Abstract:
Disclosed is a diffractive optical element which includes a diffractive grating pattern on a base plate. The diffractive grating pattern includes a plurality of phase gratings arranged in parallel lines extending along a predetermined direction to cause diffraction of an incident beam. Each of the plurality of phase gratings has an asymmetrical phase pattern. There is a phase gap, ΔP, representing a phase difference (in radians) between an end point and a beginning point of each phase pattern. The phase gap, ΔP, is substantially equal for each of the plurality of phase gratings and satisfies the relationship 
     
       0.7π&lt;|ΔP|&lt;1.2 π.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to a beam splitting optical element which divides an incident beam into a plurality of number of emitted beams, and more particularly, to a beam splitting optical element using diffractive gratings. 
     Conventionally, beam splitters using diffractive gratings have been known. In such beam splitters, linear grooves or raised portions (i.e., gratings) are formed on, for example, a glass substrate. The arrangement of the gratings determines the pattern of emitted diffracted beams. Typically, the emitted beams (±1st order beams, ±2nd order beams, . . . ) are arranged symmetrically around a central beam (i.e., a zero order diffracted beam) and, as a result, there are an odd number of diffracted beams emitted. 
     A diffraction efficiency of the conventional diffractive gratings as described above is generally in a range of 70%-85%. There is a need for a beam splitter employing diffractive gratings which has a relatively high diffraction efficiency. 
     Further, in the field of digital opto-electronics, it is particularly useful if a diffractive optical element has an even number of emitted beams having relatively similar intensities. For example, in an optical recording device accessed by a computer or an optical computer, eight bits (a byte) is a unit when data is processed. If a beam is divided into an even and desired number of beams by the beam splitter, it is advantageous since the emitted beams are used for processing the data efficiently. 
     SUMMARY OF THE INVENTION 
     It is therefore an object of the present invention to provide an improved beam splitting optical element which divides an incident beam into an even number of beams and has a higher diffraction efficiency than a conventional element. 
     For the above object, according to one aspect of the invention, there is provided a diffractive optical element, comprising a cylindrical surface provided with a diffractive grating pattern. The diffractive grating pattern includes a plurality of phase gratings arranged in parallel lines extending along a circumference of the cylindrical surface to cause diffraction of an incident beam, where a beam incident on the diffractive grating pattern is emitted as divided into a plurality of diffracted beams. Since the grating pattern is formed on a cylindrical surface, and the garatings extend along the circumference of the cylindrical surface, a mold to be used for molding the grating pattern can be made easily with use of, for example, a lathe. 
     Preferably, a surface of the optical element from which the diffracted beams are emitted is also cylindrical having a curvature that is substantially the same as a curvature of the cylindrical surface, so that the phase diffracting element has a meniscus shape and has substantially no magnifying power in total. optionally, the plurality of phase gratings are of equal width in a direction of the generatrix of the cylinder and each of the plurality of phase gratings has a continuous, nonlinear surface to cause phase differences in a wave front of the incident beam. The mold for such a grating can be made relatively easily when the lathe is used. 
     Further optionally, each of the plurality of phase gratings has an asymmetrical phase pattern, and a phase gap ΔP, representing a phase difference between an end point of each of the plurality of phase patterns and a beginning point of each of the plurality of phase patterns, in radians. The phase gap, ΔP, is substantially equal for each of the plurality of phase gratings and satisfies: 
     
         0.7π&lt;|ΔP|&lt;1.2π. 
    
     With this structure, the emitted beams (i.e., the diffracted beams) distribute asymmetrically with respect to the zero order diffracted beam, and accordingly it is possible that the number of diffracted beams can be adusted to an even number. 
     Further optionally, the plurality of phase gratings are adjusted so that each of the divided diffracted beams have substantially the same intensity and no divided beam is emitted other than the intended number of beams. As a result, an even number of diffracted beams having substantially the same intensity may be emitted from the diffractive optical element. 
     According to another aspect of the invention, there is provided a diffractive optical element, comprising a base plate provided with a diffractive grating pattern. The diffractive grating pattern includes a plurality of phase gratings arranged in parallel lines extending along a predetermined direction of the base plate to cause diffraction of an incident beam. A beam incident on the diffractive grating pattern is emitted as a plurality of diffracted beams, wherein each of the plurality of phase gratings has an asymmetrical phase pattern in a direction where the plurality of phase gratings are arranged. A phase gap ΔP, representing a phase difference between an end point of each of the plurality of phase patterns and a beginning point of each of the plurality of phase patterns, in radians, is substantially equal for each of the plurality of phase gratings and satisfies: 
     
         0.7π&lt;|ΔP|&lt;1.2π. 
    
