**1. Introduction**

218 Holograms – Recording Materials and Applications

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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

Theory of Polychromatic Reconstruction for Volume Holographic Memory 221

where **e***s*, **e***r*, **e***p*, and **e***d* are the unit direction vectors of the signal, reference, probe, and

recording wave number *kw*. In order to obtain diffraction from the grating **G**, the unit direction

is the ratio of the wave numbers of the recording and probe beams, i.e.,

= + μ

to obtain diffraction from a plane wave hologram even when the readout probe wave vector


On the contrary, when **e***p* is given, the probe wavelength λ*p* should satisfy the following

μ

Fig. 2. The possible choice of the incident angle of the probe beam to reconstruct the plane

λ

μ

μ <sup>⋅</sup> = − <sup>2</sup> <sup>2</sup> . *<sup>p</sup>* **e G G**

Hereafter, for simplicity, we assume that wavelength dispersion of the refractive index can

*<sup>p</sup>* satisfy Eq. (2), the grating will reproduce the plane wave with the unit direction

λ

λ

restriction can be derived from Eq. (1) and the relation |**e***d*|2 = 1, which is written as

μ

is not identical to the reference wave vector in the recording process, i.e.,

shown in Fig. 2. Note that there is a maximum value for the allowed

 μ

λ

*<sup>q</sup>* and the refractive index *nq*; and **G** is the grating vector normalized by the

*<sup>q</sup>* (*q* = *w*, *p*) is the wave number at the

μ= *kw*/*kp*.

*<sup>w</sup>* or **e***<sup>p</sup>* ≠ **e***r*.

(4)

*<sup>p</sup>* because the relation

*<sup>p</sup>* should be properly chosen so as to satisfy Eq. (1). Such a

+ ⋅= 0 , <sup>2</sup> *<sup>p</sup>* **e GG** (2)

can be approximated by the ratio of wavelengths. If

. *d p* **ee G** (3)

*<sup>p</sup>* is given, the incident angle of the probe beam

**G**/2 lies on the plane normal to the grating **G**, as

is uniquely determined by the set of (**e***p*, **G**).

λ

*<sup>p</sup>* that satisfy Eq. (2). Thus, it is possible

λ*<sup>p</sup>* ≠ λ

diffracted plane waves, respectively; *kq* = 2π*nq*/

wavelength

where μ

**e***p* and λ

μ

relation:

vector **e***d* expressed as

λ

vector **e***p* and the probe wavelength

be neglected, i.e., *nw* ≈ *np* ≈ *n*, and thus,

Generally, there are many combinations of **e***p* and

For example, when the probe wavelength

should be adjusted so that the vector **e***p* +

which is regarded as the Bragg degeneracy.

Note that, in this case, the wavelength ratio

wave hologram represented by **G**.

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.

Fig. 1. Concept of the nondestructive readout in the PCR method.

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.

#### **2. 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*<sup>w</sup>* is read out with a probe wavelength λ*<sup>p</sup>*, the Bragg condition is given by

$$k\_p \left(\mathbf{e}\_d - \mathbf{e}\_p\right) = k\_w \left(\mathbf{e}\_s - \mathbf{e}\_r\right) \equiv k\_w \mathbf{G}\_r \tag{1}$$

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

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

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

First we consider the diffraction from a plane-wave hologram that is formed by signal and

λ

( ) −= −≡ ( ) , *pd p ws r w kkk* **ee ee G** (1)

*<sup>w</sup>* is read out with a

of the polychromatic probe beam is essential to design a practical memory system.

Fig. 1. Concept of the nondestructive readout in the PCR method.

**2. Basic principle of the image reconstruction 2.1 Reconstruction from a plane-wave hologram** 

reference plane waves. When the grating recorded at a wavelength

*<sup>p</sup>*, the Bragg condition is given by

the multiplexing capability.

method.

probe wavelength

λ

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; *kq* = 2π*nq*/λ*<sup>q</sup>* (*q* = *w*, *p*) is the wave number at the wavelength λ*<sup>q</sup>* and the refractive index *nq*; and **G** is the grating vector normalized by the recording wave number *kw*. In order to obtain diffraction from the grating **G**, the unit direction vector **e***p* and the probe wavelength λ*<sup>p</sup>* 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

$$\left(\mathbf{e}\_p + \frac{\mu}{2}\mathbf{G}\right)\cdot\mathbf{G} = 0,\tag{2}$$

where μ is the ratio of the wave numbers of the recording and probe beams, i.e., μ = *kw*/*kp*. Hereafter, for simplicity, we assume that wavelength dispersion of the refractive index can be neglected, i.e., *nw* ≈ *np* ≈ *n*, and thus, μ can be approximated by the ratio of wavelengths. If **e***p* and λ*<sup>p</sup>* satisfy Eq. (2), the grating will reproduce the plane wave with the unit direction vector **e***d* expressed as

$$\mathbf{e}\_d = \mathbf{e}\_p + \mu \mathbf{G}.\tag{3}$$

Generally, there are many combinations of **e***p* and λ*<sup>p</sup>* 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*<sup>p</sup>* ≠ λ*<sup>w</sup>* or **e***<sup>p</sup>* ≠ **e***r*. For example, when the probe wavelength λ*<sup>p</sup>* is given, the incident angle of the probe beam should be adjusted so that the vector **e***p* + μ**G**/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*<sup>p</sup>* 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:

$$
\mu = -\frac{2\mathbf{e}\_p \cdot \mathbf{G}}{\left|\mathbf{G}\right|^2}.\tag{4}
$$

Note that, in this case, the wavelength ratio μis uniquely determined by the set of (**e***p*, **G**).

Fig. 2. The possible choice of the incident angle of the probe beam to reconstruct the plane wave hologram represented by **G**.

Theory of Polychromatic Reconstruction for Volume Holographic Memory 223

Fig. 3. Schematic diagram of (a) the recording and (b) the reconstruction schemes in the PCR

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

**e***p*; and *fs* and *fd* are the focal lengths of the Fourier transform lenses in the recording and

λ

<sup>⋅</sup> ≡ ≈ =− <sup>2</sup> <sup>2</sup> , *<sup>w</sup> pi pi*

*pi w i*

component **G***i*, respectively. Note that image degradation due to Bragg degeneracy will not

by the set of (**e***p*, **G***i*). In this case, the diffracted waves are reproduced with the direction

= + 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

by using Eqs. (9) and (10), neglecting off-Bragg diffraction. In this simulation, we assumed

**e G G**

*pi* are the Bragg-matched wavenumber and wavelength for a grating

λ

. *di p i i* **ee G** (10)

*<sup>i</sup>* and the direction vector **e***di* for each grating component **G***<sup>i</sup>*

*pi* is uniquely determined

λ

θ

*<sup>p</sup>* is the internal incident angle of

(9)

*w* is the internal half crossing angle of **e***s0* and **e***r*;

μ

occur in the PCR method because the Bragg-matched wavelength

μ

*i*

*k k*

method.

component *i*:

where *kpi* and

vector

λ

calculated the wavelength ratio

θ

reconstruction processes, respectively.

#### **2.2 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* + δ**e***si*, where **e***s0* is the central direction vector of the diverged signal beam, and δ**e***si* is the deviation vector from **e***s0*. Similarly, the corresponding diffracted component **e***di* is expressed as **e***di* = **e***d0* + δ**e***di*. In this case, the Bragg condition for each angular spectral component is given by

$$k\_p \left(\mathbf{e}\_{di} - \mathbf{e}\_p\right) = k\_w \left(\mathbf{e}\_{si} - \mathbf{e}\_r\right) \equiv k\_w \mathbf{G}\_i \,\,\,\,\tag{5}$$

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 |δ**e***si*|2 << 1 holds, then Eq. (2) can be rewritten as

$$\left(\mathbf{e}\_p + \frac{\mu}{2}\mathbf{G}\_i\right) \cdot \mathbf{G}\_i \simeq \mathbf{e}\_{d0} \cdot \mathbf{\delta}\mathbf{e}\_{si} \,. \tag{6}$$

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*<sup>p</sup>* = λ*<sup>w</sup>* and **e***<sup>p</sup>* = **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* ⋅δ**e***sj* = 0. Therefore, it is usually considered that the image 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*<sup>p</sup>* ≠ λ*<sup>w</sup>*. 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* + δ**e***pi* that satisfies

$$\left(\mathbf{e}\_{ji} + \frac{\mu}{2}\mathbf{G}\_i\right) \cdot \mathbf{G}\_i \approx \mathbf{e}\_{d0} \cdot \left(\mathbf{\hat{6}e}\_{si} + \frac{1}{\mu}\mathbf{\hat{6}e}\_{ji}\right) = \mathbf{0} \,,\tag{7}$$

where we assume that the divergence angles of the probe and signal beams are sufficiently small, and thus, the relations |δ**e***si*|2 << 1, |δ**e***pi*|2 << 1, and δ**e***pi* ⋅δ**e***si* << 1 hold. Then, the diffracted wave is reproduced as described by the following relation:

$$\mathbf{e}\_{di} = \mathbf{e}\_{pi} + \mu \mathbf{G}\_i = \mathbf{e}\_{d0} + \mathbf{\delta} \mathbf{e}\_{pi} + \mu \mathbf{\delta} \mathbf{e}\_{si} \,. \tag{8}$$

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.

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* + δ**e***si*, where **e***s0* is the central direction vector of the diverged signal beam, and δ**e***si* is the deviation vector from **e***s0*. Similarly, the corresponding diffracted component **e***di* is expressed as **e***di* = **e***d0* + δ**e***di*. In this case, the Bragg condition for each angular spectral

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 |δ**e***si*|2 << 1 holds, then Eq. (2)

<sup>0</sup> . <sup>2</sup> *<sup>p</sup> i i d si*

In order to satisfy the Bragg condition at any component *i*, the probe wave vector should be

holds. Otherwise, the obtained diffracted waves are limited to those from the components satisfying the relation **e***d0* ⋅δ**e***sj* = 0. Therefore, it is usually considered that the image information cannot be completely reproduced when the probe wavelength is different from

λ*<sup>p</sup>* = λ

However, there are two possible ways to satisfy Eq. (5) for all components even when

+ ⋅≈ ⋅ + = <sup>0</sup>

=+ = + + μ

probe beam should be expanded by using a cylindrical lens, not a spherical lens.

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

where we assume that the divergence angles of the probe and signal beams are sufficiently small, and thus, the relations |δ**e***si*|2 << 1, |δ**e***pi*|2 << 1, and δ**e***pi* ⋅δ**e***si* << 1 hold. Then, the

*<sup>w</sup>*. 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*,

μ

 μ<sup>0</sup> . *di pi i d pi si* **e e Ge δe δe** (8)

<sup>1</sup> 0 , <sup>2</sup> *pi i i d si pi* **e GGe <sup>δ</sup><sup>e</sup> <sup>δ</sup><sup>e</sup>** (7)

+ ⋅≈ ⋅

 μ

there is a probe plane wave component **e***pi* = **e***p0* + δ**e***pi* that satisfies

μ

diffracted wave is reproduced as described by the following relation:

( ) −= −≡ ( ) , *<sup>p</sup> di <sup>p</sup> w si r w i kkk* **ee ee G** (5)

**e GGe δe** (6)

*<sup>w</sup>* and **e***<sup>p</sup>* = **e***r*) so that the relation **e***d0* = **e***s0*

λ*<sup>p</sup>* ≠

**2.2 Reconstruction of the image information** 

identical to the reference wave vector (i.e.,

component is given by

can be rewritten as

the recording one.

λ

Fig. 3. Schematic diagram of (a) the recording and (b) the reconstruction schemes in the PCR method. θ*w* is the internal half crossing angle of **e***s0* and **e***r*; θ*<sup>p</sup>* is the internal incident angle of **e***p*; and *fs* and *fd* are the focal lengths of the Fourier transform lenses in the recording and reconstruction processes, respectively.

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*:

$$\mu\_i \equiv \frac{k\_w}{k\_{pi}} \simeq \frac{\mathcal{\lambda}\_{pi}}{\mathcal{\lambda}\_w} = -\frac{\mathbf{2e}\_p \cdot \mathbf{G}\_i}{\left| \mathbf{G}\_i \right|^2},\tag{9}$$

where *kpi* 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

$$\mathbf{e}\_{di} = \mathbf{e}\_p + \mu\_i \mathbf{G}\_i \,. \tag{10}$$

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*<sup>i</sup>* and the direction vector **e***di* for each grating component **G***<sup>i</sup>* by using Eqs. (9) and (10), neglecting off-Bragg diffraction. In this simulation, we assumed

Theory of Polychromatic Reconstruction for Volume Holographic Memory 225

Fig. 5. Definition of (a) the Cartesian coordinate system and (b) the elevation angle

From the symmetry of the coordinate system and the Bragg condition of Eq. (5), the

sin sin cos sin cos sin

 − − == = = 

cos cos cos cos cos cos

0. 0 2

*s r p pd p*

0 , 0 , sin , sin ,

*s s p p p p*

 αβ

βμβ

and the relations |δ**e***si*|2, |δ**e***di*|2 << 1 hold at all components *i*. Then, the deviation vectors

 ≡− ≈ + ≡ +

0 cos

**δe ee e e**

0 sin

−

*si si s si si si s si s*

 β

α

α

*di di d di p p di*

 α

 δβ

*s s p p p p*

 αβ

( )

*di*, respectively, and hereafter for simplicity, we assume that the divergence

β

0

*s*

β

0

*s*

sin sin cos cos cos 0 sin cos sin

10 ,

−

 β

≡− ≈ <sup>+</sup>

 β  π

> α*p* and β

= − 0 0 cos sin sin . *<sup>p</sup> p s* (14)

μ

α

 δβ

δα

α δβ

*p p p*

*p p p*

β

α

*<sup>k</sup>* of the unit direction vector **e***k***.** 

0 0

α

angles of the signal and diffracted beams are sufficiently small (i.e.,

δα

δα

 α δβ

*di d p di d*

**e e**

cos ,

 β

**eee e**

the azimuthal angle

α*s0* = 0, α*r* = 0, β*r* = −β*s0*, and β*d0* = −β

relations

written as

β

β

β

**Ge e**

Furthermore, we express

δ**e***si* and δ**e***di* are approximated by

and β*di* = β*d0* +δβ 0 0

0 0

0

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*,

α

0

0

δα

**δeee**

≡ +

2 sin

0 0 0 0

*s r s*

Note that if the grating component **G***0* and the wavelength ratio

≡ −= < <

 β

*s*

have only one free parameter. This is because, from Eq. (5),

β

α*<sup>k</sup>* and

(13)

(15)

*<sup>p</sup>* should hold. Thus, the direction vectors are

 αβ

 αβ

 α

*<sup>0</sup>* are given, then **e***p* and **e***d0*

*<sup>p</sup>* should satisfy

δα*si*, δβ*si*, δα*di*, δβ*di* << 1)

> β

β

β

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*<sup>w</sup>* = 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 *yd*-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.

Fig. 4. Simulated results of image reconstruction by PCR. (a) Input image and (b) the reconstructed image. The color in (b) represents the Bragg-matched wavelength of each diffracted wave. The calculation parameters are as follows: λ*w* = 532 nm; θ*<sup>w</sup>* = 30°; θ*<sup>p</sup>* = 50°; *n* = 1; and *fs* = *fd* = 100 mm.

#### **3. Theory of holographic reconstruction with polychromatic light**

#### **3.1 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

$$\mathbf{e}\_x = \frac{\mathbf{e}\_{s0} - \mathbf{e}\_r}{\left| \mathbf{e}\_{s0} - \mathbf{e}\_r \right|}, \ \mathbf{e}\_y = -\frac{\mathbf{e}\_{s0} \times \mathbf{e}\_r}{\left| \mathbf{e}\_{s0} \times \mathbf{e}\_r \right|}, \ \mathbf{e}\_z = \frac{\mathbf{e}\_{s0} + \mathbf{e}\_r}{\left| \mathbf{e}\_{s0} + \mathbf{e}\_r \right|}, \quad \left( \mathbf{e}\_{s0} \neq \pm \mathbf{e}\_r \right) \tag{11}$$

We introduce an elevation angle α*k* and azimuthal angle β*<sup>k</sup>* to specify a unit direction vector **e***k*, as is depicted in Fig. 5(b). In this case the Cartesian components of **e***k* are written as

$$\mathbf{e}\_{k} = \begin{bmatrix} e\_{kx} \\ e\_{ky} \\ e\_{kr} \end{bmatrix} = \begin{bmatrix} \cos \alpha\_{k} \sin \beta\_{k} \\ \sin \alpha\_{k} \\ \cos \alpha\_{k} \cos \beta\_{k} \end{bmatrix}. \tag{12}$$

that the input image, which was an outline character "A" with dimensions 1 cm × 1 cm,

reconstructed image was formed with spectral components ranging from 795 nm to 835 nm, and image magnification occurred in the *yd*-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

*<sup>w</sup>* = 532 nm and was reconstructed with a polychromatic

*p0* of 815 nm. From the figure, we can see that the

λ

λ

Fig. 4. Simulated results of image reconstruction by PCR. (a) Input image and (b) the reconstructed image. The color in (b) represents the Bragg-matched wavelength of each

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

( ) − ×+ <sup>=</sup> = − <sup>=</sup> ≠ ±

*k* and azimuthal angle

 β

*e k si r p di*

 β

**ee ee ee** .

sin . ,,,

**3. Theory of holographic reconstruction with polychromatic light** 

direction vectors **e***s0* and **e***r*, as shown in Fig. 5(a), whose normal bases are given by

− ×+ 0 00

**e***k*, as is depicted in Fig. 5(b). In this case the Cartesian components of **e***k* are written as

α

= = <sup>=</sup>

0 00 , ,, *s r sr s r x y z s r s r sr s r* **ee ee ee e e e ee**

α

cos sin

*kx k k*

*kz k k*

α

cos cos

α

*k ky k*

*e*

*e*

λ

β

**e** (12)

( )

*w* = 532 nm;

0

*<sup>k</sup>* to specify a unit direction vector

(11)

θ*<sup>w</sup>* = 30°; θ*<sup>p</sup>* = 50°;

diffracted wave. The calculation parameters are as follows:

shown in Fig. 4(a), was recorded at

holographic memory systems.

*n* = 1; and *fs* = *fd* = 100 mm.

