**3. Photoreactive anisotropic medium with uniaxial anisotropy**

In order to confirm the principle described in section 2, a grating was recorded in a model medium by 3D vector holography and its diffraction properties were observed. We prepared an azo-dye doped liquid crystalline material as the photoreactive anisotropic medium by mixing nematic mixture E7, a side-chain liquid crystalline polymer (SLCP), and azo-dye 4-[N-(2-hydroxyethyl)-N-ethyl]amino-4'-nitroazobenzene, more commonly known as disperse red 1 (DR1), with a weight ratio of E7 : SLCP : DR1 = 59 : 40 : 1. Here, E7 and DR1 were obtained from BDH-Merck and Ardrich, respectively. SLCP was synthesized using a poly(methyl methacrylate) backbone comprising 4-cyanophenyl benzoate side groups. The number and weight averages of SLCP used in this study were 11,700 and 32,800, respectively. The chemical structures of the three components were illustrated in Fig. 5. They were stirred at around 100°C until a homogeneous solution was obtained. The

Three-Dimensional Vector Holograms in Photoreactive Anisotropic Media 187

where 0 **c** is the normalized vector of the initial optical axis in the *xyz* -coordinate system.

**c**′ ′′ ′′ ′ 0 0 0 00 0 = ( ) cos cos , cos sin , sin

axis and the projection of 0 **c**′ onto the *x y*′ ′ -plane. The reoriented optical axis in the *xyz* ′′′ -

re ( ) ( ) ( ) ( )( ) ( ) 0 re 0 re 0 re 0 re 0 re **c**′ ′′ ′′ ′′ ′′ ′′ =+ + + + + cos cos , cos sin , sin

 θθ

simplicity, we assume that the amplitudes of the photoinduced angles are small and are proportional to the intensity of light. In addition, it has been revealed that the director of the azodye doped liquid crystalline material tilts to the perpendicular direction for the polarization azimuth. Therefore, using Equations 16 and 17, the two photoinduced angles are written as

> re 0 n 0 *C S* θ

re 0 n 1 0 2 0 ′′ ′ ′ =⋅ − *CS S*

normalized electric field vector of interference light in the *xyz* ′′′ -coordinate system, namely,

<sup>n</sup> <sup>2</sup> <sup>2</sup> 1

*x y*

′ ′ <sup>+</sup>

′ <sup>=</sup> ′

*E E*

**cR c** re F re =Θ ≡ = ( ) ′ ( ) *ccc xyz* , , cos cos , cos sin , sin ( )

( ) <sup>1</sup> , sin *<sup>z</sup>*

( ) ( ) <sup>1</sup> , sin cos *<sup>y</sup>*

Therefore, the photoinduced tilt and azimuth angles in the *xyz*-coordinate system are

re ( ) <sup>0</sup>

 θθ

*xz c*

φφ

are constants, which represent sensitivity to light intensity, and **E**n is the

0

θφ

*x y*

*E E*

φ

θ φ

φ

θ

θ

′ is the angle between the *x y*′ ′ -plane and 0 **c**′ and

 φφ

φ

In the *xyz* -coordinate system, the reoriented optical axis is given by

is the angle between the *xy* -plane and re **c** and

φ

and the projection of re **c** onto the *xy* -plane. From Equation 26, we obtain

θ

θ

Here, 0 **c**′ can be described as

coordinate system is written as

θθ

re′ is the photoinduced tilt angle and

φ

where θ0

where

where *C*

where θ

obtained as

θ and *C*φ

θ

( ) <sup>1</sup> 0 F0

<sup>−</sup> **cR c** ′ = Θ , (20)

, (21)

′ is the angle between the *x*′ -

, (22)

 θθ

re′ is the photoinduced azimuth angle. For

′ ′ = ⋅ **c E** , (23)

sin 2 cos 2 () () **c E** , (24)

**E** . (25)

 θφ

*xz c* <sup>−</sup> = , (27)

<sup>−</sup> = , (28)

*x z* , = − , (29)

φ

θ

 θ

is the angle between the *x* -axis

, (26)

θ

φ0

 φφ

mixture was sandwiched between two rubbed poly(vinyl alcohol)-coated glass substrates with 10 μm thick spacers in order to obtain a homogeneously aligned sample. By using a polarizing optical microscope, we observed that the mixture exhibited the nematic phase at room temperature. Namely, it was confirmed that the azo-dye doped liquid crystalline material with homogeneous alignment was a uniaxial anisotropic medium.

