Single holographic optical element based convolver and correlator: a theoretical study

Single holographic optical element based convolver and correlator: a theoretical study R. KANNAN,

S.V. PAPPU

A theoretical analysis is presented of a proposed multiple function single holographic optical element (MFSHOE) based simultaneous coherent optical convolver and correlator system. The proposed system offers additional advantages (for example, compactness of the total system; display of focused outputs from both the convolution and correlation channels allowing for the ability to carry out pattern recognition and signal processing operations simultaneously, etc) while preserving the beneficial features of earlier proposed systems. It is demonstrated with the support of a simulation study that the proposed system displays reasonably acceptable aberration behaviour. KEYWORDS: holographic optical elements, optical convolution, optical correlation

List of symbols

Y(s,

I’:

d)

d k

iI*

;Yc, y) g(s.y)

O(.u.y) F.G

exp

L‘$(x2+ 1 j

A (x. y )

v’)

reciprocal distance 3 n/A wavelength of light convolution complex conjugate

R(x.y)

Fourier transform operator input signal function reference function with whichf(.u,y) to be convolved/correlated output signal function Fourier transforms off and I:

Introduction

The cascading of conventional optical element (COE) based imaging systems of the type shown in Fig. I has been used extensively in the development of coherent optical signal processor (COSP) systems’-‘. The capabilities of a holographic optical element HOE) to perform more than one function at a time-\- ’ can be utilized profitably in the development of multifunction-single-HOE (MFSHOE) based COSP systems. Such systems, besides being more compact than their COE based counterparts, can provide simultaneous channels for The authors are at the Photonics Laboratories, Department Electrical Communication Engineering. Indian lnstkute of Science, Bangalore -560 012, India. Received 17 October Revised 1 February 1990.

0030-3992/90/030188-07

of 1988.

&.Y)

is

h(x.y) TH

A

Feneral light distribution used for illuminating the input reference function g(.u.y) while recording the MFSHOE general light distribution serving as reference wave during recording of the MFSHOE general light distribution used during replay from the MFSHOE impulse response complex amplitude transmittance ot the MFSHOE increment

carrying out different kinds of signal processing operations (for example, spectral analysis. pattern recognition, deblurring, etc). Recently, we have proposed and analysed an MFSHOE based COSP system for the simultaneous display of the spectrum and the image of an object’. In this paper we present a theoretical analysis of a proposed MFSHOE based simultaneous coherent optical convolver and correlator system. Simultaneous display correlation outputs General

of convolution

aspects

A schematic diagram of an MFSHOE based system is shown in Fig. 2. Following the approach taken in 0 1990

Butterworth-Heinemann

Ltd Optics

188

and

Et Laser Technology Vol22 No 3 1990

Incident plane wave

(x, 'Y, 1

IX2*Y2) dl

)j.fl a

A (x,y)

p1

R (x.yl

‘2

Fig. 1 Conventional lens based building block optical system used in cascade to generate coherent optical signal processing (COSP) systems. P, is the input plane and P, the output plane

Convolution channel

Correlation channel

B Ix,y) Fig. 2 Proposed multifunction-single-holographic optical element (MFSHOE) based simultaneous coherent optical convolver and correlator. P, is the input plane where the signal f(x, y) to be correlated or convolved with the reference function g(x, y) is placed. Pz is the plane where the proposed MFSHOE containing the matched filter and the Fresnel zone plate components is placed. P3 is the plane where the display of correlated and convolved outputs will be observed simultaneously

On-axis reference

plane

-dl2

-

wave

source

our earlier papers, we relate the input and output distributions via the superposition integral’: W,.Y,)

II

=

C

= f(-r,.yl)h(-Y3.l’3:-rl.yl)drI --‘c

4vl

(1)

Using Vander Lugt notation’ it can be shown that the II(.) of the desired MFSHOE should be ~(x~.Y~;x~.Y~)Q

[dlY(x2 -xl.yZ 7-Hb2.y2)

-yl:d2)

1 @%~3.Y3;

d3)

(2)

The impulse responses /I, and hr. corresponding to the convolution and correlation channels, must be realized in the same HOE to achieve the simultaneous display of convolution and correlation outputs in the same plane. This means, in the final analysis, the proper realization of TH for the desired HOE. In what follows we present a generalized analysis of the proposed MFSHOE. Generalized

analysis of the MFSHOE

It is evident from the very definitions for convolution and correlation that the proposed must contain the information of the reference Optics 6 Laser Technology Vol22 No 3 1990

HOE

‘6

p?