     With this optical element, a desired even number of diffracted beams, which are asymmetrically distribued with respect to the zero order beam, are obtained. 
     It shold be noted that the diffracted beams substantially consist of a desired number of beams. 
     Various examples are indicated as embodiments. In each embodiment, a predetermined error in the phase pattern is permissible. 
     Specifically, the predetermined permissible error in the phase difference may be less than 2%. 
     According to a further aspect of the invention, there is provided a diffractive optical element, comprising: a base plate having a cylindrical surface; and a diffractive grating pattern engraved on the cylindrical surface in a direction perpendicular to a generatrix of the cylindrical surface so that diffracted beams distribute in a dimension along the generatrix. 
     Since the diffractive grating pattern is formed on the cylindrical surface, a mold to be used for molding the optical element can be produced relatively easily. 
     Optionally, the grating pattern includes a plurality of phase gratings. Due to the shape of the optical element, and therefore the shape of the mold for the optical element, a complicated pattern can be employed. Accordingly, the grating can be a phase grating. When employing the phase grating, diffraction efficiency is improved. 
     Optionally or alternatively, each of the phase gratings has an asymmetrical phase pattern. As a result, the diffracted beams distribute asymmetrically with respect to a zero order diffracted beam. 
     Accordingly, by selecting an appropriate phase pattern of the phase gratings, an even number of diffracted beams can be emitted. 
     According to a further aspect of the invention, there is provided a method for producing a diffracting optical element, comprising: making a mold by (1) rotating a cylindrical metal mold about a rotation axis, and (2) moving a cutting tool to a predetermined radial distance from the rotation axis and moving the tool along the rotation axis; and applying an injection mold process with use of a master made by the steps of making the mold to make the diffracting optical element. 
     With this method, a complicated phase pattern can be formed on the mold. 
     According to a further aspect of the invention, there is provided a method for producing a mold to be used for making a diffracting optical element having a cylindrical surface with an injection mold process, comprising (1) rotating a cylindrical metal mold about a rotation axis, and (2) moving a cutting tool to a predetermined radial distance from the rotation axis and moving the tool along the rotation axis. Also with this method, a complicated pattern extending along a circumference of the mold can be formed on the circumferential surface of the mold. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a schematic enlarged perspective view of gratings formed on a beam splitter according to an embodiment of the invention; 
     FIG. 2 is a perspective view of the beam splitter according to an embodiment of the invention; 
     FIG. 3 is a perspective view illustrating a process for making a mold for the beam splitter of FIG. 2; 
     FIGS. 4 through 15 are graphs illustrating exemplary phase patterns for the beam splitter of FIG. 2; 
     FIGS. 16 through 27 are graphs showing a distribution of intensities of the diffracted beams corresponding to the exemplary phase patterns of FIGS. 4 through 15; and 
     FIG. 28 is a perspective view of the beam splitter according to an alternative embodiment of the invention. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     FIG. 2 is a perspective view of a beam splitter 10 according to an embodiment of the invention. The beam splitter 10 includes a base 11 on which a grating pattern 12 is formed. As shown in FIG. 2, the base 11 has a concave surface 11a and a convex surface 11b and is represented as a portion of a wall of a cylinder. In FIG. 2, dotted lines show an imaginary cylinder with ends C1 and C2 having equal diameters representing a diameter of the concave surface 11a. The grating pattern 12 is formed on the concave surface 11a. 
     FIG. 1 is a schematic enlarged view of the grating pattern 12. it should be noted that in FIG. 1, the grating pattern 12 is shown as formed on a flat surface, and description with reference to FIG. 1 is made as if the grating pattern 12 is formed on the flat surface. As described above, however, the grating pattern 12 is actually formed on the concave surface 11a. 
     The grating pattern 12 is formed having a plurality of identically formed phase gratings P extending linearly. Each of the phase gratings P have a predetermined length L along a y-axis direction and linearly extend in an x-axis direction. In particular, the phase gratings shown in FIG. 1 correspond to a particular numerical example (example 3) described in more detail below. Note that the y-axis direction is a direction parallel to a generatrix of the base 11 (i.e., the imaginary cylinder defined by the end circles C1 and C2) and the x-axis direction represents the curve of the concave surface 11a. 
     The grating pattern 12 and the base 11 are made of, for example, glass or a transparent resinous material. The grating pattern 12 is formed so that it divides an incident beam into a plurality of diffracted emitted beams. It should be noted that the grating pattern can be formed on a flat surface. Practically, however, due to difficulty in producing a complicated grating pattern as described below, it is preferable, to employ a cylindrical surface as a surface on which the grating pattern 12 is formed. Further, if the cylindrical surface is employed, it may be advantageous to use resinous material for the optical element, and the optical element may be formed with a molding process. 
     As shown in FIG. 1, a cross section along a Y-Z plane of each grating P has a shape which causes non-linear beams to pass therethrough along directions represented by the arrows D, E in FIG. 2. In other words, lines representative of the phase difference caused by the beam splitter 10 is similar to the cross-sectional shape of the phase gratings P. If the phase difference caused by the grating is represented by δ with respect to the lowermost portion of the surface of the phase grating P, the surface of the phase grating P is defined by a non-linear phase difference δ which varies along the y-axis direction. In other words, the phase difference δ is defined as a difference between a point on a phase grating P and a predetermined reference point on the phase grating P. In the embodiments, the predetermined reference point is determined and meets with an adjacent phase grating P with a phase gap ΔP. In particular, the phase gap ΔP should be constant for all phase gratings P and should satisfy the condition: 
     
         0.7π&lt;|ΔP|&lt;1.2π. 
    
