Implementation of uniform diffraction efficiency of partially overlapping holograms in photopolymers based on the photopolymerization of a free radical

Abstract. The exposure schedule model for uniform diffraction efficiency is extended to be suitable for the partially overlapping multiplexing method. The proposed model is based on solving an optimization problem. Fifty holograms were multiplexed using the exposure schedule calculated with the extended model. The material used in the experiment is based on the photopolymerization of a free radical. By comparing the intensity of the reconstructed images during recording with that readout after recording, the calculated exposure schedule is verified to be effective to realize the uniform diffraction efficiency for the multiplexing holographic storage.

Implementation of uniform diffraction efficiency of partially overlapping holograms in photopolymers based on the photopolymerization of a free radical 1 Introduction Holographic storage is a potential data recording technology because of its high storage capacity and fast data rate. 1 Compared with holographic photorefractive crystals, photopolymers are considered to have the most potential material for holographic storage because of their low cost, high diffraction, easy processing, and other advantages. [1][2][3] There are two kinds of photopolymer holographic recording material based on cationic ring-opening polymerization or the photopolymerization of a free radical. 4,5 In our work, the photopolymer based on the photopolymerization of a free radical is in use. The recording material has been developing and a dual-monomer photopolymer has been introduced. It is dualwavelength sensitized for increasing the storage density and capacity. 6 According to the previous work, 7 experimental tests had shown that there are usually two advantages of the dual-monomer photopolymer. First, the polymerization efficiencies of the dual-monomer are higher than those of the single-monomer ones, which also provides an easier way to increase the refractive index modulation. Second, the dualmonomer design improves the stability of the material in polymerization due to the combination of the dual monomer with the different physical and chemical property. Besides, the experimental results also have shown that the concentration ratio of POEA and NVC kept stable with few changes or fluctuation in the polymerization, which suggests that the two monomers polymerize independently without any interaction.
The mechanism of holographic recording in photopolymers based on the photopolymerization of a free radical consists of two processes. [8][9][10] One is the monomer's polymerization caused by the exposure of the interference pattern in holographic recording, which leads to monomer concentration gradient. The other is the diffusion caused by the gradient of monomers, which leads to a refractive index change, so holographic gratings are recorded inside the material. Models were introduced for the dynamics of a single-grating formation in photopolymers taking into account spatialdependent diffusion of monomers 11 and time-dependent photo-polymerization rate. 12 In the application of a large-area photopolymer for holographic storage, partially overlapping multiplexing methods, such as spatioangular multiplexing, 13 shift-multiplexing, 14 and polytopic multiplexing, 15 are applied widely. To increase the storage density, it is an effective way to apply the speckleencoded reference beam generated by a diffuser to the shiftmultiplexing method. 16,17 The dynamics of grating formation in the partially overlapping multiple recording is much more complicated including three processes: holographic recording, dark reaction, and uniform postexposure (UPE). 18,19 Different parts of one hologram experience different durations of the three processes.
However, it is very important make diffraction efficiency of all recorded holograms uniform in the application of multiplexed holography. For this purpose, the mechanism of gratings formation involving the influence of sequential recordings should be investigated. An empirical approach was proposed 20 and implemented. [21][22][23] In this method a six-order polynomial was obtained by fitting experimental data of cumulative index modulation versus cumulative exposure time by recording tens of holograms. With parameters acquired from the fitting data, exposure time of the individual hologram was calculated for uniform diffraction efficiency. [20][21][22][23] But several times of recording holograms and fitting calculation are needed to obtain the optimum time schedule realizing uniform diffraction. An experiment conducted to validate the method has been done and shown that the polynomial curve exhibited an anomalous trend in the saturation regime when it was extended from the end time of the experiment with a longer recording time, which implies that the polynomial fit does not reflect the real physical process of multiplexing. 18 Another method was proposed to identify an optimum exposure schedule for photopolymer materials governed by the nonlocal polymerization-driven diffusion model. 24 In this model, sufficient time between sequential recordings must be allowed, to wait for full monomer relaxation caused by the diffusion effect. It means that a hologram should not be recorded until the previous one finished its relaxation. However, the relaxation of a monomer often lasts several minutes. 24,25 This is not suitable for dense holographic storage in which a huge number of holograms are required to be written.
