United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency

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1 IC/99/46 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS SPATIALLY PARTIAL COHERENCE TRANSFER AND IMAGING Roman Castaneda ' Physics Department, Universidad National de Colombia Sede Medellin, A. A. 3840, Medellin, Colombia and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract In this paper the Partial Coherence Transfer Function is introduced and its relationship with the Wigner distribution function is established. It depends on the spatial coherence properties and intensity distribution of the illuminating optical field, the imaging properties of the system and the geometry of the imaging. MIRAMARE - TRIESTE May Regular Associate of the Abdus Salam 1CTP. Fax: (00574) rcastane@perseus.unalmed.edu.co

2 1. INTRODUCTION Imaging with spatially partially coherent optical fields has been extensively studied by using models based on the ABCD parameters [1], the Zernike's formula [2] and the Wigner distribution function [3-8]. Applications of these models can also be found, particularly for optical microscopy [9,10] and electron microscopy [11]. In this paper we propose an alternative description for spatially partially coherent imaging. We introduce the Partial Coherence Transfer Function, which depends on the spatial coherence properties and intensity distribution of the illuminating optical field, the imaging properties of the system and the geometry of the imaging. Its relationship with the Wigner distribution function is discussed too. 2. PARTIALLY COHERENT IMAGING Imaging with a quasimonochromatic partially coherent optical field is usually analysed by considering its cross-spectral density at four planes associated to the system, named entrance window (ew, object plane), entrance pupil (ep), exit pupil (EP) and exit window (EW, image plane) (Fig.l). Furthermore, the cross-spectral densities at the entrance window and at the entrance pupil are related through a propagation rule given first by Zernike [12]. The same occurs with the cross-spectral densities at the exit pupil and the exit window respectively. Usually, both propagations occur in the so-called Fraunhofer- Fresnel's domain [12]. But the cross-spectral densities at the entrance pupil and the exit pupil respectively will be related through a transference rule, which takes the form of a quadratic product with a function that describes the imaging properties of the system. t ew ep EP Object Plane Partially Coherent Imaging System EW Image Plane Fig. 1: Spatially partially coherent imaging

3 Accordingly, let us consider an object with transmittance t(r)= t(f]e'*^\ attached at the entrance window and illuminated with a partially coherent optical field with cross-spectral density ' ^2) = A)^'^)! ^"* 12 * s me complex degree of spatial coherence [12] of the illuminating optical field, and 7 0 (r) its intensity distribution across the object transmittance. Note that 0 < /x o (/^,r 2 ) < 1 and ^0(r,r) = 1. Then, the cross-spectral density that emerges from the entrance window will be given by Wj7 l,r 2 ) = W 0 (7 ],7 2 )t{7 ] )f(7 2 ). (2) By introducing this quantity into the Zemike's formula for the Fraunhofer-Fresnel's domain, we obtain the cross-spectral density at the entrance pupil, that is where X denotes the mean wavelength of the optical field, z the distance between the f- 1 ^ 'F J 2 2^: entrance window and the entrance pupil, q\ x; = 6 and k. { Xz) X For the analysis below, it is useful to separate the illumination properties and the geometry of the imaging from the object transmittance. Therefore, let us write eq.(3) as d\ (4) with Wo'(fj,r 2 ) = WQ^,^)^ r{, \q*\r 2 ;. Taking into account that { Xz) { Xz) - f i- -i- f" E -~ E ) t(r)= \T\ J6 z d 2 E, and WQ'^,,^2 j= JJWo^F^Fj)^ z d 2 r { d 2 r 2, the integrand in eq.(4) can be written as a function of the pupil co-ordinates as follows

4 On the other hand, the imaging properties of a system with focal length /, lateral magnification /3 and pupil function P\ ')= P^'\, with <&{ ') the wave-aberration function [12] and f = jbf', are described by the function P( "')<?* I"'; [2,10]. I X f ) Because J5 is only a scale factor, we can assume /?= 1 for the sake of simplicity and without loss of generality. Thus, the transference of the cross-spectral density from the entrance pupil to the exit pupil is given by (6) Once again, we apply the Zernike's formula in Fraunhofer-Fresnel's domain to obtain the cross-spectral density at the exit window of the imaging system, which can be expressed as w EW \ (7) where w F fc -77,, 2 -r/ 2 )=W 0 'g - + is the focussing factor [2]. Eq.(7) is valid for any plane into the Fresnel region in the image space, whose location is determined by the focussing factor. The condition = 0 F determines the plane of the best focus or diffraction focus plane [12], which is located into the Fraunhofer domain. At this plane we obtain diffraction-limited images of the object if the system is aberration free. Eq.(7) can be expressed as WW^'^') = -\ cnr(; - \q* \r 2 ; -\ )T{f} 2 )W F (77,,fj 2 ; i\,r 2 ') d\ d 2 r] 2, where the function y Xz J y Xz J y Xz J [ Xz J describes the propagation from the exit pupil to the exit window of the system. This function depends on the spatial coherence of the illumination, the pupil function of the system and the geometry of imaging.

