IV Principles of Optical Data-Processing

E. WOLF, PROGRESS IN OPTICS XIX @ NORTH-HOLLAND 1981

IV

PRINCIPLES OF OPTICAL DATA-PROCESSING BY

H. J. BUTTERWECK Eindhouen University of Technology, Eindhouen, The Netherlands

CONTENTS

$ 1. INTRODUCTION.

. . . . . . . . . . . .

.

.. .... ,

0 2. FIELD THEORY OF OPTICAL SYSTEMS. . . § 3.

.

..

PAGE

213

.

216

SYSTEM-THEORETICAL APPROACH TO COHERENT OPTICAL SIGNAL PROCESSORS . . . . . . . . . . . .

222

. . . . . .

227

§ 4. PARTIALLY COHERENT ILLUMINATION. § 5. BASIC SYSTEM CONSTRAINTS. .

.

.

,

. . . . . . . . . 232

9 6. EXAMPLES OF PHYSICAL AND ABSTRACT SYSTEMS 245

5 7. OPERATIONAL NOTATION OF OPTICAL SYSTEMS AND BASIC CASCADE EQUIVALENCES . . . . . . . 252 $ 8. OPERATIONAL ANALYSIS OF OPTICAL SYSTEMS

256

$9. SYSTEMS COMPOUNDED OF LENSES AND SECTIONS OF FREE SPACE (5%-SYSTEMS) . . , . . . . . 263 $ 10. SHIFT-INVARIANT SYSTEMS: COHERENT VERSUS

INCOHERENT ILLUMINATION. . . . . . . . . . . . .

§ 11. RELATED TOPICS.

. . .

.

268

. . . . . . . . . . . . . . . 275

REFERENCES . . . . . . . . . . . . . . . .

...... .. .

279

§

1. Introduction

In communication theory, a transmission system denotes a physical arrangement which, with or without distortion, transmits a signal from a source (transmitter) to a receiver, thereby communicating a certain amount of information. Occasionally one defines a system as the set of its constituents and their mutual arrangement (the “interior”), but in a more common approach a system is viewed as a “black box”, of which only the behaviour at the input and output terminals is studied. In earlier treatments (KUPFMULLER[1948], BAGHDADY [1961], PAPOULIS [1962]) the usually electrical signals are throughout considered as functions of time t, which implies that a system is mathematically defined as an operator

a t )= T{f(t))

(1.1)

which transforms the input signal f ( t ) into the associated output signal g ( t ) . It is easily recognized that physically realizable systems have to satisfy the fundamental constraints of causality and realness which state that g ( t ) is specified only by the past history of f ( t ) and that any real function f ( t ) is transformed into a real function g ( t ) . An important class of systems moreover satisfies the requirements of linearity and time invariance which implies that the operator T has the mathematical form of a convolution integral (PAPOULIS [19621) h(f-T)f(T)dT=

g(t)=

L-

def

-

h(T)f(t-T)dT = h(t)*f(t),

(1.2)

where the real weighting function & ( t ) denotes the impulse response of the system. If a linear, time-invariant system is excited by a harmonic signal with circular frequency w, the output is likewise harmonic with frequency w. In mathematical terms, f(t) = exp ( - i d ) is an eigenfunction of a linear, time-invariant system, as appears from g(t)=

J

0

h ( ~exp ) ( - i d + iwT) dT = h ( w ) exp ( - i d ) , 213

(1.3)

214

PRINCIPLES OF ORICAL DATA-PROCESSING

[IV, § 1

where the complex “system function” h ( w ) is the Fourier transform of the impulse response:* h ( w )=

[h ( ~exp ) ( i w ) dT

= h*(-w).

(1.4)

In the present paper, the constraints of linearity and time invariance are assumed to be satisfied throughout. This assumption excludes all effects in the realm of nonlinear optics as well as large-signal nonlinearities in electronic picture processors. Also randomly fluctuating media and fastmodulating electro-optic devices are thus left out of consideration. If a linear, time-invariant system is excited by a non-monochromatic signal the Fourier transform method applies. With f(o)and g ( w ) denoting the Fourier transforms of f(t) and g ( t ) , we then obtain the simple product relation = h(o)f(w).

(1.5)

However, most non-monochromatic signals in optics have a random character and do not admit a Fourier representation. In such cases the theory of partially coherent light applies (cf. 0 4). Thus far, the system concept was concerned only with the transformation of time signals. Mainly through the advent of two-dimensional image processing, this concept has been extended in the past decades (O’NEILL [1963], GOODMAN [1968]). Signals are, in addition to their time dependence, also considered as functions of the spatial coordinates and, as such, are processed through electronic or optical systems. From a black box point of view, an image- (or data-) processing system is then defined as any arrangement of electronic scanning devices (T.V. circuitry, twodimensional digital filters) and/or optical components (lenses, masks, gratings, holograms) which is operated between two suitably chosen reference planes. A two-dimensional light distribution in the “input plane” (the “object”) excites the system and is transformed into another light distribution in the “output plane” (the “image”). Obviously, the behaviour of electronic scanning systems differs strongly from that of purely optical systems. In an electronic system the outgoing light emerges from a built-in source (coherent or incoherent) and, as such, exhibits no correlation with the incoming light. Unlike an optical system with a strong correlation between the light disturbances (coherent or incoherent) in the two end-planes, no interference phenomena can occur *The asterisk denotes the complex conjugate.

,

IV, 8 11

INTRODUCI‘ION

215

between the input and output light distributions. In addition, electronic systems which commonly “start” with a video camera and end up with a cathode-ray tube display, process optical data only in one direction with well-defined input and output planes, whereas optical systems are inherently bidirectional processors. The significance of a general theory of optical systems reaches farther than might be expected from its primary objectives. Since any number of fictitious intermediate planes can be inserted between the input and output plane, the system under consideration can not only be split up into a number of possibly more elementary subsystems, but also can the optical field at each interior point in the system be considered by inserting the plane through that point”. This potentiality explains that even geometric-optical approximations can be elegantly derived from optical system theory (3 11).Further it has been conjectured (MENZEL,MIRANDE and WEINGARTNER [ 19731) that also the human perception of light can be adequately described with the tools of linear system analysis which then provide suitable methods for the experimental determination of the pertinent system properties. When comparing the signal transformation (“filtering”) in twodimensional space with that in time, we observe a number of significant differences (O’NEILL[1963]). Since “left”, “right”, “above” and “below” are not preferred by nature, causality has no meaningful counterpart in spatial filters. Likewise, realness is n o longer a fundamental constraintt. Linearity will remain an important restriction, but “shift invariance” (the counterpart of time invariance) which will be discussed in § 5.1, has to be considered as a special property, albeit of utmost significance. To avoid ambiguities in the presentation, the main article is concernea only with optical systems excited by light of strictly harmonic time dependence. Due to linearity and time invariance, all field quantities inside the optical system are then time-harmonic, too. Partially coherent illumination will be discussed in § 4. There we shall show that much of the formalism developed thus far also applies to the extreme case of incoherent illumination. Furthermore it appears that electronic scanning systems

* In principle, a theory of optical systems can also be set up for curved input, output, or intermediate surfaces. Difficulties with respect to the choice of suitable coordinate systems have hitherto prohibited a practical elaboration of that idea (with the incidental exception of spherical surfaces). t This is true only for coherent processors. For incoherent and electronic processors t h e input and output signals are not only real, but also nonnegative (cf. P 10).

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PRINCIPLES OF OPTlCAL DATA-PROCESSING

[IV, I 2

can be described in much the same way as incoherently illuminated systems.

0 2. Field Theory of Optical Systems 2.1. THE DATA-PROCESSING MODE

An optical system contains linear, time-invariant, source-free matter. As such, the electromagnetic field inside the system obeys Maxwell’s

equations which, for harmonic time dependence proportional to e x p ( - i d ) , read as (2.1a) curl H = -ioe E , curl E =

i w p - H.

(2.lb)

Like the complex electric and magnetic field vectors E,H, the material properties as reflected by the tensor functions E and p are dependent on position (x, y, 2). For isotropic media E and p degenerate into scalar functions E(X, y. z ) and p ( x , y, z ) which, due to potential dispersion, may also be complex functions of frequency. As indicated in Fig. 2.1 (shaded part), the interior of the optical system is characterized by a certain distribution of matter (E, p). contiguous to a vacuum region (E= E ~ p, = wo) in the vicinity of the end-planes. The latter assumption, which fairly corresponds to actual realizations of optical

\vacuum

/

Fig. 2.1. Geometry of a general optical system.

IV, I21

217

MELD THEORY OF OPTICAL SYSTEMS

systems (with “vacuum” replaced by “air”) yields a considerable simplification of the further analysis. The two end-planes carry two parallel coordinate systems xl, y1 and x2, y2. A point in either plane i (i = 1,2) will be indicated by a two-dimensional position vector r i = ( x i , y,) and a surface element dxi dy, will be shortly denoted by dr,. If a threedimensional coordinate system is required, we choose an x and y axis coinciding with the x2 and y2 axis and a z-axis pointing to the exterior of the optical system. In either reference plane we have four two-dimensional distributions of electromagnetic field quantities, viz. Ex,E,, H,, H,. The normal components E,, H, can be left out of consideration, because they are directly related to the tangential components, due to Maxwell’s equations: E, dHx/8y -dH,ldx; H, -dE,/dy -dEJdx. Concerning the tangential components a well-known theorem of resonator theory (SLATER[1954], GOUBAU[1961], BORCNIS and PAPAS [1955], KUPRADSE [1965]) states that the electromagnetic field in a cavity is uniquely determined if the tangential component of E or H is prescribed on the boundary surface. The same is true, when the tangential component E,,,, is prescribed on part of the boundary surface and Iftang is prescribed on the complementary part. Finally, on part of the boundary also an impedance boundary condition of the type Etang = Zw(Htang X n) can be imposed, where Z, and R denote the complex wall impedance and the outward normal vector, respectively. For our optical system the two end-planes plus a cylindrical surface (cf. Fig. 2.2) with an infinite radius h constitute the boundary surface. The radiation field on the infinitely remote cylindrical surface locally resembles a plane wave* with a boundary impedance Z, = Hence we can conclude that in either reference plane Etang or H,,,, can be prescribed ad libitum and that the total electromagnetic field in the optical system including the remaining quantities in the reference planes is then uniquely determined?.

-

G.

* It is tacitly assumed that the material part (E f q,,p f g o ) of the optical system has bounded axial and transverse dimensions. t It can be easily recognized that the electromagnetic field must be unique for a given E,,,, or ITtang in the reference planes. ,Assume that there are two solutions. Then the difference solution has t o satisfy Maxwell’s equation with a vanishing E,,,, or H,,,,, in the reference planes. This implies that there also the normal component of Poynting’s vector vanishes. Since then n o power is fed into the system, which has t o account for energy dissipation in the material and radiation losses, the total difference solution vanishes, too, Q.E.D. We remark, however, that this simple proof does not guarantee the existence of a field solution under the prescribed boundary conditions.

218

[IV, P 2

PRlNCIPLES OF OPTICAL DATA-PROCESSING

/

plane 1

\

plane 2

Fig. 2.2. Optical system with supplementary cylindrical surface.

This view seems to be in contradiction with the common idea that in an optical data-processing system the field is uniquely determined everywhere, when the tangential electric or magnetic field is prescribed in the input plane only. To resolve this contradiction we have to realize that in the processor “mode” the output plane is assumed to be contiguous to a source-free half-space with E = E ~ p, = po, into which electromagnetic waves are radiated. This implies that the tangential electric and magnetic fields in the output plane are linked to each other by an impedance boundary condition. However, unlike the impedance discussed above, this boundary condition is non-local: At a certain point P, Etang(P)is not only determined by Htang(P), but also by neighbouring values of Htang. As discussed further down, this generalized impedance becomes local only for fields E,;,, and H,,,, with sufficiently slow spatial variations. In an electric network analogy we can compare an optical system with a two-port (two-terminal pair network) whose electric state is described by two voltages and two currents (VAN VALKENBURG [1965]). The two-port properties find expression in two equations which implies that at either port one quantity (voltage or current) can be chosen ad libitum, thereby determining the remaining quantities. When one port is terminated with a certain impedance the voltage (or current) at the other port completely determines the electric state of the network; thus, like the situation in an optical processor, an input-output relation is established. In the data-processing mode an optical system radiates at the output plane into free space. In other words, there are no external reflections. In

IV, B 21

FIELD THEORY OF O!TICAL SYSTEMS

219

a common approximation one further assumes that free space can be replaced by another optical system without disturbing the transmission properties of the original system. Then the undisturbed output of the first system forms the excitation of the second, and a simple formalism can be developed for the cascades of optical systems. The restriction under consideration can be referred to as “absence of internal reflections”; it will be presumed to be satisfied throughout*. The tangential electric or magnetic field in the input plane forms the excitation of an optical processor. Likewise we can look at the tangential electric or magnetic field in the output plane. Since we suppose that human light perception and photographic registration is intimately associated with the intensity of the electric field, we can henceforth focus our attention on E alone. In this perspective Eta,,, in the input plane “causes” Eta,, in the output plane. With the observation that an optical system is inherently bidirectional, we can formulatet the input-output relations (VAN WEERT[1978]) c

(2.2a) (2.2b) where (2.2a) and (2.2b) pertain to the two directions of transmission 1 + 2 and 2- 1. These relations which form the basis for all theory of optical data-processing systems. are a direct consequence of the principle of superposition, valid for any linear system. The tensor functions g2, and g 12 completely reflect the data-processing properties of the system under consideration. In the following we further adopt the “scalar” approximation, in which the “cross-polarization’’ between orthogonal field components is neglected. In this approximation the x-component of E in the input plane does not excite a y-component in the output plane and vice versa. In other words, the tensors gZ, and g,, become diagonal and the optical processor corresponds to two independent systems with only x-x and y-y couplings. If we then focus our attention on one of these systems, we can *In the special case of an aperture in an otherwise opaque screen the assumptions discussed here also form part of Kirchhoff-Huygens’ principle (STRATTON [1941]). In fact, they are the basis of the Fresnel-Kirchhoff diffraction formula (BORNand WOLF[1965]). T If not stated otherwise, all integrations extend over the entire plane.

