Abstract:
The invention provides a method for producing a diffractive optical element, including forming first and second gratings, substantially in mutual registration, of at least first and second optical materials, such that for predetermined two or more wavelengths, the diffractive optical element has desired phase retardations, thereby avoiding the generation of chromatic aberrations. The invention further provides a diffractive optical element.

Description:
FIELD OF THE INVENTION 
     The present invention relates generally to optical components and systems, and specifically to diffractive optical elements and optical phase elements. 
     BACKGROUND OF THE INVENTION 
     Optical imaging systems using lenses or materials transmitting light in general have to deal with the problem of dispersion of the light. Dispersion is the change of light velocity with wavelength, and dispersion causes chromatic aberrations in the optical system by creating different refractive indices and different phase changes for different wavelengths. 
     The phase delay φ in light of wavelength λ introduced by a thickness d of material of refractive index n is given by:              φ   =         2      π     λ          (     n   -   1     )        d               (   1   )     .                                
     Diffractive systems, for directing and focusing beams of light, are well-known in the art, as described, for example, by Francis Jenkins and Harvey White in  Fundamentals of Optics , Fourth Edition, pp. 385-386 (1981), which is incorporated herein by reference. The theory of diffractive optical elements (DOEs) is further described, for example, by Nieuborg, et al., in an article entitled “Polarization-Selective Diffractive Lenslet Arrays,” published in  European Optical Society Topical Meeting Digest Series , Vol. 5 (1995), which is also incorporated herein by reference. DOEs are generally highly dispersive. 
     Zone plates are a well-known type of DOE, typically comprising concentric rings having radii proportional to the square roots of the whole numbers and a phase retardation varying by π between neighboring rings. When a beam of collimated light is incident on the zone plate, it will be diffracted to a focal point. Like other DOEs and computer-generated holograms known in the art, however, zone plates known in the art are highly chromatically dispersive. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide methods for designing and producing diffractive optical elements having a desired wavelength dispersion. 
     It is a further object of the present invention to provide optical elements designed and/or produced in accordance with these methods. 
     In one aspect of the present invention, diffractive optical elements are designed and produced so as to be substantially achromatic, i.e., so that their dispersion is effectively minimized. 
     It is another object of the present invention to provide diffractive optical elements which compensate for chromatic aberrations. Thus, in one aspect of the present invention, diffractive optical elements are applied to a refractive element, such as a lens, to reduce or remove the chromatic aberrations present because of the dispersive properties of the refractive element. 
     It is a still further object of the present invention to provide an aberration-corrected, diffractive optical element and a method for producing same for a wideband light source. 
     In preferred embodiments of the present invention, a DOE comprises first and second gratings substantially in mutual registration. The gratings are made of respective first and second materials having respective first and second refractive indices. The dimensions of the gratings, in particular their thicknesses, are chosen so as to give a desired value of wavelength dispersion and/or phase retardation. Preferably, the refractive indices and the dimensions of the gratings are chosen so as to reduce or minimize chromatic aberrations produced by the DOE or by an optical system including the DOE. Such embodiments differ from DOEs known in the art, which typically are designed for a narrow wavelength range and are highly wavelength dispersive. 
     In some preferred embodiments of the present invention, a DOE is formed on a surface of a lens by etching the first grating into the surface of the lens, overlaying the grating with the second material having a different refractive index from that of the lens, and then etching the second grating into the second material, in registration with the first grating. Alternatively, the second material may first be overlaid on the surface of the lens, and then both the first and second gratings may be etched simultaneously. The DOE preferably corrects for the chromatic dispersion of the lens. 
     In other preferred embodiments of the present invention, the DOE is formed on a surface by overlaying the surface with a layer of the first optical material, overlaying the first optical material with a layer of the second optical material, and etching the gratings in registration into both optical materials. 
     In still other preferred embodiments of the present invention, a DOE is formed by etching the first grating into a first surface of a flat plate formed from two materials having different refractive indices, and the second grating is etched in registration with the first grating into the second, opposite surface of the flat plate. DOEs in accordance with these preferred embodiments may be designed to function as substantially dispersionless diffraction gratings or diffractive focusing elements. 
     In still other preferred embodiments of the present invention, a DOE is formed by combining in registration a plurality of DOEs, each DOE of the plurality being formed as described above. 
     In some preferred embodiments of the present invention, the DOE is generated by computer calculation and fabricated, alternatively or in combination, by a lithographic process or a plasma-etch process. 
     The principles of the present invention may thus be applied both in diffractive optical systems and in hybrid systems that mix diffractive and refractive optics, using any suitable methods of design and fabrication known in the art. Although the preferred embodiments described herein relate primarily to achromatization of such optical systems, the principles of the present invention may be applied more generally to control the chromatic response of such systems in substantially any desired manner, for example, to create a desired chromatic dispersion. 
     Although preferred embodiments are largely described herein with reference to optical wavelengths of radiation, it will be appreciated that similar embodiments of the present invention may generally be constructed utilizing materials operating at non-optical wavelengths such as ultraviolet, infrared, microwave and radio wavelengths. 
     Furthermore, although preferred embodiments are described herein with reference to two layers of gratings, it will be appreciated by those skilled in the art that similar preferred embodiments may generally be constructed using additional layers of gratings. 
     There is therefore provided, in accordance with a preferred embodiment of the present invention, a method for producing a diffractive optical element, comprising forming first and second gratings, substantially in mutual registration, of respective at least first and second optical materials, such that for predetermined two or more wavelengths, the diffractive optical element has phase retardations in a desired mutual relation, thereby avoiding the generation of chromatic aberrations. 
     The invention further provides a diffractive optical element, comprising first and second optical materials in which respective first and second phase gratings are formed substantially in mutual registration, such that for predetermined wavelengths the diffractive optical element has desired phase retardations, whereby generation of chromatic aberration is avoided. 
     The invention will now be described in connection with certain preferred embodiments with reference to the following illustrative figures so that it may be more fully understood. 
    
