Applied Physics. Analysis of Optical Propagation in Thick Holographic Gratings. R. Alferness. Appl. Phys. 7, (1975) 9 by Springer-Verlag 1975

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1 Appl. Phys. 7, (1975) 9 by Springer-Verlag 1975 Applied Physics Analysis of Optical Propagation in Thick Holographic Gratings R. Alferness Electro-Optics Lab, Dept. of ECE, University of Michigan Ann Arbor, Mich , USA Received 20 Noveber 1974Accepted 24 January 1975 Abstract. A ethod of analyzing optical propagation in thick holographic gratings by decoposition of the thick aterial into thin gratings is discussed. The ethod is readily applicable to study propagation in ultiple gratings of arbitrary spatial frequency and orientation recorded in the sae thick eulsion. Applied to the double grating case, the ethod predicts strong cross-coupling between the two gratings for proper relative slope of the gratings. Results are given. nde Headings: Thick holographic gratings Optical propagation The theory of optical propagation in a holographically produced grating recorded in a thick eulsion has been developed by several authors [1--4]. When valid, Kogelnik's closed for solution using the coupled-wave approach is perhaps the ost useful [-2]. However, the ethod requires strong Bragg effects so that only the zero order and the one strong diffracted wave need be considered. The atri theory developed by Burckhardt [3] and Kaspar [4], which does not require the assuptions ade by Kogelnik and is theoretically capable of providing eact results, including the energy echange aong higher orders, requires finding the eigenvalues and eigenvectors of an infinite atri by eans of a digital coputer. n practice the atri ust be truncated, but the error introduced is sall. Kogelnik has included the possibility of sloped fi'inges while the latter two assue that the fringes are perpendicular to the plane of the eulsion. Neither of these ethods applies to a grating structure coposed of two or ore sets of fringes of arbitrary spatial frequency and orientation recorded in the sae eulsion. Such gratings are coplicated by the possibility of light diffracted fro one fringe structure subsequently being diffi'acted by a different set of fringes. We describe here a new, intuitively siple ethod for analyzing optical propagation in general grating structures holographically recorded in a thick eulsion. The approach is particularly well suited to treat several sinusoidal fringe structures of arbitrary frequency and orientation recorded in the sae eulsion. Even for the single grating case, the conceptual siplicity and ease of ipleentation of this approach offers advantage over the theories given by Burckhardt and Kaspar. While, like the atri theory, the ethod is nuerical, it requires only atri ultiplication. The ethod proceeds by decoposing the thick aterial into a series of thin slabs, each of which acts siply as a thin grating. Very general applicability can be obtained because all that need be specified are the properties of the thin gratings which ay be absorption, phase or ied. Because the effect of higher orders is included, eact results are possible and there is no need to require strong Bragg effects. Multiple fringe structures can be considered by allowing each thin grating to have several basic spatial frequencies and appropriate fringe slant factors. 1. Description of the Method For siplicity we describe the ethod for a single grating structure although etension to the ultiple grating case is straightforward. We consider the grating produced holographically, as shown in Fig. 1.

