Image of a Periodic Complex Object in an Optical System Under Partially Coherent Illumination

Image of a periodic complex object in an optical system under partially

coherent illuminationY. Ichioka and T. Suzuki

Department of Applied Physics, Faculty of Engineering, Osaka University, Yamadakami, Suita, Osaka, Japan(Received 28 June 1975; revision received 1 May 1976)

Combined effects of amplitude and phase variations on the irradiance in the image of a periodic complexobject, which has amplitude and phase distributions, have been investigated for an optical system withpartially coherent object illumination. A general expression to obtain the illuminance in the image and theimage contrast of such an object is derived by use of the concept of the effective source specifying thecoherence condition. The irradiance in the image for the diffraction-limited aberration-free optical systemilluminated with bounded effective sources that have uniform and nonuniform radiance, and also with annularillumination, have been calculated. Numerical calculations have been made to obtain the image, in whichharmonics up to the thirteenth order are taken into account. One of the results is that abrupt amplitude andphase variations are extremely influential for the appearance of the image of a periodic, low-contrast, complexobject for any mode of illumination. On the other hand, less abrupt changes of the amplitude and phase in acomplex object produce less variation of the appearance and contrast of the image for any illumination modeexcept annular illumination. Images of pure amplitude and phase objects have also been obtained by use of thegeneral treatment, and their characteristics are evaluated. It is suggested that a technique to manipulate theradiance distribution in the effective source is valuable in improving the fidelity of the image of a periodiccomplex object formed by an optical system under partially coherent illumination.

It is of importance, from the practical point of view, tostudy image formation in an optical system under par-tially coherent illumination, in which combined effectsof the amplitude and phase distributions in a complexobject on the image irradiance are considered. Knowl-edge of such combined effects could be required incases that (i) the image of a biological specimen is ob-

921 J. Opt. Soc. Am., Vol. 66, No. 9, September 1976

served in a microscope, (ii) photographic optical den-sity accompanied by a phase-changing relief image ismeasured with a microdensitometer, and (iii) partiallycoherent image processing by laser scanning' is carriedout. In the imaging processes of the systems mentionedabove, a specimen or an object with the amplitude andphase distributions is often illuminated with partially

Copyright © 1976 by the Optical Society of America 921

U

FIG. 1. Arrangement of the image forming system.

coherent light.

Although many papers have been published on imageformation with partially coherent light, little work hasbeen done about such combined effects. As far as weknow, only the pure amplitude or pure phase object hasbeen considered. 2-9 In a previous paper, 10 we derivedthe expression for the image irradiance of a sinusoidalcomplex object, and showed that the phase variation inan object was extremely influential for the appearanceof the image and the image contrast. However, becauseof the nonlinear property of the partially coherent opti-cal system, these results have provided informationonly about the combined effects of the amplitude andphase distributions in a sinusoidal complex object andwere not applicable to the evaluation of image charac-teristics for other forms of complex objects.

In this paper, to overcome this defect, our treatmentis generalized so that it can be applied to obtain theimage irradiance of any periodic complex object. Thegeneral expression derived is used to investigate com-bined effects of the amplitude and phase distributions onthe appearance of the image and its contrast for anycomplex object that has periodic structure. By repre-senting the complex transmittance of the periodic com-plex object by a Fourier series, we can also calculatethe image irradiance as a Fourier series.

For a diffraction-limited optical system, numericalcalculation of the image can be made by replacing aninfinite Fourier series by a finite series as long as wedo not deal with an object containing extremely low-spa-tial-frequency components.

We will consider two types of object models whichhave both amplitude and phase distributions, i. e., asinusoidal complex object and a trapezoidal complexobject. Adjusting object parameters in these two mod-els appropriately, we can express most complex objectsthat have periodic structures which involve abrupt andcontinuous amplitude and phase distributions. Fidelityof the image depends considerably upon combined ef-fects of the magnitude of phase variation, the spatialphase distribution, contrast of the object, and the illum-ination mode. The images of objects that have the samecontrast but different phase distributions, sometimesshow quite different appearances to an observer. Theyappear to be the images of objects that have differentcontrast and fine structures. These phenomena occurfrequently when we treat partially coherent imaging of


a low-contrast object that has large phase variations.

The image contrast, defined as the ratio of the Fou-rier coefficient in the illuminance in the image to thebias, is also calculated, and is compared with the ap-pearance of the corresponding images.

A general expression for the image of a periodic com-plex object is obtained by modification of the formulaoriginated in Ref. 1. To study the stated effects, theirradiance in the images is obtained under six differentillumination conditions.

To avoid excessive mathematical derivation, we limitour study to the case of one dimension; however, the ap-proach is easily applied to the two-dimensional case.In this paper, we mainly focus on the problem of par-tially coherent imaging of a biological specimen in amicroscope.

I. ANALYSIS

An image-forming system with four coordinate planesperpendicular to the optical axis is shown in Fig. 1.The coordinate systems are defined in the figure. Theobject 0(u, v), illuminated with the light wave radiatedfrom the primary incoherent source a, is imaged on theimage plane. If system stationarity is assumed, theimage irradiance is given by2

A(u') =Pj(x)I Jo(s)f(x - s) el i s A2dx, (1)

where y(x) is the effective source which characterizesthe state of coherence of illuminating light, o(s) is theobject spectrum, and f(s) represents coherent pupilfunction. Variables ui, v; u', v'; and x, y are expressedby reduced coordinates. 10 We may express the complextransmittance 0() of a one-dimensional periodic com-plex object as a Fourier series

O(u) =Ane.2rinxou (2)

where x0 is the fundamental spatial frequency and An isthe Fourier coefficient of component nxo. Then, theobject spectrum is given by

o(s) =ZAn5(s -nx,), (3)

where 6 denotes Dirac delta function. Substituting Eq.(3) into Eq. (1), and using the properties of Dirac deltafunction, the irradiance in the image becomes

