Binary optic lens design using flip-flop optimization

A binary optic lens is designed to achieve a predetermined effect on light propagating through the lens by specifying an initial lens design with a plurality of discontinuous subaperture regions, assigning each subaperture region a relative phase difference of 0 or .pi., and calculating the net intensity of light propagating through the lens by coherently summing the wave amplitudes for all of the subaperture regions at a given point in the image plane of the lens. The assigned phase difference for one of the subaperture regions is then changed and the intensity is recalculated by coherently summing the wave amplitudes for all of the subaperture regions at the given point. If the intensity increases over the previously calculated intensity, the changed phase difference is assigned to the selected subaperture region. The steps of changing, recalculating, and assigning are repeated for all of the subaperture regions to make one complete pass through the lens. The entire process is repeated until the calculated intensity no longer increases.

BACKGROUND OF THE INVENTION 
This invention is concerned with the design of binary optical lenses. 
The modulation of an optical wavefront by a surface relief pattern has the 
remarkable capability of generating an arbitrary image from a single 
incident plane wave. One example of such modulation in the prior art is a 
binary phase-only filter for optical correlation, where the phase relief 
pattern was generated by taking the Fourier transform of a given image 
(Flavin, et al., Correlation Experiments with a Binary Phase-Only Filter 
Implemented on a Quartz Substrate, Optical Engineering, Volume 28, Pages 
470-473 (1989). Another example involved the generation of an array of 
equal spots for optical computing devices, with the relief pattern 
generated using a Monte Carlo type iterative method, known as simulated 
annealing, to arrive at a pattern which would generate an array of spots 
(Feldman, et al., Iterative Encoding of High-Efficiency Holograms for 
Generation of Spot Arrays, Optics Letters, Volume 14, Pages 479-481 
(1989). Unlike a conventional optical surface, whose profile is 
characterized by only a few variables, a binary optical surface consists 
of a large number of discontinuous subaperture regions. The desired 
optical effect of a lens is achieved by virtue of imposing a difference in 
phase of .pi. radians between adjacent regions of the binary lens. The 
resulting interference effects between light passing through the various 
regions provides the means to produce a lens. 
Because of the large number of subaperture regions which must be analyzed 
in designing a binary optical lens, conventional ray tracing design 
techniques are cumbersome for binary optical design and may not achieve an 
optimal solution in any event. Thus it would be desirable to provide an 
alternative technique for generating an arbitrary intensity pattern 
resulting from down-field propagation of a plane wave incident on a binary 
phase relief pattern. 
SUMMARY OF THE INVENTION 
A method of designing a binary optic lens to achieve a predetermined effect 
on light propagating through the lens includes the steps of specifying an 
initial lens design with a plurality of discontinuous subaperture regions, 
assigning each subaperture region a relative phase difference of 0 or 
.pi., and calculating the net intensity of light propagating through the 
lens by coherently summing the wave amplitudes for all of the subaperture 
regions at a given point in the image plane of the lens. The assigned 
phase difference for one of the subaperture regions is then changed and 
the intensity is recalculated by coherently summing the wave amplitudes 
for all of the subaperture regions at the given point. If the intensity 
increases over the previously calculated intensity, the changed phase 
difference is assigned to the selected subaperture region. The steps of 
changing, recalculating, and assigning are repeated for all of the 
subaperture regions to make one complete pass through the lens. The entire 
process is repeated until the calculated intensity no longer increases. 
In more particular embodiments, the step of assigning each subaperture 
region a relative phase difference of 0 or .pi. may involve assigning each 
subaperture region a relative phase difference of 0, .pi., opposite that 
of the adjacent subaperture regions, or a relative phase difference of 0 
or .pi. at random. 
In another refinement, the steps of changing, recalculating, assigning, and 
repeating are repeated until the calculated intensity converges.

