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Optica Acta: International Journal of Optics

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Edge-ringing in Partially Coherent Imaging

Eric C. Kintner & Richard M. Sillitto

To cite this article: Eric C. Kintner & Richard M. Sillitto (1977) Edge-ringing in Partially CoherentImaging, Optica Acta: International Journal of Optics, 24:5, 591-605, DOI: 10.1080/713819597

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OPTICA ACTA, 1977, VOL . 24, NO. 5, 591-605

Edge-ringing in partially coherent imaging

ERIC C. KINTNERt and RICHARD M . SILLITTODepartment of Physics, University of Edinburgh

(Received 13 September 1976)

Abstract. A simple condition is shown to be necessary and sufficient tosuppress edge-ringing in optical imaging . This criterion is valid in the presenceof aberrations and apodization, and for all cases of (spatially stationary) partiallycoherent illumination . The edge-ringing tendencies of an optical system maybe assessed through the use of two new performance functions which form aFourier-transform pair . One of these performance functions is related to the` transmission cross-coefficient 'which appears in the theory of partially coherentimaging . In the coherent limit this performance function reduces to theamplitude transfer function (pupil function), and in the incoherent limit itreduces to the Optical Transfer Function (OTF) . The use of this performancefunction for the general assessment of partially coherent imaging systems issuggested .

1. IntroductionThe Optical Transfer Function (OTF) has become familiar as a performance

indicator for optical systems imaging with incoherent illumination . It is wellknown that the development of a similar indicator for systems imaging withpartially coherent illumination is frustrated by the non-linearity inherent in thisimaging process . Although partially coherent imaging is not linear with respect tothe intensity, it is linear with respect to the mutual intensity function F(xl, X2 ; 0),and it is possible to calculate the transfer of this function through an opticalsystem . However, the mutual intensity is neither readily observable nor in-tuitively meaningful, because (unlike the intensity) it is a function of all pairs ofpoints in the object rather than a function of individual points . In this situationit seems more useful to develop a performance indicator for partially coherentimaging which concentrates on a particular characteristic of the imaging process .

One undesirable feature of imaging with coherent or partially coherent illumi-nation is the phenomenon of edge-ringing, characterized by the appearance inthe image of spurious fringes near the edges of opaque objects . These fringesare due to the sharp cut-off of the amplitude transfer function (the pupil function)of an unapodized optical system, which gives rise to oscillations in the (inverse)Fourier transform (the amplitude point-spread function) . An analogous effectoccurs in communications theory when filters with very steep cut-offs are usedto transmit rapidly varying signals . Each of these effects is an example of thewell-known Gibbs phenomenon ([1] p. 209) . Photographs illustrating edge-ringing have been published by Considine [2] . One of his photographs (hisfigure 2) showed the apparent similarity between edge-ringing and Fresneldiffraction at a straight-edge ; this relationship has been discussed elsewhere[3 a] . In another of Considine's photographs (his figure 5), the resolution of a

t Present address : U.S. National Bureau of Standards, Washington, D.C. 20234, U .S.A .

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transilluminated bar chart was degraded by a factor of 3 due to edge-ringingwhen incoherent illumination was replaced by coherent illumination .

Considine pointed out that edge-ringing in coherent imaging could be sup-pressed if the optical system were apodized to make the cut-off of the amplitudetransfer function sufficiently gradual . Besides improving the resolution of asimple bar chart, such apodization would obviate the misinterpretation of complexobjects caused by the appearance of edge-ringing in the image . It should berecognized, however, that apodization would not linearize the optical imagingprocess, nor would it obviate other problems associated with coherent or partiallycoherent imaging, such as edge-shifting or the effects caused by phase shifts in theobject . Furthermore, edge-ringing tends to appear particularly in the imagingof objects with high contrast, and apodization may cause an unwelcome loss ofresolution and image brightness when objects of low contrast are imaged .

