1. Introduction

As shown earlier [1–3], the presence of two orthogonal nonlinearities — uneven exposure (or visibility of interfering beams) over the hologram field and a nonlinear dependence of diffraction efficiency on exposure leads to appearance of a limiting achievable diffraction efficiency and a change in the optimal exposure of the hologram. This effect, later called a formfactor (by analogy with the one known in gravity problems), vanishes when either of the two indicated nonlinearities disappears. The significance of this effect is confirmed by the results obtained later, which indicate the applicability of the concept of formfactor not only for holograms recorded by Gaussian beams, but also for holograms of complex images e.g. portrait holograms, because by virtue of the central limit theorem (CLT) they also have Gaussian brightness distribution statistics [4]. Furthermore, in the previous work [5] we have shown an interesting effect of the interaction of the abovementioned two nonlinearities with the nonlinearity of the photographic response of a holographic material to exposure. It turned out that the presence of the third nonlinearity leads to an increase in the maximum achievable diffraction efficiency instead of decreasing it. Other interesting effects of the formfactor were noted. For example, in [6], it was shown that the formfactor is also significant in determining the mutual influence of the nonlinear dependence of the degree of coherence of laser radiation along the direction of its propagation and the same nonlinearity of the hologram parameters over its field.

In this paper, we present new experimental results on the influence of the formfactor on the diffraction efficiency of holograms recorded in a chalcogenide glassy semiconductor and an azopolymer, which further extends the practical applicability of the formfactor effect previously discovered for volume phase holograms.

2. Volume holograms formfactor

Figure 1 shows how the local first-order diffraction efficiency changes over the entire field of a hologram with an increase in exposure. This shape is caused by the formfactor of a hologram: during the recording, the local diffraction efficiency reaches maximum in the center, then goes over it and falls, while it still grows at the edges of the hologram. This leads to a decrease in the average maximum achievable diffraction efficiency and a number of other effects as was shown in [1–6].

 

Fig. 1. The evolution of first-order diffraction efficiency of a hologram recorded by Gaussian beams. Increase of the exposure from (a) to (c) first leads to an increase in η (a), then to a dip in the center (b) and again to an increase in η (c).

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It was noted that the formfactor influence on diffraction efficiency is present in recording media only in simultaneous presence of two effects – the non-linearity of the photoresponse η_max and the inhomogeneous brightness over the hologram field E(x,y). Moreover, with the elimination of one of these two nonlinearities, the formfactor effect degenerates and the achieved diffraction efficiency is described [5] by the classical expression similar to Eq. (2), but for Ψ = Ω = 1 obtained by Kogelnik for a plane wave [7]. The degree of approximation to the classical expression is determined by the degree of approximation of one of the two aforementioned nonlinearities to a direct uniform distribution. The average diffraction efficiency η_max defined by Eq. (1) transforms into classical Eq. (2) similarly to [7] when the multiplication $\beta \cdot E(x,y)$ and V(x,y) is constant over the hologram field.

As in [4,6] the exposure of E here is given in normalized units, so that E = 1 changes the photoresponse of the holographic material leading to an increase in the sine-argument of one radian. Thus, exposure E = π/2 with β = 1 leads to η = 100% with visibility V = 1 in accordance with the expression found by Kogelnik, similar to Eq. (2) for $\Psi f({E \cdot V} )= E \cdot V$. The value Ψ determines the exposure delay to achieve the first maximum diffraction efficiency. According to the numerical simulation shown in Fig. 2(b), Ψ = 0.69 The maximum η in Fig. 2(a) is greater than the maximum in the experimentally obtained graph Fig. 2(b), since the latter on Ω was affected by the uniform intrinsic absorption of the holographic material Reoxan which was not taken into account when calculating the graphs Fig. 2(a).

 

Fig. 2. Dependence of the average diffraction efficiency ηmax on exposure. (a) Theoretical plot: the solid curve represents the diffraction efficiency according to Eq. (1) with the Gaussian form E(x,y) for determining Ψ; the dashed curve is the diffraction efficiency for the case E(x,y) = const with Ψ = 1, normalized to the maximum of the solid curve Ω = 0.41 in accordance with Eq. (2), for the convenience of comparison. (b) Experimental graph: 1st curve – nonlinear photoresponse of the refractive index Δn of the holographic material Reoxan depending on the exposure; 2nd curve – dependence of the average diffraction efficiency on exposure.