(4)

where the asterisk means complex conjugate. In Eq.(4), T(n, m) is the transmission cross coefficient" de-fined by

T(n,m) = yx)f(x +nxo)f *(x +mx0 ) dx E Tr(n,m) +iTi(n, m),(5)

where Tr(n, m) and T,(n, m) are the real and imaginaryparts of T(n, m), respectively. According to the defini-tion in Eq. (5), we obtain the relations

T7(nym)=T7(m,n) , Ti(n,m)=-Ti(m,n), (6)

Y. Ichioka and T. Suzuki 922

Au') = E EAnA* T(n, m) e 2ri (n-m)x0U,M

AMPLITUDE TRANSMITTANCE

(A)

RELATIVE PHASE SHIFT

J>\ q 0\s - B)

U

FIG. 2. Model of a sinusoidal complex object; (a) amplitudetransmittance, C +A cos27rxou, and (b) phase distribution,exp(iB cos27rjx0u).

in which y(x) is assumed to be real. Set n -m =k forall values of n and nm in Eq. (4), and put

An=R, +iIn, A*=Rm-Im,

II. OBJECT MODELS

To investigate the stated effects, two kinds of objectmodels that have both amplitude and phase distributionsare considered.

One is the complex object that has sinusoidal ampli-tude transmittance C +A cos2rx 0u and phase distributionexp(iZjC BJ cos27wjx0u) shown in Fig. 2. Then, the com-plex transmittance of the object is given by

0(u) = (C +A cos27rxu) exp(ZEBi cos2nixou) (9)

where C, A, and Bi are constants. Parameters A andC specify the object contrast and the Bj's characterizethe phase variations. This object model is adequate todescribe a sinusoidal density distribution recorded onphotographic film accompanied by the phase-changingrelief image caused by nonlinear modulation due to den-sity variations. Using the Bessel-function formula

(7)

where R, and In are the real and imaginary parts of An.Applying the relations in Eqs. (5)-(7) to Eq. (4) withthe rearranged series (n-rm =k), interchanging sumsfor n and k, and coupling exponentials, we may rewriteEq. (4) as

A(u') = S (D, cos27rkx~u' +D1 sin2wTkxou')k=O=E cos (2fkxau'-o) - (O8)k=O

whereE, =(Dk + D(Ba)ok = tan'(Da/D>),

eiBi cosirJx0u = > inJ" (B) e2lfnjxOUn

(10)

the Fourier coeff ic ient An of the complex transmittanceof the object is calculated as

A E = E 3 3 ij-J'2-j3'- 1 -lM N-J (B2 )Jj.(B3) ' * Jj,(BN)N1 Z2

3N

{CJn-. (BI) +±Ai[J,+, _ABi)- ,n-i-a(Bl)} I (11)

a= 2j2 +3j3* +NJN,

where Jn(B) is the Bessel function of the first kind andorder n. If N= 1, then

An = i'CJn(Bl) + 2Ain.1[Jn+l(B1 ) -Jn-,(Bi)]; (12)

(8b) and if N=2, then

Djp = [ [PnkTr(n, f -k) + Q,,nnTi ( -k n )] - (8c)n

D' = 3 [QnkTr(f, n - k) - PTkTi (l, n - k)], (8d)

Pn> = E1(RTI R.-k + I,, Ina-) (8e)

and

Qnh = Ek(Rn -k I-nR,,-,) , (8f)

where e0 = 1 and Ek = 2 (k •0). In Eq. (8), Dk and Dk areFourier coefficients, and E, and 0k indicate the ampli-tude and the image shift of kth harmonic in the irradi-ance in the image.

Equations (8) are the general expression for the im-age irradiance of a periodic complex object formed byan optical system under,partially coherent illumination,in which the state of coherence and the characteristicsof the image-forming system are condensed in thetransmission cross coefficient, and the object charac-teristics are described by the parameters Pn1, and Qnk,respectively. Once the shape of the object is deter-mined, Rn and I, in Eqs. (8e) and (8f) for any n can beobtained easily. Although I(u') in Eqs. (8) is repre-sented by double sums for n and k, it can be evaluatedby calculating a finite number of terms for n and k inthe case of a diffraction limited optical system.


An = 3jn"mJ.(B 2){CJn_2.(B1 ) + 2Ai[Jna2m+i(Bi) - Jn_2m i(B1)]}.

(13)

Details on the image contrast in the case of N = 1, havebeen discussed in the previous paper. lo However, forlarge N, numerical evaluation becomes impracticaleven though a large computer is used, because An isexpressed by multiple sums, consisting of a number ofcross terms of Bessel functions.

Another model is a periodic complex object with aperiod D that has trapezoidal amplitude transmittanceand phase distributions shown in Fig. 3, in which therelative phase shift is assumed to be proportional to theamplitude distribution. The object characteristics aredetermined with seven parameters, A, B, C, s1, s?,S3, and S4 in the figure, where A and C are concernedwith the contrast of the object, B with the phase varia-tion, and s 1 s 4 with the form of the object. This modelis suitable for assessment of the quality of the image ofa complex object formed by a practical image-formingsystem such as a microscope, because most fine struc-tures and characteristics in a practical object can bedescribed by combinations of seven parameters.

