Abstract:
Disclosed is a diffractive optical element which includes a diffractive grating pattern ion 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;1.2π.

Description:
This is a division of U.S. patent application Ser. No. 08/890,429, filed Jul. 9, 1997, now U.S. Pat. No. 6,021,000 the contents of which are herein incorporated in its entirety. 
    
    
     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 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 gratings 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 distributed with respect to the zero order beam, are obtained. 
     It should 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 chase 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, add (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 bean 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  11   a  and a convex surface  11   b  and is represented as a portion of a wall of a cylinder. In FIG. 2, dotted lines show an imaginary cylinder with ends C 1  and C 2  having equal diameters representing a diameter of the concave surface  11   a . The grating pattern  12  is formed on the concave surface  11   a.    
     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  11   a.    
     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 C 1  and C 2 ) and the x-axis direction represents the curve of the concave surface  11   a.    
     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 phase 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  hen 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 δ 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 form 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  11   a . 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 example 1, the phase gap ΔP is 1.00π. Note that, the phase gap ΔP is defined as a difference between the phases at the coordinate 0 and the coordinate 63. 
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 (ΔP = 1.00π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 0.00000 
                 22 
                 2.28947 
                 44 
                 2.89397 
               
               
                 1 
                 0.04000 
                 23 
                 2.51947 
                 45 
                 2.79397 
               
               
                 2 
                 0.10042 
                 24 
                 2.69518 
                 46 
                 2.74279 
               
               
                 3 
                 0.16042 
                 25 
                 2.86518 
                 47 
                 2.69279 
               
               
                 4 
                 0.21520 
                 26 
                 3.02656 
                 48 
                 2.67291 
               
               
                 5 
                 0.25520 
                 27 
                 3.14656 
                 49 
                 2.65291 
               
               
                 6 
                 0.32315 
                 28 
                 3.25191 
                 50 
                 2.66056 
               
               
                 7 
                 0.38315 
                 29 
                 3.35191 
                 51 
                 2.67056 
               
               
                 8 
                 0.44920 
                 30 
                 3.41679 
                 52 
                 2.67518 
               
               
                 9 
                 0.51920 
                 31 
                 3.48679 
                 53 
                 2.69518 
               
               
                 10 
                 0.59142 
                 32 
                 3.52997 
                 54 
                 2.72447 
               
               
                 11 
                 0.66142 
                 33 
                 3.55997 
                 55 
                 2.75447 
               
               
                 12 
                 0.75100 
                 34 
                 3.59951 
                 56 
                 2.79033 
               
               
                 13 
                 0.84100 
                 35 
                 3.59951 
                 57 
                 2.82033 
               
               
                 14 
                 0.94020 
                 36 
                 3.58770 
                 58 
                 2.86386 
               
               
                 15 
                 1.05020 
                 37 
                 3.55770 
                 59 
                 2.89386 
               
               
                 16 
                 1.17834 
                 38 
                 3.50415 
                 60 
                 2.94434 
               
               
                 17 
                 1.30834 
                 39 
                 3.44415 
                 61 
                 2.98434 
               
               
                 18 
                 1.46586 
                 40 
                 3.33870 
                 62 
                 3.03420 
               
               
                 19 
                 1.64586 
                 41 
                 3.21870 
                 63 
                 3.13420 
               
               
                 20 
                 1.84933 
                 42 
                 3.10951 
               
               
                 21 
                 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 = 0.75) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 2.34750 
                 22 
                 2.33100 
                 44 
                 6.16100 
               
               
                 1 
                 2.11350 
                 23 
                 2.54700 
                 45 
                 6.29850 
               
               
                 2 
                 2.10350 
                 24 
                 2.76700 
                 46 
                 6.38850 
               
               
                 3 
                 2.11700 
                 25 
                 2.98600 
                 47 
                 6.46600 
               
               
                 4 
                 2.13200 
                 26 
                 3.23600 
                 48 
                 6.51600 
               
               
                 5 
                 2.14700 
                 27 
                 3.57850 
                 49 
                 6.53700 
               
               
                 6 
                 2.14700 
                 28 
                 4.10850 
                 50 
                 6.48700 
               
