Theory of Polychromatic Reconstruction for Volume Holographic Memory

In volume holographic memory (van Heerden, 1963), the information is stored as a volume hologram and retrieved through the holographic reconstruction process by illuminating the hologram with a readout probe beam whose wavelength, incident angle, and wavefront should be identical to those of the reference beam used in the recording process. This requirement stems from the fact that diffraction from the volume hologram is restricted by Bragg’s law. While such a restriction is responsible for the large storage density of volume holographic memories, it also causes some obstacles for implementing practical memory systems. For example, in rewritable recording media, like photorefractive materials, illumination with a readout probe beam will rewrite the recorded hologram, destroying the stored information. Even in a photopolymer, some of the storage capacity will be wasted during the readout if some monomers still exist in the readout volume. These issues are obviously caused by the destructive probe beam having the ability to expose the recording medium in a similar manner to the recording beam. To avoid such a problem, several nondestructive readout methods have been proposed so far (Gulanyan et al., 1979; Petrov et al., 1979; Külich, 1987), where the readout is performed at a longer wavelength, outside the sensitive spectral region of the recording material. These methods can successfully reconstruct the stored image at a wavelength different from the recording one, but most of these methods may not be practical for holographic memory systems because the multiplexing capability is considerably lowered. For example, anisotropic diffraction (Petrov et al., 1979) requires a specific recording configuration and thus limits the number of multiplexed pages. A spherical probe beam method (Külich, 1987) tends to produce severe crosstalk noise from other multiplexed pages, which demands a large angular separation between two adjacent multiplexed holograms, resulting in a small storage density. Recently, we proposed another way to reconstruct an image at a different wavelength (Fujimura et al., 2007). Our method, which we call polychromatic reconstruction (PCR), utilizes a spectrally broad light source for the probe beam, as shown in Fig. 1. Each angular spectral component of the recorded gratings can be Bragg-matched with one particular wavelength within the broadband spectrum of the probe beam. Thus, the whole image can be reconstructed from the volume hologram even though the probe wavelength is very different from the recording one. On the other hand, analogous to the spherical probe beam


Introduction
In volume holographic memory (van Heerden, 1963), the information is stored as a volume hologram and retrieved through the holographic reconstruction process by illuminating the hologram with a readout probe beam whose wavelength, incident angle, and wavefront should be identical to those of the reference beam used in the recording process. This requirement stems from the fact that diffraction from the volume hologram is restricted by Bragg's law. While such a restriction is responsible for the large storage density of volume holographic memories, it also causes some obstacles for implementing practical memory systems. For example, in rewritable recording media, like photorefractive materials, illumination with a readout probe beam will rewrite the recorded hologram, destroying the stored information. Even in a photopolymer, some of the storage capacity will be wasted during the readout if some monomers still exist in the readout volume. These issues are obviously caused by the destructive probe beam having the ability to expose the recording medium in a similar manner to the recording beam. To avoid such a problem, several nondestructive readout methods have been proposed so far (Gulanyan et al., 1979;Petrov et al., 1979;Külich, 1987), where the readout is performed at a longer wavelength, outside the sensitive spectral region of the recording material. These methods can successfully reconstruct the stored image at a wavelength different from the recording one, but most of these methods may not be practical for holographic memory systems because the multiplexing capability is considerably lowered. For example, anisotropic diffraction (Petrov et al., 1979) requires a specific recording configuration and thus limits the number of multiplexed pages. A spherical probe beam method (Külich, 1987) tends to produce severe crosstalk noise from other multiplexed pages, which demands a large angular separation between two adjacent multiplexed holograms, resulting in a small storage density. Recently, we proposed another way to reconstruct an image at a different wavelength (Fujimura et al., 2007). Our method, which we call polychromatic reconstruction (PCR), utilizes a spectrally broad light source for the probe beam, as shown in Fig. 1. Each angular spectral component of the recorded gratings can be Bragg-matched with one particular wavelength within the broadband spectrum of the probe beam. Thus, the whole image can be reconstructed from the volume hologram even though the probe wavelength is very different from the recording one. On the other hand, analogous to the spherical probe beam method, the large spectral width of the polychromatic probe beam causes deterioration of the angular selectivity and results in considerable lowering of the multiplexing capability. However, unlike the case of the spherical probe beam, such a drawback can be overcome by using a selective detection method (Fujimura et al., 2010). If the proper optical component, such as a wavelength filter or grating, is additionally inserted in the imaging system, we can detect the signal image alone even though the crosstalk-noise waves are diffracted from other multiplexed pages. Therefore, PCR with the selective detection method is a promising way to achieve nondestructive readout in volume holographic memories without sacrificing the multiplexing capability. When we implement the PCR method in a holographic memory system, it is important to know the properties of the holographic reconstruction process with the polychromatic light. Indeed, the PCR method shows several notable features as compared with conventional monochromatic reconstruction. For example, the reconstructed image has a wavelength distribution that linearly shifts along the grating vector, and image magnification occurs in a direction perpendicular to the incident plane. In fact, knowledge of the required bandwidth of the polychromatic probe beam is essential to design a practical memory system. In this chapter, we develop the theory of holographic reconstruction with polychromatic light, especially from the viewpoint of its application to volume holographic memory. Based on the plane-wave expansion model, we will derive expressions for the required bandwidth, the distortion of the reconstructed image, the optimum recording configuration, the diffraction efficiency, the inter-and intra-page crosstalk noise, and the theoretical limit of the storage density. The obtained expressions show unique features of the PCR method and are very informative for constructing actual holographic memory systems utilizing the PCR method.