Smith [4] (see also Leaver and Smith [5]) has described an objective criterionfor the suppression of edge-ringing in unaberrated coherent imaging . He pointedout that if the amplitude transfer function of an optical system (i .e . the pupilfunction) can be expressed as an autocorrelation of any arbitrary function, thenthe amplitude point-spread function (the inverse Fourier transform of the pupilfunction) is everywhere real and non-negative ; this is sufficient to suppress edge-ringing . To illustrate this principle, it is instructive to note that the OpticalTransfer Function (OTF), which is the relevant transfer function for incoherentimaging, is always an autocorrelation (the autocorrelation of the pupil function) .The inverse Fourier transform of the OTF, the Point- Spread Function, is thusalways real and non-negative, and the spurious oscillations in intensity whichconstitute edge-ringing do not occur .

In this paper it is shown that the edge-ringing tendencies of an optical systemmay be assessed through the use of two new performance functions which form aFourier-transform pair . These two functions are introduced in § 2 using asimplified one-dimensional model . A simple condition is shown to be necessaryand sufficient to suppress edge-ringing . This criterion is valid in the presence ofaberrations and apodization, and for all cases of (stationary) partially coherentillumination . The behaviour of the two performance functions is illustrated in§ 3 . The arguments of the preceding sections are extended to two-dimensionalimaging systems in §4 . The relevance of the performance function in the pupilplane as a representative performance indicator for the general assessment ofpartially coherent imaging systems is discussed in § 5 .

2 . Theory of the edge-ringing performance functionsIn discussing partially coherent imaging, it is convenient to adopt a dimension-

less form of the notation which is employed by Born and Wolf [6, Chapter 10] .Let the one-dimensional pupil function (amplitude transfer function) be specifiedby

~( )exp {lira(D(e)} X

1,

0

j I>1 .

The dimensionless pupil-plane coordinate 6 is defined by the relation

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where ~ is the geometrical coordinate and a is the radius of the pupil. The realfunction t($) represents the aberrations of the pupil, and the real function d(e)can represent apodization . Where the Fraunhofer approximation is valid, theamplitude point-spread function associated with this pupil function is given bythe (inverse) Fourier transform of the pupil function

GOK(x) = f ir(e)exp {-27riex}de .

(2.2)-X

The dimensionless image-plane coordinate x is related to the geometricalcoordinate x through the equation

ax=~~x,

where f is the distance from the pupil plane to the image plane and A is the wave-length of the (quasi-monochromatic) illumination .

In the object plane (which is conjugate with the image plane), the functionF(x) describes the object, and the function J(x) describes the mutual intensityof the (spatially stationary) partially coherent illumination . For each of thesefunctions, an associated Fourier transform may be defined in the pupil plane

-CO

Hopkins [7] has shown that the real function /(6) may be identified with afictitious ' effective source ' which gives rise to the partial coherence in the objectplane through the Van Cittert-Zernike theorem .

` Edge-ringing ' implies the appearance of oscillations in the image intensitydistribution at a straight edge . Conversely, edge-ringing is absent if the imageintensity distribution at a straight edge rises monotonically from the dark regionto the bright region . The image intensity distribution in a partially coherentimaging system can be expressed as

I(x)= f f(e)IA(x,e)l 2de,-00

where00

(2.5 a)

A(x,6)= f K(x')F(x-x')exp{2iriex'}dx' .

(2.5b)-00

(Expression (2.5) was first discussed by Hopkins [7] .) The slope of the intensitydistribution is given by the first derivative of this expression

dxI(x) = 2 f /(6) Re {A*(x, e)d- A(x, 6)} de .

(2.6)

(The asterisk denotes complex conjugation .)

00F(6)= I F(x) exp {2aiix}dx, (2.3)

-00

00f(6) = f J(x) exp {2lriix} dx . (2.4)

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1

Let a straight-edge object be represented by the Heaviside step function

1 x>0,F+ (x) _

(2.7)0 x<0 .

(The subscript + indicates that the positive half-plane is bright . The reversedsituation, in which the negative half-plane is bright, will be considered shortly.)Then

xA +(x, 6) = f K(x') exp {27riex' } dx',

(2.8)- W

so

W- A+(x, e) = K(x) exp {27riex} .