     According to the beam splitter 10 of the embodiment, the cross-section along the y-axis phase gratings P have an asymmetrical shape (i.e., an asymmetrical phase distribution), and the diffracted beams are also asymmetrical with respect to a zero order diffracted beam and the incident beam is divided into an even number of emitted beams. 
     A method of forming the beam splitter 10 is now described. Since the cross-sectional shape of the phase patterns P is nonlinear and complicated, it is difficult to form a mold for the grating pattern 12 using an etching process. Accordingly, the diffractive optical element 10 is molded using a metal mold. However, if the surface on which a master pattern is formed is a flat surface, a cutting tool for forming the pattern is to be moved three dimensionally, i.e., in the x, y and z axis directions relative to the surface. Considering the size of the gratings, it may be very difficult to control the movement of the cutting tool to form the pattern precisely. 
     FIG. 3 is a perspective view illustrating a process for making a mold 30 for the beam splitter 10. Since each of the gratings has a surface which cannot be made with use of an etching or the like, the mold 30 should be used in order to make the phase gratings. The mold 30 is a cylindrical member as shown in FIG. 2, and the pattern 31, representing the grating pattern 12, is formed on the circumferential surface of the mold 30 using a cutting tool 100. The mold 30 is then used to form the beam splitter 10. When the beam splitter 10 is made, for example, a well-known injection mold method is applied using the mold 30 as a master. It should be noted that the beam splitter 10 is formed of optical plastic such as PMMA (Plymethyl methacrylate). 
     In this embodiment, as shown in FIG. 2, since the phase difference 6 of each phase pattern P is constant along the x-axis direction, the positional relation between the mold 30 and the cutting tool 100 need only be adjusted in two dimensions (i.e., the y-axis direction and the z-axis direction). Thus, the phase patterns P can be engraved accurately in a short time and at a low cost. 
     As shown in FIG. 3, the cutting tool 100 includes a lathe 20, a support 21 rotated by the lathe 20 and movable along a rotation axis thereof (i.e., movable in a y-axis direction), a sliding table 22 arranged to move perpendicular to the rotation axis of the support 21 (i.e., in a z-axis direction), and a tool 23 fixedly provided on the sliding table 22. 
     The metal mold 30 is fixed coaxially with the support 21 and is rotated in a direction Rx (corresponding to the x-axis direction of FIG. 1). Then, by appropriate movement of the support 21 along the y-axis direction and of the sliding table 22 along the z-axis direction with the metal mold 30 rotated, the mold 30 for the phase gratings P is formed. 
     The mold formed 30 is then used to from the beam splitter 10. 
     Twelve particular numerical examples of the phase grating P are now described with reference to FIGS. 4 through 27. 
     In example 1, the phase grating P is formed such that an incident beam is divided into 4 emitted beams by the beam splitter 10, in examples 2-5, the phase grating P is formed such that an incident, beam is divided into 8 emitted beams, and in examples 6-12, the phase grating P is formed such that an incident beam is divided into 16 emitted beams by the beam splitter 10. 
     In these examples, the phase grating P is designed such that: (1) intensities of each emitted beam are substantially the same, and (2) only the intended number of emitted beams are emitted. 
     In the following description, the width L (i.e., the length of in the Y-axis direction) of each phase grating P is divided into 64 co-ordinates (designated 0-63). A reference point is designated as a point at which the phase pattern P is lowest in relation to the concave surface 11a. Further, the phase difference δ for each co-ordinate is given in radians. Accordingly, the phase difference δ is more than 0. However, a height H along the z-axis direction (i.e., an actual height of the phase grating) in micrometers (μm) may be calculated, for a predetermined incident beam, using the formula: 
     
         H=δ×λ/(2π(n-1)), 
    
     where n is a refractive index of the material of the beam splitter 10 and λ is a wavelength of the incident beam. It is assumed that the beam splitter 10 is located within air whose refractive index is regarded as 1. 
     EXAMPLE 1 
     Table 1 shows data for a pattern of the phase grating P according to example 1. The data is shown graphically in FIG. 4 where a vertical axis is the phase difference δ and a horizontal axis is the coordinate along the y-axis direction. In examples 1, the phase gap ΔP is 1.00π. Note that, the phase gap ΔP is defined a difference between the phases at the coordinate 0 and the coordinate 63. 
     
                       TABLE 1______________________________________ΔP = 1.00Coord. δ  Coord.   δ Coord. δ______________________________________0      0.00000  22       2.28947 44     2.893971      0.04000  23       2.51947 45     2.793972      0.10042  24       2.69518 46     2.742793      0.16042  25       2.86518 47     2.692794      0.21520  26       3.02656 48     2.672915      0.25520  27       3.14656 49     2.652916      0.32315  28       3.25191 50     2.660567      0.38315  29       3.35191 51     2.670568      0.44920  30       3.41679 52     2.675189      0.51920  31       3.48679 53     2.6951810     0.59142  32       3.52997 54     2.7244711     0.66142  33       3.55997 55     2.7544712     0.75100  34       3.59951 56     2.7903313     0.84100  35       3.59951 57     2.8203314     0.94020  36       3.58770 58     2.8638615     1.05020  37       3.55770 59     2.8938616     1.17834  38       3.50415 60     2.9443417     1.30834  39       3.44415 61     2.9843418     1.46586  40       3.33870 62     3.0342019     1.64586  41       3.21870 63     3.1342020     1.84933  42       3.1095121     2.06933  43       2.96951______________________________________ 
    
     EXAMPLE 2 
     Table 2 shows data for a pattern of the phase grating P according to example 2. The data is shown graphically in FIG. 5. In example 2, the phase gap ΔP is 0.75π. 
     
