Out-of-phase mixed holographic gratings : a quantative analysis

Out-of-phase mixed holographicgratings : a quantative analysis

Martin Fally1,∗, Mostafa Ellabban1,2 and Irena Drevensek-Olenik3,4

1 Faculty of Physics, Nonlinear Physics, University of Vienna, Boltzmanngasse 5, A-1090Wien, Austria

2Physics Department, Faculty of Science, Tanta University, Tanta 31527, Egypt3 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1001

Ljubljana, Slovenia4 J. Stefan Institute, Jamova 39, SI-1001 Ljubljana, Slovenia

∗ Corresponding author:

[email protected]

http://nlp.univie.ac.at

Abstract: We show, that by performing a simultaneous analysis ofthe angular dependencies of the ± first and the zeroth diffraction ordersof mixed holographic gratings, each of the relevant parameters can beobtained: the strength of the phase grating and the amplitude grating,respectively, as well as a potential phase between them. Experiments on apure lithium niobate crystal are used to demonstrate the applicability of theanalysis.

© 2008 Optical Society of America

OCIS codes: (050.1950 ) Diffraction gratings;(050.7330) , (090.7330) Volume gratings;(090.0090) Holography; (090.2900) Optical storage materials.

References and links1. L. Carretero, R. F. Madrigal, A. Fimia, S. Blaya, and A. Belendez, “Study of angular responses of mixed

amplitude-phase holographic gratings: shifted Borrmann effect,” Opt. Lett. 26, 786–788 (2001).2. C. Neipp, C. Pascual, and A. Belendez, “Mixed phase-amplitude holographic gratings recorded in bleached silver

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the phase and amplitude grating in volume holograms,” Opt. Express 10, 1374–1383 (2002).4. M. A. Ellabban, M. Fally, R. A. Rupp, and L. Kovacs, “Light-induced phase and amplitude gratings in cen-

trosymmetric Gadolinium Gallium garnet doped with Calcium,” Opt. Express 14, 593–602 (2006).5. A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, “A Highly Stable Holo-

graphic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms,” Opt.Spectrosc.-USSR 103, 655–659 (2007).

6. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” AT&T Tech. J. 48(9), 2909–2947 (1969).7. E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron.

16, 173–178 (1984).8. C. Neipp, I. Pascual, and A. Belendez, “Theoretical and experimental analysis of overmodulation effects in

volume holograms recorded on BB-640 emulsions,” J. Opt. A-Pure Appl. Op. 3, 504–513 (2001).9. S. Gallego, M. Ortuno, C. Neipp, C. Garcıa, A. Belendez, and I. Pascual, “Overmodulation effects in volume

holograms recorded on photopolymers,” Opt. Commun. 215, 263–269 (2003).10. I. Drevensek-Olenik, M. Fally, and M. Ellabban, “Optical anisotropy of holographic polymer-dispersed liquid

crystal transmission gratings,” Phys. Rev. E 74, 021707 (2006).11. M. Fally, I. Drevensek-Olenik, M. A. Ellabban, K. P. Pranzas, and J. Vollbrandt, “Colossal light-induced

refractive-index modulation for neutrons in holographic polymer-dispersed liquid crystals,” Phys. Rev. Lett. 97,167803 (2006).

(C) 2008 OSA 28 April 2008 / Vol. 16, No. 9 / OPTICS EXPRESS 6528#93737 - $15.00 USD Received 11 Mar 2008; revised 17 Apr 2008; accepted 20 Apr 2008; published 23 Apr 2008

12. G. Montemezzani and M. Zgonik, “Light diffraction at mixed phase and absorption gratings in anisotropic mediafor arbitrary geometries,” Phys. Rev. E 55, 1035–1047 (1997).

13. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc.Am. 71, 811–818 (1981).

14. M. A. Ellabban, M. Bichler, M. Fally, and I. Drevensek Olenik, “Role of optical extinction in holographicpolymer-dispersed liquid crystals,” in Liquid Crystals and Applications in Optics, M. Glogarova, P. Palffy-Muhoray, and M. Copic, eds., vol. 6587, p. 65871J (SPIE Proc., 2007).

15. L. De Sio, R. Caputo, A. De Luca, A. Veltri, C. Umeton, and A. V. Sukhov, “In situ optical control and stabiliza-tion of the curing process of holographic gratings with a nematic film-polymer-slice sequence structure,” Appl.Optics 45, 3721–3727 (2006).