The real and imaginary parts of a trapezoidal com-plex object with a fundamental frequency x3 (= 1/D) arecalculated to be


Rn =[As1 sinc(2ns 1 )] + ((Asa - Cs1 ) sinc[2(B - 2ns2 +25s,)] cos[2n(B - 2ns2 - 2ns,)]

(C-A)1s 2 sin7r(B - 2ns2) +sl sin2rnsl] (C -A)(s 2 -si) sinc[t(B - 2ns2 +2ns,)] sin[7r(B - 2ns2 - 2ns,)]\+ v(B - 2ns2 + 2ns1) w(B - 2ns2+22ns 1)

+[C(s 3 -s 2)sincn(s 2 - s3) cosu(ns2 +nS3 - B.)] + ((Cs4 -As3) sinc[2(B - 2ns3 +2ns4 )] cos[K(B - 2ns4 - 2ns3)]

(A - C)[s4 sin2rns4 +s3 sinu(B - 2ns9)] (A - C)(s3 -s4 ) sinc[ (B - 2ns3 +2ns4)] sin[ 7r(B - 2ns4 - 2ns3)]\+ 7T(B - 2ns3 +2nS4) r(B - 2ns3 +2ns4)

+[A(l -s4) sincn(l -s4 ) cosrn(l +54)]

In=[-As sinc(ns1 )sin7Tns1 + ((As2 -Csi)sinc[(Bn-2ns2+2nsi)]sin[7r(B-2ns2-2nsl)]

(C-A)[s2 cos7T(B-2ns2) -s cos2vnsl] + (C-A)(s2 -sl) sinc[#(B-2ns2 +2ns)] cos[7r(B -2ns2 -2ns,)]1-I n - n I - / - b - I

7rTB - Zns 2 +Zns 1 ) T(B - Zns 2 + Zns 1)

(14a)

+[C(s2 -s3 )sinen(s2 -s3)sinr(ns2 +nS3 -B)]+ ((Cs4 -As3 ) sinc[2(B- 2ns3 +2ns4)] sin[fT(B -2ns4 -2ns3 )]

(A - C)[- S4 cos2sns4 +s5 cosr(B - 2ns9)] (A - C)(s3 -s 4)sinc['(B - 21s 33+2nS4) cos[2(B - 2ns 4 - 2ns3)]7(B - 2ns3 +2nS4) + (B - 2ns3 +2ns4),

+ [-A(1 -s 4) sincn(1 -s4)sinsn(1 +S4)3,

where sincx = sinrlx/rx. Adjusting seven object param-eters appropriately, periodic complex objects with use-ful forms are determined, as shown in Table I. By useof Eq. (14), the real and imaginary parts, Rn's and In's,of spectra of these objects are easily obtained. Theyare listed in Table I.

IfA=0andBOinEq. (11)orA=CandB0QinEq.(14), the model represents the pure phase object thathas a periodic phase variation; and if A * C and B = 0,it becomes the pure amplitude object without phasechange. Therefore, the formation of the image of apure, periodic amplitude or phase object in a partiallycoherent optical system can be considered as a specialcase of the general treatment. Of course, analysis andnumerical calculation become simpler than for a com-plex object.

TABLE I. Object models with useful forms and Fourier coef-ficients of them. R, and In mean the real and imaginary partsof Fourier coefficients A,.

OBJECT MODEL j BBJECT PARAMETERS A, (FOURIER COEFFICIENT)

A / f A, B, C, Si-P1 lR EQ. (14A)

P s2=P2. S3=P3 , S4=P4 I, EQ. (14B)

A ., F1 L A, B, C RI, 1STf I + 3RDp I + 5TH! I IN EB.(14A)

P J l s1=s

2=P1, s3=S4 5p2 In, 1sT! I + 3RD! I 1 5TH: I IN EQ.(41A)

A A, B, C, R, 2ND! I IN EQ. (14A)

P 1 s-B, S=2S53=54=B 1n 2NDI ] IN EQ. (14n)

A /A, B, C. R0, 2ND! I + 4TH! I IN EQ. (14A)

/-If 0= S2,3-=P,, S4=D In 2ND! 1 + 4TH( I IN EQ. (14a)

A A. B, C, R0 1ST! 1 + 2ND! ] + 3RD! I + 5TH I IN EC.(14A)

P s1Pi, S2-P2, s3=s4=D In: 1ST! B + 2ND I] + 3RD! ] + 5TH! I IN EQ.(1B)n

A AA A, B, C, s1=P1 RB 1ST! ] + 2ND 1 + 4THA I + 5TH[ I N EQ. (14A)

-S2.SŽ S3'P2 S45P3 In : 1STy + 2

ND I] + 4TH! I + STH! 1 iN EU1(I4n)

P1i P2 , P3 , AND P4 5B:

(14b)

III. IMAGE CONTRAST

In general, the flux transmittance of a periodic com-plex object, that has both amplitude and phase distribu-tions, and the irradiance in the corresponding imageformed by an optical system under partially coherentillumination, contain infinite numbers of harmonics.Unfortunately, there is no linear relation between thespatial frequency components in the object and the re-sultant image, because partially coherent optical sys-tems are nonlinear for either complex amplitude or ir-radiance. Moreover, the image irradiance involves,sometimes, frequency components which do not exist inthe flux transmittance of the original object due to non-linear modulation of an object. Therefore, we cannotdefine a useful measure such as the transfer function inthe incoherent case to evaluate system performance inpartially coherent illumination. Hence, to evaluate thecharacteristics of the image of a periodic complex ob-ject in a partially coherent optical system, the terms

AMPLITUDE TRANSMITTANCE

RELATIVE PHASE SHIFT

B

(a)

(b)

FIG. 3. Model of a trapezoidal gratinglike complex object; (a)amplitude transmittance, and (b) phase distribution. A, B, C,Si, S2, S3, and s4 are parameters describing the object charac-teristics. D is the period.