    
     With specific reference now to the figures in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of the preferred embodiments of the present invention only, and are presented in the cause of providing what is believed to be the most useful and readily understood description of the principles and conceptual aspects of the invention. In this regard, no attempt is made to show structural details of the invention in more detail than is necessary for a fundamental understanding of the invention, the description taken with the drawings making apparent to those skilled in the art how the several forms of the invention may be embodied in practice. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1A is a schematic, sectional illustration of a DOE in accordance with a preferred embodiment of the present invention; 
     FIG. 1B is a schematic, sectional illustration of a detail of an element of FIG. 1A; 
     FIG. 1C is a schematic, sectional illustration of a DOE with gratings aligned in a different manner; 
     FIG. 2 is a graph showing phase change as a function of wavelength generated by an element of the DOE of FIG. 1A; 
     FIG. 3 is a schematic, sectional illustration of a DOE in accordance with another preferred embodiment of the present invention; 
     FIG. 4A is a schematic, sectional illustration of a lens having a DOE formed in a surface thereof, in accordance with a preferred embodiment of the present invention; 
     FIG. 4B is a schematic, sectional illustration of a detail of an element of the lens of FIG. 4A; 
     FIG. 5A is a schematic, sectional illustration of another lens having a DOE formed in a surface thereof, in accordance with a preferred embodiment of the present invention; 
     FIG. 5B is a schematic, sectional illustration of a detail of an element of the lens of FIG. 5A; 
     FIG. 6 is a schematic, sectional illustration of the DOE of FIG. 1C for chromatic aberration correction for two wavelengths; 
     FIG. 7 is a schematic, sectional illustration of a DOE etched on two sides for chromatic aberration correction for two wavelengths, and 
     FIG. 8 is a graph showing phase changes as a function of wavelengths generated by an element of the DOE of FIG.  7 . 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     Reference is now made to FIGS. 1A and 1B, which respectively illustrate a generalized representation of a DOE  10  in sectional view and a detail showing a single element  12  thereof. DOE  10  preferably comprises a computer-generated hologram, but the description that follows is similarly applicable to other types of DOEs. DOE  10  is generally made up of a grid of elements  12 , also referred to as pixels, which are preferably rectangular when viewed face-on. 
     DOE  10  comprises a first two-dimensional grating  14  and a second two-dimensional grating  16 , in mutual registration, respectively formed from a first material  15  having a refractive index n 1  and a second material  17  having a refractive index n 2 . Thus, as seen in FIG. 1B, each element  12  includes respective sub-elements of both of gratings  14  and  16 . DOE  10  is surrounded by an ambient material  18  having a refractive index n 3 . Generally, but not exclusively, material  18  may be considered to be air or vacuum, so that n 3 =1. 
     A coherent plane wavefront of wavelength λ in material  18  is perpendicularly incident from the left onto a first side  21  of element  12  of DOE  10 . In general, the phase retardation of the wave emerging at the opposite side  23  of the DOE from region  20  and from region  22  will be different. From equation (1) the difference in phase retardation φ will be given by:              φ   =         2      π     λ          {         (       n   1     -     n   3       )          d   1       +       (       n   2     -     n   3       )          d   2         }               (   2   )                                
     wherein d 1  and d 2  are the respective substantially uniform depths of gratings  14  and  16 , for element  12 , as shown in FIG.  1 B. 
     Calculating the derivative of φ with respect to λ, and setting the derivative equal to zero, gives:                  {       [                n   1          (   λ   )              λ       -         n   1          (   λ   )       λ       ]     -     [                n   3          (   λ   )              λ       -         n   3          (   λ   )       λ       ]       }          d   1       =     
            {       [           n   2          (   λ   )       λ     -              n   2          (   λ   )              λ         ]     +     [                n   3          (   λ   )              λ       -         n   3          (   λ   )       λ       ]       }          d   2               (   3   )                                
     Equations (2) and (3) may be solved for d 1  and d 2  as follows:                d   1     =             λ   ×   φ       2      π            [       (       n   1     -     n   3       )     +         (       n   2     -     n   3       )          (       n   3     -     λ               n   3            λ         -     n   1     +     λ               n   1            λ           )         (       n   2     -     λ               n   2            λ         -     n   1     +     λ               n   1            λ           )         ]         -   1                     and             (   4   )                 d   2     =           λ   ×   φ       2      π            [       (       n   2     -     n   3       )     +         (       n   1     -     n   2       )          (       n   2     -     λ               n   2            λ         -     n   31     +     λ               n   3            λ           )         (       n   2     -     λ               n   3            λ         -     n   1     +     λ               n   1            λ           )         ]         -   1               (   5   )                                