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The obtained dependence of the η_max(E) when recording by Gaussian beams, as shown in [4,6], is in good agreement with the 2^nd experimental curve in Fig. 2(b) obtained on a holographic recording medium Reoxan [8]. The details of Reoxan preparation and its properties can be found in [9].

3. Thin holograms formfactor

New results of the formfactor influence on diffraction efficiency were also obtained for 2D (thin) holograms recorded on thin layers of azopolymers or chalcogenide glassy semiconductors (CGS). These materials showed a unique ability to register relief structures with a deep profile due to photoinduced mass transfer [10–12]. The deep surface relief (up to several micrometers) provides a wide dynamic range of the phase photoresponse of the media, which can be more than 30 rad with a grating depth of 2 µm. Holographic diffraction gratings were recorded as periodic structures of the relief phase with a period of 5 µm in converging beams according to the Leith-Upatnieks scheme [13]. The details of CGS and azopolymer thin layer obtaining and their properties can be found in [14] and [15] correspondingly. As shown in Fig. 3, the recording scheme of such relief-phase gratings is displayed.

 

Fig. 3. Holographic recording scheme of relief phase gratings. DPSS laser – diode-pumped solid-state single-mode (TEM₀₀) laser; BS – beam splitter; M – mirrors; λ/2 – half-wave plates; LD – laser diode; S – recording media; PD – silicon photodiodes.

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Holographic recording was carried out in the sensitivity range of the recording media using a wavelength of 532 nm for CGS and 473 nm for azopolymer. The convergence angle of the recording beams θ was chosen to obtain an interference pattern with a period Λ = 5.0 µm in accordance with the equation $\Lambda = {\lambda / 2} \cdot \sin ({\theta / 2})$ (θ = 6.1° for λ = 532 nm and θ = 5.4° for λ = 473 nm). Half-wave phase plates were installed so that the interfering rays had a linear polarization parallel to the incidence plane.

The distribution of light intensity in the interference pattern is converted to a spatial change in the thickness of the material during holographic recording. Exposure of the recording medium to an interference pattern with a sinusoidal intensity distribution leads to the formation of a surface relief proportional to the light intensity. The formation of a surface profile close to sinusoidal was confirmed by an atomic force microscope and was shown in [15–17]. The experimental measurements of diffraction efficiency were carried out in real time at a wavelength of λ = 650 nm (in the transparency range of the recording medium) with normal incidence of the reading beam. It is known that thin phase holograms are characterized by the presence of many diffraction orders unlike volume holograms. In this case the n-th order diffraction efficiency (η_n) for the sinusoidal phase hologram is described by the Bessel function in the form ${\eta _n} = J_n^2({{\Delta \varphi } / 2})$ [18,19], where J_n is an n-th order of the Bessel function of the first kind, Δφ – amplitude of phase contrast, which in the case of a relief phase hologram is $\Delta \varphi = {{2\pi h(n - 1)} / \lambda }$ [19–21]. Here λ denotes a wavelength of the reading beam, n is a refractive index of the medium on which the hologram is recorded and h denotes surface profile depth. Analysis of this expression shows that the maximum diffraction efficiency of the phase sinusoidal hologram is 33.9%. The theoretical dependence of diffraction efficiency in the zero, first and second orders for the relief-phase sinusoidal grating (n = 2.5, λ = 650 nm) is presented on Fig. 4(a).

 

Fig. 4. (a) Diffraction efficiency of 0, 1 and 2 orders of the phase sinusoidal grating depending on the amplitude of the phase contrast Δφ and the relief depth h (for a material with a refractive index of n = 2.5, at a wavelength of λ = 650 nm). (b) The experimentally obtained kinetics of diffraction efficiency of zero η₀ and first η₁ orders in the case of recording on a CGS

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The experimental dependence of the diffraction efficiency of 0 and 1 orders of the relief grating recorded in the scheme of Fig. 3 on the CGS layers, depending on the exposure time t, is shown in Fig. 4(b).

