Far off-Bragg reconstruction of volume holographic gratings: A comparison of experiment and theories
FFar-off-Bragg reconstruction of volume holographic gratings: a comparison ofexperiment and theories.
Matej Prijatelj, J¨urgen Klepp, Yasuo Tomita, and Martin Fally ∗ Joˇzef Stefan Institute, Jamova 39, SI 1001 Ljubljana, Slovenia University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Wien, Austria Department of Engineering Science, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan (Dated: October 18, 2018,Prijatelj-pra13˙v1)We performed light optical diffraction experiments on a nanoparticle-polymer volume holographicgrating in an angular range including also far-off-Bragg replay. A comparison of three diffractiontheories - on the same level of complexity - with our experimental results shows that the dynamicaltheory of diffraction and the first-order two-wave coupling theory using the beta-value method fitthe data very well. In contrast, the prevalent two-wave coupling theory using the K-vector closuremethod yields a poor fit with an order of magnitude worse mean squared error. These findings mustbe considered for accurate determination of coupling strength and grating thickness.
I. INTRODUCTION
It seems that after about eighty years on theories ofdiffraction from periodic structures everything shouldhave been said and done. This is even more true since
Moharam and
Gaylord in a series of papers publisheda rigorous theory of diffraction (rigorous coupled waveanalysis, RCWA), which covers nearly the entire scope ofcases that have ever been relevant [1–10]. However, forpractical purposes approximate theories are still around,which are used instead. While the optics community hasstrongly opted for the coupled-wave approach of
Kogel-nik (K-vector closure method, KVCM) [11] and much lessfor the one introduced by
Uchida (Beta-value method,BVM) [12], the neutron and X-ray communities are usingsolely the dynamical theory of diffraction (DDT) [13–19]or
Darwin’s early variant [20, 21]. Despite the fact thatthe discussion on diffraction theories for volume gratingsmight sound outworn, we will show that it is possible toexperimentally single out a set of analytic theories whichis significantly superior to others.We identified three problems in the theories whentreating the off-Bragg regime: The first is, that theKVCM misses to correctly account for energy conserva-tion, the second is that KVCM and BVM only treat thehalf-space case, i.e., a single boundary. Thus boundaryconditions for a parallel slab are not properly includedand yield an ambiguity. Finally, with the DDT, whichin principle yields the exact solution to the wave equa-tion, typically an approximation of the dispersion surfacefor small off-Bragg conditions is performed in literature[13, 19, 22], which leads to hyperbolic dispersion surfacesand is called hyperbolic approximation in what follows.It is stated by
Syms and Solymar that the differences be-tween the amplitudes derived from each of the theoriesare small [23]. However, for geometrically thin gratingswith large coupling coefficients as for our samples devia-tions in the far-off-Bragg regime are observable. ∗ [email protected] The result of this work is that the first order two-wavecoupling theory employing the BVM as well as the DDT[22] fit the experimental data extremely well, providedthat we refrain from using the hyperbolic approximationfor the latter. The KVCM fits the experimental datamuch worse.The paper is organized as follows: we start with aderivation of the relevant equations for the two-wave cou-pling theories KVCM and BVM as well as for the modalDDT without the hyperbolic approximation. Then theexperiment and the corresponding results are shown to-gether with fits to each of the theories. We also comparethe theories using proper approximations, discuss the im-plications and end with a conclusion.
