An explanation for the non-uniform grating effects during recording of diffraction gratings in photopolymers

S. Blaya; P. Acebal; L. Carretero; A. Murciano; R. F. Madrigal; A. Fimia

doi:10.1364/OE.18.000799

1. Introduction

Due to the development of holographic data storage systems, a great number of photopolymerizable systems have been developed, including commercial photopolymerizable systems (i.e. Aprillis and Inphase) with optimal properties such as high dynamic range, photosensitivity, dimensional stability, optical clarity and flatness etc... [2, 3, 4, 5]. On the other hand, noncommercial photopolymerizable systems have been widely studied due to their properties such as easy preparation, low cost, high diffraction efficiency with low energetic exposures, capability to obtain high spatial-frequency holograms and temporal stability [6]. In order to characterize the holographic response of photopolymers, energetic (temporal) variations in diffraction efficiency and angular selectivity curves are analyzed. From this experimental data, using the appropriate theory, information relating to the thickness of the grating, scattering and index modulation can be obtained as a function of various experimental parameters, i.e. composition, intensity, thickness etc... In this sense, the coupled wave theory of Kogelnik [7] has been widely used to describe these experimental results and is valid when uniform gratings are recorded with an index modulation that is proportional to the recording intensity. According to this model, in the case of the energetic variation in the diffraction efficiency of phase transmission gratings the experimental curves are explained by a sin² function of the index modulation. However, deviations from the expected responses have been experimentally observed, such as the first maximum of diffraction efficiency on the curve is lower than the second. Recently, in PVA-acrylamide photopolymers [1] these anomalous effects have been experimentally observed at low photoinitiator concentrations in materials of high thickness. According to Kogelnik’s theory these deviations from the ideal response are explained by taking into account the energetic variation of the scattering losses (noise gratings) [8].

In this work we study the response of a photopolymerizable holographic material based on a pyrromethene dye (PM567) acting as a photoinitiator and HEMA as monomer, both of them dissolved in a dry polymeric matrix of PMMA [9]. In these compositions, asymmetrical angular selectivity curves are obtained and the energetic evolution of the diffraction efficiency shows a non-ideal behavior. Thus, the aim of our study is to theoretically analyze the recorded non-uniform gratings and to explain the deviations of the experimental energetic variation of diffraction efficiency by fringe bending.

2. Experimental Procedure

In this study, we analyzed as a photopolymerizable material based on 2-hydroxiethylmethacrylate (HEMA) as a monomer (Aldrich 98 %), poly-(methylmethacrylate) (PMMA) (Average M_w 996,000) as binder (Aldrich) and 1,3,5,7,8-pentamethyl-2,6-diethylpyrromethene-difluoroborate (PM567) as photoinitiator (Exciton). All of these products were used without previous purification. The photosensitive solutions were prepared by dissolving 5 g of PMMA and PM567 in acetone (Qemical). The photopolymeric dry film was obtained by pouring this photosensitive solution onto a leveled BK7 glass and allowing it to dry under normal laboratory conditions (22°C and relative humidity of 60 %) for 20 hours. The concentration of the developed photopolymer analyzed in this work is: [PM567]=1.7 × 10^-4 mol by Kg of photosensitive solution, [HEMA]=0.22 mol by Kg of solution and [PMMA]= 13.2 ± 0.2 % by weight. Moreover the thickness of the films was around 500 μm, measured by a Mitutoyo IP 54 micrometer. Furthermore, all of the compositions show high chemical stability and do not undergo degradation, thus making it possible to use the films some months after their preparation.

Holographic gratings were recorded in symmetric geometry using the output from a p-polarized diode-pumped frequency-doubled Nd : VO ₄ laser (Coherent Verdi V5), which was spatially filtered using a microscope objective lens and a pinhole, and collimated to yield a plane-wave source of light at 532 nm. The light beam was then split into two beams which were spatially overlapped at the recording medium, resulting in an interference pattern with a spatial frequency of 1100 lines/mm and a beam intensity ratio close to 1:1. The diffracted beam intensity of a He-Ne laser (Uniphysics), p-polarized and tuned at 633 nm where the material does not absorb, was monitored in real-time with a photodetector positioned at the Bragg angle for this wavelength. Moreover, the sample was mounted on a monitorized rotation stage rotated by a computer controlled DC motor (OWIS) around an axis perpendicular to the plane defined by the two incident laser beams (located at the intersection of the beam). Once the diffraction grating was recorded, the angular response was measured using the same probe beam.

