# Numerical Model of Radical Photopolymerization Based on Interdiffusion

Authors: Shuhei Yoshida; Yosuke Takahata; Shuma Horiuchi; Hiroyuki Kurata; Manabu Yamamoto
Journal: International Journal of Polymer Science
 (2014)

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## Abstract

An accurate reaction model is required to analyze the characteristics of photopolymers. For this purpose, we propose a numerical model for radical photopolymerization. In the proposed model, elementary reactions such as initiation, propagation, and termination are considered, and we assume interdiffusion for each component in the material. We analyzed the diffraction characteristics of a radical photopolymer based on the proposed interdiffusion model with the beam propagation method. Moreover, we also performed hologram-recording experiments and evaluated the diffraction characteristics of the photopolymer medium. By comparing the numerical and experimental results, medium parameters such as reaction rate and diffusion coefficient can be estimated. We confirmed that the interdiffusion model can reproduce the experimental results and showed that the medium parameters affect the diffraction characteristics.


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## Body

## 1. Introduction

Photopolymers are photoactive organic materials with high diffraction efficiency and are low-cost and stable materials with high dynamic range; thus, photopolymers are widely used in recording holograms. The photopolymerization reaction is complex; therefore, an accurate reaction model is needed for understanding the recording mechanism and analyzing the photopolymer characteristics.Zhao and Mouroulis proposed a one-dimensional diffusion model for photopolymers [1] with some numerical results. In their simulation, the holographic grating was assumed to have a periodic structure, and the numerical analysis was based on Fourier series expansion. Following Zhao and Mouroulis, several researchers proposed reaction models based on diffusion [2–9]. In these models, the broadening of the polymer chain [2–4], shrinkage of the medium [6, 7], and dark reaction [7–9] were considered. Additionally, reaction parameters, such as kinetic constants, were estimated by using experimental results [10–15]. In these studies, models with one-dimensional structure were assumed. Therefore, it is difficult to analyze diffraction characteristics, such as angular selectivity and multiplexing, in detail. In this study, we expand the diffusion model to two dimensions to improve the analysis of diffraction characteristics. The model introduces elementary reactions such as initiation, propagation, and termination. By introducing the primary reactions, we can estimate the kinetic constants of each component.We simulated the formation of the holographic grating based on the two-dimensional interdiffusion model and used the beam propagation method (BPM) [16] to analyze the diffraction characteristics. Based on previous works [17, 18], BPM is considered effective for analyzing diffraction characteristics. By using BPM, we can analyze the diffraction characteristics of the complex holographic grating. Moreover, we also performed hologram-recording experiments. By comparing the numerical and experimental results, we estimated medium parameters, such as reaction rate and diffusion coefficient, and clarified how the medium parameters affect the diffraction characteristics.


## 2. Numerical Model

### 2.1. Chemical Kinetics

We discuss the chemical kinetics of radical polymerization and construct the reaction model. Radical polymerization proceeds following elementary reactions such as(1)I→kd2R∙R∙+M→kiR-M∙≡P∙,(2)P∙+M→kpP∙,(3)P∙+P∙→ktcPbycombinationP∙+P∙→ktd2Pbydisproportionation,(4)P∙+A→ktrP+A∙,
where I, R∙, P∙,M, and P are the initiator, primary radical, propagating radical, monomer, and polymer, respectively, and kd, ki, kp, ktc, ktd, and ktr are the reaction rate constants. Equations (1)–(4) denote the chain initiation, propagation, termination, and chain transfer, respectively. Primary radicals are produced by the decomposition of the initiators via exposure. By adding dyes to the photopolymer, the spectral sensitivity of the initiator can be adjusted. Then, the molecular weight of the propagating radical increases by binding monomers. Two propagating radicals are bound with each other. At this stage, there are two termination pathways. One is the combination reaction and the other is the disproportionation reaction. In general, radical polymerization proceeds with the abovementioned elementary reactions. Chain transfer was not considered for simplicity. Figure 1 shows the reaction path of the radical polymerization described above.Figure 1
Reaction path of radical polymerization.We assume that the initiation rate is proportional to the intensity of exposure(5)kd∝I0r,
and I0(r) is expressed as
(6)I0r=I01+Vcos⁡Kg·r,
where I0 is the exposure intensity, V is the visibility of the interference pattern, and Kg is the grating vector. Kg is defined as the wave vector difference of the signal and reference beam in the medium.


