Non-local polymerization driven diffusion based model : general dependence of the polymerization rate to the exposure intensity Non-local polymerization driven diffusion based model: general dependence of the polymerization rate to the exposure intensity

: The nonlocal diffusion model proposed by Sheridan and coworkers has provided a useful interpretation of the nature of grating formation inside photopolymer materials. This model accounts for some important experimental facts, such as the cut-off of diffraction efficiency for high spatial frequencies. In this article we examine the predictions of the model in the case of a general dependence of the polymerisation rate with respect to the intensity pattern. The effects of this dependence on the different harmonic components of the polymerisation concentration will be investigated. The influence of the visibility on the different harmonic components will also be studied. These effects are compared to the effects of varying R D and σ D .

Recently several models have been used to describe the mechanism of hologram formation inside photopolymer materials [4][5][6][7][8][9][10][11][12][13][14]. Since the first paper by Zhao and Mourolis [5] it has been clear that not only polymerization but also diffusion play an important role in the formation of the diffraction gratings. This first article proposed by Zhao et al. provides the basis for polymerization driven diffusion (PDD) models of hologram formation in photopolymers. The monomer concentration was expanded into its Fourier series and a set of coupled differential equations was solved for the different harmonic components of the monomer concentration. The different harmonic components of the polymer concentration were obtained by integrating with time the harmonic components of the monomer concentration. Finally, the harmonic components of the refractive index were assumed to be proportional to the harmonic components of the polymer concentration. The general procedure developed by Zhao et al. has been used by different authors, by introducing a Fick's law diffusion term in the equation describing the variation with time of the monomer concentration. The use of a Fourier expansion of the monomer concentration has also been a feature several models proposed in the field. Although the original model proposed by Zhao et al. was useful, there were some observed effects that the model did not include. For instance, although for high times of exposure the harmonic components of the monomer concentration tend to zero, these components have non-neglegible values for low times of exposure. Since the refractive index of the monomer is different to that of the polymer, the existence of monomer in the material for low exposure times influence the dynamical refractive index of the medium. Thus, for low exposure times the contribution of the monomer concentration to the refractive index should be included. This was studied by Aubrecht et al. [8], who proposed a modified version of Zhao's model. This contribution was also later included in a first harmonic PDD model by Neipp et al. [14] Another important fact that the Zhao models had failed to explain was the cut-off of the diffraction efficiency for high spatial frequencies. This was successfully achieved by Sheridan and co-workers, who implemented a non-local polymerization driven diffusion (NPDD) [11][12][13]. The main feature of this model is that it includes a non-local response function that takes into account the growth of polymer chains inside the photopolymer material. The model has also the particularity that for the limiting case in which the non-local effects are disregarded the model reduces exactly to the PDD proposed by Zhao et al. Previous models assumed that the polymerization rate, that is the rate of conversion of monomer in polymer by photopolymerization, had a linear dependence to the intensity pattern stored in the photopolymer material. In an article by Kwon et al. [9] this assumption was modified and a dependence on the square root of the intereference pattern was proposed. For instance, Sheridan and co-workers modified their previous model to include a general dependence [13] and in a very recent paper by S. Wu et al. [15],in which the non-local diffusion equation is solved by using a finite-difference method, a dependence of the polymerisation rate on the square root of the intensity pattern was assumed.
Zhao et al. also examined a more general dependence [16]: (1) They studied the influence of γ on the different harmonic components of the polymer concentration. In this work we present a study of the effect of γ on the NPDD model proposed by Sheridan et al. The influence of the fringe visibility on the different harmonics of the polymerisation concentration will also be analysed and compared to the effect of R D and σ D .