Basic principle of the image reconstruction 2.1 Reconstruction from a plane-wave hologram
First we consider the diffraction from a plane-wave hologram that is formed by signal and reference plane waves. When the grating recorded at a wavelength λ w is read out with a probe wavelength λ p , the Bragg condition is given by where e s , e r , e p , and e d are the unit direction vectors of the signal, reference, probe, and diffracted plane waves, respectively; k q = 2πn q /λ q (q = w, p) is the wave number at the wavelength λ q and the refractive index n q ; and G is the grating vector normalized by the recording wave number k w . In order to obtain diffraction from the grating G, the unit direction vector e p and the probe wavelength λ p should be properly chosen so as to satisfy Eq. (1). Such a restriction can be derived from Eq. (1) and the relation |e d | 2 = 1, which is written as where µ is the ratio of the wave numbers of the recording and probe beams, i.e., µ = k w /k p .
Hereafter, for simplicity, we assume that wavelength dispersion of the refractive index can be neglected, i.e., n w ≈ n p ≈ n, and thus, µ can be approximated by the ratio of wavelengths. If e p and λ p satisfy Eq. (2), the grating will reproduce the plane wave with the unit direction vector e d expressed as µ =+ . dp ee G Generally, there are many combinations of e p and λ p that satisfy Eq. (2). Thus, it is possible to obtain diffraction from a plane wave hologram even when the readout probe wave vector is not identical to the reference wave vector in the recording process, i.e., λ p ≠ λ w or e p ≠ e r . For example, when the probe wavelength λ p is given, the incident angle of the probe beam should be adjusted so that the vector e p + µG/2 lies on the plane normal to the grating G, as shown in Fig. 2. Note that there is a maximum value for the allowed λ p because the relation µ|G|/2 < 1 should hold, as is seen from Fig. 2. In this case, the trace of e p forms a circle, which is regarded as the Bragg degeneracy.
On the contrary, when e p is given, the probe wavelength λ p should satisfy the following relation: Note that, in this case, the wavelength ratio µ is uniquely determined by the set of (e p , G).

Reconstruction of the image information
A volume holographic memory usually stores two-dimensional image information. Therefore, the signal beam consists of many plane waves and creates various grating vectors after interfering with the reference plane wave. We express the i-th signal component e si as e si = e s0 + δe si , where e s0 is the central direction vector of the diverged signal beam, and δe si is the deviation vector from e s0 . Similarly, the corresponding diffracted component e di is expressed as e di = e d0 + δe di . In this case, the Bragg condition for each angular spectral component is given by where G i is the normalized grating vector for i-th component. In order to reproduce the whole image, the Bragg condition of Eq. (5) should be satisfied at all components i. If we assume that Eq. (5) is satisfied at the central signal component (i = 0) and that the divergence angle of the signal beam is sufficiently small, i.e., the relation |δe si | 2 << 1 holds, then Eq. (2) can be rewritten as In order to satisfy the Bragg condition at any component i, the probe wave vector should be identical to the reference wave vector (i.e., λ p = λ w and e p = e r ) so that the relation e d0 = e s0 holds. Otherwise, the obtained diffracted waves are limited to those from the components satisfying the relation e d0 ⋅δe sj = 0 . T h e r e f o r e , i t i s u s u a l l y c o n s i d e r e d t h a t t h e i m a g e information cannot be completely reproduced when the probe wavelength is different from the recording one. However, there are two possible ways to satisfy Eq. (5) for all components even when λ p ≠ λ w . One is Külich's approach (Külich, 1987), where a probe beam having adequate angular divergence is used instead of the plane wave. In this method, for each signal component i, there is a probe plane wave component e pi = e p0 + δe pi that satisfies µ µ where we assume that the divergence angles of the probe and signal beams are sufficiently small, and thus, the relations |δe si | 2 << 1, |δe pi | 2 << 1, and δe pi ⋅δe si << 1 hold. Then, the diffracted wave is reproduced as described by the following relation: Note that, in this method, special care should be taken about the diffraction due to the Bragg degeneracy. If several probe components simultaneously satisfy Eq. (7) for one particular signal component, its grating component will produce several diffracted waves with different direction vectors, as represented by Eq. (8). This will degrade the quality of the reconstructed image. To avoid such a situation, the angular spectral components of the probe beam should exist on only one particular plane. For example, it is preferable that the probe beam should be expanded by using a cylindrical lens, not a spherical lens.
www.intechopen.com Another way to satisfy Eq. (5) is our approach, the PCR method. This method utilizes a broadband probe beam instead of monochromatic light. The recording and readout schemes of the PCR method are illustrated in Fig. 3. In the recording process, a monochromatic signal beam bearing the image information passes through a Fourier transform lens and records a Fourier hologram in the usual way. In the readout process, the recorded hologram is readout by using a spectrally broad but spatially coherent light source, such as a super luminescent diode (SLD). All grating components satisfy the Bragg condition because the probe beam includes a spectral component that satisfies the following relation for each component i: where k pi and λ pi are the Bragg-matched wavenumber and wavelength for a grating component G i , respectively. Note that image degradation due to Bragg degeneracy will not occur in the PCR method because the Bragg-matched wavelength λ pi is uniquely determined by the set of (e p , G i ). In this case, the diffracted waves are reproduced with the direction vector µ =+ .
The characteristic feature of the PCR method is that the wavelength of each diffracted plane wave is different at different grating components G i . Therefore, the PCR method can be applied only to Fourier holograms, where one grating vector corresponds to one particular point on the object plane. Even though all diffracted waves are obtained in an image hologram and a Fresnel hologram with polychromatic light, the image cannot be reconstructed since waves with different wavelengths cannot construct a point image.
An example of the reconstructed image in the PCR method is presented in Fig. 4, where we calculated the wavelength ratio µ i and the direction vector e di for each grating component G i by using Eqs. (9) and (10), neglecting off-Bragg diffraction. In this simulation, we assumed that the input image, which was an outline character "A" with dimensions 1 cm × 1 cm, shown in Fig. 4(a), was recorded at λ w = 532 nm and was reconstructed with a polychromatic probe beam with a central wavelength λ p0 of 815 nm. From the figure, we can see that the reconstructed image was formed with spectral components ranging from 795 nm to 835 nm, and image magnification occurred in the y d -direction. Such features are considered as a consequence of using the polychromatic light for the holographic reconstruction. In the following section, we will develop a theory of holographic reconstruction with polychromatic light and investigate characteristic features of the PCR method especially in holographic memory systems.

