Trading dimensionality in signal processing

Optics & Laser Technology, Vol. 28, No. 2. pp. lOlblO7, 1996

Copyright 8 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 003Om3992/96 $15.00 + 0.00

ELSEVIER ADVANCED

0030-3992(95)00067-4

TECHNOLOGY

Trading dimensionality signal processing A. W. LOHMANN,

J. OJEDA-CASTANEDA,

in

A. SERRANO-HEREDIA

By trading one spatial dimension, in an optical processor, one can implement simple optical set-ups for signal synthesis and signal processing. We show that this concept, of trading dimensionality, has been applied advantageously in four classical optical processors. We also discuss another four applications of the same concept: the generation and parallel display of certain special functions, the synthesis of complex amplitudes with binary screens, and the generation of radially symmetric apodizers with binary sectors. KEYWORDS: optical processors, signal synthesis, apodizers, binary optics

Introduction

In the next section, we review several optical processors that trade one dimension, and which have been designed, to our knowledge, without the explicit use of the concept of trading dimensionality. In the section after, we discuss recent applications of the concept of trading dimensionality.

On one hand, optical signals are usually onedimensional due to their temporal nature. On the other hand, optical processors are able to handle data in threedimensional formats. Hence, in principle, it is possible to trade one or two dimensions in the input data of an optical processor for implementing novel architectures for signal processing.

INPUT

Y I

For example, in Fig. 1, we show the optical system commonly employed for a fan-out operation. In this example it is easy to recognize that the y-axis is traded for generating redundancy; and in this manner one can reduce the local risks of noise and distortions on the one-dimensional signal.

OUTPUT

Furthermore, as first pointed out by Cutrona’, the yaxis of the input data can be usefully traded for multiplexing several one-dimensional signals, S,(x), Iz= l,... , N In this way, one can implement a parallel spectrum analyser, as is shown in Fig. 2.

Fig. 1

Optical set-up for implementing

a fan-out

operation

Our aim here is twofold: first, we identify several optical processors that share the concept of trading one spatial dimension for performing non-conventional tasks. Secondly, we discuss several recent applications of trading dimensionality: the parallel displays and processing of certain special functions, the synthesis of one-dimensional complex amplitudes with binary objects, and the implementation of radially symmetric, analogue apodizers with binary sectors.

AWL is in Angewandte Optik, Physikalisches lnstitut der Universitlt Erlangen-Ntirnberg, Staudtstrasse 7, 91058 Erlangen, Germany. JO-C and AS-H are in the National Institute for Astrophysics, Optics and Electronics, Apdo. Postal 216, Puebla 72000, Pue., Mexico. Received 27 April 1995.

Fig. 2 Fan-in operation analyser

101

for implementing

a parallel spectrum

102

Trading dimensionality

Non-conventional

optical

in signal processing: A. W, Lohmann et al.

processors

As part of the development of optical signal processing, several efforts have been addressed for implementing optically non-conventional operations. In Table 1, we summarize four efforts that take advantage of trading one dimension for implementing non-conventional operations. The first example in Table 1 is the fan-out optical device shown in Fig. 1. It can be represented by the impulse response function, (PSF) p(x’, y’; x, v) = 6(x’ - x) exp( -i27r#y/Xf)

(1)

Table 1. Trading one spatial performing non-conventional

u(x,.Y) = S(x)@)

Traded dimension

1. Improved signal-tonoise ratio by introducing redundancy

Fan-out of S(x) along the y-axis

2. Parallel spectrum analysis’

Fan-in for multiplexing S,(x), n = 1,. , N along the y-axis

3. Space-variant impulse responses213

Fan-out of S(x) along the y-axis, and fan-in of S(x)H(y; x) along the x-axis

4. One-dimensional analogue filters with binary mask6

Analogue variations encoded as width variations along the p-axis

(2)

then the output complex amplitude is

ss S($

optical system incorporates the two previous optical setups, fan-out and fan-in, shown in Figs 1 and 2, respectively.

