Parameters extracted from fits to experimentally obtained growth curves of refractive index modulation in uncoverplated photopolymer layers.
Photopolymers were first introduced as a holographic recording material by Close
Photopolymers generally consist of a monomer, a photosensitive dye and an initiator. They can either be liquid or dry layer systems. The dry photopolymers usually contain a polymeric binder in addition to the other components. As mentioned above the first photopolymer material used for holographic recording, was reported by Close
In the eighties, Calixto  continued the work on acrylamide-based systems. The material contained acrylamide monomer, TEA as an electron donor, methylene blue photosensitizer and PVA as a binder. Blaya
While acrylamide based free-radical systems have received much attention in the literature other materials offer advantages when it comes to the development of practical storage media. Trentler
More recently, works have been carried out on the development of a commercial grade acrylate photopolymer with the same two step polymerisation , offering very fast response of the refractive index modulation with respect to the recording dosage, particularly at lower power densities, very high storage resolution, and very large refractive index modulation [25,26].
Extensive work has been carried out, in both industry and academia, on the development of these photopolymer materials and more recently on the understanding of the photochemical kinetics associated with them. In order to maximise the potential of these materials for various applications, the necessity for a physically comprehensive theoretical model of the effects which occur during photo-polymerization is becoming ever more important. Providing such a model will enable potential trends in a material’s performance to be recognized and optimised, . Such models allow simulations of ratios of various key material components to be made, yielding indications of the most suitable material compositions in order to improve material performance.
In this chapter we examine some recently published results on the Non-local Photo-polymerization Driven Diffusion (NPDD) model, [26-32]. This model provides a comprehensive theoretical representation of the processes, which occur during free radical photo-polymerization. The physically realistic model enables predictions to be made about a number of very different photopolymer materials [26-32]. We present several extensions to the previous model in particular allowing for spatially and temporally varying primary radical generation, oxygen inhibition, dark reactions and chain transfer effects. We then apply this model to analyse a number of effects observed to take place during holographic grating formation in an acrylamide/polyvinylalcohol (AA/PVA) based photopolymer and compare experimental results and the predictions of the model with the aim of characterising these effects.
The chapter is structured as follows: In Section 2 we will briefly examine some of the methods used to measure and monitor the optical performance of holographic gratings in photopolymers. In Section 3 we briefly describe the fundamental photo-kinetic processes, which occur in photopolymers during holographic exposure. Then based on these photo-kinetic processes we construct a set of first order coupled differential equations which represent the temporal and spatial variation of the concentrations of constituents of the photopolymer, which form the basis of the Non-local Photo-polymerization Driven Diffusion (NPDD) model. We then analyse some of the simulated behaviour of this model under given physical conditions to examine its physicality and present some of the predictions made which offer potential methods to improve a photopolymers holographic performance. Following these predictions, in Section 4 we then attempt to improve an acrylamide/polyvinylalcohol (AA/PVA) based photopolymer’s spatial frequency response which will increase its high density storage resolution. In Section 5 a brief conclusion is presented offering potential direction for future advancements in the area of photopolymer development.