Fig. 5. Chemical structures of (a) E7, (b) SLCP, and (c) DR1

We consider the molecular reorientation induced by irradiation with polarized light. In order to discuss the polarization state of interference light, we assume that its propagation direction is in the *xz* -plane and that the angle between the *z* -axis and the propagation direction is given by

$$
\Theta\_{\rm F} = \sin^{-1}\left[ (\sin \theta\_{\rm F}) / \overline{n} \right] / \tag{18}
$$

where *n* is the average refractive index of the medium. Here, we transform the coordinate system from *xyz* into *xyz* ′′′ as follow

$$
\begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} \cos\Theta\_{\mathrm{F}} & 0 & \sin\Theta\_{\mathrm{F}} \\ 0 & 1 & 0 \\ -\sin\Theta\_{\mathrm{F}} & 0 & \cos\Theta\_{\mathrm{F}} \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} \equiv \mathbf{R}^{-1} \left( \Theta\_{\mathrm{F}} \right) \begin{bmatrix} x \\ y \\ z \end{bmatrix}. \tag{19}
$$

In the *xyz* ′′′ -coordinate system, the orientation direction (i.e., the optical axis) in the initial state is written as

mixture was sandwiched between two rubbed poly(vinyl alcohol)-coated glass substrates with 10 μm thick spacers in order to obtain a homogeneously aligned sample. By using a polarizing optical microscope, we observed that the mixture exhibited the nematic phase at room temperature. Namely, it was confirmed that the azo-dye doped liquid crystalline

6

We consider the molecular reorientation induced by irradiation with polarized light. In order to discuss the polarization state of interference light, we assume that its propagation direction is in the *xz* -plane and that the angle between the *z* -axis and the propagation

> ( ) <sup>1</sup> F F sin sin

*n* <sup>−</sup> Θ =

where *n* is the average refractive index of the medium. Here, we transform the coordinate

*x x x y y y z z z*

′ Θ Θ ′ <sup>=</sup> ≡ Θ

−Θ Θ

In the *xyz* ′′′ -coordinate system, the orientation direction (i.e., the optical axis) in the initial

F F

0 1 0

cos 0 sin

sin 0 cos

F F

θ

, (18)

**R** . (19)

( )

1 F

−

material with homogeneous alignment was a uniaxial anisotropic medium.

51%

(a)

25%

16%

8%

(b)

(c)

system from *xyz* into *xyz* ′′′ as follow

′

direction is given by

state is written as

x

Fig. 5. Chemical structures of (a) E7, (b) SLCP, and (c) DR1

( ) <sup>1</sup> 0 F0 <sup>−</sup> **cR c** ′ = Θ , (20)

where 0 **c** is the normalized vector of the initial optical axis in the *xyz* -coordinate system. Here, 0 **c**′ can be described as

$$\mathbf{c}'\_0 = \begin{pmatrix} \cos \theta'\_0 \cos \phi'\_0 & \cos \theta'\_0 \sin \phi'\_0 & \sin \theta'\_0 \end{pmatrix} \,\tag{21}$$

where θ0 ′ is the angle between the *x y*′ ′ -plane and 0 **c**′ and φ0 ′ is the angle between the *x*′ axis and the projection of 0 **c**′ onto the *x y*′ ′ -plane. The reoriented optical axis in the *xyz* ′′′ coordinate system is written as

$$\mathbf{c}'\_{\rm re} = \left(\cos\left(\theta'\_0 + \theta'\_{\rm re}\right)\cos\left(\phi'\_0 + \phi'\_{\rm re}\right), \ \cos\left(\theta'\_0 + \theta'\_{\rm re}\right)\sin\left(\phi'\_0 + \phi'\_{\rm re}\right), \ \sin\left(\theta'\_0 + \theta'\_{\rm re}\right)\right), \tag{22}$$

where θre′ is the photoinduced tilt angle and φre′ is the photoinduced azimuth angle. For simplicity, we assume that the amplitudes of the photoinduced angles are small and are proportional to the intensity of light. In addition, it has been revealed that the director of the azodye doped liquid crystalline material tilts to the perpendicular direction for the polarization azimuth. Therefore, using Equations 16 and 17, the two photoinduced angles are written as

$$\boldsymbol{\theta}'\_{\rm re} = \mathbb{C}\_{\theta} \left| \mathbf{c}'\_{0} \cdot \mathbb{E}\_{\mathrm{n}} \right| \mathbf{S}\_{0} \,\prime \,\tag{23}$$

