## Abstract

The reaction model and the corresponding equations of PQ:DMNA/PMMA photopolymer recording at 640 nm are proposed. A series of experiments were conducted to estimate the parameters used in the equations by measuring only the dynamic behavior of the diffraction efficiency of the recorded grating. Recording the PQ:DMNA/PMMA grating can then be well-predicted and match with the experiment.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

PQ/PMMA (Phenanthrenequinone-doped polymethyl methacrylate photopolymer) is an important and intensively studied holographic recording photopolymer [1–5]. Li et al. proposed a comprehensive model to describe the photopolymerization driven diffusion model of PQ/PMMA hologram recording process [6]. Shih et al. proposed a simplified model and successfully estimate the diffraction efficiency of reflective-type PQ/PMMA volume Bragg gratings (VBGs) [7]. With reaction model, 2^{nd} order volume Bragg grating can be predicted and successfully recorded and applied to diode laser performance improvement [8].

The longest possible recording wavelength of PQ/PMMA is about in the region of green light. Ko et al. performed a series of experiments using different nitroaniline doping in PQ/PMMA which results in better hologram recording performance. Among all proposed nitroanilines, DMNA (N, N-dimethyl-4-nitroaniline) provides the highest refractive index change. Base on FTIR and mass spectrum result, DMNA does not react with PQ, MMA or PMMA [9]. Lin et al. discovered that PQ:DMNA/PMMA has the longest recording wavelength extends to red light and even in the near-infrared region [10]. A longer recording wavelength may reduce thermal load and ease thermal expansion issue while recording the hologram. Also, high power red or NIR diode lasers are more accessible. Therefore, PQ:DMNA/PMMA is a promising material for various photopolymer applications.

In this work, the reaction mechanism of PQ:DMNA/PMMA is proposed and the corresponding model and reaction rate with diffusion equations are established. A series of the experiments were performed to estimate the parameters in the equations.

## 2. Reaction model and equations

As Ref. [9] pointed out, DMNA does not react with PQ, MMA, or PMMA. However, DMNA does absorb red light. Therefore, a plausible process is that DMNA absorbs red photon to reach an excited state, DMNA*, and then transfers the energy to PQ. The rest of the reactions then follow the model proposed for PQ/PMMA [6,7]. The corresponding reactions are

*k*is the excitation rate of DMNA,

_{DMNA}*τ*is the lifetime of DMNA*,

_{DMNA}*C*is the effective energy transfer coefficient [11],

_{eff}*k*is the transfer rate from

_{PQ}^{1}PQ* to

^{3}PQ*, and

*k*is the reaction rate of the photoproduct. The excitation rate of DMNA can be written as where

_{PQ/MMA}*σ*is the absorption cross section of DMNA,

_{DMNA}*I*is the intensity of the recording light,

*λ*is the wavelength of the recording light,

*c*is speed of light and

*h*is Planck’s constant. Once PQ is excited, it quickly transfers into

^{3}PQ* [10,12]; therefore, Eq. (2) and (3) can be written as

In the following paragraphs, the square brackets represent the mole concentration of the indicated chemical components. [MMA] is assumed to be very high and considered as a constant throughout the entire recording process [13]. Similar to equations in Ref. [6] and [7], the reaction rate equations combining with diffusion equations can be written as

*D*is the diffusion constant of PQ. With the given initial conditions, numerical methods can be applied to obtain the spatial and temporal distribution of [PQ] and [PQ/MMA]. With [PQ] and [PQ/MMA], the total refractive index change, Δ

_{PQ}*n*, can be written as based on Lorentz-Lorenz formula [14,15].

_{tot}*γ*

_{PQ}and

*γ*

_{PQ/MMA}are proportional constants describing the contribution of [PQ] and [PQ/MMA] to the local Δ

*n*

_{tot}and are given in Ref [7]. As using two beam interference recording scheme, Δ

*n*

_{tot}is periodic in space yet is usually far from a simple sinusoidal function [16,17]. The first order spatial Fourier component of Δ

*n*, will be referred to as Δ

_{tot}*n*which leads to the diffraction efficiency,

*η*, of the recorded grating through couple mode theory [18].

