# Polarization Holography in 3-Indoly-Benzylfulgimide/PMMA Film

Authors: Neimule Menke; Baoli Yao; Yingli Wang; Yi Chen
Journal: Journal of Atomic, Molecular, and Optical Physics
 (2011)

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## Abstract

The photochromic and photoanisotropic properties of materials can be used in ordinary and polarization holographic recording respectively. Fulgides are well known as thermally irreversible organic photochromic compounds. And it is found that there exists photoinduced anisotropy in fulgide-doped polymeric films. In this report, a 3-indoly-benzylfulgimide/PMMA film was studied as a holographic storage media. First, the spectra and dynamics of photochromic and photoanisotropic properties of the sample are measured or calculated. Second, the diffraction efficiency (DE) dynamics at 633 nm of four kinds of different polarization holograms recorded in this sample are measured. The maximum DE value about 1% was gotten. Third, the DE spectra and DE dynamics are theoretically calculated in detail, and a good correlation of theoretically derived DE dynamic curves and the measured experimental curves was found. From the DE spectra, it is known that at the wavelengths less than 450 nm or greater than 700 nm, the nondestructive reading can be realized. The DWPS obtained in the experiments are same with the theortically deduced ones, which shows that in the orthogonal polarization holography, the polarization state of the diffracted light is orthogonal to that of the reconstruction light, which is very important to increase the SNR of the holographic storage. And all these results are applied and proved to be correct in high-density holographic image storage experiment. The area density of2×108 bits/cm2 was obtained, and the encoded data was retrieved without error.


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## Body

## 1. Introduction

Holographic optical storage becomes an important research aspect in optical storage field for its high data storage capacity and data transmission rate. Fulgides are well known as thermally irreversible organic photochromic compounds [1], which can be used as rewritable ordinary holographic recording material. But up to now, a few researches have been done on the holographic storage application of fulgide materials [2–9]. And we found that there exists photo-induced anisotropy in fulgide-doped polymeric films [10], which can be used in polarization holography application.Here, the diffracted wave polarization states (DWPS) and the diffraction efficiencies (DE) of ordinary holograms and polarization holograms recorded in a 3-indoly-benzylfulgimide/PMMA film are studied in detail. First, the photochromic and photoanisotropic properties of the sample is given, and the DE dynamics at 633 nm of parallel linearly, parallel circularly, orthogonal linearly, and orthogonal circularly polarization holograms, each of which read by four kinds of different polarized lights, are measured. Second, the DE formula and DWPS of parallel linearly, parallel circularly, orthogonal linearly, and orthogonal circularly polarization holograms in photochromic and photoanisotropic thin films are theoretically deduced; Third, using the formula deduced before, from the properties of the sample, the DE spectra and dynamics at 633 nm of four kinds of different polarization recorded holograms are theoretically calculated and compared with the experimental curves, and the DWPS obtained in the experiments are compared with the theortically deduced ones. At last, all these results have been applied in high-density holographic image storage experiment.


## 2. Material and Measurement Experiments

### 2.1. Material

The fulgide material studied here is 3-indoly-benzylfulgimide, which was synthesized by the Stobbe condensation routine [11]. The target compound of 3 mg was dissolved in a 0.l mL 10% (by weight) PMMA-cyclohexanone solution. Then, the solution was coated on a 1-mm thick K9 glass plane with a spin coater and dried in air. The thickness of the film is about 10 μm. The photochromic or photoanisotropic properties of fulgides are due to a reversible photochromic (photoisomerization) reaction that occurs between one of the colorless E-form (bleached state) and the C-form (colored state). These are the two spectrally separated photochromic forms, whose molecular formulas are shown in Figure 1.Figure 1
Molecular formula of the Indolyfulgimide and the photochromic reaction. Left,E-form; right, C-form.When the colored state is irradiated by a linearly polarized 650 nm laser, the film returns to the bleached state and photo-induced anisotropy is produced during this process. The mechanism of photo-induced anisotropy in the fulgide film is the following [10]: anisotropic absorbing molecules of fulgide are immobilized randomly in the PMMA polymeric matrix, which shows isotropic characteristic at the initial state; when the sample is irradiated by linearly polarized light photoselection of molecules take place. Molecules with long axes parallel to the exiting light polarization direction absorb the light strongly and turn to the other form very quickly, whereas those with long axes perpendicularly orientated to exiting light polarization have low absorption and stay at the initial form. As a result, special orientation of two form molecules is induced and the sample shows optical anisotropy properties macroscopically. The angular distributions of molecules in anisotropy inducing progress are shown in Figure 2. So, under the irradiation of circularly polarized light, the sample shows also optical isotropy.Figure 2
Angular distributions of molecules in anisotropy inducing progress.


#### 2.1.1. Spectra of Photochromic and Photoanisotropy Properties of the Sample

The absorption of theC-form (AC(λ)) and E-form film (AE(λ)) were measured using a UV-VIS-IR spectrophotometer (UV-3101PC, Shimadzu Inc., Japan), which were shown in Figure 3(a). And the measurement of the photo-induced dichroism is performed by measuring the transmission spectra of the film for testing light polarized parallel (T//(λ)) and perpendicular (T⊥(λ)) to the polarization direction of the exciting beam after the C-form film is excited by the linearly polarized 650 nm laser (shown in Figure 4(a)).Spectra of photochromic properties of Indolyfulgimide/PMMA film. (a) Absorption spectra of two forms. (b) Absorption difference spectrum and the corresponding refractive index changing spectrum.
(a)(b)Spectra of photoanisotropic properties of Indolyfulgimide/PMMA film excited by linearly polarized light. (a) Transmission spectra on directions parallel and perpendicular to exciting beam polarization. (b) Dichroism and birefringence spectra.
(a)(b)Figure 5
Schematic of the experimental setup for measuring transmission kinetics. PBS, polarization beam splitter;M, mirror; A1~A2, continuously adjustable attenuators; S, Shutter; He-Ne: 633 nm Helium-Neon laser; D, power meter; IW, writing beam; IT, testing beam.From Figures3(a) and 4(a), the photo-induced absorption changing spectrum ΔA(λ)=AE(λ)-AC(λ) and the photo-induced dichroism spectrum ΔAD(λ)=A⊥(λ)-A//(λ)=lg(T//(λ)/T⊥(λ)) were obtained, which are shown as solid lines in Figures 3(b) and 4(b), respectively. Assuming that ΔA and ΔAD are zero outside of the band 300~800 nm, the photo-induced refractive index changing spectrum Δn(λ)=nE(λ)-nC(λ) and the photo-induced birefringence spectrum ΔnB(λ)=n⊥(λ)-n//(λ) can be calculated according to the Kramers-Kronig relation [12](1)Δn(λ)=ln⁡102π2dp.v.∫0∞ΔA(λ′)1-λ′2/λ2dλ′,ΔnB(λ)=ln⁡102π2dp.v.∫0∞ΔAD(λ′)1-λ′2/λ2dλ′,
where nE, nC, n//, and n⊥ are the refractive indexes of E-form, C-form, and of the film excited by linearly polarized light along the photo-induced extraordinary and ordinary axes, respectively, d is the thickness of the film and p.v. denotes the Cauchy principal value of the integral. The calculated Δn, ΔnB are plotted as dot lines in Figures 3(b) and 4(b), respectively. Kramers-Kronig relation can be satisfied during all the photochromic reaction progress, so at one wavelength λ′, Δn(λ′) is proportional to ΔA(λ′) and ΔnB(λ′) is proportional to ΔAD(λ′) at different exciting time. From Figures 3(b) and 4(b), it can be seen that at 633 nm in this sample, Δn(633 nm)/ΔA(633 nm) = 0.00994 and ΔnB(633 nm)/ΔAD(633 nm) = 0.006115.


#### 2.1.2. Dynamics of Photochromic and Photoanisotropy Properties of the Sample

The transmission growing up kinetics of the sample at 633 nm were measured on the parallel and perpendicular directions to exciting beam polarization, when theC-form sample was being excited with 314 mW/cm2 and 157 mW/cm2 intensity linearly polarized 633 nm He-Ne lasers (IW), respectively, and an 1 mW/cm2 633 nm laser beam is used as the testing beam (IT), the optical setup and the results are shown in Figures 5 and 6(a). From Figure 6(a), it can be seen that the photochromic reaction of fulgide material is an optical cumulating progress, which just depending on the Exposure, so it is enough to consider just one exciting beam intensity condition for the analysis.Photoanisotropic transmission curves of Indolyfulgimide/PMMA film on the directions parallel or perpendicular to exciting beam polarization depending on the exposure or erasing time. (a) Experiment curves measured at different exiting beam intensity. (b) Simulation of experimental results. (c) Calculated curves of uniformity light.
(a)(b)(c)For the analysis, we consider a fully bistable Fulgide system, neglecting side reactions likeE-Z isomerization or aging effects. Hence, the photochemical reaction should follow the particle number balance equations
(2)Ċ=-kCC+kEE,Ė=-kEE+kCC,F=C+E
and the total concentration of fulgides molecules F is constant where E denotes the concentration of the E-isomers and C the one of the C-isomers; dots denote derivatives with respect to time t and the photoreaction rates for the annihilation of the isomers C and E are given by (3)kC,E=∫γC,E(λA)iλAdλA.
Here λA is the actinic irradiation wavelengths, iλA is the spectral density of the light intensity, and γC(λA) and γE(λA) are the photochemical reaction constants of the C and E isomers at λA, respectively, which related to the absorption cross sections σC,E via γC,E(λ)=ϕC,E(λ)·σC,E(λ)/hν, where hν is the photon energy and ϕC,E are quantum efficiencies of the photochemical reactions.The decrease ofiλ with z, described by Lambert-Beer’s law(4)diλdz=-α(z,t,λ)iλ,
where α(z,t,λ) is the local absorption coefficient in the film, which equals(5)α(λ,z,t)=σC(λ)⋅C(z,t)+σE(λ)⋅E(z,t)+αM(λ,z),
where αM is the absorption of the matrix (assumed here to be photochemically inert).Based on (2)~(5), using numerical calculation method, the experimental curves were simulated (shown as the dash lines in Figure 6(b)), and the best fitting values γ//=0.00289 cm2/mJ, γ⊥=0.0012 cm2/mJ were obtained. Because of the anisotropy of the molecules σC,E, the α and γ values of the sample are variable with the directions at same location, here just the total average values of that of all the molecules on two special directions are obtained, which are enough for calculation of the DE kinetics, and the detail process of it will not been analyzed here. In the numerical calculations, the intensity Gaussian beam distribution of the He-Ne laser beam also has been considered, which was divided as many homogenous intensity circles. Then we can calculate the photoanisotropic transmission curves of uniformity light, which is shown in the Figure 6(c).


### 2.2. Experiment Measurement of Polarization Hologram Diffraction Kinetics

#### 2.2.1. Measurement Setup

The system configuration for measuring the real-time hologram first-order diffraction kinetics of the Fulgide film is schematically illustrated in Figure7. A He-Ne laser (Melles Griot Inc., USA, 25-LHP-928, 632.8 nm, 35 mW, vertical linear polarized) is used to generate recording beams (object beam IO and reference beam IR) and readout beam (reconstruction beam IC), and a laser of diode LD (Power Technology Inc., USA, IQ2A18, 405 nm, 10 mW, vertical linear polarized) is used as the erasing light (IE) source. The He-Ne laser beam splitted into the IO, IR, and IC after beam splitter BS1 and polarization beam splitter PBS, in which IR and IC are phase conjugated (counterpropagated) beams. The diffracted light ID of IC, diffracted by the dynamic holographic grating established by the interference between the IO and IR, will be phase conjugated with the IO, whose power was real time detected by a digital power meter “D” (United Detector Technology company, USA, 11A Photometer/Radiometer, 254~1100 nm Imax = 10 mW, resolution is 0.01 nW) and a digital oscilloscope “O’’ (Tektronix company, USA, TDS3032, 300 MHz, 2.5 GS/s, 1 mV) after reflected by the BS2 (R47%). The IO and IR are symmetrically incident on the sample (Fulgide film), whose intersection angle 2θ=16.5°, so the recorded grating is a nonincline grating. Shutter S1 and S2 controls the exposure time of red and purple beams. The continuously adjustable attenuators A1~A3 are used to adjust the intensities of the waves, in this experiment IO = IR = 78.6 mW/cm2 and IC = 0.786 mW/cm2 (i.e., IO : IR : IC = 100 : 100 : 1). This insures that the subreflection gratings formed by IO and IC as well as IR and IC can be ignored. The quarter-wave plates Q1~Q4 and the polarizer P are used to change the polarization states of the waves; here four different polarization recording: parallel linearly polarization recording, parallel circularly polarization recording, orthogonal linearly polarization recording, and orthogonal circularly polarization recording were studied, every one was constructed by horizontal, vertical, left circular, and right circular polarized four kinds of lights like shown in Table 1.Table 1
DWPS and DE of different kinds of polarization holograms.
Polarisation of object lightPolarisation of reference lightPolarisation of reconstruction lightPolarization states of diffracted lights+1-order diffraction efficiencies+1−1Amplitude hologramηA,+1Phase hologramηP,+1(τ1om2)2(τ0om·J1(Ψ1om))2(τ1em2)2(τ0em·J1(Ψ1em))2(τ1om)2+(τ1em)28τ0em2·J1(Ψ1em)2+τ0om2·J1(Ψ1om)22(τ1om)2+(τ1em)28τ0em2·J1(Ψ1em)2+τ0om2·J1(Ψ1om)22(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo)22(τ0eo·sinΨ1eo)22(τ1eo)22(τ0eo·sinΨ1eo)22(τ1eo)2(τ0eo·sinΨ1eo)200Figure 7
Schematic of the experimental setup for measuring real-time polarization hologram diffraction kinetics. BS1 and BS2, beam splitters; PBS, polarization beam splitter; M1~M4, mirrors; A1~A3, continuously adjustable attenuators; P, polarizer; Q1~Q4, quarter-wave plates; S1 and S2, Shutters; D, digital power meter; O, digital oscilloscope; LD, 405 nm laser diode; He-Ne: 633 nm Helium-Neon laser; IO, object beam; IR, reference beam; IC, reconstruction beam; ID, diffracted beam; IE, erasing violet light.


#### 2.2.2. Measurement Results

In four kinds of polarization recording, and different polarization reading, the diffracted wave polarization states (DWPS) obtained in the experiments are shown in Table1 and the measured kinetic first-order diffraction efficiency (DE) curves η+1~t are shown in Figure 8. From them, the curves of the conditions when IC has same polarization state with IR were compared in Figure 9(a). The theoretical analysis of the results will be given in Section 3. It can be seen that there exists an optimal exposure about 2×78.6 mW/cm2×3.75s≈590 mJ/cm2.The diffraction efficiency kinetics curves comparison of different kinds of polarization recording and different kinds of reading in Fulgide film. (a) Parallel linearly polarization recording. (b) Parallel circularly polarization recording. (c) Orthogonal linearly polarization recording. (d) Orthogonal circularly polarization recording.
(a)(b)(c)(d)The diffraction efficiency kinetics curves comparison of different kinds of polarization recording holograms in Fulgide/PMMA film written by Gaussian beams. (a) Experimentally measured results. (b) Theoretically calculated results.
(a)(b)


## 2.1. Material

The fulgide material studied here is 3-indoly-benzylfulgimide, which was synthesized by the Stobbe condensation routine [11]. The target compound of 3 mg was dissolved in a 0.l mL 10% (by weight) PMMA-cyclohexanone solution. Then, the solution was coated on a 1-mm thick K9 glass plane with a spin coater and dried in air. The thickness of the film is about 10 μm. The photochromic or photoanisotropic properties of fulgides are due to a reversible photochromic (photoisomerization) reaction that occurs between one of the colorless E-form (bleached state) and the C-form (colored state). These are the two spectrally separated photochromic forms, whose molecular formulas are shown in Figure 1.Figure 1
Molecular formula of the Indolyfulgimide and the photochromic reaction. Left,E-form; right, C-form.When the colored state is irradiated by a linearly polarized 650 nm laser, the film returns to the bleached state and photo-induced anisotropy is produced during this process. The mechanism of photo-induced anisotropy in the fulgide film is the following [10]: anisotropic absorbing molecules of fulgide are immobilized randomly in the PMMA polymeric matrix, which shows isotropic characteristic at the initial state; when the sample is irradiated by linearly polarized light photoselection of molecules take place. Molecules with long axes parallel to the exiting light polarization direction absorb the light strongly and turn to the other form very quickly, whereas those with long axes perpendicularly orientated to exiting light polarization have low absorption and stay at the initial form. As a result, special orientation of two form molecules is induced and the sample shows optical anisotropy properties macroscopically. The angular distributions of molecules in anisotropy inducing progress are shown in Figure 2. So, under the irradiation of circularly polarized light, the sample shows also optical isotropy.Figure 2
Angular distributions of molecules in anisotropy inducing progress.