We have proposed a simplified model for grating formation in photopolymers based on the first-harmonic diffusion model 18 and extended the work to a calculation model for exposure schedules applicable to holographic recordings with common-volume (e.g., pure angle or wavelength) multiplexing methods in single-monomer photopolymers. 19 In this paper, we extend the simplified model further to be suitable for holographic recording with partially overlapping multiplexing methods, and the effect of dual-monomer is considered. Using the extended model, a time schedule for recording 50 holograms is calculated, and uniform diffraction efficiency is implemented in an experiment of shift multiplexing.

Physical Mechanism for Diffraction Efficiency of Gratings in Dual-Monomer Photopolymers
Holograms recorded in single-monomer photopolymers generally experience three processes: 18,19 holographic recording, dark reaction, and postexposure process. The index modulation, Δn, reached during holographic recording is expressed as where Δn SAT is the saturation index modulation that is related to the initial monomer concentration, τ D is the diffusion time constant, and τ P is the polymerization time constant. After the exposure stopped at t ¼ t w the gratings would still grow up for a period based on the diffusion effect, which is referred to as dark reaction. The evolution of index modulation during dark reaction is expressed as where A is related to the monomer concentration at t ¼ t w . Furthermore, the dynamics of refractive index modulation of a grating under UPE after a short holographic recording can be expressed as where t e is the start time of the postexposure process, and C is related to the monomer concentration at t ¼ t e . In multiple recordings, the effect of a sequential recording on an existing grating can be considered as UPE. Usually, the saturation refraction index of UPE and dark reaction is much lower than that of a holographic recording. 3,4 On the other hand, in the dual-monomer photopolymer, the two kinds of monomers participate in the polymerization independently, 7 without influence on each other, during holographic writing, dark reaction, and postexposure. In this way, the final refractive index modulation of the grating, ΔnðtÞ), should be expressed as where Δn 1 is the refractive index modulation of one monomer (M1), and Δn 2 is that of another one (M2). The development of both refractive index modulation Δn 1 and Δn 2 follows Eqs. (1)-(3) and can be calculated with the polymerization time constant τ P1 , τ P2 , the diffusion time constant τ D1 , τ D2 , and the saturation index modulation Δn SAT1 , Δn SAT2 , respectively. These parameters can be obtained by the data-fitting method with dark reaction and holographic writing experiments. The composition of the material for this paper are detailed in Table 1.
In the holographic writing and dark reaction experiments, the laser source is the Verdi™-V5 Diode-Pumped Lasers from Coherent Company. The wavelength of recording beams is 532 nm. The total exposure intensity is 15 μW∕ mm 2 , and the intensity ratio of object beam to reference beam is 1∶1. The angle between the two recording beams is 45 deg, and the thickness of the material is 500 μm. According to Eqs. (2) and (4), the constants τ D1 and τ D2 can be obtained directly with the experiment results of dark reaction by data-fitting processing. Then, according to Eqs. (1) and (4), the constants Δn SAT1 , Δn SAT2 , τ P1 and τ P2 can be obtained with the holographic recording and the obtained constants τ D1 and τ D2 . The calculation of datafitting processing are solved by the computer. The experimental and fitting curves are shown in Fig. 1, and the correlation coefficient is 0.999.
10 −5 for the photopolymer currently used in the experiment. The subscript 1 stands for the solid monomer NVC, and subscript 2 stands for the liquid monomer POEA here. The saturation diffraction efficiency of the material is 70.13%.