**3.1 Definition of the coordinate system** 

We introduce an elevation angle

probe beam with a central wavelength

Fig. 5. Definition of (a) the Cartesian coordinate system and (b) the elevation angle α*<sup>k</sup>* and the azimuthal angle β*<sup>k</sup>* of the unit direction vector **e***k***.** 

From the symmetry of the coordinate system and the Bragg condition of Eq. (5), the relations α*s0* = 0, α*r* = 0, β*r* = −β*s0*, and β*d0* = −β*<sup>p</sup>* should hold. Thus, the direction vectors are written as

$$\begin{aligned} \mathbf{e}\_{s0} &= \begin{bmatrix} \sin \mathcal{J}\_{s0} \\ 0 \\ \cos \mathcal{J}\_{s0} \end{bmatrix}, \quad \mathbf{e}\_r = \begin{bmatrix} -\sin \mathcal{J}\_{s0} \\ 0 \\ \cos \mathcal{J}\_{s0} \end{bmatrix}, \quad \mathbf{e}\_p = \begin{bmatrix} \cos \mathcal{a}\_p \sin \mathcal{J}\_p \\ \sin \mathcal{a}\_p \\ \cos \mathcal{a}\_p \cos \mathcal{J}\_p \end{bmatrix}, \quad \mathbf{e}\_{s0} = \begin{bmatrix} -\cos \mathcal{a}\_p \sin \mathcal{J}\_p \\ \sin \mathcal{a}\_p \\ \cos \mathcal{a}\_p \cos \mathcal{J}\_p \end{bmatrix}, \\\ \mathbf{G}\_0 &= \mathbf{e}\_{s0} - \mathbf{e}\_r = \begin{bmatrix} 2\sin \mathcal{J}\_{s0} \\ 0 \\ 0 \end{bmatrix}. \end{aligned} \tag{13}$$

Note that if the grating component **G***0* and the wavelength ratio μ*<sup>0</sup>* are given, then **e***p* and **e***d0* have only one free parameter. This is because, from Eq. (5), α*p* and β*<sup>p</sup>* should satisfy

$$
\cos \alpha\_p \sin \beta\_p = -\mu\_0 \sin \beta\_{s0} \, . \tag{14}
$$

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 |δ**e***si*|2, |δ**e***di*|2 << 1 hold at all components *i*. Then, the deviation vectors δ**e***si* and δ**e***di* are approximated by

$$\begin{aligned} \boldsymbol{\delta}\mathbf{e}\_{si} & \equiv \mathbf{e}\_{si} - \mathbf{e}\_{s0} = \boldsymbol{\delta}\boldsymbol{\alpha}\_{si} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} + \boldsymbol{\delta}\boldsymbol{\beta}\_{si} \begin{bmatrix} \cos\beta\_{s0} \\ 0 \\ -\sin\beta\_{s0} \end{bmatrix} \equiv \boldsymbol{\delta}\boldsymbol{\alpha}\_{si} \mathbf{e}\_{s\boldsymbol{\alpha}} + \boldsymbol{\delta}\boldsymbol{\beta}\_{si} \mathbf{e}\_{s\boldsymbol{\beta}}, \\\ \boldsymbol{\delta}\mathbf{e}\_{di} & \equiv \mathbf{e}\_{di} - \mathbf{e}\_{d0} = \boldsymbol{\delta}\boldsymbol{\alpha}\_{di} \begin{bmatrix} \sin\alpha\_{p}\sin\beta\_{p} \\ \cos\alpha\_{p} \\ -\sin\alpha\_{p}\cos\beta\_{p} \end{bmatrix} + \boldsymbol{\cos}\boldsymbol{\alpha}\_{p}\boldsymbol{\delta}\boldsymbol{\beta}\_{di} \begin{bmatrix} \cos\beta\_{p} \\ 0 \\ \sin\beta\_{p} \end{bmatrix} \\\ & \equiv \boldsymbol{\delta}\boldsymbol{\alpha}\_{di}\mathbf{e}\_{d\boldsymbol{\alpha}} + \boldsymbol{\cos}\boldsymbol{\alpha}\_{p}\boldsymbol{\delta}\boldsymbol{\beta}\_{di} \mathbf{e}\_{d\boldsymbol{\beta}} \end{aligned} \tag{15}$$

Theory of Polychromatic Reconstruction for Volume Holographic Memory 227

Bragg-matched wavelength does not depend only on the amplitude |δ**e***si*|, but is determined by the projection of δ**e***si* on **e***d0*. If we rewrite Eq. (19) using the Cartesian

> β β δβ

β

*s*

*p si p s p si*

2 0

<sup>=</sup> − +

tan 1 1 . 2 sin sin tan tan

*sp s p*

( )

0

 β

tan 1 1 . sin sin tan tan *p sMax sMax*

*s p s s ps*

*si si*

*y x f f*

β

δμ

α

*si*. To see these features more clearly, the spatial distributions of the Bragg-

 α

2 sin

δα

0 0

0 0

*<sup>p</sup>* = 0°), however,

matched 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

Fig. 7. The reconstructed images in the PCR method at several probe elevation angles

The input image was the same as that in Fig. 4. The calculation parameters were as follows:

*<sup>s</sup>*<sup>0</sup> = 30°; *n* = 1; *fs* = *fd* = 100 mm; and

= − <sup>+</sup> <sup>−</sup>

Δ = + + <sup>−</sup>

δ 2 holds. Equation (19) indicates that the

δβ

λ

*<sup>i</sup>* depends only on the azimuthal

*<sup>p</sup>* = 0°, which will be treated further

δ

 β

α

β

α

( )

 β

μ

0 0 0 1 1 . 2 tan tan arcsin sin *<sup>i</sup> si s s*

*<sup>p</sup>* = 0° yields

δβ *BM*) is given by

δα*si* and

*<sup>p</sup>* = 0°. Furthermore, the

α*p*.

(22)

*<sup>p</sup>* is determined from Eq. (14)

(20)

(21)

δβ*si*.

 β

where we assume that the relation|δ**e***si*|2 << |**G***0*|

μ

0

δμ

*BM p*

 λ

λ

When **e***p* lies in the *xz*-plane (i.e.,

δβ

deviation angle

in Section 3.3.

λ

at each α*p*.

*w* = 532 nm;

λ

*p0* = 815 nm;

δμ β

μ

0

Rewriting Eq. (20) using the relation of Eq. (14) at the condition

β

*i*

components in Eqs. (13) and (15) and the relation of Eq. (14), we obtain

α

*p*

− + = −

sin cos sin

ββ

Therefore, the spectral width required for the full image reconstruction (Δ

 β

The Bragg-matched wavelength generally depends on both deviation angles

α

α

( ) <sup>0</sup>

spectral width required for the probe beam has a minimum at

distortion of the reconstructed image is also smallest at

β

α δα

Fig. 6. The relation between the angular spectral component and the imaging position in the Fourier hologram. (a) The recording process and (b) the reconstruction process.

Note that the unit direction vectors **e***s*α and **e***s*β 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 (*xsi*, *ysi*) and the set of the deviation angles (δα*si*, δβ*si*) are related via

$$
\begin{pmatrix}
\delta \partial\_{si} \\
\delta \alpha\_{si}
\end{pmatrix} = \frac{1}{f\_s} \begin{pmatrix}
\chi\_{si} \\
\chi\_{si}
\end{pmatrix} \\
\text{\textsuperscript{\rot \partial}}
\tag{16}
$$

where *fs* is the focal length of the Fourier transform lens in the recording process. Similarly, the relation between the position on the reconstructed image plane (*xdi*, *ydi*) and the set of the deviation angles (δα*di*, δβ*di*) is given by

$$
\begin{pmatrix} \boldsymbol{\omega}\_{di} \\ \boldsymbol{y}\_{di} \end{pmatrix} = \boldsymbol{f}\_d \begin{pmatrix} \cos \alpha\_p \delta \beta\_{di} \\ \delta \alpha\_{di} \end{pmatrix} \tag{17}
$$

where *fd* 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 −*xsMax* ≤ *xsi* ≤ *xsMax* and −*ysMax* ≤ *ysi* ≤ *ysMax*. Therefore, the maximum deviation angles of the signal beam, δα*sMax* and δβ*sMax*, are

$$
\begin{pmatrix}
\delta \mathcal{B}\_{s\text{Max}} \\
\delta \alpha\_{s\text{Max}}
\end{pmatrix} = \frac{1}{f\_s} \begin{pmatrix}
\mathcal{X}\_{s\text{Max}} \\
\mathcal{Y}\_{s\text{Max}}
\end{pmatrix}.
\tag{18}
$$

#### **3.2 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 δ**e***si*, the difference between μ*i* and μ*<sup>0</sup>* is given by

$$
\delta\mu\_i \equiv \mu\_i - \mu\_0 \approx -\frac{2\mathbf{e}\_{d0} \cdot \mathbf{\delta}\mathbf{e}\_{si}}{\left|\mathbf{G}\_0\right|^2},\tag{19}
$$

Fig. 6. The relation between the angular spectral component and the imaging position in the

plane of the input image, as shown in Fig. 6(a). Thus, the position on the input plane (*xsi*, *ysi*)

 = <sup>1</sup> , *si si si s si*

where *fs* is the focal length of the Fourier transform lens in the recording process. Similarly, the relation between the position on the reconstructed image plane (*xdi*, *ydi*) and the set of the

> = cos , *di p di d di di*

where *fd* 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 −*xsMax* ≤ *xsi* ≤ *xsMax* and −*ysMax* ≤ *ysi* ≤ *ysMax*. Therefore, the maximum deviation angles of the

> = <sup>1</sup> . *sMax sMax sMax s sMax*

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

> μ*i* and μ

0 2

0 <sup>2</sup> , *d si*

**e δe**

*si*) are related via

*x f y*

> α δβ

δα

*x*

*<sup>0</sup>* is given by

correspond to the normal bases at the Fourier

*<sup>f</sup> <sup>y</sup>* (18)

**<sup>G</sup>** (19)

(16)

(17)

Fourier hologram. (a) The recording process and (b) the reconstruction process.

δβ

δα

α and **e***s*β

δα*si*, δβ

*di*) is given by

*x <sup>f</sup> <sup>y</sup>*

δβ

δα

δμ μ μ<sup>⋅</sup> ≡ − ≈− <sup>0</sup>

*i i*

Note that the unit direction vectors **e***s*

and the set of the deviation angles (

δα*di*, δβ

deviation angles (

signal beam,

δα

*sMax* and

δβ

**3.2 Spectral width required for the reconstruction** 

deviation vectors δ**e***si*, the difference between

*sMax*, are

where we assume that the relation|δ**e***si*|2 << |**G***0*| 2 holds. Equation (19) indicates that the Bragg-matched wavelength does not depend only on the amplitude |δ**e***si*|, but is determined by the projection of δ**e***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

$$\begin{split} \delta\mu\_{i} &= -\frac{\sin\alpha\_{p}\delta\alpha\_{si} - \cos\alpha\_{p}\sin\left(\beta\_{s0} + \beta\_{p}\right)\delta\beta\_{si}}{2\sin^{2}\beta\_{s0}}\\ &= \frac{\mu\_{0}}{2} \left[ \frac{\tan\alpha\_{p}}{\sin\beta\_{s0}\sin\beta\_{p}} \delta\alpha\_{si} - \left(\frac{1}{\tan\beta\_{s0}} + \frac{1}{\tan\beta\_{p}}\right) \delta\beta\_{si} \right]. \end{split} \tag{20}$$

Therefore, the spectral width required for the full image reconstruction (Δλ*BM*) is given by

$$
\Delta \mathcal{J}\_{\rm RM} = \mathcal{J}\_{p0} \left[ \frac{\tan \left| \alpha\_p \right|}{\sin \mathcal{J}\_{s0} \sin \left( - \mathcal{J}\_p \right)} \frac{\delta \mathcal{Y}\_{s\rm Max}}{f\_s} + \left( \frac{1}{\tan \mathcal{J}\_{s0}} + \frac{1}{\tan \mathcal{J}\_p} \right) \frac{\delta \mathcal{X}\_{s\rm Max}}{f\_s} \right]. \tag{21}$$

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*<sup>p</sup>* = 0°), however, δμ*<sup>i</sup>* 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*<sup>p</sup>* = 0°. Furthermore, the distortion of the reconstructed image is also smallest at α*<sup>p</sup>* = 0°, which will be treated further in Section 3.3.

Fig. 7. The reconstructed images in the PCR method at several probe elevation angles α*p*. The input image was the same as that in Fig. 4. The calculation parameters were as follows: λ*w* = 532 nm; λ*p0* = 815 nm; β*<sup>s</sup>*<sup>0</sup> = 30°; *n* = 1; *fs* = *fd* = 100 mm; and β*<sup>p</sup>* is determined from Eq. (14) at each α*p*.

Rewriting Eq. (20) using the relation of Eq. (14) at the condition α*<sup>p</sup>* = 0° yields

$$\delta\mu\_{i} = -\frac{\mu\_{0}}{2} \left| \frac{1}{\tan\beta\_{s0}} + \frac{1}{\tan\left[\arcsin\left(-\mu\_{0}\sin\beta\_{s0}\right)\right]} \right| \delta\beta\_{si} \,. \tag{22}$$

Theory of Polychromatic Reconstruction for Volume Holographic Memory 229

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

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

> δα*si* and δβ

image was the same as that in Fig. 4. The focal lengths of the Fourier transform lenses are (a) *fs* = *fd* = 50 mm, (b) *fs* = *fd* = 20 mm, (c) *fs* = *fd* = 10 mm. The calculation parameters were as

*<sup>s</sup>*<sup>0</sup> = 30°; and *n* = 1;

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

> ≡− ≈ + 00 0 μ

> > α

**δe** (29)

2 sin sin 0

0 0

*s si s p si p s p si*

− + = − <sup>−</sup>

cos 2 sin sin cos sin

δμ

α δα

0 2

δμ

, **δ** *di di d si i* **e ee δe G** (28)

( )

0

 β β δβ

β

0

*s*

*<sup>0</sup>*. From Eqs. (13), (15), and (20), the Cartesian

β

0 .

*si*. However, for large deviation angles, their

*si* on the reconstructed image. The input

δα*si* and δβ

proportional to the deviation angles

Fig. 9. Influence of large deviation angles

λ

**3.3 Distortion of the reconstructed image** 

reconstructed image in the PCR method.

where we neglect the product of δ**e***si* and

β δβ

> β δβ

0

*s si*

 δα

components of δ**e***di* are expressed as

μ

*di si*

*p0* = 815 nm;

From Eq. (10), the deviation vector δ**e***di* is approximated by

β

angle.

follows:

λ

*w* = 532 nm;

There is an optimum recording angle β*s0Opt* that minimizes the required spectral width for a given μ*<sup>0</sup>*, as shown in Fig. 8, which is given by

$$\mathcal{J}\_{s0\text{opt}} = \arctan\left(\frac{1}{\mu\_0}\right) \quad \text{ ( $\mu\_0 \neq 1$ )}\tag{23}$$

Then, Eq. (22) can be simplified to

$$
\delta\mathfrak{a}\_i = -\frac{1}{2} \left(\mu\_0^{-2} - 1\right) \delta\mathfrak{P}\_{si} \,. \tag{24}
$$

In this case, the required spectral width Δλ*BM* reaches a minimum and is expressed as

$$
\Delta \mathcal{A}\_{\text{BMMix}} = \mathcal{A}\_{p0} \left( \mu\_0 - \frac{1}{\mu\_0} \right) \frac{\delta \chi\_{s\text{Max}}}{f\_s}. \tag{25}
$$

However, in most cases, the allowed deviation angle δβ*sMax* is determined by the spectral width Δλ*p* of a given probe light source. Assuming that α*<sup>p</sup>* = 0°, from Eqs. (21), we obtain

$$
\delta \mathcal{B}\_{s\text{max}} = \left(\frac{1}{\tan \mathcal{B}\_{s0}} + \frac{1}{\tan \mathcal{B}\_p} \right)^{-1} \frac{\Delta \mathcal{A}\_p}{\mathcal{A}\_{p0}} \tag{26}
$$

At the optimum recording angle, δβ*sMax* reaches a maximum

$$
\delta \mathcal{B}\_{\text{sMax}} = \left(\mu\_0 - \frac{1}{\mu\_0}\right)^{-1} \frac{\Delta \mathcal{A}\_p}{\mathcal{A}\_{p0}} \tag{27}
$$

For example, if we have a probe light source whose central wavelength λ*p0* is 815 nm and whose full spectral width Δλ*p* is 50 nm, and the recording wavelength λ*<sup>w</sup>* is 532 nm, then the optimum recording angle β*s0Opt* is about 33°, and the allowed deviation angle δβ*sMax* is about 4.0°.

Fig. 8. The slope coefficient of δβ*si* in Eq. (22) is plotted as a function of the signal incident angle β*s0*. We assumed that λ*w* = 532 nm and λ*p0* = 815 nm.

μ

 μ=− − ( )

0 0

λ

Δ= −

 = ≠ 0 0 0

 μ( )

δβ <sup>2</sup> 0 <sup>1</sup> 1 .

<sup>1</sup> . *sMax*

*s x f* δ

δβ

1

−

*p*

λ

λ

α

0 0

1

−

0 0 1 *<sup>p</sup>*

*p* is 50 nm, and the recording wavelength

*p*

*s0Opt* is about 33°, and the allowed deviation angle

*si* in Eq. (22) is plotted as a function of the signal incident

λ

λ

*s p p*

 β

0

μ

1 1 tan tan

*sMax* reaches a maximum

 <sup>Δ</sup> = − 

μ

 Δ = + 

β

ax 0

 μ

*s0Opt* that minimizes the required spectral width for a

<sup>1</sup> arctan , 1 *s Opt* (23)

<sup>2</sup> *<sup>i</sup> si* (24)

*sMax* is determined by the spectral

λ

λ

*<sup>p</sup>* = 0°, from Eqs. (21), we obtain

(25)

(26)

(27)

*p0* is 815 nm and

*<sup>w</sup>* is 532 nm, then

δβ*sMax* is

*BM* reaches a minimum and is expressed as

β

δμ

*BMMin p*

 λμ

λ

*p* of a given probe light source. Assuming that

max

δβ

δβ

*sM*

For example, if we have a probe light source whose central wavelength

*s*

δβ

λ

β

δβ

*w* = 532 nm and

λ

*p0* = 815 nm.

λ

However, in most cases, the allowed deviation angle

There is an optimum recording angle

Then, Eq. (22) can be simplified to

At the optimum recording angle,

whose full spectral width Δ

the optimum recording angle

Fig. 8. The slope coefficient of

*s0*. We assumed that

In this case, the required spectral width Δ

*<sup>0</sup>*, as shown in Fig. 8, which is given by

β

given μ

width Δ

about 4.0°.

angle β λ

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.

Fig. 9. Influence of large deviation angles δα*si* and δβ*si* on the reconstructed image. The input image was the same as that in Fig. 4. The focal lengths of the Fourier transform lenses are (a) *fs* = *fd* = 50 mm, (b) *fs* = *fd* = 20 mm, (c) *fs* = *fd* = 10 mm. The calculation parameters were as follows: λ*w* = 532 nm; λ*p0* = 815 nm; β*<sup>s</sup>*<sup>0</sup> = 30°; and *n* = 1;

#### **3.3 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 δ**e***di* is approximated by

$$
\delta \mathbf{e}\_{\vec{m}} \equiv \mathbf{e}\_{\vec{m}} - \mathbf{e}\_{d0} \approx \mu\_0 \delta \mathbf{e}\_{\vec{s}} + \delta \mu\_i \mathbf{G}\_0 \, \, \, \, \, \tag{28}
$$

where we neglect the product of δ**e***si* and δμ*<sup>0</sup>*. From Eqs. (13), (15), and (20), the Cartesian components of δ**e***di* are expressed as

$$\boldsymbol{\mathfrak{S}}\mathbf{e}\_{\boldsymbol{\omega}} = \mu\_{0} \begin{bmatrix} \cos\beta\_{s0}\delta\mathcal{B}\_{\boldsymbol{s}} \\ \delta\alpha\_{s} \\ -\sin\beta\_{s0}\delta\mathcal{B}\_{\boldsymbol{s}} \end{bmatrix} - \frac{\sin\alpha\_{p}\delta\alpha\_{s} - \cos\alpha\_{p}\sin\left(\beta\_{s0} + \beta\_{p}\right)\delta\mathcal{B}\_{\boldsymbol{s}}}{2\sin^{2}\beta\_{s0}} \begin{bmatrix} 2\sin\beta\_{s0} \\ 0 \\ 0 \end{bmatrix}.\tag{29}$$

Theory of Polychromatic Reconstruction for Volume Holographic Memory 231

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

δφ

10 0

0 01 0

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

*cz*) is given by

( )

2 2

⋅+ ⋅ ≡ − − −

*p i cz p i*

**e G δe eG G δe G**

*i cz i*

<sup>+</sup>

2 2

<sup>−</sup> <sup>≡</sup> − ≈

*cz cz i i s cz*

1 0 2 sin .