220

PRINCIPLES OF OPTICAL DATA-PROCESSING

[IV, B 2

replace (2.2) by (2.3a) &(rJ =

+

1

g12(r1,r2)42(r2)dr2,

(2.3b)

where the scalar function stands for the x or y component of the electric field vector, and the scalar functions g,,, g,, describe the transmission properties in the two directions 1+ 2 and 2 + 1. In the following, the conditions for the validity of the scalar approximations are assumed to hold throughout. Apart from birefringent media which form inherently “vectorial” systems t h e approximately scalar character of most optical systems is related to the fact that the characteristic physical dimensions of the usual components considerably exceed the wavelength of light. Even in grounded glass with a very fine structure crosspolarization can hardly be observed (BASTIAANS [ 1979a1). It should be mentioned, however, that the scalar treatment of optical systems basically violates the principle of reciprocity (VAN WEERT[1978]) and, apart from a few exceptional cases, constitutes not more than an approximation.

2.2. THE RECIPROCITY THEOREM

Two different solutions of Maxwell’s equations (2.1) satisfy the reciprocity theorem (STRATTON [ 19411)

E‘”, H‘”, E(*’,H‘,’

valid for any closed surface A provided that the tensors p and E are symmetrical (which includes the degenerate case of scalar functions I*. and E ) . Physically, this condition is violated only in the presence of a static magnetic field (CASIMIR [1963]), e.g. in a Faraday rotator. As in other branches of applied physics (e.g. radio communication, electrical networks) we expect that reciprocity imposes certain restrictions upon the behaviour of an optical system when operated as a signal processor. Suppose that for the system under consideration the scalar approximation applies. Moreover the scalar input and output functions are assumed t o vary so slowly that in a spatial Fourier representation (cf. § 3) the

IV, P 21

22 1

FIELD THEORY OF OWICAL SYSTEMS

highest spatial frequencies “contained” in the signal are small compared with the wave number k = 2r/A = VANWEERT[1978] has shown that this “paraxial” approximation just implies the validity of the former scalar approximation. The fields E , H with superscripts 1 will now be associated with the transmission 1 + 2 of the optical processor. In either plane, @& is assumed to have only an x-component. Due to the slow transverse variations of E y ) ef+(’) a locally plane wave is excited in the output plane with HL1’=O and H I ” = G E P ’ . As the system is free from internal reflections the same relation holds for the input plane. If the system is operated in the opposite direction 2 -+ 1, the associated fields are indicated by a superscript 2. All plane waves travel in the opposite direction and we have a remarkable sign change: HY)= - G E F ) . With the geometry of Fig. 2.2, the two end-planes supplemented by a cylindrical surface with radius h + 00 form the closed surface in the reciprocity integral (*). Due to the asymptotic behaviour of E and H for great distances one easily estimates (SOMMERFELD [19541) that the infinitely remote cylindrical surface does not yield a contribution to (*). There remains

“6.

-1

lane 1

(E(”H(2)- E(2)H‘’))dr lx

ly

Ix

ly

1

+

LIane2

(EYJH(2) 2 Y - E(2)H(1)) 2x 2 y dr2 -- 0 .

With E , = 4 and the above relations between E, and Hy, and after division by the common factor 2=, we obtain

I

plane I

+y)+y)dr, =

I

plane 2

+k1)+k2’

dr,.

(2.4)

This important reciprocity relation valid in the scalar and paraxial approximation can be readily interpreted in terms of the characteristic functions g,, and g12 of the optical system. Assume point excitations 4:”= 6(rl - a) in plane 1 and 4:”=6(r2-b) in plane 2 . Then (2.4) yields +:“’(a)= +Y)(b).

(2.5)

On the other hand (2.3) states that $?)= 6(rl -a) causes +Y)= g21(r2,a) and &) = 6(r2- b) causes 4y)= g12(rl, b) so that (2.5) implies g12(a, 6) = gzi(b, a).

Since a and b are arbitrary, we can finally conclude that

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PRINCIPLES OF OFTICAL DATA-PROCESSING

[IV, 5 3

Hence, the transmission from a point r l in plane 1 t o a point r2 in plane 2 equals that in the opposite direction. This simple result is only valid in the paraxial approximation. If the input and output signals contain high spatial frequencies, the scalar treatment in general fails; but even if the scalar assumption is taken for granted (as in acoustical systems), (2.6) has to be properly modified (BUTIERWECK [1978]). Then it turns out that only with respect to their macroscopic structure, g,, and g,, are equal. When “viewed” through an instrument with a spatial resolution of a few wavelengths or less, one observes a considerable difference in fine structure.

0 3. System-theoretical Approach to Coherent Optical Signal Processors 3.1. INPUT-OUTPUT

RELATIONS IN SPACE AND FREQUENCY DOMAIN

In this section we consider an optical signal processor from a pure “black box” point of view. In this approach the light vibrations in the two reference planes are assumed to be describable by two scalar, complexvalued functions +l(x,, yl) and &(x,, y,) of which one plays the role of excitation and the other that of response*. Again the system is assumed to behave reflexion-free but, on the other hand, no a priori assumptions are introduced with respect to reciprocity and paraxial approximations. Henceforth, all two-dimensional signals are equivalently described in the space and frequency domain. For any signal, a Fourier transform pair +(x, y), @(X,Y ) is defined according to @ ( X Y) =

4(x, y ) exp [ W X x + Yy)] dx dy,

(3.la)

Again we mark a “space point” by a position vector r = (x, y), whereas a spatial “frequency point” is marked by a vector R = (X, Y). Then (3.1)

* As discussed in B 2, this scalar approach applies to many optical systems, exactly or approximately. On the other hand, a theory of acoustic systems is inherently scalar, with identifiable as the sound pressure.

IV, P 31

223

SYSTEM-THEORETICAL APPROACH

can be written more compactly:

J

-

@(R)= 4 ( r )exp ( G R r ) dr,

(3.2a) (3.2b)

We agree that the upper signs in the Fourier transformations refer to the transmission 1+ 2, and the lower signs to the transmission 2 + 1. This [1977]) will yield a number of double sign convention (BUTTERWECK formal advantages in the further course of this section. The ultimate reason is, however, of physical nature: we want to identify spatial frequencies with directions in three-dimensional space. Let us consider, for instance, the illumination exp [i(X,x + Y,y)] and inquire, how this two-dimensional plane wave has to be continued in three-dimensional free space. This continuation satisfies Helmholtz’s equation A 4 k2+ = 0, with k = w 6 , and obviously represents the three-dimensional plane wave (assume Xi Yo2 s k2)

+

+

exp [i(X,x

+ Y,y + Z,z)]

(*)

with

ZO=Jk2-X:- Y ; . When we now require that the wave propagates in +z-direction the positive Z, value and the upper signs in (3.1) have to be chosen, corresponding to a 1 + 2 operation. The spectral description 47r2 S(XX,, Y - Yo)of our illumination then follows from (3.la). In the 2 4 1 operation the same function corresponds to exp [-i(X,x + Y,y)] with exp [-i(X,x + Yoy + Zoz)] as three-dimensional continuation, propagating in (-2) direction. Obviously this is the conjugate of the former wave (*), and as such it has the same equiphase planes but opposite direction of propagation. If the same signs in (3.1) had been used for either operation 1 + 2 and 2 + 1, the last result would have become exp [i(X,,x + Yoy Zoz)] which corresponds to a completely different direction of propagation in three-dimensional space. In our notation the frequency pair X,, Y,) corresponds to a direction uniquely described by the “wave vector” k = (X,, Yo, +dk2- Xz - Y;) with the understanding that in the two modes 1 -+ 2 and 2-+ 1 the wave propagates in the directions k and (-k),respectively.

224

PRINCIPLES OF OPTICAL DATA-PROCESSING

[IV, D 3

In many practical problems functions +(x, y ) with rotational symmetry = m we then have $(x, y ) = +,(r). Then also the occur. With r with Fourier transform exhibits rotation symmetry: @ ( X , Y )= @,(I?) R= The Fourier transformation (3.1) then degenerates into a Hankel (or Fourier-Bessel) transformation (GOODMAN [19681) as given by

m.

1

4&) = 271.

I,

-

@,(R)J,(Rr)RdR,

(3.3b)

where J o ( . ) denotes the zero-order Bessel function. Note that (3.3) holds for either sign in (3.1). As we have learned in 9 2, the signal transformation in a linear optical processor is governed by a superposition integral. Due to (2.3) we have +2(r2) = +l(rl) =

j j

g21(r2, r1)41(r1) drl,

(3.4a)

g12(rI9r2)42(r2) dr2,

(3.4b)

for the two directions 1 + 2 and 2 + 1 . Of course, there are similar relations in the frequency domain, viz.

5

G2,(R2,Rl)@,(R1)dR1,

(3.5a)

W R J = G12(Rl,R2)@2(R2) dRz,

(3.5b)

@z(&) =

where the G-functions are coupled to the g-functions due to (3.2). W e obtain

with similar relations (only i is replaced by -i) for the second pair g,,, GI,. The highly symmetrical transformations (3.6) (which have to be carefully distinguished from ordinary Fourier transformations) will henceforth be referred to as “mixed” transformations.

IV, § 31

225

SYSTEM-THEORETICAL APPROACH

Each of the pairs g,,, g,, and GI,, G,, provides a complete characterization of the optical system. These complex-valued weighting (or Green’s) functions have simple physical interpretations: the g’s are point spreads (the counterparts of impulse responses in time domain and thus responsible for the blur in an optical system) and the G’s are wave spreads (spectral representations of responses to plane-wave excitations). In the special case (3.7a) (3.7b) for which + 1 = + 2 , we have a “through connection” or an “identity system”. Notice that g,, and G,, form a mixed pair according to (3.6) and not a Fourier transformation pair!

3.2. CASCADES AND INVERSE SYSTEMS

If two systems are placed in cascade, the output of the first system forms the input of the second (cf. Fig. 3.1). Considering only one direction of transmission we then have

and, after elimination of the intermediate signal +,(r2),

1

2

3

Fig. 3.1. Cascade of two optical systems

226

PRINCIPLES OF OPTICAL DATA-PROCESSING

[IV, P 3

with g,,(r,, rl) =

j

g32(r3,r2)g21(r2, r l ) dr2.

(3.11)

We notice that, apart from some important exceptions to be discussed below, a change of the order of the individual systems in a cascade also changes the properties of the overall system. Further, the cascade formulas can be extended to more than two subsystems. For n systems (3.11) then becomes an (n- 1)-fold integral in which integrations are carried out over all intermediate planes with surface elements dr,, dr,, . . . dr,. Finally, we consider the question whether for a given output signal an associated input signal can be found. Mathematically, this amounts to the solution of an integral equation. If and only if it can be solved (in D 5 a number of explicitly solvable cases is compiled), the inverse signal transformation can be written in the form

Although these equations must not be read as cause - effect - relations (the transmission remains in the direction 1 +. 2) we can associate with them a fictitious inverse optical system whose weighting functions are gil(rl, r2) and Gi1(R1,R2) and which operates in the 2 -+1 direction. The question whether the inverse system can be realized belongs to the domain of system synthesis and can, for the moment being, remain unanswered. The cascade of the original and the (fictitious) inverse system yields, of course, the identity system. Since this is one of the few cascades where the order of the subsystems may be interchanged, we obtain with (3.7) and (3.11) the two equivalent relations

(3.13b) Similar relations can be derived for the pair G,,, G i l and also for the

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PARTIALLY COHERENT ILLUMINATION

221

functions corresponding to the 2 + 1 direction. Unlike the relations (3.12a) where also the signals &, &,occur, (3.13) should be viewed as the definition* of the point spread of the inverse system and as the starting point for its evaluation. §

4. Partially Coherent Illumination

4.1. SPECTRAL TREATMENT OF PARTIAL COHERENCE

In the preceding sections time-harmonically illuminated optical systems have been the objects of field-theoretical (Q2) and system-theoretical (Q3) investigations. In this section we temporarily exchange timeharmonic signals for signals associated with stationary random processes. In optical terms, we deal with partially coherent light including the extreme cases of complete incoherence and complete coherence. In our treatment we adopt the modern frequency-domain approach developed by MANDEL and WOLF[1976] and BASTIAANS [1977]. This approach saves a number of necessary quasi-monochromaticity assumptions of former [1964]). theories (BERANand PERRANT Resuming the lines of 9 1, we consider a single-input-single-output, linear, time-invariant system with impulse response h(t) and system function h(o).If a random signal f ( t ) excites such a system, the convolution integral (1.2) remains valid for the determination of the output signal g ( t ) . Since, however, a random signal does not possess an ordinary Fourier transform, the simple product relation (1.5) in the frequency domain becomes meaningless. On the other hand we are, in general, interested in statistical averages rather than in the detailed structures of the pertinent time functions (this is certainly true for optical signals of which, due to the high frequencies involved, not more than the meansquare values can be measured). Such a statistical average is the autocorrelation function Ef(T) = ( f ( t ) f ( t - T ) ) ,

where ( ) denotes ensemble or (due to ergodicity) time averaging. From (1.2) we can easily derive the autocorrelation function of the output signal *The two equations (3.13) are analogous to the two definitions of t h e inverse of a matrix A, viz. AA-' = 1 and A - ' A = 1 with 1 the unit matrix. These relations are equivalent, to be sure, but their numerical elaboration leads to a different set of equations.

228

g ( t ) from that

[IV, 0 4

PRINCIPLES OF OPTICAL DATA-PROCESSING

of the input signal f(t) as

c, ( 7 )= h(7)* Sy(7) * h ( - T ) , where * denotes convolution with respect to the time shift variable T. Notice that this relation involves only deterministic functions and easily admits Fourier transformation. With the power spectra S,(w) and S f ( w ) defined as the Fourier transforms of the pertinent autocorrelation functions we simply obtain S , ( O ) = lh(w)12 S f ( W ) ,

(4.1)

i.e. the power spectrum of the output signal is obtained from that of the input signal through multiplication by the squared magnitude of the system function. The necessary generalization to multiple-input-multiple-output systems is plain sailing. Assume that a system with N inputs is excited by functions $ ? ( t ) (j = 1 , 2 , . . . N) and that these are transformed into N output functions &?‘(t) (I = 1 , 2 , . . . N) such that through a generalization of (1.2) N

&‘“‘(t)=

1hii(t)*4?(t),

j-

1

(1

= 1, 2 , .

. . N).