$$\phi\_{\rm re}' = \mathbb{C}\_{\phi} \left| \mathbf{c}\_0' \cdot \mathbf{E}\_n \right| \left[ \mathbf{S}\_1 \sin \left( 2\phi\_0' \right) - \mathbf{S}\_2 \cos \left( 2\phi\_0' \right) \right], \tag{24}$$

where *C*θ and *C*φ are constants, which represent sensitivity to light intensity, and **E**n is the normalized electric field vector of interference light in the *xyz* ′′′ -coordinate system, namely,

$$\mathbf{E}\_n = \frac{1}{\sqrt{\left|E\_x'\right|^2 + \left|E\_y'\right|^2}} \begin{vmatrix} E\_x'\\ E\_y'\\ 0 \end{vmatrix}.\tag{25}$$

In the *xyz* -coordinate system, the reoriented optical axis is given by

$$\mathbf{c}\_{\rm re} = \mathbf{R} \left( \Theta\_{\rm F} \right) \\
\mathbf{c}\_{\rm re}' \equiv \begin{pmatrix} c\_{x'} & c\_{y'} & c\_z \end{pmatrix} = \begin{pmatrix} \cos\theta\cos\phi, \ \cos\theta\sin\phi, \ \sin\theta \end{pmatrix}, \tag{26}$$

where θ is the angle between the *xy* -plane and re **c** and φ is the angle between the *x* -axis and the projection of re **c** onto the *xy* -plane. From Equation 26, we obtain

$$\theta(x, z) = \sin^{-1} c\_{z, \prime} \tag{27}$$

$$\phi(x, z) = \sin^{-1}\left(c\_y / \cos \theta\right),\tag{28}$$

Therefore, the photoinduced tilt and azimuth angles in the *xyz*-coordinate system are obtained as

$$
\theta\_{\text{re}}\left(\mathbf{x},\;z\right) = \theta - \theta\_0.\tag{29}
$$

Three-Dimensional Vector Holograms in Photoreactive Anisotropic Media 189

0

(c)

0

first-order diffraction efficiencies

−45 −30 −15 0 15 30 45

θin (deg)

Incident angle

Fig. 7. Angular dispersion of the diffraction efficiency for (a) *p*-, (b) *s*-, and (c) *q*+-polarized probe beam. Red and blue plots represent the measured positive and negative first-order diffraction efficiencies. Solid and dashed lines represent the calculated positive and negative

10

20

30

40

50

10

20

30

Diffraction efficiency (%)

40

50 0

(b)

10

20

30

40

50

(a)

$$
\phi\_{\rm re} \begin{pmatrix} \mathbf{x} \ \mathbf{z} \end{pmatrix} = \phi - \phi\_0 \, \mathrm{d} \tag{30}
$$

Fig. 6. Schematic illustration of the experimental setup (HWP, half-wave plate; PBS, polarizing beam splitter; M, mirror). Circles represent rotary stage in the incident plane

#### **4. Observed diffraction properties**

Figure 6 schematically illustrates the experimental setup for the holographic recording and the reconstruction. A frequency doubled, Nd-doped yttrium aluminum garnet laser with an operating wavelength of 532 nm was used for recording the hologram. This is because the azo-dye effectively reacts to irradiation with green light. The LP recording beam was divided into *s*- and *p*- polarizations using a polarizing beam splitter, and the two beams intersected on the sample as illustrated in Fig. 6. The intensity of the two recording beams was 900 mW/cm2, respectively. The reconstruction properties were probed by an LP He-Ne laser with a wavelength of 633 nm. The intensity of diffracted beams was measured by a photodiode detector while varying the incident angle of the probe beam. All measurements were conducted at room temperature.

Figure 7 presents the measured positive and negative first-order diffraction efficiencies for 1 2 θ θ =− = ° 1.5 (i.e., F θ = 0 ), 0 θ = 0 , and 0 φ = π 2 . The ±1st-order diffraction efficiencies η± were defined as the ratio of the intensity of the diffracted beam to the total intensity of the transmitted light. As seen in Fig. 7, strong diffraction appeared in the ±1st-order when the probe beam was *p*- and *s*-polarized, respectively. When the probe beams was *q*+-polarized, relatively strong diffraction was observed in both the orders. The results shown in Fig. 7 clearly indicate that the diffraction efficiency depends on the incident angle of the probe beam θin . It is also important to remember that the angular dependence is asymmetry for in θ = 0 even though F θ = 0 . Figure 8 illustrates observed polarization states of the diffracted beams in the case of in θ = 0 . This result implies that the grating diffracts *p*- and *s*-polarized components of incident light as *s*- and *p*-polarized light, respectively. Here, the *p*- and *s*-polarizations were defined for the *xz* -plane in this chapter. This property with polarization discrimination is useful for applying to optical elements such as a polarizing beam splitter with a thin thickness.

