**3.1 Introduction**

The two main criteria are compactness and compatibility with planar PC circuit. PC structure was emerged as the best candidate to meet both requirements. PC slab waveguide is highly birefringent and compact; moreover, the fabrication process of the asymmetric loaded PC slab waveguide and integrated planar optical circuits are compatible.

Fig. 3(a) shows the schematic of the proposed polarization rotator. It consists of a single defect line PC slab waveguide. The geometrical asymmetry that is required to couple two orthogonal polarizations to each other was introduced to the upper layer of the defect line. The upper layer is made of the same material as the slab layer etched asymmetrically with

Silicon based polarization rotators are more attractive in the sense that fabrication process is more compatible with the complementary metal-oxide semiconductor (CMOS) technology. Chen and et al. introduced silicon slanted rib waveguide for polarization rotation (Chen et al., 2003). Deng and et al. implemented slanted wall in Si by wet etching of Silicon <100>; thus, the side wall angle (52o) was not a flexible parameter. The total length of the fabricated device was more than 3 mm which was considered bulky (Deng, 2005). Moreover, the fabrication process of slanted-wall ridge waveguide is not compatible with planar optics

Recently, Wang and Dai proposed Si nanowire based polarization rotator with asymmetrical cross section, depicted in Fig. 2. The side wall is vertical; thus, it could be realized utilizing dry etching, reactive ion etching (RIE) (Wang & Dai, 2008). They were able to design asymmetric si nanowire device as small as 10 µm. Single mode guiding is required to avoid multimode interface that leads to lower polarization conversion efficiency. However, single mode silicon nanowires are so small that makes the fabrication very difficult and challenging. The fabrication tolerance is very small; thus, the proposed structure is not a robust device in the sense that small fabrication error could diminish the performance of the device. Moreover, to achieve a compact polarization rotator, the height of the loading (h) is 240 nm that is almost half of the thickness of the nanowire (H=500 nm) leading to a huge

Fig. 2. The sketch of the cross section of asymmetric Si nanowire for polarization rotation

Having studied the existing polarization rotators, the major issues are either size or complexity of the structure. To tackle these issues, a PC based polarization rotator is

The two main criteria are compactness and compatibility with planar PC circuit. PC structure was emerged as the best candidate to meet both requirements. PC slab waveguide is highly birefringent and compact; moreover, the fabrication process of the asymmetric

Fig. 3(a) shows the schematic of the proposed polarization rotator. It consists of a single defect line PC slab waveguide. The geometrical asymmetry that is required to couple two orthogonal polarizations to each other was introduced to the upper layer of the defect line. The upper layer is made of the same material as the slab layer etched asymmetrically with

loaded PC slab waveguide and integrated planar optical circuits are compatible.

circuit.

coupling loss.

application (Wang & Dai, 2008).

introduced in the following section.

**3. PC based polarization rotator** 

**3.1 Introduction** 

respect to z-axis (propagation direction). Power conversion reversal happens at half beat lengths along the line. In order to avoid power conversion reversal and synchronize the coupling, the upper layer that is half beat length long is alternated on either side of the zaxis with the given period. The proposed structure is described as periodic asymmetric loaded PC slab waveguide. Because of the large birefringence of PC structures, the PC based polarization rotator is expected to be very compact as opposed to periodic asymmetric loaded rib waveguide. Compact structure requires smaller number of loading layers; hence the radiation loss at the junctions between different sections will be reduced.

<sup>(</sup>b)

Fig. 3. The sketch of (a) periodic asymmetric loaded triangular PC slab waveguide (b) asymmetric loaded PC slab waveguide.

Due to the compactness of the structure, a rigorous numerical method, 3D-FDTD can be employed for analysis and simulation. However, for preliminary and quick design an analytical method that provides the approximate values of the structural parameters is preferred. Coupled-mode theory is a robust and well-known method for the analysis of perturbed waveguide structures. Thus, the coupled-mode theory based on semi-vectorial modes was developed for PC structures (Bayat et al., 2009). However, the frequency band of

Photonic Crystal for Polarization Rotation 301

Maximize BG of PC by opt. w/a using PWEM

> **Hybrid mode analysis**

**3D-FDTD & SFT**

**For arbitrary hole shape PC**

(6.a)

(6.b)

**1. Optimization of Thickness for Wideband operation**

**3D-FDTD**

Fig. 4. The flow chart of the design methodology of PC slab waveguide based polarization

PC slab waveguide and top loaded layer are represented by a, w, t and tup, respectively. The top cladding layer is asymmetric with respect to the z-axis (propagation direction) and alternates periodically throughout the propagation direction to synchronize the coupling between the two polarizations. The vector wave equation for the transverse electric field (x-y and z are the transverse and propagation directions, respectively) is given by

2 2 2

2 2 2

where, n is the refractive index distribution of the waveguide and <sup>2</sup> *t* is the transverse

2 2

*<sup>t</sup>* 2 2 *x y* 

The vector properties are manifested on the right hand side of equ. (6.a), equ. (6.b); which indicates that the two orthogonal polarizations may be coupled to each other as a result of

Two approaches including normal mode analysis of the asymmetric loaded PC slab waveguide and coupled mode theory based on semi-vectorial modes were employed to design the polarization rotator structure. Both methods are explained in the following

2 2 2 *<sup>y</sup>* 1 1 *ty y x y <sup>E</sup> n n E nkE E <sup>E</sup> z y n n xy y* 

2

2 2 2 1 1 *<sup>x</sup> tx x x <sup>y</sup> <sup>E</sup> n n E nkE E <sup>E</sup> z x xx n n y* 

2 22

2 22

**3. Design verification 2'.Design using 3D-FDTD**

Max overlap for fast & slow modes

**Operating frequencyband**

**PWEM analysis**

**t**

Yes

No

**Coupled-mode Theory**

**2. Preliminary design**

**Semi-vectorial modal analysis using BPM**

rotator.