(2.9)

Inserting these expressions into equation (2.6) gives

dx I+(x)2 f /(6) Re

{f K*(x')K(x) exp {2aii(x x')} dx'} de . (2.10)

Now, since /(6) is real by definition, and hence J( - x) =J* (x), it follows that

dx I+(x) - 2Re {K(x)JK*(x') [S/()exP

6{ 27rii(x' x) } de l dx' }

I=2 Re 1

-00

K(x) I K*(x')J(x' - x) dx'

=2Re{K(x)C JGOl

GO

=2RefK(x) C f K(x-x')J(x')dx'j .

(2.11)J0

(In the first step, the inversion of equation (2.4) is invoked, and in the last step,the variable of integration is changed from x' to x - x' .) The requirement thatedge-ringing be suppressed in this case implies that this derivative must be non-negative for all x.

A similar argument may be applied to the reversed Heaviside step function

0 x>0,F_(x)=

(2.12)1 x<0,

to give for the slope of the resulting image distribution

dx_

-00

I-(x)=-2Re{K(x) [K(x_f

x ')J(x')dx']*} .

(2.13)

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For edge-ringing to be absent in this case, this derivative must be everywhere

(Note that this addition is a mathematical operation suggested by the definitionof Q(x) ; it does not correspond to any physical experiment .) For edge-ringingto be suppressed, that is, for the conditions on each of equations (2 .14) and (2 .15)to be satisfied, it is evidently necessary that

Re {Q (x) } > 0 .

(2.18)

It will now be demonstrated that satisfying condition (2 .18) is sufficient tosuppress edge-ringing ; that is, if condition (2 .18) is satisfied, then the derivatives

where since= sin 6/6 . The functions I+(x) and I_(x) are plotted in figure 1 (a),and the function [I+(x) -I_(x)] is plotted in figure 1 (b) . The edge-ringingoscillations are very much more pronounced on the bright side of the edge than onthe dark side. (In Fresnel diffraction, there are no oscillations on the dark side ;however, the present physical model is based on Fraunhofer diffraction . Theoscillations on the dark side can be observed in a photograph of edge-ringingpublished by Leaver and Smith [5] .) If the pupil function is apodized to reduceedge-ringing, it can be expected that the oscillations on the dark side, which arein any case very small, will become negligible . Therefore, as the function Q(x)approaches a form satisfying condition (2 .18), the residual oscillations to the rightof the origin in figure 1 (b) are, for practical purposes, due entirely to the behaviourof the function I+(x) ; those on the left are due entirely to the function I(x).

in each of equations (2 .14) and (2 .15) are everywhere non-negative . Considerthe behaviour of an unaberrated and unapodized pupil function ;

1

161 51,_*'M _

0

iei>1 .(2.19)

In the limiting case of coherent illumination, one can show that

I±(x) - l ± 1 2f since deJ

2,

(2.20)7T 0

non-positive. Therefore, if edge-ringing is to be absent in both cases, the twoquantities

dx I+ (x)=2Re{K(x)C f K(x-x')J(x')dx']*} (2.14)

andJ0

-71_(x)=2 Re{K(x) [JK(x_x')J(x')dx'] * }

(2.15)

must be non-negative for all x

l

.

-00

Consider now a new function Q(x) defined by

Q(x)- K(x)[SK(x -x')J(x')dx']* . (2.16)-00

Clearly, adding equations (2 .14) and (2 .15) gives

dx [I+(x)-I_(x)]=2 Re {Q(x) } .

(2.17)

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q

x

(b)

Figure 1 . (a) Intensity distributions in the images of forward and reversed coherentlyilluminated straight edges . (b) A combination of these functions (see text) .

(This point has been discussed at greater length by Kintner [3 b] .) Consequently,condition (2 .18) implies the two separate conditions

TXI+(x) ~ 0 and -

dxI_(x) ~ 0 ;

thus condition (2 .18) is both necessary and sufficient for the suppression of edge-ringing . It is interesting that the behaviour of Im {Q(x)} has no bearing on theedge-ringing properties of an optical system .

