**3.3.1 Design of the polarization rotator using 3D-FDTD modal analysis**

#### **3.3.1.1 PWEM analysis**

The first step of the design of the polarization rotator structure is to calculate the frequency band over which lossless propagation takes place for both x-polarized and y-polarized waves and then proceed with the design using both normal mode analysis and coupled mode theory within the aforementioned frequency band. The asymmetric loaded PC slab waveguide shown in Fig. 3(b) was first simulated using PWEM to obtain the band diagram and the frequency band of the modes. The thickness of the PC slab structure plays an important role on the polarization dependent guiding (Bayat et al., 2007). It is possible to obtain maximum overlap between x-polarized and y-polarized waves by optimizing the thickness of the PC slab waveguide.The same design methodology was employed to design the thickness of the asymmetric loaded PC slab waveguide.

Fig. 6(a) shows the super-cell for the asymmetric loaded triangular PC slab waveguide. By including several unit cells in horizontal plane, the defect lines in the super-lattice structure are isolated. In PWEM analysis, the definition of the TE-like and TM-like waves is based on the symmetry planes of the modes. The dominant components of TM-like mode (*Hy*, *Ez*, *Ex*) and the non-dominant components (*Ey*, *Hz*, *Hx*) have even and odd symmetry w.r.t. y=0 plane, respectively. Similarly, the dominant components of TE-like mode (*Ey*, *Hz*, *Hx*) and the non-dominant components (*Hy*, *Ez*, *Ex*) have even and odd symmetry to y=0 plane, respectively. The thickness of the top loaded layer, tup, is an important design parameter. Heuristically speaking, the larger the thickness of the upper layer is, the stronger the geometrical asymmetry is leading to a more compact device. However, to comply

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

In this section the results of the design using both coupled mode theory and normal mode

The first step of the design of the polarization rotator structure is to calculate the frequency band over which lossless propagation takes place for both x-polarized and y-polarized waves and then proceed with the design using both normal mode analysis and coupled mode theory within the aforementioned frequency band. The asymmetric loaded PC slab waveguide shown in Fig. 3(b) was first simulated using PWEM to obtain the band diagram and the frequency band of the modes. The thickness of the PC slab structure plays an important role on the polarization dependent guiding (Bayat et al., 2007). It is possible to obtain maximum overlap between x-polarized and y-polarized waves by optimizing the thickness of the PC slab waveguide.The same design methodology was employed to design

Fig. 6(a) shows the super-cell for the asymmetric loaded triangular PC slab waveguide. By including several unit cells in horizontal plane, the defect lines in the super-lattice structure are isolated. In PWEM analysis, the definition of the TE-like and TM-like waves is based on the symmetry planes of the modes. The dominant components of TM-like mode (*Hy*, *Ez*, *Ex*) and the non-dominant components (*Ey*, *Hz*, *Hx*) have even and odd symmetry w.r.t. y=0 plane, respectively. Similarly, the dominant components of TE-like mode (*Ey*, *Hz*, *Hx*) and the non-dominant components (*Hy*, *Ez*, *Ex*) have even and odd symmetry to y=0 plane, respectively. The thickness of the top loaded layer, tup, is an important design parameter. Heuristically speaking, the larger the thickness of the upper layer is, the stronger the geometrical asymmetry is leading to a more compact device. However, to comply

**3.3.1 Design of the polarization rotator using 3D-FDTD modal analysis** 

the thickness of the asymmetric loaded PC slab waveguide.

coefficient for regions 1 and 2 inside a unit cell.

**3.3 Design of the polarization rotator** 

analysis are presented.

**3.3.1.1 PWEM analysis** 

with the fabrication constrains the upper limit of the thickness of the loaded layer is restricted to tup=0.2a.

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 be narrow band.

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 conversion application.

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 using 3D-FDTD simulation is presented.

