**3.1 Spatial- and polarization-modules**

As shown in figure 8, the spatial- and polarization- module (SPM) is composed of two spatial walk-off polarizers (SWPs), a 45° Faraday rotator (FR), and a 45° half wave-plate (H). For easy understanding, an orthogonal *x*-*y*-*z* coordinate is introduced, symbol ⊕ represent an unpolarized light, and symbols and represent *h*-polarized (*y*-polarized) and *v*polarized (*x*-polarized) lights, respectively.

Fig. 8. Structure and operation characteristic for (a), (b) SPMy and (c), (d) SPMx.

From figure 8(a), when an unpolarized light is incident into the SWP1 in +*z* direction, the transmitted light is divided into two orthogonally polarized components, *h*- and *v*polarized lights, respectively. These two lights then pass through the 45° FR and 45° H. Therefore, their statuses of polarization (SOPs) are rotated 90° in total. Continuing their journey, they enter the SWP2 and are recombined together with a lateral shift *L* in −*y* direction. On the other hand, Fig. 8(b) shows that when an unpolarized light is incident into the SWP2 in −*z* direction, the transmitted light is similarly divided into two orthogonally polarized components, *v*- and *h*-polarized lights. These two lights then sequentially pass through the same H and FR. Their SOPs are rotated −45° by the H and +45° by the FR. Because Faraday rotator is a nonreciprocal element, their SOPs are rotated 0° in total. Therefore, the *v*-polarized light transmits the SWP1 directly and the *h*-polarized light transmits the SWP1 with a lateral shift 2*L* in +*y* direction. In Figs. 8(a) and 8(b), because the shifts of transmitted light of the SPM are in *y*-direction, this operation type SPM is defined as SPMy. Based on the same principle, when the SPMy is clockwise rotated 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 type SPM is defined as SPMx.

#### **3.1.1 Parallel connection of two SPMs**

292 Holograms – Recording Materials and Applications

As shown in figure 8, the spatial- and polarization- module (SPM) is composed of two spatial walk-off polarizers (SWPs), a 45° Faraday rotator (FR), and a 45° half wave-plate (H). For easy understanding, an orthogonal *x*-*y*-*z* coordinate is introduced, symbol ⊕ represent

(a) (b)

(c) (d)

From figure 8(a), when an unpolarized light is incident into the SWP1 in +*z* direction, the transmitted light is divided into two orthogonally polarized components, *h*- and *v*polarized lights, respectively. These two lights then pass through the 45° FR and 45° H. Therefore, their statuses of polarization (SOPs) are rotated 90° in total. Continuing their journey, they enter the SWP2 and are recombined together with a lateral shift *L* in −*y* direction. On the other hand, Fig. 8(b) shows that when an unpolarized light is incident into the SWP2 in −*z* direction, the transmitted light is similarly divided into two orthogonally polarized components, *v*- and *h*-polarized lights. These two lights then sequentially pass through the same H and FR. Their SOPs are rotated −45° by the H and +45° by the FR. Because Faraday rotator is a nonreciprocal element, their SOPs are rotated 0° in total. Therefore, the *v*-polarized light transmits the SWP1 directly and the *h*-polarized light transmits the SWP1 with a lateral shift 2*L* in +*y* direction. In Figs. 8(a) and 8(b), because the shifts of transmitted light of the SPM are in *y*-direction, this operation type SPM is defined as SPMy. Based on the same principle, when the SPMy is clockwise rotated

Fig. 8. Structure and operation characteristic for (a), (b) SPMy and (c), (d) SPMx.

represent *h*-polarized (*y*-polarized) and *v*-

**3. Holographic-type optical circulators** 

an unpolarized light, and symbols and

polarized (*x*-polarized) lights, respectively.

**3.1 Spatial- and polarization-modules** 

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

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

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 expressed as

$$
\begin{bmatrix} \mathbf{x}\_{h(2n-1)} \\ \mathbf{x}\_{v(2n-1)} \end{bmatrix} = \begin{bmatrix} \mathbf{2}(n-1)L \\ \mathbf{2}(1-n)L \end{bmatrix} \text{ (for an odd port)} \tag{11}
$$

and

$$
\begin{bmatrix} \mathbf{x}\_{h(2n)} \\ \mathbf{x}\_{v(2n)} \end{bmatrix} = \begin{bmatrix} 2nL \\ 2(2-n)L \end{bmatrix}' \text{ (for an even port)}\tag{12}
$$

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 zshape.

Polarization-Selective Substrate-Mode

forward and (b) backward transmissions.

SPMy and SPMx.