2 30 R. Alferness D i z=o Ol. Fig. 1. Construction of single thick grating by interferring plane waves R and S. The eulsion thickness is D The fil eulsion of thickness D is perpendicular to the z ais and is assued to be of infinite etent in the and y directions. We liit the analysis to one lateral diension. Plane wave R of aplitude a 1 incident at angle 01 (angles easured inside the eulsion and the sign convention is that of Goodan [7]) with respect to the z ais and S of aplitude a 2 at 02 interfere to produce a grating throughout the volue of the eulsion. The eposure E = ~ ( is the total intensity and 9 is the eposure tie) can be written as [5] E(, z)= ~o{1 + r cos [2~z f( - ~)]}, (1) where f = (sin01- sin02)2, 2 is the free space wavelength, z=z tanq~, ~b--( )2 is the slant of the fringes, o = a 2 + a 2 and r = 2al az(a 2 + a2). For any plane z = constant, the fringes are sinusoidally odulated with frequency f. However, because of fringe slant (when the construction bea directions are not syetrical about the z ais) the fringes at z are shifted by ~ relative to those at z = 0 which introduces a relative phase shift of -2rcf~. (We stress that f is the frequency of the fringes in any z = constant plane and not the inverse of the perpendicular distance between fringe planes.) We eaine the properties of optical propagation in the thick grating of Fig. 1 by decoposing it into a series of thin slabs of thickness A z. The agnitude z of A z ust be sufficiently sall so that each slab acts as a thin grating. A reasonable upper liit on A z to eet this condition is that for the given readout wavelength and grating frequency, Kogelnik's Q factor (Q~2~2Az f2) [2] for the thin grating be on the order of one. Each of the thin gratings, therefore, ehibits no Bragg effects and its angular distribution of diffracted light is accurately predicted by thin grating theory. We also assue that the actual diffraction process takes place discretely at the planes z=naz, n= 1, 2... To deterine the aplitude transittance of each of the thin gratings that results fro the eposure given in (l), we ust specify the type of recording aterial. Here we consider a pure phase aterial (for eaple, dichroated gelatin). We also assue a linear relationship between the total inde of refraction change A n, relative to the bulk eulsion inde no, and the eposure E, A n=e. The phase shift resulting fro the inde change for a plane wave passing through the thin grating at arbitrary angle Oi is (2TEA na z))~cosoi. Therefore, the aplitude transittance resulting fro the eposure given by (1) for a plane wave at Oi is [6] T a (, z) = ep [(] 2 ~ ~ o A z)~cos 01) 9 {1 + rcos [2 Tc f ( - Xz)]}] 9 Using a failiar epansion [6], the aplitude transittance of the n th grating can be written as T A (, n) = ep {j27c z o A z2cos Oi} q= --cjo (j)q Jq(b) ep[j2rcqf( -,)], where Jq(b) is the qth order Bessel function, b = (21rA z'crlo))~ cos01 = (2~A znl)2 cos 0 i and, = n A z tan~b, nl = rrlo is the inde odulation of the grating. The n TM thin grating can therefore be represented by an infinite superposition of aplitude gratings of spatial frequency qf with diffraction aplitudes given by the Bessel function of order q. The eponential outside the suation sign in (3) represents an uniportant coon phase factor and will be dropped. Note that the diffraction aplitudes depend upon the thickness of the thin slab and the wavelength and direction of the incident plane wave as well as upon the inde odulation nl. Also the qth frequency coponent is shifted in phase by - 2~z qfn A z tan~b because of fringe slant. The effect of this phase shift is iportant because it is the eans by which angular selectivity about the Bragg angle results. (2) (3)