II. DIFFRACTION FROM A SINUSOIDALVOLUME GRATING
In what follows we summarize the different approachesto solve the wave equation for sinusoidal transmission vol-ume gratings and give the results for the diffraction am-plitudes with a particular emphasis on the far-off-Braggregime, which is subject to our experiments.We start with the simplest set of assumptions. Spaceis divided into three distinct regions: the input region(free space), the grating region (periodic material), andthe output region (free space). A sketch of the geome-try is provided in Fig. 1. The grating be lossless, onedimensional (modulated only along the x direction), si-nusoidal, isotropic, and of the pure phase-type. Thus itcan be described by n ( x ) = n + n cos ( Kx ) , (1)where n is the mean refractive index of the material un-der investigation, n the refractive-index modulation and K the spatial frequency of the grating. The scalar waveequation (Helmholtz equation), for the polarization stateperpendicular to the plane of incidence ( s or H mode po- a r X i v : . [ phy s i c s . op ti c s ] J un ⃗ q S n = n = n ( x ) ⃗ q R ⃗ K x z ⃗ k θ B ̂ N Θ FIG. 1. (Color online) Illustration of the geometry and theboundary condition. (cid:126)k , (cid:126)q R , (cid:126)q S are the wavevectors of the in-cident wave in the input region, the forward diffracted wave,and the diffracted wave in the grating region. (cid:126)K, ˆ N, Θ , θ B denote the grating vector, the sample surface normal unitvector, the angle of incidence (external) and the Bragg angle(in the medium). larization) in such a medium is (cid:104) (cid:126) ∇ + ( n ( x ) k ) (cid:105) E ( x, z ) = 0 , (2)where k = 2 π/λ with λ the free space wavelength oflight and E ( x, z ) the normalized electric field amplitude.It was shown that an exact solution of the wave equa-tion can be obtained by solving an infinite number ofcoupled linear differential equations of first order underappropriate boundary conditions [1] (RCWA). In the rig-orous treatment of the problem the coupled wave analysisand the modal approach are completely equivalent, i.e.,lead to identical results [2]. For thick volume hologramsdiscussed here only two diffracted beams of considerablefield amplitudes exist at the same time, e.g., the firstand zeroth diffraction order [24]. In addition, the ampli-tudes strongly depend on how well the Bragg condition,2 β sin θ B = K , is fulfilled. Here θ B is the Bragg angleand β = k n the propagation constant in the material.A commonly defined off-Bragg parameter [11] is ϑ = K (cid:18) sin θ − K β (cid:19) , (3)where θ is the angle of incidence. Our interest is to com-pare three common diffraction theories with experimentaldata obtained for a wide range of angles, particularly inthe far-off-Bragg regime. In contrast, usually only thevery vicinity of the Bragg peak is considered and a lin-earization of ϑ is performed (see also appendix B), a sim-plification which is not applicable in our case. In what follows we give the relevant equations for each of the threetheories that cover also the far-off-Bragg regime. A. First-order two-wave coupling: Kogelnik’sapproach (KVCM)
The basic idea of Kogelnik’s coupled wave approach isto solve Eq. (2) by the ansatz E ( x, z ) = R ( z ) exp ( ı(cid:126)q R · (cid:126)x ) + S ( z ) exp ( ı(cid:126)q S · (cid:126)x ) , (4)i.e., the sum of two waves whose amplitudes vary whenpropagating through the sinusoidal refractive index pat-tern. By inserting Eq. (4) into Eq. (2) two coupled dif-ferential equations for the amplitudes R ( z ) , S ( z ) resultthat can be solved for appropriate boundary conditions.In most cases, second order derivatives can be neglectedbecause the amplitudes are slowly varying functions ascompared to the exponentials in Eq. (4) (slowly varyingenvelope approximation). This simplifies the system ofdifferential equations to first order on the expense thatthe boundary conditions are to some extent ambiguous(we end up with a half-space case instead of a slab, i.e.,the output region is not considered). Coupling of thewaves arises via the Bragg condition, which relates thewavevector (cid:126)q S of the diffracted beam to the wavevector (cid:126)q R of the forward diffracted - frequently called also ’trans-mitted’ [25] - beam. At this point Kogelnik introduces therelation (cid:126)q S = (cid:126)q R ± (cid:126)K. (5)While this is a correct choice from a mathematical pointof view [26], it gives physically meaningful results only ifthe Bragg condition is fulfilled exactly. When going off-Bragg, Eq. (5) predicts either a wavelength change (sortof ’inelastic’ scattering) that does, in fact, not occur, ora refractive index that strongly depends on the off-Braggparameter. Furthermore, the direction of the diffractedbeam’s wavevector is not correctly predicted for the off-Bragg case [27]. A wavevector diagram for the off-Braggcase using Kogelnik’s choice (KVCM) is shown in Fig. 2.The diffraction efficiency for the first order, the quantitymeasured in our experiment, is given by η K = SS ∗ = (cid:20) ν K sinc (cid:113) ν K + ξ K (cid:21) (6) ν K = κ dc R (7) ξ K = ϑ d c R (8)with κ = n π/λ the coupling coefficient, d the hologramthickness, c R = cos θ and ∗ denotes the complex conju-gate. FIG. 2. (Color online) Wavevector diagram for the KVCM(Kogelnik) in off-Bragg position with β = | (cid:126)q R | (cid:54) = | (cid:126)q S | accord-ing to Ref. [11]. ˆ N the surface normal unit vector. B. First-order two-wave coupling: Uchida’sapproach (BVM)