re ( ) <sup>0</sup>

optical rail

sample

HWP HWP

2 . The ±1st-order diffraction efficiencies

PBS

. (30)

*x*

He-Ne laser

*z*

η±

θ= 0

 *x z* , = − φ φ

φ

HWP

HWP

M

Fig. 6. Schematic illustration of the experimental setup (HWP, half-wave plate; PBS, polarizing beam splitter; M, mirror). Circles represent rotary stage in the incident plane

Figure 6 schematically illustrates the experimental setup for the holographic recording and the reconstruction. A frequency doubled, Nd-doped yttrium aluminum garnet laser with an operating wavelength of 532 nm was used for recording the hologram. This is because the azo-dye effectively reacts to irradiation with green light. The LP recording beam was divided into *s*- and *p*- polarizations using a polarizing beam splitter, and the two beams intersected on the sample as illustrated in Fig. 6. The intensity of the two recording beams was 900 mW/cm2, respectively. The reconstruction properties were probed by an LP He-Ne laser with a wavelength of 633 nm. The intensity of diffracted beams was measured by a photodiode detector while varying the incident angle of the probe beam. All measurements

Figure 7 presents the measured positive and negative first-order diffraction efficiencies for

were defined as the ratio of the intensity of the diffracted beam to the total intensity of the transmitted light. As seen in Fig. 7, strong diffraction appeared in the ±1st-order when the probe beam was *p*- and *s*-polarized, respectively. When the probe beams was *q*+-polarized, relatively strong diffraction was observed in both the orders. The results shown in Fig. 7 clearly indicate that the diffraction efficiency depends on the incident angle of the probe beam

in . It is also important to remember that the angular dependence is asymmetry for in

of incident light as *s*- and *p*-polarized light, respectively. Here, the *p*- and *s*-polarizations were defined for the *xz* -plane in this chapter. This property with polarization discrimination is useful for applying to optical elements such as a polarizing beam splitter with a thin thickness.

= 0 . Figure 8 illustrates observed polarization states of the diffracted beams in

= 0 . This result implies that the grating diffracts *p*- and *s*-polarized components

φ = π

= 0 , and 0

M

Nd:YAG(2ω)

**4. Observed diffraction properties** 

were conducted at room temperature.

θ = 0 ), 0 θ

1 2 θ

θ

 θ

even though F

the case of in

=− = ° 1.5 (i.e., F

θ

θ

PBS

M

M

Fig. 7. Angular dispersion of the diffraction efficiency for (a) *p*-, (b) *s*-, and (c) *q*+-polarized probe beam. Red and blue plots represent the measured positive and negative first-order diffraction efficiencies. Solid and dashed lines represent the calculated positive and negative first-order diffraction efficiencies

Three-Dimensional Vector Holograms in Photoreactive Anisotropic Media 191

The calculated diffraction efficiencies for the experimental conditions, described in section 4, were shown in Fig. 7. The parameters used this calculation are as follows: o *n* = 1.52 ,

diffraction efficiencies were well explained with the use of the theoretical model and the FDTD calculation. This fact suggests that the 3D vector hologram is actually recorded in the azo-dye doped liquid crystalline material. The calculated distribution of the photoinduced tilt and azimuth angles were illustrated in Fig. 10. It is clearly known that a slanted grating is formed based on the spatially varying state of polarization. Figure 11 illustrates the calculated distribution of electric fields in and around the hologram when the incident beam is *p*- and *s*-polarized. Here, the AR layers were not implemented in this calculation to accentuate the air-hologram boundaries. For the *p*-polarization, the *s*-component was gradually generated in the hologram and propagated to the positive first-order direction. In contrast, the *p*-component was gradually generated in the hologram and propagated to the negative first-order direction when the incident light was *s*-polarized. The calculated results are consistent with the observed property of polarization conversion shown in Fig. 8. This

Λ*<sup>x</sup>* Λ*<sup>x</sup>*

*x x*

re and (c) their value

<sup>−</sup> = × cm2/W. As seen in Fig. 7, the observed

+3.6°

(c)

−3.6°

0.0°

φ

**5.2 Analysis of observed diffraction properties** 

<sup>−</sup> = × cm2/W, and <sup>2</sup> *C* 4.8 10

fact also demonstrates the accuracy of the theoretical model.

