**2. Polarization-selective substrate-mode volume holograms**

#### **2.1 Conventional polarization-selective substrate-mode volume holograms**

Figure 1 shows the structure of the conventional polarization-selective substrate-mode hologram [Huang, 1994] which is composed of four volume holograms, input grating coupler (HI), polarization beam splitter hologram (HPBS), output grating couplers (HOS and HOP), and two substrates. An unpolarized light is incident on HI normally, and is diffracted into HPBS at a special angle. The output diffraction lights of HPBS are split into *s*- and *p*components which are perpendicular to each other. These two components are then total internal reflected (TIR) at the base of the substrate and are diffracted and coupled out normally by HOS and HOP, respectively. Therefore, the *s*- and *p*-polarized lights are successfully separated.

Fig. 1. Schematic representation of the conventional polarization-selective substrate-mode volume hologram.

In this structure, HI, HPBS, HOS, and HOP are actually transmission-type phase volume holograms and can be designed according to the coupled-wave theory [Kogelnik, 1969]. For a transmission-type phase volume hologram, as shown in Fig. 2, the relation between the diffraction efficiencies of *s*- and *p*-components can be written as

$$
\hbar \eta\_{s,p} = \sin^2 \nu\_{s,p} \tag{1}
$$

where the modulation parameters for *s*- and *p*-components, υ*s* and υ*<sup>p</sup>* , are given as

$$\upsilon\_s = \frac{\pi N\_1}{\left(\cos \theta\_{r1} \cos \theta\_{r2}\right)^{1/2}},\tag{2}$$

$$\boldsymbol{\upsilon}\_p = \boldsymbol{\upsilon}\_s \cos(\theta\_{r2} - \theta\_{r1}) \,\text{.}\tag{3}$$

and

284 Holograms – Recording Materials and Applications

in several optical systems, such as optical sensing, optical data storage, imaging system, and switching network. In 2003, the PSVHs was firstly proposed to replace crystal-type SWPs in a four-port optical ciculator [Chen et al., 2003]. In the application, these PSVHs are consequently termed as holographic spatial walk-off polarizer (HSWP). Due to the introduction of HSWPs, the fabricated four-port optical circulator has advantages of polarization-independence, compactness, high isolation, low polarization mode dispersion,

However, the feasibility of conventional PSVHs is usually limited by the finite refractive index modulation strength of a recording material. The common solution is to increase the thickness of the recording material, in order to compensate the shortage of the refractive index modulation strength in the phase modulation term. However, under the thickness condition of thick material, the distortion effect of interference fringe is worsened. An ideal holographic recording condition hinges on the thin thickness of a recording material with a high refractive index modulation strength. Actually these cannot be completed in both respects. To overcome the problems, based on the coupled-wave theory and the structure of substrate-mode hologram, a special design of PSVHs was proposed with a relatively large splitting angle near 90° [Chen et al., 2008]. With this design, a low refractive index modulation strength is required, which can be easily achieved with common recording

materials. In addition, this design should bear all merits of conventional PSVHs.

As the design of optical communication systems becomes more and more complex, an optical circulator with many input and output ports has become highly desirable. However, the port numbers for presently most commercial optical circulators are limited. In 2004, based on holographic spatial- and polarization-modules (HSPMs), two kinds design of holographic-type multi-port optical circulator were also proposed [Chen et al., 2004; Chen et al., 2004]. The HSPM is consisted of two HSWPs, an half-wave plate (H), and a Faraday rotator (FR). The merits of these designs include polarization-independence, compactness, high isolation, low polarization mode dispersion, and easy fabrication. Furthermore, the

Accordingly, this chapter devotes to introduce the polarization-selective substrate-mode volume hologram in several respects and its novel applications in design of optical circulator. The second section, according to coupled-wave theory, will clearly describe the principle and characteristic of conventional PSVHs; a modified design method of PSVHs will also be described to overcome the shortage in refractive index modulation strength. The third section will introduce the applications of PSVH to optical circulators. The principle and operation characteristic of a four-port optical circulator will be introduced. The following context will introduce the principles and operation characteristic of holographic spatial- and polarization-modules (HSPMs) and their applications to multi-port optical circulators. All the design details will be described and their characteristic will be discussed.

low cost, and easy fabrication.

number of port can be scaled up easily.

