Patent Publication Number: US-8531747-B2

Title: Hologram, hologram data generation method, and exposure apparatus

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a hologram, a hologram data generation method, and an exposure apparatus. 
     2. Description of the Related Art 
     A projection exposure apparatus has conventionally been employed to fabricate a micro-patterned semiconductor device such as a semiconductor memory or a logic circuit by using photolithography (printing). The projection exposure apparatus projects and transfers a circuit pattern formed on a reticle (mask) onto a substrate such as a wafer via a projection optical system. 
     A resolution R of the projection exposure apparatus is given by: 
                   R   =       k   1     ×     λ   NA               (   1   )               
where λ is the exposure light wavelength, NA is the numerical aperture of the projection optical system, and k 1  is a process constant determined by, for example, a development process.
 
     The shorter the exposure light wavelength or the higher the NA of the projection optical system, the better the resolution. However, it might be difficult to further shorten the current exposure light wavelength because the transmittance of a glass material generally decreases as the exposure light wavelength shortens. It might be also difficult to further increase the NA of the projection optical system available at present because the depth of focus decreases in inverse proportion to the square of the NA of the projection optical system, and because it might be difficult to design and manufacture lenses to form a high-NA projection optical system. 
     Under the circumstances, there have been proposed resolution enhanced technologies (RETs) of improving the resolution by decreasing the process constant k 1 . One of these RETs is the so-called modified illumination method (or oblique illumination method). 
     The modified illumination method generally inserts an aperture stop, which has a light-shielding plate on the optical axis of an optical system, in the vicinity of the exit surface of an optical integrator which forms a uniform surface light source, thereby obliquely irradiating a reticle with exposure light. 
     The modified illumination method includes, for example, an annular illumination method and quadrupole illumination method that are different in the aperture shape of an aperture stop (i.e., the shape of the light intensity distribution). There has also been proposed another modified illumination method which uses a computer generated hologram (CGH) in place of an aperture stop, in order to improve the use efficiency (illumination efficiency) of the exposure light. 
     Along with an increase in the NA of the projection optical system, a polarized illumination method which controls the polarization state of exposure light is also required to increase the resolution of the projection exposure apparatus. The polarized illumination method illuminates a reticle with, for example, S-polarized light alone, which has a component in the circumferential direction of concentric circles about the optical axis. A contrast of the image to be formed might be enhanced by using the S-polarized light alone. 
     In recent years, there has been proposed a technique which exploits both the modified illumination method (the formation of a light intensity distribution having a desired shape, e.g., a quadrupolar shape) and the polarized illumination method (i.e., polarization state control). 
     For example, Japanese Patent Laid-Open No. 2006-196715 discloses a technique which implements both the modified illumination method and polarized illumination method using a light beam conversion element composed of a plurality of pairs of a form birefringence region and a diffraction region. Japanese Patent Laid-Open No. 2006-196715 describes controlling the polarization state using the form birefringence region and the shape (i.e., a reconstructed image) of the light intensity distribution at the predetermined plane using the diffraction region. The number of the pairs depends on kinds of polarization states formed on the predetermined plane. 
     U.S. Pat. No. 7,265,816 (or Japanese Patent Laid-Open No. 2006-5319) discloses a technique which can control the balance among four poles of a quadrupolar light intensity distribution typically formed by the modified illumination method and the polarized illumination method. U.S. Pat. No. 7,265,816 refers, after converting four circularly polarized lights into four linearly polarized lights different from each other with a quarter wave plate, to change the light intensity distribution at a predetermined plane by controlling the balance with using four separated CGHs which function as a diffractive optical element corresponding to each linearly polarized light. 
     A CGH design technique is disclosed in “Iterative method applied to image reconstruction and to computer-generated holograms”, OPTICAL ENGINEERING, Vol. 19, No. 3, May/June 1980, 297-305. 
     The conventional technique requires a plurality of the separated CGHs to form a reconstructed image composed of a plurality of polarization states, and the number of separated CGHs to be required depends on the number of a variety of polarization states. 
     When a plurality of CGHs combined with each other are used, an irradiance variation might occur in a reconstructed image if an optical integrator cannot sufficiently correct the intensity distribution of the incident light (for example, if the light impinges on only some of these CGHs). 
     SUMMARY OF THE INVENTION 
     The present invention provides a hologram which can reduce the irradiance variation. 
     According to an aspect of the present invention, a hologram which forms a light intensity distribution on a predetermined plane by using an incident light is provided. The hologram includes a plurality of cells configured to control both a phase of a first polarized light component, in a first polarization direction, of the incident light and a phase of a second polarized light component in a second polarization direction perpendicular to the first polarization direction. The plurality of cells are designed to form a portion, in an overlap region in which a first light intensity distribution region formed on the predetermined plane by the first polarized light component and a second light intensity distribution region formed on the predetermined plane by the second polarized light component are superposed on each other, having a polarization state different from the first and second polarized light components. A phase difference between the phase of the first polarized light component and the phase of the second polarized light component is a constant value in the portion, and the phase of the first polarized light component is diffused in the portion. 
     Further features of the present invention will become apparent from the following description of exemplary embodiments with reference to the attached drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  illustrates a hologram. 
         FIG. 1B  illustrates a plurality of cells of the hologram. 
         FIG. 2A  illustrates a first light intensity distribution region. 
         FIG. 2B  illustrates a second light intensity distribution region. 
         FIG. 2C  illustrates an overlap region. 
         FIG. 2D  illustrates a portion in the overlap region. 
         FIG. 2E  illustrates an exemplary target image. 
         FIG. 3  illustrates a flowchart of a method for generating a computer generated hologram. 
         FIG. 4  illustrates a flowchart of a method for generating a computer generated hologram whose target image is symmetric. 
         FIG. 5  is a perspective view showing a cell structure. 
         FIG. 6  is a perspective view showing a cell structure. 
         FIG. 7  is a perspective view showing a cell structure having a three-dimensional structure which generates form birefringence. 
         FIGS. 8A and 8B  illustrate light intensity distributions divided for X and Y-polarizations. 
         FIG. 8C  illustrates a phase difference between X and Y-polarizations on a predetermined plane. 
         FIGS. 8D and 8E  illustrate the phase distributions of a computer generated hologram designed for X and Y-polarizations. 
         FIGS. 8F and 8G  illustrate reconstructed images obtained by using computer generated holograms with the phase distributions shown in  FIGS. 8D and 8E , respectively. 
         FIGS. 8H and 8I  illustrate phase distributions of reconstructed images ( FIGS. 8F and 8G ), respectively. 
         FIG. 8J  illustrates a phase difference between  FIGS. 8H and 8I . 
         FIGS. 8K and 8L  are magnified figures of the left-upper part of  FIGS. 8D and 8E , respectively. 
         FIG. 8M  is a chart showing the thickness of a computer generated hologram obtained by integrating the computer generated hologram compatible with the X-polarization target image and compatible with the Y-polarized light target image. 
         FIG. 9A  illustrates an exemplary target image. 
         FIGS. 9B and 9C  illustrate light intensity distributions divided for X and Y-polarizations, respectively. 
         FIGS. 9D and 9E  illustrate phase differences between X and Y-polarizations on the predetermined plane. 
         FIGS. 9F and 9G  illustrate phase symmetries on the predetermined plane. 
         FIGS. 9H and 9I  illustrate phase distributions of a computer generated hologram designed for X and Y-polarization, respectively. 
         FIGS. 9J and 9K  illustrate reconstructed images obtained by using computer generated holograms with the phase distributions shown in  FIGS. 9H and 9I , respectively. 
         FIGS. 9L and 9M  illustrate phase distributions of reconstructed images ( FIGS. 9H and 9I ), respectively. 
         FIG. 9N  illustrates a phase difference between  FIGS. 9L and 9M . 
         FIG. 10A  illustrates an exemplary target image for a computer generated hologram. 
         FIGS. 10B and 10C  illustrate light intensity distributions divided for X and Y-polarizations, respectively. 
         FIGS. 10D and 10E  illustrate phase differences between X and Y-polarizations on the predetermined plane. 
         FIGS. 10F and 10G  illustrate phase symmetries on the predetermined plane. 
         FIGS. 10H and 10I  illustrate phase distributions of a computer generated hologram designed for X and Y-polarization, respectively. 
         FIGS. 10J and 10K  illustrate reconstructed images obtained by using computer generated holograms with the phase distributions shown in  FIGS. 10H and 10I , respectively. 
         FIGS. 10L and 10M  illustrate phase distributions of reconstructed images ( FIGS. 10H and 10I ), respectively. 
         FIG. 10N  illustrates a phase difference between  FIGS. 10L and 10M . 
         FIG. 11  illustrates the arrangement of an exposure apparatus according to one aspect of the present invention. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     Exemplary embodiments according to the present invention will be described below with reference to the attached drawings. The same reference numerals denote the same members throughout the drawings, and a repetitive description thereof will not be given. 
       FIG. 1A  illustrates a hologram  100 . The hologram  100  is, for example, a computer generated hologram. 
     As shown in  FIG. 1A , the hologram  100  forms a light intensity distribution (i.e., reconstructed image) LI on a predetermined plane PS, for example, located at an aperture position of an illumination system for a lithography tool, by modulating a phase distribution of the wavefront of the incident light. A shape of the light intensity distribution LI is not limited to the shape illustrated in  FIG. 1A . 