### 2.2. Interdiffusion Model

In this section, we explain the interdiffusion model. Figure2 shows the phenomenological reaction model for photopolymers.Figure 2
Photopolymer reaction model.The signal and reference beams generate the interference pattern in the medium, and polymerization proceeds in the bright area. Polymerization causes concentration gradients; therefore, each component diffuses in response to the gradients. Because the refractive index of each component differs, the hologram is recorded as the refractive index distribution in the photopolymer medium.In this study, we assumed that the photopolymerization proceeds via the diffusion and reaction of each component. The diffusion and reaction of each component can be described as a generalized mass action system [19–21] using the formulation
(7)∂Xi∂t=∑j=1Nαij∏k=1dXjgijk-∑j=1Nβij∏k=1dXjhijk+∇·Di∇Xi,
where d is the number of components and subscript i=1,2,…,5 represents the initiator, primary radical, propagating radical, monomer, and polymer, respectively. N represents the number of production reaction terms. Xi and Di are the concentration and diffusion coefficients of each component, respectively. αi and βi are the reaction rate constants, and the terms gijk and hijk represent the interactive effect between Xi and Xj. The first and second term of the right-hand side of the equation indicate the increase and decrease of the material, and the third term denotes diffusion. We also assumed that the polymer does not diffuse in the medium. Since molecular weight of the polymer chains is high, the polymer hardly diffuses on the short time-scale. However, on the long timescale, diffusion of the polymer might cause degradation of the hologram because of collapse of the grating form. The parameters in (7) are listed in Tables 1 and 2. The reaction rates of each component are decided by the reaction kinetics of the elementary reactions.Table 1
Parameters of the interdiffusion model (j = 1).
i
α
i
1
g
i
11
g
i
12
g
i
13
g
i
14
g
i
15
β
i
1
h
i
11
h
i
12
h
i
13
h
i
14
h
i
15
1
0
0
0
0
0
0
k
d
1
0
0
0
0
2
2kd
1
0
0
0
0
k
i
0
1
0
1
0
3
k
i
0
1
0
1
0
k
t
0
0
2
0
0
4
0
0
0
0
0
0
k
i
0
1
0
1
0
5
k
t
0
0
2
0
0
0
0
0
0
0
0Table 2
Parameters of the interdiffusion model (j = 2).
i
α
i
2
g
i
21
g
i
22
g
i
23
g
i
24
g
i
25
β
i
2
h
i
21
h
i
22
h
i
23
h
i
24
h
i
25
1
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
k
p
0
0
1
1
0
5
0
0
0
0
0
0
0
0
0
0
0
0We estimate parameters, such as reaction rate constants and diffusion coefficients, by using experimental results. The refractive index distribution of the medium is determined by using the Lorentz-Lorenz equation(8)n=1+2ϕ1-ϕ,ϕ=∑ini2-1ni2+2ϕi,
where ϕi is the volume fraction and ni is the refractive index of the ith component. The volume fraction ϕi is
(9)ϕi=wi/di∑iwi/di,
where wi is the mass fraction and di is the density of each component. Equation (7) is discretized for numerical analysis, and it is formulated with the alternating direction implicit method. In the simulation, materials cannot move over the boundary; therefore we employed the Neumann boundary condition
(10)∂Xi∂n=0,r∈∂Ω,
where n is the normal unit vector at the boundary.