Definition of the coordinate system
In this section, we introduce a coordinate system that allows for a more quantitative discussion of the PCR method. A Cartesian coordinate system is defined here using unit direction vectors e s0 and e r , as shown in Fig. 5(a), whose normal bases are given by We introduce an elevation angle α k and azimuthal angle β k to specify a unit direction vector e k , as is depicted in Fig. 5 Note that if the grating component G 0 and the wavelength ratio µ 0 are given, then e p and e d0 have only one free parameter. This is because, from Eq. (5), α p and β p should satisfy α βµβ =− 00 cos sin sin .
p ps Furthermore, we express α si , β si , α di , and β di , as α si = α s0 + δα si , β si = β s0 + δβ si , α di = α d0 + δα di , and β di = β d0 + δβ di , respectively, and hereafter for simplicity, we assume that the divergence angles of the signal and diffracted beams are sufficiently small (i.e., δα si , δβ si , δα di , δβ di << 1) and the relations |δe si | 2 , |δe di | 2 << 1 hold at all components i. Then, the deviation vectors δe si and δe di are approximated by Note that the unit direction vectors e sα and e sβ correspond to the normal bases at the Fourier plane of the input image, as shown in Fig. 6(a). Thus, the position on the input plane (x si , y si ) and the set of the deviation angles (δα si , δβ si ) are related via δβ δα where f s is the focal length of the Fourier transform lens in the recording process. Similarly, the relation between the position on the reconstructed image plane (x di , y di ) and the set of the deviation angles (δα di , δβ di ) is given by αδ β δα where f d is the focal length of the Fourier transform lens in the reconstruction process. Furthermore, hereafter we assumed that the dimensions of the input image are in the ranges −x sMax ≤ x si ≤ x sMax and −y sMax ≤ y si ≤ y sMax . Therefore, the maximum deviation angles of the signal beam, δα sMax and δβ sMax , are δβ δα

Spectral width required for the reconstruction
In order to reconstruct the image information, an adequate spectral width is needed for the probe beam to satisfy the Bragg condition at every grating component. In this section, we will theoretically estimate such a spectral width. When we rewrite Eq. (9) using the deviation vectors δe si , the difference between µ i and µ 0 is given by where we assume that the relation|δe si | 2 << |G 0 | 2 holds. Equation (19) indicates that the Bragg-matched wavelength does not depend only on the amplitude |δe si |, but is determined by the projection of δe si on e d0 . If we rewrite Eq. (19) using the Cartesian components in Eqs. (13) and (15) and the relation of Eq. (14), we obtain . 2s i n s i n t a n t a n Therefore, the spectral width required for the full image reconstruction (∆λ BM ) is given by The Bragg-matched wavelength generally depends on both deviation angles δα si and δβ si . When e p lies in the xz-plane (i.e., α p = 0°), however, δµ i depends only on the azimuthal deviation angle δβ si . To see these features more clearly, the spatial distributions of the Braggmatched wavelength are illustrated in Fig. 7. In this simulation, we calculated the reconstructed image with the Bragg-matched wavelength assuming that the grating recorded at 532 nm was read out at around 815 nm. As is predicted from Eq. (21), the spectral width required for the probe beam has a minimum at α p = 0°. Furthermore, the distortion of the reconstructed image is also smallest at α p = 0°, which will be treated further in Section 3.3. The input image was the same as that in Fig. 4. The calculation parameters were as follows: www.intechopen.com