               
                 7 
                 2.10350 
                 29 
                 4.68100 
                 51 
                 6.38100 
               
               
                 8 
                 1.96350 
                 30 
                 5.10100 
                 52 
                 6.17100 
               
               
                 9 
                 1.32750 
                 31 
                 5.39450 
                 53 
                 5.85000 
               
               
                 10 
                 0.36750 
                 32 
                 5.57450 
                 54 
                 5.47000 
               
               
                 11 
                 0.07000 
                 33 
                 5.70650 
                 55 
                 5.22600 
               
               
                 12 
                 0.00000 
                 34 
                 5.79650 
                 56 
                 5.06600 
               
               
                 13 
                 0.02550 
                 35 
                 5.84250 
                 57 
                 4.99800 
               
               
                 14 
                 0.12550 
                 36 
                 5.88250 
                 58 
                 4.94800 
               
               
                 15 
                 0.23800 
                 37 
                 5.89250 
                 59 
                 4.93550 
               
               
                 16 
                 0.40800 
                 38 
                 5.89250 
                 60 
                 4.92550 
               
               
                 17 
                 0.65600 
                 39 
                 5.89650 
                 61 
                 4.91000 
               
               
                 18 
                 0.96600 
                 40 
                 5.90650 
                 62 
                 4.89000 
               
               
                 19 
                 1.34000 
                 41 
                 5.91450 
                 63 
                 4.70250 
               
               
                 20 
                 1.72000 
                 42 
                 5.96450 
               
               
                 21 
                 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 = 0.99π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 0.00000 
                 22 
                 5.53800 
                 44 
                 5.95500 
               
               
                 1 
                 0.04000 
                 23 
                 5.69800 
                 45 
                 5.74500 
               
               
                 2 
                 0.08700 
                 24 
                 5.81200 
                 46 
                 5.57400 
               
               
                 3 
                 0.16700 
                 25 
                 5.87200 
                 47 
                 5.43400 
               
               
                 4 
                 0.24600 
                 26 
                 5.88400 
                 48 
                 5.33000 
               
               
                 5 
                 0.35600 
                 27 
                 5.85400 
                 49 
                 5.23000 
               
               
                 6 
                 0.45900 
                 28 
                 5.78500 
                 50 
                 5.11400 
               
               
                 7 
                 0.63900 
                 29 
                 5.70500 
                 51 
                 4.94400 
               
               
                 8 
                 0.88800 
                 30 
                 5.63700 
                 52 
                 4.63400 
               
               
                 9 
                 1.31800 
                 31 
                 5.63700 
                 53 
                 4.11400 
               
               
                 10 
                 1.99700 
                 32 
                 5.67800 
                 54 
                 3.52200 
               
               
                 11 
                 2.65700 
                 33 
                 5.79800 
                 55 
                 3.20200 
               
               
                 12 
                 3.03300 
                 34 
                 5.95300 
                 56 
                 3.03800 
               
               
                 13 
                 3.30300 
                 35 
                 6.12300 
                 57 
                 2.95800 
               
               
                 14 
                 3.52200 
                 36 
                 6.28800 
                 58 
                 2.94800 
               
               
                 15 
                 3.71200 
                 37 
                 6.41800 
                 59 
                 5.98500 
               
               
                 16 
                 3.91200 
                 38 
                 6.51300 
                 60 
                 2.96900 
               
               
                 17 
                 4.15200 
                 39 
                 6.55300 
                 61 
                 2.98900 
               
               
                 18 
                 4.41000 
                 40 
                 6.53000 
                 62 
                 3.03300 
               
               
                 19 
                 4.73000 
                 41 
                 6.47000 
                 63 
                 3.09800 
               
               
                 20 
                 5.03600 
                 42 
                 6.35300 
               
               
                 21 
                 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 = 0.99π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 1.57900 
                 22 
                 2.15900 
                 44 
                 2.38400 
               
               
                 1 
                 1.63900 
                 23 
                 2.39900 
                 45 
                 2.11400 
               
               
                 2 
                 1.66700 
                 24 
                 2.60700 
                 46 
                 1.93200 
               
               
                 3 
                 1.69700 
                 25 
                 2.81700 
                 47 
                 1.81200 
               