## 2.1. Chemical Kinetics

We discuss the chemical kinetics of radical polymerization and construct the reaction model. Radical polymerization proceeds following elementary reactions such as(1)I→kd2R∙R∙+M→kiR-M∙≡P∙,(2)P∙+M→kpP∙,(3)P∙+P∙→ktcPbycombinationP∙+P∙→ktd2Pbydisproportionation,(4)P∙+A→ktrP+A∙,
where I, R∙, P∙,M, and P are the initiator, primary radical, propagating radical, monomer, and polymer, respectively, and kd, ki, kp, ktc, ktd, and ktr are the reaction rate constants. Equations (1)–(4) denote the chain initiation, propagation, termination, and chain transfer, respectively. Primary radicals are produced by the decomposition of the initiators via exposure. By adding dyes to the photopolymer, the spectral sensitivity of the initiator can be adjusted. Then, the molecular weight of the propagating radical increases by binding monomers. Two propagating radicals are bound with each other. At this stage, there are two termination pathways. One is the combination reaction and the other is the disproportionation reaction. In general, radical polymerization proceeds with the abovementioned elementary reactions. Chain transfer was not considered for simplicity. Figure 1 shows the reaction path of the radical polymerization described above.Figure 1
Reaction path of radical polymerization.We assume that the initiation rate is proportional to the intensity of exposure(5)kd∝I0r,
and I0(r) is expressed as
(6)I0r=I01+Vcos⁡Kg·r,
where I0 is the exposure intensity, V is the visibility of the interference pattern, and Kg is the grating vector. Kg is defined as the wave vector difference of the signal and reference beam in the medium.


## 2.2. Interdiffusion Model

In this section, we explain the interdiffusion model. Figure2 shows the phenomenological reaction model for photopolymers.Figure 2
Photopolymer reaction model.The signal and reference beams generate the interference pattern in the medium, and polymerization proceeds in the bright area. Polymerization causes concentration gradients; therefore, each component diffuses in response to the gradients. Because the refractive index of each component differs, the hologram is recorded as the refractive index distribution in the photopolymer medium.In this study, we assumed that the photopolymerization proceeds via the diffusion and reaction of each component. The diffusion and reaction of each component can be described as a generalized mass action system [19–21] using the formulation
(7)∂Xi∂t=∑j=1Nαij∏k=1dXjgijk-∑j=1Nβij∏k=1dXjhijk+∇·Di∇Xi,
where d is the number of components and subscript i=1,2,…,5 represents the initiator, primary radical, propagating radical, monomer, and polymer, respectively. N represents the number of production reaction terms. Xi and Di are the concentration and diffusion coefficients of each component, respectively. αi and βi are the reaction rate constants, and the terms gijk and hijk represent the interactive effect between Xi and Xj. The first and second term of the right-hand side of the equation indicate the increase and decrease of the material, and the third term denotes diffusion. We also assumed that the polymer does not diffuse in the medium. Since molecular weight of the polymer chains is high, the polymer hardly diffuses on the short time-scale. However, on the long timescale, diffusion of the polymer might cause degradation of the hologram because of collapse of the grating form. The parameters in (7) are listed in Tables 1 and 2. The reaction rates of each component are decided by the reaction kinetics of the elementary reactions.Table 1
Parameters of the interdiffusion model (j = 1).
i
α
i
1
g
i
11
g
i
12
g
i
13
g
i
14
g
i
15
β
i
1
h
i
11
h
i
12
h
i
13
h
i
14
h
i
15
1
0
0
0
0
0
0
k
d
1
0
0
0
0
2
2kd
1
0
0
0
0
k
i
0
1
0
1
0
3
k
i
0
1
0
1
0
k
t
0
0
2
0
0
4
0
0
0
0
0
0
k
i
0
1
0
1
0
5
k
t
0
0
2
0
0
0
0
0
0
0
0Table 2
Parameters of the interdiffusion model (j = 2).
i
α
i
2
g
i
21
g
i
22
g
i
23
g
i
24
g
i
25
β
i
2
h
i
21
h
i
22
h
i
23
h
i
24
h
i
25
1
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
k
p
0
0
1
1
0
5
0
0
0
0
0
0
0
0
0
0
0
0We estimate parameters, such as reaction rate constants and diffusion coefficients, by using experimental results. The refractive index distribution of the medium is determined by using the Lorentz-Lorenz equation(8)n=1+2ϕ1-ϕ,ϕ=∑ini2-1ni2+2ϕi,
where ϕi is the volume fraction and ni is the refractive index of the ith component. The volume fraction ϕi is
(9)ϕi=wi/di∑iwi/di,
where wi is the mass fraction and di is the density of each component. Equation (7) is discretized for numerical analysis, and it is formulated with the alternating direction implicit method. In the simulation, materials cannot move over the boundary; therefore we employed the Neumann boundary condition
(10)∂Xi∂n=0,r∈∂Ω,
where n is the normal unit vector at the boundary.