Controllable material parameters are initial [PQ] and initial [DMNA] which will be respectively referred as [PQ]_{0} and [DMNA]_{0}. Controllable exposure parameters are *I* and exposure time, *t*_{ex}. The unknown yet required parameters in the equations are *σ _{DMNA}*,

*k*,

_{PQ/MMA}*D*,

_{PQ}*C*, and

_{eff}*τ*which cannot be directly measured. However, the unknown parameters may be evaluated by

_{DMNA}*η*or Δ

*n*of the samples. The dynamic behavior of Δ

*n*and several measurable parameters are defined and shown in Fig. 1. ${\left. {\frac{{d\Delta n}}{{dt}}} \right|_{t \to 0}}$ indicates the increasing rate of Δ

*n*at the very beginning of the exposure when diffusion term can be neglected in Eq. (9). For simplicity, this parameter will be referred as $\Delta {\dot{n}_0}$. When exposure terminates, Δ

*n*reaches Δ

*n*and then gradually increases as an inverted exponential decay from Δ

_{ex}*n*to Δ

_{ex}*n*where the chemical reaction and diffusion are both terminated.

_{f}*t*

_{1/e}is the time when Δ

*n*reaches ${n_f} - {{({{n_f} - {n_{ex}}} )} / e}$ after

*t*, In this work, $\Delta {\dot{n}_0}$ and

_{ex}*t*

_{1/e}are utilized to evaluate the unknown parameters.

## 3. Experiments

At 25°C, the saturation concentrations of PQ and DMNA in MMA solution are experimentally estimated to be 1 wt% and 1.68 wt%, respectively. A two-step polymerization [10,19] is adopted to prepare the samples. The thickness of the sample is determined by the mold and is set to be 2 mm. The curing temperature is 45°C. After curing, the volume reduces by about 28%. Note that [PQ] and [DMNA] are presented in the unit of wt% for the convenient of fabrication. In the calculation, mol/cm^{3} is used. With the volume reduction, 1 wt% [PQ] equals 6.67 × 10^{−5} mol/cm^{3} and 1 wt% [DMNA] equals 8.36 × 10^{−5} mol/cm^{3}. Other wt% value can be converted proportionally since MMA dominates the total mass.

Figure 2 shows the absorption spectra of PQ/PMMA, DMNA/PMMA, and PQ:DMNA/PMMA with saturated [PQ]_{0} and [DMNA]_{0} measured by a spectrometer (Hitachi U-4100). From 620 to 650 nm, PQ/PMMA does not have observable absorption. Yet, both DMNA/PMMA and PQ:DMNA/PMMA have absorption with similar trend in this range as shown in the insert of Fig. 2. Interestingly, the absorption coefficient spectrum of PQ:DMNA/PMMA is about twice of DMNA/PMMA. Clearly, the presence of PQ enhances the absorption. Highly effective energy transfer from DMNA* to PQ can indeed lead to higher effective absorption. Thus, the plausible reaction is as described above and the effective *σ*_{DMNA} is estimated to be 5.44 × 10^{−22} cm^{2} at 640 nm.