When the illumination is either fully coherent or fully incoherent, the functionQ(x) reduces to simple and familiar forms . In the coherent limit, the mutualintensity J(x) is effectively constant, so

Q,,(x)=K(x) f K*(x')Io dx'=constant K(x) ;

(2.21)-00

that is, Q 0(x) corresponds to the amplitude point-spread function . In theincoherent limit, J(x) corresponds to a delta function, so

MQ,(x)=K(x) I K*(x')Io8(x')dx'=Io1K(x)12 ;

(2.22)

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= f

oo

Edge-ringing in partially coherent imaging

597

that is, Qi(x) is proportional to the Point-Spread Function (PSF) . Intuitively,therefore, Q(x) has the superficial appearance of a spread function characterizingan optical system .

Further insight into the behaviour of Q(x) may be gained by investigating itsFourier transform in the pupil plane

Q(6) = f Q(x) exp {2lriex} dx .-00

This transform can be obtained from equation (2.16) by using modifications ofequations (2.2) and (2.4), and changing variables where appropriate

ooQ(e) = f f K(x)K*(x - x')J*(x') exp {27riex}dx'dx

Cff fM_*_ (e+6')exp {-27ri(e+6 ')x}d(e+e')J K*(x-x')J*(x')

= f ~( )

( + ') *( ')d ' .-oo

In the coherent limit, the ' effective source ' /() corresponds to a deltafunction, so

Qje)= I Io8( ')

( + ') '*( ')d '=constant .

(),

(2.25)-00

that is, Q(e) is proportional to the amplitude transfer function (pupil function) .In the coherent limit, f(6) is constant, so

Q#)=Io f '( + ') '*( ') d '=Io OTF( ) ;-Co

x exp {2-,riix } dx' dx

f[J*(x')exp {-2lrie'x'}dx'J

(V+e')

x[-co[IK*(x_x')exp { - 2rrie'(x-x')}dxl dd'

(2.24)

(2.26)

thus Qj(e) corresponds to the Optical Transfer Function . Therefore, notsurprisingly, Q(6) has the superficial appearance of a transfer function for anoptical system .

As is shown in Born and Wolf [6, § 10.5 .3], the behaviour of a partially coherentimaging system is completely characterized by the transmission cross-coefficient,which describes the response of the system in terms of the mutual intensity. Fora one-dimensional system, this function is defined as

'T(e1 ;62)= f fW)-*'(61+0 '*(62+0 dc' .

(2.27)- Co

A comparison of equation (2.24) with equation (2.27) shows that Q(e) is a particularsubset of g (e1 ; 62)

Q(e) _67-(e

(2.28)

O.A .

2 R

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An interesting corollary of the edge-ringing condition (2.18) is obtained fromthe definition of Q(6) in equation (2.24) . Let /(e) represent a uniform sourceof finite size, i .e .

constant 161,6e" ,

lei > 6es-

(2.29)0

Since .*(6) vanishes for 161 > 1, the integral in equation (2.24) is unaffected byany change in the source size within the range 1 6es < oo . Since edge-ringingis known to vanish in the incoherent limit Ses--~co, it follows that edge-ringingmust vanish whenever ees > 1, even though the optical system may remainpartially coherent and non-linear . It is often suggested that the presence ofedge-ringing is a diagnostic symptom of partially coherent imaging. The aboveresult shows that where edge-ringing appears the system must be partiallycoherent and non-linear, but that the converse is not true : the absence of edge-ringing does not guarantee a linear imaging process .

Smith [4 a] showed that edge-ringing in coherent imaging would not occur ifthe pupil function _*"(6) could be expressed as an autocorrelation . This impliesthat the amplitude point-spread function K(x), the inverse Fourier transform ofi(6), is real and non-negative . The above arguments show that Smith'scriterion is unnecessarily restrictive because it implies the vanishing of Im {Q(x)}and therefore permits no aberrations in the imaging process. The new edge-ringing performance function, with its associated edge-ringing criterion, ac-curately gauges the edge-ringing properties of an optical system in the presenceof aberrations and apodization, and for all cases of partially coherent illumination .