lx

v(x’,y’) =

P(x’,Y’;x, YMX,Y) dx dv

=

(3)

In this fashion the signal becomes redundant, and less sensitive to noise and distortions, along the /-axis. In other words, one dimension is traded for immunity to noise. The second example in Table 1 is the parallel optical analyser depicted in Fig. 2. This device was first suggested by Cutrona’. In our notation the PSF of this optical processor is p(~,y’; x,y) = 6(y’ - y) exp(-i2Wx)

In this example, at the first stage, we note a fan-out operation with the PSF. ~1 (x’,~‘; XO,YO)= 6(x’ -

x0)

exp(-iW’y0lW

as in (1). Hence, the complex amplitude at the input of the first stage, UO(XO,~O) = S(x0) 6(yo), is mapped at the input of the second stage as 03 U/(X’,y’) =

11 = $7

PI(~‘,Y’;~o,YO)~O(~O,YO) dxo dye (8)

This input is multiplied by an optical mask, whose amplitude transmittance is H(x’, y’). Thus, the complex amplitude just behind the mask is U(X’,y’) = H(x’, y’)S(x’)

Sri(x) KY - nd)

(7)

(4)

In (4) v = x’/Xfdenotes a spatial frequency variable. Hence, if the input complex amplitude is a set of 2N+ 1 signals U(X,Y) = 2

for

Applications

In (1) the symbol S( .) denotes the physical implementation of Dirac’s delta function. If the input signal, u(x,y), is confined to the x-axis, namely

dimension tasks

(5)

n=-N

then the y-axis is exploited for multiplexing the N signals which are separated by a distance d. Hence, the output complex amplitude is

(9)

The second optical stage is represented, as (4), by the PSF pz(x,y;x’,y’)

= S(JJ -

y’) exp(-i27rxx’/Xf)

(10)

ss 00

v(w’) = =

P(V,Y’;x, YMX,Y)dx dy

-&(.,

6(y’ - nd)

(6)

n=-N

In other words, one dimension is traded for multiplexing 2Nf 1 signals, which are, in parallel, Fourier transformed. The third example in Table 1 was proposed independently by Goodman et aL2 and Marks et a1.3for implementing a space-variant processor. The optical setup employed by these two research groups is schematically shown in Fig. 3. It is to be noted that the

Fig. 3

Optical system for implementing

a space-variant

processor

103

Trading dimensionality in signal processing: A. W. Lohmann et al. Table 2. Recent dimensionality

The output complex amplitude is now 00 +LY) = =

P~(x,Y;

AY’)~+‘,.Y’)~(~‘)

dx’

y) JMH(x',

II

dy’

exp( -i27rxx’/Xf)S(x’)

dx’ (11)

Along the y-axis, x = 0, (11) becomes co H(x’,y)S(x’) s -cc

dx’

(12)

From (12) it is apparent that the input signal S(x’) is mapped into the output signal ~(0, y), through a spacevariant PSF H(x’, y); the latter is implemented by using an optical mask between the fan-in and the fan-out operations. In other words, a two-dimensional mask is employed to trade one dimension for implementing a one-dimensional, space-variant, impulse response. A similar set-up is used for optical matrix multiplication4. Finally, in Table 1, we consider a useful approach for synthesizing an apodized, one-dimensional, PSF by using a two-dimensional binary screen. According to Jacquinot and Roizen-Dossier’ this optical technique was proposed by Couder and Jacquino6. Their proposal can be described as follows. For the present application it is convenient to consider the following, two-dimensional, binary mask with complex amplitude transmittance

(13) which is employed as the pupil function in the optical processor in Fig. 4. For this application the PSF is, of course CCI p(y, pu)exp[i2+v

R WV)/2 =S J

P(-%Y) =

I!