*d d*

0 0

θ

re and (b)

φ

Fig. 10. Calculated distribution of (a)

(a) (b)

*z z*

<sup>e</sup> *n* = 1.75 , <sup>2</sup> *C* 4.9 10 θ

Fig. 8. Polar plots of polarization states for incident and diffraction beams. The incident beam, indicated as solid lines, is (a) *p*-, (b) *s*-, and (c) *q*+-polarized. Red and blue plots represent measured data for positive and negative first-order diffraction beams

#### **5. Theoretical characterization**

#### **5.1 FDTD method**

In order to characterize the observed results mentioned above, optical behaviour of the periodical distribution of the dielectric tensor, which was described in section 3, were calculated by employing the FDTD method. This method imposes no limitations on the dimension of the object and the incident conditions since Maxwell's equations are directly solved numerically by the calculus of finite differences in the algorithm (Yee, 1966; Teflova & Gedne, 2005). The FDTD method can analyze anisotropic media such as liquid crystals (Kriezis & Elston, 1999, 2000; Titus et al., 2001; Scharf, 2007). In addition, it has been reported that the FDTD method is suitable for characterizing anisotropic gratings recorded by vector holography (Oh & Escuti, 2006, 2007; Ono et al. 2008a, 2008b).

Figure 9 schematically illustrates the model for the FDTD calculation in this study. We set the calculation conditions as follows: the wavelength of the incident light is 633 nm, the size of the analytical area ( *x* -direction × *z* -direction) is 162 μm × 15 μm, the grid spacing is 20 nm × 20 nm, and the time step is 0.047 fs. As the boundary condition, the perfectly matched layer (PML) with the thickness of 1 μm was used (Teflova & Gedne, 2005). The glass substrates and the alignment layers were ignored and antireflection (AR) layers were installed at the air-hologram boundaries for reducing the effect of multiple interference (Southwell, 1983). The diffraction properties in the far field were calculated by Fourier transforming each component of the electric field at the output plane.

Fig. 9. Schematic layout of the analytical area for the FDTD calculation (AR, anti-reflection coatings; PML, perfectly matched layer)

#### **5.2 Analysis of observed diffraction properties**

190 Holograms – Recording Materials and Applications

(a) (b) (c)

−1st

Fig. 8. Polar plots of polarization states for incident and diffraction beams. The incident beam, indicated as solid lines, is (a) *p*-, (b) *s*-, and (c) *q*+-polarized. Red and blue plots represent measured data for positive and negative first-order diffraction beams

In order to characterize the observed results mentioned above, optical behaviour of the periodical distribution of the dielectric tensor, which was described in section 3, were calculated by employing the FDTD method. This method imposes no limitations on the dimension of the object and the incident conditions since Maxwell's equations are directly solved numerically by the calculus of finite differences in the algorithm (Yee, 1966; Teflova & Gedne, 2005). The FDTD method can analyze anisotropic media such as liquid crystals (Kriezis & Elston, 1999, 2000; Titus et al., 2001; Scharf, 2007). In addition, it has been reported that the FDTD method is suitable for characterizing anisotropic gratings recorded by vector

Figure 9 schematically illustrates the model for the FDTD calculation in this study. We set the calculation conditions as follows: the wavelength of the incident light is 633 nm, the size of the analytical area ( *x* -direction × *z* -direction) is 162 μm × 15 μm, the grid spacing is 20 nm × 20 nm, and the time step is 0.047 fs. As the boundary condition, the perfectly matched layer (PML) with the thickness of 1 μm was used (Teflova & Gedne, 2005). The glass substrates and the alignment layers were ignored and antireflection (AR) layers were installed at the air-hologram boundaries for reducing the effect of multiple interference (Southwell, 1983). The diffraction properties in the far field were calculated by Fourier

Fig. 9. Schematic layout of the analytical area for the FDTD calculation (AR, anti-reflection

air AR

PML

hologram

air AR

+1st

**5. Theoretical characterization** 

holography (Oh & Escuti, 2006, 2007; Ono et al. 2008a, 2008b).

transforming each component of the electric field at the output plane.

output

input

coatings; PML, perfectly matched layer)