Holograms -Recording Materials and Applications 228
There is an optimum recording angle β s0Opt that minimizes the required spectral width for a given µ 0 , as shown in Fig. 8, which is given by Then, Eq. (22) can be simplified to In this case, the required spectral width ∆λ BM reaches a minimum and is expressed as However, in most cases, the allowed deviation angle δβ sMax is determined by the spectral width ∆λ p of a given probe light source. Assuming that α p = 0°, from Eqs. (21), we obtain 1 max 00 11 tan tan At the optimum recording angle, δβ sMax reaches a maximum 1 ax 0 00 For example, if we have a probe light source whose central wavelength λ p0 is 815 nm and whose full spectral width ∆λ p is 50 nm, and the recording wavelength λ w is 532 nm, then the optimum recording angle β s0Opt is about 33°, and the allowed deviation angle δβ sMax is about 4.0°. So far, assuming the small deviation angles of the signal beam, we derived the expression for the Bragg-matched wavelength and found that the wavelength shift is simply proportional to the deviation angles δα si and δβ si . However, for large deviation angles, their relation cannot be expressed by Eq. (20) and becomes nonlinear. Examples for the reconstructed images at large deviation angles are shown in Fig. 9. In this simulation, the focal length of the Fourier transform lenses was varied from 50 mm to 10 mm while the input image size was kept constant, which resulted in an increase of the maximum deviation angle from 5.7° to 30°. A larger angular spectrum of the signal beam results in greater deviation from the linear relationship and requires a probe beam of much broader spectrum. Moreover, considerable distortion of the reconstructed image occurs at large deviation angle.

Distortion of the reconstructed image
In a thin hologram, it is well-known that magnification or reduction of the reconstructed image occurs during readout at a wavelength different from the recording one (Champagne, 1967). The PCR method, in contrast, produces a directionally-stretched image depending on the readout configuration, as shown in Fig. 7. Such distortion is likely to cause errors in retrieving data from volume holograms, but if the property of the distortion is known and predictable, the stored information can be completely recovered after the image processing of the distorted image. In this section, we will derive an expression for the distortion of the reconstructed image in the PCR method. From Eq. (10), the deviation vector δe di is approximated by where we neglect the product of δe si and δµ 0 . From Eqs. (13), (15), and (20), the Cartesian components of δe di are expressed as  (29) www.intechopen.com

Holograms -Recording Materials and Applications 230
On the other hand, δe di is also expressed by Eq. (15) using e dα and e dβ . Rewriting Eq. (29) using normal bases of e dα and e dβ yields Comparing the coefficients in (30) with those in Eq. (15), we obtain the relation between the deviation angles (δα si , δβ si ) and (δα di , δβ di ): From Eqs. (16) and (17), we can finally derive the expression for a transfer matrix between the object and reconstructed image plane, that is, When α p = 0° or β p = −90°, non-diagonal components in the transfer matrix become zero and the image can be reconstructed without any tilting. However, the distortion becomes minimal at α p = 0°, as was seen in Fig. 7.

Multiplexing with crystal rotation
In this section, we investigate the multiplexing capability in the PCR method. While many multiplexing methods have been proposed so far, we adopt peristrophic multiplexing (Curtis et al., 1994) and rotation of the crystal angle (hereafter, the latter will be called crystal angle multiplexing). The multiplexing configuration considered here is shown in Fig. 10. Note that other multiplexing methods are also possible in principle, but most of them are probably not suitable for the PCR method because they require a complicated system to read a target page. For example, ordinary angle multiplexing, which varies the incident angle of the reference beam, will also require moving the imaging system. This is because e d0 will be pointed in a different direction at each multiplexed page, as expressed by Eq. (10). The situation is similar in the case of the wavelength multiplexing method. Peristrophic multiplexing and crystal angle multiplexing, in contrast, do not require any additional movement other than the crystal rotation. In addition, the imaging properties are almost unchanged with each multiplexed page because the recording angle is constant. Thus, these multiplexing methods are suitable for holographic memory systems employing PCR. In the peristrophic and crystal angle multiplexing methods, recording of another page is performed after crystal rotation by the proper angle. In order to retrieve the stored information without crosstalk, the rotation angle should be sufficiently large so that the other multiplexed holograms cannot produce noise diffracted waves that would disturb the detection of the target signal image. On the other hand, the theoretical limit of the storage density is inversely proportional to the angular separation between the adjacent multiplexed holograms. Therefore, the angular selectivity (i.e., how small the angular separation can be made) is an important figure of merit determining the total storage density of the system. In the following sections, we will derive an expression of the minimum rotation angle for each multiplexing method. Fig. 10. Configuration of crystal angle multiplexing combined with peristrophic rotation. Peristrophic multiplexing and crystal angle multiplexing are performed by rotating the crystal around the z-axis and an axis inclined at φ cz from the y-axis, respectively.