(Haung et al., 1992):

differential operator defined as:

geometrical asymmetry.

subsections.

the modes of the asymmetric loaded PC slab waveguide must be determined prior to the coupled mode analysis. Plane wave expansion method (PWEM) was employed for modal analysis of the asymmetric loaded PC slab waveguide. For coupled mode analysis, the semivectorial modes of the asymmetric loaded PC slab waveguide, Fig. 3 (b), were calculated using semi-vectorial beam propagation method (BPM) of RSOFT, version 8.1. Coupled mode theory was employed to calculate the cross coupling between x-polarized and ypolarized waves. To simplify the problem for analytical calculations, instead of circular-hole PC pattern, square-hole PC pattern was employed. The coupled mode theory is an approximate method that provides an estimation of the structural parameters. The combination of the coupled mode theory and PWEM provides the frequency band over which low loss high efficiency polarization rotation is expected to be achieved.

Although, coupled mode theory presents a quick and efficient design methodology; it poses a tedious and error-prone process of discretization along the propagation direction for more complicated geometries such as circular hole PC slab. Thus, another design methodology based on vector-propagation characteristics of normal modes of asymmetric loaded PC slab waveguide was implemented. It can be employed for any arbitrary shape PC slab structure. Normal mode analysis of asymmetric loaded PC slab waveguide provides with fast and slow normal modes of the structure; so that, the half-beat length can be easily calculated (Mrozowki, 1997). The vector-propagation characteristics of normal modes of the asymmetric loaded PC slab waveguide were calculated using 3D-FDTD analysis combined with spatial fourier transform (SFT) of the electric field along the propagation direction. Both coupled-mode analysis and normal mode analysis led to the same results for square shape PC slab waveguide based polarization rotator. To verify the design parameters obtained using coupled mode theory and normal mode analysis, polarization rotator structure was simulated using 3D-FDTD.

Fig. 4 presents the flow chart of the design. It shows the design consists of three main steps. In the first step, the operational frequency band is determined using PWEM analysis. In this step, the thickness of the asymmetric loaded PC slab waveguide is optimized to provide maximum frequency band over which highly efficient polarization conversion is expected to take place. In the second step, coupled mode theory is employed for preliminary design of the polarization rotator. The outputs of this step are the length of top loaded layers (half-beat length) and total number of top loaded layers. Finally, to verify the design parameters obtained using coupled mode theory the 3D-FDTD simulation is performed. Coupled mode theory was developed for square hole geometries. For circular hole PC structure, another design methodology based on 3D-FDTD was developed, shown by 2' in the figure. As it was explained before, 3D-FDTD analysis of asymmetric loaded PC slab waveguide combined with SFT was employed to obtain the vector-propagation characteristics of the normal modes (slow and fast modes) of the structure. The accuracy of this method can be examined by 3D-FDTD simulation of the polarization rotator structure. In Sec. 3.2, the design methodology has been elaborated. The design and simulation results are presented in Sec. 3.3.

#### **3.2 Theory**

The schematic of the asymmetric square-hole PC slab polarization rotator is shown in Fig. 3(a). In this structure, the unit cell, the width of the square holes, the thicknesses of silicon

the modes of the asymmetric loaded PC slab waveguide must be determined prior to the coupled mode analysis. Plane wave expansion method (PWEM) was employed for modal analysis of the asymmetric loaded PC slab waveguide. For coupled mode analysis, the semivectorial modes of the asymmetric loaded PC slab waveguide, Fig. 3 (b), were calculated using semi-vectorial beam propagation method (BPM) of RSOFT, version 8.1. Coupled mode theory was employed to calculate the cross coupling between x-polarized and ypolarized waves. To simplify the problem for analytical calculations, instead of circular-hole PC pattern, square-hole PC pattern was employed. The coupled mode theory is an approximate method that provides an estimation of the structural parameters. The combination of the coupled mode theory and PWEM provides the frequency band over

Although, coupled mode theory presents a quick and efficient design methodology; it poses a tedious and error-prone process of discretization along the propagation direction for more complicated geometries such as circular hole PC slab. Thus, another design methodology based on vector-propagation characteristics of normal modes of asymmetric loaded PC slab waveguide was implemented. It can be employed for any arbitrary shape PC slab structure. Normal mode analysis of asymmetric loaded PC slab waveguide provides with fast and slow normal modes of the structure; so that, the half-beat length can be easily calculated (Mrozowki, 1997). The vector-propagation characteristics of normal modes of the asymmetric loaded PC slab waveguide were calculated using 3D-FDTD analysis combined with spatial fourier transform (SFT) of the electric field along the propagation direction. Both coupled-mode analysis and normal mode analysis led to the same results for square shape PC slab waveguide based polarization rotator. To verify the design parameters obtained using coupled mode theory and normal mode analysis, polarization rotator