The only difference between Uchida’s and Kogelnik’sapproach is the choice of the diffracted wave vector as (cid:126)q S = (cid:126)q R ± (cid:126)K + ∆ q ˆ N . (9)The BVM ensures energy conservation, i.e., | (cid:126)q S | = | (cid:126)q R | = β , by introducing a phase-mismatch parameter ∆ q .The resulting wavevector diagram for the off-Bragg case(∆ q (cid:54) = 0) can be seen in Fig. 3. From geometrical rea-soning the phase mismatch amounts to ∆ q = β ( c R − c S )with c S = (cid:112) cos θ + 2 ϑ/β = c R √ X and a parame-ter X = ϑ/ ( βc R ), which is convenient for approximationsdiscussed later (Appendix B). Solving Eq. (2) with thewavevector choice of Eq. (9) yields for the first orderdiffraction efficiency η B = c R c S (cid:20) ν B sinc (cid:113) ν B + ξ B (cid:21) (10) ν B = κ d √ c R c S (11) ξ B = ∆ q d . (12)These equations that have been derived, e.g., in Ref. [28]look quite similar to the ones obtained for Kogelnik’sapproach. Uchida ’s seminal paper already suggested theBVM [12]. However, it was somewhat hidden by thesecond important topic addressed: the attenuation of thegrating modulation along the sample depth. The latter isnot included in the present treatment in order to enablea direct comparison of the theories.
FIG. 3. (Color online) Wavevector diagram for the BVM(Uchida) in off-Bragg position with β = | (cid:126)q R | = | (cid:126)q S | accordingto Ref. [12]. C. Two-wave modal approach: Dynamical theoryof diffraction (DDT)
The modal approach to solve Eq. (2) is well knownin solid state physics for electrons in a periodic potential(band structure, e.g., [29, 30]), and also for diffraction ofX-rays (e.g.,[16, 18, 19, 31]) or neutrons [13] by crystallattices. In volume holography it has been much lessprominent [22, 32, 33]. The reason might be that it canbe usefully applied only for highly idealised gratings [22].In the modal approach the solution is taken rigorously inthe form of a sum of m Bloch waves (eigen-modes) witha grating periodic amplitude function E (cid:126)k E m ( (cid:126)x ) = (cid:88) (cid:126)k E m(cid:126)k e ı(cid:126)k(cid:126)x , (13) E m(cid:126)k e ıs (cid:126)K = E m(cid:126)k + s (cid:126)K with s ∈ Z . Fourier transforming Maxwell’s equations and insertingEqs. (13) and (1) yields a system of coupled algebraicequations for the amplitudes E m(cid:126)k . Details can be foundin Appendix A and Ref. [22]. For sinusoidal volume holo-grams we know that only four of the amplitudes - two foreach eigen-mode - have appreciable magnitude and thusa linear system of two equations for each eigen-mode re-mains. To obtain consistent and non-trivial solutions tothe systems, conditions for the magnitudes of the per-mitted wavevectors in the grating region arise:(2 βκ ) = (cid:0) β − | (cid:126)q R | (cid:1) (cid:0) β − | (cid:126)q S | (cid:1) . (14)The latter is the decisive equation in the DDT and repre-sents the so-called dispersion surface, i.e., the surface ofpermitted wavevectors. The dispersion surface and thewave vectors of the eigen-modes in the grating region areshown in Fig. 4. These permitted wave vectors can eas-ily be found geometrically by following the three stepsbelow:1. Starting from the origin of the reciprocal space (cid:126) k at the(external) angle of incidence Θ.2. Phase matching at the boundary requires thatthe tangential compontents of the wavevectors areidentical, i.e., k ,x = q R,x . Thus draw a line alongthe sample surface normal ˆ N and find the points ofintersection A, B with the dispersion surface (typi-cally called ’tie-points’, two in the transmission ge-ometry)3. The permitted wavevectors of propagation arenow constructed by forming the vectors −−→ AK = (cid:126)q (0) S , −−→ BK = (cid:126)q (1) S , −→ A (cid:126)q (0) R , −→ B (cid:126)q (1) R The circles with radius β around the reciprocal latticepoints (cid:126) , (cid:126)K represent the surface of permitted wavevec-tors in the medium without a grating viz. κ = 0. Theyapproximate the permitted wavevectors of propagationin a grating very well unless in the very vicinity of theBragg condition. This