Finally, the fourth section is conclusion.

**2. Polarization-selective substrate-mode volume holograms** 

**2.1 Conventional polarization-selective substrate-mode volume holograms** 

Figure 1 shows the structure of the conventional polarization-selective substrate-mode hologram [Huang, 1994] which is composed of four volume holograms, input grating coupler (HI), polarization beam splitter hologram (HPBS), output grating couplers (HOS and HOP), and two substrates. An unpolarized light is incident on HI normally, and is diffracted

$$N\_1 = \frac{n\_1 d}{\mathcal{A}}.\tag{4}$$

*N*1 is the effective index modulation in which λ is the reconstruction wavelength, *d* is the thickness of the recording material, and *n*1 is the refractive index modulation. θ*r*1 and θ*<sup>r</sup>*2 are corresponding angles of the reconstruction and the diffraction beams in the recording material, respectively.

Polarization-Selective Substrate-Mode

θ

thickness of the recording material.

volume hologram (I).

diffraction angle

Volume Holograms and Its Application to Optical Circulators 287

material and substrate. In this way, the diffracted beam is totally reflected at point B and hits the grating again at point C. This beam is totally reflected at point C, and the reflected beam from point C satisfies the Bragg condition of the grating. The propagation direction of the reflected beam is in parallel to that of the beam diffracted by the grating at point A. Because the structure of the grating at point C is the same as that at point A, the diffracted beam at point C will be in parallel to the input beam at point A; that is, the output beam passes normally through the substrate (channel 2). The detail of the beam propagation at point C is shown in the upper right circle of Fig. 3. Consequently, two orthogonally polarized parallel beams with the separation of length AC=2*d*(tan*θr*2) can be obtained in which *d* is the

Fig. 3. Schematic representation of the proposed polarization-selective substrate-mode

common optical systems for the purpose of more compactness.

In addition, in the same principle, we can properly choose the substrate with its refractive index equally that of the recording material (*ns*=*nf*). Under this condition, the light propagation details in Fig. 3 change as shown in Fig. 4. The light separated distance becomes AC=2*t*(tan*θr*2) in which *t* is the thickness of the substrate. In generally, the structure in Fig. 3 is suitable for integrated optical systems, and that in Fig. 4 can be applied in

θ

*<sup>c</sup>* at the interface of recording

**2.2 Alternative design of polarization-selective substrate-mode volume holograms**  Shown in Fig. 3 is a schematic representation of the proposed polarization-selective substrate-mode volume hologram [Chen et al., 2008] which is consisted of a transmissiontype phase volume holographic grating and a substrate. Its grating structure is designed in such a way that either of the *s*- or *p*-polarized component of a normal incident beam at A is transmitted straight through the grating and the substrate (channel 1) while the other orthogonally polarized component is completely diffracted into the substrate with a large

*r*2 which is larger than the critical angle

Fig. 2. Reconstruction geometry of the phase volume hologram: S, *s*-polarization field; P, *p*polarization field.

In the case of normal incident, *i*.*e*. θ*<sup>r</sup>*1=0°, eqs. (2) and (3) can be reduced as

$$\upsilon\_s = \frac{\pi N\_1}{\left(\cos \theta\_{r2}\right)^{1/2}},\tag{5}$$

$$
\upsilon\_p = \upsilon\_s \cos(\theta\_{r2}).\tag{6}
$$

Accordingly, the design of HI requires η*s*=η*<sup>p</sup>*≥90% that can be solved by eqs. (1)-(6). The function of HPBS requires η*s*=100% and η*p*=0 or η*s*=0 and η*<sup>p</sup>*=100%. The functions of HOP and HOS require η*p*=100% (η*s*=0) and η*s*=100% (η*<sup>p</sup>*=0), respectively.