-cc

+

.w)l dv d/L

Applications

Traded dimension

5. Parallel display of special functions7)8

The generating function is encoded along the y-axis

6. Synthesis of ‘phase only’ signals9

Binary interferograms with ‘phase detours’ along the y-axis

7. Synthesis of onedimensional complex amplitudes”~‘2

Generalized slit: phase detours as displacements along the y-axis, amplitude as widths along the y-axis

8. Radial analogue filters with binary masksI

Analogue variations encoded as angular widths

v=-R

Along the x-axis, y = 0, the PSF in (14) can be written as

p=-W(v)/2

(14)

SOURCE

J

R

P(X,Y= 0) =

W(V) exp(i2rxv)

dv

(15)

LIP0

Note that (15) represents the generation of an apodized, one-dimensional impulse response by using an analogue, spatial filter W(V), which was implemented with the boundaries (along the p-axis) of a binary screen. In other words, one dimension of the pupil plane is traded for encoding analogue variations while using a binary screen. The results summarized in Table 1 were invented, to our knowledge, without using the concept of trading dimensionality. In what follows we discuss some examples that were suggested by employing the concept of trading dimensionality. Recent

exp[ i27r(xv + YI_L)] dp dv

POINT

of trading

-co

-cc

v(O1.Y)=

applications

applications

In this section we indicate how to take advantage of the concept of trading dimensionality. For our first application, in Table 2, we consider the optical set-up in Fig. 5 that is a Fourier transformer.

INPUT

OUTPUT

Fig. 4 Optical set-up for synthesizing impulse responses

apodized, one-dimensional

Fig. 5 Fourier transformer functions

for parallel display of the Bessel

104 Parallel

Trading dimensionality display

of special

in signal processing: A. W. Lohmann et al.

functions

If the complex amplitude transmittance in the input is U(X,y), then the complex amplitude transmittance in the output is 00 rXV,P) =

4% Y) ss x exF[-i2?r(xu + Y,u)]dx dY

(16)

The above relationship can be suitably modified to contain the generating function of certain special functions, used in mathematical physics, as follows7s. We trade the Y-axis to encode variations of a certain functionf(x). In other words, the amplitude transmittance of the input is the binary curve 4x> Y) = S[Y -f(x)1

(17)

Hence, the complex amplitude distribution at the output is obtained, by substituting (17) in (16) to give G(v, P) =

co exp [-i27r(xv +f(x)p)] s --3c

dx

(18) Fig. 6

In some problems related to aberrated wavefronts, one obtains the Airy function that is defined as 4(3C)-1/3t]

= Pg

/m

exp[-i(cx3 + tx)] dx

Parallel display of the Airy function

of variable arguments

where J,(. ) denotes the ordinary Bessel function of the first kind and order m, then (23) can be written as (19)

--w By simple comparison between (18) and (19) we note that by setting

f(x) = ox3

(20)

where CYis an arbitrary constant, (18) becomes 00 fi(V,P) =

exp[-i(2~~~x3y

+ 27rvx)] dx

s = 211;U6~~~)-‘i3A[(671~~~-1/32?i~]

(21)

It is apparent from (21) that along the line p = R, the complex amplitude is proportional to the Airy function with independent variable V. In other words, the Fraunhofer diffraction in (21) displays, in parallel, the Airy function with variable parameter R, as shown in Fig. 6.

It is apparent from (25) that the Fraunhofer diffraction pattern consists of a series of equidistant lines along the v-axis, as shown in Fig. 5. Each bright line has a variation proportional to the Bessel function of order m. In other words, by suitable trading of the Y-axis, we are able to display in parallel the ordinary Bessel functions of the first kind, with variable order m. The same idea can be applied to display, also in parallel, the first derivatives of the Laguerre polynomials’. In some recent publications we have the use of the unconventional processor shown in Fig. 7. The pupil aperture of this optical processor employs, along the

INPUT

Now, we consider another possibility. By setting f(x) = c sin(27rx/d)

(22)

Equation (18) becomes C(V,P) =

O”exp[-i27r(xv+ap s --oo

sin(27rx/d))] dx

(23)

If we employ Jacobi’s identity exp[-i27rap sin(2rx/d)]