     The hologram  100  controls a phase of a first polarized light component (e.g., a first linearly polarized light component IL 1 ), in a first polarization direction (e.g., X-polarization), of an incident light. The first linearly polarized light component IL 1  forms a first light intensity distribution region on the predetermined plane PS. 
     The hologram  100  also controls a phase of a second polarized light component (e.g., a second linearly polarized light component IL 2 ), in a second polarization direction (e.g., Y-polarization) perpendicular to the first polarization direction. The second linearly polarized light component IL 2  forms a second light intensity distribution region on the predetermined plane PS. 
     The hologram  100  includes a plurality of cells  110  (e.g., a collection of cells  110 ) as shown in  FIG. 1B  to control phases of the first and second linearly polarized light components. 
     The plurality of cells might be designed so that there is an overlap region in which the first light intensity distribution region formed on the predetermined plane PS and the second light intensity distribution region formed on the predetermined plane PS are superposed on each other. 
     For example, when the first linearly polarized light component IL 1  forms the first light intensity distribution region LI 1  as shown in  FIG. 2A , and the second linearly polarized light component IL 2  forms the second light intensity distribution region LI 2  as shown in  FIG. 2B , an overlap region MA is shown in  FIG. 2C . Hereinafter, for ease of reference, MA 1 , MA 2 , MA 3 , and MA 4  are referred to as a first, second, third, and fifth quadrant, respectively. 
     The plurality of cells  110  might be designed so that a portion PA is formed in the overlap region MA. In  FIG. 2D , an overlap region MA 1  is divided into three portions of PA 1 , PA 2 , and PA 3  in light of a polarization state. And an overlap region MA 2  is divided into PA 4 , PA 5 , and PA 6 , an overlap region MA 3  is divided into PA 7 , PA 8 , and PA 9 , and an overlap region MA 4  is divided into PA 10 , PA 11 , and PA 12 . And at least one of the portions from PA 1  to PA 12  shown in  FIG. 2D  has a polarization state different from the first and second linearly polarized light components. It is possible that all of the portions in the overlap region MA have polarization states different from the first and second linearly polarized light components. 
     A phase difference between the first linearly polarized light component and the second linearly polarized light component in each portion (e.g., PA 1 ) might be a constant value. A value of the phase difference in each portion might be similar to or different from each other. The value of the phase difference can be selected from, but is not limited to, −π/2, 0, π/2, π, π/4, −π/4, 3π/4, and −3π/4. 
     The plurality of cells  110  might be designed to form such portions in the overlap regions (e.g., MA 1 ). A phase difference between the phase of the first linearly polarized light component and the phase of second linearly polarized light component might be a constant value in the each portion. It is possible that the phase difference of each of the portions adjacent to each other might be different from or similar to each other. 
     The phase of the first linearly polarized light component is diffused in the portion (e.g., PA 1 ). The diffused phase means that a phase distribution on the predetermined plane PS has a random phase. In other words, the diffused phase means that the phase distribution has a plurality of spatial frequencies. Regarding a reconstructed image, the phase distribution having the plurality of spatial frequencies means that the reconstructed image is formed by a light emitted from every point of the hologram. When the phase difference between the phase of the first polarized light component and the phase of second polarized light component is a constant value in a portion, if the phase of the first polarized light component is diffused in the portion, the phase of the second polarized light component is also diffused in the portion. 
     To form a polarization state different from the first and second linearly polarized light components in the portion, the hologram  100  might be designed in consideration of a phase difference between the first and second linearly polarized light components on the predetermined plane PS, or amplitude ratio (intensity rate) between the first and second linearly polarized light components on the predetermined plane PS, or both. 
     For example, when an incident light is a linearly polarized light in a polarization direction of +45° with respect to an X-axis, the polarization direction of the linearly polarized light is changed in a range from more than 0° to less than +90° with respect to an X-axis by controlling the amplitude ratio (the intensity rate). And also, the polarization direction of the linearly polarized light is changed in a range from less than 0° to more than −90° with respect to an X-axis by giving the phase difference of π and controlling the amplitude. 
     When an incident light is a linearly polarized light in a polarization direction of +45° with respect to an X-axis, the polarization direction of the linearly polarized light is converted into a circularly polarized light by giving the phase difference of π/2 or −π/2 between the first and second linearly polarized light components on the predetermined plane PS. 
       FIG. 2E  shows an example of the light intensity distribution LI as explained in an example 1. Each polarization state on the predetermined plane PS is indicated by an arrow in  FIG. 2E . 
     Shapes of the first light intensity distribution region LI 1  shown in  FIG. 2A  and the second light intensity distribution LI 2  shown in  FIG. 2B  is changed according to the light intensity distribution LI. Polarization states of portions PA 1  to PA 12  in the overlap region MA isn&#39;t limited to the disclosed types as shown in  FIG. 2E , and might also be changed. 
     A data generation method used to manufacture a computer generated hologram  100  is described next. 
     First, a desired light intensity distribution LI on the predetermined plane PS in  FIG. 1  is divided into a first light (e.g., X-polarization) intensity distribution region LI 1  formed by a first linearly polarized light component and a second light (e.g., Y-polarization) intensity distribution region LI 2  formed by a second linearly polarized light component perpendicular to the first linearly polarized light component. 
     Second, a phase difference between a phase of the first linearly polarized light component in the first light intensity distribution region LI 1  and a phase of the second linearly polarized light component in the second light intensity distribution region LI 2 . The phase difference is determined in accordance with a relationship between a polarization direction of the incident light and a polarization state of the light intensity distribution LI to be formed on the predetermined plane PS so that a portion which has a designed polarization state is formed in the overlap region MA. 
     Third, a first hologram data to form the first light intensity distribution region LI 1  and a second hologram data to form the second light intensity distribution region LI 2  is generated under an admissibility condition of diffusing the phase of the first linearly polarized light component (e.g., X-polarization) while maintaining the phase difference which is determined. The first and second hologram data can be called as a hologram data for X and Y-polarizations. When a phase distribution of the first linearly polarized light component on the predetermined plane PS is diffused under maintaining the phase difference, a phase distribution of the second linearly polarized light component on the predetermined plane PS is also diffused. The phase of the first linearly polarized light component might be diffused in the first light intensity distribution region LI 1  entirely, and a diffused area might be optionally limited to the overlap region MA. 
     Finally, the hologram data for the first and second linearly polarized light components (e.g., X and Y-polarizations) are integrated with each other. 
     When the second light intensity distribution region is related to the first light intensity distribution region in symmetry for a light intensity distribution and a phase distribution, the second hologram data might be easily obtained in consideration of the symmetry after the first hologram data is generated. 
     The hologram data generation method is described in detail in examples below. 
     The hologram  100  which gives different phase distributions to the wavefronts of the first linearly polarized light component (e.g., X-polarization) and the second linearly light component (e.g., Y-polarized light) is explained in detail below. 
       FIG. 5  is a perspective view showing a cell structure which forms the hologram  100 . As shown in  FIG. 5 , the hologram  100  might include a plurality of rectangular cells  110 . As will be described later, the dimensions and arrangements of the plurality of cells  110  are set such that the X-polarization is in phase with, or π out of phase from the Y-polarized light, in each overlap region MA in which the first light intensity distribution LI 1  and the second light intensity distribution LI 2  are superposed on each other. 
     To give different phase distributions to the wavefronts of X-polarization and Y-polarized light, the hologram  100  might independently control the wavefronts in the respective polarization directions. For example, in order to form two phase levels for each of the X- and Y-polarizations, it is possible to give binary phases to the wavefronts in each of the two polarization directions. For this purpose, four types of cell structures might be included in the hologram  100 . Each of cells  110   a  to  110   d  shown in  FIG. 5  has a cell structure of one of these four types. The hologram  100  is formed by arraying cells  110  of four types in a tetragonal lattice pattern. 
     As shown in  FIG. 5 , the plurality of cells  110  include anisotropic media  112  which change the polarization state of an incident light, and isotropic media  114  which do not change the polarization state of the incident light. More specifically, the cell  110   a  includes an anisotropic medium  112  alone, the cell  110   b  includes an anisotropic medium  112  and isotropic medium  114 , and each of the cells  110   c  and  110   d  includes an isotropic medium  114  alone. That the isotropic media  114  do not change the polarization state of the incident light means herein that they do not change the polarization state of the incident light compared to the anisotropic media  112 . For this reason, this embodiment assumes that a medium in which the difference between their refractive indices with respect to X-polarization and Y-polarized light is 0 (inclusive) to 0.001 (inclusive) in an isotropic medium. 
     The plurality of cells  110  might include an anisotropic cell which has an anisotropic medium configured to change a polarization state of the incident light as described above. 
     The steps among the cells  110   a  to  110   d  in the Z direction can be represented by using a refractive index n of the isotropic medium  114 , a refractive index n x  of the anisotropic medium  112  with respect to X-polarization, and a refractive index n y  of the anisotropic medium  112  with respect to Y-polarized light. This embodiment exemplifies a case in which n=n x &gt;n y . 
     To configure a two-phase-level computer generated hologram as the hologram  100 , a cell to shift a phase of π is necessary. To attain this state, thicknesses H 1  of the anisotropic medium  112  of the cell  110   a  and the isotropic medium  114  of the cell  110   c  need satisfy: 
     
