Vol. 2, No. 5/May 1985/J. Opt. Soc. Am. A 743

Effect of polarizers used as masks on the perfect-lensaperture

Biswajit Chakraborty

Department of Applied Physics, Optics & Optoelectronics Section, University of Calcutta, 92, A.P.C. Road,Calcutta 700 009, India

Received August 1, 1984; accepted December 1984

The different zones of the lens aperture of a perfect lens are masked by polarization devices. The central circularzone of the lens aperture is masked by one polarizer, and the outer annular zone is masked by another polarizer.These two polarizers are at two different orientations. An analyzer is placed at the output side. It is seen that the

optical transfer function (OTF) of such a system depends on the state of polarization of the input beam, the orien-

tation of the analyzer placed at the output side, the radius of the central-circular-zone mask, and the orientationof the masks. In this paper, variation of the OTF of such a system with the orientation of the outer-annular-zonemask is studied.

INTRODUCTION

The imaging qualities of an optical system under incoherentillumination depend on its optical transfer function (OTF).In order to change the nature of the image obtained by anoptical system, it is thus necessary to modify the OTF of thesystem. The practical utility of the OTF has been discussedby Smith' and Kubota and Asakura.2 The following showsthe different systems in which the study of OTF's has revealedgreat practical importance. The study of the OTF of anapodized optical system is well known. 3 The opposite ofapodization, i.e., superresolution, 3 has also added new interestfor suitably modifying the imaging qualities of the opticalsystem. The effects of central circular phase coating havebeen studied by Osterberg and Wilkins,4 5 and the effects ofsemitransparent and phase annuli have been studied byThompson. 6 Lit7 has shown that the suitable use of phasecoating may markedly improve the resolution of the lens. Ahalf-wave centrally coated lens is found to be useful for pho-tography of extended objects, when the coating is made on theO.9a portion of the lens, where a is the radius of the perfect-lens aperture. 8 The use of polarization masks was found tobe effective for the modification of image qualities of an op-tical system. Chakraborty and Mukherjee 9 have studied theeffects of using polarization masks on different zones of theperfect lens. They have shown that by suitable choice of thestate of polarization of the input beam and with proper or-ientation of the polarization masks, one can simulate the ef-fects of different phase and amplitude coatings at differentportions of the lens.

In the present paper, we have considered a diffraction-limited optical system with zonal masking of a perfect-lensaperture. Two polarizers at two different orientations, oneat the central circular zone and the other at the outer annularzone, are used for masking. The pupil transmission functionof the optical system is made dependent on the state of po-larization of the input beam. Consequently, the diffractionspread produced by the system will depend on the input po-larization. Now, by placing an analyzer at the output, the twoorthogonal components of the diffracted vector may be madeto interfere, yielding a redistribution of intensity in the dif-

fraction pattern. The OTF of such system is studied with theorientation of the outer-annular-zone mask, keeping the or-ientation of the central-circular-zone mask and the outputanalyzer fixed.

THEORY

Let the input beam be elliptically polarized and be repre-sented by the Jones vector'0

b eia (1)

The azimuth t and the ellipticity X of the input beam aregiven by the following equations:

tan 2 = tan 2,y cos 6,

sin 2x = -(sin 2,y)sin 6, (2)

where y = tan-lb/a.Let the central zone of radius e of the perfect lens be masked

by a polarizer, whose transmission axis makes an angle 0 withthe horizontal direction. The polarizer may be representedby a 2 X 2 matrix, given by"

P(O) = cos20 sin 0 cos 0

sin 0 cos 0 sin2 01 (3)

The vector wave illuminating the central zone of the radiuse is given by

Ecentral = P(O)Jin

= (a cos 0 + b ei6 sin 0)|s 6Isin

(4)

The outer annular zone extending from radius e to 1 is maskedby another polarizer, with the transmission axis at an anglea to the horizontal direction.

Similarly, the wave illuminating this annular zone will begiven by

Eannular = (a cos a + b eia sin a) cos al.sin a

(5)

Now the vector amplitude distribution in the Fraunhofer

0740-3232/85/050743-04$02.00 © 1985 Optical Society of America

Biswajit Chakraborty

744 J. Opt. Soc. Am. A/Vol. 2, No. 5/May 1985

diffraction plane resulting from the central zone of radius Emay be written as

Jcentral = (a cos 6 + b ei6 sin 0)wE2 J,(EX) Cos 6(EX) sin

(6)

Here X = K<,, K is the propagation constant, and so is theradial coordinate in the observation plane. J,(Z) is the Besselfunction of the first kind of order one.