3 . Illustrations of the edge-ringing performance functionsFigures 2-7 illustrate the behaviour of the function Q(x) and its Fourier

transform Q(e) for several representative examples of one-dimensional opticalsystems. These systems may incorporate apodization, as well as defocusing orspherical aberration . Thus the pupil function takes either of two forms

(1-blel)exp{2-iae 2} lel,<1,1"(0 =

(Defocusing)

(3 .1 a)0

,>1,

or

(1 -blel) exp {2iriae 4 } lel S 1,_

(Spherical( )

0

161> 1

Aberration)

(3.1 b)

The parameter a specifies the amount of aberration and the parameter b specifiesthe degree of apodization. (The triangular function which results when b = 1and a = 0 is, of course, the autocorrelation of a rectangle function .) The effectivesource function is simply

,f(S)

=

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Edge-ringing in partially coherent imaging

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~'s= 0

0.5

0

I - I

0

1.0

2.0

x~' 0

IIIIIIE -

1 .0

2 .0

3.0

x

-1 .0 -

r1.0

2.0

3.0

-1 .0

Figure 2. The performance functions Q( st) and Q(x) for several representative one-dimensional optical systems . No aberration, unapodized : a=0, b=0.

1 .0-

X -

'Cr 0

I

I

E - 1 .0

x

x

2 .0

b&"3 .0

1 .0

2.0

3.0

x

- 1 .0

Figure 3 . The performance functions Q(6) and Q(x) for several representative one-dimensional optical systems . No aberration, apodized : a=0, b=1 .

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1 .0

0I.0

0 -0.20.5

t

2.0

-1 .0

Figure 4 . The performance functions Q(6) and Q(x) for several representative one-dimensional optical systems . Defocused, unapodized : a=0-75, b=O .

M

2.0 3.0

Figure 5 . The performance functions Q(6) and Q(x) for several representative one-dimensional optical systems . Defocused, apodized : a-0-75, b=1.

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x!9` 00.2

0.5 .

1 .0

2.0

1 .0

Figure 6 . The performance functions Q(~) and Q(x) for several representative one-dimensional optical systems . Spherical aberration, unapodized : a=0 .75, b=0 .

C

x

2 .0 3.0

-1 .0

Figure 7 . The performance functions Q(~) and Q(x) for several representative one-dimensional optical systems . Spherical aberration, apodized: a=0.75, b=1 .

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In these examples, the function Q(6) was calculated by numerical integrationfrom equation (2.24), here re-written as

00Q(e) = C.. I r'(6')Y(e+fl Y*(6') de' .

(3 .3)-X

The function Q(x) was then obtained by a numerical Fourier transformation .The normalization constant C,z was chosen so that Q(0) __ 1 . For convenience inillustrating these examples, Q(x) was arbitrarily re-normalized so that its maximumabsolute value is equal to 1, i .e. IQ(x)I ma,X -1 .

Each of the six sets of plots in figures 2-7 represents one hypothetical pupilfunction with specified aberration and apodization . For each pupil function,four different values of the coherence parameter 6es are represented by the fourdifferent curves . The upper plot shows the modulus of the function QW .In the coherent limit this is the modulus of the pupil function, and in the in-coherent limit it is equivalent to the Modulation Transfer Function (MTF) .The two lower plots show the imaginary and real parts of the function Q(x) .Again, in the coherent limit Q(x) reduces to the amplitude point-spread function,and in the incoherent limit it corresponds to the Point-Spread Function (PSF)which is necessarily real and non-negative . (The PSF is equivalent to theLine-Spread Function in this one-dimensional model .)

By comparing these plots, one may readily see some of the features whichhave been predicted . In the first set (figure 2), the transfer function in thecoherent limit (the pupil function) has a sharp cut-off which gives rise to negativeexcursions in the real part of Q(x), that is, edge-ringing . On the other hand,the incoherent transfer function (the OTF) decreases smoothly, and as requiredthe real part of Q(x) is non-negative . Two other curves, representing twodegrees of partial coherence, are also plotted . These seem to indicate thatedge-ringing is not significant when S es > 0 .5 .