       
         
           
             
               
                 
                   
                     H 
                     1 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
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                         λ 
                         
                           
                             n 
                             x 
                           
                           - 
                           
                             n 
                             y 
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         λ 
                         
                           n 
                           - 
                           
                             n 
                             y 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     A thickness H 2  of the isotropic medium  114  of the cell  110   b , that is, a difference H 2  between the thickness of the cell  110   c  and that of the cell  110   b  or  110   d  (a difference H 2  between the thickness of the isotropic medium  114  of the cell  110   c  and that of the isotropic medium  114  of the cell  110   d ) need satisfy: 
     
       
         
           
             
               
                 
                   
                     H 
                     2 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       λ 
                       
                         n 
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Assuming X-polarization which impinges on the cell  110   c  as a reference, X-polarization which impinges on the cell  110   a  is in phase with the reference. Also, assuming Y-polarized light which impinges on the cell  110   c  as a reference, Y-polarized light which impinges on the cell  110   a  is π out of phase from the reference. 
     Assuming X-polarization which impinges on the cell  110   c  as a reference, X-polarization which impinges on the cell  110   b  is π out of phase from the reference. Also, assuming Y-polarized light which impinges on the cell  110   c  as a reference, Y-polarized light which impinges on the cell  110   b  is in phase with the reference. 
     Assuming X-polarization which impinges on the cell  110   c  as a reference, X-polarization which impinges on the cell  110   d  is π out of phase from the reference. Also, assuming Y-polarized light which impinges on the cell  110   c  as a reference, Y-polarized light which impinges on the cell  110   d  is also π out of phase from the reference. 
     In this manner, the computer generated hologram can give binary phases to the wavefronts in the two polarization directions respectively by using the cell structures of four types (cells  110   a  to  110   d ) shown in  FIG. 5 . In other words, the cell structures of four types shown in  FIG. 5  exemplify four combinations of phases given to the wavefronts of the X-polarization and Y-polarized light, that is, (0, π), (π, 0), (0, 0), and (π, π). 
     A case in which n x =n=1.6 and n y =1.4 will be exemplified as a concrete numerical example. In this case, letting X be the wavelength of the incident light, the thicknesses H 1  and H 2  are 2.5λ and 0.833λ, respectively, that fall within few multiples of the wavelength λ. These values are realistic as the thicknesses of the cells of a computer generated hologram. 
     In one example, the anisotropic media  112  might include an anisotropic layer. The anisotropic media  112  of all cells may have identical optic axis directions. If the anisotropic media  112  of all the cells shown in  FIG. 5  have identical optic axis directions, at least one cell (the cell  110   b  in this embodiment) includes an anisotropic medium  112  formed by the anisotropic layer, and an isotropic medium  114  formed by an isotropic layer. 
     The optic axis means herein an axis along a direction in which, because the refractive indices in all directions perpendicular to a propagation direction of an incident light are constant in the anisotropic medium  112 , no birefringence occurs even if non-polarized light impinges on the anisotropic cell so that ordinary and extraordinary rays match each other or have a minimum deviation if any. 
     In another example, the anisotropic media  112  might be included in an anisotropic cell. The anisotropic media  112  of respective cells may have different optic axis directions.  FIG. 6  is a perspective view showing a cell structure which forms a hologram  100  including anisotropic cells. If the cells of the hologram  100  do not have identical optic axis directions, that is, the optic axis direction of each cell is selected freely, each cell of four types can be formed from an anisotropic medium  112  or isotropic medium  114  alone. In other words, each cell may not be formed by a combination of an anisotropic medium  112  or isotropic medium  114 . In this case, the hologram  100  includes a first anisotropic cell  110   a   0 , a second anisotropic cell  110   b   0 , a first isotropic cell  110   c   0 , and a second isotropic cell  110   d   0 , as shown in  FIG. 6 . The first anisotropic cell  110   a   0  and second anisotropic cell  110   b   0  are made of, for example, birefringent materials. The direction of an optic axis OA 1  of the first anisotropic cell  110   a   0  is different from that of an optic axis OA 2  of the second anisotropic cell  110   b   0 , and, for example, they intersect at right angles with each other, as shown in  FIG. 6 . 
     As described above, the plurality of cells  100  might include anisotropic cells including an anisotropic medium configured to change a polarization state of the incident light, and isotropic cells including an isotropic medium configured not to change a polarization state of the incident light. 
     Note that a function of setting light components in two polarization directions of the incident light to be in phase with, or π out of phase from each other, in the cells  110   a   0  to  110   d   0  of four types of the computer generated hologram  100  shown in  FIG. 6  is the same as in the cells  110   a  to  110   d  of four types of the computer generated hologram  100  shown in  FIG. 5 . 
     Thicknesses (the thicknesses in the Z direction) h 1  of the first anisotropic cell  110   a   0  and second anisotropic cell  110   b   0 , a thickness h 2  of the first isotropic cell  110   c   0 , and a thickness h 3  of the second isotropic cell  110   d   0  can be represented by using the following three refractive indices (first to third refractive indices). The first refractive indices are a refractive index n E  of the first anisotropic cell  110   a   0  with respect to X-polarization, and a refractive index n E  of the second anisotropic cell  110   b   0  with respect to Y-polarized light. The second refractive indices are a refractive index no of the first anisotropic cell  110   a   0  with respect to Y-polarized light, and a refractive index no of the second anisotropic cell  110   b   0  with respect to X-polarization. The third refractive indices are refractive indices n of the first isotropic cell  110   c  and second isotropic cell  110   d . This embodiment exemplifies a case in which n O &gt;n E . 
     To configure a two-step computer generated hologram  100 , a phase shift of π is necessary. To attain this state, the thicknesses h 1  of the first anisotropic cell  110   a   0  and second anisotropic cell  110   b   0  need satisfy: 
     