Photonic Crystal for Polarization Rotation 309



Fig. 7. Cross-section of electromagnetic field components obtained by 3D-FDTD analysis of asymmetric loaded PC slab waveguide at f=604.5 GHz (a/λ=0.267) with t=0.8a, w=0.6a,

To calculate birefringence and Lπ, it is required to obtain the vector-propagation characteristics of normal modes of the asymmetric loaded PC slab waveguide, Fig. 3(b), using 3D-FDTD simulation. To calculate the birefringence of the structure, the effective frequency-dependent indices of refraction of the structure for both *Ex* and *Ey* components are calculated using SFT analysis of transverse electric field components along the propagation direction at the center point of the defect line (x=0, y=0). For 3D-FDTD simulation, the input is a single frequency sinusoidal with guassian distribution in space. As

(c) (d)

y (mm)



0

0.05

0.1

0.15

(a) (b)

y (mm)


0

0.05 0.1 0.15

x (mm)

x (mm)





0

0.2

0.4

0.6

0.8

x (mm)

x (mm)

tup=0.2a, nsi=3.48 (a) *Ex* (b) *Ey* (c) *Hx* (d) *Hy.*

**3.3.1.2 3D-FDTD modal analysis** 



y (mm)

y (mm)



0

0.05

0.1

0.15



0

0.05

0.1

0.15

Fig. 6. (a) The supercell of the asymmetric loaded PC slab waveguide for PWEM analysis. The band diagram for the asymmetric loaded PC slab waveguide obtained by PWEM for (b) t=0.6a, tup=0.2a and w=0.6a (c) t=0.8a, tup=0.2a and w=0.6a.

(a)

(b)

(c)

Fig. 6. (a) The supercell of the asymmetric loaded PC slab waveguide for PWEM analysis. The band diagram for the asymmetric loaded PC slab waveguide obtained by PWEM for (b)

t=0.6a, tup=0.2a and w=0.6a (c) t=0.8a, tup=0.2a and w=0.6a.

Fig. 7. Cross-section of electromagnetic field components obtained by 3D-FDTD analysis of asymmetric loaded PC slab waveguide at f=604.5 GHz (a/λ=0.267) with t=0.8a, w=0.6a, tup=0.2a, nsi=3.48 (a) *Ex* (b) *Ey* (c) *Hx* (d) *Hy.*

#### **3.3.1.2 3D-FDTD modal analysis**

To calculate birefringence and Lπ, it is required to obtain the vector-propagation characteristics of normal modes of the asymmetric loaded PC slab waveguide, Fig. 3(b), using 3D-FDTD simulation. To calculate the birefringence of the structure, the effective frequency-dependent indices of refraction of the structure for both *Ex* and *Ey* components are calculated using SFT analysis of transverse electric field components along the propagation direction at the center point of the defect line (x=0, y=0). For 3D-FDTD simulation, the input is a single frequency sinusoidal with guassian distribution in space. As

Photonic Crystal for Polarization Rotation 311

of a/λ=0.252 is calculated and plotted in Fig. 9. The input wave is y-polarized or TE-like wave. Thus, the dominant component of electric field is *Ey*. The normalized SFT spectrum of both *Ey* and *Ex* components of electric field are plotted in Fig. 9. The SFT diagram of *Ex* has only one peak verifying PWEM diagram that only y-polarized wave is guided at

(a)

SFT (Ex )

SFT (Ey )

4 4.5 5 5.5 6 6.5

1/ g (mm-1)

(b) Fig. 8. The normalized SFT spectrum of transverse electric field components of asymmetric loaded PC slab waveguide for (a) TE-like wave input and (b) TM-like wave input (w=0.6a,

0

0.2

0.4

0.6

Normalized SFT (a.u.)

t=0.8a and tup=0.2a, λ=500 μm).

0.8

1

this frequency.

time proceeds and wave propagates along z-direction inside the defect line, the wave evolves into the modes of the structure. Thus, in the steady state case by applying SFT to the field distribution along the propagation direction at any point inside the defect line, the propagation characteristics of the modes can be obtained.

In our design example, the structural parameters of the PC polarization rotator are determined by assigning the normalized central frequency of the fundamental mode, 0.265, to the operating frequency. For example for f=600 GHz corresponding to the normalized frequency of 0.265, the unit cell size would be 132.5 μm (a= 0.265λ). Fig. 6 (c) shows that the overlap between x-polarized and y-polarized guiding lay within the frequency band of 0.258-0.267. The normalized SFT diagram for the input normalized frequency of a/λ=0.265 is calculated and plotted in Fig. 8. Fig. 8(a) and 8(b) correspond to TE-like and TM-like excitations, respectively. For TE-like excitation, at the input *Ey* is the dominant component; however, as the wave proceeds, the input evolves into the normal modes of structure that are depicted by the two dominant peaks in SFT spectrum. In SFT spectrum, the horizontal

axis is the 1/λg where *<sup>g</sup> neff* . λ and neff are free space wavelength and effective refractive