Fig. 4 presents the flow chart of the design. It shows the design consists of three main steps. In the first step, the operational frequency band is determined using PWEM analysis. In this step, the thickness of the asymmetric loaded PC slab waveguide is optimized to provide maximum frequency band over which highly efficient polarization conversion is expected to take place. In the second step, coupled mode theory is employed for preliminary design of the polarization rotator. The outputs of this step are the length of top loaded layers (half-beat length) and total number of top loaded layers. Finally, to verify the design parameters obtained using coupled mode theory the 3D-FDTD simulation is performed. Coupled mode theory was developed for square hole geometries. For circular hole PC structure, another design methodology based on 3D-FDTD was developed, shown by 2' in the figure. As it was explained before, 3D-FDTD analysis of asymmetric loaded PC slab waveguide combined with SFT was employed to obtain the vector-propagation characteristics of the normal modes (slow and fast modes) of the structure. The accuracy of this method can be examined by 3D-FDTD simulation of the polarization rotator structure. In Sec. 3.2, the design methodology has been elaborated. The design and simulation results

The schematic of the asymmetric square-hole PC slab polarization rotator is shown in Fig. 3(a). In this structure, the unit cell, the width of the square holes, the thicknesses of silicon

which low loss high efficiency polarization rotation is expected to be achieved.

structure was simulated using 3D-FDTD.

are presented in Sec. 3.3.

**3.2 Theory** 

Fig. 4. The flow chart of the design methodology of PC slab waveguide based polarization rotator.

PC slab waveguide and top loaded layer are represented by a, w, t and tup, respectively. The top cladding layer is asymmetric with respect to the z-axis (propagation direction) and alternates periodically throughout the propagation direction to synchronize the coupling between the two polarizations. The vector wave equation for the transverse electric field (x-y and z are the transverse and propagation directions, respectively) is given by (Haung et al., 1992):

$$\frac{\partial^2 E\_\mathbf{x}}{\partial z^2} + \nabla\_\mathbf{l}^2 E\_\mathbf{x} + n^2 k^2 E\_\mathbf{x} = -\frac{\partial}{\partial \mathbf{x}} \left( E\_\mathbf{x} \frac{1}{n^2} \frac{\partial n^2}{\partial \mathbf{x}} \right) - \frac{\partial}{\partial \mathbf{x}} \left( E\_y \frac{1}{n^2} \frac{\partial n^2}{\partial y} \right) \tag{6.a.}$$

$$\frac{\partial^2 E\_y}{\partial x^2} + \nabla\_t^2 E\_y + n^2 k^2 E\_y = -\frac{\partial}{\partial y} \left( E\_x \frac{1}{n^2} \frac{\partial n^2}{\partial x} \right) - \frac{\partial}{\partial y} \left( E\_y \frac{1}{n^2} \frac{\partial n^2}{\partial y} \right) \tag{6.b}$$

where, n is the refractive index distribution of the waveguide and <sup>2</sup> *t* is the transverse differential operator defined as:

$$
\nabla\_t^2 = \frac{\partial^2}{\partial \mathbf{x}^2} + \frac{\partial^2}{\partial \mathbf{y}^2}
$$

The vector properties are manifested on the right hand side of equ. (6.a), equ. (6.b); which indicates that the two orthogonal polarizations may be coupled to each other as a result of geometrical asymmetry.

Two approaches including normal mode analysis of the asymmetric loaded PC slab waveguide and coupled mode theory based on semi-vectorial modes were employed to design the polarization rotator structure. Both methods are explained in the following subsections.

Photonic Crystal for Polarization Rotation 303

the polarization dependence of wave propagation has been partially taken into account; thus, the coupling between the two x-polarized and y-polarized waves can be modeled

In a triangular lattice PC structure, cross-section varies along the propagation direction within one unit cell. Employing square holes instead of circular holes simplifies the problem of modeling of such structures. According to Fig. 5, the unit cell can be divided into two regions with designated coupling coefficients. Thus, the problem boils down to calculating the coupling coefficients for regions 1 and 2. Semi-vectorial BPM (BPM package of RSOFT) was employed to calculate the semi-vectorial modes of the asymmetric PC slab waveguide shown in Fig. 3(b). The output of BPM analysis were the profile and the propagation constants of the x-polarized and y-polarized modes of the asymmetric loaded PC slab waveguide that were used to calculate the coupling coefficients of the x-polarized and ypolarized waves. Assuming that the profile of the total transverse field in the asymmetric

> ˆ ˆ () (, ) () (, ) *<sup>x</sup> <sup>y</sup> j z <sup>j</sup> <sup>z</sup> E Ex Ey a ze xye a ze xye x y xx y y*

vectorial solution of wave equation for x-polarized and y-polarized waves, respectively. *β<sup>x</sup>* and *βy* are propagation constants along x and y directions, respectively. Substituting equ. (8)

respectively, and assuming that the amplitude of the field are slowly varying along z-

