Two-point resolution of a microscope: effect of nonuniformand nonsymmetric illumination using a semitransparentir-phase annular aperture

V. P. Nayyar and N. K. Verma

This paper presents a theoretical study of the effects of partial coherence (via the effect of condenser aper-ture) on the resolution of two pinholes by a semitransparent and r-phase annular aperture microscope. Theillumination used is nonuniform and nonsymmetric. The response of the resolution has been studied inthree discrete planes of illumination. It has been seen that by making the central region of the r-phase an-nular aperture completely transparent, best resolution is obtained in the plane of illumination s = 7r/2. Theoscillatory behavior of the limiting resolution has also been displayed graphically.

Introduction

Despite the fact that a centrally obstructed circularaperture enhances the degree of resolution of an opticalsystem, the common use of an annular aperture with alarge portion of the aperture blocked is discouraged inmost optical instruments. There are two reasons forthis: first, a loss of light energy in the image in pro-portion to the area of the aperture which is blocked and,second, image quality deterioration because of un-wanted enhanced secondary maxima. Thompson'proposed the use of a semitransparent and ir-phaseannular aperture for improving the resolving power andfocal depth of optical instruments. This type of aper-ture consists of a semitransparent central region whoseamplitude transmittance only is varied, and the relativephase between the central and annular region is fixedas r-difference. This type of aperture if used as anobjective lens of a microscope will overcome the firstproblem by minimizing the loss of irradiance by thesemitransparent central region, and if the central regionis made fully transparent (by increasing the value of thereal part of complex amplitude transmittance, t tounity), best resolution is attained without sacrificing anylight energy. Recently Nayyar2'3 studied the variationof a semitransparent and 7r-phase annular aperture withthe degree of coherence employing various resolutioncriteria.

V. P. Nayyar is with Punjabi University, Physics Department,Patiala-147002, India. N. K. Verma is with Thapar College of En-gineering, Physics Department, Patiala-147001, India.

Received 6 April 1976.

The variation in the resolving power resulting fromthe coherence effects is quite significant in optical mi-croscopy employing various modes of illumination.The Kohler and critical illuminations are usually em-ployed. In the former illumination, the amplitudedistribution in the object plane is kept constant whilethe phase varies, and the reverse is true for the lattermode of illumination. However, it has been shown4

that the complex degree of coherence of the light inci-dent upon the object plane of a microscope is the samewhether critical or Kbhler illumination is used. Sinceit is evident that the resolving power depends only onthe degree of coherence of the light incident upon theobject and on the properties of the microscope objective,the limit of resolution is independent of the mode ofillumination. In most of the systems, it is adequate toconsider the illumination over the objective apertureto be uniform and symmetric. Sometimes, however, itis desirable to modify the image characteristics by theasymmetry and nonuniformity of illumination (e.g.,apodization).

In this communication, we have studied the variationof the limit of resolution with the ratio of the numericalapertures of the illuminating condenser (aligned toproduce nonsymmetric and nonuniform illumination)and of the semitransparent and 7r-phase annular aper-ture objective. This mode of illumination will coun-teract pronouncement of secondary maximum as aconsequence of the use of annular aperture and thusrestores the contrast in the image with improved reso-lution. The mode of illumination has been assumed tobe critical, and a modified version 5 of the Rayleigh cri-terion used for incoherent sources has been applied.

Theory

We consider two pinholes PI(xI,yI) and P2(x2,y2) inthe object plane. When these pinholes are illuminated

2460 APPLIED OPTICS / Vol. 16, No. 9 / September 1977

via an auxiliary lens system under conditions of uniformillumination, the irradiance distribution at any pointP' in the image plane of the microscope objective(semitransparent and 7r-phase annular aperture)

I(P') = [K(v1)]2 + [K(v2 )] + 2(PlP2)[K(vI)J[K(V 2 )J, (1)

where

K(v) = - 1 [2J1(v) l( + t)2J 1 (Ev)] (2)is the intensity response of the aberration free objectiveaperture. is the obstruction ratio (<1) of the aperture,and t' is the real part of the complex amplitude trans-mittance of the semitransparent region. AP1,P2 ) is thecomplex degree of coherence between the points P1 andP2, and

P = k [(x - X1)2 + (y - yl)2

1/2 no sin0O,1

V2 = k(x - X2)2

+ (y -Y2)21/

2no sinOol'

where no sinko is the numerical aperture of the objective,and (x,y) are the coordinates of a point in the objectplane.