### 2.1.1. Spectra of Photochromic and Photoanisotropy Properties of the Sample

The absorption of theC-form (AC(λ)) and E-form film (AE(λ)) were measured using a UV-VIS-IR spectrophotometer (UV-3101PC, Shimadzu Inc., Japan), which were shown in Figure 3(a). And the measurement of the photo-induced dichroism is performed by measuring the transmission spectra of the film for testing light polarized parallel (T//(λ)) and perpendicular (T⊥(λ)) to the polarization direction of the exciting beam after the C-form film is excited by the linearly polarized 650 nm laser (shown in Figure 4(a)).Spectra of photochromic properties of Indolyfulgimide/PMMA film. (a) Absorption spectra of two forms. (b) Absorption difference spectrum and the corresponding refractive index changing spectrum.
(a)(b)Spectra of photoanisotropic properties of Indolyfulgimide/PMMA film excited by linearly polarized light. (a) Transmission spectra on directions parallel and perpendicular to exciting beam polarization. (b) Dichroism and birefringence spectra.
(a)(b)Figure 5
Schematic of the experimental setup for measuring transmission kinetics. PBS, polarization beam splitter;M, mirror; A1~A2, continuously adjustable attenuators; S, Shutter; He-Ne: 633 nm Helium-Neon laser; D, power meter; IW, writing beam; IT, testing beam.From Figures3(a) and 4(a), the photo-induced absorption changing spectrum ΔA(λ)=AE(λ)-AC(λ) and the photo-induced dichroism spectrum ΔAD(λ)=A⊥(λ)-A//(λ)=lg(T//(λ)/T⊥(λ)) were obtained, which are shown as solid lines in Figures 3(b) and 4(b), respectively. Assuming that ΔA and ΔAD are zero outside of the band 300~800 nm, the photo-induced refractive index changing spectrum Δn(λ)=nE(λ)-nC(λ) and the photo-induced birefringence spectrum ΔnB(λ)=n⊥(λ)-n//(λ) can be calculated according to the Kramers-Kronig relation [12](1)Δn(λ)=ln⁡102π2dp.v.∫0∞ΔA(λ′)1-λ′2/λ2dλ′,ΔnB(λ)=ln⁡102π2dp.v.∫0∞ΔAD(λ′)1-λ′2/λ2dλ′,
where nE, nC, n//, and n⊥ are the refractive indexes of E-form, C-form, and of the film excited by linearly polarized light along the photo-induced extraordinary and ordinary axes, respectively, d is the thickness of the film and p.v. denotes the Cauchy principal value of the integral. The calculated Δn, ΔnB are plotted as dot lines in Figures 3(b) and 4(b), respectively. Kramers-Kronig relation can be satisfied during all the photochromic reaction progress, so at one wavelength λ′, Δn(λ′) is proportional to ΔA(λ′) and ΔnB(λ′) is proportional to ΔAD(λ′) at different exciting time. From Figures 3(b) and 4(b), it can be seen that at 633 nm in this sample, Δn(633 nm)/ΔA(633 nm) = 0.00994 and ΔnB(633 nm)/ΔAD(633 nm) = 0.006115.


### 2.1.2. Dynamics of Photochromic and Photoanisotropy Properties of the Sample

The transmission growing up kinetics of the sample at 633 nm were measured on the parallel and perpendicular directions to exciting beam polarization, when theC-form sample was being excited with 314 mW/cm2 and 157 mW/cm2 intensity linearly polarized 633 nm He-Ne lasers (IW), respectively, and an 1 mW/cm2 633 nm laser beam is used as the testing beam (IT), the optical setup and the results are shown in Figures 5 and 6(a). From Figure 6(a), it can be seen that the photochromic reaction of fulgide material is an optical cumulating progress, which just depending on the Exposure, so it is enough to consider just one exciting beam intensity condition for the analysis.Photoanisotropic transmission curves of Indolyfulgimide/PMMA film on the directions parallel or perpendicular to exciting beam polarization depending on the exposure or erasing time. (a) Experiment curves measured at different exiting beam intensity. (b) Simulation of experimental results. (c) Calculated curves of uniformity light.
(a)(b)(c)For the analysis, we consider a fully bistable Fulgide system, neglecting side reactions likeE-Z isomerization or aging effects. Hence, the photochemical reaction should follow the particle number balance equations
(2)Ċ=-kCC+kEE,Ė=-kEE+kCC,F=C+E
and the total concentration of fulgides molecules F is constant where E denotes the concentration of the E-isomers and C the one of the C-isomers; dots denote derivatives with respect to time t and the photoreaction rates for the annihilation of the isomers C and E are given by (3)kC,E=∫γC,E(λA)iλAdλA.
Here λA is the actinic irradiation wavelengths, iλA is the spectral density of the light intensity, and γC(λA) and γE(λA) are the photochemical reaction constants of the C and E isomers at λA, respectively, which related to the absorption cross sections σC,E via γC,E(λ)=ϕC,E(λ)·σC,E(λ)/hν, where hν is the photon energy and ϕC,E are quantum efficiencies of the photochemical reactions.The decrease ofiλ with z, described by Lambert-Beer’s law(4)diλdz=-α(z,t,λ)iλ,
where α(z,t,λ) is the local absorption coefficient in the film, which equals(5)α(λ,z,t)=σC(λ)⋅C(z,t)+σE(λ)⋅E(z,t)+αM(λ,z),
where αM is the absorption of the matrix (assumed here to be photochemically inert).Based on (2)~(5), using numerical calculation method, the experimental curves were simulated (shown as the dash lines in Figure 6(b)), and the best fitting values γ//=0.00289 cm2/mJ, γ⊥=0.0012 cm2/mJ were obtained. Because of the anisotropy of the molecules σC,E, the α and γ values of the sample are variable with the directions at same location, here just the total average values of that of all the molecules on two special directions are obtained, which are enough for calculation of the DE kinetics, and the detail process of it will not been analyzed here. In the numerical calculations, the intensity Gaussian beam distribution of the He-Ne laser beam also has been considered, which was divided as many homogenous intensity circles. Then we can calculate the photoanisotropic transmission curves of uniformity light, which is shown in the Figure 6(c).


## 2.1.1. Spectra of Photochromic and Photoanisotropy Properties of the Sample

The absorption of theC-form (AC(λ)) and E-form film (AE(λ)) were measured using a UV-VIS-IR spectrophotometer (UV-3101PC, Shimadzu Inc., Japan), which were shown in Figure 3(a). And the measurement of the photo-induced dichroism is performed by measuring the transmission spectra of the film for testing light polarized parallel (T//(λ)) and perpendicular (T⊥(λ)) to the polarization direction of the exciting beam after the C-form film is excited by the linearly polarized 650 nm laser (shown in Figure 4(a)).Spectra of photochromic properties of Indolyfulgimide/PMMA film. (a) Absorption spectra of two forms. (b) Absorption difference spectrum and the corresponding refractive index changing spectrum.
(a)(b)Spectra of photoanisotropic properties of Indolyfulgimide/PMMA film excited by linearly polarized light. (a) Transmission spectra on directions parallel and perpendicular to exciting beam polarization. (b) Dichroism and birefringence spectra.
(a)(b)Figure 5
Schematic of the experimental setup for measuring transmission kinetics. PBS, polarization beam splitter;M, mirror; A1~A2, continuously adjustable attenuators; S, Shutter; He-Ne: 633 nm Helium-Neon laser; D, power meter; IW, writing beam; IT, testing beam.From Figures3(a) and 4(a), the photo-induced absorption changing spectrum ΔA(λ)=AE(λ)-AC(λ) and the photo-induced dichroism spectrum ΔAD(λ)=A⊥(λ)-A//(λ)=lg(T//(λ)/T⊥(λ)) were obtained, which are shown as solid lines in Figures 3(b) and 4(b), respectively. Assuming that ΔA and ΔAD are zero outside of the band 300~800 nm, the photo-induced refractive index changing spectrum Δn(λ)=nE(λ)-nC(λ) and the photo-induced birefringence spectrum ΔnB(λ)=n⊥(λ)-n//(λ) can be calculated according to the Kramers-Kronig relation [12](1)Δn(λ)=ln⁡102π2dp.v.∫0∞ΔA(λ′)1-λ′2/λ2dλ′,ΔnB(λ)=ln⁡102π2dp.v.∫0∞ΔAD(λ′)1-λ′2/λ2dλ′,
where nE, nC, n//, and n⊥ are the refractive indexes of E-form, C-form, and of the film excited by linearly polarized light along the photo-induced extraordinary and ordinary axes, respectively, d is the thickness of the film and p.v. denotes the Cauchy principal value of the integral. The calculated Δn, ΔnB are plotted as dot lines in Figures 3(b) and 4(b), respectively. Kramers-Kronig relation can be satisfied during all the photochromic reaction progress, so at one wavelength λ′, Δn(λ′) is proportional to ΔA(λ′) and ΔnB(λ′) is proportional to ΔAD(λ′) at different exciting time. From Figures 3(b) and 4(b), it can be seen that at 633 nm in this sample, Δn(633 nm)/ΔA(633 nm) = 0.00994 and ΔnB(633 nm)/ΔAD(633 nm) = 0.006115.


## 2.1.2. Dynamics of Photochromic and Photoanisotropy Properties of the Sample

The transmission growing up kinetics of the sample at 633 nm were measured on the parallel and perpendicular directions to exciting beam polarization, when theC-form sample was being excited with 314 mW/cm2 and 157 mW/cm2 intensity linearly polarized 633 nm He-Ne lasers (IW), respectively, and an 1 mW/cm2 633 nm laser beam is used as the testing beam (IT), the optical setup and the results are shown in Figures 5 and 6(a). From Figure 6(a), it can be seen that the photochromic reaction of fulgide material is an optical cumulating progress, which just depending on the Exposure, so it is enough to consider just one exciting beam intensity condition for the analysis.Photoanisotropic transmission curves of Indolyfulgimide/PMMA film on the directions parallel or perpendicular to exciting beam polarization depending on the exposure or erasing time. (a) Experiment curves measured at different exiting beam intensity. (b) Simulation of experimental results. (c) Calculated curves of uniformity light.
(a)(b)(c)For the analysis, we consider a fully bistable Fulgide system, neglecting side reactions likeE-Z isomerization or aging effects. Hence, the photochemical reaction should follow the particle number balance equations
(2)Ċ=-kCC+kEE,Ė=-kEE+kCC,F=C+E
and the total concentration of fulgides molecules F is constant where E denotes the concentration of the E-isomers and C the one of the C-isomers; dots denote derivatives with respect to time t and the photoreaction rates for the annihilation of the isomers C and E are given by (3)kC,E=∫γC,E(λA)iλAdλA.
Here λA is the actinic irradiation wavelengths, iλA is the spectral density of the light intensity, and γC(λA) and γE(λA) are the photochemical reaction constants of the C and E isomers at λA, respectively, which related to the absorption cross sections σC,E via γC,E(λ)=ϕC,E(λ)·σC,E(λ)/hν, where hν is the photon energy and ϕC,E are quantum efficiencies of the photochemical reactions.The decrease ofiλ with z, described by Lambert-Beer’s law(4)diλdz=-α(z,t,λ)iλ,
where α(z,t,λ) is the local absorption coefficient in the film, which equals(5)α(λ,z,t)=σC(λ)⋅C(z,t)+σE(λ)⋅E(z,t)+αM(λ,z),
where αM is the absorption of the matrix (assumed here to be photochemically inert).Based on (2)~(5), using numerical calculation method, the experimental curves were simulated (shown as the dash lines in Figure 6(b)), and the best fitting values γ//=0.00289 cm2/mJ, γ⊥=0.0012 cm2/mJ were obtained. Because of the anisotropy of the molecules σC,E, the α and γ values of the sample are variable with the directions at same location, here just the total average values of that of all the molecules on two special directions are obtained, which are enough for calculation of the DE kinetics, and the detail process of it will not been analyzed here. In the numerical calculations, the intensity Gaussian beam distribution of the He-Ne laser beam also has been considered, which was divided as many homogenous intensity circles. Then we can calculate the photoanisotropic transmission curves of uniformity light, which is shown in the Figure 6(c).


## 2.2. Experiment Measurement of Polarization Hologram Diffraction Kinetics

### 2.2.1. Measurement Setup

The system configuration for measuring the real-time hologram first-order diffraction kinetics of the Fulgide film is schematically illustrated in Figure7. A He-Ne laser (Melles Griot Inc., USA, 25-LHP-928, 632.8 nm, 35 mW, vertical linear polarized) is used to generate recording beams (object beam IO and reference beam IR) and readout beam (reconstruction beam IC), and a laser of diode LD (Power Technology Inc., USA, IQ2A18, 405 nm, 10 mW, vertical linear polarized) is used as the erasing light (IE) source. The He-Ne laser beam splitted into the IO, IR, and IC after beam splitter BS1 and polarization beam splitter PBS, in which IR and IC are phase conjugated (counterpropagated) beams. The diffracted light ID of IC, diffracted by the dynamic holographic grating established by the interference between the IO and IR, will be phase conjugated with the IO, whose power was real time detected by a digital power meter “D” (United Detector Technology company, USA, 11A Photometer/Radiometer, 254~1100 nm Imax = 10 mW, resolution is 0.01 nW) and a digital oscilloscope “O’’ (Tektronix company, USA, TDS3032, 300 MHz, 2.5 GS/s, 1 mV) after reflected by the BS2 (R47%). The IO and IR are symmetrically incident on the sample (Fulgide film), whose intersection angle 2θ=16.5°, so the recorded grating is a nonincline grating. Shutter S1 and S2 controls the exposure time of red and purple beams. The continuously adjustable attenuators A1~A3 are used to adjust the intensities of the waves, in this experiment IO = IR = 78.6 mW/cm2 and IC = 0.786 mW/cm2 (i.e., IO : IR : IC = 100 : 100 : 1). This insures that the subreflection gratings formed by IO and IC as well as IR and IC can be ignored. The quarter-wave plates Q1~Q4 and the polarizer P are used to change the polarization states of the waves; here four different polarization recording: parallel linearly polarization recording, parallel circularly polarization recording, orthogonal linearly polarization recording, and orthogonal circularly polarization recording were studied, every one was constructed by horizontal, vertical, left circular, and right circular polarized four kinds of lights like shown in Table 1.Table 1
DWPS and DE of different kinds of polarization holograms.
Polarisation of object lightPolarisation of reference lightPolarisation of reconstruction lightPolarization states of diffracted lights+1-order diffraction efficiencies+1−1Amplitude hologramηA,+1Phase hologramηP,+1(τ1om2)2(τ0om·J1(Ψ1om))2(τ1em2)2(τ0em·J1(Ψ1em))2(τ1om)2+(τ1em)28τ0em2·J1(Ψ1em)2+τ0om2·J1(Ψ1om)22(τ1om)2+(τ1em)28τ0em2·J1(Ψ1em)2+τ0om2·J1(Ψ1om)22(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo)22(τ0eo·sinΨ1eo)22(τ1eo)22(τ0eo·sinΨ1eo)22(τ1eo)2(τ0eo·sinΨ1eo)200Figure 7
Schematic of the experimental setup for measuring real-time polarization hologram diffraction kinetics. BS1 and BS2, beam splitters; PBS, polarization beam splitter; M1~M4, mirrors; A1~A3, continuously adjustable attenuators; P, polarizer; Q1~Q4, quarter-wave plates; S1 and S2, Shutters; D, digital power meter; O, digital oscilloscope; LD, 405 nm laser diode; He-Ne: 633 nm Helium-Neon laser; IO, object beam; IR, reference beam; IC, reconstruction beam; ID, diffracted beam; IE, erasing violet light.


### 2.2.2. Measurement Results

In four kinds of polarization recording, and different polarization reading, the diffracted wave polarization states (DWPS) obtained in the experiments are shown in Table1 and the measured kinetic first-order diffraction efficiency (DE) curves η+1~t are shown in Figure 8. From them, the curves of the conditions when IC has same polarization state with IR were compared in Figure 9(a). The theoretical analysis of the results will be given in Section 3. It can be seen that there exists an optimal exposure about 2×78.6 mW/cm2×3.75s≈590 mJ/cm2.The diffraction efficiency kinetics curves comparison of different kinds of polarization recording and different kinds of reading in Fulgide film. (a) Parallel linearly polarization recording. (b) Parallel circularly polarization recording. (c) Orthogonal linearly polarization recording. (d) Orthogonal circularly polarization recording.
(a)(b)(c)(d)The diffraction efficiency kinetics curves comparison of different kinds of polarization recording holograms in Fulgide/PMMA film written by Gaussian beams. (a) Experimentally measured results. (b) Theoretically calculated results.
(a)(b)


## 2.2.1. Measurement Setup

The system configuration for measuring the real-time hologram first-order diffraction kinetics of the Fulgide film is schematically illustrated in Figure7. A He-Ne laser (Melles Griot Inc., USA, 25-LHP-928, 632.8 nm, 35 mW, vertical linear polarized) is used to generate recording beams (object beam IO and reference beam IR) and readout beam (reconstruction beam IC), and a laser of diode LD (Power Technology Inc., USA, IQ2A18, 405 nm, 10 mW, vertical linear polarized) is used as the erasing light (IE) source. The He-Ne laser beam splitted into the IO, IR, and IC after beam splitter BS1 and polarization beam splitter PBS, in which IR and IC are phase conjugated (counterpropagated) beams. The diffracted light ID of IC, diffracted by the dynamic holographic grating established by the interference between the IO and IR, will be phase conjugated with the IO, whose power was real time detected by a digital power meter “D” (United Detector Technology company, USA, 11A Photometer/Radiometer, 254~1100 nm Imax = 10 mW, resolution is 0.01 nW) and a digital oscilloscope “O’’ (Tektronix company, USA, TDS3032, 300 MHz, 2.5 GS/s, 1 mV) after reflected by the BS2 (R47%). The IO and IR are symmetrically incident on the sample (Fulgide film), whose intersection angle 2θ=16.5°, so the recorded grating is a nonincline grating. Shutter S1 and S2 controls the exposure time of red and purple beams. The continuously adjustable attenuators A1~A3 are used to adjust the intensities of the waves, in this experiment IO = IR = 78.6 mW/cm2 and IC = 0.786 mW/cm2 (i.e., IO : IR : IC = 100 : 100 : 1). This insures that the subreflection gratings formed by IO and IC as well as IR and IC can be ignored. The quarter-wave plates Q1~Q4 and the polarizer P are used to change the polarization states of the waves; here four different polarization recording: parallel linearly polarization recording, parallel circularly polarization recording, orthogonal linearly polarization recording, and orthogonal circularly polarization recording were studied, every one was constructed by horizontal, vertical, left circular, and right circular polarized four kinds of lights like shown in Table 1.Table 1
DWPS and DE of different kinds of polarization holograms.
Polarisation of object lightPolarisation of reference lightPolarisation of reconstruction lightPolarization states of diffracted lights+1-order diffraction efficiencies+1−1Amplitude hologramηA,+1Phase hologramηP,+1(τ1om2)2(τ0om·J1(Ψ1om))2(τ1em2)2(τ0em·J1(Ψ1em))2(τ1om)2+(τ1em)28τ0em2·J1(Ψ1em)2+τ0om2·J1(Ψ1om)22(τ1om)2+(τ1em)28τ0em2·J1(Ψ1em)2+τ0om2·J1(Ψ1om)22(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1Mm2)2(τ0Mm·J1(Ψ1Mm))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo2)2(τ0eo·J1(Ψ1eo))2(τ1eo)22(τ0eo·sinΨ1eo)22(τ1eo)22(τ0eo·sinΨ1eo)22(τ1eo)2(τ0eo·sinΨ1eo)200Figure 7
Schematic of the experimental setup for measuring real-time polarization hologram diffraction kinetics. BS1 and BS2, beam splitters; PBS, polarization beam splitter; M1~M4, mirrors; A1~A3, continuously adjustable attenuators; P, polarizer; Q1~Q4, quarter-wave plates; S1 and S2, Shutters; D, digital power meter; O, digital oscilloscope; LD, 405 nm laser diode; He-Ne: 633 nm Helium-Neon laser; IO, object beam; IR, reference beam; IC, reconstruction beam; ID, diffracted beam; IE, erasing violet light.