3. Theoretical Procedure

The analysis of the angular selectivity curves of the recorded diffraction gratings provides quantitative information relating to the response of the materials. In this sense, the thickness of the grating and the index modulation can be obtained by the use of the Kogelnik’s theory. However, in the case of non-uniform gratings this theoretical treatment cannot be applied [7]. Due to absorption effects and shrinkage, non-uniform gratings may be recorded and as a consequence models that take into account effects such as fringe bending and the decrease in index modulation must be used [10, 11, 12, 13]. So, in this study we apply previous theoretical treatments [13, 14] based on the model of Uchida [10] while introducing the possibility of bending of the fringe planes [11]. Assuming, as is usually observed in photopolymers, that pure phase transmission gratings are recorded, the refractive index of the recorded unslanted grating is given by:

n (x, z) = n_{0} + n_{1} (z) \cos (K x + Δφ (z))

where n ₀ is the average refractive index, n ₁(z) the index modulation, K = 2π/Λ, and Λ the grating period. It is assumed that the index modulation n ₁(z) is given by the following equation:

n_{1} (z) = n_{10} \exp (\frac{- 2 α_{r} z}{\cos (θ_{r}^{'})})

where n ₁₀ is the index modulation given at z = 0, α_r is the absorption coefficient at the recording wavelength, and θ́_r is the incidence angle of the recording beam inside the medium at this wavelength. Finally, the term Δφ(z) in equation 1 is the fringe bending which is given by equation 3, where a ₀,a ₁,a ₂,a ₃ are constants and l = z/d, (d is the thickness of the grating along the propagation direction z (effective thickness)). Note that this expression is one of the two cases analyzed in reference [11]. However, it can be easily demonstrated that for arbitrary coefficients both cases are equivalent and the description of the recorded grating is the same.

Δφ (z) = \frac{2 π}{n_{0}} (a_{0} + a_{1} l + a_{2} l (2 - l) + a_{3} l (3 - 3 l + l^{2}))

Therefore, as previously described [10, 11, 13, 14], taking into account that during the reconstruction stage two waves propagate inside the material, the diffracted (S) and the transmitted (R) beam, then by solving the corresponding Helmholtz equation, the following coupled differential equation system is obtained:

c_{R} \frac{d R (z)}{d z} + (α + α_{s}) R (z) = F_{R} S (z)

c_{S} \frac{d S (z)}{d z} + (α + α_{s}) S (z) = F_{S} R (z)

where z ∈ [0,d] and the coupling functions F_R and F_S are given by:

F_{R} = - i \frac{π}{λ_{r}} n_{1} (z) \exp (i Δ φ (z)) \exp (iΔ k_{z} z)

F_{S} = - i \frac{π}{λ_{r}} n_{1} (z) \exp (- i Δ φ (z)) \exp (- i Δ k_{z} z)

here Δk_z is the z component of the vector Δ⃗k, defined by:

Δ \vec{k} = {\vec{k}}_{R} - {\vec{k}}_{S} - \vec{K}

and

Δ k_{z} = K \cos (ϕ + θ_{r} - θ_{0})

where, λ_r is the reconstruction wavelength and θ_r is the corresponding angle of incidence inside the material, θ ₀ is the Bragg angle, α is the absorption coefficient, α_s is a coefficient related to the scattering losses, ϕ is the angle of the grating vector and c_R and c_S are given by:

c_{R} = \cos θ_{r}

c_{S} = - \cos (2 ϕ - 2 θ_{0} + θ_{r})

Finally, the diffraction efficiency, η, for unslanted gratings (given by equation 12 where S*(d) is the conjugate complex of S(d)) and is obtained by numerically solving the system of coupled differential equations 4 and 5 at different incidence angles taking into account the boundary conditions R(0) = 1 and S(0) = 0.