3 Optical Propagation in Thick Holographic Gratings 31 To find the effect of the n TM slab upon an incident plane wave of arbitrary spatial frequency we use the results of thin grating theory. Consistent with such theory the effect of each thin slab can be divided into two parts. First, the propagation of the incident wave over the interval A z which is described by the free space transfer function [7]. Second, the separation of the incident wave into the various diffracted waves at z = n A z as dictated by the aplitude transittance function in (3). Therefore, for an input plane wave of spatial frequency f~, aplitude Ai and phase ~i at z=(n-1)az, the field at z=naz(i.e., out of the n TM grating) is A(, naz)=ep(j~b~)a~ "ep(-j2~zqf n) ~. (j)qjq(b) q----oo -ep {-(J2rc~ 9Az) (1-22 f2f~ l (4). ep fj 2 ~z( f + q f)]. For each plane wave incident upon the n th slab there is a countable infinity of plane waves of spatial frequency (f + q f) produced. The aplitude of the qth wave is the product of the input aplitude and the strength of the qth grating frequency coponent [Jq(b) in this case]. The phase shift of each wave is given by three factors. The ter ep(-2~zqfn) in (4) represents the phase shift for the qth order of the grating that results fro the lateral shift of the n TM grating. The ter ep {[(]2rcn o A z)2] (1-22 f2fz} gives the spatial frequency dependent phase shift due to propagation over the longitudinal distance A z. n addition, because the eulsion is a phase aterial, there is the phase factor (j)q. Because each plane wave output fro the n th slab is input to the (n + 1) slab, (4) shows that for an incident plane wave of frequency f at z=0 only plane waves of spatial frequency (f + q f) are present inside the thick grating. Suppose we illuinate the thick grating at z = 0 with a plane wave of spatial frequency f and aplitude Ai. We represent the aplitudes of each of the possible plane waves that eit fro the n TM grating by (o) A 1 A n: ", linteger_>0 Az i, where A~ is the total aplitude of the plane wave of spatial frequency [f~-(l+l)f2] for l odd and (f~ + f~2) for l even. Because (4) is valid for an input plane wave of arbitrary spatial frequency, we can relate the total aplitude of each of the plane waves at z=naz to its value at z=(n- 1)Az by A, =,H A,_ 1, (5) where the atri of coupling coefficients,h~j, deterined fro (4) describe the aplitude and phase shift effected by the n TM thin grating in diffracting the jth plane wave into the k th one.,hkj is non zero only if the thin grating has an aplitude frequency coponent equal to the difference between the input and output spatial frequencies. The functional for of the coupling coefficients which we have derived for a pure phase aterial, can always be deterined fro thin grating theory and depend upon the type of gratingphase, aplitude or ied and upon the assued relationship between eposure and fil response. The subscript n on the coupling atri ephasizes the fact that because of fringe slant the coupling coefficients depend upon the longitudinal position of the thin grating slab. The coupling atri for the pure phase case is epressed eplicitly in Appendi A. f we now use the initial condition at z = 0 to write then the aplitudes at z = na z are A, =,H... 1H A 0. (6) The aplitude of each of the plane waves at the eit plane z = D are given by (6) for n= N = DA z. 2. Results for a Single and Double Grating Generally for practical eposure values the arguent b of the Jq (b) is sufficiently sall so that for large q the Jq(b) are negligibly sall, and the H atrices can be truncated with little loss in accuracy. We have written a siple coputer progra to perfor the atri ultiplication of (6). To check the accuracy of the results predicted by this approach for a single grating, we copare our results with those of Kogelnik. Using the atri theory, Kaspar has shown that Kogelnik's results are accurate for gratings of low odulation and large thickness [4]. We therefore

4 32 R. Alferness 00 D 5r 1 r,q 1 r )- U Z LL w Z O ~.s 0,~ F Fig. 3. Construction of double grating structure by interferring plane waves R and $1 and then R and S 2 u. a i 100 u6 2':, 2~ '1 :~2 3'3 z 5 0 o READOUT ANGLE o Fig, 2. Diffraction efficiency of the first order versus the angle of the readout bea for single grating. The construction beas are at 0 ~ and 30 ~ )~ = ~t and D = 50 lain e~ apply this ethod to a grating constructed on a 50 g eulsion with construction beas at 0 ~ and 30 ~ and 2= gin. The inde of refraction odulation nl used is that which according to Kogelnik would give 100 % diffraction into the first order for readout at the Bragg angle (30~ n Fig. 2 we plot the calculated diffraction efficiency of the first order versus readout angle. The results are in good agreeent with those predicted by Kogelnik for the case of the readout wave polarized perpendicular to the plane of incidence (i.e., along the grating lines). The intensity of the higher orders was found to be essentially zero. To deonstrate the