light wavelength is referred to herein as a fundamental lattice of that type. In light of Equation (14), fundamental lattices are created by those sets of direct lattice primitive vectors, {a_n}, for which the quantity β′·A′·A·β is minimized. Such sets of direct lattice primitive vectors are often geometrically identifiable by those primitive vectors, a_n, of minimal length that, to within the constraints of the lattice symmetry, are most nearly orthogonal to one another. Table 1 and Table 2 summarize exemplary fundamental direct lattice matrices, A_f, and their corresponding minimum sets of reciprocal lattice vectors, {k_n}, (expressed as the D×(D+1) matrix, K_f) for all two-dimensional and three-dimensional Bravais lattice types. Table 3 and Table 4 summarize the wavelength-normalized sizes, a/λ, of such lattices in terms of their unit cell aspect ratios η, ξ, and inclination angle, γ, where appropriate. Finally, FIG. 17 shows intensity plots of several exemplary two-dimensional and three-dimensional fundamental lattices.

[0321]

TABLE 1
Direct Lattice and Wave Vector Matrices for the Five
Fundamental 2D Optical Bravais Lattices
Lattice
Type	Af	Kfa
obliqueb	[ a b ⁢ ⁢ cos ⁢ ⁢ γ 0 b ⁢ ⁢ sin ⁢ ⁢ γ ]	π sin 2 ⁢ γ ⁡ [ 1 / a + cos ⁢ ⁢ α / b ( 1 - 2 ⁢ ⁢ sin 2 ⁢ ⁢ γ ) / a + cos ⁢ ⁢ γ / b 1 / α + cos ⁢ ⁢ γ / b sin ⁢ ⁢ γ / b sin ⁢ ⁢ 2 ⁢ ⁢ γ / a + - sin ⁢ ⁢ γ / b sin ⁢ ⁢ γ / b ]
hex- agonal	[ a - a / 2 0 3 ⁢ a / 2 ] ⁢	π ⁡ [ 2 3 ⁢ a 4 3 ⁢ a 2 3 ⁢ a 2 3 ⁢ a 0 2 3 ⁢ a ]
centered rect- angular	[ a / 2 a / 2 b / 2 - b / 2 ] ⁢	π ⁡ [ a b 2 + 1 a a b 2 - 1 a a b 2 - 1 a 0 - 2 / b 2 / b ]
primi- tive rect- angular	[ a 0 0 b ]	π ⁡ [ 1 / a - 1 / a 1 / a 1 / b 1 / b - 1 / b ]
square	same as primitive rectangular, except b = a
aKf is a 2 × 3 matrix, each column of which gives one of the wave vectors, kn, (in Cartesian coordinates) in the minimum plane wave set
bγ is the angle between the a and b sides of the conventional unit cell

[0322]

TABLE 2
Direct Lattice and Wave Vector Matrices for the Fourteen Fundamental 3D Optical Bravais Lattices
Lattice Type	Af	Kfa
triclinic	general: [a1 a2 a3]	use Eqs. (1), (7), and (9)
trigonalb	[ a a ⁢ ⁢ cos ⁢ ⁢ γ a ⁢ ⁢ cos ⁢ ⁢ γ 0 a ⁢ ⁢ sin ⁢ ⁢ γ a ⁢ ⁢ cos ⁢ ⁢ γ ⁢ ⁢ tan ⁡ ( γ / 2 ) 0 0 a ⁢ ⁢ 1 - cos 2 ⁢ γ cos 2 ⁡ ( γ / 2 ) ]
centered monoclinicb	[ b ⁢ ⁢ cos ⁢ ⁢ γ b ⁢ ⁢ cos ⁢ ⁢ γ b ⁢ ⁢ cos ⁢ ⁢ γ - a + a + a b ⁢ ⁢ sin ⁢ ⁢ γ - b ⁢ ⁢ sin ⁢ ⁢ γ b ⁢ ⁢ sin ⁢ ⁢ γ c c c ] / 2	π a + b ⁢ ⁢ cos ⁢ ⁢ γ ⁡ [ 1 sin 2 ⁢ γ + ( b ⁢ ⁢ cos ⁢ ⁢ γ c ) 2 1 sin 2 ⁢ γ + ( b ⁢ ⁢ cos ⁢ ⁢ γ c ) 2 b 2 ⁢ ⁢ sin ⁢ ⁢ 2 ⁢ γ 2 ⁢ c 2 + a b ⁢ ⁢ sin ⁢ ⁢ γ b 2 ⁢ ⁢ sin ⁢ ⁢ 2 ⁢ γ 2 ⁢ c 2 - a + 2 ⁢ b ⁢ ⁢ cos ⁢ ⁢ γ b ⁢ ⁢ sin ⁢ ⁢ γ c ⁢ ⁢ cot ⁢ ⁢ γ b ⁢ ⁢ sin ⁢ ⁢ γ + a c c ⁢ ⁢ cot ⁢ ⁢ γ b ⁢ ⁢ sin ⁢ ⁢ γ - a + 2 ⁢ b ⁢ ⁢ cos ⁢ ⁢ γ c ⁢ ⁢ ⁢ cos ⁢ ⁢ 2 ⁢ γ sin 2 ⁢ γ + ( b ⁢ ⁢ cos ⁢ ⁢ γ c ) 2 cos ⁢ ⁢ 2 ⁢ γ sin 2 ⁢ γ + ( b ⁢ ⁢ cos ⁢ ⁢ γ c ) 2 b 2 ⁢ ⁢ sin ⁢ ⁢ 2 ⁢ γ 2 ⁢ c 2 + a + 2 ⁢ b ⁢ ⁢ cos ⁢ ⁢ γ b ⁢ ⁢ sin ⁢ ⁢ γ b 2 ⁢ ⁢ sin ⁢ ⁢ 2 ⁢ γ 2 ⁢ c 2 - a b ⁢ ⁢ sin ⁢ ⁢ γ c ⁢ ⁢ cot ⁢ ⁢ γ b ⁢ ⁢ sin ⁢ ⁢ γ - a c c ⁢ ⁢ cot ⁢ ⁢ γ b ⁢ ⁢ sin ⁢ ⁢ γ + a + 2 ⁢ b ⁢ ⁢ cos ⁢ ⁢ γ c ] ⁢
simple monoclinicb	[ a b ⁢ ⁢ cos ⁢ ⁢ γ 0 0 b ⁢ ⁢ sin ⁢ ⁢ γ 0 0 0 c ]	π sin 2 ⁢ γ ⁡ [ 1 / a + cos ⁢ ⁢ α / b ( 1 - 2 ⁢ ⁢ sin 2 ⁢ ⁢ γ ) / a + cos ⁢ ⁢ γ / b 1 / a + cos ⁢ ⁢ γ / b 1 / a + cos ⁢ ⁢ γ / b sin ⁢ ⁢ γ / b sin ⁢ ⁢ 2 ⁢ γ / a + sin ⁢ ⁢ γ / b - sin ⁢ ⁢ γ / b sin ⁢ ⁢ γ / b sin 2 ⁢ γ c sin 2 ⁢ γ c sin 2 ⁢ γ c - sin 2 ⁢ γ c ]
hexagonal	[ a - a / 2 0 0 3 ⁢ a / 2 0 0 0 c ]	π ⁡ [ 2 3 ⁢ a - 4 3 ⁢ a 2 3 ⁢ a 2 3 ⁢ a 2 3 ⁢ a 0 - 2 3 ⁢ a 2 3 ⁢ a 1 / c 1 / c 1 / c - 1 / c ]
face-centered orthorhombic	[ a - a 0 0 0 - b c c - c ]	π ⁡ [ 0 - 2 / a 2 / a 0 - 2 / b 0 0 2 / b c a 2 - c b 2 + 1 c c a 2 - c b 2 - 1 c c a 2 - c b 2 - 1 c c a 2 - c b 2 + 1 c ]
body-centered orthorhombic	[ - a a a b - b b c c - c ] ⁢   / 2	π ⁡ [ 1 / a 1 / a - 1 / a - 1 / a 1 / b - 1 / b 1 / b - 1 / b 1 / c - 1 / c - 1 / c 1 / c ]
base-centered orthorhombic	[ a / 2 a / 2 0 b / 2 - b / 2 0 0 0 c ] ⁢	π ⁡ [ a b 2 + 1 a a b 2 - 1 a a b 2 - 1 a a b 2 + 1 a 0 - 2 / b 2 / b 0 1 / c 1 / c 1 / c - 1 / c ]
simple orthorhombic	[ a 0 0 0 b 0 0 0 c ]	π ⁡ [ 1 / a - 1 / a 1 / a 1 / a 1 / b 1 / b - 1 / b 1 / b 1 / c 1 / c 1 / c - 1 / c ]
centered	same as body-centered orthorhombic, except b = a
tetragonal
simple	same as simple orthorhombic, except b = a
tetragonal
face-centered cubic	[ 0 - a a a 0 - a a a 0 ] / 2	π a ⁡ [ 0 - 2 2 0 - 2 0 0 2 1 - 1 - 1 1 ]
body-centered	same as body-centered orthorhombic, except c = b = a
cubic
simple cubic	same as simple orthorhombic, except c = b = a
aKf is a 3 × 4 matrix, each column of which gives one of the wave vectors, kn, (in Cartesian coordinates) in the minimum plane wave set
bγ is the angle between the a and b sides of the conventional unit cell

[0323]

TABLE 3
Wavelength-Normalized Sizes of the Five Fundamental
Two-Dimensional Optical Bravais Lattices
	Lattice Type	a/λa
	obliqueb	1 2 ⁢ ⁢ sin 2 ⁢ ⁢ γ ⁡ [ 1 + 1 η 2 + 2 ⁢ ⁢ cos ⁢ ⁢ γ η ] 1 / 2
	hexagonal	2/3
	centered	(1 + 1/η2)/2
	rectangular
	primitive	{square root over (1 + 1/η2)}/2
	rectangular
	square	1/{square root over (2)}
	aη ≡ b/a and ξ ≡ c/a are the aspect ratios of the conventional unit cell
	bγ is the angle between the a and b sides of the conventional unit cell

[0324]

TABLE 4
Wavelength-Normalized Sizes of the Fourteen Fundamental
Three-Dimensional Optical Bravais Lattices
Lattice Type	a/λa
triclinic or trigonal	use Eqs. (1), (7), (9), and (10)
centered monoclinicb	1 2 ⁢ ( 1 + η ⁢ ⁢ cos ⁢ ⁢ γ ) [ ( 1 sin 2 ⁢ ⁢ γ + ( η ⁢ ⁢ cos ⁢ ⁢ γ ξ ) 2 ) 2 + ( 1 η ⁢ ⁢ sin ⁢ ⁢ γ + η 2 ⁢ ⁢ sin ⁢ ⁢ 2 ⁢ γ 2 ⁢ ξ 2 ) 2 + ( 1 ξ + ξ ⁢ ⁢ cot ⁢ ⁢ γ η ⁢ ⁢ sin ⁢ ⁢ γ ) ) 2 ] 1 / 2
simple monoclinicb	1 2 ⁢ ⁢ sin 2 ⁢ ⁢ γ ⁡ [ 1 + 1 η 2 + 2 ⁢ ⁢ cos ⁢ ⁢ γ η + ( sin 2 ⁢ ⁢ γ ξ ) 2 ] 1 / 2
hexagonal	{square root over (4/9 + 1/(2ξ)2)}
face-centered	{square root over (4/η2 + (ξ − ξ/η2 + 1/ξ)2)}/2
orthorhombic
base-centered	{square root over ((1 + 1/η2)2 + 1/ξ2)}/2
orthorhombic
body-centered or	{square root over (1 + 1/η2 + 1/ξ2)}/2
simple orthorhombic
centered or simple	{square root over (2 + 1/ξ2)}/2
tetragonal
face-centered cubic	{square root over (5)}/2
body-centered or	{square root over (3)}/2
simple cubic
aη ≡ b/a and ξ ≡ c/a are the aspect ratios of the conventional unit cell
bγ is the angle between the a and b sides of the conventional unit cell

[0325] The characteristics of fundamental optical lattices makes them useful for a variety of microscopy applications. For example, a fundamental lattice contains the minimum possible number of D(D+1) non-zero spatial frequencies, k_n−k_m, and the corresponding points in reciprocal space are separated by the largest distances, |b_i−b_j|, possible for the given lattice type. Therefore, the signal, S(x)=I(x)O(x), that results from the interaction of a fundamental lattice having an intensity distribution, I(x), with an object of interest having a spatial distribution, O(x), of signal-producing entities yields well-separated D-dimensional regions in reciprocal space that encode the spatial frequencies of Õ(k) out to a reciprocal distance, |k_n−k_m|, beyond the traditional limit of 2k. The D-dimensional regions encoding the spatial frequencies of the object of interest can be recovered by recording a series of images at different phases using structured illumination microscopy techniques, for example, as described in M. Ben-Levi and E. Peleg, U.S. Pat. No. 5,867,604; J. T. Frohn, et al., Proc. Natl. Acad. Sci. USA 97, 7232-7236 (2000); P. T. C. So, et al., J. Opt. Soc. Am. A 18, 2833-2845 (2001); and G. Cragg, et al., U.S. Pat. No. 6,255,642 B1, all of which are incorporated herein by reference. Also, because a fundamental lattice permits the creation of a basis, e(x,t), that has the most rapid spatial variation possible between adjacent maxima and minima, such lattices can be used effectively in nonlinear multiple field techniques such as stimulated emission depletion (“STED”), for example, as described in S. W. Hell and J. Wichmann, Opt. Lett. 19, 780-782 (1994); or S. Hell and J. Wichmann, U.S. Pat. No. 5,731,588, both of which are incorporated herein by reference. Such lattices also can be used effectively in coherent anti-Stokes Raman scattering (“CARS”) microscopy, for example, as described in J. Cheng, et al., Biophys. J. 83, 502-509 (2002), which is incorporated herein by reference. The use of a fundamental lattice can also enhance imaging speed in microscopy when such a lattice incorporates a basis of discrete excitation maxima, because the N-dimensional primitive cell that must be scanned to form a complete image is of the smallest possible size.