## 3. Numerical and Experimental Results

### 3.1. Experimental Setup

Figure3 shows the experimental configuration for evaluating the photopolymer medium.Figure 3
Experimental configuration.We used the second harmonic of yttrium aluminum garnet laser (532 nm) for recording and the laser diode (670 nm) for readout. We performed two-beam interference recording using the 532 nm laser and monitored the change in diffraction efficiency in real time using the 670 nm laser. The signal and reference beam formed a 60° angle. By using a laser of a different wavelength for readout, we could monitor the diffraction characteristics in real time. The experimental conditions are listed in Table3.Table 3
Experimental conditions.
Exposure intensityI0
10 mW/cm2
Incident angle of signalθs
30°
Incident angle of referenceθr
−30°
Wavelength for recordingλ
532 nm
Medium thickness
25μm
Visibility
1.0
Beam diameter for recordingφ
5 mm
Beam diameter for monitorφ
1 mmWe used a highly sensitive (S>10cm2/J), 25 μm thick, photopolymer medium tin for the experiment. The sample medium was fabricated by injecting the photopolymer between 1 mm glass substrates. The shrinkage ratio of the medium is 0.1% order. The composition and property of the photopolymer are listed in Table 4. This sample was provided by Daicel Corporation. Detailed information of the material cannot be revealed because of nondisclosure agreement.Table 4
Composition and properties of the photopolymer medium.
Component
Mass fraction
Density [g/cm3]
Refractive index
Initiator
5.0 wt%
1.078
1.525 (in solvent)
Monomer
45 wt%
1.183
1.485
Polymer
—
1.277
1.518
Solvent
25 wt%
1.106
1.502
Plasticizer
25 wt%
0.965
1.430


### 3.2. Simulation Scheme

We used BPM [10] for the numerical analysis of the diffraction characteristics of the holographic gratings. BPM is based on steady-state Maxwell’s equations. The electromagnetic field is computed with the following equation:
(11)2jnrk0∂ϕ∂z-∂2ϕ∂z2=Pϕ,P=∂2∂x2+k02n2-nr2TE,n2∂∂x1n2∂∂x+k02n2-nr2TM,
where nr is the reference refractive index, k0 is the wave number in free space, and n is the refractive index of the medium. This equation is developed from steady-state Maxwell’s equations and slowly varying the envelope approximation. In the conventional BPM, the ∂2φ/∂z2 term is neglected to transform (11) into a parabolic partial differential equation. In this study, however, we do not neglect the ∂2φ/∂z2 term; (11) is split into forward and backward wave equations and is expanded as high-order Pade series for accurate calculations [22]. The transparent boundary condition [23] is adopted as the absorption boundary condition. The electromagnetic field is calculated by numerically solving (11) in the z-direction.Figure4 shows the virtual analytical region of the numerical analysis. The holographic grating is located in the analytical region, which is surrounded by the absorption boundary.Figure 4
Analytical region.The incident plane wave enters the analytical region from on the left side. The incident beam is diffracted by the holographic grating, and the diffracted beam is separated from the transmitting beam using the fast Fourier transform. Diffraction efficiency is defined as the ratio of the intensity of the diffracted and incident beam.