DESCRIPTION OF THE INVENTION 
It is an outstanding feature of this invention to provide a new technique 
for generating a binary optical lens design. The inventive approach 
utilizes a merit function which is defined to be the intensity calculated 
by taking the coherent sum of the wave amplitudes from each subaperture of 
the binary lens. Each term in this sum is assigned a plus or a minus sign 
depending on whether that subelement is initially designated to have a 
zero or .pi. radians phase. Starting with some distribution of minus signs 
(corresponding to .pi. phase difference regions) the intensity at a given 
point in the image plane is evaluated. Then, stepping through the binary 
optical surface, each term is changed in sign (i.e., is "flipped" from 
zero to .pi. radians phase, or vice versa) and the effect of this flip on 
the merit function (i.e., the focal plane intensity at the given point) is 
noted. If the change increases the value of the merit function, the change 
is retained. Otherwise, the term at that point is "flopped", i.e., 
returned to its former state. Each succeeding term is then considered, by 
flipping its sign and evaluating the merit function. One pass is completed 
when all terms have been so examined. 
FIG. 1 is a schematic representation illustrating the operation of a binary 
lens. The binary lens 100 consists of an optical material which is divided 
into multiple subelements, as indicated by the alternately shaded 
subelements. A plane wave, represented by the rays 104, 106, and 108, will 
be diffracted by the lens because the subelements have relative 
thicknesses designed to impose a difference of one half wavelength in the 
phase of the portions of the plane wave passing through adjacent 
subelements of the lens. In this manner, the plane wave, after passing 
through the lens, illuminates an image plane 110 with some nonuniform 
intensity pattern 112. 
The design method of this invention begins by dividing an optical surface, 
such as the optical surface 200 illustrated in schematic profile in FIG. 
2, into a large number of cells. A field point 214 a distance F from the 
optical surface and at a height x is selected and the field amplitude du 
is determined from a given cell in the optical surface 
EQU du=.tau.(x')e.sup.-i2.pi.r/.lambda. dx' 1) 
where x' is the position of the cell in the optical surface, t is the 
amplitude transmission function, and r is the distance from that cell to 
the field point at x 
##EQU1## 
The total image field amplitude u(x) at the point x is a superposition of 
the incremental fields from all the cells in the optical surface 
##EQU2## 
where .sub..tau.j is the amplitude transmission for the jth cell 
EQU .tau.j=e.sup.i.phi.j =.+-.1 4) 
where the transmission phase takes on the values of either 0 or .pi.. 
The field given by Equation 3) is complex, so its real and imaginary parts 
are computed separately 
EQU u(x)=U+iV 5) 
EQU where 
EQU U=.tau.j cos(2.pi.rj/.lambda.) 6) 
EQU V=-.tau.j sin(2.pi.rj/.lambda.) 7) 
Since the arguments of the sine and cosine are fixed for any geometry and 
do not depend on any surface relief configuration, they may be 
pre-computed and stored in arrays. Thus the U and V are easily calculated 
by either adding or subtracting the jth cosine or sine term, depending on 
whether that cell has a zero or .pi. phase. 
The intensity I at the point x is 
EQU I=U.sup.2 +V.sup.2 8) 
The complete intensity in the field plane is determined by changing x and 
repeating the summations over the optical plane cells. 
The intensity at the point x is the single number I given by Equation 8). 
In order to evaluate the intensity at that point if the phase on one cell 
were to change its state, it is not necessary to perform the complete sum 
over the optical surface. Instead, all that is required is to subtract 
from U and V that term corresponding to the cell that is changed and add 
the opposite state. Thus, to change the kth cell, 
EQU U.fwdarw.U-.tau..sub.k cos(2.pi.r.sub.k /.lambda.)+(-.tau..sub.k 
cos(2.pi.r.sub.k /.lambda.)) 9) 
EQU V.fwdarw.V+.tau..sub.k sin(2.pi.r.sub.k /.lambda.)-(-.tau..sub.k 
sin(2.pi.r.sub.k /.lambda.)) 10) 
With the new U and V the new intensity is easily evaluated by Equation 8). 