       
         
           
             
               
                 
                   
                     h 
                     1 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       λ 
                       
                         
                           n 
                           O 
                         
                         - 
                         
                           n 
                           E 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     To form a wavefront matching the one obtained at the refractive index no of the first anisotropic cell  110   a   0  with respect to Y-polarized light and the refractive index no of the second anisotropic cell  110   b   0  with respect to X-polarization, the thickness h 2  of the first isotropic cell  110   c   0  need satisfy: 
     
       
         
           
             
               
                 
                   
                     h 
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                           n 
                           O 
                         
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                         1 
                       
                       
                         n 
                         - 
                         1 
                       
                     
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                       h 
                       1 
                     
                   
                 
               
               
                 
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                   5 
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     Also, to form a wavefront matching the one obtained at the refractive index n E  of the first anisotropic cell  110   a   0  with respect to X-polarization and the refractive index n E  of the second anisotropic cell  110   b   0  with respect to Y-polarized light, the thickness h 3  of the second isotropic cell  110   d   0  need satisfy: 
     
       
         
           
             
               
                 
                   
                     h 
                     3 
                   
                   = 
                   
                     
                       
                         
                           n 
                           E 
                         
                         - 
                         1 
                       
                       
                         n 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       h 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     A case in which n O =1.6, n E =1.4, and n=1.5 will be exemplified as a concrete numerical example. In this case, letting λ be the wavelength of the incident light, the thicknesses h 1 , h 2 , and h 3  are 2.5λ, 2λ, and 3λ, respectively, that fall within few multiples of the wavelength λ. These values are realistic as the thicknesses of the cells of a computer generated hologram. 
     Each of the first anisotropic cell  110   a   0  and second anisotropic cell  110   b   0  may be formed from a diffraction grating (three-dimensional structure) which generates form birefringence. In other words, the anisotropic medium may include one of a birefringent material and a three-dimensional structure which generates form birefringence.  FIG. 7  is a perspective view showing a cell structure which forms a hologram including anisotropic cells formed from diffraction gratings which generate form birefringence. The hologram  100  shown in  FIG. 7  includes a first anisotropic cell  110   a   1 , second anisotropic cell  110   b   1 , first isotropic cell  110   c   1 , and second isotropic cell  110   d   1 . 
     Each of the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1  is formed from a diffraction grating which generates form birefringence, as described above. Each of the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1  is formed, for example, from a one-dimensional diffraction grating having a periodic structure with a pitch P smaller than the wavelength of the incident light in order to prevent the generation of diffracted light components of orders other than the 0th order. 
     The first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1  are configured such that the direction of the pitch of the periodic structure of the first anisotropic cell  110   a   1  is different from that of the second anisotropic cell  110   b   1 . This makes it possible to attain a cell which advances the wavefront of X-polarization from that of Y-polarized light, and a cell which retards the wavefront of X-polarization from that of Y-polarized light. 
     Japanese Patent Laid-Open No. 2006-196715 discloses a diffraction grating made of quartz as an example of the diffraction grating which generates form birefringence. According to Japanese Patent Laid-Open No. 2006-196715, when quartz has a refractive index of 1.56 with respect to a wavelength of 193 nm, and the duty ratio of the diffraction grating in the form birefringence region is 1:1 (=0.5), a refractive index n ⊥  of the diffraction grating in the direction of the pitch is 1.19, and a refractive index n II  of the diffraction grating in a direction perpendicular to the pitch is 1.31. 
     Even when each anisotropic cell is formed from a diffraction grating which generates form birefringence, thicknesses h 1 ′ of the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1  need satisfy equation (4) upon substituting h 1 ′ for h 1 . Likewise, a thickness h 2 ′ of the first isotropic cell  110   c   1  need satisfy equation (5) upon substituting h 2 ′ for h 2 , and a thickness h 3 ′ of the second isotropic cell  110   d   1  need satisfy equation (6) upon substituting h 3 ′ for h 3 . 
     A case in which the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1  are made of quartz compatible with a wavelength λ=193 nm will be exemplified as a concrete numerical example. The refractive index of the quartz is assumed to be 1.56, a refractive index n ⊥  of the diffraction grating in the direction of the pitch is assumed to be 1.19, and a refractive index n II  of the diffraction grating in a direction perpendicular to the pitch is assumed to be 1.31, as described above. To obtain the thicknesses h 1 ′ of the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1 , the thickness h 2 ′ of the first isotropic cell  110   c   1 , and the thickness h 3 ′ of the second isotropic cell  110   d   1  using equations (4) to (6), it is only necessary to substitute n ⊥  for n E  and substitute n II  for n O . In this case, the thicknesses h 1 ′ of the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1  are 4.17λ from equation (4). This value is equal to the thickness of a λ/2 plate as one type of wave plate. From equations (5) and (6), the thickness h 2 ′ of the first isotropic cell  110   c   1  and the thickness h 3 ′ of the second isotropic cell  110   d   1  are 1.41λ and 2.31λ, respectively, that are smaller than the thicknesses h 1 ′ of the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1 . In this manner, the thicknesses h 1 ′ of the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1 , the thickness h 2 ′ of the first isotropic cell  110   c   1 , and the thickness h 3 ′ of the second isotropic cell  110   d   1  fall within the thickness of the λ/2 plate. 4.17λ is realistic as the thickness of the cell of a computer generated hologram. 
     This embodiment has exemplified a two-phase-level computer generated hologram as the hologram  100 , so the hologram  100  can be formed from anisotropic cells having one thickness and isotropic cells having two thicknesses. However, the present invention is not particularly limited to the two-phase-level computer generated hologram, and is applicable to a computer generated hologram of a multiple of steps more than two phase levels which is formed from anisotropic cells having more than one thickness, and isotropic cells having more than two thicknesses. In this embodiment, the one-dimensional diffraction grating is used as the diffraction grating which generates form birefringence, and a two-dimensional diffraction grating may be used. The phase of the computer generated hologram is not limited to quantized levels (i.e., discrete levels), and the phase of the hologram might be changed continuously by changing the thickness of each cell continuously. 
