Fresnel Diffraction in Phase-Space

Fresnel diffraction in phase-space

R. Palomino1,2, M. de Icaza-Herrera2, V. M. Castaño2

1 Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, A.P. 1152 Puebla, Pue., México 2 Instituto de Física, UNAM, A.P. 1-1010 Querétaro, 76000, México

Abstract: Fresnel Diffraction is worked out in phase space by 2. Mathematical background simple calculations based on the Wigner Distribution Functions 2.1. The Cornu’s integrals for the transfer functions of an array of ideal lenses, which allow the simulation of a composite material and of a continuous Fresnels integrals are defined as: medium. s Key words: Wigner-distribution-function Ð fresnel-diffraction  pt2  C(s )= ∫ cos   dt (1)  2  0 s  pt2  1. Introduction S(s )= ∫ sin  dt . (2)  2  0 One of the classical problems of optics is the elegant solu- being customary to define tion to the diffraction by an aperture derived by Augustin 2 Jean Fresnel, from whose formulation the so-called Fres- s itp =≡+2 (3) nel integrals emerge naturally, constituting not only a BCS()sdtsis∫e () (). must in any optics course but also a powerful tool in many 0 areas of Physics. Marie Alfred Cornu, another pioneer of As it is well known in optics, the locus of this function is modern optics, devised a rather elegant and ingenuous the so-called Cornu’s spiral, which is used in various prob- geometrical description of the Fresnel integrals, which, in lems in optics. addition to representing a tool for calculating numerical values, also stresses out some of the very fundamentals of optics, related to trajectories, diffraction intensities and 2.2. The Wigner distribution function the like, opening the way to physical optics in a modern The so-called Wigner Distribution Function (WDF) is a sense. phase-space mathematical tool, originally designed in the With the emerging of computers, graphical methods such 1920s for dealing with quantum mechanics problems [3]. In as the beautiful spiral discovered by Cornu [1] have lost the last few decades, however, photon, X-ray and electron interest in the eyes of the specialists in optics. The funda- optics have found interesting applications of this and other mental physics behind, however, deserves further explora- related phase-space functions [3Ð8]. tion under the light of the new mathematical tools, special- In general, for a pure quantum system, described by the ly those developed with the aid of phase space functions and wave function Y( p) of its momentum p, the Wigner Func- group theory, for they have brought exciting possibilities tion (WDF) is defined as: into optics, not only for achieving a deeper understanding of some classical problems, but also for studying modern   n∞ 1n optical devices, such as fiber optics. Accordingly, this arti- Fd()q, p=   ∫ yΨΨ ( q+ y )*(±) q y  ph cle shows an alternative derivation of Cornu’s integrals and ± ∞ spiral, purely from phase space considerations, which also 2i ⋅⋅exp ± (py ) . enables us to recover the wave distribution from the inte- h grals themselves. Where, as usual, h is Planck’s constant divided by 2p and n the number of freedom’s degrees of the system. The main properties and mathematical characteristics of the WDF, as well as their relation to other functions (Cohens, ambiguity, wavelets, etc.) have been reviewed elsewhere [3Ð11] Received 19 June 2000; accepted 2 September 2000. being here only relevant to point out that the WDF is indeed Correspondence to: V. M. Castaño a phase-space distribution, since it depends, as observed in Fax: ++52-1-56234165 the above equation (1) on both spatial and momentum coor- E-mail: [email protected] dinates.

It is customary, however, to define the WDF of a function f, which represents a wave function or any other physical quantity, as: ∞ =  + u  u Wf (x,w) ∫ f  x  f * x ±  ± ∞ 2 2 ⋅ exp {± iw ⋅ u} d 2u, (4) where x and w are the position and momentum vectors. The kernel of the WDF is, of course:  u  u H ( p, u) = f p + f * p ± . (5) f  2   2 

3. Cornu’s integrals and the Wigner distribution function

Let us consider a physical medium formed by an array of tiny lenses, as schematically depicted in figure 1. This mod- el can describe nearly any composite material, that is, a physical system formed by a continuous matrix in which a collection of “lenses”, i.e. particles with different scattering Fig. 1. Schematical representation of a “crystal” character- properties as compared to that of the matrix, are embedded. ized by the lattice parameter a. By changing the periodicity, the size of the “lenses” and by introducing “defects” in the array, it is possible to simulate many real situations. Where Wf is the WDF before the layer Dz. For the case It is straightforward to calculate [4], in terms of a “Phys- of having a plane wave incident over that layer, the result- ical Optics” language, the transfer function fL of an ideal lens ing WDF is: (of focal distance f placed in the position qkx, qky of the crys- Wf ( p,w) talline structure: TP = 2 ∑ 4 p AA* k, l fL (x, y; qk) = f (x, y)   i k  q2 ± q2  z   ⋅ i ko 2 + 2 (6) ⋅ exp  o  l k +  p ± D w ⋅ (q ± q ) × exp ± [(x ± qkx ) (y ± qky ) ] ,   k l  2 f   f  2 k0   k  q ± q     where x, y and qkx, qky are the spatial and momentum spac- × o k l + Dz d   p ±  1 ±  w . (9) es coordinates respectively, ko the wave vector of the inci-  f 2  {   dent wave and f(x, y) is the distribution before the lens. To calculate the whole contribution of a layer of thickness Dz, Obviously, what matters, from a practical standpoint, is all one has to do is, according to the principle of super- to calculate the original function, either as a function f of the position, to add over all the lenses contained within that spatial coordinates, f(x) or as its Fourier transform F(w). To layer: do this, let us consider the kernel of the WDF and its relation to the original WDF [12]: g (x, y, z) = ∑ g(x, y) T  w  w k H (x,w) = f x + f * x ± f  2   2   i k  ⋅ exp ± o [(x ± q )2 + (y ± q )2 ] . (7) ∞  2 f kx ky  1 = ∫ Wf (x, K) exp {iw ⋅ K} dK. (10) 4p2 By applying the definition of equation (4), the WDF of ± ∞ equation (7) is: By evaluating Hf in x = w/2, one gets:  u  W ( p,w) = ∑ g (u) g * (0) = Hg , u , (11) fTP   k, l 2 = i k  q2 ± q2    since f (0) f *(0) = Hf(0,0), then f *(0) H f (0, 0) and ⋅ exp  o l k +  p ± Dz w ⋅ (q ± q )  ×    k l  = δ  f  2 k0  thus f (0) (H f (0, 0)) exp i . Substituting in equation (11):    × Dz + ko Dz qk ± ql  u  Wf  p ± w,w  p ± w ±  . Hg  , u    2 iδ k0 f k0 2 g (u) = e . (12) H (0, 0) (8) g R. Palomino et al., Fresnel diffraction in phase-space 39