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

*da z j e a z e nka ze a ze dz*

*xx y y*

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

*j e a z e nka ze a ze dz*

*yy x x*

*a z e a ze e*

2 22 2 2 22 2

*t x ave x t y ave y*

*e nk e e nk e*

*a z e a ze e*

*<sup>x</sup> xx x t x x x x x x*

2 22 2

2 2

2 2

*yy yx n n*

( )0 ( )0

1 1 () ( ) () ( )

*yy y t y y y y y y*

*xx xy n n*

1 1 () ( ) () ( )

2 2

 

> 

, (10)

*j z*

*n n*

2 22 2

2 2

*j z*

*n n*

, (8)

 and *<sup>y</sup> <sup>j</sup> <sup>z</sup> e* ,

(9.a)

(9.b)

are x- and y-components of electric field of the semi-

more accurately using coupled mode analysis.

loaded PC slab waveguide is represented as following:

 and (,) *<sup>y</sup> <sup>j</sup> <sup>z</sup> ye xye*

direction (propagation direction); the following equation is obtained:

*y*

*da z*

into equ. (6) and multiplying both side of equ. (6.a), and equ. (6.b) by *<sup>x</sup> <sup>j</sup> <sup>z</sup> e*

Where ( , ) *<sup>x</sup> <sup>j</sup> <sup>z</sup> xe xye*

Where:

Where, 2

*ave*

 *y <sup>x</sup>* ,

By invoking the following assumption:

*x y*

,

 #### **3.2.1 Normal mode analysis using 3D-FDTD**

The first approach to design the polarization rotator structure is to calculate vectorpropagation characteristics of normal modes (fast and slow modes) of the asymmetric loaded PC slab waveguide (Fig. 3(b)). The half-beat length and the total number of the loaded layers can be calculated using equ. (1). To obtain modal characteristics of the fast and slow modes of the asymmetric loaded PC slab waveguide, 3D-FDTD method is employed. The propagation constants of x-polarized and y-polarized waves are extracted from 3D-FDTD simulation results using SFT of the electric field along the propagation direction. However, first the frequency band over which slow and fast modes are guided must be determined so that 3D-FDTD simulation could be performed over the aforementioned frequency band. PWEM is employed to obtain the band diagram of the asymmetric loaded PC slab waveguide. To calculate the birefringence of the structure, the effective frequencydependent index of refraction of the normal modes is to be calculated. To obtain the aforementioned data, the accurate dispersion analysis is carried out, which is based on SFT of the electromagnetic field distribution in the PC slab waveguide along the propagation direction at any point on defect line cross section, the plane normal to the propagation direction (*y*, z). To employ SFT, it is assumed that the electromagnetic field in the PC slab waveguide can be expressed as a modal expansion at the normal plane, as following:

$$E\_{\alpha}(\mathbf{x}, y, z) = \sum\_{n,m} E\_{n,m,\alpha} e^{j\beta\_{n,m,\alpha}z} \tag{7}$$

where *En m*, , and *n m*, , represent the electric field component and the propagation constant of the *(n,m)th* mode at frequency ω. The peaks of the SFT spectrum describe the propagating modes of the structure. These peaks are independent of the location, (*x0* , *y0*) and the electromagnetic field components. The effective refractive indices of the modes can be determined by locating these peaks.

Having determined the effective refractive indices of the fast and slow modes, Lπ can be calculated as well as the total number of top loaded layers. The design methodology presented in this subsection requires finding the vector-propagation characteristics of normal modes of asymmetric loaded PC slab waveguide. It is a very general methodology and can be extended to any air hole geometry of PC slab structure as opposed to coupledmode theory that is more efficient for simple air hole geometry PC structures such as square hole PC slab based polarization rotator. In the following subsection, coupled-mode theory based on semi-vectorial modes of the structure is presented.

#### **3.2.2 Coupled-mode theory**

Huang and Mao employed similar coupled mode theory based on the scalar modes to analyze polarization conversion in a periodic loaded rib waveguide. In a PC slab waveguide, the propagation characteristics strongly depend on the polarization of the propagating wave leading to a large birefringence (Genereux et al., 2001). However, scalar modal analysis completely ignores the polarization dependence of the wave propagation; thus, it is too simplified to represent the wave propagation inside a PC slab waveguide. Here, coupled mode theory based on semi-vectorial modes of a PC structure was developed to analyze the asymmetric loaded PC slab waveguide. Using semi-vectorial modal analysis,

The first approach to design the polarization rotator structure is to calculate vectorpropagation characteristics of normal modes (fast and slow modes) of the asymmetric loaded PC slab waveguide (Fig. 3(b)). The half-beat length and the total number of the loaded layers can be calculated using equ. (1). To obtain modal characteristics of the fast and slow modes of the asymmetric loaded PC slab waveguide, 3D-FDTD method is employed. The propagation constants of x-polarized and y-polarized waves are extracted from 3D-FDTD simulation results using SFT of the electric field along the propagation direction. However, first the frequency band over which slow and fast modes are guided must be determined so that 3D-FDTD simulation could be performed over the aforementioned frequency band. PWEM is employed to obtain the band diagram of the asymmetric loaded PC slab waveguide. To calculate the birefringence of the structure, the effective frequencydependent index of refraction of the normal modes is to be calculated. To obtain the aforementioned data, the accurate dispersion analysis is carried out, which is based on SFT of the electromagnetic field distribution in the PC slab waveguide along the propagation direction at any point on defect line cross section, the plane normal to the propagation direction (*y*, z). To employ SFT, it is assumed that the electromagnetic field in the PC slab

waveguide can be expressed as a modal expansion at the normal plane, as following:

based on semi-vectorial modes of the structure is presented.

where *En m*, ,

and

**3.2.2 Coupled-mode theory** 

 *n m*, ,

be determined by locating these peaks.

