Volume zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 111. nnmber 1.2 CHEMICAL PHYSICS LEM ’ERS 26 October 1984 SPACIAL HARMONIC ANALYSIS OF TRANSIENT OPTICAL GRATING Taihyun CHANG, Hongdoo KIM and Hyuk YU Departrmn t of C7remistry, Uttiversity of Wisconsin. Madison. Wisconsitt 53 106. USA Rreceived 22 June 1984; in final form 23 July 1984 A novel modification of the forced Rayleigh scattering method for determining the translational diffusion coefficient and lifetime of photochromic moieties is effected through imposition of a nonsinusoidal transient optical sating flOt.2) to a condensed medium sample. A multiple~rder diffraction pattern is observed and subsequent analysis of the exponential decays of higher+xder diffraction spots reveals that they originate from the higher-order spa&l Fourier components of TOG. 1. Introduction A transient optical grating method, variously called forced Rayfeigh scattering (FRS) [I], IloIograph~c relaxation spectroscopy [2] or holographic grating technique [3j, has been used to study a number of thermal [ 1,4,5] and mass [2,3,6-g] diffusion prob- lems. An essential feature of the method is to deter- mine the relaxation time 7 of an exponential decay of the optical field diffracted by a TOG as it is erased by a thermal and/or mass diffusion process. Also, in order to extract appropriate coefficients of the diffusive processes, one must vary the ~~lar3~teristic distance d of the diffusion and observe the dependence of 7 on d, which for tflis technique is the period of the sinusoidal optical grating that is variable through crossing angle of the two writing beams to induce a TOG. We describe here a method whereby different grating spacings, d, can bc sampled at a fixed crossing angle. The novel feature in this instance is to perform spatial harmonic analysis of TOG in order to extract the coefficient of mass diffusion. in the FRS experiment to study mass diffusion, p~~otocilroIl~c probes are dispersed in the condensed medium; they may be photochromic dye mole&es or macromolecules iabeted by such dye molecules. A har- monic concentration profile of photochromicalIy ex- cited probe molecules is obtained upon illumination of the sample for a short time by two writing beams with wavelength h at a given crossing angle 8, which may be represented as C(x,O) = $4 [I f- cos(KIX)J, 01 where A is the amplitude of the photoc~lro~cally ex- cited state concentration and x1 is proportional to the spatial frequency of the harmonic concentration profile d-l, i.e. Kl = 27rK’ = (47r/h) sin(B/2), where the fringe spacing ci is given as (2) d = h/2 sin(t)/2). (3) In expressing the initial concentration profde (I), we have ignored the Gaussian profile of the writing beams. In doing so, the validity of the argument to be devel- oped below is not affected whiIe it allows some simpli- fication. Since we deal with one-dimensional diffusion, the time dependent concentration profile with the initial condition (1) is readily derived as C(X,f) = &4 11 + COS(K& exp(--nfDQ], (4) where D is the diffusion coefficient of the photochromic probe. The Fraunhofer diffraction pattern of the sinus- oidal amplitude grating represented by the transmission function (4) is proportional to its spatial Fourier trans- form [IO], c(K ,t) = $4 {6(K) + 71 eXp(-K$)[&(K - K1) + 6(K + K1)]), (5) 63 0 009-2614/84/S 03.00 0 Elsevier Science Publishers B.V. ~North-Holland Physics Publ~~in~ Division)