[0326] Sparse Lattices

[0327] Although fundamental lattices are useful for many applications, non-fundamental optical lattices, which are of larger periodicity than the fundamental lattice for any given type, can also be useful in microscopy. For example, the individual signals from adjacent intensity maxima can be more easily differentiated in larger period optical lattices, particularly in three-dimensional lattices, because the distances between such maxima can be greater than in fundamental lattices, which can be helpful in overcoming resolution problems during three-dimensional detection. Similarly, non-fundamental lattices can be used to avoid a high density of intensity maxima, where such a density could be excessively perturbative of certain samples. A larger period lattice can also be used to decrease the density of intensity maxima in order to cover the desired field of view at the desired frame rate with a given detector having a fixed number of elements.

[0328] Optical lattices that have greater periodicity than fundamental lattices, referred to herein as sparse lattices, can be selected for any desired lattice type by applying Equations (9b), (10), (12), and (13) as described above to yield the D+1 wavevectors, k_n, arising from direct lattice primitive vector sets, {a_n}, and corresponding matrices, A, for which the quantity β′·A′·A·β is larger than the minimum value associated with the fundamental lattice. The selection of a sparse lattice of a given type (as characterized by the Bravais symmetry, aspect ratios η and ξ, and inclination angles γ and δ of the lattice) therefore reduces to the selection of additional valid primitive vector sets, {a_n}. Because any such set, {a_n}, must span an entire direct lattice, any two such sets, {a_In} and {a_IIn}, must be integral combinations of one another, so that:

[0329] $\begin{array}{cc}\begin{array}{cc}{a}_{\mathrm{IIm}}=\sum _{n=1}^D{c}_{\mathrm{mn}}{a}_{\mathrm{In}}& m=1\dots D\end{array}& \left(15a\right)\\ \begin{array}{cc}{a}_{\mathrm{Im}}=\sum _{n=1}^D{d}_{\mathrm{mn}}{a}_{\mathrm{IIn}}& m=1\dots D\end{array}& \left(15b\right)\end{array}$
or, in matrix form:
A_II′=CA_I′  (16a)
A_I′=DA_II′  (16b)

[0330] Thus, the selection of a sparse lattice further reduces to the selection of valid combination matrices, C, that allow new direct lattice matrices, A_II, to be generated from known direct lattices, A_I, including, for example, the fundamental direct lattice matrices, A_f, given in Table 1 and Table 2.

[0331] Equation (16a) and Equation (16b) collectively indicate that C⁻¹=D. Therefore, the matrix C⁻¹ has integral elements, as do the matrices C and D. However, the determinant of any matrix with integral elements must be an integer. Furthermore, for any matrix, |C⁻¹|=1/|C|. Accordingly, the only valid combination matrices are those for which:
|C|=±1.  (17)

[0332] Equation (17) provides for the convenient selection of different sparse lattices and their D+1 constituent wavevectors. Such selection can be achieved by:

• • [0333] a) cycling through different integer possibilities for each of the elements of the combination matrix C and evaluating |C| in each case;
  • [0334] b) determining whether each of the resulting matrices satisfies Equation (17);
  • [0335] c) applying matrices that satisfy Equation (17) to Equation (16a) along with known direct lattice matrices A_I to select additional matrices A_II;
  • [0336] d) applying Equations (9b), (10), (12), and (13) to A_II, to select the corresponding wavevectors, {k_n}, and using Equation (14) to find the wavelength-normalized lattice size for the given η, ξ, and γ.

[0337] Although sparse lattices can only be selected in discrete sizes, the density of valid solutions for such lattices, particularly in three dimensions, is such that a sparse lattice usually can be selected to approximate many desired sizes larger than that of the fundamental lattice. FIGS. 8A and 10A show exemplary sparse lattices in two-dimensional and three-dimensional, respectively. FIG. 18 illustrates direct lattice primitive vectors and resulting intensity plots to provide for comparison between a fundamental simple cubic lattice and a related sparse lattice having a periodicity that is √{square root over (59/3)}=4.435 times larger than the periodicity of the fundamental lattice.

[0338] New valid combination matrices, C, can be generated from any product of known combination matrices or their inverses. Also, because Equation (17) was derived without reference to any specific Bravais symmetry, a valid combination matrix, C, can be applied to any type of direct lattice matrix, A, during the selection of new sparse lattices.

[0339] As the choice of direct lattice primitive vectors, a_n, extends to lattice points successively farther from the origin, the number of valid lattice matrices, A, and hence the likely number of differently sized sparse lattices for any given type, increases rapidly. Even when only relatively short primitive vectors are considered (e.g., C_mn=0, ±1, ±2, ±3 for two dimensions and C_mn=0, ±1 for three dimensions, as used in Tables 5 and 6, respectively), many lattice sizes exist, and the number of different sizes generally increases as lattice symmetry decreases.

[0340] In three dimensions, increasingly sparse lattices can be understood to arise when the four wavevectors k₀, k₁, k₂, k₃ in the minimum set all lie near the surface of a single cone, as shown in FIG. 19A. In accordance with the arguments presented in connection with FIG. 7, the result of this wavevector relationship is a pattern that approximates a 2D lattice in the x-y plane perpendicular to the axis, ê_z, of the cone, and that varies slowly along ê_z, as illustrated in FIG. 19B and FIG. 19C, respectively. As with the two-dimensional lattices depicted in FIG. 8B and FIG. 8C, the periodicity of such a sparse lattice within the x-y plane increases as the cone angle decreases.

[0341]

TABLE 5
Periodicities of Two-Dimensional Sparse Lattices Generated
With Combinations of Fundamental Primitive Vectors
	Number of Sparse	Five Smallest a/λ,	Largest a/λ,
Lattice Typea	Lattice Sizes	a = b	a = b
square	10	1/{square root over (2)}, {square root over (5/2)}, 5/{square root over (2)},	{square root over (1105/2)}
		{square root over (65/2)}, {square root over (85/2)}
primitive	15	reduces to square for a = b
rectangular
centered	20	reduces to square for a = b
rectangular
hexagonal	12	⅔, 2/{square root over (3)}, 2{square root over (7/3)}	14{square root over (133)}/3
		2{square root over (91)}/3, {square root over (133/3)}
afor simplicity, only lattice types with fixed unit cell angles are considered here

[0342]

TABLE 6
Periodicities of Three-Dimensional Sparse Lattices Generated With
Combinations of Fundamental Primitive Vectors
Lattice	Number of Sparse	Five Smallest a/λ,	Largest a/λ,
Typea	Lattice Sizes	a = b = c	a = b = c
simple	cubic (a = b = c): 10 tetragonal (a = b): 29 orthorhombic: 59	{square root over (3)}/2, {square root over (11)}/2, {square root over (19)}/2, {square root over (27)}/2, {square root over (35)}/2	203 2
base-	orthorhombic: 76	N.A.	N.A.
centered
face- centered	cubic (a = b = c): 60 orthorhombic: 148	{square root over (5)}/2, 3/2, {square root over (11)}/2, {square root over (21)}/2, {square root over (29)}/2	6761 2
body-	cubic (a = b = c): 33	{square root over (3)}/2, 1, {square root over (2)},	{square root over (689)}
centered	tetragonal (a = b): 113	{square root over (11)}/2, {square root over (5)}
	orthorhombic: 236
hexagonal	(a = c): 47 (a ≠ c): 50	5/6, 7/6, 57/6, 65/6, 85/6	8157 2
afor simplicity, only lattice types with fixed unit cell angles are considered here

[0343] As wavevectors of a sparse lattice approach the state of lying on a single conic surface, the wavevector differences approach zero in the direction of the conic axis, such that (k_m−k_n)·ê_z→0, for all values of m and n. As a result, Equation (10) indicates that all valid reciprocal lattice primitive vectors, b_n, lie near the x-y plane. When expressed in this basis, B′ has only two effectively non-zero columns. Equation (9a) and Equation (9b) imply that the direct lattice primitive vectors, a_n, comprising A=2π(B′)⁻¹ must also lie in this plane. Thus, direct lattice primitive vectors, a_n, giving rise to increasingly sparse lattices can be geometrically identified by selecting vectors of increasing length that lie progressively nearer to, but not exactly upon, a single plane.

[0344] Composite Lattices

[0345] As discussed in conjunction with FIGS. 8A and 10A, sparse lattices contain only one spatial frequency in each of the D+1 directions defined by the possible combinations of D wavevectors from the D+1 element minimum set. Thus, for sparse lattices of increasing size, the resolution and potential confinement of the intensity maxima (with the appropriate basis) becomes increasingly degraded in these directions. More than D+1 plane waves can be used to provide high spatial frequency content in the excitation field in all directions and more symmetric confinement of the intensity maxima regardless of the wavelength-normalized lattice size, for example, as shown in FIGS. 8D and 10D. Such lattices that are composed of more than D+1 plane waves are referred to herein as composite lattices.

[0346] To construct composite lattices, the Bravais symmetry and periodicity of the lattice must be preserved as additional plane waves are added to an existing set. The selection procedure described above for sparse lattices can typically produce multiple direct lattice matrices, A, and associated wavevector sets, {k_n}, yielding lattices of identical size and symmetry, but exhibiting differing spatial frequencies, k_n−k_m, that define the confinement of the excitation, [(k_n−k_m)·ê_j]_max, (i.e., the intensity maxima) in any direction, ê_j, within each primitive cell. FIG. 20A and FIG. 20B show two such lattices of identical size and symmetry that exhibit different directional excitation confinement. Composite lattices can be constructed by superimposing all of the unique plane waves from two or more such “size-degenerate” sparse lattices. FIG. 20C shows the composite lattice that results from the superimposition of the two sparse lattices depicted in FIG. 20A and FIG. 20B.

[0347] Any such combination of size-degenerate sparse lattices can be used to generate a composite lattice. However, composite lattices of high excitation confinement and symmetry can be formed systematically where the lattice sets satisfy the equation:
A_II=MA_I,  (18)
where:
M·M′=I.  (19)

[0348] For such cases, Equations (9b), (18), and (19) can be combined to yield B_II=MB_I and, with the additional use of Equation (12), β_II=Mβ_I. Since β_II′·A_II′·A_II·β_II=β_I′·A_I′·A_I·β_I for such cases, the sparse lattices represented by A_I and A_II are indeed, by virtue of Equation (14), size degenerate.

[0349] Combining these results with Equations (10) and (13) yields k_IIn=Mk_In. Thus, the geometric effect of the transformation matrix, M, on the direct lattice, reciprocal lattice, and constituent wave vectors is identical. This characteristic may also be observed from the fact that the orthogonality condition expressed in Equation (19) is satisfied by all single and multiplicative combinations of reflections and rotations of the lattice. A composite lattice of increased confinement and basis symmetry within each primitive cell can therefore be created by applying one or more such transformations to the wavevectors of an existing lattice and then superposing all new wavevectors in the transformed set with the original ones. Repeated application of this procedure can further increase the confinement and basis symmetry of a composite lattice, until all possible wavevectors resulting from such transformations are uncovered. Thus, like sparse lattices, composite lattices can have larger primitive cells than fundamental lattices. However, composite lattices can also have greater confinement of individual intensity maxima than sparse lattices of the same Bravais symmetry.