     When d 1  and d 2  obey the above equations for a given wavelength λ and phase retardation φ, element  12  of DOE  10  will have a dispersion at the given wavelength substantially equal to zero. If the equations are solved for a central wavelength, for example, λ=550 nm, element  12  will be substantially achromatic over a range of wavelengths surrounding the central wavelength. Assuming gratings  14  and  16  to comprise FK54 and SF59 Schott glasses, respectively, with thicknesses d 1 =5.63 μm and d 2 =2.28 μm, and w 1 =w 2 =1 μm, it has been found that φ varies by only about ±2π/30 over the range of wavelengths from 470 to 685 nm. Since DOE  10  comprises a plurality of elements  12 , each element achromatized in accordance with the above equations, DOE  10  will be substantially achromatic over the range of wavelengths from 470 to 685 nm. 
     FIG. 1C illustrates a slight modification of the alignment of the first and second gratings  14  and  16 . While in FIG. 1A, the gratings  14  and  16  are aligned ‘back-to-back’ and contact each other, according to FIG. 1C the gratings  14  and  16  are aligned with an air gap n 3  between them. The calculation of the phase retardation φ as a function of the wavelength λ is the same as with respect to the embodiment of FIGS. 1A and 1B. 
     FIG. 2 is a graph of φ versus λ for element  12  of DOE  10 , assuming the above values of grating thicknesses and refractive indices. The relative phase retardation φ is plotted on axis  30  as a function of the wavelength λ of the incident radiation, plotted on axis  32 . 
     The graph of FIG. 2 demonstrates that there are three general regions for the variation of φ with λ: a dispersive region  34 , corresponding to values of λ less than about 470 nm, where φ is generally increasing with λ; a generally non-dispersive region  36  corresponding to values of λ between about 470 nm and 685 nm, where φ is generally constant with λ; and a dispersive region  38 , corresponding to values of λ greater than about 685 nm, where φ is generally decreasing with λ. 
     It will thus be appreciated that the preferred embodiment of the present invention described with reference to FIGS. 1A and 1B corresponds to region  36  of the graph of FIG.  2 . It should also be understood that the general form and properties of the graph of FIG. 2 will be applicable to elements generally represented by FIG. 1B, with values chosen for d 1 , d 2 , n 1  and n 2  that are generally different from the values for the preferred embodiment of the present invention represented by FIG.  1 A. 
     Furthermore, by suitable choices of materials  15  and  17  and control of thicknesses d 1  and d 2 , the shape and inflection points of the graph of FIG. 2 may be varied, so that the graph will have desired values and slopes at certain predetermined design wavelengths. 
     FIG. 3 schematically illustrates a DOE  40 , comprising a substantially achromatic zone plate, in accordance with another preferred embodiment of the present invention, shown in sectional view. 
     DOE  40  comprises a first refractive material  42 , preferably Schott glass SF59, and a second refractive material  44 , preferably Schott glass FK54, having a common surface  46 . DOE  40  is generally circular, with a diameter of the order of 4 mm. Materials  42  and  44  have respective gratings  48  and  50  etched into their respective outer surfaces  52  and  54 , to maximum respective depths 2.28 μm and 5.63 μm. Gratings  48  and  50  are generated according to a computer program, incorporated herein below as Appendix A, for calculating the results of equations (4) and (5) so that the DOE functions as a substantially achromatic zone plate of focal length 1 cm for light of all wavelengths from 470 nm to 685 nm. The depths at any specific position are illustrated schematically in FIG.  3  and are given precisely by the computer program. 
     Gratings  48  and  50  are preferably etched into outer surfaces  52  and  54 , using photolithographic techniques known in the art. Most preferably, a dual surface mask aligner is used to ensure that the gratings are substantially mutually registered. Alternatively or additionally, the gratings may be molded into the surfaces using masters, which are themselves produced using methods of photolithography, as is likewise known in the art. 
     FIGS. 4A and 4B schematically show a generalized representation of a lens  60  having a DOE  62  overlaid on a surface  64  thereof, in accordance with another preferred embodiment of the present invention. Lens  60 , made of any suitable optical glass known in the art, typically has chromatic aberration, with relative phase shift of transmitted waves generally decreasing with increasing wavelength. FIG. 4A is a side, sectional view illustrating the DOE on surface  64 . FIG. 4B shows a sectional element  66  of the lens of FIG.  4 A. DOE  62  comprises a plurality of elements  66 . 
     Lens surface  64  is overlaid with a first coating  70  having refractive index n 1  and a second coating  74  having refractive index n 2 , preferably using vacuum deposition techniques. Gratings  68  and  72  are then etched into the coated layers, preferably using photolithographic techniques. Element  66  comprises a region  78  and a region  76 . Region  78  comprises a section of grating  68 , of thickness d 1  and width w 1 , overlaid by a section of grating  72 , of thickness d 2  and width w 1 . d 1  and d 2  are set so that the phase retardation of element  66  lies in region  34  of FIG. 2, corresponding to a general increase of phase retardation φ with wavelength λ. Segment  76  preferably comprises a section of grating  68  of generally negligible thickness, overlaid by a section of grating  72  of generally negligible thickness. 