Fig. 3 Recording geometries used in realizing the various components of the proposed MFSHOE. (a) Generalized recording geometry for the realization of MFSHOE. P, is the input plane where the reference function g(x, y) will be placed and Pz is the plane where the desired MFSHOE will be recorded by choosing the appropriate A(x, y) and R(x, y) as will be shown. (b) Recording geometry for realizing the matched filter (MF) component. P, is the plane where the function g(x, y) is to be located. P, is the plane where the quasi-Fourier transform (QFT) of g(x, y) is to be recorded. 0, is the off-set angle for the reference wave. (c) Recording geometry for realizing the Fresnel zone plate (FZP) component using an on-axis plane reference wave. P, is the plane where the required point source is to be located. P, is the plane where the FZP is to be recorded.

function g(x,y) that is to be convolved/correlated with any input signal functionf(x,y). In order to realize the HOE, an arbitrary complex light distribution A(x,y) modulated by the information represented byg(x,y) is mixed with a mutually coherent off-axis reference light distribution R(x,y) as shown in Fig. 3a. Then the TH of the HOE thus realized will be given by the expression,

189

TH a [A(xl._v,)g(xl.yl)

* Y(xz.yz:dt)l

TH(x:.Y~)

RQ2.y:)

a

Y(x~._v~:

dj - d6)

G(d,xllA.

+ IA(x,._v,)g(x,.y,) * Y(.r2.y2: d,)j*R(xZ,yl)

(3)

Let the HOE thus realized be located in plane PJ of the reconstruction geometry shown in Fig. 2. Let flx._v) located in plane P,, be illuminated by light of arbitrary distribution B(x.y). Then the output distribution that will appear in plane P3 is given by the expression.

d,yJA)

exp( -jkd6Ax2)

(8)

TH(x~.)I:) a Y(.v,._v~: dj - db) G*(d,xJA.d,y2/A)

exp(-jkd6Av2) The convolved o&3.Y3)=

(9)

output corresponding to (8) is = Y(xZ.y2: dJ + dj + d, - dh) -= G(d,xzlA. d,yJA)

[d~Y(X~ -xl,y? Tn(x?.‘~)

-yl;d,)]

@Y(xj.ys:

F(dzxz/A.dzyzlA)

Our aim is to make O(x!._vs) appear simultaneously as part convolution (designated by 0v(x3,y3) and part correlation (designated by 0J~j.y~)). If the exact Fourier transform (EFT) off(x,y) can be displayed in plane PJ. then we recast (4) as

F(d>xJA.dly,lA)

Y(x,.y3;

exp (- jkdj(xlxJ dr,dy?

+ y?y~)

dv& (10) } d, + dJ + dj - d6) = 1 and dl = dl. (IO) + ytvz)

If Y(xz,yl: becomes

d3)

b I d.vI

(5)

It is evident that (5) will represent .F[FG] or .‘F [FG*J if T,, is made to contain the information of .Y [g(x,y)I. Under such a situation our HOE. though resembling the Vander Lugt filters, requires the use of two reference waves while recording (see Appendix I). Such an HOE is not only cumbersome to make but will exhibit poor SNR. Therefore we proceed to answer the following questions: (i) How can one realize the MFSHOE using only a single reference wave? (2) What should be the nature of the wavefronts A(x,y). R(x,y) and B(x.y) in order to realize the MFSHOE?