index of the propagating mode, respectively. For example, in Fig. 8(a) the SFT spectrum of *Ey* has a peak at 1/λg=5.16 where coincides with one the peaks of SFT spectrum of *Ex* component. The corresponding effective refractive index neff or nf (refractive index of fast normal mode) would be 2.58. The other peak of SFT spectrum of *Ex* component correspond to x-polarized or slow mode. It has been located at 1/λg=5.52 resulting in neff or ns (refractive index of slow normal mode) of 2.76. It is seen that the asymmetric loaded layer has induced a large birefringence. Having determined the effective refractive indices of the two normal modes, the half-beat length or Lπ can be calculated using equ. (1). For the two refractive indices calculated above at λ=500 μm, half-beat length (Lπ) would be 1.39 mm which is equivalent to 10.5a, a is the unit cell of PC slab. The same values are obtained for nf and ns from graph 7(b) which is plotted for TM-like input wave.

Another important parameter that can be extracted from modal analysis is polarization rotation angel, *φ*; which is tilted angel of optical axes with respect to Cartesian coordinate. The following expression is used to calculate the polarization rotation angel:

$$\varphi = \tan^{-1}(\frac{\text{abs}(SFT(E\_x))}{\text{abs}(SFT(E\_y))})\Big|\_{\text{@ }\text{peak}, TE-\text{like}} = \tan^{-1}(\frac{\text{abs}(SFT(E\_y))}{\text{abs}(SFT(E\_x))})\Big|\_{\text{@ }\text{peak}, TM-\text{like}}\tag{19}$$

Value of polarization rotation angel (*φ*) for above example is 6.5o. Polarization rotation angel is important in determining the total number of loaded layers required to achieve 90o polarization rotation. For this example, at the end of fourth loaded layer, the polarization of input wave should be rotated by 104o that exceeds 90o. To compensate the extra rotation angel, the length of the last top loaded layer can be increased. Thus, the normal mode analysis provides with the length and total numbers of top loaded layers.

Below normalized frequency of a/λ=0.257, only y-polarized wave is guided; thus, no polarization rotation is expected as x-polarized wave is not guided. To verify it, normalized SFT diagram of asymmetric loaded PC slab waveguide for the input normalized frequency

time proceeds and wave propagates along z-direction inside the defect line, the wave evolves into the modes of the structure. Thus, in the steady state case by applying SFT to the field distribution along the propagation direction at any point inside the defect line, the

In our design example, the structural parameters of the PC polarization rotator are determined by assigning the normalized central frequency of the fundamental mode, 0.265, to the operating frequency. For example for f=600 GHz corresponding to the normalized frequency of 0.265, the unit cell size would be 132.5 μm (a= 0.265λ). Fig. 6 (c) shows that the overlap between x-polarized and y-polarized guiding lay within the frequency band of 0.258-0.267. The normalized SFT diagram for the input normalized frequency of a/λ=0.265 is calculated and plotted in Fig. 8. Fig. 8(a) and 8(b) correspond to TE-like and TM-like excitations, respectively. For TE-like excitation, at the input *Ey* is the dominant component; however, as the wave proceeds, the input evolves into the normal modes of structure that are depicted by the two dominant peaks in SFT spectrum. In SFT spectrum, the horizontal

index of the propagating mode, respectively. For example, in Fig. 8(a) the SFT spectrum of *Ey* has a peak at 1/λg=5.16 where coincides with one the peaks of SFT spectrum of *Ex* component. The corresponding effective refractive index neff or nf (refractive index of fast normal mode) would be 2.58. The other peak of SFT spectrum of *Ex* component correspond to x-polarized or slow mode. It has been located at 1/λg=5.52 resulting in neff or ns (refractive index of slow normal mode) of 2.76. It is seen that the asymmetric loaded layer has induced a large birefringence. Having determined the effective refractive indices of the two normal modes, the half-beat length or Lπ can be calculated using equ. (1). For the two refractive indices calculated above at λ=500 μm, half-beat length (Lπ) would be 1.39 mm which is equivalent to 10.5a, a is the unit cell of PC slab. The same values are obtained for nf and ns

Another important parameter that can be extracted from modal analysis is polarization rotation angel, *φ*; which is tilted angel of optical axes with respect to Cartesian coordinate.