The general non-local diffusion model
The one-dimensional non-local response diffusion equation, which governs the change of monomer concentration, u(x,t), with time is [11]: In this equation D(x,t) is the diffusion constant, G(x,x') is the non-local response function and F(x',t') represents the rate of polymerisation at point x' and time t'.
The monomer concentration is assumed to be of the form: while the non-local response function G(x,x') will be supposed to have a Gaussian form: The square root of the variance σ characterizes the length scale over which the nonlocal effect is significant.
In this work we will assume the following general dependence for the polymerization rate: And κ is a fixed constant.
The rate of polymerization will now be supposed to have the following series expansion: The diffusion constant will also be expanded in a Fourier series as: As it has been done in previous papers [5,[11][12][13] we will initially only retained the first two harmonic components of the Fourier series.
To calculate the two harmonic components of this expansion we will determine the values of the maximum and minimum diffusion coefficients in the polymer, D max and D min respectively. Therefore D 0 and D 1 can be calculated as [5]: D 0 =(D max +D min )/2 (10) D 0 =(D max -D min )/2 (11) Since the diffusion coefficient will also be considered to decay with time it has the following dependence: By using Eqs. (13,14) and (10,11) the final expression of the diffusion coefficient can be calculated as: If Eqs. (7) and (15) are substituted in Eq. (2) the more general NPDD model will be obtained.

Dimension-less equations
As has been done in a previous paper by S. Wu et al. [15] it is interesting to set Eq. (2) in dimensional-less form.
The same definitions as those in Ref. [15] will be taken: R D is a parameter that measures the relative strength between the mechanisms of diffusion and polymerization. If we define τ D as τ D =1/DK 2 and τ P as 1/F 0 . τ D and τ P express the characteristic time of the mechanisms of diffusion and polymerization, respectively. Therefore for values of R D >1 diffusion dominates over polymerization, whereas for values of R D <1 polymerization is the dominant process. On the other hand σ D is a parameter that controls the non-locality effects in the dimensionless non-local response function G D (x D ,x D ') defined as: In dimension-less variables, Eq. (2) takes the form: The non-dimensional polymerization rate is expressed as: The non-dimensional diffusion constant now has the form: Finally, by substituting the Fourier expansion of Eq. (22) in Eq. (21) and retaining only, the first four harmonic terms the following four dimensionless coupled monomer rate equations can be found as: With the following definitions: Ch S i =exp(-i 2 σ D /2) (30) The resulting concentration of polymerised monomer can be obtained as: therefore the four orders of the polymerisation spatial-harmonic components are:

Numerical results and discussion
In this section, numerical results about the influence of the dimensionless variables on the polymerisation characteristics of the recorded holograms are presented. It is also interesting to study the final patterns stored in the holograms. To do this Eqs. (24)-(27) and (32)-(35) were solved for a high value of the parameter t D to allow the steady states of all the harmonic components of the polymer and monomer concentration to be reached. From Figs. 1 and 2 it is reasonable that a value of t D =20 is sufficient for this purpose.  [13,15]. It can also be appreciated that higher values of σ D also imply that the nonlinearity is reduced, an effect which is also noted in [15]. The influence of γ on the ratios N 2 /N 1 and N 3 /N 1 has also been investigated. Figure 6 shows the absolute value of ratio N 2 /N 1 as a function of the parameter γ for different values of σ D with a fixed value of R D =1. In this case, the dependence of the ratio N 2 /N 1 resembles straight lines for all values of σ D . Although it can be appreciated that for low values of the parameter γ, the nonlinearity is increased, the effect of varying γ in the range of 0.5≤γ≤1 is not as important as is the effect of R D in the range 0.01≤R D ≤100, these ranges assumed for real materials [17]. This can also be seen in Fig. 7 where the absolute value of ratio N 3 /N 1 is represented as a function of γ. Nonetheless, in this case, the slope of the curve corresponding to a value of σ D =0 is higher than that of Fig. 6. For values of σ D equal to 1 and 0.5 the value of N 3 can be disregarded with respect to the value of N 1 .

Conclusions
The influence of the parameter γ, which quantifies the nonlinear dependence of the polymerisation rate on the exposure intensity, upon the harmonic components of the polymerisation and monomer concentration has been investigated. Furthemore the effect of the visibility on the different harmonic components of the polymer concentration has also been examined. The effect of decreasing the visibility is basically to reduce the pattern stored in the hologram, with a consequent diminution of the values of all the harmonic components of the polymer concentration higher than zero. It has also been demonstrated that a decreasing value of γ contributes to a decrease in the values of the first harmonic components of the polymer and monomer concentration. In addition the effect of R D on the distribution profiles of the final polymer concentration stored in the hologram has been examined. It was found that the higher the values of R D , the more the final polymer distribution resembles a sinusoidal pattern.