However, in order to satisfy the requirements of HOP and HOS, the parameters υ*s* and υ*p* stand on the following conditions: (1) υ*s*=[*m*+(1/2)]π and υ*p*=*m*π (for η*s*=100% and η*<sup>p</sup>*=0); (2) υ*s*=*m*π and υ*p*=[*m*-(1/2)]π (for η*s*=0 and η*<sup>p</sup>*=100%), where *m* is a positive integer. Under these conditions, the values of related parameters *m*, θ*<sup>r</sup>*2, and *N*1 are listed in Table 1. In order to fulfill the required TIR inside the substrates, only conditions at *m*=1 are valid. However, the feasibility of fabricating these elements is usually limited by the finite refractive index modulation strength *n*1 of a recording material. Therefore, an alternative design method is described below to overcome this drawback.


Table 1. Related parameters for condition (1) (η*s*=100% and η*p*=0) and condition (2) (η*<sup>s</sup>*=0 and η*<sup>p</sup>*=100%).

Fig. 2. Reconstruction geometry of the phase volume hologram: S, *s*-polarization field; P, *p*-

( ) 1 1/2 2

π*N*

*r*

( ) <sup>2</sup> cos .

 θ

θ

cos *<sup>s</sup>*

υυ

η*s*=η

However, in order to satisfy the requirements of HOP and HOS, the parameters

η

*s*=[*m*+(1/2)]

fulfill the required TIR inside the substrates, only conditions at *m*=1 are valid. However, the feasibility of fabricating these elements is usually limited by the finite refractive index modulation strength *n*1 of a recording material. Therefore, an alternative design method is

η*p*=0 or η*s*=0 and

υ

η

*s*=100% (

υ

*<sup>r</sup>*1=0°, eqs. (2) and (3) can be reduced as

,

*<sup>p</sup>*=0), respectively.

π and υ*p*=*m*π (for η

1 2 3 4 5

Condition (1) 48.2° 36.9° 31.0° 27.3° 24.6° Condition (2) 60.0° 41.4° 33.5° 29.0° 25.8°

Condition (1) 1.22 2.24 3.24 4.24 5.24 Condition (2) 0.707 1.73 2.74 3.74 4.74

*s*=100% and

η

θ

η

<sup>=</sup> (5)

*ps r* = (6)

*<sup>p</sup>*=100%), where *m* is a positive integer. Under these

*m* 

η

*<sup>r</sup>*2, and *N*1 are listed in Table 1. In order to

*p*=0) and condition (2) (

*<sup>p</sup>*≥90% that can be solved by eqs. (1)-(6). The

*<sup>p</sup>*=100%. The functions of HOP and

*s*=100% and

υ*s* and υ*p*

η

η*<sup>s</sup>*=0

*<sup>p</sup>*=0); (2)

θ

*s*=100% and

η

polarization field.

In the case of normal incident, *i*.*e*.

Accordingly, the design of HI requires

*p*=100% (

stand on the following conditions: (1)

*p*=[*m*-(1/2)]

η

*s*=0) and

η

π (for η*s*=0 and

conditions, the values of related parameters *m*,

described below to overcome this drawback.

Table 1. Related parameters for condition (1) (

function of HPBS requires

η

HOS require

υ*s*=*m*π and υ

> θ*r*2

*N*<sup>1</sup>

and η

*<sup>p</sup>*=100%).