*y x*

Value of polarization rotation angel (*φ*) for above example is 6.5o. Polarization rotation angel is important in determining the total number of loaded layers required to achieve 90o polarization rotation. For this example, at the end of fourth loaded layer, the polarization of input wave should be rotated by 104o that exceeds 90o. To compensate the extra rotation angel, the length of the last top loaded layer can be increased. Thus, the normal mode

Below normalized frequency of a/λ=0.257, only y-polarized wave is guided; thus, no polarization rotation is expected as x-polarized wave is not guided. To verify it, normalized SFT diagram of asymmetric loaded PC slab waveguide for the input normalized frequency

@ , @ ,

(19)

*y x peak TE like peak TM like*

The following expression is used to calculate the polarization rotation angel:

1 1

analysis provides with the length and total numbers of top loaded layers.

( ( )) ( ( )) tan ( )) tan ( )) ( ( )) ( ( ))

*abs SFT E abs SFT E abs SFT E abs SFT E*

. λ and neff are free space wavelength and effective refractive

propagation characteristics of the modes can be obtained.

from graph 7(b) which is plotted for TM-like input wave.

axis is the 1/λg where *<sup>g</sup> neff*

of a/λ=0.252 is calculated and plotted in Fig. 9. The input wave is y-polarized or TE-like wave. Thus, the dominant component of electric field is *Ey*. The normalized SFT spectrum of both *Ey* and *Ex* components of electric field are plotted in Fig. 9. The SFT diagram of *Ex* has only one peak verifying PWEM diagram that only y-polarized wave is guided at this frequency.

Fig. 8. The normalized SFT spectrum of transverse electric field components of asymmetric loaded PC slab waveguide for (a) TE-like wave input and (b) TM-like wave input (w=0.6a, t=0.8a and tup=0.2a, λ=500 μm).

Photonic Crystal for Polarization Rotation 313

(a) (b)

the structure shown in Fig. 3(b) obtained by semi-vectorial 3D BPM analysis

high P.C.E. within the frequency band of the defect mode (0.258-0.267).

Fig. 11. Power exchange between the x-polarized and y-polarized wave versus the

0.275 obtained by coupled-mode analysis.

propagation length (t=0.8a, tup=0.2a, w=0.6a, a=132.5 μm, nsi=3.48) for a/λ=0.255, 0.265 and

(t=0.8a, tup=0.2a, w=0.6a, a=132.5 μm, nsi=3.48 and λ=500 μm).

Fig. 10. The profile of (a) Ex and (b)Ey components of x-polarized and y-polarized modes of

For a/λ=0.265 (λ=0.5 mm), 96% efficiency at z=7.2 mm (millimeter) was achieved. It is expected that by increasing or decreasing the normalized frequency, the power conversion efficiency reduce. To achieve high power conversion efficiency, the last silicon brick (top loaded layer) was no flipped around z-axis. In other word, the length of the last silicon brick was larger than 10a as it was predicted by normal mode analysis method, as well. The P.C.E. for a/λ=0.275 and 0.255 is larger than 75% at z=7.2 mm. Thus, it is expected to have a very

Fig. 9. The normalized SFT spectrum of transverse electric field components of asymmetric loaded PC slab waveguide for TE-like wave input (w=0.6a, t=0.8a and tup=0.2a, a/λ0=0.252).

In next section, the design of the polarization rotator using coupled-mode theory based on semi-vectorial modes is presented.

#### **3.3.1.3 Design of the polarization rotator using coupled-mode theory**

In this section, coupled mode theory discussed earlier was employed to design the asymmetrically loaded PC polarization rotator. In order to employ the coupled mode theory, first the semi-vectorial modes of the asymmetrically loaded PC slab waveguide, Fig. 3(b), must be calculated. Semi-vectorial BPM was employed for semi-vectorial modal analysis of the structure. The normalized electric field for x-polarized (TM-like) and ypolarized (TE-like) waves for the normalized frequency of a/λ=0.265 are shown in Fig. 10(a) and 10(b), respectively. It shows that the electric field distribution is asymmetric in both vertical and lateral directions as a result of the geometrical asymmetry. The propagation constants of the corresponding modes were calculated using semi-vectorial BPM simulation, as well. The effective refractive indices of x-polarized and y-polarized waves were 2.6567 and 2.5007, respectively. A big birefringence was observed as expected in PC slab waveguide structure. For aforementioned parameters, the coupling coefficients of the periodic asymmetric loaded PC polarization rotator (shown in Fig. 3(b)) were calculated using equ. (13) for both regions of 1 and 2, depicted in Fig. 5. Using equ. (15), the loading period was calculated, 10.8a. The value of half-beat length, Lπ, computed using coupled-mode theory and normal mode analysis were 10.8a and 10.5a, respectively. Thus, both methods deliver the same results that is a proof of the effectiveness of them. Fig. 11 shows the power exchange between the two polarizations along the propagation distance for a/λ=0.275, 0.265 and 0.255, λ0=500 μm (600GHz). The length of each top loaded layer is 10a.