Considering the order of magnitude of Δn is generally less than 10 −4 , in the condition of small index modulation, diffraction efficiency η is propotional to the square of refractive index Δn: Equations (1)-(5) together with all necessary parameters fitted from experimental data constitute the fundamentals of our exposure schedule model.

Exposure Schedule Model for Partially
Overlapping Multiplexing Take a shift multiplexing recording process for an example, the scheme is demonstrated in Fig. 2 for realizing uniform diffraction efficiency with partially overlapping multiplexing. As Fig. 2(a) shows, the shift interval is δ. Every hologram is divided into r zones, r ¼ d∕δ:r is also defined as the degree of multiplexing, and d is the size of a single hologram. In this way, as Fig. 2(b) shows, the whole recording area for total N holograms marked by i is divided into N þ r − 1 zones marked by j. The total recording time is t N , t N ¼ T. And for recording the i'th hologram, the recording time is from t i−1 to t i .
The scheme shown in Fig. 2 can depict the situation of every zone in the holographic storage. Generally, each zone experiences holographic recording, UPE, and dark reaction sequentially, but some characteristics are found out in the three processes. For the holographic recording process, the initial monomer concentration in the zones 1 ≤ j ≤ r is different from that in the zones r < j ≤ N þ r − 1 because the latter experiences pre-exposure before recording begins. For the UPE and dark reaction processes, because the other holograms are overlapping with the N'th hologram in the zones N þ 1 ≤ j ≤ N þ r − 1, there is no dark reaction, but the UPE lasts to the end in these zones.
According to the discussion above, the whole holographic storage can be divided into three different ranges of zone number j, 1 ≤ j ≤ r, r þ 1 ≤ j ≤ N, and N þ 1 ≤ j ≤ N þ r − 1. Having investigated the dynamics of index modulation in each zone carefully and through somewhat tedious derivation, we obtained for each range a special mathematical expression for the contribution of the j'th zone to the refractive index modulation of the i'th hologram, Δn ij . In the first range, the contribution of monomer M1 to Δn ij , Δn ð1Þ ij , can be written as:  In the intermediate range, ðr < j ≤ NÞ; and in the last range By changing the subscript "1" to "2" in Eqs. (6)-(8), we obtain the correspondent expressions for monomer M2. And then, using Eqs. (4) and (5), the contribution of the j'th zone to the diffraction efficiency of the i'th hologram can be obtained. The derivation of Eqs. (6)-(8) will be detailed in the Appendix.
Our purpose is to calculate an exposure schedule, t 1 − t 0 ; t 2 − t 1 ; : : : ; t N − t N−1 , (t N ¼ T), following which the diffraction efficiencies of total N holograms are uniform. Furthermore, the uniform diffraction efficiency should be as high as possible. This calculation for the maximum average is equivalent to the following optimization problem: where C 1 is a constant. Equation (10) means the diffraction efficiency of the i'th hologram occupying r zones from j ¼ i to j ¼ i þ r − 1 is equal to a common value C 1 , so the diffraction efficiencies can be uniform. Additionally, there is another restriction according to the physical condition, that is, the refractive index modulation of each zone should not exceed the permitted range of the material: X j i¼j−rþ1 ½Δn 1 ði; jÞ þ Δn 2 ði; jÞ ≤ C 2 ðΔn SAT1 þ Δn SAT2 Þ; i ¼ 1; 2; 3; : : : ; N: The left hand of Eq. (11) represents the refractive index modulation of the j'th zone overlapped by r holograms from i ¼ j − r þ 1 to i ¼ j, and the coefficient C 2 ðC 2 ≤ 1Þ in the right hand is an adjustable controlling constant, but it does not need to make each zone's refractive index modulation also uniform.