The vector amplitude distribution resulting from the an-nular zone may be written in the same way and is given by

Jannular = (a cos a + b e6 sin a)

X 7 J,(X) - 2Jl(EX)l lcos al.X (EX) Jisinag (7)

Now, if there is no analyzer at the output, the vector amplitudedistribution in the image of a point will be given by

I A cosS 6+ B cos a IJ = Jcentral + Jannular =A s I + B ,s (8)I A sin + B sin aI

where

(EX)

and

B=(acosa+beiasina) [J J,(X) E2 J, (EX)X (X

The intensity distribution at the Fraunhofer diffractionplane is therefore (Tr(JXJ*)). Here J* represents the Her-mitian conjugate of J, X represents the Kronecker product,and the angle brackets represent the time average.

If, however, an analyzer is placed at the output at an azi-muth ,. the vector amplitude distribution at the image of apoint is given by

J(X) =Icos2 sin ,Bcos A cos 0 + B cos a (10)Isin cos /3 sin2 j A sin 0 + B sin a

The intensity point spread function (IPSF) of the lens in-cluding the polarization masks and the analyzer is thereforegiven by

I(X), = (Tr[J(X)XJ* (X)]),=IA cos(,- 0) + B cos( - 12- 11

Now if we assume that the input beam is linearly horizon-tally polarized ( = 0, b = 0) and if we let = 7r/4 and l = 7r/2,then, from Eqs. (9) and (11), the IPSF may be expressed (ig-noring the constant photometric factor a27r2) as

I(X, a) = J (2) COS2 a sin2 a

J,2(fX)+ E4(0.25 + cos2 a sin2 a-cos a sin a)

J(J(EX)+ J(X) (EX) E2(cos a sin a - 2 sin2 a cos2 a).

(12)

The OTF of the system is given by the Fourier transform ofthe IPSF at the image of the point and hence can be writtenas

T(w, a, E) = cos 2 a sin2 aA(w, 1, 1)+ E4(0.25 + cos2 a sin 2 a - cos a sin a)A(w, E, e)+ E2(cos a sin a - 2 sin2 a cos2 a) A(w, 1, e).

where

A(w, El, E2) = E12 COS' [ 2 + 1

2- E22

2wel

+ E2 2 1o[- ~W2 + E22 - El2 1

2w+2

(13)

'(W__+_____-____ (w2+ E1

2- E2

2)

21/2

2w I 4w2 E,2 I(W2 + E22 - El 2 (w 2 + E22 E,2 )2 11/2

-Z E2 2w - 4w 2E22 J

when Iel - E21 < IwI < E1 + E21,= WE2 , when I wI < el - 21,= 0, when Iwl > El + E 21.

Here, A(w, el, 2) gives the common area between two over-lapping circles as a function of their radii El, E2 and the sepa-ration w between their centers. The normalized OTF may,therefore, be expressed as

F(w, a, E) = (/Ao) cos2 a sin 2 aA(w, 1, 1) + 4(0.25+ cos 2 a sin2

a - cos a sin a)A(w, E, E)

+ E2(cos a sin a - 2 sin 2a cos2 a)A(w, 1, )], (15)

where

Ao = T(0, a, e),= r cos2 a sin2 a

+ 7 6(0.25 + cos2a sin2 a - cos a sin- a)+ 7rE4(cos a sin a - 2 sin2 a cos2 a). (16)

From Eqs. (9) and (15), it is seen that the OTF of the systemdepends on the state of polarization of the input beam and thevalues of e, a, , and /3. It is interesting to note that the OTFof the system depends on three convolution terms: A(w, 1, 1)represents the convolution of a unit circle with respect to itself,A(w, E, E) represents the convolution of a circle of radius E withrespect to itself, and (w, 1, E) represents the convolution ofa unit circle with another choice of radius E. The OTF for acertain orientation of the outer-annular-zone mask is thereforegiven by the linear combination of the three convolutionterms, the weight of each term being dependent on a.

COMPUTATION AND DISCUSSIONS

Expression (15) can be looked on as the ratio of the contrastin the image to that in the object. The unmodulated back-ground luminance has an equal effect on both the numeratorand the denominator and hence cancels out. This expressionwas evaluated for a = 0, -45°, -60°, -75°, -90° and for E= 0.30, 0.50, and 0.707.