924 J. Opt. Soc. Am., Vol. 66, No. 9, September 1976 Y. Ichioka and T. Suzuki 924

(a) (b) (c)

1] -1 xCOHERENT PAR. COHERENT PAR. COHERENT

r(x)=l r(x)=l

PAR. COHERENT PAR. COHERENT

ANNULAR ILLUMI. r(x)=x2

(d) (e)

xPAR. COHERENT

r(x)=-x2+1

(f)FIG. 4. Six effective sources.

image contrast and image shift have been applied, andare defined as the ratio of the Fourier coefficient of kthharmonic to the bias in the irradiance in the image

CTk=EEk/Eo, 0e =tan'(D'/Dk) (15)

lated for any reduced spatial frequency less than 2.Image contrast can be plotted in a way similar to the op-tical transfer function of an incoherent optical system.It should be noted that the image contrast and its shiftserve to specify only the characteristics of the image ofthe stated complex object in a partially coherent opticalsystem and not to describe the system performance. Inspite of this restriction, they are still useful for evalu-ating the effects of nonlinearity of the system and non-linear modulation of the object.

IV. NUMERICAL CALCULATION

Using relations in Eqs. (8) and (15), we calculatethe image illuminance of a periodic complex object andimage contrast in the optical system under partially co-herent illumination. Let us consider a diffraction-limited aberration-free optical system and complex ob-jects of periodic structures illuminated with boundedand annular effective sources. Hereafter, we onlytreat a symmetrical pupil function around the opticalaxis. That is, the pupil function of the image formingsystem is

f(x)=1 for IxJ|c

= 0 otherwise . (16)

Referring to Eqs. (8), CTk and Sk may be specified bythe object and system characteristics and are calcu-

I

The effective sources are specified bytions depicted in Fig. 4, which are

six configura-

(coherent source);

for IxI-0.5 (R)

= 0 otherwise,

for IxJ'1.0 (R)

(bounded effective source with uniform radiance);

(bounded effective source with uniform radiance);

= 0 otherwise,

for 0. 95 (R')- I x| C 1. 0 (R) (annular source);

= 0 otherwise,

(e) y(x)=x 2 for Ix I ;1.0 (R)

= 0 otherwise,

(f) y(x)=-x 2 +l for Ixi 51.0 (R)

= 0 otherwise

(bounded effective source with nonuniform radiance);

(bounded effective source with nonuniform radiance),

where R is the size of the effective source and is equiv-alent to the ratio of the condenser numerical apertureto the objective numerical aperture. R plays an im-portant role in describing the coherent condition of theilluminating light.

Dk and Dk' in Eq. (8) have to be calculated as infiniteseries of the product of the transmission cross coeffi-cient T(n, n - k) and the combined terms of the real andimaginary parts of the object spectrum, in which thetransmission cross coefficient T(n, n - k) for any n andk, can be evaluated by the overlapping area of threemutually shifted functions y(x), f (x+nxo), and f*{x+ (n- k)x0}. Fortunately, the numerical calculation of theabove infinite series can be made by taking sums of fi-nite terms for limited n unless the extremely low-spa-


tial-frequency component is treated, because the trans-mission cross coefficient has nonzero value for n, k(positive) and spatial frequency x0 satisfying 0 kxo - 2,I(n-k)xo+1 1_R, and Inx 0 -1I-R, otherwise it be-comes zero.

V. APPEARANCE AND CONTRAST OF IMAGE

A. Square gratinglike complex object

A square gratinglike object that has amplitude andphase distributions is used to examine how the abruptamplitude and phase variations in the object affected theimage appearance and contrast. Object parameters areset to be s 1 = S2 = 0. 25 and s3 = S4 = 0. 75. The fundamen-tal frequency of the object is selected as x0= 0. 18, and


(a)

(b)

y(x) = 6(x)

y(x) = 1

(c) y(x) = 1

(d) y (x =1

A-S A-A.7 A-A.5 A-ACS1 XD-0.18 COHERENT

INCREASE OF AMPLITUDE CONTRAST

FIG. 5. Images of square gratinglike complex objects thathave different amplitude and phase variations illuminated withcoherent source in Fig. 4(a). Object parameter A specifyingthe amplitude contrast is changed, from the left to the right,as 1.0, 0. 7, 0. 5, and 0 and the parameter B specifying thephase variation is changed, from the bottom to the top, as 0,'sr, 7r, and 37r, respectively.

a computer calculation has been made to get the imageirradiance containing harmonics up to the eleventh or-der for integers in the range In! c 25.

Figures 5-10 show the images under six coherenceconditions specified by effective sources depicted inFig. 4. In every figure, 13 images are drawn for theirone period, and are arranged in accordance withchanges of parameters A and B specifying the objectamplitude contrast and the relative phase shift, respec-tively. They are, from left to right, 1 (pure phase ob-ject), 0. 7, 0. 5, and 0 for A, and from the bottom to thetop, 0 (pure amplitude object), 21r, 7r, and 37r for B, re-spectively. Further analysis shows that, for the dif-fraction-limited aberration-free optical system, imagesfor B= 37T coincide with those for B= 1FT. This arrange-

inLl_

-D/2 -D/4 D/4 D/2A-1 A.0.7 A5-.5 A-SC-1 X.o".18 BOUNDED SOURCE R_-U.


FIG. 7. Same as Fig. 5 but with the bounded effective sourcewith uniform radiance with R= 1.0 in Fig. 4(c).

ment facilitates systematic evaluation of combined ef-fects of amplitude and phase variations on the appear-ances of the images.

Figure 11 shows the image contrast of the images ofthe square gratinglike complex objects under six illu-mination conditions shown in Fig. 4, in which curvesfor the fundamental and second harmonic componentsare plotted for the objects whose parameters are A0. 25, C = 1, and B= O. FT, and X, where the sign of

the image contrast of the second harmonics is re-versed. The image contrast of the second harmoniccomponent provides information about nonlinear effectsdue to the partially coherent optical system and aboutnonlinear modulation of the object, because there is noharmonic components of the even order in the idealimage of the square gratinglike complex object.