In the case of the layer Dz, one can repeat the above inver- α (l ± ux) sion procedure to obtain the function, which yields: = AA* iδ  p  p f (u) 2 e   B(s) α 2 a α (± l ± ux ) H (x, w)  i k     p f = ∑ exp  0 q ± x ± w   l   α AA* l 2 ( f ± Dz 2  (l ± uy) ⋅ p  2  B(s) α . (19) i k   w (± l ± uy ) × ∑ exp ± 0 q ± x +  p 2 ( f ± z  k  2  l  D  (13) By defining: 4. Concluding remarks

k0 w w a = , P+ = x + , P± = x ± (14) The approach described here allows to deduce the Cornu’s ( f ± Dz) 2 2 integrals from fundamental considerations in phase space, the exponential functions can be re-written as: enhancing the physics involved, for the trajectories in phase space can be regarded as a generalized “geometrical” optics, iα 2 iα 2 (ql ± P± ) ± (qk ± P+ ) ∑ e 2 ∑ e 2 . (15) as has been demonstrated previously [6]. It is also important l k to point out the possibility of recovering the wave function f (x) or its Fourier transform F(u) from the Cornu’s integrals By taking the limit when the lattice parameter a of the in a very simple way, as shown in equation (16). The meth- array (fig. 1) is very small as compared to the wavelength od also allows to calculate the optical properties of compos- of the incoming wave, that is, in the case of having a con- ite materials as well as those of a continuous medium, by tinuous instead of a composite medium, the summations of conveniently choosing the parameters which define the lat- equation (10) become integrals: tice of fig. 1. This, along with computational advantages of iα 2 ± (qk ± P+ ) ∆q ∆q the WDF, which have been described elsewhere [4], open ∑ e 2 x y a2 interesting opportunities for explaining, and even predict- k ing, the optical behavior of complex media. Na iα 2 2 ± (q ± P+ ) 1 2 x x ≈ ∫ e dqx a ± Na References 2 Na [1] Hecht E, Zajak A: Optics. pp. 378Ð382. Addison-Wesley Pub- iα 2 2 ± (q ± P+ ) lishing Co, Tokyo 1974 1 2 y y ⋅ ∫ e dqy . (16) [2] Wigner EP: On the Quantum Correction for Thermodynamic a ± Na Equilibrium. Phys. Rev. 40 (1932) 749Ð759 2 [3] Castaño VM: A generalized display of the Contrast Transfer Function in Transmission Electron Microscopy. Optik 81 We have basically deduced Cornu’s integrals, since, by (1988) 35Ð37 making the following change of variables: [4] Gutiérrez-Castrejón R, Castaño VM: Analysis of optical systems through intermediate functions. Optik 91 (1992) Na 24Ð28 2 iα 2 ± [qx ± P± x ] [5] Vázquez G, Acosta D, Castaño VM: Two dimensional analy- 1 2 ∫ e dqx sis of the contrast transfer function in high resolution trans- a ± Na mission electron microscopy. Optik 89 (1992) 181Ð183 2 [6] Castaño VM, Santiago P, Gómez A: Image displacements in α   transmission electron microscopy: a phase space approach. Na ± P π  ± x  iπ t 2 Optik 95 (1993) 31Ð33 2 dt = 1 p ∫ e 2 [7] Gómez A, Santiago P, Castaño VM: The Van Cittert-Zernike a α theorem revisited: alternative approaches in the small angle α  Na  approximation for electron diffraction. Optik 95 (1993) 59Ð62 ± ± P± x p  2  [8] Reyes J, Rodríguez R, Cotorogea M, Castaño VM: Applica- α  Na  tion of Wigner Distribution Function to Fiber Optics. Optik 1 p  ± P± x  110 (1999) 305Ð308 = B(s) p 2 . (17) α α  Na  [9] Castaño VM, Gómez A: A phase space approach to X-ray a  ± ± P± x  p 2 optics. J. X-Ray Sci. Technol. 4 (1993) 67Ð76 [10] Cohen L: Generalized Phase-Space Distribution Functions. J. One can re-write the integrals as: Math. Phys. 7 (1966) 781Ð786 2 2 [11] Papoulis A: Ambiguity function in Fourier optics. J. Opt. Soc. a(qx Ð PÐx) = pt . (18) Am. 64 (1974) 779Ð788 [12] Palomino R: Aplicaciones de las técnicas de espacio fase a la Finally, by using the method described before to calculate caracterización ultrasónica de materiales. M.Sc. thesis (Phys- f (u) from the Kernel of the WDF, one obtains: ics), pp 45Ð53, U.N.A.M., Mexico City 1998