### 3.3. Parameter Fitting

We used the Levenberg-Marquardt algorithm (LMA) [24, 25] for parameter fitting. LMA is a nonlinear numerical optimization method and combines the Gauss-Newton algorithm and Gradient descent. LMA searches for parameters that minimize the assessment function. Generally, chi-square distribution χ2 is used as the assessment function
(12)χ2a=∑iyi-fxi;aσi2=y-fa2,
where a=(kd,ki,kp,kt,D1,D2,D3,D4) is the parameter vector, xi and yi are the given independent and dependent variables, σi is the standard deviation, f is the model function,  represents the Euclidean distance, y=(y1/σ1,y2/σ2,…), and f=(f(x1;a)/σ1,f(x2;a)/σ2,…). LMA is used iteratively. The increment of parameters δ is obtained by solving the following system of linear equations:
(13)H+λdiag⁡Hδ=∇χ2a=JTχ2a=-2y-fa·∂fa∂aT,
where represents the inner product, J is the Jacobian matrix, H is the Hessian matrix
(14)H=∂2χ2(a)∂a12∂2χ2(a)∂a1∂a2⋯∂2χ2(a)∂a1∂an∂2χ2(a)∂a2∂a1∂2χ2(a)∂a22⋯∂2χ2(a)∂a2∂an⋮⋮⋱⋮∂2χ2(a)∂an∂a1∂2χ2(a)∂an∂a2⋯∂2χ2(a)∂an2,
and λ is a fudge factor. Typically, the second-order derivative terms of the matrix elements in the Hessian matrix are neglected for simplicity and stability [26]
(15)∂2χ2(a)∂ai∂aj=2∂fa∂ai·∂fa∂aj-y-fa·∂2fa∂ai∂aj≈2∂fa∂ai·∂fa∂aj.
Then, the Hessian matrix can be approximated as
(16)H≈JTJ.
Figure 5 shows the LMA procedure.Figure 5
LMA procedure.