(11)

Equation (I I) represents the convolved output centred around ( - A d61dJ, 0). that appears when the condition d, + d4 + d5 - d6 = 0 is satisfied. The correlated output corresponding to (9) is given by Or(X3.Y.I) a

a Y(x2.y2; d3 + d4 + d6 - d,)

-m

G*(d,xJli,

MFSHOE

Instead of considering (4) under the regime of an EFT, let us consider it under the regime of a quasiFourier transform (QFT). It means that we can let (-rz.yz: d,)

II

exp{-jkdj(x2(x,-y) hzdyz

If Y (x2, yz; d3 + d4 + dh - d5) = I and d, = dl, then (12) becomes n Ovb3,~3)

a Yb3,y3:

d3)

II

__f~Xl.Yl)

Equation (13) represents the cross-correlation output centred around (A d61dJ, 0) that appears when the condition d, + d4 + dh - d5 = 0 is satisfied.

cc

Y(xz,yz;d,

+ d,) Tc,(x~,y?)

II

-=

F(d+JA.,

d&A)

exp i -.Wdxzx,+

drz dyz

(~3.y~: d,) _wd\

(7)

It is found convenient to use a spherical reference wave. which is represented by R = Y(xz.+ A,yZ: d6). Then the TH for obtaining the convolution and correlation channels, respectively will be

190

(12)

(6)

Then (5) will be modified as a

Y(x>,y~; 4)

= B(.r,.y,)Y(x,,y,;dz) em _f(x~.y~)exp ~-j~d~(x+~+y~YdI dx,dy, = Y(xz,y2; d.t) F(dzxzlA. d&A)

W,.Y3)

d,y21k)

F(dzxz/A.dzyrlA)

-t YLy3

of the

d3)

(4)

du I 4v I

Realization

Y(x3.y3:

d,)

Since our objective is to obtain the simultaneous display of convolution and correlation. it means that the conditions d3 + dd + dS - dh = 0 and dj + da + de - d5 = 0 should be satistied simultaneously. It means that dS = dh and d4 = - d3. The condition dS = dh means that the curvature of the reference wave at the hologram plane should be equal to the curvature offered by the quadratic phase factor of the QFT term ofg(x.y) at the same Optics 8 Laser Technology Vol22 No 3 1990

plane. Hence the distribution at plane P, in Fig. 3a will have to be Y*(xl.y~:d,)g(xI.yl). Therefore we can write (6) as Y(x,,yz:

d,)

II

a ~(~1~~l)Y(xl.yl;~2)f(XI.~l) ei”, { - jkdz(x>r, +yLvl) 1 dxtdy, = Y *(XI. yz: d3)F(d2xllA. d&A) (14)

Since the quadratic phase factor appearing on the right-hand side of (14) is negative, the fulfillment of the condition portrayed in (14) can be accomplished in practice only by using an additional optical element in plane Pz. besides the hologram containing .F [g(x,y)}. The term within the integral of (14) can result in a QFf of the form

when B(x,.y,) = Y(xl.yl;dl). Therefore the distribution that will appear in plane P? is given by (Y*(xz.yz: d3 + dz)] I Y(x~._v~;~~) F(dzx,lA. d2yJA) t

(15)

The first term in the above expression represents the phase term that needs to be taken care of by the additional optical element to be located in plane PI besides the FT hologram ofg(x,y); and it is clear from this term that the additional element should be a Fresnel zone plate (FZP). The ideal way to realize a desired MFSHOE. therefore. is to incorporate the additional FZP in the QFT hologram. Thus, it can be seen that the desired MFSHOE can be realized by choosing the wavefronts A(x.y), R(x.y). and B(.r.y) appropriately.