Apodization sufficient to satisfy Smith's criterion in the coherent limit hasbeen included in an unaberrated pupil function to produce the second set of plots(figure 3) . The apodization has reduced the high frequency response, but inreturn edge-ringing has been entirely eliminated .

In the third set of plots (figure 4), some defocusing has been introduced in anunapodized pupil function . Though the aberration does not affect the modulusof the pupil function, it clearly reduces the high frequency response of the systemin the incoherent limit . This implies as a consequence the broadening of thecentral maximum in the function Q(x), in this case the PSF . Away from theincoherent limit, the aberration has led to an increase in the size of the negativeexcursions of Re {Q(x)l ; this signifies increased edge-ringing effects . Also,Im {Q(x)} no longer vanishes in all cases, as it had before .

The aberration and apodization which were considered separately in thepreceding sets of plots are combined in the next set of plots (figure 5) . Theseshow that the apodization which suffices to suppress edge-ringing for an unaber-rated pupil is no longer sufficient for the defocused pupil . Nevertheless, edge-ringing (as evidenced by the negative excursions of Re {Q(x)1) has been reducedconsiderably .

To show the effects of a different type of aberration, similar calculations werecarried out using a pupil function with spherical aberration . The fifth set ofplots (figure 6) shows that without apodization this aberration also aggravates

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and

00A(x,y ;e,,7)= If K(x',y')F(x-x',y-y')exp{2iri(ex'+-qy')}dx'dy' . (4 .1 b)

Let the coordinate system be defined so that a straight-edge object lies alongthe y-axis. (Note that this implies a specific orientation of the straight edge in

Edge-ringing in partially coherent imaging

603

the problem of edge-ringing . However, when apodization is included (figure 7),edge-ringing appears to be reduced more effectively than in the case of the de-focused pupil . The greater effectiveness of apodization in this situation is easilyexplained . Compared with defocusing, spherical aberration tends to distort thewavefront most strongly near the periphery of the pupil, and it is this region whichis most affected by the apodization illustrated here . This suggests that an opticaldesigner attempting to reduce the effects of edge-ringing through the use ofapodization should be relatively more tolerant of higher-order aberrations .

The images of square-wave test patterns have also been computed for each ofthe optical systems described above [3 b], and these plots confirm the remarksalready made .

4 . The edge-ringing performance functions in two dimensionsThe arguments of § 2 are easily extended to two-dimensional optical systems .

In two dimensions, the imaging equation (2.5) becomes

where

I(x,y) = If /(e,'7)JA(x,Y ;e,,))I2ded,1, (4.1 a)

A similar argument holds for the derivative of I_(x, y) .Now let the two-dimensional extensions of the functions Q(x) and Q(e) be

defined by

Q(x,y)=-K(x,y)C f f K(x-x',y-y')J(x',y')dx'dy']*

(4.5)C

oo

Q(e,_7)== 11 /(e','7') (e +661,,q+,,') x'*( ','q')dd'dq' .

(4.6)

the object plane .) Then the object function is, following equation (2.7),

1 1 x>0,F+(x,Y) = (4.2)

0 x<0 .Thus

m x

A+(x,y ;e,,7)= f f K(x',y')exp {27Ti(ex'+iy')}dx'dy',

(4.3)-00-CO

and, following equation (2.10),

dx I+(x,Y)= 2 Re{ ff f(6,,7)A*(x,y ;6,q)W-A(x,y ;e,,7)ded-I

= 2 Re{JK(x,

y") I f f K(x-x',y y')J(x', y') dx' dy'J* dy"} .

(4.4)-co o

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Then, as in condition (2.18), edge-ringing will be suppressed if

Re{

f Q(x, y) dy} >' O .

(4.7)~

The integral in condition (4.7) arises because the straight edge is essentially aone-dimensional object . The integrated function on the left-hand side of therelation corresponds to a line-spread function . Since a specific orientation ofthe straight edge has been assumed, condition (4.7) does not guarantee freedomfrom edge-ringing for other orientations unless some additional condition isimposed, e.g. circular symmetry of the function Q(x, y) .

