Multibeam optical system

[0002] The present invention relates to a multibeam optical system that divides a laser beam emitted from a laser source into a plurality of beams and forms a plurality of beam spots on an object surface. Particularly, the invention relates to the optical system that employs a diffractive beam-dividing element to divide a laser beam emitted from a laser source.

[0003] The multibeam optical system needs a beam-dividing element that divides a laser beam emitted from a laser source into a plurality of beams to form a plurality of beam spots on the object surface.

[0004] A conventional multibeam optical system has employed a prism-type beam splitter as the beam-dividing element, which comprises a plurality of prism blocks cemented to one another. The cemented faces of the prism blocks are provided with multi-layer coatings having the desired reflecting properties, respectively.

[0005] However, when employing a prism-type beam splitter, since each one of the multi-layer coatings can divide an incident beam only into two separate beams, the number of prism blocks corresponding to the required number of separate beams must be cemented to one another. Further, when cementing one block to another block, an angle error between two cemented face unavoidably arises. Accordingly, when a large number of separate beams are required, the deviations of the beam spots on the object surface tend to become large due to an accumulation of positional errors between the cemented prism blocks.

[0006] Recently, a diffractive beam-dividing element has become used in place of a prism-type beam splitter. Since the diffractive beam-dividing element is made of a single block that is not cemented, it does not generate any positional error even when the large number of the separate beams are required.

[0007] With employing the diffractive beam-dividing element, however, since the diffraction angle of a light beam varies depending upon the wavelength thereof, the same order diffracted beam may be separated to form a plurality of beam spots in different positions on the object surface, in case a light source emits a light beam having a plurality of peak wavelengths.

[0008] For example, an argon laser, which is used as a light source of a laser photo plotter or the like, has a plurality of peak wavelengths in the ultraviolet and visible regions. Therefore, in order to avoid the above defects, it has been required to use a filter for passing a beam component of a selected peak wavelength. Thus, the beam components of peak wavelengths other than the selected peak wavelength are cut off by the filter, which results in low energy efficiency.

[0009] Further, even if a beam emitted from a light source has a single peak wavelength, in case a peak wavelength of a beam actually emitted from a light source fluctuates or varies, a beam spot pitch on a surface to be exposed is changed.

[0010] It is therefore an object of the present invention to provide an improved multibeam optical system capable of avoiding the defect such as a separation of the same order diffracted beam or a variation of the beam spot pitch caused by the wavelength dependence of a diffractive beam-dividing element employed therein.

[0011] For the above object, according to the present invention, there is provided an improved multibeam optical system that includes a compensating optical system, which is afocal and consists of a first group and a second group, arranged at the position where beams divided by a diffractive beam-dividing element are incident thereon. The compensating optical system has a characteristic such that the angular magnification thereof is inversely proportional to the wavelength of the incident beam.

[0012] With this construction, the same order diffracted beams of the respective wavelengths diffracted by the diffractive beam-dividing element exit at the different diffraction angles, and the diffracted beams are incident on the first group of the compensating optical system. Since a diffraction angle of the diffractive beam-dividing element increases as a wavelength becomes longer, an incident angle on the compensating optical system increases as a wavelength becomes longer. On the other hand, when the angular magnification of the compensating optical system is inversely proportional to the wavelength, the ratio of an incident angle on the compensating optical system to an exit angle therefrom decreases as a wavelength becomes longer. Therefore, the angular difference among the diffracted beam caused by the wavelength dependence of the diffractive beam-dividing element can be reduced when the beams transmit the compensating optical system.

[0013] In another aspect of the invention, the following conditions (1) and (2) are satisfied to counterbalance the angular difference of the diffracted beams with the variation of the angular magnification of the compensating optical system: $\begin{array}{cc}{v}_1=\frac{f_1+f_2}{f_1}·v_{\mathrm{DOE}}& \left(1\right)\\ v_2=-\frac{f_1+f_2}{f_2}·v_{\mathrm{DOE}}& \left(2\right)\end{array}$

[0014] where ν₁is the Abbe number of the first group, f₁is the focal length of the first group, ν₂is the Abbe number of the second group, f₂is the focal length of the second group, and ν_DOEis a dispersive power of the diffractive beam-dividing element, which corresponds to an Abbe number of a refractive lens.