The recording and reading scheme is shown in Fig. 3. The laser is a single longitudinal mode 200 mW 640 nm diode-pumped solid-state laser (LASOS Lasertechnik GmbH) with a Faraday isolator to ensure the stability of the output. M_{1} and M_{2} are plane mirrors for alignment. The beam size is reduced 5× to increase the intensity by L_{1} and L_{2} which has focal lengths 125 mm and 25 mm, respectively. The beam is then split by a 50/50 beam splitter (Thorlabs BS010). By carefully aligning M_{3} and M_{4}, the two beams collinearly counter-propagate between these mirrors. In other words, the recorded grating is a reflective volume Bragg grating. The beam radius on the sample is about 0.025 cm. The recorded grating period, Λ equals *λ*/2*n* or 215 nm since PMMA has a refractive index about 1.488 at 640 nm. The PQ:DMNA/PMMA sample is sandwiched by two wedge prisms (Thorlabs BSF2550) will silicon oil as index matching fluid and is placed on the optical path of the two beams. An optical shutter (Thorlabs SHB1 T) is in between BS and M_{4} and is controlled by a computer. By briefly closing the shutter for 0.2 sec each time, the diffracted light can then be measured by a powermeter (Ophir PD300-3W). *η* of the sample can then be acquired. In the first few 100s sec of the exposure when *η* changes rapidly, the sampling period is set to be about 10–30 sec. As *η* change slows down, the sampling period is extended to 60 sec or even longer. The dynamic behavior of Δ*n* can be calculated using couple mode theory. Several examples are shown in Fig. 4. To reach the highest *η* or the highest Δ*n*, longer *t _{ex}* and higher

*I*are needed as shown in Figs. 4(a) and 4(b). Figure 4(c) is obtained from Fig. 4(a) using coupled mode theory and similarly for Fig. 4(d). All the $\Delta {\dot{n}_0}$ curves have a trend similar to Fig. 1. Therefore, $\Delta {\dot{n}_0}$ and

*t*

_{1/e}can be acquired.

$\Delta {\dot{n}_0}$ at different recording fluences and [DMNA]_{0} is shown in Fig. 5(a). $\Delta {\dot{n}_0}$ as functions of [DMNA]_{0} with the two lowest recording fluences is plotted in Fig. 5(b). $\Delta {\dot{n}_0}$ is linearly proportional to the [DMNA]_{0}. In other words, the excitation of PQ is more likely to be energy transferring from DMNA* rather than the nonlinear optics effect of DMNA suggested in Ref. [9]. The mechanism of $\Delta {\dot{n}_0}$ reduction at higher recording fluence is beyond the scope of this work and requires further investigation. Quenching of DMNA may be the cause.

Since ^{3}PQ* quickly reacts with MMA, the chemical reaction ceases right after the exposure is ended. The diffusion of PQ then dominates Δ*n*. *D*_{PQ} can be written as Ref [7] suggested as

As in Figs. 4(c) and 4(d), experiment with different *t*_{ex} and [DMNA]_{0} were performed. *t*_{1/e} is solely dependent on [DMNA]_{0} as shown in Fig. 6. Therefore, *t*_{1/e} can be fitted with

*y*

_{1}=5699.69 s,

*β*=1.37 × 10

^{−5}mol/cm

^{3}and

*y*

_{0}=284.03 s.

*D*

_{PQ}can be rewritten as

Taking the time derivative of Eq. (12), *C*_{eff}, *k*_{PQ/MMA}, and $\Delta {\dot{n}_0}$ can be related by using Eqs. (9) and (11). Therefore, using $\Delta {\dot{n}_0}$ to evaluate *C*_{eff}, *k*_{PQ/MMA} is practical. Figure 7(a) shows $\Delta {\dot{n}_0}$ as a function of *I* with 1 wt% [PQ]_{0} with different [DMNA]_{0}. With given *γ*_{PQ} and *γ*_{PQ/MMA}, *C _{eff}*=55 cm

^{3}/s·mol and

*k*

_{PQ/MMA}=120 s

^{−1}can be found by fitting.

The nominal absorbed photon density *ϕ*_{abs} within the sample can be written as

*N*is Avogadro constant,

_{A}*d*is the thickness of the sample.

*C*is found depends on

_{eff}*ϕ*

_{abs}as shown in Fig. 7(b). With

*ϕ*less than 3 × 10

_{abs}^{22}#/cm

^{3}, the trend of

*C*is not directly depends on [PQ]

_{eff}_{0}and [DMNA]

_{0}which also suggest the reaction is not from nonlinear optical effect of DMNA. Since

*C*describes the transfer efficiency from DMNA* to PQ, higher

_{eff}*ϕ*

_{abs}leads to better transfer is expected. The higher

*ϕ*

_{abs}is, the more DMNA is excited. Therefore, PQ can be more efficiently excited in a microscope perspective.