η = \frac{∣ c_{S} ∣}{c_{R}} S (d) S^{*} (d)

4. Results and Discussion

Pyrromethene dyes have been widely used as solid-state dye lasers due to their high energy conversion, long lifetime and operational photostability [15, 16]. In photopolymerizable systems, it has been shown that it is possible to use pyrromethene dyes as photoinitiators in cationic polymerization reactions with a photoacid generator based on a naphthalimide derivative [17, 18, 19, 20] and also in radical reactions with benzophenone [21, 22]. In the case of photopolymerizable holographic recording materials, these compounds have photoinitiated radical polymerization reactions of acrylates in PMMA [23, 9, 24] or acrylamide in PVA [25]. Finally in 3D optical storage pyrromethene dyes have been used as a fluorescent compound in a polymeric host, resulting in a fluorescence modulation which is based on the diffusion of dye within the material [26].

It has been previously demonstrated that a photopolymerizable poly-methylmethacrylatehost system based on 2-hydroxiethylmethacrylate and photoinitiated by a pyrromethene dye can be used as holographic recording material [9]. With these materials, at low intensity, diffraction efficiencies of around 60 % with exposures of ≈ 1 J/cm ² can be reached; however, at high intensity poorer responses are obtained. In figure 1 the energetic variation of the diffraction efficiency at different recording intensities is plotted. As can be seen at an intensity of 2 mW/cm ² the best diffraction efficiencies reached a value of around 25 %. However, for recording intensities of 24 and 99 mW/cm ², the diffraction efficiencies were lower (3 % and 1 % respectively). Moreover, at all the intensities, inhibition periods were observed due to the effects of oxygen during the photoinitiation processes. As can be deduced from figure 1 as the intensity rose the inhibition period increases (1.35, 1.36 and 1.75 J/cm ² which respectively correspond to intensities of 2, 24 and 99 mW/cm ²).

The experimental and theoretical curves are shown in figure 2. The fitted curves were obtained by calculating the diffraction efficiency from equation 12. For this, the differential equation system given by expressions 4 and 5 were numerically solved by varying the free parameters a ₀,a ₁,a ₂,a ₃,n ₁₀,d,α_r,α and α_s. The optimal values of this set of free parameters were obtained from the analysis of the regression coefficient (r ²) [27]. Table 1 lists the values of the corresponding set of optimal free parameters obtained at different intensities. Therefore, it can be deduced from figure 2, that good agreement between theory and experiment was obtained as the regression coefficient values show. Moreover, it is observed that at higher intensity worse fittings were obtained. Regarding the values reached, the index modulation at z = 0 (n ₁₀) decreased as the intensity increased, which is related to the lower diffraction efficiencies reached at higher intensities. Furthermore, it is important to point out that the effective thickness of the grating (d) did not present a defined tendency versus the intensity, thus the average value was around 300 μm. Finally, the absorption coefficient at the recording wavelength (α_r) diminished as the intensity increased. This parameter is related to the concentration of photoinitiator, which, as a consequence of the photobleaching process, depends on the intensity [28]. As the intensity increases the photobleaching rate increases, therefore the concentration of photoini-tiator and absorption coefficient will be lower [28]. According to equation 2, this result implies that at higher intensity the variation of the index modulation along z (non-uniformity of the grating) will be lower. Thus, in figure 3-a the variation of the index modulation as a function of the material thickness is represented. It can be seen that at low intensity, higher index modulations are observed with greater variation throughout the thickness of the material, and n ₁₀ is constant at higher intensities. In order to analyze the effect of the bending given by the parameters a ₀,a ₁,a ₂,a ₃, in figure 3-b the values of Δψ(z) (equation 3) obtained using these set of parameters were plotted. Hence, at low intensity fringe bending is quasi-linear throughout the thickness with negative variations. And at higher thickness, a non-linear behavior is obtained with a positive maximum around 50 μm. Finally, it is important to point out that the observed experimental detuning of the Bragg angle and the displaced diffraction efficiency maximum of around 60 % at lower intensities in the angular responses can be justified by the results displayed in figures 3-a and 3-b.