     Radiation passing through lens  60  is incident on surface  64 , passes through DOE  62  and exits to the right. The chromatic aberration of lens  60  creates a general decrease of phase retardation φ with wavelength λ of radiation passing through the lens and incident on surface  64 . To the right of surface  64 , element  66  creates an increase in phase retardation φ with wavelength λ. Each of elements  66  of DOE  62  is designed, based on the equations (4) and (5), so that the net resultant phase retardation φ is approximately zero. Thus, the addition of DOE  62  to lens  60  substantially reduces the chromatic aberration of the lens  60 . 
     FIGS. 5A and 5B schematically illustrate another optical element  80  in sectional view, in accordance with a preferred embodiment of the present invention. FIG. 5B shows a sectional detail of an element  82  of FIG.  5 A. Optical element  80  preferably comprises a lens  84 , similar to lens  60 , formed of a material having refractive index n 1 . A DOE  86 , comprising a plurality of elements  82 , is formed on a surface  88  of lens  84  to correct the chromatic aberration of the lens. DOE  86  serves to compensate for chromatic aberration in lens  85 , so that optical element  80  is substantially achromatic, like lens  60 , as described above. 
     To form element  82  of DOE  86 , zone  90  of lens  84  is etched to a depth d, into surface  88 . Zone  90  has a width w 2  preferably of the order of 1 μm. Superimposed to a depth d 2  on the non-etched section of surface  88  is a second zone  92 , formed of a second material having refractive index n 2 . Zones  90  and  92  are in mutual registration, thus forming element  82 . 
     Element  80  is preferably fabricated by first molding and/or grinding lens  84 , and then coating lens surface  88  with a layer of material  94  to a depth greater than or equal to d 2 , preferably by vacuum deposition, or using another technique known in the art. Depths d 2  for region  92  and d 1 +d 2  for region  90  are then etched into the coated lens surface for each of the plurality of elements  82 , preferably by photolithography. 
     Alternatively, element  80 , including DOE  86 , may be formed using precision molding techniques known in the art. In this case, lens  84  and material  94  preferably comprise optical plastics, as are known in the art. 
     It will be appreciated that while the preferred embodiments of thee present invention described above have used DOEs fabricated to conform to region  34  of FIG. 2, in order to correct chromatic aberration in the base lenses caused by a decrease in phase retardation with wavelength, other preferred embodiments of the present invention can be fabricated to conform to region  38  of FIG. 2, in order to correct chromatic aberration in the base lenses caused by an increase in phase retardation with wavelength. 
     Although the above preferred embodiments comprise transmissive optical elements operating in the visible domain, it will be appreciated by those skilled in the art that the principles of the present invention may be applied to produce other types of elements, such as reflective elements. Furthermore, diffractive optical elements may be produced in accordance with the principles of the present invention for use in other regions of the electromagnetic spectrum, as well as with other forms of radiation, such as ultrasonic radiation. 
     The term “optical element” as used in the present patent application and in the claims, is therefore to be understood to include elements for controlling a beam of radiation of any applicable type and/or wavelength. 
     Conventional computer-generated holograms (DOEs) operate at the specific wavelength for which they were designed. Operating at another wavelength will thus cause chromatic aberration. Since many potential DOE applications require the simultaneous use of more than one wavelength, correction of the chromatic aberration is required. 
     Chromatic aberration correction for two wavelengths will now be described with reference to FIG. 6, illustrating one pixel of the two aligned DOEs  14 ,  16 . The phase retardation φ, of light with wavelength λ propagating through that pixel, is:                    [         n   1          (   λ   )       -       n   g          (   λ   )         ]          d   1       +       [         n   2          (   λ   )       -       n   g          (   λ   )         ]          d   2         =       1     2      π          λ                 φ             (   6   )                                
     wherein n 1 (λ), n g (λ), n 2 (λ) are the refractive indices of the materials along the light path, and d 1  and d 2  are the etched depths. 
     When light with two different wavelengths λ 1  and λ 2  is propagated through the pixel, the phase retardations φ 1  and φ 2  of the two wavelengths are given by the ma equation:                  (     n   -     n   g       )        d     =       1     2      π          λ                 φ             (   7   )                                
     wherein:        n   =     [             n   1          (     λ   1     )               n   2          (     λ   1     )                   n   1          (     λ   2     )               n   2          (     λ   2     )             ]               n   g     =     [             n   g          (     λ   1     )               n   g          (     λ   1     )                   n   g          (     λ   2     )               n   g          (     λ   2     )             ]             d   =     [           d   1               d   2           ]               λ                 φ     =     [             λ   1          (       φ   1     +         m   1     ·   2        π       )                   λ   2          (       φ   2     +         m   2     ·   2        π       )             ]                            