2. Optical testing of photopolymers for holography
In the study of holographic recording materials it is common to record gratings in photosensitive materials, such as photopolymers, and to then optically examine the resulting gratings. The gratings produced are often electromagnetically modelled using Kogelnik’s two-wave coupled wave theory. Kogelnik’s two-wave coupled wave theory, , describes the efficiency with which thick holograms can diffract incident light. Analytic expressions for both the angular and wavelength dependence of the diffraction efficiency as the incident light deviates from the Bragg condition are derived. Thus the dependence of the diffraction efficiency,
Rearranging Eq (1) enables an expression for the temporally varying refractive index modulation,
By monitoring the amount of light diffracted from a weak probe beam during exposure,
During the grating recording process the evolution of the grating is monitored in real time. One of the main advantages of many photopolymer materials is that they are self-processing and thus, non-latent, , therefore the diffractive scattering properties are immediately available as the grating is being formed. This allows the evolution of the grating to be monitored by replaying the grating as it is being recorded using a probe laser with a wavelength, which lies outside the absorption spectrum of the photosensitizer used. This ensures that the probing does not affect the fabrication process. In the set-up presented in Figure 1 the probe laser is operating at
Examining Eq (1) it is clear that the on-Bragg replay angle of the probe beam,
In the above we have emphasised the recording of a single unslanted gratings and the capture of growth curves. However, the photosensitive sample can also be mounted on a rotation and translation stage (as shown in the inset of Figure 1), enabling the analysis of the angular response of the grating and the recording of grating arrays, . In Holographic Data Storage (HDS) where the storage of multiple pages of information is required, these holograms (gratings) are angularly multiplexed on top of each other within the same volume of the holographic medium. It is the ability to achieve this which makes photopolymer materails an attractive media for optical storage.
It must also be noted that using this optical setup, the temporal variation in the absorbance of the photo-sensitiser can be examined. As the recording beams which are used to record the grating, transmit through the photopolymer sample, they can be collected in the photo-detectors shown in Figure 1. Thus enabling key material parameters related to the photo-absorption kinetics and sensitivity of the photopolymer, to be examined.
In the next section we will examine the photo-kinetic and photo-physical behaviour of these photopolymer materials. It is these reactions which are the basis of the theoretical models which are used as photopolymer material optimisation tools and are the fundamental building blocks of the Non-local Photo-polymerisation Driven Diffusion (NPDD) model [11,26-34].
3. Photo-kinetic behaviour
Let us begin this section with a review of the kinetic models of photopolymerisation which have been presented in the literature.
3.1. Review of kinetic models
The photochemical processes, which are present during photopolymerization, are complex [11,26-34,38-43], however an understanding of these processes is of utmost importance if a practical model is to be developed. In a recent review,  many of the assumptions made in developing photochemical models of free radical photo-polymerisation were discussed, [38-43]. A number of physical effects not included in the current models were listed, which indicated a lack of physicality under certain exposure conditions. Following the appearance of this review, a series of papers were published [30-32] which addressed many of these issues and provided a model containing a consistent set of chemical reaction equations to take into account many of these effects. These effects included;
Removal of the steady state approximation for macroradical concentration,
inclusion of spatially and temporally non-local polymer chain growth,
inclusion of time varying photon absorption,
simultaneously including the effects of both primary, i.e. -, and bimolecular, i.e. -, termination,
inclusion of the changes in the polymerization kinetic constants caused by increased viscosity, and finally
inclusion of polymerization inhibiting effects.
The resulting Non-local Photo-polymerisation Driven Diffusion (NPDD) model was then experimentally verified by applying it to study (a) normalised transmission curves, and (b) growth curves of refractive index modulation for both short and continuous holographic exposures, in two significantly different free radical photopolymer materials [23,30-32]. The quality of the fits obtained to both photopolymer materials, indicated the versatility and applicability of the NPDD model.
In the past number of years, extensive work has been presented in the literature to describe the time varying absorption effects, which occur in photopolymer materials during exposure, [21,28,29]. In all cases the aim has been to improve the understanding of the photo-kinetics occurring in these materials, and critically to enable accurate predictions of the generation of primary radicals. A model of photosensitiser behaviour proposed by Carretero
We will now examine the methods used to extend the NPDD model in by more accurately representing the temporal and spatial variation of the photosensitiser concentration and the associated temporal and spatial generation and removal of primary radicals. As a result the number of approximations made in modelling the photo-initiation kinetics are significantly reduced. Thus a more physically accurate representation of the photo-polymerization kinetics is produced. Crucially, the increased physicality of the proposed model enables a more accurate analysis of the process of inhibition.
3.2. Reaction mechanisms
The kinetic model presented in this analysis is based upon the following four reaction processes.