, (,,) *<sup>j</sup> n m <sup>z</sup> n m n m E xyz E e*

constant of the *(n,m)th* mode at frequency ω. The peaks of the SFT spectrum describe the propagating modes of the structure. These peaks are independent of the location, (*x0* , *y0*) and the electromagnetic field components. The effective refractive indices of the modes can

Having determined the effective refractive indices of the fast and slow modes, Lπ can be calculated as well as the total number of top loaded layers. The design methodology presented in this subsection requires finding the vector-propagation characteristics of normal modes of asymmetric loaded PC slab waveguide. It is a very general methodology and can be extended to any air hole geometry of PC slab structure as opposed to coupledmode theory that is more efficient for simple air hole geometry PC structures such as square hole PC slab based polarization rotator. In the following subsection, coupled-mode theory

Huang and Mao employed similar coupled mode theory based on the scalar modes to analyze polarization conversion in a periodic loaded rib waveguide. In a PC slab waveguide, the propagation characteristics strongly depend on the polarization of the propagating wave leading to a large birefringence (Genereux et al., 2001). However, scalar modal analysis completely ignores the polarization dependence of the wave propagation; thus, it is too simplified to represent the wave propagation inside a PC slab waveguide. Here, coupled mode theory based on semi-vectorial modes of a PC structure was developed to analyze the asymmetric loaded PC slab waveguide. Using semi-vectorial modal analysis,

, ,

represent the electric field component and the propagation

(7)

 

, ,

 

**3.2.1 Normal mode analysis using 3D-FDTD** 

the polarization dependence of wave propagation has been partially taken into account; thus, the coupling between the two x-polarized and y-polarized waves can be modeled more accurately using coupled mode analysis.

In a triangular lattice PC structure, cross-section varies along the propagation direction within one unit cell. Employing square holes instead of circular holes simplifies the problem of modeling of such structures. According to Fig. 5, the unit cell can be divided into two regions with designated coupling coefficients. Thus, the problem boils down to calculating the coupling coefficients for regions 1 and 2. Semi-vectorial BPM (BPM package of RSOFT) was employed to calculate the semi-vectorial modes of the asymmetric PC slab waveguide shown in Fig. 3(b). The output of BPM analysis were the profile and the propagation constants of the x-polarized and y-polarized modes of the asymmetric loaded PC slab waveguide that were used to calculate the coupling coefficients of the x-polarized and ypolarized waves. Assuming that the profile of the total transverse field in the asymmetric loaded PC slab waveguide is represented as following:

$$E = E\_x \hat{\mathbf{x}} + E\_y \hat{\mathbf{y}} \quad = a\_x(z)e\_x(\mathbf{x}, y)e^{-j\beta\_x z} + a\_y(z)e\_y(\mathbf{x}, y)e^{-j\beta\_y z} \tag{8}$$

Where ( , ) *<sup>x</sup> <sup>j</sup> <sup>z</sup> xe xye* and (,) *<sup>y</sup> <sup>j</sup> <sup>z</sup> ye xye* are x- and y-components of electric field of the semivectorial solution of wave equation for x-polarized and y-polarized waves, respectively. *β<sup>x</sup>* and *βy* are propagation constants along x and y directions, respectively. Substituting equ. (8) into equ. (6) and multiplying both side of equ. (6.a), and equ. (6.b) by *<sup>x</sup> <sup>j</sup> <sup>z</sup> e* and *<sup>y</sup> <sup>j</sup> <sup>z</sup> e* , respectively, and assuming that the amplitude of the field are slowly varying along zdirection (propagation direction); the following equation is obtained:

$$\begin{aligned} &-j2\beta\_x e\_x \frac{da\_x(z)}{dz} + a\_x(z)\nabla\_t^2 e\_x + n^2 k^2 a\_x(z) e\_x - \beta\_x^2 a\_x(z) e\_x = \\ &-a\_x(z) \frac{\partial}{\partial \mathbf{x}} (e\_x \frac{1}{n^2} \frac{\partial n^2}{\partial \mathbf{x}}) - a\_y(z) e^{-j\Lambda z} \frac{\partial}{\partial \mathbf{x}} (e\_y \frac{1}{n^2} \frac{\partial n^2}{\partial y}) \end{aligned} \tag{9.a}$$

$$\begin{aligned} &-j2\beta\_y e\_y \frac{da\_y(z)}{dz} + a\_y(z)\nabla\_t^2 e\_y + n^2 k^2 a\_y(z) e\_y - \beta\_y^2 a\_y(z) e\_y = \\ &-a\_y(z) \frac{\partial}{\partial y} (e\_y \frac{1}{n^2} \frac{\partial n^2}{\partial y}) - a\_x(z) e^{i\Lambda x} \frac{\partial}{\partial y} (e\_x \frac{1}{n^2} \frac{\partial n^2}{\partial x}) \end{aligned} \tag{9.b}$$