                       TABLE 2______________________________________ΔP = 0.75Coord. δ  Coord.   δ Coord. δ______________________________________0      2.34750  22       2.33100 44     6.161001      2.11350  23       2.54700 45     6.298502      2.10350  24       2.76700 46     6.388503      2.11700  25       2.98600 47     6.466004      2.13200  26       3.23600 48     6.516005      2.14700  27       3.57850 49     6.537006      2.14700  28       4.10850 50     6.487007      2.10350  29       4.68100 51     6.381008      1.96350  30       5.10100 52     6.171009      1.32750  31       5.39450 53     5.8500010     0.36750  32       5.57450 54     5.4700011     0.07000  33       5.70650 55     5.2260012     0.00000  34       5.79650 56     5.0660013     0.02550  35       5.84250 57     4.9980014     0.12550  36       5.88250 58     4.9480015     0.23800  37       5.89250 59     4.9355016     0.40800  38       5.89250 60     4.9255017     0.65600  39       5.89650 61     4.9100018     0.96600  40       5.90650 62     4.8900019     1.34000  41       5.91450 63     4.7025020     1.72000  42       5.9645021     2.05100  43       6.06100______________________________________ 
    
     EXAMPLE 3 
     Table 3 shows data for a pattern of the phase grating P according to example 3. The data is shown graphically in FIG. 6. In example 3, the phase gap ΔP is 0.99π. 
     
                       TABLE 3______________________________________ΔP = 0.99Coord. δ  Coord.   δ Coord. δ______________________________________0      0.00000  22       5.53800 44     5.955001      0.04000  23       5.69800 45     5.745002      0.08700  24       5.81200 46     5.574003      0.16700  25       5.87200 47     5.434004      0.24600  26       5.88400 48     5.330005      0.35600  27       5.85400 49     5.230006      0.45900  28       5.78500 50     5.114007      0.63900  29       5.70500 51     4.944008      0.88800  30       5.63700 52     4.634009      1.31800  31       5.63700 53     4.1140010     1.99700  32       5.67800 54     3.5220011     2.65700  33       5.79800 55     3.2020012     3.03300  34       5.95300 56     3.0380013     3.30300  35       6.12300 57     2.9580014     3.52200  36       6.28800 58     2.9480015     3.71200  37       6.41800 59     2.9580016     3.91200  38       6.51300 60     2.9690017     4.15200  39       6.55300 61     2.9890018     4.41000  40       6.53000 62     3.0330019     4.73000  41       6.47000 63     3.0980020     5.03600  42       6.3530021     5.32600  43       6.16300______________________________________ 
    
     EXAMPLE 4 
     Table 4 shows data for a pattern of the phase grating P according to example 4. The data is shown graphically in FIG. 7. In example 4, the phase gap ΔP is 0.99π. 
     
                       TABLE 4______________________________________ΔP = 0.99Coord. δ  Coord.   δ Coord. δ______________________________________0      1.57900  22       2.15900 44     2.384001      1.63900  23       2.39900 45     2.114002      1.66700  24       2.60700 46     1.932003      1.69700  25       2.81700 47     1.812004      1.73900  26       3.04600 48     1.746005      1.74400  27       3.25600 49     1.756006      1.73000  28       3.49200 50     1.825007      1.70000  29       3.71200 51     1.965008      1.55400  30       3.90500 52     2.265009      1.25400  31       4.02500 53     2.8250010     0.71100  32       4.06700 54     3.4780011     0.25100  33       4.04700 55     3.8780012     0.05000  34       3.97200 56     4.0950013     0.00000  35       3.84200 57     4.2350014     0.04100  36       3.70100 58     4.3470015     0.12100  37       3.57100 59     4.4370016     0.28200  38       3.44500 60     4.5040017     0.50200  39       3.33500 61     4.5540018     0.80200  40       3.22000 62     4.6260019     1.15200  41       3.09000 63     4.6860020     1.52900  42       2.9060021     1.87900  43       2.65600______________________________________ 
    
     EXAMPLE 5 
     Table 5 shows data for a pattern of the phase grating P according to example 5. The data is shown graphically in FIG. 8. In example 5, the phase gap ΔP is 1.00π. 
     