Then we obtain

... where

is called the (cross-)correlation function” which, for j = k , degenerates into the autocorrelation function. In optics, where the signals 4 are functions of position r, i j k ( 7 ) = (&rj, t ) & * ( r k , t - 7))is usually referred to as the (mutual) coherence function. The frequency-domain equivalent of (4.3) reads as

j=l k=l

* Notice that (4.4) also applies in the case of complex time functions like the “analytic signals’’ (BORN and WOLF [1965]).

IV, § 41

229

PARTIAFLY COHERENT ILLUMINATION

with

the (mutual) power spectrum which for j = k degenerates into the (real, non-negative) auto-power spectrum. An optical signal processor is a system with an infinite number of inputs and an infinite number of outputs. The discrete variables 1, j in (4.2) then become the continuous variables r,, rl in (3.4), hij(w)becomes gZ1(r2,r l ) (which is also a function of frequency w ) , and the sums pass into integrals. The continuous analogue of (4.5) can then be written as S(r& r;, w ) =

II

g2!(r$,r;, w)S(r;, ry, o ) g z l ( r i , ry, w ) dr; dry. (4.7)

A number of conclusions can be drawn from this fundamental relation. First we recognize the importance of second-order statistics: the output mutual power spectrum can be uniquely determined from the input mutual power spectrum provided that the system properties are known. The measurable total intensity at a certain point r2 in the output plane then follows from (4.7) by consideration of r2 = r; = r;. With the inverse of (4.6) we obtain:

-

intensity at r2 = mean-square value of 4(r2, t)

‘I

= S(r2, r2, 0 ) = -

27T

S(r2, r2, w ) dw.

(4.8)

Further, we notice that (4.7) is linear in the mutual power spectrum. As such, it is the generalization of (3.4) for the case of partially coherent illumination. It is important to recognize that the time-harmonic analysis of an optical system is entirely sufficient for the prediction of its behaviour under partially coherent illumination. With the exception of narrow-band excitation (for which S ( r ; , fl,w ) resembles a Dirac pulse around w = w , , ) , we need, however, have disposal of the complete frequency characteristics of the system, as given by gZ1(r2,r l , w ) . Anyway, due to the “universality” of the optical system functions g2,(r2, r l , w ) we can conclude that the system properties are inherently independent of the light coherence and that, more particularly, two systems equivalent for coherent illumination are equivalent for any degree of partial coherence.

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DV, 5 4

4 . 2 . INCOHERENT ILLUMINATION

The important degeneracy of a vanishing mutual power spectrum for r\ # ry corresponds to incoherent illumination.W e then have S(r\, r;, o)= p(ri, w ) 6(r; -r;)

(4.9)

for the input power spectrum, where p(rl, o)denotes the (auto-) power spectrum* at the point rl. Insertion into (4.7) yields

which relation allows the conclusion that incoherent light in general does not remain incoherent: the light vibrations at two points in the output plane receive contributions from all points in the input plane and hence exhibit a certain degree of correlation. Often, one is merely interested in the auto-power spectrum in the output plane:

This famous relation is usually referred to as “superposition of power” for incoherent illumination. We note, however, the difference of symbols: S(r2, r2, w ) is a true auto-power spectrum, whilst p(rl, w ) was only shortly denoted as such. Strictly speaking, S(rl, r l , w ) is infinite with a finite S(r2, r2, w ) , which puts the poor efficiency of incoherently illuminated systems into evidence. Actually, a finite “correlation area” is required in the input plane, in order to produce a nonvanishing response in the output plane?. Formally (4.11) exhibits a certain resemblance with the superposition integral (3.4) for the time-harmonic case. The main difference is that the input and output signals as well as the weighting function lg21(2are real

* Strictly speaking, the auto-power spectrum of incoherent light according to (4.9) becomes infinite. However, we do not wish to introduce a new name for p which has most properties in common with the power spectrum. ? The necessity for a finite correlation area is also revealed by dimensional considerations. With ( 3 . 4 ) the dimension of g is (area)-’ and that of lgI2 is (area)-2. If, as is done in many textbooks, p in (4.t1) is replaced by S, we obviously need an additional constant multiplier with the dimension of an area. This is just the correlation area as mentioned above.

IV, 5 41

PARTlALLY COHERENT ILLUMINATION

231

and positive”. Since the positiveness is difficult to translate into the frequency domain the spectral counterpart (3.5) of (3.4) is seldom constructed in the incoherent case. An important exception is the shiftinvariant system discussed in § 5. The “modulation transfer function” introduced for these systems will be treated in § 10.2. For two systems arranged in cascade, caution has to be used when applying (4.11). After the light has passed the first system, incoherent light has become partially coherent, and the general formula (4.7) applies. If this transformation incoherence + partial coherence would not take place, no imaging with incoherent light would be possible. This implies that in the relation (3.11) for the cascade of two systems g must not be replaced by 1gI2 for incoherent illumination?. Rather, one has first to employ (3.11) with the complex g-functions and take the squared magnitude after that. 4.3. COHERENT ILLUMINATION

An interesting special situation occurs when an optical system is illuminated by a point source located at r l = a. Then we have to insert

S ( r i , r;l, w ) = 6(r{ - a, r ; - a)q(w)

(with q ( o ) 2 0 )

(4.12)

into (4.7) and thereby obtain (4.13) with t ( r , 0) = g2,(r, a,

w)JiGJ.

(4.14)

Light, whose mutual power spectrum can be factorized according t o (4.13) is referred to as coherent, although not necessarily monochromatic. It has the remarkable property that it remains coherent after passage through linear systems. Indeed, if the input power spectrum can be

* The present theory applies also to electronic scanning systems if they provide a linear intensity mapping. Of course, the transformation from incoherent into partially coherent light, as arises in pure optical systems, does not occur here. A possible combination of blur and nonlinear behaviour leads to complications which are beyond the scope of this article. t The situation is different when in the intermediate plane a rotating diffuser is inserted which removes the partial coherence.

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factorized according to (4.13) we obtain from (4.7)

and the output power spectrum can be likewise factorized: S(r;, r;, w ) = t2(r;, w)tT(r’l, o) with t2(r2,w ) =

1

gZ1(r2,r l rw)tl(rl, a) drl.

(4.15)

(4.16)

Hence, the function t(r, w ) transforms like our former function &(r) according to (3.4) so that we are justified to state that coherent light propagates in much the same way as strictly time-harmonic light or, more rigorously, as deterministic, Fourier-transformable light pulses. For (4.16) would exactly apply in the latter case with t(r, o)denoting the Fourier transforms of the signals under consideration. In this section we have abandoned the consistent space-frequency approach of the previous time-harmonic treatment. The reason is twofold: First, partial coherence is only a collateral subject within the framework of optical system theory, and second, the dual of incoherent light, the spatially stationary light, is of minor practical significance. However, in connection with the somewhat more specialized case of transmission of incoherent light through shift-invariant systems we shall resume the space-frequency dualism in FI 10. §

5. Basic System Constraints

5.1. SINGLE CONSTRAINTS

After this “stochastic” intermezzo we return to strictly harmonic time dependence and resume the line of P 3. In this connection we consider a number of basic restrictions which can be imposed upon the system behaviour and which can be expressed in terms of conditions for the weighting functions g,,, g12. As these are linked to the corresponding frequency functions GZ1,GI, via the mixed transformations (3.6) each condition applying to the g’s has a frequency pendant applying to the G’s. The pertinent checks for the correctness of these various interrelations are left to the reader. At the end of this section some physical

IV, § 51

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233

systems are considered which satisfy the constraints in question with a certain degree of accuracy. The reciprocity condition forms a link between the two directions of transmission, as formulated by gZl(r2, r d = g d r 1 , r2Ir

(5.la)

G21(R2, R I )= Gi,(Ri, R2).

(5.lb)

In 5 2 we have shown that systems filled with isotropic material d o approximately satisfy this condition. Notice that we profit by the double sign convention introduced in (3.2). With a uniform sign (5.la) would have transformed into the asymmetrical condition GZ1(R2,R , ) = Gi2(-Ri7 - R J . The remaining conditions discussed in this section apply to a single direction of transmission. which henceforth will be chosen in the 1 + 2 mode. This does not exclude that the same condition holds for the 2 + 1 direction, too. In that case the condition under consideration will be said t o be satisfied completely. Next we consider losslessness” which expresses equality of input and output signal “energies”: (5.2a) (5.2b) When we insert (3.4a) in (5.2a) and require that the resulting identity holds for all input signals 41(rl), we obtain (with the same steps in the frequency domain)

I

g21(r2,r d g ? ~ h r;) , dr2 = 6( r l - r 3 G21(R2, RI)G?I(Rz,Ri)dR,= W - R ; ) ,

(5.3a)

(5.3b)

as necessary and sufficient condition for the validity of (5.2). We note, in passing, that if (5.3) is satisfied for all temporal frequencies w, even the

* This signal-theoretical definition has not an immediate physical meaning. Physical losslessness involves the time average of Poynting’s vector E XH,but it can be shown that in the scalar, paraxial approximation its normal component is proportional to 1&1*. See also VAN WEERT[1980].

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time-space integrals of the squared input and output signals are equal (in the case of light pulses), or the space integrals of the temporal meansquare values are equal (in the case of “power” signals including partially coherent signals). For partially coherent illumination* the pertinent proof can be given with the aid of (4.7). A concomitant property of utmost importance is easily found for lossless systems: they admit an explicit determination of their inverse systems. Comparison of (5.3) and (3.13) reveals that

The next property to be discussed is symmetry. Owing t o the mixed transformation formulas (3.6) we have to distinguish two types. Spatial symmetry is defined by

whereas spectral symmetry is defined by

Spatial and spectral symmetry are not mutually excluding properties. Indeed, a number of important systems are spatially and spectrally symmetric (cf. B 6). In order to give an idea about the physical meaning of spatial (spectral) symmetry we can state that the transmission from a space (frequency) point A in the input plane to a point B in the output plane equals that from the projection of B on the input plane to the projection of A on the output plane. A subclass of spatially symmetric systems is formed by the spreadless systems, whereas the shift-invariant systems are a subclass of the spec-

* This result holds also for incoherent illumination. This seems to be in contradiction with the poor efficiency observed in connection with (4.11).The paradox is resolved by the fact that incoherent light has an enormous blur in comparison with coherent light so that the integral of the autopower spectrum ultimately remains the same. Also note in this connection that, due to (5.3) the integral of l g 2 , ( r 2 , r,)12 over r z becomes infinite, which effect compensates the seemingly poor efficiency due to (4.1 1).

IV, § 51

235

BASIC SYSTEM CONSTRAINTS

trally symmetric systems. Their characteristic functions are given by (5.7a) (5.7b)

1

M 2 , ( R )= mZ1(r)exp (-iR

*

r) d r

(5.7c)

r) dr

(5.8~)

for spreadlessness, and

=

hZl(r)exp (-iR

for shift invariance. Similar relations hold for the 2 + 1 mode provided )replaced by (+i) in ( 5 . 7 ~ )and (5.8~). that (4is We first discuss the behaviour of these important systems in the space domain. Inserting (5.7a) in (3.4a) yields

42(r)= m21(r)41(r)

(5.9)

for a spreadless system, i.e. a local relation between the input and output signal, without blurring effects. Such a system therefore acts as a (amplitude and/or phase) modulator. For a shift-invariant system we obtain the spatial convolution

42b)= h21(r) * 41(r), i.e. 42(4=

1

~ ( r 2 - r 1 ) 4 1 ( rdrl ~

(5.10)

from (5.8a) and (3.4a) which reflects the inherent property that if &(rl) is replaced by &(rl -a) ( a is a constant vector), the corresponding output signal changes from &(rJ to 42(r2- a). Hence, the image is shifted over the same (vectorial) distance a as the object, without change of its form. and PARIS This is in contrast with shift-variant image formation (LOHMANN [1965]), where the structure of the image depends upon the location of the object.

236

[IV, P 5

PRINCIPLES OF OPTICAL DATA-PROCESSING

If we compare the spatial behaviour of these systems with their spectral behaviour, as given by (5.7b) and (5.8b), we have (5.11)

for the spreadless and shift-invariant system, i.e. the signal transformation for one system in the space domain has the same character as that for the other in the frequency domain. The two types of systems are said to be dual in a wide sense (PAPOULIS [1968b]), whereas we speak about strictsense duality if moreover the functions m2,(r) and H21(R)(and herewith Mzl(R) and hZ1(r))are similar with respect to their mathematical structure. Due to this definition, free space and a lens are, within their respective approximations, dual in the strict sense (cf. Q 6). Besides the inherent spatial symmetry of spreadless systems and the inherent spectral symmetry of shift-invariant systems the complementary symmetry condition can also be satisfied. Obviously this occurs, when h21(r) or Mzl(R) are even functions. When a shift-invariant system is illuminated with a plane wave exp (iR * r ) insertion into (5.10) yields H21(R)exp (iR r ) for the output signal which allows the conclusion that exp (iR r ) is an “eigenfunction” of a shift-invariant system with the “system function” H2,(R) as proportionality factor (eigenvalue). Indeed, this property forms the background for the simple product relation (5.12) in the frequency domain, which states that there n o frequency “mixing” takes place. Dual statements hold for the spreadless system. Either type of systems is of practical importance: spreadless systems are always associated with small axial dimensions (transparencies and thin lenses belong t o this category), whereas shift-invariant systems are often required for optical filtering (deblurring, matched filtering, differentiation, contrast improvement etc.), but also, albeit in special forms, readily provided by nature (free space and, with some restriction, diffractionlimited imaging). Numerous examples, including the problems encountered with incoherent illumination will be treated in forthcoming sections. The last restriction to be discussed in this section is that of rotation invariance. Unlike the former restrictions it applies only to genuinely two-dimensional systems. In analogy to shift invariance, it means that with an object rotation round the origin r = O an equal-angle image

-

-

IV, 9: 51

BASIC SYSTEM CONSTRAINTS

231

rotation is associated, with the image structure maintained. First considering a general system and introducing polar coordinates x = r cos a, y = r sin a, we rewrite (3.4a) as (5.13) If now rotation invariance is required, the response to a point source at r I = r,, a,=O has to be rotated by an angle a, if the source moves to rl = ro, a t = a,, i.e. gz1Jr2, r,, a2,0 ) becomes g21.p(r2,r,, a 2 - al, 0 ) = g21.p(r2,r,, az,a l ) . This implies that g21.p= q21(r2r

r17

ff2-

a,)

(5.14)

is a function of a 2 - a 1 only so that the integration over a 1 in (5.13) degenerates into a convolution. Assume now that the rotated object is proportional to the original object, then, on account of linearity and rotation invariance, the same statement holds for the image with an equal proportionality factor. Functions with this property have the general shape exp (irna)f(r),

rn

= 0, *l, &2,. . . .