[0350] However, the Bravais symmetry of a lattice will be preserved only for transformations A_I→A_II and subsequent superpositions that satisfy Equation (16a) as well as Equations (18) and (19). In other words, Bravais symmetry is only preserved by transformations that map a lattice onto itself. FIG. 21 illustrates one example of this characteristic of composite lattices. The intensity within each cell of the initial simple orthorhombic fundamental lattice depicted in FIG. 21A and FIG. 21B (where η=0.66 and ξ=2.0) is asymmetric and exhibits poor confinement in several directions. Because the orthorhombic crystal group is characterized by invariance under reflections about the principal planes only, FIGS. 21C and 21D show that when the plane waves resulting from all reflection combinations are superimposed (i.e., eight plane waves in total), the resulting composite lattice maintains the orthorhombic nature of the original lattice, while imposing similar symmetry and improved confinement of the excitation (i.e., on the intensity maxima of the lattice) within the basis.

[0351] The comparatively poor intensity confinement that then remains along the z-axis, as shown in FIG. 21D, can be rectified by superimposing an additional composite set upon the existing set (16 plane waves total as shown in FIG. 21D). The additional composite set can be obtained by rotating the existing set through 90° about the [100] axis of the lattice. The wavevectors that result from such a transformation are illustrated in FIG. 21E. However, as such a transformation is not orthorhombic invariant, the orthorhombic symmetry is destroyed thereby, and the result is a simple tetragonal composite lattice, as shown in FIG. 21F, for which the rotation is allowed. Continuing through all combinations of 90° rotations about the three principal axes, the 48 wavevectors for which are shown in FIG. 21G, similarly destroys the tetragonal symmetry of the composite lattice and eventually leads to the composite body-centered cubic structure shown in FIG. 21H, which has tightly-confined, widely-spaced intensity maxima.

[0352] Maximally Symmetric Composite Lattices

[0353] A composite lattice is referred to herein as a maximally symmetric composite lattice if it contains all of the unique wavevectors that can be obtained by applying every allowed symmetry transformation to a sparse lattice's D+1 original wavevectors. A maximally symmetric composite lattice exhibits the highest spatial frequency content and greatest basis symmetry for any optical lattice of its type. Of the five Bravais lattice types that can be formed in two dimensions, the hexagonal type has the most allowed symmetry operations and hence yields maximally symmetric composite lattices of the greatest intensity confinement and basis symmetry, as shown, for example, in FIG. 8D. Similarly, the hexagonal type three-dimensional lattice also yields maximally symmetric composite lattices of the greatest confinement and basis symmetry in the [001] plane. However, the greatest overall confinement and basis symmetry in all directions can be achieved by the three three-dimensional lattices of the cubic crystal group, since they permit the same reflection and rotation symmetry operations to be performed in or about all three principal planes. FIG. 10D shows a three-dimensional intensity plot for an exemplary maximally symmetric body-centered cubic lattice of the cubic crystal group.

[0354] Maximally symmetric cubic lattices can exhibit near-optimal three-dimensional confinement of the excitation at the intensity maxima of the lattice, even when the periodicity of the original sparse lattice is far greater than that of the fundamental lattice. FIGS. 22A. 22B, 22C, and 22D and FIGS. 22E, 22F, 22G, and 22H illustrate this trait for a particular body-centered cubic lattice by depicting a sparse lattice and its maximally symmetric composite lattice, respectively. For the sparse lattice shown in FIG. 22A, the confinement of the intensity maxima, [(k_n−k_m)·ê_j]_max, varies from 0.057k to 0.762k along the axes, ê_j=ê_{Δk min} 2200 and ê_j=ê_{Δk max} 2201, leading to considerable asymmetry in the basis, which is depicted in FIGS. 22B, 22C, and 22D. The corresponding maximally symmetric composite lattice depicted in FIGS. 22E, 22F, 22G, and 22H has 96 plane waves propagating over a much greater range of directions, so its excitation confinement exhibits less variation (1.78k to 2.00k) and more closely approaches the theoretical maximum of 2k in all directions.

[0355] Consequently, the selection of maximally symmetric square or hexagonal lattices can be especially appropriate for two-dimensional microscopy applications where a basis is selected to form a confined intensity maximum within each primitive cell. Likewise, the selection of maximally symmetric two-dimensional cubic or hexagonal lattices can be especially appropriate for analogous three-dimensional microscopy applications.

[0356] Subset Lattices

[0357] Occasionally, however, constraints other than excitation confinement can be of paramount concern in the selection of an optical lattice (e.g., arrangement of the excitation/detection optics, optimization of polarization in a specific direction within the basis). In such circumstances, it can be desirable to select a lattice having a different symmetry, different aspect ratios, and/or having fewer plane waves relative to the maximally symmetric composite lattice of any given type. The latter non-maximally symmetric lattices are referred to herein as subset lattices, as long as they still consist of N>D+1 plane waves, because they are formed by a composite subset of waves from a maximally symmetric composite set.

Design and Construction of the Basis for an Optical Lattice

[0358] To apply optical lattices to optical microscopy, it is necessary to identify not only the appropriate lattice types for a given application and the wavevector sets, {k_n}, with which they can be produced, but also design an appropriate basis field, e(x,t), (which, after all, will ultimately determine the relationship between the sample of interest and the multidimensional images obtained) for the application. As stated previously, this basis field, e(x,t), depends on both the complex electric field, e_n, and the wavevector, k_n, of every plane wave that contributes to a given lattice.

[0359] The Basis of Maximal Intensity with Optimized Polarization at a Single Point

[0360] One specific basis useful for optical microscopy consists of a single dominant, tightly confined intensity maximum within each primitive cell as exemplified by the lattices shown in FIG. 8D, FIG. 10D, and FIG. 22F. This basis as applied to any lattice can serve as a multi-point excitation field for massively parallel multifocal microscopy. Thus, an entire two- or three-dimensional region spanning many periods of the lattice within a sample of interest can be imaged by scanning the lattice and basis relative to the sample over only the dimensions of one primitive cell. In contrast, a confocal microscope imaging the same region would be required to scan over the entire region sequentially.

[0361] It is also useful for optical microscopy to control the polarization properties of the basis field, e(x,t). Either alone or in combination with polarization sensitive detection, such control can yield additional insights into the nature of reflective, absorptive, or scattering samples. Furthermore, the absorption of light by molecular markers (e.g., fluorescent molecules) within biological systems, and thus the ultimate signal comprising an image, can depend upon the orientations of these molecules relative to the polarization of the excitation light. Thus, control of the polarization of the excitation light can yield information concerning the former.

[0362] A basis can be created satisfying the twin goals of having concentrated intensity at a single point and having substantial control of the polarization at this same point by maximizing a particular desired electric field state, e_d(x_d,t_d)≡e_d, defined with respect to a unit vector, ê_p, at a given time, t_d, and a selected point, x_d, within the primitive cell. Considering each constituent plane wave individually, ê_p and k_n, together define a natural orthonormal coordinate system, {ê_kn,ê_pkn⊥, ê_kn⊥}, for representing e_n(x,t) as:
e_n(x,t)=e_n[cos χexp(iθ)ê_kn⊥+sin χexp(iψ)ê_pkn⊥]exp[i(k_n·x−ωt)],  (20)
where 0≦χ≦π/2, −π≦θ≦π, −π≦ψ≦π, and:
ê_kn=k_n/|k_n|,  (21a)
ê_pkn⊥=(ê_p×ê_kn)/sin Ω_n  (21b)
ê_kn⊥=ê_kn×ê_pkn⊥=(ê_p−cos Ω_nê_kn)/sin Ω_n,  (21c)
Ω_n=cos⁻¹(ê_p′·ê_kn), and  (21d)
ê_kn⊥′·ê_p=sin Ω_n.  (21e)
Equation (20) exploits the orthogonality condition expressed in Equation (5). Similarly, e_d can be represented in the related orthonormal coordinate system {ê_p,ê_pkn⊥,ê_p⊥} as:
e_d=(ê_p′·ê_d)ê_p+(ê_pkn⊥′·e_d)ê_pkn⊥+(ê_p⊥′·e_d)ê_p⊥,  (22)
where:
ê_p⊥=ê_pkn⊥×ê_p=(ê_kn−cos Ω_nê_p)/sin Ω_n,  (23a)
ê_kn⊥′·ê_p⊥=−cos Ω_n.  (23b)

[0363] The unit vector, ê_pkn⊥, common to both coordinate systems lies along the intersection of the planes orthogonal to ê_p and ê_kn, and Ω_n is the angle between ê_p and ê_kn.

[0364] Since the physical electric field is given by the real part of e_n(x,t), e_d can be maximized by finding the value of ê_n for each plane wave, consistent with Equation (5), that maximizes the projection of e_n(x,t) at x_d and t_d onto e_d (i.e., Re{e_n*(x_d,t_d)′·e_d}). By manipulating Equations (20)-(23), it can be shown that:
Re{e_n*(x_d,t_d)′·e_d}=Re{[sin Ω_n(ê_p′·e_d)−cos Ω_n(ê_p⊥′·e_d)][e_n*(x_d,t_d)′·ê_kn⊥]+(ê_pkn⊥′·e_d)[e_n*(x_d,t_d)′·ê_pkn⊥]}  (24)
As a result, the selection of this basis can be reduced to the matter of finding the components of e_n(x_d,t_d) (and thus e_n) along ê_pkn⊥ and ê_kn⊥ that maximize the quantity represented in Equation (24).

[0365] Maximal Intensity with Optimized Linear Polarization

[0366] An example that can help to explain the process of finding the plane wave fields e_n that optimize the e_d component of the basis, e(x,t), at the position, x_d, and the time, t_d, involves maximizing the linear polarization of e(x_d,t_d) in a desired direction ê_p. For such a case:
e_d=Eê_p,  (25)
which, when taken in conjunction with Equation (20) and Equation (24), yields:
Re{e_n*(x_d,t_d)′·e_d}=e_nE sin Ω_n cos χ cos(k_n·x_d−ωt_d+θ).  (26)

[0367] Since Ω_n is fixed in Equation (21d), the expression in Equation (26) is maximized for χ=0 and θ=−(k_n·x_d−ωt_d). Thus, in light of Equation (20):
e_n(x,t)=e_nê_kn⊥exp[i(k_n·(x−x_d)−ω(t−t_d))] (where e_d is linear)  (27)

[0368] As represented geometrically in FIG. 23, Equation (27) thus implies that linear polarization along the ê_p axis 2300 is maximized when: (1) each plane wave is linearly polarized along the axis defined by the intersection of the plane 2301 that is perpendicular to the wavevectors, k_n, 2302 and the plane 2303 on which ê_p and k_n both lie; and (2) the phase is selected so that the real part of the field, e_n, 2304 is a maximum at x_d and t_d.

[0369] To conclude this basis-selection example, the relative plane wave amplitudes e_n must still be selected. This can be accomplished in practice simply by only selecting plane waves of equal amplitude, such that:
e_n=e_oê_kn⊥exp[−i(k_nx_d−ωt_d)], and, for e_d linear, |e_n|=e_o.  (28)

[0370] The resulting basis can have a strong intensity maximum at x=x_d within the primitive cell and good contrast with respect to the background intensity elsewhere in the cell, as is the case for the examples shown in FIG. 8D and FIG. 10D. For these examples, the corresponding wavevectors, k_n, and the electric field vectors, e_n(x_d, t), are shown in FIG. 24A and FIG. 24B, respectively at eight different times t_q=t_d+2πq/8ω, q=0 . . . 7. The length of each field vector is proportional to its real amplitude at the time, t_q. FIG. 24C shows a key relating the time, t_q, to the grayscale shade of each field vector for all plots of this type.

[0371] Because of the requirement that k_n·e_n=0, the strength of the intensity maxima for the particular basis represented by Equation (28) depends upon the lattice type and its distribution of wavevectors, k_n, with the strongest maxima occurring for lattices having k_n more nearly orthogonal to ê_p. The polarization purity at the maxima and the likely intensity contrast elsewhere in the primitive cell also depend upon these same factors. FIG. 8D and FIG. 11A illustrate these points, as FIG. 8D shows strong intensity maxima and high contrast with ê_p=ê_z perpendicular to the plane of all k_n, and FIG. 11A shows weaker maxima and reduced contrast with ê_p=ê_x parallel to the plane of all k_n.

[0372] FIG. 25 illustrates a similar example for a three-dimensional hexagonal lattice (with ξ=2). For ê_p∥ê_x≡[100], as illustrated in FIG. 25A-C, the directions of the fields, e_n, vary sufficiently such that the relative intensities of the field components, |e(x)*·ê_x|² 2500, |e(x)·ê_y|² 2501, and |e(x)·ê_z|² 2502 are only (|e(x)·ê_x|²)_max/(|e(x)·ê_y|²)_max≈6.08 and (|e(x)·ê_x|²)_max/(|e(x)·ê_z|²)_max≈3.07, leading to reduced contrast in the overall intensity |e(x)|². On the other hand, as illustrated in FIGS. 25D-F, for ê_p∥ê_z≡[001], the field directions, e_n, are more nearly aligned, yielding (|e(x)·ê_z|²)_max/(|e(x)·ê_x|²)_max=(|e(x)·ê_z|²)_max/(|e(x)·ê_y|²)_max≈21.3, and superior overall contrast.