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18. K. Sutter and P. Gunter, “Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine dopedwith 7,7,8,8-tetracyanoquinodimethane,” J. Opt. Soc. Am. B 7, 2274–2278 (1990).

19. F. Kahmann, “Separate and simultaneous investigation of absorption gratings and refractive-index gratings bybeam-coupling analysis,” J. Opt. Soc. Am. A 10, 1562–1569 (1993).

20. M. Fally, “Separate and simultaneous investigation of absorption gratings and refractive-index gratings by beam-coupling analysis: comment,” J. Opt. Soc. Am. A 23, 2662–2663 (2006).

1. Introduction

Recently, volume holographic gratings with a modulation of both, the absorption coefficient andthe refractive index, have attracted attention in various materials such as silver halide emulsions[1, 2, 3], doped garnet crystals [4] or materials with colloidal color centers[5] . The simplesttheoretical description (two-wave-coupling theory) of light propagation in an isotropic mediumwith a periodic modulation of the (complex) dielectric constant was already given long agoby Kogelnik [6]. Considering periodic phase and amplitude modulations, the grating types aretreated to be in phase. Later Guibelalde generalized the equations to be valid for out-of-phasegratings [7]. The quantity of major interest usually is the (first order) diffraction efficiency η 1,defined as the ratio of powers between the diffracted beam and the incoming beam. For thecase of high diffraction efficiencies (above 50%) or even for overmodulated gratings [8, 3, 9],the so called ’transmission efficiency’ η0, i.e., more correctly termed as zero order diffrac-tion efficiency, was also employed for characterization of the grating parameters. It was sug-gested, that by measuring the diffraction and transmission efficiency it is possible to evaluatethe refractive-index modulation n1 and the absorption constant modulation α1 if one assumesin-phase gratings [1].

In this article we show how the shape of the angular dependencies for the ± first and zeroorder diffraction efficiencies depend characteristically on the parameters n 1,α1 and the phaseϕ between them. We generalize the formulae given in Ref. [1] to the case of out-of-phasegratings and demonstrate at two experimental examples that the analysis is applicable. This isimportant, as up to now the evaluation of mixed gratings including a phase was only conductedby beam-coupling experiments, an interferometric technique which is more demanding froman experimental point of view.

2. Diffraction efficiencies of zero and ± first order

According to Refs. [6, 7] a plane wave propagating in a (thick) medium with a one dimensionalperiodic complex dielectric constant, composed of its real part n(x) = n 0 + n1 cos(Kx) andimaginary part α(x) = α0 +α1 cos(Kx+ ϕ), yields outgoing complex electric field amplitudesfor the (zero order) forward diffracted R0 and (first order) diffracted R±1 waves. These dependcharacteristically on the following parameters: the mean absorption constant α 0, the thicknessd of the grating, the dephasing ϑ due to the deviation from Bragg’s law and the complex


coupling constant κ± = n1π/λ − iα1/2e±iϕ = κ1 − iκ2e±iϕ . Further, K denotes the spatialfrequency of the grating, n0 the mean refractive index of the medium, and ϕ a possible phaseshift between the refractive-index and absorption grating. The goal of an experiment is to extractthe grating parameters n1,α1,ϕ by varying the dephasing, e.g., through measuring the angularresponse of η0 = R0R∗

0/I and η±1 = R±1R∗±1/I where ∗ denotes the complex conjugate and I

the incident intensity. For simplicity in calculations and as the most often used experimentalsetup we assume a symmetrical geometry, i.e., that the grating vector and the surface normalare mutually perpendicular. A schematic of the setup is shown in Fig. 1.

z

x

+2 ΘB

-2 ΘB

η0

η+1

η-1

e:

α(x)n(x)

Θ

I

ϕ Λ/(2π)

Λ

K

Fig. 1. Schematic of the setup. The angular dependence of the zeroth η0(Θ) and ± firstdiffraction orders η±1(Θ) from a mixed grating is measured by rotating the sample aroundan axis perpendicular to the grating vector �K . Θ denotes angles outside the medium. Note,that we are within the thick grating regime, where only two beams are propagating simul-taneously: the zero order together with either the −1st or the +1st order.