Hence, an input function with an angular dependence according to exp(irna) is transformed into a similar function at the output. Note, however, that the radial functions need not be equal so that in general the total function is no eigenfunction. With rn = 0 we find that functions 4 with rotational symmetry retain this property after passage through a rotation-invariant system. Since a rotation in the space domain r corresponds to an equal-angle rotation in the frequency domain R, there is no sense to seek for the dual system: a rotation-invariant system is its own dual. Reciprocity, losslessness, and the various forms of symmetry have hitherto been defined in a rather abstract manner. We shall now link these constraints to certain physical or geometrical properties of an optical system. Moreover we present illustrating examples and counterexamples, for which the conditions defining the above constraints are satisfied or violated. As shown in 8 2.2, most optical systems satisfy the reciprocity condition (2.6) in the paraxial approximation. Beyond that approximation, only special systems, like free space, remain reciprocal in the sense of (2.6). On the other hand, non-reciprocity in the paraxial region has to be sought

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[IV, § 5

among the devices operating with static magnetic fields. So, the cascade of a 90"-Faraday rotator and a 90" reciprocal rotator (consisting of an optically active material) constitutes a scalar, nonreciprocal system. Due to 0" and 180" rotations in the two opposite transmission modes the pertinent system functions have different signs and, as such, describe a nonreciprocal 180" phase shifter. In general, losslessness can be shown to be equivalent t o absence of energy dissipation within the optical system (VAN WEERT[1980]). In mathematical terms, losslessness then implies realness of E(X, y, 2). If, however, the spatial frequencies involved are so high that nonuniform plane waves are excited, attenuation need not be associated with energy dissipation. This is the case, e.g., when free space is excited with spatial frequencies exceeding the wave number k . According to P 6.1, we thus obtain the lowpass characteristic of free space. In the low-frequency region, where the above dissipation mechanism applies, typical lossless representatives are formed by the dissipation-free phase modulators, whereas dissipative films act as lossy amplitude modulators with a modulation function smaller than unity. The formal conditions for symmetry can always be satisfied by suitable structural symmetries. Let a fictitious intermediate plane be inserted midway between the input and output plane and let, for the moment being, the x-y-plane of the coordinate system be shifted to this intermediate plane. Let furthermore the system be filled with an isotropic medium with a scalar dielectric constant E(X, y, z ) so that reciprocity applies. Then we have spatial symmetry if E ( X , y, z ) = E ( X , y, - 2 ) (i.e. mirror symmetry with respect to the intermediate plane) and we have spectral symmetry if E ( X , y, z ) = E ( - x , -y, -z) (i.e. symmetry with respect to the origin). Finally, rotation symmetry is obtained if E ( X , y, z ) is a rotationally symmetric function, depending on (x"+ y") and z. The proof for the spatially symmetrical case follows from the simple observation that, due to the special distribution of E, the transmission from a point A in plane 1 to a point B in plane 2 equals that from the projection A' of A onto plane 2 to the projection B' of B onto plane 1 which, due to reciprocity, again equals that from B' to A'. So, the definition for spatial symmetry is fulfilled. A similar reasoning applies to the realization of spectral symmetry. Attention should be paid to the fact that reasoning of this sort does not admit the conclusion that "black-box-symmetry" necessarily implies structural symmetry. There is some evidence, that this need not be true!

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BASIC SYSTEM CONSTRAINTS

239

5.2. CONSERVATION LAWS

The constraints discussed above are fundamental in the sense that most of them are preserved in cascade combinations and under system inversion. (In fact, only symmetry is not per se preserved in cascades.) This finds expression in the following theorems.

Theorem 1. In a cascade of optical systems, each of which satisfies the condition of reciprocity, losslessness, spreadlessness, shift invariance, or rotation invariance, the overall system satisfies that condition, too. Theorem 2 . If an optical system admits system inversion and if it satisfies the condition of reciprocity, losslessness, symmetry, spreadlessness, shift invariance, or rotation invariance, the inverse system satisfies that condition, too. (In the case of reciprocity system inversion involves both directions (1 + 2 and 2 -+ I).) The proofs of the various aspects of these theorems can be readily furnished through combination of the pertinent system constraints with the cascade and inversion relations (3.11), (3.13); they are left to the reader. Rather, we want to discuss a special consequence of the conservation of reciprocity in cascades. If two reciprocal systems which are realized as cascades of reciprocal components and which differ in their structure and their components have been proven to be equivalent (i.e. to have equal weighting functions g, G ) in one direction, they are also equivalent in the opposite direction. Therefore, from the equivalence proofs for the two directions one (in general the easiest) may be chosen ad libitum. For example, if a certain cascade of free space, lens, free space (which are reciprocal components*) appears to be equivalent to an abstract, ideally imaging system (which occurs, when the “lens law” is satisfied) in one direction, the same is automatically true for the opposite direction. For further, less trivial applications of this principle cf. 0 7. With respect to cascades a special result holds for spreadless and shift-invariant systems. Not only are the two properties preserved but also may the order in which the individual systems are arranged be interchanged without any influence upon the overall system properties. We

* Note

system.

that in our approach a section of free space is always considered as a separate

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PRINCIPLES OF OWICAL DATA-PROCESSING

[IV, 5 5

then have with (5.9) and (5.12) m,,(r)= mn.,-i(r) . . . m32(r)m21(r),

(5.15a)

Hn,,(R)= Hn,n-l(R) . . H32(R)H21(R)

(5.1%)

for the cascade of ( n - 1) spreadless or shift-invariant systems. In the complementary domains these ordinary products are transformed into convolution products. Due to the commutativity of both types of products the order of arrangement of the individual systems is arbitrary.

5.3. MULTIPLE CONSTRAINTS

Many practical systems exactly or approximately satisfy a number of the foregoing constraints simultaneously. The resulting properties, some of which interesting and surprising, are now discussed. First we combine reciprocity and (complete) losslessness. Then (5.1) and (5.4) combine to form the relations gbI(r1, r2) = gTAr1, r2),

(5.16a)

GLI(R1, R2) = GTZ(R1, R2).

(5.16b)

Thus, the transmission of the “inverse” system is the complex conjugate of the reverse system. As can be concluded from Fig. 5.1 this amounts to the following property: if an excitation 4 1 ( r l ) in plane 1 causes a response 42(r2) in plane 2, then 4:(r2) in plane 2 causes 4T(r,) in plane 1 (cf. Fig. 5.2). Anticipating a result of the next section we mention that free space is approximately lossless and reciprocal. Then the above property can be visualized by a point source in plane 1 exciting a divergent wave in plane 2, while its conjugate there produces a wave now converging towards plane 1 (Fig. 5.3). Since, however, the losslessness condition is not exactly fulfilled, the wave does not converge to an exact point, but only to a “focal” region with dimensions in the order of a wavelength*. If, for the moment being, harmonic time dependence is abandoned and

* In holography one encounters a practical application of the principle illustrated in Fig. 5.3. After illumination of a developed hologram with the reference wave one obtains on the observer’s side besides the wanted virtual image (representing the original wave front) also an unwanted real image (due t o the complex conjugate wavefront).

241

BASIC SYSTEM CONSTRAINTS

’

original system

2

inverse system

Fig. 5.1. Relation between the inverse and reverse transmission in a lossless, reciprocal system.

Fig. 5.2. Two compatible signal pairs for the opposite transmission directions in a lossless, reciprocal system.

Fig. 5.3. Divergent and convergent waves in free space. The illuminations in the right-hand planes are each other’s complex conjugates.

real polychromatic signals i i ( r i ,t ) ( i = 1,2) with spectral components in a certain frequency band are considered and if, moreover, the system is lossless and reciprocal in that frequency band, we can draw the following conclusion from Fig. 5.2: when &(rl, t) excites i2(r2, t ) , then i 2 ( r 2 ,-t) excites &(rl, - t ) . This result originates in the simple fact that under time reversal the Fourier transform of a real function of time changes into its

242

PRINCIPLES OF OPTICAL DATA-PROCESSING

[IV, 0 5

complex conjugate. Hence we can conclude that joint fulfilment of losslessness and reciprocity implies time reuersibility. Combination of reciprocity with other constraints yields some formal simplification of the signal-processing description. Jointly symmetrical, reciprocal systems are characterized by (cf. (5.1), ( 5 . 5 ) , (5.6))

g2,(r2, r J

=

g12(r2,r l )

G 2 1 ( R 2R, l ) = G 1 2 ( R 2R,) ,

for spatial symmetry,

(5.17a)

for spectral symmetry,

(5.17b)

which implies that for these functions the indices 1 2 , 2 1 may be dropped (this is not true for the complementary functions). A spatially symmetrical, reciprocal system is characterized by one symmetrical space transmission function g(r2,r l )= g ( r l ,r2), whilst for a spectrally symmetrical, reciprocal system we have G ( R 2 ,R,)= G(Rl, R2) applying in either direction of transmission. For the special cases of spreadless and shift-invariant systems we can conclude that the transmission properties are completely described by the bidirectional functions m ( r )= m21(r)= m12(r) and H ( R )= = H 1 2 ( R ) On . the other hand, we have in the complementary domains M2,(R)= M 1 2 ( - R )and h21(r)= h12(-r) so that, except for even functions, there the specification of a transmission function must always be accompanied by a reference direction of transmission. Combination of reciprocity and rotation invariance yields with (5.14) q2,(r2,

r17

a2-

= q,,(r,, r 2 ,

a1 - a 2 )

or

q21(r2,r l , a)= q 1 2 ( r l ,r2, -a).

(5.18a)

If the functions are even with respect to a,we have the simpler result q21(r27

rl, a)= q 1 2 ( r 1 , r 2 , a).

(5.18b)

Even functions of a imply that the response to a point source at rl = r,, a1= a, is angularly symmetric with respect to the source, and the system can be said to be rotation-free. This condition can be expected to be met in most rotation-invariant systems of practical interest. The remainder of this section is concerned with one direction (1 -+ 2) only; so reciprocity is no longer involved. First, we consider the combination of shift and rotation invariance. Shift invariance implies validity of (5.8a):

g21(r2,r l ) = h 2 1 b - r l ) .

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BASIC SYSTEM CONSTRAINTS

243

Rotation invariance implies that rotation symmetry of a signal is preserved. As the point source 6 ( r , ) has to be reckoned among the rotationally symmetrical signals its response hZ1(r2)must belong to that class, too: hz1(r) = hZl,&) = h21,p(J=7).

(5.19)

and the condition (5.14) for rotation invariance is obviously met. With the knowledge that every motion of a two-dimensional figure (a “rigid body” in kinematics) can be considered as a succession of a rotation round a fixed point 0 and a translation (cf. Fig. 5.4),we realize that shift- and rotation-invariant systems are invariant with respect to any figure motion. What has to be kept constant during the motion are merely the mutual distances between all points of the figure: distortions and magnifications are thus excluded. From Fig. 5.4 we can further conclude that any motion can also be interpreted as a rotation round some point 0’. In this view, shift invariance plus rotation invariance round a fixed point is equivalent to

Fig. 5.4. General motion of a two-dimensional figure interpreted (i) as a rotation B-B‘ round 0 followed by a translation B’-B”, (ii) as a pure rotation B-B” round 0’.

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PRINCIPLES OF OVTICAL DATA-PROCESSING

[IV, § 5

rotation invariance round all points. Apparently this again implies shift invariance, since a shift can be viewed as a rotation round an infinitely remote point*. For a shift- and rotation-invariant system the relation between the point spread h21(r) and the system function H21(R),both rotationally symmetric, is given by a Hankel transformation (3.3). When also the illumination is rotationally symmetric, the image has this property, too, and with (RI= R (5.12) passes into (5.21) If now @,.,(R) is given by a 6-function at R = A, the same holds true for @2,p(R).Such a 6-function corresponds to a Bessel function J,(Ar) in the space domain; hence J,(Ar) is an eigenfunction of a shift- and rotationinvariant system with H21,p(R)as proportionality factor (eigenvalue). The dual of the above shift- and rotation-invariant system is a spreadless system with rotationally symmetric modulation function rn(r). This seems, however, to have less practical significance and therefore deserves no further consideration. On the other hand, the combination of losslessness with spreadlessness or shift invariance occurs frequently; it is characterized by the condition (rn2,(r)1= 1

for spreadlessness,

(5.22a)

IH21(R)I =1

for shift invariance,

(5.22b)

which follows from (5.2) combined with (5.9) or (5.12). Apparently a lossless and spreadless system constitutes a pure phase modulator, while the corresponding shift-invariant system represents what is called an “all-pass’’ in electrical engineering. If IR( does not exceed the wave number k = 27r/A, free space belongs to this class (cf. P 6). It is important to note that the conditions (5.22) are hardly to translate into the complementary domains, i.e. there are no simple conditions for MZ1(R)and h21(r).

* In either point of view, a two-dimensional rigid body is characterized by three degrees of freedom. Starting from a certain initial position, the pure rotation is given by the x-y-coordinates of the rotation center plus the rotation angle, and in the combined rotation-translation the rotation angle plus the displacements in the x and y directions characterize the motion quantitatively.

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P

245

6. Examples of Physical and Abstract Systems

In this section we first analyze some basic structures which are in common use as building blocks of optical systems. In the second part we compile several abstract systems with desirable properties without as yet investigating their realizability.

6.1. PHYSICAL SYSTEMS

The probably most basic optical “component” is a section of free space with length d . For the sake of convenience, we assume that the index of refraction n equals unity (vacuum), otherwise the wave number k = 06 occurring in forthcoming formulas has to be multiplied by n. The fact that free space is included in the catalogue of optical components (a strange idea in classical optics) is typical of the system-theoretical approach of optics. Free space is reciprocal, shift- and rotation-invariant and, as such, can be described by a rotationally symmetric system function H ( R )= H J R ) with a signal transformation obeying (5.12). On the other hand, H ( R ) can also be interpreted as the eigenvalue (proportionality factor) pertaining to the eigenfunction exp (iR r ) = exp [i(Xx + Y y ) ] .This two-dimensional plane wave propagates as exp [i(Xx + Yy + Zz)] in three-dimensional space, where

Z = J k 2 - X 2 - y2,

k

= 2r/A =

06,

and z denotes the distance of a field point from the input plane. For X 2 + Y 2 < k 2 this plane wave propagates in +z-direction, whereas it decays exponentially for X 2 + Y 2 >k 2 . For this to be true the square root for Z has obviously to be taken in the first quadrant. In the output plane z = d we then obtain the field distribution exp (iZd) exp [i(Xx + Y y ) ] which yields the system function H ( R )= H,(R) = exp (iZd) =exp ( i d J k 2 - X 2 - Y 2 )= exp (id-).