The effect of nonuniform and nonsymmetric illumi-nation can be seen through the complex degree of co-herence at the object plane. Now according to the vanCittert theorem, the degree of coherence for any pair ofpoints in the plane of the objective is the same as due toan incoherent source filling the condenser aperture. Tocalculate the complex degree of coherence across thepoints P1 and P2, we would evaluate the normalizedcomplex amplitude at the corresponding point P1 in afar-field diffraction pattern produced by the condenseraperture centered on P2 .

The expression for the far-field diffraction patternof a circular aperture with nonuniform and nonsym-metric illumination is of the form6

ip exp(-ikr)a 2 2r (1E(r,O,0) = X Jo Jo

exp[iv't cos(, - ')] -(dedo', (4)

where

=-[(xI -x 2)2 +(Y1 - Y2)2 1/2 n,' sinO,'. (5)

n,' sinO,' is the numerical aperture of the condenser onthe side of the microscope objective; p,k' are the polarcoordinates in the condenser aperture; a is the radiusof the lens aperture; and r,O,b are the spherical polarcoordinates of the field point. f ,0') is an arbitraryfunction representing the aperture illumination, and t= p/a, the reduced aperture coordinate.

For an illumination of the type7

- f(I') = 1 -2 cos20, (6)the expression for the normalized amplitude would beof the form

E (i) =§ 3 .JJ ) + 1 J3(') + 1 JP') 2 l3 4 v' 4 v' 2 v' I

(P1P2) 2 [3 J(mv 2 ) + J3 (mvI2 ) (1 + 2 2)mv2 mv12

(8)

wheren,' sinO,'

m =no singo

P12 = v'/m = 27r/Xo[(xi -X2)2 + (Yi - Y2)2

]/2 no sino. (9)

Equation (1) holds only if P' is very close to the geo-metrical image of P1 and P2. If the points P1(xi,yi) andP2(x2 ,y2) are considered to lie on the X axis at equaldistances from the origin, then by considering the in-tensity I(P') only along the X axis, i.e., along the linejoining the geometrical image, we can rewrite Eq. (1)as

I(P') = K(v + b)]2 + K(v - b)]2

+ 2,u(m,2b)[K(v + b)I[K(v-b], (10)

where

K(P ± b) = 1--+E2(l2+Jt')X { ( i b) -E2(J + t') 2 [(v i b)]|

2 J(2mb) J3 (2mb) 1y.(r,2b) = -1 3 2mb + b (1 + 2 cos2) I.jim)3 L2mb 2mbi

(11)

(12)

v is the optical coordinate measured from the middlepoint of Pi', P2' defined by

0

C

CD

0)CG.E

= 0.0

01= 00 = "/4

-0 = /2

1 2 3 1 4Ratio of the numerical apertures, m

(7)

Thus the degree of coherence at the object plane isgiven by

Fig. 1. Oscillatory variation of limiting separation 2b8 with the ratioof the numerical apertures m for three planes of illumination. The

value e = 0.0 corresponds to the clear circular objective aperture.

September 1977 / Vol. 16, No. 9 / APPLIED OPTICS 2461

e = 0.50t' = 0.0

- = 0.0

-0 = /4

1 2 3 4Ratio of the numerical apertures, m

5

C= 0.50t' = 0.60

0= 0.0-0 = /4

-0= /2

Fig. 2. The limiting separation 2b plotted against the ratio of thenumerical apertures m for three values of the planes of illumination.

Value of e is 0.50 and that of t', 0.0.

5 e = 0.50t' = 1.0

0=0.0

= "/2

4

C

C

coCILCDco

Ei

0 1 2 3 4Ratio of the numerical apertures, m

Fig. 4. Same as in Fig. 2 with value oft' = 1.0.

Fig. 3. Same as in Fig. 2 with value of t' = 0.6.