Slightly adapting the convenient notation from Ref. [1] the efficiencies for transmission grat-ings can easily be calculated to yield

η±1(θ ) = A(θ )κ2

1 + κ22 ±2κ1κ2 sinϕ

2z

(cosh

[z1/2Dcosψ

]− cos

[z1/2Dsinψ

])(1)

η0(θ ) = A(θ )((z+ ϑ 2)cosh

[z1/2Dcosψ

]+(z−ϑ 2)cos

[z1/2Dsinψ

]

+ 2cosϕ|cosϕ |ϑz1/2

{sinψ sinh

[z1/2Dcosψ

]− cosψ sin

[z1/2Dsinψ

]})(2)

with the abbreviations A(θ ) = exp{−2α0D}, D = d/cosθ and

ϑ = K(sinθ − sinθB) (3)

z ={[ϑ 2 +4(κ2

1 −κ22 )]2 +[8κ1κ2 cosϕ ]2

}1/2(4)

2ψ = arccos

(−ϑ 2 +4(κ2

1 −κ22)

z

). (5)

Here, θB denotes the Bragg angle (inside the medium). Eq. (1) and Eq. (2) are valid for θ ≥ 0;for θ ≤ 0 the angles and phase-shifts are replaced by their negative values, i.e., η ±1(−θ ) =η∓1(θ ) and η±1(−ϕ) = η∓1(ϕ). Note, that Eq. (1) is identical to Eq. (11) from Ref. [7]. Em-ploying Eq. (1) and Eq. (2) we now study the particular case of α 0d = 1 and κ2 = α0/2, i.e.,maximal grating strength for the amplitude contribution [6]. We vary the strength of the phase


grating between κ1 = κ2/4,κ2,4κ2 with different phase angles ϕ = 0,π/4,π/2,3π/4 betweenthe grating types. The angular dependencies of the zero order and ± first order diffraction effi-ciencies are depicted in Fig. 1(a)-(d).

At this point let us summarize the main characteristic features occurring in the diffractionefficiencies at the example for ϕ = π/4 to obtain a qualitative understanding of the curve shapesand their dependency on the ratio of κ1/κ2:

• Zero order diffraction efficiency η0(θ )

– The curves are symmetric with respect to normal incidence, i.e., θ = 0.

– Neither the minima nor the maxima of the curve are located at the Bragg angle,except for κ1 = 0 or κ2 = 0 or ϕ = π/2. In general the position and the height ofthe minima or maxima depend in a complex way on κ 1,κ2,ϕ and even the meanabsorption constant α0 (see discussion for α0d � 1).

– For κ1 < κ2 the curve at the Bragg angle extends more to the region above the meanabsorption curve (dash-dot line, first term in Eq. (1) and Eq. (2) ) than below, i.e.,as a simple approximation η max

0 + ηmin0 > 2A(θB). The same is true vice versa for

κ1 > κ2

– Note, that for |θ | � θB the curve resides below the mean absorption curve, for|θ | � θB above

• Diffraction efficiency

– The maximum value of the diffraction efficiency differs for η −1(θB) and η+1(θB);in our case η−1(θB) < η+1(θB).

– The curves are symmetric with respect to θB, i.e., η1(θB + x) = η1(θB − x) exceptfor their different mean absorption A(θB ± x).

– Note, that despite κ1 < κ2 the diffraction efficiency η−1(θB,κ2/4) > η−1(θB,κ2)for the minus first diffraction order, whereas it is vice versa for the plus first diffrac-tion order, i.e., η+1(θB,κ2/4) < η+1(θB,κ2)

Next we would like to point out the difference between the curves for various ϕ values. Figure2(c) shows a unique case which is most instructive. For ϕ = π/2 the coupling constant κ =κ1±κ2,∈ R . Thus a maximum difference between η−1 and η+1 is obtained, culminating in thefull depletion of η−1 if κ1 = κ2 (see appendix). Finally, we want to draw the attention to the caseof ϕ = 3π/4 > π/2. Then η±1 gives identical results as for ϕ −π/2. The zero order diffractionefficiency η0, however, approaches the mean absorption curve for |θ | � θ B from above in thecase of ϕ < π/2 and contrary from below for ϕ > π/2 . Considering these arguments it isobvious, that only a simultaneous fit of all diffraction data, i.e., zero and ± first order diffractedintensities, allows to extract the decisive parameters κ1,κ2,ϕ . On the other hand these curvesare therefore fingerprints of the relation between the parameters. The following recipe can helpin judging about the general situation (for α0d ≈ 1):

• Check η±1: if their magnitudes differ, this is a fingerprint that mixed gratings exist thatare out of phase (ϕ = 0). The ratio η+1/η−1 at the Bragg position obtains a maximumvalue for ϕ = π/2 and for κ1 = κ2 [4].