[0015] Further, it is preferable that the compensating optical system substantially satisfies the following conditions (3), (4) and (5): $\begin{array}{cc}\frac{f_1}{v_1}=-\frac{f_2}{v_2}& \left(3\right)\\ f_1=f_2& \left(4\right)\\ v_1=-v_2=2v_{\mathrm{DOE}}.& \left(5\right)\end{array}$

[0016] Each of the first group and the second group may be an element having reflecting surfaces of a positive power on which a diffractive lens structure is formed. In such a case, the conditions (1), (2) and (5) can be satisfied without difficulty. Alternatively, each of the first and second groups may be a composite element of a positive refractive lens and a diffractive lens structure.

[0017] The first group is preferably located at a position where the distance from the diffractive beam-dividing element is equal to the focal length f₁of the first group.

[0018] The multibeam optical system of the present invention is usually applied to a multibeam scanning optical system, however it can be applied to other systems as a matter of course.

[0019] FIG. 1 shows a multibeam scanning optical system embodying the invention;

[0020] FIGS. 2 and 3 are partial enlarged views of a compensating optical system in FIG. 1 showing the principle of the invention; and

[0021] FIG. 4 shows an exemplary positive power reflector of the first group of the compensating optical system.

[0022] A multi-beam optical system embodying the present invention will be described hereinafter by referring to the accompanying drawings. FIG. 1 shows a multibeam scanning optical system embodying the invention; and FIGS. 2 and 3 show the principle of the invention. At first the construction of the embodiment will be described with reference to FIG. 1, and then the principle of the invention will be described with reference to FIGS. 2 and 3. In the drawings, curved mirrors, which are disposed in the optical system, are represented as if they are light transmittable elements like lenses in order to provide a clear understanding of an optical path. Therefore, the optical system is developed to make the optical axis straight in the drawings. Further, optical element such as a lens and a mirror are shown as thin lenses in the drawings.

[0023] As shown in FIG. 1, a multi-beam scanning optical system embodying the invention comprises a laser source 1, abeam expander 2, a diffractive beam-dividing element 3, a compensating optical system 4, a converging mirror 5, a multi-channel modulator 6, a collimator lens 7, a polygonal mirror 8 as a deflector, an fθ lens 9 as a scanning lens and an object surface 10 to be exposed, which are arranged in this order from left in FIG. 1.

[0024] Since the polygon mirror 8 rotate about the rotation axis that is perpendicular to the sheet of FIG. 1 to deflect the beams in a vertical direction (i.e., up-and-down direction) in FIG. 1, the vertical direction in FIG. 1 is referred to as a main scanning direction. Further, since the object surface 10 moves in a direction perpendicular to the sheet of FIG. 1 to form a two-dimensional image thereon, the direction perpendicular to the sheet of FIG. 1 is referred to as an auxiliary scanning direction.

[0025] The laser source 1 is a multiline laser source such as an argon laser having a plurality of peak wavelengths. The beam expander 2 adjusts the diameter of the laser beam emitted from the laser source 1. The diffractive beam-dividing element 3 diffracts the parallel beam from the beam expander 2 to divide it into a plurality of laser beams outputting at different angles. Further, since the diffractive beam-dividing element 3 diffracts the incident laser beams of the respective wavelengths at different angles, the same order diffracted beams of the respective wavelengths exit at the different diffraction angles from the diffractive beam-dividing element 3.