Critique

of the

proposed

ultimately govern the aberrated behaviour of the proposed system can be identified by remembering the facts that, (a) the object wave used in the realization of the FZP component is Y(x.y; 2d). (b) the reference wave is an on-axis plane wave, (c) the reconstruction wave (as derived through the reconstruction from the MF component) is Y(x.y; d) exp (j2rrprx). Therefore, the phase error introduced (which is given by the familiar expression for third-order aberration”) can be shown to be given by the following expression. *+r {

6/d3) +fp3(8x,cos&‘d3)

-+I~(-

-~p2(2x:cos2~,/‘d3)

-fP’

(2xf/d3)

1

(16)

and where p 2=x2+y’andx,=dtan81 d = d, = d2 = d,: thus revealing that the aberrations of the proposed MFSHOE based system will depend ultimately only on d and 8,. A simulation study has been carried out to evaluate the performance of the proposed system under the situations where the MFSHOE functions either as a classical Vander Lugt matched filter (VMF)’ or a phase only filter (POF)“. The letter ‘G’ shown in Fig. 4 is the chosen input function and it is written in a 32 X 32 sub-matrix of a 64 X 64 array, while the HOE is written in a 64 X 64 array. Thus, the physical dimensions of the HOE is four times that of the object. It should be noted that the region of interest in the correlation plane is small compared with the size of HOE, and its aberrations are object independent since they arise solely due to the FZP. Gaussian noise of different variances has been added to the input

system

In view of the fact that the proposed MFSHOE based system envisages the replacement of several optical elements with only one element. one can expect rather unacceptable aberrations in the system. That this is not the case will be demonstrated through a simulation study. The generalized recording geometry used in the realization of the proposed MFSHOE is shown in Fig. 3a. The matched filter (MF) component of it is realized by using the recording arrangement shown schematically in Fig. 3b. where 8, represents the off-set angle for the reference wave. This component itself is not expected to contribute to the aberrated behaviour of the total system. since the wavefronts A(x._r) and B(x.y) used respectively. during the realization and subsequent use of the MFSHOE. are tacitly assumed to be identical. The Fresnel zone plate (FZP) component of the MFSHOE is recorded using an on-axis plane reference wave as shown in Fig. 3c. As such it cannot also contribute any aberrations on its own. However. it will contribute to the aberrated behaviour of the total system for the simple reason that the off-set reconstructed waves’ output from the MF component will transilluminate the FZP. Under this situation the parameters that Optics Vol22

Et Laser Technology No 3 1990

Fig. 4 Letter ‘G’ chosen in the simulation study

to represent

the input function

f(x. y)

191

Peak intensity obtained with MFSHOE under the situation of its functioning as WvlF or POF SR =

(17) Peak intensity obtained under the situation of pure VMF or POF

The peak intensity to be used in the denominator (17) is found to be 17.23 for the character G. Computer plots of the autocorrelation of G corresponding to the three situations mentioned above (that is, regarding the functions of MFSHOE as a VMF) are obtained using practically useful values of d = 45 cm and 13, = 6”; and with these values a Strehl ratio of 0.54 is found. The computer plots so obtained are displayed in Fig. 5. The close resemblance between Figs 5b and SC is interpreted

fig. 5 Computer plot of autocorrelation of ‘G’ under different situations of the classical Vander Lugt type matched filter (VMF). (a) Pure VMF. (b) VMF with noise. Noise variance of 9 has been used to get this plot as it was found to be the optimum value to discriminate comfortably the autocorrelation peak. (c) VMF with noise and aberrations. This plot is obtained by adding noise variance of 9 and aberrations corresponding to the situation of d = 45 cm, 8, = 6” and Strehl

ratio SR = 0.54

functionJx,y) placed in plane PI of Fig. 2, and the phase errors (and hence the aberrations) calculated for various values of d and 0, have been introduced at plane Pz. The signal power is calculated by summing all the contributions from the object and the sum total is 31.5. It is found that a noise variance of 9 gave best results and hence that value is used for obtaining the autocorrelation plots shown in Figs 5 and 6. A 64 X 64 2-D FFT computation has been made to obtain the complex amplitude distributions at planes P2 and PJ. Autocorrelations have been computed under the situations of, (a) pure VMF; (b) VMF with noise; and (c) VMF with noise and aberrations. The last item corresponds to the situation of the use of the proposed MFSHOE. In the context of the present study the Strehl ratio (SR) can be expressed as,