latter situation is shown in thecloseup of the dispersion surface in the lower part of Fig.4. The moduli of the permitted wave vectors of propaga-tion in the grating region are obtained by using Bragg’scondition (cid:126)q S = (cid:126)q R ± (cid:126)K and solving Eq. (14) for (cid:126)q R (cid:12)(cid:12)(cid:12) (cid:126)q ( m ) R (cid:12)(cid:12)(cid:12) = β (cid:16) ϑ + β ± (cid:112) ϑ + (2 κ ) (cid:17) . (15)This means, that for each propagation direction un-der consideration ( (cid:126)q = (cid:126) , (cid:126)K ) there are two eigen-modes m = 0 , Ewald - a sort ofbirefringence.Usually, the region in the vicinity of the Bragg condi-tion is approximated, i.e., the spheres around the recip-rocal lattice points (cid:126) , (cid:126)K with radius β are approximatedby planes (asymptotic gray lines in Fig. 4) and the dis-persion surfaces are hyperbolae. We disregard this sim-plification so that the theory can be applied for the far-off-Bragg region, too. For neutrons the correspondingequations for the diffracted intensities were establishedin Ref. [34, 35]. The amplitudes E m(cid:126)k are determined bytaking into account the boundary conditions and can beexpressed by the grating parameters. Then the first or-der diffraction efficiency according to the DDT takes theform η D = (2 κ ) (2 κ ) + ϑ sin (cid:18)
12 ∆ qd (cid:19) (16)where ∆ q = q (1) Rz − q (0) Rz . This is the function used for thefits to the experimental data in Sec. III. ⃗ k ⃗ q S ( ) ⃗ q S ( ) ⃗ ⃗ K ⃗ q R ( ) ⃗ q R ( ) ̂ N k β Θ B A
FIG. 4. (Color online) Dispersion surface in the off-Bragggeometry. Top: overview including the four permitted wavevectors (cid:126)q (0);(1) R ; S . The orange lines represent part of the dis-persion surface. An approximate version of the latter - asusually assumed in literature - is shown in dotted gray linestyle. Bottom: A closeup of the dispersion surface in thevicinity of the Bragg condition with the mismatch ∆ q . ∆ q isthe difference between the gray asymptotic lines (viz spheres)along the ˆ N -direction, i.e., horizontal. III. EXPERIMENTAL & RESULTS
The diffraction experiments were performed on a SiO nanoparticle-polymer grating [36] with a grating spacingΛ = 2 π/ | (cid:126)K | = 500 nm using an s -polarized He-Ne laserbeam at a wavelength λ of 632 . y − axis while measuring the zero and first L P BE BSPD r PD +1 PD SD ̂ N y − ⃗ K ⃗ K ∥ y FIG. 5. (Color online) Experimental setup scheme:L,P,BE,D,BS,S,PD denote the He-Ne laser, polarizer, beamexpander, diaphragm, beamsplitter, sample and photodiodes,respectively. The amplifier is symbolized by the triangle(top). Sample geometry for measuring the diffraction effi-ciency (left bottom) and the background (right bottom), ro-tation axis was always y . order diffracted powers, i.e., rocking curves are recorded.To get reliable values in the far-off-Bragg regime we mea-sured the diffraction powers using Si-photodiodes whichwere plugged into a light amplifier allowing linear am-plification in the 10 -order signal range. To ensure mostreliable results we took at least five power values at eachangular position θ thus being able to evaluate a standarderror of the mean. Furthermore, a background measure-ment was conducted by first rotating the sample by 90 ◦ around the sample surface normal, and then recording arocking curve again, this time the grating vector (cid:126)K beingparallel to the rotation axis y . By attaching an opaquecircular mask (diameter 5 mm) to the sample’s front-surface, it was assured that the incident beam passed theidentical sample volume for both, the diffraction experi-ment and the background measurement. The results ofthe diffraction experiments are shown in Fig. 6. A simul-taneous least-squares fit to the weighted and background-corrected diffraction efficiency for the minus and plus firstorder were performed in the angular range | θ − θ B | (cid:38) . χ K value being nearlyan order of magnitude larger than other approaches. Themost striking difference is observed in the positions of theminima. In Fig. 7 the experimental minimum positionsare compared to those obtained from the theories. Fromthe angular dependence of the residuals we conclude thatthe KVCM strongly deviates in the far-off-Bragg region. IV. DISCUSSION
