The range of exposure for which the holographic reciprocity law holds in photopolymers, is mainly dependent on the light exposure intensity and polymerization rate between photo-initiator and monomers. Matching this is the key to improving performance. Characterization of the dependence on diffraction efficiency of the volume transmission gratings on holographic reciprocity matching of TI/PMMAs under different milliseconds with different thickness (1-3mm) has been carried out for the novel high-sensitive TI/PMMA polymers. Diffraction gratings can be recorded in TI/PMMAs under 20ms with the exposure intensity of 115mW/cm2. The physical and chemical mechanism under and after single shot exposure is analyzed which can be divided into three parts, namely, photo-induced polymerization, dark diffusion of photosensitive molecules, and counter-diffusion of photoproducts. Holographic properties of TI/PMMAs of different thickness (1-3mm) under different shingle-shot durations and repetition rates are investigated in detail as well. The diffraction efficiency reaches 67% with the response time of 15.69s. By this way, volume holographic gratings with no reciprocity failure can be recorded under multi-pulse exposure, with high grating strength and rapid sensitivity in TI/PMMAs, which indicates the volume holographic memories have the potential for recording and storing transient information in life and in the military.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
TI/PMMA polymers are made up of photosensitive initiator titanocene (Irgacure 784, BASF) (TI), thermal initiator 2,2-azo-bisisobutyrolnitrile (AIBN) and monomer methyl methacrylate (MMA) . A detail research of the influence of different concentration of initiators has been investigated with an optimized doping concentration of 4.0wt% TIs and 2.0wt% AIBNs . Compared to traditional PQ/PMMAs [3,4], TI molecules are more soluble in MMA liquid and more sensitive at the range of visible light than PQs . This kind photosensitive molecules have been used in different kind of photopolymers [6,7]. It can absorb photons to form isomer and produce photo-cleavage to connect surrounding monomers . By doping TIs into PMMA matrix, the composite exhibits better holographic performances with the characteristics of negligible shrinkage and high density storage [2,9,10]. Meanwhile, the holographic performances of TI/PMMAs have been presented in our previous researches with the maximum diffraction efficiency of 74% and the response time of 20s under green-light continuous wave (CW) exposure [11,12]. It is indicated that TIs dispersed PMMA system is a more feasible material in future volume data storage applications.
Nowadays, many researches focus on ultrafast holography due to the fact that there is more and more transient information at the fast pace of life [13,14]. As for holographic data storage of ultrafast recordings, PQ/PMMAs and PVA/AAs have been demonstrated that photopolymers can be available to realize ultrafast holographic storage under nanosecond pulsed exposure [15–18]. PQ/PMMAs exhibit higher storage ability but low sensitivity under pulsed exposure [15–17], while PVA/AAs have fast response time but larger shrinkage and hygroscopicity . However, a holographic phenomenon called reciprocity failure occurs under ultra-short time exposure, such as nanosecond, microsecond, etc. Holographic reciprocity means that exposure time and exposure intensity can be directly proportional to each other and the total exposure flux is unchanged, but this law is failure when the recording time is ultra-short, eventually causes a dramatic reduction of the grating strength [19,20]. This phenomenon is occurred in silver-halide , photorefractive polymers  and photopolymers . It is indicated that solving the holographic reciprocity failure becomes the key factor on ultrafast holographic data storage.
The first part of our article is mainly describe the holographic reciprocity law under short exposure duration, which is analyzed under the condition on single pulse exposure. Therefore, a theoretical model and experiments under milliseconds is proposed to demonstrate the holographic reciprocity matching in TI/PMMAs after one single shot exposure. Then, we investigate the holographic characteristics under multi shots exposure, which is aimed at explore further on our TI/PMMAs in ultrafast holographic recording. Here, we introduce the dark diffusion process after exposure. At last, we represent a preliminary theoretical model to analysis the dynamic process under multi-pulse exposure and further research direction in the future due to the internal complex multistage reactions in photopolymers. An internal dynamic diffusion analysis before and after single shot exposure is proposed according to the nonlocal photo-polymerization driven diffusion (NPDD) model [24–27] which is mainly used to explain the cumulative diffraction efficiency under multi shots exposure of different time intervals. This work presents a feasibility and improvement of ultrafast holographic storage on TI/PMMAs with holographic reciprocity matching.