### 3.4. Results

Figure6 shows the time variation of the diffraction efficiency. Solid and dashed curves express the experimental and simulation results, respectively. Horizontal and vertical axes correspond to the exposure time and diffraction efficiency. Diffraction efficiency increased at the start of exposure and saturated with time, and it was almost saturated after 100 s. There are differences between the experimental and simulation results. By focusing on the short-time scale after the start of exposure (t<5s), we identified an occurrence of a minor peak in the experimental results. This phenomenon may be caused by the refractive index contrast transition [27].Time variation of diffraction efficiency.
(a)
(b)Medium parameters are estimated by fitting the experimental and simulation results using LMA. The estimated medium parameters are listed in Table5.Table 5
Estimated medium parameters.
Parameter
Value
k
d
7.277 ± 5.307 (×10−2) [1/s]
k
i
1.576 ± 3.007 [1/s]
k
p
1.463 ± 0.1661 (×10−1) [1/s]
k
t
4.916 ± 1.755 (×10−1) [1/s]
D
1
7.143 ± 9.668 (×10−12) [cm2/s]
D
2
~0 [cm2/s]
D
3
~0 [cm2/s]
D
4
7.259 ± 20.83 (×10−12) [cm2/s]The root mean squared error (RMSE) between experimental and simulation results is 9.454 × 10−3. The diffusion coefficient of the primary radical D2~0 indicates that the primary radical reacts with the monomer immediately after production. Furthermore the propagating radical D3~0 indicates that diffusion of propagating radical is inhibited by a rapid increase in molecular weight of polymer chains by propagation. Next, we evaluate the angular selectivity based on the estimated parameters. Figure 7 shows the experimental and simulation angular selectivity after 100 s exposure time. The horizontal axis Δθ expresses the angle gap between the recording and readout angle.Figure 7
Angular selectivity.The full-width-at-half-maximum of the experimental data is 2.44 × 10−3 rad. Simulation well reproduced the experimental results. The experimental results were also reproduced with a sinc2-like function, which is expected from coupled wave analysis (CWA) [28]. The refractive index modulation Δn, which is estimated from the experimental results with CWA, is Δn=(6.018±0.05127)×10-3. Bragg nulls clearly occur in the experimental results and thus absorption by the medium is low [29]. The simulation results and CWA slightly differ from the experimental results. This can be caused by the difference between each grating form.The space average concentration of each component in time is shown in Figure8. The monomer consumption rate, that is, the polymer generation rate, depends on the concentration of the propagating radical. Therefore, at the beginning of the exposure, the diffraction efficiency did not increase (Figure 6).Figure 8
Time variation of the concentration.Figure9 shows the time variation of the refractive index distribution. The horizontal axis is the x-coordinate normalized with the grating period Λ and the vertical axis is the refractive index n.Figure 9
Refractive index distribution in time.The results show that the higher harmonic components increase with exposure time. The higher harmonic components increase proportionally to the ratio between the diffusion coefficient and polymerization rate. The grating form can be approximated as Fourier series expansion:(17)nx=a0+∑k=1∞akcos⁡2πΛkx.
Fourier coefficients ak(k≤10) which is estimated from the simulation results are listed in Table 6.Table 6
Fourier coefficients.
k
a
k
0
1.483
1
6.343 × 10−3
2
−3.467 × 10−3
3
1.708 × 10−3
4
−1.030 × 10−3
5
6.774 × 10−4
6
−4.702 × 10−4
7
3.358 × 10−4
8
−2.462 × 10−4
9
1.828 × 10−4
10
−1.388 × 10−4As shown in the Table6, the grating has high-harmonic components. The first higher harmonic component a1=6.343×10-3 closes to Δn=6.018×10-3 which is estimated from the experimental results with CWA. Consequently, the interdiffusion model can be used to estimate the medium parameters.


## 3.1. Experimental Setup

Figure3 shows the experimental configuration for evaluating the photopolymer medium.Figure 3
Experimental configuration.We used the second harmonic of yttrium aluminum garnet laser (532 nm) for recording and the laser diode (670 nm) for readout. We performed two-beam interference recording using the 532 nm laser and monitored the change in diffraction efficiency in real time using the 670 nm laser. The signal and reference beam formed a 60° angle. By using a laser of a different wavelength for readout, we could monitor the diffraction characteristics in real time. The experimental conditions are listed in Table3.Table 3
Experimental conditions.
Exposure intensityI0
10 mW/cm2
Incident angle of signalθs
30°
Incident angle of referenceθr
−30°
Wavelength for recordingλ
532 nm
Medium thickness
25μm
Visibility
1.0
Beam diameter for recordingφ
5 mm
Beam diameter for monitorφ
1 mmWe used a highly sensitive (S>10cm2/J), 25 μm thick, photopolymer medium tin for the experiment. The sample medium was fabricated by injecting the photopolymer between 1 mm glass substrates. The shrinkage ratio of the medium is 0.1% order. The composition and property of the photopolymer are listed in Table 4. This sample was provided by Daicel Corporation. Detailed information of the material cannot be revealed because of nondisclosure agreement.Table 4
Composition and properties of the photopolymer medium.
Component
Mass fraction
Density [g/cm3]
Refractive index
Initiator
5.0 wt%
1.078
1.525 (in solvent)
Monomer
45 wt%
1.183
1.485
Polymer
—
1.277
1.518
Solvent
25 wt%
1.106
1.502
Plasticizer
25 wt%
0.965
1.430


