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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 9 — Apr. 28, 2008
  • pp: 6528–6536
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Out-of-phase mixed holographic gratings : a quantative analysis

Martin Fally, Mostafa Ellabban, and Irena Drevenšek-Olenik  »View Author Affiliations


Optics Express, Vol. 16, Issue 9, pp. 6528-6536 (2008)
http://dx.doi.org/10.1364/OE.16.006528


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Abstract

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

© 2008 Optical Society of America

1. Introduction

Recently, volume holographic gratings with a modulation of both, the absorption coefficient and the refractive index, have attracted attention in various materials such as silver halide emulsions [1

1. L. Carretero, R. F. Madrigal, A. Fimia, S. Blaya, and A. Beléndez, “Study of angular responses of mixed amplitude-phase holographic gratings: shifted Borrmann effect,” Opt. Lett. 26, 786–788 (2001). [CrossRef]

, 2

2. C. Neipp, C. Pascual, and A. Beléndez, “Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials,” J. Phys. D Appl. Phys. 35, 957–967 (2002). [CrossRef]

, 3

3. C. Neipp, I. Pascual, and A. Beléndez, “Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms,” Opt. Express 10, 1374–1383 (2002). [PubMed]

], doped garnet crystals [4

4. M. A. Ellabban, M. Fally, R. A. Rupp, and L. Kovács, “Light-induced phase and amplitude gratings in centrosymmetric Gadolinium Gallium garnet doped with Calcium,” Opt. Express 14, 593–602 (2006). [CrossRef] [PubMed]

] or materials with colloidal color centers[5

5. A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, “A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms,” Opt. Spectrosc.-USSR 103, 655–659 (2007). [CrossRef]

]. The simplest theoretical description (two-wave-coupling theory) of light propagation in an isotropic medium with a periodic modulation of the (complex) dielectric constant was already given long ago by Kogelnik [6

6. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” AT&T Tech. J. 48(9), 2909–2947 (1969).

]. Considering periodic phase and amplitude modulations, the grating types are treated to be in phase. Later Guibelalde generalized the equations to be valid for out-of-phase gratings [7

7. E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173–178 (1984). [CrossRef]

]. 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 the case of high diffraction efficiencies (above 50%) or even for overmodulated gratings [8

8. C. Neipp, I. Pascual, and A. Beléndez, “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). [CrossRef]

, 3

3. C. Neipp, I. Pascual, and A. Beléndez, “Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms,” Opt. Express 10, 1374–1383 (2002). [PubMed]

, 9

9. S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun. 215, 263–269 (2003). [CrossRef]

], the so called ’transmission efficiency’ η 0, i.e., more correctly termed as zero order diffraction efficiency, was also employed for characterization of the grating parameters. It was suggested, that by measuring the diffraction and transmission efficiency it is possible to evaluate the refractive-index modulation n 1 and the absorption constant modulation α 1 if one assumes in-phase gratings [1

1. L. Carretero, R. F. Madrigal, A. Fimia, S. Blaya, and A. Beléndez, “Study of angular responses of mixed amplitude-phase holographic gratings: shifted Borrmann effect,” Opt. Lett. 26, 786–788 (2001). [CrossRef]

].

In this article we show how the shape of the angular dependencies for the ± first and zero order diffraction efficiencies depend characteristically on the parameters n 1,α 1 and the phase φ between them. We generalize the formulae given in Ref. [1

1. L. Carretero, R. F. Madrigal, A. Fimia, S. Blaya, and A. Beléndez, “Study of angular responses of mixed amplitude-phase holographic gratings: shifted Borrmann effect,” Opt. Lett. 26, 786–788 (2001). [CrossRef]

] to the case of out-of-phase gratings and demonstrate at two experimental examples that the analysis is applicable. This is important, as up to now the evaluation of mixed gratings including a phase was only conducted by beam-coupling experiments, an interferometric technique which is more demanding from an experimental point of view.

2. Diffraction efficiencies of zero and ± first order

According to Refs. [6

6. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” AT&T Tech. J. 48(9), 2909–2947 (1969).

