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Programmable wavefront generation using two binary phase spatial light modulators
__ __ EB
2%
I March 1996
OPTICS COMMUNICATIONS
ELSEYIER
Optics Communications
124 ( 1996) 345-353
Full length article
Programmable wavefront generation using two binary phase spatial light modulators G.G. Yang, S.E. Broomfield Depnrtment
ofEngineeringScience.
University qj’ Oxford, O.xfiwd, OX1 3P.I. UK
Received 3 July 1995
Abstract A programmable wavefront generation system using two binary phase spatial light modulators is described. This a a spacevariant system with high diffraction efficiency. The amplitude and phase of a wavefront can be modulated with the system. It is
experimentally implemented with two ferroelectric liquid crystal SLMs. A programmable optical crossbar interconnection based on this scheme is also discussed. The properties of the system are analysed.
1. Introduction
There is interest in spatial light modulators ( SLMs) for a variety of applications including the general manipulation of wavefronts, reconfigurable optical interconnects, beam steering and shaping. Usually the SLM is restricted to perform wavefront modulation of either amplitude or phase, but not both. The advantage of phase modulation is that no intrinsic loss is associated with it. Phase-only SLMs may be subdivided into those for which the modulation can be continuously varied and those for which modulation is achieved by switching between two discrete phase states, commonly separated by 7~. In this paper we are concerned with the latter, binary phase spatial light modulators (BPSLMs), because they are well suited to demonstrate high performance with the available devices. It is usual for the output of the modulator to be viewed in the Fourier plane of a SLM/positive lens combination. As a consequence, the output of the binary phase hologram has inversion symmetry and is restricted to be space-invariant. If an output which has inversion symmetry is not required, then there is a loss 0030.4018/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDfOO30.4018(95)00602-8
system
of diffraction efficiency Q, defined as the fraction of power incident as required in the Fourier plane relative to the total power transmitted by the SLM. Inversion symmetry may be broken by introducing a fixed binary phase hologram [ 1,2] but in this case Q is not increased. Alternative, an asymmetric output may be generated by increasing the number of phase levels [ 3,4]. Results obtained in this way increase qd but the output remains space-invariant. It has been pointed out by Bartelt [5] and Yang [6,7] that the restriction of space-invariance can be removed by effectively cascading two phase masks in two Fourier-transforming stages. The amplitude and phase of a wavefront can be modulated with the combined system. In operation Bartelt’s scheme [5] was designed to produce the required amplitude modulation by the first Fourier-transforming stage and the required phase modulation by the second. It was implemented using fixed phase filters with continuous phase modulation specified at N X N sampling points, when the input and output signals are also specified at N X N sampling points. Yang’s scheme [7] uses two binary phase holograms each with N’ X N2 sampling points.
346
G. G. Yang. S. E. Broomjield / Optics Communications
The latter is easier to implement using the available technology and can realize general space-variant operations because there is more freedom to modulate the wavefront. In this paper we consider a similar binary phase scheme [7] except that fixed phase holograms are replaced by binary phase SLMs and the design method to achieve a given wavefront is based on simultaneous, iterative optimization of both SLMs. The former gives programmable wavefront generation, the latter improves performance. The system is demonstrated by producing an asymmetric output. Simulation of its performance as a reconfigurable optical interconnect is also presented. Both utilize the space-variant performance of the system. The properties of our scheme are analyzed in Section 2. In Section 3 the performance is simulated for a system with two binary phase SLMs each with 128 X 128 pixels, and the experimental results are presented for a system comprising two ferroelectric liquid crystal spatial light modulators (FLC SLMs). In Section 4 a programmable optical crossbar interconnection system using this scheme is described. Results and conclusions are presented in Section 5.
124
(I 996) 345-353
nth plane designated P, is c,~(x,, Yn), the transmission of the nth plane due to the SLM is b,(x,, y,) and the output of the nth plane is a,( x,, Y,) . Thus we have a,(~> Y,) =bn(x,,
Yn)c,(x,> Y,J .
(1)
The basic transform relation between adjacent cascaded stages under the scalar, paraxial approximation is
Xexp
-_ie(-w,,-,+y,,v,t-,) M (2)
where A is the wavelength,fis the focal length of the lens and A is a constant. The relationship between the input and the output of the system is described by %(X,9 Y3) 0Z =A 1111 -x
c,(~,,~,,)~,(x,,YI)B~(x~--xI,)~~-?!,)
2. The two binary phase SLM system
(3)
The system is shown in Fig. 1. The system consists of two cascaded Fourier transforms (stage 1 and 2) with an input stage (stage 0) which for convenience is also presented as a Fourier transform. The first BPSLM is at the front focal plane P, of the lens L,, the second BPSLM is at the back focal plane P2 of the same lens. The second lens L2 forms the output at its back focal plane P,. We define the following notation for the nth stage (n = 0, 3) : the complex amplitude input to the
where B, is the Fourier transform of the second BPSLM, b,. The impulse response of this system is x h(X”, Y& x3, Y3) =
It is clear that the difference output plane. system. This response of a
II --cc
h(x,,
PI stage 0
Fig.
P?
sl%2 stage
I
Output stage
I. The two binary phase SLM system.
2
Y?-Y,)
the impulse response does not depend on of the coordinates between the input and Therefore, the system is a space-variant is to be contrasted with the impulse single-phase SLM system which is
h(x,, YO;xj,Y3) =B(xo-x,,
SLMI
Y,> &(x.~-.x,,
YO-~3)
,
(5)
where B is the Fourier transform of the SLM which can be with binary, multi-level or continuous phase modulation. It only depends on the position difference, and is therefore a space-invariant system.
G. G. Yang. S. E. Brownfield / Optics Communications
Table I The schemes of configurations Scheme
c
?‘,, 1 =
,=I
(
341
of two binary phase SLM system.
6,. BPSLMI
c2, Input of hz
hz, BPSLM2
d2, Input to L3
(11. Output
real and even (symmetric) real and asymmetric real and even (symmetric) real and asymmetric
real and even (symmetric) complex and hermitian real and even (symmetric) complex and hermitian
real and even (symmetric) real and even (symmetric) real and asymmetric real and asymmetric
real and even (symmetric) complex and hermitian real and asymmetric complex and asymmetric
real and even
Assume the input function is a point at xo=~,,=O, i.e. the complex amplitude of its Fourier transform C, (x,, y, ) = I. This is the case for generating a desired pattern by using a computer-generated hologram. Two Fourier transforms are performed by the system. The system characteristic can be analyzed according to the Fourier theorem. As shown in Fig. I, the first BPSLM b, (s,. v, ) is transformed to c-,(x7, yZ) at plane P, by lens L,. Then the lens .& transforms the function c2(x7, y?) b1~.x2.y2) to form the output at plane Pi. Schemes for producing outputs with different properties using this system are shown in Table 1. The first scheme in the table produces a real symmetric output with symmetric patterns on the two BPSLMs. Real asymmetric patterns can be generated by an asymmetric BPSLM 1 and a symmetric BPSLM2, shown as the second scheme. With an asymmetric BPSLM2, complex hermitian or complex asymmetric outputs can be achieved with a symmetric or asymmetric BPSLMl, respectively. It indicates that all wavefronts with real symmetric. complex hermitian, real asymmetric and complex asymmetric properties can be generated with the proposed system using binary phase modulation. The system can be analyzed in a discrete form in a similar way to that described earlier. Assume that the rzth plane consists of N,, X N,, square pixels with side length of A,,. Eq. (2) now becomes c,,(x,,.