                       TABLE 5______________________________________ΔP = 1.00Coord. δ  Coord.   δ Coord. δ______________________________________0      0.00000  22       5.13656 44     5.808121      0.03494  23       5.21307 45     5.603232      0.08372  24       5.20778 46     5.379183      0.17593  25       5.13788 47     5.124604      0.29127  26       4.97072 48     4.888075      0.41957  27       4.73558 49     4.689506      0.57022  28       4.48842 50     4.489527      0.77908  29       4.32697 51     4.297598      1.04067  30       4.24668 52     4.051689      1.33889  31       4.24576 53     3.8012410     1.70048  32       4.27485 54     3.5678211     2.04910  33       4.39394 55     3.3477712     2.38318  34       4.58241 56     3.1943613     2.68329  35       4.84203 57     3.0836814     2.97144  36       5.19737 58     3.0112015     3.26819  37       5.51068 59     2.9710716     3.58289  38       5.77602 60     2.9639117     3.93565  39       5.94409 61     2.9898718     4.28787  40       6.03755 62     2.9992719     4.60094  41       6.07922 63     3.1375020     4.85049  42       6.0567421     5.03591  43       5.96950______________________________________ 
    
     EXAMPLE 6 
     Table 6 shows data for a pattern of the phase grating P according to example 6. The data is shown graphically in FIG. 9. In example 6, the phase gap ΔP is 1.01π. 
     
                       TABLE 6______________________________________ΔP = 1.01Coord. δ  Coord.   δ Coord. δ______________________________________0      0.00000  22       9.01142 44     7.398841      0.11909  23       9.28050 45     7.477932      0.38167  24       9.40910 46     7.703023      0.86576  25       9.51318 47     8.327114      1.60085  26       9.57577 48     8.880195      2.24994  27       9.42985 49     8.919276      2.70552  28       8.82495 50     8.639377      3.15461  29       8.55903 51     8.243458      3.79220  30       8.57712 52     7.869549      4.65129  31       8.64120 53     7.4986210     5.33787  32       8.69029 54     6.9002211     5.83696  33       8.68438 55     6.0993012     6.33905  34       8.73447 56     5.5298913     6.84314  35       9.13856 57     5.2089714     7.22122  36       9.91164 58     4.8515715     7.25031  37       10.12573                            59     4.3306516     6.79540  38       10.16132                            60     3.6637417     6.30948  39       10.14541                            61     3.2778218     6.15257  40       10.08499                            62     3.1334219     6.17165  41       9.91408 63     3.1775020     6.27975  42       8.3196721     7.34883  43       7.42876______________________________________ 
    
     EXAMPLE 7 
     Table 7 shows data for a pattern of the phase grating P according to example 7. The data is shown graphically in FIG. 10. In example 7, the phase gap ΔP is 0.98π. 
     
                       TABLE 7______________________________________ΔP = 0.98Coord. δ  Coord.   δ Coord. δ______________________________________0      0.00000  22       9.18279 44     8.335231      0.10112  23       9.81372 45     7.857332      0.21482  24       10.31935                            46     7.253753      0.46317  25       10.74935                            47     6.513114      0.87331  26       11.07071                            48     5.960115      1.39192  27       11.23482                            49     5.558476      1.73829  28       11.08046                            50     5.078897      1.78828  29       10.43581                            51     4.474008      1.74891  30       9.96051 52     4.015309      1.68984  31       9.86263 53     3.8721810     1.76142  32       9.91072 54     3.9017711     2.02072  33       10.11877                            55     4.0490212     2.55159  34       10.68425                            56     4.2005613     3.28166  35       11.42907                            57     4.2177414     3.83542  36       11.68461                            58     4.0015515     4.34223  37       11.60267                            59     3.5631116     4.99540  38       11.38748                            60     3.2491517     5.83822  39       11.05666                            61     3.1179818     6.55497  40       10.65121                            62     3.0884519     7.10304  41       10.11445                            63     3.0930020     7.68791  42       9.4300521     8.41221  43       8.81092______________________________________ 
    
     EXAMPLE 8 
     Table 8 shows data for a pattern of the phase grating P according to example 8. The data is shown graphically in FIG. 11. In example 8, the phase gap ΔP is 1.14π. 
     
                       TABLE 8______________________________________ΔP = 1.14Coord. δ  Coord.   δ Coord. δ______________________________________0      0.00000  22       8.14593 44     6.389761      0.35675  23       8.06924 45     6.159632      0.59145  24       7.78458 46     5.822473      1.02421  25       7.15265 47     5.418434      1.76643  26       6.44611 48     4.987835      2.45950  27       5.97778 49     4.506066      2.97905  28       5.63530 50     4.058567      3.46447  29       5.27806 51     3.783508      4.30512  30       4.81668 52     3.742289      5.26163  31       4.14179 53     4.1544910     5.78634  32       3.46774 54     5.0098311     6.07644  33       3.08316 55     5.3481312     6.23928  34       2.88663 56     5.3987813     6.33414  35       2.84806 57     5.2276414     6.27698  36       3.03808 58     4.6092315     6.11553  37       3.51615 59     3.7774516     5.97524  38       4.17024 60     3.4690417     5.92432  39       4.54980 61     3.3976618     6.05341  40       4.92638 62     3.4117419     6.40250  41       5.62633 63     3.5918520     7.41097  42       6.2729221     8.03059  43       6.49224______________________________________ 
    
     EXAMPLE 9 
     Table 9 shows dicta for a pattern of the phase grating P according to example 9. The data is shown graphically in FIG. 12. In example 93, the phase gap ΔP is 0.86π. 
     