The optimization model can be solved numerically by the computer. The routing time for calculating an exposure schedule is typically 6.2 s. Using the fitted parameters obtained from Fig. 1, and taking the controlling constant C 2 ¼ 0.5 and multiplexing degree r ¼ 6, an exposure schedule for 51 holograms was calculated and shown in Fig. 3. The total exposure time is T ¼ 531 s. The exposure time for the sequentially recorded holograms is rising smoothly except for the 51st. It is reasonable because the diffraction efficiencies of the former 50 holograms are compensated in dark reaction and UPE processes after holographic recording. But the 51st hologram only experiences the holographic recording process without any additional compensation, so the exposure time for recording the 51st hologram is much longer than the others.

Experiment of Multiplexing Storage
To verify the calculated schedule, a multiplexing experiment with shift multiplexing has been done, and the setup is shown in Fig. 4. The laser source and other experiment conditions are the same as those in the holographic recording and dark reaction experiments mentioned above except the intensity ratio of object beam to reference beam is 1∶2 in the multiplexing experiment. The object beam is modulated by the spatial light modulator (SLM). An image is located to the SLM, making the object to be recorded. Then, through Lens L2, the Fourier transform of the object is obtained on the plane of recording media. The reference beam, modulated by a diffuser, is imaged by inserting Lens L1, to the plane of recording media, so that the reference wave is phaseonly modulated. The diffuser is a speckle phase shifter with the speckle size of 50 μm. The shifting interval δ is 0.25 mm leading to a multiplexing degree r ¼ 6, since the hologram spot on the material is 1.5 mm.
Using the calculated exposure schedule shown in Fig. 3, 51 holograms were recorded in a dual-monomer photopolymer disc. The actual total recording time was 570 s, a little longer than the calculated one due to the time consumption of the stage rotating and program routine.
The reconstructed images are captured by a CCD camera whose automatic gain and background light control function is closed, so the gray scale of the image can reflect the image intensity. To monitor the multiplexing recording process, each hologram was read out immediately after its recording finished, and after the 51 hologram recordings finished, all the holograms were reconstructed once again.
The reconstructed image of the 51st hologram is degraded due to excessive exposure for its overlong recording time; nevertheless, others were reconstructed successfully. So, the calculated exposure schedule shown in Fig. 3 is actually suitable for recording 50 holograms. The reconstructed images for holograms No. 23 and No. 46 are shown in Fig. 5. Although the noise increased, as shown in Fig. 5(c) and 5(d), the intensity of the two reconstructed images became closer than in the monitoring process, as shown in Fig. 5(a) and 5(b). It is noticed that the reconstructed images after recording are darker than those in monitoring because the scattering noise gratings rise during the whole recording process, which makes the transmittance of the whole recording area fall down. 26,27 Therefore, all the recorded holograms are influenced. Similarly, other reconstructed images after recording are also darker than those in monitoring.
In order to estimate the experimental result quantitatively, we use the average gray scale of an image as a measure of its intensity, which is proportional to its diffraction efficiency. The gray scale of every image is normalized by that of the first one, which is referred to as normalized intensity. The normalized intensity of the reconstructed images read out immediately after recording, and the final reconstructed images are shown in Fig. 6. The curve for the images read out immediately is rising all the time as shown in Fig. 6(a). It is reasonable because the more recording time is offered, the higher diffraction efficiency is gotten before the saturation.   In contrast, the curve for the final ones, as Fig. 6(b) shows, is quite close to a straight line. It means the previous recorded holograms in relatively low diffraction efficiency were compensated by dark reaction and UPE for different time following the calculated schedule in Fig. 3, and finally, diffraction efficiency of all the recorded holograms is more uniform.

Conclusion
We extended the exposure schedule model of uniform diffraction efficiency in a single-monomer photopolymer to that in dual monomers with partially overlapping multiplexing method and proposed an optimization algorithm to calculate the schedule. The extended model is validated by a preliminary experiment. By using a shift-multiplexing method for 50 holograms in a dual-monomer photopolymer material with the calculated exposure schedule, the intensity of the final reconstructed images became fairly uniform. The proposed model is surely suitable for other partially overlapping multiplexing, such as the phase-shift multiplexing method. 28 Additionally, it is noticed that the final reconstructed images are influenced by the scattering noise gratings and become darker than those in monitoring. We will discuss it and raise the quality of the recorded holograms in our future work.