[0026] The parallel laser beams divided by the diffractive beam-dividing element 3 are, respectively, incident on the compensating optical system 4. The compensating optical system 4 consists of a first group 4athat has a negative chromatic dispersion and a second group 4bthat has a positive chromatic dispersion. The first group 4ais composed of a concave mirror and a diffractive lens structure having a positive power formed on the concave mirror, as indicated in FIG. 4. Similarly, the second group 4bis composed of a concave mirror and a diffractive lens structure having a negative power formed on the concave mirror. Each of the first and second groups 4aand 4bhas a positive power as a whole. The compensating optical system 4 in an afocal optical system and the angular magnification is inversely proportional to the wavelength of the incident beam to compensate an angular difference of the same order diffracted beams caused by the wavelength dependence of the diffractive beam-dividing element 3. The same order diffracted beams of the respective wavelengths are adjusted to travel along the same optical path after the beams passing through the compensating optical system 4. Abbe numbers and focal lengths of the first and second groups 4aand 4bof the compensating optical system 4 are determined to achieve the compensating function, as described below. The first group 4ais located such that the distance from the diffractive beam-dividing element 3 to the front principal point thereof is equal to the focal length f₁of the first group 4a. The second group 4bis located such that the distance from the rear principal point of the first group 4ato the front principal point of the second group 4bis equal to the sum of the focal length f₁of the first group 4aand the focal length f₂of the second group 4b.

[0027] The parallel beams of the respective diffraction orders passing through the compensating optical system 4 are converged by the converging mirror 5. The convergent beams are aligned such that the chief rays are parallel to one another. The multi-channel modulator 6 such as an acousto-optic modulator (AOM) is located at beam waist position of the converged beams. The multi-channel modulator 6 is provided with a plurality of channels each of which changes a direction of the converged laser beam in response to the input ultrasonic wave. The detail construction of the multichannel modulator is not illustrated because the multichannel modulator is a device that is in general use. In a channel, when the ultra sonic wave is applied to a medium, a diffraction grating is formed by a compression wave caused in the medium. The diffracted beam emerges as the modulated beam, and the non-diffracted beam is cut off by a shading plate. The channels of the multichannel modulator 6 are independently controlled to independently modulate the laser beams, i.e., to independently turn ON/OFF the laser beams, respectively.

[0028] The collimator lens 7 is arranged such that the front focal point thereof is coincident with the multichannel modulator 6 on a center axis of the beams, i.e., an optical axis of the system. The modulated beams are converged into parallel beams whose beam axes intersect at the rear focal point of the collimator lens 7.

[0029] The polygonal mirror 8 is arranged such that the reflecting surface is located at the rear focal point of the collimator lens 7. The polygonal mirror 8 rotates about the rotation axis to deflect the laser beams in the main scanning direction while keeping angular difference among the laser beams of the respective diffraction orders.

[0030] The deflected laser beams are converged by the fθ lens 9 to form a plurality of beam spots aligned in the auxiliary scanning direction at equal intervals on the object surface 10. The beam spots simultaneously scan in the main scanning direction at a constant speed as the polygonal mirror 8 rotates.

[0031] Details of the compensating optical system 4 will be explained by referring to FIGS. 2 and 3. In this embodiment, the respective one of the first and second groups 4aand 4bcomprises a concave mirror on which a diffractive lens structure is formed.

[0032] The diffractive lens structure has an advantage in its suitability for compensating the angular difference caused by the wavelength dependence of the diffractive beam dividing element 3. The dispersive power ν_DOEof the diffractive lens structure, which corresponds to an Abbe number of a refractive lens, is −3.453. That is, the diffractive lens structure shows a relatively large dispersive power as compared with a refractive lens, which allows the compensating optical system 4 to correct the chromatic dispersion of the diffractive beam-dividing element even if it is too large to be corrected by a refractive lens.

[0033] Further, when the diffractive lens structure is formed on the reflecting surface as described above, the chromatic dispersion in the compensating optical system 4 occurs only by the diffractive lens structures thereof, the dispersion shows a linear relationship with the wavelength of the light beam, which is suitable to cancel the chromatic dispersion caused by the diffractive beam-dividing element 3 as it shows also a linear relationship with the wavelength of the light beam.

[0034] In FIGS. 2 and 3, the laser beam of the first wavelength λ_Aand the chief ray thereof are illustrated by solid lines, the laser beam of the second wavelength λ_Band the chief ray thereof are illustrated by dotted lines. Assuming that the first wavelength λ_Ais longer than the second wavelength λ_B.

[0035] Since the diffractive beam-dividing element 3 has a negative chromatic dispersion, the diffraction angle of the beam of the first wavelength λ_Ais larger than that of the second wavelength λ_B, as shown in FIG. 3. Therefore, a height (a distance from the optical axis Ax) of an incident point and an incident angle on the first group 4aof the compensating optical system 4 increase as the wavelength becomes longer.

[0036] The first group 4aof the compensating optical system 4 functions to equate the intersection heights of the same order diffracted beams (particularly, the chief rays) having the respective wavelengths when the beams are incident on the second group 4b. Accordingly, the first group 4amust have a negative dispersion (1/ν₁<0) to increase a deviation angle as a wavelength becomes longer. Next, the value ν₁will be found.

[0037] In FIG. 2, with respect to the chief ray of the first wavelength λ_A,

[0038] u_Ais an angle of the chief ray to the optical axis Ax when the ray is incident on the first group 4a;

[0039] h_1Ais a height of an incident point when the ray is incident on the first group 4a;

[0040] u′_Ais an angle of the chief ray to the optical axis Ax when the ray exits the first group 4a;

[0041] φ_1Ais a power of the first group 4a; and

[0042] f_1Ais a focal length of the first group 4a.

[0043] In the same manner, with respect to the chief ray of the second wavelength λ_B,

[0044] u_Bis an angle of the chief ray to the optical axis Ax when the ray is incident on the first group 4a;

[0045] h_1Bis a height of an incident point when the ray is incident on the first group 4a;

[0046] u′_Bis an angle of the chief ray to the optical axis Ax when the ray exits the first group 4a;

[0047] φ_1Bis a power of the first group 4a; and

[0048] f_1Bis a focal length of the first group 4a.

[0049] Assuming that the chief ray of the first wavelength λ_Ais parallel to the optical axis Ax after a reflection by the first group 4aand the chief rays of the first and second wavelengths λ_Aand λ_Bare incident at the same position on the second group 4b. In this case, the angle u′_Aof the chief ray of the first wavelength λ_Ais given by the equation (1):

[0050] u′_A=u_A+h_1Aφ1A  (1)

[0051] Since the chief ray of the first wavelength λ_Ais parallel to the optical axis Ax, u′_A=0, and then the equation (1) is converted to

[0052] u_A=−h_1Aφ1A  (2)

[0053] On the other hand, the angle u′_Bof the second wavelength λ_Bis given by the equation (3):

[0054] $\begin{array}{cc}{u}_B^{\prime }=u_B+h_{1B}{\varphi }_{1B}=\frac{{\lambda }_B}{{\lambda }_A}·u_A+\frac{{\lambda }_B}{{\lambda }_A}·h_{1A}·{\varphi }_{1B}& \left(3\right)\end{array}$

[0055] Substitution of the equation (2) into the equation (3) yields the equation (4). $\begin{array}{cc}\begin{array}{c}{u}_B^{\prime }=-\frac{{\lambda }_B}{{\lambda }_A}·h_{1A}·{\varphi }_{1A}+\frac{{\lambda }_B}{{\lambda }_A}·h_{1A}·{\varphi }_{1B}\\ =\frac{{\lambda }_B}{{\lambda }_A}·h_{1A}\left({\varphi }_{1B}-{\varphi }_{1A}\right)\\ =\frac{{\lambda }_B}{{\lambda }_A}·h_{1A}\left(\frac{1}{f_{1B}}-\frac{1}{f_{1A}}\right)\end{array}& \left(4\right)\end{array}$

[0056] Further, h_1A=−u_A×f_1Aas shown in FIG. 2, the equation (4) is converted to $u_B^{\prime }=-\frac{{\lambda }_B}{{\lambda }_A}·u_A·f_{1A}·\frac{f_{1A}-f_{1B}}{f_{1A}{f}_{1B}}$

[0057] Assuming that f_1A−f_1B=−Δf₁, and f_1B=f₁, $u_B^{\prime }=\frac{{\lambda }_B}{{\lambda }_A}·u_A·\frac{\Delta f_1}{f_1}$

[0058] Since the value of Δf represents a longitudinal chromatic aberration and it is equal to −f/ν, the above equation is converted to $\begin{array}{cc}{u}_B^{\prime }=-\frac{{\lambda }_B}{{\lambda }_A}·\frac{u_A}{v_1}& \left(5\right)\end{array}$

[0059] Further, according to FIG. 2, a condition to compensate the difference Δh₁between h_1Aand h_1Bis represented as follows: $\begin{array}{cc}\begin{array}{c}\Delta h_1=\left(u_A-u_B\right)·f_1=u_B^{\prime }·\left(f_1+f_2\right)\\ =\left(u_A-\frac{{\lambda }_B}{{\lambda }_A}·u_A\right)·f_1=u_B^{\prime }·\left(f_1+f_2\right)\\ =\frac{{\lambda }_A-{\lambda }_B}{{\lambda }_A}·u_A·f_1=u_B^{\prime }·\left(f_1+f_2\right)\end{array}& \left(6\right)\end{array}$

[0060] On the basis of the equations (5) and (6), $\begin{array}{c}{u}_B^{\prime }=-\frac{{\lambda }_B}{{\lambda }_A}·\frac{u_A}{v_1}\\ =\frac{f_1}{f_1+f_2}·\frac{{\lambda }_A-{\lambda }_B}{{\lambda }_A}·u_A\end{array}$

[0061] It is converted to the equation (7) to solve about ν₁as follows: $\begin{array}{cc}{v}_1=\frac{{\lambda }_B}{{\lambda }_B-{\lambda }_A}·\frac{f_1+f_2}{f_1}& \left(7\right)\end{array}$

[0062] The equation (7) can be converted to the equation (8) because λ_B/(λ_B−λ_A) is equal to the equivalent Abbe number ν_DOEof a diffractive optical element. $\begin{array}{cc}{v}_1=\frac{f_1+f_2}{f_1}·v_{\mathrm{DOE}}& \left(8\right)\end{array}$

[0063] The equation (8) is a condition of the Abbe number ν₁required for the first group 4ato equate the heights of the diffracted beams of the wavelengths λ_Aand λ_Bwhen they are incident on the second group 4b. As described above, since the compensating optical system 4 is an afocal optical system as a whole, both of the focal lengths f₁and f₂take positive values. Accordingly, the Abbe number ν₁is required to be negative. A glass lens has a positive value in the Abbe number, it is impossible to satisfy the equation (8) when the first group 4aconsists of glass lenses only. Therefore, the first group 4ais formed as a combination of a reflection mirror that has no chromatic dispersion and a diffractive lens structure whose equivalent abbe number is negative.

[0064] An incident angle on the second group 4bof a predetermined order diffracted beam (particularly, a chief ray) deflected by the first group 4aincreases as a wavelength becomes shorter. The second group 4bhas a function to direct the same order diffracted beams that are incident at the same position into the same direction. Accordingly, the second group 4bmust have a positive dispersion (1/ν₂>0), to increase a deviation angle as a wavelength becomes shorter. Next, the value ν₂will be found.

[0065] In FIG. 2, with respect to the chief ray of the first wavelength ν_A,

[0066] u″_Ais an angle of the chief ray to the optical axis Ax when the ray exits the second group 4b; and

[0067] φ_2Ais a power of the second group 4b.

[0068] In the same manner, with respect to the chief ray of the second wavelength λ_B,

[0069] u″_Bis an angle of the chief ray to the optical axis Ax when the ray exits the second group 4b; and

[0070] φ_2Bis a power of the second group 4b.

[0071] In this case, the angle u″_Aof the chief ray of the first wavelength λ_Ais given by the equation (9):

[0072] u″_A=u′_A+h_2φ2A  (9)

[0073] Since the chief ray of the first wavelength λ_Ais parallel to the optical axis Ax, u′_A=0, and then the equation (9) is converted to

[0074] u″_A=h_2φ2A  (10)

[0075] On the other hand, the angle u″_Bof the second wavelength λ_Bis given by the equation (11):

[0076] u″_B=u′_B+h_2φ2B  (11)

[0077] In order to coincide the rays of both of the wavelengths λ_Aand λ_B, it is required to satisfy u″_A=u″_B. Therefore, the required condition (12) is obtained based on the equation (10) and (11).

[0078] u′_B+h_2φ2B=h_2φ2A  (12)

[0079] This equation is converted to the equation (13) to solve about u′_B, $\begin{array}{cc}\begin{array}{c}{u}_B^{\prime }=h_2{\varphi }_{2A}-h_2{\varphi }_{2B}\\ =h_2\left({\varphi }_{2A}-{\varphi }_{2B}\right)\end{array}& \left(13\right)\end{array}$

[0080] Combining the equations (4), (13) and substituting h₂=h_1Afor the combined equation yields the equation (14). $\begin{array}{cc}{h}_2·\left({\varphi }_{2A}-{\varphi }_{2B}\right)=\frac{{\lambda }_B}{{\lambda }_A}·h_{1A}·\left({\varphi }_{1B}-{\varphi }_{1A}\right){\varphi }_{2A}-{\varphi }_{2B}=-\frac{{\lambda }_B}{{\lambda }_A}·\left({\varphi }_{1B}-{\varphi }_{1A}\right)& \left(14\right)\end{array}$

[0081] Here, ${\varphi }_{1A}-{\varphi }_{1B}=\frac{1}{f_{1A}}-\frac{1}{f_{1B}}=\frac{f_{1B}-f_{1A}}{f_{1A}}·f_{1B}$

[0082] and assuming that f_1B−f_1A=Δf₁=−f₁/ν₁, and f_1A≈f₁≈f_1B,

[0083] φ_1A−φ_1B≈f₁/f₁²≈−1/(ν₁f₁)  (15)

[0084] Similarly,

[0085] φ_2A−φ_2B≈−1/(ν₂f₂)  (16)

[0086] Substitution of the equations (15) and (16) into the equation (14) yields the equation (17). $\begin{array}{cc}-\frac{1}{v_2·f_2}=\frac{{\lambda }_B}{{\lambda }_A}·\frac{1}{v_1·f_1}& \left(17\right)\end{array}$

[0087] As the equation (17) is converted and is solved about ν₂, $v_2=-\frac{{\lambda }_A}{{\lambda }_B}·\frac{f_1}{f_2}·v_1$

[0088] and when the equation (7) is substituted therein, $\begin{array}{cc}{v}_2=-\frac{{\lambda }_A}{{\lambda }_B}·\frac{f_1}{f_2}·\frac{{\lambda }_B}{{\lambda }_B-{\lambda }_A}·\frac{f_1+f_2}{f_1}=-\frac{{\lambda }_A}{{\lambda }_B-{\lambda }_A}·\frac{f_1+f_2}{f_2}& \left(18\right)\end{array}$

[0089] The equation (18) can be converted to the equation (19) because λ_B/(λ_B−λ_A) is equal to the equivalent Abbe number ν_DOEof a diffractive optical element. $\begin{array}{cc}{v}_2=-\frac{f_1+f_2}{f_2}·v_{\mathrm{DOE}}& \left(19\right)\end{array}$

[0090] The equation (19) is a condition of ν₂required to the second group 4bto direct the exiting directions of the beams of the wavelengths λ_Aand λ_Bthat are incident on the second group 4bat different incident angles.

[0091] The above descriptions are directed to the chief rays of the wavelengths λ_Aand λ_B. In addition to the conditions (8) and (19), the compensating optical system 4 is required to coincide convergence or divergence of the beams of the wavelength λ_Aand λ_Bwith each other in order to compensate the effect of the chromatic dispersion of the diffractive beam-dividing element 3 in the optical path that is the side of the object surface 10 than the compensating optical system 4. For this purpose, the beams of the wavelengths λ_Aand λ_Bshould be parallel beams when they are reflected by the second group 4b.

[0092] As shown in FIG. 3, the beams of the wavelengths λ_Aand λ_Bare focused at the points where the distances from the first group 4aare equal to the focal lengths f_1Aand f_1B, respectively. And then, the beams are incident on the second group 4bas divergent beams. Therefore, a sum of focal lengths f_1A, f_2Aand a sum of focal lengths f_1B, f_2Bare required to be equal to a principal distance f₁+f₂between the principal points of the first group 4aand the second group 4bin order to collimate the beams of the wavelengths λ_Aand λ_B. Further, the compensating optical system 4 should be afocal to both of the wavelengths λ_Aand λ_B. These conditions are expressed by the following equations (20). $\begin{array}{cc}{f}_{1A}+f_{2A}=f_{1B}+f_{2B}{f}_{2B}-f_{2A}=f_{1A}-f_{1B}\Delta f_2=-\Delta f_1-f_2/v_2=f_1/v_1& \left(20\right)\end{array}$

[0093] A combination of values of f₁and f₂that satisfies the conditions (8), (19) and (20) is found by substituting the equations (8) and (19) into the equation (20) as follows:

[0094] f₁=f₂  (21)

[0095] Further, a combination of values of ν₁and ν₂that satisfies the conditions (8), (19) and (20) is found by substituting the equation (20) into the equations (8) and (19) as follows:

[0096] ν₁=−ν₂=2ν_DOE  (22)

[0097] Next, the relationship between angular magnifications β_A, β_Bof the compensating optical system 4 at the wavelengths λ_A, λ_Band the wavelengths λ_A, λ_Bper se will be verified.

[0098] The focal length f_1Bof the first group 4aat the second wavelength λ_Bcan be converted as follows from the equation (22). $\begin{array}{cc}\begin{array}{c}{f}_{1B}=f_{1A}+\Delta f_1\\ =f_{1A}-f_1/v_1\\ =f_{1A}\left(1-1/v_1\right)\\ =f_{1A}·\left(1-\frac{{\lambda }_B-{\lambda }_A}{2{\lambda }_B}\right)\\ =f_{1A}·\left(\frac{{\lambda }_A+{\lambda }_B}{2{\lambda }_B}\right)\end{array}& \left(23\right)\end{array}$

[0099] Similarly, the focal length f_2Bof the second group 4bat the second wavelength λ_Bcan be converted as follows from the equation (22). $\begin{array}{cc}\begin{array}{c}{f}_{2B}=f_{2A}+\Delta f_2\\ =f_{2A}-f_2/v_2\\ =f_{2A}\left(1-1/v_2\right)\\ =f_{2A}·\left(1-\frac{{\lambda }_B-{\lambda }_A}{2{\lambda }_A}\right)\\ =f_{2A}·\left(\frac{{\lambda }_A+{\lambda }_B}{2{\lambda }_A}\right)\end{array}& \left(24\right)\end{array}$

[0100] Accordingly, the ratio β_A/β_Bbetween the angular magnifications β_A, β_Bof the compensating optical system 4 at the wavelengths λ_A, λ_Bcan be obtained from the equations (23) and (24). $\begin{array}{cc}\begin{array}{c}\frac{{\beta }_A}{{\beta }_B}=\frac{f_{1A}}{f_{2A}}·\frac{f_{2B}}{f_{1B}}\\ =\frac{f_{1A}·f_{2A}·\frac{{\lambda }_A+{\lambda }_B}{2{\lambda }_A}}{f_{2A}·f_{1A}·\frac{{\lambda }_A+{\lambda }_B}{2{\lambda }_B}}\\ ={\lambda }_B/{\lambda }_A\end{array}& \left(25\right)\end{array}$

[0101] The equation (25) shows that the angular magnification of the compensating optical system 4 satisfying the condition (22) is inversely proportional to the wavelength of the incident beam.