As already said in the introduction, typically the dis-persion surface in the DDT is approximated in the vicin-ity of the Bragg condition so that the dispersion surfacesare hyperbolae. This of course is not applicable for thefar-off-Bragg region. On the other hand, a direct com-parison of analytical formulae is only possible if properapproximations are applied, which is done in AppendixB.Let us start our discussion with mentioning that ∆ q (appearing in DDT) is equivalent to ∆ q (appearing inthe BVM) in the limit of zero coupling ( κ = 0). In thewavevector diagram (Fig. 3) a background refractive in-dex n is implicitely assumed to describe the permittedwavevectors of propagation in the medium. On the otherhand, we know that at the exact Bragg condition a waveexperiences a different refractive index within the limits n ± n . This is only reflected in the DDT, where - inthe frame of the two mode case - two waves with slightlydifferent refractive indices propagate towards each of thetwo reciprocal lattice points (cid:126) (cid:126)K shown in Fig. 4.The value of ∆ q is the difference between the two cross-ing asymptotic gray lines (i.e. actually circles) along thesample surface normal shown in the closeup of Fig. 4.In Ref. [27] the effective thickness of the grating - forthe same sample - was estimated to be d = 58 . ± . µ mby fitting Eq. (10) to the angular dependence of thediffraction efficiency shown in Fig. 6. The discrepancyto the value determined in the present work originatesfrom three facts: (i) here we used experimentally mea-sured values weighted by their standard error of the mean(no weighting in [27]), (ii) the angular range was ex-tended to the far-off-Bragg region which has particularinfluence on the obtained thickness via the phase func-tion through its minima, and (iii) the background wasthoroughly subtracted. Browsing through the literatureof evaluating the refractive-index modulation of volumeholographic gratings, we find mainly two approaches: a)measuring a rocking curve and calculating n and d byusing the KVCM in the linearized version and b) justassuming the grating thickness to be the measured me-chanical sample thickness and calculating n from thediffraction efficiency at the (supposed to be) Bragg an-gle. While it is evident that the latter can only serve asa rough estimate, the results of the former are usuallytaken as serious parameters. When fitting of the datashown in Fig. 6 (without weighting) is performed on thisbasis, i.e., less thorough, values around the Bragg peakcontribute the most to the result. This paradoxicallyleads to fitting parameters which are much closer to theones obtained by BVM and DDT (deviations of about1% for n and d ). The latter, however, is only due to thefact that we have an almost ideal grating very accuratelydescribed by Eq. ( refeq:grating).After all we are left with the question: are we able todecide if any of the theories is superior? Our experimentand the sample under investigation provide almost ideal -0.1 -0.05 0.0 0.05 0.1 0.15Off-Bragg parameter ϑ (rad)543210 l og ( D i ff r a c t i o n e ff i c i e n c y η ) n K =(2 . ± . × − d K =(56 . ± . µmχ K =7 . × n B =(2 . ± . × − d B =(56 . ± . µmχ B =1 . × n D =(2 . ± . × − d D =(56 . ± . µmχ D =1 . × θ B =0 . rad ˆ= ϑ ≡ KVCMBVMDDTData: η +1 FIG. 6. (Color online) Dependence of the first order diffraction efficiency on the Off-Bragg parameter ϑ (see Eq. (3)) for ananoparticle-polymer grating. The plus and minus first order data (symbols) were fitted using either the KVCM (green line),BVM (red line) or the DDT (black line). Fitting results including the weighted χ are given in the inset. For further details,see text.
15 10 5 0 5 10 15Order of Minimum121086420 R e s i d u a l s ( m r a d ) KVCMBVMDDT
FIG. 7. (Color online) Comparison of the minimum positionsbetween the experimental data and the three models. Thedeviation of the measured minimum positions from the threefitted models (residuals) is shown. conditions, i.e., the assumptions about the grating takenin the models are fulfilled. We identify the KVCM as lessreliable due to the significantly higher χ K (cid:29) χ B,D (seeFig. 6). A decision on the basis of χ alone to recom- mend either the BVM or the DDT is not evident. How-ever, when it comes to less perfect gratings, e.g., taperwith an attenuation of the grating modulation along thesample depth [37] due, for example, to non-neglible linearabsorption during recording [38], there is no known wayto apply the modal approach that leads to the DDT. Incontrast, it has been successfully proven that the prob-lem can even be analytically treated using a coupled waveanalysis [12]. V. CONCLUSION
Diffraction by a one dimensional volume holographicgrating with a relatively thick film (about 57 micron) andhigh refractive- index modulation ( n about 0 . c R /c S in the amplitude. We finally comment that froma practical point of view most of the gratings are notthat perfect, and thus the BVM including the attenua-tion along sample depth as discussed in Ref. [12] shouldbe the first choice. ACKNOWLEDGMENTS
This work was supported by the Austrian Science Fund(P-20265), the Slovenian-Austrian bilateral programme¨OAD-WTZ (SI 07/2011) and by the Ministry of Educa-tion, Culture, Sports, Science, and Technology of Japan(Grant No. 23656045).
