The optical transfer function of a perfect lens with polarisation masks

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1978 J. Opt. 9 251

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Original article J. Optics (Paris), 1978, vol. 9, no 4, pp. 251-254

DEPARTMENT OF APPLIED PHYSICS. CALCUTTA UNIVERSITY,

92. Acharya Prafulla Chandra Road. Culcutta-9 ilndiu)

THE OPTICAL TRANSFER FUNCTION OF A PERFECT LENS

WITH POLARISAT1,ON MASKS A. K. CHAKRABORTY, B. MONDAL ADHIKARI, R. ROY CHOUDHURY

Mors CLBS : KEY WORDS : Fonction de transfert Transfer function Filtre polarisant Polarization mask

Fonction de transfert d’un objectif parfait diaphragm& par des filtres polarisants

RESUME : Nous avons ttudii. la fonction de transfert (FTO) S-UMMARY : The optical transfer function (OTF) of a perfect d’un objectif parfait dont I’ouverture est diaphragmte par des lens with zonal masking of the lens aperture with polarisation filtres polarisants annulaires. Nous avons dtmontrk que la FT0 devices has been studied. It is seen that the OTF of such system d’un tel systeme est une fonction continue de l’etat de polarisation can be varied continuously by varying state of polarisation of the du faisceau d’entree et egalement de l’orientation de I’analyseur input beam and also by changing the orientation of the analyser plact a la sortie. placed at the output side.

1. - INTRODUCTION

The imaging qualities of an optical system depend upon its optical transfer function (OTF). In order to change the nature of the image obtained by an optical system it is thus necessary to modify the OTF of the system. The modification of OTF is necessary for a number of purposes. The use of apodisers at the pupil of an optical system for side-lobe suppression is well-known [l]. The question of super-resolution has also added new interest for suitably modifying the imaging qualities of the optical system. Several authors have studied the effect of phase-coatings on the lens aperture. Osterberg and Wilkins [2, 31 have considered the effects of circular central phase-coat- ing. The effects of semi-transparent and phase annuli have been studied by Thompson [4]. Lit [5] has shown that the suitable use of phase-coating may markedly improve the resolution of the lens. Recently, Chakra- borty and Mukherjee [6] have studied the effects of using polarisation masks on different zones of a perfect lens. They have shown that by suitable choice of the state of polarisation of the input beam and with proper orientation of the polarisation masks one can simulate the effects of different phase and amplitude coatings at different portions of the lens.

In the present article, we have studied the variation of OTF of a perfect lens having polarisation masks at

different zones with the orientation of the analyser at the output.

2. - THEORY

The principle of the method proposed is based on the fact that if the pupil transmission function of an optical system is made dependent on the state of polarisation of the incoming beam the diffraction spread produced by the system will also depend on the input,polarisation. Besides, by the use of an ana- lyser at the output, the two orthogonal components of diffracted vector wave may be made to interfere yielding a redistribution of ,intensity in the diffraction pattern. In order to establish the principle it would suffice to consider the situation where the two portions of a lens are masked by two polarisers at two different orientations and input beam is polarised. Effectively similar situation is obtained if retardation plates or rotators are used as polarisation masks.

Let us assume that the input beam is elliptically polarised and is represented by the Jones vector,

252 A. K. CHAKRABORTY et al. J . Optics (Paris), 1978, vol. 9, no 4

The azimuth t+b and the ellipticity x of the input beam are given by the following equations

( 2 ) { tan 2 rc/ = tan 2 tl cos 6 sin 2 rc/ = sin 2 c( cos 6

where a = tan- bjn. The central zone of radius E is masked by a polariser whose transmission axis makes an angle 8 with the X-axis (figure I ) , which may be represented by the projection matrix P(@ given by

cos2 e (3) p(e) =

sin Q cos 0 sin e COS e sin2 e

Y

FIG. 1.

The vector wave illuminating the central zone of radius E will, therefore, be given by

= p(e) J~

Similarly, if the outer annular zone extending from the radius E to 1 is masked by a polariser having its transmission axis at an angle a with the X-axis, the wave illuminating this annular zone may be written as,

( 5 ) E , = (a cos a + h eis sin a) cos a

The vector amplitude distribution in the Franhaufer diffraction plane due to the central zone of radius E may be written as

~ , ( k & ~ ) COS e (6 ) J, = (a cos 8 + h ei6 sin e) I T E ~ ~ ksp I sin e I where J , ( Z ) is the Bessel function of first rank, k is the propagation constant and p is the radial co-ordinate in the observation plane.

Similarly, the vector amplitude distribution due to the annular zone may be given by,

(7) J , = (a cos tl + b eis sin a) x

J I P

X IT- - m'- Jl(kEP)3 , I ;P,"," 1 kW

Now, if there is no analyser at the output, the vector amplitude distribution in the image of a point will be given by,

A cos e + B cos tl (8) J = J, + J, = A sin e + B sin a

A = (a cos e + h eis sin e ) m 2 - where,

J l (kv) kEP

and (9) B = (a cos a + h ei6 sin a) x

The intensity distribution at the Franhaufer diffrac- tion plane is, therefore, given by

(10) I ( p ) = ( Tr (J x J*) )

where X indicates the kronecher product and the sharp brackets represent the time-average.