**5.1 FDTD method** 

The calculated diffraction efficiencies for the experimental conditions, described in section 4, were shown in Fig. 7. The parameters used this calculation are as follows: o *n* = 1.52 , <sup>e</sup> *n* = 1.75 , <sup>2</sup> *C* 4.9 10 θ <sup>−</sup> = × cm2/W, and <sup>2</sup> *C* 4.8 10 φ <sup>−</sup> = × cm2/W. As seen in Fig. 7, the observed diffraction efficiencies were well explained with the use of the theoretical model and the FDTD calculation. This fact suggests that the 3D vector hologram is actually recorded in the azo-dye doped liquid crystalline material. The calculated distribution of the photoinduced tilt and azimuth angles were illustrated in Fig. 10. It is clearly known that a slanted grating is formed based on the spatially varying state of polarization. Figure 11 illustrates the calculated distribution of electric fields in and around the hologram when the incident beam is *p*- and *s*-polarized. Here, the AR layers were not implemented in this calculation to accentuate the air-hologram boundaries. For the *p*-polarization, the *s*-component was gradually generated in the hologram and propagated to the positive first-order direction. In contrast, the *p*-component was gradually generated in the hologram and propagated to the negative first-order direction when the incident light was *s*-polarized. The calculated results are consistent with the observed property of polarization conversion shown in Fig. 8. This fact also demonstrates the accuracy of the theoretical model.

Fig. 10. Calculated distribution of (a) θre and (b) φre and (c) their value

Three-Dimensional Vector Holograms in Photoreactive Anisotropic Media 193

probe beam was *p*- and *s*-polarized. We attribute the reason for the discrepancy to a change of the Bragg angle owing to the difference of the refractive index for the two

θ

θin (deg)

<sup>−</sup>1 for the *p*-polarization were

θin (deg)

−15 0 15 30 45

*d* (μm) 10 7 5 η+

> η+1

−45 −30 −15 0 15 30 45

max was remarkably influenced by the polarization state of

max at Λ = *<sup>x</sup>* 2 μm were approximately 21° and 2° when the

Λ*<sup>x</sup>* = 2, 4, 8 μm

in (deg) Incident angle

for the *p*-polarized probe beam calculated by varying the film

= =− ° 12.5 on *d* in the case of Λ = *<sup>x</sup>* 1 μm . As seen in

η

in (deg) Incident angle

<sup>+</sup> for *p*-polarization at (a) Λ = *<sup>x</sup>* 1 μm and (b) Λ = *<sup>x</sup>* 10 μm

Diffraction efficiency (%)

<sup>−</sup>1 for the *s*-polarization reached about 100% around

40 50

Diffraction efficiency (%)

max varied by Λ*<sup>x</sup>*

± ± 1 in 1 max ( ) ≡ ( ) were almost the same for respective Λ*<sup>x</sup>* . However,

polarizations. This is one of a characteristic of the 3D vector hologram.

θ

<sup>−</sup> for *s*-polarization

<sup>−</sup>1 at in max θ θ

η

(a) (b)

*d* (μm) 10 7 5

Incident angle

−15 0 15 30 45

θ

η

η

Fig. 13. Angular dependence of the diffraction efficiency calculated by varying Λ*<sup>x</sup>* . (a)

thickness *d* . It is clearly shown in Fig. 14 that the angular dependence of the diffraction efficiency decreases with increasing Λ*x* (i.e., Λ ) and decreasing *d* . This is understood by considering the difference of thick and thin gratings, similarly common diffraction gratings in isotropic media (Kogelnik, 1969; Hariharan, 1996). Figure 15 illustrates dependence of

*d* = 22 μm and then decreased with increasing *d* , respectively. These curves were well fitted sinusoidal functions similarly transmission type thick gratings in anisotropic media. In

+1 for the *s*-polarization and

θ

θ

(a) (b)

Λ*<sup>x</sup>* = 2, 4, 8 μm

Incident angle

η

+1 for the *p*-polarization and

η

η<sup>+</sup>1 in ( ) θ

−45 −30 −15 0 15 30 45

ηθ

 ηθ

Diffraction efficiency (%)

for *p*-polarization. (b)

Figure 14 presents

at in max θ θ

η

Diffraction efficiency (%)

Fig. 14. Angular dependence of

Fig. 15,

= =° 24.7 and

contrast, it was confirmed that

nearly equal to zero, respectively.