For a = 0 and -90°, the plane of vibration of the lightcoming from the outer annular zone will be perpendicular andparallel, respectively, to the transmission axis of the analyzerat the output. In both the cases, we get the transfer functions(Fig. 1) for the celebrated Airy diffraction disk at the Fraun-hofer diffraction plane that is due to the central circular zone.For such a focused aberration-free system, the transferfunction is given by the overlapping area of the aperture,

Biswajit Chakrab~orty

(14)

Vol. 2, No. 5/May 1985/J. Opt. Soc. Am. A 745

0.9

90.kI

O.3

0.1

0.4

Fig. 1. Transfer functions for a = 0°, -900, 6 = 450, f3 = 900.

0.9

0.8

0.7

frequency detail. At a = -45°, the system behaves as acentral r phase coating.9 It is interesting to note that, at a= -75° and e = 0.707 (the areas of the central circular zoneand the outer annular zone are equal), the OTF has a negativevalue (Fig. 4), which signifies that there are contrast reversalswithin a certain range of spatial frequency, and the contrastfalls to zero between regions of sign change. With radiallyoriented lines (spoke target), this 1800 phase change can beobserved. This corresponds to the OTF of a defocused lenswithout aberration. A gradual attenuation of contrast and

6 0.30 '-60

0.9 6- 0.50

6 - 0.707

0.8

0.7

t060.5-

a3

6=0.30

's. .50

cL - -45

0.8

g 3-

Fig. 3. Transfer functions for a = -600, 9 450, /3= 900.

0.5

.

0.4

0.3

0.21

0.!

0.8

0.7

0.4

Fig. 2. Transfer functions for a =-45°, 0 = 450, / = 900.I

shifted linearly against itself. The OTF drops to zero whenw is equal to the diameter (2E) of the central circular zone.

For both the curves (Figs. 2 and 3), with a=- 450, -600and e = 0.30, 0.50, there is an increase in contrast for thelow-frequency end of the curve, with a corresponding loss incontrast at the high-frequency end. But, for E = 0.707, thereis an increase in contrast at the high-frequency end of thecurve, and a peak near the resolution limit of the system isobtained. This can best be interpreted by a correspondingcentrally obstructed system." The high-spatial-frequencyresponse of the system is improved at the expense of low-

6 -0.30 dL -- 75

e 0.707

, _,-

Fig. 4. Transfer functions for a = -750, 0 = 450, / = 900.

Biswajit Chakraborty

746 J. Opt. Soc. Am. A/Vol. 2, No. 5/May 1985

contrast reversals is obtained for increasing spatial frequencywhen the defocusing error is severe. From the nature of thetransfer-function curves, we find that the system can be use-fully applied to the contrast enhancement in a certain rangeof spatial frequency. The improvement of contrast of thedegraded image, by subtracting the focused and defocusedimage of the image, is well known.' 2 Hence if the imagesobtained for e = 0.30 and e = 0.707 (at a = -75°) are sub-tracted from each other, the gain in the difference image willbe maximum at w = 0.40.

Similarly if the images obtained with a = -75° and a =-60° (for e = 0.707) are subtracted from each other, the dif-ference image will show maximum gain at w - 0.90. Here, thegain of the peak value over the gain at w = 0.50 is approxi-mately 15%.

ACKNOWLEDGMENTS

The author is indebted to A. K. Chakraborty for his valuablesuggestions and discussions and thankful to S. Dutta of theComputer Centre, Calcutta University, for his untiring co-

operation. Finally, the author is grateful to the UniversityGrants Commission for granting him a research fellowshipthat enabled him to undertake this work.

REFERENCES

1. F. D. Smith, Appl. Opt. 2, 335 (1963).2. H. Kubota and T. Asakura, Appl. Opt. 1, 284 (1962).3. P. Jacquinot, and B. Roizen-Dossier, in Progress in Optics III,

E. Wolf, ed. (North-Holland, Amsterdam, 1964), p. 31.4. H. Osterberg and J. E. Wilkins, J. Opt. Soc. Am. 39, 353

(1949).5. J. E. Wilkins, J. Opt. Soc. Am. 40, 220 (1950).6. B. J. Thompson, J. Opt. Soc. Am. 55, 145 (1965).7. J. W. Y. Lit, J. Opt. Soc. Am. 61, 297 (1971).8. P. K. Katti and M. Singh, Opt. Acta, 20, 959 (1973).9. A. K. Chakraborty and H. Mukherjee, J. Opt. (India) 5, 71

(1976).10. W. A. Shurcliff and S. S. Ballard, Polarized Light (Affiliated

East-West, New Delhi, 1971).11. E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley,

New York, 1963).12. G. L. Rogers, Noncoherent Optical Processing (Wiley, New York,

1977).

Biswajit Chakraborty