     Note that although this embodiment has exemplified only the cell structure of the hologram  100 , it may not be easy to bond materials having different properties, as shown in  FIGS. 5 ,  6 , and  7 . Furthermore, if each anisotropic cell is formed from a diffraction grating which generates form birefringence, as shown in  FIG. 7 , the diffraction grating floats in the air, which might be hard to maintain. In view of this, in practice, the above-described anisotropic cells and isotropic cells might be formed on a substrate made of, for example, quartz. 
     In the  FIG. 5 , a direction along the optical axis for anisotropic media  112  of every cell coincides with each other. Therefore, all anisotropic media  112  in hologram  100  might be formed on a substrate made of the same anisotropic media, and all isotropic media  114  might be formed on a substrate made of the same isotropic media. More specifically, the anisotropic substrate is located at the upper-side of the plurality of cells  110 , and the isotropic substrate is located at the bottom-side of the plurality of cells  110 . In the  FIG. 5 , the anisotropic media and the isotropic media are in contact with each other, but they also might be separated from each other along Z-direction. 
     In this embodiment, it is described that the incident light is linearly polarized light including X-polarized light and Y-polarized light having the same amplitude assuming a case in which the hologram  100  forms a light intensity distribution including X-polarized light and Y-polarized light at the same ratio as in annular illumination. A hologram optionally can be designed to be compatible with the formation of a light intensity distribution including X-polarization and Y-polarization at different ratios by using polarized light including X-polarized light and Y-polarized light having different amplitudes as the incident light in order to obtain high efficiency. A partially coherent light can be used as the incident light. Circularly polarized light or elliptic polarization can also be used as the incident light, and in that case the thickness of each cell of the hologram  100  might be required to change. 
     Example 1 
     A detailed design example of a computer generated hologram as the hologram  100  will be explained below with reference to a flowchart of  FIG. 3 . 
     An example to design the light intensity distribution LI shown in  FIG. 2E  will be explained. 
     This target image illustrated in  FIG. 2E  includes a right and left circularly polarized light and a linearly polarized light along a circumferential direction of concentric circles (i.e., corresponding to S-polarization). The S-polarization means each pixel on the predetermined plane is formed by the linearly polarized light, and the direction of the polarized light for each pixel is along the circumferential direction of concentric circles. 
     Referring to  FIG. 3 , in step S 1002 , the target image is divided into a light intensity distribution formed by X-polarization and another light intensity distribution formed by Y-polarization. More specifically, values are applied into each light intensity distribution region for X and Y-polarizations LI 1  and LI 2  in  FIGS. 2A and 2B .  FIGS. 8A and 8B  show the light intensity distribution divided for X and Y-polarizations respectively. In  FIGS. 8A and 8B , a color closer to white indicates a higher intensity, and that closer to black indicates a lower intensity. 
     In the portion formed by S-polarization, the method to divide the light intensity distribution depends on the angle of S-polarization. In the portion formed by circularly polarized light, the light intensity distribution is divided on even for X and Y-polarizations. 
     In step S 1004 , the phase difference between X and Y-polarizations on the predetermined plane is determined. More specifically, values are applied into each portion of MA in  FIG. 2D .  FIG. 8C  shows the phase difference under the condition where the incident light is a linearly polarized light and the polarization direction with respect to the X-axis is +45°. The phase difference in the portion formed by right circularly polarized light is π/2. The phase difference in the portion formed by left circularly polarized light is −π/2. The phase difference in the portion formed by S-polarization in the second and fourth quadrant is 0 because the S-polarization can be formed by adjusting the ratio of intensity for X and Y-polarizations of the incident light. The phase difference in the portion formed by S-polarization in the first and third quadrant is π because the S-polarization cannot be formed only by adjusting the ratio of intensity for X and Y-polarizations of the incident light. Then, to reverse the phase of one of the X and Y-polarizations included in the incident light is required. 
     Therefore, the target image ( FIG. 2E ) includes four portions whose constant value as the phase difference is selected from 0, π/2, π and −π/2 as shown in  FIG. 8C . 
     In step S 1006 , hologram data for X and Y-polarizations are generated under an admissibility condition of diffusing the phase of X-polarization while maintaining the phase difference as shown in  FIG. 8C . Maintaining the phase difference might be required only in the overlap region. There is no limitation for the phase difference in an area other than the overlap region. Therefore any phase difference can be chosen in the area other than the overlap region. 
     The phase difference in the overlap region is limited to maintain the determined value, but a phase itself is not limited. Therefore hologram data can be generated with optimization using iterative Fourier transform (i.e., Gerchberg-Saxton algorithm). More specifically, the admissibility condition of diffusing the phase means that random distribution might be used for the initial data for hologram data. In each iterative calculation step, hologram data for X and Y-polarizations are generated separately, then the phase in the overlap region on the predetermined plane might be shifted to maintain the phase difference condition shown in  FIG. 8C . The shift of the phase might be executed so that an amount of shifting the phase for X and Y-polarizations becomes smaller. It might not be required to maintain the condition strictly at the initial period for optimization, and shifting the phase to a direction according to the determined condition might be required. 
       FIGS. 8D and 8E  show one example of generated hologram data for X and Y-polarizations, that is the  FIGS. 8D and 8E  show phase distributions of the holograms. An area of white in  FIGS. 8D and 8E  indicates the phase is π, and the other area indicates the phase is 0.  FIGS. 8F and 8G  show reconstructed images with generated hologram data ( FIGS. 8D and 8E .)  FIGS. 8F and 8G  correspond to the light intensity distribution divided for X and Y-polarizations ( FIGS. 8A and 8B .) In  FIGS. 8F and 8G , a color closer to white indicates a higher intensity, and that closer to black indicates a lower intensity. 
       FIG. 8H  shows the phase distribution on the predetermined plane for X-polarization. The phase distribution is diffused in each portion, in each overlap region and in the light intensity distribution region for X polarization. And also the phase distribution for Y-polarization is diffused as shown in  FIG. 8I . 
       FIG. 8J  shows the phase difference between X and Y-polarizations on the predetermined plane ( 8 H and  8 I). Considering a phase periodic property (i.e., π is equal to −π), it is understood that the phase difference illustrated in  FIG. 8J  corresponds to the phase difference illustrated in  FIG. 8C . 
     The phase difference shown in  FIG. 8C  is the condition for generating the hologram data. Therefore, the phase difference calculated from the actual generated hologram data has an error as shown in  FIG. 8J . 
     The phase other than the overlap region is replaced with 0 in  FIGS. 8H ,  8 I, and  8 J in order to emphasize the phase in the overlap region. Actually, the phase other than the overlap region is composed of not only 0 but also any other values. 
     The diffused phase on the predetermined plane as shown in  FIGS. 8H and 8I  means that the reconstructed image is formed by a light emitted from every point of the hologram. In the case that the reconstructed image is formed by the light emitted from every point of the hologram, image quality can be higher due to the high diffractive efficiency. The diffused phase in the computer generated hologram can be realized by controlling the phase difference and not controlling the phases of X- and Y-polarizations in the design method. 