## 2.2.2. Measurement Results

In four kinds of polarization recording, and different polarization reading, the diffracted wave polarization states (DWPS) obtained in the experiments are shown in Table1 and the measured kinetic first-order diffraction efficiency (DE) curves η+1~t are shown in Figure 8. From them, the curves of the conditions when IC has same polarization state with IR were compared in Figure 9(a). The theoretical analysis of the results will be given in Section 3. It can be seen that there exists an optimal exposure about 2×78.6 mW/cm2×3.75s≈590 mJ/cm2.The diffraction efficiency kinetics curves comparison of different kinds of polarization recording and different kinds of reading in Fulgide film. (a) Parallel linearly polarization recording. (b) Parallel circularly polarization recording. (c) Orthogonal linearly polarization recording. (d) Orthogonal circularly polarization recording.
(a)(b)(c)(d)The diffraction efficiency kinetics curves comparison of different kinds of polarization recording holograms in Fulgide/PMMA film written by Gaussian beams. (a) Experimentally measured results. (b) Theoretically calculated results.
(a)(b)


## 3. Theoretical Analysis

### 3.1. Background

For isotropy recording materials, only when the object lightO⃗ and reference light R⃗ have components with same polarization state, there the intensity grating exists and the holograms can be recorded, called ordinary holograph. But for photo-anisotropy materials, even if the O⃗ and R⃗ have orthogonal polarization states, the holograms also can be recorded, because that at this time although the intensity of the superposed light is a constant, but its polarization state changes with the phase difference between O⃗ and R⃗. Photo-anisotropy materials can record the polarization state of exciting beam, so the holographic gratings can be recorded, called polarization holography, in which not only the intensity and phase signals of O⃗ can be stored (the photo-anisotropy is depending on the exposure), but also its polarization state can be stored.The holographic recording progress is a reaction progress between the exciting interference light and the material. So, first the properties of the exciting interference light will be analyzed and then the reaction of the sample with the light will be studied.


#### 3.1.1. The Properties of Exiting Interference Pattern of Object Light and Reference Light

Choosexozplane as incidence plane, the O⃗ incidents into the recording plane xoy with a very small incident angle θ, and the R⃗ has the same incident angle (shown in Figure 10(a)), so the phase components of O⃗ and R⃗ on the x-axis are φO=k·x·sinθ and φR=-k·x·sinθ, respectively, and the phase difference of them is(6)Δφ=φO-φR=2⋅k⋅x⋅sinθ,(k=2πλ).Sketch map (a) of the holographic recording setup and (b) of the superposing of two orthogonal circularly polarized lights.
(a)(b)(A) Parallel Polarization State
It is known that in the interference field of two coherent wavesO⃗ and R⃗ with the same polarization, the polarization state is constant and same with before E⃗1=O⃗1+R⃗1=O⃗1(1+const·e-iΔφ), but the intensity is periodically modulated corresponding to Δφ as I1=O2+R2+2·O·R·cosΔφ. At the condition that the amplitudes of O⃗ and R⃗ are same, that is,  O=R=A, the interference field distribution of two special examples of it are drawn in Figures 11(a) and 11(c), including the parallel linearly polarization and parallel circularly polarization.The distributions of the interference fields in four kinds of polarization holography storage and the corresponding molecular distributions and orientations.
(a)
Parallel linearly(b)
Orthogonal linearly(c)
Parallel circularly(d)
Orthogonal circularly(B) Orthogonal Polarization State
In the interference field of two coherent waves with orthogonal polarization, that is,R⃗2·O⃗2*=O⃗2·R⃗2*=0, the intensity is constant I2=O2+R2, but the polarization state is periodically modulated; here two special examples of which are analyzed, including the orthogonal linearly polarization and orthogonal circularly polarization.(B1) Orthogonal Linearly Polarization
SupposeO⃗ and R⃗ are, respectively, horizontal and vertical linearly polarized, so:
(7)E⃗3=O⃗3+R⃗3=O⋅[10]⋅eiφo+R⋅[01]⋅eiφR=eiφO[OR⋅e-iΔφ]→O=R=AA⋅eiφO⋅[1e-iΔφ].
It shows that the superposed light of two orthogonal linearly polarized lights is an elliptical polarized light, whose azimuth α3 is
(8)α3=12⋅arctan⁡[(2ORO2-R2)cos⁡Δφ]=O=R=A{π42kπ-π2<Δφ<2kπ+π2,-π42kπ-π2<Δφ<2kπ+3π2,k∈Z
and whose ellipticity ε is
(9)ε3=arctan(L2L1)=12arcsin(sin2β⋅sinΔφ)→O=R=Aarctan(L2L1)=Δφ2,
where L1, L2 indicate the long and short axes, respectively and β  indicates the amplitude angle (tanβ=R/O). It can be seen that the azimuth and ellipticity of interference field are periodically changing along the Δφ with period 2π. The interference field distribution at the condition O=R=A is drawn in Figure 11(b).(B2) Orthogonal Circularly Polarization
SupposeO⃗ and R⃗ are, respectively, right and left circularly polarized, so:
(10)E⃗4=O⃗4+R⃗4=O⋅[1212i]eiφo+R⋅[12-12i]⋅eiφR=12[O+R⋅e-iΔφ(O-R⋅e-iΔφ)i]⋅eiφO.
It is clear that the superposed light of two orthogonal circularly polarized lights is also an elliptical polarized light, whose ellipticity will not change with the Δφ, that is, the long and short axes are always (O+R) and (O-R), respectively. When O=R=A, the short axis is zero, that is, the linearly polarized light. The azimuth of the light is α4=-Δφ/2. Figure 10(b) is a single sketch map showing the superposing of two orthogonal circularly polarized lights. The interference field distribution at the condition O=R=A is drawn in Figure 11(d).


#### 3.1.2. Analysis of the Properties of Different Kinds of Polarization Holograms and Their DWPS and DE

Under the irradiation of the interference field, the photochromic reaction will be induced in the fulgide sample, so the space modulation of the distribution ofE-form and C-form molecules will be caused by the intensity modulation of optical field; photo-anisotropy will be induced during the photochromic reaction under the irradiation of noncircularly polarized light, so the space modulation of the orientations (tropism) of E-form and C-form molecules will be caused by the polarization state modulation of optical field.Here, we just consider four kinds of polarization holographic storage, including the parallel linearly, parallel circularly, orthogonal linearly, and orthogonal circularly polarized holograms. Figure11 is a single sketch map showing the space modulation of the distributions and orientations (optical axis) of E-form and C-form molecules in these four kinds of polarization holographic storage under the linearity recording condition when O=R=A. The arrows and ellipses on the horizontal line indicate the periodical distributions of intensity magnitudes and polarization states of the recording field corresponding to the different phase differences Δφ along the x-axis. The rectangle frames under them indicate the distribution and orientation of the molecules under the corresponding light irradiations, where the black and white colors indicate the C-form and E-form molecules, respectively, the arrows indicate the direction of induced optical axes, and the star flower (*) means the sample is isotropy at the place.From Figure11, it can be seen that in the parallel circularly polarized hologram just ordinary hologram exits, and in the orthogonal polarized condition, just polarization hologram exits, but in the parallel linearly polarized holography, the ordinary hologram and polarization hologram will be recorded together.The underside, the Jones matrixes of four kinds of polarization hologram, and their DWPS and DE are analyzed under linearly recording condition. The DE are calculated using the diffraction theory of two-dimensional grating, because the thickness of the fulgide film is very small (10μm); the recorded hologram can be considered as plane hologram.Suppose the Jones coordinate is(x,y), so the Jones matrix of the hologram is(11)T(x,y)=[τxx⋅eiΨxxτxy⋅eiΨxyτyx⋅eiΨyxτyy⋅eiΨyy],
where τ and Ψ indicate the light amplitude transmissions and light phase transmission function of the sample, and when read by the reconstruction light C⃗, the diffracted light D⃗ of the hologram will be(12)D⇀=T(x,y)⋅C⃗=[τxx⋅eiΨxxτxy⋅eiΨxyτyx⋅eiΨyxτyy⋅eiΨyy]⋅[CxCy]=[τxx⋅eiΨxx⋅Cx+τxy⋅eiΨxy⋅Cyτyx⋅eiΨyx⋅Cx+τyy⋅eiΨyy⋅Cy](whereC⃗=[CxCy]=[Cx0eiφxCy0eiφy]).For simplify, the condition whenO=R=1 is chosen and analysed here. And the 45° contrarotated coordinate system is defined as (X,Y), shown in Figure 12(a). In Figure 12(b), the photoanisotropic amplitude transmission curves are shown. The (τe-τo) indicate the photo-induced anisotropy of the sample under the irradiation of linearly polarized light at some exposure, (ne-no) indicate the corresponding birefringence, τM=(τe+τo)/2 indicate the amplitude transmission of the sample for nonpolarized light or circularly polarized light at this time (at the area of light strips in the interference field), τm indicates the amplitude transmission of the sample before the illumination of light (at the area of dark strips in the interference field, isotropy), τE and τC indicate the amplitude transmission of the E-form and C-form sample, respectively, nM, nm, nE, and nC indicate the corresponding refractive index of the sample, Ψi=k0·ni·d(i=e,o,M,m,E,C), and it is defined that τ0ij=(τi+τj)/2, τ1ij=(τi-τj)/2, Ψ0ij=(Ψi+Ψj)/2, Ψ1ij=(Ψi-Ψj)/2(i,j=e,o,M,m,E,C). In the Figure 12(b), the linearity recording area is given, that is, the linearity recording area of the increasing curve of τe, where it is insured that the light amplitude transmission changing of the sample has the direct ratio with the exposure no matter using what kind of polarization holographic recording and reading.(a) The definition of coordinate system. (b) The photo-induced dichroism curve of the sample depending on the exposure.
(a)(b)(1) Parallel Linearly Polarization Recording
From Figure11, it can be seen that the photo-induced optical axis is along the y-axis, so
(13)T1=[τx⋅eiΨX00τy⋅eiΨY].
Under the linearity recording condition,
(14)τx=τ0om+τ1om⋅cos⁡Δφ,τy=τ0em+τ1em⋅cos⁡Δφ,Ψx=Ψ0om+Ψ1omcos⁡Δφ,Ψy=Ψ0em+Ψ1emcos⁡Δφ.
For the amplitude recording material, Ψx=Ψy=Ψ0. If the hologram is reconstructed with C⃗, which has same incident angle with R⃗, using (12) the D⃗ can be calculated as
(15)D⃗A1=TA1⋅C⇀=eiΨ0⋅[τ0om+τ1omcos⁡Δφ00τ0em+τ1em⋅cos⁡Δφ]⋅[CxCy]⋅e-i(Δφ/2)=eiΨ0⋅[τ0om⋅Cxτ0em⋅Cy]⋅e-i(Δφ/2)+eiΨ0⋅12[τ1om⋅Cxτ1em⋅Cy]⋅e-i(3Δφ/2)+eiΨ0⋅12[τ1om⋅Cxτ1em⋅Cy]⋅ei(Δφ/2).
So, the first-order DE of it would be
(16)ηA,±1=14⋅(τ1om⋅|Cx|)2+(τ1em⋅|Cy|)2|Cx|2+|Cy|2=14⋅(τ1om2⋅|Cx|2|Cx|2+|Cy|2+τ1em2⋅|Cy|2|Cx|2+|Cy|2).
IfR⃗ (vertical linearly polarized) itself is used as C⃗, the η±1=(τ1em/2)2 and the DWPS are the same as C⃗ (shown in Table 1). If a horizontal polarized light is used as C⃗, the η±1=(τ1om/2)2 and the DWPS are the same as C⃗ (shown in Table 1). But at other stations the diffracted lights are the superposed light of the diffracted lights of vertical and horizontal polarized light components of C⃗; when reconstruct with circularly polarized light, the diffracted light will be ellipse polarized, whose long axes are along the vertical direction, and DE is the average value of that of horizontal and vertical polarized lights. For the phase recording material, τx=τy=τ0, using the same method the DWPS and DE of it can be deduced, in which four typical ones are written in Table 1.
At the nonlinearly recording condition, the grating of the amplitude and refractive index will no longer be a single sine function distributing, but will be a superposition of a lot of different sine function distribution(17)τi=τ0i+∑n=1∞τni⋅cos⁡(n⋅Δφ),(i=x,y),ψi=ψ0i+∑n=1∞ψni⋅cos⁡(n⋅Δφ),(i=x,y).
So when reconstruct with horizontal or vertical polarized light, the polarization states of the diffracted lights will not change, and the ±1-order diffraction efficiencies of the amplitude grating and phase grating will be, respectively, ηA,±1=τ1i2/4 and ηP,±1=(τ0i·J1(Ψ1i))2(i=x,y). But when reconstruct with circularly polarized light, the diffracted lights are the superposed light of the diffracted lights of horizontal and vertical polarized light component of the reconstruction beam, so it will be ellipse polarized lights, whose long axes are along the vertical direction, and the diffraction efficiencies are the their average value.(2) Parallel Circularly Polarization Recording
From Figure11, it can be seen that the sample is isotropy at this condition, so
(18)T2=[τ⋅eiΨ00τ⋅eiΨ].
Under the linearity recording condition,(19)τ=τ0Mm+τ1Mm⋅cos⁡Δφ,Ψ=Ψ0Mm+Ψ1Mmcos⁡Δφ.
Using (12), the first-order DWPS and DE can be calculated when it is read by bragg angle entered arbitrary polarized light, in which the four typical ones were written in Table 1. At the nonlinearly recording condition, the same as the parallel linearly polarization holography, the grating will no longer be a single sine function distributing. But the sample is isotropy, that is, Tx=Ty. So the polarization state of the diffracted lights are always the same as the C⃗, and the diffraction efficiency is invariable with the polarization state of the C⃗, ηA,±1=τ12/4, ηP,±1=(τ0·J1(Ψ1))2.(3) Orthogonal Linearly Polarization Recording
For the orthogonal linearly polarization hologram, the superposed light is elliptically polarized light, whose long axes haveα=±π/4 angle between the x-axis, that is, along the X, Y axes. So first the Jones vector of the interference field Equation (7) is turned into X,Y coordinate
(20)E⃗3|O=R=A=A⋅eiφO⋅[cos⁡αsinα-sinαcos⁡α]⋅[1e-iΔφ]=α=π/42A⋅eiφO⋅[cos⁡Δφ2i⋅sinΔφ2].
The optical anisotropy induced by elliptically polarized light should have direct ratio with the intensity difference betweenX,Y two directions I3X-I3Y=2A2cosΔφ. So, the amplitude transmission Jones matrix of the holograph can be written as
(21)T3(X,Y)=[τX⋅eiΨX00τY⋅eiΨY],
where
(22)τX=τ0eo+τ1eo⋅cos⁡Δφ,τY=τ0eo-τ1eo⋅cos⁡Δφ,ΨX=Ψ0eo+Ψ1eo⋅cos⁡Δφ,ΨY=Ψ0eo-Ψ1eo⋅cos⁡Δφ.
And then the Jones matrix of the holograph is turned into usually usedx,y coordinate:
(23)T3(x,y)=[cos⁡α-sinαsinαcos⁡α]⋅[τX⋅eiΨX00τY⋅eiΨY]⋅[cos⁡αsinα-sinαcos⁡α]=α=π/412[τX⋅eiΨX+τY⋅eiΨYτX⋅eiΨX-τY⋅eiΨYτX⋅eiΨX-τY⋅eiΨYτX⋅eiΨX+τY⋅eiΨY].
Using (12), the first-order DWPS and DE can be calculated when it is read by bragg angle entered lights, in which the four typical ones were written in Table 1. It can be deduced that the x and y components of the ±1-order diffracted lights are reversed comparing to the C⃗, so their long axes are symmetry distributed on the X-axis with the long axis of C⃗, where four special conditions exit: When the C⃗ is vertical, horizontal, right circularly or left circularly polarized light, the polarization state of the diffracted light is orthogonal to that of C⃗. The polarization of the ±2 order diffracted lights are same with the reconstruction light. The ±1-order DE of amplitude and phase holograms are respectively (τ1eo/2)2 and (τ0eo·J1(Ψ1eo))2.
At the nonlinearly recording condition, (τe-τo) and (ne-no) will not have direct ration with the exposure, but the anisotropy distribution will not change, so the diffraction efficiency equations at the linearly recording condition also can be used.(4) Orthogonal Circularly Polarization Recording
From Section3.1.1, it can be seen that the superposed light field of two orthogonal circularly polarized lights (the same intensity) is a linearly polarized light, and the intensity will not change with Δφ, so every where in the sample its amplitude transmission matrix in the coordinate of the its main axes should be
(24)T4(e,o)=[τe⋅eiΨe00τo⋅eiΨo].
The optical axis of the superposed light is changing withΔφ, which has Δφ/2 angle with the X-axis to counter-clockwise direction, so the angle with the x-axis is α=Δφ/2+π/4. So, the Jones matrix of the recording material in the x,y coordinate will be(25)T4(x,y)=[τesin2Δφ2⋅eiΨe+τo⋅co⁡s2Δφ2⋅eiΨo-12⋅sinΔφ⋅(-τe⋅eiΨe+τo⋅eiΨo)-12⋅sinΔφ⋅(-τe⋅eiΨe+τo⋅eiΨo)τeco⁡s2Δφ2⋅eiΨe+τo⋅sin2Δφ2⋅eiΨo].Using (12), it can be deduced that, for the orthogonal circularly polarization holographic recording, there just exist the 0-order and ±1-order diffracted lights. The polarization state of the 0-order diffracted light is same with that of C⃗, and the polarization state of the +1 and −1-order diffracted lights are always right and left circularly polarized, respectively, which are same with that of O⃗ and R⃗. When the C⃗ is an arbitrary polarized light, it has the 0-order and ±1-order diffracted lights, whose left circularly polarized light component will diffract the +1-order light, and the right circularly polarized light component will diffract the −1-order light. The ±1-order DE of amplitude holograph are respectively ηA,+1=(τ1eo·Cl0/C0)2 and ηA,-1=(τ1eo·Cr0/C0)2, that of the phase holograph are, respectively ηP,+1=(τ0eo·sinΨ1eo·Cl0/C0)2 and ηP,-1=(τ0eo·sinΨ1eo·Cr0/C0)2, where the C0, Cl0, Cr0 are the amplitudes of C⃗ and of its left and right circularly polarized light components. In which, when the C⃗ is left circularly polarized light (same with R⃗), only the 0-order and +1 order diffracted lights exist, and the polarization state of the +1 order diffracted light is orthogonal to that of C⃗, and ηA,+1=(τ1eo)2, ηA,-1=0, ηP,+1=(τ0eo·sinΨ1eo)2 and ηP,-1=0. When the C⃗ is right circularly polarized light (orthogonal with R⃗), only the 0-order and −1 order diffracted lights exist, and the polarization state of the −1-order diffracted light is also orthogonal to that of C⃗, and ηA,+1=0, ηA,-1=(τ1eo)2, ηP,+1=0 and ηP,-1=(τ0·sinΨ1eo)2, which are written in Table 1. At the nonlinearly recording condition, the same as the orthogonal linearly polarization holographic recording, the anisotropy grating distribution will not change, so the diffraction efficiency equations at the linearly recording condition also can be used.


#### 3.1.3. DWPS and DE Formulas of Different Kinds of Hologram

In Table1, the DWPS and DE of parallel linearly, parallel circularly, orthogonal linearly and orthogonal circularly polarization holographs at the linearly, recording condition are summarized, when reconstructed with linearly polarized lights and circularly polarized lights. Usually the holograms are mixed hologram, whose DE approximately equals to the summation of that of the amplitude grating and phase grating: η1=ηa1+ηp1.It can be seen that the DWPS obtained in the experiments are the same as the theortically deduced ones. And in the orthogonal polarization holography, if reconstructed with the reference light, the polarization state of the diffracted light is orthogonal to that of the reconstruction light, which is very important to increase the SNR of the holographic storage. Because that the polarization state of the scattered noise light is almost the same as that of the reconstruction light, which can be filtered by using the polarizer and quarter-wave plate.


### 3.2. Theoretical Calculations of DE of Fulgide Film

In this section, the DE spectra and DE dynamic curves at 633 nm of different kinds of holograms recorded in the sample will be calculated from the properties of the sample written in Section2, by using the theory written in Section 3.1. In Section 2, it is given that for fulgide materials the photochromic and photo-induced anisotropy phenomenon are optical accumulating progresses, so the holographic recording progress is also an accumulating progress, which just depending on the exposure (average exposure E̅=2·I0·t), so it is enough to consider just one exciting beam intensity condition. For the simplicity, the object refractive ratio (ORR) is chosen as 1 : 1, that is, O=R=A=(I0)1/2.


#### 3.2.1. Theoretical Calculation of DE(633 nm) ~E̅ Curves on the Exposure of Uniformity Beams

(1) The Calculation of DE in the Parallel Polarization Recording Condition
For parallel polarization recording condition, the optical intensity distribution should be(26)I(Δφ)=O2+R2+2⋅O⋅R⋅cos⁡Δφ=2⋅I0⋅(1+cos⁡Δφ).
So at recording time t, the exposure (E) distribution on the sample is
(27)E(Δφ,t)=I(Δφ)⋅t=2⋅I0⋅(1+cos⁡Δφ)⋅t.
So, for different kinds of C⃗, the T(C⃗,Δφ,t)~Δφ of the parallel linearly and parallel circularly recorded holograms at different t in the sample can be obtained from T~E curves shown in Figure 6(c)(28)T(Δφ,t)=T(E(Δφ))={T//(2⋅I0⋅(1+cos⁡Δφ)⋅t)parallellinearlyrecording,O⃗//R⃗//C⃗T⊥(2⋅I0⋅(1+cos⁡Δφ)⋅t)parallellinearlyrecording,O⃗//R⃗⊥C⃗(T//+T⊥)(2⋅I0⋅(1+cos⁡Δφ)⋅t)2parallelcircularlyrecording.(a) The Calculation of DE of the Amplitude Gratings (ηA)
The amplitude transmission changing curves on the phase differenceτ~Δφ at t will be
(29)τ(Δφ,t)=[T(Δφ,t)]1/2.
Every τ~Δφ grating curves at different t can be spread into Thaler progression, that is, simulated with the formula
(30)τ=τ0+∑n=110τn⋅cos⁡(n⋅Δφ).
Such the different order amplitude transmission gratings’ amplitude values τn(n=0,1,…,10) at different t can be obtained. So the changing curves of τn(n=0,1,…,10) on the recording exposure τn~E̅ can be obtained, and then the different order DE changing curves of amplitude gratings on the recording exposure (ηA,±n~E̅) can be calculated as
(31)η0=τ02,ηA,±n=τn24,(n=1,2,…,10).(b) The Calculation of DE of the Phase Gratings (ηP)
From theT(t)~Δφ, the A(t)~Δφ curves can be calculated
(32)A(Δφ,t)=log⁡(1T(Δφ,t)).
And then every A~Δφ grating curves at different t can be spread into Thaler progression, that is, simulated with the underside formula
(33)A=A0+∑i=110Ai⋅cos⁡(i⋅Δφ).
So, the different order absorption gratings’ amplitude values Ai(i=0,1,…,10) for different kinds of R⃗ at different t can be obtained. From Section 2, it is known that, at one wavelength, between the ΔA and Δn exit direct ratio relationship. And then after multiplexed with responding ratio value, the different order phase gratings’ amplitude values ni(i=0,1,…,10) for different polarization R⃗ at different t can be got
(34)ni=consti⋅Ai.
So, the different order DE changing curves of phase gratings on recording exposure (ηA,±n~E̅) can be calculated. Here only the first-order DE is discussed, From Section 3.1, it is known that only the n1 will induce the first-order diffracted light, and ηP,±1=(τ0·J1(k·n1·d))2.(c) The Calculation of DE of the Mixed Gratings (ηtotal)
Theηtotal can be calculated by submitting the ηA and ηP: η+1=ηA,+1+ηP,+1.(d) The Calculation Results of DE of Fulgide Film in Parallel Polarization Recording
For example, in the parallel linearly polarization recording parallel polarization reading condition (C⃗//O⃗//R⃗), when I0=50 mW/cm2, from the T//~E curve shown in Figure 6(c), the T//(t)~Δφ of the holographic grating at different recording times t can be calculated, from which the τ//(t)~Δφ can be obtained, which is shown in Figure 13.
Everyτ//~Δφ curve was spread into Thaler progression, that is was simulated with (30), then the τn (n=0,1,…,10) on the direction of the exiting beam polarization at different t were obtained; for example, in the Figure 14 two typical amplitude transmission grating curves (sine distribution and negative triangle wave function) at different t and their simulated curves are given. So, the τn~E̅  (n=0,1,…,10) curves can be got, from which the ηA,±n~E̅(n=0,1,…,10) curves can be calculated, from which the first-order DE (ηA,±1) is chosen and shown in Figure 15 as dash line 2. It can be seen that even if the ORR = 1 : 1, because the nonlinear recording the DE will also drop down. The optimal exposure is about 1 J/cm2, and at the uniformity light recording condition, the maximum ηA,±1 is 2%. From the T//(t)~Δφ curves, the A//(t)~Δφ curves can be calculated. After every A//(t)~Δφ curve is spread into Thaler progression, the Ai(i=0,1,…,10) on the direction of the exiting beam polarization at different t can be obtained. In Section 2, it is given that Δn(633 nm)/ΔA(633 nm) = 0.00994. So after the Ai multiplexed with 0.00994, the ni,// (i=0,1,…,10) at different t can be got, from which the ηP,±1~E̅ can be calculated, which is shown in Figure 16. The DE of the mixed holograph ηtotal can be obtained by submitting that of the amplitude grating and phase grating, shown in Figure 16. It can be seen that at the uniformity light recording condition, the optimal exposure is about 1 J/cm2, and, the maximum η±1 is about 3.56%.It can be seen that ηA,±1 and ηP,±1 nearly has direct ratio in parallel linearly recording hologrpahy, the decreasing rate of ηP,±1 is a little bit slower than that of ηA,±1, which is ignored in other conditions in parallel polarization recording, and just the ηA,±1 is calculated as before and ηtotal is took as 1.89 times of the ηA,±1.
Using the same method shown before, fromT⊥ ~E curve shown in Figure 6(c), the ηA,±n~E̅ curves for the C⃗, which has perpendicular polarization to that of the exciting beams (C⃗⊥O⃗//R⃗), are calculated, from which the ηA,±1 is shown in Figure 15. Comparing with the condition of C⃗//O⃗//R⃗, just the changing rate is slow down. When reconstruct with circularly polarized light, the diffracted lights are the superposed light of the diffracted lights of vertical and horizontal polarized light component of the C⃗, so it will be ellipse polarized lights, whose long axes are along the vertical direction and the DE are the average values of that of the vertical and horizontal polarized lights, which is shown in Figure 15.
In the parallel circularly polarization holograph (ORR = 1 : 1), theT distribution is the same for any kinds of polarization state lights, so the DE are also the same. Using the same method shown before, from the T(t)~Δφ curves of parallel circularly recording condition shown in (28), the ηA,±n~E̅ curves of parallel circularly recording holograms are calculated and shown in Figure 15.Figure 13
The amplitude transmission changing curves depending on the phase difference on the direction of the exiting beam polarization at different recording time.Amplitude transmission grating curves and their simulating curves (a)t=2 s; (b) t=160 s.
(a)(b)Figure 15
The first-order diffraction efficiencies of the amplitude gratings in parallel polarization holograms.Figure 16
The first-order diffraction efficiencies of the amplitude grating and phase grating in the parallel linearly recorded mixing holograph.(2) The Calculation of DE in the Orthogonal Polarization Recording Condition
From Section3.1, it is known that in orthogonal polarization recording condition at nonlinear recording area, the DE equations of the linearly recording area shown in Table 1 also can be used. From Figure 6(c), the τe~E and τo~E curves of sample under uniformity light irradiation can be calculated. Then from them using the DE formulas, the ηA,+1~E̅ curves of the orthogonal linearly polarization holography and orthogonal circularly polarization holography can be calculated, which are shown in Figure 17.
It can be seen that the optimal exposure is 1.2 J/cm2, and the maximum amplitude grating DE are, respectively, 0.4% and 1.6%. The decreasing rates are quicker than parallel polarization holography. In Section 2, it is given that in this sample ΔnB(633 nm)/ΔAD(633 nm) = 0.006115. The ηP,+1, the same as the condition in parallel polarization holography, nearly has direct ratio with the ηA,+1, the ηtatol in mixing holograph is about 1.28 times of ηA,+1.Figure 17
The first-order diffraction efficiencies of the amplitude gratings in orthogonal polarization holograms.


#### 3.2.2. The Calculation of DE ~E̅ Curves on the Exposure of the Gaussian Beams

Because in experiment, the writing and reading beams are 633 nm He-Ne Gaussian beams, so this condition was considered in the calculation below. The light pattern is divided into many intensity homogenous circles, whose intensities and areas areIi(i=1⋯n) and Si(i=1⋯n), respectively. Using Ii(i=1⋯n), from the ηtotal~E̅ curves calculated in Section 3.2.1, the DE of every circle at different time ηi,t(i=1⋯n) can be calculated. When read by the original R⃗, the diffracted light power PD,t can be got by submitting that of all the circles PD,i,t=ηi,t·Ii·Si, and then by dividing by the power of C⃗ (PC), the total DE of the sample for the Gaussian reading beam ηtotal,Gaussian,t=PD,t/PC can be obtained. The calculated DE~t curves of different kinds of polarization haplographies recorded in this sample, when the writing beams are 633 nm He-Ne Gaussian beams, whose average intensities are 78.6 mW/cm2 (same with the experiment values) and the original R⃗ is used as C⃗, are shown in Figure 9(b).It can be seen that the maximum values’ ratio of the measured values is basically coincide with the theoretically analyzed one. Only for parallel linearly polarization hologram, in the orthogonal linearly polarization reconstruction condition (C⃗⊥O⃗//R⃗), the diffraction efficiency is lower than parallel linearly polarization reconstruction condition (C⃗//O⃗//R⃗). It is because that, in calculation the affection of C⃗ is not considered, which can be proved to be very small when the IC=IO/100=IR/100 comparing to the affection of nonlinear absorption of the film, whose detail calculation progress will not be given here, where it also can be deduced that the reading beam affection is larger in C⃗⊥O⃗//R⃗ condition than in C⃗//O⃗//R⃗ condition.And the theoretical results are larger than the experimental results, and the reaction is quicker (optimal exposures are about 590 mJ/cm2 and 430 mJ/cm2, respectively, in experimental results and theoretical results); this may be caused by: (1) the sample is not homogeneous, the density is different at different area, (2) the incidence angles of beamsθ in the experiment are about 8.2°, so the intensities of them on the sample plane will be cosθ≈0.9898  times of the values used in calculation, (3) the photoreaction rate constants used in calculation is a little bit larger than real ones.And it can be seen that no matter during the ordinary holograph recording process in photochromic media, or during the polarization holograph recording process in photo-induced anisotropy media, there exits an overshooting peak in the diffraction efficiency, which then decays to a lower permanent level or also to zero. From the theoretical analyses, it can be deduced that it is caused by the diminishing of fringe contrast mainly caused by the nonlinear saturation effects of photoisomerization process and photo-induced anisotropy process. In experiment, there also exit the diminishing of fringe contrast caused by a photochemically active readout beam and unequal intensities of object and reference waves. It can be theoretically calculated that the effects of them show very smaller than that of the nonlinear saturation effects, which will not be given here.


#### 3.2.3. The DE Spectra of Different Kinds of Holography in Fulgide Film

Suppose that the holographic recording is a linearity recording, from the spectra ofΔA, Δn, ΔAD, and ΔnB, shown in Figures 3(b) and 4(b), using the DE formulas of different kinds of polarization holographies, shown in Table 1, the DE spectra of the ordinary holograms (i.e., parallel circularly polarization recording) and polarization holograms (orthogonal linearly and orthogonal circularly polarization recording) can be calculated, which are shown in Figure 18, where the solid line, dot dash lines and dash lines indicate the ηtotal, ηA, and ηP, respectively. It can be seen that at 450 nm and 700 nm the diffraction efficiencies are higher, while the absorption is very small, where the recorded information can be read out without any photochromic reaction, that is, the nondestructive reconstruction can be realized.Diffraction efficiency spectra of the (a) Ordinary holography, (b) Orthogonal linearly polarization holography, and (c) Orthogonal circularly polarization holography.
(a)(b)(c)


## 3.1. Background

For isotropy recording materials, only when the object lightO⃗ and reference light R⃗ have components with same polarization state, there the intensity grating exists and the holograms can be recorded, called ordinary holograph. But for photo-anisotropy materials, even if the O⃗ and R⃗ have orthogonal polarization states, the holograms also can be recorded, because that at this time although the intensity of the superposed light is a constant, but its polarization state changes with the phase difference between O⃗ and R⃗. Photo-anisotropy materials can record the polarization state of exciting beam, so the holographic gratings can be recorded, called polarization holography, in which not only the intensity and phase signals of O⃗ can be stored (the photo-anisotropy is depending on the exposure), but also its polarization state can be stored.The holographic recording progress is a reaction progress between the exciting interference light and the material. So, first the properties of the exciting interference light will be analyzed and then the reaction of the sample with the light will be studied.


### 3.1.1. The Properties of Exiting Interference Pattern of Object Light and Reference Light

Choosexozplane as incidence plane, the O⃗ incidents into the recording plane xoy with a very small incident angle θ, and the R⃗ has the same incident angle (shown in Figure 10(a)), so the phase components of O⃗ and R⃗ on the x-axis are φO=k·x·sinθ and φR=-k·x·sinθ, respectively, and the phase difference of them is(6)Δφ=φO-φR=2⋅k⋅x⋅sinθ,(k=2πλ).Sketch map (a) of the holographic recording setup and (b) of the superposing of two orthogonal circularly polarized lights.
(a)(b)(A) Parallel Polarization State
It is known that in the interference field of two coherent wavesO⃗ and R⃗ with the same polarization, the polarization state is constant and same with before E⃗1=O⃗1+R⃗1=O⃗1(1+const·e-iΔφ), but the intensity is periodically modulated corresponding to Δφ as I1=O2+R2+2·O·R·cosΔφ. At the condition that the amplitudes of O⃗ and R⃗ are same, that is,  O=R=A, the interference field distribution of two special examples of it are drawn in Figures 11(a) and 11(c), including the parallel linearly polarization and parallel circularly polarization.The distributions of the interference fields in four kinds of polarization holography storage and the corresponding molecular distributions and orientations.
(a)
Parallel linearly(b)
Orthogonal linearly(c)
Parallel circularly(d)
Orthogonal circularly(B) Orthogonal Polarization State
In the interference field of two coherent waves with orthogonal polarization, that is,R⃗2·O⃗2*=O⃗2·R⃗2*=0, the intensity is constant I2=O2+R2, but the polarization state is periodically modulated; here two special examples of which are analyzed, including the orthogonal linearly polarization and orthogonal circularly polarization.(B1) Orthogonal Linearly Polarization
SupposeO⃗ and R⃗ are, respectively, horizontal and vertical linearly polarized, so:
(7)E⃗3=O⃗3+R⃗3=O⋅[10]⋅eiφo+R⋅[01]⋅eiφR=eiφO[OR⋅e-iΔφ]→O=R=AA⋅eiφO⋅[1e-iΔφ].
It shows that the superposed light of two orthogonal linearly polarized lights is an elliptical polarized light, whose azimuth α3 is
(8)α3=12⋅arctan⁡[(2ORO2-R2)cos⁡Δφ]=O=R=A{π42kπ-π2<Δφ<2kπ+π2,-π42kπ-π2<Δφ<2kπ+3π2,k∈Z
and whose ellipticity ε is
(9)ε3=arctan(L2L1)=12arcsin(sin2β⋅sinΔφ)→O=R=Aarctan(L2L1)=Δφ2,
where L1, L2 indicate the long and short axes, respectively and β  indicates the amplitude angle (tanβ=R/O). It can be seen that the azimuth and ellipticity of interference field are periodically changing along the Δφ with period 2π. The interference field distribution at the condition O=R=A is drawn in Figure 11(b).(B2) Orthogonal Circularly Polarization
SupposeO⃗ and R⃗ are, respectively, right and left circularly polarized, so:
(10)E⃗4=O⃗4+R⃗4=O⋅[1212i]eiφo+R⋅[12-12i]⋅eiφR=12[O+R⋅e-iΔφ(O-R⋅e-iΔφ)i]⋅eiφO.
It is clear that the superposed light of two orthogonal circularly polarized lights is also an elliptical polarized light, whose ellipticity will not change with the Δφ, that is, the long and short axes are always (O+R) and (O-R), respectively. When O=R=A, the short axis is zero, that is, the linearly polarized light. The azimuth of the light is α4=-Δφ/2. Figure 10(b) is a single sketch map showing the superposing of two orthogonal circularly polarized lights. The interference field distribution at the condition O=R=A is drawn in Figure 11(d).


### 3.1.2. Analysis of the Properties of Different Kinds of Polarization Holograms and Their DWPS and DE

Under the irradiation of the interference field, the photochromic reaction will be induced in the fulgide sample, so the space modulation of the distribution ofE-form and C-form molecules will be caused by the intensity modulation of optical field; photo-anisotropy will be induced during the photochromic reaction under the irradiation of noncircularly polarized light, so the space modulation of the orientations (tropism) of E-form and C-form molecules will be caused by the polarization state modulation of optical field.Here, we just consider four kinds of polarization holographic storage, including the parallel linearly, parallel circularly, orthogonal linearly, and orthogonal circularly polarized holograms. Figure11 is a single sketch map showing the space modulation of the distributions and orientations (optical axis) of E-form and C-form molecules in these four kinds of polarization holographic storage under the linearity recording condition when O=R=A. The arrows and ellipses on the horizontal line indicate the periodical distributions of intensity magnitudes and polarization states of the recording field corresponding to the different phase differences Δφ along the x-axis. The rectangle frames under them indicate the distribution and orientation of the molecules under the corresponding light irradiations, where the black and white colors indicate the C-form and E-form molecules, respectively, the arrows indicate the direction of induced optical axes, and the star flower (*) means the sample is isotropy at the place.From Figure11, it can be seen that in the parallel circularly polarized hologram just ordinary hologram exits, and in the orthogonal polarized condition, just polarization hologram exits, but in the parallel linearly polarized holography, the ordinary hologram and polarization hologram will be recorded together.The underside, the Jones matrixes of four kinds of polarization hologram, and their DWPS and DE are analyzed under linearly recording condition. The DE are calculated using the diffraction theory of two-dimensional grating, because the thickness of the fulgide film is very small (10μm); the recorded hologram can be considered as plane hologram.Suppose the Jones coordinate is(x,y), so the Jones matrix of the hologram is(11)T(x,y)=[τxx⋅eiΨxxτxy⋅eiΨxyτyx⋅eiΨyxτyy⋅eiΨyy],
where τ and Ψ indicate the light amplitude transmissions and light phase transmission function of the sample, and when read by the reconstruction light C⃗, the diffracted light D⃗ of the hologram will be(12)D⇀=T(x,y)⋅C⃗=[τxx⋅eiΨxxτxy⋅eiΨxyτyx⋅eiΨyxτyy⋅eiΨyy]⋅[CxCy]=[τxx⋅eiΨxx⋅Cx+τxy⋅eiΨxy⋅Cyτyx⋅eiΨyx⋅Cx+τyy⋅eiΨyy⋅Cy](whereC⃗=[CxCy]=[Cx0eiφxCy0eiφy]).For simplify, the condition whenO=R=1 is chosen and analysed here. And the 45° contrarotated coordinate system is defined as (X,Y), shown in Figure 12(a). In Figure 12(b), the photoanisotropic amplitude transmission curves are shown. The (τe-τo) indicate the photo-induced anisotropy of the sample under the irradiation of linearly polarized light at some exposure, (ne-no) indicate the corresponding birefringence, τM=(τe+τo)/2 indicate the amplitude transmission of the sample for nonpolarized light or circularly polarized light at this time (at the area of light strips in the interference field), τm indicates the amplitude transmission of the sample before the illumination of light (at the area of dark strips in the interference field, isotropy), τE and τC indicate the amplitude transmission of the E-form and C-form sample, respectively, nM, nm, nE, and nC indicate the corresponding refractive index of the sample, Ψi=k0·ni·d(i=e,o,M,m,E,C), and it is defined that τ0ij=(τi+τj)/2, τ1ij=(τi-τj)/2, Ψ0ij=(Ψi+Ψj)/2, Ψ1ij=(Ψi-Ψj)/2(i,j=e,o,M,m,E,C). In the Figure 12(b), the linearity recording area is given, that is, the linearity recording area of the increasing curve of τe, where it is insured that the light amplitude transmission changing of the sample has the direct ratio with the exposure no matter using what kind of polarization holographic recording and reading.(a) The definition of coordinate system. (b) The photo-induced dichroism curve of the sample depending on the exposure.
(a)(b)(1) Parallel Linearly Polarization Recording
From Figure11, it can be seen that the photo-induced optical axis is along the y-axis, so
(13)T1=[τx⋅eiΨX00τy⋅eiΨY].
Under the linearity recording condition,
(14)τx=τ0om+τ1om⋅cos⁡Δφ,τy=τ0em+τ1em⋅cos⁡Δφ,Ψx=Ψ0om+Ψ1omcos⁡Δφ,Ψy=Ψ0em+Ψ1emcos⁡Δφ.
For the amplitude recording material, Ψx=Ψy=Ψ0. If the hologram is reconstructed with C⃗, which has same incident angle with R⃗, using (12) the D⃗ can be calculated as
(15)D⃗A1=TA1⋅C⇀=eiΨ0⋅[τ0om+τ1omcos⁡Δφ00τ0em+τ1em⋅cos⁡Δφ]⋅[CxCy]⋅e-i(Δφ/2)=eiΨ0⋅[τ0om⋅Cxτ0em⋅Cy]⋅e-i(Δφ/2)+eiΨ0⋅12[τ1om⋅Cxτ1em⋅Cy]⋅e-i(3Δφ/2)+eiΨ0⋅12[τ1om⋅Cxτ1em⋅Cy]⋅ei(Δφ/2).
So, the first-order DE of it would be
(16)ηA,±1=14⋅(τ1om⋅|Cx|)2+(τ1em⋅|Cy|)2|Cx|2+|Cy|2=14⋅(τ1om2⋅|Cx|2|Cx|2+|Cy|2+τ1em2⋅|Cy|2|Cx|2+|Cy|2).
IfR⃗ (vertical linearly polarized) itself is used as C⃗, the η±1=(τ1em/2)2 and the DWPS are the same as C⃗ (shown in Table 1). If a horizontal polarized light is used as C⃗, the η±1=(τ1om/2)2 and the DWPS are the same as C⃗ (shown in Table 1). But at other stations the diffracted lights are the superposed light of the diffracted lights of vertical and horizontal polarized light components of C⃗; when reconstruct with circularly polarized light, the diffracted light will be ellipse polarized, whose long axes are along the vertical direction, and DE is the average value of that of horizontal and vertical polarized lights. For the phase recording material, τx=τy=τ0, using the same method the DWPS and DE of it can be deduced, in which four typical ones are written in Table 1.
At the nonlinearly recording condition, the grating of the amplitude and refractive index will no longer be a single sine function distributing, but will be a superposition of a lot of different sine function distribution(17)τi=τ0i+∑n=1∞τni⋅cos⁡(n⋅Δφ),(i=x,y),ψi=ψ0i+∑n=1∞ψni⋅cos⁡(n⋅Δφ),(i=x,y).
So when reconstruct with horizontal or vertical polarized light, the polarization states of the diffracted lights will not change, and the ±1-order diffraction efficiencies of the amplitude grating and phase grating will be, respectively, ηA,±1=τ1i2/4 and ηP,±1=(τ0i·J1(Ψ1i))2(i=x,y). But when reconstruct with circularly polarized light, the diffracted lights are the superposed light of the diffracted lights of horizontal and vertical polarized light component of the reconstruction beam, so it will be ellipse polarized lights, whose long axes are along the vertical direction, and the diffraction efficiencies are the their average value.(2) Parallel Circularly Polarization Recording
From Figure11, it can be seen that the sample is isotropy at this condition, so
(18)T2=[τ⋅eiΨ00τ⋅eiΨ].
Under the linearity recording condition,(19)τ=τ0Mm+τ1Mm⋅cos⁡Δφ,Ψ=Ψ0Mm+Ψ1Mmcos⁡Δφ.
Using (12), the first-order DWPS and DE can be calculated when it is read by bragg angle entered arbitrary polarized light, in which the four typical ones were written in Table 1. At the nonlinearly recording condition, the same as the parallel linearly polarization holography, the grating will no longer be a single sine function distributing. But the sample is isotropy, that is, Tx=Ty. So the polarization state of the diffracted lights are always the same as the C⃗, and the diffraction efficiency is invariable with the polarization state of the C⃗, ηA,±1=τ12/4, ηP,±1=(τ0·J1(Ψ1))2.(3) Orthogonal Linearly Polarization Recording
For the orthogonal linearly polarization hologram, the superposed light is elliptically polarized light, whose long axes haveα=±π/4 angle between the x-axis, that is, along the X, Y axes. So first the Jones vector of the interference field Equation (7) is turned into X,Y coordinate
(20)E⃗3|O=R=A=A⋅eiφO⋅[cos⁡αsinα-sinαcos⁡α]⋅[1e-iΔφ]=α=π/42A⋅eiφO⋅[cos⁡Δφ2i⋅sinΔφ2].
The optical anisotropy induced by elliptically polarized light should have direct ratio with the intensity difference betweenX,Y two directions I3X-I3Y=2A2cosΔφ. So, the amplitude transmission Jones matrix of the holograph can be written as
(21)T3(X,Y)=[τX⋅eiΨX00τY⋅eiΨY],
where
(22)τX=τ0eo+τ1eo⋅cos⁡Δφ,τY=τ0eo-τ1eo⋅cos⁡Δφ,ΨX=Ψ0eo+Ψ1eo⋅cos⁡Δφ,ΨY=Ψ0eo-Ψ1eo⋅cos⁡Δφ.
And then the Jones matrix of the holograph is turned into usually usedx,y coordinate:
(23)T3(x,y)=[cos⁡α-sinαsinαcos⁡α]⋅[τX⋅eiΨX00τY⋅eiΨY]⋅[cos⁡αsinα-sinαcos⁡α]=α=π/412[τX⋅eiΨX+τY⋅eiΨYτX⋅eiΨX-τY⋅eiΨYτX⋅eiΨX-τY⋅eiΨYτX⋅eiΨX+τY⋅eiΨY].
Using (12), the first-order DWPS and DE can be calculated when it is read by bragg angle entered lights, in which the four typical ones were written in Table 1. It can be deduced that the x and y components of the ±1-order diffracted lights are reversed comparing to the C⃗, so their long axes are symmetry distributed on the X-axis with the long axis of C⃗, where four special conditions exit: When the C⃗ is vertical, horizontal, right circularly or left circularly polarized light, the polarization state of the diffracted light is orthogonal to that of C⃗. The polarization of the ±2 order diffracted lights are same with the reconstruction light. The ±1-order DE of amplitude and phase holograms are respectively (τ1eo/2)2 and (τ0eo·J1(Ψ1eo))2.
At the nonlinearly recording condition, (τe-τo) and (ne-no) will not have direct ration with the exposure, but the anisotropy distribution will not change, so the diffraction efficiency equations at the linearly recording condition also can be used.(4) Orthogonal Circularly Polarization Recording
From Section3.1.1, it can be seen that the superposed light field of two orthogonal circularly polarized lights (the same intensity) is a linearly polarized light, and the intensity will not change with Δφ, so every where in the sample its amplitude transmission matrix in the coordinate of the its main axes should be
(24)T4(e,o)=[τe⋅eiΨe00τo⋅eiΨo].
The optical axis of the superposed light is changing withΔφ, which has Δφ/2 angle with the X-axis to counter-clockwise direction, so the angle with the x-axis is α=Δφ/2+π/4. So, the Jones matrix of the recording material in the x,y coordinate will be(25)T4(x,y)=[τesin2Δφ2⋅eiΨe+τo⋅co⁡s2Δφ2⋅eiΨo-12⋅sinΔφ⋅(-τe⋅eiΨe+τo⋅eiΨo)-12⋅sinΔφ⋅(-τe⋅eiΨe+τo⋅eiΨo)τeco⁡s2Δφ2⋅eiΨe+τo⋅sin2Δφ2⋅eiΨo].Using (12), it can be deduced that, for the orthogonal circularly polarization holographic recording, there just exist the 0-order and ±1-order diffracted lights. The polarization state of the 0-order diffracted light is same with that of C⃗, and the polarization state of the +1 and −1-order diffracted lights are always right and left circularly polarized, respectively, which are same with that of O⃗ and R⃗. When the C⃗ is an arbitrary polarized light, it has the 0-order and ±1-order diffracted lights, whose left circularly polarized light component will diffract the +1-order light, and the right circularly polarized light component will diffract the −1-order light. The ±1-order DE of amplitude holograph are respectively ηA,+1=(τ1eo·Cl0/C0)2 and ηA,-1=(τ1eo·Cr0/C0)2, that of the phase holograph are, respectively ηP,+1=(τ0eo·sinΨ1eo·Cl0/C0)2 and ηP,-1=(τ0eo·sinΨ1eo·Cr0/C0)2, where the C0, Cl0, Cr0 are the amplitudes of C⃗ and of its left and right circularly polarized light components. In which, when the C⃗ is left circularly polarized light (same with R⃗), only the 0-order and +1 order diffracted lights exist, and the polarization state of the +1 order diffracted light is orthogonal to that of C⃗, and ηA,+1=(τ1eo)2, ηA,-1=0, ηP,+1=(τ0eo·sinΨ1eo)2 and ηP,-1=0. When the C⃗ is right circularly polarized light (orthogonal with R⃗), only the 0-order and −1 order diffracted lights exist, and the polarization state of the −1-order diffracted light is also orthogonal to that of C⃗, and ηA,+1=0, ηA,-1=(τ1eo)2, ηP,+1=0 and ηP,-1=(τ0·sinΨ1eo)2, which are written in Table 1. At the nonlinearly recording condition, the same as the orthogonal linearly polarization holographic recording, the anisotropy grating distribution will not change, so the diffraction efficiency equations at the linearly recording condition also can be used.


### 3.1.3. DWPS and DE Formulas of Different Kinds of Hologram

In Table1, the DWPS and DE of parallel linearly, parallel circularly, orthogonal linearly and orthogonal circularly polarization holographs at the linearly, recording condition are summarized, when reconstructed with linearly polarized lights and circularly polarized lights. Usually the holograms are mixed hologram, whose DE approximately equals to the summation of that of the amplitude grating and phase grating: η1=ηa1+ηp1.It can be seen that the DWPS obtained in the experiments are the same as the theortically deduced ones. And in the orthogonal polarization holography, if reconstructed with the reference light, the polarization state of the diffracted light is orthogonal to that of the reconstruction light, which is very important to increase the SNR of the holographic storage. Because that the polarization state of the scattered noise light is almost the same as that of the reconstruction light, which can be filtered by using the polarizer and quarter-wave plate.


## 3.1.1. The Properties of Exiting Interference Pattern of Object Light and Reference Light

Choosexozplane as incidence plane, the O⃗ incidents into the recording plane xoy with a very small incident angle θ, and the R⃗ has the same incident angle (shown in Figure 10(a)), so the phase components of O⃗ and R⃗ on the x-axis are φO=k·x·sinθ and φR=-k·x·sinθ, respectively, and the phase difference of them is(6)Δφ=φO-φR=2⋅k⋅x⋅sinθ,(k=2πλ).Sketch map (a) of the holographic recording setup and (b) of the superposing of two orthogonal circularly polarized lights.
(a)(b)(A) Parallel Polarization State
It is known that in the interference field of two coherent wavesO⃗ and R⃗ with the same polarization, the polarization state is constant and same with before E⃗1=O⃗1+R⃗1=O⃗1(1+const·e-iΔφ), but the intensity is periodically modulated corresponding to Δφ as I1=O2+R2+2·O·R·cosΔφ. At the condition that the amplitudes of O⃗ and R⃗ are same, that is,  O=R=A, the interference field distribution of two special examples of it are drawn in Figures 11(a) and 11(c), including the parallel linearly polarization and parallel circularly polarization.The distributions of the interference fields in four kinds of polarization holography storage and the corresponding molecular distributions and orientations.
(a)
Parallel linearly(b)
Orthogonal linearly(c)
Parallel circularly(d)
Orthogonal circularly(B) Orthogonal Polarization State
In the interference field of two coherent waves with orthogonal polarization, that is,R⃗2·O⃗2*=O⃗2·R⃗2*=0, the intensity is constant I2=O2+R2, but the polarization state is periodically modulated; here two special examples of which are analyzed, including the orthogonal linearly polarization and orthogonal circularly polarization.(B1) Orthogonal Linearly Polarization
SupposeO⃗ and R⃗ are, respectively, horizontal and vertical linearly polarized, so:
(7)E⃗3=O⃗3+R⃗3=O⋅[10]⋅eiφo+R⋅[01]⋅eiφR=eiφO[OR⋅e-iΔφ]→O=R=AA⋅eiφO⋅[1e-iΔφ].
It shows that the superposed light of two orthogonal linearly polarized lights is an elliptical polarized light, whose azimuth α3 is
(8)α3=12⋅arctan⁡[(2ORO2-R2)cos⁡Δφ]=O=R=A{π42kπ-π2<Δφ<2kπ+π2,-π42kπ-π2<Δφ<2kπ+3π2,k∈Z
and whose ellipticity ε is
(9)ε3=arctan(L2L1)=12arcsin(sin2β⋅sinΔφ)→O=R=Aarctan(L2L1)=Δφ2,
where L1, L2 indicate the long and short axes, respectively and β  indicates the amplitude angle (tanβ=R/O). It can be seen that the azimuth and ellipticity of interference field are periodically changing along the Δφ with period 2π. The interference field distribution at the condition O=R=A is drawn in Figure 11(b).(B2) Orthogonal Circularly Polarization
SupposeO⃗ and R⃗ are, respectively, right and left circularly polarized, so:
(10)E⃗4=O⃗4+R⃗4=O⋅[1212i]eiφo+R⋅[12-12i]⋅eiφR=12[O+R⋅e-iΔφ(O-R⋅e-iΔφ)i]⋅eiφO.
It is clear that the superposed light of two orthogonal circularly polarized lights is also an elliptical polarized light, whose ellipticity will not change with the Δφ, that is, the long and short axes are always (O+R) and (O-R), respectively. When O=R=A, the short axis is zero, that is, the linearly polarized light. The azimuth of the light is α4=-Δφ/2. Figure 10(b) is a single sketch map showing the superposing of two orthogonal circularly polarized lights. The interference field distribution at the condition O=R=A is drawn in Figure 11(d).


## 3.1.2. Analysis of the Properties of Different Kinds of Polarization Holograms and Their DWPS and DE

Under the irradiation of the interference field, the photochromic reaction will be induced in the fulgide sample, so the space modulation of the distribution ofE-form and C-form molecules will be caused by the intensity modulation of optical field; photo-anisotropy will be induced during the photochromic reaction under the irradiation of noncircularly polarized light, so the space modulation of the orientations (tropism) of E-form and C-form molecules will be caused by the polarization state modulation of optical field.Here, we just consider four kinds of polarization holographic storage, including the parallel linearly, parallel circularly, orthogonal linearly, and orthogonal circularly polarized holograms. Figure11 is a single sketch map showing the space modulation of the distributions and orientations (optical axis) of E-form and C-form molecules in these four kinds of polarization holographic storage under the linearity recording condition when O=R=A. The arrows and ellipses on the horizontal line indicate the periodical distributions of intensity magnitudes and polarization states of the recording field corresponding to the different phase differences Δφ along the x-axis. The rectangle frames under them indicate the distribution and orientation of the molecules under the corresponding light irradiations, where the black and white colors indicate the C-form and E-form molecules, respectively, the arrows indicate the direction of induced optical axes, and the star flower (*) means the sample is isotropy at the place.From Figure11, it can be seen that in the parallel circularly polarized hologram just ordinary hologram exits, and in the orthogonal polarized condition, just polarization hologram exits, but in the parallel linearly polarized holography, the ordinary hologram and polarization hologram will be recorded together.The underside, the Jones matrixes of four kinds of polarization hologram, and their DWPS and DE are analyzed under linearly recording condition. The DE are calculated using the diffraction theory of two-dimensional grating, because the thickness of the fulgide film is very small (10μm); the recorded hologram can be considered as plane hologram.Suppose the Jones coordinate is(x,y), so the Jones matrix of the hologram is(11)T(x,y)=[τxx⋅eiΨxxτxy⋅eiΨxyτyx⋅eiΨyxτyy⋅eiΨyy],
where τ and Ψ indicate the light amplitude transmissions and light phase transmission function of the sample, and when read by the reconstruction light C⃗, the diffracted light D⃗ of the hologram will be(12)D⇀=T(x,y)⋅C⃗=[τxx⋅eiΨxxτxy⋅eiΨxyτyx⋅eiΨyxτyy⋅eiΨyy]⋅[CxCy]=[τxx⋅eiΨxx⋅Cx+τxy⋅eiΨxy⋅Cyτyx⋅eiΨyx⋅Cx+τyy⋅eiΨyy⋅Cy](whereC⃗=[CxCy]=[Cx0eiφxCy0eiφy]).For simplify, the condition whenO=R=1 is chosen and analysed here. And the 45° contrarotated coordinate system is defined as (X,Y), shown in Figure 12(a). In Figure 12(b), the photoanisotropic amplitude transmission curves are shown. The (τe-τo) indicate the photo-induced anisotropy of the sample under the irradiation of linearly polarized light at some exposure, (ne-no) indicate the corresponding birefringence, τM=(τe+τo)/2 indicate the amplitude transmission of the sample for nonpolarized light or circularly polarized light at this time (at the area of light strips in the interference field), τm indicates the amplitude transmission of the sample before the illumination of light (at the area of dark strips in the interference field, isotropy), τE and τC indicate the amplitude transmission of the E-form and C-form sample, respectively, nM, nm, nE, and nC indicate the corresponding refractive index of the sample, Ψi=k0·ni·d(i=e,o,M,m,E,C), and it is defined that τ0ij=(τi+τj)/2, τ1ij=(τi-τj)/2, Ψ0ij=(Ψi+Ψj)/2, Ψ1ij=(Ψi-Ψj)/2(i,j=e,o,M,m,E,C). In the Figure 12(b), the linearity recording area is given, that is, the linearity recording area of the increasing curve of τe, where it is insured that the light amplitude transmission changing of the sample has the direct ratio with the exposure no matter using what kind of polarization holographic recording and reading.(a) The definition of coordinate system. (b) The photo-induced dichroism curve of the sample depending on the exposure.
(a)(b)(1) Parallel Linearly Polarization Recording
From Figure11, it can be seen that the photo-induced optical axis is along the y-axis, so
(13)T1=[τx⋅eiΨX00τy⋅eiΨY].
Under the linearity recording condition,
(14)τx=τ0om+τ1om⋅cos⁡Δφ,τy=τ0em+τ1em⋅cos⁡Δφ,Ψx=Ψ0om+Ψ1omcos⁡Δφ,Ψy=Ψ0em+Ψ1emcos⁡Δφ.
For the amplitude recording material, Ψx=Ψy=Ψ0. If the hologram is reconstructed with C⃗, which has same incident angle with R⃗, using (12) the D⃗ can be calculated as
(15)D⃗A1=TA1⋅C⇀=eiΨ0⋅[τ0om+τ1omcos⁡Δφ00τ0em+τ1em⋅cos⁡Δφ]⋅[CxCy]⋅e-i(Δφ/2)=eiΨ0⋅[τ0om⋅Cxτ0em⋅Cy]⋅e-i(Δφ/2)+eiΨ0⋅12[τ1om⋅Cxτ1em⋅Cy]⋅e-i(3Δφ/2)+eiΨ0⋅12[τ1om⋅Cxτ1em⋅Cy]⋅ei(Δφ/2).
So, the first-order DE of it would be
(16)ηA,±1=14⋅(τ1om⋅|Cx|)2+(τ1em⋅|Cy|)2|Cx|2+|Cy|2=14⋅(τ1om2⋅|Cx|2|Cx|2+|Cy|2+τ1em2⋅|Cy|2|Cx|2+|Cy|2).
IfR⃗ (vertical linearly polarized) itself is used as C⃗, the η±1=(τ1em/2)2 and the DWPS are the same as C⃗ (shown in Table 1). If a horizontal polarized light is used as C⃗, the η±1=(τ1om/2)2 and the DWPS are the same as C⃗ (shown in Table 1). But at other stations the diffracted lights are the superposed light of the diffracted lights of vertical and horizontal polarized light components of C⃗; when reconstruct with circularly polarized light, the diffracted light will be ellipse polarized, whose long axes are along the vertical direction, and DE is the average value of that of horizontal and vertical polarized lights. For the phase recording material, τx=τy=τ0, using the same method the DWPS and DE of it can be deduced, in which four typical ones are written in Table 1.
At the nonlinearly recording condition, the grating of the amplitude and refractive index will no longer be a single sine function distributing, but will be a superposition of a lot of different sine function distribution(17)τi=τ0i+∑n=1∞τni⋅cos⁡(n⋅Δφ),(i=x,y),ψi=ψ0i+∑n=1∞ψni⋅cos⁡(n⋅Δφ),(i=x,y).
So when reconstruct with horizontal or vertical polarized light, the polarization states of the diffracted lights will not change, and the ±1-order diffraction efficiencies of the amplitude grating and phase grating will be, respectively, ηA,±1=τ1i2/4 and ηP,±1=(τ0i·J1(Ψ1i))2(i=x,y). But when reconstruct with circularly polarized light, the diffracted lights are the superposed light of the diffracted lights of horizontal and vertical polarized light component of the reconstruction beam, so it will be ellipse polarized lights, whose long axes are along the vertical direction, and the diffraction efficiencies are the their average value.(2) Parallel Circularly Polarization Recording
From Figure11, it can be seen that the sample is isotropy at this condition, so
(18)T2=[τ⋅eiΨ00τ⋅eiΨ].
Under the linearity recording condition,(19)τ=τ0Mm+τ1Mm⋅cos⁡Δφ,Ψ=Ψ0Mm+Ψ1Mmcos⁡Δφ.
Using (12), the first-order DWPS and DE can be calculated when it is read by bragg angle entered arbitrary polarized light, in which the four typical ones were written in Table 1. At the nonlinearly recording condition, the same as the parallel linearly polarization holography, the grating will no longer be a single sine function distributing. But the sample is isotropy, that is, Tx=Ty. So the polarization state of the diffracted lights are always the same as the C⃗, and the diffraction efficiency is invariable with the polarization state of the C⃗, ηA,±1=τ12/4, ηP,±1=(τ0·J1(Ψ1))2.(3) Orthogonal Linearly Polarization Recording
For the orthogonal linearly polarization hologram, the superposed light is elliptically polarized light, whose long axes haveα=±π/4 angle between the x-axis, that is, along the X, Y axes. So first the Jones vector of the interference field Equation (7) is turned into X,Y coordinate
(20)E⃗3|O=R=A=A⋅eiφO⋅[cos⁡αsinα-sinαcos⁡α]⋅[1e-iΔφ]=α=π/42A⋅eiφO⋅[cos⁡Δφ2i⋅sinΔφ2].
The optical anisotropy induced by elliptically polarized light should have direct ratio with the intensity difference betweenX,Y two directions I3X-I3Y=2A2cosΔφ. So, the amplitude transmission Jones matrix of the holograph can be written as
(21)T3(X,Y)=[τX⋅eiΨX00τY⋅eiΨY],
where
(22)τX=τ0eo+τ1eo⋅cos⁡Δφ,τY=τ0eo-τ1eo⋅cos⁡Δφ,ΨX=Ψ0eo+Ψ1eo⋅cos⁡Δφ,ΨY=Ψ0eo-Ψ1eo⋅cos⁡Δφ.
And then the Jones matrix of the holograph is turned into usually usedx,y coordinate:
(23)T3(x,y)=[cos⁡α-sinαsinαcos⁡α]⋅[τX⋅eiΨX00τY⋅eiΨY]⋅[cos⁡αsinα-sinαcos⁡α]=α=π/412[τX⋅eiΨX+τY⋅eiΨYτX⋅eiΨX-τY⋅eiΨYτX⋅eiΨX-τY⋅eiΨYτX⋅eiΨX+τY⋅eiΨY].
Using (12), the first-order DWPS and DE can be calculated when it is read by bragg angle entered lights, in which the four typical ones were written in Table 1. It can be deduced that the x and y components of the ±1-order diffracted lights are reversed comparing to the C⃗, so their long axes are symmetry distributed on the X-axis with the long axis of C⃗, where four special conditions exit: When the C⃗ is vertical, horizontal, right circularly or left circularly polarized light, the polarization state of the diffracted light is orthogonal to that of C⃗. The polarization of the ±2 order diffracted lights are same with the reconstruction light. The ±1-order DE of amplitude and phase holograms are respectively (τ1eo/2)2 and (τ0eo·J1(Ψ1eo))2.
At the nonlinearly recording condition, (τe-τo) and (ne-no) will not have direct ration with the exposure, but the anisotropy distribution will not change, so the diffraction efficiency equations at the linearly recording condition also can be used.(4) Orthogonal Circularly Polarization Recording
From Section3.1.1, it can be seen that the superposed light field of two orthogonal circularly polarized lights (the same intensity) is a linearly polarized light, and the intensity will not change with Δφ, so every where in the sample its amplitude transmission matrix in the coordinate of the its main axes should be
(24)T4(e,o)=[τe⋅eiΨe00τo⋅eiΨo].
The optical axis of the superposed light is changing withΔφ, which has Δφ/2 angle with the X-axis to counter-clockwise direction, so the angle with the x-axis is α=Δφ/2+π/4. So, the Jones matrix of the recording material in the x,y coordinate will be(25)T4(x,y)=[τesin2Δφ2⋅eiΨe+τo⋅co⁡s2Δφ2⋅eiΨo-12⋅sinΔφ⋅(-τe⋅eiΨe+τo⋅eiΨo)-12⋅sinΔφ⋅(-τe⋅eiΨe+τo⋅eiΨo)τeco⁡s2Δφ2⋅eiΨe+τo⋅sin2Δφ2⋅eiΨo].Using (12), it can be deduced that, for the orthogonal circularly polarization holographic recording, there just exist the 0-order and ±1-order diffracted lights. The polarization state of the 0-order diffracted light is same with that of C⃗, and the polarization state of the +1 and −1-order diffracted lights are always right and left circularly polarized, respectively, which are same with that of O⃗ and R⃗. When the C⃗ is an arbitrary polarized light, it has the 0-order and ±1-order diffracted lights, whose left circularly polarized light component will diffract the +1-order light, and the right circularly polarized light component will diffract the −1-order light. The ±1-order DE of amplitude holograph are respectively ηA,+1=(τ1eo·Cl0/C0)2 and ηA,-1=(τ1eo·Cr0/C0)2, that of the phase holograph are, respectively ηP,+1=(τ0eo·sinΨ1eo·Cl0/C0)2 and ηP,-1=(τ0eo·sinΨ1eo·Cr0/C0)2, where the C0, Cl0, Cr0 are the amplitudes of C⃗ and of its left and right circularly polarized light components. In which, when the C⃗ is left circularly polarized light (same with R⃗), only the 0-order and +1 order diffracted lights exist, and the polarization state of the +1 order diffracted light is orthogonal to that of C⃗, and ηA,+1=(τ1eo)2, ηA,-1=0, ηP,+1=(τ0eo·sinΨ1eo)2 and ηP,-1=0. When the C⃗ is right circularly polarized light (orthogonal with R⃗), only the 0-order and −1 order diffracted lights exist, and the polarization state of the −1-order diffracted light is also orthogonal to that of C⃗, and ηA,+1=0, ηA,-1=(τ1eo)2, ηP,+1=0 and ηP,-1=(τ0·sinΨ1eo)2, which are written in Table 1. At the nonlinearly recording condition, the same as the orthogonal linearly polarization holographic recording, the anisotropy grating distribution will not change, so the diffraction efficiency equations at the linearly recording condition also can be used.


## 3.1.3. DWPS and DE Formulas of Different Kinds of Hologram

In Table1, the DWPS and DE of parallel linearly, parallel circularly, orthogonal linearly and orthogonal circularly polarization holographs at the linearly, recording condition are summarized, when reconstructed with linearly polarized lights and circularly polarized lights. Usually the holograms are mixed hologram, whose DE approximately equals to the summation of that of the amplitude grating and phase grating: η1=ηa1+ηp1.It can be seen that the DWPS obtained in the experiments are the same as the theortically deduced ones. And in the orthogonal polarization holography, if reconstructed with the reference light, the polarization state of the diffracted light is orthogonal to that of the reconstruction light, which is very important to increase the SNR of the holographic storage. Because that the polarization state of the scattered noise light is almost the same as that of the reconstruction light, which can be filtered by using the polarizer and quarter-wave plate.


## 3.2. Theoretical Calculations of DE of Fulgide Film

In this section, the DE spectra and DE dynamic curves at 633 nm of different kinds of holograms recorded in the sample will be calculated from the properties of the sample written in Section2, by using the theory written in Section 3.1. In Section 2, it is given that for fulgide materials the photochromic and photo-induced anisotropy phenomenon are optical accumulating progresses, so the holographic recording progress is also an accumulating progress, which just depending on the exposure (average exposure E̅=2·I0·t), so it is enough to consider just one exciting beam intensity condition. For the simplicity, the object refractive ratio (ORR) is chosen as 1 : 1, that is, O=R=A=(I0)1/2.


### 3.2.1. Theoretical Calculation of DE(633 nm) ~E̅ Curves on the Exposure of Uniformity Beams

(1) The Calculation of DE in the Parallel Polarization Recording Condition
For parallel polarization recording condition, the optical intensity distribution should be(26)I(Δφ)=O2+R2+2⋅O⋅R⋅cos⁡Δφ=2⋅I0⋅(1+cos⁡Δφ).
So at recording time t, the exposure (E) distribution on the sample is
(27)E(Δφ,t)=I(Δφ)⋅t=2⋅I0⋅(1+cos⁡Δφ)⋅t.
So, for different kinds of C⃗, the T(C⃗,Δφ,t)~Δφ of the parallel linearly and parallel circularly recorded holograms at different t in the sample can be obtained from T~E curves shown in Figure 6(c)(28)T(Δφ,t)=T(E(Δφ))={T//(2⋅I0⋅(1+cos⁡Δφ)⋅t)parallellinearlyrecording,O⃗//R⃗//C⃗T⊥(2⋅I0⋅(1+cos⁡Δφ)⋅t)parallellinearlyrecording,O⃗//R⃗⊥C⃗(T//+T⊥)(2⋅I0⋅(1+cos⁡Δφ)⋅t)2parallelcircularlyrecording.(a) The Calculation of DE of the Amplitude Gratings (ηA)
The amplitude transmission changing curves on the phase differenceτ~Δφ at t will be
(29)τ(Δφ,t)=[T(Δφ,t)]1/2.
Every τ~Δφ grating curves at different t can be spread into Thaler progression, that is, simulated with the formula
(30)τ=τ0+∑n=110τn⋅cos⁡(n⋅Δφ).
Such the different order amplitude transmission gratings’ amplitude values τn(n=0,1,…,10) at different t can be obtained. So the changing curves of τn(n=0,1,…,10) on the recording exposure τn~E̅ can be obtained, and then the different order DE changing curves of amplitude gratings on the recording exposure (ηA,±n~E̅) can be calculated as
(31)η0=τ02,ηA,±n=τn24,(n=1,2,…,10).(b) The Calculation of DE of the Phase Gratings (ηP)
From theT(t)~Δφ, the A(t)~Δφ curves can be calculated
(32)A(Δφ,t)=log⁡(1T(Δφ,t)).
And then every A~Δφ grating curves at different t can be spread into Thaler progression, that is, simulated with the underside formula
(33)A=A0+∑i=110Ai⋅cos⁡(i⋅Δφ).
So, the different order absorption gratings’ amplitude values Ai(i=0,1,…,10) for different kinds of R⃗ at different t can be obtained. From Section 2, it is known that, at one wavelength, between the ΔA and Δn exit direct ratio relationship. And then after multiplexed with responding ratio value, the different order phase gratings’ amplitude values ni(i=0,1,…,10) for different polarization R⃗ at different t can be got
(34)ni=consti⋅Ai.
So, the different order DE changing curves of phase gratings on recording exposure (ηA,±n~E̅) can be calculated. Here only the first-order DE is discussed, From Section 3.1, it is known that only the n1 will induce the first-order diffracted light, and ηP,±1=(τ0·J1(k·n1·d))2.(c) The Calculation of DE of the Mixed Gratings (ηtotal)
Theηtotal can be calculated by submitting the ηA and ηP: η+1=ηA,+1+ηP,+1.(d) The Calculation Results of DE of Fulgide Film in Parallel Polarization Recording
For example, in the parallel linearly polarization recording parallel polarization reading condition (C⃗//O⃗//R⃗), when I0=50 mW/cm2, from the T//~E curve shown in Figure 6(c), the T//(t)~Δφ of the holographic grating at different recording times t can be calculated, from which the τ//(t)~Δφ can be obtained, which is shown in Figure 13.
Everyτ//~Δφ curve was spread into Thaler progression, that is was simulated with (30), then the τn (n=0,1,…,10) on the direction of the exiting beam polarization at different t were obtained; for example, in the Figure 14 two typical amplitude transmission grating curves (sine distribution and negative triangle wave function) at different t and their simulated curves are given. So, the τn~E̅  (n=0,1,…,10) curves can be got, from which the ηA,±n~E̅(n=0,1,…,10) curves can be calculated, from which the first-order DE (ηA,±1) is chosen and shown in Figure 15 as dash line 2. It can be seen that even if the ORR = 1 : 1, because the nonlinear recording the DE will also drop down. The optimal exposure is about 1 J/cm2, and at the uniformity light recording condition, the maximum ηA,±1 is 2%. From the T//(t)~Δφ curves, the A//(t)~Δφ curves can be calculated. After every A//(t)~Δφ curve is spread into Thaler progression, the Ai(i=0,1,…,10) on the direction of the exiting beam polarization at different t can be obtained. In Section 2, it is given that Δn(633 nm)/ΔA(633 nm) = 0.00994. So after the Ai multiplexed with 0.00994, the ni,// (i=0,1,…,10) at different t can be got, from which the ηP,±1~E̅ can be calculated, which is shown in Figure 16. The DE of the mixed holograph ηtotal can be obtained by submitting that of the amplitude grating and phase grating, shown in Figure 16. It can be seen that at the uniformity light recording condition, the optimal exposure is about 1 J/cm2, and, the maximum η±1 is about 3.56%.It can be seen that ηA,±1 and ηP,±1 nearly has direct ratio in parallel linearly recording hologrpahy, the decreasing rate of ηP,±1 is a little bit slower than that of ηA,±1, which is ignored in other conditions in parallel polarization recording, and just the ηA,±1 is calculated as before and ηtotal is took as 1.89 times of the ηA,±1.
Using the same method shown before, fromT⊥ ~E curve shown in Figure 6(c), the ηA,±n~E̅ curves for the C⃗, which has perpendicular polarization to that of the exciting beams (C⃗⊥O⃗//R⃗), are calculated, from which the ηA,±1 is shown in Figure 15. Comparing with the condition of C⃗//O⃗//R⃗, just the changing rate is slow down. When reconstruct with circularly polarized light, the diffracted lights are the superposed light of the diffracted lights of vertical and horizontal polarized light component of the C⃗, so it will be ellipse polarized lights, whose long axes are along the vertical direction and the DE are the average values of that of the vertical and horizontal polarized lights, which is shown in Figure 15.
In the parallel circularly polarization holograph (ORR = 1 : 1), theT distribution is the same for any kinds of polarization state lights, so the DE are also the same. Using the same method shown before, from the T(t)~Δφ curves of parallel circularly recording condition shown in (28), the ηA,±n~E̅ curves of parallel circularly recording holograms are calculated and shown in Figure 15.Figure 13
The amplitude transmission changing curves depending on the phase difference on the direction of the exiting beam polarization at different recording time.Amplitude transmission grating curves and their simulating curves (a)t=2 s; (b) t=160 s.
(a)(b)Figure 15
The first-order diffraction efficiencies of the amplitude gratings in parallel polarization holograms.Figure 16
The first-order diffraction efficiencies of the amplitude grating and phase grating in the parallel linearly recorded mixing holograph.(2) The Calculation of DE in the Orthogonal Polarization Recording Condition
From Section3.1, it is known that in orthogonal polarization recording condition at nonlinear recording area, the DE equations of the linearly recording area shown in Table 1 also can be used. From Figure 6(c), the τe~E and τo~E curves of sample under uniformity light irradiation can be calculated. Then from them using the DE formulas, the ηA,+1~E̅ curves of the orthogonal linearly polarization holography and orthogonal circularly polarization holography can be calculated, which are shown in Figure 17.
It can be seen that the optimal exposure is 1.2 J/cm2, and the maximum amplitude grating DE are, respectively, 0.4% and 1.6%. The decreasing rates are quicker than parallel polarization holography. In Section 2, it is given that in this sample ΔnB(633 nm)/ΔAD(633 nm) = 0.006115. The ηP,+1, the same as the condition in parallel polarization holography, nearly has direct ratio with the ηA,+1, the ηtatol in mixing holograph is about 1.28 times of ηA,+1.Figure 17
The first-order diffraction efficiencies of the amplitude gratings in orthogonal polarization holograms.


### 3.2.2. The Calculation of DE ~E̅ Curves on the Exposure of the Gaussian Beams

Because in experiment, the writing and reading beams are 633 nm He-Ne Gaussian beams, so this condition was considered in the calculation below. The light pattern is divided into many intensity homogenous circles, whose intensities and areas areIi(i=1⋯n) and Si(i=1⋯n), respectively. Using Ii(i=1⋯n), from the ηtotal~E̅ curves calculated in Section 3.2.1, the DE of every circle at different time ηi,t(i=1⋯n) can be calculated. When read by the original R⃗, the diffracted light power PD,t can be got by submitting that of all the circles PD,i,t=ηi,t·Ii·Si, and then by dividing by the power of C⃗ (PC), the total DE of the sample for the Gaussian reading beam ηtotal,Gaussian,t=PD,t/PC can be obtained. The calculated DE~t curves of different kinds of polarization haplographies recorded in this sample, when the writing beams are 633 nm He-Ne Gaussian beams, whose average intensities are 78.6 mW/cm2 (same with the experiment values) and the original R⃗ is used as C⃗, are shown in Figure 9(b).It can be seen that the maximum values’ ratio of the measured values is basically coincide with the theoretically analyzed one. Only for parallel linearly polarization hologram, in the orthogonal linearly polarization reconstruction condition (C⃗⊥O⃗//R⃗), the diffraction efficiency is lower than parallel linearly polarization reconstruction condition (C⃗//O⃗//R⃗). It is because that, in calculation the affection of C⃗ is not considered, which can be proved to be very small when the IC=IO/100=IR/100 comparing to the affection of nonlinear absorption of the film, whose detail calculation progress will not be given here, where it also can be deduced that the reading beam affection is larger in C⃗⊥O⃗//R⃗ condition than in C⃗//O⃗//R⃗ condition.And the theoretical results are larger than the experimental results, and the reaction is quicker (optimal exposures are about 590 mJ/cm2 and 430 mJ/cm2, respectively, in experimental results and theoretical results); this may be caused by: (1) the sample is not homogeneous, the density is different at different area, (2) the incidence angles of beamsθ in the experiment are about 8.2°, so the intensities of them on the sample plane will be cosθ≈0.9898  times of the values used in calculation, (3) the photoreaction rate constants used in calculation is a little bit larger than real ones.And it can be seen that no matter during the ordinary holograph recording process in photochromic media, or during the polarization holograph recording process in photo-induced anisotropy media, there exits an overshooting peak in the diffraction efficiency, which then decays to a lower permanent level or also to zero. From the theoretical analyses, it can be deduced that it is caused by the diminishing of fringe contrast mainly caused by the nonlinear saturation effects of photoisomerization process and photo-induced anisotropy process. In experiment, there also exit the diminishing of fringe contrast caused by a photochemically active readout beam and unequal intensities of object and reference waves. It can be theoretically calculated that the effects of them show very smaller than that of the nonlinear saturation effects, which will not be given here.


### 3.2.3. The DE Spectra of Different Kinds of Holography in Fulgide Film

Suppose that the holographic recording is a linearity recording, from the spectra ofΔA, Δn, ΔAD, and ΔnB, shown in Figures 3(b) and 4(b), using the DE formulas of different kinds of polarization holographies, shown in Table 1, the DE spectra of the ordinary holograms (i.e., parallel circularly polarization recording) and polarization holograms (orthogonal linearly and orthogonal circularly polarization recording) can be calculated, which are shown in Figure 18, where the solid line, dot dash lines and dash lines indicate the ηtotal, ηA, and ηP, respectively. It can be seen that at 450 nm and 700 nm the diffraction efficiencies are higher, while the absorption is very small, where the recorded information can be read out without any photochromic reaction, that is, the nondestructive reconstruction can be realized.Diffraction efficiency spectra of the (a) Ordinary holography, (b) Orthogonal linearly polarization holography, and (c) Orthogonal circularly polarization holography.
(a)(b)(c)


## 3.2.1. Theoretical Calculation of DE(633 nm) ~E̅ Curves on the Exposure of Uniformity Beams

(1) The Calculation of DE in the Parallel Polarization Recording Condition
For parallel polarization recording condition, the optical intensity distribution should be(26)I(Δφ)=O2+R2+2⋅O⋅R⋅cos⁡Δφ=2⋅I0⋅(1+cos⁡Δφ).
So at recording time t, the exposure (E) distribution on the sample is
(27)E(Δφ,t)=I(Δφ)⋅t=2⋅I0⋅(1+cos⁡Δφ)⋅t.
So, for different kinds of C⃗, the T(C⃗,Δφ,t)~Δφ of the parallel linearly and parallel circularly recorded holograms at different t in the sample can be obtained from T~E curves shown in Figure 6(c)(28)T(Δφ,t)=T(E(Δφ))={T//(2⋅I0⋅(1+cos⁡Δφ)⋅t)parallellinearlyrecording,O⃗//R⃗//C⃗T⊥(2⋅I0⋅(1+cos⁡Δφ)⋅t)parallellinearlyrecording,O⃗//R⃗⊥C⃗(T//+T⊥)(2⋅I0⋅(1+cos⁡Δφ)⋅t)2parallelcircularlyrecording.(a) The Calculation of DE of the Amplitude Gratings (ηA)
The amplitude transmission changing curves on the phase differenceτ~Δφ at t will be
(29)τ(Δφ,t)=[T(Δφ,t)]1/2.
Every τ~Δφ grating curves at different t can be spread into Thaler progression, that is, simulated with the formula
(30)τ=τ0+∑n=110τn⋅cos⁡(n⋅Δφ).
Such the different order amplitude transmission gratings’ amplitude values τn(n=0,1,…,10) at different t can be obtained. So the changing curves of τn(n=0,1,…,10) on the recording exposure τn~E̅ can be obtained, and then the different order DE changing curves of amplitude gratings on the recording exposure (ηA,±n~E̅) can be calculated as
(31)η0=τ02,ηA,±n=τn24,(n=1,2,…,10).(b) The Calculation of DE of the Phase Gratings (ηP)
From theT(t)~Δφ, the A(t)~Δφ curves can be calculated
(32)A(Δφ,t)=log⁡(1T(Δφ,t)).
And then every A~Δφ grating curves at different t can be spread into Thaler progression, that is, simulated with the underside formula
(33)A=A0+∑i=110Ai⋅cos⁡(i⋅Δφ).
So, the different order absorption gratings’ amplitude values Ai(i=0,1,…,10) for different kinds of R⃗ at different t can be obtained. From Section 2, it is known that, at one wavelength, between the ΔA and Δn exit direct ratio relationship. And then after multiplexed with responding ratio value, the different order phase gratings’ amplitude values ni(i=0,1,…,10) for different polarization R⃗ at different t can be got
(34)ni=consti⋅Ai.
So, the different order DE changing curves of phase gratings on recording exposure (ηA,±n~E̅) can be calculated. Here only the first-order DE is discussed, From Section 3.1, it is known that only the n1 will induce the first-order diffracted light, and ηP,±1=(τ0·J1(k·n1·d))2.(c) The Calculation of DE of the Mixed Gratings (ηtotal)
Theηtotal can be calculated by submitting the ηA and ηP: η+1=ηA,+1+ηP,+1.(d) The Calculation Results of DE of Fulgide Film in Parallel Polarization Recording
For example, in the parallel linearly polarization recording parallel polarization reading condition (C⃗//O⃗//R⃗), when I0=50 mW/cm2, from the T//~E curve shown in Figure 6(c), the T//(t)~Δφ of the holographic grating at different recording times t can be calculated, from which the τ//(t)~Δφ can be obtained, which is shown in Figure 13.
Everyτ//~Δφ curve was spread into Thaler progression, that is was simulated with (30), then the τn (n=0,1,…,10) on the direction of the exiting beam polarization at different t were obtained; for example, in the Figure 14 two typical amplitude transmission grating curves (sine distribution and negative triangle wave function) at different t and their simulated curves are given. So, the τn~E̅  (n=0,1,…,10) curves can be got, from which the ηA,±n~E̅(n=0,1,…,10) curves can be calculated, from which the first-order DE (ηA,±1) is chosen and shown in Figure 15 as dash line 2. It can be seen that even if the ORR = 1 : 1, because the nonlinear recording the DE will also drop down. The optimal exposure is about 1 J/cm2, and at the uniformity light recording condition, the maximum ηA,±1 is 2%. From the T//(t)~Δφ curves, the A//(t)~Δφ curves can be calculated. After every A//(t)~Δφ curve is spread into Thaler progression, the Ai(i=0,1,…,10) on the direction of the exiting beam polarization at different t can be obtained. In Section 2, it is given that Δn(633 nm)/ΔA(633 nm) = 0.00994. So after the Ai multiplexed with 0.00994, the ni,// (i=0,1,…,10) at different t can be got, from which the ηP,±1~E̅ can be calculated, which is shown in Figure 16. The DE of the mixed holograph ηtotal can be obtained by submitting that of the amplitude grating and phase grating, shown in Figure 16. It can be seen that at the uniformity light recording condition, the optimal exposure is about 1 J/cm2, and, the maximum η±1 is about 3.56%.It can be seen that ηA,±1 and ηP,±1 nearly has direct ratio in parallel linearly recording hologrpahy, the decreasing rate of ηP,±1 is a little bit slower than that of ηA,±1, which is ignored in other conditions in parallel polarization recording, and just the ηA,±1 is calculated as before and ηtotal is took as 1.89 times of the ηA,±1.
Using the same method shown before, fromT⊥ ~E curve shown in Figure 6(c), the ηA,±n~E̅ curves for the C⃗, which has perpendicular polarization to that of the exciting beams (C⃗⊥O⃗//R⃗), are calculated, from which the ηA,±1 is shown in Figure 15. Comparing with the condition of C⃗//O⃗//R⃗, just the changing rate is slow down. When reconstruct with circularly polarized light, the diffracted lights are the superposed light of the diffracted lights of vertical and horizontal polarized light component of the C⃗, so it will be ellipse polarized lights, whose long axes are along the vertical direction and the DE are the average values of that of the vertical and horizontal polarized lights, which is shown in Figure 15.
In the parallel circularly polarization holograph (ORR = 1 : 1), theT distribution is the same for any kinds of polarization state lights, so the DE are also the same. Using the same method shown before, from the T(t)~Δφ curves of parallel circularly recording condition shown in (28), the ηA,±n~E̅ curves of parallel circularly recording holograms are calculated and shown in Figure 15.Figure 13
The amplitude transmission changing curves depending on the phase difference on the direction of the exiting beam polarization at different recording time.Amplitude transmission grating curves and their simulating curves (a)t=2 s; (b) t=160 s.
(a)(b)Figure 15
The first-order diffraction efficiencies of the amplitude gratings in parallel polarization holograms.Figure 16
The first-order diffraction efficiencies of the amplitude grating and phase grating in the parallel linearly recorded mixing holograph.(2) The Calculation of DE in the Orthogonal Polarization Recording Condition
From Section3.1, it is known that in orthogonal polarization recording condition at nonlinear recording area, the DE equations of the linearly recording area shown in Table 1 also can be used. From Figure 6(c), the τe~E and τo~E curves of sample under uniformity light irradiation can be calculated. Then from them using the DE formulas, the ηA,+1~E̅ curves of the orthogonal linearly polarization holography and orthogonal circularly polarization holography can be calculated, which are shown in Figure 17.
It can be seen that the optimal exposure is 1.2 J/cm2, and the maximum amplitude grating DE are, respectively, 0.4% and 1.6%. The decreasing rates are quicker than parallel polarization holography. In Section 2, it is given that in this sample ΔnB(633 nm)/ΔAD(633 nm) = 0.006115. The ηP,+1, the same as the condition in parallel polarization holography, nearly has direct ratio with the ηA,+1, the ηtatol in mixing holograph is about 1.28 times of ηA,+1.Figure 17
The first-order diffraction efficiencies of the amplitude gratings in orthogonal polarization holograms.


## 3.2.2. The Calculation of DE ~E̅ Curves on the Exposure of the Gaussian Beams

Because in experiment, the writing and reading beams are 633 nm He-Ne Gaussian beams, so this condition was considered in the calculation below. The light pattern is divided into many intensity homogenous circles, whose intensities and areas areIi(i=1⋯n) and Si(i=1⋯n), respectively. Using Ii(i=1⋯n), from the ηtotal~E̅ curves calculated in Section 3.2.1, the DE of every circle at different time ηi,t(i=1⋯n) can be calculated. When read by the original R⃗, the diffracted light power PD,t can be got by submitting that of all the circles PD,i,t=ηi,t·Ii·Si, and then by dividing by the power of C⃗ (PC), the total DE of the sample for the Gaussian reading beam ηtotal,Gaussian,t=PD,t/PC can be obtained. The calculated DE~t curves of different kinds of polarization haplographies recorded in this sample, when the writing beams are 633 nm He-Ne Gaussian beams, whose average intensities are 78.6 mW/cm2 (same with the experiment values) and the original R⃗ is used as C⃗, are shown in Figure 9(b).It can be seen that the maximum values’ ratio of the measured values is basically coincide with the theoretically analyzed one. Only for parallel linearly polarization hologram, in the orthogonal linearly polarization reconstruction condition (C⃗⊥O⃗//R⃗), the diffraction efficiency is lower than parallel linearly polarization reconstruction condition (C⃗//O⃗//R⃗). It is because that, in calculation the affection of C⃗ is not considered, which can be proved to be very small when the IC=IO/100=IR/100 comparing to the affection of nonlinear absorption of the film, whose detail calculation progress will not be given here, where it also can be deduced that the reading beam affection is larger in C⃗⊥O⃗//R⃗ condition than in C⃗//O⃗//R⃗ condition.And the theoretical results are larger than the experimental results, and the reaction is quicker (optimal exposures are about 590 mJ/cm2 and 430 mJ/cm2, respectively, in experimental results and theoretical results); this may be caused by: (1) the sample is not homogeneous, the density is different at different area, (2) the incidence angles of beamsθ in the experiment are about 8.2°, so the intensities of them on the sample plane will be cosθ≈0.9898  times of the values used in calculation, (3) the photoreaction rate constants used in calculation is a little bit larger than real ones.And it can be seen that no matter during the ordinary holograph recording process in photochromic media, or during the polarization holograph recording process in photo-induced anisotropy media, there exits an overshooting peak in the diffraction efficiency, which then decays to a lower permanent level or also to zero. From the theoretical analyses, it can be deduced that it is caused by the diminishing of fringe contrast mainly caused by the nonlinear saturation effects of photoisomerization process and photo-induced anisotropy process. In experiment, there also exit the diminishing of fringe contrast caused by a photochemically active readout beam and unequal intensities of object and reference waves. It can be theoretically calculated that the effects of them show very smaller than that of the nonlinear saturation effects, which will not be given here.


## 3.2.3. The DE Spectra of Different Kinds of Holography in Fulgide Film

Suppose that the holographic recording is a linearity recording, from the spectra ofΔA, Δn, ΔAD, and ΔnB, shown in Figures 3(b) and 4(b), using the DE formulas of different kinds of polarization holographies, shown in Table 1, the DE spectra of the ordinary holograms (i.e., parallel circularly polarization recording) and polarization holograms (orthogonal linearly and orthogonal circularly polarization recording) can be calculated, which are shown in Figure 18, where the solid line, dot dash lines and dash lines indicate the ηtotal, ηA, and ηP, respectively. It can be seen that at 450 nm and 700 nm the diffraction efficiencies are higher, while the absorption is very small, where the recorded information can be read out without any photochromic reaction, that is, the nondestructive reconstruction can be realized.Diffraction efficiency spectra of the (a) Ordinary holography, (b) Orthogonal linearly polarization holography, and (c) Orthogonal circularly polarization holography.
(a)(b)(c)


## 4. Application in Holographic Image Storage

### 4.1. Holographic Image Storage Setup

The property difference of different polarization holograms have be seen in real image holographic storage, whose optical set up is shown in Figure19. The same as the measurement set up, the 633 nm He-Ne laser (3 mW, random polarized) and 405 nm diode laser are used as recording and erasing light sources, respectively; shutter S1 and S2 used to control their exposure times; quarter-wave plates Q1~Q3 and attenuators A1~A2 are used to change the polarization states and intensities of the recording (object light O and reference light R), reading (C) and diffracted (D) waves. We know that in the orthogonal polarization holography, when C-beam has the same polarization state with R-beam, the D-beam has orthogonal polarization with C-beam, while the scattering noise has similar polarization with C-beam. So using the Q3 and polarizer P in front of L3, the scattering noise can be filtered in orthogonal polarization holography. Located on the front focus plane of Fourier transformation lens L3, spatial light modulator (SLM) loads the encoded data image from computer PC1. And the diaphragm D1, located on the special frequency spectrum plane of 4 f system composed of L3 and L4, is used to filter high-order diffractive waves of grating structure of SLM, so on the back focus plane of L4 the object image (OI) without grating structure can be obtained. In the R-beam path, lens L6 and L7 comprise another 4 f system, and rotatable mirror M2 and the sample, located separately on the focus planes of L6 and L7, are used to realize angle multiplexing. L5 and L8 compose another 4 f system, L5 performs the Fourier transformation of the object image (OI) at the position of the Fulgide film (sample), and Lens L8 performs the inverse Fourier transform at the plane of the CCD camera to yield the diffracted image. Four different polarization recordings like listed in Table 1 are stored with this set up.Figure 19
Experimental setup of polarization holographic image storage. PBS, polarization beam splitter;M1~M4, mirrors; L1~L8, lens; A1 and A2, continuously adjustable attenuators; D1 and D2: diaphragm; P, polarizer; Q1~Q3, quarter-wave plates; S1 and S2, shutters; LD, 405 nm laser diode; He-Ne: 633 nm Helium-Neon laser; O-beam, object beam; R-beam, reference beam; C-beam, reconstruction beam; D-beam, diffracted beam; E-beam, erasure violet light.


### 4.2. Holographic Image Storage Results

Figure20 shows the comparison of retrieval diffractive images with different polarization recording waves on the fulgide film, when reconstruction beams have same polarization state with reference beams. It can be seen that these results are same with the measured results shown above. When the intensities of recording wave and readout wave are certain, parallel linear polarization recording obtains the highest diffraction efficiency as well as high scattering noise; parallel circular polarization recording obtains high diffraction efficiency and scattering noise; orthogonal linear polarization recording obtains low scattering noise and diffraction efficiency; orthogonal circular polarization recording obtains higher diffraction efficiency and high signal-to-noise ratio. So, orthogonal circular polarization recording is chosen as the method of polarization holographic data storage on the fulgide film and Fourier transformation holographic method is chosen to get high storage density.Results of holographic storage experiments with different polarization recording waves on Fulgide film: (a) parallel linear polarization recording; (b) parallel circular polarization recording; (c) orthogonal linear polarization recording; (d) orthogonal circular polarization recording.
(a)(b)(c)(d)Figure21 shows the experimental results of orthogonal circular polarization holographic optical data storage on Fulgide film by Fourier transformation holographic method. The images are separately stored file, encoded binary monochromic image, retrieval diffractive image, decoded result, retrieved file, and measurement of size of holographic image. The retrieval diffractive image that is clear is processed by decoding procedure, and the obtained retrieved file is the same as the stored file. In the experiment, data size of each stored holography is 81×61 bits, and the size of holography is 60 μm×42 μm. So the storage area density of 2 × 108 bits/cm2 is obtained. The nonhomogeneity and flaws of the surface of sample, uncoaxiality of optical elements and uncertainty of adjustment brings out some distortion of the diffractive image and error codes that can be reduced to the minimum by designing reasonable encoding and decoding procedures.Results of orthogonal circular polarization holographic optical data storage on BR-D96N film by Fourier transformation holographic method. (a) Stored file. (b) Encoded binary monochromic image. (c) Retrieval diffractive image. (d) Decoded result. (e) Retrieved file. (f) Measurement of size of hologram (one grid of the scale corresponds to 10μm).
(a)(b)(c)(d)(e)(f)


## 4.1. Holographic Image Storage Setup

The property difference of different polarization holograms have be seen in real image holographic storage, whose optical set up is shown in Figure19. The same as the measurement set up, the 633 nm He-Ne laser (3 mW, random polarized) and 405 nm diode laser are used as recording and erasing light sources, respectively; shutter S1 and S2 used to control their exposure times; quarter-wave plates Q1~Q3 and attenuators A1~A2 are used to change the polarization states and intensities of the recording (object light O and reference light R), reading (C) and diffracted (D) waves. We know that in the orthogonal polarization holography, when C-beam has the same polarization state with R-beam, the D-beam has orthogonal polarization with C-beam, while the scattering noise has similar polarization with C-beam. So using the Q3 and polarizer P in front of L3, the scattering noise can be filtered in orthogonal polarization holography. Located on the front focus plane of Fourier transformation lens L3, spatial light modulator (SLM) loads the encoded data image from computer PC1. And the diaphragm D1, located on the special frequency spectrum plane of 4 f system composed of L3 and L4, is used to filter high-order diffractive waves of grating structure of SLM, so on the back focus plane of L4 the object image (OI) without grating structure can be obtained. In the R-beam path, lens L6 and L7 comprise another 4 f system, and rotatable mirror M2 and the sample, located separately on the focus planes of L6 and L7, are used to realize angle multiplexing. L5 and L8 compose another 4 f system, L5 performs the Fourier transformation of the object image (OI) at the position of the Fulgide film (sample), and Lens L8 performs the inverse Fourier transform at the plane of the CCD camera to yield the diffracted image. Four different polarization recordings like listed in Table 1 are stored with this set up.Figure 19
Experimental setup of polarization holographic image storage. PBS, polarization beam splitter;M1~M4, mirrors; L1~L8, lens; A1 and A2, continuously adjustable attenuators; D1 and D2: diaphragm; P, polarizer; Q1~Q3, quarter-wave plates; S1 and S2, shutters; LD, 405 nm laser diode; He-Ne: 633 nm Helium-Neon laser; O-beam, object beam; R-beam, reference beam; C-beam, reconstruction beam; D-beam, diffracted beam; E-beam, erasure violet light.


## 4.2. Holographic Image Storage Results

Figure20 shows the comparison of retrieval diffractive images with different polarization recording waves on the fulgide film, when reconstruction beams have same polarization state with reference beams. It can be seen that these results are same with the measured results shown above. When the intensities of recording wave and readout wave are certain, parallel linear polarization recording obtains the highest diffraction efficiency as well as high scattering noise; parallel circular polarization recording obtains high diffraction efficiency and scattering noise; orthogonal linear polarization recording obtains low scattering noise and diffraction efficiency; orthogonal circular polarization recording obtains higher diffraction efficiency and high signal-to-noise ratio. So, orthogonal circular polarization recording is chosen as the method of polarization holographic data storage on the fulgide film and Fourier transformation holographic method is chosen to get high storage density.Results of holographic storage experiments with different polarization recording waves on Fulgide film: (a) parallel linear polarization recording; (b) parallel circular polarization recording; (c) orthogonal linear polarization recording; (d) orthogonal circular polarization recording.
(a)(b)(c)(d)Figure21 shows the experimental results of orthogonal circular polarization holographic optical data storage on Fulgide film by Fourier transformation holographic method. The images are separately stored file, encoded binary monochromic image, retrieval diffractive image, decoded result, retrieved file, and measurement of size of holographic image. The retrieval diffractive image that is clear is processed by decoding procedure, and the obtained retrieved file is the same as the stored file. In the experiment, data size of each stored holography is 81×61 bits, and the size of holography is 60 μm×42 μm. So the storage area density of 2 × 108 bits/cm2 is obtained. The nonhomogeneity and flaws of the surface of sample, uncoaxiality of optical elements and uncertainty of adjustment brings out some distortion of the diffractive image and error codes that can be reduced to the minimum by designing reasonable encoding and decoding procedures.Results of orthogonal circular polarization holographic optical data storage on BR-D96N film by Fourier transformation holographic method. (a) Stored file. (b) Encoded binary monochromic image. (c) Retrieval diffractive image. (d) Decoded result. (e) Retrieved file. (f) Measurement of size of hologram (one grid of the scale corresponds to 10μm).
(a)(b)(c)(d)(e)(f)


## 5. Conclusions

From the analysis of diffracted wave polarization states in the different kinds of polarization holograms, it can be seen that in the orthogonal polarization holography, if reconstructed with the reference light, the polarization state of the diffracted light is orthogonal to that of the reconstruction light, which is very important to increase the SNR of the holographic storage.From the dynamic curve calculation of the film, it can be seen that, the decrease of the diffraction efficiencies at the high exposure is caused by the nonlinear recording of the sample, and the effect of the reading beam is very small. The maximum values of the diffraction efficiencies at 633 nm of the parallel linearly, parallel circularly, orthogonal linearly, and orthogonal circularly polarized holograms recorded in 3-indoly-benzylfulgimide/PMMA film are 3.56%, 3.026%, 0.49%, and 1.96%, respectively. In the condition of the Gaussian beam, the experimentally measured values are smaller than the theoretical calculated ones, but the ratio between the diffraction efficiencies of different kinds of polarization holographs is nearly the same as the calculated ones. The maximum measured diffraction efficiency of the parallel linearly polarization hologram is 1%.From the diffraction efficiency spectrum, it can be seen that at the wavelengths less than 450 nm or greater than 700 nm the nondestructive reading can be realized.The results obtained in holographic image storage proved that the results obtained in calculation and measurement experiment are correct. In Fourier transformation orthogonal circular polarization recording, high SNR and high density of 2 × 108 bits/cm2 are obtained.



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*Source: 509507-2011-05-23.xml*