**δe G G** (33)

displacement vector of the grating component (δ**e***cz*) is approximated by

δφ

δμ

*cz*

*cz*

δφ

δμ

<sup>⋅</sup> ≈ −

2

= −

α

0

*s*

sin , sin

*p cz*

*d cz*

0 2 0

**e δe G**

> δφ β

φ

*cz* from the *y*-axis, respectively.

0

β δφ

*cz* around the *z*-axis (peristrophic rotation), the

(34)

crystal around the z-axis and an axis inclined at

When the crystal is rotated by a small angle

**3.4.1 Peristrophic multiplexing** 

ratio after the peristrophic rotation (

On the other hand, δ**e***di* is also expressed by Eq. (15) using **e***d*α and **e***d*β. Rewriting Eq. (29) using normal bases of **e***d*α and **e***d*βyields

$$\boldsymbol{\mathfrak{G}}\mathbf{e}\_{di} = \left(\frac{\mu\_0}{\cos\alpha\_p}\boldsymbol{\delta}\alpha\_{si}\right)\mathbf{e}\_{d\alpha} + \left(\mu\_0\frac{\tan\alpha\_p}{\tan\beta\_p}\boldsymbol{\delta}\alpha\_{si} + \cos\alpha\_p\boldsymbol{\delta}\beta\_{si}\right)\mathbf{e}\_{d\beta}.\tag{30}$$

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*):

$$
\begin{pmatrix}
\cos\alpha\_p \delta \mathcal{B}\_{\dot{\alpha}} \\
\delta \alpha\_{\dot{\alpha}}
\end{pmatrix} = \begin{pmatrix}
\cos\alpha\_p & \mu\_0 \frac{\tan\alpha\_p}{\tan\beta\_p} \\
0 & \frac{\mu\_0}{\cos\alpha\_p}
\end{pmatrix} \begin{pmatrix}
\delta \mathcal{B}\_{\dot{\alpha}} \\
\delta \alpha\_{\dot{\alpha}}
\end{pmatrix}.
\tag{31}
$$

From Eqs. (16) and (17), we can finally derive the expression for a transfer matrix between the object and reconstructed image plane, that is,

$$\mathbf{T}\begin{pmatrix} \mathbf{x}\_{di} \\ \mathbf{y}\_{di} \end{pmatrix} = \mathbf{T} \begin{pmatrix} \mathbf{x}\_{si} \\ \mathbf{y}\_{si} \end{pmatrix}, \quad \mathbf{T} \equiv \frac{f\_d}{f\_s} \begin{pmatrix} \cos\alpha\_p & \mu\_0 \frac{\tan\alpha\_p}{\tan\beta\_p} \\ & \tan\beta\_p \\ & 0 & \frac{\mu\_0}{\cos\alpha\_p} \end{pmatrix}. \tag{32}$$

When α*<sup>p</sup>* = 0° or β*<sup>p</sup>* = −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*<sup>p</sup>* = 0°, as was seen in Fig. 7.

#### **3.4 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.

#### **3.4.1 Peristrophic multiplexing**

230 Holograms – Recording Materials and Applications

 = ++ 

α

*p*

 β

0

 δα

**δe e e** (30)

α

*p*

β

μ

α μ

*p*

*<sup>p</sup>* = −90°, non-diagonal components in the transfer matrix become zero and

cos tan , .

0

cos

tan

0

α

*p*

0

α and **e***d*β

 α δβ

δβ

δα

α

*p*

β

μ

**T T** (32)

cos

tan

0

α

*p*

cos .

. Rewriting Eq. (29)

(31)

β

On the other hand, δ**e***di* is also expressed by Eq. (15) using **e***d*

δα

δα*di*, δβ*di*):

 = ≡ 

*di si s*

*x x f y y f*

α

*p p*

cos tan

 μ

0 tan

*di si d si p si d*

Comparing the coefficients in (30) with those in Eq. (15), we obtain the relation between the

α μ

cos

<sup>=</sup>

0

cos tan .

From Eqs. (16) and (17), we can finally derive the expression for a transfer matrix between

*di si p d*

the image can be reconstructed without any tilting. However, the distortion becomes

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

*p p di p si di si*

α and **e***d*βyields

δα*si*, δβ*si*) and (

μ

0

α

α δβ

δα

the object and reconstructed image plane, that is,

*<sup>p</sup>* = 0°, as was seen in Fig. 7.

using normal bases of **e***d*

deviation angles (

When α

minimal at

*<sup>p</sup>* = 0° or

α

β

**3.4 Multiplexing with crystal rotation** 

When the crystal is rotated by a small angle δφ*cz* around the *z*-axis (peristrophic rotation), the displacement vector of the grating component (δ**e***cz*) is approximated by

$$
\boldsymbol{\delta\mathbf{e}}\_{cz} \equiv \begin{pmatrix} \mathbf{1} & -\boldsymbol{\delta\phi}\_{cz} & \mathbf{0} \\ \boldsymbol{\delta\phi}\_{cz} & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{1} \end{pmatrix} \mathbf{G}\_i - \mathbf{G}\_i \doteq \begin{bmatrix} \mathbf{0} \\ \mathbf{2}\sin\mathcal{J}\_{s0}\boldsymbol{\delta\phi}\_{cz} \\ \mathbf{0} \end{bmatrix}.\tag{33}$$

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

$$\begin{split} \delta\mu\_{cz} &= -\frac{2\mathbf{e}\_p \cdot (\mathbf{G}\_i + \mathbf{\delta}\mathbf{e}\_{cz})}{\left|\mathbf{G}\_i + \mathbf{\delta}\mathbf{e}\_{cz}\right|^2} - \left(-\frac{2\mathbf{e}\_p \cdot \mathbf{G}\_i}{\left|\mathbf{G}\_i\right|^2}\right) \\ &= -\frac{2\mathbf{e}\_{d0} \cdot \mathbf{\delta}\mathbf{e}\_{cz}}{\left|\mathbf{G}\_0\right|^2} \\ &= -\frac{\sin\alpha\_p}{\sin\beta\_{s0}}\delta\phi\_{cz} \end{split} \tag{34}$$

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

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

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

Next, let us consider the angular selectivity in the crystal angle multiplexing, where the

1 0 cos 0

φ δφ

φ δφ

*cz cy*

− − <sup>−</sup> **δe G G** (39)

δμ

<sup>≡</sup> − ≈

cos sin 1 2 sin cos

φ

*cz* is the peristrophic rotation angle.

*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

0 1 sin 0 .

*cz cy cz cy s cz cy*

*cz*, as shown in Fig. 10. Note that the inclination

0

 φ δφ

β

*cy*) and the displacement vector of **e***di* after

δφ

*cy cz cy i i*

 φ δφ

**3.4.2 Crystal angle multiplexing with peristrophic rotation** 

whose spectral width is 40 nm.

angle φ

crystal is rotated around an axis inclined at

angle multiplexing (δ**e***cy*) is approximated by

φ δφ

the crystal rotation (δ**e***dcy*) are expressed as

In this case, the shift of the wavelength ratio (

δφ

*cz*. We also

the reconstructed image obtained while varying the peristrophic rotation angle

multiplexing without any adverse influence on the multiplexing capability.

where last equality is obtained by substituting Eqs. (13) and (33). The displacement vector of **e***di* after the peristrophic rotation (δ**e***dcz*) is expressed as

$$\begin{split} \delta \mathbf{\dot{e}}\_{dc} &= \left[ \mathbf{e}\_p + (\mu\_i + \delta \boldsymbol{\mu}\_{cz}) \left( \mathbf{G}\_i + \delta \mathbf{e}\_{cz} \right) \right] - \left( \mathbf{e}\_p + \mu\_i \mathbf{G}\_i \right) \\ &= \mu\_0 \delta \mathbf{e}\_{cz} + \delta \boldsymbol{\mu}\_{cz} \mathbf{G}\_0 \\ &= -2 \, \delta \phi\_{cz} \begin{bmatrix} \sin \alpha\_p \\ \cos \alpha\_p \sin \beta\_p \\ 0 \end{bmatrix}, \end{split} \tag{35}$$

where we use Eqs. (13), (14), (33), and (34) to derive the last equality. If we rewrite Eq. (35) using normal bases of **e***d*α and **e***d*β 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:

$$
\begin{pmatrix}
\delta \mathfrak{x}\_{diz} \\
\delta \mathfrak{y}\_{diz}
\end{pmatrix} = 2\mu\_0 \sin \mathcal{B}\_{s0} f\_d \begin{pmatrix}
\tan \mathcal{a}\_p \\
\tan \mathcal{B}\_p \\
\hline 1
\end{pmatrix} \delta \phi\_{cz}.
\tag{36}
$$

For simplicity, we assume that α*<sup>p</sup>* = 0. Then, from Eqs. (20), (32), (34), and (36), the Braggmatched wavelength λ*BMcz* can be expressed as a function of the imaging position (*xd*, *yd*) and the rotation angle δφ*cz*:

$$\lambda\_{\text{gMax}}\left(\mathbf{x}\_{d}, y\_{d}, \delta\phi\_{cz}\right) = \lambda\_{p0} - \frac{\lambda\_{p0}}{2} \left(\frac{1}{\tan\beta\_{s0}} + \frac{1}{\tan\beta\_{p}}\right) \frac{\mathbf{x}\_{d}}{f\_{d}}$$

$$\left(-\mathbf{x}\_{s\text{Max}} \frac{f\_{d}}{f\_{s}} \le \mathbf{x}\_{d} \le \mathbf{x}\_{s\text{Max}} \frac{f\_{d}}{f\_{s}}\right) \tag{37}$$

$$-\mu\_{0} y\_{s\text{Max}} \frac{f\_{d}}{f\_{s}} + 2\mu\_{0} \sin\beta\_{s0} f\_{d} \delta\phi\_{cz} \le y\_{d} \le \mu\_{0} y\_{s\text{Max}} \frac{f\_{d}}{f\_{s}} + 2\mu\_{0} \sin\beta\_{s0} f\_{d} \delta\phi\_{cz}$$

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 *yd* 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

$$\mathcal{L}\phi\_{cz\text{Min}} = \frac{\mathcal{Y}\_{s\text{Max}}}{\sin\mathcal{B}\_{s0}f\_s}.\tag{38}$$

This equation does not include the wavelength ratio μ*<sup>0</sup>*. 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*<sup>0</sup>* in the *yd* direction. This is because the amount of spatial shift is also increased by a factor of μ*<sup>0</sup>*, as shown in Eq. (36).

where last equality is obtained by substituting Eqs. (13) and (33). The displacement vector of

( )( )

**δee G δe eG**

≡ ++ + −+ 

*dcz p i cz i cz p i i*

α

*p*

sin 2 cos sin , 0

*cz p p*

 

 β

where we use Eqs. (13), (14), (33), and (34) to derive the last equality. If we rewrite Eq. (35)

similarly to Section 3.3, we obtain the shift amount of the imaging position due to the

 β

*dicz p*

<sup>=</sup>

0 0

( )

α

*p*

tan tan 2 sin . <sup>1</sup>

β

α

*p*

*BMcz* can be expressed as a function of the imaging position (*xd*, *yd*) and

*p d*

0

β

≈− +

 

*s d cz*

cos

δφ

*<sup>p</sup>* = 0. Then, from Eqs. (20), (32), (34), and (36), the Bragg-

*s pd*

 β

μ

δφ

*x*

 β δφ

*czMin*) is expressed as

*<sup>0</sup>*. Therefore, the readout at a longer

*<sup>0</sup>* in the *yd* direction. This is

*<sup>0</sup>*, as shown in Eq. (36).

<sup>=</sup> (38)

μ

μ

μ

and compare the coefficients with those in Eq. (15),

(35)

(36)

(37)

.

μ δμ

> δμ

*cz cz*

0 0

**δe G**

 α

= −

μ

*<sup>f</sup> <sup>y</sup>*

**e***di* after the peristrophic rotation (δ**e***dcz*) is expressed as

α and **e***d*β

> δ

*x*

δ

*dicz*

α

*BMcz d d cz p*

μ

minimum angular separation in peristrophic multiplexing (

This equation does not include the wavelength ratio

though the reconstructed image was magnified by a factor of

because the amount of spatial shift is also increased by a factor of

 δφ

( ) <sup>0</sup>

 β δφ

− + ≤≤ +

0

 λ

*d d*

*s s*

*czMin*

δφ

1 1 , , 2 tan tan

λ

*x y <sup>f</sup>*

*f f x xx f f*

− ≤≤

0 00 0 00

*f f y f <sup>y</sup> y f f f*

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 *yd* 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

2 sin 2 sin

*sMax s d cz d sMax s d cz*

*d d sMax d sMax s s*

> μ

0 . sin *sMax*

β

*y*

wavelength does not affect the angular selectivity in the peristrophic multiplexing, even

*s s*

*f*

μ

using normal bases of **e***d*

For simplicity, we assume that

μ λ

δφ*cz*:

λ

peristrophic rotation:

matched wavelength

the rotation angle

μ

≈ +

δφ

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.

#### **3.4.2 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 (δ**e***cy*) is approximated by

$$\boldsymbol{\mathfrak{G}}\mathbf{e}\_{cy} \equiv \begin{pmatrix} 1 & 0 & \cos\phi\_{cz}\delta\phi\_{cy} \\ 0 & 1 & \sin\phi\_{cz}\delta\phi\_{cy} \\ -\cos\phi\_{cz}\delta\phi\_{cy} & -\sin\phi\_{cz}\delta\phi\_{cy} & 1 \end{pmatrix} \mathbf{G}\_{i} - \mathbf{G}\_{i} = \begin{bmatrix} 0 \\ 0 \\ -2\sin\beta\_{s0}\cos\phi\_{cz}\delta\phi\_{cy} \end{bmatrix}. \tag{39}$$

In this case, the shift of the wavelength ratio (δμ*cy*) and the displacement vector of **e***di* after the crystal rotation (δ**e***dcy*) are expressed as

Theory of Polychromatic Reconstruction for Volume Holographic Memory 235

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

*BM* in Eq. (25), as the probe spectral width Δ

*p cz s x*

 φ *f*

λ

φ

*BMcy* at each rotation

*cz* is assumed to be

δφ

*cy* is the

λ*<sup>p</sup>*. The

<sup>0</sup> tan 1 . tan cos *s sMax*

β

 = + 

β

λ*p*, then

δφ*cyMin* is

(45)

the full image reconstruction, Δ

given by

angle δφ

zero.

width of 40 nm was used.

λ

δφ

*cyMin*

Fig. 12. The imaging location *xd* and the Bragg-matched wavelength

be reconstructed by the probe beam. The peristrophic rotation angle

Fig. 13. The reconstructed images after the crystal rotation around the *y*-axis.

rotation angle. The polychromatic light with a central wavelength of 815 nm and a spectral

*cy*. The reconstructed image field is limited by the probe spectral band Δ

dotted part of the Bragg-matched line corresponds to the portion of the image that will not

$$\delta\boldsymbol{\mu}\_{\rm cy} \simeq -\frac{2\mathbf{e}\_{d0} \cdot \delta\mathbf{e}\_{\rm cy}}{\left|\mathbf{G}\_{0}\right|^{2}} = \frac{\cos\alpha\_{p}\cos\beta\_{p}}{\sin\beta\_{s0}}\cos\phi\_{\rm c}\delta\phi\_{\rm c} \tag{40}$$

and

$$\boldsymbol{\mathfrak{ds}}\_{\rm{dcy}} = \mu\_0 \boldsymbol{\mathfrak{ds}}\_{\rm{cy}} + \boldsymbol{\mathfrak{ds}}\_{\rm{cy}} \mathbf{G}\_0 = 2 \cos \alpha\_p \cos \varphi\_{\rm{cz}} \boldsymbol{\mathfrak{ds}}\_{\rm{ccy}} \begin{bmatrix} \cos \beta\_p \\ 0 \\ \sin \beta\_p \end{bmatrix} \tag{41}$$

respectively. Then, the imaging shift due to the rotation of the crystal angle multiplexing is written as

$$
\begin{pmatrix}
\delta \mathbf{x}\_{\mathrm{d}\boldsymbol{y}} \\
\delta \mathbf{y}\_{\mathrm{d}\boldsymbol{y}}
\end{pmatrix} = f\_d \begin{pmatrix}
\mathbf{2} \cos \alpha\_p \cos \phi\_{\boldsymbol{x}} \delta \phi\_{\boldsymbol{c}} \\
\mathbf{0}
\end{pmatrix}.
\tag{42}
$$

Thus, the reconstructed image will shift in the *xd* direction regardless of α*<sup>p</sup>*. Assuming that α*<sup>p</sup>* = 0, we obtain the Bragg-matched wavelength λ*BMcy* as a function of the imaging position (*xd*, *yd*) and the rotation angles δφ*cy* and φ*cz*:

$$\mathcal{A}\_{\text{BMy}}\left(\mathbf{x}\_{d'},\mathbf{y}\_{d'},\mathfrak{G}\boldsymbol{\phi}\_{\text{cy}},\boldsymbol{\phi}\_{\text{cz}}\right) = \mathcal{A}\_{p0} - \frac{\mathcal{A}\_{p0}}{2} \left(\frac{1}{\tan\beta\_{s0}} + \frac{1}{\tan\beta\_{p}}\right) \frac{\mathbf{x}\_{d}}{f\_{d}} + \frac{\mathcal{A}\_{p0}}{\tan\beta\_{s0}} \cos\phi\_{\text{cz}} \boldsymbol{\delta}\boldsymbol{\phi}\_{\text{cy}}$$

$$\begin{pmatrix} -\mathbf{x}\_{s\text{Max}} & \frac{f\_{d}}{f\_{s}} + 2f\_{d}\cos\phi\_{\text{cz}}\boldsymbol{\delta}\boldsymbol{\phi}\_{\text{cy}} \le \mathbf{x}\_{d} \le \mathbf{x}\_{s\text{Max}}\\ & -\mu\_{0}y\_{s\text{Max}} & \frac{f\_{d}}{f\_{s}} \le y\_{d} \le \mu\_{0}y\_{s\text{Max}} \frac{f\_{d}}{f\_{s}} \end{pmatrix} . \tag{43}$$

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 Δλ*<sup>p</sup>*, the minimum angular separation (δφ*cyMin*) is given by

$$\delta\phi\_{\text{cylin}} = \min\left[\frac{\mathbf{x}\_{s\text{Max}}}{\cos\phi\_{\text{cz}}f\_{s}}, \frac{\tan\mathcal{B}\_{s0}}{2\cos\phi\_{\text{cz}}\mathcal{A}\_{p0}}\left(\Delta\mathcal{J}\_{p} + \mathcal{J}\_{p0}\left(\frac{1}{\tan\mathcal{B}\_{s0}} + \frac{1}{\tan\mathcal{B}\_{p}}\right)\frac{\mathbf{x}\_{s\text{Max}}}{f\_{s}}\right)\right],\tag{44}$$

0 0

sin *d cy p p cy cz cy*

α

2 cos cos . <sup>0</sup>

 φ δφ

2 tan tan tan

ββ

*p p d*

≈+ =

respectively. Then, the imaging shift due to the rotation of the crystal angle multiplexing is

*dicy p cz cy*

1 1 ,, , cos

*BMcy d d cy cz p cz cy*

2 cos 2 cos

*d d sMax d sMax s s*

 μ

*sMax d cz cy d sMax d cz cy*

 − + ≤≤ +

*f f y yy f f*

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 Bragg-

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

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

*cyMin*) is given by

0

β

 φλ λ

0 0 0 tan 1 1 min , , cos 2 cos tan tan *sMax s sMax*

β

*cz s cz p s p s*

*x x f f*

λ λ

<sup>=</sup> Δ+ +

<sup>=</sup>

α

λ

 ≈− + + 

α

*s*

β

 β

cos

2 cos cos 0 ,

0 0

*x*

*s pd s*

 ϕ δφ

**δe δe G** (41)

 φ δφ

**<sup>G</sup>** (40)

cos

*p*

β

*p*

α

φ δφ

*BMcy* as a function of the imaging position

 λ

φ δφ

*p0* and its full spectral width is Δ

 β

 β

.