Volume Holograms and Its Application to Optical Circulators 295

(a) (b)

Fig. 11. Structure and operation characteristic of the connected SPMy and SPMx for (a)

Fig. 12. Operation characteristic for an unpolarized light shuttled between the connected

Shown in Fig. 13 is the design of proposed holographic-type 4-port polarizationindependent optical circulator. In the design, based on a previously described SPM, two polarization-selective substrate-mode volume holograms are applied to implement the function of spatial walk-off polarizers. They are consequently termed as holographic spatial walk-off polarizers (HSWPs). The two identical HSWPs face the opposite directions as shown in the figure. Besides the SPM, this optical circulator also consists of four reflection prisms (RPs) and six polarization-beam splitters (PBSs). If an input beam is normally incident on HSWP1 from Port 1, as shown in Fig. 13(a), then the *s*- polarized component passes through HSWP1 directly and the *p*- polarized component also passes through HSWP1

**3.2 Four-port polarization-independent optical circulator** 

Fig. 10. Operation characteristic for an unpolarized light shuttled between the two sides of two connected SPMxs.

#### **3.1.2 Orthogonal connection of two SPMs**

Similarly, as shown in figures 11(a) and 11(b), a SPMy and a SPMx are sequentially connected, *i*.*e*. orthogonal connection of two SPMs. In Fig. 11(a), when an unpolarized light is incident into the module in +*z* direction, the transmitted unpolarized light is spatially shifted *L* in +*x* and −*y* directions, respectively. On the other hand, Fig. 11(b) shows that when an unpolarized light is incident into the module in −*z* direction, the *h*-polarized light transmits the module with a lateral shift 2*L* in +*y* direction and the *v*-polarized light transmits the module with a lateral shift 2*L* in −*x* direction. Consequently, Fig. 12 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 along two slanted lines *y*=*x* and *y*=*x*-2, respectively. The corresponding (*x*, *y*) coordinates of the *h*- and *v*polarized components at two sides of the module can be expressed as

$$
\begin{bmatrix}
\mathbf{x}\_{h(2n-1)} & \mathbf{y}\_{h(2n-1)} \\
\mathbf{x}\_{v(2n-1)} & \mathbf{y}\_{v(2n-1)}
\end{bmatrix} = \begin{bmatrix}
(n-1)L & (n-1)L \\
(1-n)L & (1-n)L
\end{bmatrix}' \text{ (for an odd port)}\tag{13}
$$

and

$$
\begin{bmatrix}
\mathbf{x}\_{h(2n)} & \mathbf{y}\_{h(2n)} \\
\mathbf{x}\_{v(2n)} & \mathbf{y}\_{v(2n)}
\end{bmatrix} = \begin{bmatrix}
n\boldsymbol{L} & (n-2)\boldsymbol{L} \\
(2-n)\boldsymbol{L} & -n\boldsymbol{L}
\end{bmatrix}. \text{ (for an even port)}\tag{14}
$$

Accordingly, the module can sequentially guide and separate the forward and backward transmitted lights in another z-shape.

Fig. 10. Operation characteristic for an unpolarized light shuttled between the two sides of

Similarly, as shown in figures 11(a) and 11(b), a SPMy and a SPMx are sequentially connected, *i*.*e*. orthogonal connection of two SPMs. In Fig. 11(a), when an unpolarized light is incident into the module in +*z* direction, the transmitted unpolarized light is spatially shifted *L* in +*x* and −*y* directions, respectively. On the other hand, Fig. 11(b) shows that when an unpolarized light is incident into the module in −*z* direction, the *h*-polarized light transmits the module with a lateral shift 2*L* in +*y* direction and the *v*-polarized light transmits the module with a lateral shift 2*L* in −*x* direction. Consequently, Fig. 12 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 along two slanted lines *y*=*x* and *y*=*x*-2, respectively. The corresponding (*x*, *y*) coordinates of the *h*- and *v*-

( 1) ( 1) , (1 ) (1 )

( 2) . (2 )

Accordingly, the module can sequentially guide and separate the forward and backward

(for an odd port) (13)

(for an even port) (14)

polarized components at two sides of the module can be expressed as

*x y n Ln L x y nL nL*

− − <sup>=</sup> − −

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

(2 1) (2 1) (2 1) (2 1)

(2 ) (2 ) (2 ) (2 )

*hn hn vn vn*

transmitted lights in another z-shape.

− − − −

*hn hn vn vn*

two connected SPMxs.

and

**3.1.2 Orthogonal connection of two SPMs** 

Fig. 11. Structure and operation characteristic of the connected SPMy and SPMx for (a) forward and (b) backward transmissions.

Fig. 12. Operation characteristic for an unpolarized light shuttled between the connected SPMy and SPMx.