               
                 4 
                 1.73900 
                 26 
                 3.04600 
                 48 
                 1.74600 
               
               
                 5 
                 1.74400 
                 27 
                 3.25600 
                 49 
                 1.75600 
               
               
                 6 
                 1.73000 
                 28 
                 3.49200 
                 50 
                 1.82500 
               
               
                 7 
                 1.70000 
                 29 
                 3.71200 
                 51 
                 1.96500 
               
               
                 8 
                 1.55400 
                 30 
                 3.90500 
                 52 
                 2.26500 
               
               
                 9 
                 1.25400 
                 31 
                 4.02500 
                 53 
                 2.82500 
               
               
                 10 
                 0.71100 
                 32 
                 4.06700 
                 54 
                 3.47800 
               
               
                 11 
                 0.25100 
                 33 
                 4.04700 
                 55 
                 3.87800 
               
               
                 12 
                 0.05000 
                 34 
                 3.97200 
                 56 
                 4.09500 
               
               
                 13 
                 0.00000 
                 35 
                 3.84200 
                 57 
                 4.23500 
               
               
                 14 
                 0.04100 
                 36 
                 3.70100 
                 58 
                 4.34700 
               
               
                 15 
                 0.12100 
                 37 
                 3.57100 
                 59 
                 4.43700 
               
               
                 16 
                 0.28200 
                 38 
                 3.44500 
                 60 
                 4.50400 
               
               
                 17 
                 0.50200 
                 39 
                 3.33500 
                 61 
                 4.55400 
               
               
                 18 
                 0.80200 
                 40 
                 3.22000 
                 62 
                 4.62600 
               
               
                 19 
                 1.15200 
                 41 
                 3.09000 
                 63 
                 4.68600 
               
               
                 20 
                 1.52900 
                 42 
                 2.90600 
               
               
                 21 
                 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 1.00λ. 
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 (ΔP = 1.00π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 0.00000 
                 22 
                 5.13656 
                 44 
                 5.80812 
               
               
                 1 
                 0.03494 
                 23 
                 5.21307 
                 45 
                 5.60323 
               
               
                 2 
                 0.08372 
                 24 
                 5.20778 
                 46 
                 5.37918 
               
               
                 3 
                 0.17593 
                 25 
                 5.13788 
                 47 
                 5.12460 
               
               
                 4 
                 0.29127 
                 26 
                 4.97072 
                 48 
                 4.88807 
               
               
                 5 
                 0.41957 
                 27 
                 4.73558 
                 49 
                 4.68950 
               
               
                 6 
                 0.57022 
                 28 
                 4.48842 
                 50 
                 4.48952 
               
               
                 7 
                 0.77908 
                 29 
                 4.32697 
                 51 
                 4.29759 
               
               
                 8 
                 1.04067 
                 30 
                 4.24668 
                 52 
                 4.05168 
               
               
                 9 
                 1.33889 
                 31 
                 4.24576 
                 53 
                 3.80124 
               
               
                 10 
                 1.70048 
                 32 
                 4.27485 
                 54 
                 3.56782 
               
               
                 11 
                 2.04910 
                 33 
                 4.39394 
                 55 
                 3.34777 
               
               
                 12 
                 2.38318 
                 34 
                 4.58241 
                 56 
                 3.19436 
               
               
                 13 
                 2.68329 
                 35 
                 4.84203 
                 57 
                 3.08368 
               
               
                 14 
                 2.97144 
                 36 
                 5.19737 
                 58 
                 3.01120 
               
               
                 15 
                 3.26819 
                 37 
                 5.51068 
                 59 
                 2.97107 
               
               
                 16 
                 3.58289 
                 38 
                 5.77602 
                 60 
                 2.96391 
               
               
                 17 
                 3.93565 
                 39 
                 5.94409 
                 61 
                 2.98987 
               
               
                 18 
                 4.28787 
                 40 
                 6.03755 
                 62 
                 2.99927 
               
               
                 19 
                 4.60094 
                 41 
                 6.07922 
                 63 
                 3.13750 
               
               
                 20 
                 4.85049 
                 42 
                 6.05674 
               
               
                 21 
                 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 = 1.01π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 0.00000 
                 22 
                 9.01142 
                 44 
                 7.39884 
               