#### **2.2 Alternative design of polarization-selective substrate-mode volume holograms**

Shown in Fig. 3 is a schematic representation of the proposed polarization-selective substrate-mode volume hologram [Chen et al., 2008] which is consisted of a transmissiontype phase volume holographic grating and a substrate. Its grating structure is designed in such a way that either of the *s*- or *p*-polarized component of a normal incident beam at A is transmitted straight through the grating and the substrate (channel 1) while the other orthogonally polarized component is completely diffracted into the substrate with a large diffraction angle θ*r*2 which is larger than the critical angle θ*<sup>c</sup>* at the interface of recording material and substrate. In this way, the diffracted beam is totally reflected at point B and hits the grating again at point C. This beam is totally reflected at point C, and the reflected beam from point C satisfies the Bragg condition of the grating. The propagation direction of the reflected beam is in parallel to that of the beam diffracted by the grating at point A. Because the structure of the grating at point C is the same as that at point A, the diffracted beam at point C will be in parallel to the input beam at point A; that is, the output beam passes normally through the substrate (channel 2). The detail of the beam propagation at point C is shown in the upper right circle of Fig. 3. Consequently, two orthogonally polarized parallel beams with the separation of length AC=2*d*(tan*θr*2) can be obtained in which *d* is the thickness of the recording material.

Fig. 3. Schematic representation of the proposed polarization-selective substrate-mode volume hologram (I).

In addition, in the same principle, we can properly choose the substrate with its refractive index equally that of the recording material (*ns*=*nf*). Under this condition, the light propagation details in Fig. 3 change as shown in Fig. 4. The light separated distance becomes AC=2*t*(tan*θr*2) in which *t* is the thickness of the substrate. In generally, the structure in Fig. 3 is suitable for integrated optical systems, and that in Fig. 4 can be applied in common optical systems for the purpose of more compactness.

Polarization-Selective Substrate-Mode

substrate.

0.0

Fig. 5. The relation of *x* v.s. diffraction efficiencies,

0.1

0.2

0.3

0.4

0.5

Diffraction efficiency (X100%)

0.6

0.7

0.8

0.9

1.0

Volume Holograms and Its Application to Optical Circulators 289

Shown in Fig. 6 is a preliminary measurement result of a fabricated element for *p*polarized input signal. The thickness *t* of the substrate is 1.50mm, and the light separating distance is about 26mm. The technique of shorter wavelength construction for longer wavelength reconstruction is applied for the fabrication of the holographic polarization selective element. A 532nm solid-state laser was applied as the exposure light source. Silver-halide recording material (VRP-M, Slavich) is used for the fabrication of this element designed with *θr*2=83.5° for 632.8nm. The related recording material parameters of *n* and *d*, before and after post-processing are measured by an optical thin film analysis system (Model: nkd-6000TM, aquila Instruments Ltd.) The measured parameters are *nf*1=1.60 (@ 532nm), *nf*2=1.66 (@ 632.8nm), *d*1=5.70μm, and *d*2=5.35μm. Therefore, the ideal value of phase modulation *n*1*d* is 0.11μm. In addition, in order to easy the operation, a right-angle prism with specification of 150×150×50mm is introduce for the exposure light guiding. Some castor oil (n=1.48, @20°C) is used as index-matching oil. Due to the large diffraction angle, a BaSF2 glass substrate (*ns*=1.66, Producer: Schott Glaswerke and Schott Glass Technologies) with the same refractive index of recording material is used avoiding the reflection at the interface of recording material and

0.0 0.5 1.0 1.5

η*s* and η

π*n1 d /* λ

*a* = ( cos

θ*r2*=85o

**S**

θ*r2* )1/2

**P**

*p*, considering *θr2*=85°.

Fig. 4. Schematic representation of the proposed polarization-selective substrate-mode volume hologram (II).

According to eqs. (1), (5), and (6), the relation between the diffraction efficiencies of *s*- and *p*components can be rewritten as