[0373] When the basis represented by Equation (28) is considered in conjunction with Equation (3) and Equation (21), the following result is implicated:

[0374] $\begin{array}{cc}e\left(x_p,t\right)·{\hat{e}}_p=\mathrm{Re}\left\{e_o\sum _{n=0}^N\sin {\Omega }_n\exp \left[i\left({\mathrm{kx}}_p\cos {\Omega }_n-\mathrm{i\omega }t+{\phi }_n\right)\right]\right\}.& \left(29\right)\end{array}$

[0375] Specifically, when a basis is selected with plane waves of equal amplitude, the highest spatial frequencies along the axis, ê_p, have the least contribution to the total confinement of the intensity maxima. This characteristic is depicted in FIG. 10D, where, despite the symmetric wavevector distribution represented by FIG. 24B, the intensity maxima are less confined along the desired polarization direction. This characteristic can be compensated for by altering the aspect ratio of the optical lattice, where such alteration is permitted by the Bravais symmetry, to increase the wavevector components, k_n·ê_p, along the desired polarization direction. However, as explained above, such alteration will simultaneously reduce the polarization purity and intensity contrast.

[0376] Conversely, the polarization purity and the intensity contrast can be improved at the expense of the spatial frequency distribution and the resulting confinement. FIG. 26 illustrates such a scenario, where the wavevector cone angle of the two-dimensional lattice depicted in FIG. 11A is reduced in order to increase the projection of each constituent plane wave field, e_n, in a desired direction, ê_p∥ê_x. Thus, for a lattice with the basis represented by Equation (28), a trade-off occurs between the linear polarization purity and the degree of confinement along the polarization axis. The appropriate balance between the polarization purity and the degree of confinement can be selected on a case-by-case basis in light of the needs of each particular microscopy application.

[0377] For any two-dimensional lattice, there exists a particularly symmetric basis of linear polarization wherein ê_p is along the axis of the cone on which all the constituent wavevectors lie. As a result, all spatial frequencies, k_m−k_n, contribute equally to the superposition in Equation (3). In the limit where all the wavevectors lie in a single plane, the highest possible confinement of the intensity maxima within the plane can be achieved, along with excellent contrast, by selecting the basis of Equation (28) with ê_p perpendicular to the plane. This is the case because, as shown in FIG. 8D, all plane waves exhibit optimal constructive interference at such maxima.

[0378] The confinement for a basis of predominately linear polarization can also be improved, as suggested by Equation (29), by selecting plane waves of amplitude e_n=e_o/sin Ω_n so that the projection e_n·ê_p of each plane wave field is identical:
e_n=(e_o/sin Ω_n)ê_kn⊥exp[−i(k_n·x_d−ωt_d)] (for e_d linear, |e_n|=e_o/sin Ω_n)  (30)
For example, the field vectors and resulting bases represented by Equation (28) and Equation (30) can be compared for a simple cubic lattice of an intensity period, √{square root over (11)}λ/2, as shown in FIGS. 27A-B and FIGS. 27C-D, respectively. Specifically, plane waves are shown with equal amplitudes in FIG. 27A and with equal projections in FIG. 27C. FIG. 27B and FIG. 27D illustrate the resulting confinement symmetry, as opaque surfaces of at least 50% maximum intensity, for plane waves of equal amplitude, and with equal projections, respectively. Although the confinement symmetry can be improved by selecting plane waves with equal projections, the intensity contrast across each primitive cell is significantly reduced, due to less efficient destructive interference at points away from x=x_d.

[0379] Maximal Intensity with Optimized Circular Polarization

[0380] The confinement can also be enhanced by selecting a desired electric field polarization state, e_d, at the intensity maxima that corresponds to the desired symmetry. For example, optimization of circular, rather than linear, polarization at the maxima can be represented by:
e_d=Eexp(iθ)(ê_pkn⊥+iê_p⊥)/√{square root over (2)}≡Eê_L,  (31)
which, in conjunction with Equation (20) and Equation (24), yields:
Re{e_n*(x_d,t_d)′·e_d}=e_nE[−cos Ω_n cos χ sin(k_n·x_d−ωt_d+θ−θ)+sin χ cos(k_nx_d−ωt_d+ψ−θ)]/√{square root over (2)}≡G(χ,θ,ψ).  (32)

[0381] Equation (32) is maximized for ∂G/∂χ=∂G/∂θ=∂G/∂ψ=0, ∂²G/∂χ²<0, ∂²G/∂θ²<0, and ∂²G/∂ψ²<0. Such maximization occurs when tan χ=1/|cos Ω_n|, θ=−(k_n·x_d−ωt_d)−(cos Ω_n/|cos Ω_n|)π/2+θ, and ψ=−(k_n·x_d−ωt_d)+θ. Therefore, by Equation (20):

[0382] $\begin{array}{cc}{e}_n\left(x,t\right)=\frac{e_n\left(-i\cos {\Omega }_n{\hat{e}}_{\mathrm{kn}\perp }+{\hat{e}}_{\mathrm{pkn}\perp }\right)}{\sqrt{1+{\cos }^2{\Omega }_n}}\exp \left[i\left(k_n·\left(x-x_d\right)-\omega \left(t-t_d\right)+\vartheta \right)\right]\left(\mathrm{for}{e}_d\mathrm{circular}\right).& \left(33\right)\end{array}$

[0383] A simple basis with good intensity contrast can be selected when all plane waves are assigned equal amplitudes:

[0384] $\begin{array}{cc}{e}_n=\frac{e_o\left(-i\cos {\Omega }_n{\hat{e}}_{\mathrm{kn}\perp }+{\hat{e}}_{\mathrm{pkn}\perp }\right)}{\sqrt{1+{\cos }^2{\Omega }_n}}\exp \left[-i\left(k_n·x_d-\omega t_d-\vartheta \right)\right]\left(\mathrm{for}{e}_d\mathrm{circular},e_n=e_o\right)& \left(34\right)\end{array}$

[0385] FIG. 28A shows the field vectors that result from applying Equation (34) to the simple cubic lattice depicted in FIG. 27. The field vectors indicate that the ê_L component of the total field at the intensity maxima is optimized when the plane wave polarization ellipticity varies according to cos Ω_n, i.e., from circular for k_n parallel to ê_p to linear for k_n perpendicular to ê_p. The corresponding opaque surfaces of 50% intensity, as shown in FIG. 28B, demonstrate that nearly spherical intensity confinement can be obtained while simultaneously maintaining an intensity contrast (as represented by the translucent surfaces of 15% intensity), which is better than that illustrated in the linear polarization example in FIG. 27B.

[0386] The selection of a basis of circularly polarized intensity maxima yields photons in a single state of angular momentum. This can be useful for microscopy applications, because such a basis is better attuned to generating signal from particular quantum constituents within a sample based on specific selection rules for transitions between quantum mechanical states. However, for fields described by Equation (34), the ellipticity of each field, e_n, must be manipulated in addition to its orientation and phase. For this reason, it is simpler to maximize circular polarization at the location, x_d, and the time, t_d, subject to the constraint that each constituent field is linearly polarized, such that:
e_n(x,t)=e_n[cos χê_kn⊥+sin χê_pkn⊥]exp[i(k_n·x−ωt+θ)].  (35)

[0387] When considered in conjunction with Equations (24) and (31), Equation (35) yields:
Re{e_n*(x_d,t_d)′·e_d}=ê_nE[−cos Ω_n cos χ sin(k_n·x_dωt_d+θ−θ)+sin χ cos(k_n·x_d−ωt_d+θ−θ)]/√{square root over (2)}≡H(χ,θ).  (36)

[0388] The absolute maximum among the solutions of ∂H/∂χ=∂H/∂θ=0, ∂²H/∂χ²<0, and ∂²H/∂θ²<0 occurs when χ=π/2 and θ=−(k_n·x_d−ωt_d)+θ. When all plane wave fields are selected with equal amplitudes, Equation (35) yields:
e_n=e_oê_pkn⊥exp[−i(k_n·x_d−ωt_d−θ)] (for e_d circular, e_n linear, and |e_n|=e_o).  (37)

[0389] By analogy to the linear polarization case shown in FIG. 23, the circular polarization perpendicular to ê_p is maximized at the location, x_d, when each linear field, e_n, is aligned in its allowed plane to lie along the intersection of this plane (⊥k_n) with the plane of the desired circular polarization (⊥ê_p). FIG. 29A illustrates this trait where the e_n based on Equation (37) are plotted for the same simple cubic lattice shown in FIG. 27 and FIG. 28. The resulting basis, shown in FIG. 29B, retains the spherically symmetric confinement of intensity maxima evidenced by the more general case of Equation (34), shown in FIG. 28B, but with somewhat reduced contrast with respect to the background due to the linear constraint on each constituent e_n.

[0390] Generalized Parameter Optimization to Construct a Basis

[0391] The basis construction methods discussed in connection with Equations (20)-(37) have the virtue of simplicity, because they depend upon the individual optimization of each plane wave field, e_n, after the wavevectors, k_n, which dictate the properties of the lattice, have already been determined. Secondary optimization of other parameters such as polarization purity and intensity contrast relative to the background can then be performed by varying the desired polarization state, e_d, or the lattice parameters, as illustrated in FIG. 25 and FIG. 26, respectively. In the most general terms, however, the complete electromagnetic field of the lattice, as described in Equation (3), depends upon both the wavevectors, k_n, and the fields, e_n, of the constituent plane waves. As a result, optimization of any desired property, Υ, (e.g., maximization of the gradient, Υ=∇(e(x,t)·ê_i)·ê_j, of the ê_i component of the electric field in the ê_j direction) can alternatively be performed by finding simultaneous critical values, ∂Υ/∂v_l=0 and ∂²Υ/∂v_l²<0, for any subset, {v_l}, of the components of these wavevectors, k_n, and fields, e_n, or functions thereof under certain specified constraints (e.g., lattice symmetry, relative amplitudes |e_n|, etc.).

Exemplary Applications

[0392] Selection of an Appropriate Lattice and Basis for Microscopy

[0393] The analysis detailed herein suggests that, in a common approach to optical lattice microscopy in which a lattice having intense, highly confined maxima is to be scanned over the dimensions of a primitive cell to generate a complete two-dimensional or three-dimensional image of a sample, care should be taken to appropriately select the lattice symmetry, periodicity, orientation, number of constituent wavevectors, and basis for a particular application. The lattice symmetry should be maximized to the extent possible in light of the sample geometry, the detection strategy, and the desired polarization purity. Cubic and hexagonal lattices, for example, are often useful for this purpose. To enhance the optical confinement, many wavevectors, k_n, can be selected from the associated maximally symmetric composite set in accordance with the experimental constraints of the application (e.g., occlusion by an objective nosepiece, excessive aberration through a cover glass).

[0394] The wavelength-normalized periodicity can be selected from the discrete possibilities associated with the aforementioned sparse lattice analysis in order to balance considerations such as the ease of signal detection from the individual maxima, the degree of unwanted excitation elsewhere, the desired field of view, and the imaging speed. The plane wave fields, e_n, that compose the basis should be selected to achieve the intended signal generation mechanism within the sample. For conventional linear polarization (e.g., birefringence) or fluorescence anisotropy measurements (e.g., determining the orientation of individual molecular fluorophores), the bases represented by Equation (28) and Equation (30) often can be helpful. The basis represented by Equation (28) can be preferred when polarization purity, |e(x_d)·ê_p|²/|e(x_d)|², at the maxima is paramount, and the basis represented by Equation (30) can be preferred when symmetric confinement of the excitation and consequent excitation resolution takes precedence over the increase in background emission (which can be partially removed via pinhole filtering) and related photobleaching. When the total intensity, |e(x_d)|², is responsible for the desired signal (as in the common case in which a large ensemble of randomly distributed fluorophores exist within each detection region), the basis indicated by Equation (34) can be particularly useful, because it combines near-optimal confinement for cubic lattices in all directions with excellent contrast between the intensity maxima and the background elsewhere in the primitive cells.

[0395] Non-Scanning, Massively Parallel Localized Optical Measurements

[0396] Lattices of intense, highly confined intensity maxima can also be applied in lieu of scanning to fixed point statistical biophysical techniques, such as, for example, fluorescence correlation spectroscopy (“FCS”), as described, for example, in S. T. Hess, et al., Biochemistry, 41, 697-705 (2002), fluorescence photobleaching recovery (“FPR” or “FRAP”), as described, for example, in J. Blonk, et al., J. Microsc., 169, 363-374 (1993), and single molecule detection/fluorescence resonant energy transfer, (“SMD/FRET”), as described, for example, in S. Weiss, Science, 283, 1676-1683 (1999) and B. Schuler, et al., Nature, 419, 743-747 (2002), all of which are incorporated herein by reference. In FCS, fluctuations in the signal arising from a population of molecules diffusing in and out of a single light excitation focus are analyzed to infer the local diffusion rate of said molecules, which in turn can be indicative of changes in the molecular weight due to ligand/receptor binding. In FPR, a light excitation focus is used to photobleach fluorescent molecules in a localized area, and the time for the fluorescence to recover in the bleached area is indicative of the diffusion rate of molecules in the bleached region.