               
                 1 
                 0.11909 
                 23 
                 9.28050 
                 45 
                 7.47793 
               
               
                 2 
                 0.38167 
                 24 
                 9.40910 
                 46 
                 7.70302 
               
               
                 3 
                 0.86576 
                 25 
                 9.51318 
                 47 
                 8.32711 
               
               
                 4 
                 1.60085 
                 26 
                 9.57577 
                 48 
                 8.88019 
               
               
                 5 
                 2.24994 
                 27 
                 9.42985 
                 49 
                 8.91927 
               
               
                 6 
                 2.70552 
                 28 
                 8.82495 
                 50 
                 8.63937 
               
               
                 7 
                 3.15461 
                 29 
                 8.55903 
                 51 
                 8.24345 
               
               
                 8 
                 3.79220 
                 30 
                 8.57712 
                 52 
                 7.86954 
               
               
                 9 
                 4.65129 
                 31 
                 8.64120 
                 53 
                 7.49862 
               
               
                 10 
                 5.33787 
                 32 
                 8.69029 
                 54 
                 6.90022 
               
               
                 11 
                 5.83696 
                 33 
                 8.68438 
                 55 
                 6.09930 
               
               
                 12 
                 6.33905 
                 34 
                 8.73447 
                 56 
                 5.52989 
               
               
                 13 
                 6.84314 
                 35 
                 9.13856 
                 57 
                 5.20897 
               
               
                 14 
                 7.22122 
                 36 
                 9.91164 
                 58 
                 4.85157 
               
               
                 15 
                 7.25031 
                 37 
                 10.12573 
                 59 
                 4.33065 
               
               
                 16 
                 6.79540 
                 38 
                 10.16132 
                 60 
                 3.66374 
               
               
                 17 
                 6.30948 
                 39 
                 10.14541 
                 61 
                 3.27782 
               
               
                 18 
                 6.15257 
                 40 
                 10.08499 
                 62 
                 3.13342 
               
               
                 19 
                 6.17165 
                 41 
                 9.91408 
                 63 
                 3.17750 
               
               
                 20 
                 6.27975 
                 42 
                 8.31967 
               
               
                 21 
                 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 = 0.98π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 0.00000 
                 22 
                 9.18279 
                 44 
                 8.33523 
               
               
                 1 
                 0.10112 
                 23 
                 9.81372 
                 45 
                 7.85733 
               
               
                 2 
                 0.21482 
                 24 
                 10.31935 
                 46 
                 7.25375 
               
               
                 3 
                 0.46317 
                 25 
                 10.74935 
                 47 
                 6.51311 
               
               
                 4 
                 0.87331 
                 26 
                 11.07071 
                 48 
                 5.96011 
               
               
                 5 
                 1.39192 
                 27 
                 11.23482 
                 49 
                 5.55847 
               
               
                 6 
                 1.73829 
                 28 
                 11.08046 
                 50 
                 5.07889 
               
               
                 7 
                 1.78828 
                 29 
                 10.43581 
                 51 
                 4.47400 
               
               
                 8 
                 1.74891 
                 30 
                 9.96051 
                 52 
                 4.01530 
               
               
                 9 
                 1.68984 
                 31 
                 9.86263 
                 53 
                 3.87218 
               
               
                 10 
                 1.76142 
                 32 
                 9.91072 
                 54 
                 3.90177 
               
               
                 11 
                 2.02072 
                 33 
                 10.11877 
                 55 
                 4.04902 
               
               
                 12 
                 2.55159 
                 34 
                 10.68425 
                 56 
                 4.20056 
               
               
                 13 
                 3.28166 
                 35 
                 11.42907 
                 57 
                 4.21774 
               
               
                 14 
                 3.83542 
                 36 
                 11.68461 
                 58 
                 4.00155 
               
               
                 15 
                 4.34223 
                 37 
                 11.60267 
                 59 
                 3.56311 
               
               
                 16 
                 4.99540 
                 38 
                 11.38748 
                 60 
                 3.24915 
               
               
                 17 
                 5.83822 
                 39 
                 11.05666 
                 61 
                 3.11798 
               