$$
\eta\_s = \sin^2(\frac{\chi}{a}) \tag{7a}
$$

and

$$
\eta\_p = \sin^2(ax),
\tag{7b}
$$

where

$$
\chi = \frac{\pi n\_1 d}{\mathcal{A}},
\tag{8a}
$$

and

$$a = \left(\cos \theta\_{r2}\right)^{1/2}.\tag{8b}$$

It is obvious from eqs. (7a) and (7b) that the diffraction efficiencies of *s*- and *p*- components oscillate in the form of a sine square function asynchronously of which the primitive periods are *Ts*=*a*π and *Tp*=π/*a*, respectively. Therefore, when *θr*2 has a large diffraction angle near 90°, the parameter *a* has a relative small value. This condition results a smaller value of *Ts* and a larger value of *Tp*, and the peak values of *s*- and *p*- diffraction efficiencies leave far away each other. The smaller value of *Ts* means a smaller required phase modulation. Therefore, in the condition of a small phase modulation value *n*1*d*, we can obtain a desired result of *ηs*=100% and *ηp*~0 and complete the purpose of polarization beam splitting effectively. Shown in Fig. 5 is the relation of diffraction efficiencies v.s. *x* considering *θr2*=85°. It is obviously that when the value of *x* equals 0.46, corresponding to an effective index modulation *N*1=0.15, we can obtain *ηs*=100% and *ηp*≅1.89%.

Fig. 4. Schematic representation of the proposed polarization-selective substrate-mode

η

η

*x* π

modulation *N*1=0.15, we can obtain *ηs*=100% and *ηp*≅1.89%.

According to eqs. (1), (5), and (6), the relation between the diffraction efficiencies of *s*- and *p*-

<sup>2</sup> sin ( ), *<sup>s</sup> x a*

<sup>2</sup> sin ( ), *<sup>p</sup>*

<sup>1</sup> , *n d*

1/2

λ

<sup>2</sup> (cos ) . *<sup>r</sup> a* = θ

It is obvious from eqs. (7a) and (7b) that the diffraction efficiencies of *s*- and *p*- components oscillate in the form of a sine square function asynchronously of which the primitive periods are *Ts*=*a*π and *Tp*=π/*a*, respectively. Therefore, when *θr*2 has a large diffraction angle near 90°, the parameter *a* has a relative small value. This condition results a smaller value of *Ts* and a larger value of *Tp*, and the peak values of *s*- and *p*- diffraction efficiencies leave far away each other. The smaller value of *Ts* means a smaller required phase modulation. Therefore, in the condition of a small phase modulation value *n*1*d*, we can obtain a desired result of *ηs*=100% and *ηp*~0 and complete the purpose of polarization beam splitting effectively. Shown in Fig. 5 is the relation of diffraction efficiencies v.s. *x* considering *θr2*=85°. It is obviously that when the value of *x* equals 0.46, corresponding to an effective index

= (7a)

= *ax* (7b)

<sup>=</sup> (8a)

(8b)

volume hologram (II).

and

where

and

components can be rewritten as

Shown in Fig. 6 is a preliminary measurement result of a fabricated element for *p*polarized input signal. The thickness *t* of the substrate is 1.50mm, and the light separating distance is about 26mm. The technique of shorter wavelength construction for longer wavelength reconstruction is applied for the fabrication of the holographic polarization selective element. A 532nm solid-state laser was applied as the exposure light source. Silver-halide recording material (VRP-M, Slavich) is used for the fabrication of this element designed with *θr*2=83.5° for 632.8nm. The related recording material parameters of *n* and *d*, before and after post-processing are measured by an optical thin film analysis system (Model: nkd-6000TM, aquila Instruments Ltd.) The measured parameters are *nf*1=1.60 (@ 532nm), *nf*2=1.66 (@ 632.8nm), *d*1=5.70μm, and *d*2=5.35μm. Therefore, the ideal value of phase modulation *n*1*d* is 0.11μm. In addition, in order to easy the operation, a right-angle prism with specification of 150×150×50mm is introduce for the exposure light guiding. Some castor oil (n=1.48, @20°C) is used as index-matching oil. Due to the large diffraction angle, a BaSF2 glass substrate (*ns*=1.66, Producer: Schott Glaswerke and Schott Glass Technologies) with the same refractive index of recording material is used avoiding the reflection at the interface of recording material and substrate.

Fig. 5. The relation of *x* v.s. diffraction efficiencies, η*s* and η*p*, considering *θr2*=85°.

Polarization-Selective Substrate-Mode

0

Maximum index modulation strength

Satisfied thickness

halide material.

Fig. 7. The theoretical relation of *ER*2 v.s. *θr*2.