(6.1)

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The corresponding point spread follows from (3.3) as

h,(r) =

I,

1

”

a

= --

ad

H,(R)J~WW d~ (6.2a)

{exp [ik-]/(2~-)},

which integral was first evaluated by Sommerfeld (cf. WATSON[19661). This h,(r) is conveniently approximated in two steps. First, differentiating with respect to d and neglecting a term small for d > > 2 ~ /=k A, one obtains

k d exp[ikdr2+d2] h,(r)=-27ri&GF J7T-Z

9

(6.2b)

whereupon the small-angle (“paraxial”) approximation r << d yields

h,(r)

== k

exp (ikr2/2d).

(6.2~)

Notice that a constant phase factor exp(ikd) (which only might be interesting in interferometric applications, where t h e absolute phase shift of free space is involved) has been dropped in ( 6 . 2 ~ )The . widely used result ( 6 . 2 ~ )is usually referred to as the Fresnel approximation of free-space propagation; its validity has been the object of various investigations (PAPOULIS [1968a1, GOODMAN [19681, MAITHIJSSE and HAMMER [1975]). If not stated otherwise it is henceforth taken for granted. The system function pertinent to the Fresnel approximation is found by . obtain Hankel transformation of ( 6 . 2 ~ )We d H,(R) = exp (-i - R 2 ) , 2k

(6.3)

which (curiously enough) is the low-frequency approximation of (6.1). We observe that, due to [H,(R)I= 1, free space is lossless in the Fresnel approximation, whereas the exact result (6.1) reflects a low-pass transmission characteristic, in which all frequencies R > k are attenuated and, in fact, completely lost if kd >> 1. For large distances d, but finite extent of the input illumination, the Fresnel diffraction passes into Fraunhofer diffraction. Inserting ( 6 . 2 ~ into )

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(5.10) then yields

k

b2(r2) = -[exp [ik Ir2- r l ~ z / 2 d l d J l ~dr, rd 2md k exp (ikr:/2d) 2rrid

=-

exp (-ikr2

- rl/d)dJl(rl)dr,.

(6.4)

Apart from a quadratic phase factor (which can be avoided by looking on an output sphere instead of an output plane) one “sees” the Fourier transform of the input signal dJ1(rl). An important class of optical components which, moreover, is relatively easy to manufacture, is formed by the modulators (spreadless systems). Its simplest representative is an aperture in an otherwise opaque screen, with a modulating function m(r) =

1 0

in the aperture, elsewhere.

It can be easily extended to a multiple-aperture modulator which, for instance, is met in the half-tone realization of a transparency with continuous gray-shades. An approximate representative of the phase modulators which, moreover, is in widespread use, is provided by the lens. With reference to Fig. 6.1, we assume that between the two reference planes 1-2, a distance d o apart, a dielectric material with constant refractive index n, but variable thickness d ( r )5 do is inserted. Under the condition that the input signal and the thickness function d(r) do not vary too rapidly (i.e. that they contain only low frequencies) we obtain a local input-output relation

1

,vacuum

’1

refractive index n (constant)

Fig. 6.1. Geometry of a phase modulator.

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Fig. 6 . 2 . A thin, plano-convex lens.

that is determined as if a plane wave was normally incident. Neglecting reflections at the boundary surfaces we then obtain the phase delay

knd(r)+ k [ d , - d ( r ) ] = k [ ( n - l ) d ( r ) f d , l and the modulating function

m ( r )= exp [ik(n - l ) d ( r ) ] , where we have dropped the (uninteresting) constant phase factor exp (ikd,). Apart from the proportionality factor k ( n - 1 ) the thickness function d ( r ) then determines the local phase delay. For a spherical lens d ( r ) = dp(r) is rotationally symmetric and follows a quadratic law in the paraxial approximation ( r < rmax<< p in Fig. 6.2). Simple geometrical considerations reveal that for a plano-convex* lens with curvature radius p the local thickness becomes (apart from an additive constant)

dp(r ) = -r2/2p which leads to the two characteristic functions

m(r) = m,(r) = exp (-ikr2/2f),

(6.6a) (6.6b)

with

f = p/(n - 1 ) in that approximation. In § 8 it will turn out that f equals the focal distance of the lens. For r = r,,, (cf. Fig. 6.2) the phase function attains

* For other types (double-convex, positive meniscus, double- and plano-concave, negative meniscus) of lenses cf. GOODMAN [1968]. Notice that for an overall concave lens f becomes negative.

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the maximum absolute value kr$,,/2f which remains constant for r > r,,,. The pertaining mathematically inconvenient modulation function is usually replaced by m = 0 for r > r,,,, corresponding to a window modulator cascaded with the idealized lens according to (6.6). This rough approximation is guided by the geometric-optical idea that “rays” travelling past the lens are definitely leaving the optical system and do no longer contribute to the output signal. In a final approximation (the “thin-lens” model) one completely neglects the finite lateral lens dimensions. This model is mathematically described by (6.6) for all values of r and R. We have to keep in mind, however, that (6.6) is the result of a number of rather simplifying assumptions (spreadlessness, normal plane wave incidence, weakly curved lens surfaces, infinite lateral dimensions, absence of reflections) so that in certain optical systems this model possibly predicts the lens behaviour with insufficient accuracy.

6.2. ABSTRACT SYSTEMS

Next we treat the Fourier transformer, which has, in fact, furnished the name “Fourier optics” for the modern field of optical signal processing. It is characterized by the pair of functions

k

g21(r2, r1) =

exp (-i-

k r2 rl), d

-

d d G21(R2,R,) = 27rik exp (-i R2 R1), ~

(6.7a) (6.7b)

which leads to the input-output relation

in the space domain and a corresponding relation in the frequency domain. In rough terms, the output signal indeed represents the Fourier transform of the input signal. However, a scaling constant with the dimension (length)-2 is required to provide a mapping of the frequency domain of the usual Fourier transformation into the output space domain. This constant equals k/d in (6.8), where the length d is a characteristic

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constant of the Fourier transformer under consideration and the wave number k occurs in connection with its actual realization (cf. 0 8.1). The proportionality factor kl(2rrid) in front of the integral is SO chosen that the Fourier transformer becomes lossless. We further assume that g,, = g,, and G,, = G,, which implies reciprocity. Moreover, the Fourier transformer is (spatially and spectrally) symmetrical, and rotation-invariant, but neither shift-invariant nor spreadless. Notice that a shift of the input signal merely introduces a linear phase factor in the output signal which remains invisible in case only the output intensify is observed*. This property manages to detect certain frequencies in undetermined regions of the input plane. In 0 8 it will be shown that a Fourier transformer can be simply realized as a cascade of a lens and two sections of free space. While the real-time realization of a time-domain Fourier transformation is prohibited due to the required knowledge of the future values of the input signal, no causality requirement impedes a spatial Fourier transformation. A system of equal theoretical and practical importance is formed by the magnifier. Its characteristic functions are given by g21(r2, rJ = t S(rl - tr2L

(6.9a) (6.9b)

which implies M r ) = t&(tr),

(6.10)

and a similar relation in the frequency domain. Obviously the magnifier is a generalization of the identity system (through connection) into which it degenerates for t = 1. For It( < 1 it furnishes a size magnification and for ltl>1 a size reduction. Furthermore, for t < O the scene is inverted. The occurrence of the proportionality factor t in (6.10) guarantees the losslessness of the magnifier. Reduction of the linear dimensions by a factor t reduces the surface dimensions by t2, but simultaneously increases the intensity (=square of the amplitude) by t2. Again we assume that reciprocity applies, i.e. g,, = g,, and GI, = Gzl. This implies that (6.10) is also valid in the direction 2- 1. Then upscaling in one direction means downscaling in the opposite direction and vice

* Note that the output intensity can also be interpreted as the Fourier transform of the autocorrelation function of the input signal.

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versa so that a specification of the magnification t has to be associated with a reference direction. How this is accomplished formally, will be discussed in § 7. Although it constitutes a device with a simple mathematical description, a magnifier with I t \ # 1 does not satisfy the constraints of spatial and spectral symmetry (and herewith those of shift invariance and spreadlessness). However, it is rotation-invariant.

6.3. CASCADES, INVERSIONS, AND DUALITIES OF ELEMENTARY SYSTEMS

When two or more sections of free space, lenses or magnifiers are arranged in cascade the resulting system is again equivalent to a section of free space, a lens or a magnifier. Only in the case of Fourier transformers another type of system can be (but need not be) created after cascading (cf. § 7.2). For two sections of free space with lengths d , and d , the resulting free space has the length d , + d , . Notice that this trivial result is also found formally by applying the Fresnel approximation (6.3)! For two lenses in cascade the reciprocal focal distances l/fl and l/f2 (the “powers”) have to be added, while for two cascaded magnifiers the magnifications have to be multiplied. On account of the losslessness of the four basic optical systems, inversion according to (5.4)can be easily accomplished. For free space with length d inversion again leads to free space, now with length ( - d ) . Thus, retrieval of the input signal of a free-space section amounts to transmission of the output signal through a fictitious free space with corresponding negative length. Notice, however, that this is true only in the Fresnel approximation. If this is abandoned, system inversion for d sufficiently large is completely impossible because the high frequencies IR\> k are lost. For a lens with focal distance f system inversion leads to a focal distance (-f). Hence, a convex lens is transformed into a concave lens (and vice versa) and their cascade provides an identity system. Inversion of a Fourier transformer changes its characteristic length into its negative value ( d + - d ) , while in the case of a magnifier t is transformed into l/t. Finally we remark that free space and lens are dual in the strict sense (compare (6.6a) with (6.3) and (6.6b) with (6 .2 ~ )). The dual of a magnifier with magnification t is a magnifier with magnification l l t (which is also its inverse). The Fourier transformer is its own dual.

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7. Operational Notation of Optical Systems and Basic Cascade Equivalences

7.1. AN OPERATIONAL NOTATION

Most optical systems can be viewed as cascades of more or less elementary systems, among which the four basic components of the last section. In order to facilitate the analysis of such cascades we now introduce an operational notation*. In this notation, a system is represented by a Gothic symbol with the pertaining transmission functions ad libitum added between brackets. A general system is denoted by Wgzl(r2, r J r g12(r,, 4

1

or

WGzi(Rz, RI),G12(R1, R2)I, where the first notation applies to the space domain and the second to the frequency domain. The inverse system is represented by W and the identity equation G a s @ , means equivalence of the systems Ga and 8, in either direction. Two systems Ga[gzi(rZ7r l ) , glArl, rZ)l and ab[g32(r3, 4 , g2,(r2, 4 1 , which are cascaded so that abis to the right of @a, form the new system aC=(By,@$with @c[g3l(r3, rl), g13(rl?r3)1*

Clearly, plane no. 1 is at the left, plane no. 3 at the right, while the intermediate plane 2 disappears when only the overall system GC is viewed. The new system function g3, is determined according to (3.11) with a corresponding formula for g13. Special systems admit simplifications. Reciprocal systems require only one g-function, as in W&,

rA1,

where g obviously applies to either direction. Integration over r l is required for the 14 2 transmission, and integration over r2 for the 2 4 1 transmission. Henceforth, all systems are assumed t o be reciprocal. *This operational notation should be viewed as a continuation and refinement of a method proposed by VAN DER LUGT[1966]. The present analysis follows BUTTERWECK [1977].

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Subclasses of reciprocal systems are indicated by special symbols. So, @ [ H ( R ) ]and m[m(r)] denote shift-invariant and spreadless systems, and G ( d ) ,L?(d),3 ( d ) , X ( t : 1) represent the four systems of the preceding section, viz. free space, lens*, Fourier transformer and magnifier. The characteristic quantities d (length of free space, focal distance [formerly denoted by f], and scaling constant of Fourier transformation, respectively) and t (magnification) are added between brackets. For some reciprocal systems it has to be explicitly stated to which direction the quantity inside brackets refer. This is the case, for instance, with the magnifier, where the notation t : 1 indicates that the physical dimensions of the left-hand scene are It1 times those of the right-hand scene so that (6.10) applies. The same is true if shift-invariant systems are to be described in the space domain and spreadless systems in the frequency domain. Then, according to h2*(r)= hI2(-r) and M21(R)= M I 2 ( - R ) ,we can write

@[h,,(r)I

or

@[G(r)I,

YJ2[M2,(R)]

or

YJl[fi(R)],

with the understanding that the pertinent functions have to be mirrored for the (t) direction. From the above considerations we would expect that shift-invariant and spreadless systems are advantageously described by the functions H ( R ) and rn ( r ) , respectively. Indeed, the modulation function rn (r) completely characterizes a spreadless system, independent of the transmission direction. The situation is, however, more complicated for a shift-invariant system. With the double sign convention in the Fourier relations (3.2) we have established a one-to-one correspondence between R and a direction in three-dimensional space (cf. B 3.1). As is illustrated in Fig. 7.1, where the hatching indicates the pass-band of a spatial narrow-band filter, the reversed system passes a different wave direction. Thus, for a shiftinvariant system, the specification of the system function H ( R ) has to be accompanied by a labelling (1,2) of the two reference planes. This way the spatial orientation of the system is fixed. When the system has to be reversed or, what amounts to the same result, the system is viewed from “behind the paper”, the reference planes are interchanged and H ( R ) is replaced by H ( - R ) .

* In % ( d ) the finite,dimensions of the lens pupil are not taken into account. If desired a window modulator can be cascaded with 2 ( d ) .

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‘passed “ray” Fig. 7.1. Change of the pass-band of a narrow-band spatial filter due to reversal.

