1. Introduction

The parameters that govern the behaviour of photopolymers may be characterised using different diffusion models [1–6]. The parameters obtained depend on the chemical composition used, each chemical composition being optimized for each particular application. Photopolymer applications include grating couplers [7], focusing gratings [8], optical interconnects [9], optical data storage [10], holographic filters [11], acoustooptical deflectors and acoustooptical modulators [12], since these materials have several attractive advantages. In recent years there has been an increase in holographic data storage applications and new companies have developed photopolymer disks for information storage [10, 13–19]. In order to achieve high capacity holographic memories the thickness of the material must be 500 µm or more. In this way the material may be competitive with current storage information devices [10]. The high thickness allows us to obtain narrow angular scans and multiple holograms can be recorded overlapped at each location by using hologram-multiplexing methods such as angle, wavelength, phase-code, fractal, peristrophic, and shift multiplexing [10]. In order to optimize the absorbent photopolymers for data storage, low concentrations of dye and monomer are needed to obtain low noise (scattered light) and large values of effective optical thickness. However, in order to obtain high values of dynamic range (M#) we need to record many holograms in the same volume with high concentrations of dye and monomer [10]. Therefore, the optimum balance between these characteristics must be found. Nevertheless it is interesting to remark that multiplexing methods can be used to record many diffractive elements in the same volume [20], for example lenses with different focal lengths, applications that are very interesting for many displays.

Using polyvinylalcohol/acrylamide (PVA/AA) based photopolymer it is easy to obtain layers with a physical thickness of around 1 mm [21–22]. The composition of this type of layers has been optimized to record holographic gratings. When modeling the behaviour of these layers using conventional models, thick layers present two important drawbacks. In the first place, the difference between the effective optical thickness and the physical thickness in the material due to the absorption of light inside the material [23], and in the second, the growth of scattered for long recording exposures [21], usually higher than 150 mJ/cm². To solve the first problem 3-D analysis has been proposed by our research group taking into account the variation in the absorption of light in the material versus exposure time [24]. In this paper we use this analysis to describe the behaviour of the material in real time and to study different exposure intensities both theoretically and experimentally.

2. Theoretical model

Let us now reintroduce the model used to analyze the grating formation [24]. We propose the use of the following two equations to describe 3-dimensional polymerization

where n is the refractive index, [M] the monomer concentration, [P] the polymer concentration, D is the difusivity, γ indicates the relationship between the intensity and the polymerization rate, k_R, and I(x,z,t) is the recording intensity attenuated by the dye inside the material, as defined in previous studies [24]

where I ₀ is the average intensity, Λ is the grating period (0.88 µm in this work), and α is the coefficient of the light attenuation.

The absorption of light inside the material as a function of time can be described in a first approximation as follows

The initial value of α(α(0)=α ₀) can be obtained if the transmittance and the physical thickness of the layer are known. In our material for layers with a physical thickness of around 900 µm the transmittance is around 0.5%. In others words, the values of α ₀ for this composition are around 0.006 µm^-1 [24]. β is a constant that determines the influence on the intensity as the dye is consumed and depends on many factors (chemical composition of the material, temperature, humidity, etc.). The absorption decay depends on the dye and the intensity used, and K_α defines the rate at which the dye is consumed. In this paper we use β=0.5 and K_α=0.005 cm² m W^-1 s^-1 in our fittings [24].

The Trommsdorff effect describes the evolution of the polymerization rate with time in three steps, firstly the induction period, secondly the autoacceleration, and in the last place the polymerization decreases due to the limitation of the diffusion [25]. We assume that the two first processes are very fast, so we only take into account the decrease in the polymerization rate due to the Trommsdorff effect after the process is initiated. It is given by

where φ is the attenuation coefficient of the polymerization rate.

The diffusivity is limited by the presence of polymer, then

If the thickness of the material is around 1 mm and the period of the grating recorded is lower than 1 µm, then we can assume that

Using this approximation the monomer diffusion along the z axis can be disregarded, and eqs. (3) and (4) can be solved in a similar manner to that used for the 2-D case.

Our simple approach to solve these equations in order to obtain qualitative information about the behaviour of the material in depth is as follows: The photopolymer film is divided into G different sub-films each of thickness d_g; the total thickness of the photopolymer, d, is given by the sum of the thicknesses of the different sub-films

where G is the number of sub-films used. Then we can obtain the Fourier refractive index expansion for each sub-grating.