*n*DCG<0.0

500

1000

1500

*ER*2

2000

2500

3000

Volume Holograms and Its Application to Optical Circulators 291

gelatin and silver-halide material are not excess 0.08 and 0.03, respectively. Therefore, the condition of finite phase modulation *n*1*d* will cause these elements hard to be realized by conventional method. This situation is especially serious in the near infrared for optical communications. The improved method not only can solve the problem but also has all merits of conventional substrate-mode volume holograms such as compactness, plane

80 81 82 83 84 85

θ*r2*

Conventional method Improved method

8 *n*SH<0.03 *n*DCG<0.0

<sup>8</sup>*n*SH<0.03

Diffraction angle

*θr2* 48.19° 60° *θr2* (*ηs*, *ηp*) (100%, 0) (0, 100%) (*ηs*, *ηp*) *n*1*d*/*λ* 1.22 0.71 *n*1*d*/*λ*

Table 2. Comparisons for the improved method and conventional method; *n*DCG : maximum index modulation strength of DCG; *n*SH : maximum index modulation strength of silver-

*<sup>d</sup>*/*<sup>λ</sup>* >15.31 >40.82 >8.84 >23.57 >1.85 >4.92

8 *n*SH<0.03 *n*DCG<0.0

structure, easily light collimation, easily fabrication, and low cost.

Fig. 6. Transmission image for *p*-polarized input signal.

In fabrications, the value of refractive index modulation *n*1 relates the exposure time. Therefore, according to eqs. 8(a) and 8(b), knowing the reconstruction wavelength, the recording material thickness, and the diffraction angle, the refractive index modulation *n*<sup>1</sup> can be obtained

$$m\_1 = \frac{T\_s}{2} = \frac{\lambda \left(\cos \theta\_{r2}\right)^{1/2}}{2d}.\tag{9}$$

In the previously mentioned case, the estimated refractive index modulation *n*1 is 0.02 that can be obtained by controlling the exposure time experimentally.

In addition, according to eqs. (7a) and (7b), the extinction ratio (ER) of channel 1 and channel 2 can be defined as

$$ER\_1 = \frac{1 - \eta\_p}{1 - \eta\_s} \,\prime \tag{10a}$$

and

$$\hat{\sigma}\_{2}\hat{\sigma}\_{2} = \frac{\eta\_{s}^{2}}{\eta\_{p}^{2}} = \left[\sin(\frac{\pi\cos\theta\_{r2}}{2})\right]^{-4}.\tag{10b}$$

According to eq. (10a), *ER*1 >>1 can be obtained easily with this design. From eq. (10b), it is obvious that *ER*2 is related with the diffraction angle *θr*2. Therefore, *ER*2 has a lager value as *θr*2 is larger. Shown in Fig. 7 is the relation of *ER*2 v.s. *θr*2. It can be seen that the value of *ER*<sup>2</sup> is larger than 1000, when *θr*2 is larger than 83.5°. In the same mentioned case, the diffraction efficiencies of the *s*- and *p*-components are about 83% and 5%, and the calculated extinction ratio of channel 1 and channel 2 are 5.58 and 275, respectively. The preliminary experimental results show the validity of the proposed method. The experimental errors mainly come from the optical setup, the process of optical exposure, and the postprocessing.

In addition, the conventional polarization-selective substrate-mode volume holograms are designed with *θr*2=48.19° and 60°. The comparisons of the improved method and conventional method are listed in Table 2. Considering the commercial holographic recording materials, the maximum values of refractive index modulation of dichromated

In fabrications, the value of refractive index modulation *n*1 relates the exposure time. Therefore, according to eqs. 8(a) and 8(b), knowing the reconstruction wavelength, the recording material thickness, and the diffraction angle, the refractive index modulation *n*<sup>1</sup>

1

can be obtained by controlling the exposure time experimentally.