• Check η0: if η0(θ = 0) < A(0) then |ϕ | < π/2 and else vice versa

• If ηmax0 + ηmin

0 > 2A(θB), the absorptive component is dominating and else vice versa.


0.0

0.1

0.2κ1 = κ2/4:

κ1 = κ2:

κ1 = 4 κ2:

A(θ)

(a)

η 0,η ±1

ϕ=0

0.0

0.1

0.2

η0

min

η0

max(b)

ϕ=π/4

0.0

0.1

0.2 (c)

ϕ=π/2

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.30.0

0.1

0.2 (d)

θ [rad]

ϕ=3π/4

Fig. 2. Angular dependence of the zeroth and first orders diffraction efficiencies for increas-ing grating strength of the phase grating contribution. For all graphs: α0 = 2.5×104,α0d =1, κ2 = α0/2. Red thin lines: κ1 = κ2/4, blue lines: κ1 = κ2 and green thick lines:κ1 = 4κ2with ϕ = 0 (a), ϕ = π/4 (b), ϕ = π/2 (c), and ϕ = 3π/4. The dash-dot line indicatesthe mean absorption curve A(θ ) . θ = 0,±θB are marked by vertical lines. Note, that forϕ = π/2,κ1 = κ2 the minus first order Bragg peak completely disappears and the zeroorder peaks, too (shown in (c), blue lines)


• For overmodulated phase gratings another feature of the diffraction efficiencies be-comes prominent: the side minima near the Bragg peak are lifted to nonzero values (forϕ = π/2). This striking feature can already be understood in the case ϕ = 0 wherewe simply add up the pure absorptive and the pure phase grating. The positions of

the sth side minima are then given by ϑ (1)s = 2[(sπ/D)2 − κ2

1 ]1/2 (phase grating) and

ϑ (2)s = 2[(sπ/D)2 +κ2

2 ]1/2 (absorption grating). Thus, their minima considerably deviatefrom each other for κ1,2 ≈ sπ/D. Recalling, that κ2 = α1/2 ≤ α0/2 such a situation willpractically occur if κ1 � π/D, i.e., for s > 1 (overmodulated phase gratings exist). Thisis realized in various systems (see e.g., [2, 8, 10] but did not deserve proper attention.

Finally, we would like to recall that for ϕ → ϕ +π the complex coupling parameters κ ± → κ∓are interchanged and thus the η±1 → η∓1. For η0 the term in the second line of Eq. 2 changessign because of cosϕ → cos(ϕ + π) = −cosϕ .

3. Experimental and discussion

The investigations were performed on a pure congruently melted lithium niobate crystal (thick-ness: 5mm). Holographic transmission gratings were prepared by a standard two-wave mixingsetup using an argon-ion laser at a recording wavelength of λ p = 351 nm. Two plane waveswith equal intensities and parallel polarization states (s-polarization) were employed as record-ing beams under a crossing angle of 2ΘB = 20.21◦ (outside the medium) corresponding to agrating period of 1000 nm where the polar c-axis is lying in the plane of incidence. The totalintensity of the writing beams was 9 mW/cm2. HPDLC samples were fabricated from a UVcurable mixture prepared from commercially available constituents as previously reported inliterature [10]. The grating period was 1216 nm, the grating thickness about 30μm [11]. Afterholographic recording we postcured the sample by illuminating it homogeneously with one ofthe UV writing beams.