2. Materials and methods
In our work, we prepared a new kind of photopolymer by dispersing cationic photo-initiator titanocene into the host matrix, MMA. This photopolymer material was prepared by three mixed components, namely, MMA, AIBN and TI . The MMAs not only act as the host monomer to be photo-polymerized during exposure, but exist as the matrix to be solidified in the whole polymer system, while, the AIBNs are used to polymerize monomers during the heating process called as the thermo-initiator. Further, the TIs, which are regarded as the photo-initiator, induce monomers to be polymerized in exposing process. In the preparations, a three-step polymerization method is proposed, which can be divided into pre-polymerization, high temperature polymerization and low temperature polymerization. Firstly, the photo-initiator TIs and thermo-initiator AIBNs were dissolved into MMA matrix with an optimized concentration of 4.0wt% and 2.0wt% in the pre-polymerization process, respectively. The mixture was stirred for 24h with a stationary temperature of 40°C, which aimed at removing the nitrogen produced by thermo-decomposition of AIBNs and making the components mix evenly. Then the viscoid liquid mixture was filtered and moved into the incubator with the baking temperature of 72°C for the second step; i.e. high temperature polymerization. Finally, the mixed solution was solidified under 45°C for 48h to complete the low temperature polymerization. Samples were classified with different thicknesses (1-3 mm) after polishing and ready for milliseconds ultrafast holographic measurement. According to the Kogelnik theory , the real optical thickness of 1mm, 2mm and 3mm TI/PMMAs are 650μm, 990μm and 1450μm.
The ultrafast holographic installation is a modified two-wave coupling interference system with green-light recording, which can be depicted in . Two beams with the same optical path, exposure intensity and polarization state are interfered with the surface and interior of samples to record the holographic gratings. Here, we add the electronic controlling shutter (ECS) with the precision of 20 milliseconds on each beam path before interference. By this way, we can control the single exposure duration, exposure interval and exposure times. The milliseconds ultrafast holographic measurement can be achieved by the method above.
3. Results and discussions
3.1 theoretical analysis of TI/PMMAs on holographic reciprocity law
The photo-polymerization of TI/PMMAs is depicted as follows
There are two main step in this process, namely, photo-initiation and polymerization. When light acts on TIs, the photosensitive molecule (TI) absorbs the photon to transfer into the excited state, that is, the time of the photosensitive molecule to reach the excited state is a random variable T, and its distribution function can be expressed as
Where D represents the exposure density. Also, the polymerization rate and reduction rate are also random variables which obey exponential distribution. Their distribution functions can be depicted as, respectively:
The probability density functions of F(t), P(t) and R(t) are f(t), p(t) and r(t), respectively. We assume that event H is used to generate polymerizations with monomers after excitation. Therefore, when the exposure power density is D and the exposure time is t, the reciprocity law density can be expressed as
Here, the implementation of H is divided into two steps, the step forward is the first excitation and polymerization chance of the photosensitive molecule, the event A, while the step backward is the second occurrence of its excitation and polymerization, the event B, then H = A + B. Event A is decided by event T and kp, that is, , since T and kp are independent with each other, the probability density of A is depicted as the convolution of the two parameter above, expressed as,
Therefore, if the polymerization occurs before the excited state molecules jump to the ground state, that is, , then A = T. While if the excited state photosensitive molecules are reduced to the ground state, that is, , the whole process returns to the initial state, resulting in reciprocity failure. The distribution function of B is M(T). According to the total probability formula, the distribution function can be written as:
When the condition is satisfied with ,
When it meets the condition with , the pre-formed excited state photosensitive molecule produces a reduction, then an event B occurs in a new cycle. If the B event occurs, then the condition is still needed. It can be expressed as
According to the formula,
M(t) can be written as:
The key factor is transformed to focus on , in which Kr and KP satisfy the exponential distribution. According to their distribution functions P(T) and R(T), the average can be obtained and used to instead of KP and Kr. Therefore,
According to the relationship between random variables, H = A + B, reciprocity law matching relation can be expressed as
The expression of u(t) is:
The simulations are depicted in Fig. 1 and Fig. 2 with the polymerization rate and reduction rate of 104 and 10−2 with the magnitude of second, respectively . Figure 1 described the influence of exposure intensity on holographic reciprocity failure. From the simulation results we can see, to enlarge the matching range of reciprocity law, enhancing the exposure intensity is an available method. The largest range is from 10−3 to 102 under 10J/m2. This is mainly due to the increased availability of photons can promote the polymerization rate of excited photosensitive molecules.