=

2 J,(27rap) m=-IX x exp( -i2_lrmx/d)

(24)

Fig. 7 fan-out

Non-conventional operation

optical processor for generating

the

105

Trading dimensional@ in signal processing: A. W. Lohmann et al.

v-axis, a narrow slit, which is located at p = Q. Hence, the pupil function is (26)

P(V>CL)= b(P - fl) The impulse response is

(27)

~(-5 Y) = 6(x) exp(i2rfiY)

It is apparent from (27) that, except for a phase factor, the impulse response, p (x, Y), produces images along the x-axis while it takes the Fourier transform, for a fixed spatial frequency, along the Y-axis in a similar way to the fan-out processor in Fig. 1. In other words, if the input complex amplitude is u(x, Y), then the output complex amplitude is Y(X,Y)=

Js’ -lx

Complex

amplitude

2

U(X,Y) =

Cmexp[i2r(y-f(x))44

lil=-CX3

+

2

C,

exp[ih(y - g(-x))nldl

It is straightforward to show that, in this application, the output complex amplitude is

= {2&d exp( i2&Y)}

u(x, Y’) dy

x exp [-i2nfl (f(x) + g(x)) /2]

(28)

synthesis

The second application in Table 2 is based on the fundamental idea that an interferogram is an image hologram5. Hence, one can generate new wavefronts by recording a binary interferogram with the aid of a computer plotter9. A one-dimensional wavefront is, of course, a onedimensional only-phase signal. Consequently, for generating one-dimensional only-phase signals one records binary, grating-like interferograms of the form

(29) M=--03

Note that in (29) we have encoded a phase variationf(x) as a detour of the Y-axis. That is, the functionf(x) encodes the desired phase profile as a distortion of the interferogram.

(32)

x cos [27rfl(f(x) - g(x)) /2]

It is apparent from (32) that by suitably generating the phase detoursf(x) and g(x), one can generate a onedimensional complex amplitude signal whose phase variation is [f(x) +g(x)]/2, and with amplitude variation proportional to the cosine of &[f(x) -g(x)]. In other words, the Y-axis is now suitably traded for encoding phase variations (as the middle phase detour of two binary interferograms), as well as for encoding amplitude variations (as the difference phase detour of the two interferograms). Similar results can be obtained by using other approaches1’s’2. Synthesis

of radial apodizers

with

binary

sectors

The fourth application of Table 2 is a three-dimensional extension of the proposal of Couder and Jacquinot in Fig. 4, also described in the fourth line of Table 1. For discussing this extension it is convenient to write the three-dimensional impulse response of the classical 4-f,

If this binary pattern is placed in the input of the optical processor in Fig. 7 then, according to (28) and (29), the output complex amplitude is an only-phase signal. That is

OUTPUT

u(x,Y) = exp(i27rRY) 2 C, exp[-i27rf(x)m/d] !?I=--03 X

exp[-i2r(fl-

= (Col:xp(i2&Y))

(31)

II=--0c

+ exp[-i27rRg(x)]}

This unconventional mapping finds the interesting applications listed in lines 2 and 3 of the Table 2, which are discussed next.

M J

synthesis

For the third application in Table 2, we consider that the input consists of two binary, grating-like interferograms, as in (29). That is

00

s x exp(-i2nRYT

signal

v(x,y> = (Gdexp(i2~Q)) ~exp[-i2~Rf(x)l

p(x - x’, Y - Y’)u(x’,Y’) dx’ dy’

= exp(i2nRY)

Signal

one-dimensional array illuminator at z = d2/4X. For further information the reader is referred to Ref. 10.

m/d)y’] dy’ exp[-i2&S(x)]

(30)

In other words, the Y-coordinate is traded for encoding (as phase-detours) one-dimensional only-phase signals. Figure 8 shows our experimental verification for synthesizing a phase-grating, at z = 0, which produces a

Fig. 8 Synthesis illuminator

of a phase grating for a Lohmann

array

106 POINT

Trading dimensionality SOURCE

PUPIL

in signal processing: A. W, Lohmann et al.