Appendix A: Derivation of the dispersion surfaceEq. (14)
We follow the derivations given by
Batterman and Cole for x-rays [19] and
Russell for light [22].We start with the Helmholtz equation (2). The refrac-tive index from Eq. (1) is a real quantity and periodicwith the reciprocal lattice vector (cid:126)K = ( K, , n = ε ∈ R . Hence, ε ( (cid:126)x ) = ε + 12 (cid:88) g ∈ N ,g> ε g (cid:16) e ı (cid:126)K g (cid:126)x + e − ı (cid:126)K g (cid:126)x (cid:17) , (A1) where (cid:126)K g ≡ g (cid:126)K, g ∈ N . Further, it is known that so-lutions to the wave equation for periodic media have aparticular form E m ( (cid:126)x ) = (cid:88) (cid:126)k E m(cid:126)k e ı(cid:126)k(cid:126)x (A2)where E m(cid:126)k are functions with the periodicity of themedium. The E m ( (cid:126)x ) are eigen-modes of the wave equa-tion labelled by m (indicating different energies num-bered in increasing order) and are called Bloch waves.By inserting Eqs. (A1) and (A2) into Eq. (2) we endup with an infinite number of algebraic equations for theinfinite number of coefficients E m(cid:126)k : (cid:88) (cid:126)k ( ε k − | (cid:126)k | ) E m(cid:126)k e ı(cid:126)k(cid:126)x + k (cid:88) g ∈ N ,g> ε g (cid:16) e ı ( (cid:126)k + (cid:126)K g ) (cid:126)x + e ı ( (cid:126)k − (cid:126)K g ) (cid:126)x (cid:17) E m(cid:126)k = 0 (cid:88) (cid:126)k ( β − | (cid:126)k | ) E m(cid:126)k + k (cid:88) g ∈ N ,g> ε g ( E m(cid:126)k + (cid:126)K g + E m(cid:126)k − (cid:126)K g ) = 0 . (A3)This is effectively the Fourier transformed version of thewave equation in a periodic medium.Assuming that only two of the coefficients considerablydiffer from zero - say E m(cid:126)q , E m(cid:126)q + (cid:126)K - the system of equationsreduces to( β − | (cid:126)q | ) E m(cid:126)q + k ε E m(cid:126)q + (cid:126)K = 0 for (cid:126)k = (cid:126)qk ε E m(cid:126)q + ( β − | (cid:126)q + (cid:126)K | ) E m(cid:126)q + (cid:126)K = 0 for (cid:126)k = (cid:126)q + (cid:126)K Nontrivial solutions to this system of equations requirethe determinant of the coefficient-matrix E m(cid:126)k , i.e., thecharacteristic polynomial, to vanish. This leads to aquartic equation in | (cid:126)q | ( β − | (cid:126)q | )( β − | (cid:126)q + (cid:126)K | ) = (cid:18) k ε (cid:19) , (A4)which gives the moduli q ( m ) of the m permitted wavevec-tors for propagation in a periodic medium. The latteris Eq. (14) when using the relations κ = k n / , ε =2 n n and the notation (cid:126)q = (cid:126)q R , (cid:126)q + (cid:126)K = (cid:126)q S .Finally, we find the total field in the grating as a sum over all eigen-modes E ( (cid:126)x ) = (cid:88) m v m (cid:88) (cid:126)k = (cid:126)q ( m ) ,(cid:126)q ( m ) + (cid:126)K E m(cid:126)k e ı(cid:126)k(cid:126)x , (A5)where the coefficients v m must be determined consideringthe boundary conditions. We express the ratio E m(cid:126)q E m(cid:126)q + (cid:126)K = 2 κϑ ± (cid:112) ϑ + (2 κ ) , (A6)and assume that the incident wave at the entrance bound-ary has an amplitude equal to unity so that Eq. (A5)yields conditional equations for the v m : v = E (cid:126)q + (cid:126)K E (cid:126)q + (cid:126)K E (cid:126)q − E (cid:126)q E (cid:126)q + (cid:126)K v = − v E (cid:126)q + (cid:126)K E (cid:126)q + (cid:126)K . The field-amplitude of the waves travelling towards thereciprocal lattice point (cid:126)K is E − ( (cid:126)x ) = v E (cid:126)q + (cid:126)K e ı ( (cid:126)q (0) + (cid:126)K ) (cid:126)x + v E (cid:126)q + (cid:126)K e ı ( (cid:126)q (1) + (cid:126)K ) (cid:126)x (A7)so that for the geometry used in the experiment dis-cussed here (Fig. 5) the diffracted amplitude E − ( z = d )amounts to E − = ( | (cid:126)q (0) | − β )( | (cid:126)q (1) | − β )2 βκ ( | (cid:126)q (1) | − | (cid:126)q (0) | ) × (cid:16) e ıq (0) z d − e ıq (1) z d (cid:17) . (A8)Using the definition η = | E − ( z = d ) / | we end up withEq. (16). Appendix B: ’Proper’ approximation in the DDT forcomparison with KVCM and BVM