Meanwhile, we examined the effect of different polymerization rate on holographic reciprocity law failure, as shown in Fig. 2. It is indicated that faster polymerization rate can also expand the matching range. The polymerization rate represents the velocity of grating formation, which is decided by the intrinsic sensitivity of photo-initiator. It means more photo-initiator will be excited to form refractive index modulated grating by polymerization process if the polymerization rate of photo-initiator is shorter than the exposure duration. When the exposure duration is determined, the photo-initiator will be excited at a lower probability if the polymerization rate grows, which states a negation exponent. Figure 2(d) is a fitting to depict the trend of factor differences and matching range. These two approaches can be effective ways to avoid holographic reciprocity failure while recording gratings in photopolymers. Moreover, the reciprocity failure occurred after more than 100 seconds is due to the insufficient amount of photosensitive molecules. Besides, we found that a new match was produced after the failure of the reciprocity law with a lower reciprocity coefficient, as shown in the left part of Fig. 2. This demonstrated that gratings could be recorded in photopolymers under pulsed exposure, but with a severe decline on grating strength. Therefore, we decided to examine if the holographic reciprocity matching happened when the pulse duration was over the magnitude of 10−3 s in the next section.
3.2 Experimental results of TI/PMMAs on holographic reciprocity matching
In the experiment, we investigated the volume holographic grating strength under millisecond exposure according to the simulations on largest matching range of 10−3~102 s. Here, we defined the grating strength as the square root of diffraction efficiency. A 532nm CW laser beam was used to record holographic gratings. All the experiments were carried out under the pulse exposure intensity of 115 mW/cm2. We achieved the millisecond time recording by using the ECS with the precision of 20 milliseconds on each shot. The shutter speed limitation is 20ms, hence we can get the single pulse duration of 20ms, 50ms, 100ms, 200ms and 500ms by control the shutter speed. To compare the holographic reciprocity, we examined the grating strength under one shot with different pulse durations, as shown in Fig. 3.
The one-shot duration was set up from 20 ms to 500 ms. With the increment of exposure duration, the diffraction efficiency enhancement was improved according to Fig. 3. Meanwhile, the gradient in the monomer and other chemical compound concentration will be higher with the development of exposure duration, therefore the effects of diffusion are more relevant. It was mainly due to the longer exposure time contributed to more excited photosensitive molecules, which would do more contributions to the polymerization with monomers. Meanwhile, to examine the holographic reciprocity, we compared the original grating strength after exposure with no enhancement of dark diffusion, as depicted in Fig. 4. It was indicated that TI/PMMAs can match holographic reciprocity law under the exposure range of milliseconds, which had a good general coherence to theoretical results. The reciprocity matching coefficient in thinner TI/PMMAs was higher than that of thicker TI/PMMAs due to the thinner ones demonstrating better photon sensitivity, namely, thinner TI/PMMAs exhibited better holographic performance under the condition of pulsed exposure system. These measurements provide a support on ultrafast holographic recordings in milliseconds with no loss of the mismatch between exposure intensity and illumination durations. Figure 4(d) is a fitting to depict the trend of coefficient with different thicknesses.
Though volume transmission gratings are able to be recorded in TI/PMMAs in a single-shot pulse without reciprocity law failure, we still try to investigate further on holographic performances under multi-shot exposure to analyze more details on the diversity of ultrafast milliseconds holographic recording. A method of multi-shot exposure in millisecond durations is proposed in this section. To analyze the holographic properties influenced by multi-shot, two aspects, dark diffusion enhancement and real-time recording were examined, respectively.