               
                 18 
                 6.55497 
                 40 
                 10.65121 
                 62 
                 3.08845 
               
               
                 19 
                 7.10304 
                 41 
                 10.11445 
                 63 
                 3.09300 
               
               
                 20 
                 7.68791 
                 42 
                 9.43005 
               
               
                 21 
                 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 
               
             
             
               
                   
               
               
                 (Δ P = 1.14) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
               
                  0 
                 0.00000 
                 22 
                 8.14593 
                 44 
                 6.38976 
               
               
                  1 
                 0.35675 
                 23 
                 8.06924 
                 45 
                 6.15963 
               
               
                  2 
                 0.59145 
                 24 
                 7.78458 
                 46 
                 5.82247 
               
               
                  3 
                 1.02421 
                 25 
                 7.15265 
                 47 
                 5.41843 
               
               
                  4 
                 1.76643 
                 26 
                 6.44611 
                 48 
                 4.98783 
               
               
                  5 
                 2.45950 
                 27 
                 5.97778 
                 49 
                 4.50606 
               
               
                  6 
                 2.97905 
                 28 
                 5.63530    50 
                 4.05856 
               
               
                  7 
                 3.46447 
                 29 
                 5.27806 
                 51 
                 3.78350 
               
               
                  8 
                 4.30512 
                 30 
                 4.81668 
                 52 
                 3.74228 
               
               
                  9 
                 5.26163 
                 31 
                 4.14179 
                 53 
                 4.15449 
               
               
                 10 
                 5.78634 
                 32 
                 3.46774 
                 54 
                 5.00983 
               
               
                 11 
                 6.07644 
                 33 
                 3.08316 
                 55 
                 5.34813 
               
               
                 12 
                 6.23928 
                 34 
                 2.88663 
                 56 
                 5.39878 
               
               
                 13 
                 6.33414 
                 35 
                 2.84806 
                 57 
                 5.22764 
               
               
                 14 
                 6.27698 
                 36 
                 3.03808 
                 58 
                 4.60923 
               
               
                 15 
                 6.11553 
                 37 
                 3.51615 
                 59 
                 3.77745 
               
               
                 16 
                 5.97524 
                 38 
                 4.17204 
                 60 
                 3.46904 
               
               
                 17 
                 5.92432 
                 39 
                 4.54980 
                 61 
                 3.39766 
               
               
                 18 
                 6.05341 
                 40 
                 4.92638 
                 62 
                 3.41174 
               
               
                 19 
                 6.40250 
                 41 
                 5.62633 
                 63 
                 3.59185 
               
               
                 20 
                 7.41097 
                 42 
                 6.27292 
               
               
                 21 
                 8.03059 
                 43 
                 6.49224 
               
               
                   
               
             
          
         
       
     
     EXAMPLE 9 
     Table 9 shows data for a pattern of the phase grating P according to example 9. The data is shown graphically in FIG.  12 . In example 9, the phase gap ΔP is 0.86π. 
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 9 
               
             
             
               
                   
               
               
                 (ΔP = 0.86π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 0.00000 
                 22 
                 10.41512 
                 44 
                 11.31028 
               
               
                 1 
                 0.02139 
                 23 
                 10.45448 
                 45 
                 11.04914 
               
               
                 2 
                 0.24455 
                 24 
                 10.66319 
                 46 
                 10.70583 
               
               
                 3 
                 0.63197 
                 25 
                 11.05519 
                 47 
                 10.49409 
               
               
                 4 
                 1.25970 
                 26 
                 11.52474 
                 48 
                 10.31520 
               
               
                 5 
                 1.93395 
                 27 
                 11.90449 
                 49 
                 10.09816 
               
               
                 6 
                 2.45426 
                 28 
                 12.17922 
                 50 
                 9.41989 
               
               
                 7 
                 2.94875 
                 29 
                 12.38914 
                 51 
                 8.18725 
               
               
                 8 
                 3.67947 
                 30 
                 12.57125 
                 52 
                 7.75399 
               
               
                 9 
                 4.63407 
                 31 
                 12.71915 
                 53 
                 7.51219 
               