To obtain an equation similar to the form of Eqs. (6)and (10) also for the DDT, we approximate ∆ q using Eq.(15): q ( m ) Rz − q Rz ≈ (cid:112) β ( ϑ + β cos θ ) (cid:34) ± (cid:112) ϑ + (2 κ ) ϑ + β cos θ ) (cid:35) , ∆ q ≈ (cid:112) β (cid:115) ϑ + (2 κ ) ϑ + βc R . (B1)Then the functional form of the diffraction efficiency -an amplitude term A multiplied by an oscillatory phaseterm sin Φ - looks similar to that of Eqs. (6) and (10)yielding η D = A D sin Φ D (B2) A D = (2 κ ) ϑ + (2 κ ) = (2 κ ) ( βc R X ) + (2 κ ) = ν K ξ K + ν K (B3)Φ D = (cid:114)
11 + X (cid:113) ( βc R X ) + (2 κ ) d c R = (cid:114)
11 + X (cid:113) ν K + ξ K . (B4)For the experimental angular range θ B -0.155. . . θ B +0.166radian (angles in the medium) and κ = 2 . × − µ m − (as obtained from the fit) the deviation of Eq. (B2) fromEq. (16) is less than about 0.3%. Rewriting Eq. (6) forcomparison in terms of X, ν K , ξ K , we have η K = A K sin Φ K (B5) A K = A D (B6)Φ K = Φ D √ X. (B7)Thus, the phase term of the KVCM differs from that ofthe DDT by a factor √ X while the amplitudes areidentical. We proceed with Eq. (10) in terms of X whichyields η B = A B sin Φ B (B8) A B = 1 √ X ϑ (rad)12345678910111213141516 Φ ( θ ) / π : O r d e r o f M i n i m u m θ B KVCMBVMDDT 12 Φ ( θ ) / π FIG. 8. (Color online) Φ( θ ) for each approach according toEqs. (B4),(B7),(B10). Roots of the diffraction efficiency oc-cur at angles for which Φ( θ s ) = sπ , i.e, where the dottedhorizontal lines intersect the functions. × (2 κ ) (2 κ ) + ( βc R (1 − √ X )) √ X (B9)Φ B = d c R × (cid:115) [ βc R (1 − √ X )] + (2 κ ) √ X . (B10)Despite the formal similarity of the expressions Eqs. (6)and (10) for KVCM and BVM, respectively, we need totake a further approximation for a direct comparison. Werecall that we are interested in the far-off-Bragg region,where the theories might significantly differ one another.Then ϑ (cid:29) (2 κ ) and we neglect κ in Eqs. (B2) - (B10) incomparison to ϑ and X . The approximate phase func-tions then readΦ ,D = | X | (cid:114)
11 +
X βc R d ,B = | − √ X | βc R d ,K = | ϑ | d c R = | X | βc R d . (B13)Higher order minimum positions are given by Φ( θ s ) = sπ with s ∈ N . Fig. 8 shows Φ( θ ) for each of the theories.BVM and DDT lead to identical results in the phasefunction up to second order in θ − θ B , whereas KVCM isequivalent only in the very vicinity of the Bragg condi-tion. For the amplitude part of the diffraction efficiency,the situation is different. Assuming again ϑ (cid:29) κ (forthe far-off-Bragg angular range) we get A ,D = (cid:18) κβc R X (cid:19) (B14) ϑ (rad)3.503.253.002.752.50 l og [ A ( θ ) ] θ B A D ,A K A B A ,D A ,B FIG. 9. (Color online) The amplitude functions A ( θ ) - Eqs.(B3),(B6),(B9) - and A ( θ ) - Eqs. (B14) - in the off-Braggregion for each model. A ,B = 1 √ X (2 κ ) ( βc R [1 − √ X ]) √ XA ,K = A ,D . In Fig. 9 the amplitude functions A ( θ ) are shown. Thistime the DDT is identical to the KVCM and differs fromBVM, namely by a factor √ X . The relative devia-tion of 1 −√ XA B /A D is less than 0 .