δφ

(42)

(43)

*<sup>p</sup>*. Assuming that

*cy* in Fig. 12. The

λ*<sup>p</sup>*, the

(44)

β

sin

2 cos cos

*dcy cy cy p cz cy*

*d*

( ) 0 0

0 0

*d d*

*f f x f xx f f f*

− ≤≤

*x y <sup>f</sup>*

λ

*s s*

0 2

0 0

*dicy*

Thus, the reconstructed image will shift in the *xd* direction regardless of

0

φ δφ

matched wavelength in Eq. (43) is plotted at several rotation angles

λ

*x <sup>f</sup> <sup>y</sup>*

δ

δ

δφ*cy* and φ*cz*:

μ

*<sup>p</sup>* = 0, we obtain the Bragg-matched wavelength

 δφ φ

the central wavelength of the probe beam is

δφ

*cyMin p p*

φ

minimum angular separation (

δφ

(*xd*, *yd*) and the rotation angles

λ

 δμ

<sup>⋅</sup> ≈− = **<sup>e</sup> <sup>δ</sup><sup>e</sup>**

δμ

μ

and

written as

α

Fig. 13.

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, Δλ*BM* in Eq. (25), as the probe spectral width Δλ*p*, then δφ*cyMin* is given by

$$\mathcal{L}\phi\_{cyMin} = \left(1 + \frac{\tan\mathcal{B}\_{s0}}{\tan\mathcal{B}\_p}\right)\frac{\mathcal{X}\_{s\text{Max}}}{\cos\phi\_{cx}f\_s}.\tag{45}$$

Fig. 12. The imaging location *xd* and the Bragg-matched wavelength λ*BMcy* at each rotation angle δφ*cy*. The reconstructed image field is limited by the probe spectral band Δλ*<sup>p</sup>*. The dotted part of the Bragg-matched line corresponds to the portion of the image that will not be reconstructed by the probe beam. The peristrophic rotation angle φ*cz* is assumed to be zero.

Fig. 13. The reconstructed images after the crystal rotation around the *y*-axis. δφ*cy* is the rotation angle. The polychromatic light with a central wavelength of 815 nm and a spectral width of 40 nm was used.

Theory of Polychromatic Reconstruction for Volume Holographic Memory 237

( ) sin( ) sinc

( ) 0 0 rect , *p p*

is the spectral intensity of the probe beam whose central wavelength, full spectral width,

rect 1 2 1 2 .

Equation (49) indicates that the off-Bragg diffraction will occur at Δ**K** around the origin. The

( ) <sup>2</sup> , , *<sup>i</sup>*

holds at any component of the off-Bragg vector Δ**K**. Thus, hereafter, we assume that Eq. (53) represents an upper limit of |Δ*Ki*| in which the grating **G***<sup>i</sup>* + δ**e***Goff* can produce off-Bragg

Before considering the crosstalk noise, let us first discuss the prpperties of signal diffraction

*x s off p*

Δ =

Δ

*off p pi off offMax I Id* πδλ

 ∝ +

δλ

*y w*

Δ

*z*

From Eq. (49), the intensity of the signal diffracted wave at wavelength

δλ

*K K k K*

δα*soff* = δβ*soff* = δφ*cy* = δφ

0 0 2 sin

β δλ λ

( ) <sup>2</sup> sinc , *off*

λ δλ

0 <sup>0</sup> 2 sin *w p offMax n L w sx* λ λ

β

0

0 .

 λ

<sup>=</sup> (56)

*i K i xyz <sup>L</sup>* π

 < = = <sup>&</sup>gt;

*x x*

*p p*

<sup>−</sup> <sup>=</sup> Δ Δ

λ λ

1 12

*x*

0 12

*x*

 λ

*<sup>p</sup>*, and *Ip0*, respectively; and rect(*x*) is the rectangle function, defined

Δ= = (53)

*x*

*I*

λ

*p*

λ

( )

*I*

diffracted intensity will become zero when the relation

is the sinc function;

and intensity are

diffraction.

as

where

as

λ*p0*, Δλ

**3.5.1 Properties of signal diffraction** 

from a target grating **G***i*. In this case, we can set

Bragg vector in Eq. (48) can be simplified to

*x*

*<sup>x</sup>* <sup>≡</sup> (50)

(51)

(52)

*cz* = 0°. Then the off-

λ*pi* + δλ (54)

(55)

*off* is written

Note that when the wavelength ratio μ*<sup>0</sup>* 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.

#### **3.5 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***<sup>i</sup>* + δ**e***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 Δ**K** can be written as

$$\Delta \mathbf{K} \equiv k\_w \left( \mathbf{G}\_i + \delta \mathbf{e}\_{\mathrm{Coff}} \right) - \frac{2 \pi n\_p}{\lambda\_{pi} + \delta \lambda\_{\mathrm{off}}} \left( \mathbf{e}\_{di} - \mathbf{e}\_p \right) = k\_w \left( \delta \mathbf{e}\_{\mathrm{Coff}} + \frac{\delta \vec{\lambda}\_{\mathrm{off}}}{\lambda\_{p0}} \mathbf{G}\_0 \right) \tag{46}$$

where we also assumed that the relation δλ*off* << λ*pi* holds. The grating deviation vector δ**e***Goff* can be divided into two classes according to the origin. One is related to the grating that is formed by a different signal component **e***si* + δ**e***soff*. The other corresponds to the grating recorded in a different multiplexed page. Thus, δ**e***Goff* can be expressed as

$$
\delta \mathbf{\dot{e}}\_{\text{Goff}} = \mathbf{\dot{o}} \mathbf{e}\_{\text{s} \text{f}} + \mathbf{\dot{o}} \mathbf{e}\_{\text{cy}} + \mathbf{\dot{o}} \mathbf{e}\_{\text{cz}} \,\tag{47}
$$

where δ**e***cz* and δ**e***cy* are the displacement vectors due to the crystal rotation, defined by Eqs. (33) and (39), respectively. Then the Cartesian components of Eq. (46) can be written as

$$
\begin{bmatrix}
\Delta K\_x \\
\Delta K\_y \\
\Delta K\_z \\
\Delta K\_z
\end{bmatrix} = k\_w \begin{bmatrix}
0 & \cos \mathcal{J}\_{s0} & 0 & 0 & 2 \sin \mathcal{J}\_{s0} \\
1 & 0 & 0 & 2 \sin \mathcal{J}\_{s0} & 0 \\
0 & -\sin \mathcal{J}\_{s0} & -2 \cos \mathcal{J}\_{sz} \sin \mathcal{J}\_{s0} & 0 & 0 \\
\end{bmatrix} \begin{bmatrix}
\delta \alpha\_{\text{s\text{\textquotedblleft}}} \\
\delta \beta\_{\text{s\text{\textquotedblleft}}} \\
\delta \phi\_x \\
\delta \lambda\_{\text{s\textquotedblright}} / \lambda\_{\text{p\textquotedblleft}}
\end{bmatrix},
\tag{48}
$$

where δα*soff* and δβ*soff* are the off-Bragg deviation angles that are related to δ**e***soff* through Eq. (15). If we assume that the modulation of the grating is weak enough to validate the Born approximation, the intensity of the off-Bragg diffracted wave can be expressed as (Barbastathis & Psaltis, 2000)

$$I\_{\rm eff} \approx \left[ \text{sinc}\left(\frac{\Delta K\_x L\_x}{2}\right) \cdot \text{sinc}\left(\frac{\Delta K\_y L\_y}{2}\right) \cdot \text{sinc}\left(\frac{\Delta K\_z L\_z}{2}\right) \right]^2 I\_p \left(\mathcal{J}\_{\rm pi} + \delta \mathcal{S}\_{\rm eff}\right) d\mathcal{J}\_s \tag{49}$$

where *Lx*, *Ly*, and *Lz* are the hologram dimensions in the *x*, *y*, and *z* directions, respectively;

$$\text{sinc}(\mathfrak{x}) \equiv \frac{\sin(\mathfrak{x})}{\mathfrak{x}} \tag{50}$$

is the sinc function;

236 Holograms – Recording Materials and Applications

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

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-

the situation where a grating component **G***<sup>i</sup>* + δ**e***Goff* produces a diffracted wave whose

( ) ( ) <sup>0</sup>

≡ + − −≈ +

can be divided into two classes according to the origin. One is related to the grating that is formed by a different signal component **e***si* + δ**e***soff*. The other corresponds to the grating

where δ**e***cz* and δ**e***cy* are the displacement vectors due to the crystal rotation, defined by Eqs. (33) and (39), respectively. Then the Cartesian components of Eq. (46) can be written as

0 cos 0 0 2 sin

*x s s soff y w s cy z s cz s cz*

(15). If we assume that the modulation of the grating is weak enough to validate the Born approximation, the intensity of the off-Bragg diffracted wave can be expressed as

> *y y x x z z off p pi off K L K L K L <sup>I</sup> I d*

sinc sinc sinc , <sup>222</sup>

Δ Δ <sup>Δ</sup> ∝ ⋅⋅ +

where *Lx*, *Ly*, and *Lz* are the hologram dimensions in the *x*, *y*, and *z* directions,

<sup>Δ</sup> Δ = Δ −−

0 sin 2 cos sin 0 0

0 0

1 0 0 2 sin 0 ,

<sup>2</sup> , *<sup>p</sup> off*

+

, **δe***Goff soff cy cz* = ++ **δe δe δe** (47)

0

β

*soff* are the off-Bragg deviation angles that are related to δ**e***soff* through Eq.

 β

( ) <sup>2</sup>

λ δλ

*pi off p*

**ΔK G δe e e δe G** (46)

λ*pi* + δλ

*w i Goff di p w Goff*

*n k k* π

> δλ*off* << λ

0 0

 φβ

λ δλ

recorded in a different multiplexed page. Thus, δ**e***Goff* can be expressed as

β

β

δφ

*<sup>0</sup>* is unity (i.e., the readout wavelength is the same as

λ*w* and λ

*off* and **e***di*, respectively. In this case, the off-

δλ

λ

*pi* holds. The grating deviation vector δ**e***Goff*

0

*pi* and consider

0

(48)

(49)

*off p*

 λ

*soff*

δα

δβ

δφ

δφ

δλ λ

*cyMin* becomes zero. This is because we

μ

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

Note that when the wavelength ratio

wavelength and unit direction vector are

where we also assumed that the relation

Bragg vector Δ**K** can be written as

*K K k K*

where

δα*soff* and

respectively;

δβ

(Barbastathis & Psaltis, 2000)

**3.5 Off-Brag diffraction** 

the recording one), the required angular separation

always required, regardless of the hologram dimension.

$$I\_p\left(\mathcal{A}\right) = \frac{I\_{p0}}{\Delta \mathcal{A}\_p} \text{rect}\left(\frac{\mathcal{A} - \mathcal{A}\_{p0}}{\Delta \mathcal{A}\_p}\right) \tag{51}$$

is the spectral intensity of the probe beam whose central wavelength, full spectral width, and intensity are λ*p0*, Δλ*<sup>p</sup>*, and *Ip0*, respectively; and rect(*x*) is the rectangle function, defined as

$$\text{rect}\left(\mathbf{x}\right) = \begin{cases} 1 & \left|\mathbf{x}\right| < 1/2 \\ 1/2 & \left|\mathbf{x}\right| = 1/2 \\ 0 & \left|\mathbf{x}\right| > 1/2 \end{cases} \tag{52}$$

Equation (49) indicates that the off-Bragg diffraction will occur at Δ**K** around the origin. The diffracted intensity will become zero when the relation

$$\left|\Delta K\_i\right| = \frac{2\pi}{L\_i} \quad \left(i = \propto, y, z\right) \tag{53}$$

holds at any component of the off-Bragg vector Δ**K**. Thus, hereafter, we assume that Eq. (53) represents an upper limit of |Δ*Ki*| in which the grating **G***<sup>i</sup>* + δ**e***Goff* can produce off-Bragg diffraction.

#### **3.5.1 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-Bragg vector in Eq. (48) can be simplified to

$$
\begin{bmatrix}
\Delta \mathcal{K}\_x \\
\Delta \mathcal{K}\_y \\
\Delta \mathcal{K}\_z
\end{bmatrix} = k\_w \begin{pmatrix}
2 \sin \mathcal{B}\_{s0} \, \delta \mathcal{A}\_{\text{eff}} \left/ \mathcal{A}\_{p0} \right. \\
0 \\
0
\end{pmatrix}.
\tag{54}
$$

From Eq. (49), the intensity of the signal diffracted wave at wavelength λ*pi* + δλ*off* is written as

$$I\_{\rm eff} \approx \text{sinc}^2 \left(\frac{\pi \delta \mathcal{k}\_{\rm off}}{\delta \mathcal{k}\_{\rm off\_{\rm Max}}}\right) I\_p \left(\mathcal{k}\_{\rm pi} + \delta \mathcal{k}\_{\rm off}\right) d\mathcal{k}\_{\rm J} \tag{55}$$

where

$$\mathcal{\delta}\mathcal{\mathcal{X}}\_{\text{offMax}} = \frac{\mathcal{X}\_w \mathcal{X}\_{p0}}{2n\_w \sin \mathcal{B}\_{s0} L\_x} \tag{56}$$

Theory of Polychromatic Reconstruction for Volume Holographic Memory 239

*sMax si soff sMax*

 δα

*sMax si soff sMax*

 δβ

*p p pi off p*

*pi off pi*

 δλ δβ

 δα δα

− ≤+ ≤

δβ

− ≤+ ≤

 λ δλ δλ

⇔− − ≤ ≤− +

λ

λ*pi* and λ

δα

δα δα

δβ

δβ δβ

λ

*pi* is the difference between

targeting position (*xsi*, *ysi*) within the input image.

where

wave.

δβ

set δλ

Note that

δα

δλ

δλ

λ

*si sMax soff si sMax*

*si sMax soff si sMax*

Δ Δ − ≤+ + ≤+

*p p*

00 0 2 2

Δ Δ

Fig. 14. Schematic diagram of the intra-page crosstalk noise. The off-Bragg diffracted waves coming from other input positions will reach the same position as the signal diffracted

*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

( )

*soffMax mono n Ln L w sx w sz* λ

*soffMax mono n Lw y*

*w*

0 0

 λ

> β

<sup>=</sup> (67)

λ

min , . cos sin *w w*

β

*soffMax(mono)* are usually much smaller than

δα

= (66)

*soffMax(mono)* and

δα

*sMax* and

δβ

*sMax* and

δβ

Our goal in this section is to find an upper limit of the off-Bragg deviation angles (

*off* = 0. Then, the maximum off-Bragg deviation angles

( )

δβ

δβ δα

obtained from Eqs. (60), (61), and (62):

*soffMax(mono)* and

satisfy Eqs. (63) and (64), respectively.

, 2 2

 δλ

*p p*

 δα

> δα δα

δβ

 δβ δβ

⇔− − ≤ ≤− + (63)

⇔− − ≤ ≤− + (64)

 λ

 λ  λ

*p0*. Note that the above ranges depend on the

(65)

δα*soff*,

*soffMax(mono)* are

is the half spectral width of the signal diffracted wave. Therefore, a target grating **G***<sup>i</sup>* produces the signal diffraction with a certain spectral width which depends on the hologram dimension *Lx*. The total intensity of the signal diffracted wave (*Idif*) can be calculated by integrating Eq. (55). The result is given by

$$I\_{\rm df} \approx \underbrace{\int \text{sinc}^2 \left( \frac{\pi \mathcal{S} \mathcal{S}\_{\rm eff}}{\mathcal{S} \mathcal{S}\_{\rm eff\text{Max}}} \right) I\_p \left( \mathcal{A}\_{\rm pi} + \mathcal{S} \mathcal{A}\_{\rm off} \right) d \left( \mathcal{S} \mathcal{A}\_{\rm off} \right) \tag{57} \\ \approx \frac{\delta \mathcal{S}\_{\rm off\text{Max}}}{\Delta \mathcal{A}\_p} I\_{p \text{s}} \tag{57}$$

where we assumed that the probe spectral width Δλ*p* is much larger than δλ*offMax*, meaning that *Ip*(λ) is considered to be a constant over the region where the integrand is appreciable, and we used the relation

$$\int\_{--}^{\pi} \text{sinc}^2 \left( \pi \right) d\mathfrak{x} = \pi \,. \tag{58}$$

Note that if Δλ*<sup>p</sup>* → 0, then *Idif* → *Ip0*. Thus, the diffraction efficiency in the PCR method is a factor of δλ*offMax*/Δλ*<sup>p</sup>* 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.