Peristrophic multiplexing
When the crystal is rotated by a small angle δφ cz around the z-axis (peristrophic rotation), the displacement vector of the grating component (δe cz ) is approximated by 0 10 0 10 2 s i n .
Note that we neglected the product of small quantities. Due to the small rotation of the grating vectors, the Bragg-matched wavelength and the unit direction vector of each diffracted wave are also changed according to Eqs. (9) and (10). The shift of the wavelength ratio after the peristrophic rotation (δµ cz ) is given by where last equality is obtained by substituting Eqs. (13) and (33). The displacement vector of e di after the peristrophic rotation (δe dcz ) is expressed as where we use Eqs. (13), (14), (33), and (34) to derive the last equality. If we rewrite Eq. (35) using normal bases of e dα and e dβ and compare the coefficients with those in Eq. (15), similarly to Section 3.3, we obtain the shift amount of the imaging position due to the peristrophic rotation: For simplicity, we assume that α p = 0. Then, from Eqs. (20) 2t a n t a n . 2s i n 2s i n Note that the inequality in Eq. (37) represents the location of the reconstructed image. The peristrophic rotation causes the reconstructed image to shift in the y d direction while keeping the spatial distribution of the Bragg-matched wavelength unchanged. In order to record another page without crosstalk, the rotation angle should be large enough to shift the noise image away from the signal imaging area. Thus, from the inequality in Eq. (37), the minimum angular separation in peristrophic multiplexing (δφ czMin ) is expressed as This equation does not include the wavelength ratio µ 0 . Therefore, the readout at a longer wavelength does not affect the angular selectivity in the peristrophic multiplexing, even though the reconstructed image was magnified by a factor of µ 0 in the y d direction. This is because the amount of spatial shift is also increased by a factor of µ 0 , as shown in Eq. (36).

Theory of Polychromatic Reconstruction for Volume Holographic Memory 233
In order to see these behaviours more clearly, in Fig. 11, we show the simulated results of the reconstructed image obtained while varying the peristrophic rotation angle δφ cz . We also present results for the conventional monochromatic readout for comparison. Note that the off-Bragg diffraction was also neglected in the monochromatic case, but here, in order to see the influence of the peristrophic rotation on the Bragg condition, the probe light was assumed to be quasi-monochromatic with a small spectral width of 1 nm, instead of taking the finite hologram dimension into account. Although a part of the image faded away due to the Bragg selectivity in the quasi-monochromatic case, the rotation angles required for multiplexing are almost the same in both cases. Therefore, the use of polychromatic light in the PCR method has only a small influence on the multiplexing capability in the peristrophic multiplexing method, and the PCR method could employ peristrophic multiplexing without any adverse influence on the multiplexing capability. Fig. 11. Influence of the peristrophic rotation on the reconstructed image obtained with (a) conventional quasi-monochromatic light whose central wavelength is 532 nm and whose spectral width is 1 nm, and (b) polychromatic light whose central wavelength is 815 nm and whose spectral width is 40 nm. δφ cz is the peristrophic rotation angle.

Crystal angle multiplexing with peristrophic rotation
Next, let us consider the angular selectivity in the crystal angle multiplexing, where the crystal is rotated around an axis inclined at φ cz , as shown in Fig. 10. Note that the inclination angle φ cz results from the crystal rotation due to the peristrophic multiplexing. In a similar way to the previous section, the displacement vector G i due to a small rotation in the crystal angle multiplexing ( In this case, the shift of the wavelength ratio (δµ cy ) and the displacement vector of e di after the crystal rotation (δe dcy ) are expressed as www.intechopen.com In contrast to peristrophic multiplexing, the crystal angle multiplexing leads to a change in the Bragg-matched wavelength as well as a shift of the imaging location. The Braggmatched wavelength in Eq. (43) is plotted at several rotation angles δφ cy in Fig. 12. The dotted part of the Bragg-matched line in the figure corresponds to the portion of the image that is not reconstructed by the probe beam because the Bragg-matched wavelength is beyond the probe spectral band. Such behaviour can be clearly seen in Fig. 13. When the solid part of the Bragg-matched line lies inside the signal imaging area, the noise diffraction will spatially overlap with the signal image and cause severe crosstalk. Therefore, the angular separation between adjacent holograms should be sufficiently large so that the solid part will be shifted away from the signal imaging area. If we assume that the central wavelength of the probe beam is λ p0 and its full spectral width is ∆λ p , the minimum angular separation (δφ cyMin ) is given by where min[a, b] is a minimum function yielding the smaller value of a and b. Note that the former parameter in the minimum function is the rotation angle required for separating the noise and signal images spatially, and the latter corresponds to that required for spectral separation. The above equation implies that an unnecessarily large spectral width increases the minimum angular separation needlessly. Thus, if we take the spectral width required for the full image reconstruction, ∆λ BM in Eq. (25)  Note that when the wavelength ratio µ 0 is unity (i.e., the readout wavelength is the same as the recording one), the required angular separation δφ cyMin becomes zero. This is because we assumed that the hologram dimension is large enough to neglect off-Bragg diffraction. In other words, if we employ the PCR method, the angular separation specified by Eq. (45) is always required, regardless of the hologram dimension.