Clearly, another condition which is sufficient (but not necessary) to satisfycondition (4.7) is

Re {Q(x, y) } > 0 (for all x, y) .

(4.8)

While this condition is somewhat more strict than condition (4 .7), it obviatesthe problem posed by the specific orientation of the straight-edge object .

5 . The use of the performance function 0(6, ,0 as a general performance indicator forpartially coherent imagingIt has already been remarked in § 2 that the function Q behaves somewhat

like a spread function characterizing the optical system, and that similarly thefunction Q behaves somewhat like a transfer function . This behaviour has beenillustrated in § 3 . Since no general performance indicator has yet becomeestablished for partially coherent imaging (in the way that the Optical TransferFunction has become established for incoherent imaging), it is appropriate toconsider whether the function Q(e,,q) could serve as such a performance indicator .

As a performance indicator for incoherent imaging, the Optical TransferFunction fulfills two important conditions . First, it completely characterizesthe intensity response of an incoherent imaging system ; and secondly, it presentsobjective information about the optical system in an intuitively meaningful form .Since the partially coherent imaging process is non-linear, the function QV, ))cannot qualify as a transfer function because it does not fulfil the first condition .Only the unwieldy function g (61,% ; ~2, "72), which specifies the transfer of themutual intensity through the optical system, can satisfy this requirement . Never-theless, although no simple function can fulfill the first condition, the functionQ(6,,q ) fulfils the second condition in several respects

(1) it is simple ;

(2) it is independent of the object ;

(3) it is a particular subset of the more complicated function .T (61, 171 ; e2, 12)which does satisfy the first condition ;

(4) it reduces to the amplitude transfer function (pupil function) for coherentimaging and to the Optical Transfer Function for incoherent imaging,and it forms a continuous transition between these two limits ;

(5) it completely characterizes the imaging process with respect to oneimportant feature, namely, edge-ringing .

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Edge-ringing in partially coherent imaging

605

Therefore, while it must always be remembered that partially coherentimaging is non-linear in intensity and that the function Q(6, ,q) does not completelycharacterize the imaging process, this function may prove useful for describingthe general performance of a partially coherent imaging system .

On montre qu'il existe une condition necessaire et suffisante simple pour supprimerles anneaux de bord de plage dans l'imagerie optique . Ce critere est valable en presenced'aberrations et d'apodisation et pour tous les cas d'eclairage partiellement coherent(spatialement stationnaire) . Les tendances d'un systeme optique a former des anneauxde bord de plage peuvent etre precisees en introduisant deux nouvelles fonctions de per-formance qui forment une paire de transformees de Fourier . Une de ces fonctionsde performance est reliee au `coefficient de transmission' qui apparait dans la theorie del'imagerie partiellement coherente . A la limite, dans le cas coherent, cette fonction deperformance se reduit a la fonction transfert d'amplitude (fonction pupillaire) et dans lalimite incoherente, elle redonne la fonction de transfert optique . On suggere l'utilisationde cette fonction de performance pour 1'evaluation, dans le cas general, des systemesformant des images en eclairage partiellement coherent .

REFERENCES

[1] BRACEWELL, R. N ., 1965, The Fourier Transform and its Applications (McGraw-Hill) .[2] CONSIDINE, P . S., 1966, Y. opt. Soc. Am ., 56, 1001 .[3] (a) KINTNER, E. C ., 1975, Optica Acta, 22, 235 ; (b) 1975, Ph.D . Thesis, University of

Edinburgh.[4] (a) SMITH, R . W., 1971, Optics Commun ., 4, 157 ; (b) 1972, Ibid ., 6, 8 ; (c) 1973, Ibid.,

9, 61 .[5] LEAVER, F. G., and SMITH, R . W., 1973, Optik, 39,156 .[6] BORN, M., and WOLF . E., 1959, Principles of Optics (Pergamon Press) .[7] HOPKINS, H. H ., 1953, Proc. R. Soc . A, 217, 408 ; 1957, J. opt . Soc. Am., 47, 508 .

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