L Fig. 6 Computer plots of autocorrelation of ‘G’ under different situations of the Horner type phase only filter (POF). (a) Pure POF. (b) POF with noise. Noise variance of 9 has been added to get this plot as it was found to be the optimum value to discriminate comfortably the autocorrelation peak. (c) POF with noise and aberrations. This plot is obtained by adding noise variance of 9 and aberrations corresponding to the situation of d = 45 cm, 0, = 6” and Strehl ratio SR = 0.54

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as a clear indication of the fact that the proposed MFSHOE is not inferior to VMF with respect to its signal-to-noise discrimination capabilities. A similar conclusion is drawn from the autocorrelation plots shown in Fig. 6 and which are obtained under the situation of the MFSHOE functioning as a POF. Therefore. the following conclusions are drawn from the simulation study. (i) The performance of the proposed MFSHOE system matches well with that of the existing systems. (ii) The MFSHOE is more tolerant to aberrations under VMF situation rather than under a POF situation. (iii) A better Strehl ratio (z 0.8) can be obtained by making d large and 8, small. The variation of SR (calculated using (17)) as a function of d for various values of 8, is shown in Fig. 7 and the corresponding numerical data is given in Table 1. The proposed MFSHOE is somewhat similar to the lensless matched filter (LLMF) proposed some years ago by Bage and Beddoes13. The significant difference, however. is that their FZP component was meant only for displaying the focused output from the correlation channel, whereas ours is intended for obtaining focused outputs from both the convolution and correlation channels. Thus, the I

.o

.s

Finally. it may be mentioned that the FZP component of the MFSHOE can also be realized using the off-axis technique; and its implications are discussed in Appendix II.

Summaty

and conclusions

A theoretical analysis of a simultaneous coherent optical convolver and cot-relator system based on the use of a multiple-function-single-HOE (MFSHOE) is presented. It is demonstrated that, (a) the proposed system exhibits beneficial features (for example, focused outputs from both the convolution and correlation channels) in addition to preserving those exhibited by the systems proposed by others”-‘4; and (b) the aberration behaviour of the proposed system is reasonably good.

Appendix

I

Case for requiring two reference waves for obtaining the desired MFSHOE.

r

The .F [g(x,y)] must appear as one of the terms in TH, which means that we must record g(x,y) as an FT hologram. For convolution TH a G(d,x2/A. d,y?l,l) R*. It implies, therefore. that ~(x,y) has to be a plane wave. Substituting this in (5) we get

1

0.8

proposed MFSHOE based system offers additional beneficial features while preserving those offered by the systems proposed previously”- 4.

0.6

I:

5 Ov(x3.y3)

z Y

2

m

0.4

Y

-6fy , , l

0.2

0

10

25

55

40 Distance

70

(cm)

Variation of Strehl ratio obtained with the proposed Fig. 7 MFSHOE as a function of distance D for various values of angle 0,

Table 1. Strehl ratio variation with d and 0, for the proposed MFSHOE

85

15 30 45 60 75

II G(x~d,lA.yzd,;:) F(dzx,/A.,d~,/A) exp 1-jkd~(x~x3

Wz.yr;

dz + d3)

R*(x2.yz) +YSY~)~~YI~YI

(18)

for the situation of d3 = d2 = d,. Now if we want the cross-correlation we look at channel 2 with TH a G*(dlx21A. d,yJA) R(xz. yz). Therefore we get = Wx3.~3;

= Y(xz.

d3)

yz: dz + d,)

II

81 Peg) 0

6

12

18

0.08 0.2 1 0.38 0.54 0.9 1

0.06 0.26 0.54 0.72 0.87

0.06 0.19 0.35 0.51 0.71

0.07 0.17 0.30 0.4 1 0.53

Optics 8 Laser Technology Vol 22 No 3 1990

d3)