general applicability of this approach we give soe new results for the case of a grating ade with two fringe structures. As an interesting case we consider the grating in Fig. 3 ade by an incoherent superposition of interferring plane waves $1 and R and then S 2 and R. There is no inter- Fig. 4. Diffraction efficiency of the first (o) and cross-coupled () orders versus the relative inde odulation nln ~ for a double grating constructed as in Fig. 3. n o is the inde odulation which would give 100% diffraction efficiency in the first order in the absence of the second grating. The two gratings have equal inde odulation, R is in the z direction, 01 = 30 ~ 02 = - 25 ~ D = 24 g and 2 = p ference between S 1 and S 2. There are two distinct but overlapping fringe patterns recorded throughout the volue of the eulsion. Therefore each thin grating has two basic frequency coponents and corresponding lateral shift factors due to fringe slope as well as higher order haronics of each basic frequency. The possible plane waves inside the eulsion therefore include those that are diffracted by one fringe structure and then subsequently by the other. The coupling atrices are epanded in a straightforward way to include such coupling, Note that each

5 Optical Propagation in Thick Holographic Gratings 33 fringe structure has the construction bea R in coon. We apply this ethod for a read-out bea incident at the angle of S~. we find as a result that, in addition to a strong diffracted bea in the direction of R, there is also a strong diffracted bea in the direction of $2. Because the readout bea S~ is at the Bragg angle of one fringe structure and its diffracted bea R is at the Bragg angle of the other structure, we ight epect this doubly diffracted wave, which we call the cross-coupled order, to becoe quite Acknowledgeents. thank Professors E.N. Leith and J. Upatnieks and S.Case for helpful discussions during the course of this work. The generous support of the National Science Foundation (Grant No. GK-42361) is also greatly appreciated. Appendi A We show below the for of.h for a single thick grating in a pure phase aterial. ej~~ J0(b0),H = {-je j~~ J_ dbo)e -j~'~176 l jej~o Jl(bo)e+JO,(,~ JeJ~' Jl(bl)eJO~(") -JeJ~2 J-l(b2)e-J~'l(") 'l e j~l Jo(bO - ej~2 j_2(bz)e-ja2(.) -- e j~' J2(bl)e j~ e j~z J0(b2) strong. ndeed, it does. n Fig. 4 we plot the diffraction efficiency of the first and cross-coupled orders versus grating inde odulation nl. Each fringe structure was assued to have the sae inde odulation. The cross-coupled order becoes strong at the epense of the first order which would be 100% for nln ~ = 1 if the second fringe structure was not present. The presence of the second grating liits the aiu diffraction efficiency of the first order; however, the cross-coupled order can becoe 100% efficient (i.e. all the incident light is diffracted into this order) for sufficiently high inde odulation. This rather interesting result deonstrates the iportance of cross-coupling of energy between uliple gratings recorded in the sae thick eulsion and rules out the possibility of treating the effect of each grating independently. t also shows the fleibility of the above ethod in treating optical propagation in ultiple grating structures. Other results such as the effect of offsetting one fringe structure with respect to the other (such that they no longer have a coon construction bea and cross-coupling is reduced) and of varying the relative odulations of the two structures as well as the angular selectivity curves in such cases will be reported in a subsequent paper. where fl= f~-(l+l)f2 [fi + l f~2 2~Azn 1 bl ;d1 _,~ft 2 2~A_znol1 ~ --,~ V f12 as(n ) = - 2=qf n tan~b, odd 1 even where 2 is the free space wavelength of the readout wave, fi is its spatial frequency, ~ is the slope of the fringes with respect to the z ais and no is the bulk refractive inde of the eulsion. References 1. E.N.Leith, A.Koza, J.Upatnieks, J.Marks, N.Massey: Appl. Opt. 5, 1303 (1966) 2. H.Kogelnik: Bell Syst. Tech. J. 48, 2909 (1969) 3. C.B.Burckhardt: J. Opt. Soc. A. 56, t502 (1966) 4. F.G.Kaspar: J. Opt. Soc. A. 63, 37 (1973) 5. R.J.Collier, C.B.Burckhardt, L.H.Lin: Optical Holography, (Acadeic Press, New York 1971) p J.Upatnieks, C.D.Leonard: BM J. Res. Devel. 527 (1970) 7. J.W.Goodan: ntroduction to Fourier Optics (McGraw-Hill, New York 1968) p. 54