Careful observation of Figs. 5-11 reveals the follow-ing significant facts and phenomena related to combinedeffects in the specified illumination modes:

B-32 I

B."

CD

(-I

B-S

A-l A-0.7 A-0.5C-B Xo-.18 BOUNDED SOURCE R-0.5


-D/2 -D/4 0 D/4 D/2AS-

FIG. 6. Same as Fig. 5 but With the bounded effective sourcewith uniform radiance with R=0.5 in Fig. 4(b).


FIG.4(d).

A-1

L + B-3 V2

B-fl

1

CF-

02:C

C

1)CA

02

B-S

-D/2 -D/4 5 D/4 D/2A-0.7 A-,.5 A50

C-B XSo.A18 ANNULAR ILLUMINATION


8. Same as Fig. 5 but with the annular source in Fig.


B=31T/2 IC

F-

B= or -

C,

B=Tr/2 <

B=O_/ 7�

-1

I

B'W 2

B-3TV2 1

F-

B-Tl

C; 2:

B-/ -

J- B-3m'2

B- -n

B- rV2

B-

A-1 A-0, 7

C 1 XSoA.18 BOUNDED SOURCE RB- r(X)IX2


1

0

F-

CD

I 1 u>O ,

Oa(u) = ) 2(1 +AeiB) u=O 0

(AeiB u<O.

(17)

This object has a complex amplitude Fourier spectrumgiven by

(18)

0 -

-D/2 -D/4 0 D/4 0/2A-0.5 A-S

FIG. 9. Same as Fig. 5 but with the bounded effective sourcewith nonuniform radiance distribution specified by y(x) =x2 inFig. 4(e).

(i) In the image formation of the diffraction-limitedaberration-free optical system under coherent, nearcoherent, and annular illumination, phase change is ex-tremely influential for the appearances of the images ofthe pure phase object and the low-amplitude contrastobject that have sharp boundaries of phase. Most no-ticeable phenomena are generation of heavy ringing andsharp notches or peaks at the location of sharp bound-aries.

(ii) There is a rapid decrease in the amount of ring-ing in the image and a monotonic decrease in edge gra-dient with decreasing coherence, as would be expected.On the contrary, a decrease in the depth of the notchesis insensitive to decreasing coherence.

(iii) Increase of amplitude contrast and phase con-trast in the complex object serve to increase the overallimage contrast in any illumination condition. Increaseof phase variation serves to sharpen the edge appear-ance in partially coherent illumination.

(iv) The structures of the images of the complex ob-jects illuminated with the bounded effective source withnonuniform radiance distribution specified by y(x)=x2are much improved from the point of view of high-fi-delity imaging of a complex object. There are no steepirradiance peaks or sharp notches at the location ofsharp boundaries of amplitude and/or phase. On theother hand, the bounded source with nonuniform radi-ance distribution specified by y (x) = - x2+ 1 does notserve to improve the image quality.

The phenomena mentioned above may be illustratedby examining the features in the images of the edge ob-jects, that have sharp boundaries of amplitude and/orphase, illuminated with the annular source with varioussource sizes.

Let us consider the edge object referred to as thestep function, positioned with the edge at the origin.The complex transmittance of this object is representedby the function


+ (1+ A 2 - 2A cosB) ln2(-+ R)]

ILnr

(21)

B-3'o'2 1

0nO

B-nV <U

B-U

-D/2 -D/ 0 0 DD/2A-1 A-S.7 A-0.5 A2O

C-1 Xo-.S18 BOUNDED SOURCE R-i r(X)_-X2.1

INCREASE OF AMPLITUDE CONTRAST -FIG. 10. Same as Fig. 5 but with the bounded effective sourcewith nonuniform radiance distribution specified by yx =- x2+ 1in Fig. 4(f).


.. 1.where u and x are expressed by reduced coordinate, andA and B are constants related to the amplitude contrastand phase variation in the edge object, respectively.Now, consider that the edge object, illuminated with thelight radiated from two point sources located at x=Rand x= - R (annular source in one dimension), is imagedby the diffraction-limited aberration-free optical sys-tem whose pupil function is specified by Eq. (16). Theradiance of the effective source is given by

y(X) = 2[6(x-R) + 6(x+R)] . (19)Substitution of the relations of Eqs. (16), (18), and (19)into Eq. (1) yields the irradiance of the diffraction-limited image of the edge object. This may be given by

15 (u') = (1/4ir 2 )(7r2 (1 + A 2 + 2A cosB) - 27T(A2 - 1)

x [Si(X2) - Si(XO)] + (1 +A2 - 2A cosB)

X{[Si(X 2 ) - Si(X1)]2 + [Ci(X 2 ) - Ci(X 1 )]2 }) , (20)

where u' is also expressed by the reduced coordinateand Si(X) and Ci(X) are the sine and cosine integral func-tions, and where Xl = 27iu'(- 1 +R) and X2 = 27ru'(1 +R).Equation (20) gives the irradiance in the diffraction-limited image of the edge object under annular illumina-tion at all values of u' excepting u'=0, since Ci(O)=- co.The image irradiance at u' = 0 may be evaluated fromthe integral of the complex amplitude of the object spec-trum within the diffraction-limited aperture. 12 By doingso, the image irradiance at u' = 0 is obtained by

I,(u'= 0) = [7 [r2(1+A2+ 2A cosB)

o,, (x) = I [ (1 + Ae'96 W - (I - Ae")il7rx] ,

-

COHERENT A-O.25C-D.0

2 FUNDAMENTAL

1.0REDUCED SPATIAL FRE(

(a)