The grating characteristics of the samples was analyzed by monitoring the angular depen-dencies of the ± first and zero order diffraction efficiencies. For this purpose the samples werefixed on an accurately controlled rotation stage with an accuracy of ±0.001 ◦) and facultatively(HPDLC) in a heating chamber. In the case of LiNbO3 we used a single considerably reducedreadout beam at λr = λp = 351 nm and s-polarization, whereas for the HPDLC a He-Ne laserbeam at a readout wavelength of λr = 543 nm and p-polarization state was employed. Figure3 shows the experimental curves for the 0.,±1. diffraction orders from a grating recorded innominally pure congruently melted LiNbO3 . According to the recipe given above we imme-diately can diagnose mixed out-of-phase refractive-index and amplitude gratings, because theη+1 > η−1. Further by inspecting the zero order diffraction we come to know that the phase0 < ϕ < π/2. The effects in the zero order are not so prominent for two reasons: the overalldiffraction efficiency is very small and the phase grating is dominant because the zero order dif-fraction curve extends mostly to values below the mean absorption curve (dash-dot line in Fig.3). A simultaneous fit of Eq. 1 and Eq. 2 to the measured data yielded the following parameters:n1 = (3.01±0.04)×10−6,α1 = 8.18±0.48m−1,ϕ = 1.027±0.059,α0 = 118±1.7m−1 witha reduced chi-square value of 1.89× 10−7 . From this value and Fig. 3 it is obvious that theequations excellently fit the data.

Finally, we intend to demonstrate the usability of the (qualitative) analysis employing an ex-ample with strong overmodulation and high extinction: holographic polymer-dispersed liquidcrystals (H-PDLCs). Only recently was a preliminary beam-coupling analysis of such a systemconducted, a task which is not simple from an experimental point of view [14], in particular ifthe experiments should be carried out under high temperatures or application of external elec-tric fields. Figure 4 shows the diffraction curves from a grating in a HPDLC at an elevated


0.0685 0.0690 0.0695

η0

+θB

η+1

θ [rad]-0.0695 -0.0690 -0.0685

0.0000

0.0025

0.0050

0.0075

0.3000

0.3025

0.3050

0.3075

η0

η-1

-θB

Fig. 3. Angular dependence of the diffraction efficiencies for the zero and ± first orders of agrating recorded in a pure LiNbO3 sample. The unequal values of the first order diffractionefficiencies are an impressive signature for the existence of mixed phase and amplitudegratings that are out of phase. The zero order diffraction efficiency at the Bragg angles showonly a slight asymmetry with respect to θB because the diffraction efficiencies are small(< 1%) and the phase grating is by far dominating. The solid lines show a simultaneous fitto η±1 and η0. The dashed-dot line indicates the mean absorption curve.

temperature. We can understand the major characteristic features as follows: The liquid crystal(LC) component in an HPDLC is highly birefringent. Statistical alignment of the LC-droplets ofabout the light wavelength’s size leads to strong scattering, i.e., extinction which can be treatedsimilar to absorption provided that multiple scattering does not play an essential role. HPDLCsbasically consist of alternating regions with high and low concentration of LCs embedded in apolymer matrix. Thus, these periodically varying scatterers act as extinction gratings. In addi-tion, of course, also the refractive index is strongly modulated (at least via the density changes).Therefore, HPDLCs are typical examples of mixed gratings. Furthermore, it is well known inliterature that the light-induced refractive-index changes are extremely high and strong over-modulation occurs (see e.g. [10]). Such an example is shown in Fig. 4. From the experimentaldata we conclude, that combined refractive-index and extinction gratings are produced. This isconsistent with our previous beam-coupling measurements [14]. However, we do not dare todecide about a possible phase between them. A quantitative evaluation is not possible for thiscase as we are aware of the fact, that in HPDLCs the gratings are anisotropic and thus the ba-sic equations of Ref. [6] should be replaced by the full equations given by Montemezzani andZgonik [12]. In addition, the gratings are usually rather inhomogeneous across the sample butmight be considerably improved upon further efforts during recording [15]. The non-zero min-ima in the diffracted beams partially might originate from overmodulation as discussed abovebut mainly from the inhomogeneity of the gratings and a profile perpendicular to the grating


-0.2 -0.1 0.0 0.1 0.2

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

+θB-θB

η 0,±1

θ [rad]

α0d=3κ1d=2.55κ2d=0.15ϕ=0

0.0

2.5x10-6

5.0x10-6

7.5x10-6

1.0x10-5

1.2x10-5

Inte

nsity

[arb

. u.]