, 7

7. E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173–178 (1984). [CrossRef]

] a plane wave propagating in a (thick) medium with a one dimensional periodic complex dielectric constant, composed of its real part n(x)=n 0+n 1 cos(Kx) and imaginary part α(x)=α 0+α 1 cos(Kx+φ), yields outgoing complex electric field amplitudes for the (zero order) forward diffracted R̂0 and (first order) diffracted R̂±1 waves. These depend characteristically on the following parameters: the mean absorption constant α 0, the thickness d of the grating, the dephasing ϑ due to the deviation from Bragg’s law and the complex coupling constant κ ±=n 1 π/λ- 1/2e ±=κ 1- 2 e ±. Further, K denotes the spatial frequency of the grating, n 0 the mean refractive index of the medium, and φ a possible phase shift between the refractive-index and absorption grating. The goal of an experiment is to extract the grating parameters n 1,α 1,φ by varying the dephasing, e.g., through measuring the angular response of η 0= 0 * 0/I and η ±1= ±1 * ±1/I where * denotes the complex conjugate and I the incident intensity. For simplicity in calculations and as the most often used experimental setup we assume a symmetrical geometry, i.e., that the grating vector and the surface normal are mutually perpendicular. A schematic of the setup is shown in Fig. 1.

Fig. 1. Schematic of the setup. The angular dependence of the zeroth η 0(Θ) and ± first diffraction orders η ±1(Θ) from a mixed grating is measured by rotating the sample around an axis perpendicular to the grating vector K⃗. Θ denotes angles outside the medium. Note, that we are within the thick grating regime, where only two beams are propagating simultaneously: the zero order together with either the -1st or the +1st order.

Slightly adapting the convenient notation from Ref. [1

1. L. Carretero, R. F. Madrigal, A. Fimia, S. Blaya, and A. Beléndez, “Study of angular responses of mixed amplitude-phase holographic gratings: shifted Borrmann effect,” Opt. Lett. 26, 786–788 (2001). [CrossRef]

] the efficiencies for transmission gratings can easily be calculated to yield

η±1(θ)=A(θ)κ12+κ22±2κ1κ2sinφ2z(cosh[z12Dcosψ]cos[z12Dsinψ])
(1)
η0(θ)=A(θ)((z+ϑ2)cosh[z12Dcosψ]+(zϑ2)cos[z12Dsinψ]
+2cosφcosφϑz12{sinψsinh[z12Dcosψ]cosψsin[z12Dsinψ]})
(2)

with the abbreviations A(θ)=exp{-2α 0 D}, D=d/cosθ and

ϑ=K(sinθsinθB)
(3)
z={[ϑ2+4(κ12κ22)]2+[8κ1κ2cosφ]2}12
(4)
2ψ=arccos(ϑ2+4(κ12κ22)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()=η ∓1(φ). Note, that Eq. (1) is identical to Eq. (11) from Ref. [7

7. E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16, 173–178 (1984). [CrossRef]

]. Employing Eq. (1) and Eq. (2) we now study the particular case of α 0 d=1 and κ 2=α 0/2, i.e., maximal grating strength for the amplitude contribution [6

6. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” AT&T Tech. J. 48(9), 2909–2947 (1969).

]. We vary the strength of the phase grating between κ 1=κ 2/4,κ 2,4κ 2 with different phase angles φ=0,π/4,π/2,3π/4 between the grating types. The angular dependencies of the zero order and ± first order diffraction efficiencies are depicted in Fig. 1(a)-(d).

At this point let us summarize the main characteristic features occurring in the diffraction efficiencies at the example for φ=π/4 to obtain a qualitative understanding of the curve shapes and their dependency on the ratio of κ 1/κ 2:

  • Zero order diffraction efficiency η 0(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 κ 1=0 or κ 2=0 or φ=π/2. In general the position and the height of the minima or maxima depend in a complex way on κ 1, κ 2,φ and even the mean absorption constant α 0 (see discussion for α 0 d≫1).
    • For κ 1<κ 2 the curve at the Bragg angle extends more to the region above the mean absorption curve (dash-dot line, first term in Eq. (1) and Eq. (2)) than below, i.e., as a simple approximation ηmax 0+ηmin 0>2A(θB). The same is true vice versa for κ1>κ 2
    • 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., η 1(θB+x)=η 1(θB-x) except for their different mean absorption A(θB±x).
    • Note, that despite κ 1<κ 2 the diffraction efficiency η -1(θB,κ 2/4)>η -1(θB,κ2) for the minus first diffraction order, whereas it is vice versa for the plus first diffraction order, i.e., η +1(θB,κ2/4)<η +1(θB,κ2)