     In step S 1008 , the hologram data for X and Y-polarizations generated in step S 1006  ( FIGS. 8D and 8E ) are integrated. 
       FIGS. 8K and 8L  are magnified figures of the left-upper part of  FIGS. 8D and 8E , respectively.  FIG. 8M  illustrates the thickness of the structure to realize the phase distribution obtained by integrating  FIGS. 8K and 8L . In  FIG. 8M , reference numerals  110   a   1 ,  10   b   1 ,  110   c   1 , and  110   d   1  correspond to cell structures shown in  FIG. 7 , respectively. In other words,  FIG. 8M  is a chart showing the thickness of a computer generated hologram which integrates the computer generated hologram compatible with the X-polarization target image ( FIG. 8K ), and that compatible with the Y-polarized light target image ( FIG. 8L ). Although the computer generated hologram shown in  FIG. 8M  has the cell structure shown in  FIG. 7 , it may optionally have the cell structure shown in  FIG. 5  or  6 . 
     In  FIG. 8M , the density represents the thickness of each cell (in the Z direction). A color closer to white indicates a larger thickness, and that closer to black indicates a smaller thickness. The numerical values shown in  FIG. 8M  indicate the thicknesses of the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1 , and the thicknesses of the first isotropic cell  110   c   1  and second isotropic cell  110   d   1  in the computer generated hologram  100  (unit: μm). Note that the numerical values shown in  FIG. 8M  exemplify a case in which the first anisotropic cell  110   a   1  and second anisotropic cell  110   b   1  are made of quartz having a refractive index of 1.56 with respect to a wavelength of 193 nm. 
     The structure of the whole hologram area can be generated by the same process. 
     There are four combinations of the phases of the X-polarization and Y-polarized light in the computer generated hologram, that is, (0, π), (π, 0), (0, 0), and (π, π). The cell structure shown in  FIG. 5 ,  6 , or  7  can be used for a computer generated hologram compatible with these four phase combinations. In other words, a computer generated hologram which forms the target image shown in  FIG. 2E  has the cell structure shown in  FIG. 5 ,  6 , or  7 . 
     The cell structure of a computer generated hologram compatible with four phase combinations will be shown in detail. For example, if the combination of the phases of the X-polarization and Y-polarized light is (0, π), the cell  110   a  shown in  FIG. 5 , the cell  110   a   0  shown in  FIG. 6 , or the cell  110   a   1  shown in  FIG. 7  is adopted. If the combination of the phases of the X-polarization and Y-polarized light is (π, 0), the cell  110   b  shown in  FIG. 5 , the cell  110   b   0  shown in  FIG. 6 , or the cell  110   b   1  shown in  FIG. 7  is adopted. If the combination of the phases of the X-polarization and Y-polarized light is (0, 0), the cell  110   c  shown in  FIG. 5 , the cell  110   c   0  shown in  FIG. 6 , or the cell  110   c   1  shown in  FIG. 7  is adopted. If the combination of the phases of the X-polarization and Y-polarization is (π, π), the cell  110   d  shown in  FIG. 5 , the cell  110   d   0  shown in  FIG. 6 , or the cell  110   d   1  shown in  FIG. 7  is adopted. 
     When a hologram is formed by combining a plurality of CGHs such as described in the background of the present invention, an irradiance variation might occur in the reconstructed image if the optical integrator cannot sufficiently correct the intensity distribution of the incident light (for example, if the light impinges on only some of these CGHs.) According to the example 1, the illumination variation can be decreased. 
     When a plurality of separated CGHs are combined, unnecessary diffracted light might be generated due to structural discontinuity at the boundary between the separated-CGHs. According to the example 1, a deterioration of the reconstructed image due to the unnecessary diffracted light can be reduced. 
     Example 2 
     A detailed design example of the hologram  100  is described next with reference to a flowchart of  FIG. 4 . The light intensity distribution LI including phase distribution is line symmetric in the example 2. 
     An example that is line symmetric with respect to the line y=x (wherein x means the polarization direction of the X-polarization IL 1 , and γ means the second polarization direction of Y-polarization IL 2 ) is described here. The hologram can be designed so that a phase distribution of the Y-polarization intensity distribution is equal to a phase distribution which is realized by flipping a phase distribution of the X-polarization IL 1  intensity distribution along the axis. In other words, the hologram data for Y-polarization can be obtained by flipping the hologram data for X-polarization with respect to the line of Y=X. 
     The example describes y=x as the line symmetric axis. When the line symmetric axis is y=−x, the hologram will be designed by using the similar technique. 
       FIG. 9A  shows the target image which has the line symmetric along the axis y=x. This target image is formed by S-polarization. 
     Referring to  FIG. 4 , in step S 2002 , the target image is divided into a light intensity distribution formed by X-polarization and another light intensity distribution formed by Y-polarization.  FIGS. 9B and 9C  show the light intensity distribution divided for X and Y-polarizations. 
     In step S 2004 , the phase difference between X and Y-polarizations on the predetermined plane is determined.  FIG. 9D  shows the phase difference under a condition where the incident light is a linearly polarized light and the polarization direction with respect to the X-axis is +45°. The phase difference in the second and fourth quadrant is 0, and the phase difference in the first and third quadrant is π. 
     In step S 2006 , phase symmetry between a phase distribution of the X-polarization and a phase distribution of the Y-polarization is determined.  FIG. 9E  is a more general figure of the phase difference of the target image formed by S-polarization as shown in  FIG. 9D . After generating hologram data for X-polarization, hologram data for Y polarization is obtained by flipping the generated hologram data for X polarization along the axis Y=X.  FIG. 9F  shows one example of phase symmetry for X-polarization under considering the above.  FIG. 9G  shows phase symmetry for Y-polarization obtained by flipping  FIG. 9F . The difference between  FIGS. 9F and 9G  corresponds to  FIG. 9E . 
     In step S 2008 , hologram data for X-polarization are generated under an admissibility condition of diffusing the phase of X-polarization while maintaining the phase symmetry shown in  FIG. 9F . The above described method can be used to generate the hologram data. In each iterative calculation step, the phase in the overlap region on the predetermined plane might be shifted to maintain the phase symmetry condition. The shift of the phase might be executed so that an amount of shifting the phase becomes smaller. Two possible methods for calculating the diffuser could be a phase-mean method and a complex-amplitude-mean method. The input to both methods is the un-normalized diffuser complex-amplitude coming from the present loop of the diffuser-optimization algorithm. The output is the single diffuser phase that will be forwarded to the next loop. The process continues until the loop terminates. In the phase-mean method, the mean and difference of the two input phases are calculated. If the difference is less than Pi, the mean is used as the output phase. If the difference is greater than Pi, Pi is added to or subtracted from the mean so that the output phase is between −Pi and Pi. In the complex-amplitude-mean method, the diffuser phase is calculated from the mean of the two input complex-amplitudes. The output phase is the phase of the mean complex-amplitude. 
       FIG. 9H  shows one example of generated hologram data for X-polarization.  FIG. 9J  shows reconstructed images obtained by using generated hologram data ( FIG. 9H ).  FIG. 9J  corresponds to the light intensity distribution for X-polarization in shown  FIG. 9B .  FIG. 9L  shows phase distributions for X-polarization on the predetermined plane. The phase distributions are diffused in each portion, each overlap region and the light intensity distribution region for X polarization.  FIG. 9L  corresponds to the phase symmetry  FIG. 9F . 