T=63°

Fig. 4. Zeroth and ± first diffraction orders of a strongly overmodulated grating in HPDLCat 63◦ Celsius. Λ = 1.2μm, λ=543 nm, d = 30 μm (same sample as used for the inves-tigations in Ref. [10]. The lower graphs show a simulation according to Eq. 1 and Eq. 2.Note, that here the mean extinction is already rather high. We do not expect that a fit couldbe successful for at least three reasons: (1) as in HPDLCs anisotropic gratings are formed,for the basic equations the full theory of Montemezzani and Zgonik should be employed[12]. (2) It can be noticed, that around θ = 0 more than two waves are propagating in themedium. Therefore, also the two-wave coupling theory is not fully applicable. Instead arigorous coupled wave analysis should be performed [13]. (3) The gratings are expected tobe inhomogeneous and non-sinusoidal [11], thus not completely fulfilling the requirements

vector [16]. However, a qualitative understanding of the changes occuring during heating orapplying an electric field can still be read off from the diffraction curves like those shown inRef. [10].

4. Remarks and conclusion

The above discussed analysis is easily applicable for α0d ≈ 1,κ1 ≈ κ2 and η1(θB)/η0(θB) �0.01, so that with the chosen example of LiNbO3 above we are already at the limit. If onegrating type is dominant the analysis still remains valid, however, the resulting values for ϕ andthe smaller component result in quite large errors.

We would like to draw the attention to the fact, that for α0 � 1 the absorptive grating strengthis considerably limited, so that in general the zero order diffraction will not feel the Bragg dif-fraction. On the other hand, for α0d � 1, the forward diffracted beam will exhibit a maximumnear the Bragg position, a fact which is well known in x-ray optics (anomalous transmission),


see e.g. [17].We would like to point out, that the analysis of only the first diffraction orders cannot give the

full information on all relevant parameters [4]. However, it is sufficient to use the± first togetherwith the zero order diffraction and to avoid more demanding beam-coupling (interferometric)experiments. A prospective phase between the grating and the interference pattern[18, 19, 20],however, cannot be determined by simple diffraction experiments.

We further would like to emphasize, that the limitations of the coupled wave equations ac-cording to Ref. [6] should be kept in mind when employing Eq. 1 and Eq. 2, e.g., it is assumedthat the gratings are planar, purely sinusoidal and isotropic (for anisotropic gratings the theorygiven in Ref. [12] should be employed), α1 ≤ α0 (for violation of this condition see [5]) andonly two beams are kept in the coupling scheme. If the latter is not applicable the theory ofrigorous coupled waves has to be applied [13], naturally with an increase of the number ofcoupling constants between the beams and thus with loss of simplicity.

Here, we provided a quite simple and also fast technique to analyze the properties of out-of-phase mixed gratings, in particular for overmodulated gratings, quantitatively. The characteri-zation of the diffraction properties is a prerequisite for further successful developments in thefield of diffractive optics, optical technologies and photonics, such as the design and fabrica-tion of novel nanostructured materials (photonic crystals) by a light-induced modulation of thecomplex refractive index.

Acknowledgment

We are grateful to Profs. Th. Woike and M. Imlau for providing the LiNbO 3 sample. Financiallysupported by the Austrian Science Fund (P-18988) and the OAD in the frame of the STCprogram Slovenia-Austria (SI-A4/0708). We acknowledge continuous support by E. Tillmannsby making one of his labs available to us.

Appendix

For the particular case of ϕ = π/2 the diffraction efficiencies read:

η±1(θ ) = A(θ )z−14(κ1 ±κ2)2 sin2(

z1/2D/2)

=

= A(θ )z−1r±14(κ21 −κ2

2 )sin2(

z1/2D/2)

(6)

η0 = A(θ )z−1[ϑ 2 +4(κ2

1 −κ22 )cos2

(z1/2D/2

)](7)

z = ϑ 2 +4(κ21 −κ2

2 ).

It’s interesting to note, that for this case the diffracted and forward diffracted beams havethe functional dependence of pure phase gratings with an effective coupling constant of2[κ2

1 −κ22 ]1/2. The amplitude of the diffracted beams, however, is enhanced or diminished by a

multiplication with or division by r = (κ1 −κ2)/(κ1 + κ2), respectively. Therefore, it’s easy tosee that for κ1 = κ2 the curves shown in Fig. 2 (c) arise.


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Out-of-phase mixed holographic gratings : a quantative analysis