Fig. 1. Energetical variation of the diffraction efficiency (η) at different recording intensities.

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Table 1. Optimal parameters calculated from the fittings of the experimental angular selectivity curves obtained at different recording intensities. In all cases a0 = as = a= 0.

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Fig. 2. Theoretical (lines) and experimental (symbols) angular selectivity curves obtained at different recording intensities.

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Once the angular selectivity curves have been analyzed, we can relate these results to the corresponding energetic variations of the diffraction efficiency (figure 1). As is well-known [13] and can be deduced from figure 2 due to the fringe bending and non-uniform refractive index profile a Bragg angle mismatch is observed. That is at 2 mW/cm ² at the Bragg angle the diffraction efficiency is around 25 %, whereas the maximum is around 60 %. Therefore, in order to analyze the energetic variation of the diffraction efficiency, in addition to the index modulation building up, the dynamics of the fringe bending must be taken into account. In this sense, an explanation of the non-ideal behavior of the diffraction efficiency in high thickness PVA-acrylamide photopolymers has recently been published [1]. In this study it was experimentally observed that at low photoinitiator concentrations deviations from the ideal response are produced, and these are explained the energetic variation of the scattering losses. This hypothesis is verified by the one beam transmittance experiments, where as well as the photobleaching of the dye, noise gratings are formed [8]. However, the anomalous diffraction efficiency curves obtained with the material used in our study, i.e. figure 1, can not be explained by the hypothesis given in reference [1] due to the following reasons:

Fig. 3. Variation the index modulation (a) and fringe bending (b), versus the thickness of the grating. Data was obtained from the fitting of the angular selectivity curves.

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In reference [1], Kogelnik’s theory is used, taking into account that the index modulation and absorption coefficient (scattering) vary as a function of the energetic exposure. However, in our study non-uniform gratings were obtained due to different reasons such as shrinkage or photopolymerization and diffusion processes.
Moreover, with this material the transmittance curves are mainly given by the photo-bleaching process due to the high transmittance values reached.

So, in our study we propose an alternative theoretical description for non-ideal energetic diffraction efficiency curves which can only be applied when non-uniform gratings are recorded. For this, we modified the model described in the theoretical section, (which is valid for a fixed energy), by introducing the dependency on energy in several parameters. Therefore, by assuming a stationary state, we solved the corresponding coupled differential equations at each of the different energy values. Thus, we assumed that due to the building up of the grating, the index modulation at z = 0 (which appears in equation 2) depends on the energy and is given by equation 13, where n _10f corresponds to the value obtained from the fit of the respective angular selectivity curve (the index modulation reached), Β (cm ²/mJ) is a parameter related to the rate of grating formation and E_inh is the energetic inhibition period.

\begin{matrix} n_{10} (E) = n_{10 f} (1 - \exp (- β (E - E_{inh}))) E > E_{inh} \\ n_{10} (E) = 0 E < E_{inh} \end{matrix}

Furthermore, we assume that the fringe bending varies during recording, thus Δψ(z) instead of being given by equation 3 is written as:

Δφ (z, E) = \frac{2 π}{n_{0}} (a_{0} + A_{1} l + A_{2} l (2 - l) + A_{3} l (3 - 3 l + l^{2}))

where the parameters A ₁, A ₂ and A ₃ are given by:

\begin{matrix} A_{1} & = & a_{1} (1 - \exp (- β_{1} (E - E_{0}))) & E > E_{0} \\ A_{2} & = & a_{2} (1 - \exp (- β_{2} (E - E_{0}))) & E > E_{0} \\ A_{3} & = & a_{3} (1 - \exp (- β_{3} (E - E_{0}))) & E > E_{0} \\ A_{1} & = & A_{2} = A_{3} = 0 & E < E_{0} \end{matrix}

here a ₁, a ₂ and a ₃ are the parameters obtained from the fit of the respective angular selectivity curve, Β_i (cm ²/mJ) the rate of variation of the i-parameter, E ₀ (mJ/cm ²) is related to the energetic period at which the bending of the grating begins, and it is assumed that E ₀ > E_inh.