     The elements of the n matrix are the refractive indices of the dispersive materials for the two wavelengths. The elements of the n g  matrix are the refractive indices of the intermediate material for the two wavelengths. The elements of the d matrix are the etched thickness of the two DOEs. The elements of the λφ matrix are the required phases for the two wavelengths, multiplied by the wavelengths m 1  and m 2  are some arbitrary integers. The phase values can be the same or different, and consequently, the DOE behavior will be the same or different for both wavelengths. 
     Solving equation (7) for the required etched depths d in the two DOEs, we obtain:              d   =       1     2      π              (     n   -     n   g       )       -   1          λ                 φ             (   8   )                                
     The etched depths of the two DOEs calculated by equation (7) cause the suitable phase retardations φ 1  and φ 2  for the two wavelengths simultaneously. 
     For investigation of the overall behavior of DOE thickness, equation (8) is expressed explicitly:                  d   1     =           [         n   2          (     λ   2     )       -       n   g          (     λ   2     )         ]              λ   1          φ   1         2      π         -       [         n   2          (     λ   1     )       -       n   g          (     λ   1     )         ]              λ   2          φ   2         2      π                     [         n   1          (     λ   1     )       -       n   g          (     λ   1     )         ]          [         n   2          (     λ   2     )       -       n   g          (     λ   2     )         ]                   [         n   1          (     λ   2     )       -       n   g          (     λ   2     )         ]          [         n   2          (     λ   1     )       -       n   g          (     λ   1     )         ]                      
            d   2     =           [         n   1          (     λ   2     )       -       n   g          (     λ   2     )         ]              λ   1          φ   1         2      π         -       [         n   1          (     λ   1     )       -       n   g          (     λ   1     )         ]              λ   2          φ   2         2      π                     [         n   1          (     λ   1     )       -       n   g          (     λ   1     )         ]          [         n   2          (     λ   2     )       -       n   g          (     λ   2     )         ]                   [         n   1          (     λ   2     )       -       n   g          (     λ   2     )         ]          [         n   2          (     λ   1     )       -       n   g          (     λ   1     )         ]                         (   9   )                                
     The minimum absolute thickness is achieved when the denominator of the equation (9) is set to a maximum. The denominator consists of two expressions; each expression is made up of two multiples. 
     For normal dispersive materials where the refractive index is higher for shorter wavelength, n(λ) is a decreasing function. If the intermediate material is air, n g  equals 1. Assuming that λ 2 &gt;λ 1 , each of the two expressions consists of multiples of a high value and a low value. The maximum value of the denominator occurs when one material will have a high dispersion and the other material will have a low dispersion. 
     This method can be expanded for more than two wavelengths. More degrees of freedom, and consequently more flexibility when using more than two wavelengths, can be obtained when other configurations of pixel shapes with multi-layers etching are used. 
     FIG. 7 shows one pixel, consisting of two aligned, etched elements with an etched layer on two sides of each element. In this configuration, the phase retardation of light with wavelength λ propagating through that pixel is:              Δ   =           [         n   1          (   λ   )       -       n   2          (   λ   )         ]          d   1       +       [         n   g          (   λ   )       -       n   2          (   λ   )         ]          d   2       +       [         n   g          (   λ   )       -       n   3          (   λ   )         ]          d   3       +       [         n   4          (   λ   )       -       n   3          (   λ   )         ]          d   4         =       1     2      π          λ                 φ               (   10   )                                
     Four similar equations can be written for four different wavelengths. Solving these equations for the d values will give the required thickness in each layer of the two substrates. The phase values for each wavelength can be the same or different, as mentioned earlier. 
     Adding more layers of different materials adds more degrees of freedom and enables the possibility of designing the DOE for more wavelengths. 
     As an example, the thicknesses of a binary DOE for two wavelengths was calculated. A binary DOE has two levels only, causing two phase retardations: 0 or π. The two wavelengths are λ 1 =543.5 nm and λ 2 =632.8 nm of the He—Ne laser. The materials for the two DOEs are BK7 glass and As 2 S. The intermediate material is air. The corresponding refractive indices of the materials for the wavelengths are shown in Table 1: 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 The Refractive Indices of the DOE Material for Two Wavelengths 
               