ANNULAR ILLUMINATION R-S.DA 0.25C-D.O

\ \D-'1t2 D- n

1.0 2.D0 "REDUCED SPATIAL FREQUENCY

BOUNDED SOURCE R-U.S T(DI-A-0.25C-D.O

X" FUNDAMENTAL

2.0 0o 0 1.D 2.0]UENCY REDUCED SPATIAL FREQUENCY

(b)

Xo

Xo

REDUCED SPATIAL FREQUENCY

(c)

BOUNDED SOURCE R-D.D T(X)I-X20 1A-0.25C-S.D

\ FUNDAMENTAL

REDUCED SPATIAL

(d) (e) (f)FIG. 11. Effects of phase variations on the image contrast for the fundamental and second harmonic components in the image ofthe square gratinglike complex object illuminated by six different effective sources in Fig. 4. Illuminating light sources to obtaincontrast curves are; (a) the coherent source in Fig. 4(a); (b) the bounded effective source with uniform radiance with R = 0.5 inFig. 4(b); (c) the bounded effective source with uniform radiance with R=1.0 in Fig. 4(c); (d) the annular source in Fig. 4(d); (e)the bounded effective source with nonuniform radiance specified by y(X) = x2; and (f) the bounded effective source with nonuniformradiance specified by y(x)=- x2 +1 in Fig. 4(f).

If R=0, Eqs. (20) and (21) give the image irradianceunder central coherent illumination.

Generation of heavy ringing and sharp notches in co-herent imaging can be interpreted by Gibbs phenomenonin the Fourier integral theory and illustrating diagramin Fig. 12. Figure 12 shows effects of phase changeson the appearance in the image of edge objects thathave the same amplitude contrast, under coherent il-lumination [R = 0 in Eqs. (20) and (21)]. Patterns on theright- and left-hand sides in Fig. 12 indicate the be-havior of the amplitude component of the diffraction-limited images of the edge objects and that of the cor-responding image irradiance. The complex transmit-tance of the edge object is given by putting A= 0. 87 inEq. (17). Phase parameter B is changed for the figuresfrom the top to the bottom as 7T, 2ir, and 0. It is evidentfrom Fig. 12 that the notch goes down as the amplitudeof the step increases and the maximum ringing and thedeepest notches occur when B= or. That is, the ampli-tude of oscillation in ringing and the depth of the notchincrease as the object amplitude contrast decreasesand phase change is close upon or. The amount of oscil-lation in ringing is proportional to that of the step.A generation of the notch which goes to zero is due tothe negative amplitude in the object. Thus, the diffrac-tion-limited images of complex object that have sharpboundaries of the amplitude and/or phase contain heavyringing and sharp dark lines (i. e., notches) at the loca-tion of sharp boundaries in the object. Quantitative de-pendence of these effects is clearly observed in theimage structure diagram in Fig. 5.


Figure 13 shows the effect of variation of the sourcesize in the annular source on a change in the appearanceof the image of the pure phase object that has the sharpphasestepof or. For this object, putting A= 1 and B= or in

IRRADIANCE AMPLITUDE

B-TV2

A-O .87 C-1. D

D-D

FIG. 12. Generation of the notches in the images of the edgeobjects that have sharp boundaries of amplitude and phase un-der coherent illumination. The parameter A specifying ampli-tuLde contrast in Eq. (17) is 0. 87 for every figure, and thoparameter B is Or, P21r, and 0 for the top, middle, and bottomfigures.


D

Xo' �20

L___1_

.5

-1. 0 0 1.0

.0.

R=0. 735

-1.0 0 1.0

-1.0 0 1.0

.245

-1.0 0 1.0

- 0

(COHERENT)

-1,0 -0.5 0 0.5 1.0REDUCED DISTANCE

FIG. 13. Variation of the irradiance in the images of the purephase object [A = 1 and B = 7r in Eq. (17)] that has sharp bound-ary of phase as a change in the source size R in the annularsource.

Eqs. (20) and (21), the image irradiance is given by

Ie(U') = (l/1r2){[Si(X2) - Si(X1)12 + [Ci(X2) - Ci(X)12}, (22)

i(u'= 0) = (l/7r2)In2[(1 +R)/(1-R)] . (23)

Figure 14 illustrates that variation of the source size inannular source leads to a change in the image irradianceat u'=0. From Fig. 14, image irradiance at u'=0 in-creases abruptly as the source size is close upon unity,and it becomes infinity at R = 1. Under these illumina-tion condition, the sharp boundaries in the object maybe encountered as the bright lines. The irradiancepeaks at the location of sharp boundaries in the imagestructure diagram in Fig. 8 may be caused by thisphenomenon.

Increase of the source size in the annular source cor-responds to shift of the object spectrum in the pupilplane. If the source size exceeds unity, zero spatialfrequency component is blocked out, and the resultantimage appears in dark background. If the source sizeexceeds unity but is not so large, the sharp boundariesin the object can be observed as bright lines in darkbackground. This situation is similar to that in Schlier-en method which obtains information about phase ob-jects. Thus, the annular source, whose source size

is close upon unity, is worthy to obtain informationabout sharp boundaries of amplitude and/or phase in thecomplex object. However, it is not expected to get ahigh-fidelity image under such illumination condition,owing to the heavy shift of the object spectrum passingthrough the pupil.

The image of the edge object illuminated with thebounded effective source with uniform radiance can beevaluated from integrating the image irradiance inEqs. (20) and (21) over the source area. A rapid de-crease in the amount of ringing in the image with de-creasing coherence may result from cancellation of theoscillating terms such as the sine and cosine integralfunctions during integration.