Where: *y <sup>x</sup>* ,

By invoking the following assumption:

$$\begin{aligned} \nabla\_t^2 e\_x + (n^2 k^2 - \mathcal{J}\_{ave}^2) e\_x &= 0 \\ \nabla\_t^2 e\_y + (n^2 k^2 - \mathcal{J}\_{ave}^2) e\_y &= 0 \end{aligned} \tag{10}$$

Where, 2 *x y ave* ,

Photonic Crystal for Polarization Rotation 305

where as, the refractive index profile is different as depicted in Fig. 5 leading to different values of coupling coefficients for regions 1 and 2. If the cross-coupling coefficients in both regions 1 and 2 were assumed to be equal (κxy= κyx= κ), the coupled-mode equations could be solved analytically as presented in equ. (14) below. Nonetheless, numerical methods could be easily implemented for general cases where the cross-coupling coefficients were not equal. Given the exact analytical solution as *A(z)=MA(0)*; where *A* is a column vector for

> cos( ) cos( / 2)sin( ) sin( / 2)sin( ) 1,2 sin( / 2)sin( ) cos( ) cos( / 2)sin( ) *i i i i i i i i*

> > 2 2 *i ii*

*i* 

 (14)

 

 

(15)

, (18)

*xy yx k k* ; where the imaginary parts were very small. Numerical,

*i i i i i i i i z j z j z M i j z zj z*

> 2 *xxi yyi*

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

The ± signs correspond to the alternative sections of the periodic loading. *z1* and *z2* are the length of regions 1 and 2 shown in Fig. 5. Assuming that w is the width of a square hole, *z1*

 *z1=a-w* (16.a)

 *z2=w* (16.b) Having set *M1* and *M2* as the transfer matrix of regions 1 and 2, the transfer matrix for one

 *M±=M1±.M*2±, (17)

1 2

Thus, the length of one top silicon brick is *Lπ* and the top cladding layer alternates periodically throughout the propagation length. The simulation results revealed that for our

and analytical solutions of the coupled mode theory, equ. (12.a) and (8.b), give us almost the same results. From equ. (14) the preliminary value of the loading period before employing

In next section, first the band structure of the asymmetric loaded PC slab waveguide is calculated using PWEM. Therefore, frequency band over which the polarization rotator can

the numerical method to solve the coupled mode equation, equ. (12), was calculated.

The loading period or half-beat length can be approximated as follows:

*L* *i* 

 

coefficients *ax* and *ay*; the transfer matrix (*M*) is expressed as following:

and *z2* are determined as following:

structure, *xy yx k k* and \*

be operated is determined.

unit cell is obtained:

*i*

A simplified form of equ. (9) is obtained:

$$\begin{split} & -j2\,\beta\_x e\_x \frac{da\_x(z)}{dz} + (\beta\_{\text{ave}}^2 - \beta\_x^2) a\_x(z) e\_x = \\ & -a\_x(z) \frac{\partial}{\partial x} (e\_x \frac{1}{n^2} \frac{\partial n^2}{\partial x}) - a\_y(z) e^{-j\lambda x} \frac{\partial}{\partial x} (e\_y \frac{1}{n^2} \frac{\partial n^2}{\partial y}) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \tag{11.a} \\ & -j2\,\beta\_y e\_y \frac{da\_y(z)}{dz} + (\beta\_{\text{ave}}^2 - \beta\_y^2) a\_y(z) e\_y = \\ & -a\_y(z) \frac{\partial}{\partial y} (e\_y \frac{1}{n^2} \frac{\partial n^2}{\partial y}) - a\_x(z) e^{j\lambda x} \frac{\partial}{\partial y} (e\_x \frac{1}{n^2} \frac{\partial n^2}{\partial x}) \end{split} \tag{11.b}$$

Multiplying both sides of equ. (11.a), and equ. (11.b) by \* *xe* and \* *ye* (\*- conjugate), respectively and integrating over the cross-section, the following coupled mode equations are obtained:

$$\begin{aligned} \frac{da\_x(z)}{dz} &= -j\kappa\_{xx}a\_x(z) - j\kappa\_{xy}a\_y(z) \\ \frac{da\_y(z)}{dz} &= -j\kappa\_{yy}a\_y(z) - j\kappa\_{yx}a\_x(z) \end{aligned} \tag{12}$$

Where:

$$\kappa\_{\rm xx} = \frac{(\beta\_{\rm ane}^2 - \beta\_x^2) \iint e\_x^\* \, e\_x dx dy + \iint e\_x^\* \, \frac{\partial}{\partial \mathbf{x}} (e\_x \, \frac{1}{n^2} \frac{\partial n^2}{\partial \mathbf{x}}) dx dy}{2 \, \beta\_x \iint e\_x^\* \, e\_x dx dy} \tag{13.a}$$

$$\kappa\_{xy} = \frac{e^{-j\Delta x} \iint e\_{\mathbf{x}}^{\*} \frac{\partial}{\partial \mathbf{x}} (e\_{y} \frac{1}{n^{2}} \frac{\partial n^{2}}{\partial y}) d\mathbf{x} dy}{2 \,\beta\_{\mathbf{x}} \iint e\_{\mathbf{x}}^{\*} e\_{\mathbf{x}} d\mathbf{x} dy} \tag{13.b}$$