               
                 10 
                 5.28644 
                 32 
                 12.78723 
                 54 
                 7.14200 
               
               
                 11 
                 5.75541 
                 33 
                 12.73151 
                 55 
                 6.22557 
               
               
                 12 
                 6.29385 
                 34 
                 12.67358 
                 56 
                 5.34703 
               
               
                 13 
                 7.22466 
                 35 
                 12.59483 
                 57 
                 4.96071 
               
               
                 14 
                 8.10410 
                 36 
                 12.48228 
                 58 
                 4.61458 
               
               
                 15 
                 8.59831 
                 37 
                 12.25170 
                 59 
                 4.07851 
               
               
                 16 
                 8.96738 
                 38 
                 11.83332 
                 60 
                 3.42295 
               
               
                 17 
                 9.36434 
                 39 
                 11.38850 
                 61 
                 3.06471 
               
               
                 18 
                 9.85078 
                 40 
                 11.15996 
                 62 
                 2.90472 
               
               
                 19 
                 10.26310 
                 41 
                 11.13878 
                 63 
                 2.69100 
               
               
                 20 
                 10.46203 
                 42 
                 11.22934 
               
               
                 21 
                 10.45751 
                 43 
                 11.33304 
               
               
                   
               
             
          
         
       
     
     EXAMPLE 10 
     Table 10 shows data for a pattern of the phase grating P according to example 10. The data is shown graphically in FIG.  13 . In example 10, the phase gap ΔP is 1.07π. 
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 10 
               
             
             
               
                   
               
               
                 (ΔP = 1.07π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 1.88359 
                 22 
                 6.11453 
                 44 
                 0.00000 
               
               
                 1 
                 2.20660 
                 23 
                 5.47716 
                 45 
                 0.11027 
               
               
                 2 
                 2.55505 
                 24 
                 4.92379 
                 46 
                 0.55334 
               
               
                 3 
                 2.96194 
                 25 
                 4.32768 
                 47 
                 1.03784 
               
               
                 4 
                 3.50884 
                 26 
                 3.82595 
                 48 
                 1.31994 
               
               
                 5 
                 4.12613 
                 27 
                 3.35523 
                 49 
                 1.57085 
               
               
                 6 
                 4.76767 
                 28 
                 2.90596 
                 50 
                 1.87626 
               
               
                 7 
                 5.47822 
                 29 
                 2.32512 
                 51 
                 2.43060 
               
               
                 8 
                 6.13673 
                 30 
                 1.74405 
                 52 
                 3.02974 
               
               
                 9 
                 6.68710 
                 31 
                 1.41813 
                 53 
                 3.35112 
               
               
                 10 
                 7.20112 
                 32 
                 1.33906 
                 54 
                 3.36775 
               
               
                 11 
                 7.89495 
                 33 
                 1.50709 
                 55 
                 3.30493 
               
               
                 12 
                 9.04040 
                 34 
                 1.88758 
                 56 
                 3.22809 
               
               
                 13 
                 9.57330 
                 35 
                 2.18736 
                 57 
                 3.38151 
               
               
                 14 
                 9.71868 
                 36 
                 2.27226 
                 58 
                 3.63778 
               
               
                 15 
                 9.62977 
                 37 
                 2.04002 
                 59 
                 3.99318 
               
               
                 16 
                 9.28356 
                 38 
                 1.53403 
                 60 
                 4.32386 
               
               
                 17 
                 8.61084 
                 39 
                 1.01003 
                 61 
                 4.56700 
               
               
                 18 
                 8.05756 
                 40 
                 0.69457 
                 62 
                 4.81525 
               
               
                 19 
                 7.66091 
                 41 
                 0.47837 
                 63 
                 5.24823 
               
               
                 20 
                 7.25906 
                 42 
                 0.31341 
               
               
                 21 
                 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 = 1.04π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 5.72434 
                 22 
                 3.36629 
                 44 
                 2.81337 
               
               
                 1 
                 5.63396 
                 23 
                 3.48561 
                 45 
                 3.34012 
               
               
                 2 
                 5.50930 
                 24 
                 3.42313 
                 46 
                 3.80482 
               
               
                 3 
                 5.27261 
                 25 
                 3.12300 
                 47 
                 4.35758 
               