3.3 Experimental results on holographic performances with different experimental conditions
Grating strength stored in TI/PMMAs under multi-shot exposure with different single-shot durations and repetition rates were investigated. Firstly, the influence of pulse repetition rates was examined. Grating strengths under 40 time exposures with the single-shot duration of 20 ms of different repetition rate (1~20 Hz) in TI/PMMAs were examined, as shown in Fig. 5. The TI/PMMA has a significant holographic property called dark diffusion enhancement process (DDEP), when means the grating will be enhanced after exposure in the darkness with no additional exposure. It is noteworthy that this process has also been observed in several other photopolymer materials [29,30]. Here, we use the diffraction efficiency enhancement to depict the DDEP of TI/PMMAs under pulsed exposure, which can be defined as the ratio of maximum DDEP diffracted intensity to the initial value just after exposure. We depicted the diffraction efficiency enhancement of different TI/PMMA thicknesses under different repetition rates exposure, as shown in Fig. 5(d). It indicated that the accumulation of grating strength was enhanced by shortening the pulse interval, i.e. increasing the repetition rates. However, the experimental results of 1mm TI/PMMAs were quite strange, which could be described as significant fluctuation with the increment of repetition rates. It was mainly because thinner sample possessed higher sensitivity, this led to a faster formation speed of volume grating. If the multi pluses interval was long, a brief dark diffusion process was occurred in the exposure area of samples. Thinner samples could contribute a faster diffusion effect due to their higher sensitivity. The whole diffusion process was mainly made up of two parts, photo-initiator diffusion and counter-diffusion of photoproducts, which was covered from the theoretical model in . The photo-initiator diffusion could improve the grating strength, with the time passing in the darkness, this process was faded because the concentration gradient of photo-initiator tended to be dynamic equilibrium. At this circumstance, counter-diffusion of short-chain photoproducts occurred which had a severe decline on grating strength. If the exposure interval was short, the photo-initiators would be excited continuously, the photo-initiator diffusion dominated the process, which indicated that there was a decrease in the probability on counter-diffusion of photoproducts. With the cumulative number of exposures, the photoproducts formed long chains, and stabilized the volume grating. This is why the repetition rates have a greater influence on thinner samples.
The experimental results indicated that higher repetition rate could promote the dark diffusion enhancement in TI/PMMAs after exposure. Meanwhile, 3mm thick materials supplied sufficient photo-initiator which could prolong the dark diffusion process. However, thicker samples might cause more serious scattering effect, that was why 1mm TI/PMMA reached the highest grating strength of 15%, while 3mm TI/PMMA got the largest enhancement in dark diffusion process of 90% on diffraction efficiency. Further, the temporal evolution of diffraction efficiency under different single-shot duration (20~500 ms) of 1Hz repetition rate in TI/MMAs was examined, as shown in Fig. 6. We controlled the exposure of 40 pulses on each duration measurement.
In Fig. 6(a), it described the diffraction efficiency of 1mm TI/PMMAs. There is a slight increase in the DDEP then followed by a negative trend. The main reason for this phenomenon are as follows: Thinner samples exhibit better sensitivity which means the gratings formation speed is faster compared to thicker ones. When the exposure flux is determined, the real-time diffraction efficiency value of 1mm (46%) is higher than that of 3mm (10%). However, there are also fewer photo-initiators in thinner samples caused an attenuation on diffraction efficiency enhancement in the DDEP. When the photo-initiators are consumed completely in the exposure area, the counter-diffusion of photoproducts dominate the dark diffusion process, which eventually leads a decline in diffraction efficiency. Here, the dark diffusion rate consists of two main part, photo-initiator diffusion can enhance the diffraction efficiency, while counter-diffusion of photoproducts is to weaken the grating strength.
Longer exposure duration could excite more photosensitive molecules, which did more contribution to the grating formation. This explanation was also applied in multi-shot exposure. In the experiment, we got the maximum diffraction efficiency in 1mm TI/PMMAs of 46%, while the largest enhancement in dark diffusion process reached 20% in 3mm samples. In this section, we examined and analyzed the influence of repetition rate and single-shot duration of pulse on holographic performances, which indicated that the holographic performances under multi-pulse exposure exhibited better improvement on the condition of longer duration and faster repetition.
3.4 Experimental results on holographic performances in TI/PMMAs under long time pulse exposure
This section was proposed to analyze the maximum grating formation ability of TI/PMMAs under long time exposure. In the experiment, different single-shot duration was used to examined the grating formation process with the same dark interval of 30 ms, as shown in Fig. 7. From the experimental results we could see, longer single-shot duration not only enhances the grating strength, but prolongs the response time in TI/PMMAs. However, we demonstrated that there was no reciprocity law failure while exposing under milliseconds exposure. The main reason for the decline in diffraction efficiency of different single-shot duration was due to the dark diffusion mechanism of TI/PMMAs. There were three steps while the sample was exposed by one shot exposure, photo-induced polymerization, dark diffusion of photosensitive molecules and counter-diffusion of photoproducts, respectively. Shorter duration would cause less diffusion of photosensitive molecules and more counter-diffusion of photoproducts, which eventually led to a severe decline in diffraction efficiency. Compared to our previous work of 3mm TI/PMMAs, the diffraction efficiency and response time was 74% and 20s under CW exposure . The diffraction efficiency reached 67% with the response time of 15.69s in multi-shot long-time exposure. It was indicated that the diffraction efficiency and response time could be regulated by controlling the single-shot duration to meet different demands.