             
          
           
               
                 Glass 
                 λ 1  = 543.5 nm 
                 λ 2  = 632.8 nm 
               
               
                   
               
               
                 BK7 
                 n 1 (λ 1 ) = 1.51885 
                 n 1 (λ 2 ) = 1.51509 
               
               
                 As 2 S 
                 n 2 (λ 1 ) = 2.71445 
                 n 2 (λ 2 ) = 2.60615 
               
               
                   
               
             
          
         
       
     
     Equation 6 was used to calculate the DOE thickness for all combinations of phases for the two wavelengths. The optimized integral numbers m 1  and m 2 , that give the minimal overall thickness value of the combined DOE, were chosen for these calculations. 
     Positive and negative thickness values mean adding to, or etching the substrate, respectively. Considering the maximum positive value as the zero point, and all other values negative in reference to it, therefore microlithography etching process can be applied. 
     The following Table 2 contains the results of the calculations of the thicknesses d 1  and d 2  in microns for the two materials for all phase-value combinations. Table 2 also shows the integral numbers m 1  and m 2  and the minimal thicknesses d′ 1  and d′ 2  after optimization. It can be seen that the optimization process diminishes the overall thickness by one order of magnitude. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 The Required Etched Depths of the Combined DOE 
               
               
                 for Different Phase Combinations of the Two Wavelengths, 
               
               
                 before and after Optimization 
               
             
          
           
               
                 φ 1  [rad] 
                 φ 2  [rad] 
                 m 1   
                 m 2   
                 d 1  [10 −6  m] 
                 d 2  [10 −6  m] 
                 m 1   
                 m 2   
                 d′ 1  [10 −6  m] 
                 d′ 2  [10 −6  m] 
               
               
                   
               
               
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 0 
                 π 
                 0 
                 0 
                 10.9064 
                 −3.3007 
                 −3 
                 −3 
                 −1.8786 
                 −0.3825 
               
               
                 π 
                 0 
                 0 
                 0 
                 −8.7755 
                 2.843 
                 −3 
                 −2 
                 0.2523 
                 −0.8689 
               
               
                 π 
                 π 
                 0 
                 0 
                 2.1308 
                 −0.4864 
                 −1 
                 −1 
                 −2.1308 
                 0.4864 
               
               
                   
               
             
          
         
       
     
     FIG. 8 shows the simulated response of the DOE for different wavelengths, where λ 1  and λ 2  are the wavelengths for which it was designed. The phase retardations required in this case are n for the two wavelengths. It can be seen that the phase retardations for these wavelengths are n as required, but the phase changes are sharp for other wavelengths. Simulation shows that for a light source with a bandwidth of 2 nm, the phase retardation changes are less than 1%. All other wavelengths will suffer strong chromatic aberration and degradation in efficiency. Other phase combinations show similar behavior. 
     The response of the phase values to errors in fabrication is calculated by:                Δ                 φ     =         2      π     λ          {         [         n   1          (   λ   )       -       n   g          (   λ   )         ]        Δ                   d   1       +       [         n   2          (   λ   )       -       n   g          (   λ   )         ]        Δ                   d   2         }               (   11   )                                
     This equation shows that the error response is similar to, or less than, that in other types of DOEs, because it is a function of the two independent variables d 1  and d 2 . 
     It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrated embodiments and that the present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. 
     