Now we explain the reason why the images of squaregratinglike complex objects still contain sharp dips atthe location of sharp boundaries in the object illuminatedwith the bounded effective source with uniform radiance(R = 1). To do so, we examine the image irradiance atthe location of the sharp boundary (u'= 0) of the purephase object, that has the phase step of 7T, under corre-sponding illumination condition. This can be given byintegrating Eq. (22) over the source area (I RI < 1), i. e.,

IR=D(U'= °)f= f In'__R) dR . (24)

The definite integral in Eq. (24) has the value of r2Then, IR=1(U'= 0)= 3. This value corresponds to thevalue of the dip in the image of the pure phase object(A= 1 and B= 1r) on the right-hand side of the second rowin Fig. 7.

From the fact that the annular source with R = 1 en-ables to enhance the edge structure, light radiated fromthe source rim of the bounded effective source men-tioned above ought to enhance the image irradiance atthe location of the sharp boundaries. On the other hand,that from the inner part of the source serves to producethe irradiance dip at the corresponding position (u'= 0)as shown in Fig. 14. In a quadrature, two effects maycompensate each other. In the present case, theeffect of the former is less than that of the latter. As aresult, the sharp irradiance dips would occur at u' = 0.This consideration and comparison of image structurediagrams in Figs. 7 and 8 lead to a conclusion to obtain

... .I

1t

s -S SIIo

1.01t-

0. sI-

0 0.5 1.0 1.5 2.0

SOURCE SIZE

R

FIG. 14. Variation of the irradiance at the location of sharpboundary (u' = 0) in the image of the pure phase object used inFig. 13 as a function of the source size in the annular source.


1 . 5

Y. Ichioka and T. Suzuki

B-31V2

I -Tr

1

C

Bar/2 LL

C>

B-O

-D/2 I 0/2A-1.0 A-0.7 A-0.5 A-AC-IA. XI-.18 COHERENT


FIG. 15. Images of the trapezoidal gratinglike complex objectilluminated with the coherent source in Fig. 4(a). ParametersA and B describe the amplitude and phase contrast in the objectmodel in Fig. 3, where C=1.

the method to improve the quality of the image of thecomplex object, that has sharp boundaries of amplitudeand/or phase. If the radiance in the inner part of thebounded source is reduced in a proper manner, main-taining high level around the source rim, the resultantimage would provide the desirable image structureswithout sharp irradiance peaks or dips at the location ofthe sharp boundaries. Nonuniform radiance distributionspecified by y(x)=x2 depicted in Fig. 4(e) is such an ex-ample. Structures of the images in Fig. 9 obtained byusing this source clarify that such nonuniform radiancedistribution in the effective source gives desirable ef-fects on suppression of generation of the irradiancepeaks or dips.

As a matter of course, it is not expected to diminishthe irradiance dips in partially coherent imaging whenthe complex object is illuminated with the effectivesource with nonuniform radiance distribution specifiedby y(x)=-x 2 +1. The images in Fig. 10 show that useof such effective source is meaningless to improve theimage fidelity.

It is known that an optimum apodization to minimizethe ringing effect can be attained by manipulation of thepupil function. 13, 14 The facts mentioned above indicatethat an optimum apodization to suppress the ringing ef-fect and the irradiance peaks or dips at the location ofthe sharp boundaries in the object can be also made bychanging radiance of the light source used in a nonlinearsystem. The optimum radiance distribution, Vm(x), ofsuch an effective source is obtained by solving the fol-lowing integral equation, i. e.,

If1 2' /'1+R,2 J ym(R) ln i - R) dR =.

The dips appeared in the image structure diagramsin Figs. 5-7 looks like the first derivative of the sharpboundaries of amplitude and/or phase. However, itseems that they result from the effect of negative ampli-tude caused by the abrupt and fairly large phase change


at the edge rather than the effect of the first derivativeof the sharp boundaries, because there is no sharp dipsin the images of the pure amplitude objects that havesharp boundaries as shown in the bottom rows in Figs.6 and 7.

B. Trapezoidal gratinglike complex object

Figures 15 and 16 show the images of the complex ob-jects that have trapezoidal amplitude and phase varia-tions formed by an optical system illuminated with co-herent source and the bounded effective source withuniform radiance distribution in Figs. 4(a) and 4(c), re-spectively. Form factors of the object in Fig. 3 are de-termined as s, = 0. 1, s 2 = 0. 3, s 3 = 0. 5, S4 = 0. 8, andC = 1. The fundamental frequency is selected as x0=0. 18. The parameter A specifying the amplitude con-trast of the object are changed, from the left to theright in the image structure diagram, as 1. 0 (purephase object), 0. 7, 0. 5, and 0 (pure amplitude object)and the parameter B specifying the phase variation ofthe object are changed, from the bottom to the top, as 0(pure amplitude object), Ear, fr, and 3rr, respectively.These numerical results reveal that the continuousphase change in the complex object has little influenceon the image appearance and the image contrast de-pends only upon the object amplitude contrast in any il-lumination condition.

We also calculated the images of the sawtoothed ob-jects that have amplitude and phase variations under sixillumination conditions. There is no significantphenom-enon present in the image structures that is notalready essentially present in the image structures ofthe square gratinglike complex object and the trapezoi-dal gratinglike complex object.