  • Check η ±1: if their magnitudes differ, this is a fingerprint that mixed gratings exist that are out of phase (φ≠0). The ratio η +1/η -1 at the Bragg position obtains a maximum value for φ=π/2 and for κ 1=κ 2 [4

    4. M. A. Ellabban, M. Fally, R. A. Rupp, and L. Kovács, “Light-induced phase and amplitude gratings in centrosymmetric Gadolinium Gallium garnet doped with Calcium,” Opt. Express 14, 593–602 (2006). [CrossRef] [PubMed]

    ].
  • Check η 0: if η 0(θ=0)<A(0) then |φ|<π/2 and else vice versa
  • If η max 0+η min 0>2A(θB), the absorptive component is dominating and else vice versa.
  • For overmodulated phase gratings another feature of the diffraction efficiencies becomes prominent: the side minima near the Bragg peak are lifted to nonzero values (for φπ/2). This striking feature can already be understood in the case φ=0 where we 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[(/D)2-κ 2 1]1/2 (phase grating) and ϑ (2) s=2[(/D)2+κ 2 2]1/2 (absorption grating). Thus, their minima considerably deviate from each other for κ 1,2/D. Recalling, that κ 2=α 1/2≤α 0/2 such a situation will practically occur if κ 1π/D, i.e., for s>1 (overmodulated phase gratings exist). This is realized in various systems (see e.g., [2

    2. C. Neipp, C. Pascual, and A. Beléndez, “Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials,” J. Phys. D Appl. Phys. 35, 957–967 (2002). [CrossRef]

    , 8

    8. C. Neipp, I. Pascual, and A. Beléndez, “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). [CrossRef]

    , 10

    10. I. Drevenšek-Olenik, M. Fally, and M. Ellabban, “Optical anisotropy of holographic polymer-dispersed liquid crystal transmission gratings,” Phys. Rev. E 74, 021707 (2006). [CrossRef]

    ] but did not deserve proper attention.
Fig. 2. Angular dependence of the zeroth and first orders diffraction efficiencies for increasing grating strength of the phase grating contribution. For all graphs: α 0=2.5×104,α 0 d=1, κ 2=α 0/2. Red thin lines: κ 1=κ 2/4, blue lines: κ 1=κ 2 and green thick lines:κ 1=4κ 2 with φ=0 (a), φ=π/4 (b), φ=π/2 (c), and φ=3π/4. The dash-dot line indicates the mean absorption curve A(θ). θ=0,±θB are marked by vertical lines. Note, that for φ=π/2,κ 1=κ2 the minus first order Bragg peak completely disappears and the zero order peaks, too (shown in (c), blue lines)

Finally, we would like to recall that for φφ+π the complex coupling parameters κ ±κ are interchanged and thus the η ±1η ∓1. For η 0 the term in the second line of Eq. 2 changes sign because of cosφ→cos(φ+π)=-cosφ.

3. Experimental and discussion

The investigations were performed on a pure congruently melted lithium niobate crystal (thickness: 5mm). Holographic transmission gratings were prepared by a standard two-wave mixing setup using an argon-ion laser at a recording wavelength of λp=351 nm. Two plane waves with equal intensities and parallel polarization states (s-polarization) were employed as recording beams under a crossing angle of 2ΘB=20.21° (outside the medium) corresponding to a grating period of 1000 nm where the polar c-axis is lying in the plane of incidence. The total intensity of the writing beams was 9 mW/cm2. HPDLC samples were fabricated from a UV curable mixture prepared from commercially available constituents as previously reported in literature [10

10. I. Drevenšek-Olenik, M. Fally, and M. Ellabban, “Optical anisotropy of holographic polymer-dispersed liquid crystal transmission gratings,” Phys. Rev. E 74, 021707 (2006). [CrossRef]

]. The grating period was 1216 nm, the grating thickness about 30µm [11

11. M. Fally, I. Drevenšek-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). [CrossRef] [PubMed]

]. After holographic recording we postcured the sample by illuminating it homogeneously with one of the UV writing beams.