#### **3.2 Four-port polarization-independent optical circulator**

Shown in Fig. 13 is the design of proposed holographic-type 4-port polarizationindependent optical circulator. In the design, based on a previously described SPM, two polarization-selective substrate-mode volume holograms are applied to implement the function of spatial walk-off polarizers. They are consequently termed as holographic spatial walk-off polarizers (HSWPs). The two identical HSWPs face the opposite directions as shown in the figure. Besides the SPM, this optical circulator also consists of four reflection prisms (RPs) and six polarization-beam splitters (PBSs). If an input beam is normally incident on HSWP1 from Port 1, as shown in Fig. 13(a), then the *s*- polarized component passes through HSWP1 directly and the *p*- polarized component also passes through HSWP1

Polarization-Selective Substrate-Mode

efficiencies, i.e.,

with

η

*p*>90% and

η

η

condition are given as [Kogelnik, 1969]

*<sup>s</sup>* ≅0% and

η

*i*

η

Volume Holograms and Its Application to Optical Circulators 297

different from theoretical values, the transmittances of two orthogonally polarized

In Port 1 2 3 4

1 14.26a 2.09b 47.91c 14.18c

2 11.91c 14.26a 2.02b 47.91c

3 28.24c 14.15c 14.26a 2.09b

4 2.02b 28.24c 44.92c 14.26a

Table 3. Associated parameters (in Decibels) for the prototype 4-port optical circulator with

Since the fabricated HSWPs have no anti-reflection coatings, there is about 4% reflection loss at each boundary. If they are anti-reflection coated, then the reflection losses should be decreased to 0.1%. In addition, if the holographic exposure and the post-processing procedure are controlled more accurately, the HSWPs may have the theoretical diffraction

performance of this 4-port optical circulator can be enhanced greatly for demands of a

wavelength shift with respect to the central wavelength *λ*r, the diffraction efficiencies of a transmission-type phase volume hologram for the *s-* and *p*-polarization states near the Bragg

> 222 2 2 sin ( ) (1 / ) *i*

ν ξ

ξ ν

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

ξ

*K*

design the central wavelength at other ranges for optical communications.

*i*

2 , 8 cos *<sup>f</sup> <sup>d</sup> K d*

*n* λ

<sup>2</sup> 4 ( ) sin . <sup>2</sup> *f d*

λ

π

*r n*

Substituting the experimental conditions *n*1=0.054, *d*=17μm, *λ*r=1300nm, *θr*2=60°, and *n*f2=1.48 (at *λ*r=1300nm) into eq. (15), the theoretical curves of diffraction efficiencies versus wavelengths for the HSWP is shown in Fig. 14. It is obvious that the bandwidth with

π

2

 θ

θ

*<sup>s</sup>*≅0% at 1300nm central wavelength is as large as 20nm. It is also possible to

commercial device. Moreover, if *K* and Δλ are the magnitude of grating vector *K*

wavelength 1300 nm; aReturn losses; bInsertion losses; cIsolations.

Out Port

*<sup>p</sup>* ≅100%. Under these two possible improved conditions, the

( *i sp* = , ), (15)

−Δ <sup>=</sup> (16a)

= ⋅ (16b)

and the

components are slightly different in the routes of Port 2→Port 3 and Port 4→Port 1.

after two diffractions and two total internal reflections. Next, these two orthogonally polarized components pass through FR and H. Their state of polarization (SOP) are rotated a total of 90°, +45° by FR and +45° by H. For easy understanding, a circle with a bisecting line is used to represent the associated SOP of the light after propagating through each component. Symbols and represent the electric field lies in the planes perpendicular (*s*polarization) and parallel (*p*-polarization) to the paper plane respectively, and the symbol ⊕ represents the light beam has both *s*- and *p*-polarized components. The beams finally enter HSWP2 and then recombine together with the similar diffraction and total internal reflection effects in HSWP1 and reach Port 2.

Fig. 13. Structure and operation principles of the 4-port polarization independent optical circulator.

On the other hand, if an input beam is incident normally on HSWP2 from Port 2 as shown in Fig. 13(b), then the *s*-polarized component passes through HSWP2 directly, and the *p*polarized component also passes through HSWP2 after two diffractions and two total internal reflections. These two orthogonally polarized components pass through H and FR. Their SOPs are rotated -45° by H and +45° by FR, a total of 0°. The *s*-polarized component passes through HSWP1 and is reflected by three PBSs and one PR, and enters Port 3. The *p*polarized component is diffracted and total internal reflected similarly in HSWP1 and propagates through one RP, one PBS, and another RP. Finally, it arrives at Port 3 and recombines with the *s*-polarized component. Two other similar operations for the routes of Port 3→Port 4 and Port 4→Port 1 can be done with the introduction of additional RPs and PBSs, as shown in Fig. 13(c) and 13(d) respectively.