7% in the rangediscussed.To summarize: For the diffraction efficiency • the phase term of the DDT agrees to that of theBVM excellently; the KVCM differs by a factor of(1 + X ) − / , while • the amplitude term of the DDT is equal to that ofKVCM; whereas now that of the BVM differs by afactor of (1 + 2 X ) / . While the first statement is fully confirmed by the ex-perimental data and can be seen in Figs. 6 and 7, thesecond statement can hardly be verified from the dataset.One reason could be that the side wings of the amplitudefunction are also influenced by absorption, an issue notconsidered here. Approximate theories and their interrelation
As said above, the equation for the dispersion sur-face Eq. (14) discussed in nearly any publication is ap-proximated to form hyperbolic sheets in the vicinity ofthe Bragg incidence, i.e., the quartic equation in β istransformed in a quadratic one with the approximation β − ( (cid:126)q ) ≈ ( β − | (cid:126)q | )2 β . Then the magnitudes of thepermitted wavevectors are given by (cid:12)(cid:12)(cid:12) (cid:126)q ( m ) R (cid:12)(cid:12)(cid:12) = β + 12 (cid:16) ϑ ± (cid:112) ϑ + (2 κ ) (cid:17) . (B15)The approximate version of the dispersion surface isshown in Fig. 4 as dotted grey lines. Such an approxi-mation leads to the identical diffraction efficiency for theDDT and the KVCM as given by Eq. (6) [22, 39]. Usingthe approximation of Eq. (B1) and truncating the expan-sion of the phase correction factor (1+ X ) − / ≈ O [ X ]in Eq. (B2) with the constant term, we arrive at the sameresult.In addition, the off-Bragg parameter is frequently lin-earized as in Ref. [11], so that ϑ ( θ ) = K cos θ B ( θ − θ B )which then should give even worse results and anyhow isvalid only close to the Bragg angle θ B . [1] M. G. Moharam and T. K. Gaylord, J. Opt. Soc. Am. , 811 (1981).[2] T. K. Gaylord and M. G. Moharam, Appl. Phys. B ,1 (1982).[3] M. G. Moharam and T. K. Gaylord, J. Opt. Soc. Am. , 187 (1982).[4] M. G. Moharam and T. K. Gaylord, J. Opt. Soc. Am. , 1385 (1982).[5] M. G. Moharam and T. K. Gaylord, J. Opt. Soc. Am. , 451 (1983).[6] M. G. Moharam and T. K. Gaylord, J. Opt. Soc. Am. , 1105 (1983).[7] T. K. Gaylord and M. G. Moharam, Proc. IEEE , 894(1985).[8] M. G. Moharam and T. K. Gaylord, J. Opt. Soc. Am. A , 1780 (1986).[9] M. G. Moharam, D. A. Pommet, E. B. Grann, andT. K. Gaylord, J. Opt. Soc. Am. A , 1077 (1995).[10] M. G. Moharam, E. B. Grann, D. A. Pommet, andT. K. Gaylord, J. Opt. Soc. Am. A , 1068 (1995).[11] H. Kogelnik, Bell Syst. Tech. J. , 2909 (1969).[12] N. Uchida, J. Opt. Soc. Am. , 280 (1973). [13] H. Rauch and D. Petrascheck, in Neutron diffraction ,edited by H. Dachs (Springer-Verlag, Berlin HeidelbergNew York, 1978), vol. 6 of
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