$$\kappa\_{yy} = \frac{(\beta\_{ave}^2 - \beta\_y^2) \iint e\_y^\* \, e\_y dx dy + \iint e\_y^\* \cdot \frac{\partial}{\partial y} (e\_y \frac{1}{n^2} \frac{\partial n^2}{\partial y}) dx dy}{2 \beta\_y \iint e\_y^\* \, e\_y dx dy} \tag{13.c}$$

$$\kappa\_{\rm yx} = \frac{e^{j\Lambda z} \iint e\_y^\* \frac{\partial}{\partial y} (e\_x \frac{1}{n^2} \frac{\partial n^2}{\partial x}) dx dy}{2 \,\beta\_y \iint e\_y^\* \, e\_y dx dy} \tag{13.4}$$

*κxx* and *κyy* are the self-coupling coefficients; whereas, *κxy* and *κyx* refer to cross-coupling coefficients. In equ. (13.a), and equ. (13.c), the second terms were negligible in comparison with the first terms. The coupling coefficients must be solved for both regions 1 and 2 (see Fig. 5), using equ. (13). The distribution of the electric fields in both regions are the same;

2 2

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

*dz*

*y*

*dz*

Multiplying both sides of equ. (11.a), and equ. (11.b) by \*

*y*

*dz*

 

*xy*

 

*yx*

*xx*

*yy*

*dz da z*

*da z*

*da z j e a ze*

 

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

*j e a ze*

 

*<sup>x</sup> x x ave x x x*

*xx y y*

*y y ave y y y*

*yy x x*

*a z e a ze e*

*a z e a ze e*

2 2

2 2

*xe* and \*

2

(13.a)

(13.b)

(13.c)

(13.d)

2

2

*y y n*

2

*x x n*

*j z*

*n n*

(11.a)

(11.b)

(12)

*ye* (\*- conjugate),

*j z*

*n n*

2 2

2 2

*yy y n n x*

*xx xy n n*

1 1 () ( ) () ( )

2 2

1 1 () ( ) () ( )

respectively and integrating over the cross-section, the following coupled mode equations

( ) () ()

*<sup>x</sup> xx x xy y*

2 2\* \*

 

*j z*

*da z <sup>j</sup> a z <sup>j</sup> a z*

( ) () ()

\*

<sup>2</sup> \* 2

*<sup>n</sup> e e e dxdy x y n e e dxdy*

<sup>1</sup> .( )

\*

\*

<sup>2</sup> \* 2 \*

*<sup>n</sup> e e e dxdy y x n e e dxdy*

<sup>1</sup> .( )

*e e dxdy*

*y yy*

*x xx*

*e e dxdy*

*x xx*

<sup>1</sup> ( ) . .( )

2 . *ave x x x x x*

2 .

<sup>1</sup> ( ) . .( )

2 .

2 .

*κxx* and *κyy* are the self-coupling coefficients; whereas, *κxy* and *κyx* refer to cross-coupling coefficients. In equ. (13.a), and equ. (13.c), the second terms were negligible in comparison with the first terms. The coupling coefficients must be solved for both regions 1 and 2 (see Fig. 5), using equ. (13). The distribution of the electric fields in both regions are the same;

*y yy*

*y x*

 

*ave y y y y y*

 

2 2\* \*

*j z*

*x y*

 

*yy y yx x*

*j a z j a z*

 

 

*<sup>n</sup> e e dxdy e e dxdy*

*<sup>n</sup> e e dxdy e e dxdy*

A simplified form of equ. (9) is obtained:

are obtained:

Where:

where as, the refractive index profile is different as depicted in Fig. 5 leading to different values of coupling coefficients for regions 1 and 2. If the cross-coupling coefficients in both regions 1 and 2 were assumed to be equal (κxy= κyx= κ), the coupled-mode equations could be solved analytically as presented in equ. (14) below. Nonetheless, numerical methods could be easily implemented for general cases where the cross-coupling coefficients were not equal. Given the exact analytical solution as *A(z)=MA(0)*; where *A* is a column vector for coefficients *ax* and *ay*; the transfer matrix (*M*) is expressed as following:

$$M\_{i\pm} = \begin{pmatrix} \cos(\Omega\_i z\_i) - j\cos(\phi\_i \text{ / 2})\sin(\Omega\_i z\_i) & \mp j\sin(\phi\_i \text{ / 2})\sin(\Omega\_i z\_i) \\ \mp j\sin(\phi\_i \text{ / 2})\sin(\Omega\_i z\_i) & \cos(\Omega\_i z\_i) + j\cos(\phi\_i \text{ / 2})\sin(\Omega\_i z\_i) \end{pmatrix} \quad i = 1, 2 \tag{14}$$

