**1. Introduction**

294 Photonic Crystals – Innovative Systems, Lasers and Waveguides

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Due to the unique guiding properties of photonic crystal (PC) structure, such as sharp low loss bends, it is considered one of the main contenders of a compact optical integrated circuit (OIC). PC is foreseen as the next generation of hybrid photonic-electronic integrated circuit. However, one of the main issues in implementation of a PC based OIC is its strong polarization dependence guiding behavior.

The components of optical integrated circuit exhibit strong polarization dependence behavior which translates into their random response to random polarizations. One approach to render the polarization sensitivity of an optical integrated circuit is to eliminate the randomness of the input polarization by splitting it into two orthogonal polarizations (TE, TM) and rotating one of the polarizations; thus, single polarization is realized on the chip (Barwicz et al., 2007). The focus of this chapter is to implement PC based polarization rotator which is capable of rotating the polarization of light to an arbitrary angle. The large birefringence in PC structure leads to a small optical path difference between the two polarizations which can result in realization of ultra-compact polarization rotators. Ease of fabrication and its compatibility with integrated PC technology is considered another main advantage of the PC based polarization rotator. This chapter is organized as follows. In Sec.2, an overview on the passive polarization rotators is given. In Sec. 3, a novel polarization rotator structure is introduced and designed. Fabrication and characterization of the PC based polarization rotator are discussed in Sec. 4 and Sec. 5, respectively.

#### **2. An overview on passive polarization rotators**

Passive polarization rotator structures are mostly composed of geometrically asymmetric structures where the symmetry of the structure is disturbed so that two orthogonal polarizations could be coupled to each other. Imposing asymmetry into a symmetric waveguide structure leads to a perturbation in the primary waveguide axes, depicted in Fig.1, where Ef and Es are projected fields on fast and slow axes called fast and slow normal modes, respectively. They travel with different speeds resulting in phase delay between the two components. For the phase delay of 180o, the power conversion between the two components has reached to its maximum and the propagating distance is called the half-beat length Lπ, defined as (Mrozowski, 1997):

$$L\_{\pi} = \frac{\pi}{\beta\_s - \beta\_f} = \frac{\pi}{(n\_s - n\_f)k\_0} = \frac{\lambda}{2(n\_s - n\_f)}\tag{1}$$

Photonic Crystal for Polarization Rotation 297

Employed full vectorial analysis based on versatile finite element beam propagation method (VFEBPM) to improve the design and reduce the polarization conversion length to 400 μm at operating wavelength of 1.55 µm (Obayya et al., 2000). Due to the huge size of the device, the design could not be verified with the aid of rigorous numerical methods, three

(a)

(b) (c) (d)

Lπ (d) the output field after another Lπ, *Eo2* (*Es* and *Ef* are the slow and fast modes,

loaded rib waveguide (b) normal modes of perturbed optical axis (*Es*

rotator was designed and fabricated (El-Refaei & Yevick, 2003).

respectively).

Fig. 1. The sketch of asymmetric loaded rib waveguide and its normal modes (a) asymmetric

Based on the same idea, tilted waveguiding polarization converter was introduced first by Heidrich (Heidrich et al., 1992). The first device was implemented by laterally tilted InP/GaInAsP rib waveguide on stepped substrate. The total length of the device was more than 7 mm . Later on by improving the design using coupled mode theory and BPM, the more compact device with the total length of 0.9 mm was implemented by Van der Tol. Besides being bulky, the device undertakes huge coupling loss at the junctions between the adjacent periodic sections. To eliminate the loss, single section devices were proposed (Tzolov & Fontaine, 1996; Huang et al., 2000; Rahman et al., 2001). By using angled single section waveguides in InP/InGaAs material system, another type of short polarization

the input electrical field,*Ei* , and the rotated output field, *Eo1*, after propagation distance of

*<sup>+</sup>*, *Ef <sup>+</sup>*), (*Es -*, *Ef --*) (c) dimensional finite difference time domain method (3D-FDTD).