[0397] Currently, FCS, FPR, and SMD/FRET are commonly implemented using the single focus of a confocal microscope, but these techniques can benefit from the massive parallelism and reduced volume at each maximum that is made possible by optical lattice excitation, as opposed to single focus, confocal microscopy. Indeed, the improved intensity confinement and reduction in extraneous photobleaching outside the maxima, which is made possible by the use of optical lattices can significantly enhance sensitivity in SMD studies and permit far longer observations times in a given region with fewer photophysically-induced artifacts.

[0398] The improved polarization purity that is possible at lattice excitation maxima can be useful for fluorescence anisotropy and single molecule orientation and conformation measurements and can be advantageous compared to the intensity maximum created with confocal microscopy. For example, FIG. 15 compares surface plots of 50 percent maximum intensity within a simple cubic lattice and surface plots of 50 percent maximum intensity within the focal region of a high NA objective lens. FIG. 30 shows the relative intensities and spatial distributions of each of the three orthogonal field components. In the confocal microscopy case, significant side lobes of z-polarization exist on either side of, and partially within, the desired x-polarized central maximum, as shown in FIG. 30A and FIG. 30B. Such side lobes result from a large angle of refraction as marginal rays pass through the objective, which causes rotation of the polarization vector. In contrast, in the lattice microscopy case, secondary maxima of y and z polarization are comparatively smaller, as shown in FIG. 30D and FIG. 30E, because the polarization of each plane wave can be individually manipulated to optimize its contribution to the desired x-polarization at the desired point x_d. Furthermore, because the secondary lattice maxima are centered at nodes of the desired field e(x)·ê_x, their contribution to the detected signal can be more readily removed via spatial filtering (e.g., with a pinhole filter). For maximally symmetric lattices of the cubic crystal group, the constituent plane waves propagate in directions, k_n, spread over the entire angular sphere, so there is no significant asymmetry in the polarization purity as a function of the desired polarization direction, ê_p. In contrast, for the single focus lens case, the polarization purity is worst for ê_p=ê_z, because the wavevector distribution is centered about the optical axis.

[0399] Lithography

[0400] Optical lattices having confined intensity maxima can also be useful for photolithography. As shown in FIG. 31, two-dimensional and three-dimensional lattice structures with a selectable wavelength-normalized periodicity can be rapidly produced in photosensitive materials 3100. With spatially defined excitation, exposed regions 3101 occur at the intensity maxima 3102, the size of which is controlled by adjusting the total exposure dose. Lattice structures can be useful for photonic bandgap applications, or as two-dimensional and three-dimensional holographic filters analogous to the diffractive notch filters currently produced by standing wave methods.

[0401] Referring to FIG. 31C, exposure of the sample at multiple points in each unit cell of the optical lattice can also be used to create more complicated diffractive structures with a basis of specifically tailored scattering regions 3103. As shown in FIG. 32, repeated patterns of any aperiodic structure, which are useful in many lithographic applications, can be created when an optical lattice can be scanned relative to a photosensitive material and the amplitude of the total excitation field, e(x,t), can be fully modulated from point to point. As with microscopy applications, the complete region of interest can be exposed by scanning over the dimensions of a single primitive cell. As shown in FIG. 33A, a three-dimensional lattice can be used to vertically confine the exposure of a planar photoresist layer (for cases where an undercut profile is desired). As shown in FIG. 33B, a two-dimensional lattice can be used to create uniform, high aspect ratio structures.

[0402] By utilizing the bases described above, the excitation in each cell of an optical lattice can be confined to nearly the diffraction limit in all desired directions, with minimal unintended exposure elsewhere. Furthermore, the exposed region can be precisely tailored using trepanning operations. Additionally, excitation can be performed through multiple lenses of low or moderate numerical aperture. This trait is beneficial because, as line-widths are further reduced and excitation wavelengths are further shortened in photolithography processes, aberration-free high numerical aperture lenses for traditional widefield lithography are increasingly difficult to design.

[0403] Although three-dimensional translucent blocks are depicted in FIGS. 31-33, any photosensitive material can be physically altered via lattice excitation. Such materials include thin-films and fields of surface-adsorbed molecules (whether monolayer or sub-monolayer). In particular, lattice excitation can be useful to improve the areal density and sensitivity in existing array-based assays for high throughput screening and combinatorial chemistry.

[0404] Multiphoton Microscopy and Lithography

[0405] Another issue of significance for both photolithography and microscopy is the non-zero background intensity remaining even for bases designed to maximize the excitation at a single point in each primitive cell. Such background results in unintentional exposure of surrounding areas in photolithography and unnecessary photobleaching of molecular fluorophores in microscopy. Although these effects are reduced when optical lattices are used as compared with conventional linear (e.g., confocal) microscopy, nonlinear excitation of the lattice and the sample can further reduce the background. For example, application of two-photon absorption as an exemplary nonlinear technique is shown in FIG. 34. The peak absorption wavelength, λ_abs, of the photosensitive material or molecular fluorophores of the sample dictates the excitation confinement and ultimate spatial resolution during either linear excitation (i.e., when λ_source=λ_abs) or during two-photon excitation (i.e., when λ_source=2λ_abs), and hence is chosen for the comparison of one- and two-photon lattice excitation in a maximally symmetric body-centered cubic lattice, having a period=√{square root over (5)}λ_source, with conventional, single focus two-photon excitation (NA=1.2) in water, as shown in FIG. 34.

[0406] As expected, the dependence of the excitation strength in two-photon absorption on the square of the intensity or on the fourth power of the field amplitude, |e(x)|⁴ results in a far lower excitation background away from the lattice maxima than in the case of single photon absorption (where the excitation strength is proportional to |e(x)|²), as seen in the lower regions of the planar slice surface plots in FIGS. 34B, 34C, and 34D. However, due to the longer initial wavelength, λ_source, the confinement of the two-photon excitation maxima is inferior to that of the single photon maxima for any given material of peak absorption, λ_abs. Thus, nonlinear excitation generally should be used to reduce the excitation background only in applications where such reduction takes precedence over achieving the highest possible resolution.

[0407] The small cross-sections of many nonlinear processes also often demand ultrashort pulsed excitation with high peak power to achieve sufficient signal. However, a temporal pulse of width, Δt, introduces a frequency spread, Δω≈1/Δt, that is inconsistent with the perfect coherence necessary for the creation of ideal, infinite lattices. The significance of this effect can be estimated by replacing each continuous constituent plane wave of a given lattice with a wave packet having a sharp leading edge and a tail of exponential decay length, τ, where the leading edge of each packet is assumed to arrive at the same position, x_d, at the same time, t_d. Mathematically, the fields are given by
e_n(x,t)=0, for t−t_d<k_n·(x−x_d)/ω_o and  (38a)

[0408] $\begin{array}{cc}\begin{array}{c}{e}_n\left(x,t\right)=e_n\exp \left\{\left[k_n·\left(x-x_d\right)/{\omega }_o\tau -\left(t-t_d\right)/\tau \right]/2\right\}\\ \exp \left\{i\left\{k_n·\left(x-x_d\right)-{\omega }_o\left(t-t_d\right)\right]\right\}\\ =e_n\exp \left\{\left(i+1/2{\omega }_o\tau \right)\left[k_n·\left(x-x_d\right)-{\omega }_o\left(t-t_d\right)\right]\right\},\\ \mathrm{for}t-t_d>k_n·\left(x-x_d\right)/{\omega }_o\end{array}& \left(38b\right)\end{array}$
Equation (38b) is a solution of the wave equation, and reduces to the continuous plane wave result as τ→∞. Furthermore, the frequency spectrum of |e_n(ω)|², has a Lorentzian lineshape having full-width-at-half-maximum equal to 1/τ and a center frequency, ω_o, characteristic of a lifetime-limited process. Superposition of all wave packets then gives the effect of the pulsed excitation on the overall lattice field. Integrating over an entire pulse to determine the time-averaged intensity yields, with the aid of Equation (38),

[0409] $\begin{array}{cc}\begin{array}{c}{\langle {e\left(x\right)}^2\rangle }_t=\left(1/\tau \right){\int }_{-\infty }^{\infty }{\sum _{n=0}^N{e}_n\left(x,t\right)}^{2}dt\\ =\sum _{m=0}^N\sum _{n=0}^N\left(e_m^{*}·e_n\right)\exp \left[i\left(k_n-k_m\right)·\left(x-x_d\right)\right]\\ \exp \left[-\left(k_n-k_m\right)·\left(x-x_d\right)/2{\omega }_o\tau \right]\end{array}& \left(39\right)\end{array}$
Three-dimensional isosurfaces of the time averaged two-photon excitation strength |e(x)|²_t² for the body-centered cubic lattice shown in FIG. 34 are plotted in FIG. 35A, where the free-space wavelength, λ₀=2πc/ω₀, is assumed to equal 800 nm and the decay length, τ, is assumed to equal 10⁻¹⁴ sec (i.e., ω_oτ=31.3). Corresponding linear grayscale images showing the two-photon excitation strength in the xy-plane for τ=10⁻¹⁴, τ=2·10⁻¹⁴, and τ=5·10⁻¹⁴ sec are given in FIGS. 35B, 35C, and 35D, respectively.

[0410] Comparison of Equations (6) and (39) and inspection of FIG. 35 reveal two primary effects of the pulsed excitation. First, the excitation strength across the lattice is modified by an envelope having a width, Δw, that is on order of cτ. Second, the spatial frequencies of the excitation at any point, x, decay fastest along the direction of the radial vector, x−x_d, which causes the lattice maxima to become increasingly elongated in this direction with increasing distance from the center of the envelope. However, two-photon excitation in living cells is commonly performed with pulses that are longer than 10⁻¹³ sec, so that these effects become significant (and hence require compensation of the effective excitation PSF) only when the desired field of view is of order 30 μm or larger.

[0411] Optical Trapping

[0412] Yet another application of optical lattices of confined intensity maxima is optical trapping of matter by the periodic potential wells in the lattice. For a spherical dielectric particle of radius, a, and refractive index, n_p, in a medium of refractive index, n_med, the force due to a monochromatic electromagnetic field in the electrostatic limit (ka<<1) has two contributions: one attractive term due to the gradient of the field:

[0413] $\begin{array}{cc}{\langle F_{\nabla }\left(x\right)\rangle }_t={\langle \left[p\left(x,t\right)·\nabla \right]e\left(x,t\right)\rangle }_t=\left(\frac{{\varepsilon }_{\mathrm{rel}}-1}{{\varepsilon }_{\mathrm{rel}}+2}\right)\frac{a^3}{4}\nabla {e\left(x,t\right)}^2,{\varepsilon }_{\mathrm{rel}}={\left(n_p/n_{\mathrm{med}}\right)}^2,& \left(40a\right)\end{array}$
and one repulsive term due to Rayleigh scattering:

[0414] $\begin{array}{cc}\begin{array}{c}{\langle F_{\mathrm{scat}}\left(x\right)\rangle }_t=\frac{n_{\mathrm{med}}}{c}\sigma {\langle S\left(x,t\right)\rangle }_t\\ =\frac{n_{\mathrm{med}}}{3}{k}^4{a}^6{\frac{{\varepsilon }_{\mathrm{rel}}-1}{{\varepsilon }_{\mathrm{rel}}+2}}^2\mathrm{Re}\left\{e\left(x\right)\times h*\left(x\right)\right\},\end{array}& \left(40b\right)\end{array}$
where σ is the scattering cross-section of the particle, S(x,t) is the Poynting's vector and, for the lattice electric field, e(x,t), of Equation (3), the corresponding magnetic field, h(x,t), is given by:

[0415] $\begin{array}{cc}h\left(x,t\right)=n_{\mathrm{med}}\sum _{n=0}^N\left(k_n\times e_n\right)\exp \left(i{k}_n·x\right)/k.& \left(41\right)\end{array}$

[0416] Components of the trapping force in the xy, xz, and yz planes are plotted in FIG. 36 for a single trap at the focus of a water-immersion microscope objective having a NA=1.2 and an optical axis, ê_z. Also plotted in FIG. 36 are components of the trapping force in the xy, xz, and yz planes for one of the multiple traps formed by the intensity maxima of a maximally symmetric body-centered cubic lattice of intensity period √{square root over (26)}λ. It is assumed that ka=0.8, n_p=1.6, and x-polarization at the trap is optimized in both cases.