In the above set of chemical equations,
3.3. Primary radical production
As can be seen in Eq (4), the initiation process involves two steps: The
In order to do this, we assume that the following photochemical reactions, take place upon illumination of a photopolymer layer sensitised with a xanthene or thiazine type photosensitiser , with an appropriate wavelength. These are as follows,
In these equations
In order to use the proposed rate equations, it is first necessary to convert the exposure intensity
The rate of production of the excited state photosensitiser, appearing in Eq (8a) can then be represented by (s-1), where
3.4. Model development
In the case of holographic illumination, i.e. to record a holographic grating, there is a spatial distribution of irradiance, which in our case is typically cosinusoidal. In this case the incident intensity is represented as, where
As in the previous analysis, [30-32] it is assumed that the effect of inhibition during exposure is due solely to the initially dissolved oxygen present within the photopolymer layer. The non-uniform recording irradiance causes concentration gradients of oxygen as it is consumed in inhibitory reactions. This then results in the diffusion of oxygen from the dark non-illuminated regions to the bright illuminated regions. As oxygen molecules are small compared to the other material components which constitute the photopolymer layer, it can be assumed that the oxygen is relatively free to diffuse rapidly, resulting in a one-dimensional standard diffusion equation for the concentration of inhibitor,
where in this equation
The equation governing the concentration of primary radicals, including the new term for primary radical generation, is given by
Including both types of termination mechanism (primary and bimolecular) and the effects of inhibition, the equation governing macroradical concentration is then
where the squared term represents the effects of bimolecular termination. The generation term in this equation previously appears as the removal term due to macroradical initiation in Eq (15).
The non-uniform irradiance creates monomer concentration gradients, and as a result monomer diffuses from the dark regions to the monomer depleted exposed regions. This results in a spatial polymer concentration distribution, which provides the modulation of refractive index in the material, i.e., the holographic grating. We represent the monomer concentration using the following 1D diffusion equation,
The equation governing the polymer concentration is
Since all the above equations presented in Eqs (9–13), (15-17) and (19), depend upon the spatial distribution of the exposing intensity, they will all be periodic even functions of
As in previous analysis the Fourier series expansion of the monomer and polymer harmonics involves use of the non-local response parameter
3.5. Model simulations
In order to examine the general behaviour of this model, we now generate a number of theoretical simulations and analyse their predictions. In all theoretical simulations presented here, it is assumed that time varying viscosity effects are negligible and therefore,
12 spatial concentration harmonics are retained in the simulations, solved using the initial conditions presented in Eq (20) with
As can be observed from Figure 2, the spatial sinusoidal variation in the exposing interference pattern causes a rapid consumption of the ground state dye in the bright illuminated regions. As the exposure time increases the sinusoidal variation of the dye concentration is distorted and the width of the non-illuminated dark bands narrows. This loss in sinusoidal fidelity results in a spatial production of primary radicals, as shown in Figure 3, which deviates significantly from the sinusoidal primary radical generation which would be generated using the term presented in Eq (3). Subsequently, this yields a non-linear material response, as the number of polymer chains initiated are not simply generated in direct proportion to the exposing interference pattern. This is an important prediction of the model, which agrees well with experimental observation.
Using the same parameter values used to generate Figures 2 and 3, Figure 4 shows a simulation of the amplitudes of the first two concentration harmonics of the monomer,
Figure 5, shows a simulation of growth curves of refractive index modulation with varying values of the concentration of initially dissolved oxygen,
When comparing the experimental results obtained using the optical setup described above with the theoretical predictions generated by the extended model, it became clear that when using the model as presented, the trend of increased inhibition times,
In a previous paper published by the authors , it was found that by cover-plating or sealing the photopolymer layer with glass slides, the inhibition times observed during exposure compared with the uncoverplated or unsealed layers, were significantly reduced. These effects were attributed to the removal or reduction of oxygen diffusing in from the surrounding environment, which was replacing or replenishing the oxygen consumed during exposure. It must be noted at this point, that the experimental data examined here were uncoverplated photopolymer layers, which were subject to this potential external oxygen diffusion.