## 3.2. Simulation Scheme

We used BPM [10] for the numerical analysis of the diffraction characteristics of the holographic gratings. BPM is based on steady-state Maxwell’s equations. The electromagnetic field is computed with the following equation:
(11)2jnrk0∂ϕ∂z-∂2ϕ∂z2=Pϕ,P=∂2∂x2+k02n2-nr2TE,n2∂∂x1n2∂∂x+k02n2-nr2TM,
where nr is the reference refractive index, k0 is the wave number in free space, and n is the refractive index of the medium. This equation is developed from steady-state Maxwell’s equations and slowly varying the envelope approximation. In the conventional BPM, the ∂2φ/∂z2 term is neglected to transform (11) into a parabolic partial differential equation. In this study, however, we do not neglect the ∂2φ/∂z2 term; (11) is split into forward and backward wave equations and is expanded as high-order Pade series for accurate calculations [22]. The transparent boundary condition [23] is adopted as the absorption boundary condition. The electromagnetic field is calculated by numerically solving (11) in the z-direction.Figure4 shows the virtual analytical region of the numerical analysis. The holographic grating is located in the analytical region, which is surrounded by the absorption boundary.Figure 4
Analytical region.The incident plane wave enters the analytical region from on the left side. The incident beam is diffracted by the holographic grating, and the diffracted beam is separated from the transmitting beam using the fast Fourier transform. Diffraction efficiency is defined as the ratio of the intensity of the diffracted and incident beam.


## 3.3. Parameter Fitting

We used the Levenberg-Marquardt algorithm (LMA) [24, 25] for parameter fitting. LMA is a nonlinear numerical optimization method and combines the Gauss-Newton algorithm and Gradient descent. LMA searches for parameters that minimize the assessment function. Generally, chi-square distribution χ2 is used as the assessment function
(12)χ2a=∑iyi-fxi;aσi2=y-fa2,
where a=(kd,ki,kp,kt,D1,D2,D3,D4) is the parameter vector, xi and yi are the given independent and dependent variables, σi is the standard deviation, f is the model function,  represents the Euclidean distance, y=(y1/σ1,y2/σ2,…), and f=(f(x1;a)/σ1,f(x2;a)/σ2,…). LMA is used iteratively. The increment of parameters δ is obtained by solving the following system of linear equations:
(13)H+λdiag⁡Hδ=∇χ2a=JTχ2a=-2y-fa·∂fa∂aT,
where represents the inner product, J is the Jacobian matrix, H is the Hessian matrix
(14)H=∂2χ2(a)∂a12∂2χ2(a)∂a1∂a2⋯∂2χ2(a)∂a1∂an∂2χ2(a)∂a2∂a1∂2χ2(a)∂a22⋯∂2χ2(a)∂a2∂an⋮⋮⋱⋮∂2χ2(a)∂an∂a1∂2χ2(a)∂an∂a2⋯∂2χ2(a)∂an2,
and λ is a fudge factor. Typically, the second-order derivative terms of the matrix elements in the Hessian matrix are neglected for simplicity and stability [26]
(15)∂2χ2(a)∂ai∂aj=2∂fa∂ai·∂fa∂aj-y-fa·∂2fa∂ai∂aj≈2∂fa∂ai·∂fa∂aj.
Then, the Hessian matrix can be approximated as
(16)H≈JTJ.
Figure 5 shows the LMA procedure.Figure 5
LMA procedure.