[0417] The trapping potential between any two points x and x_ref is then given by:

[0418] $\begin{array}{cc}U\left(x\right)=-{\int }_{x_{\mathrm{ref}}}^x\left[{\langle F_{\nabla }\left(x\right)\rangle }_t+{\langle F_{\mathrm{scat}}\left(x\right)\rangle }_t\right]·ds,& \left(42\right)\end{array}$
which is plotted in FIG. 37 for the same cases and conditions as used for the traps illustrated in FIG. 36.

[0419] FIGS. 36 and 37 reveal several advantages of trapping with a maximally symmetric optical lattice as opposed to a single lens, single focus system. First, the greater confinement of the lattice intensity maxima due to the inclusion of wavevectors from a wider range of directions results in moderate enhancement (≈1.3×) of the peak trapping force in the xy plane, and significant enhancement (≈2.7×) of the gradient contribution to the trapping force along the axis, ê_z. Thus, although the depth of trapping potential minimum for a given peak intensity are similar in the two scenarios, the potential well is significantly narrower in the lattice case, indicating closer confinement to a specific desired position for a dielectric particle of a given kinetic energy. Second, due to the symmetry of the wavevector distribution, the scattering force in the lattice case is zero, whereas in the lens case the overall energy flow S(x) shifts the potential minimum along ê_z and reduces the energy necessary for the particle to escape the trap. Indeed, Equations (40) suggest that for a sufficiently large particle, the scattering force dominates in single lens illumination, and the particle will not be confined. Finally, lattice excitation allows an arbitrarily large number of particles to be trapped simultaneously with continuously adjustable spacing in two-dimensions, or with a discrete set of possible periods for a three-dimensional lattice of given symmetry and aspect ratios.

[0420] Optical lattice trapping can be used in the conventional fashion to manipulate whole cells or organelles within single cells, as well as to measure intracellular mechanical properties (e.g., effective bending moduli) and the force output of intracellular molecular motors. In addition, however, the multiplicity of traps afforded by lattice excitation can be uniquely used to immobilize a free-floating cell at many points across its interior (e.g., using an infrared wavelength which is not absorbed by the cell) to facilitate imaging with a second lattice at a wavelength chosen to excite the desired target. This eliminates the issues of phase aberrations in beams passing obliquely through the substrate and reflections of beams incident upon the substrate (as discussed below) that must be addressed in the lattice excitation of cultured cells.

[0421] Beyond the realm of biology, a dilute beam of individual atoms (e.g., chromium) can be trapped, cooled, and then deposited on a surface using a two-dimensional lattice. This approach has been used with one or two crossed standing waves to create gratings of period of the close order of λ/2 and linewidths as small as 15 nm (see, e.g., G. Timp, et al., Phys. Rev. Lett., 69, 1636 (1992), which is incorporated herein by reference). As with the lithography examples described herein with reference to FIGS. 31C and 32B, extension to two-dimensional composite lattices permits the creation of arrays of controllable period with a repeated arbitrary pattern in each primitive cell, except at a resolution dictated, not by λ, but rather by the degree to which the transverse atomic motion has been damped by the lattice. Similarly, atomic cooling in two-dimensional and three-dimensional lattices, which has been used in quantum electronics experiments (e.g., for Bose-Einstein condensation and quantum computing) and typically is performed with fundamental lattices, may also benefit from the narrower trapping potentials and decoupling of the potential depth from the periodicity possible with sparse, composite lattices in general and maximally symmetric composite lattices in particular. As these applications involve selection rules sensitive to specific components of the electromagnetic field, they may also benefit from the basis construction methods outlined above.

[0422] Stimulated Emission Depletion Microscopy

[0423] Until now the methods of molecular excitation that have been considered, including multiphoton excitation, limit spatial resolution in microscopy to the order of ˜λ_abs/(2n_med) dictated by the peak absorption wavelength, λ_abs, of the molecular species and the refractive index, n_med, of its surrounding medium. Although much can be learned about the structure and function of living cells at this resolution, extension of optical microscopy, and lattice microscopy in particular, well beyond λ_abs/(2n_med) can provide a unique non-invasive tool for the study of dynamic intracellular processes in the critical range between macromolecules and organelles.

[0424] Additionally, until now the construction of bases that optimize a particular polarization state and hence the intensity at a specific point and time within each primitive cell have been emphasized. However, other types of useful bases also exist and also can be useful for lattice microscopy applications.

[0425] One method adaptable to optical lattice microscopy that can provide significant superresolution and that exemplifies the use of bases other than those founded upon intensity maximization at a point is stimulated emission depletion microscopy (“STED”), described, for example, by S. W. Hell and J. Wichmann, Opt. Lett., 19, 780-782 (1994); and S. Hell and J. Wichmann, U.S. Pat. No. 5,731,588, which are both incorporated herein by reference. As shown in FIG. 38, a brief excitation pulse near the peak absorption wavelength, λ_abs, of a given molecular species can raise such molecules that are near the focal point of a lens to an excited quantum state. On a time scale that is short compared to the fluorescence lifetime of this state, a second pulse near the peak emission wavelength, λ_depl, can then be sent in order to return a fraction of the excited molecules to the ground state via stimulated emission. The molecules remaining in the excited state after the passage of the depletion pulse can be confined to a region 3804 that is small compared to λ_abs if: (1) the secondary depletion field 3800 is tailored to have a central node 3801 at the maximum of the primary excitation field 3802 with surrounding depletion sidebands 3803; and (2) the relative strengths of the two fields are adjusted accordingly. Such an operation yields an effective PSF consistent with superresolution microscopy.

[0426] Therefore, in order to use optical lattice microscopy in conjunction with STED, at least two lattices are needed: an excitation-lattice having intensity maxima as discussed above, and a depletion lattice, as shown in FIG. 39, having a node immediately surrounded by a strong depletion field in each primitive cell.

[0427] One exemplary way to create the depletion lattice is to superimpose multiple sub-lattices that have the same periodicity but bases with intensity maxima in different directions relative to a common nodal point. The complete depletion lattice and basis shown in FIG. 39 results from the superposition of three such sub-lattices. The first sub-lattice, as shown in FIGS. 40A, 40B, 40C, 40D, and 40E, is a fundamental body-centered cubic (BCC) lattice having an intensity period, τ_depl=√{square root over (3)}λ_depl/2, with a basis given by e_d∥ê_p∥ê_x≡[100] in Equation (21) and Equation (28), and each e_n phase shifted by an amount, Δφ_n=−k_n·(ê_y+ê_z)τ_depl/2, to align a nodal point within the primitive cell of the first sub-lattice with a maximum of the excitation lattice. The second sub-lattice, shown in FIGS. 40F, 40G, 40H, 40I, and 40J, is a fundamental BCC lattice obtained by rotating the wavevectors, k_n, of the first lattice by 90° about ê_z, and having a basis given by e_d parallel to ê_p∥[010]≡ê_y in Equation (21) and Equation (28), and with each e_n being phase shifted by an amount, Δφ=−k_n·(ê_z+ê_z)τ_depl/2, to align a nodal point of the second sub-lattice with the corresponding nodal point of the first sub-lattice. The third sub-lattice, shown in FIGS. 40K, 40L, 40M, 40N, and 40O, is a maximally symmetric simple cubic lattice having the same period with a basis given by e_d parallel to ê_p∥[001]≡ê_z in Equation (21) and Equation (28), and each ê_n being phase shifted by and amount, Δφ_n=−(k_n·ê_z)τ_depl/2, to align a nodal point of the third sub-lattice with those of the other sub-lattices.

[0428] To apply optical lattices to STED, the depletion lattice should be of the same Bravais crystal group as the excitation lattice (although of a longer wavelength). The depletion lattice should also have the same defining parameters, η, ξ, and γ, and the same absolute periodicity as the excitation lattice or some harmonic thereof, (i.e., τ_exc=mτ_depl, m=1, 2, . . . ), so that some subset of the nodes of the depletion lattice can be aligned with the maxima of the excitation lattice. Thus, the wavelength-normalized periodicities of the two lattices (which only exist in discrete steps) should be selected to match the Stokes shift, λ_depl/λ_abs, of the target molecular species. An example is shown in FIG. 41, where a maximally symmetric BCC excitation lattice of intensity period, τ_exc=√{square root over (5)}λ_abs, (indicated by translucent surfaces of 50% of maximum intensity in FIG. 41A and intensity plots in FIGS. 41B, 41C, and 41D) is used in conjunction with the depletion lattice of FIG. 39 and a species of assumed Stokes shift, λ_depl/λ_abs=√{square root over (5/3)}≈1.29. With a depletion pulse of sufficient strength and duration, the result is a final STED lattice (as shown by the opaque surfaces of 50% intensity in FIG. 41A and by the intensity plots in FIGS. 41E, 41F, and 41G) of the same symmetry and periodicity as the excitation lattice, but with emission regions confined on a scale small compared to that of the wavelength. A complete super-resolution image can be generated by scanning such a lattice over the dimensions of a single primitive cell, as described for previous examples.

[0429] “Optical Lattice STED” microscopy has several advantages compared to conventional, single focus STED microscopy. First, lattice-based STED microscopy offers massive parallelism for vastly improved imaging speed. Imaging speed is an issue of even greater importance in STED microscopy than in standard, linear fluorescence microscopy, because the size of (and hence the amount of signal from) each post-depletion emission region is so small. Second, the depletion basis of lattice-based STED microscopy offers improved coverage of the desired depletion shell, as well as improved sharpness in the depletion field near the node compared to conventional STED microscopy. Finally, lattice-based STED microscopy offers reduced photobleaching as compared to full-field, single focus lens excitation. Photobleaching is an issue of particular significance for STED microscopy, because the small emission region arising in STED microscopy requires that the illuminated fluorophores undergo many more excitation cycles for a given volume of sample to be imaged than in the convention, single focus microscopy.

[0430] The above-described method of superimposing multiple lattices to induce a desired optical response from the sample is not limited to STED. Indeed, any multiple wavelength optical technique can be similarly applied to lattice microscopy, including four wave mixing, coherent anti-Stokes Raman scattering (CARS), and multi-wavelength, multiphoton absorption.

[0431] Total Internal Reflection Microscopy

[0432] An alternative method capable of achieving resolution beyond λ_abs/(2n_med) in living cells is total internal reflection microscopy (“TIRM”), described, for example, by D. Axelrod, et al., J. Microsc., 129, 19-28 (1983), which is incorporated herein by reference. In TIRM, a specimen in a medium of a refractive index, n_med, in contact with a planar substrate of a higher refractive index, n_sub, is illuminated with light originating within the substrate and incident upon the medium at an angle, θ>θ_critical≡sin⁻¹(n_med/n_sub), with respect to the vector, ê_z, that is perpendicular to the substrate/medium interface. As a result, the light is totally reflected from the interface and back into the substrate. However, on the specimen/medium side of the interface, an electromagnetic field is created that propagates parallel to the interface, but that decreases exponentially in amplitude with increasing distance from the interface. This field can excite fluorescence within that portion of the specimen in closest proximity to the interface and can therefore be used to produce images containing information from a significantly narrower section (e.g., ˜100 nm) of the specimen than is possible with confocal microscopy, and with much lower photobleaching elsewhere.

[0433] Conventional TIRM uses uniform, widefield illumination, so that spatial resolution in the plane of the interface is dictated by the ˜λ_abs/(2n sin θ_max)=λ_abs/(2·NA) detection resolution of the objective lens used to collect the signal. In prism-based implementations of TIRM, this objective is usually located opposite the substrate (i.e., the prism), due to working distance constraints and the attendant spherical aberration introduced when the prism is in the beam path of the collected light. Thus, n=n_med, and resolution is limited by the refractive index of the medium (e.g., water, in which n_med=1.33). In through-the-objective based implementations of TIRM, the rear pupil of the objective is illuminated only at its periphery, creating a thin, conical illumination shell incident upon at interface at a cone half-angle beyond the critical angle. However, because an ultra-high numerical aperture objective (e.g., NA>1.4) is used to achieve the necessary cone angle, the same objective, when used for detection, gives n=n_sub over the path of the collected signal, and hence resolution n_sub/n_med times greater (e.g., 1.14× for n_med=1.33 and n_sub=1.52 (i.e., cover glass)) than when detection occurs from the medium side.

[0434] Practical considerations involving lens design and the properties of high refractive index liquids typically preclude extending the detection resolution beyond the limits of NA<1.45 and n_sub<1.6. However, an appropriate optical lattice, particularly with a basis of well-separated, highly confined intensity maxima, can contribute excitation resolution in TIRM adding to the overall system resolution.