as seen in Fig. 13. In addition,

the probe beam. For example,

Fig. 11. Calculated electric fields for the (a) *p*- and (b) *s*-polarized probe beam. *Ex* and *Ey* are the real part of the *x* - and *y* -components for the electric field vector

#### **5.3 Simulated diffraction properties**

This section investigates the relationship between various parameters of the 3D vector hologram and its diffraction properties through the theoretical model. Figure 12 presents the calculated results of the grating structure for varying Λ*<sup>x</sup>* . The grating pitch in the grating Λ and the slanted angle Θ<sup>S</sup> , which is the angle between the *x*-axis and the grating vector, are given by

$$
\Lambda = \Lambda\_z \cos \Theta\_\mathbf{S} = \Lambda\_z \cos \left( \tan^{-1} \frac{\Lambda\_z}{\Lambda\_\chi} \right) \tag{31}
$$

where Λ*z* is the grating pitch for the *z*-axis. In this case, Λ and Λ*z* are determined by the distribution of *S*2 and *S*3 as described in Equations 23 and 24. We confirmed that Λ and Λ*z* decreased with decreasing Λ*<sup>x</sup>* . It should also be pointed out that Λ ≅Λ Δ = *z x n* 2.31 μm for relatively large Λ*x* (namely, for 1 2 θ θ= << 1 ).

Fig. 12. Dependence of (a) Λ , Λ*<sup>z</sup>* , and (b) ΘS on Λ*<sup>x</sup>*

Figure 13 illustrates angular dispersion of η<sup>±</sup>1 calculated for Λ = *<sup>x</sup>* 2, 4, 8 μm when the probe beam is *p*- and *s*-polarized. The maximum values of the diffraction efficiency

*Ex Ey Ex Ey* (a) (b)

hologram

*d*

*z*

*x*

**5.3 Simulated diffraction properties** 

for relatively large Λ*x* (namely, for 1 2

Λ

Λ*z*

0 5 10 15 20

Fig. 12. Dependence of (a) Λ , Λ*<sup>z</sup>* , and (b) ΘS on Λ*<sup>x</sup>*

Figure 13 illustrates angular dispersion of

(a) (b)

0.0

0.5

1.0

Λ, Λ*z* (μm)

1.5

2.0

2.5

air

Fig. 11. Calculated electric fields for the (a) *p*- and (b) *s*-polarized probe beam. *Ex* and *Ey* are

This section investigates the relationship between various parameters of the 3D vector hologram and its diffraction properties through the theoretical model. Figure 12 presents the calculated results of the grating structure for varying Λ*<sup>x</sup>* . The grating pitch in the grating Λ and the slanted angle Θ<sup>S</sup> , which is the angle between the *x*-axis and the grating vector, are given by

<sup>S</sup> cos cos tan *<sup>z</sup> z z*

<sup>−</sup> <sup>Λ</sup> Λ=Λ Θ =Λ

where Λ*z* is the grating pitch for the *z*-axis. In this case, Λ and Λ*z* are determined by the distribution of *S*2 and *S*3 as described in Equations 23 and 24. We confirmed that Λ and Λ*z* decreased with decreasing Λ*<sup>x</sup>* . It should also be pointed out that Λ ≅Λ Δ = *z x n* 2.31 μm

> θ θ= << 1 ).

1

0

Λ*<sup>x</sup>* (μm) Λ*<sup>x</sup>* (μm)

η

probe beam is *p*- and *s*-polarized. The maximum values of the diffraction efficiency

30

60

ΘS (deg)

90

*x*

<sup>Λ</sup> , (31)

0 5 10 15 20

<sup>±</sup>1 calculated for Λ = *<sup>x</sup>* 2, 4, 8 μm when the

Λ*x*

the real part of the *x* - and *y* -components for the electric field vector

air

ηθ ηθ ± ± 1 in 1 max ( ) ≡ ( ) were almost the same for respective Λ*<sup>x</sup>* . However, θ max varied by Λ*<sup>x</sup>* as seen in Fig. 13. In addition, θ max was remarkably influenced by the polarization state of the probe beam. For example, θ max at Λ = *<sup>x</sup>* 2 μm were approximately 21° and 2° when the probe beam was *p*- and *s*-polarized. We attribute the reason for the discrepancy to a change of the Bragg angle owing to the difference of the refractive index for the two polarizations. This is one of a characteristic of the 3D vector hologram.