       
         
               
               
             
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
           
               
                   
                 APPENDIX A 
               
               
                   
                   
               
             
             
               
                   
                 The following program listing, written in PASCAL programming 
               
             
          
           
               
                 language, is incorporated herein as a part of the present patent 
               
               
                 application, as a portion of the description of the best mode for carrying 
               
               
                 out the invention: 
               
               
                 {$N+} 
               
               
                 uses graph,crt; 
               
               
                 const 
               
             
          
           
               
                   
                 Pi=3.14159; 
               
             
          
           
               
                   
                 lamda0=0.55e−6; 
               
               
                   
                 A01=2.4549347; 
               
               
                   
                 A11=−8.3372034e−3; 
               
               
                   
                 A21=1.6841270e−2; 
               
               
                   
                 A31=5.0168527e−4; 
               
               
                   
                 A41=−1.4413749e−5; 
               
               
                   
                 A51=2.0771351e−6; 
               
               
                   
                 A02=1.0; 
               
               
                   
                 A12=0; 
               
               
                   
                 A22=0; 
               
               
                   
                 A32=0; 
               
               
                   
                 A42=0; 
               
               
                   
                 A52=0; 
               
               
                   
                 A03=2.718928; 
               
               
                   
                 A13=2.2108077e−2; 
               
               
                   
                 A23=1.0592509e−2; 
               
               
                   
                 A33=1.0816965e−4; 
               
               
                   
                 A43=−1.6472538e−6; 
               
               
                   
                 A53=5.9240991e−7; 
               
               
                   
                 A01=2.4498259; 
               
               
                   
                 A11=1.0128610e−2; 
               
               
                   
                 A21=1.8753684e−2; 
               
               
                   
                 A31=1.1999618e−3; 
               
               
                   
                 A41=8.8610291e−5; 
               
               
                   
                 A51=9.8139193e−6; 
               
               
                   
                 A03−2.9177579; 
               
               
                   
                 A13=−1.1483287e−2; 
               
               
                   
                 A23=3.3825845e−2; 
               
               
                   
                 A33=2.5277439e−3; 
               
               
                   
                 A43=−1.7332899e−4; 
               
               
                   
                 A53=2.1465274e−5; 
               
               
                   
                 A03=3.5278149; 
               
               
                   
                 A13=1.7049614e−2; 
               
               
                   
                 A23=4.2895039e−2; 
               
               
                   
                 A33=1.9248178e−3; 
               
               
                   
                 A43=7.5388918e−5; 
               
               
                   
                 A53=1.3032008e−5; 
               
               
                   
                 A01=2.2718929; 
               
               
                   
                 A11=−1.0108077e−2; 
               
               
                   
                 A21=1.0592509e−2; 
               
               
                   
                 A31=2.0816965e−4; 
               
               
                   
                 A41=7.6472538e−6; 
               
               
                   
                 A51=4.9240991e−7; 
               
             
          
           
               
                   
                 type 
               
             
          
           
               
                   
                 dat=file of real; 
               
               
                   
                 ar=array[1..25] of real; 
               
             
          
           
               
                   
                 var 
               
             
          
           
               
                   
                 mode,graphdriver:integer; 
               
               
                   
                 deltax, aperture, Radius,max_phase,f:real; 
               
               
                   
                 lens:dat; 
               
               
                   
                 name:string; 
               
             
          
           
               
                   
                 function modd(a,b:real):real; 
               
               
                   
                 begin 
               
             
          
           
               
                   
                 modd:=a−b*trunc(a/b); 
               
             
          
           
               
                   
                 end; 
               
               
                   
                 function power(x,n:real):extended; 
               
               
                   
                 var 
               
             
          
           
               
                   
                 po:extended; 
               
             
          
           
               
                   
                 begin 
               
             
          
           
               
                   
                 if n&lt;&gt;0 then po:=exp(n*1n(x)); 
               
               
                   
                 if n=0 then po:=1; 
               
               
                   
                 power:=po; 
               
             
          
           
               
                   
                 end; 
               
               
                   
                 function n (lamda,A0,A1,A2,A3,A4,A5:real):extended; 
               
               
                   
                 var 
               
             
          
           
               
                   
                 n1:extended; 
               
             
          
           
               
                   
                 begin 
               
             
          
           
               
                   
                 lamda:=lamda*1e6; 
               
               
                   
                 n1:=A0; 
               
               
                   
                 n1:=n1+A1*power(lamda,2); 
               
               
                   
                 n1:=n1+A2/power(lamda,2); 
               
               
                   
                 n1:=n1+A3/power(lamda,4); 
               
               
                   
                 n1:=n1+A4/power(lamda,6); 
               