                       TABLE 9______________________________________ΔP = 0.86Coord. δ  Coord.   δ Coord. δ______________________________________0      0.00000  22       10.41512                            44     11.310281      0.02139  23       10.45448                            45     11.049142      0.24455  24       10.66319                            46     10.705883      0.63197  25       11.05519                            47     10.494094      1.25970  26       11.52474                            48     10.315205      1.93395  27       11.90449                            49     10.098166      2.45426  28       12.17922                            50     9.419897      2.94875  29       12.38914                            51     8.187258      3.67947  30       12.57125                            52     7.753999      4.63407  31       12.71915                            53     7.5121910     5.28644  32       12.78723                            54     7.1420011     5.75541  33       12.73151                            55     6.2255712     6.29385  34       12.67358                            56     5.3470313     7.22466  35       12.59483                            57     4.9607114     8.10410  36       12.48228                            58     4.6145815     8.59831  37       12.25170                            59     4.0785116     8.96738  38       11.83332                            60     3.4229517     9.36434  39       11.38850                            61     3.0647118     9.85078  40       11.15996                            62     2.9047219     10.26310 41       11.13878                            63     2.6910020     10.46203 42       11.2293421     10.45751 43       11.33304______________________________________ 
    
     EXAMPLE 10 
     Table 10 shows data for a pattern of the phase grating P according to examples 10. The data is shown graphically in FIG. 13. In example 10, the phase gap ΔP is 1.07π. 
     
                       TABLE 10______________________________________ΔP = 1.07Coord. δ  Coord.   δ Coord. δ______________________________________0      1.88359  22       6.11453 44     0.000001      2.20660  23       5.47716 45     0.110272      2.55505  24       4.92379 46     0.553343      2.96194  25       4.32768 47     1.037844      3.50884  26       3.82595 48     1.319945      4.12613  27       3.35523 49     1.570856      4.76767  28       2.90596 50     1.876267      5.47822  29       2.32512 51     2.430608      6.13673  30       1.74405 52     3.029749      6.68710  31       1.41813 53     3.3511210     7.20112  32       1.33906 54     3.3677511     7.89495  33       1.50709 55     3.3049312     9.04040  34       1.88758 56     3.2280913     9.57330  35       2.18736 57     3.3815114     9.71868  36       2.27226 58     3.6377815     9.62977  37       2.04002 59     3.9931816     9.28356  38       1.53403 60     4.3238617     8.61084  39       1.01003 61     4.5670018     8.05756  40       0.69457 62     4.8152519     7.66091  41       0.47837 63     5.2482320     7.25906  42       0.3134121     6.74087  43       0.09924______________________________________ 
    
     EXAMPLE 11 
     Table 11 shows data for a pattern of the phase grating P according to example 11. The data is shown graphically in FIG. 14. In example 11, the phase gap ΔP is 1.04π. 
     
                       TABLE 11______________________________________ΔP = 1.04Coord. δ  Coord.   δ Coord. δ______________________________________0      5.72434  22       3.36629 44     2.813371      5.63396  23       3.48561 45     3.340122      5.50930  24       3.42313 46     3.804823      5.27261  25       3.12300 47     4.357584      4.88795  26       2.69584 48     4.969805      4.30602  27       2.35180 49     5.692876      3.64948  28       1.99120 50     6.462427      3.08115  29       1.63443 51     7.237848      2.55867  30       1.17193 52     7.818499      1.97143  31       0.85687 53     8.3050010     1.34005  32       0.75565 54     8.7997111     0.79516  33       0.92786 55     9.4498112     0.40111  34       1.14320 56     9.9326513     0.10653  35       1.34150 57     10.1575114     0.00000  36       1.34215 58     10.0603515     0.05143  37       1.13101 59     9.8089016     0.39145  38       0.74260 60     9.4886117     0.97952  39       0.50082 61     9.2776918     1.48361  40       0.45241 62     9.1167819     1.93317  41       0.64103 63     8.9858720     2.36975  42       1.0851121     2.92970  43       2.01522______________________________________ 
    
     EXAMPLE 12 
     Table 12 shows data for a pattern of the phase grating P according to examples 12. The data is shown graphically in FIG. 15. In example 12, the phase gap ΔP is 0.98π. 
     
                       TABLE 12______________________________________ΔP = .98Coord. δ  Coord.   δ Coord. δ______________________________________0      1.54498  22       5.12490 44     4.589321      1.53807  23       5.76799 45     4.085402      1.47715  24       6.51557 46     3.648503      1.24424  25       7.20367 47     3.385584      0.73233  26       7.74924 48     3.325675      0.22242  27       8.19234 49     3.458766      0.00000  28       8.70042 50     3.703857      0.04708  29       9.36852 51     3.916948      0.38968  30       9.97109 52     3.976029      1.19776  31       10.29019                            53     3.8351110     1.79235  32       10.33327                            54     3.2046911     1.98643  33       10.09636                            55     2.5427912     1.97353  34       9.53795 56     2.3883613     1.83361  35       8.96804 57     2.4914614     1.66370  36       8.57712 58     2.8580415     1.65178  37       8.20621 59     3.5231416     1.84488  38       7.75680 60     4.0822117     2.25997  39       7.15388 61     4.3963118     2.82905  40       6.55447 62     4.5553919     3.44314  41       6.03255 63     4.6089920     4.01223  42       5.5551521     4.55932  43       5.07323______________________________________ 
    
