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

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.


I. 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 [1,2,3], doped garnet crystals [4] or materials with colloidal color centers [5] . 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]. 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]. 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 [3,8,9], 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].
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] 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.

II. DIFFRACTION EFFICIENCIES OF ZERO AND ± FIRST ORDER
According to Refs. [6,7] 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 diffractedR 0 and (first order) diffracted R ±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 − iα 1 /2e ±iϕ = κ 1 − iκ 2 e ±iϕ . 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 =R 0R * 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 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 Figure 1.
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 (dashdot line, first term in Equations (1) and (2) ) than below, i.e., as a simple approximation η max The same is true vice versa for κ 1 > κ 2 -Note, that for |θ| ≪ θ B the curve resides below the mean absorption curve, for |θ| ≫ θ B above • Diffraction efficiency -The maximum value of the diffraction efficiency differs for η −1 (θ B ) and η +1 (θ B ); in our case -The curves are symmetric with respect to θ B , i.e., for the minus first diffraction order, whereas it is vice versa for the plus first diffraction order, i.e., η +1 ( Next we would like to point out the difference between the curves for various ϕ values. Figure 2(c) shows a unique case which is most instructive. For ϕ = π/2 the coupling constant κ = κ 1 ± κ 2 , ∈ R . Thus a maximum difference between η −1 and η +1 is obtained, culminating in the full depletion of η −1 if κ 1 = κ 2 (see appendix). Finally, we want to draw the attention to the case of ϕ = 3π/4 > π/2. Then η ±1 gives identical results as for ϕ − π/2. The zero order diffraction efficiency η 0 , however, approaches the mean absorption curve for |θ| ≫ θ B from above in the case of ϕ < π/2 and contrary from below for ϕ > π/2 . Considering these arguments it is obvious, that only a simultaneous fit of all diffraction data, i.e., zero and ± first order diffracted intensities, allows to extract the decisive parameters κ 1 , κ 2 , ϕ. On the other hand these curves are therefore fingerprints of the relation between the parameters. The following recipe can help in judging about the general situation (for α 0 d ≈ 1): • 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].

III. 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 λ p = 351 nm. Two plane waves with equal intensities and parallel polarization states (s-polarization) were employed as recording beams under a crossing angle of 2Θ B = 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/cm 2 . HPDLC samples were fabricated from a UV curable mixture prepared from commercially available constituents as previously reported in literature [10]. The grating period was 1216 nm, the grating thickness about 30µm [11]. 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 LiNbO 3 we used a single considerably reduced readout beam at λ r = λ p = 351 nm and s-polarization, whereas for the HPDLC a He-Ne laser beam at a readout wavelength of λ r = 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 LiNbO 3 . 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 Figure 3). A simultaneous fit of Equations 1 and 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 Figure 3 it is obvious that the equations excellently fit the data.
Finally, we intend to demonstrate the usability of the (qualitative) analysis employing an example with strong overmodulation and high extinction: holographic polymer-dispersed liquid crystals (H-PDLCs). Only recently was a preliminary beam-coupling analysis of such a system conducted, a task which is not simple from an experimental point of view [14], in particular if the experiments should be carried out under high temperatures or application of external electric fields. Figure 4 shows the diffraction curves from a grating in a HPDLC at an elevated 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 of about the light wavelength's size leads to strong scattering, i.e., extinction which can be treated similar to absorption provided that multiple scattering does not play  [10]. The lower graphs show a simulation according to Equations 1 and 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 an essential role. HPDLCs basically consist of alternating regions with high and low concentration of LCs embedded in a polymer matrix. Thus, these periodically varying scatterers act as extinction gratings. In addition, 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 in literature that the light-induced refractive-index changes are extremely high and strong overmodulation occurs (see e.g. [10]). Such an example is shown in Figure 4. From the experimental data we conclude, that combined refractive-index and extinction gratings are produced. This is consistent with our previous beam-coupling measurements [14]. However, we do not dare to decide about a possible phase between them. A quantitative evaluation is not possible for this case as we are aware of the fact, that in HPDLCs the gratings are anisotropic and thus the basic equations of Ref. [6] should be replaced by the full equations given by Montemezzani and Zgonik [12]. In addition, the gratings are usually rather inhomogeneous across the sample but might be considerably improved upon further efforts during recording [15]. The non-zero minima in the diffracted beams partially might originate from overmodulation as discussed above but mainly from the inhomogeneity of the gratings and a profile perpendicular to the grating vector [16]. However, a qualitative understanding of the changes occuring during heating or applying an electric field can still be read off from the diffraction curves like those shown in Ref. [10].

IV. REMARKS AND CONCLUSION
The above discussed analysis is easily applicable for α 0 d ≈ 1, κ 1 ≈ κ 2 and η 1 (θ B )/η 0 (θ B ) 0.01, so that with the chosen example of LiNbO 3 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 α 0 ≪ 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].
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 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 [18,19,20], 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] should be kept in mind when employing Equations 1 and 2, e.g., it is assumed that the gratings are planar, purely sinusoidal and isotropic (for anisotropic gratings the theory given in Ref. [12] should be employed), α 1 ≤ α 0 (for violation of this condition see [5]) 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], naturally with an increase of the number of coupling constants between the beams and thus with loss of simplicity.