#### **3.5.2 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* + δ**e***soff*. In this case, we can set δφ*cy* = δφ*cz* = 0°, and then the off-Bragg vector in Eq. (48) can be rewritten as

$$
\begin{bmatrix}
\Delta K\_x\\\Delta K\_y\\\Delta K\_z
\end{bmatrix} = k\_w \begin{bmatrix}
\cos \mathcal{J}\_{s0} \delta \mathcal{B}\_{s\mathcal{H}} + 2 \sin \mathcal{J}\_{s0} \, \delta \mathcal{A}\_{s\mathcal{H}} / \mathcal{A}\_{p0} \\
\delta \alpha\_{s\mathcal{H}} \\
\end{bmatrix}.
\tag{59}
$$

The off-Bragg diffraction will occur only when a set of (δα*soff*, δλ*off* ) satisfies all of the following inequalities:

$$-\frac{\lambda\_w}{n\_w \cos \mathcal{B}\_{s0} L\_x} \le \delta \mathcal{B}\_{s0\%} + 2 \tan \mathcal{B}\_{s0} \frac{\delta \mathcal{X}\_{\circ \text{f}}}{\lambda\_{p0}} \le \frac{\lambda\_w}{n\_w \cos \mathcal{B}\_{s0} L\_x} \,\text{,}\tag{60}$$

$$-\frac{\lambda\_w}{n\_w L\_y} \le \delta \alpha\_{\text{soff}} \le \frac{\lambda\_w}{n\_w L\_y} \, \, \, \tag{61}$$

$$-\frac{\lambda\_w}{n\_w \sin \beta\_{s0} L\_z} \le \delta \beta\_{s0f} \le \frac{\lambda\_w}{n\_w \sin \beta\_{s0} L\_z}.\tag{62}$$

On the other hand, the variable ranges of δα*soff*, δβ*soff*, and δλ*off* are also restricted by the input image size or the probe spectrum, which are given by

is the half spectral width of the signal diffracted wave. Therefore, a target grating **G***<sup>i</sup>* produces the signal diffraction with a certain spectral width which depends on the hologram dimension *Lx*. The total intensity of the signal diffracted wave (*Idif*) can be

> ( )( ) <sup>2</sup> <sup>0</sup> sinc , *off offMax dif p pi off off p*

that *Ip*(λ) is considered to be a constant over the region where the integrand is appreciable,

( ) <sup>2</sup> sinc . *x dx*

<sup>∞</sup>

grating component **G***i* can diffract only the limited spectral component around

Such a reduction in the diffraction efficiency is an unavoidable drawback of the PCR

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* +

cos 2 sin

 Δ + Δ = Δ −

*y w soff z s soff*

δβ

λ

λ

β

β δβ

*x s soff s off p*

sin

δα

β δβ

0 0 0 0 2 tan , cos cos *w w off soff <sup>s</sup> n L w sx p w sx n L*

 β

, *w w soff nL nL w y w y*

0 0 . sin sin *w w soff nL nL w sz w sz*

 −≤≤ δα

> δβ

δα*soff*, δβ*soff*, and

δλ

− ≤+ ≤ (60)

 λ

> β

δλ

− ≤≤ (62)

 λ

*I Id I*

λ δλ

∝ +≈

*offMax p*

λ

π

*<sup>p</sup>* → 0, then *Idif* → *Ip0*. Thus, the diffraction efficiency in the PCR method is

0 00

 β δλ λ

0

*<sup>p</sup>* smaller than that in monochromatic readout. This is because the

 δλ

<sup>Δ</sup> (57)

δλ

*p* is much larger than

−∞ <sup>=</sup> (58)

*cz* = 0°, and then the off-Bragg vector in Eq. (48) can be

δα*soff*, δλ

λβ

 λ .

(61)

*off* are also restricted by the input

λ

δλ

*offMax*, meaning

λ*pi*.

(59)

*off* ) satisfies all of the

calculated by integrating Eq. (55). The result is given by

∞

−∞

and we used the relation

λ

**3.5.2 Intra-page crosstalk noise** 

δ**e***soff*. In this case, we can set

following inequalities:

δλ*offMax*/Δλ

Note that if Δ

a factor of

method.

rewritten as

where we assumed that the probe spectral width Δ

πδλ

δλ

δφ*cy* = δφ

The off-Bragg diffraction will occur only when a set of (

λ

image size or the probe spectrum, which are given by

On the other hand, the variable ranges of

β

*K K k K*

$$\begin{aligned} -\delta\alpha\_{s\text{Max}} &\leq \delta\alpha\_{si} + \delta\alpha\_{s\text{off}} \leq \delta\alpha\_{s\text{Max}}\\ \Leftrightarrow -\delta\alpha\_{si} - \delta\alpha\_{s\text{Max}} &\leq \delta\alpha\_{s\text{off}} \leq -\delta\alpha\_{si} + \delta\alpha\_{s\text{Max}} \end{aligned} \tag{63}$$

$$\begin{aligned} -\delta\beta\_{s\text{Max}} &\leq \delta\beta\_{si} + \delta\beta\_{s\text{off}} \leq \delta\beta\_{s\text{Max}}\\ \Leftrightarrow -\delta\beta\_{si} - \delta\beta\_{s\text{Max}} &\leq \delta\beta\_{s\text{off}} \leq -\delta\beta\_{si} + \delta\beta\_{s\text{Max}} \end{aligned} \tag{64}$$

$$\begin{split} \mathcal{\lambda}\_{p0} - \frac{\Delta \mathcal{\lambda}\_{p}}{2} &\leq \mathcal{\lambda}\_{p0} + \delta \mathcal{\lambda}\_{pi} + \delta \mathcal{\lambda}\_{\text{\textquotedblleft}} \leq \mathcal{\lambda}\_{p0} + \frac{\Delta \mathcal{\lambda}\_{p}}{2} \\ \Longleftrightarrow -\delta \mathcal{\lambda}\_{pi} - \frac{\Delta \mathcal{\lambda}\_{p}}{2} &\leq \delta \mathcal{\lambda}\_{\text{\textquotedblright}} \leq -\delta \mathcal{\lambda}\_{pi} + \frac{\Delta \mathcal{\lambda}\_{p}}{2} \end{split} \tag{65}$$

where δλ*pi* is the difference between λ*pi* and λ*p0*. Note that the above ranges depend on the targeting position (*xsi*, *ysi*) within the input image.

Fig. 14. Schematic diagram of the intra-page crosstalk noise. The off-Bragg diffracted waves coming from other input positions will reach the same position as the signal diffracted wave.

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(mono)* are obtained from Eqs. (60), (61), and (62):

$$
\delta \alpha\_{s\!\!\!/\!\!/\!Max\!\!/\!(m\!\!/\!moron)} = \frac{\hat{\mathcal{A}}\_w}{n\_w L\_y} \tag{66}
$$

$$\delta \mathcal{B}\_{s\text{offMax}\{\text{mwo}\}} = \min \left[ \frac{\mathcal{A}\_w}{n\_w \cos \mathcal{B}\_{s0} L\_x}, \frac{\mathcal{A}\_w}{n\_w \sin \mathcal{B}\_{s0} L\_z} \right]. \tag{67}$$

Note that δα*soffMax(mono)* and δβ*soffMax(mono)* are usually much smaller than δα*sMax* and δβ*sMax* and satisfy Eqs. (63) and (64), respectively.

Theory of Polychromatic Reconstruction for Volume Holographic Memory 241

Fig. 15. The input image area suffering from intra-page crosstalk noise in (a) the

*p0* = 532 nm, Δ

. 2 cos 2 cos sin 2 cos 2 cos sin *soff w w soff cy cz w cz s z nL nL cz w cz s z*

δφ λ

δβ

*cz* that is large enough to avoid inter-page crosstalk noise. Since the variable

− − ≤ ≤− + (73)

*off* ) satisfies all of the above inequalities. First, let us consider Eq. (72) to find the

*soff* is expressed as Eq. (63), the off-Bragg diffraction will occur whenever

00 00 . 2 sin 2 sin 2 sin 2 sin *sMax si w sMax si w cz s w sy n L s w sy n L*

δφ

The above inequality is an expression that is applicable at one particular position (*xsi*, *ysi*) within the input image. In order to avoid crosstalk at every position, the peristrophic

δφ

β

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

δα

δα*si* ≤ δα δα

− − ≤≤ + (74)

*czMin(off)*; that is,

 λ

δφ

 β

0 0 , sin 2 sin *sMax w*

*s w sy n L*

*<sup>p</sup>* = 50 nm). The calculation parameters are as follows:

0 0

 δα

φ

*<sup>p</sup>* = 0 nm) and (b) the PCR method

 λ

 φβ

> λ

ββ

= + (75)

*sMax*. The right side in Eq. (75) consists of two

*czMin* obtained in Section 3.4.1; and the

δα*soff* , δβ*soff* , δφ*cy* ,

β*<sup>s</sup>*<sup>0</sup> = 30°;

> δφ*cz*

λ*<sup>w</sup>* = λ

Note again that the off-Bragg diffraction will occur only when the set of (

 λ

( )

*czMin off*

+ −

ββ

*cz* should be set larger than

δφ

> δα*sMax* ≤

parts: one is the minimum peristrophic rotation angle

λ

λ

 φβ

conventional monochromatic case (

φ

δφ

satisfies the following inequality:

δα

δφ

where we used the relation −

 δα

*p0* = 815 nm, Δ

*nw* = 1; *fs* = 100 mm; and *Ly* = *Lz* = 10 mm.

λ

δβ

(λ

δφ*cz* , δλ

*<sup>w</sup>* = 532 nm,

rotation angle

rotation angles

δα

range of

Next, we move to the polychromatic case. In this case, the variable range of δλ*off* is appreciable owing to the large spectral width Δλ*<sup>p</sup>*, as expressed in Eq. (65). On the other hand, in order to satisfy Eqs. (60) and (62) at the same δβ*soff*, there should exist a common region in the two inequalities; that is,

$$\begin{cases} -\frac{\mathcal{\lambda}\_w}{n\_w \sin \mathcal{\mathcal{B}}\_{s0} L\_z} \le -2 \tan \mathcal{\mathcal{B}}\_{s0} \frac{\partial \mathcal{\lambda}\_{\text{off}}}{\mathcal{\lambda}\_{p0}} + \frac{\mathcal{\lambda}\_w}{n\_w \cos \mathcal{\mathcal{B}}\_{s0} L\_x} \\ -2 \tan \mathcal{\mathcal{B}}\_{s0} \frac{\partial \mathcal{\lambda}\_{\text{off}}}{\mathcal{\lambda}\_{p0}} - \frac{\mathcal{\lambda}\_w}{n\_w \cos \mathcal{\mathcal{B}}\_{s0} L\_x} \le \frac{\mathcal{\lambda}\_w}{n\_w \sin \mathcal{\mathcal{B}}\_{s0} L\_z} \\ \iff \left| \frac{\partial \mathcal{\lambda}\_{\text{off}}}{\mathcal{\lambda}\_{p0}} \right| \le \frac{\mathcal{\lambda}\_w}{2 n\_w \tan \mathcal{\mathcal{B}}\_{s0}} \left( \frac{1}{\cos \mathcal{\mathcal{B}}\_{s0} L\_x} + \frac{1}{\sin \mathcal{\mathcal{B}}\_{s0} L\_z} \right). \end{cases} \tag{68}$$

The off-Bragg diffraction will occur at λ*pi* + δλ*off* only when δλ*off* satisfies both Eqs. (65) and (68). However, because Δλ*p* is sufficiently large, every δλ*off* within Eq. (68) will always satisfy Eq. (65). Therefore, Eq. (68) could be considered a sufficient condition for δλ*off*. Similarly, if we choose a proper δλ*off* from Eq. (68), every δβ*soff* within Eq. (62) will always satisfy Eq. (60). Thus, Eq. (62) is considered to be a sufficient condition for δβ*soff*. Consequently, the variable ranges of δα*soff* and δβ*soff* are determined by Eqs. (61) and (62), respectively. Thus the maximum off-Bragg deviation angles δα*soffMax(PCR)* and δβ*soffMax(PCR)* are given by

$$
\delta \alpha\_{\text{sgfMax(PCR)}} = \frac{\mathcal{A}\_w}{n\_w L\_y} \,\,\,\,\tag{69}
$$

$$
\delta \mathcal{B}\_{s\text{offMax}\{\text{PCR}\}} = \frac{\mathcal{A}\_w}{n\_w \sin \mathcal{B}\_{s0} L\_z}. \tag{70}
$$

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 (*xs*, *ys*) at several values of *Lx*. While the input image area suffering from the crosstalk noise shrinks with increasing *Lx* in the conventional monochromatic case, that in the PCR method is unchanged.

#### **3.5.3 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***<sup>i</sup>* + δ**e***soff* + δ**e***cz* + δ**e***cy*. Similar to the previous section, we obtain the set of inequalities given by

$$-2\tan\beta\_{s0}\frac{\delta\mathcal{U}\_{\text{off}}}{\lambda\_{p0}} - \frac{\lambda\_w}{n\_w\cos\beta\_{s0}L\_x} \le \delta\beta\_{s0} \le -2\tan\beta\_{s0}\frac{\delta\lambda\_{\text{off}}}{\lambda\_{p0}} + \frac{\lambda\_w}{n\_w\cos\beta\_{s0}L\_x},\tag{71}$$

$$\frac{\delta \alpha\_{\rm soff}}{2 \sin \beta\_{s0}} - \frac{\lambda\_w}{2n\_w \sin \beta\_{s0} L\_y} \le \delta \phi\_{cz} \le \frac{\delta \alpha\_{\rm soff}}{2 \sin \beta\_{s0}} + \frac{\lambda\_w}{2n\_w \sin \beta\_{s0} L\_y},\tag{72}$$

λ

0 0 00

*w w off s w sz p w sx off w w*

β

*n L n L*

λ

00 0

*p w sx w sz*

δλ

1 1 . 2 tan cos sin

*off* only when

δλ

δβ

0 . sin *w*

β

*n LL*

cos sin

*n Ln L*

0 00 0

δβ

*soffMax(PCR)* and

*soffMax PCR n L w sz*

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 (*xs*, *ys*) at several values of *Lx*. While the input image area suffering from the crosstalk noise shrinks with increasing *Lx* in the conventional monochromatic case, that in the PCR method

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***<sup>i</sup>* + δ**e***soff* + δ**e***cz* + δ**e***cy*. Similar to the previous section, we obtain the set of inequalities given by

δβ

δφ

0 0 0 0 2 tan 2 tan , cos cos *off w w off*

00 00 , 2 sin 2 sin 2 sin 2 sin *soff w w soff cz s w sy n L s w sy n L*

*<sup>p</sup> n L w sx p w sx n L*

− − ≤≤ − + (71)

δα

0 0

*s soff s*

λ

λ

ββ

 β

( ) , *<sup>w</sup> soffMax PCR n Lw y*

λ

λ

⇔ ≤ <sup>+</sup>

ββ

*p w s sx sz*

δβ

λβ

 λ

> β

 β

δλ

δβ

*soff* are determined by Eqs. (61) and (62), respectively. Thus the

 λ δλ*off* is

(68)

*<sup>p</sup>*, as expressed in Eq. (65). On the other

*soff*, there should exist a common

*off* satisfies both Eqs. (65) and

δλ

*soff*. Consequently, the variable

*off*. Similarly, if

*off* within Eq. (68) will always satisfy

*soff* within Eq. (62) will always satisfy Eq. (60).

*soffMax(PCR)* are given by

<sup>=</sup> (70)

 λ

> β

δλ

λ

ββ

− ≤≤ + (72)

 λ

 β

= (69)

Next, we move to the polychromatic case. In this case, the variable range of

2 tan sin cos

λβ

λ

λ*pi* +δλ

*p* is sufficiently large, every

δα

( )

δα

δβ

Eq. (65). Therefore, Eq. (68) could be considered a sufficient condition for

*off* from Eq. (68), every

Thus, Eq. (62) is considered to be a sufficient condition for

− −≤

− ≤ − +

appreciable owing to the large spectral width Δ

 

 

The off-Bragg diffraction will occur at

δλ

δβ

(68). However, because Δ

δα

*soff* and

**3.5.3 Inter-page crosstalk noise** 

δλ

λ

β

> δα

maximum off-Bragg deviation angles

we choose a proper

ranges of

is unchanged.

region in the two inequalities; that is,

hand, in order to satisfy Eqs. (60) and (62) at the same

2 tan

δλ

λ

λ

0

β

*off w*

δλ

*s*

β

λ

Fig. 15. The input image area suffering from intra-page crosstalk noise in (a) the conventional monochromatic case (λ*<sup>w</sup>* = λ*p0* = 532 nm, Δλ*<sup>p</sup>* = 0 nm) and (b) the PCR method (λ*<sup>w</sup>* = 532 nm, λ*p0* = 815 nm, Δλ*<sup>p</sup>* = 50 nm). The calculation parameters are as follows: β*<sup>s</sup>*<sup>0</sup> = 30°; *nw* = 1; *fs* = 100 mm; and *Ly* = *Lz* = 10 mm.

$$-\frac{\delta\mathcal{\boldsymbol{\beta}}\_{\mathrm{s}\boldsymbol{\text{f}}}}{2\cos\phi\_{\boldsymbol{\text{c}}}}-\frac{\mathcal{\boldsymbol{\lambda}}\_{\boldsymbol{w}}}{2n\_{\boldsymbol{w}}\cos\phi\_{\boldsymbol{\text{c}}z}\sin\mathcal{\boldsymbol{\beta}}\_{\boldsymbol{s}0}L\_{z}}\leq\delta\boldsymbol{\phi}\_{\boldsymbol{\text{c}}}\leq-\frac{\delta\mathcal{\boldsymbol{\beta}}\_{\boldsymbol{s}\boldsymbol{\text{f}}}}{2\cos\phi\_{\boldsymbol{\text{c}}z}}+\frac{\mathcal{\boldsymbol{\lambda}}\_{\boldsymbol{w}}}{2n\_{\boldsymbol{w}}\cos\phi\_{\boldsymbol{\text{c}}z}\sin\mathcal{\boldsymbol{\beta}}\_{\boldsymbol{s}0}L\_{z}}.\tag{73}$$

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:

$$-\frac{\delta \alpha\_{s\text{Max}} + \delta \alpha\_{si}}{2\sin\mathcal{J}\_{s0}} - \frac{\lambda\_w}{2n\_w\sin\mathcal{J}\_{s0}L\_y} \le \delta \phi\_{\text{ex}} \le \frac{\delta \alpha\_{s\text{Max}} - \delta \alpha\_{si}}{2\sin\mathcal{J}\_{s0}} + \frac{\lambda\_w}{2n\_w\sin\mathcal{J}\_{s0}L\_y}.\tag{74}$$

The above inequality is an expression that is applicable at one particular position (*xsi*, *ysi*) 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,

$$\delta\phi\_{cz\text{Min(off)}} = \frac{\delta\alpha\_{s\text{Max}}}{\sin\mathcal{J}\_{s0}} + \frac{\mathcal{A}\_w}{2n\_w\sin\mathcal{J}\_{s0}L\_y},\tag{75}$$

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

Theory of Polychromatic Reconstruction for Volume Holographic Memory 243

*s w*

δβ δβ

where we used the relation of Eq. (20) in the derivation. From Eqs. (73) and (81), the off-

( )

<sup>−</sup> <sup>+</sup> − ≤

 φ

δβ δβ

( )

*s w*

δφ

*p w sx*

tan cos

( )

 φ

tan 1 ,

 φ

*s si sMax cy cyMin mono p cz*

The above inequality is an expression that is applicable at one particular position (*xsi*, *ysi*) within the input image. In order to avoid crosstalk at every position, the crystal rotation

*cyMin(PCR)*; that is,

0 ( ) ( )

β δβ

 = + + 

> δβ*si* ≤ δβ

β

tan cos *s sMax cyMin PCR cyMin mono p cz*

usually much smaller than the former, and thus the influence of the off-Bragg diffraction can

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 *VK1bit*, the

> 1 , *KG*

where *VKG* 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

*K bit*

*bit*

*<sup>V</sup> <sup>N</sup> V*

0

*si sMax soff*

*n L*

 β

*cy* satisfies the following inequality:

( )

δφ

δφ

*cyMin mono cy*

 λ

*p w sx*

0

δβ

*n L*

 β

( )

δφ

.