7.2. CASCADE EQUIVALENCES

We are now prepared to discuss a number of basic equivalences of system cascades, which facilitate the analysis of complicated systems. How these tools are applied to actual systems, will be treated in the next section. In the present section we confine ourselves to a statement of the various equivalences and a brief discussion of their implications. The pertinent proofs, which are throughout easy to construct, have been given [1977]) K and are omitted here. We only want to elsewhere ( B ~ R W E C recall that, due to the reciprocity of all building blocks, all equivalence proofs need to be given only for one (the “easiest”) direction of transmission. All equivalences to be discussed in this section can be traced back to five identities: (7.la) (7.2a) (7.3a) (7.4a)

(7.5) The first three relations are concerned with the interaction of a magnifier X with a Fourier transformer 8, a general shift-invariant (9)or a spreadless (!LR) system. We observe that 5Z is “absorbed” by a Fourier transformer (thereby changing its scaling constant from d to rd), whereas

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it is “pushed” through a shift-invariant and a spreadless system (thereby again changing the scaling of their characteristic functions). The next result (7.4a) states that a Fourier transformer can be pushed through a shift-invariant system thereby transforming it into the dual spreadless system, or vice versa. It can be traced back to the well-known “convolution theorem” which states that the Fourier transform of a convolution of two functions equals the product of their Fourier transforms. In this interpretation, @ corresponds to (spatial) convolution and YJI corresponds to (spatial) multiplication. Finally, (7.5) is a special result for the family 2,8,G, which does not admit a generalization towards more general systems. It is easily found by working out the first integral in (6.4) and properly identifying the resulting individual terms. It states that a freespace section G can be replaced by two lenses (which are concave for d >O) and a Fourier transformer in cascade. These results admit some modifications and specializations. First, from (7.la) we have

8(4)%d2) = % ( - ( 4 / d 2 ) : I),

Xdi)8(d2)8(d3) X-didJdJ.

(7.lb) (7.1~)

A cascade of two (or, more generally, an even number of) Fourier transformers thus constitutes a magnifier, whereas three (or, more generally, an odd number of) Fourier transformers again equals a Fourier transformer. We remark that (7.lb) is easily found from (7.la) through “post-multiplying’’ (7. l a ) by the inverse system f’J(-d)which cancels f’J(d). Finally ( 7 . 1 ~ )is found through application of (7.la) and (7.lb) to its lefthand side. Next, we apply (7.2a) and (7.3a) to the special shift-invariant and spreadless systems G and 2. We then find

E(t:l)G(d)= G ( t 2 d ) 5 ( t :l),

(7.2b)

X ( t : 1)2(d)=2(r2d)Z(t: 1).

(7.3b)

Again, the magnifier re-scales the adjacent systems by transforming d into t2d. By viewing (7.4a) from “behind the paper”, H ( R ) is transformed into H ( - R ) , while 8 and X9. as spatially symmetric systems remain unchanged. We thus are led to:

??(d)@[H(-R)l=m[ H(: r)]%d).

(7.4b)

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Some minor notational changes further yield

Xd)lDl[m(r)I=@[ m(i R)]5(d), DmCm (r)B:(d)=5(d)@[ m

(- $ R)].

(7.4c) (7.4d)

Application of (7.4) to I! and 6 leads to

tY(dJWd2) E I!(d?/dJ%di),

(7.4e)

Wd2)5(dJ

(7.4f)

%di)Wd?/dd.

Hitherto, 6 was described in terms of R and Vl in terms of r. In rare cases also a description with interchanged roles is required. For instance, with

@[H(R)I = HL(r)I on e obtains

in (7.2a), or with

mn[m(r)I = lDln[fi(~>I one obtains

in (7.3a). Finally (7.4a) reads as

Q[K(r)]%(d) -S(d)W[4r2 $6(-$

R)].

All results follow from well-known scaling theorems of Fourier theory. § 8.

Operational Analysis of Optical Systems

8.1. ACTUAL REALIZATIONS OF FOURIER TRANSFORMER AND MAGNIFIER

First we apply the general results of the previous section to analyze two well-known realizations of a Fourier transformer. Observe that the operational notation throughout obviates the need for illustrations.

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In the cascade z(d)G(d)Z(d) we have two lenses, each of which is placed in the focal plane of the other. With (7.5) and the observation that 2 ( d ) and its inverse 2 ( - d ) cancel each other, we have

E(d)G(d)2(d ) 3 2 ( d ) 2 ( - d ) % ( d ) 2 ( - d ) 2 ( d ) g(d).

(8.1)

The dual of this realization is the more familiar cascade G(d)2(d)G(d), i.e. one lens operated between its two focal planes. As g(d) is its own dual, we expect again a Fourier transformer % ( d ) as equivalent. In a formal proof, we cascade G(d)g(d)G(d)from the left with the identity system %(-d)%(d)and then push % ( d ) through the cascade. With (7.4e) and (8.1) we obtain:

Wd)Wd)Wd) %:(-d)%(d)G(d)2(d)G(d)

-%:(-d)2(d)G(d)2(d)g(d)

27- d ) % . ( d ) X d ~) % ( d ) .

(8.2)

Note that the realizations according to (8.1) and (8.2) provide a new proof for the losslessness and rotation invariance of a Fourier transformer: it is based upon the knowledge that these constraints are preserved in cascades and that they are satisfied by the individual components G and 2. The “classical” optical system is a lens providing a sharp imaging. In where the two lengths a, b are our notation this reads as G(u)C(f)G(b), related to the focal distance f according to the lens law

With (7.5) we write

G(aP(f)G(b) 2(-a)%(a)W- a)C(fR(- b)%(b)W--b ) and state that, due to (8.3) and the additivity of the lens powers, the three central lenses cancel out. Then the two adjacent Fourier transformers are combined into a magnifier (cf. (7.lb)) and finally the lens is pushed through the magnifier (cf. (7.3b)):

G(a)2!Cf)G(b) =2 ( - a ) g ( a ) g ( b ) 2 ( -b ) - B ( - a ) 2 ( =B(-; a : 1)2(--)2(-b)=X(-b2 a

-:

U

b

: l)2(- b )

: 1)2(f- b).

(8.4a)

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In the last step, we have combined the two lenses at the right, making use of the lens law (8.3).This way we have derived the well-known result that the cascade G2G satisfying (8.3) provides a sharp imaging with a quadratic phase error. Hence, an ideal magnifier (with a negative magnification) is constructed by placing a correcting convex lens with focal distance b - f > O at the right end:

6(a)8(f)Q(b)~(b-n~ : 1). ~(-~

(8.4b)

Note that the left-hand focal plane of the correcting lens and the right-hand focal plane of the main lens coincide.

8.2. FOURIER FILTERING

A shift-invariant system is conveniently synthesized as a modulator inserted between two Fourier transformers (thereby transforming the former into its dual). With ( 7 . 4 ~ )and the fact that g ( d ) and g ( - d ) cancel each other, we then have:

This relation is the basis of Fourier filtering. In the intermediate plane where the modulator 2R is inserted, the Fourier transform of the left-hand input signal is manipulated in some way or another. Inverse Fourier transformation then yields a filtered version of the input signal with the properly rescaled modulation function rn acting as system function of the resulting shift-invariant system. In the ultimate realization of (8.5) the Fourier transformers are synthesized according to (8.2). A minor flaw occurs due to the unrealizability of g ( - d ) in (8.5) which would require an G ( - d ) . Instead we replace g ( - d ) by g ( d ) and so obtain

i.e. a shift-invariant system followed by an uninteresting inversion (Fig. 8.1).

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Imagine now that the modulator is not carefully adjusted in the central “Fourier plane”, but undergoes an axial displacement 6 to the right (cf. Fig. 8.1). Then we can advantageously utilize the notion of a negative length of free space. We first go to the right of the original Fourier plane, insert YJJ1 there, and then go back to the left. The resulting cascade can then be described as

where (7.4e) has been applied. Clearly, the effect of the displacement 6 is reflected by two fictitious lenses arranged on either side of the wanted 6. Thereby the shift-invariance property is lost. The right-hand lens is not disturbing, when only the output intensity is measured, but the left-hand lens introduces errors for illuminations with sufficiently large linear dimensions. Since, like all equivalences discussed in this and the previous section, the identity (8.7) applies to all degrees of coherence, we can state an interesting result for incoherent illumination: then also the left-hand lens does not exert any influence upon the system properties, with the result that the modulator can be placed ad libitum between the two lenses in Fig. 8.1! 8.3. INSERTION OF A MODULATOR IN FRONT OF A FOCUS; ABERRATION ERRORS

Assume now that a convergent beam is focused on the axis ( r = 0) and that a modulator \%R[m(r)]is inserted in front of the focus, at a distance d. We can then apply the notion of a negative-length free-space section, moving backwards over a distance ( - d ) , inserting Yh!, and going forward to the former focus. With (7.5) we obtain:

The three central components commute so that 2 ( d ) and %-d) cancel each other. Further, with ( 7 . 4 ~ )and the eventual annihilation of the pair 3(-d), g(d)we have

G(-d>n[rn( r ) ] G ( d= ) 2 ( d ) @ [m(-

d

R)]2(-d),

(8.8)

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.

input plane

.-. -. -----

[IV, 9: 8

-.-

output plane

Fig. 8.1. Standard realization of a shift-invariant system with inversion. Included is an unwanted shift 6 of the modulator away from the exact “Fourier plane”.

from which we can, incidentally, conclude that this configuration is its own dual. Our above assumption that without ?m the light was focused on the axis, amounts to a left-hand illumination of our system with a fictitious point source S ( r ) which, of course, remains undisturbed by the first lens 13(d). If also the phase distortion due to 5 2 - d ) can be left out of consideration (which is allowed with pure intensity measurements), we “view” the point spread of @ at the system output. This is determined as the inverse Fourier transform of the system function H(R) = m(-(d/k)R), yielding

Hence, apart from the quadratic phase factor due to 13(-d) we “see” the properly scaled Fourier transform of the modulation function m ( r ) . We conclude from (8.9) that the pattern h(r) is enlarged with increasing distance d. The most obvious application of this result is found, when m ( r ) represents the exit pupil of an otherwise ideally imaging system. For a circular aperture with radius a we have (8.10)

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and (8.9) becomes the “Airy” pattern

ka J,(kar/d) h ( r )= h,( r ) = 27rd r

(8.1 1)

In D 9.2 we shall show that the result (8.9) and its application (8.11) also holds true if a modulator is inserted in any intermediate plane of a perfectly imaging system even if such a plane is separated from the focus by one or more lenses. An actual optical system does not form an exact focus in the hypothetical absence of the window modulator YJl. Due to spherical aberrations the illumination in the modulator plane exhibits phase errors that can be represented by an additional phase modulator with modulation function exp [ib(r)] in tandem with the pupil modulation function (8.10). Notice that such a fictitious phase modulator can be elegantly compensated along holographic way by recording exp[ib(r)] on an off-axis hologram and using its “conjugate” wave exp [-ib(r)] for compensation (LEITH [1977]). Without compensation, the real function m ( r ) of (8.10) is converted into a complex function, whose absolute value is now given by (8.10). This implies that the “energy” of m ( r ) (i.e. the integral of its squared modulus) remains unchanged and that, due to Parseval’s theorem, the same is true for the point spread h ( r ) . But there is some indication that h ( r ) becomes “broader” after addition of the phase distortion. To see that we determine the value of Ih(r)l at the origin r = 0, which according to (8.9), is found as

Among all functions m ( r ) with equal absolute value the constant-phase function yields the maximum value of the last integral and, hence, of Ih(0)l. (Particularly when the phase variations of m(r) exceed 27r, the different contributions to the integral will tend to interfere in a destructive way.) Since the integral of Ih(r)JZhas been found to be independent of the phase function, the constant-phase m ( r ) thus leads to the strongest concentration of lh(r)l around the origin. The beneficial effect of suitable amplitude modulators in the exit pupil (apodization) is discussed in D 10.1.

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8.4. SOME PHENOMENA IN FREE-SPACE PROPAGATION

Assume that a free-space section of length d is subsequently illuminated by an arbitrary function f ( r ) and its complex conjugate f * ( r ) transmitted through a Fourier transformer % ( - d ) . We maintain that t h e observed moduli at t h e system output are equal in both cases. With g ( r ) denoting the output signal for the first illumination, as illustrated by

- f(r)

-

G(d)

K(r)

we have the situation f*(r)

% ( - d ) S ( d ) E-

i*(r)

G(-d)C(-d)G(-d)G(d)

for the second illumination, where use was made of (8.2). Observe that in the last identity G ( - d ) and 6 ( d ) cancel each other and that at the output of the left-hand G ( - d ) the function g * ( r ) appears. For, if f ( r ) is transformed into g ( r ) by G ( d ) , then f * ( r ) is transformed into g*(r) by the (conjugate) system G ( - d ) . Since lg*(r)l= Ig(r)l and 2 ( - d ) merely introduces an additional phase factor, we have proved the above assertion. Application of this result to various functions f ( r ) reveals that the function pairs sin ( k a x l d ) sin ( k a y l d ) 2 kyld exp [ - a ( x 2 + y')]

lead to the same output intensities ( p , (x) denotes the rectangular pulse with p , ( x ) = 1 for Ixl
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class, that under a certain condition self-imaging can take place, for which the output signal equals the input signal (“Talbot effect”). With the free-space transmission function

we see that for all spatial frequencies

R = { p m d , v-4

p, v = O , 1 , 2 , . . . ,

IRl2 = ( p 2 + v2)47rk/d we obtain H ( R )= 1. Thus, for all periodic functions whose fundamental frequency equals w d (or a multiple of that) all harmonics are transmitted the same way and n o (linear) signal distortion occurs. The reader is invited to verify that the difference between the distances from an image point to the corresponding object point and to its “neighbour” point (a period distant) then becomes a quarter wavelength (or a submultiple of that). Only under this somewhat surprising condition constructive and destructive interference take care for the fact that an image point “receives” light only from the corresponding source point.

0 9. Systems Compounded of Lenses and Sections of Free Space (53.5-systems)

In this section we pay special attention t o those important systems which are built up of only lenses and sections of free space (“5%-systems”). Since we have shown ( 5 8.1) that Fourier transformers and magnifiers can be realized as 2G-systems (i.e. 3 and 5Z form subsets of the set of %-systems), we can as well speak about E g E - sy ste m s. This class of systems has a number of interesting properties.