If the PBSs are located accurately in the configurations of Fig. 13(b) and 13(d), there will be no optical path difference between *s-* and *p*-polarizations for any route. Hence, this optical circulator can function as a polarization-independent 4-port optical circulator without polarization mode dispersion (PMD).

Listed in Table 3 are parameters for a prototype of 4-port optical circulator which were estimated from the diffraction efficiencies of fabricated HSWPs and the transmittances of FR and H. The diffraction efficiencies of η*s* and η*<sup>p</sup>* are 3% and 90%, respectively; the transmittances of FR and H are 0.95 and 0.97, respectively. Because η*s* and η*<sup>p</sup>* are slightly


different from theoretical values, the transmittances of two orthogonally polarized components are slightly different in the routes of Port 2→Port 3 and Port 4→Port 1.

Table 3. Associated parameters (in Decibels) for the prototype 4-port optical circulator with wavelength 1300 nm; aReturn losses; bInsertion losses; cIsolations.

Since the fabricated HSWPs have no anti-reflection coatings, there is about 4% reflection loss at each boundary. If they are anti-reflection coated, then the reflection losses should be decreased to 0.1%. In addition, if the holographic exposure and the post-processing procedure are controlled more accurately, the HSWPs may have the theoretical diffraction efficiencies, i.e., η*<sup>s</sup>* ≅0% and η*<sup>p</sup>* ≅100%. Under these two possible improved conditions, the performance of this 4-port optical circulator can be enhanced greatly for demands of a commercial device. Moreover, if *K* and Δλ are the magnitude of grating vector *K* and the wavelength shift with respect to the central wavelength *λ*r, the diffraction efficiencies of a transmission-type phase volume hologram for the *s-* and *p*-polarization states near the Bragg condition are given as [Kogelnik, 1969]

$$\eta\_i = \frac{\sin^2(\sqrt{\nu\_i^2 + \xi^2})}{(1 + \xi^2 \;/\; \nu\_i^2)} \quad (i = s, \; p \;). \tag{15}$$

with

296 Holograms – Recording Materials and Applications

after two diffractions and two total internal reflections. Next, these two orthogonally polarized components pass through FR and H. Their state of polarization (SOP) are rotated a total of 90°, +45° by FR and +45° by H. For easy understanding, a circle with a bisecting line is used to represent the associated SOP of the light after propagating through each

polarization) and parallel (*p*-polarization) to the paper plane respectively, and the symbol ⊕ represents the light beam has both *s*- and *p*-polarized components. The beams finally enter HSWP2 and then recombine together with the similar diffraction and total internal reflection

(a) (b)

(c) (d) Fig. 13. Structure and operation principles of the 4-port polarization independent optical

On the other hand, if an input beam is incident normally on HSWP2 from Port 2 as shown in Fig. 13(b), then the *s*-polarized component passes through HSWP2 directly, and the *p*polarized component also passes through HSWP2 after two diffractions and two total internal reflections. These two orthogonally polarized components pass through H and FR. Their SOPs are rotated -45° by H and +45° by FR, a total of 0°. The *s*-polarized component passes through HSWP1 and is reflected by three PBSs and one PR, and enters Port 3. The *p*polarized component is diffracted and total internal reflected similarly in HSWP1 and propagates through one RP, one PBS, and another RP. Finally, it arrives at Port 3 and recombines with the *s*-polarized component. Two other similar operations for the routes of Port 3→Port 4 and Port 4→Port 1 can be done with the introduction of additional RPs and

If the PBSs are located accurately in the configurations of Fig. 13(b) and 13(d), there will be no optical path difference between *s-* and *p*-polarizations for any route. Hence, this optical circulator can function as a polarization-independent 4-port optical circulator without

Listed in Table 3 are parameters for a prototype of 4-port optical circulator which were estimated from the diffraction efficiencies of fabricated HSWPs and the transmittances of FR

η

*<sup>p</sup>* are 3% and 90%, respectively; the

*<sup>p</sup>* are slightly

η*s* and η

η*s* and

transmittances of FR and H are 0.95 and 0.97, respectively. Because

and represent the electric field lies in the planes perpendicular (*s*-

component. Symbols

circulator.

PBSs, as shown in Fig. 13(c) and 13(d) respectively.

polarization mode dispersion (PMD).

and H. The diffraction efficiencies of

effects in HSWP1 and reach Port 2.

$$\xi = \frac{-\Delta\lambda K^2 d}{8\pi n\_{f2}\cos\theta\_d},\tag{16a}$$

$$K = \left(\frac{4\pi n\_{f2}}{\mathcal{A}\_r}\right) \cdot \sin\frac{\theta\_d}{2}.\tag{16b}$$

Substituting the experimental conditions *n*1=0.054, *d*=17μm, *λ*r=1300nm, *θr*2=60°, and *n*f2=1.48 (at *λ*r=1300nm) into eq. (15), the theoretical curves of diffraction efficiencies versus wavelengths for the HSWP is shown in Fig. 14. It is obvious that the bandwidth with η*p*>90% and η*<sup>s</sup>*≅0% at 1300nm central wavelength is as large as 20nm. It is also possible to design the central wavelength at other ranges for optical communications.