as follows

FUNCTION

p(O,O,z) =

d12exp (-i7rzn2/2) X

J

0.5 e(c)

exp [-i27r(XzR2/2)<] d< (38)

-0.5

Note that we are able to generate an analogue spatial filter Q(c)>,by suitably choosing borders of the binary sectors in (35). In our application we consider that 1 WP2/2)

(394

=

1 + (7r/27J2 [(p/q2 Fig. 9 Optical processor for implementing binary sectors

radial apodizers with

- 0.51 2

or equivalently 1 e(r) = 1 + (7r(/2T)2

optical set-up in cylindrical coordinates

~(6 $34 =

J2~“B(h 4 exp(-i7h2) 0

0

x exp[i27rprcos(8 - $)]p dp de

(33)

In (33) Sz is the maximum value of p; p(p, 0) represents the pupil function in polar coordinates (p, O), and z > 0 denotes the axial displacement from the paraxial focus z = 0, as shown in Fig. 9. The axial impulse response is obtained by setting r = 0, and 4 = 0 in (33) to give p(0, 0, z) = 27rl”{&-‘“p(P>B) x exp ( -irXzp2)p

where T denotes the light throughput, normalized with respect to that of the clear aperture. Figure 10 shows the PSF that is generated by using the binary sectors in Fig. 9, which synthesize an annular Lorentzian apodizer. The irradiance distribution along the optical axis decreases as ]~(O,O,Z)~~= Aexp [-4XR2T]zl]

where A is a constant factor. Figure 11 shows our first experimental verifications of the above result. For more details see Ref. 14.

do} dp

(34)

It is apparent from (34) that the radial variation of the pupil function is obtained by taking an angular average. Consequently, we can trade the f3coordinate for encoding width variations of a binary screen, and in this manner obtain analogue spatial filters, which will shape the axial irradiance distributiont3.

a>

In other words, we consider that the pupil function is composed of 2M angular sectors in the form of the pie, or a daisy flower (see Fig. 9). Every sector has a binary aperture whose borders follow the function W(p2/2). Then, the pupil function is (35) It is straightforward to obtain, by substituting (35) in (34) that the axial impulse response becomes p(O,O,z) = r~(l/M)/nW(p2/2)exp(-inizp2)p

dp

0

m=l

J

b)

R

= 27r

0

W(p2/2) exp ( -iTXzp2)p

dp

(36)

This result can conveniently be written as a Fourier transform, by using the change of variable <=

(P2W2)

-

0.5,

c!(c) = VP2/4

(37)

(40)

Fig. 10 PSF associated with the mask in Fig. 9: (a) in-focus impulse response; (b) out-of-focus impulse response

Trading dimensionality in signal processing: A. W. Lohmann et al. 10 11 12

13

14

107

Arrizbn, V., Ojeda-Castaieda, J. Multilevel phase gratings for arrav illuminators. ADDSOvt. 33 (1994) 5925-5931 Lohhann, A.W., dje%-Ca&ied& J. Computer generated holography: novel procedure, Opt Comm, 103 (1993) 181- 184 Lohmaan, A.W., Ojeda-Castaieda, J., Serrano-Heredia, A. Synthesis of I-D complex amplitudes using Young’s experiment, Opt Lett, 19 (1994) 55-57 Lohmann, A.W., Ojeda-Castaieda, J., Serrano-Heredia, A. Synthesis of analog apodizers with binary angular sectors, Appl Opt, 34 (1994) 317-322 Ojeda-Castaiieda, J., Tepichin, E., Diaz, A. Arbitrarily high focal depth with quasioptimum real and positive transmittance apodizer, Appl Opt, 28 (1989) 2666-2670

Biographies

0.40

;

4000

60.00 2

Fig. 11

Axial

‘1’

I-':-

0.00

irradiance

(cm)