Off-Brag diffraction
So far, we considered only the diffracted wave satisfying the Bragg condition. In reality, however, owing to the finite hologram dimension, the grating produces non-Bragg-matched diffracted waves, referred to as off-Bragg diffraction. In this section, we consider the off-Brag diffraction in the PCR method and derive expressions for crosstalk noise.
We assume that a set of (G i , e p , e di ) satisfies the Bragg condition at λ w and λ pi and consider the situation where a grating component G i + δe Goff produces a diffracted wave whose wavelength and unit direction vector are λ pi + δλ off and e di , respectively. In this case, the off-Bragg vector ∆K can be written as where we also assumed that the relation δλ off << λ pi holds.
where δe cz and δe cy are the displacement vectors due to the crystal rotation, defined by Eqs.
where δα soff and δβ soff are the off-Bragg deviation angles that are related to δe soff through Eq.
is the spectral intensity of the probe beam whose central wavelength, full spectral width, and intensity are λ p0 , ∆λ p , and I p0 , respectively; and rect(x) is the rectangle function, defined as () 11 2 r e c t 12 12.
01 2 Equation (49) indicates that the off-Bragg diffraction will occur at ∆K around the origin. The diffracted intensity will become zero when the relation holds at any component of the off-Bragg vector ∆K. Thus, hereafter, we assume that Eq. (53) represents an upper limit of |∆K i | in which the grating G i + δe Goff can produce off-Bragg diffraction.

Properties of signal diffraction
Before considering the crosstalk noise, let us first discuss the prpperties of signal diffraction from a target grating G i . In this case, we can set δα soff = δβ soff = δφ cy = δφ cz = 0°. Then the off- where we assumed that the probe spectral width ∆λ p is much larger than δλ offMax , meaning that I p (λ) is considered to be a constant over the region where the integrand is appreciable, and we used the relation Note that if ∆λ p → 0, then I dif → I p0 . Thus, the diffraction efficiency in the PCR method is a factor of δλ offMax /∆λ p smaller than that in monochromatic readout. This is because the grating component G i can diffract only the limited spectral component around λ pi .
Such a reduction in the diffraction efficiency is an unavoidable drawback of the PCR method.

Intra-page crosstalk noise
In this section, we will consider the intra-page crosstalk noise which is derived from the unwanted diffraction coming from the same page but a different grating component, G i + δe soff . In this case, we can set δφ cy = δφ cz = 0°, and then the off-Bragg vector in Eq. (48) The off-Bragg diffraction will occur only when a set of (δα soff , δλ off ) satisfies all of the where δλ pi is the difference between λ pi and λ p0 . Note that the above ranges depend on the targeting position (x si , y si ) within the input image. Our goal in this section is to find an upper limit of the off-Bragg deviation angles (δα soff , δβ soff ). Such maximum angles specify an input image area suffering from intra-page crosstalk noise, as shown in Fig. 14. First, we will consider the conventional monochromatic case because it can be understood more easily. Since the probe beam is monochromatic, we can set δλ off = 0. Then, the maximum off-Bragg deviation angles δα soffMax(mono) and δβ soffMax ( The off-Bragg diffraction will occur at λ pi + δλ off only when δλ off satisfies both Eqs. (65) The above equations imply that the intra-page crosstalk noise in the PCR method does not depend on the hologram dimension L x . This feature of the PCR method is clearly shown in Fig. 15, where the normalized diffracted intensity is plotted as a function of the position (x s , y s ) at several values of L x . While the input image area suffering from the crosstalk noise shrinks with increasing L x in the conventional monochromatic case, that in the PCR method is unchanged.

Inter-page crosstalk noise
In this section, we will consider the inter-page crosstalk noise which is caused by the unwanted diffraction coming from a grating component in another multiplexed page, i.e., G i + δe soff + δe cz + δe cy . Similar to the previous section, we obtain the set of inequalities given by Note again that the off-Bragg diffraction will occur only when the set of (δα soff , δβ soff , δφ cy , δφ cz , δλ off ) satisfies all of the above inequalities. First, let us consider Eq. (72) to find the rotation angle δφ cz that is large enough to avoid inter-page crosstalk noise. Since the variable range of δα soff is expressed as Eq. (63), the off-Bragg diffraction will occur whenever δφ cz satisfies the following inequality: The above inequality is an expression that is applicable at one particular position (x si , y si ) within the input image. In order to avoid crosstalk at every position, the peristrophic rotation angles δφ cz should be set larger than δφ czMin(off) ; that is, where we used the relation −δα sMax ≤ δα si ≤ δα sMax . The right side in Eq. (75) consists of two parts: one is the minimum peristrophic rotation angle δφ czMin obtained in Section 3.4.1; and the other is the contribution from the off-Bragg diffraction. However, the latter is usually much smaller than the former, and thus the influence of the off-Bragg diffraction can be neglected in the peristrophic multiplexing. Moreover we should also note that Eq. (75) is applicable in both the monochromatic case and the PCR method. This is consistent with the result in Fig. 11.
Next, we will focus on Eqs. (71) and (73) to find the rotation angle δφ cy that is large enough to avoid crosstalk noise. Before considering the case of PCR, we will again treat the conventional monochromatic readout first. In this case, we can set δλ off = 0. Thus, Eq. (71) On the other hand, the variable range of δβ soff is also limited by the input image size, as is expressed in Eq. (64). Since the off-Bragg diffraction will occur only when δβ soff satisfies both Eqs. (64) and (79), the sufficient condition for δβ soff to produce the off-Bragg diffraction is given by where max[a, b] is a maximum function yielding the larger value between a and b. If we assume that α p = 0 and take the spectral width required for the full image reconstruction where we used the relation of Eq. (20) in the derivation. From Eqs. (73) and (81), the off-Bragg diffraction will occur whenever δφ cy satisfies the following inequality: The above inequality is an expression that is applicable at one particular position (x si , y si ) within the input image. In order to avoid crosstalk at every position, the crystal rotation angles δφ cy should be set larger than δφ cyMin(PCR) ; that is, where we used the relation −δβ sMax ≤ δβ si ≤ δβ sMax . The right side in Eq. (83) consists of the minimum crystal rotation angle δφ cyMin obtained in Section 3.4.2 and the contribution from the off-Bragg diffraction δφ cyMin (mono) . Similarly to the peristrophic rotation, the latter is usually much smaller than the former, and thus the influence of the off-Bragg diffraction can be ignored also in the crystal angle multiplexing.