*sMax*. The right side in Eq. (83) consists of the

= (84)

*cyMin* obtained in Section 3.4.2 and the contribution from

*cyMin(mono)*. Similarly to the peristrophic rotation, the latter is

 λ ,

(81)

(82)

(83)

( )

−+ + − ≤

tan cos

0

*soff si sMax*

≤− + − +

β

δφ

δβ δβ

0

≤ + +

β

β

δφ

*p cz*

*s si sMax*

tan 2 cos

β

δβ δβ

0

β

β

tan <sup>1</sup>

0

β

β

tan <sup>1</sup>

δβ*sMax* ≤

δφ

δφ

storage capacity of the holographic memory (*Nbit*) is expressed as

tan 2 cos

+

tan <sup>1</sup>

tan <sup>1</sup>

δφ

*cy* should be set larger than

δφ

be ignored also in the crystal angle multiplexing.

δβ

Bragg diffraction will occur whenever

angles

δφ

where we used the relation −

the off-Bragg diffraction

**3.6 Storage capacity** 

minimum crystal rotation angle

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) is simplified to

$$\left| \delta \mathcal{B}\_{s\mathcal{G}} \right| \leq \frac{\lambda\_w}{n\_w \cos \beta\_{s0} L\_x}. \tag{76}$$

Note that every δβ*soff* within Eq. (76) always satisfies Eqs. (64). From Eqs. (73) and (76), the off-Bragg diffraction will occur whenever δφ*cy* satisfies the following inequality:

$$\left|\delta\phi\_{cy}\right| \leq \frac{\lambda\_w}{2n\_w\cos\phi\_{cz}} \left(\frac{1}{\cos\mathcal{J}\_{s0}L\_x} + \frac{1}{\sin\mathcal{J}\_{s0}L\_z}\right). \tag{77}$$

Conversely, in order to avoid inter-page crosstalk noise, the crystal rotation angle should be set larger than δφ*cyMin(mono)*; that is,

$$\mathcal{S}\phi\_{\text{cyMin(mmo)}} = \frac{\lambda\_w}{2n\_w \cos \phi\_z} \left(\frac{1}{\cos \mathcal{J}\_{s0}L\_x} + \frac{1}{\sin \mathcal{J}\_{s0}L\_z}\right). \tag{78}$$

Now, let us return to the case of the PCR method. Due to the wide spectral width Δλ*<sup>p</sup>*, the restriction of δβ*soff* in Eq. (71) is considerably relaxed as compared with the monochromatic case; that is,

$$\begin{cases} \tan \mathcal{B}\_{s0} \frac{\left(2 \mathcal{\delta} \mathcal{\lambda}\_{pi} - \Delta \mathcal{\lambda}\_{p}\right)}{\mathcal{\lambda}\_{p0}} - \frac{\mathcal{\lambda}\_{w}}{n\_{w} \cos \mathcal{B}\_{s0} L\_{x}} \le \mathcal{\delta} \mathcal{\beta}\_{s0} \\\\ \mathcal{\delta} \mathcal{B}\_{s0} \le \tan \mathcal{B}\_{s0} \frac{\left(2 \mathcal{\delta} \mathcal{\lambda}\_{pi} + \Delta \mathcal{\lambda}\_{p}\right)}{\mathcal{\lambda}\_{p0}} + \frac{\mathcal{\lambda}\_{w}}{n\_{w} \cos \mathcal{B}\_{s0} L\_{x}} \end{cases} \tag{79}$$

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

$$\begin{cases} \max \left[ \tan \beta\_{s0} \frac{\left( 2 \delta \lambda\_{pi} - \Delta \lambda\_{p} \right)}{\lambda\_{p0}} - \frac{\lambda\_{w}}{n\_{w} \cos \beta\_{s0} L\_{x}}, -\delta \beta\_{s} - \delta \beta\_{s \text{Max}} \right] \leq \delta \beta\_{s0} \\\\ \delta \beta\_{s0} \leq \min \left[ \tan \beta\_{s0} \frac{\left( 2 \delta \lambda\_{pi} + \Delta \lambda\_{p} \right)}{\lambda\_{p0}} + \frac{\lambda\_{w}}{n\_{w} \cos \beta\_{s0} L\_{x}}, -\delta \beta\_{si} + \delta \beta\_{s \text{Max}} \right]' \end{cases} \tag{80}$$

where max[*a*, *b*] is a maximum function yielding the larger value between *a* and *b*. If we assume that α*<sup>p</sup>* = 0 and take the spectral width required for the full image reconstruction Δλ*BM* of Eq. (25) as the probe spectral width Δλ*<sup>p</sup>*, then Eq. (80) can be modified to

$$\begin{cases} -\left(1+\frac{\tan\mathcal{B}\_{s0}}{\tan\mathcal{B}\_p}\right)(\delta\mathcal{B}\_{si}+\delta\mathcal{B}\_{shx}) - \frac{\mathcal{A}\_w}{n\_w\cos\mathcal{B}\_{s0}L\_x} \le \delta\mathcal{B}\_{s0} \\\\ \delta\mathcal{B}\_{s\mathcal{Y}} \le -\left(1+\frac{\tan\mathcal{B}\_{s0}}{\tan\mathcal{B}\_p}\right)(\delta\mathcal{B}\_{si}-\delta\mathcal{B}\_{shx}) + \frac{\mathcal{A}\_w}{n\_w\cos\mathcal{B}\_{s0}L\_x} \end{cases} \tag{81}$$

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:

$$\begin{cases} \left(1 + \frac{\tan \mathcal{B}\_{s0}}{\tan \mathcal{B}\_p} \right) \frac{(\delta \mathcal{B}\_{si} - \delta \mathcal{B}\_{s \text{Max}})}{2 \cos \phi\_{cz}} - \delta \phi\_{c y \text{Min(mwoo)}} \le \delta \phi\_{cy} \\\\ \delta \phi\_{cy} \le \left(1 + \frac{\tan \mathcal{B}\_{s0}}{\tan \mathcal{B}\_p} \right) \frac{(\delta \mathcal{B}\_{si} + \delta \mathcal{B}\_{s \text{Max}})}{2 \cos \phi\_{cz}} + \delta \phi\_{c y \text{Min(mwoo)}} \end{cases} \tag{82}$$

The above inequality is an expression that is applicable at one particular position (*xsi*, *ysi*) 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,

$$
\delta\phi\_{\text{cyMin\{PCR\}}} = \left(1 + \frac{\tan\beta\_{s0}}{\tan\beta\_p}\right) \frac{\delta\mathcal{B}\_{s\text{Max}}}{\cos\phi\_{cz}} + \delta\phi\_{\text{cyMin\{muno\}}}\tag{83}
$$

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.

#### **3.6 Storage capacity**

242 Holograms – Recording Materials and Applications

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.

avoid crosstalk noise. Before considering the case of PCR, we will again treat the conventional

δφ

λ

0 . cos *w soff n L w sx* λ

*soff* within Eq. (76) always satisfies Eqs. (64). From Eqs. (73) and (76), the

0 0

1 1 . 2 cos cos sin

*soff* in Eq. (71) is considerably relaxed as compared with the monochromatic

0 0

*p w sx*

*pi p <sup>w</sup>*

*n L*

 β

λ

= +

 β

0 0

cos .

cos

δβ δβ

*n L*

 β

*soff* is also limited by the input image size, as is

δβ δβ

*<sup>p</sup>*, then Eq. (80) can be modified to

*soff* to produce the off-Bragg diffraction is

 δβ

δβ

,

λ

 β

δβ

(78)

*cy* satisfies the following inequality:

β

1 1 . 2 cos cos sin

≤ +

*cy n LL w cz s x s z*

Conversely, in order to avoid inter-page crosstalk noise, the crystal rotation angle should be

*cyMin mono n LL w cz s x s z*

( )

λ

*pi p w s soff p w sx*

0 0

 λ

δβ

0 0

*p w sx*

*<sup>p</sup>* = 0 and take the spectral width required for the full image reconstruction

*n L*

 β

λ

*pi p <sup>w</sup> soff s si sMax*

*n L*

 β

λ

*s si sMax soff*

φβ

φβ

δφ

≤ (76)

*off* = 0. Thus, Eq. (71) is simplified to

*cy* that is large enough to

(77)

λ*<sup>p</sup>*, the

*soff* satisfies both

(79)

(80)

Next, we will focus on Eqs. (71) and (73) to find the rotation angle

δβ

*w*

*w*

Now, let us return to the case of the PCR method. Due to the wide spectral width Δ

λ

( )

λ

0

 β

expressed in Eq. (64). Since the off-Bragg diffraction will occur only when

 + Δ ≤ +

2

( )

 + Δ ≤ + −+ 

λ

*pi p <sup>w</sup>*

min tan , cos

0 0

− Δ − −− ≤

*p w sx*

 λ

where max[*a*, *b*] is a maximum function yielding the larger value between *a* and *b*. If we

λ

δλ

δβ

 λ

− Δ − ≤

λ

monochromatic readout first. In this case, we can set δ

δφ

*cyMin(mono)*; that is,

δφ

( )

0

β

tan

δβ

0

β

*BM* of Eq. (25) as the probe spectral width Δ

2

δλ

Eqs. (64) and (79), the sufficient condition for

On the other hand, the variable range of

δβ

α

2

δλ

tan

( )

λ

max tan , cos

 λ

2

δλ

0

β

*soff s*

Note that every

set larger than

restriction of

case; that is,

given by

assume that

Δλ δβ

δφ

δβ

off-Bragg diffraction will occur whenever

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 *VK1bit*, the storage capacity of the holographic memory (*Nbit*) is expressed as

$$N\_{bit} = \frac{V\_{\&G}}{V\_{\&1bit}}\text{,}\tag{84}$$

where *VKG* 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

Theory of Polychromatic Reconstruction for Volume Holographic Memory 245

0

 β δα

 β

8 cos sin 2 .

Next, let us consider this in combination with peristrophic multiplexing. In this case, crystal

*n n nn n*

max

*n*

δφ

*w cz s s s cyMax*

δφ

0 max max

 δβ δφ

φ

( ) max

, 2, 1, 0, 1, 2, , sin

*czMin* is the minimum angular separation in peristrophic multiplexing of Eq. (38); *n*

δα

*czMax* is a maximum peristrophic rotation angle. If we take the summation of cos

max

β

0 , sin *s*

0 max max

max 0 0

( ) max max

1

0

0 max <sup>0</sup> max 8 sin 2 2 sin sin cos

 β

*w s s cyMax s czMax s czMax*

 β

*s n*

β

approximately holds. Then, *VKG* after crystal angle multiplexing in combination with

*s s s s s s*

<sup>=</sup> + − <sup>+</sup>

*s*

= = =− − − (90)

*w cz s cy s s*

δφ

*k d dd*

max max

δα

δβ

δβ

(89)

 δβ

*cyMax s s*

δα

*cyMax s s*

*czn*; that is,

max max

δα

− −−

max max

≥ (91)

( ) ( )

<sup>+</sup> (93)

( )

 δα  φ

*s s* (95)

 φ

 β δβ

(94)

δα

 β

max max

*n n*

sin 1 cos 1 sin sin

φ*czn*

(92)

( )

*V dV*

φ

*KG cz cz K*

=

=

=

angle multiplexing is performed at each inclined angle

3

3

*k*

cos sin 2

φ

0

*czMax*

max 0

β

*s*

sin

δα

max

β

*n*

*s*

<sup>=</sup>

0

*s*

β

2 sin ~ sin cos ,

φ

*czMax czMax*

sin *s*

= −

*czMax* = π/2, we obtain

3 \_ <sup>0</sup> 0 max 8 sin sin 2 . *V k KG total w s* =

π

δα

−

*sMax* << 1 and the relation

*czMax*

φ

peristrophic multiplexing, which is denoted by *VKG\_total*, is expressed as

( )

φ φ

β δβ δφ

 φ

δα

β

0 max

*s*

β

*s*

δα

max

*n KG total KG cz czn n n*

*k*

=− =

*V V*

max \_ \_ 3

> φ*cyMax* = φ

δα

2 sin

δα

2 sin

δα

φ

*s*

β

is an integer; and *nmax* is the maximum integer that satisfies the relation

*s*

δα φ

\_

φ

*czn czMin*

 δφ

φ

where

where φ

δφ

over all *n*, we obtain

where we assumed that

In particular, if we set δ

max max

φ

*n n*

=− =−

max max

*czn*

cos cos

cos

*n n n n s*

sin

that probe beam. Such a grating vector should be removed from *VKG*, 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 *VK1bit* and *VKG* 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*<sup>p</sup>* = 0° in this section. First, let us consider *VKG* 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

$$\mathbf{R}\begin{pmatrix}\phi\_{cy},\phi\_{cz}\end{pmatrix} = \begin{pmatrix}\cos\phi\_{cy}\cos\phi\_{cz} & \cos\phi\_{cy}\sin\phi\_{cz} & \sin\phi\_{cy} \\ -\sin\phi\_{cz} & \cos\phi\_{cz} & 0 \\ -\sin\phi\_{cy}\cos\phi\_{cz} & -\sin\phi\_{cy}\sin\phi\_{cz} & \cos\phi\_{cy}\end{pmatrix} \tag{85}$$

The grating vector *kw*(**G***<sup>0</sup>* + δ**e***si*) operated on by that matrix is written as

$$\mathbf{R}\left(\boldsymbol{\phi}\_{\rm yr},\boldsymbol{\phi}\_{\rm z}\right)k\_{w}\left(\mathbf{G}\_{0}+\boldsymbol{\Theta}\mathbf{e}\_{\rm si}\right) = 2k\_{w}\sin\beta\_{s0}\begin{bmatrix}\cos\phi\_{\rm cy}\cos\phi\_{\rm z} \\ -\sin\phi\_{\rm z} \\ -\sin\phi\_{\rm cy}\cos\phi\_{\rm z} \end{bmatrix} + k\_{w}\begin{bmatrix}\cos\phi\_{\rm cy}\sin\phi\_{\rm z} \\ \cos\phi\_{\rm z} \\ -\sin\phi\_{\rm cy}\sin\phi\_{\rm z} \end{bmatrix}\boldsymbol{\delta}\boldsymbol{\alpha}\_{\rm s} \tag{86}$$

$$+k\_{w}\begin{bmatrix}\cos\mathcal{J}\_{s0}\cos\phi\_{\rm cy}\cos\phi\_{\rm cr} - \sin\mathcal{J}\_{s0}\sin\phi\_{\rm cy} \\ -\cos\mathcal{J}\_{s0}\sin\phi\_{\rm cr} \\ -\cos\mathcal{J}\_{s0}\sin\phi\_{\rm cy}\cos\phi\_{\rm cr} - \sin\mathcal{J}\_{s0}\cos\phi\_{\rm cy} \end{bmatrix}\boldsymbol{\delta}\boldsymbol{\beta}\_{\rm s}.$$

If above grating is further rotated around the *y*-axis by a small angle δφ*cy*, we obtain the differential grating vector (δ**K**G) resulting from small shifts of δα*s*, δβ*s*, and δφ*cy*:

$$\begin{aligned} \mathbf{\delta K}\_{G} &= k\_{w} \left( \delta \mathbf{\alpha}\_{s} \delta \mathbf{e}\_{K\alpha} + \delta \beta\_{s} \delta \mathbf{e}\_{K\beta} + \delta \mathbf{\sigma}\_{\eta} \delta \mathbf{e}\_{K\eta} \right), \\ \mathbf{\delta \mathbf{e}}\_{K\alpha} &= \begin{bmatrix} \cos \phi\_{\eta} \sin \phi\_{cz} \\ \cos \phi\_{cz} \\ -\sin \phi\_{\eta} \sin \phi\_{cz} \end{bmatrix}, \; \mathbf{\delta \mathbf{e}}\_{K\mu} = \begin{bmatrix} \cos \beta\_{s0} \cos \phi\_{\eta} \cos \phi\_{z} - \sin \beta\_{s0} \sin \phi\_{\eta} \\ -\cos \beta\_{s0} \sin \phi\_{cz} \\ -\cos \beta\_{s0} \sin \phi\_{\eta} \cos \phi\_{z} - \sin \beta\_{s0} \cos \phi\_{\eta} \end{bmatrix}, \\ \; \mathbf{\delta \mathbf{e}}\_{K\eta} &= -2 \sin \beta\_{s0} \begin{bmatrix} \sin \phi\_{\eta} \cos \phi\_{z} \\ 0 \\ \cos \phi\_{\eta} \cos \phi\_{z} \end{bmatrix}. \end{aligned} \tag{87}$$

Thus, the differential volume in frequency space is given by

$$\begin{split}dV\_{\boldsymbol{k}} &= \boldsymbol{k}\_{w}^{\;\;3} \left[ \left( \boldsymbol{\mathsf{\dot{s}}} \mathbf{e}\_{\boldsymbol{k}\boldsymbol{\alpha}} \times \boldsymbol{\mathsf{\dot{s}}} \mathbf{e}\_{\boldsymbol{k}\boldsymbol{\beta}\boldsymbol{s}} \right) \cdot \boldsymbol{\mathsf{\dot{s}}} \mathbf{e}\_{\boldsymbol{k}\boldsymbol{\alpha}} \right] \boldsymbol{\delta} \boldsymbol{\alpha}\_{s} \boldsymbol{\delta} \boldsymbol{\mathcal{B}}\_{s} \boldsymbol{\delta} \boldsymbol{\phi}\_{\boldsymbol{\alpha}} \\ &= \boldsymbol{k}\_{w}^{\;\;3} \cos \boldsymbol{\phi}\_{\boldsymbol{c}z} \sin 2 \boldsymbol{\mathcal{B}}\_{s0} \boldsymbol{\delta} \boldsymbol{\alpha}\_{s} \boldsymbol{\delta} \boldsymbol{\mathcal{B}}\_{s} \boldsymbol{\delta} \boldsymbol{\phi}\_{\boldsymbol{c}y} \,. \end{split} \tag{88}$$

When we assumed that δα*s*, δβ*s*, and δφ*cy* are in the ranges −δα*sMax* < δα*<sup>s</sup>* < δα*sMax*, −δβ*sMax* < δβ*<sup>s</sup>* < δβ*sMax*, and −δφ*cyMax* < δφ*cy* < δφ*cyMax*, respectively, then *VKG* after crystal angle multiplexing around the axis inclined at φ*cz*, which is denoted by *VKG\_*φ*cz*, can be written as

that probe beam. Such a grating vector should be removed from *VKG*, because the data stored there cannot be read out. Therefore, the theoretical limit of the storage capacity is

Although *VK1bit* and *VKG* are generally different in different multiplexing methods, we will consider here the crystal angle multiplexing combined with the peristrophic rotation discussed

α

φ

φ

0 0 0 0 0

β

cos cos cos sin sin cos sin cos sin cos sin cos

cos cos cos sin

 

*cy cz cy cz*

*cy cz cy cz*

 φ

sin cos sin sin

− −

*s cy cz s cy*

*s cy cz s cy*

δα*s*, δβ*s*, and δφ*cy*:

0 0 0 0 0

β

**R** (85)

cos cos cos sin sin , sin cos 0 . sin cos sin sin cos

 = −

− −

*cy cz cy cz cy*

 φφ

*cy cz cy cz cy*

 φφ

φφ

 φ

φφ

βφφ

*w s cz*

− −

,

cos , cos sin ,

βφφ

βφφ

δα δβ δφ

> δα*sMax* <

*cy cz s cy cz s cy*

*cy cz s cy cz s cy*

cos sin cos cos cos sin sin

sin sin cos sin cos sin cos

<sup>−</sup> <sup>=</sup> = − − −−

<sup>−</sup>

βφφ

*<sup>p</sup>* = 0° in this section. First, let us

*cz*. The matrix that

δα

. *si* δβ

*cy*, we obtain the

(86)