In order to represent this process in the model, an additive term representing the replenishing of inhibiting oxygen from the outside surrounding air, into the material layer, was included. Therefore, Eq (13) was revised and became,
In order to illustrate these effects Figure 6 shows a simulation of the behavior of the oxygen concentration with varying values of the replenishing constant,
Implementing the appropriate Fourier series expansion to Eq (21) under the same initial conditions, the model is then applied to the experimental growth curves recorded in uncoverplated layers, yielding much more accurate fits to the data. Figure 7 shows a subset of this data with the corresponding fits obtained using the model. Some of the parameter values which were obtained from the fits to various intensities are,
The most significant values extracted from the fits are presented in Table 1 along with the parameter search ranges, which were used to obtain a best fit between experimental and theoretical prediction. These search ranges are typical of the valued presented in the literature for similar photopolymer materials, [42,48,51]. The Mean Squared Error (MSE) between the fit and the data are also included to indicate the quality of the fits.
|-||0.1 - 5.0||0.1 - 9.0||1.0 – 12.0||-||-|
As can be observed from Figure 7, the fit quality is very good and the model predicts the observed trend, that a reduction in the exposure intensity causes an increase in the inhibition period due to (i) initially dissolved oxygen and (ii) oxygen diffusion into the material from the surrounding air. It can also be seen that there is a reduction in the propagation and termination rates with increasing exposure intensities. This is most likely due to the increased viscosity effects, which occur due to increased conversion of monomer to polymer . This is consistent with the results obtained from the previous model, [30-32]. It must also be noted at this point that the estimates obtained for the rates of propagation and termination are slightly higher than those reported in the previous published work by the authors, [30-32]. This is as a result of a more physically accurate description of the primary radical generation introduced by the model development. However, the estimated values extracted from the fits still remain well within the accepted ranges presented in the literature for similar photopolymer materials.
In order to verify the necessity for the inclusion of the additive oxygen replenishing term in Eq (21), several growth curves of refractive index modulation were recorded in coverplated layers. These growth curves were recorded under the same conditions as the uncoverplated layers presented in Figure 7. Figure 8 shows experimental growth curves recorded at an exposure intensity of
As can be observed from the figure there is a significant reduction in the inhibition period, from
|-||0.1 - 5.0||0.1 - 9.0||1.0 - 12.0||-||-|
In this section, further developments of the Non-local Photo-polymerization Driven Diffusion (NPDD) model, were presented. For the first time, the spatial and temporal variations in primary radical generation were included. These extensions provided a more physically comprehensive theoretical representation of the processes, which occur during free radical photo-polymerization. A clearer more physical representation of the reactions, which take place during the photo-initiation stages, was also provided, including the spatial and temporal consumption and regeneration of the photosensitiser and the reactions between the excited dye molecules and the co-initiator. Simulations were presented, which highlight the loss of sinusoidal fidelity of the primary radical generation. This behaviour deviates from that which was previously predicted in the literature. Subsequently, this change in the spatial generation of primary radicals has a substantial effect on the distribution of the polymer chains formed and hence, on the resulting refractive index modulation recorded.
The model was then further extended to incorporate the effect of oxygen diffusion from outside the material layer by including a rate of oxygen replenishing. This allowed accurate modelling of the inhibition effects, which dominate the start of grating growth. The results obtained were consistent with previous studies where cover-plating techniques were used.
In the following section, we will examine the effects of adding chain transfer agents to an AA/PVA photopolymer material in order to reduce the average polymer chain length grown during grating fabrication. This will then cause a reduction in the extent of the non-local chain growth of these polymer chains and thus reduce the fall off in refractive index modulation at higher spatial frequencies.