Polarization-Selective Substrate-Mode

circulator.

which can be expressed as

Volume Holograms and Its Application to Optical Circulators 299

(a) (b)

(c) (d)

However, in the design of Fig. 15, the optical path of the *p*- component is larger than that of the *s*-component. This optical path difference might cause polarization mode dispersion (PMD) to blur the transmission signal. Therefore, in order to solve the PMD problem, the original optical guiding paths in Fig. 15 must be changed. As shown in Fig. 16, two different guiding modules can be introduced for the odd and even ports, respectively, which are composed of PBSs and RPs. The designs of these two guiding modules with specifications (*Length*×*Width*) of (4*n*-3)*L*×0.31(*n*-1)*L* and (4*n*-4)*L*×0.31(*n*-1)*L* for an odd and an even port are shown in Fig. 17(a) and (b), respectively. These guiding modules are located at (*x*Mj, *z*Mj)

(2 1) (2 1) [*x z Mn Mn* − − ] [2(1 ) ( 2 4) ], = − −− *nL n L* (for an odd port) (19)

(2 ) (2 ) [*x z nL n L Mn Mn* ] [2 (2 4) ], = + (for an even port) (20)

Fig. 15. Structure and operation principles of the proposed multi-port optical quasi-

Fig. 14. Calculated diffraction efficiencies of the HSWP versus wavelength at 1300 nm central wavelength.

#### **3.3 Multi-port polarization-independent optical quasi-circulator**

Based on two connected SPMxs, it is obvious that if some polarization-beam splitters (PBSs), and reflection prisms (RPs) are introduced appropriately at the corresponding positions of the *h*- and *v*-components, a multi-port optical quasi-circulator can be obtained. Shown in Fig. 15 is an optical quasi-circulator with 2n-ports consisting of a pair of HSPMxs, PBSs, and RPs. According to equations (11) and (12), the introduced PBSs and RPs at the *j*-th port are located at (*x*PBSj, *z*PBSj) and (*x*RPj, *z*RPj), which can be expressed as

$$
\begin{bmatrix} x\_{PBS(2n-1)} & z\_{PBS(2n-1)} \\ x\_{RP(2n-1)} & z\_{RP(2n-1)} \end{bmatrix} = \begin{bmatrix} 2(n-1)L & (-2n-4)L \\ 2(1-n)L & (-2n-4)L \end{bmatrix} \tag{17}
\\
\text{(for an odd port)}\tag{17}
$$

$$
\begin{bmatrix} \varkappa\_{PBS(2n)} & \varkappa\_{PBS(2n)}\\ \varkappa\_{RP(2n)} & \varkappa\_{RP(2n)} \end{bmatrix} = \begin{bmatrix} 2nL & (2n+4)L\\ 2(2-n)L & (2n+4)L \end{bmatrix}, \text{ (for an even part)}\tag{18}
$$

where *n* is a positive integer. Figure 15(a), (b), (c), and (d) show the routes of port 1→port 2, port 2→port 3, port 3→port 4, and port (2n-1)→port 2n, respectively. In these figures, symbols ◪ and ◢ represent a PBS and a RP. Other propagation routes can be obtained based on the similar principle.

**P**

**12501260127012801290130013101320133013401350**

**Wavelength** λ **(nm)**

2( 1) ( 2 4) , 2(1 ) ( 2 4)

2 (2 4) , 2(2 ) (2 4)

where *n* is a positive integer. Figure 15(a), (b), (c), and (d) show the routes of port 1→port 2, port 2→port 3, port 3→port 4, and port (2n-1)→port 2n, respectively. In these figures, symbols ◪ and ◢ represent a PBS and a RP. Other propagation routes can be obtained based

Fig. 14. Calculated diffraction efficiencies of the HSWP versus wavelength at 1300 nm

Based on two connected SPMxs, it is obvious that if some polarization-beam splitters (PBSs), and reflection prisms (RPs) are introduced appropriately at the corresponding positions of the *h*- and *v*-components, a multi-port optical quasi-circulator can be obtained. Shown in Fig. 15 is an optical quasi-circulator with 2n-ports consisting of a pair of HSPMxs, PBSs, and RPs. According to equations (11) and (12), the introduced PBSs and RPs at the *j*-th port are located at (*x*PBSj, *z*PBSj) and (*x*RPj, *z*RPj), which can be expressed

**3.3 Multi-port polarization-independent optical quasi-circulator** 

*x z nL nL x z nL n L*

− −− <sup>=</sup> − −−

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

(2 1) (2 1) (2 1) (2 1)

− − − −

(2 ) (2 ) (2 ) (2 )

*PBS n PBS n RP n RP n*

*PBS n PBS n RP n RP n*

**S**

(for an odd port) (17)

(for an even port) (18)

**0**

central wavelength.

on the similar principle.

as

**10**

**20**

**30**

**40**

**50**

**Diffraction efficiency (%)**

**60**

**70**

**80**

**90**

**100**

Fig. 15. Structure and operation principles of the proposed multi-port optical quasicirculator.