The refractive index modulation, n ₁(t), depends on the refractive indexes of the different components and following Aubrecht et al. [26], —with a minor typographical correction (6 appears in the denominator of the first part of the right hand side of the eq. (9) instead of 3 as has been shown in previous papers [24, 27, 28])—, the refractive index modulation can be written as follows

where n_p is the polymer refractive index, n_m is the monomer refractive index, n_b is the binder refractive index and n_dark is the refractive index of the layer before exposure.

In the material used in this study the different refractive indexes take the following values n_m=1.486, n_b=1.474 and n_dark=1.478. These values were obtained using the Lorentz-Lorenz equation and the method used is described in references 24 and 26. The calculations are based on refractometer measurements using water solutions.

One of the problems involved in working theoretically with thick layers (where the refractive index profile is attenuated in depth) is calculating the diffraction efficiency around the Bragg angle. It is common in holography to express the profile recorded in the grating in terms of the refractive index. For this study we assume that a refractive index profile stored in the sub-layer g, n_g(z), has the form [29–30]

where K=2π/Λ. In our material n ₃≤n ₁/8 and n ₃ can be neglected [23, 24]. Therefore they do not affect the values of the higher harmonics in the diffraction efficiency around the first Bragg angle. In this paper we obtain the refractive index profile in each sub-layer using the diffusion model and then we relate the harmonic components of the refractive index to the Fourier components of the dielectric permittivity (ε), provided that n ₁≪n ₀, in order to calculate an algorithm based on Rigorous Coupled Wave analysis (RCW) proposed by Moharam and Gaylord [31]. In this case

where ε _±i are the harmonics of the Fourier expansion of the dielectric permittivity. The method used in this paper is described partially in previous papers [23, 29], but now we don’t assume exponential the attenuation of the index in depth. In this paper the values of the refractive index profile are given by the model in each sub-grating. To apply the boundary conditions we assume as in previous papers that there are no reflected orders between the sub-layers inside the material.

3. Experimental

The photopolymerizable solution was prepared, as in previous papers [21, 22], adding yellowish eosin (the dye, 9.00×10^-5 M), together with a solution of acrylamide (the monomer, 0.34 M) and triethanolamine (the co-initiator, 0.15 M) to a PVA solution (the binder, 13.30% weight/volume). In this study the solutions were prepared using a conventional magnetic stirrer, under red light and in standard laboratory conditions (temperature, pressure, relative humidity). If the viscosity is too high to use a magnetic stirrer, we stir the solution slowly by hand to prevent the formation of air bubbles and then apply a vacuum atmosphere to both the prepared solution and the coated molds so that any eventual air bubbles are completely eliminated [20]. The solutions are deposited, by gravity, in polystyrene circular molds and left in the dark to allow the water to evaporate, while recording in laboratory conditions (temperature and relative humidity) during the process. When a high percentage of the water content has evaporated (around 88%) [20], the “dry” material is removed from the mold, cut into squares and deposited, without the need of adhesive, onto the surface of glass plates measuring 6.5×6.5 cm². The plates are then ready for exposure, which takes place immediately. On each plate we record four gratings with a spatial frequency of 1125 lines/mm and diameter of 1.5 cm, the experimental set-up used to recorded non-slanted gratings can be seen in previous works [20, 28].

4. Results and discussion

4.1 Behaviour of thick layers in real time

We now present and discuss the fittings of the experimental data taking into account the attenuation of light inside the material. In Fig. 1 we present the fitting of a layer with a physical thickness of around 900 µm and initial transmittance of 0.5% (white dots) for the recording wavelength (514 nm). These data can be obtained before the recording processes using a spectrometer and a conventional micrometer. The grating was recorded using an incident intensity of 6 mW/cm². The parameters obtained with the fitting are D=2×10^-10 cm²/s, k_R=0.018 cm mW^-1/2 s^-1, n_p=1.506 and φ=0.005 cm²mW^-1µm^-1. We can see in Figure 1 that with this model the behaviour of the 800 µm and 1000 µm layers is similar during the first 15 seconds. This occurs because no grating is recorded at a depth of over 700 µm inside the layer. We applied the model to simulate different cases of physical thickness using the same parameters (diffusivity, polymerization rate, etc.). We believe it is interesting to note the similar behaviour of the layers with a thickness of over 600 µm. This effect is due to the limitation of the optical thickness by Beer’s law and to the fact that in last sub-layers of the material no gratings are recorded. This phenomenon cannot be predicted using 1-dimensional models. In these models if higher thickness are studied the grating strengths are higher too [24], and to fit one layer correctly it is necessary to determine the effective optical thickness beforehand. In other words you need to fit the angular response first [23]. For this value of absorption, the results using a 1-dimensional model and the model used in this work are similar only when the thickness of the layer is less than 500 µm.