               
                 4 
                 4.88795 
                 26 
                 2.69584 
                 48 
                 4.96980 
               
               
                 5 
                 4.30602 
                 27 
                 2.35180 
                 49 
                 5.69287 
               
               
                 6 
                 3.64948 
                 28 
                 1.99120 
                 50 
                 6.46242 
               
               
                 7 
                 3.08115 
                 29 
                 1.63443 
                 51 
                 7.23784 
               
               
                 8 
                 2.55867 
                 30 
                 1.17193 
                 52 
                 7.81849 
               
               
                 9 
                 1.97143 
                 31 
                 0.85687 
                 53 
                 8.30500 
               
               
                 10 
                 1.34005 
                 32 
                 0.75565 
                 54 
                 8.79971 
               
               
                 11 
                 0.79516 
                 33 
                 0.92786 
                 55 
                 9.44981 
               
               
                 12 
                 0.40111 
                 34 
                 1.14320 
                 56 
                 9.93265 
               
               
                 13 
                 0.10653 
                 35 
                 1.34150 
                 57 
                 10.15751 
               
               
                 14 
                 0.00000 
                 36 
                 1.34215 
                 58 
                 10.06035 
               
               
                 15 
                 0.05143 
                 37 
                 1.13101 
                 59 
                 9.80890 
               
               
                 16 
                 0.39145 
                 38 
                 0.74260 
                 60 
                 9.48861 
               
               
                 17 
                 0.97952 
                 39 
                 0.50082 
                 61 
                 9.27769 
               
               
                 18 
                 1.48361 
                 40 
                 0.45241 
                 62 
                 9.11678 
               
               
                 19 
                 1.93317 
                 41 
                 0.64103 
                 63 
                 8.98587 
               
               
                 20 
                 2.36975 
                 42 
                 1.08511 
               
               
                 21 
                 2.92970 
                 43 
                 2.01522 
               
               
                   
               
             
          
         
       
     
     EXAMPLE 12 
     Table 12 shows data for a pattern of the phase grating P according to example 12. The data is shown graphically in FIG.  15 . In example 12, the phase gap ΔP is 0.98π. 
     
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 12 
               
             
             
               
                   
               
               
                 (ΔP = 0.98π) 
               
             
          
           
               
                 Coord. 
                 δ 
                 Coord. 
                 δ 
                 Coord. 
                 δ 
               
               
                   
               
             
          
           
               
                 0 
                 1.54498 
                 22 
                 5.12490 
                 44 
                 4.58932 
               
               
                 1 
                 1.53807 
                 23 
                 5.76799 
                 45 
                 4.08540 
               
               
                 2 
                 1.47715 
                 24 
                 6.51557 
                 46 
                 3.64850 
               
               
                 3 
                 1.24424 
                 25 
                 7.20367 
                 47 
                 3.38558 
               
               
                 4 
                 0.73233 
                 26 
                 7.74924 
                 48 
                 3.32567 
               
               
                 5 
                 0.22242 
                 27 
                 8.19234 
                 49 
                 3.45876 
               
               
                 6 
                 0.00000 
                 28 
                 8.70042 
                 50 
                 3.70385 
               
               
                 7 
                 0.04708 
                 29 
                 9.36852 
                 51 
                 3.91694 
               
               
                 8 
                 0.38968 
                 30 
                 9.97109 
                 52 
                 3.97602 
               
               
                 9 
                 1.19776 
                 31 
                 10.29019 
                 53 
                 3.83511 
               
               
                 10 
                 1.79235 
                 32 
                 10.33327 
                 54 
                 3.20469 
               
               
                 11 
                 1.98643 
                 33 
                 10.09636 
                 55 
                 2.54279 
               
               
                 12 
                 1.97353 
                 34 
                 9.53795 
                 56 
                 2.38836 
               
               
                 13 
                 1.83361 
                 35 
                 8.96804 
                 57 
                 2.49146 
               
               
                 14 
                 1.66370 
                 36 
                 8.57712 
                 58 
                 2.85804 
               
               
                 15 
                 1.65178 
                 37 
                 8.20621 
                 59 
                 3.52314 
               