     Tables 13and 14 show the output intensity for the emitted beams of the beam splitter 10 in each of the above twelve examples as a relative intensity when the intensity of the incident beam is defined as 1. Further, an effective intensity represents a sum of the intensities of the intended emitted beams as a percentage of the incident beam. As explained above, the intended emitted beams are, for example, in example 1, the four emitted beams of order -1 to +2, or in example 2, the eight emitted beams of order -3 to +4. 
     FIGS. 16-27 show the data of Tables 13 and 14 graphically, the horizontal axis represents the order of the emitted diffracted beam and the vertical axis represents the intensity of each order where the intensity of the incident beam is defined as 1. 
     
                       TABLE 13______________________________________Order Ex. 1    Ex. 2   Ex. 3  Ex. 4 Ex. 5  Ex. 6______________________________________-10   0.00097  0.00396 0.00162                         0.00142                               0.00142                                      0.00123-9    0.00113  0.00850 0.00503                         0.00526                               0.00136                                      0.00568-8    0.00126  0.00762 0.00528                         0.00273                               0.00244                                      0.00207-7    0.00207  0.00171 0.00186                         0.00440                               0.00068                                      0.05752-6    0.00257  0.00093 0.00023                         0.00048                               0.00513                                      0.05869-5    0.00201  0.00156 0.00039                         0.00091                               0.00011                                      0.05867-4    0.00874  0.00584 0.00316                         0.00175                               0.00007                                      0.05900-3    0.01013  0.11779 0.11830                         0.11805                               0.12016                                      0.05797-2    0.00226  0.11917 0.11781                         0.11865                               0.11955                                      0.05872-1    0.22965  0.11753 0.11965                         0.11949                               0.12001                                      0.059280     0.23019  0.11745 0.11878                         0.11879                               0.12056                                      0.059751     0.23039  0.11841 0.11876                         0.11903                               0.12045                                      0.059702     0.22923  0.11717 0.11962                         0.11942                               0.11997                                      0.059073     0.00231  0.11643 0.11770                         0.11859                               0.11957                                      0.058754     0.01049  0.11663 0.11824                         0.11794                               0.12017                                      0.057985     0.00898  0.00126 0.00320                         0.00174                               0.00004                                      0.059116     0.00213  0.00462 0.00046                         0.00092                               0.00019                                      0.058617     0.00282  0.00004 0.00022                         0.00056                               0.00550                                      0.058828     0.00225  0.00227 0.00184                         0.00444                               0.00073                                      0.057509     0.00137  0.00003 0.00521                         0.00279                               0.00263                                      0.0025310    0.00133  0.00397 0.00499                         0.00527                               0.00158                                      0.00658Effec. 91.95%   94.06%  94.89% 95.00%                               96.04% 93.91%______________________________________ 
    
     
                       TABLE 14______________________________________Order Ex. 7    Ex. 8   Ex. 9  Ex. 10                               Ex. 11 Ex. 12______________________________________-10   0.00165  0.00019 0.00024                         0.00004                               0.00052                                      0.00020-9    0.00089  0.00053 0.00002                         0.00106                               0.00117                                      0.00017-8    0.00457  0.00028 0.00292                         0.00098                               0.00028                                      0.00092-7    0.06045  0.05997 0.06068                         0.06070                               0.06086                                      0.06125-6    0.06056  0.06019 0.06117                         0.06027                               0.06086                                      0.06103-5    0.06008  0.06018 0.06126                         0.06076                               0.06076                                      0.06091-4    0.06037  0.05995 0.06077                         0.06038                               0.06097                                      0.06116-3    0.06089  0.06015 0.06083                         0.06078                               0.06078                                      0.06105-2    0.06033  0.06017 0.06074                         0.06065                               0.06085                                      0.06115-1    0.06028  0.06058 0.06070                         0.06044                               0.06101                                      0.061220     0.06020  0.06056 0.06061                         0.06092                               0.06116                                      0.061071     0.06023  0.06050 0.06051                         0.06082                               0.06092                                      0.061052     0.06022  0.06060 0.06043                         0.06087                               0.06099                                      0.061183     0.06041  0.06108 0.06063                         0.06111                               0.06106                                      0.061154     0.06087  0.06074 0.06066                         0.06119                               0.06100                                      0.060955     0.06037  0.06055 0.06053                         0.06127                               0.06107                                      0.061106     9.06008  0.06092 0.06091                         0.06057                               0.06129                                      0.060937     0.06055  0.06035 0.06083                         0.06116                               0.06104                                      0.061028     0.06041  0.06129 0.06032                         0.06099                               0.06120                                      0.061189     0.00463  0.00144 0.00115                         0.00113                               0.00173                                      0.0011110    0.00086  0.00187 0.00028                         0.00021                               0.00035                                      0.00026Effec 96.63%   96.78%  97.16% 97.29%                               97.58% 97.74%______________________________________ 
    
     As shown in table 13 and 14, the effective intensity of the intended emitted beams is more than 91% in each example and reaches as high as 97.74%. 
     Tables 15 and 16 show the intensity of the intended emitted beams as a percentage of an ideal value. For instance, in example 1, the ideal value of each emitted beam is 0.25 (4 emitted beams are desired) where the intensity of the incident beam is 1, however, the actual intensity of the emitted beam of -1 order is 0.22965 so that the percentage of the -1 order beam is 92%. Similarly, the ideal values are 0.125 in examples 2-4 (8 emitted beams are desired) and 0.0625 in examples 6-12 (16 emitted beams are desired). 
     