     In step S 2010 , hologram data for Y-polarization is obtained by flipping the hologram data for X-polarization shown in  FIG. 9H .  FIG. 9I  shows the hologram data for Y-polarization.  FIG. 9K  shows reconstructed images obtained by using the generated hologram data shown in  FIG. 9I .  FIG. 9K  corresponds to the light intensity distribution for Y-polarization shown in  FIG. 9C .  FIG. 9M  shows phase distributions for Y-polarization on the predetermined plane.  FIG. 9N  shows the phase difference between X and Y-polarizations on the predetermined plane ( 9 L and  9 M).  FIG. 9N  corresponds to the phase difference between the phases for X- and Y-polarization shown in  FIG. 9D . 
     In step S 2012 , the hologram data for X and Y-polarizations generated in step S 2008  and S 2010  shown in  FIGS. 9H and 9I  are integrated with each other. The above described method can be used to generate the hologram data. 
     The phase state in the overlap region might be determined on the basis of a light intensity ratio between X and Y-polarizations. It is required to consider speckles for X and Y-polarizations to maintain the light intensity ratio because constructed images obtained by using the generated hologram is formed by speckles. The speckles for X and Y-polarizations might be similar to each other because the hologram data for X and Y-polarizations are symmetric each other. The speckles similar to each other can enhance degree of polarization of the reconstructed image. 
     In the design method described in the example 2, the light intensity ratio of the target image is always 1:1 because the symmetric axis in the method is y=x. Then, the light intensity ratio of the incident light is also always 1:1. Therefore, a degree of +45° as the polarization direction with respect to the X-axis of the incident light can always available. 
     Example 3 
     The case of the light intensity distribution LI including phase distribution is four times rotational symmetric will be explained below. A flowchart shown in  FIG. 4  is available for the example 3. 
     The hologram comprising the plurality of cells can be designed so that a phase distribution of the Y-polarization light intensity distribution is equal to a phase distribution which is realized by rotating a phase distribution of the X-polarization light intensity distribution by an angle of 90 degrees. In other words, the hologram data for Y-polarization can be obtained by rotating the hologram data for X-polarization by an angle of 90 degrees. 
       FIG. 11A  shows the target image which is four times rotational symmetric. This target image is formed by S-polarization. 
     Referring to  FIG. 4 , in step S 2002 , the target image is divided into a light intensity distribution formed by X-polarization and another light intensity distribution formed by Y-polarization.  FIGS. 10B and 10C  show the light intensity distribution divided for X and Y-polarizations, respectively. 
     In step S 2004 , the phase difference between X and Y-polarizations on the predetermined plane is determined.  FIG. 10D  shows the phase difference under a condition where the incident light is a linearly polarized light and the polarization direction with respect to the X-axis is +45°. The phase difference in the second and fourth quadrant is 0, and the phase difference in the first and third quadrant is π. 
     In step S 2006 , phase symmetry between a phase distribution of the X-polarization and a phase distribution of the Y-polarization is determined.  FIG. 10E  is a more general figure of the phase difference of the target image formed by S-polarization shown in  FIG. 10D . After designing hologram data for X-polarization, a hologram data for Y polarization is obtained by rotating the designed hologram data for X polarization by clockwise 90°.  FIG. 10F  shows one example of phase symmetry for X-polarization with considering the above.  FIG. 10G  shows phase symmetry for Y-polarization obtained by rotating the phase symmetry shown in  FIG. 10F . The difference between  FIGS. 10F and 10G  corresponds to  FIG. 10E . 
     In step S 2008 , hologram data for X-polarization are generated under an admissibility condition of diffusing the phase of X-polarization while maintaining the phase symmetry shown in  FIG. 10F . To generate the hologram data, above described method will be available. In each iterative calculation step, the phase in the overlap region on the predetermined plane might be shifted to maintain the phase symmetry condition. The shift of the phase might be executed so that an amount of shifting the phase becomes smaller. 
       FIG. 10H  shows one example of generated hologram data for X-polarization.  FIG. 10J  shows reconstructed images obtained by using the generated hologram data shown in  FIG. 10H .  FIG. 10J  corresponds to the light intensity distribution for X-polarization shown in  FIG. 10B .  FIG. 10L  shows phase distributions for X-polarization on the predetermined plane. The phase distributions are diffused in each portion, in each overlap region and in the light intensity distribution region for X polarization.  FIG. 10L  corresponds to the phase symmetry shown in  FIG. 10F . 
     In step S 2010 , hologram data for Y-polarization is obtained by rotating the hologram data for X-polarization shown in  FIG. 10H  according to information of the symmetry.  FIG. 10I  shows the hologram data for Y-polarization.  FIG. 10K  shows reconstructed images obtained by using the generated hologram data shown in  FIG. 10I .  FIG. 10K  corresponds to the light intensity distribution for Y-polarization shown in  FIG. 10C .  FIG. 10M  shows phase distributions for Y-polarization on the predetermined plane.  FIG. 10N  shows the phase difference between X and Y-polarizations on the predetermined plane shown in  FIGS. 10L and 10M . It is understood that  FIG. 10N  corresponds to the phase difference  FIG. 10D . 
     In step S 2012 , the hologram data for X and Y-polarizations generated in  FIGS. 10H and 10I  in step S 2008  and S 2010  are integrated with each other. To generate the hologram data, above described method will be available. 
     In the examples regarding symmetric targets, the described targets comprise only S-polarization, but target images might comprise not only S-polarization but also circularly polarized light and so on. In this case, phase symmetry on the predetermined plane is different from the phase symmetry shown in  FIG. 9F  or  10 F. 
     A calculation time by using the symmetric data generating method shown in  FIG. 4  might be half of the time by using the general data generating method shown in  FIG. 3  because second hologram data will be obtained by using the symmetric data after the first hologram data is generated. 
     The hologram may be also optionally generated with the flowchart shown in  FIG. 3  even when the required target image is symmetric. 
     In the above described examples, the design method for the target images whose portions in the overlap region MA formed by a linear polarization of which has a polarization direction different from the X and Y-polarizations or a circular polarization is explained. The target image may also optionally comprise an elliptic polarization. In this case, the phase difference to form a required target image on the predetermined plane PS may be selected from a value which is different from 0, π/2, π and −π/2. 
     Because the conventional arts require separated CGHs of types in a number equal to that of polarization directions of the target image, it might not be easy for the conventional arts to continuously change the polarization direction in each pixel. In contrast, this embodiment according to the present invention can provide a computer generated hologram which can continuously change the polarization direction in each pixel, as described above. 
     Although this embodiment has exemplified a case in which the computer generated hologram includes few cells, a light intensity distribution with a desired shape and polarization state can be formed even by increasing the number of cells of the computer generated hologram. Increasing the number of cells of the computer generated hologram may make it possible to decrease the sizes of pixels which divide the light intensity distribution (target image), thus forming a uniform light intensity distribution. 
     Example 4 
     An exposure apparatus  1  to which the hologram  100  is applied will be explained below with reference to  FIG. 11 .  FIG. 11  illustrates an exemplary arrangement of the exposure apparatus  1 . 