[0102] Next, a numerical example of the compensating optical system 4 that satisfies the conditions (21) and (22) will be described.

[0103] In the present example, the focal lengths f₁, f₂and the Abbe numbers ν₁, ν₂Of the first and second groups 4a, 4bare fixed as follows:

[0104] The first group 4a: f₁=120, ν₁=2ν_DOE

[0105] The second group 4b: f₂=120, ν₂=−2ν_DOE

[0106] It should be noted that a resultant total focal length f of a composite optical element, which consists of a reflecting surface and a diffractive lens structure formed thereon, is defined as a resultant total value of the component f_refdue to the power of a curved reflecting surface and the component f_difdue to the power caused by the diffracting lens structure as expressed by the equation (26). $\begin{array}{cc}\frac{1}{f}=\frac{1}{f_{\mathrm{dif}}}+\frac{1}{f_{\mathrm{ref}}}& \left(26\right)\end{array}$

[0107] Further, a resultant total Abbe number ν of such a composite optical element has the following relationship as shown in the equation (27) with the focal lengths f, f_ref, f_dif, the Abbe number of a reflecting mirror ν_ref(=∞), and an equivalent Abbe number ν_DOE. $\begin{array}{cc}\begin{array}{c}\frac{1}{f·v}=\frac{1}{f_{\mathrm{dif}}·v_{\mathrm{DOE}}}+\frac{1}{f_{\mathrm{ref}}·v_{\mathrm{ref}}}\\ =\frac{1}{f_{\mathrm{dif}}·v_{\mathrm{DOE}}}\end{array}& \left(27\right)\end{array}$

[0108] Substitution of f=f₁=120, ν=ν₁=2ν_DOEinto the equation (27) yields the diffractive component f_1difof the focal length of the first group 4a.

[0109] f_1dif=240

[0110] In the same manner, substitution of f=f₂=120, ν=ν₂=−2ν_DOEinto the equation (27) yields the diffractive component f_2difof the focal length of the second group 4b.

[0111] f_2dif=−240

[0112] On the other hand, substitution of f_dif=f_1dif=240, f=f₁=120 into the equation (26) yields the reflecting component f_1refof the focal length of the first group 4a.

[0113] f_1ref=240

[0114] In the same manner, substitution of f_dif=f_2dif=−240, f=f₂=120 into the equation (26) yields the reflecting component f_2refof the focal length of the second group 4b.

[0115] f_2ref=80

[0116] The following TABLE provides a summary of specification of the compensating optical system 4 of the numerical example.

[0117] With this setting, since the compensating optical system 4 satisfies the conditions (8), (19) and (20), the system 4 perfectly compensates an angular difference of the same order diffracted beams caused by the wavelength dependence of the diffractive beam-dividing element 3.

[0118] As described above, according to the present invention, an angular difference among the same order diffracted beams having different peak wavelengths caused by the wavelength dependence of a diffractive beam-dividing element can be compensated by the chromatic dispersion of the compensating optical system. Therefore, the same order diffracted beams diffracted at different angles can be directed in the same direction at the same position.

[0119] Accordingly, the defect such as a separation of the same order diffracted beam or a variation of the beam spot pitch due to difference of wavelength can be avoided while employing a diffractive beam-dividing element that is in no need of cementing and is able to divide the incident beam with high accuracy.

[0120] The present disclosure relates to the subject matter contained in Japanese Patent Application No. 2000-1192, filed on Jan. 7, 2000, which is expressly incorporated herein by reference in its entirety.