Thus, at the Bragg angle, by introducing n ₁₀(E) and Δφ(z,E), for each value of energy with a stepsize of 1 mJ/cm ², the system of coupled differential equations 4 and 5 may be solved, obtaining the diffraction efficiency at the Bragg angle as a function of the exposure. So, by varying the parameters Β, Β_i and E ₀ we simulated the experimental curves obtained at different intensities for this composition. Figure 4 shows the corresponding theoretical and experimental curves and as can be seen a qualitative description of the energetic variation of the diffraction efficiency can be realized. At low intensity a good concordance between theory and experiment is obtained; however, at higher intensities the exact shape of the second peak of the diffraction efficiency can only be qualitatively described. Moreover in table 2 the parameters obtained from these theoretical curves are shown. As can be seen, the energetic rate of grating formation is higher at low intensities and the E ₀ parameter is lower.

Table 2. Theoretical parameters calculated from the fittings of the experimental energetic variations of the diffraction efficiency obtained at different recording intensities.

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5. Conclusions

In this study we analyzed the holographic properties of a photopolymerizable system based on a poly-methylmethacrylate host and 2-hydroxiethylmethacrylate photoinitiated by a pyrromethene dye (PM567). For this, we studied at different recording intensities the inhibition period observed in the recorded energetic variations of the diffraction efficiency and the respective angular selectivity curves. Furthermore, from the theoretical analysis of the angular selectivity curves, the fringe bending of the planes, non-uniform refractive index modulation and thickness of the grating were obtained. Moreover, it is important to point out that with these compositions non-ideal energetic variations of the diffraction efficiency were observed. In this study we proposed that these non-ideal curves can be explained by effects related to the non-uniform gratings recorded. Hence, by introducing the energetic variation of the index modulation and the fringe bending in the theoretical treatment of Kubota and Uchida a qualitative description of the experimental energetic diffraction efficiency curves can be obtained using the data extracted from the corresponding angular selectivity curves.

Fig. 4. Experimental (symbols) and theoretical (lines) energetic variation of the diffraction efficiency (η) at different recording intensities

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Acknowledgments

The authors acknowledge support from project FIS2006-09319 and FIS2009-11065 of Minis-terio de Ciencia y Tecnología (CICYT) of Spain.

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Intensity	α_r	a₁	a₂	a₃	n ₁₀	d(cm)	r ²
(mW/cm ²)	(cm^-1)					Effective thickness
2	13	-0.66	0.3	-0.3	0.00079	0.0315	0.9993
24	0.5	-1.28	-1.7	2.0	0.00029	0.0325	0.9901
99	0.5	-1.80	-1.0	1.7	0.00019	0.0250	0.9841

Intensity	β	β ₁	β ₂	β ₃	E_inh	E ₀
(mW/cm²)	(cm²/mJ)	(cm²/mJ)	(cm²/mJ)	(cm²/mJ)	(mJ/cm²)	(mJ/cm²)
2	0.0021	0.006	0.04	0.0045	1340	1460
24	0.0008	0.029	0.005	0.022	1416	2004
99	0.0005	0.021	0.0004	0.0033	1587	2430

Intensity	α_r	a₁	a₂	a₃	n ₁₀	d(cm)	r ²
(mW/cm ²)	(cm^-1)					Effective thickness
2	13	-0.66	0.3	-0.3	0.00079	0.0315	0.9993
24	0.5	-1.28	-1.7	2.0	0.00029	0.0325	0.9901
99	0.5	-1.80	-1.0	1.7	0.00019	0.0250	0.9841

Intensity	β	β ₁	β ₂	β ₃	E_inh	E ₀
(mW/cm²)	(cm²/mJ)	(cm²/mJ)	(cm²/mJ)	(cm²/mJ)	(mJ/cm²)	(mJ/cm²)
2	0.0021	0.006	0.04	0.0045	1340	1460
24	0.0008	0.029	0.005	0.022	1416	2004
99	0.0005	0.021	0.0004	0.0033	1587	2430

An explanation for the non-uniform grating effects during recording of diffraction gratings in photopolymers

Abstract

1. Introduction

2. Experimental Procedure

3. Theoretical Procedure

4. Results and Discussion

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Tables (2)

Equations (15)

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