[0435] Consider, for example, a subset lattice oriented so that one of the lattice plane families (described by a set Miller indices) is parallel to the substrate/medium interface, and that includes only those plane waves of the corresponding maximally symmetric lattice, or some subset thereof, whose wavevectors, k_m, originate in the substrate (i.e., from z<0) and are totally internally reflected at the interface (at z=0):
e_m^sub(x,t)=e_m^subexp[i(k_m·x−ωt)]  (43a)
θ_m≡cos⁻¹(k_m·ê_z/k_sub)>θ_critical≡sin⁻¹(n_med/n_sub), θ_m<π/2  (43b)
Each such plane wave creates an interfacial plane wave whose field amplitude decays exponentially with increasing distance from the substrate/medium interface:
e_m^med(x,t)=e_m^medexp[−z/δ(θ_m)]exp{i[(k_m·(ê_i)_m)(x·(ê_i)_m)−ωt]}, where:  (44a)

[0436] $\begin{array}{cc}\delta \left({\theta }_m\right)=\frac{\left(n_{\mathrm{med}}/n_{\mathrm{sub}}\right){\lambda }_{\mathrm{med}}}{2\pi \sqrt{{\sin }^2{\theta }_m-{\left(n_{\mathrm{med}}/n_{\mathrm{sub}}\right)}^2}},& \left(44b\right)\end{array}$
(ê_s)_m=(k_m×ê_z)/(k_sub sin θ_m), (ê_i)_m=ê_z×(ê_s)_m, and (ê_para)_m=(ê_s)_m×k_m/k_sub.  (44c)

[0437] The field, e_m^med, of each interfacial wave is related to the field, e_m^sub, of its corresponding substrate wave by the complex Fresnel coefficients for transmission at the interface:
e_m^sub=[(ê_s)_m·e_m^med](ê_s)_m/t_s+[ê_z·e_m^med](ê_para)_m/t_para,  (45a)

[0438] $\begin{array}{cc}{t}_s=\frac{2\cos {\theta }_m}{\cos {\theta }_m+i\sqrt{{\sin }^2{\theta }_m-{\left(n_{\mathrm{med}}/n_{\mathrm{sub}}\right)}^2}},\mathrm{and}& \left(45b\right)\\ t_{\mathrm{para}}=\frac{2\left(n_{\mathrm{med}}/n_{\mathrm{sub}}\right)\cos {\theta }_m}{{\left(n_{\mathrm{med}}/n_{\mathrm{sub}}\right)}^2\cos {\theta }_m+i\sqrt{{\sin }^2{\theta }_m-{\left(n_{\mathrm{med}}/n_{\mathrm{sub}}\right)}^2}}.& \left(45c\right)\end{array}$

[0439] A planar array of isolated yet individually confined intensity maxima can be created at the interface by using, for example, the basis construction methods of Equation (24) or, specifically, the bases leading to Equations (28), (34), or (37), to determine the field, e_m^med, for each interfacial wave that achieves the desired basis. Equations (45) can then be used to determine the field, e_m^sub, of each corresponding original plane wave within the substrate. Thus, for example, the two-dimensional array of intensity maxima shown in FIG. 42A is created via superposition of the interfacial plane waves with the wavevectors, k_m^med=(k_m·(ê_i)_m)(ê_i)_m, and the fields, e_m^med, represented in FIG. 42B, which in turn exist due to excitation within the substrate using plane waves with wavevectors, k_m, and fields, e_m^sub, as represented in FIG. 42C. The intensity within these maxima decays exponentially with increasing distance from the interface as shown in FIG. 42D.

[0440] Note that, as demonstrated in FIG. 42A and as implied by Equations (3) and (44), the separation and transverse confinement of the intensity maxima within the medium are the same as if the lattice and basis were constructed entirely within the substrate. Thus, even though the specimen is within the medium, the transverse excitation resolution in optical lattice TIRM is determined by the substrate refractive index, n_sub. Materials that are substantially transmissive at visible wavelengths and that have exceptionally high refractive indices (e.g., gallium nitride, GaN, with n_sub=2.45 or gallium phosphide, GaP, with n_sub=3.4) can therefore be used to attain excitation resolution well in excess of the detection resolution of conventional TIRM.

[0441] Increasing the substrate refractive index, n_sub, also results in a smaller critical angle, θ_critical, which can permit a greater number of plane waves within the maximally composite set to be included in the excitation for a given lattice. As shown in FIG. 43, for example, the same lattice type (i.e., a simple cubic lattice), periodicity in substrate wavelengths (√{square root over (35)}λ_sub/2), and basis (e.g., having a polarization maximized along ê_z) is used as in FIG. 42, except that the refractive index of the substrate is increased from n_sub=2.45 to n_sub=3.4. As a result, all 24 substrate-originating plane waves of the maximally symmetric lattice are totally internally reflected and hence can be included, as shown in FIG. 43C, rather than only 16 as shown in FIG. 42C. As demonstrated by comparing FIG. 43A to FIG. 42A, this leads to improved intensity contrast relative to the background in the plane of the interface.

[0442] As indicated by Equation (44b), it is also possible to exploit a higher substrate index, n_sub, to decrease the evanescent decay length of the lattice maxima, and hence create an even sharper excitation band within the sample. For example, FIG. 44B shows the improvement in the axial confinement of the excitation when the lattice consists of the same 16 substrate wavevectors, k_m, and the same interfacial basis as in FIG. 42D, except that a higher substrate index of refraction, n_sub=3.4, rather than n_sub=2.45 is used. The axial confinement seen in FIG. 44B is also superior to that obtained when n_sub=3.4 using all 24 waves, as shown in FIG. 43D, because the 8 waves omitted in FIG. 44 have the smallest angles of incidence, θ_m, and, hence, according to Equation (44b), the longest decay lengths, δ(θ_m), within the medium. However, the omission of the eight plane waves also reduces the intensity contrast as shown in FIG. 44A, demonstrating a trade-off that exists when three-dimensional lattices having totally internally reflected constituent plane waves at multiple angles of incidence, θ_m, are applied to TIRM.

[0443] All the wavevectors, k_m^med, of any TIRM-generated interfacial optical lattice, by definition, lie in the plane of the interface. Thus, as discussed previously in reference to FIG. 8D and FIG. 11A, the constraint, k_m^med·e_m^med=0, implies that strong intensity maxima, high intensity contrast, and good polarization purity can be achieved for the basis that exists when all interfacial waves are polarized perpendicular to the interface, i.e., when e_m^med is parallel to ê_z for every wave. From Equation (45a), this occurs when every corresponding plane wave within the substrate is polarized within its plane of incidence (i.e., when the plane waves are p-polarized, and e_m^sub∥(ê_para)_m).

[0444] In TIRM applications such as molecular orientation measurements within cellular membranes, bases of other preferred polarization may be needed. FIG. 45A, for example, shows the intensity distribution at the interface for the same lattice, same 16 wavevectors, and same substrate refractive index n_sub=2.45 as in FIG. 42A, except with a basis chosen to optimize the polarization along the axis, ê_x, at the intensity maxima rather than along the axis, ê_z. Increasing the substrate refractive index to increase the number of allowed wavevectors and therefore the intensity contrast can be particularly effective for bases such as this that have significant desired polarization within the plane of the interface. For example, FIG. 45B shows the contrast improvement under the same conditions as in FIG. 45A, except with n_sub=3.4 and 24 constituent wavevectors, that is, with the same lattice conditions as in FIG. 43, except with a different basis.

[0445] A key advantage of applying two- or three-dimensional sparse composite lattices to TIRM is that, unlike standing wave TIRM (as described, for example, in G. Cragg, et al., U.S. Pat. No. 6,255,642 B1, which is incorporated herein by reference), the confinement of the intensity maxima at the interface can be decoupled from their separation. Therefore, substrates having exceptionally high refractive indices, n_sub, (e.g., gallium nitride, which has an index of refraction n_sub=2.45 for light having photon energies of 2.5 eV) can be used to improve the confinement and associated transverse excitation resolution, while the lattice periodicity can be chosen to insure that, even at such high n_sub, the individual maxima remain resolvable at the more limited resolution of the detection optics, which is a necessary precondition for their collective use as a multi-point excitation array in microscopy.

[0446] When a high substrate index, n_sub, is used, detection of the signal with a high NA objective lens on the medium side of the interface (e.g., NA=1.2 for a good quality water immersion objective) eliminates the need to compensate for aberrations that can be introduced upon passage of the signal through the substrate. The overall transverse resolution is then dominated by the excitation resolution, which is on the order of λ_abs/(2·n_sub), rather than the detection resolution, which is on the order of λ_abs/(2·NA). However, even when the through-the-objective configuration is used to both excite the lattice and to detect the signal, and the substrate index, n_sub, is substantially lower (e.g., when n_sub=1.52 as when typical cover glass is used for the substrate), the contribution of the excitation resolution provided by the lattice results in slight improvement in the overall transverse resolution compared to conventional TIRM with uniform illumination, assuming the collected light is additionally spatially filtered at the image plane. This effect is analogous to the transverse resolution improvement seen in confocal microscopy compared to widefield microscopy resulting from the combination of point excitation with point detection.

[0447] The narrow excitation band within the medium occurring in either conventional or lattice-based TIRM should not be confused with high axial resolution, since axially separated features at the same transverse location are not individually resolved. However, three-dimensional imaging, including high axial resolution, in close proximity to the interface can be implemented in conventional TIRM by recording multiple two-dimensional images at different angles of incidence in the range θ_critical<θ<π/2 and processing the resulting data in light of the known decay length, δ(θ), (e.g., as given by Equation (44b)) in each image. This technique can also be applied to TIRM with two-dimensional lattice excitation, since all the substrate wavevectors, k_m, share the same cone angle, θ_m, and hence the lattice type (although not the periodicity) is preserved when all θ_m (with θ_critical<θ<π/2) are changed in unison. FIG. 46B and FIG. 46D compare the intensity decay within the medium for a particular two-dimensional lattice (n_med=1.33, n_sub=3.4) when sin θ_m=1 and sin θ_m=1.2 sin θ_critical, respectively. Although the intensity can be made to penetrate arbitrarily far within the medium as θ_m→θ_critical, the transverse resolution simultaneously degrades, as seen by comparing FIG. 46A and FIG. 46C, until it is dictated by the refractive index of the medium, n_med, rather than that of the substrate, n_sub.

[0448] This same multi-angle axial imaging method can also be applied to TIRM with three-dimension lattice excitation, provided that all substrate wavevectors, k_m, are rotated so that sin θ_m changes by the same factor for each wavevector, and the interfacial symmetry is thereby preserved. However, since any three-dimensional lattice contains wavevectors, k_m, of at least two different incident angles, θ_m, the total possible range of adjustment will be less than in the two-dimensional case, because both (θ_m)_max<π/2 and (θ_m)_min>θ_critical are required.

[0449] Multidimensional Localization

[0450] Another superresolution technique that can benefit from its adaptation to optical lattice microscopy is multidimensional localization, as described, for example, in E. Betzig, Opt. Lett., 20, 237-239 (1995), which is incorporated herein by reference. As shown in FIG. 47A, in multidimensional localization, a plurality of discrete emitters (e.g., individual fluorescent marker molecules within a cell) that are not well resolved in S spatial dimensions in a given optical system are imaged while P independent optical properties that can be expected to vary among the emitters (e.g., polarization or wavelength of the emission) are simultaneously measured at each point. If the resolution in the P dimensions defined by the measured properties is sufficiently high, then, for a given number of emitters within an S dimensional volume defined by the spatial resolution of the system, the emitters can be individually isolated in the S+P multidimensional space formed by the spatial coordinates and the measured parameters, as shown in FIG. 47B. Once isolated, methods such as that described in N. Bobroff, Rev. Sci. Instrum., 57, 1152 (1986), which is incorporated herein by reference, can be used to determine the location of each emitter to an accuracy that, with sufficient signal strength and knowledge of the spatial point spread function of the system, can be far better than the system resolution. Projection of these results back into the spatial dimensions then results in an S-dimensional image of the emitters at a resolution equal to this accuracy, as shown in FIG. 47C.

[0451] Optical lattice microscopy can contribute to this process because the volumetric point spread function of the excitation, as demonstrated in FIG. 15, can be significantly smaller than in single focus methods such as confocal microscopy. Thus, a greater spatial density of emitters can be isolated for a given resolution in the measurement of the P independent optical parameters. Furthermore, greater accuracy in the spatial localization, and hence greater resolution in the final image of the isolated emitters, can be achieved for a given signal-to-noise ratio from each emitter.

[0452] Since the accuracy of the localization and ultimate spatial resolution improve as this signal-to-noise ratio increases, optical lattice microscopy can also contribute through its ability to collect signal from numerous tightly confined volumes simultaneously. This can be applied to: (1) improve the isolation process by improving the signal-dependent resolution in one or more of the P measured parameters; (2) improve the localization process after isolation and hence the final spatial resolution at a given imaging speed; and (3) increase the imaging speed at a chosen spatial resolution.