## 3.4. Results

Figure6 shows the time variation of the diffraction efficiency. Solid and dashed curves express the experimental and simulation results, respectively. Horizontal and vertical axes correspond to the exposure time and diffraction efficiency. Diffraction efficiency increased at the start of exposure and saturated with time, and it was almost saturated after 100 s. There are differences between the experimental and simulation results. By focusing on the short-time scale after the start of exposure (t<5s), we identified an occurrence of a minor peak in the experimental results. This phenomenon may be caused by the refractive index contrast transition [27].Time variation of diffraction efficiency.
(a)
(b)Medium parameters are estimated by fitting the experimental and simulation results using LMA. The estimated medium parameters are listed in Table5.Table 5
Estimated medium parameters.
Parameter
Value
k
d
7.277 ± 5.307 (×10−2) [1/s]
k
i
1.576 ± 3.007 [1/s]
k
p
1.463 ± 0.1661 (×10−1) [1/s]
k
t
4.916 ± 1.755 (×10−1) [1/s]
D
1
7.143 ± 9.668 (×10−12) [cm2/s]
D
2
~0 [cm2/s]
D
3
~0 [cm2/s]
D
4
7.259 ± 20.83 (×10−12) [cm2/s]The root mean squared error (RMSE) between experimental and simulation results is 9.454 × 10−3. The diffusion coefficient of the primary radical D2~0 indicates that the primary radical reacts with the monomer immediately after production. Furthermore the propagating radical D3~0 indicates that diffusion of propagating radical is inhibited by a rapid increase in molecular weight of polymer chains by propagation. Next, we evaluate the angular selectivity based on the estimated parameters. Figure 7 shows the experimental and simulation angular selectivity after 100 s exposure time. The horizontal axis Δθ expresses the angle gap between the recording and readout angle.Figure 7
Angular selectivity.The full-width-at-half-maximum of the experimental data is 2.44 × 10−3 rad. Simulation well reproduced the experimental results. The experimental results were also reproduced with a sinc2-like function, which is expected from coupled wave analysis (CWA) [28]. The refractive index modulation Δn, which is estimated from the experimental results with CWA, is Δn=(6.018±0.05127)×10-3. Bragg nulls clearly occur in the experimental results and thus absorption by the medium is low [29]. The simulation results and CWA slightly differ from the experimental results. This can be caused by the difference between each grating form.The space average concentration of each component in time is shown in Figure8. The monomer consumption rate, that is, the polymer generation rate, depends on the concentration of the propagating radical. Therefore, at the beginning of the exposure, the diffraction efficiency did not increase (Figure 6).Figure 8
Time variation of the concentration.Figure9 shows the time variation of the refractive index distribution. The horizontal axis is the x-coordinate normalized with the grating period Λ and the vertical axis is the refractive index n.Figure 9
Refractive index distribution in time.The results show that the higher harmonic components increase with exposure time. The higher harmonic components increase proportionally to the ratio between the diffusion coefficient and polymerization rate. The grating form can be approximated as Fourier series expansion:(17)nx=a0+∑k=1∞akcos⁡2πΛkx.
Fourier coefficients ak(k≤10) which is estimated from the simulation results are listed in Table 6.Table 6
Fourier coefficients.
k
a
k
0
1.483
1
6.343 × 10−3
2
−3.467 × 10−3
3
1.708 × 10−3
4
−1.030 × 10−3
5
6.774 × 10−4
6
−4.702 × 10−4
7
3.358 × 10−4
8
−2.462 × 10−4
9
1.828 × 10−4
10
−1.388 × 10−4As shown in the Table6, the grating has high-harmonic components. The first higher harmonic component a1=6.343×10-3 closes to Δn=6.018×10-3 which is estimated from the experimental results with CWA. Consequently, the interdiffusion model can be used to estimate the medium parameters.


## 4. Conclusion

We simulated the formation of holographic gratings in photopolymers using the interdiffusion model. We considered the elementary reactions of initiation, propagation, and termination. We analyzed the diffraction characteristics in detail by high-order BPM and estimated medium parameters, such as reaction rates and diffusion coefficients, with LMA. The proposed model and analytical methodology can be used to quantify the medium parameters. We believe that the results will provide useful information for developing holographic recording materials.



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*Source: 243895-2014-12-10.xml*