I24 ( 1996) 345-353
c
U,.,(F1- 1)
(symmetric) real and asymmetric complex and hermitian complex and asymmetric
where 0 denotes convolution. The function c,,(x,,. y,,) with continuous variable may be written in terms of its samples c,,,(n) c,Z(&, Y,!) = c,,,(n) (7)
Xexp
(pr+qs)
-j z (
n I
i
sine
(&I
~4~~
(8) if 1
_ 44-r
N,,p I
(9)
v
Defining G,~4~,~( n,
12 -
I ) =
exp
(pr+qs)
(10) Eq. (8) becomes
, = I
1
C,,,(F2)=
c c
G,>y’.$(~~, n- l)a,.,(F2- 1)
(11)
@sinc(G~;~sinc(YA;_ Xcxp
-jr__
N--l(rx,+sy,,)
‘) ,
).=I
v=,
This can be expressed in matrix form:
where the propagaticln matrix G is a four-dimensional matrix. The whole cascaded system can be expressed in 3 matrix
hrni
A,=G?2xb2XG2,
as
Xb, xG,,,xCo,
(13)
whcrc b, and bl represent the states of the binary phase SLMs. It can also be described as A i = T,,, x G, >
( 14)
where T,,, is the transform matrix T,,,=Gq2xb2xG2,
xb, xG,o,
( 15)
the elements 01 T3,, are Txo(k, 1; r, s 1 = C ,“I
C
&(P> 4)G,,,,,(3>
2)
<,--I
We now consider the number of degrees of freedom required of the system to find the solution of b, and bZ, in the tirst case, to achieve a given transform T3ri, and in the second case, to produce a target output A, with a given input C,,. Assume the two SLMs have the same pixel size d and number M of pixels along its each side. i.e. A, = A2 = A ; N, = N2 =M. Also assume that the sampling number along each side of the input and output planes is N,, = Nj = N. b, and b, can be solved by using Eq. ( 16). The equation can be resolved into two equations for real and imaginary parts. The equation is a set of 2XN’XN’ linear equations with 2 XMXM unknown variables (MXM for 6, and and M X M for bz ) . If M = N’, i.e. the number of sampling points in the SLMs is equal to the product of the number of sampling points in the input and output planes, we have a unique solution of the set of equations. Therefore, the system can perform any given wavefront transformation, if the two SLMs have continuous phase modulation. In the general two-dimensional case, to achieve a space-variant operation from an input wavefront specified by N’ sampled values to an output wavefront specified by N’ sampled values, the number of sampling points of the transform matrix which is represented by a hologram with amplitude and phase modulation should equal the product of the number of
sampling points at the input and output plane, i.e. N’ X N’. It implies that to achieve a space-variant transform with N’ X N’ degrees of freedom, it is necessary for the hologram to have the same numbers of degrees of freedom. The complex transmission values of the BPSLMs are only + 1 and -1. however. This restricts the wavefront transforms which the system can achieve. In general it is impossible to find the exact solution of Eq. ( 16) for large N. An optimisation algorithm has to be used to find the closest approximate solution. Therefore, in the sense of an approximate solution general transformations with binary phase SLMs can still be achieved. An optical crossbar interconnection system between N X N input and N X N output arrays using the scheme is presented as an example of applications in Section 4. To produce a target output A, with a given input C,,, the equation (14) which consists of 2 XNXN linear equations and 2 XMXM unknown variables can be directly applied. If M=N, a unique solution may be found. As discussed above, for a large number of N only a closest approximate solution can be reached by optimization. It is noted that the condition of M= N’ mentioned above is a sufficient but not a necessary condition. Thus even when M=N, the system is still capable of generating some complex wavefronts. This will be discussed in Section 3.
3. The performance
of the two-BPSLM
system
In order to demonstrate the system performance, several patterns shown in Fig. 2 are selected as the target functions for the calculation. To provide a comparison,
b)
0 d)
0
Fig. 2. The target patterns for simulations.
C)
G. G. Yang. S. E. Broornjeld / Optics Communications
124 (1996) 345-353
349
Table 2 The simulation results and comparison. Target pattern
Calculated
NU
Compared
(%)
(%‘o)
(%)
(%)
a b c d e
81.2 81.6 73.8 76.2 75.0
3.2 0.5 1.0
41.1 in [3] 40.4in 131 40.0 in [2]
7.3 in [3] 3.0 in [3]
q,
I)
12- la;(k I) 1212
(17)
1
k.l where a3 (k, 1) is the actual amplitude output and a\ (k, I) is the target amplitude output. First of all, SLMl and SLM2 are set to random binary phase values and the cost function C is evaluated. One of the two SLMs is then randomly selected, and the phase of one randomly selected pixel is changed by rr from its current state. The new cost function is calculated. If the cost is reduced, the change is accepted, otherwise rejected. The process is then repeated. After about 10 N2 iterations the process converges to a near optimum solution, where N is the number of pixels in one dimension of an SLM. Simulation results for the target patterns of Fig. 2 are given in Table 2 in which the non-uniformity of the output is defined as NU=
zMax - lrvlin fix,,+Lli” ’
(18)
Fl
n-l
-
-
hut
Polarizer
Compare NU
where &,, and Zrvlinare maximum and minimum peak intensities respectively. This shows that about 75% efficiency is obtained by using the two BPSLM system with 128 X 128 pixels for generating the patterns. This is a remarkable increase of the results obtained with a single BPSLM which are shown for comparison in Table 2. This comparison shows that not only is the diffraction efficiency higher, but the uniformity is improved with the two-BPSLM system. Experimental results were obtained using the arrangement shown in Fig. 3. This is equivalent to the stages 1 and 2 of Fig. 1. The binary phase SLM is created by placing a FLC SLM between crossed linear polarizers. These are arranged so that incident electric field vector of the radiation bisects the angle between which the director of the FLC SLM pixels are switched. This produces output states which are of the same intensity, but differ by rr phase [ 91. In the cascaded system only three polarizers are required. The FLC SLM has 128 X 128 square pixels with size length 165 pm and the aperture size of the SLM is 21 mm square. The pixellated Fourier transform of SLM 1 at plane P;needs to match the dimension of SLM2. To achieve this, the Fourier plane image formed at plane Pi is scaled by lenses L2and L3 to be imaged again on plane P2.With
most of the patterns are the asymmetric patterns used in Refs. [ 1,2]. A pixel is treated as a point in the calculation. The direct binary search algorithm [ 81 is used in the optimal design phase. The cost function is defined as:
c= c [ In,(k,
q,
SLM 1
_ Polarizer
0 L2
Fig. 3. The experimental
optical system.f,
= 200 mm& =f4= 250 mm and the objective L2 is 10 X
G.G. Yang, S.E. Broomjield/Optics
Communications
124 (1996) 345-353
Table 3 The predicted and experimental
result
Diffraction efficiency Non-uniformity of the spots
Fig. 4. Exprrimrntal outputpatterncreated by the cascaded BPSLM system
Iwo-
A = 632.8 nm.,f= 200 mm of L, the arrangement forms a side length of a pixel A, of about 6 pm. The magnification to I70 km (the pixel side length plus the pixel space ofthe SLM) is achieved by using a 10 X objective L2 and a lens L, v= 250 mm). Then a lens Lj with focal lengthf‘= 250 mm is used as a Fourier transform lens. This output pattern is displayed at output plane P,. The calculated designs are loaded to the two SLMs from a computer. The asymmetric pattern shown in Fig. 2e is generated in the experiment. Fig. 4 is the experimental output when the optimized binary phase patterns shown in Fig. 5 are loaded on the two SLMs. Black and white in Fig. 5 correspond to 0 and rphase shift. A comparison between the predicted and experimental results is given in Table 3.