9.1. EQUIVALENT “CIRCUITS”

We first show that every $%-system can be replaced by an equivalent system of at most three components from the catalogue 2,G,g,E. Borrowing an appropriate term of the electrical engineers community, we will henceforth speak about an equivalent “circuit”. First let us prove that a general 5%-system containing a total of N

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89

lenses+sections of free space and to be denoted by GN,can always be replaced by a cascade 282 containing not more than two lenses and a Fourier transformer. We assert then:

2(d 1 ) 8 ( d 2 ) 2 ( d 3 )

@N

>

(9.1)

where the d, ( i = 1 , 2 , 3 ) have to be properly adapted to @”. We prove (9.1) by induction and assume that it is valid* for a certain N . Then we add from the right-hand side a further component and prove that the new @)N+l again can be represented in the form (9.1). Addition of a lens 2 ( d 4 ) preserves the general form (9.1) in a trivial way: 2(d4) and 2 ( d 3 )can be directly combined into a new lens. Addition of a free-space section G(d4) and reduction to the form (9.1) is somewhat more laborious. For convenience, we omit the characteristic constants and only write 2,G,%,2. Then we have @N+ 1

= @NG =

2826

from (9.1). Subsequently we apply 2=(the dual of (7.5)) obtaining @N+1=2WWG=2@@Z and then push the right-hand 8 to the left ’ j to a magnifier E: and combine it with the left-hand f @N+1

= 222G.

With G=282 (cf. (7.5)) we further obtain @N+l=2E2@2and finally push X to the right and absorb it in 8: @,+I

= 22X$j%? = ,282,

Q.E.D.

Likewise we can show that any 2G-system possesses also the following equivalents: @N = (9.2) @jN

= G2G,

(9.3)

@N

=

Gi%,

(9.4)

@jN

= 2G8,

(9.5)

@jN

= G28,

(9.6)

= 2Gc.

(9.7) The equivalence (9.3) which states that a general 2G-system can always be replaced by a single lens operated between two appropriate “reference @N

*With (7.5) it is valid for N = 1, if = 9.

(31,

=G. With d,= -d, and d 2 A 0 it is also valid if

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planes” is well known from geometrical optics (O’NEILL [1963]). On the other hand, (9.1) is most suitable to derive the general expression for the point spread of an 26-system. With (6.6) and (6.7) and the cascade formula (3.11) we easily obtain

for the point spread of GN according to (9.1). W e conclude that a general 26-system has an exponential point spread with a quadratic form in r l and r 2 as exponent. The factor in front of the exponential function guarantees the losslessness of the overall system. From the dual equivalent circuit (9.4) it follows that an expression similar to (9.8) also applies in the frequency domain. Finally we note that the proofs for the validity of the various equivalent circuits need be given for only one circuit, since all these cascades can be easily transformed into each other. Like their electric counterparts, the different equivalent circuits exist, however, only for almost every 26-system in the sense that one or more of the characteristic constants might degenerate (become zero or infinite) in special cases.

9.2. MODULATORS IN BG-SYSTEMS

With the notation of the inverse system and the equivalent circuits of the preceding paragraph we are able to generalize a result of 0 8.3 which stated that, apart from some quadratic phase factor, insertion of a modulator into a converging beam produces the Fourier transform of the modulating function in the focal plane. We now prove that this result also holds if a modulator is inserted somewhere in. a perfectly imaging system illuminated by a point source. A perfectly imaging 26-system transforms a point illumination into a point image. Like the simple lens, obeying the “lens law” (8.3), we then have the equivalent circuit (8.4) consisting of a magnifier E followed by a lens 2 representing a quadratic phase aberration. This system to be denoted by @ is now conceptually split up into two parts G1 and G2with the understanding that the modulator 2R is inserted between them:

G&

= G +G 1 r n 2 .

Instead of the last notation, we can describe the system with modulator in

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another way: We first go through the whole system @, then go back to the modulator plane (which involves system inversion a;), insert the modulator 9Jl and finally move through a2to the output. In symbols, the whole system then reads as

@W2rn2.

@ and

(9.9)

a2are next replaced by equivalent circuits: (9.10)

according to (9.1). Inversion of CS2 implies inversion of the individual components of the equivalent circuit and reversal of order. This implies @lrn@2

= @@$Jm2 X( t : 1)2(d1)2(-d4)8(-d3)2(-~2)rn2(~*)%(d3)2(d~).

Now 2 ( d 2 ) absorbs 2 ( - d 2 ) , like %(d3) is absorbed into 8 ( - d 3 ) after having transformed 2X into a shift-invariant system 6.Omitting all characteristic constants we then obtain

@,rn@,

= Z2@2

(9.11)

which - apart from the magnifier E - equals the former result (8.8). Hence a modulator inserted in an arbitrary intermediate plane of a perfectly imaging system produces the properly scaled Fourier transform of its modulation function in the output plane. If the modulator represents a circular aperture (e.g. due to finite lens dimensions), we get an Airy pattern. Such an aperture can always be transformed t o another plane whereby, due to rescaling, its diameter in general is changed. Especially it can be transformed to the conceptual beginning or end* of the system there forming the entrance or exit pupil. If two modulators YX1, are inserted into a perfectly imaging system, similar considerations as those which led to (9.11) now yield

n2

@Irn1@2rn2@3

=X52 &2&2

(9.12)

for the overall system, where @ , @ 2 @ 3 represents the original system and YJll and YJ, are transformed into and Q2. If the intermediate lens between 6,and 6,were absent, 6, and 6, would be in cascade which

* The “beginning” and the “end” of a system are formed by arbitrary boundary planes (e.g. the tangential planes of the “first” and “last” lens) and have to be carefully distinguished from the input and output planes.

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amounts to multiplication of their system functions Hl(R) and H2(R) which, apart from scaling factors, equal the modulation functions of YJll If and n2then represent apertures one of them will and n2. completely cover the other (reckoning with the pertaining scaling constants) and thus completely determines the system’s behaviour. This is what geometrical optics predicts: The finite extent of the entrance or exit pupil is found by geometrically projecting the smallest aperture of the system onto the entrance or exit plane (GOODMAN [1968]). The presence of the intermediate lens in (9.12) makes this result, however, not perfectly correct. Only when the apertures are rather large compared to the wavelength, 6, and 6, are low-pass filters with rather high cut-off frequencies or, what amounts to the same, with narrow point spreads. Point illumination of (9.12) from the left then causes an almost-point illumination of the intermediate lens, which gives rise to a negligible phase distortion. A similar reasoning applies when more than two apertures form part of the optical system.

n1

9.3. SYSTEMS CONTAINING CYLINDRICAL LENSES

A component of practical importance that renders a number of interesting signal transformations possible and that was not treated hitherto, is the cylindrical lens. If the axis of the cylinder is oriented parallel to the y-axis, the modulation function becomes y-independent and is given by

m ( r )= exp (-ikx2/2f).

(9.13a)

In the frequency domain we obtain (9.13b) Systems containing sections of free space, spherical lenses and cylindrical lenses of arbitrary orientations are no longer rotation-invariant and, as such, not describable by the simple expression (9.8) for the point spread of a general EG-system. In that expression the occurrence of )r112,r l r2, and lr2)2 reflects the rotation invariance of the overall system, with the pleasant result that not more than three constants d l , dZ,d3 describe the system behaviour completely. When cylindrical lenses are added to the catalogue of components, the quadratic form (9.8) is maintained, t o be sure, but also mixed terms as xIy1,x2y1occur and the total system is

268

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described by 10 coefficients. Our operational notation seems to offer n o longer specific advantages and the analysis of systems with cylindrical lenses will in general be carried out with the aid of the cascade formulas (3.11). However, it is our strong feeling that a systematic theory of such generalized 5%-systems which would yield more insight into their behaviour has still to be developed. Probably, van der Lugt’s formalism (VAN DER LUGT[1966]) might render a suitable starting point in this direction.

5 10. Shift-invariant Systems: Coherent Versus Incoherent Illumination On account of their practical usefulness, the shift-inuariant systems deserve special consideration. Only shift-invariant systems handle all “portions” of the spatial input signal in an equal manner, thus making possible numerous types of signal processing: low-pass, high-pass, band-pass and band-suppressing filtering, spatial differentiation (GORLITZand LANZL [1975], BUTTERWECK and WIERSMA [1977]), contrast improvement, phase[1935]), matched filtering, pattern multiplicacontrast methods (ZERNIKE tion, image deblurring (GOODMAN [1968]). In contrast with their shiftvariant counterparts, such systems are also amenable to a strikingly simple realization. According to (8.5), any modulator inserted between two Fourier transformers constitutes a shift-invariant system. In this section we first review the coherent behaviour of shift-invariant systems. In the further course a spectral analysis of incoherent illumination is developed with special emphasis upon low-pass filtering.

10.1. COHERENT ILLUMINATION

The behaviour of a shift-invariant system with strictly time-harmonic illumination was studied in P 5.1. With the shorthand notations +l(rl) = +in(rl), +2(r*) = +out(r*),

hz1(r)= h(r),

&(R)

= H ( R )=

I

h(r) exp (-iR

*

r) dr,

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the main results (5.10) and (5.12) can be rewritten as

(10. l a ) ( 10.1b)

Hence, in space domain the signal transformation in a shift-invariant system is governed by a (two-dimensional) convolution of the input signal with the point spread h ( r ) , whereas in the frequency domain an ordinary multiplication with the system function H ( R ) is involved. Next, (8.5) reveals that a modulator with modulation function m ( r ) is transformed by two adjacent Fourier transformers ’fJ(d),’fJ(-d) into a shift-invariant system with a system function

H ( R )= m ( t R ) .

(10.2)

If n o constraints are imposed upon m ( r ) , any desirable system function H ( R ) can obviously be realized, and herewith any point spread h(r). The situation changes when e.g. only amplitude modulation can be performed in the “Fourier plane”. Then m ( r ) is positive real and so is H(R). As the inverse Fourier transform of a positive function, the point spread h ( r ) is then subject to severe restrictions. We note that this formal problem is well known from communication theory where h ( r ) is replaced by the autocorrelation function, whose Fourier transform is the positive power spectrum. For the practical implications of the constraints under consideration we refer to the next section, where this problem is reconsidered, albeit in a different physical context. Another restriction of practical interest appears when m ( r ) = 0 for Irl> r,. Such a restriction is associated with diffraction-limited imaging, where the exit pupil with aperture radius r, plays the role of the modulator. With (10.2), H ( R ) then vanishes for IR(> kr,/d and we can speak about a low-pass filtering. The cases m ( r ) = 1 for Irl< ro as well as m ( r )= exp (ib(r)) were extensively discussed in 0 8.3, where we reached the conclusion that a varying b(r) due to spherical aberration errors in general broadens the point spread h(r) and so deteriorates the imaging properties. On the other hand, manipulation of ( m ( r ) (for lr(
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PRINCIPLES OF OPTICAL DATA-PROCESSING

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5 10

Fig. 10.1. About apodization of a diffraction-limited system.

inverse Fourier transform (corresponding to the “line spread”) and therefore exhibits no overshoot after transmission of a black-white jump. Also the width of the pertaining h,(x) (if properly defined) becomes smaller than that of h , ( x ) . For optimization of apodization procedures we further refer to PAPOULIS [1968a].

10.2. INCOHERENT ILLUMINATION

Like coherent light, there is another extreme case within the framework of partially coherent light, that is only realizable in an approximate manner: the incoherent light. What it distinguishes from its coherent counterpart is the property that the idealization degenerates after passage through linear systems, i.e. incoherent light becomes partially coherent after linear transformations. For incoherent illumination, we have a linear mapping of intensities. It is governed by (4.11) and will now be specialized for the shift-invariant system. With the new notations* p ( r J = lin(ri), S(rz, r2) = lout(r2), gzl(rz, r l ) = h(r2-r1),

we obtain I O u h )= lh(r)12*Mr)y

(10.3)

i.e. the output “intensity” is found as the convolution of the input “intensity” with the squared modulus of the point spread h(r). In a

* The reader is warned against the different dimensions of I , , and I,,,,, as was discussed in 54.2.

SHIFT-INVARIANT SYSTEMS

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27 1

comparison with (10.1) for coherent illumination, we immediately observe that only positive, real quantities are involved in the incoherent signal transformation. This fact has an important implication for the manner in which the frequency counterpart of (10.3) is usually formulated. As the Fourier transforms of positive functions assume their [1968]), they are advantagemaximum values at the origin (GOODMAN ously normalized with respect to that value. In our case, this yields

5

f ( R )= I(r) exp (-iR

*

r) d r / j I ( r ) dr,

(h(r)I2exp (-iR

*

r) dr/]lh(r)12 dr,

(10.4)

(10.5)

with the indices “in” and “out” to be added to f(R) and I ( r ) . Clearly, f ( R )and R ( R ) assume unity values for R = 0, which cannot be exceeded for any R # 0. With these normalized spectral functions (10.3) is transformed into i0”,(R) = H(R)iin(R).

(10.6)

While fin(R)and fout(R)are commonly known as the normalized input and output spectra, R ( R ) is usually referred to as the complex optical transfer function (OTF). Its squared modulus Ik(R)12 is called the modulation transfer function (MTF).Like the system functions of time-domain filters, f i ( R ) can be experimentally determined through excitation by harmonic functions (“eigenfunctions”). For that purpose, sinusoidal test patterns” are in use for which we can easily derive that the ratio of the relative fluctuations (I,,,=- I~”)/(ImaX I ~ , )at the output and the input directly equals Ifi(R)I, while the shift between the location of the maxima determines arg fi(R). The spatial frequency of the sine pattern equals the modulus of R and the normal to the equiphase lines determines its direction. Since H(R) is the normalized Fourier transform of lh(r)I2= h(r)h*(r)it can also be written as

+

E.T(R)

*

= {WR) H*(-R))I{H(R)* H*(-R)),=,,

(10.7)

which is the normalized autocorrelation of H(R)or (except for the scaling constant) the normalized autocorrelation of the modulation function m(r)

* On account of 1 2 0 , these test patterns contain a constant bias term besides the desired sinusoidal function.