( )1 2 2

= = (9)

<sup>−</sup> <sup>=</sup> <sup>−</sup> (10a)

<sup>−</sup> = = (10b)

 θ

*d*

cos . 2 2

*Ts <sup>r</sup> <sup>n</sup>*

1

*ER*

2 2

η

η

*ER*

*p*

λ

In the previously mentioned case, the estimated refractive index modulation *n*1 is 0.02 that

In addition, according to eqs. (7a) and (7b), the extinction ratio (ER) of channel 1 and

<sup>1</sup> , <sup>1</sup> *p s*

η

η

2 4

*s r*

According to eq. (10a), *ER*1 >>1 can be obtained easily with this design. From eq. (10b), it is obvious that *ER*2 is related with the diffraction angle *θr*2. Therefore, *ER*2 has a lager value as *θr*2 is larger. Shown in Fig. 7 is the relation of *ER*2 v.s. *θr*2. It can be seen that the value of *ER*<sup>2</sup> is larger than 1000, when *θr*2 is larger than 83.5°. In the same mentioned case, the diffraction efficiencies of the *s*- and *p*-components are about 83% and 5%, and the calculated extinction ratio of channel 1 and channel 2 are 5.58 and 275, respectively. The preliminary experimental results show the validity of the proposed method. The experimental errors mainly come from the optical setup, the process of optical exposure, and the post-

In addition, the conventional polarization-selective substrate-mode volume holograms are designed with *θr*2=48.19° and 60°. The comparisons of the improved method and conventional method are listed in Table 2. Considering the commercial holographic recording materials, the maximum values of refractive index modulation of dichromated

 πθ

2

cos sin( ) . <sup>2</sup>

Fig. 6. Transmission image for *p*-polarized input signal.

can be obtained

and

processing.

channel 2 can be defined as

gelatin and silver-halide material are not excess 0.08 and 0.03, respectively. Therefore, the condition of finite phase modulation *n*1*d* will cause these elements hard to be realized by conventional method. This situation is especially serious in the near infrared for optical communications. The improved method not only can solve the problem but also has all merits of conventional substrate-mode volume holograms such as compactness, plane structure, easily light collimation, easily fabrication, and low cost.

Fig. 7. The theoretical relation of *ER*2 v.s. *θr*2.


Table 2. Comparisons for the improved method and conventional method; *n*DCG : maximum index modulation strength of DCG; *n*SH : maximum index modulation strength of silverhalide material.

Polarization-Selective Substrate-Mode

type SPM is defined as SPMx.

backward transmissions.

expressed as

and

shape.

SPMs.

**3.1.1 Parallel connection of two SPMs** 

Volume Holograms and Its Application to Optical Circulators 293

90° with respect to +*z* axis (viewing from SWP1 to SWP2), the shifts of transmitted light of the SPM are in *x*-direction, as shown in Figs. 8(c) and 8(d). Accordingly, this operation

As shown in figures 9(a) and 9(b), two SPMxs are connected, *i*.*e*. parallel connection of two

(a) (b) Fig. 9. Structure and operation characteristic of two connected SPMxs for (a) forward and (b)

In Fig. 9(a), when an unpolarized light is incident into the module in +*z* direction, the transmitted unpolarized light is spatially shifted 2*L* in +*x* direction. On the other hand, Fig. 9(b) shows that when an unpolarized light is incident into the module in −*z* direction, the *h*polarized light transmits the module directly and the *v*-polarized light transmits the module with a lateral shift 4*L* in −*x* direction. Consequently, Fig. 10 shows that when an unpolarized light is shuttled between the two sides of the module, the *h*- and *v*-polarized components are separated in two opposite directions gradually in *x*-*z* plane at *y*=0. The corresponding *x* coordinates of the *h*- and *v*-polarized components at two sides of the module can be

2( 1) , 2(1 )

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

where subscripts *h* and *v* denote the *h*- and *v*-polarized components, (2*n*-1) and (2*n*) indicate the port numbers, and *n* is a positive integer. Accordingly, the module can sequentially guide and separate the forward and backward transmitted lights in a z-

(for an odd port) (11)

(for an even port) (12)

(2 1) (2 1)

(2 ) (2 )

*h n v n*

− −

*x n L x n L*

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

*x nL x n L* <sup>=</sup> <sup>−</sup>

*h n v n*