The grating characteristics of the samples was analyzed by monitoring the angular dependencies of the ± first and zero order diffraction efficiencies. For this purpose the samples were fixed 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 reduced readout beam at λr=λp=351 nm and s-polarization, whereas for the HPDLC a He-Ne laser beam at a readout wavelength of λr=543 nm and p-polarization state was employed. Figure 3 shows the experimental curves for the 0.,±1. diffraction orders from a grating recorded in nominally pure congruently melted LiNbO3. According to the recipe given above we immediately can diagnose mixed out-of-phase refractive-index and amplitude gratings, because the η +1>η -1. Further by inspecting the zero order diffraction we come to know that the phase 0<φ<π/2. The effects in the zero order are not so prominent for two reasons: the overall diffraction efficiency is very small and the phase grating is dominant because the zero order diffraction 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: n 1=(3.01±0.04)×10-6,α 1=8.18±0.48m-1,φ=1.027±0.059,α 0=118±1.7m-1 with a reduced chi-square value of 1.89×10-7. From this value and Fig. 3 it is obvious that the equations excellently fit the data.

Fig. 3. Angular dependence of the diffraction efficiencies for the zero and ± first orders of a grating recorded in a pure LiNbO3 sample. The unequal values of the first order diffraction efficiencies are an impressive signature for the existence of mixed phase and amplitude gratings that are out of phase. The zero order diffraction efficiency at the Bragg angles show only 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 fit to η ±1 and η 0. The dashed-dot line indicates the mean absorption curve.
Fig. 4. Zeroth and ± first diffraction orders of a strongly overmodulated grating in HPDLC at 63° Celsius. Λ=1.2µm, λ=543 nm, d=30 µm (same sample as used for the investigations 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 could be 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 the medium. Therefore, also the two-wave coupling theory is not fully applicable. Instead a rigorous coupled wave analysis should be performed [13]. (3) The gratings are expected to be inhomogeneous and non-sinusoidal [11], thus not completely fulfilling the requirements

4. Remarks and conclusion

The above discussed analysis is easily applicable for α 0 d≈1,κ 1κ 2 and η 1(θB)/η 0(θB)≿0.01, so that with the chosen example of LiNbO3 above we are already at the limit. If one grating type is dominant the analysis still remains valid, however, the resulting values for φ and the smaller component result in quite large errors.

We would like to draw the attention to the fact, that for α0≪1 the absorptive grating strength is considerably limited, so that in general the zero order diffraction will not feel the Bragg diffraction. On the other hand, for α 0 d≫1, the forward diffracted beam will exhibit a maximum near the Bragg position, a fact which is well known in x-ray optics (anomalous transmission), see e.g. [17

17. B. W. Batterman and H. Cole, “Dynamical Diffraction of X Rays by Perfect Crystals,” Rev. Mod. Phys. 36, 681–717 (1964). [CrossRef]

].

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

4. M. A. Ellabban, M. Fally, R. A. Rupp, and L. Kovács, “Light-induced phase and amplitude gratings in centrosymmetric Gadolinium Gallium garnet doped with Calcium,” Opt. Express 14, 593–602 (2006). [CrossRef] [PubMed]

]. However, it is sufficient to use the ± first together with the zero order diffraction and to avoid more demanding beam-coupling (interferometric) experiments. A prospective phase between the grating and the interference pattern[18

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

, 19

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

, 20

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

], however, cannot be determined by simple diffraction experiments.

We further would like to emphasize, that the limitations of the coupled wave equations according to Ref. [6

6. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” AT&T Tech. J. 48(9), 2909–2947 (1969).

] should be kept in mind when employing Eq. 1 and Eq. 2, e.g., it is assumed that the gratings are planar, purely sinusoidal and isotropic (for anisotropic gratings the theory given in Ref. [12

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

] should be employed), α 1α 0 (for violation of this condition see [5

5. A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, “A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms,” Opt. Spectrosc.-USSR 103, 655–659 (2007). [CrossRef]

]) and only two beams are kept in the coupling scheme. If the latter is not applicable the theory of rigorous coupled waves has to be applied [13

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

], naturally with an increase of the number of coupling 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 characterization of the diffraction properties is a prerequisite for further successful developments in the field of diffractive optics, optical technologies and photonics, such as the design and fabrication of novel nanostructured materials (photonic crystals) by a light-induced modulation of the complex refractive index.