The theoretical dependence of diffraction efficiency shown in Fig. 4(a) actually represents a case of a linear increase in the depth of the relief (linear photoresponse). The absence of zero values on the curves in Fig. 4(b) in comparison with the calculated curves in Fig. 4(a) can be explained by the same effect of the formfactor as in [1–6]. In the case of inhomogeneous intensity distribution during exposure, the depth of the grating profile will be equally inhomogeneous over the hologram field and will vary depending on the intensity at the recording location. To study the effect of illumination inhomogeneity on diffraction efficiency during the formation of a relief grating (i.e. the influence of the formfactor), unexpanded laser beams were used, which are well described by the Gaussian function. The interference of such beams leads to the formation of an interference pattern with a Gaussian distribution of the depth of intensity modulation and, therefore, to the formation of a holographic grating with a Gaussian distribution of the depth of the surface topography. The diameter of the unexpanded beam was 1.5 mm; the surface intensity at the point of incidence of two beams of the same intensity was 4.3 W/cm². The study of the profile of the resulting grating using atomic force microscopy showed the presence of a well-developed surface relief with sinusoidal profile and depth of about 2.5 µm. According to the formula $\Delta \varphi = {{2\pi h(n - 1)} / \lambda }$ for surface relief gratings, this value corresponds to a phase modulation depth of 36.2 radians in the case of CGS layers for which the refractive index at a wavelength of 650 nm is 2.5.

Figure 5(a) shows the interference patterns of two coherent beams with a Gaussian intensity distribution and Fig. 5(b) with uniform intensity.

 

Fig. 5. Intensity profiles in interference patterns: (a) when two beams are mixed with a Gaussian intensity distribution; (b) when two beams are mixed with a uniform intensity distribution.

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Figure 6 shows the experimental kinetics of first-order diffraction efficiency of recorded gratings using beams with uniform intensity and beams with a Gaussian intensity distribution. As in Fig. 2, the first-order diffraction efficiency η₁ of a grating recorded by Gaussian beams does not exceed 8%, and the first-order diffraction efficiency η₁ of a grating recorded by homogeneous beams reaches its theoretical maximum of 33.9%. In addition, the position of the maximum diffraction efficiency of the second grating (Gaussian beams) does not coincide with the position of the maximum of the first grating (uniform beams), but lags behind it. This indicates the presence of a formfactor that turned out to be equal to Ψ = 0.75 in the experiment on recording on a CGS.

 

Fig. 6. The dependences of the first-order diffraction efficiency η₁ for the recorded gratings using homogeneous beams (red) and beams with a Gaussian intensity distribution (blue).

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The maximum of the second grating does not coincide with the maximum of the first. Also, observing experimentally the distribution of diffraction efficiency over the field of the recorded hologram with increasing exposure Fig. 7, one can observe a picture similar to that calculated in Fig. 1.

 

Fig. 7. The evolution of the first-order diffraction pattern of the hologram recorded using Gaussian beams a) on the azopolymer and b) on the CGS. Rings with zero diffraction efficiency, similar to Fig. 1, are clearly visible. The appearance and increase in the number of rings with increasing exposure is seen, similarly to Fig. 1.

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4. Conclusion

The results obtained in the work indicate that the formfactor effect, discovered earlier and described in [1–6] for volume media, also applies to thin holograms. Thus, it can be argued that this effect is present in all holograms, regardless of their type and nature of diffraction. However, due to the difference between the values of the formfactor for the Raman-Nath diffraction and for that for the Bragg diffraction, the formfactor for the diffraction on surface gratings on a CGS turned out to be slightly larger Ψ = 0.75 than that for the corresponding volume phase media, e.g. Reoxan Ψ = 0.69, on which the effect of the formfactor was first noticed. The number of rings in the image of the diffraction orders reconstructed by the hologram allows to determine the photoresponse of the holographic material directly in the process of recording the hologram by the Gaussian beams without involving additional equipment and technologies.