2462 APPLIED OPTICS / Vol. 16, No. 9 / September 1977

t

CM

.2Cea)

.)

2.

5

. 404

C0

aa)

cC

.E

3

2.5

Ratio of the numerical apertures, rm

v = [(2ir)/(Xo)] xno sinto, (13)

and coordinate x is measured from the middle point ofP1 P2, and 2b is the optical separation of points P1' andP 2' given by

2b = v12 =- [(xI -x 2 )2 + (YI - y2)2 1'1 2 -no sin~o. (14)

,no

In order to obtain the critical resolution, Eq. (10) isiteratively solved by varying the separation 2b until theRayleigh criterion (according to which resolution isobtained when the intensity at the dip is 26.5% less thanthe intensity of the weaker maxima in the resultantintensity curve) is satisfied. During each iteration theminimum intensity (Im) of the central dip and the peakintensity of either maximum (Ip) is evaluated. Theratio Im/Ip is plotted against the corresponding valuesof 2b; and the separation which gives rise to the value0.735 is the Rayleigh limit of resolution.

Results and Discussion

The limiting separation has been obtained as afunction of the numerical aperture ratio m for threedifferent planes of illumination ( = 0,7r/4, 7r/2) in theobject space. The obstruction ratio of the objective lenshas been taken to be 0.0 and 0.50. The case e = 0.0corresponds to the clear circular aperture, whereas thevalue = 0.50 refers to the case when 25% of the totalarea is occupied by the central region. Values of thereal part of the complex amplitude transmittance t' ofthe central region have been chosen to be 0.0, 0.6, and1.0. The value t' = 0.0 corresponds to the opaque an-nular objective aperture. For each value of t', threecurves have been plotted, each corresponding to onevalue of 0. Other than the resolution criterion, thelimiting separation depends on the ratio m, values of eand t', as well as on the plane of illumination. In Figs.1-4, we have plotted the limiting separation against the

ratio of the numerical apertures m for three discretevalues of 0. The horizontal lines in Figs. 1-4 correspondto the limiting separation for a uniformly illuminatedcondenser with a numerical aperture identical to thatof the imaging lens (m = 1).

It is seen that the curves displaying the variation ofcritical resolution with m (whose values in some regionsgive rise to conditions of partial coherence) exhibit theoscillatory behavior of 2b5, i.e., for a particular curve,the limiting separation 2b, has a principal maximumvalue at m = 0 (which corresponds to conditions ofcomplete coherence), then it decreases to the lowestminimum, after passing through this value, it increasesand attains a local maximum, further decreases to ahigher local minimum, and so on. The critical resolu-tion 2b, oscillates around the value corresponding toincoherent illumination (at m = 1).

It is evident that for a particular value of e and t', thevalue of the lowest minima decreases as we go from theplane of illumination q = 0 to 0 = Ir/2. The value of themaximum attainable degree of resolution increasesconsiderably as the real part of the complex amplitudetransmittance t' increases from 0.0 to 1.0, e.g., for a clear7r-phase annular aperture (t' = 1.0, e = 0.50), the mini-mum value of 2b, comes out to be 2.48 in the = 7r/2plane.

REFERENCES1. B. J. Thompson, J. Opt. Soc. Am. 55, 145 (1965).2. V. P. Nayyar, Opt. Commun. 9, 377 (1973).3. V. P. Nayyar, Opt. Commun. 11, 27 (1974).4. H. H. Hopkins and P. M. Baraham, Proc. Phys. Soc. B. 63, 737

(1950).5. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,

1965).6. M. K. Hu, J. Opt. Soc. Am. 53, 261 (1963).7. This illumination is not circularly symmetric, but it does have

symmetry with respect to two principal planes. In the plane 0 =0, the illumination is (1- 2), and in the plane 0 = r/2, the illu-mination is constant.

K. N. Liou of University of Utah-left-and K. D. Krebs of Johannes Gutenberg University, photographed by F. S. Harris, Jr. of NASALangley at the August 1976 Garmisch-Partenkirchen Symposium on Radiation in the Atmosphere.

September 1977 / Vol. 16, No. 9 / APPLIED OPTICS 2463