SLM 1 Fig. 5. SLM pixel array used to create asymmetric
Predicted
Measured
75% I%
58% 6%
The diffraction efficiency of the experiment is lower than the theoretical prediction. Firstly, this is due to the aberration of the optical system. As there are many optical components in the system, it is difficult to align the system perfectly. Another important factor is attributed to the diffraction effect not included in the optimization process. All of the above computations are based on treating a pixel as a point, i.e. the finite size of a pixel and aperture of the optical system were not taken into account. The error of the reconstructed image due to diffraction of finite aperture is called leakage in the computer-generated hologram [lo]. Eq. (7) describes the amplitude distribution at the back focal plane of a lens after the light passes through a SLM at the front focal plane. There are two sine functions in the equation, sinc(x,N,,A,lAf) sinc(y,,N,/Af) due to the aperture effect and sinc(x,,A,,lhf) sinc(y,A,,lAfl due to the pixellation effect of the SLM. The latter is a smooth function over the full aperture, and only causes a small deviation from the result. The former, however, causes a large error, because the sine function varies rapidly over the Fourier plane. However, the function sinc(x,,N,,A,lAf) sinc(y,N,,A,lhf) becomes sine(p) sine(q), according to the Eq. (9), which is non-zero only when p = q = 0, and is not revealed unless it is oversampled. The leakage from parts of the diffraction
SLM 2 output of Fig. 4, where white and black correspond
to 0 and v phase shift
G. G. Yang, S.E. Broomfield
/ Optics Communications
pattern is not controlled by the optimization algorithm. The claim that this causes lower diffraction efficiency can be verified by using a large pixel size on the second SLM. This was done in the following way. The optimisation was performed for the generation of an off-axis spot assuming 64 X 64 pixels in each SLM. Then the optimised pattern for the first SLM was duplicated four times to occupy the whole of the SLM ( 128 X 128 pixels) _The optimized pattern on the second SLM was expanded to occupy the whole of the SLM by using larger pixels which consist of 2 X 2 pixels in the same state. The effect of sidelobes from the sine function is reduced in this case. Four duplicated patterns appear at the output plane. The measured total diffraction efficiency is about 76% which is consistent with the theoretical prediction. Then the performance in which the function of sinc(x,~N,,A,IAj) sinc(yJVnAJhj) is included in the computation is evaluated by using the SLM patterns shown in Fig. 5. This is done by oversampling to 256 X 256 instead of 128 X 128 sampling points. 63% diffraction efficiency and 4% non-uniformity result. All above discussions confirm that the difference between the predicted and experimental results is due to the leakage problem. In practice, the problem can be overcome by introducing the sine function into the optimization process by oversampling, say 256 X 256 or 512 X 5 12 sampling points are taken in the computation, this is feasible if high-performance computer facilities are available. We will apply the oversampling technique in the optimization of an optical interconnection system in next section.
4. The programmable one-to-one optical interconnection system using two BPSLMs A typical example for which shift variant performance is essential is the programmable optical crossbar one-to-one interconnection system. Several schemes for optical interconnection have been proposed [ 11,121. However, most of them suffer from the drawback of low efficiency and large spot size. Our system with two BPSLMs shown in Fig. 6 can achieve high diffraction efficiency with diffraction-limited spot size. As mentioned above, in order to realize some complex space-variant operations, N’ X N* degrees of freedom have to be employed, if the input and output are spec-
I24 (1996) 345-353
BPSLMI
Fig. 6. The one-to-one RPSLMs.
351
BPSLMZ
optical interconnection
system using two
ified at N2 sampling points. For simplicity, we shall discuss the one dimensional case, i.e. N 1D light sources are connected to N 1D detectors with one-to-one connectivity. We develop a matrix form to describe the interconnection system. The intensity input can be expressed by a matrix as co= (c?, c;, . . . . co,) )
(19)
where
I
(20)
The desired intensity output A, is A, = (aif, a;, . . ., a;) ,
(21)
where a: is also a column vector. The position of a nonzero element in each u’ corresponds to the position of a detector which is connected to the ith light source. The interconnection can also be described by Eq. ( 14) in which T is the interconnection transform. The intensity input and desired output matrices for N = 4 in our simulation are taken to be 1 i/p
[ Gl
=
0 i 00
0 1 0 0
0
0 0
I
10’ 00 1
(22)
The system connects the first light source to the first detector, the second to the third, the third to the fourth
1.52
G. G. Yang. S. E. Broomfield / Optics Communications
and the fourth to the second. detined as
The cost function
is
(23) To overcome the leakage problem, the aperture effect due to the sine function sinc(x,N,,A.lAf) sinc(y,,N,,A,!/hf) can be included in the optimization process by taking 32 and 128 sampling points, instead of 16 and 64 points at the second SLM plane for the N = 4 and 8 interconnection system respectively. Then the optimized actual output for N= 4 is /0.816 0.080 [A,1 ;,cma~= 0.005
0.074 0.016 0.036
0.105 0.728 0.245
0.0021 0.175 0.747 I ’
\0.055
0.750
0.110
0.065)
I-
(241 its corresponding is 0.049 with
0.80
0.01
124 (I 996) 345-353
spot size in order to efficiently couple light into a fibre due to the small aperture of each facet hologram. Our scheme does not have this limitation as the full aperture is used for all inputs. A spot size of around a few microns diameter can be obtained, which is two orders of magnitude better than the multiple-facet hologram system. This makes the scheme attractive for interconnecting between a large number of nodes with programmable capability. However, the cross-talk in the system is still high, about 20%. This is due to the fact that binary phase modulation limits the degrees of freedom of the system. If the signal to be interconnected is binary, a threshold detector might be used to suppress the cross-talk. Each point in the input plane is projected to a given coordinate according to a required transform in the above scheme. Therefore, the proposed optical interconnection system is also an optical coordinate transform system. It also has the same advantages mentioned above over the optical coordinate system implemented by a multiple-facet hologram [ 131.