Appendix

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

η±1(θ)=A(θ)z14(κ1±κ2)2sin2(z12D2)=
 =A(θ)z1r±14(κ12κ22)sin2(z12D2)
(6)
η0=A(θ)z1[ϑ2+4(κ12κ22)cos2(z12D2)]
z=ϑ2+4(κ12κ22).
(7)

It’s interesting to note, that for this case the diffracted and forward diffracted beams have the functional dependence of pure phase gratings with an effective coupling constant of 2[κ 2 1-κ 2 2]1/2. The amplitude of the diffracted beams, however, is enhanced or diminished by a multiplication with or division by r=(κ 1κ 2)/(κ 1+κ 2), respectively. Therefore, it’s easy to see that for κ 1=κ 2 the curves shown in Fig. 2 (c) arise.

Acknowledgment

We are grateful to Profs. Th. Woike and M. Imlau for providing the LiNbO3 sample. Financially supported by the Austrian Science Fund (P-18988) and the ÖAD in the frame of the STC program Slovenia-Austria (SI-A4/0708). We acknowledge continuous support by E. Tillmanns by making one of his labs available to us.

References and links

1.

L. Carretero, R. F. Madrigal, A. Fimia, S. Blaya, and A. Beléndez, “Study of angular responses of mixed amplitude-phase holographic gratings: shifted Borrmann effect,” Opt. Lett. 26, 786–788 (2001). [CrossRef]

2.

C. Neipp, C. Pascual, and A. Beléndez, “Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials,” J. Phys. D Appl. Phys. 35, 957–967 (2002). [CrossRef]

3.

C. Neipp, I. Pascual, and A. Beléndez, “Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms,” Opt. Express 10, 1374–1383 (2002). [PubMed]

4.

M. A. Ellabban, M. Fally, R. A. Rupp, and L. Kovács, “Light-induced phase and amplitude gratings in centrosymmetric Gadolinium Gallium garnet doped with Calcium,” Opt. Express 14, 593–602 (2006). [CrossRef] [PubMed]

5.

A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, “A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms,” Opt. Spectrosc.-USSR 103, 655–659 (2007). [CrossRef]

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). [CrossRef]

8.

C. Neipp, I. Pascual, and A. Beléndez, “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). [CrossRef]

9.

S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun. 215, 263–269 (2003). [CrossRef]

10.

I. Drevenšek-Olenik, M. Fally, and M. Ellabban, “Optical anisotropy of holographic polymer-dispersed liquid crystal transmission gratings,” Phys. Rev. E 74, 021707 (2006). [CrossRef]

11.

M. Fally, I. Drevenšek-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). [CrossRef] [PubMed]

12.

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

13.

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

14.

M. A. Ellabban, M. Bichler, M. Fally, and I. Drevenšek Olenik, “Role of optical extinction in holographic polymer-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). [CrossRef]

15.

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

16.

N. Uchida, “Calculation of diffraction efficiency in hologram gratings attenuated along the direction perpendicular to the grating vector,” J. Opt. Soc. Am. 63, 280–287 (1973). [CrossRef]

17.

B. W. Batterman and H. Cole, “Dynamical Diffraction of X Rays by Perfect Crystals,” Rev. Mod. Phys. 36, 681–717 (1964). [CrossRef]

18.

K. Sutter and P. Günter, “Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” J. Opt. Soc. Am. B 7, 2274–2278 (1990). [CrossRef]

19.

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

20.

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

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.7330) Diffraction and gratings : Volume gratings
(090.0090) Holography : Holography
(090.2900) Holography : Optical storage materials
(090.7330) Holography : Volume gratings

ToC Category:
Diffraction and Gratings

History
Original Manuscript: March 11, 2008
Revised Manuscript: April 17, 2008
Manuscript Accepted: April 20, 2008
Published: April 23, 2008

Citation
Martin Fally, Mostafa Ellabban, and Irena Drevensek-Olenik, "Out-of-phase mixed holographic gratings : a quantative analysis," Opt. Express 16, 6528-6536 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6528


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References

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