To obtain .F(F. G) we must eliminate the quadratic phase factor. It means that we should have R(x2.yz) = Y(x2.yz: d2 + d3). Then the output is given by 3c O”(.Y3.Y3) = Y(,~3,_v3;d3)f(xI.yI) II -m (19) g(x3 -xI.y3 -yl)hidyl

Or(x3.~3)

D b-4

= Vx37y3:

F(dzx,/A.dtv,/~)R(xz.yz) exp 1- jkd3(xzx3 + yIv3)tdyidyi

(20)

To obtain .F(F.G*) we must eliminate the quadratic phase factor which means that we must have R(xz.yz) = Y*(x~.y~: dz + d3) for the realization of the correlation off and g. Therefore the output is given by

193

a2

Or(X3.Y3)

=

II

-m

qx3._Y3:

d3)f(x,._v,)

g*(xl -x;.yl

-yJ)dr,dy, (21)

for d3 = d2 = d,. It is thus evident that two reference waves, one converging and the other diverging but both of the same radius of curvature, are required for the realization of the desired MFSHOE.

Appendix

II

Computation with off-auk zone plate Under the condition of off-axis zone plate we have the object wave Y(x,y; M), the reference wave exp (j27$3zx). and the reconstructing wave Y(x.y; d) exp G2npIx); where PI = sin 8,/h and pZ = sin &/,I. At plane PZ we have the complex amplitude distribution F(u, v) G*(x.y: d) Y*(.r,_v: d) exp {CjZnlA)(sin 8, - sin e2)_y). For small angle sin 8 = 0; therefore we get F(u V)G*(rr. V)Y*(.Y._Kd) exp((j2nlA)

(e, - e2)d

By adjusting 6, and 8: appropriately we can arrive at the following conclusions. By having larger values for 0, - 0? we can separate the channels from the dc term easily: thus facilitating the use of a large spacebandwidth product signal Information.

Acknowledgements This work was done under the projects funded by the Department of Electronics and Ministry of

Human Resource Development, Government of India. The help received from S. Regunathan and M.R. Ravi in obtaining the computer plots is gratefully appreciated. References Vander Lugt, A.B. ‘A review of optical data-processing techniques’, Opr Acra 15 (1968) l-33 Goodman, J.W. Introduction to Fourier Optics, McGraw Hill. New York (1968) Goodman, J.W. ‘Operations achievable with coherent optical information processing systems’. Proc IEEE 65 (1977) 29-38 Vander Lugi, A.B. ‘Operational notation for the analysis and synthesis of optical-data processing systems’. Proc IEEE 54(1966) loss-lo63 5 Close, D.H. ‘Holographic optical elements’. Opr Eng 14 (1975) 408-419 6 Pappu, S.V. ‘Holographic optical elements: State of the art review’. Parts I and II. Opr Laser Technol 21 (1989) 305-313; 365-376

7 Solymar, L., Cooke, D.J. Volume Holography and Volume Gratings, Academic Press, New York (1981) 8 Kannan, R., Pappu, S.V. ‘Simultaneous display of image and spectrum in the same plane: A generalized analysis’. Opr Loser Gchnol 20 (1958) 205-209 9 Vander Lugt, A.B. Signal detection by complex spatial tiltering’. IEEE 7ium fn/orm Theory IT 10 (1964) 139-145 IO Meier, R.W. ‘Magnification and third order aberrations in holography’. J Opr Sot ..lm 5.5 (1965) 987-992 II Homer, J. ‘Light utilization in optical correlators’. App Opt 21 (1982)4511-4514 I’ Caulfield, H.J. ‘Role of Homer efliciency in the optimization of spatial filters for optical pattern recognition’. &J Opr 21 (1982) 4391-4392 13 Bage, M.J., Beddoes, M.P. ‘Lensless matched filter: Operating principle. sensitivity to spectrum shift. and third order holographic abrrrations’, J 0~1 Sot Am 63 (1973) X330-2839 14 Vander Lugt, A.B. ‘Practisal consideration for the use of spatial carrier frequency filters’. ,4pp Opt 5 (1966) 1760-1765

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