               
                 16 
                 1.84488 
                 38 
                 7.75680 
                 60 
                 4.08221 
               
               
                 17 
                 2.25997 
                 39 
                 7.15388 
                 61 
                 4.39631 
               
               
                 18 
                 2.82905 
                 40 
                 6.55447 
                 62 
                 4.55539 
               
               
                 19 
                 3.44314 
                 41 
                 6.03255 
                 63 
                 4.60899 
               
               
                 20 
                 4.01223 
                 42 
                 5.55515 
               
               
                 21 
                 4.55932 
                 43 
                 5.07323 
               
               
                   
               
             
          
         
       
     
     Tables 13 and 14 show the output intensity for the emitted beams of the beams 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 
                 0.05928 
               
               
                 0 
                 0.23019 
                 0.11745 
                 0.11878 
                 0.11879 
                 0.12056 
                 0.05975 
               
               
                 1 
                 0.23039 
                 0.11841 
                 0.11876 
                 0.11903 
                 0.12045 
                 0.05970 
               
               
                 2 
                 0.22923 
                 0.11717 
                 0.11962 
                 0.11942 
                 0.11997 
                 0.05907 
               
               
                 3 
                 0.00231 
                 0.11643 
                 0.11770 
                 0.11859 
                 0.11957 
                 0.05875 
               
               
                 4 
                 0.01049 
                 0.11663 
                 0.11824 
                 0.11794 
                 0.12017 
                 0.05798 
               
               
                 5 
                 0.00898 
                 0.00126 
                 0.00320 
                 0.00174 
                 0.00004 
                 0.05911 
               
               
                 6 
                 0.00213 
                 0.00462 
                 0.00046 
                 0.00092 
                 0.00019 
                 0.05861 
               
               
                 7 
                 0.00282 
                 0.00004 
                 0.00022 
                 0.00056 
                 0.00550 
                 0.05882 
               
               
                 8 
                 0.00225 
                 0.00227 
                 0.00184 
                 0.00444 
                 0.00073 
                 0.05750 
               
               
                 9 
                 0.00137 
                 0.00003 
                 0.00521 
                 0.00279 
                 0.00263 
                 0.00253 
               
               
                 10 
                 0.00133 
                 0.00397 
                 0.00499 
                 0.00527 
                 0.00158 
                 0.00658 
               
             
          
           
               
                 Effec. 
                 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 
                 0.06122 
               
               
                 0 
                 0.06020 
                 0.06056 
                 0.06061 
                 0.06092 
                 0.06116 
                 0.06107 
               
               
                 1 
                 0.06023 
                 0.06050 
                 0.06051 
                 0.06082 
                 0.06092 
                 0.06105 
               
               
                 2 
                 0.06022 
                 0.06060 
                 0.06043 
                 0.06087 
                 0.06099 
                 0.06113 
               
               
                 3 
                 0.06041 
                 0.06108 
                 0.06063 
                 0.06111 
                 0.06106 
                 0.06115 
               
               
                 4 
                 0.06087 
                 0.06074 
                 0.06066 
                 0.06119 
                 0.06100 
                 0.06095 
               
               
                 5 
                 0.06037 
                 0.06055 
                 0.06053 
                 0.06127 
                 0.06107 
                 0.06110 
               
               
                 6 
                 0.06008 
                 0.06092 
                 0.06091 
                 0.06057 
                 0.06129 
                 0.06093 
               
               
                 7 
                 0.06055 
                 0.06035 
                 0.06083 
                 0.06116 
                 0.06104 
                 0.06102 
               
               
                 8 
                 0.06041 
                 0.06129 
                 0.06032 
                 0.06099 
                 0.06120 
                 0.06118 
               
               
                 9 
                 0.00463 
                 0.00144 
                 0.00115 
                 0.00113 
                 0.00173 
                 0.00111 
               
               
                 10 
                 0.00086 
                 0.00187 
                 0.00028 
                 0.00021 
                 0.00035 
                 0.00026 
               
             
          
           
               
                 Effec 
                 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 
               
               
                   
               
               
                 Order 
                 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 Δ 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 examples 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 axis of about 40 μm and a maximum height H along the z-axis of about 1 μm. 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  11   a , 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.