cost is C= 0.058. For N= 8 the cost 5. Discussion
0.06
0.07
0.00
0.01
0.02
0.03
0.06
0.03
0.04
0.02
0.75
0.03
0.01
0.06
0.02
0.01
0.07
0.01
0.01
0.76
0.09
0.02
0.03
0.80
0.04
0.03
0.02
0.00
0.03
0.04
0.04
0.09
0.78
0.01
0.01
0.02
0.00
0.04
0.01
0.00
0.01
0.78
0.03
0.01
0.04
0.11
0.08
0.02
0.05
0.02
0.00
0.01
0.01
0.80
0.04
0.07
0.77
0.00
0.03
0.01
0.02
0.04
(21 The results show that high efficiency can also be achieved with the two-BPSLM system used for optical interconnection. It is independent of the dimension of the interconnection. This is the advantage over the optical crossbar interconnection system based on a vectormatrix-multiplication scheme [ 111, in which the efficiency is inversely dependent on the dimension of the interconnection. A recent scheme for implementing crossbar switching is to use a multiple-facet hologram [ 121. Each input illuminates a facet hologram which directs light into the desired direction. The main drawback of the scheme is that it is difficult to obtain a small
and conclusions
The system operation mechanism can be considered in the following way. The first SLM generates a desired amplitude distribution at the second SLM plane, the second SLM provides a desired phase modulation. The two SLMs act like a sandwich hologram in which one produces a amplitude modulation and another a phase modulation as indicated in Ref. [ 51. Under our optimisation process the two SLMs both contribute to the phase and amplitude modulation. Since the optimisation moves between the two SLMs randomly, there is no way to tell which one contributes to the amplitude or phase modulation. This provides more freedom to optimize the system. From the Fourier expansion it is easy to deduce that only 40% diffraction efficiency can be obtained with one binary phase hologram. However, it is not obvious that 80% efficiency can be achieved with a two-SLM system. In order to understand the operation mechanism, the simulation was made by optimising an onaxis focal spot with a normal incident plane wave as the input. We look at the amplitude distribution at the plane P2 in Fig. 1. Due to the symmetric property of the first BPSLM the amplitude at P2 is a real function
G. G. Yang, S. E. Broomfield / Optics Communications 124 (1996) 345-353
with phase values of 0 and n: The second BPSLM compensates all the phase variation at plane Pz. Thus the input to the lens L2 is a wavefront with uniform phase. This is why the lens L2 can focus the light into a spot. The reduction of total efficiency from 100% to 80% is due to the non-uniform intensity distribution at plane P,. It is well known that only a plane wave with uniform intensity and phase can be focused by a lens with 100% efficiency. Secondly, the simulation for an off-axis focused spot was made. The phase distribution at plane P2 before BPSLMZ is a complex distribution. Careful examination of the wavefront behind BPSLM2 shows that its phase differences between adjacent pixels are constant. This means that the wavefront behind SLM2 is an oblique plane wave. Because a complex wavefront can be decomposed into a set of plane waves, according to the theory an angular spectrum of a plane wave, we are convinced that the two BPSLM system can generate an complex wavefront. The required computation time in optimising the system is considerably longer than the time required for a single binary phase hologram, or for two binary holograms in four-phase level hologram systems in which the pointwise Fourier transform [ 141 can be used during the optimization process. The optimisition of the cascaded Fourier transform system described here takes several hours run on a Convex machine which is a parallel processing high speed computer. This is because a full 2D Fourier transform has to be taken from P, to P2 even when only a single pixel is changed at SLMl. The pointwise Fourier transform can only be used at the second stage from plane P, to P,. The system is similar to the four-level phase SLM system [4] in that two binary phase SLMs in cascade are used and around 80% diffraction efficiency can be reached in these two systems. However, the second SLM is put at the image plane of the first SLM in the four-phase level system, instead of at the Fourier plane in our system. The advantages of the four-phase level system are high diffraction efficiency, ease of computation and implementation. However, the significant difference between the two systems is the property of spatial variance. As shown in Eq. (5)) the conventional single-phase-only holograms, such as a phase-only matched filter performs a space-invariant operation.
353
Thus only space-invariant operations can be realized with the single phase-only holograms, no matter how many phase levels are used in the phase holograms. We conclude that the programmable wavefront generation system with two binary phase SLMs cascaded at Fourier transform planes is a high diffraction efficiency and space-variant system. The amplitude and phase of a wavefront can be modulated with the system. It is experimentally implemented by two FLC SLMs. The experimental results are consistent with theoretical prediction. A programmable optical crossbar interconnection system can also be based on the two BPSLM scheme.
Acknowledgement The authors would like to thank Professor Paige for stimulating discussions and valuable tions. The UK Engineering and Physical Research Council (EPSRC) and DRA( MOD) porting this work is also acknowledged.
E.G.S. sugges-
Science for sup-
References [l] M.A.A. Neil and E.G.S. Paige, Proc. 4th Int. Conf. on Holographic Systems, Components and Applications, IEE Conf. Publication 378 (1993) 85-90. [2] M.S. Kim and C.C. Guest, Appl. Optics 32 (1993) 678. [3] G.J. Swanson and W.B. Veldkamp, Opt. Eng. 28 ( 1989) 605. 141 S.E. Broomfield, M.A.A. Neil and E.G.S. Paige. Electronics Letters 29 (1993) 1661. [5] H. Bartelt, Appl. Optics 23 (1984) 1499. [6] G.G. Yang, D.L. Lian, J.J. Zhang, J.P. Chen and Y.P. Ho, Optics Len. 16 (1991) 162. [7] G.G. Yang, Proc. 3rd Int. Conf. on Holographic Systems, Components and Applications, IEE Conf. Publication 342 (1991) 45-49. [S] M.A. Seldwitz, J.B. Allebach and D.W. Sweenney, Appl. Optics 28 (1987) 2788. [9] S.E. Broomfield, M.A.A. Neil, E.G.S. Paige and G.G. Yang, Electron. Lett. 28 (1992) 26. [lo] B.K. Jennison and J.D. Alleback, J. Opt. Sot. Am. 6 (1989) 234. [ 111 A.A. Sawchuk and B.K. Jenkins, Proc. SPIE 625 (1986) 143. [ 121 K.S. Urguhart and S.H. Lee, Appl. Optics 33 ( 1994) 3670. [ 131 0. Bryngdahl, J. Opt. Sot. Am. 64 (1974) 1092. [ 141 M.P. Damws, R.J. Dowling, P. McKee and D. Wood, Appl. Optics 30 (1991) 2685.
2%
I March 1996
OPTICS COMMUNICATIONS
ELSEYIER
Optics Communications
124 ( 1996) 345-353
Full length article
Programmable wavefront generation using two binary phase spatial light modulators G.G. Yang, S.E. Broomfield Depnrtment
ofEngineeringScience.
University qj’ Oxford, O.xfiwd, OX1 3P.I. UK
Received 3 July 1995
Abstract A programmable wavefront generation system using two binary phase spatial light modulators is described. This a a spacevariant system with high diffraction efficiency. The amplitude and phase of a wavefront can be modulated with the system. It is
experimentally implemented with two ferroelectric liquid crystal SLMs. A programmable optical crossbar interconnection based on this scheme is also discussed. The properties of the system are analysed.
1. Introduction
There is interest in spatial light modulators ( SLMs) for a variety of applications including the general manipulation of wavefronts, reconfigurable optical interconnects, beam steering and shaping. Usually the SLM is restricted to perform wavefront modulation of either amplitude or phase, but not both. The advantage of phase modulation is that no intrinsic loss is associated with it. Phase-only SLMs may be subdivided into those for which the modulation can be continuously varied and those for which modulation is achieved by switching between two discrete phase states, commonly separated by 7~. In this paper we are concerned with the latter, binary phase spatial light modulators (BPSLMs), because they are well suited to demonstrate high performance with the available devices. It is usual for the output of the modulator to be viewed in the Fourier plane of a SLM/positive lens combination. As a consequence, the output of the binary phase hologram has inversion symmetry and is restricted to be space-invariant. If an output which has inversion symmetry is not required, then there is a loss 0030.4018/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDfOO30.4018(95)00602-8
system
of diffraction efficiency Q, defined as the fraction of power incident as required in the Fourier plane relative to the total power transmitted by the SLM. Inversion symmetry may be broken by introducing a fixed binary phase hologram [ 1,2] but in this case Q is not increased. Alternative, an asymmetric output may be generated by increasing the number of phase levels [ 3,4]. Results obtained in this way increase qd but the output remains space-invariant. It has been pointed out by Bartelt [5] and Yang [6,7] that the restriction of space-invariance can be removed by effectively cascading two phase masks in two Fourier-transforming stages. The amplitude and phase of a wavefront can be modulated with the combined system. In operation Bartelt’s scheme [5] was designed to produce the required amplitude modulation by the first Fourier-transforming stage and the required phase modulation by the second. It was implemented using fixed phase filters with continuous phase modulation specified at N X N sampling points, when the input and output signals are also specified at N X N sampling points. Yang’s scheme [7] uses two binary phase holograms each with N’ X N2 sampling points.
346
G. G. Yang. S. E. Broomjield / Optics Communications
The latter is easier to implement using the available technology and can realize general space-variant operations because there is more freedom to modulate the wavefront. In this paper we consider a similar binary phase scheme [7] except that fixed phase holograms are replaced by binary phase SLMs and the design method to achieve a given wavefront is based on simultaneous, iterative optimization of both SLMs. The former gives programmable wavefront generation, the latter improves performance. The system is demonstrated by producing an asymmetric output. Simulation of its performance as a reconfigurable optical interconnect is also presented. Both utilize the space-variant performance of the system. The properties of our scheme are analyzed in Section 2. In Section 3 the performance is simulated for a system with two binary phase SLMs each with 128 X 128 pixels, and the experimental results are presented for a system comprising two ferroelectric liquid crystal spatial light modulators (FLC SLMs). In Section 4 a programmable optical crossbar interconnection system using this scheme is described. Results and conclusions are presented in Section 5.
124
(I 996) 345-353
nth plane designated P, is c,~(x,, Yn), the transmission of the nth plane due to the SLM is b,(x,, y,) and the output of the nth plane is a,( x,, Y,) . Thus we have a,(~> Y,) =bn(x,,
Yn)c,(x,> Y,J .
(1)
The basic transform relation between adjacent cascaded stages under the scalar, paraxial approximation is
Xexp
-_ie(-w,,-,+y,,v,t-,) M (2)
where A is the wavelength,fis the focal length of the lens and A is a constant. The relationship between the input and the output of the system is described by %(X,9 Y3) 0Z =A 1111 -x
c,(~,,~,,)~,(x,,YI)B~(x~--xI,)~~-?!,)
2. The two binary phase SLM system
(3)
The system is shown in Fig. 1. The system consists of two cascaded Fourier transforms (stage 1 and 2) with an input stage (stage 0) which for convenience is also presented as a Fourier transform. The first BPSLM is at the front focal plane P, of the lens L,, the second BPSLM is at the back focal plane P2 of the same lens. The second lens L2 forms the output at its back focal plane P,. We define the following notation for the nth stage (n = 0, 3) : the complex amplitude input to the
where B, is the Fourier transform of the second BPSLM, b,. The impulse response of this system is x h(X”, Y& x3, Y3) =
It is clear that the difference output plane. system. This response of a
II --cc
h(x,,
PI stage 0
Fig.
P?
sl%2 stage
I
Output stage
I. The two binary phase SLM system.
2
Y?-Y,)
the impulse response does not depend on of the coordinates between the input and Therefore, the system is a space-variant is to be contrasted with the impulse single-phase SLM system which is
h(x,, YO;xj,Y3) =B(xo-x,,
SLMI
Y,> &(x.~-.x,,
YO-~3)
,
(5)
where B is the Fourier transform of the SLM which can be with binary, multi-level or continuous phase modulation. It only depends on the position difference, and is therefore a space-invariant system.
G. G. Yang. S. E. Brownfield / Optics Communications
Table I The schemes of configurations Scheme
c
?‘,, 1 =
,=I
(
341
of two binary phase SLM system.
6,. BPSLMI
c2, Input of hz
hz, BPSLM2
d2, Input to L3
(11. Output
real and even (symmetric) real and asymmetric real and even (symmetric) real and asymmetric
real and even (symmetric) complex and hermitian real and even (symmetric) complex and hermitian
real and even (symmetric) real and even (symmetric) real and asymmetric real and asymmetric
real and even (symmetric) complex and hermitian real and asymmetric complex and asymmetric
real and even
Assume the input function is a point at xo=~,,=O, i.e. the complex amplitude of its Fourier transform C, (x,, y, ) = I. This is the case for generating a desired pattern by using a computer-generated hologram. Two Fourier transforms are performed by the system. The system characteristic can be analyzed according to the Fourier theorem. As shown in Fig. I, the first BPSLM b, (s,. v, ) is transformed to c-,(x7, yZ) at plane P, by lens L,. Then the lens .& transforms the function c2(x7, y?) b1~.x2.y2) to form the output at plane Pi. Schemes for producing outputs with different properties using this system are shown in Table 1. The first scheme in the table produces a real symmetric output with symmetric patterns on the two BPSLMs. Real asymmetric patterns can be generated by an asymmetric BPSLM 1 and a symmetric BPSLM2, shown as the second scheme. With an asymmetric BPSLM2, complex hermitian or complex asymmetric outputs can be achieved with a symmetric or asymmetric BPSLMl, respectively. It indicates that all wavefronts with real symmetric. complex hermitian, real asymmetric and complex asymmetric properties can be generated with the proposed system using binary phase modulation. The system can be analyzed in a discrete form in a similar way to that described earlier. Assume that the rzth plane consists of N,, X N,, square pixels with side length of A,,. Eq. (2) now becomes c,,(x,,.
I24 ( 1996) 345-353
c
U,.,(F1- 1)
(symmetric) real and asymmetric complex and hermitian complex and asymmetric
where 0 denotes convolution. The function c,,(x,,. y,,) with continuous variable may be written in terms of its samples c,,,(n) c,Z(&, Y,!) = c,,,(n) (7)
Xexp
(pr+qs)
-j z (
n I
i
sine
(&I
~4~~
(8) if 1
_ 44-r
N,,p I
(9)
v
Defining G,~4~,~( n,
12 -
I ) =
exp
(pr+qs)
(10) Eq. (8) becomes
, = I
1
C,,,(F2)=
c c
G,>y’.$(~~, n- l)a,.,(F2- 1)
(11)
@sinc(G~;~sinc(YA;_ Xcxp
-jr__
N--l(rx,+sy,,)
‘) ,
).=I
v=,
This can be expressed in matrix form:
where the propagaticln matrix G is a four-dimensional matrix. The whole cascaded system can be expressed in 3 matrix
hrni
A,=G?2xb2XG2,
as
Xb, xG,,,xCo,
(13)
whcrc b, and bl represent the states of the binary phase SLMs. It can also be described as A i = T,,, x G, >
( 14)
where T,,, is the transform matrix T,,,=Gq2xb2xG2,
xb, xG,o,
( 15)
the elements 01 T3,, are Txo(k, 1; r, s 1 = C ,“I
C
&(P> 4)G,,,,,(3>
2)
<,--I
We now consider the number of degrees of freedom required of the system to find the solution of b, and bZ, in the tirst case, to achieve a given transform T3ri, and in the second case, to produce a target output A, with a given input C,,. Assume the two SLMs have the same pixel size d and number M of pixels along its each side. i.e. A, = A2 = A ; N, = N2 =M. Also assume that the sampling number along each side of the input and output planes is N,, = Nj = N. b, and b, can be solved by using Eq. ( 16). The equation can be resolved into two equations for real and imaginary parts. The equation is a set of 2XN’XN’ linear equations with 2 XMXM unknown variables (MXM for 6, and and M X M for bz ) . If M = N’, i.e. the number of sampling points in the SLMs is equal to the product of the number of sampling points in the input and output planes, we have a unique solution of the set of equations. Therefore, the system can perform any given wavefront transformation, if the two SLMs have continuous phase modulation. In the general two-dimensional case, to achieve a space-variant operation from an input wavefront specified by N’ sampled values to an output wavefront specified by N’ sampled values, the number of sampling points of the transform matrix which is represented by a hologram with amplitude and phase modulation should equal the product of the number of
sampling points at the input and output plane, i.e. N’ X N’. It implies that to achieve a space-variant transform with N’ X N’ degrees of freedom, it is necessary for the hologram to have the same numbers of degrees of freedom. The complex transmission values of the BPSLMs are only + 1 and -1. however. This restricts the wavefront transforms which the system can achieve. In general it is impossible to find the exact solution of Eq. ( 16) for large N. An optimisation algorithm has to be used to find the closest approximate solution. Therefore, in the sense of an approximate solution general transformations with binary phase SLMs can still be achieved. An optical crossbar interconnection system between N X N input and N X N output arrays using the scheme is presented as an example of applications in Section 4. To produce a target output A, with a given input C,,, the equation (14) which consists of 2 XNXN linear equations and 2 XMXM unknown variables can be directly applied. If M=N, a unique solution may be found. As discussed above, for a large number of N only a closest approximate solution can be reached by optimization. It is noted that the condition of M= N’ mentioned above is a sufficient but not a necessary condition. Thus even when M=N, the system is still capable of generating some complex wavefronts. This will be discussed in Section 3.
3. The performance
of the two-BPSLM
system
In order to demonstrate the system performance, several patterns shown in Fig. 2 are selected as the target functions for the calculation. To provide a comparison,
b)
0 d)
0
Fig. 2. The target patterns for simulations.
C)
G. G. Yang. S. E. Broornjeld / Optics Communications
124 (1996) 345-353
349
Table 2 The simulation results and comparison. Target pattern
Calculated
NU
Compared
(%)
(%‘o)
(%)
(%)
a b c d e
81.2 81.6 73.8 76.2 75.0
3.2 0.5 1.0
41.1 in [3] 40.4in 131 40.0 in [2]
7.3 in [3] 3.0 in [3]
q,
I)
12- la;(k I) 1212
(17)
1
k.l where a3 (k, 1) is the actual amplitude output and a\ (k, I) is the target amplitude output. First of all, SLMl and SLM2 are set to random binary phase values and the cost function C is evaluated. One of the two SLMs is then randomly selected, and the phase of one randomly selected pixel is changed by rr from its current state. The new cost function is calculated. If the cost is reduced, the change is accepted, otherwise rejected. The process is then repeated. After about 10 N2 iterations the process converges to a near optimum solution, where N is the number of pixels in one dimension of an SLM. Simulation results for the target patterns of Fig. 2 are given in Table 2 in which the non-uniformity of the output is defined as NU=
zMax - lrvlin fix,,+Lli” ’
(18)
Fl
n-l
-
-
hut
Polarizer
Compare NU
where &,, and Zrvlinare maximum and minimum peak intensities respectively. This shows that about 75% efficiency is obtained by using the two BPSLM system with 128 X 128 pixels for generating the patterns. This is a remarkable increase of the results obtained with a single BPSLM which are shown for comparison in Table 2. This comparison shows that not only is the diffraction efficiency higher, but the uniformity is improved with the two-BPSLM system. Experimental results were obtained using the arrangement shown in Fig. 3. This is equivalent to the stages 1 and 2 of Fig. 1. The binary phase SLM is created by placing a FLC SLM between crossed linear polarizers. These are arranged so that incident electric field vector of the radiation bisects the angle between which the director of the FLC SLM pixels are switched. This produces output states which are of the same intensity, but differ by rr phase [ 91. In the cascaded system only three polarizers are required. The FLC SLM has 128 X 128 square pixels with size length 165 pm and the aperture size of the SLM is 21 mm square. The pixellated Fourier transform of SLM 1 at plane P;needs to match the dimension of SLM2. To achieve this, the Fourier plane image formed at plane Pi is scaled by lenses L2and L3 to be imaged again on plane P2.With
most of the patterns are the asymmetric patterns used in Refs. [ 1,2]. A pixel is treated as a point in the calculation. The direct binary search algorithm [ 81 is used in the optimal design phase. The cost function is defined as:
c= c [ In,(k,
q,
SLM 1
_ Polarizer
0 L2
Fig. 3. The experimental
optical system.f,
= 200 mm& =f4= 250 mm and the objective L2 is 10 X
G.G. Yang, S.E. Broomjield/Optics
Communications
124 (1996) 345-353
Table 3 The predicted and experimental
result
Diffraction efficiency Non-uniformity of the spots
Fig. 4. Exprrimrntal outputpatterncreated by the cascaded BPSLM system
Iwo-
A = 632.8 nm.,f= 200 mm of L, the arrangement forms a side length of a pixel A, of about 6 pm. The magnification to I70 km (the pixel side length plus the pixel space ofthe SLM) is achieved by using a 10 X objective L2 and a lens L, v= 250 mm). Then a lens Lj with focal lengthf‘= 250 mm is used as a Fourier transform lens. This output pattern is displayed at output plane P,. The calculated designs are loaded to the two SLMs from a computer. The asymmetric pattern shown in Fig. 2e is generated in the experiment. Fig. 4 is the experimental output when the optimized binary phase patterns shown in Fig. 5 are loaded on the two SLMs. Black and white in Fig. 5 correspond to 0 and rphase shift. A comparison between the predicted and experimental results is given in Table 3.
SLM 1 Fig. 5. SLM pixel array used to create asymmetric
Predicted
Measured
75% I%
58% 6%
The diffraction efficiency of the experiment is lower than the theoretical prediction. Firstly, this is due to the aberration of the optical system. As there are many optical components in the system, it is difficult to align the system perfectly. Another important factor is attributed to the diffraction effect not included in the optimization process. All of the above computations are based on treating a pixel as a point, i.e. the finite size of a pixel and aperture of the optical system were not taken into account. The error of the reconstructed image due to diffraction of finite aperture is called leakage in the computer-generated hologram [lo]. Eq. (7) describes the amplitude distribution at the back focal plane of a lens after the light passes through a SLM at the front focal plane. There are two sine functions in the equation, sinc(x,N,,A,lAf) sinc(y,,N,/Af) due to the aperture effect and sinc(x,,A,,lhf) sinc(y,A,,lAfl due to the pixellation effect of the SLM. The latter is a smooth function over the full aperture, and only causes a small deviation from the result. The former, however, causes a large error, because the sine function varies rapidly over the Fourier plane. However, the function sinc(x,,N,,A,lAf) sinc(y,N,,A,lhf) becomes sine(p) sine(q), according to the Eq. (9), which is non-zero only when p = q = 0, and is not revealed unless it is oversampled. The leakage from parts of the diffraction
SLM 2 output of Fig. 4, where white and black correspond
to 0 and v phase shift
G. G. Yang, S.E. Broomfield
/ Optics Communications
pattern is not controlled by the optimization algorithm. The claim that this causes lower diffraction efficiency can be verified by using a large pixel size on the second SLM. This was done in the following way. The optimisation was performed for the generation of an off-axis spot assuming 64 X 64 pixels in each SLM. Then the optimised pattern for the first SLM was duplicated four times to occupy the whole of the SLM ( 128 X 128 pixels) _The optimized pattern on the second SLM was expanded to occupy the whole of the SLM by using larger pixels which consist of 2 X 2 pixels in the same state. The effect of sidelobes from the sine function is reduced in this case. Four duplicated patterns appear at the output plane. The measured total diffraction efficiency is about 76% which is consistent with the theoretical prediction. Then the performance in which the function of sinc(x,~N,,A,IAj) sinc(yJVnAJhj) is included in the computation is evaluated by using the SLM patterns shown in Fig. 5. This is done by oversampling to 256 X 256 instead of 128 X 128 sampling points. 63% diffraction efficiency and 4% non-uniformity result. All above discussions confirm that the difference between the predicted and experimental results is due to the leakage problem. In practice, the problem can be overcome by introducing the sine function into the optimization process by oversampling, say 256 X 256 or 512 X 5 12 sampling points are taken in the computation, this is feasible if high-performance computer facilities are available. We will apply the oversampling technique in the optimization of an optical interconnection system in next section.
4. The programmable one-to-one optical interconnection system using two BPSLMs A typical example for which shift variant performance is essential is the programmable optical crossbar one-to-one interconnection system. Several schemes for optical interconnection have been proposed [ 11,121. However, most of them suffer from the drawback of low efficiency and large spot size. Our system with two BPSLMs shown in Fig. 6 can achieve high diffraction efficiency with diffraction-limited spot size. As mentioned above, in order to realize some complex space-variant operations, N’ X N* degrees of freedom have to be employed, if the input and output are spec-
I24 (1996) 345-353
BPSLMI
Fig. 6. The one-to-one RPSLMs.