4. Improving the spatial frequency response of photopolymers
The non-local spatial response function presented in Eq (18) (in Section 3.4) represents the effect of a chain initiation at location
The introduction of a chain transfer agent acts to reduce the average molecular weight of polymer chains grown during free radical polymerization. Therefore a chain transfer agent (CTA) can provide a practical method to reduce the non-local response length. In this section an extended NPDD model is presented, which includes the chain transfer reactions and all major photochemical processes.
4.1. Chain transfer mechanism
In many polymerization systems, the average polymer weight is observed to be lower than predicted by the chain transfer reaction [29,52-55]. Generally, the chain transfer process causes the premature termination of a growing macro-radical chain and arises because of the presence of CTA . Due to this reaction, a new radical is produced which is referred to in this work as a re-initiator. This re-initiator reacts with a monomer molecule to initiate a new growing macro-radical chain. Therefore we can write the chain transfer reactions as,
The polymerization rate,
This quantifies the effect of the various chain transfer reactions on the number-average degree of polymerization. [
which will be discussed in detail in next section.
4.2. Model development
In order to begin to examine the effects of the presence of CTA on the material non-local response length,, (the actual physical length of the nonlocal response parameter, units of nanometres) we introduce a rate equation governing the CTA concentration:
It should be noted that, in the following analysis, we only consider chain transfer to chain-transfer agent, i.e., the chain transfer constants for monomer and initiator are assumed negligible. To further simplify the analysis in this work, we assume that
The equation governing the re-initiator concentration can be given by,
Furthermore the chain transfer and re-initiation reactions effect the variation of macro-radical, [
As before the concentrations of the components of the photopolymer and the amended equations appearing here in Eqs (27–30) will be periodic even functions of
For brevity we will assume here that harmonics of order greater than
These coupled equations along with those presented in Section 3 are then solved under the initial conditions,
4.3. Numerical results
We will now examine some of the predictions of the extended model presented in this section. Unless otherwise stated all kinetic parameter values are assigned appropriate values, which are typical for the AA/PVA photopolymer material examined. The simulations are performed retaining four spatial concentration harmonics and the coupled differential equations are solved using the initial conditions given in Eq. (35). In all cases,
As different types of chain transfer agents will exhibit different kinetic behaviours, which result in variations in the polymerization rate and therefore changes to the number-average degree of polymerization,
In order to demonstrate the relationship between the non-local response length, , and the refractive index modulation,
Figure 11 shows the saturation refractive index modulation, , plotted as a function of the grating spatial frequency, for the same grating parameter values, as those used in the generation of Figure 10. These results predict that lowervalues lead to a significant improvement in the high spatial frequency response of the material and therefore a reduction in the high spatial frequency roll-off observed experimentally. This is an important prediction of the NPDD model and motivates the study of the feasibility of applying chain transfer agents in free radical based photopolymer materials.
4.4. Experimental results
In this subsection, we aim to demonstrate and compare the effects of a number of CTAs on the spatial frequency response of the various photopolymer material compositions under examination. Following the analysis presented in Section 4.3, we aim to show that a reduction in average polymer chain length results in the predicted reduction in the non-local chain length and hence an increase in the material’s response at high spatial frequencies. In order to achieve this, experimentally obtained growth curves and the saturation values of refractive index modulation,
The three material compositions examined are; (i) a standard acrylamide/polyvinylalcohol photopolymer [30-34], (ii) the standard acrylamide/polyvinylalcohol photopolymer with the addition of 0.96 g of sodium formate (CTA-1) [29,32,32] and (iii) the standard acrylamide/polyvinylalcohol photopolymer with 0.53 ml of 1-mercapto-2-propanol (CTA-2)..For each sample of these material composition examined, several growth curves were measured using a recording intensity of 1 mW/cm2, at a recording wavelength of
Applying the extended NPDD model presented in the previous subseciton, the experimental growth curve data was fit using a least squares algorithm, in which the Mean Square Error (MSE) between the prediction of the model and the experimental data was minimized to extract key material parameters. In order to carry out the fitting process, search ranges of typical parameter values based on data presented in the literature were used [30-34,48]. The model was then applied to analyse the temporal variation of the refractive index modulation
The parameters estimated for the compositions studied are presented in Table 3 (standard AA/PVA), Table 4 (CTA-1) and Table 5 (CTA-2). For each spatial frequency the saturation refractive index modulation value,, is provided in the first column of each table.