     The exposure apparatus  1  is a projection exposure apparatus which transfers the pattern of a reticle  20  onto a wafer  40  by the step &amp; scan scheme. However, the exposure apparatus  1  can adopt the step &amp; repeat scheme or another exposure scheme. 
     As shown in  FIG. 11 , the exposure apparatus  1  includes an illumination apparatus  10 , a reticle stage (not shown) for supporting the reticle  20 , a projection optical system  30 , and a wafer stage (not shown) for supporting the wafer  40 . 
     The illumination apparatus  10  illuminates the reticle  20  on which a circuit pattern to be transferred is formed, and includes a light source  16  and illumination optical system  18 . 
     The light source  16  is, for example, an excimer laser such as an ArF excimer laser with a wavelength of about 193 nm or a KrF excimer laser with a wavelength of about 248 nm. However, the light source  16  is not particularly limited to an excimer laser, and may be, for example, an F 2  laser with a wavelength of about 157 nm or a mercury lamp with a narrow wavelength range. 
     The illumination optical system  18  illuminates the reticle  20  with light from the light source  16 , and performs modified illumination on the reticle  20  in a predetermined polarization state while ensuring a predetermined illuminance. In this example, the illumination optical system  18  includes a light extension optical system  181 , beam shaping optical system  182 , polarization controller  183 , phase controller  184 , exit angle saving optical element  185 , relay optical system  186 , multibeam generation unit  187 , polarization state adjusting unit  194 , and the hologram  100 . The illumination optical system  18  also includes a relay optical system  188 , aperture  189 , zoom optical system  190 , multibeam generation unit  191 , aperture stop  192 , and irradiation unit  193 . 
     The light extension optical system  181  deflects light from the light source  16  to guide it to the beam shaping optical system  182 . The beam shaping optical system  182  shapes the section of the light from the light source  16  into a desired shape by converting the horizontal to vertical ratio of the section of the light from the light source  16  into a desired value (e.g., by changing the sectional shape from a rectangle to a square). The beam shaping optical system  182  forms a light beam with a size and an angle of divergence which are required to illuminate the multibeam generation unit  187 . 
     The polarization controller  183  includes, for example, a linear polarizer and has a function of removing unnecessary polarized light components. It is possible to efficiently convert light from the light source  16  into desired linearly polarized light by minimizing polarized light components removed (shielded) by the polarization controller  183 . 
     The phase controller  184  converts the linearly polarized light obtained by the polarization controller  183  into circularly polarized light by giving a phase difference of λ/4 to it. 
     The exit angle saving optical element  185  includes, for example, an optical integrator (e.g., a fly-eye lens or fiber bundle including a plurality of microlenses), and outputs the light at a predetermined angle of divergence. 
     The relay optical system  186  converges the light which emerges from the exit angle saving optical element  185  on the multibeam generation unit  187 . The relay optical system  186  adjusts the exit surface of the exit angle saving optical element  185  and the incident surface of the multibeam generation unit  187  to hold the Fourier transform relationship (the relationship between the object plane and the pupil plane or that between the pupil plane and the image plane). 
     The multibeam generation unit  187  includes an optical integrator (e.g., a fly-eye lens or fiber bundle including a plurality of microlenses) for uniformly illuminating the polarization state adjusting unit  194  and computer generated hologram  100 . The exit surface of the multibeam generation unit  187  forms a light source surface including a plurality of point light sources. The light which emerges from the multibeam generation unit  187  impinges on the polarization state adjusting unit  194  as circularly polarized light. 
     The polarization state adjusting unit  194  converts the circularly polarized light obtained by the phase controller  184  into linearly polarized light having a desired polarization direction by giving a phase difference of λ/4 to it. The light which emerges from the polarization state adjusting unit  194  impinges on the computer generated hologram  100  as linearly polarized light. 
     More specifically, in one example, the incident light generated from the light source  16  might include X and Y-polarizations, and an amplitude of X-polarization might be equal to an amplitude of Y-polarization. 
     The hologram  100  forms a light intensity distribution (e.g., a light intensity distribution LI as shown in  FIG. 1 ) at the position of the aperture  189  via the relay optical system  188 . The hologram  100  can take any of the above-described forms, and a detailed description thereof will not be given here. 
     The aperture  189  has a function of passing only a light intensity distribution formed by the hologram  100 . The computer generated hologram  100  and aperture  189  are set to hold the Fourier transform relationship. 
     The zoom optical system  190  enlarges a light intensity distribution formed by the hologram  100  at a predetermined magnification, and projects it onto the multibeam generation unit  191 . 
     The multibeam generation unit  191  is inserted on the pupil plane of the illumination optical system  18 , and forms, on its exit surface, a light source image (effective light source distribution) corresponding to the light intensity distribution formed at the position of the aperture  189 . In this example, the multibeam generation unit  191  includes an optical integrator such as a fly-eye lens or cylindrical lens array. The aperture stop  192  is inserted near the exit surface of the multibeam generation unit  191 . 
     The irradiation unit  193  includes, for example, a condenser optical system and illuminates the reticle  20  with an effective light source distribution formed on the exit surface of the multibeam generation unit  191 . 
     The reticle  20  has a circuit pattern and is supported and driven by the reticle stage (not shown). Diffracted light generated by the reticle  20  is projected onto the wafer  40  via the projection optical system  30 . Since the exposure apparatus  1  is of the step &amp; scan scheme, it transfers the pattern of the reticle  20  onto the wafer  40  by scanning them. 
     The projection optical system  30  projects the pattern of the reticle  20  onto the wafer  40 . The projection optical system  30  can be a dioptric system, catadioptric system, or catoptric system. 
     The wafer  40  is a substrate onto which the pattern of the reticle  20  is projected (transferred), and is supported and driven by the wafer stage (not shown). However, it is also possible to use a glass plate or another substrate in place of the wafer  40 . The wafer  40  is coated with a resist. 
     As described above, the computer generated hologram  100  does not give a phase distribution to the wavefront of light polarized in a single direction, but two-dimensionally gives different phase distributions to the wavefronts of both X-polarization and Y-polarization. This makes it possible to form a light intensity distribution LI almost without generating any loss in light amount. 
     In exposure, light emitted by the light source  16  illuminates the reticle  20  by the illumination optical system  18 . The light which bears the information of the pattern of the reticle  20  forms an image on the wafer  40  by the projection optical system  30 . The illumination optical system  18  used for the exposure apparatus  1  can suppress any illumination variation and loss in light amount, and form a light intensity distribution with a desired shape and polarization state by the hologram  100 . Hence, the exposure apparatus  1  can provide high-quality devices (e.g., a semiconductor device, an LCD device, an image sensing device (e.g., a CCD), and a thin-film magnetic head) with a high throughput and a good economical efficiency. 
     While the present invention has been described with reference to exemplary embodiments, it is to be understood that the invention is not limited to the disclosed exemplary embodiments. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions.