Null reconstruction of orthogonal circular polarization hologram with large recording angle

An’an Wu; Guoguo Kang; Jinliang Zang; Ying Liu; Xiaodi Tan; Tsutomu Shimura; Kazuo Kuroda

doi:10.1364/OE.23.008880

1. Introduction

In conventional holography, the amplitude and phase variations between the signal and reference wave are recorded in an interference pattern [1–3]. However, in the polarization holography, it employs waves with two different polarizations for recording information. In this case, not only the amplitude and phase but also the polarization states are to be recorded on the polarization-sensitive materials [4].

For a long time, the polarization holography hasn’t attracted much interest due to the lack of a systematic theory. Although some theories for polarization holography have been proposed by several authors [5–12], unfortunately, most of them are quite complicated for the reason that the response of the recording material is described as a function of the canonical form of the polarization state. Moreover, they assumed the signal and reference waves propagate almost in parallel.

In the research of polarization holography, some interesting phenomena were found. For example, the null reconstruction to be investigated in this paper was experimentally observed by Todorov in 1986 and theoretically explained by Huang using paraxial coupled mode theory in 1995. Prof. Shiuan Huei Lin of National Chiao Tung University also found this phenomenon under paraxial approximation [13]. Their polarization hologram was recorded by orthogonal circularly polarized waves. In the reconstruction process, by reading out the hologram with the orthogonal reference wave, the diffraction efficiency was zero. Their experiment and theory were based on the paraxial approximation, i.e., the cross angle between the signal and reference wave was very small, which indeed leads to the null reconstruction. However, in addition to the small cross angle, some other factors that results in the null reconstruction were found recently [14]. As a result, the null reconstruction of polarization hologram can also be observed with a large recording angle, successfully breaking the bonds of paraxial approximation. In this paper, we studied the null reconstruction in the case of the polarization volume hologram that was recorded with a large cross angle. Without the paraxial approximation, whether the polarization state of the signal wave can be reconstructed faithfully or not will also be discussed in this paper.

In recent years, a new tensor theory of polarization holography was proposed without the paraxial limitation. In this theory, the polarization hologram is expressed as the tensor product or dyadic product of the total electric field. It is applicable to any geometry of interfering beams, a large crossing angle, and a skew geometry with arbitrary polarizations [14]. Thanks to the development of theory, a boost in the research of polarization holography is on its way [15, 16].

Thus, in this paper, we first present the new tensor theory of polarization holography and derive the reconstruction characteristics of the polarization volume hologram that was recorded by the orthogonal circularly polarized waves. Then, in Section 3, we describe experimental studies to verify our theoretical results. Both the reconstruction characteristics of the hologram that read out by the original reference wave and orthogonal reference wave will be presented. Finally, some conclusions together with the potential applications of the null reconstruction, especially in the holographic data storage are given in Section 4.

2. Theoretical explanation of null reconstruction without paraxial approximation

The tensor theory of the polarization holography is based on the photo-induced change of the dielectric tensor of the polarization-sensitive materials. The recording process of polarization holography is shown in Fig. 1(a). We consider the polarization hologram is recorded by superposing a signal wave that is a plane wave incident at angle of θ_O with a reference wave that is a plane wave incident at angle θ_R. The signs of θ_O and θ_R are opposite. The vector amplitudes of the signal wave and reference wave are O and R respectively. The electric field can be written as

E = O e x p (i k_{O} \cdot r) + R e x p (i k_{R} \cdot r) .

where r is the position vector,

k_{O}

and

k_{R}

are the wave vectors of the signal wave and reference wave, respectively.

Fig. 1 Schematic diagram of polarization holography: (a) recording, (b) reconstruction.

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In polarization holography, the hologram is recorded by the modulation of periodic polarization states on the polarization-sensitive medium. In this case, the dielectric coefficient should be expressed as a tensor. Based on the molecular model of materials in Ref [14], the dielectric tensor can be written as

[ε] = (n_{0}^{2} + α | E |^{2}) 1 + β (E E^{*} + E^{*} E)

where n₀ is the refractive index of material before the exposure, 1 is the 2nd-order unit tensor. EE* and E*E are tensor product or dyadic product. The scalar α and β are coefficients of the scalar and tensor components of the photo-induced change in the dielectric tensor, and refer to the coefficients for intensity and polarization holograms, respectively.

During the reconstruction process, as shown in Fig. 1(b), the reconstructed wave S is obtained by the illumination of the reading wave F. For simplicity, we assumed the signal and reference waves have the common incident plane and the Bragg condition is satisfied, that is

\begin{array}{l} k_{F} = k_{R}, k_{S} = k_{O}, \\ θ_{F} = θ_{R}, θ_{S} = θ_{O} . \end{array}

By using the coupled wave theory for vector wave, the reconstructed wave S can be written as

\begin{array}{l} S \propto β (R^{*} \cdot F) O + {α (O \cdot R^{*}) F + β (O \cdot F) R^{*} \\ - [(α (O \cdot R^{*}) F + β (O \cdot F) R^{*}) \cdot {\hat{k}}_{O}] {\hat{k}}_{O}} \end{array}

where

{\hat{k}}_{O}

is the unit wave vector of the signal wave.

In the following, we will study the reconstruction characteristics of the polarization volume hologram that was recorded by the orthogonal circularly polarized waves. Here, we define vectors s and p is orthogonal polarization vectors associated with the signal wave and reference wave in the coordinate system as shown in Fig. 1.

s_{j} = (\begin{array}{l} 0 \\ 1 \\ 0 \end{array}), p_{j} = (\begin{array}{l} \cos θ_{j} \\ 0 \\ \sin θ_{j} \end{array})

where j = O, R, F, S.

In the orthogonal circular polarization hologram, regardless of the signal wave is left-handed circular polarization and the reference wave is right-handed circular polarization or the signal wave is right-handed circular polarization and the reference wave is left-handed circular polarization, the conclusion will be the same. Hence, we only consider the first case. We define the intensity of the signal wave and reference wave as 1. The signal wave and the reference wave can be written as

O = \frac{1}{\sqrt{2}} (s_{O} + i p_{O}), R = \frac{1}{\sqrt{2}} (s_{R} - i p_{R}) .

We use a reference wave with an arbitrary elliptic polarization state to read out the hologram. The reference wave with an arbitrary elliptic polarization state can be expressed as

F = l s_{F} + m e^{i φ} p_{F}

where l and m are constant, φ is the phase difference between the

s_{F}

and

p_{F}

.

From the Eq. (4), we can get

\begin{array}{l} S \propto {α l [1 - \cos (θ_{O} - θ_{R})] + 2 β l + i β m e^{i φ} [1 + \cos (θ_{O} - θ_{R})]} s_{O} \\ + {α m e^{i φ} \cos (θ_{O} - θ_{R}) [1 - \cos (θ_{O} - θ_{R})] + i β l [1 + \cos (θ_{O} - θ_{R})] \\ - β m e^{i φ} [1 + \cos^{2} (θ_{O} - θ_{R})} p_{O} . \end{array}

In order to obtain the null reconstruction, we can get Eqs. (9) and (10)

α l [1 - \cos (θ_{O} - θ_{R})] + 2 β l + i β m e^{i φ} [1 + \cos (θ_{O} - θ_{R})] = 0,

α m e^{i φ} \cos (θ_{O} - θ_{R}) [1 - \cos (θ_{O} - θ_{R})] + i β l [1 + \cos (θ_{O} - θ_{R})] - β m e^{i φ} [1 + \cos^{2} (θ_{O} - θ_{R})] = 0.

From Eqs. (9) and (10), we can get the necessary conditions of null reconstruction.

l = m, φ = \frac{π}{2} .

It means that the reference wave we use to read out the hologram is left-handed circular polarization. It is orthogonal to the original reference wave. Besides l = m and φ = π/2,

\cos (θ_{O} - θ_{R}) = 1

or

α + β = 0

has to be satisfied to achieve the null reconstruction. When

\cos (θ_{O} - θ_{R}) = 1

, namely

θ_{O} = θ_{R} = 0^{\circ}

, the reconstructed wave is disappeared. This is the case of null reconstruction under the paraxial approximation. However, without the paraxial approximation, the condition α + β = 0 should be satisfied. The values of α and β varies with the increase of exposure. Therefore, only when the exposure reaches a certain level, at this point, the polarization and intensity holograms attain a balance, the null reconstruction occurs.

To sum up, the null reconstruction can be achieved when we use the orthogonal reference wave to read out the orthogonal circular polarization hologram under the condition that α + β = 0 is satisfied. This is different from the null reconstruction in the case of paraxial approximation.

In addition, we can get the conditions that the polarization states of signal wave can be reconstructed faithfully without the paraxial approximation as following,

l = - m, φ = \frac{π}{2}, α + β = 0.

Hence, we can conclude that the polarization states can be reconstructed faithfully when we use the original reference wave to read out the orthogonal circular polarization hologram as long as α + β = 0 is satisfied.

3. Experimental investigation

We have experimentally investigated the null reconstruction and faithful reconstruction of polarization volume hologram based on the orthogonal circularly polarized waves without paraxial approximation. According to Eqs. (11) and (12), when we use the original reference wave and orthogonal reference wave use to read out the hologram respectively, we can get

\begin{array}{l} S \propto [β + \frac{1}{4} (α + β) (1 + \cos^{2} (θ_{O} - θ_{R})) - \frac{1}{2} (α - β) \cos (θ_{O} - θ_{R})] l \\ + \frac{1}{4} (α + β) (1 - \cos^{2} (θ_{O} - θ_{R})) r, \end{array}

S^{'} \propto \frac{1}{4} (α + β) (1 - \cos^{2} (θ_{O} - θ_{R})) l + \frac{1}{4} (α + β) {(1 - \cos (θ_{O} - θ_{R}))}^{2} r

where l and r represent the left- and right-handed circularly polarized component of the reconstructed wave, respectively. All the intensities of the recording wave and reconstruction wave are defined as 1.

The intensity is more easily to be measured. We divide the reconstructed wave into left-handed circularly polarized component and right-handed circularly polarized component. Then we measure the intensity respectively.

When we use the original reference wave to reconstruct, from Eq. (13), the intensity of the two components can be written as

I_{L} \propto {[β + \frac{1}{4} (α + β) (1 + \cos^{2} (θ_{O} - θ_{R})) - \frac{1}{2} (α - β) \cos (θ_{O} - θ_{R})]}^{2} I_{F},

I_{R} \propto \frac{1}{16} {(α + β)}^{2} {(1 - \cos^{2} (θ_{O} - θ_{R}))}^{2} I_{F}

Where

I_{L}

and

I_{R}

are the intensity of the left- and right-handed circularly polarized components of the reconstructed wave, respectively,

I_{F}

is the intensity of the reference wave.

When we use the orthogonal reference wave to reconstruct, from Eq. (14), the intensity of the two components can be written as

I_{L}^{'} \propto \frac{1}{16} {(α + β)}^{2} {(1 - \cos^{2} (θ_{O} - θ_{R}))}^{2} I_{F}^{'},

I_{R}^{'} \propto \frac{1}{16} {(α + β)}^{2} {(1 - \cos (θ_{O} - θ_{R}))}^{4} I_{F}^{'}

where

I_{L}^{'}

,

I_{R}^{'}

and

I_{F}^{'}

are defined as the same as

I_{L}

,

I_{R}

and

I_{F}

.

Equations (15)-(18) show the tendency of the intensity of each component of the reconstructed wave theoretically. Based on these equations, when α + β = 0 is satisfied, the intensities of these component can be expressed as

I_{L} \propto β^{2} {[1 + \cos (θ_{O} - θ_{R})]}^{2} I_{F},

I_{R} = 0,

I_{L}^{'} = 0,

I_{R}^{'} = 0.

The experiment setup is shown in Fig. 2. A laser with a wavelength of 532 nm was used for recording and reconstructing. The laser was split into signal wave and reference wave by using the polarization beam splitter (PBS1). The polarization state of the signal wave was fixed to left-handed circular polarization by using a polarizer (P1) and a quarter-wave plate (QWP1). The polarization state of the reference wave was fixed to right-handed circular polarization by using P2 and QWP2. The signal wave and reference wave are plane waves with an intensity ratio at 1:1. They were incident in the common plane with the crossing angle of $θ_{O} - θ_{R} \approx 42^{\circ}$ . Under the paraxial approximation, the cross angle must be smaller than 10 degrees. Therefore, the cross angle in this experiment conforms to the large recording angle. The hologram was recorded on a 2-mm thick polarization-sensitive medium composed of 9, 10-phenanthrenequinone-doped poly-methyl methacrylate (PQ-PMMA). The characteristics of PQ-PMMA have been described in Ref [17]. and [18]. In the reconstruction process, we stopped exposure and probed after different exposure times. We used the original and orthogonal reference wave to reconstruct respectively. The reconstructed wave was split to left- and right-handed circularly polarized components by using QWP3 and PBS2. Then we can measure the intensity respectively. With these two components, we can analyze the reconstructed wave that we got from the reconstruction process.

Fig. 2 Experimental setup of polarization holography based on the orthogonal circularly polarized wave: HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarization beam splitter; P, polarizer.

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Figure 3(a) shows the measured intensity of the left- and right-handed circularly polarized components of the reconstructed wave that was reconstructed by the original reference wave. As shown in Fig. 3(a), at about 18 seconds, the right-handed circularly polarized component of the reconstructed wave decreases to zero. At this point the reconstructed wave only have the left-handed circularly polarized component, it means that the reconstructed wave is the left-handed circularly polarized wave. Therefore, the faithful reconstruction was achieved, and thereof we believe that α + β = 0 was satisfied at this moment.

Fig. 3 Reconstructed by the original reference wave (a) and orthogonal reference wave (b).

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Figure 3(b) shows the measured intensity of the left- and right-handed circularly polarized components of the reconstructed wave that reconstructed by the orthogonal reference wave. At the same time as the faithful reconstruction, both the intensity of the left- and right-handed circularly polarized components of the reconstructed wave decrease to zero. It means that the reconstructed wave is disappeared when we use the orthogonal reference wave to reconstruct.

From the above, the coefficients α and β related to the material change with the different exposure time. With a large recording angle, when the values of α and β are satisfied α + β = 0, that is the polarization and intensity hologram achieve a balance, the faithful and null reconstruction can be achieved. The intensity of each component is consistent with the Eqs. (19)-(22).The tendency of the intensity of each component is conform to the Eqs. (15)-(18). Therefore, the experimental results verify the theoretical derivation.

4. Conclusion

We investigated the null reconstruction and faithful reconstruction in the case of the polarization volume hologram that was recorded with a large cross angle. Just when the polarization and intensity hologram reach a balance, namely, the coefficients related to the material are satisfied α + β = 0, the null reconstruction and faithful reconstruction can be realized by the illumination of the orthogonal reference wave and original reference wave. Under the paraxial approximation, the small cross angle limits its practical application. However, in this paper, the large cross angle does not have this limit. As a consequence, the null reconstruction will have various potential applications, such as a potential enhancement in optical storage capacity for volume holograms. Holographic data storage is promising because of its large storage capacity and high transfer rate [19, 20]. In holographic data storage, the storage capacity can be improved by null reconstruction. We can record two holograms at the same position with the same Bragg condition. One is recorded by using the signal wave that is left-handed circular polarization and the reference wave with right-handed circular polarization. The other one is recorded by the waves with the polarizations that are mutual exchange in the first case. In the reconstruction process, we can use right-handed circularly polarized wave and left-handed circularly polarized wave to read out the holograms, respectively. The right-handed circularly polarized wave can read out the first hologram faithfully. But the second hologram cannot be read out because of the null reconstruction. It is a similar situation when we use the left-handed circularly polarized wave to read out the holograms. Hence, the two holograms can be read out independently. With this method, considering the storage capacity, it is simple and effective to double the storage density. In the following work we will extend the application of null reconstruction and investigate the polarization hologram recorded by more complex polarization states, i.e., elliptical polarizations.

Acknowledgments

This work is supported by a grant from the National Natural Science Foundation of China (Grant No. 61475019 and 61205053). We thank Prof. Shiuan Huei Lin of National Chiao Tung University for providing the materials.

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Null reconstruction of orthogonal circular polarization hologram with large recording angle

Abstract

1. Introduction

2. Theoretical explanation of null reconstruction without paraxial approximation

3. Experimental investigation

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (22)

Optics Express