(87)

(88)

 

φ

*cy* around the *y*-axis, is expressed as

 φ

 φ

> φφ

 φφ

 βφ

 βφ

> βφ

 βφ

δα*<sup>s</sup>* < δα*sMax*, −δβ*sMax* <

*cz*, can be written as

φ

*cyMax*, respectively, then *VKG* after crystal angle

 φ φ

δφ

ultimately determined by the readout wavelength not the recording one.

consider *VKG* after the crystal angle multiplexing around the axis inclined at

The grating vector *kw*(**G***<sup>0</sup>* + δ**e***si*) operated on by that matrix is written as

*cy cz cz cz*

φφ

, 2 sin sin cos

+= − +

*kk k*

β

*k*

*K s cz K s s cz*

 δϕ

 β

If above grating is further rotated around the *y*-axis by a small angle

( )

 β

sin cos

 

cos cos

3

δφ*cy* <

*k*

=

φ

Thus, the differential volume in frequency space is given by

*dV k*

δα*s*, δβ*s*, and

δφ*cyMax* <

multiplexing around the axis inclined at

φ

*cy c*

 φ

 φ*cy cz*

*z*

( ) <sup>3</sup>

= ×⋅ 

α

φ

δφ

δφ

φ

<sup>0</sup> cos sin 2 . *K w K s K s Kcy s s cy*

*cy* are in the ranges −

*cz*, which is denoted by *VKG\_*

 β

*w cz s s s cy*

**δe δe δe**

 β δα δβ δφ

differential grating vector (δ**K**G) resulting from small shifts of

*G w s K s s K s cy Kcy*

 δβ

α

= ++

**δK δe δe δe**

φ

 φ

> φ

0

**δe** 0 .

β

2 sin

*Kcy s*

= −

*k*

δα

φ

φ

**δe δe**

α

When we assumed that

*sMax*, and −

δβ*<sup>s</sup>* < δβ *cy cz w si w s cz w cz si*

+ −

 φ

φφ

*cz* around the *z*-axis, and subsequently rotates at

in Section 3.4. Furthermore, for simplicity, we assume

( )

( ) ( ) 0 0

φφ

rotates at

φ

φ φ

**R G δe**

$$\begin{split} V\_{K\_{\rm KG\_{\rm Qtz}}} \left( \phi\_{\rm cr} \right) &= \int dV\_{K} \\ &= k\_{w} \, ^{3} \cos \phi\_{\rm cr} \sin 2 \mathcal{J}\_{\rm s0} \int\_{-\delta \phi\_{\rm yMax}}^{\delta \phi\_{\rm ykn}} d\delta \phi\_{\rm y} \int\_{-\delta \theta\_{\rm znm}}^{\delta \phi\_{\rm znm}} d\delta \mathcal{S} \alpha\_{s} \int\_{-\delta \theta\_{\rm znm}}^{\delta \phi\_{\rm znm}} d\delta \mathcal{S} \theta\_{s} \\ &= 8 k\_{w} \, ^{3} \cos \phi\_{\rm z} \, \sin 2 \mathcal{J}\_{\rm s0} \delta \alpha\_{s} \text{max} \, \delta \theta\_{s} \text{max} \, \delta \phi\_{\rm yMax} \, . \end{split} \tag{89}$$

Next, let us consider this in combination with peristrophic multiplexing. In this case, crystal angle multiplexing is performed at each inclined angle φ*czn*; that is,

$$\phi\_{c\boldsymbol{n}} = n\delta\phi\_{c\text{Min}} = n\frac{\delta\alpha\_{s\text{max}}}{\sin\beta\_{s0}} \quad \left(n = -n\_{\text{max}}, \dots, -2, -1, 0, 1, 2, \dots n\_{\text{max}}\right),\tag{90}$$

where δφ*czMin* is the minimum angular separation in peristrophic multiplexing of Eq. (38); *n* is an integer; and *nmax* is the maximum integer that satisfies the relation

$$
\phi\_{cz\text{Max}} \ge n\_{\text{max}} \frac{\delta \alpha\_{s\text{max}}}{\sin \mathcal{B}\_{s0}} \,\prime \tag{91}
$$

where φ*czMax* is a maximum peristrophic rotation angle. If we take the summation of cosφ*czn* over all *n*, we obtain

$$\begin{split} \sum\_{n=-n\_{\text{max}}}^{n\_{\text{max}}} \cos \phi\_{\text{cm}} &= \sum\_{n=-n\_{\text{max}}}^{n\_{\text{max}}} \cos \left( n \frac{\delta \alpha\_{\text{max}}}{\sin \beta\_{s0}} \right) \\ &= \frac{\cos \left( \frac{\delta \alpha\_{\text{max}}}{2 \sin \beta\_{s0}} \right)}{\sin \left( \frac{\delta \alpha\_{\text{max}}}{2 \sin \beta\_{s0}} \right)} \sin \left[ \frac{\delta \alpha\_{s\text{max}}}{\sin \beta\_{s0}} (n\_{\text{max}} + 1) \right] - \cos \left[ \frac{\delta \alpha\_{\text{max}}}{\sin \beta\_{s0}} (n\_{\text{max}} + 1) \right] \\ &\sim \frac{2 \sin \beta\_{s0}}{\delta \alpha\_{s\text{max}}} \sin \phi\_{c\text{Max}} - \cos \phi\_{c\text{Max}} \end{split} \tag{92}$$

where we assumed that δα*sMax* << 1 and the relation

$$\boldsymbol{\phi}\_{c\text{:Max}} \sim \frac{\delta \alpha\_{s\text{:max}}}{\sin \mathcal{J}\_{s0}} (n\_{\text{:max}} + 1) \tag{93}$$

approximately holds. Then, *VKG* after crystal angle multiplexing in combination with peristrophic multiplexing, which is denoted by *VKG\_total*, is expressed as

$$\begin{split} V\_{\text{KG\\_total}} &= \sum\_{n=-n\_{\text{max}}}^{n\_{\text{max}}} V\_{\text{KG\\_oxz}} \left( \phi\_{\text{cm}} \right) \\ &= 8k\_w \, ^3 \sin 2\beta\_{s0} \delta \mathcal{\beta}\_{s\text{max}} \delta \phi\_{\text{qMax}} \left( 2 \sin \beta\_{s0} \sin \phi\_{\text{cM\text{x}}} - \delta \alpha\_{s\text{max}} \cos \phi\_{\text{cM\text{x}}} \right) \end{split} \tag{94}$$

In particular, if we set δφ*cyMax* = φ*czMax* = π/2, we obtain

$$V\_{\rm KG\\_total} = 8\pi k\_w \, ^\circ \sin \beta\_{s0} \sin 2\beta\_{s0} \delta \beta\_{s\,\max} \,. \tag{95}$$

Theory of Polychromatic Reconstruction for Volume Holographic Memory 247

where we assume that the second term at the right side of Eq. (104) is much smaller than the

0

*K bit PCR sMax wyz s p*

for the PCR method is much larger than that for monochromatic readout. If we rewrite Eq.

*offMax* in Eq. (56), we obtain

*LLL* π

3 3

16 sin tan

4 sin 2

*w yz s*

<sup>=</sup> <sup>+</sup>

*s0* = 30°, *nw* = 1,

*<sup>p</sup>* = −50°. The resultant storage capacities *Nbit(mono)* and *Nbit(PCR)* are 3.2 Tbit and 4.3

3

8 *BM*

λ

λ*BM* / δλ

[ ]

0

β

*s p*

δβ

 β

*cy*. From Eq. (43), the amount of shift of the Bragg-

<sup>=</sup> . (109)

 β

δβ

0 0

ξ ξ<sup>=</sup> <sup>+</sup> (107)

> 1 1 tan tan

*x y z offMax*

δλ

1 min ,1 *w xyz s s bit mono sMax*

0

β

β

= +

*w s*

 β

*L L*

1( )

monochromatic readout. The resultant storage capacity in each readout method is

*n LLL*

2

*n LL*

β

Gbit, respectively. Therefore, the storage capacity in the PCR method decreases by more

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

As seen in Fig. 12, the Bragg-matched wavelength at a certain imaging position (*xd*, *yd*) will

0

β

λ

tan *p cy xfix cz cy s*

0 cos

φ δφ

*w*

λ

*w*

λ

*K bit PCR*

0 16 sin 1 1 . tan tan

δβ

ββ

δβ

<sup>Δ</sup> <sup>=</sup> (106)

(105)

*sMax* instead. Due to this feature, *VK1bit*

*offMax* larger than that in the

(108)

*sMax* =4.0°, and *Lx* = *Ly* = *Lz* = 10

first term and can be neglected. Then *VK1bit* for the PCR method is given by:

3

λ

π

δλ

*V*

( ) 3

π

( ) 2

*p0* = 815 nm,

we derive theoretical limit of the storage capacity after the improvement.

δφ

\_

*cy\_xfix* is expressed as

δλ π

*bit PCR*

λ

*N*

1( )

Equation (105) does not include *Lx* but depends on

Therefore, *VK1bit* for the PCR method is a factor of Δ

*BM* in Eq. (21) and

*N*

*<sup>w</sup>* = 532 nm,

(105) using Δ

For example, if

β

mm, then

λ

than two orders of magnitude.

**4. Selective detection method** 

**4.1 Basic concept and principle** 

matched wavelength

change after the crystal rotation of

δλ

λ

*<sup>n</sup> <sup>V</sup>*

We find that *VKG\_total* increases with increasing maximum deviation angle δβ*sMax*. Note that *VKG* is, of course, independent of the readout scheme, and thus, Eq. (95) can be applied to both the monochromatic case and the PCR method.

Next, we move on to *VK1bit*. Let δα*s1bit*, δβ*s1bit*, and δφ*cy1page* denote the angular separations needed to distinguish two different bits or pages. Then *VK1bit* can be written as

$$V\_{K1bit} = k\_w \, ^3 \cos \phi\_{cz} \sin 2\beta\_{s0} \delta \alpha\_{s1bit} \delta \beta\_{s1bit} \delta \phi\_{cy1page} \,. \tag{96}$$

In the monochromatic case, we can adopt δα*soffMax(mono)* in Eq. (66), δβ*soffMax(mono)* in Eq. (67), and δφ*cyMin(mono)* in Eq. (78) as δα*s1bit*, δβ*s1bit*, and δφ*cy1page*, respectively:

$$
\delta \alpha\_{s1bit \text{(mono)}} = \frac{\mathcal{J}\_w}{n\_w L\_y} \tag{97}
$$

$$
\delta \mathcal{B}\_{s1bit \text{(mwo)}} = \frac{\mathcal{A}\_w}{n\_w} \min \left[ \frac{1}{\cos \mathcal{B}\_{s0} L\_x}, \frac{1}{\sin \mathcal{B}\_{s0} L\_z} \right] \tag{98}
$$

$$\delta\phi\_{\text{cy4}\text{рас(mwo)}} = \frac{\mathcal{\lambda}\_w}{2n\_w \cos\phi\_z} \left(\frac{1}{\cos\mathcal{B}\_{s0}L\_x} + \frac{1}{\sin\mathcal{B}\_{s0}L\_z}\right) \tag{99}$$

Then, *VK1bit* for the monochromatic case is given by

$$V\_{K1hit(mano)} = \frac{8\pi^3}{L\_x L\_y L\_z} \left(1 + \min\left[\xi\_\prime 1/\xi^\varepsilon\right]\right),\tag{100}$$

where

$$\xi \equiv \frac{\tan \beta\_{s0} L\_z}{L\_x}. \tag{101}$$

On the other hand, in the case of the PCR method, we similarly adopt δα*soffMax(PCR)* in Eq. (69), δβ*soffMax(PCR)* in Eq. (70), and δφ*cyMin(PCR)* in Eq. (83) as δα*s1bit*, δβ*s1bit*, and δφ*cy1page*, respectively:

$$
\delta \alpha\_{s1bit(PCR)} = \frac{\mathcal{A}\_w}{n\_w L\_y} \tag{102}
$$

$$\delta \mathcal{B}\_{s1hit\{PR\}} = \frac{\mathcal{\lambda}\_w}{n\_w \sin \mathcal{B}\_{s0} L\_z} \tag{103}$$

$$\begin{split} \delta \boldsymbol{\phi}\_{\text{cy1}\text{page}(\text{PCR})} &= \left( 1 + \frac{\tan \beta\_{s0}}{\tan \beta\_p} \right) \frac{\delta \mathcal{B}\_{s\text{Max}}}{\cos \phi\_{cz}} + \delta \boldsymbol{\phi}\_{\text{cy1}\text{page}(\text{mouse})} \\ &= \left( 1 + \frac{\tan \beta\_{s0}}{\tan \beta\_p} \right) \frac{\delta \mathcal{B}\_{s\text{Max}}}{\cos \phi\_{cz}}, \end{split} \tag{104}$$

*VKG* is, of course, independent of the readout scheme, and thus, Eq. (95) can be applied to

*s1bit*, and

<sup>1</sup> 01 1 1 cos sin 2 . *V k K bit w cz s s bit s bit cy page* =

δα

*s1bit*, and

1( )

 β δα δβ δφ

δφ

*s bit mono n Lw y*

*s bit mono n LL <sup>w</sup> sx sz*

*cy page mono n LL w cz s x s z*

<sup>8</sup> 1 min ,1 , *K bit mono*

<sup>0</sup> tan . *s z x*

*L* β *L*

*cyMin(PCR)* in Eq. (83) as

*w*

<sup>0</sup> sin *w*

*s sMax cy page PCR cy page mono p cz*

> *s sMax p cz*

β

 φ

 φ δφ

λ

λ

*xyz*

1( )

*s bit PCR n Lw y*

*s bit PCR n L w sz*

0 1( ) 1( )

β δβ

β

β δβ

β

tan cos

0

tan 1 , tan cos

= +

*LLL* π

ξ

On the other hand, in the case of the PCR method, we similarly adopt

δφ

δα

1( )

tan <sup>1</sup>

 = + + ≈ + 

δβ δφ

*w*

λ

1 1 min , cos sin

<sup>=</sup>

β

= +

( ) [ ] <sup>3</sup>

ξ ξ

2 cos cos sin

φβ

*soffMax(mono)* in Eq. (66),

*cy1page*, respectively:

0 0

 β

0 0 1 1

 β

≡ (101)

δα*s1bit*, δβ

= (102)

<sup>=</sup> (103)

δβ

(96)

(98)

(100)

δα

*s1bit*, and

(104)

*soffMax(PCR)* in Eq.

δφ*cy1page*,

*soffMax(mono)* in Eq. (67),

*cy1page* denote the angular separations

δβ

(99)

= (97)

*sMax*. Note that

We find that *VKG\_total* increases with increasing maximum deviation angle

needed to distinguish two different bits or pages. Then *VK1bit* can be written as

φ

δα

*w*

*w*

λ

λ

δα*s1bit*, δβ

3

δα*s1bit*, δβ

1( )

1( )

*V*

1( )

δβ

δφ

*soffMax(PCR)* in Eq. (70), and

δφ

Then, *VK1bit* for the monochromatic case is given by

both the monochromatic case and the PCR method.

In the monochromatic case, we can adopt

Next, we move on to *VK1bit*. Let

*cyMin(mono)* in Eq. (78) as

and δφ

where

(69), δβ

respectively:

where we assume that the second term at the right side of Eq. (104) is much smaller than the first term and can be neglected. Then *VK1bit* for the PCR method is given by:

$$V\_{K1hi(PCR)} = \frac{16\pi^3 n\_w \sin \beta\_{s0}}{\mathcal{A}\_w L\_y L\_z} \left(\frac{1}{\tan \beta\_{s0}} + \frac{1}{\tan \beta\_p} \right) \delta \beta\_{s0x} \,. \tag{105}$$

Equation (105) does not include *Lx* but depends on δβ*sMax* instead. Due to this feature, *VK1bit* for the PCR method is much larger than that for monochromatic readout. If we rewrite Eq. (105) using Δλ*BM* in Eq. (21) and δλ*offMax* in Eq. (56), we obtain

$$V\_{K1bit\text{(PR)}} = \frac{8\pi^3}{L\_x L\_y L\_z} \frac{\Delta \mathcal{J}\_{\text{BM}}}{\text{\textdegree \\$}\_{\text{offMax}}} \tag{106}$$

Therefore, *VK1bit* for the PCR method is a factor of Δλ*BM* / δλ*offMax* larger than that in the monochromatic readout. The resultant storage capacity in each readout method is

$$N\_{bit(mno)} = \frac{16\pi n\_w \,^3 L\_x L\_y L\_z}{\lambda\_w^{\,3}} \frac{\sin^3 \beta\_{s0} \tan \beta\_{s0}}{1 + \min\left[\underline{\xi}, 1/\underline{\xi}\right]} \delta \beta\_{s\text{Max}} \tag{107}$$

$$N\_{bit(PCR)} = \frac{4\pi n\_w \,^2 L\_y L\_z}{\lambda\_w^{2}} \frac{\sin 2\mathcal{J}\_{s0}}{\left(\frac{1}{\tan \mathcal{J}\_{s0}} + \frac{1}{\tan \mathcal{J}\_p}\right)}\tag{108}$$

For example, if λ*<sup>w</sup>* = 532 nm, λ*p0* = 815 nm, β*s0* = 30°, *nw* = 1, δβ*sMax* =4.0°, and *Lx* = *Ly* = *Lz* = 10 mm, then β*<sup>p</sup>* = −50°. The resultant storage capacities *Nbit(mono)* and *Nbit(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.

#### **4. 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.

#### **4.1 Basic concept and principle**

As seen in Fig. 12, the Bragg-matched wavelength at a certain imaging position (*xd*, *yd*) will change after the crystal rotation of δφ*cy*. From Eq. (43), the amount of shift of the Braggmatched wavelength δλ*cy\_xfix* is expressed as

$$
\delta \mathcal{S}\_{cy\\_xfix} = \frac{\mathcal{X}\_{p0}}{\tan \mathcal{B}\_{s0}} \cos \phi\_{cz} \delta \phi\_{cy\\_xn} \tag{109}
$$

Theory of Polychromatic Reconstruction for Volume Holographic Memory 249

Fig. 17. The concept of the selective detection method. The dotted part of the Bragg-matched

line corresponds to the portion of the image that will not be detected by the imager.

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 *TLVF* is represented by

$$T\_{\rm LVF} \left( \propto\_{d'} \mathcal{A} \right) = \text{rect} \left( \frac{\mathcal{A} - \mathcal{A}\_{\rm RM} \left( \propto\_{d'} y\_{d'}, 0, \phi\_{ex} \right)}{\Delta \mathcal{A}\_{\rm UV}} \right), \tag{110}$$

where λ is the wavelength; Δλ*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.