If, however, an analyser is introduced at the output at an azimuth p, the vector amplitude distribution at the image of a point is given by,

(11) J ( P 3 P ) =

cos2 p sin cos P A cos 8 + B cos a = I sin p cos p sin2 P / I A sin 0 + B sin a

'

The intensity at the image plane or the point spread function of the lens including the polarisation masks and the analyser will, therefore, be given by,

(12) I(P, P ) = ( Tr [J(P, 1) x J* (P , m1 ) = I ( A cos 0 + B cos a) cos +

+ ( A sin e + B sin a) sin p l 2 = I A COS ( p - e ) + B COS ( p - a) 1' .

Now, if we assume that the input beam is linearly polarised along X-axis (h = 0) and if we let 8 = n/4 and 2 = - n/4, then apart from a constant photometric factor, the PSF may be given by

J. Optics (Paris), 1978, vol. 9, no 4 A. K. CHAKRABORTY et al. 253

where, X = k p , a new co-ordinate in the image plane. The OTF of the system may, therefore, be given by,

T(to, /I), E ) = (cos - sin /S)’ ~ ( w , 1, I ) + 4 E’ sin’ /?d(w, E , E ) + 4 c2 sin’ /?(cos /? - sin /?) d(w, 1, E )

= TIE’ , if I (o 1 I - E~ I = 0 , if 1 (11 1 > (c1 + E’ ) .

The normalised value of the OTF may, therefore, be expressed as,

F((!,. p. I : ) = - [(cos /I - sin p)’ d(w, 1, 2) + 4 e4 sin’ /?A(w, E , E ) + 4 E’ sin’ /?(cos - sin /?) ~ ( o , I , E ) ] 1

A 0

(1 5 )

where (16) A , = cos /I) - sin P)’ + 4 sin’ /? + 4 m’ sin’ /?(cos /? - sin /?) .

E:0,707

E x 0 5

075 - E.03

(3=3O0

1J - FIG. 2.

I The OTF of the optical system considered will,

therefore, be given by the Fourier transform of intensity distribution Z(p, p) at the image of a point. It is seen from Eq. (9) that A and B depend upon the state of polarisation of the input beam and also on the aportioning of the lens aperture, that is on the value of E. The OTF of the system, therefore, depends on (i) the state of polarisation of the input beam, (ii) the value of E (iii) on the orientation of the masks, that is, on the values of a and 0 and (iv) on the orientation of the analyser, /?.

J . Oprics (Par l s i . 1978, vol. 9, no 4

F I G 3

It is interesting to note that the OTF of the system depends upon three convolution terms. d(o, 1. 1) represents the convolution of a unit circle with respect to itself. Similarly, d(o, E , E ) is the convolution of a circle of radius E with respect to itself and d(o, 1, E ) represents the convolution of a unit circle with another circle of radius E . The OTF for a certain orientation of the analyser is, therefore, given by the linear

21

254 A. K. CHAKRABORTY et al. J . Optics (Paris), 1978, vol. 9, no 4

combination of these three convolution terms, the weightage of each term being dependent on the orien- tation of the analyser.

3. - COMPUTATION AND DISCUSSIONS

We have computed the optical transfer function of a perfect lens masked at the central and the outer zones by two polarisers, having azimuths along 7c/4 and - n/4 respectively with X-axis. The computa- tions have been made for three different values of E ,

FIG. 4.

viz., 0.3, 0.5, 0.707 for different orientations (p) of the analyser. It is seen that for p = 300, the spatial fre- quency response is positive for all values of E consi- dered (figure 2). For f i = 600 and for f l = 90°, the OTF assumes negative values which signifies that there is a contrast reversal within certain range of spatial frequency (figure 3 and figure 4 ) . It must be mentioned that when equals 900, the system behaves as a lens having a central x-phase-coating as was shown by Chakraborty and Mukherjee [6] .

From the nature of the OTF, some interesting appli- cations of these systems may be envisaged. The improvement of contrast of the degraded image by subtracting the focussed and defocussed image of the same is well known. From the nature of the OTF for E = 0.3 and E = 0.707 in case of = 900, it is clear that if the images obtained in these two cases are subtracted from one another there is a conside- rable improvement in the high frequency contents of the degraded image with a consequent improve- ment of contrast.

* * *

REFERENCES

[l] JACQUINOT (P,) et al. - Progress in Optics, 3 (ed. E. Wolf),

[2] OSTERBERG (H.), WILKINS (J. E.). - J . Opt. Soc. Am., 1949,

[3] WILKINS (J. E.). - J . Opt. Soc. Am., 1950, 40, 220. [4] THOMPSON (B. J.). - J . Opt. Soc. Am., 1965, 55, 145. [S] LIT (J. W. Y . ) . - J . Opt. Soc. Am., 1971, 61, 297. [6] CHAKRABORTY (A. K.). MUKHERJEE (H.). - J . Optics ( India)

North-Holland, Amsterdam, 1964, p. 31.

39, 553.

1976, 5. 71.

(Manuscript received 22 february 1978.)

0 Masson, Paris, 1978