C. Sinusoidal complex object

Figure 17 shows dependence of the phase variation onthe irradiance in the image of a sinusoidal complex ob-ject formed by an optical system illuminated with the

B-3r42

1

B-r Trl:E

B-I2

-D/2 0 D/2

A-.1. A-0.7 A-0.5 A-A

C-1.0 Xa-.18 BOUNDED SOURCE R-1 r()-l

INCREASE OF AMPLITUDE CONTRAST -

,I-

C,

"I

FIG. 16. Same as Fig. 13 but with the bounded effectivesource with uniform radiance with R =1 in Fig. 4(c).


a0.8r

-B=O

U,REDUCED DISTANCE

FIG. 17. Images of the sinusoidal complex objects formed byan optical system illuminated with the bounded effective sourcewith uniform radiance with R= 0.5 in Fig. 4(b). The complextransimttance of the object is (1 +1cos27rx0u) exp(iB cos27rxou).

bounded effective source with uniform radiance (R = 0. 5).The complex amplitude of the original object is(1 + - cos27x~u) exp(iB cos27Tx0u). Individual curves aredepicted for half-period of the resultant images of thecomplex objects with different phase distribution. Fromthis figure, it becomes apparent that the image struc-ture of the complex object that has sinusoidal amplitudeand phase variations is distorted and the bright parts inthe image are sharpened as the phase variation in-creases. This fact tells us that, in the course of micro-densitometry of the photographic density accompaniedby the phase changing relief image, phase variation dueto the relief image must be compensated optically if wewant to measure the precise density of the specimen.

VI. SUMMARY AND CONCLUSION

By use of the Fourier-series analysis and the objectmodels of the trapezoidal gratinglike complex object andthe sinusoidal complex object, we have derived the gen-eral expression to obtain the image irradiance of anyperiodic complex object in an optical system under par-tially coherent illumination. Numerical results haveclarified following significant phenomena in partiallycoherent imaging due to combined effects of amplitudeand phase variations in the original object.

(i) The images of the complex objects that have sharpboundaries of amplitude and/or phase provide, some-times, quite different appearances to an observer in ac-cordance with difference of coherence conditions andphase variations in the objects, even though the ampli-tude contrast in the objects is the same. Therefore, whenwe observe the image of a low amplitude contrast com-plex object that has sharp boundaries, we must take intoaccount that the resultant images tend to provide spuri-ous appearance to an observer.

(ii) In partially coherent imaging of the complex ob-ject without sharp boundary of amplitude and/or phase,phase variation has little influence in changing the imagestructure and contrast.

(iii) Phase changes in the object become less im-portant in changing the image structure as the ampli-tude contrast of the object increases. This conclusionis valid independent of the degree of coherence of the

W=. 5 P-0,5


illumination. This fact strongly suggests that when theimage of the low-amplitude contrast complex object,such as a biological specimen, is observed in a micro-scope, it is desirable to dye the specimen if possible.

(iv) The structure of the image becomes obvious atlow-amplitude and phase contrast levels in the object.An increase in object amplitude or phase contrast thenserves only to increase the overall contrast of theimage. There are no image structure features pre-sented in the image of the highest contrast amplitudeand phase objects which are not already essentiallypresent in the image of the low-contrast amplitude andphase object.

(v) The bounded effective source whose radiance dis-tribution is of the form y(x) = x2 is worthy to suppressthe ringing effect and notches in the images of the com-plex objects due to a nonlinear system. Effect of thissource seems to be equivalent to that of the apodizedaperture to minimize ringing. This fact suggests thatthere are better effective sources to provide high-fideli-ty image in partially coherent imaging of a complex ob-ject.

(vi) Annular illumination is useful to maintain high-image contrast to the incoherent resolution limit and todetect the sharp boundaries of amplitude and phase inthe complex object. However, it is inadequate to formhigh-fidelity image of the complex object.

We mainly focused on the problem of evaluation of theimage of a biological specimen observed in a micro-scope in this paper. The results are also applicableto other problems in microscopy. For instance, the re-sults are useful during inspection of an integrated cir-cuit in a microscope. In addition, we would like topoint out that the results of this paper indicate that ap-propriate consideration must be paid to coherence ef-fects on measurement of photographic optical densitywith a microdensitometer because variation of densityin an emulsion is accompanied by a phase-changingrelief.

Moreover, the results obtained also apply to partiallycoherent image processing by changing laser systems,because the image processing operations which can beimplemented in a conventional partially coherent opti-cal system such as that analyzed here can also be im-plemented by a scanning laser. The detector spatialsensitivity profile in the laser scanning system playsthe same role as the radiance distribution of the inco-herent source in the system treated in this paper.

We also point out that the results are applicable to theproblem of image evaluation in electron microscopy tosome extent, because in such a field observation of thelow-amplitude contrast complex object with the highdegree of coherence of illumination is a very importantsubject.

1D. Kermisch, J. Opt. Soc. Am. 65, 887 (1975).2H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).3B. J. Thompson, "Image Formation with Partially Coherent

Light, "Progress in Optics, edited by E. Wolf, Vol. VII(North-Holland, Amsterdam, 1969).


4R. J. Becherer and G. B. Parrent, Jr., J. Opt. Soc. Am.57, 1479 (1967).

5R. E. Swing and J. R. Clay, J. Opt. Soc. Am. 57, 1180(1967).

6M. De and S. C. Som, J. Opt. Soc. Am. 53, 779 (1963).7 R. Barakat, Opt. Acta 17, 337 (1970).8 M. De and P. K. Mondal, Opt. Acta 17, 397 (1970).9 R. E. Kinzly, J. Opt. Soc. Am. 55, 1002 (1965).

10Y. Ichioka, K. Yamamoto, and T. Suzuki, J. Opt. Soc. Am.65, 892 (1975).

"M. Born and E. Wolf, Principles of Optics, 2nd ed. (Perga-mon, New York, 1964), p. 530.

12K. G. Birch, Opt. Acta 17, 43 (1970).13p. Jacquinot and B. Roizen-Dossier, "Apodisation, " in Ref.

3, Vol. III, p. 31.R4K. Yamamoto (private communication).

932 J. Opt. Soc. Am., Vol. 66, No. 9, September 1976 Copyright � 1976 by the Optical Society of America 932

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Image of a Periodic Complex Object in an Optical System Under Partially Coherent Illumination