 

Fig. 1. Diffraction efficiency (DE) versus exposure time for one grating with a physical thickness of 900 µm, the experimental data (white dots) and the simulations using the theoretical model for different thicknesses (1000 µm, 800 µm, 600 µm, 300 µm and 100 µm).

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Fig. 2. Diffraction efficiency (DE) as a function of replay angle for the grating presented in Figure 1 (900 µm of physical thickness and spatial frequency of 1125 lines/mm). Experimental (o), algorithm using diffusion model (—).

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When we take into account the values of refractive index obtained in each sub-layer using the model to calculate the diffraction efficiency with an algorithm based on RCW, we can see the smoothing of the secondary lobes in the angular scan due to attenuation of the refractive index modulation in depth [23, 29] (Fig. 1). We can see in Fig. 2 how the diffusion model presented predicts the smoothing of the angular response and diffraction efficiency of the thick layers. In this case the effective optical thickness [23] of the layer is 690±20 µm and the initial value of the refractive index modulation is 3.10×10^-4.

4.2 Gratings recorded with different recording intensities

The predictions of the model were evaluated for different values of recording intensity. The results obtained for 2 mW/cm² and 20 mW/cm² are presented in Fig. 3. For these intensities we need to expose the layer for around 60 and 5 seconds, respectively, to obtain maximum diffraction efficiency at the Bragg angle. We can see that the effective optical thickness is similar for these two intensities plotted in Fig. 3.

 

Fig. 3. Refractive index modulation, n ₁, as function of the thickness and the exposure time for two different recording intensities: 2 mW/cm² and 20 mW/cm²

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Nevertheless, when an experimental analysis is done, differences between the layers recorded using different intensities are observed. In Fig. 4 the angular responses for two different recording intensities, 2 mW/cm² and 20 mW/cm², respectively, are shown.

 

Fig. 4. Angular responses for two different recording intensities: 2 mW/cm² and 20 mW/cm².

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We can see that the central lobe is narrower when higher intensities are used. Then for these high intensities the effective optical thickness of the gratings is greater. In order to fit the experimental data we use the algorithm described in reference 23. We obtain 570±20 µm (intensity of 2 mW/cm²) and 880±20 µm (intensity of 20 mW/cm²) as the effective optical thickness of the gratings, and the refractive index modulation near the surface is 3.05×10^-4 and 4.10×10^-4, respectively. The whole study done in the second half of the paper is summarized in Table 1.

Table 1. Parameters of the gratings recorded using different intensities.

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The explanation of the differences between the effective optical thicknesses described in Table I can be related to two factors. One is the monomer diffusion in the z direction. For low recording intensities we need longer exposures (around 60 seconds) to obtain the maximum diffraction efficiency. Therefore monomer diffusion in the z direction plays an important role in grating formation. The other is the low values of recording light in the last sub-layers of the material for small intensities. The layers with a physical thickness of around 900 µm present transmittances of around 0.5% for the recording wavelength. In other words, the recording intensity in the last sub-layer of the material is 10 µW/cm² (whereas the recording intensity in the first sub-layer is 2 mW/cm²). We checked this effect using similar material and observed that under intensities of 70 µW/cm² no gratings are recorded after 100 mJ/cm2 of exposure. If an attenuation of light of 0.006 µm^-1 is considered for a recording intensity of 2 mW/cm², then the intensity inside the material at a depth of over 540 µm is lower than 70 µW/cm². Therefore, we believe this to be one of the causes of the small effective optical thickness obtained using an intensity of 2 mW/cm². Consequently, the model cannot explain the differences between the optical thickness for different recording intensities and cannot be applied for weak recording intensities. This problem can be solved introducing a recording intensity cutoff value in the model, for intensities lower than this value no gratings can be recorded, this value must be determined experimentally for each dye and photopolymer composition.

5. Conclusions

Good fittings have been obtained using the diffusion model and the algorithm based on RCW for one particular recording intensity. This model permits diffraction efficiency to be fitted as a function of time without previously determining the effective optical thickness of the layer. It is only necessary to know the absorption and physical thickness of the layer. Nevertheless, for large changes in the recording intensities the model has its limitations. The experimental dependence of the effective optical thickness on recording intensity in this type of materials has been shown.