However, in the design of Fig. 15, the optical path of the *p*- component is larger than that of the *s*-component. This optical path difference might cause polarization mode dispersion (PMD) to blur the transmission signal. Therefore, in order to solve the PMD problem, the original optical guiding paths in Fig. 15 must be changed. As shown in Fig. 16, two different guiding modules can be introduced for the odd and even ports, respectively, which are composed of PBSs and RPs. The designs of these two guiding modules with specifications (*Length*×*Width*) of (4*n*-3)*L*×0.31(*n*-1)*L* and (4*n*-4)*L*×0.31(*n*-1)*L* for an odd and an even port are shown in Fig. 17(a) and (b), respectively. These guiding modules are located at (*x*Mj, *z*Mj) which can be expressed as

$$\begin{bmatrix} \mathbf{x}\_{M(2n-1)} & z\_{M(2n-1)} \end{bmatrix} = \begin{bmatrix} 2(1-n)L & (-2n-4)L \end{bmatrix} \text{ (for an odd port)}\tag{19}$$

$$\begin{bmatrix} \mathbf{x}\_{M(2n)} & z\_{M(2n)} \end{bmatrix} = \begin{bmatrix} 2n\mathbf{L} & (2n+4)\mathbf{L} \end{bmatrix} \text{ (for an even part)} \tag{20}$$

Polarization-Selective Substrate-Mode

Volume Holograms and Its Application to Optical Circulators 301

(a) (b)

Use the fabricated HSWPs mentioned in Section 3.2, a prototype of 6-port polarizationindependent optical quasi-circulator for 1300nm can be assembled. In addition to a pair of HSPMs, it needs another eight polarization-beam splitters and ten reflection prisms to complete the function of this 6-port optical quasi-circulator. The characteristic parameters of

In Port 1 2 3 4 5 6 1 14.26a 4.18b >25.36c >25.36c >25.36c >25.36c 2 >26.92c 14.26a 3.90b >26.92c >26.92c >26.92c 3 >25.36c >25.36c 14.26a 4.18b >25.36c >25.36c 4 >26.92c >26.92c >26.92c 14.26a 3.90b >26.92c 5 >25.36c >25.36c >25.36c >25.36c 14.26a 4.18b Table 4. Associated parameters (in Decibels) for the prototype 6-port optical quasi-circulator

In order to solve the PMD problem, two different guiding modules composed of PBSs and RPs can be appropriately introduced for the odd and even ports, respectively. However, if more compact modules are desired, these guiding devices should become smaller simultaneously. The result will increase the difficulty of device assembling. Expediently, we can increase the beam splitting distance L (*L*=2*t*tan*θd*) by increasing the thickness of the substrate to reduce the assembling difficulty. Another reliable method is to operate this

Based on two connected SPMy and SPMx, it is obvious that if reflection prisms (RPs) and polarization beam-splitters (PBSs) are introduced appropriately to guide the light beams in and out of the module, an alternative multi-port optical quasi-circulator can also be obtained. Only one RP should be added at port 1 and port 2, separately. For other ports, each port needs two RPs and one PBS. According to eqs. (13) and (14), the introduced RPs and PBS at the *j*-th

Out Port

Fig. 17. PBSs and RPs guiding modules for (a) the odd ports; (b) the even ports.

this prototype device are estimated and listed in Table 4.

device beginning with a high number port.

with wavelength 1300 nm; aReturn losses; bInsertion losses; cIsolations.

**3.4 Improved multi-port polarization-independent optical quasi-circulator** 

port are located at (*x*RP1j, *y*RP1j), (*x*RP2j, *y*RP2j) and (*x*PBSj, *y*PBSj), which can be expressed as

where *n* is a positive integer larger than 1. The coordinate in equation (19) corresponds to the center of the RP (in red color) in the odd-port guiding module; the coordinate in equation (20) corresponds to the center of the PBS (in green color) in the even-port guiding module. When the guiding modules are appropriately introduced, the optical path differences between the *h*- and *v*-components can be reduced to zero. Therefore, the PMD problem can be solved. Fig. 16(a), (b), (c), and (d) show the routes of port 1→port 2, port 2→port 3, port 3→port 4, and port (2n-1)→port 2n, respectively. Other propagation routes can be obtained based on the similar principle.