#### **4.2 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), (72), and (73). Now, due to the transmission band of the LVF, the variable range of δλ*off* should be modified to

$$-\frac{\Delta\mathcal{J}\_{\rm LVF}}{2} \le \delta\mathcal{J}\_{\rm eff} \le \frac{\Delta\mathcal{J}\_{\rm LVF}}{2}.\tag{111}$$

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 *TLVF* is represented by

> ( ) ( ) , ,0, , rect , *BMcy d d cz*

λλ

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

Fig. 16. Configuration for the selective detection method. A special wavelength filter, like a

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

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

> . 2 2 *LVF LVF off*

δλΔ Δ

*off* ) satisfies all of inequalities in Eqs. (71), (72), and (73). Now, due to the

δλ

 λ

*off* should be modified to

− ≤≤ (111)

seen in Section 3.5.3, the off-Bragg diffraction will occur only when the set of (

λ

band-pass linear variable filter, is inserted at the reconstructed image plane.

wavelength lies outside the transmission band of the LVF.

transmission band of the LVF, the variable range of

**4.2 Theoretical description of the selective detection method** 

*x y T x*

transmission band is proportional to the spatial shift of the illuminated position.

*LVF*

λ

<sup>−</sup> <sup>=</sup> <sup>Δ</sup>

 φ

*LVF* is the full width of the transmitting band; and rect(*x*) is the

(110)

δα*soff* , δβ*soff* ,

*LVF d*

λ

is the wavelength; Δ

where λ

δφ*cy* , δφ*cz* , δλ λ

Fig. 17. The concept of the selective detection method. The dotted part of the Bragg-matched line corresponds to the portion of the image that will not be detected by the imager.

Theory of Polychromatic Reconstruction for Volume Holographic Memory 251

[ ] [ ]

2

<sup>Δ</sup> Δ≡ +

*F LLL F* π

*LVF x y z LVF*

*soffMax(SelDet)* in Eq. (117), and

*cy1page* in Eq. (96), respectively, then *VK1bit* for the selective detection

[ ] <sup>3</sup>

 ξ

λ

1 1

−

1 8 1 min , , *LVF*

<sup>Δ</sup> = + Δ Δ

*LVF*

λ

3

*LVF*. On the other hand, when Δ

*LLL* π

Comparing this with *VK1bit(PCR)* in Eq. (106), we find that the storage capacity will be

*VK1bit(SelDet)* becomes identical to *VK1bit(mono)*. Therefore, the storage capacity in the PCR method

should be noted that, in this case, the diffraction efficiency will also decrease with

( )( ) <sup>2</sup>

*di p pi off off p*

λ δλ

In order to avoid such reduction in the diffraction efficiency, by considering the spectral

<sup>16</sup> <sup>1</sup> 1 min 2 , . <sup>2</sup> *K bit SelDet*

From Eqs. (95) and (118), the resultant storage capacity in the selective detection method is

[ ]

λ

*w xyz LVF s s bit selDet s*

1 min ,

*F*

*w LVF*

( ) 3 max

<sup>Δ</sup> <sup>=</sup> <sup>Δ</sup>

3

*xyz*

*LLL* π

*I Id I*

<sup>8</sup> . <sup>2</sup> *LVF*

*LVF*. This is because the assumption used to derive Eq. (57) is no longer valid,

*off LVF*

<sup>Δ</sup> <sup>∝</sup> + ≈ <sup>Δ</sup> (121)

 δλ

ξ ξ

] ≤ 1 holds, we see that the storage capacity at Δ

[ ]

λ

ξ

*LVF*

<sup>+</sup> <sup>Δ</sup>

<sup>8</sup> sin sin 2 .

*offMax p*

δλ*off* ≤ Δλ*LVF*/2:

λ*LVF*/2 ≤

sinc .

δλ

λ

*x y z offMax*

δλ

*offMax*

δφ

*F*

λ

(119)

λ*LVF* >> 2

δλ

λ

0

δλ

λ

λ*LVF* = 2

= + (122)

[ ]

λ

 ξ δβ

0 0

*F*

ββ λ

 ξ

<sup>Δ</sup> <sup>=</sup> (120)

λ*LVF* << 2

*cyMin(SelDet)* in Eq. (114) as

(118)

δλ*offMax*

*offMax* is satisfied,

*LVF*. However, it

*offMax*. In this case,

λ*LVF* =

(123)

δβ

[ ]

is the improvement factor of the storage capacity. When the condition Δ

1( )

will be completely recovered if we use an LVF having a sufficiently small Δ

πδλ

δλ

*offMax* of the signal diffracted wave, we should set Δ

*K bit SelDet*

λ

*LVF*

λ

*F*

*V*

2

1( )

*offMax* will reach about half that of the monochromatic case.

3

λ

*n LLL F*

λ*BM*/Δλ

and the integration range should be modified to -Δ

*LVF*

λ

Δ

Δ−

2

λ

*LVF*

*V*

ξ, 1/ξ

π

If we adopt

δα*s1bit*, δβ

where

δα

*s1bit*, and

*V*

method is written as

*soffMax(SelDet)* in Eq.(116),

δφ

1( )

*K bit SelDet*

holds, Eq. (118) can be simplified to

improved by a factor of 2Δ

λ

decreasing Δ

width

2δλ δλ

*VK1bit(SelDet)* becomes

Since the relation 0 < min[

*N*

Note that the variable range of δλ*off* is now independent of the targeting position (*xsi*, *ysi*) within the input image because of the transmitting property of the LVF. In this case, the restriction of δβ*soff* in Eq. (71) should be changed to

$$\left|\delta\beta\_{s\circ f}\right| \leq \tan\beta\_{s0}\frac{\Delta\dot{\lambda}\_{\rm LVF}}{\lambda\_{p0}} + \frac{\lambda\_w}{n\_w\cos\beta\_{s0}L\_x}.\tag{112}$$

From Eqs. (73) and (112), we find that the off-Bragg diffraction will occur in the range

$$\left|\delta\phi\_{cy}\right| \leq \frac{\tan\beta\_{s0}}{2\cos\phi\_{cz}} \frac{\Delta\lambda\_{\rm LVF}}{\lambda\_{p0}} + \frac{\lambda\_w}{2n\_w\cos\phi\_{cz}} \left(\frac{1}{\cos\beta\_{s0}L\_x} + \frac{1}{\sin\beta\_{s0}L\_z}\right) \tag{113}$$

Therefore, in order to avoid inter-page crosstalk noise, the crystal rotation angle should be set larger than δφ*cyMin(SelDet)*; that is,

$$\begin{split} \delta\phi\_{\text{cylMin(SelDet)}} &= \frac{\tan\beta\_{s0}}{2\cos\phi\_{cz}} \frac{\Delta\lambda\_{\text{LVF}}}{\lambda\_{p0}} + \frac{\lambda\_w}{2n\_w\cos\phi\_{cz}} \left( \frac{1}{\cos\beta\_{s0}L\_x} + \frac{1}{\sin\beta\_{s0}L\_z} \right) \\ &= \frac{\Delta\lambda\_{\text{LVF}}}{2\Delta\lambda\_{\text{RM}}} \delta\phi\_{\text{cylMin(PCR)}} + \delta\phi\_{\text{cylMin(mmw)}} \end{split} \tag{114}$$

where δφ*cyMin(mono)* and δφ*cyMin(PCR)* are the minimum crystal rotation angle in the monochromatic case and the PCR method, respectively; and we use Eq. (21) to derive the second equality. If we ignore the δφ*cyMin(mono)*, we see that δφ*cyMin(SelDet)* is improved by a factor of 2Δλ*BM*/Δλ*LVF* than δφ*cyMin(PCR)*. Note that the probe spectral width Δλ*<sup>p</sup>* need not be equal to the spectral width required for the full image reconstruction Δλ*BM* because Eq. (114) no longer includes Δλ*p*.

Furthermore, the intra-page crosstalk noise will be also suppressed if Δλ*LVF* is sufficiently small. Since the off-Bragg diffraction will occur only when δβ*soff* satisfies both the inequalities in Eqs. (60) and (62) under the range of Eq. (111), the restriction of δβ*soff* should be written as

$$\left| \delta \mathcal{B}\_{s\mathcal{G}} \right| \leq \min \left[ \tan \mathcal{B}\_{s0} \frac{\Delta \mathcal{A}\_{\rm LVF}}{\mathcal{A}\_{p0}} + \frac{\mathcal{A}\_w}{n\_w \cos \mathcal{B}\_{s0} L\_x}, \frac{\mathcal{A}\_w}{n\_w \sin \mathcal{B}\_{s0} L\_z} \right]. \tag{115}$$

Therefore, the maximum off-Bragg deviation angles δα*soffMax(SelDet)* and δβ*soffMax(SelDet)* are obtained as follows:

$$
\delta \alpha\_{\text{soffMax(SelfDet)}} = \frac{\mathcal{A}\_w}{n\_w L\_y} \tag{116}
$$

$$\delta \mathcal{J}\_{\text{soffMax(sidlet)}} = \min \left[ \tan \mathcal{J}\_{s0} \frac{\Delta \mathcal{J}\_{\text{lVF}}}{\lambda\_{p0}} + \frac{\lambda\_w}{n\_w \cos \mathcal{J}\_{s0} L\_x}, \frac{\lambda\_w}{n\_w \sin \mathcal{J}\_{s0} L\_z} \right]. \tag{117}$$

If we adopt δα*soffMax(SelDet)* in Eq.(116), δβ*soffMax(SelDet)* in Eq. (117), and δφ*cyMin(SelDet)* in Eq. (114) as δα*s1bit*, δβ*s1bit*, and δφ*cy1page* in Eq. (96), respectively, then *VK1bit* for the selective detection method is written as

$$\left| V\_{K1\text{bit}\left(\text{SelfDet}\right)} = \frac{1}{F\left[\Delta\mathcal{J}\_{\text{LVF}}\right]} \frac{8\pi^3}{L\_x L\_y L\_z} \left\{ 1 + \min\left[ \frac{\xi}{F\left[\Delta\mathcal{J}\_{\text{LVF}}\right]}, \frac{F\left[\Delta\mathcal{J}\_{\text{LVF}}\right]}{\xi} \right] \right\},\tag{118}$$

where

250 Holograms – Recording Materials and Applications

within the input image because of the transmitting property of the LVF. In this case, the

λ

Δ

λ

tan 1 1 2 cos 2 cos cos sin

Therefore, in order to avoid inter-page crosstalk noise, the crystal rotation angle should be

 φ

() ( )

*cyMin PCR cyMin mono*

monochromatic case and the PCR method, respectively; and we use Eq. (21) to derive the

*cyMin(mono)*, we see that

*cyMin(PCR)*. Note that the probe spectral width Δ

 δφ

, <sup>2</sup>

*s LVF w*

tan 1 1 2 cos 2 cos cos sin

 φ

=+ +

 λ

<sup>Δ</sup> ≤+ +

 λ

0 0 tan . cos *LVF w*

0 0 0

 β

0 0 0

Δ

*n LL*

 β

*cyMin(PCR)* are the minimum crystal rotation angle in the

δφ

0 00

*p w sx w sz n Ln L*

δα

0 00

*p w sx w sz n Ln L*

= + (117)

*LVF w w*

*LVF w w*

λ

min tan , . cos sin

Δ

λλ

min tan , . cos sin

 Δ ≤ +

λλ

( ) *<sup>w</sup> soffMax SelDet n Lw y*

λ

*cz p w cz s x s z*

*cz <sup>p</sup> n LL w cz s x s z*

*<sup>p</sup> n L w sx*

 λ

> β

0

From Eqs. (73) and (112), we find that the off-Bragg diffraction will occur in the range

 β

*s LVF w*

*off* is now independent of the targeting position (*xsi*, *ysi*)

≤ + (112)

 β

. (113)

 β

λ

δβ

λ

 λ

*soffMax(SelDet)* and

 λ

= (116)

ββ

ββ

*cyMin(SelDet)* is improved by a factor

λ

δβ

*<sup>p</sup>* need not be equal to

*soff* satisfies both the

δβ

*LVF* is sufficiently

(115)

*soffMax(SelDet)* are

*soff* should

*BM* because Eq. (114) no

(114)

δλ

*soff s*

*soff* in Eq. (71) should be changed to

δβ

0

*cyMin(SelDet)*; that is,

( )

δφ

*cyMin SelDet*

*cyMin(mono)* and

second equality. If we ignore the

λ*p*.

> δβ

βλ

φλ

0

βλ

φλ

δφ

= +

*LVF*

λ

λ

Δ

δφ

Δ

*BM*

δφ

Furthermore, the intra-page crosstalk noise will be also suppressed if Δ

inequalities in Eqs. (60) and (62) under the range of Eq. (111), the restriction of

0

λ

β

δα

β

( ) 0

*soffMax SelDet s*

the spectral width required for the full image reconstruction Δ

small. Since the off-Bragg diffraction will occur only when

*soff s*

Therefore, the maximum off-Bragg deviation angles

Note that the variable range of

*cy*

δφ

δφ

δφ

δβ

restriction of

set larger than

where

of 2Δλ*BM*/Δλ*LVF* than

δφ

longer includes Δ

be written as

obtained as follows:

δβ

$$F\left[\Delta\mathcal{J}\_{\rm LVF}\right] \equiv \left(\frac{\Delta\mathcal{J}\_{\rm LVF}}{2\,\text{\mathcal{S}\delta}\_{\rm y\mathcal{H}\text{ax}}} + 1\right)^{-1} \tag{119}$$

is the improvement factor of the storage capacity. When the condition Δλ*LVF* >> 2δλ*offMax* holds, Eq. (118) can be simplified to

$$V\_{K1bit\text{\textquotedblleft}\text{Sell\textquotedblright}\text{\textquotedblright}} = \frac{8\pi^3}{L\_x L\_y L\_z} \frac{\Delta\mathcal{J}\_{LV}}{2\mathcal{S}\mathcal{J}\_{\text{offMax}}}.\tag{120}$$

Comparing this with *VK1bit(PCR)* in Eq. (106), we find that the storage capacity will be improved by a factor of 2Δλ*BM*/Δλ*LVF*. On the other hand, when Δλ*LVF* << 2δλ*offMax* is satisfied, *VK1bit(SelDet)* becomes identical to *VK1bit(mono)*. Therefore, the storage capacity in the PCR method will be completely recovered if we use an LVF having a sufficiently small Δλ*LVF*. However, it should be noted that, in this case, the diffraction efficiency will also decrease with decreasing Δλ*LVF*. This is because the assumption used to derive Eq. (57) is no longer valid, and the integration range should be modified to -Δλ*LVF*/2 ≤ δλ*off* ≤ Δλ*LVF*/2:

$$I\_{di} \approx \int\_{-\frac{\Delta\mathcal{J}\_{\rm UF}}{2}}^{\frac{\Delta\mathcal{J}\_{\rm UF}}{2}} \operatorname{sinc}^2 \left(\frac{\pi \mathcal{S}\mathcal{J}\_{\rm eff}}{\delta\mathcal{J}\_{\rm eff\Delta x}}\right) I\_p \left(\mathcal{J}\_{pi} + \delta\mathcal{J}\_{\rm eff}\right) d\left(\delta\mathcal{J}\_{\rm eff}\right) \approx \frac{\Delta\mathcal{J}\_{\rm VF}}{\Delta\mathcal{J}\_p} I\_{p0} \,. \tag{121}$$

In order to avoid such reduction in the diffraction efficiency, by considering the spectral width δλ*offMax* of the signal diffracted wave, we should set Δλ*LVF* = 2δλ*offMax*. In this case, *VK1bit(SelDet)* becomes

$$V\_{K1bit\text{(SellDet)}} = \frac{16\pi^3}{L\_{\text{x}}L\_{y}L\_{z}} \left\{ 1 + \min\left[2\underline{\xi}, \frac{1}{2\underline{\xi}}\right] \right\}.\tag{122}$$

Since the relation 0 < min[ξ, 1/ ξ] ≤ 1 holds, we see that the storage capacity at Δλ*LVF* = 2δλ*offMax* will reach about half that of the monochromatic case.

From Eqs. (95) and (118), the resultant storage capacity in the selective detection method is

$$N\_{bit\text{(selfDet)}} = \frac{8\pi n\_w \,^3 L\_x L\_y L\_z}{\lambda\_w^3} \frac{F\left[\Delta\mathcal{J}\_{\text{LVF}}\right] \sin\mathcal{J}\_{s0} \sin 2\mathcal{J}\_{s0}}{1 + \min\left[\frac{\xi}{F\left[\Delta\mathcal{J}\_{\text{LVF}}\right]}, \frac{F\left[\Delta\mathcal{J}\_{\text{LVF}}\right]}{\xi}\right]} \delta\mathcal{J}\_{s\text{max}}.\tag{123}$$

Theory of Polychromatic Reconstruction for Volume Holographic Memory 253

A larger input image (*xsMax*, *ysMax*) or smaller focal length *fs* require a larger spectral width

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

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

Due to the wide spectral width of the probe beam, the hologram dimension *Lx* 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

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

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,

Barbastathis, G. & Psaltis, D., (2000). Volume holographic multiplexing methods, In:

*Holographic data storage*, Coufal, H.J., Psaltis, D., & Sincerbox, G.T., pp. 21-62,

problem can be solved by using the selective detection method mentioned below.

**Selective detection method and the achievable storage capacity** 

*<sup>p</sup>* = 0°), it becomes independent of *ysMax*. Furthermore, the condition

image distortion, where the image is magnified by the wavelength ratio

*BM* to reconstruct the whole image. Generally the required spectral width depends on both directions *xsMax* and *ysMax*, but if the signal, reference, and probe beams lie in the same plane

α

*<sup>p</sup>* = 0° minimizes the

*0* in the *yd*-

μ

**The required spectral width and image distortion** 

direction, but is unchanged in the *xd*-direction.

**Multiplexing method** 

each multiplexed page. **Diffraction efficiency** 

**Intra- and Inter-page crosstalk noise** 

theoretical limit for the monochromatic case.

Springer, 978-3540666912, New York.

method.

simultaneously.

**6. References** 

Δλ

(α

In most cases, δβ*sMax* is limited by the spectral width Δλ*<sup>p</sup>* of a given probe light source. Thus, if Eq. (26) is substituted into Eq. (123), we obtain

$$N\_{bit(self)} = \frac{16\pi n\_w \,^3 L\_x L\_y L\_z}{\lambda\_w^3} \frac{F\left[\Delta\lambda\_{\rm{LVF}}\right]}{1 + \min\left[\frac{\xi}{F\left[\Delta\lambda\_{\rm{LVF}}\right]}, \frac{F\left[\Delta\lambda\_{\rm{LVF}}\right]}{\xi}\right]} \frac{\sin^3 \mathcal{J}\_{s0}}{\xi} \frac{\Delta\lambda\_p}{\lambda\_{p0}}.\tag{124}$$

For example, if λ*<sup>w</sup>* = 532 nm, λ*p0* = 815 nm, Δλ*<sup>p</sup>* = 50 nm, β*s0* = 30°, *nw* = 1, *Lx* = *Ly* = *Lz* = 10 mm, and Δλ*LVF* = 2δλ*offMax* = 0.043 nm, then the resultant storage capacity *Nbit(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 Δλ*LVF* in Fig. 18. The improvement starts at Δλ*LVF* = 100 nm, then the storage capacity increases linearly with increasing Δλ*LVF*, and finally, it asymptotically approaches the theoretical limit of the monochromatic case. Note that the kink observed at Δλ*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).

Fig. 18. Storage capacity as a function of transmission bandwidth Δλ*LVF*. The calculation parameters are as follows: λ*w* = 532 nm, λ*p0* = 815 nm, Δλ*<sup>p</sup>* = 50 nm, β*s0* = 30°, *nw* = 1, and *Lx* = *Ly* = *Lz* = 10 mm.

#### **5. 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.