3.5 Preliminary theoretical analysis of the dynamic mechanism under and after single-shot exposure
In this section, we proposed a preliminary analysis for the physical and chemical mechanism under and after single-shot exposure. As shown in Figs. 5-7, the maximum grating strength became weak with the decrease of single-shot duration on the condition of holographic reciprocity matching, which was mainly because the dark enhancement process was influenced by single-shot duration while multi-pulse exposure. The whole polymerization and diffusion process is included three parts, photo-induced polymerization, dark diffusion of photosensitive molecules and counter-diffusion of photoproducts, which can be described as [4,24–27]:
Where is the molar absorption coefficient, d represents the thickness of sample, is the quantum yield, I0 is the exposure intensity, V is the light visibility and depicts polymerization rate constant. TI means the original concentration of TIs before exposure, while Product0 and TI0 represent the concentration of photoproduct and TIs after exposure. The sample firstly is exposed under single-shot light with the duration of , described as Eq. (15). Then the exposure area begins to occur dark diffusion process as depicted in Eq. (16)-(17). Because a photosensitive molecule can polymerize to form a photoproduct, namely, Product0 = TI0. This process is described in Fig. 8.
In ultrafast duration exposure, the refractive index grating is firstly formed by the generation of photoproducts in the bright region, as shown in Fig. 8(a). Then, the exposure is switched off, the exposure area starts to diffuse in the darkness. TIs become the first kind component to diffuse due to the consumption in the exposure area causes a higher concentration gradient difference. Obviously, the difference of TIs at the boundary of grating is larger than other position, which can contribute to the enhancement of grating strength. Finally, the other component, photoproducts, begins to counter-diffuse since the improving increment of concentration gradient between the bright and dark region. It is a slow process because the photoproducts have chain structures and concentration difference in the grating boundary of photoproducts are not significant compared with other exposure position. Meanwhile, the boundary difference of TIs is also declined due to the fact that un-reacted TIs diffuse in from the external exposure area. Therefore, the third step, counter-diffusion of photoproducts, can make an attenuation on grating strength. Throughout the whole process, the second step is the main process to enhance the grating strength, the duration of which is determined by single-shot duration and intensity. Therefore, if the time interval between every two pulses is too long, the grating strength enhancement will grow at a very slow speed, as shown in Fig. 8(d). Besides, this is only a preliminary theoretical analysis since we don’t consider the complex impediment of different molecules in the process of dark diffusion, which will be further investigated in our future research.
This work proposed a probability calculation on holographic reciprocity law failure, which indicated that the exposure duration of 10−3~102 time magnitude in photopolymers could match the holographic reciprocity law without any loss of grating strength. TI/PMMAs have been experimentally demonstrated that it could match the holographic reciprocity under the exposure duration over 20ms. Meanwhile, the reciprocity matching coefficient in thinner TI/PMMAs was higher than that of thicker TI/PMMAs due to the thinner ones owned better photon sensitivity. In order to study more comprehensively on the holographic performance of materials in millisecond pulses, we examined the holographic performances of different thick TI/PMMAs (1-3mm) under multi shots exposure with different single-shot durations and repetition rates. The experimental results indicated that higher repetition rate could promote the dark diffusion enhancement in TI/PMMAs after exposure, while longer exposure duration could excite more photosensitive molecules, which did more contribution to the grating formation. Also, the material thickness influenced the grating strength and the response time. By recording gratings under multi-shot exposure, we could regulate the grating recording ability and sensitivity, shorter duration and thinner samples contributed to quicker sensitivity while longer duration and thicker ones were able to improve the grating strength, from which we got the optimized diffraction efficiency of 67% or response time of 3.25s. Finally, we made a preliminary theoretical analysis on the influence of time intervals and exposure durations. The whole polymerization and diffusion process under and after single-shot exposure can be included by three parts, photo-induced polymerization, dark diffusion of photosensitive molecules and counter-diffusion of photoproducts. The second step is the main process to enhance the grating strength, while the third step, counter-diffusion of photoproducts, can make an attenuation on grating strength. This is why the grating strength and response time are influenced by the time intervals and exposure durations. All these measurements and analysis can provide a new kind research approach under milliseconds with the holographic reciprocity law matching in TI/PMMAs to control the volume holographic grating strength and sensitivity.
National Basic Research Program of China (2013CB328702); the National Natural Science Foundation of China (11374074).
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