Fig. 16. Structure and operation principles of the proposed multi-port optical quasicirculator without polarization mode dispersion.

where *n* is a positive integer larger than 1. The coordinate in equation (19) corresponds to the center of the RP (in red color) in the odd-port guiding module; the coordinate in equation (20) corresponds to the center of the PBS (in green color) in the even-port guiding module. When the guiding modules are appropriately introduced, the optical path differences between the *h*- and *v*-components can be reduced to zero. Therefore, the PMD problem can be solved. Fig. 16(a), (b), (c), and (d) show the routes of port 1→port 2, port 2→port 3, port 3→port 4, and port (2n-1)→port 2n, respectively. Other propagation routes

(a) (b)

(c) (d)

Fig. 16. Structure and operation principles of the proposed multi-port optical quasi-

circulator without polarization mode dispersion.

can be obtained based on the similar principle.

Fig. 17. PBSs and RPs guiding modules for (a) the odd ports; (b) the even ports.

Use the fabricated HSWPs mentioned in Section 3.2, a prototype of 6-port polarizationindependent optical quasi-circulator for 1300nm can be assembled. In addition to a pair of HSPMs, it needs another eight polarization-beam splitters and ten reflection prisms to complete the function of this 6-port optical quasi-circulator. The characteristic parameters of this prototype device are estimated and listed in Table 4.


Table 4. Associated parameters (in Decibels) for the prototype 6-port optical quasi-circulator with wavelength 1300 nm; aReturn losses; bInsertion losses; cIsolations.

In order to solve the PMD problem, two different guiding modules composed of PBSs and RPs can be appropriately introduced for the odd and even ports, respectively. However, if more compact modules are desired, these guiding devices should become smaller simultaneously. The result will increase the difficulty of device assembling. Expediently, we can increase the beam splitting distance L (*L*=2*t*tan*θd*) by increasing the thickness of the substrate to reduce the assembling difficulty. Another reliable method is to operate this device beginning with a high number port.

#### **3.4 Improved multi-port polarization-independent optical quasi-circulator**

Based on two connected SPMy and SPMx, it is obvious that if reflection prisms (RPs) and polarization beam-splitters (PBSs) are introduced appropriately to guide the light beams in and out of the module, an alternative multi-port optical quasi-circulator can also be obtained. Only one RP should be added at port 1 and port 2, separately. For other ports, each port needs two RPs and one PBS. According to eqs. (13) and (14), the introduced RPs and PBS at the *j*-th port are located at (*x*RP1j, *y*RP1j), (*x*RP2j, *y*RP2j) and (*x*PBSj, *y*PBSj), which can be expressed as

Polarization-Selective Substrate-Mode

high potential in optical communications.

National Chiao Tung University, Taiwan, ROC.

**4. Conclusion** 

**5. Acknowledgment** 

**6. References** 

Volume Holograms and Its Application to Optical Circulators 303

In Port 1 2 3 4 5 1 14.26a 3.26b >20.46c >37.65c >54.85c 2 >54.85c 14.26a 3.26b >20.46c >37.65c 3 >37.65c >54.85c 14.26a 3.26b >20.46c 4 >54.85c >54.85c >54.85c 14.26a 3.26b Table 5. Associated parameters (in Decibels) for the prototype 5-port optical quasi-circulator

Compared with the design in Section 3.3, only the second HSPM is rotated 90° clockwise in this improved device. So this design still has all the advantages of the previous one. In addition, because two orthogonally polarized components have the same numbers of diffractions and total internal reflections in this design, their optical path lengths are all the same. Consequently, only fewer PBSs and RPs are required to guide the light beams in and out of the module. Hence the optical configuration is simpler and it is easier to be assembled. Moreover, the optical paths and the optical elements are not restricted in the same plane as the previous design in Section 3.3. So the light leakages producing by the unideal diffraction efficiencies of the HSWPs can not enter any port. The crosstalk between any ports can be avoided. Hence this design has higher isolations. In this device, polarization-selective substrate-mode volume holograms are used to replace conventional crystal spatial walk-off polarizers. Accordingly, compared with conventional quasicirculators, this design has merits of compactness, easy fabrication, and low-cost. So it has

In this chapter, polarization-selective substrate-mode volume holograms were introduced which are applied in several novel designs of optical circulator. Alternative design method of polarization-selective substrate-mode volume holograms was also introduced for overcoming the finite refractive index limit in practical holographic recording materials. The described optical circulators have advantages of polarization-independent, compactness, high isolation, low polarization mode dispersion, easy fabrication, and low cost. In addition, the port number of the proposed multi-port device can be expanded easily. High application potential of these devices in optical communications is expected. Their commercialization

These researches were supported partially by grants from the National Science Council, Taiwan, ROC, under contracts NSC 94-2215-E-035-003, NSC 94-2215-E-035-009, NSC 96- 2221-E-035-052, NSC 97-2221-E-035-016-MY3, and the Lee & MTI Center for Networking at

Ramaswami, R.; Sivarajan, K.; & Sasaki, G. (2009). Optical Networks: A Practical Perspective. 3rd ed. Morgan Kaufmann, ISBN 9780123740922, San Francisco, USA

finally relies on available high-performance holographic recording materials.

with wavelength 1300 nm; aReturn losses; bInsertion losses; cIsolations.