Fig. 13. Angular dependence of the diffraction efficiency calculated by varying Λ*<sup>x</sup>* . (a) η+ for *p*-polarization. (b) η<sup>−</sup> for *s*-polarization

Figure 14 presents η<sup>+</sup>1 in ( ) θ for the *p*-polarized probe beam calculated by varying the film thickness *d* . It is clearly shown in Fig. 14 that the angular dependence of the diffraction efficiency decreases with increasing Λ*x* (i.e., Λ ) and decreasing *d* . This is understood by considering the difference of thick and thin gratings, similarly common diffraction gratings in isotropic media (Kogelnik, 1969; Hariharan, 1996). Figure 15 illustrates dependence of η+1 at in max θ θ = =° 24.7 and η<sup>−</sup>1 at in max θ θ = =− ° 12.5 on *d* in the case of Λ = *<sup>x</sup>* 1 μm . As seen in Fig. 15, η+1 for the *p*-polarization and η<sup>−</sup>1 for the *s*-polarization reached about 100% around *d* = 22 μm and then decreased with increasing *d* , respectively. These curves were well fitted sinusoidal functions similarly transmission type thick gratings in anisotropic media. In contrast, it was confirmed that η+1 for the *s*-polarization and η<sup>−</sup>1 for the *p*-polarization were nearly equal to zero, respectively.

Fig. 14. Angular dependence of η<sup>+</sup> for *p*-polarization at (a) Λ = *<sup>x</sup>* 1 μm and (b) Λ = *<sup>x</sup>* 10 μm

Three-Dimensional Vector Holograms in Photoreactive Anisotropic Media 195

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Fig. 15. Dependence of (a) η<sup>+</sup> at in θ = ° 24.7 and (b) η<sup>−</sup> at in θ =− ° 12.5 on the film thickness. Red and blue lines are the calculated results for *p*- and *s*-polarizations

#### **6. Conclusions**

In this chapter, the principle of 3D vector holography has been described. We considered interference of two polarized plane waves with coherency in anisotropic media, and formulated a type of 3D vector hologram as spatial distribution of the dielectric tensor based on the polarization state of interference light. Optical behaviour of the periodic distribution was simulated by employing the FDTD method and the result was indeed consistent with the observed diffraction properties of the holographic grating, recorded in an azo-dye doped liquid crystalline medium with uniaxial optical anisotropy. Hence, we confirmed that various periodic structures, consisted of spatial distribution of optical anisotropy, could be formed by means of the 3D vector holography. Furthermore, detailed diffraction properties of the holographic grating were studied with the use of the FDTD method. As a result, it was revealed that the 3D vector hologram exhibit various useful characteristics including high diffraction efficiencies, angular dispersion, polarization dependence, and polarization conversion. We believe that the 3D vector holography realizes fabrication of higher-order structures of optical anisotropy and that these structures can be applied to diffractive optical elements with multiple functions, 3D optical memories, anisotropic photonic crystals and so on.

#### **7. References**


Diffraction efficiency (%)

η<sup>−</sup> at in θ

> 0 10 20 30 Thickness *d* (μm)

=− ° 12.5 on the film thickness.

*s*

*p*

(a) (b)

*p*

Diffraction efficiency (%)

Fig. 15. Dependence of (a)

**6. Conclusions** 

**7. References** 

Thickness *d* (μm)

Red and blue lines are the calculated results for *p*- and *s*-polarizations

η<sup>+</sup> at in θ

optical memories, anisotropic photonic crystals and so on.

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**Part 2** 

**Holographic Data Storage** 


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**9** 

*Taiwan* 

**Diffraction Property of** 

Yeh-Wei Yu and Ching-Cherng Sun

*National Central University* 

**Collinear Holographic Storage System** 

The collinear storage system was proposed by Optware [1,2]. It is a coaxially aligned optical structure for signal and reference beams, which are encoded simultaneously by the same spatial light modulator (SLM) and the two beams interfere with each other in the recording medium through a single objective lens. The system has been proven a capability of large storage capacity, high transfer rate, short access time, and besides, it is compatible with existing disc storage systems such as CDs and DVDs [3-4]. Recent report of the collinear Volume holographic storage (VHS) has performed a storage density as high as 270 Gbits/inch2 [5]. Many advantages are proposed, including uniform shift selectivity in both radial and tangential directions, a fairly large wavelength shift and a fairly large tilt

Fig. 1-1. The collinear holographic storage system proposed by Optware [2].

**1. Introduction** 

tolerance [6].

**1.1 The collinear volume holographic storage system** 