Examining the estimated values for
In Table 3 the mean non-local response length in the standard AA/PVA material is estimated to be approximately ≈ 61.3 nm. This value agrees well with the previous estimates reported in the literature [29-34]. For the material containing sodium formate (CTA-1), the corresponding value is ≈ 47.6 nm, as given in Table 4, which corresponds to a reduction ~22.3% in the mean non-local response length,. In Table 5, the value obtained for the material with 1-mercapto-2-propanol (CTA-2) is ≈ 39.8 nm. This corresponds to a ~35.0%, reduction in the mean value. Therefore significant improvements in the high spatial frequency responses with the addition of CTA can be clearly observed. CTA-2 is more effective in reducing the average polymer chain length. Furthermore, in very high spatial frequencies, the results consistently indicate an accompanying improvement in the magnitude of the saturation refractive index modulation. Based on the analysis above it is clear that shortening the polymer chains through the addition of chain transfer agents improves this photopolymer’s spatial frequency response.
The improvements in the high spatial frequency response can clearly be observed. In order to demonstrate the improvements in AA/PVA material performance, numerical fits were carried out to the growth curves of all the three testing samples, for the 3000 lines/mm spatial frequency case. The results are presented in Figure 13 with the associated error bars, indicating the reproducibility of the experimental results.
In this section the NPDD model was extended to examine the effects of the addition of CTAs to an AA/PVA photopolymer. As the NPDD model predicts, a reduction in average polymer chain length grown during polymerisation will reduce the negative smearing effects which are caused by non-local polymer chain growth. As CTAs are known to reduce the average polymer chain length grown during polymerisation, a number of various types and concentrations of CTA were added to reduce these detrimental effects. Then using the NPDD model and fitting growth curves of refractive index modulation at various spatial frequencies for each of the compositions under examination, estimates of key material parameters were obtained. All results verified that the use of CTAs caused a substantial increase in the response of the AA/PVA photopolymer’s spatial frequency response, particularly at higher spatial frequencies, which is in line with the NPDD model predictions.
In this chapter we briefly reviewed some of the developments made in the area of photopolymer material development. We also examined some of the extensions which have been made to the Non-local Photo-polymerisation Driven Diffusion (NPDD) model in order to increase its physicality, with the aim of producing a tool for photopolymer material optimization. A detailed understanding of the photochemical and photo-physical effects which take place during and after holographic exposure is of extreme importance in order to achieve such a tool. Some of the recent developments which have been made in order to achieve this were illustrated and their implications examined. Among these were the temporal and spatial primary radical, the effects of oxygen inhibition, non-local polymer chain growth and the addition of chain transfer agents to improve the spatial frequency response. As various photopolymer compositions have diverse chemical and structural characteristics, knowledge of the characteristics required when choosing these components will offer an informed choice to yield specific improvements in material performance. The implications of the predictions presented here suggest that there are many ways in which improvements can be made.
There still remain a number of effects which have not been included into the NPDD model which would increase its physicality and therefore make it a more powerful tool. One of these which has not been included here, is the addition of time varying viscosity effects. When polymerisation occurs, densification and crosslinking occurs, resulting in a reduction in the material’s fractional free volume. This reduction causes an increase in the viscosity of the material which restricts the rate at which polymerisation proceeds. Accurate modelling of these effects would allow for more precise recording schedules for subsequent holographic exposures to be determined and would enable optimum concentration ratios of the main constituents of the material to be found.
The authors acknowledge the support of the Irish Research Council for Science, Engineering and Technology through the Empower Postdoctoral research scholarship. The authors also acknowledge the support of Enterprise Ireland and Science Foundation Ireland through the national development plan.