Storage capacity
Since a single point on the input image corresponds to one particular plane wave in the recording medium, a Fourier hologram stores the information in such a way that one data bit is stored at one localized point in frequency space, and thus, nonlocally in real space. However, due to the off-Bragg diffraction, one data point in frequency space will reproduce an image with a finite dimension in the reconstructed image plane. In order to distinguish two different bits, corresponding data points in frequency space must maintain a certain interval large enough to avoid crosstalk. Therefore, one data bit in real space actually occupies a finite volume in frequency space. If such a volume is denoted by V K1bit , the storage capacity of the holographic memory (N bit ) is expressed as where V KG is the volume in frequency space which the grating vector can attain through multiplexing. Note that, if the readout wavelength is different from the recording one, as is in the present case, there exists a grating vector that cannot satisfy the Bragg condition with that probe beam. Such a grating vector should be removed from V KG , because the data stored there cannot be read out. Therefore, the theoretical limit of the storage capacity is ultimately determined by the readout wavelength not the recording one. Although V K1bit and V KG are generally different in different multiplexing methods, we will consider here the crystal angle multiplexing combined with the peristrophic rotation discussed in Section 3.4. Furthermore, for simplicity, we assume α p = 0° in this section. First, let us consider V KG after the crystal angle multiplexing around the axis inclined at φ cz . The matrix that rotates at φ cz around the z-axis, and subsequently rotates at φ cy around the y-axis, is expressed as The grating vector k w (G 0 + δe si ) operated on by that matrix is written as If above grating is further rotated around the y-axis by a small angle δφ cy , we obtain the differential grating vector (δK G ) resulting from small shifts of δα s , δβ s , and δφ cy : When we assumed that δα s , δβ s , and δφ cy are in the ranges −δα sMax < δα s < δα sMax , −δβ sMax < δβ s < δβ sMax , and −δφ cyMax < δφ cy < δφ cyMax , respectively, then V KG after crystal angle multiplexing around the axis inclined at φ cz , which is denoted by V KG_φcz , can be written as www.intechopen.com For example, if λ w = 532 nm, λ p0 = 815 nm, β s0 = 30°, n w = 1, δβ sMax =4.0°, and L x = L y = L z = 10 mm, then β p = −50°. The resultant storage capacities N bit (mono) and N bit(PCR) are 3.2 Tbit and 4.3 Gbit, respectively. Therefore, the storage capacity in the PCR method decreases by more than two orders of magnitude.

Selective detection method
In the previous section, we see that the PCR method causes a significant decrease in the storage capacity. However, such a problem can be overcome if we employ the selective detection method together with PCR (Fujimura et al., 2010). The method is based on the selective detection of a target signal image that is submerged in noise waves. By inserting a suitable wavelength separator into the reconstructed image plane, we can retrieve the stored information without crosstalk even if the angular separation is not large enough to suppress the noise diffraction. In this section, we will explain how to remove the crosstalk noise, and we derive theoretical limit of the storage capacity after the improvement.

Basic concept and principle
As seen in Fig. 12 (109) www.intechopen.com

Holograms -Recording Materials and Applications 248
If we take advantage of this difference of the Bragg-matched wavelength, it should be possible to detect the signal image alone, even if the noise images overlap with the target signal image. For example, let us consider inserting a special wavelength filter into the reconstructed image plane as shown in Fig. 16, whose transmittance T LVF is represented by ,, 0 , ,r e c t , where λ is the wavelength; ∆λ LVF is the full width of the transmitting band; and rect(x) is the rectangle function defined in Eq. (52). Note that the transmission spectrum of this filter depends on the illuminated location on the filter; that is, the optical waves passing through different spatial positions will undergo different spectral filtering by this filter. Such a wavelength filter is known as a linear variable filter (LVF) since the spectral shift of the transmission band is proportional to the spatial shift of the illuminated position. Fig. 16. Configuration for the selective detection method. A special wavelength filter, like a band-pass linear variable filter, is inserted at the reconstructed image plane.
The principle of the selective detection is shown in Fig. 17. Since the transmitting wavelength of the LVF coincides with the spectral dispersion of the target signal image, every diffracted wave that constitutes the signal image can go through the LVF and will be detected by the imager. On the contrary, the LVF will reject the noise diffracted wave whose wavelength lies outside the transmission band of the LVF.