Where βs and βf are the propagation constants of slow and fast modes, and *ns* and *nf* are corresponding effective refractive indices, respectively. The process of polarization rotator in geometrically asymmetric structure can be explained with more details as follows. According to Fig. 1(b), the transverse component of the normal modes of for example the asymmetric loaded rib waveguide (Shani et al., 1991) can be expressed as following, where *φ* is the optical perturbation angle and *Es* and *Ef* are slow and fast modes, respectively:

$$E\_s = \cos\varphi \hat{\mathbf{x}} - \sin\varphi \hat{\mathbf{y}} \tag{2.a}$$

$$E\_f = \sin\varphi \hat{\mathbf{x}} + \cos\varphi \hat{\mathbf{y}} \tag{2.b}$$

Assuming that the input wave is x-polarized, it can be expressed as the combination of Es and Ef as following:

$$E\_i = \cos\varphi \, e^{-j\beta\_s z} E\_s + \sin\varphi \, e^{-j\beta\_f z} E\_f \tag{3}$$

At half-beat length long, the slow and fast modes become out of phase resulting in destructive interference between the two modes. Thus, at z=Lπ, the total field become:

$$E\_{o1} = \cos\varphi E\_s - \sin\varphi E\_f \tag{4}$$

Substituting equ. (2) into equ. (4) results in:

$$E\_{o1} = \cos 2\varphi \hat{\mathbf{x}} - \sin 2\varphi \hat{\mathbf{y}} \tag{5}$$

Thus, at z=Lπ the input wave has been rotated by 2φ with respect to x-axis; depicted in Fig. 1 (c). To avoid the reversal of power conversion and synchronize the power conversion, where φ<45o, the top loaded layer of the asymmetric loaded layer rib waveguide must be inverted w.r.t the center of rib waveguide at z=Lπ where z is the propagation direction, Fig. 1(a). At the end of the next brick, the polarization of the input signal has been rotated by 4φ; depicted in Fig 1(d). The top loaded layers will be arranged periodically and repeated until the total phase shift becomes 90o (Snyder & Love, 1983).

In single section polarization rotator structures, φ is adjusted to 45o; so that, 90o polarization rotation could be achieved by only one section.

Several structures of longitudinally variable passive polarization rotators have been reported in literature including: periodic asymmetric loaded rib waveguide (Shani et al., 1991), periodic tilted waveguiding section (Heidrich et al., 1992), periodically loaded strip waveguide (Mertens et al., 1998), cascaded bend waveguides (Van Dam et al., 1996; Liu et al., 1997, 1998a, 1998b) and a mode-evolution-based polarization rotator structure (Watts et al., 2005).

Periodic asymmetric loaded rib waveguide was experimentally demonstrated by Shani. The asymmetric loading of the waveguide would perturb the axes of the primary waveguide. By periodically alternating the loaded layer in longitudinal direction the polarization conversion or rotation will be accumulated coherently. The total length of the device was more than 3 mm. Haung and Mao employed coupled mode theory based on scalar modes to analyze the structure theoretically (Haung & Mao, 1992). Later on, Obayya and at el.

Where βs and βf are the propagation constants of slow and fast modes, and *ns* and *nf* are corresponding effective refractive indices, respectively. The process of polarization rotator in geometrically asymmetric structure can be explained with more details as follows. According to Fig. 1(b), the transverse component of the normal modes of for example the asymmetric loaded rib waveguide (Shani et al., 1991) can be expressed as following, where *φ*

> cos sin ˆ ˆ *E xy <sup>s</sup>*

sin cos ˆ ˆ *E x <sup>f</sup>* 

Assuming that the input wave is x-polarized, it can be expressed as the combination of Es

cos sin *<sup>f</sup> <sup>s</sup> j z j z E eE e E i s <sup>f</sup>* 

At half-beat length long, the slow and fast modes become out of phase resulting in destructive interference between the two modes. Thus, at z=Lπ, the total field become:

> <sup>1</sup> *E EE o s* cos sin

<sup>1</sup> cos2 sin 2 ˆ ˆ *E xy <sup>o</sup>* 

Thus, at z=Lπ the input wave has been rotated by 2φ with respect to x-axis; depicted in Fig. 1 (c). To avoid the reversal of power conversion and synchronize the power conversion, where φ<45o, the top loaded layer of the asymmetric loaded layer rib waveguide must be inverted w.r.t the center of rib waveguide at z=Lπ where z is the propagation direction, Fig. 1(a). At the end of the next brick, the polarization of the input signal has been rotated by 4φ; depicted in Fig 1(d). The top loaded layers will be arranged periodically and repeated until

In single section polarization rotator structures, φ is adjusted to 45o; so that, 90o polarization

Several structures of longitudinally variable passive polarization rotators have been reported in literature including: periodic asymmetric loaded rib waveguide (Shani et al., 1991), periodic tilted waveguiding section (Heidrich et al., 1992), periodically loaded strip waveguide (Mertens et al., 1998), cascaded bend waveguides (Van Dam et al., 1996; Liu et al., 1997, 1998a, 1998b) and a mode-evolution-based polarization rotator structure (Watts

Periodic asymmetric loaded rib waveguide was experimentally demonstrated by Shani. The asymmetric loading of the waveguide would perturb the axes of the primary waveguide. By periodically alternating the loaded layer in longitudinal direction the polarization conversion or rotation will be accumulated coherently. The total length of the device was more than 3 mm. Haung and Mao employed coupled mode theory based on scalar modes to analyze the structure theoretically (Haung & Mao, 1992). Later on, Obayya and at el.

 

(3)

(2.a)

*y* (2.b)

*<sup>f</sup>* (4)

(5)

is the optical perturbation angle and *Es* and *Ef* are slow and fast modes, respectively:

and Ef as following:

et al., 2005).

Substituting equ. (2) into equ. (4) results in:

the total phase shift becomes 90o (Snyder & Love, 1983).

rotation could be achieved by only one section.

Employed full vectorial analysis based on versatile finite element beam propagation method (VFEBPM) to improve the design and reduce the polarization conversion length to 400 μm at operating wavelength of 1.55 µm (Obayya et al., 2000). Due to the huge size of the device, the design could not be verified with the aid of rigorous numerical methods, three dimensional finite difference time domain method (3D-FDTD).

Fig. 1. The sketch of asymmetric loaded rib waveguide and its normal modes (a) asymmetric loaded rib waveguide (b) normal modes of perturbed optical axis (*Es <sup>+</sup>*, *Ef <sup>+</sup>*), (*Es -*, *Ef --*) (c) the input electrical field,*Ei* , and the rotated output field, *Eo1*, after propagation distance of Lπ (d) the output field after another Lπ, *Eo2* (*Es* and *Ef* are the slow and fast modes, respectively).

Based on the same idea, tilted waveguiding polarization converter was introduced first by Heidrich (Heidrich et al., 1992). The first device was implemented by laterally tilted InP/GaInAsP rib waveguide on stepped substrate. The total length of the device was more than 7 mm . Later on by improving the design using coupled mode theory and BPM, the more compact device with the total length of 0.9 mm was implemented by Van der Tol. Besides being bulky, the device undertakes huge coupling loss at the junctions between the adjacent periodic sections. To eliminate the loss, single section devices were proposed (Tzolov & Fontaine, 1996; Huang et al., 2000; Rahman et al., 2001). By using angled single section waveguides in InP/InGaAs material system, another type of short polarization rotator was designed and fabricated (El-Refaei & Yevick, 2003).

Photonic Crystal for Polarization Rotation 299

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

(a)

(b)

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

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

the radiation loss at the junctions between different sections will be reduced.

asymmetric loaded PC slab waveguide.

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 circuit.

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 coupling loss.

Fig. 2. The sketch of the cross section of asymmetric Si nanowire for polarization rotation application (Wang & Dai, 2008).

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 introduced in the following section.