With the result in Fig. 7(b), *C _{eff}* can be fitted with

*C*

_{0}=55 cm

^{3}/s·mol,

*C*

_{1}=−51.25 cm

^{3}/s·mol, and

*ϕ*

_{0}=2.09 × 10

^{22}#/cm

^{3}. The splitting of

*C*with higher

_{eff}*ϕ*may be caused by the quenching of DMNA and require further study.

_{abs}As described above, DMNA absorbs a photon and then transfers the energy to PQ; therefore, higher [DMNA] and [PQ] leads to shorter *τ*_{DMNA} which is found directly related to the product of [PQ]_{0} and [DMNA]_{0} as shown in Fig. 8. *τ*_{DMNA} can be fitted with

*τ*

_{1}=1400 s,

*ζ*=2.52 × 10

^{−9}mol

^{2}/cm

^{6}and

*τ*

_{0}=232 s.

With the above experiments and discussions, the parameters used in the equations are obtained and summarized in Table 1. Thus, the behavior of PQ:DMNA/PMMA can be fully simulated. The simulation results agree well with the experimental results with various controllable parameters as shown in Fig. 9 and Fig. 10. As described above, to reach highest possible Δ*n _{f}*, higher

*I*and longer

*t*are needed. With long

_{ex}*t*, Δ

_{ex}*n*can be obtained with different [PQ]

_{0}and [DMNA]

_{0}by the model as shown in Fig. 11. To reach higher Δ

*n*, [PQ]

_{f}_{0}also should be as high as possible. However, [PQ]

_{0}is limited to about 1 wt% by the saturation concentration. The higher [DMNA]

_{0}, the faster the grating is recorded. Yet, fast grating formation results in higher spatial order of refractive index and lead to slightly lower

*η*as shown in Fig. 10 and Fig. 11. At any given [DMNA]

_{0}, Δ

*n*is linearly depending on [PQ]

_{f}_{0}. On the contrary, with constant [PQ]

_{0}and [DMNA]

_{0}>3 × 10

^{−5}mol/cm

^{3}, Δ

*n*is almost independent to [DMNA]

_{f}_{0}. Overall speaking, DMNA plays the role of “sensitizer” which absorbs the photon then transfers the energy to PQ yet does not involve chemical reaction with any other molecules. In practical applications, shorter

*t*is crucial. Therefore, both [PQ]

_{ex}_{0}and [DMNA]

_{0}should reach saturation concentrations. Recording intensity

*I*should be as high as possible.

## 4. Conclusion

This work establishes a reaction model and the corresponding reaction rate with diffusion equations of PQ:DMNA/PMMA, a new photopolymer for hologram at the recording wavelength of 640 nm. Through a series of experiments, the five unknown parameters, *σ*_{DMNA}, *k*_{PQ/MMA}, *D*_{PQ}, *C*_{eff}, *and τ*_{DMNA} in the equations are obtained solely using the diffraction efficiency measurement. With the knowledge of the equations and parameters, the PQ:DMNA/PMMA recording process can be well-predicted and match with the experiment. To reach higher final Δ*n*, higher initial PQ concentration is required. Longer exposure time also leads to higher final Δ*n.* Higher recording light intensity gives faster growth rate of Δ*n*. Higher initial DMNA concentration does give higher growth rate of Δ*n* or faster grating formation However, fast grating formation may dominate and lead to high spatial order of grating and lead to slightly lower final Δ*n*.

## Funding

Ministry of Science and Technology, Taiwan (MOST 108-2221-E-008-085-MY3).

## Acknowledgment

The author would like to thank Prof. Lin, Shiuan-Huei for his generous help and valuable discussion.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

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