Theoretical description of the selective detection method
In this section, we will see the influence of the LVF on the properties of the crosstalk noise, and we formulate the achievable storage capacity in the selective detection method. As was seen in Section 3.5.3, the off-Bragg diffraction will occur only when the set of (δα soff , δβ soff , δφ cy , δφ cz , δλ off ) satisfies all of inequalities in Eqs. (71) Comparing this with V K1bit(PCR) in Eq. (106), we find that the storage capacity will be improved by a factor of 2∆λ BM /∆λ LVF . On the other hand, when ∆λ LVF << 2δλ offMax is satisfied, V K1bit(SelDet) becomes identical to V K1bit(mono) . Therefore, the storage capacity in the PCR method will be completely recovered if we use an LVF having a sufficiently small ∆λ LVF . However, it should be noted that, in this case, the diffraction efficiency will also decrease with decreasing ∆λ LVF . This is because the assumption used to derive Eq. (57) Since the relation 0 < min[ξ, 1/ ξ] ≤ 1 holds, we see that the storage capacity at ∆λ LVF = 2δλ offMax will reach about half that of the monochromatic case. From Eqs. (95) and (118) For example, if λ w = 532 nm, λ p0 = 815 nm, ∆λ p = 50 nm, β s0 = 30°, n w = 1, L x = L y = L z = 10 mm, and ∆λ LVF = 2δλ offMax = 0.043 nm, then the resultant storage capacity N bit(selDet) is 1.3 Tbit. Therefore, in this case, the storage capacity in the selective detection method reaches 40 percent of the theoretical limit for the monochromatic case. Finally, the storage capacity in Eq. (124) is plotted as a function of ∆λ LVF in Fig. 18. The improvement starts at ∆λ LVF = 100 nm, then the storage capacity increases linearly with increasing ∆λ LVF , and finally, it asymptotically approaches the theoretical limit of the monochromatic case. Note that the kink observed at ∆λ LVF = 0.06 nm is derived from the minimum function; that is to say, from that point on, the storage capacity further increases due to the reduction of the intra-page crosstalk noise δβ soffMax in Eq. (117). parameters are as follows: λ w = 532 nm, λ p0 = 815 nm, ∆λ p = 50 nm, β s0 = 30°, n w = 1, and L x = L y = L z = 10 mm.

Summary
We have developed the theory of holographic reconstruction with polychromatic light. In particular, focusing on its application to holographic memory, the required spectral width, distortion of the reconstructed image, diffraction efficiency, intra-and inter-page crosstalk, and storage capacity were investigated. The obtained results are summarized below. www.intechopen.com

Theory of Polychromatic Reconstruction for Volume Holographic Memory 253
The required spectral width and image distortion A larger input image (x sMax , y sMax ) or smaller focal length f s require a larger spectral width ∆λ BM to reconstruct the whole image. Generally the required spectral width depends on both directions x sMax and y sMax , but if the signal, reference, and probe beams lie in the same plane (α p = 0°), it becomes independent of y sMax . Furthermore, the condition α p = 0° minimizes the image distortion, where the image is magnified by the wavelength ratio µ 0 in the y ddirection, but is unchanged in the x d -direction.

Multiplexing method
Crystal angle multiplexing used in combination with peristrophic multiplexing is the most suitable multiplexing method in polychromatic reconstruction (PCR). Other multiplexing methods are also possible in principle, but most of them require a complicated system to read a target page because the diffracted wave e d0 will be pointed in a different direction at each multiplexed page.

Diffraction efficiency
Since the grating component can diffract only a limited spectral component, the diffraction efficiency, which is defined here as the ratio of the diffracted power by one grating component to the total power of the input probe beam, is much smaller than in the conventional monochromatic readout. This may be an unavoidable drawback of the PCR method.

Intra-and Inter-page crosstalk noise
Due to the wide spectral width of the probe beam, the hologram dimension L x does not influence the intra-page crosstalk noise, and thus the input image area suffering from the intrapage crosstalk noise is slightly enlarged. On the other hand, the inter-page crosstalk noise is a crucial problem in the PCR method. The angular Bragg selectivity is greatly degraded, and thus, the storage capacity decreases by more than two orders of magnitude. However, this problem can be solved by using the selective detection method mentioned below.

Selective detection method and the achievable storage capacity
Taking advantage of the wavelength difference after the crystal rotation, we can detect the signal image alone, even if the noise images overlap with the target signal image. The storage capacity when employing the selective detection method reached 40 percent of the theoretical limit for the monochromatic case. PCR is a unique and useful method for holographic memory systems. While other nondestructive readout methods have failed to achieve a high storage capacity, we theoretically proved that PCR, when used in combination with the selective detection method, enables us to achieve both nondestructive readout and a high storage capacity, simultaneously. Barbastathis, G. & Psaltis, D., (2000). Volume holographic multiplexing methods, In: Holographic data storage, Coufal, H.J., Psaltis, D., & Sincerbox, G.T., pp. 21-62, Springer, 978-3540666912, New York.