               
                   
                 n1:=n1+A5/power(lamda,8); 
               
               
                   
                 n1:=sqrt(n1); 
               
               
                   
                 n:=n1; 
               
             
          
           
               
                   
                 end; 
               
               
                   
                 function 
               
             
          
           
               
                 dn(lamda,n,A0,A1,A2,A3,A4,A5:real):extended; 
               
             
          
           
               
                   
                 var 
               
             
          
           
               
                   
                 n1,n2:extended: 
               
             
          
           
               
                   
                 begin 
               
             
          
           
               
                   
                 n2:=2*A1*1e+12*lamda; 
               
               
                   
                 n2:=n2−2*A2*1e−12*power(lamda,−3); 
               
               
                   
                 n2:=n2−4*A3*1e−24*power(lamda,−5); 
               
               
                   
                 n2:=n2−6*A4*1e−36*power(lamda,−7); 
               
               
                   
                 n2:=n2−8*A5*1e−48*power(lamda,−9); 
               
               
                   
                 dn:=n2/(2*n); 
               
             
          
           
               
                   
                 end; 
               
               
                   
                 procedure thickness (lamda0,phi:real;var d1,d2:real); 
               
               
                   
                 var 
               
             
          
           
               
                   
                 n1,n2,n3,dn1,dn2,dn3, 
               
               
                   
                 a,b,c,dd1,dd2,dd3:real; 
               
             
          
           
               
                   
                 begin 
               
             
          
           
               
                   
                 n1:=n(lamda0,A01,A11,A21,A31,A41,A51); 
               
               
                   
                 n2:=n(lamda0,A02,A12,A22,A32,A42,A52); 
               
               
                   
                 n3:=(lamda0,A03,A13,A23,A33,A43,A53); 
               
               
                   
                 dn1:=dn(lamda0,n1,A01,A11,A21,A31,A41,A51); 
               
               
                   
                 dn2:=dn(lamda0,n2,A02,A12,A22,A32,A42,A52); 
               
               
                   
                 dn3:=dn(lamda0,n3,A03,A13,A23,A33,A43,A53); 
               
               
                   
                 a:=dn1*lamda0; 
               
               
                   
                 b:=dn2*lamda0; 
               
               
                   
                 c:=dn3*lamda0; 
               
               
                   
                 dd1:=n2−n1+a−b; 
               
               
                   
                 dd2:=n3−n2+b−c; 
               
               
                   
                 d1:=(n1−n2)+(n3−n2)*dd1/dd2; 
               
               
                   
                 d2:=(n1−n2)*dd2/dd1+(n3−n2); 
               
               
                   
                 d1:=phi*lamda0/(2*Pi*d1); 
               
               
                   
                 d2:=phi*lamda0/(2*Pi*d2); 
               
             
          
           
               
                   
                 { 
                 write1n(‘dl=’,d1); 
               
               
                   
                   
                 write1n(‘d2=’,d2); 
               
               
                   
                   
                 write1n(‘phi=’,phi); 
               
               
                   
                   
                 read1n;} 
               
             
          
           
               
                   
                 end; 
               
               
                   
                 procedure combine; 
               
               
                   
                 var 
               
             
          
           
               
                   
                 i,j:integer; 
               
               
                   
                 phase,rad,x,y,d1,d2:real; 
               
             
          
           
               
                   
                 begin 
               
             
          
           
               
                   
                 assign(lens,‘lens.dat’); 
               
               
                   
                 rewrite(lens); 
               
               
                   
                 for j:=−127 to 128 do 
               
               
                   
                 for i:=−127 to 128 do 
               
               
                   
                 begin 
               
             
          
           
               
                   
                 x:=deltax*i; 
               
               
                   
                 y:=deltax*j; 
               
               
                   
                 rad:=sqr(x)+sqr(y); 
               
               
                   
                 phase:=(2*Pi/lamda0)*(max_phase−rad/(2*f); 
               
               
                   
                 phase:=modd(phase,2*Pi); 
               
               
                   
                 thickness(lamda0,phase,d1,d2); 
               
               
                   
                 write(lens,d1); 
               
               
                   
                 write(lens,d2); 
               
             
          
           
               
                   
                 end; 
               
             
          
           
               
                   
                 close(lens); 
               
             
          
           
               
                   
                 end; 
               
               
                   
                 begin 
               
             
          
           
               
                   
                 f:=1e−3; 
               
               
                   
                 aperture:=4e−4; 
               
               
                   
                 Radius:=aperture*sqrt(2); 
               
               
                   
                 deltax:=aperture/256; 
               
               
                   
                 max_phase:=sqr(Radius)/(2*f); 
               
               
                   
                 combine; 
               
             
          
           
               
                   
                 end.