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in the “Fourier plane”. From (10.7) we also note the Hermitian property A ( - R ) = fi*(R). From the foregoing we conclude that not every (Hermitian) function of R is realizable as an OTF f i ( R ) .If and only if it has a positive Fourier transform or (what amounts t o the same) if it can be written as an autocorrelation, such a function is permitted. From the viewpoint of filter synthesis this rules out a great number of interesting frequency responses. If a function f i ( R )is permitted, it can be written in the form (10.7) in an infinite number of ways. This can be easily seen in the space domain where the transition lhI2+ h is associated with an arbitrary phase factor”. In certain (but not all) cases this degree of freedom can be exploited to make H ( R ) positive real. If that succeeds, Fourier filtering can be performed with a pure amplitude modulator. An important “natural” shift-invariant system is formed by diffractionlimited imaging. As discussed in previous sections, we then have m ( r )= 0 for Irl> r, or, what amounts t o the same, H ( R )= 0 for (RI>R,. Autocorrelation according to (10.7) then broadens the frequency characteristics and makes that the low-pass cut-off frequency pertaining to the incoherent system function f i ( R )equals 2R,. At a first glance this seems to improve the imaging quality and this the more, as the effects of autocorrelation resemble that of apodization (cf. Fig. 10.1; the autocorrelation of a rectangle becomes a triangle). Moreover, the positiveness of the incoherent point spread prevents overshoot. A thorough investigation (GOODMAN [1968]) reveals, however, that the belief that incoherent imaging in general is superior to coherent imaging, has insufficient foundation. 10.3. LOW-PASS FILTERS

The problems around diffraction-limited imaging will now be generalized to more general low-pass filtering. On the one hand this problem is important enough to deserve a separate discussion and on the other hand it illustrates general principles of incoherent filtering. Our [1962]. For treatment follows the ingenious lines developed by LUKOSZ sake of brevity, we confine ourselves to one-dimensional low-pass filters, where the vectors r , R are replaced by x (position) and X (spatial frequency). The cut-off frequency of the filter is denoted by X , so that

* WALTHER [1963] has shown that this arbitrariness is strongly restricted if h is bandlimited. This occurs when the modulation function m ( r ) in the Fourier plane has finite “dimensions”.

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fi(X)=O for IXl>X,. Moreover, Ih(x)l' is assumed to be even which implies realness of fi(X). With these premises in mind we now derive an upper bound for fi(X) in the pass-band (XI< X,. One of the features of incoherent filtering is that with any positive input signal a positive output signal is associated. We shall choose a particularly simple input signal which, in fact, contains not more than one free parameter. The requirement of a positive output signal then yields a necessary condition for R(X). As illustrated in Fig. 10.2 (left) the input signal lin(x) is given by the function 1 + C cos X,x which is sampled at the equidistant points xk = 2rk/NXo with k = 0, *l, +2, . . . and N an integer to be specified in due course. Without sampling, the spectrum of the above signal would consist of three lines at the frequencies 0, *X,, but due to sampling we obtain the periodic repetition of this line triplet, with a period NX, in the frequency domain (PAPOULIS [1962]). Assume now that the cut-off frequency X, is below (N-l)Xo. Then only the central triplet is transmitted by the filter, attended by a modification of the line intensities. While the ratio of the line heights at X = *Xo and at X = 0 originally equals C/2,this becomes after passage through the filter fi(X0)C/2. In time domain we obtain at the output lout(x)= IDC,out X (1 + Cfi(X,) cos X,x) which remains positive if IR(X,)l 5 1/C. Under which condition is the input signal positive? We consider the marginal condition, viz. lin?O where the equality sign holds for some x. This amounts to the requirement that the smallest sampling value be zero. For N even, this yields C = 1, but for N odd we obtain C cos (.rrlN)= 1. The latter case (N even yields a trivial result) leads after combination with 2 0 to the above condition for Iout

Ifi(xo)l 5 cos (dNL

(10.8)

which is valid for X c 5 ( N - l)Xo or

X,ZX,/(N- 1).

(10.9)

At a first glance (10.8) and (10.9) seem to apply only for N odd. That this result is also valid for N even, can, however, be easily concluded when the sample points are shifted by half a sampling distance (from 2.rrk/NXo to 2 4 k +i)/NX,) so that the minimum sampling value again becomes zero for C cos (r/N) = 1. Thus, for N = 3 , 4 , 5 , . . . corresponding to X,, 2 XJ2, X, 2 X,/3, X, 2 X,/4,.. . we have ~fi~~cos(r/3)=~,)fi~~cos(~/4)=

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Fig. 10.2. A suitable input signal li,(x) and its Fourier transform for N = 7.

cos (.rr/5)= 0.81, . . . , respectively. Combining these inequalities then yields the “Lukosz bound’’ as depicted in Fig. 10.3. Only transmission characteristics Ifi(X)) which are below this bound are permitted. Of course, this condition is only necessary, i.e. not every function satisfying it can be realized. In fact, realizability can only be checked via the sign of the inverse Fourier transform of fi(X). From a synthesis viewpoint, permitted functions fi(X) can, moreover, always be constructed through autocorrelation of an arbitrary function H(X), according to (10.7). A fundamental, permitted function fi(X), which is obtained by autocorrelation of a rectangular H(X) is the triangular characteristic fitri(X), as inserted in Fig. 10.3. Apart from scaling constants, its inverse Fourier transform equals (sin xlx)’ which becomes zero in an infinite number of points, viz. x = n r ( n # 0). Since any deviation from the triangular characteristic threatens to destroy the positiveness of the inverse Fourier transform, we are justified to call the triangular function marginally permitted. It is now logical to ask which functions Afi(X) superimposed upon the triangular function fitri(X)are absolutely forbidden. The answer is that the inverse Fourier transform of Afi(X) must not be negative, where that of &(X) vanishes; otherwise the sum would be locally negative. So we have to solve a nice problem of sampling theory, viz. to look for those functions Afi(X) whose inverse Fourier transforms have nonnegative I

Fig. 10.3. Lukosz bound for the system function of a low-pass filter.

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sampling values at the equidistant zeros of (sin x/x)*. As we demonstrate below one readily reaches the conclusion that AR(X) + AR(XC-X), i.e. the OTF plus its mirrored version, must be negative for all X. In other words, if at some frequency X, the total function R ( X ) exceeds fit"(X), it has to remain below fitriat the mirror frequency (X,-X) by at least the same amount! In the mean, G(X)has to be excesses in some regions have to be (over-) compensmaller than fitri(X); sated in others. This implies that the average value of R ( X ) cannot exceed 4 ( = t h e average value of Rtri(X)).Notice, however, that all these restrictions are only necessary; fulfilment does not guarantee realizability ! We sketch the proof of the above statement. First we notice that sampling in the x-domain corresponds to periodic repetition in the X-domain; the period turns out to become X,. As Afi(X) extends from -X, to +X,, we get an overlapping due to this repetition process such that Afi(X,- X) has to be superimposed upon Afi(X). Positive sampling values now imply that the Fourier series expansion m

Afi(X)+Afi(X,-X)=c,+

1 C, cos(2n.rrXIXc)

n=l

contains positive coefficients c,, cz, c 3 . . . (c, may become negative because at x = 0, (sin XIX)' does not vanish). This again implies that the above function attains its maximum values at X = 0 and X = X,, where it becomes c,+ c, + c2+ c3+ * * . On the other hand it must vanish there, because we must not deviate from &(X) at these end points. Hence it is negative in the interior O < X < x , Q.E.D.

0 11. Related Topics In the present article, general optical data-processing systems have been studied under various aspects. In retrospect, however, we are aware that our description is far from complete. As an exhaustive discussion of the missing aspects would fill at least another article, we cannot do more than mention their existence. Optical systems were hitherto described in wave-theoretical terms only. Another possible description, albeit of restricted validity, involves geometric-optical principles. This applies particularly t o 2%-systems, where our three characteristic parameters (viz. the elements of one of the equivalent circuits) find their counterpart in the elements of the 2 x 2 ray

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§ 11

transformation matrix (DESCHAMPS [1972], O’NEILL [1963]). As its determinant equals unity, this matrix likewise contains three free parameters. An agreeable property of that description is that system cascading amounts to simple matrix multiplication. The fourth system parameter is the price to be paid for this computational advantage. Recently the Wigner distribution has proved to be a useful tool in [1978a1, WOLF [19781). This simuloptical system analysis (BASTIAANS taneous space-frequency formalism* is based upon wave theory, t o be sure, but its results are most conveniently interpreted in geometric-optical terms. In fact, the Wigner distribution constitutes a modern approach to derive geometrical optics from wave optics in the limit of extremely small wavelengths (BASTIAANS [1979b1). Quantum electrodynamics and light propagation in terms of photons likewise belong to the vocabulary of modern optics. These descriptions have found widespread application in the theory of partial coherence (PERINA [1972]), with the remarkable result that the final conclusions are in formal agreement with those of classical wave theory. While the present article deals with deterministic signals and systems, stochastic aspects can also be taken into account. Noise, like the granularity of a photographic film, has to be considered as a stochastic signal, but also test patterns can be treated as such (O’NEILL [1963]). Speckle is due to unpredictable system fluctuations (dust particles etc.). Finally, grounded glass is an example of a special optical system (a modulator) that has a meaningful description only on a stochastic basis. Rotating grounded glass is a convenient means to convert coherent or partially coherent light into almost incoherent light. In our treatment optical signals were represented throughout by continuous, two-dimensional functions, i.e. to every position r = ( x , y) a signal value was assigned. For several problems it is more appropriate to use a discrete-space description, in which the signals are represented by two-dimensional arrays of numbers. Then the fundamental superposition integral (3.4) is converted into a sum. If, for the moment being, only one-dimensional signals are taken into account, we have a transformation

* Also without the Wigner formalism and occasionally without proper awareness combined space-frequency argumentations are in common use. As an example, consider a diffraction grating of finite dimensions illuminated by a plane wave. The field behind the grating is then usually decomposed into diffracted “bundles” or “beams”, which e.g. are “lost” if they travel past the following lens. The notion of a beam with a direction (frequency) and a position (space) is such a mixed concept alien to rigorous Fourier theory. Also the well-known space-bandwidth product (LOHMANN [1967]) belongs t o this category.

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of an input “vector” into an output “vector”, for which the system properties can be represented by a matrix. Cascading then finds expressing in matrix multiplication, system inversion means matrix inversion. Losslessness involves a unitary matrix, and symmetry a symmetrical matrix. The conservation theorems of 0 5.2 are then interpretable in terms of well-known matrix properties. The practical significance of a discrete-space representation is two-fold: First, halftone realizations of continuous-tone objects are inherently discrete. When, moreover, the measurement of the output signal is performed only at the corresponding array points, we have, in fact, an entirely discrete signal-and-system representation. In a second case discreteness is less tangible. We mean the possible representation of a band-limited signal by its sampling values. If the system (e.g., as in the shift-invariant case) does not produce additional higher frequencies, the input and output signal can be likewise sampled, and we obtain a fictitious discrete system”. A complete optical system theory has to reckon with the vectorial nature of light. This implies that the pertinent catalogue of optical components needs to comprise polarizers, quarter-wavelength plates etc., which manipulate the state of polarization of the incoming light. At present, it seems that only components without lateral parameter variations have been studied, which transform normally incident plane waves in normally outgoing plane waves. Fundamental work in this direction has been done by JONES [1956]. An important aspect of optical system theory is that of synthesis. Whereas we have extensively discussed the realization of shift-invariant systems (cf. 0 lo), that of shift-variant systems was hardly touched upon. In fact, at present no general synthesis procedure is available?, and the question arises whether it would be very useful (FRANCOIS and CARLSON [1979]). The variety of these systems is so vast, that every subclass asks for a specific treatment. We only mention systems which involve a geometric distortion and which have been studied by BRYNGDAHL [1974]. In this context we like to mention that not all practical systems fit into our approach which is essentially based upon cascade connections of simple *Also series expansions of the input and output signal in terms of given sets of (orthogonal) functions lead to discretizations of optical systems. t This assertion applies only to two-dimensional systems with point spreads g2,(r2,r l ) . For one-dimensional systems the pertinent point spread g,,(x,, xl) can be relatively simply realized with two-dimensional means (GOODMAN[1977]).

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components. This approach excludes e.g. volume holograms, but also systems containing mirrors and beam-splitters (LOHMANN and RHODES [ 19781). Arrangements using these components add new possibilities to the solution of a number of synthesis problems. Concerning synthesis, we have to deal with two basic questions. First, are there any fundamental restrictions to be imposed upon the point spread of an optical system? Certainly, the familiar causality constraint of time-domain filtering has no spatial counterpart. For incoherent systems, we have the restriction that all space functions including the point spread, are real and nonnegatiue. On the other hand, there is sufficient evidence for the belief, that coherent processing is unrestricted in the sense that any well-behaving complex point spread function can, in principle, be realized. Secondly, which systems (i.e. which point spreads) are actually required? It seems that the main interest is directed towards the easily realizable shift-invariant systems. Besides the systems producing geometric distortions as discussed above and some very special systems (like the Mellin transformer required for pattern recognition (GOODMAN [1977])) the class of the weakly shift-variant (i.e. the almost shift-invariant) systems will probably attract the most attention. For further information cf. GOODMAN [1977], CASASENT and PSALTIS [1978]. A comprehensive study of optical systems has also to include a number of (semi-) technological aspects. We only mention the practical realization of a modulator. For m ( r ) = Im(r)l< 1 we have an amplitude modulator with a positive real modulation function which, due to passivity, cannot exceed unity. A photographic realization of such a modulator with continuous shades of gray meets considerable difficulties due to the nonlinear characteristics of the photographic emulsion. With a highresolution film one can simulate “gray” by a “half-tone’’ technique that utilizes only the two levels m = 0 (black) and m = 1 (white). The pertinent pattern which can be drawn and reproduced with high precision can be optimized in the sense that after proper low-pass filtering exactly the required continuous function appears (BASTIAANS [1978bl)). For circularly symmetric functions another two-level simulation applies making use of concentric rings (WIERSMA [1978]). More severe problems are envisaged if a phase modulator has to be realized. In the present state of the art, bleaching and etching processes have not yet led to satisfactory, well-reproducible results (VANDER LUGT [1974]). Due to the ingenious method of V A N DER LUGT[1964] phase

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variations can, however, be modulated upon a high-frequency carrier thereby transforming the desired phase modulator into an amplitude modulator. Using holographic methods, complex light distributions* can be directly recorded on photographic film (provided that the linear part of its characteristics is used) and thereby transformed into modulation functions m ( r ) . Another way which ultimately leads to two-level modand ulators, is that of computer holography as proposed by LOHMANN PARIS[1967]. We finally note that a lens as a special phase modulator can be simulated by a Fresnel-zone plate which consists of concentric transparent rings (PAPOULIS [1968a]).

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