The flip-flop optimization scheme for determining the two level binary 
phase surface consists of the following algorithm. From some starting 
point, such as, for example, all cells being set to zero phase, the 
intensity pattern is evaluated. The state of each cell is then changed 
separately and the intensity evaluated using the above approach. If the 
new intensity is closer to the desired intensity pattern, then the new 
state is retained, otherwise the old state is restored. All cells in the 
optical surface are likewise tested. This constitutes one pass. The 
process is repeated until a complete pass makes no more changes in the 
intensity pattern. At this point the optimization is complete. 
In applying the technique of the invention to several specific lens 
designs, a unique solution has been found in three or four passes from a 
variety of starting distributions of phase. Unlike conventional lens 
design methods (such as CODE V), no derivatives are used and no ray 
tracing is performed. The merit function may be expanded to include 
off-axis image points by adding the intensity at focus for plane waves 
incident at given field angles. Another advantage of the inventive method 
is that it seems to find the globally best configuration for a given set 
of requirements. 
The power of the inventive technique may be illustrated by the improvement 
which can be achieved in diffraction efficiency. An illustrative classical 
lens is represented in FIGS. 3 and 4, where FIG. 3 is a plot of the 
aperture plane position versus the refractive depth of the lens and FIG. 4 
is a plot of the calculated theoretical irradiance for the lens on the 
vertical axis as a function of position on the image plane. The efficiency 
for this lens is approximately 40%. If a corresponding binary optical 
structure is considered as being divided into 1,000 subapertures, and if 
initially all of the subapertures are considered to have the same phase 
(i.e., either 0 or .pi.), the plot of the aperture plane position versus 
etch depth appears as in FIG. 5 and the calculated irradiance is plotted 
in FIG. 6. After a single pass through all of the subapertures of the 
design using the flip-flop optimization technique of the present 
invention, the binary optic profile depicted in FIG. 7 resulted, with a 
calculated irradiance as shown in FIG. 8. After only two passes of the 
flip-flop technique, the design converged, with a profile as indicated in 
FIG. 9 and the calculated irradiance shown in FIG. 10. 
Another example of the power of the inventive technique is depicted in 
FIGS. 11-16. Here, the initial configuration, as shown in profile (FIG. 
11) and calculated irradiance (FIG. 12), was for a binary lens with 1,000 
subapertures of alternating phase. A single pass through the flip-flop 
optimization technique yielded the profile of FIG. 13, with an irradiance 
pattern as shown in FIG. 14. The flip-flop optimization technique 
converged after only three passes, with the resulting profile and 
irradiance of FIGS. 15 and 16. 
An additional illustration of the power of this invention compares the 
binary lens design which results with an analogous Fresnel lens. FIGS. 17 
and 18 depict the profile and irradiance pattern for a classical lens, 
while FIG. 19 and 20 provide analogous information for a Fresnel lens, 
which exhibits a diffraction efficiency, as is well know, of approximately 
41% relative to the classical lens. The flip-flop designed binary lens, 
however, as shown in FIGS. 21 and 22, exhibits a diffraction efficiency of 
approximately 54%. In addition, it can also be seen that the method of 
this invention produces an irradiance pattern with reduced side lobes. 
In conclusion, the flip-flop optimization technique of this invention is a 
robust method which produces rapid convergence, with the final solution 
differing only slight even with widely different initial phase patterns 
for the subapertures of the lens. Surprisingly, binary lenses designed 
with this method exhibit higher diffraction efficiencies than two level 
binary optic Fresnel lenses. 
The preferred embodiments of this invention have been illustrated and 
described above. Modifications and additional embodiments, however, will 
undoubtedly be apparent to those skilled in the art. Furthermore, 
equivalent elements may be substituted for those illustrated and described 
herein, parts or connections might be reversed or otherwise interchanged, 
and certain features of the invention may be utilized independently of 
other features. Consequently, the exemplary embodiments should be 
considered illustrative, rather than inclusive, while the appended claims 
are more indicative of the full scope of the invention.