Minimizing Deviations from Ideal Lattice Behavior

[0453] In an actual experimental setting, a number of factors can contribute to perturb both the wavevectors, k_n, and fields, e_n, of the idealized plane wave set for a given lattice and basis. It is therefore instructive to estimate their respective impact on the lattice and basis properties and resulting performance for the applications enumerated above. Strategies can then be developed for their amelioration.

[0454] Reflection and Refraction at Interfaces

[0455] A common scenario in lithography involves exposure of a photosensitive layer (e.g., a photoresist layer, such as, for example, polymethylmethacrylate (“PMMA”)) atop a substrate (e.g., silicon), where the photosensitive layer and the substrate have dissimilar optical properties. Similarly, the imaging of cultured cells occurs near a fluid/substrate interface of dissimilar optical properties (e.g., water and glass). At any such interface between regions of dissimilar optical properties, an incident plane wave is generally split into two plane waves: a first plane wave that is specularly reflected back into the originating medium of the wave, and a second plane wave that is transmitted from the first region at an angle of refraction into the other region. The ratio of this division is determined by the Fresnel coefficients for that interface.

[0456] For those waves of a lattice originating in a substrate having a refractive index, n_sub, and transmitted through a planar interface to reach the sample located in a medium of refractive index, n_med, the wave properties of the wavevectors, k_m, and the fields, e_m, within the sample can match those dictated by the lattice and basis construction methods described above if the original substrate wave is created with wavevectors, k_m^sub, and fields, e_m^sub, such that:
k_m^sub=k[sin θ_m^sub(ê_para)_m+cos θ_m^subê_z] and  (46a)
e_m^sub=((ê_s)_m·e_m)(ê_s)_m/t_s+((ê_para))_m·e_m)ê_para^sub)_m/t_para, where:  (46b)
θ_m≡cos⁻¹(k_m·ê_z/k), θ_m^sub=sin⁻¹[(n_med/n_sub)sin θ_m] (Snell's law)  (46c)
(ê_s)_m=(k_m×ê_z)/k sin θ_m, (ê_para)_m=(ê_s)_m×k_m/k, and (ê_para^sub)_m=(ê_s)_m×k_m^sub/k.  (46d)

[0457] Here the axis, ê_z, is defined perpendicular to the interface, and t_s, t_para are the complex Fresnel coefficients for transmission through the interface for the (ê_s)_m (s-polarized) and (ê_para^sub)_m (p-polarized) components of the substrate field, respectively. If (n_sub/n_med)sin θ_m^sub>1, then the substrate wave is totally reflected at the interface, and cannot contribute to the lattice at the sample.

[0458] For each wave of the lattice originating within the medium of the sample, a reflected wave is created at the interface that, in general, alters the symmetry and periodicity of the lattice as well as the basis. For example, as shown in FIG. 48, a two-dimensional lattice having all its constituent wavevectors on a cone of half-angle, θ, oriented with the cone axis perpendicular to the interface(s) is transformed into a three-dimensional lattice having a single spatial frequency of 2k cos θ along this axis. If the interfaces are parallel to a mirror plane of the lattice, however, the wavevectors of the reflected waves are also within the maximally symmetric composite set, and hence the symmetry and periodicity is preserved, although the basis is still perturbed, as shown in FIG. 49.

[0459] The lattice and interface properties can be tailored in several different ways to minimize the amplitudes of the reflected waves and hence their effect on the basis. First, the lattice can be designed so that some or all of the initial wavevectors are incident on the interface at angles close to reflection minima for their polarizations. In particular, plane waves used to create a two-dimensional lattice in a dielectric medium of refractive index n_med=Re{n_med} perpendicular to a dielectric substrate of refractive index n_sub=Re{n_sub} (e.g., in the case of a two-dimensional lattice within a specimen surrounded by an aqueous environment having n_med=1.33, cultured on a glass surface having n_sub=1.52) will exhibit no reflections if the plane wave wavevectors are all incident at an angle equal to Brewster's angle, θ_B=tan⁻¹ (n_sub/n_med), with respect to the vector normal of the interface plane, ê_z, and if the initial basis is selected to maximize linear polarization perpendicular to the interface as per Equation (28) (so that the fields, e_n, are within the plane of incidence for each wave). Second, to reduce reflections at multiple and/or larger angles of incidence, such as is desired for a three-dimensional lattice or a two-dimensional lattice of optimal intensity confinement, a single or multilayer anti-reflection (AR) film can be applied to the interface. For example, a MgF₂ film of refractive index, n_f=1.38, and thickness, h=n_mn_fλ_m/4√{square root over (n_f²−(n_m sin θ)²)}, reduces the amplitude |r| of the reflection coefficient of a p-polarized wave incident at θ=80° to the vector, ê_z, normal to a water-glass interface from 0.438 to 0.089, and that of an s-polarized wave from 0.539 to 0.029. Thus, the basis can remain relatively unperturbed when it is translated normal to an AR coated interface, as shown in FIGS. 50C and 50F, as compared with the substantial variations due to differing degrees of constructive and destructive interference between the incident and reflected waves as observed with an uncoated water/glass interface as shown in FIGS. 50B and 50E. Third, in those cases where it is impractical to design an AR coating sufficiently effective at all angles of the incident wavevectors of the lattice, it is always possible to simply omit those plane waves exhibiting excessive reflection, although with the concomitant loss of intensity confinement characteristic of subset lattices.

[0460] In those cases where the sample medium and the substrate are both substantially transparent (e.g., the water/glass interface common to cultured cells), an alternative approach can be used to produce an unperturbed, translationally invariant lattice. First, the lattice is oriented relative to the interface such that for every incident plane wave originating within the sample medium having wavevector,
k_m=k[sin θ_m(ê_para)_m+cos θ_mê_z],  (47)
and having the initial basis-dictated field, e_m, a reflected wave having a wavevector,
k_m^r=k[sin θ_m(ê_□)_m−cos θ_mê_z],  (48a)
e_m^r=r_s((ê_s)_m·e_m)(ê_s)_m−r_para((ê_i)_m·e_m)(ê_i)_m+r_para(ê_ze_m)ê_z, and (ê_i)_m=ê_z×(ê_s)_m  (48b)
is created that propagates in the same direction, k_m^r, as some other wavevector, k_n, of the corresponding maximally symmetric composite lattice (where r_s, r_para are the complex Fresnel reflection coefficients at the interface for s- and p-polarized waves, respectively). This reflected wave is then superimposed with a wave originating in the substrate with a wavevector, k_m^sub, and field, e_m^sub, given by:
k_m^sub=k[sin θ_m^sub(ê_para)_m−cos θ_m^subê_z], sin θ_m^sub=(n_med/n_sub)sin θ_m  (49a)
e_m^sub=−((ê_s)_m·e_m^r)(ê_s)_m/t_s−((ê_para^r)_m·e_m^r)(ê_para^sub)_m/t_para or  (49b)
e_m^sub=((ê_s)_m·(e_n−e_m^r))(ê_s)_m/t_s−((ê_para^r)_m·(e_n−e_m^r))(ê_para^sub)_m/t_para,  (49c)
(ê_para^r)_m=(ê_s)_m×k_m^r/k,  (50)
so that the resulting refracted wave transmitted into the sample medium also propagates in the direction, k_n, with a field, e_n^t, such that the combined plane wave, e_m^r+e_n^t, field in the medium is either zero (for Equation (49b)) or equal to the ideal basis-dictated field, e_n, associated with the wavevector, k_n, in the absence of the substrate (for Equation 49c)).

[0461] Alternatively, interfacial reflections can be exploited to improve the intensity confinement at the lattice maxima, particularly near opaque substrates, where the reflected plane waves contribute lattice wavevectors that would otherwise originate within the substrate and therefore would be inaccessible. In such cases, the basis should be selected so that the electric field of each reflected wave approximates the field required to optimize the basis polarization at the desired positions of the lattice maxima. However, because the phase relationship between each incident and reflected wave is fixed by the Fresnel reflection coefficients and hence the properties of the interface, in general a different basis polarization will be needed to achieve optimal confinement and intensity strength at every distance from the interface, as demonstrated in FIG. 51.

[0462] For a given basis having a desired polarization at the confined intensity maxima, the strength and contrast relative to the background can be further optimized if the electric field of each incident wave, given as,
e_m(x,t)=e_o{sin χexp(iη_s)(ê_s)_m+cos χexp(iη_p)[sin θ_m(ê_i)_m+cos θ_mê_z]}·exp[i(k_m·x−ωt)]  (51)
is selected not to maximize its projection, Re{e_m*(x_d,t_d)′·e_d}, on the desired polarization state, e_d, at x_d and t_d as in Equation (24) and the specific examples given by Equations (28), (34), and (37), but rather the projection, Re{[(e_m(x_d,t_d)+e_m^r(x_d,t_d))*]′·e_d}, of the superposition of this wave with its reflected counterpart, which is given by:
e_m^r(x,t)=e_o{r_s sin χexp(iη_s)(ê_s)_m+r_p cos χexp(iη_p)[sin θ_m(ê_i)_m−cos θ_mê_z]}·exp[i(k_m^r·x−ωt)]  (52)

[0463] Because the reflection coefficients, r_s and r_p, are generally complex, in the specific case where linear polarization is optimized (i.e., when e_d=E_oê_d), the projection of the superimposed incident and reflected waves on the desired polarization state is given by:
Re{[(e_m(x_d,t_d)+e_m^r(x_d,t_d))*]′·e_d}≡G(χ,η_s,η_p)=e_oE_o{sin χ((ê_s)_m·ê_d)[cos(φ_m+η_s)+Re(r_s)cos(φ_m^r+η_s)−Im(r_s)sin(φ_m^r+η_s)]+cos χ[(sin θ_m(ê_z·ê_d)+cos θ_m((ê_para)_m·ê_d))cos(φ_m+η_p)+(sin θ_m(ê_z·ê_d))−cos θ_m((ê_para)_m·ê_d))(Re(r_p)cos(φ_m^r+η_p)−Im(r_p)sin(φ_m^r+η_p))]}  (53a)
where: φ_m=k_m·x_d−ωt_d and φ_m^r=k_m^r·x_d−ωt_d.  (53b)

[0464] Application of the maximization conditions, ∂G/∂η_s=∂G/η_p=∂G/∂χ=0, then yield, respectively:

[0465] $\begin{array}{cc}{\eta }_s=-{\tan }^{-1}\left[\frac{\sin {\phi }_m+\mathrm{Re}\left(r_s\right)\sin {\phi }_m^r+\mathrm{Im}\left(r_s\right)\cos {\phi }_m^r}{\cos {\phi }_m+\mathrm{Re}\left(r_s\right)\cos {\phi }_m^r-\mathrm{Im}\left(r_s\right)\sin {\phi }_m^r}\right]+n_s\pi ,& \left(54a\right)\end{array}$
η_p=−tan⁻¹{[(sin θ_m(ê_z·ê_d)+cos θ_m((ê_para)_m·ê_d))sin φ_m+(sin θ_m(ê_z·ê_d)−cos θ_m((ê_para)_m·ê_d))(Re(r_p)sin φ_m^r+Im(r_p)cos φ_m^r)]/[(sin θ_m(ê_z·ê_d)+cos θ_m((ê_para)_m·e_d))cos φ_m+(sin θ_m(ê_z·ê_d)−cos θ_m((ê_para)_m·ê_d))(Re(r_p)cos φ_m^r−Im(r_p)sin φ_m^r)]}+n_pπ,  (54b)
χ=tan⁻¹{[((ê_s)_m·ê_d)(cos(φ_m+η_s)+Re(r_s)cos((φ_m^r+η_s)−Im(r_p)sin(φ_m^r+η_s))]/[(sin θ_m(ê_z·ê_d)+cos θ_m((ê_para)_m·e_d))cos(φ_m+η_p)+(sin θ_m(ê_z·ê_d)−cos θ_m((ê_para)_m·e_d))(Re(r_p)cos(φ_m^r+η_p)−Im(r_p)sin(φ_m^r+η_p))]}+n_χπ,  (54c)
where n_s, n_p, and n_χ are even or odd, based on the additional maximization conditions ∂²G/∂η_s²<0, ∂²G/∂η_p²<0, and ∂²G/∂χ²<0:
cos(φ_mη_s)+Re(r_s)cos(φ_m^r+η_s)−Im(r_s)sin(φ_m^r+η_s)>0,  (55a)
(sin θ_m(ê_z·ê_d)+cos θ_m((ê_para)_m·ê_d))cos((φ_m+η_p)+(sin θ_m(ê_z·ê_d)−cos θ_m((ê_para)_m·ê_d))(Re(r_p)cos((φ_m^r+η_p)−Im(r_p)sin((φ_m^r+η_p))>0, and  (55b)
G(χ,η_s,η_p)>0.  (55c)