351
BPSLMZ
optical interconnection
system using two
ified at N2 sampling points. For simplicity, we shall discuss the one dimensional case, i.e. N 1D light sources are connected to N 1D detectors with one-to-one connectivity. We develop a matrix form to describe the interconnection system. The intensity input can be expressed by a matrix as co= (c?, c;, . . . . co,) )
(19)
where
I
(20)
The desired intensity output A, is A, = (aif, a;, . . ., a;) ,
(21)
where a: is also a column vector. The position of a nonzero element in each u’ corresponds to the position of a detector which is connected to the ith light source. The interconnection can also be described by Eq. ( 14) in which T is the interconnection transform. The intensity input and desired output matrices for N = 4 in our simulation are taken to be 1 i/p
[ Gl
=
0 i 00
0 1 0 0
0
0 0
I
10’ 00 1
(22)
The system connects the first light source to the first detector, the second to the third, the third to the fourth
1.52
G. G. Yang. S. E. Broomfield / Optics Communications
and the fourth to the second. detined as
The cost function
is
(23) To overcome the leakage problem, the aperture effect due to the sine function sinc(x,N,,A.lAf) sinc(y,,N,,A,!/hf) can be included in the optimization process by taking 32 and 128 sampling points, instead of 16 and 64 points at the second SLM plane for the N = 4 and 8 interconnection system respectively. Then the optimized actual output for N= 4 is /0.816 0.080 [A,1 ;,cma~= 0.005
0.074 0.016 0.036
0.105 0.728 0.245
0.0021 0.175 0.747 I ’
\0.055
0.750
0.110
0.065)
I-
(241 its corresponding is 0.049 with
0.80
0.01
124 (I 996) 345-353
spot size in order to efficiently couple light into a fibre due to the small aperture of each facet hologram. Our scheme does not have this limitation as the full aperture is used for all inputs. A spot size of around a few microns diameter can be obtained, which is two orders of magnitude better than the multiple-facet hologram system. This makes the scheme attractive for interconnecting between a large number of nodes with programmable capability. However, the cross-talk in the system is still high, about 20%. This is due to the fact that binary phase modulation limits the degrees of freedom of the system. If the signal to be interconnected is binary, a threshold detector might be used to suppress the cross-talk. Each point in the input plane is projected to a given coordinate according to a required transform in the above scheme. Therefore, the proposed optical interconnection system is also an optical coordinate transform system. It also has the same advantages mentioned above over the optical coordinate system implemented by a multiple-facet hologram [ 131.
cost is C= 0.058. For N= 8 the cost 5. Discussion
0.06
0.07
0.00
0.01
0.02
0.03
0.06
0.03
0.04
0.02
0.75
0.03
0.01
0.06
0.02
0.01
0.07
0.01
0.01
0.76
0.09
0.02
0.03
0.80
0.04
0.03
0.02
0.00
0.03
0.04
0.04
0.09
0.78
0.01
0.01
0.02
0.00
0.04
0.01
0.00
0.01
0.78
0.03
0.01
0.04
0.11
0.08
0.02
0.05
0.02
0.00
0.01
0.01
0.80
0.04
0.07
0.77
0.00
0.03
0.01
0.02
0.04
(21 The results show that high efficiency can also be achieved with the two-BPSLM system used for optical interconnection. It is independent of the dimension of the interconnection. This is the advantage over the optical crossbar interconnection system based on a vectormatrix-multiplication scheme [ 111, in which the efficiency is inversely dependent on the dimension of the interconnection. A recent scheme for implementing crossbar switching is to use a multiple-facet hologram [ 121. Each input illuminates a facet hologram which directs light into the desired direction. The main drawback of the scheme is that it is difficult to obtain a small
and conclusions
The system operation mechanism can be considered in the following way. The first SLM generates a desired amplitude distribution at the second SLM plane, the second SLM provides a desired phase modulation. The two SLMs act like a sandwich hologram in which one produces a amplitude modulation and another a phase modulation as indicated in Ref. [ 51. Under our optimisation process the two SLMs both contribute to the phase and amplitude modulation. Since the optimisation moves between the two SLMs randomly, there is no way to tell which one contributes to the amplitude or phase modulation. This provides more freedom to optimize the system. From the Fourier expansion it is easy to deduce that only 40% diffraction efficiency can be obtained with one binary phase hologram. However, it is not obvious that 80% efficiency can be achieved with a two-SLM system. In order to understand the operation mechanism, the simulation was made by optimising an onaxis focal spot with a normal incident plane wave as the input. We look at the amplitude distribution at the plane P2 in Fig. 1. Due to the symmetric property of the first BPSLM the amplitude at P2 is a real function
G. G. Yang, S. E. Broomfield / Optics Communications 124 (1996) 345-353
with phase values of 0 and n: The second BPSLM compensates all the phase variation at plane Pz. Thus the input to the lens L2 is a wavefront with uniform phase. This is why the lens L2 can focus the light into a spot. The reduction of total efficiency from 100% to 80% is due to the non-uniform intensity distribution at plane P,. It is well known that only a plane wave with uniform intensity and phase can be focused by a lens with 100% efficiency. Secondly, the simulation for an off-axis focused spot was made. The phase distribution at plane P2 before BPSLMZ is a complex distribution. Careful examination of the wavefront behind BPSLM2 shows that its phase differences between adjacent pixels are constant. This means that the wavefront behind SLM2 is an oblique plane wave. Because a complex wavefront can be decomposed into a set of plane waves, according to the theory an angular spectrum of a plane wave, we are convinced that the two BPSLM system can generate an complex wavefront. The required computation time in optimising the system is considerably longer than the time required for a single binary phase hologram, or for two binary holograms in four-phase level hologram systems in which the pointwise Fourier transform [ 141 can be used during the optimization process. The optimisition of the cascaded Fourier transform system described here takes several hours run on a Convex machine which is a parallel processing high speed computer. This is because a full 2D Fourier transform has to be taken from P, to P2 even when only a single pixel is changed at SLMl. The pointwise Fourier transform can only be used at the second stage from plane P, to P,. The system is similar to the four-level phase SLM system [4] in that two binary phase SLMs in cascade are used and around 80% diffraction efficiency can be reached in these two systems. However, the second SLM is put at the image plane of the first SLM in the four-phase level system, instead of at the Fourier plane in our system. The advantages of the four-phase level system are high diffraction efficiency, ease of computation and implementation. However, the significant difference between the two systems is the property of spatial variance. As shown in Eq. (5)) the conventional single-phase-only holograms, such as a phase-only matched filter performs a space-invariant operation.
353
Thus only space-invariant operations can be realized with the single phase-only holograms, no matter how many phase levels are used in the phase holograms. We conclude that the programmable wavefront generation system with two binary phase SLMs cascaded at Fourier transform planes is a high diffraction efficiency and space-variant system. The amplitude and phase of a wavefront can be modulated with the system. It is experimentally implemented by two FLC SLMs. The experimental results are consistent with theoretical prediction. A programmable optical crossbar interconnection system can also be based on the two BPSLM scheme.
Acknowledgement The authors would like to thank Professor Paige for stimulating discussions and valuable tions. The UK Engineering and Physical Research Council (EPSRC) and DRA( MOD) porting this work is also acknowledged.
E.G.S. sugges-
Science for sup-
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