120.00

160W

( OPTICAL AXIS )

distribution

for the PSF in Fig. 10

Conclusions We have indicated that unconventional optical processors can be implemented, by suitably trading one or two dimensions in a three-dimensional optical system. The use of this concept, of trading dimensionality, is identified in eight different optical processors. We have discussed several recent uses of the concept for the parallel display of certain special functions, the synthesis of one-dimensional complex signals, and the generation of radial apodizers with angular sectors. Acknowledgements We are indebted to one anonymous reviewer for comments. We thank Juan G. Ibarra, Alejandro Landa and J. Gustav0 Ramirez for their helpful assistance. References Cutrona, L.J. Recent developments in coherent optical technology. In Optical and Electra-Optical Information Processing (Ed, Tippet J.T.) MIT Press, Cambridge (1965) Goodman, J.W., Kelhnan, P., Hansen, E.W. Linear space-variant optical processing of 1D signals, Appl Opt, 16 (1977) 733-738 Marks, R.J., Walkup, J.F., Hagler, M.O. Space variant processing of 1-D signals, Appl Opt, 16 (1977) 739-745 Heinz, R.A., Artman, J.O., Lee, S.H. Matrix multiplication by optical methods, Appl Opf, 9 (1970) 2161-2168 Jacquinot, P., Roizen-Dossier, B. Apodisation. In Progress in Optics III (Ed. Wolf, F.), North Holland, Amsterdam (1964) Couder, A., Jacquinot, P. CR Acad Sci Paris 208 (1939) 1639 Lohmann, A.W., Ojeda-Castaiieda, J., Serrano-Heredia, A. Bessel functions: parallel display and processing, Opt Left, 19 (1994) 55-57 Lohmann, A.W., Ojeda-Castafieda, J., Ibarra, J.G. Airy function and Laguerre polynomials: optical display and processing, Op/ Comm, 109 (1994) 361-367 Ojeda-Castaiieda, J., Arrizbn, V. Synthesis of 1-D phase profiles with optical variable path, Microwave and Opt Tech Left, 5 (1992) 429-432

Adolf W. Lohmann received a BS degree and a PhD degree from the Technical University of Braunschweig, in Germany. He worked in the Royal Institute of Technology in Sweden (1958-1959), at IBM in USA (1961- 1967), as a Professor of Applied Physics and Information Science in the University of California in La Jolla (1967- 1973), and as a Professor of Applied Optics in the University of Erlangen-Ntirnberg, in Germany (1973-1993); where he is Emeritus Professor (1993). Currently, he is at the Weizmann Institute in Israel, as Erna and Jakob Michael visiting Professor. He has made contributions in physical optics and information science, including optical signal processing, holography, astronomical optics, spectrometry, optics in biology, interferometry and particle acceleration by laser light. He is the recipient of Federal Medal of Merit (Germany, 1981), the SPIE President award (1983), the OSA’s Max Born Award (1984) and the OSA’s CEK Mees Medal (1987).

J. Ojeda-Castaiieda earned a BS degree in Physics from the National Autonomous University of Mexico in 1972, and a PhD degree, in Physics, from the J. J. Thomson Laboratory, University of Reading, UK. From 1982 to 1984, he was a visiting scientist in the Physikalisches Institut der Universitgt ErlangenNiirnberg. He has been Visiting Professor in the Institute of Optics in Madrid (1980), in the Department of Optics of the University of Valencia (1988, 1989, 1992), and the Department of Physics of the Autonomous University of Barcelona (1933), all in Spain; as well as in the Center of Research in Optics, in La Plata, Argentina (1986). He is a full professor at the National Institute for Astrophysics, Optics and Electronics, where he acts as General Director. He is a fellow of SPIE and a fellow of OSA.

received a BS degree in physics from Instituto Tecnol6gico de Monterrey, a MS degree from Centro de Investigacibn Cientifica y Estudios Superiores de Ensenada, and a PhD from Instituto National de Astrofisica, Optica y Electrhnica (INAOE), all in Mkxico. Presently, under a leave of absence from INAOE, he is at Penn State University as a postdoctoral fellow. Alfonso Serrano-Heredia