*i*

$$
\Omega\_i = \sqrt{\delta\_i^2 + \kappa\_i^2}
$$

$$
\delta\_i = \frac{\kappa\_{xxi} - \kappa\_{yyi}}{2} \tag{15}
$$

$$
\tan(\wp\_i / 2) = \frac{\kappa\_i}{\delta\_i}
$$

The ± signs correspond to the alternative sections of the periodic loading. *z1* and *z2* are the length of regions 1 and 2 shown in Fig. 5. Assuming that w is the width of a square hole, *z1* and *z2* are determined as following:

$$z\_1 \mathbf{\bar{a}} \cdot \mathbf{w} \tag{16.\mathbf{a}}$$

$$z\_2 \exists w \tag{16.b}$$

Having set *M1* and *M2* as the transfer matrix of regions 1 and 2, the transfer matrix for one unit cell is obtained:

$$M\_{\underline{x}} \equiv M\_{1\uparrow\downarrow}, M\_{2\sqcup\nu} \tag{17}$$

The loading period or half-beat length can be approximated as follows:

$$L\_{\pi} \approx \frac{\pi}{\Omega\_1 + \Omega\_2},\tag{18}$$

Thus, the length of one top silicon brick is *Lπ* and the top cladding layer alternates periodically throughout the propagation length. The simulation results revealed that for our structure, *xy yx k k* and \* *xy yx k k* ; where the imaginary parts were very small. Numerical, and analytical solutions of the coupled mode theory, equ. (12.a) and (8.b), give us almost the same results. From equ. (14) the preliminary value of the loading period before employing the numerical method to solve the coupled mode equation, equ. (12), was calculated.

In next section, first the band structure of the asymmetric loaded PC slab waveguide is calculated using PWEM. Therefore, frequency band over which the polarization rotator can be operated is determined.

Photonic Crystal for Polarization Rotation 307

with the fabrication constrains the upper limit of the thickness of the loaded layer

The band diagrams for two different slab thicknesses t=0.6a and t=0.8a are obtained by PWEM and plotted in Fig. 6(b) and (c), respectively. The thickness of the top loaded layer, width of the PC squares and refractive index of silicon are tup=0.2a, w=0.6 and nsi=3.48, respectively. There are two modes depicted by dotted and solid lines. The mode graphed by dotted line resembles an index-guided mode except the mini-stop band observed at the zone boundary. We will call it index-guided mode. The other mode depicted by solid line will be called Bloch mode. The index-guided mode is considered to be y-polarized wave for which the dominant electric field component is in y-direction. On the other hand, the Bloch mode is considered x-polarized wave. For t=0.6a, the index-guided mode crosses the Bloch mode and is folded back at the zone boundary at a/λ=0.28. The Bloch mode touches the zone boundary (Κ) at a/λ=0.259 and crosses TE-like PC slab modes at a/λ=0.274. Since the indexguided mode and Bloch mode cross each other, the difference between the effective refractive indices of the two modes, which is proportional to 1/Lπ, varies with frequency significantly. Thus, the polarization converter made of a PC slab with t=0.6a is expected to

On the other hand, by increasing the thickness of the slab to t=0.8a, the index-guided mode has been pushed down to the lower frequencies. The index-guided mode depicted by dotted line in Fig. 6(c) has folded back at the zone boundary at a/λ=0.243. Within the frequency band of the Bloch mode, 0.257-0.267, the two bands are parallel; thus, the variation of the difference between the effective indices of the two modes and Lπ with frequency are negligible. In this diagram, the index-guided and Bloch modes also correspond to fast and slow modes, respectively. At normalized frequency of a/λ=0.267, Bloch mode crosses TE-like PC slab mode. In 3D-FDTD simulations, the central normalized frequency of a/λ=0.265 is assigned to f=600 GHz resulting in the unit cell size of a=132.5 µm. The crosssection of *Ex*, *Ey*, *Hx* and *Hy* components for TE-like input at a/λ=0.267 (f=604.5 GHz) are plotted in Fig. 7. Graphs verify the presence of PC slab modes. *Ey* and *Hx* components of PC slab modes own even symmetry w.r.t. y=0 plane as opposed to *Ex* and *Hy* components where have odd symmetry. Therefore, PC slab modes are TE-like verifying the PWEM analysis. Moreover, the field distribution inside the defect line indicates that *Ey* and *Hx* are the dominant components of electric and magnetic field, respectively indicating that TE-like wave at the input excites y-polarized wave. All four components of y-polarized mode have even symmetry with respect to y=0 and x=0 planes; although, the minor components of input wave, TE-like, (*Ex* and *Hy*) have odd symmetry w.r.t y=0 plane. Thus, TE-like input wave has been evolved into the mode of the asymmetric loaded PC slab waveguide. By further increasing the thickness of PC slab waveguide, higher order modes will be pushed down inside the bandgap; which is not suitable for our application. Having compared the band diagram for t=0.6a and t=0.8a; it is seen that t=0.8a suits better for the polarization

Now that the overlap between x-polarized and y-polarized guiding are defined, 3D-FDTD simulation for extracting the modal characteristics of asymmetric loaded PC slab waveguide will be limited to the aforementioned frequency band. In following, the modal analysis

is restricted to tup=0.2a.

be narrow band.

conversion application.

using 3D-FDTD simulation is presented.

Fig. 5. Top view of the asymmetrically loaded PC based polarization rotator. The top cover layer is marked by the dark solid line in the figure. κ1 and κ2 represent the cross-coupling coefficient for regions 1 and 2 inside a unit cell.

### **3.3 Design of the polarization rotator**

In this section the results of the design using both coupled mode theory and normal mode analysis are presented.