5 3 TRANSFERENCE OF SPATIALLY PARTIAL COHERENCE It is convenient to express eq.(7) in terms of the centre and difference co-ordinates [7]: (9) So, we have f, = -^D, f 2 = " A - ^f D, F,'= F A ' + - F D ' and F 2 ' = r A - - r' D. Furthermore, q '^ J ' e = e and d 2^d 2^2 -d 2^Ad 2^D because the Jacobean of the co-ordinate transformation is equal one [8]. Therefore, eq.(7) can be expressed as Y (i Y From eqs.(8) and (10) it is straightforward that WA %f) 2 -,T' A +-r' D,r' A --?»)= (10). * -, r, with [ 1 1 \ f \ \ f 1 \ k - % F+li _n -if -n P<f+^f P*<f-if e" 7 '" d 2 c t ^A^ TSD ''I >SA TfiD 'h \ r \ SA T TSD r 3^ ^o According to eq.(ll), W F \0,0; F A ',<f 4 ) is a space-frequency description of the behaviour of the optical system under spatially partial illumination. Indeed, it exhibits the mathematical form of a certain Wigner distribution function [3-5,8], where the involved values of the ~ ( I- \ pupil function are weighted by W F <f A + -<f D1, % A D. Following Bastiaans [3], it represents the amplitude of a ray passing through the position t, A in the exit pupil and having a spatial frequency - JL - /, corresponding to its direction toward the exit window A z plane in paraxial approach (Fraunhofer-Fresnel's domain). (ID " SD

6 To obtain W F \r\ x,ri 2 ; r' A,E, A ) only the weighting values into the integral should be changed ~ f ^ by shifting the function W F \ A + ^D1, t; A - t, D by T] x,r\ 2, as occurs in eq.(l 1). For this reason, eq.(ll) is not a Wigner distribution function as usually defined, but behaves in a similar way. Fig. 2: Illustrating the Partial Coherence Transfer Function The integration domain in eq.(ll) is determined taking into account the size of the exit pupil and the size of the region, inside which W F will take its main values. If the region is greater than or of the order of the exit pupil, the integration domain will be the exit pupil.

7 Otherwise, the integration domain will be the region. The so defined integration domain determines the set of radiator pairs, which contribute for the values of W F. From eqs.(10) and (11), the cross-spectral density at the exit window plane can be written as (12) It is apparent that W F (77,,77, ; r" A,E, A ) plays the role of a Transfer Function, which takes the spectral information of the object at the entrance pupil of the system, r(<f j, and transfers it to the exit pupil (Fig. 2). It depends on: Spatial coherence properties and intensity distribution of the illuminating optical field. Imaging properties of the system. Geometry of the imaging. For this reason we call it the {Spatially) Partial Coherence Transfer Function. Acknowledgments This work was done within the framework of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The author expresses his thanks to the Abdus Salam ICTP for support. REFERENCES 1. Friberg, A.T. (Ed.). Selected papers on Coherence and Radiometry. SPIE Milestone Series Vol. MS 69, Castaneda, R. and F.F. Medina. Partially coherent imaging with Schell-model beams. Opt. Laser Tech. 29 (1997) Bastiaans, M.J. The Wigner distribution function and Hamilton's characteristics of a geometric optical system. Opt. Comm. 30 (1979) Bastiaans, M.J. The Wigner distribution function of partially coherent light. Opt. Acta 28(1981) Bastiaans, M.J. Application of the Wigner distribution function to partially coherent light. J. Opt. Soc. Am. A 3 (1986) Lohmann, A.W., J. Ojeda-Castaneda and N. Streibl. The influence of wave aberrations on the Wigner distribution. Opt. Appl. 13 (1983) Brenner, K.H. and J. Ojeda-Castaneda. Ambiguity function and Wigner distribution function applied to partially coherent imagery. Opt. Acta No.2 (1984) Castaneda, R. On the relationship between the cross-spectral density and the Wigner distribution function. J.Mod.Opt. 45 (1998) Courjon, D, D. Charraut and G. Bou Debs. Bilinear transfer in microscopy. J. Mod. Opt. 34 (1987)

8 10. Glindemann, A. Measurement of the degree of coherence in conventional microscopes. Proc. SPIE 1028 (1989), Medina, F.F. and G. Pozzi. Spatial coherence of anisotropic and astigmatic sources in interference electron microscopy and holography.!. Opt. Soc. Am. A 7 (1990) Born, M and E. Wolf. Principles of Optics. 5. ed. Pergamon Press, Oxford, 1975.