Appendix: The Derivation of Eqs. (6)-(8)
We have proposed a simplified model 18,19 from fundamental principles of monomer diffusion and photo-polymerization. 12 where u 0 ðtÞ stands for the average of momomer concentration, u 1 ðtÞ is the amplitude of monomer concentration modulated distribution. m is the modulation of interference fringes. By solving Eq. (13), u 0 ðtÞ is expressed as: The refractive index modulation is calculated by Eq. (15) (Ref. 3): where C n is a proportional constant. For the holographic recording, u 0 ðtÞ is related to the start time of recording. According to the physical model for partially overlapping multiplexing shown in Fig. 2, the start time is t j−r in the range of j > r, so Eq. (14) is modified as follows: where U is the monomer concentration before holographic recording. It is noticed that u ij1 ðt i−1 Þ is 0 because holographic recording is just starting at the time t i−1 for the i'th hologram. With this conditon, by Eqs. (13) and (16), u 1ij ðtÞ in different range of j can be gotten in the range t ∈ ½t i−1 ; t.
Then, with Eqs. (15) and (17) or Eqs. (15) and (19), the refractive index modulation in the recoding process can be depicted as: ; ; j ∈ ðr;N þr−1; where Δn SAT is expressed as Also, because holographic recording for the i'th hologram is not happening before the time t i−1 , the refractive index modulation attributed by the i'th hologram Δn ij ðt i−1 Þ is 0. For the UPE, the modulaton of the interference fringes m is 0. And Eq. (13) is modified as: then u 1ij ðtÞ can be gotten in the range t ∈ ½t i ; t.
where u 1ij ðt i Þ can be gotten by Eqs. (17) and (18) in a different range of j at the time t i , the end of the holographic recording. Then, with Eqs. (15) and (23), the refractive index modulation in UPE process can be gotten in the range t ∈ ½t i ; t j in the condition of j ≅ N, or in the range ∈ ½t i ; T in the conditon of j > N according to the model shown in Fig. 2.
As discussed above, considering the different expression of u 1ij ðtÞ shown in Eqs. (17) and (18) in a different range of j at the time t i , the index modulation in UPE process is divided into three parts: Δn ij ðTÞ − Δn ij ðt i Þ For the dark reaction, the terms 1∕τ P and m are 0 because of no exposure in this process. So Eq. (13) is modified as: then u 1ij ðtÞ can be gotten in the range t ∈ ½t j ; t.
where u 1ij ðt j Þ can be obtained by Eqs. (17) and (24) in the range j ∈ ½1; r, or Eqs. (18) and (24) in the range j ∈ ðr; N þ r − 1 at the time t j , the end of UPE. Then, with Eqs. (15) and (32), the refractive index modulation in dark reaction process can be gotten in the range t ∈ ½t j ; T, according to the model shown in Fig. 2. Besides, it is noticed that there is no dark reaction in the range of N ≅ j ≅ N þ r − 1, so the expression of the index modulation in dark reaction is also divided into three parts: Δn ij ðTÞ − Δn ij ðt j Þ ¼ 0; Then, by changing the representation "τ P " to "τ P1 " and "τ D " to "τ D1 ", the expression of the contribution of monomer M1 to Δn ij , Δn ð1Þ ij , in the range of 1 ≤ j ≅ r as shown in Eq. (6), can be obtained by adding Eqs. (19), (25), and (30). The expression of Eq. (7) in the range of r < j ≅ N can be obtained by adding Eqs. (20), (26), and (31). And the expression of Eq. (8) can be obtained by adding Eqs. (20), (27), and (32).