Out Port

$$\begin{bmatrix} \begin{array}{cccc} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \end{array} \end{array} \end{array} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \end{array} \end{array} \end{array} \end{array} \end{ \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \end{array} \end{array} \end{array} \end{bmatrix} \end{bmatrix} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} (n-1)L \end{array} \end{array} \begin{array}{c} \begin{array}{c} (n-1)L \end{array} \end{array} \end{bmatrix} \end{bmatrix} \end{bmatrix} \end{cases} = \begin{bmatrix} (n-1)L & (1-n)L \\ (1-n)L & (n-1)L \end{bmatrix} \end{bmatrix} \end{cases} \tag{21}$$
 
$$\begin{bmatrix} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} (n-1)L \end{array} \end{array} \end{array} \end{bmatrix} \begin{array}{c} \begin{array}{c} (n-1)L \end{array} \end{array} \end{bmatrix} \end{bmatrix} \end{)} \end{)} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} (n-1)L \end{array} \end{array} \end{bmatrix} \end{bmatrix} \end{)} \tag{21}$$

$$\begin{bmatrix} \mathbf{x}\_{RP1(2n)} & \mathbf{y}\_{RP1(2n)} \\ \mathbf{x}\_{RP2(2n)} & \mathbf{y}\_{RP2(2n)} \\ \mathbf{x}\_{PRS(2n)} & \mathbf{y}\_{PRS(2n)} \end{bmatrix}\_{z=L} = \begin{bmatrix} nL & (2-n)L \\ (2-n)L & nL \\ nL & nL \end{bmatrix}\_{z=L} \tag{22}$$

These equations are still valid for port 1 and port 2 to determine the position of its associated RP. Shown in Fig. 18 is a 5-port polarization-independent optical quasi-circulator consisting of a pair of HSPMx and HSPMy, 3 PBSs, and 8 RPs. Figures 18(a) and 18(b) show the routes of port 1→port 2 and port 4→port 5, respectively. Based on the same principle, other propagation and expanded routes can also be obtained.

Fig. 18. Structure and operation principles of the proposed 5-port polarization-independent optical quasi-circulator.

A 5-port polarization-independent optical quasi-circulator for 1300nm can be assembled with the fabricated HSWPs. In addition to a pair of orthogonal HSPMs, it needs another three PBSs and eight RPs to complete the function of this device. Its associated losses and isolation values are estimated and listed in Table 5. The isolation values are in the range from 20 to 54dB. The return losses and the insertion losses are about 14dB and about 3dB, respectively. Return losses mainly come from the interface reflections that influence the isolation values directly. If the applied HSWPs are anti-reflection coated and are fabricated under accurate fabrication processes, the return losses could be over 50 dB and the diffraction efficiencies may reach theoretical values, i.e., *η*h=0% and *η*v=100%. Under these two improved conditions, the performance of this 5-port optical quasi-circulator can be enhanced greatly with isolation values larger than 51dB and insertion losses smaller than 0.9dB.


Table 5. Associated parameters (in Decibels) for the prototype 5-port optical quasi-circulator with wavelength 1300 nm; aReturn losses; bInsertion losses; cIsolations.

Compared with the design in Section 3.3, only the second HSPM is rotated 90° clockwise in this improved device. So this design still has all the advantages of the previous one. In addition, because two orthogonally polarized components have the same numbers of diffractions and total internal reflections in this design, their optical path lengths are all the same. Consequently, only fewer PBSs and RPs are required to guide the light beams in and out of the module. Hence the optical configuration is simpler and it is easier to be assembled. Moreover, the optical paths and the optical elements are not restricted in the same plane as the previous design in Section 3.3. So the light leakages producing by the unideal diffraction efficiencies of the HSWPs can not enter any port. The crosstalk between any ports can be avoided. Hence this design has higher isolations. In this device, polarization-selective substrate-mode volume holograms are used to replace conventional crystal spatial walk-off polarizers. Accordingly, compared with conventional quasicirculators, this design has merits of compactness, easy fabrication, and low-cost. So it has high potential in optical communications.