                       TABLE 15______________________________________0rder   Ex. 1  Ex. 2    Ex. 3                        Ex. 4  Ex. 5                                    Ex. 6______________________________________-7                                       92%-6                                       94%-5                                       94%-4                                       94%-3             94%      95%  94%    96%  93%-2             95%      94%  95%    96%  94%-1      92%    94%      96%  96%    96%  95%0       92%    94%      95%  95%    96%  96%1       92%    95%      95%  95%    96%  96%2       92%    94%      96%  96%    96%  95%3              93%      94%  95%    96%  94%4              93%      95%  94%    96%  93%5                                        95%6                                        94%7                                        94%8                                        92%Δ 0%     2%       2%   2%     0%   4%______________________________________ 
    
     
                       TABLE 16______________________________________Order Ex. 7   Ex. 8   Ex. 9 Ex. 10 Ex. 11 Ex. 12______________________________________-7    97%     96%     97%   97%    97%    98%-6    97%     96%     98%   96%    97%    98%-5    96%     96%     98%   97%    97%    97%-4    97%     96%     97%   97%    98%    98%-3    97%     96%     97%   97%    97%    98%-2    97%     96%     97%   97%    97%    98%-1    96%     97%     97%   97%    98%    98%0     96%     97%     97%   97%    98%    98%1     96%     97%     97%   97%    97%    98%2     96%     97%     97%   97%    98%    98%3     97%     98%     97%   98%    98%    98%4     97%     97%     97%   98%    98%    98%5     97%     97%     97%   98%    98%    98%6     96%     97%     97%   97%    98%    97%7     97%     97%     97%   98%    98%    98%8     97%     98%     97%   98%    98%    98%Δ 1%      2%      1%    2%     1%     1%______________________________________ 
    
     As shown in Tables 15 and 16, the intensities of the intended emitted beams are in a range of 92%-98% in all examples. Further, a difference A between maximum and minimum percentage values is at most 4%. Accordingly, in these examples, the energy of the incident beam is effectively equally divided among the intended emitted beams. 
     The intensities shown in the above Tables and Figures represent ideal values. In considering some errors in the beam diffractive element, the efficiency will be reduced. In particular, if the error in the efficiency is to be under 10%, the permissible error in the pattern of the phase grating P is about 2%. For example, if the refractive index n of the environment is 1.5, the incident beam has a wavelength λ of 488 nm and is to be divided into 8 emitted beams every 0.0125 rad, the pattern has a length L along the y-axis of about 40 μm and a maximum height H along the z-axis of about 1 μμm. Here, the height H is defined as a difference between the highest and the lowest points in the phase pattern P. Thus, the permissible error in the height is only 0.02 μm. 
     However, the permissible error range can be extended. For example, if a difference between the refractive indexes of the phase grating P and the environment is decreased, the size of the phase pattern can be increased. That is, if the difference between refractive indexes is smaller, the height H can be larger. Accordingly, the required processing precision of the beam splitter 10 can be reduced. In a particular case, the concave surface 11a, including the. grating pattern 12, may be covered with a liquid layer having a refractive index that is almost equal to that of the grating pattern 12. 
     As described above, in each embodiment, the grating pattern is asymmetrical in the direction where the gratings are aligned. Thus the diffracted beams are not symmetrical with respect to zero order beam, and an even number of diffracted beams are generated. Further, the diffraction efficiency is raised by forming the grating pattern to have multi-level phase distribution, and energy of the incident beam is efficiently used. 
     It should be noted that the phase patterns described above should be optimized so that the diffracted beams consist substantially of a desired number (even number) of beams, and the desired number of beams have substantially the same intensities. 
     Although the structure and operation of a beam splitter is described herein with respect to the preferred embodiments, many modifications and changes can be made without departing from the spirit and scope of the invention. 
     For example, as an alternative, the phase patterns P may be formed as substantially indenting into the base 11 (as shown in FIG. 28) rather than as substantially protruding from the base 11(as shown in FIG. 2). For examples given above, the alternative forms for the phase patterns P can be obtained if the reference point (i.e., the 0 point) remains the same but each of the phase differences δ are defined as negative values. In other words, the cross-section of the phase pattern in the Y-Z plane can be considered to be rotated about the Y-axis by 180 degrees to produce a mirror image of the phase pattern. 
     The present disclosure relates to subject matter contained in Japanese Patent Application Nos. HEI 08-198271, filed on Jul. 9, 1996, and HEI 08-198272, filed on Jul. 9, 1996, which are expressly incorporated herein by reference in their entirety.