Defining the power conversion efficiency (P.C.E.) as following:

$$P.C.E. = \frac{P\_{TM}}{P\_{TM} + P\_{TE}} \times 100 = \frac{a\_x^2}{a\_x^2 + a\_y^2} \times 100\tag{20}$$

2 3 4 5 6

1/ g (mm-1)

Fig. 9. The normalized SFT spectrum of transverse electric field components of asymmetric loaded PC slab waveguide for TE-like wave input (w=0.6a, t=0.8a and tup=0.2a, a/λ0=0.252).

In next section, the design of the polarization rotator using coupled-mode theory based on

In this section, coupled mode theory discussed earlier was employed to design the asymmetrically loaded PC polarization rotator. In order to employ the coupled mode theory, first the semi-vectorial modes of the asymmetrically loaded PC slab waveguide, Fig. 3(b), must be calculated. Semi-vectorial BPM was employed for semi-vectorial modal analysis of the structure. The normalized electric field for x-polarized (TM-like) and ypolarized (TE-like) waves for the normalized frequency of a/λ=0.265 are shown in Fig. 10(a) and 10(b), respectively. It shows that the electric field distribution is asymmetric in both vertical and lateral directions as a result of the geometrical asymmetry. The propagation constants of the corresponding modes were calculated using semi-vectorial BPM simulation, as well. The effective refractive indices of x-polarized and y-polarized waves were 2.6567 and 2.5007, respectively. A big birefringence was observed as expected in PC slab waveguide structure. For aforementioned parameters, the coupling coefficients of the periodic asymmetric loaded PC polarization rotator (shown in Fig. 3(b)) were calculated using equ. (13) for both regions of 1 and 2, depicted in Fig. 5. Using equ. (15), the loading period was calculated, 10.8a. The value of half-beat length, Lπ, computed using coupled-mode theory and normal mode analysis were 10.8a and 10.5a, respectively. Thus, both methods deliver the same results that is a proof of the effectiveness of them. Fig. 11 shows the power exchange between the two polarizations along the propagation distance for a/λ=0.275, 0.265 and 0.255, λ0=500 μm (600GHz). The length of each top

2

(20)

2 2 ... 100 100 *TM <sup>x</sup> TM TE x y*

*P P a a*

*P a PCE*

0

0.2

0.4

0.6

Normalized SFT (a.u.)

semi-vectorial modes is presented.

loaded layer is 10a.

0.8

1

SFT (Ey )

SFT (Ex )

**3.3.1.3 Design of the polarization rotator using coupled-mode theory** 

Defining the power conversion efficiency (P.C.E.) as following:

Fig. 10. The profile of (a) Ex and (b)Ey components of x-polarized and y-polarized modes of the structure shown in Fig. 3(b) obtained by semi-vectorial 3D BPM analysis (t=0.8a, tup=0.2a, w=0.6a, a=132.5 μm, nsi=3.48 and λ=500 μm).

For a/λ=0.265 (λ=0.5 mm), 96% efficiency at z=7.2 mm (millimeter) was achieved. It is expected that by increasing or decreasing the normalized frequency, the power conversion efficiency reduce. To achieve high power conversion efficiency, the last silicon brick (top loaded layer) was no flipped around z-axis. In other word, the length of the last silicon brick was larger than 10a as it was predicted by normal mode analysis method, as well. The P.C.E. for a/λ=0.275 and 0.255 is larger than 75% at z=7.2 mm. Thus, it is expected to have a very high P.C.E. within the frequency band of the defect mode (0.258-0.267).

Fig. 11. Power exchange between the x-polarized and y-polarized wave versus the propagation length (t=0.8a, tup=0.2a, w=0.6a, a=132.5 μm, nsi=3.48) for a/λ=0.255, 0.265 and 0.275 obtained by coupled-mode analysis.

Photonic Crystal for Polarization Rotation 315

frequency band of 0.258-0.267. 3D-FDTD simulation results show that the P.C.E higher than 90% is realized within the frequency band of 0.258-0.267; over which the defect mode lies.

(a) (b)

(c) (d)

is much sharper than the slope of the rise of *ax2(z)*. More importantly, *ay*

the input plane (t=0.8a, tup=0.2a, nsi=3.48, a/λ=0.265, λ=500 µm).

to 3D-FDTD simulation (Fig. 16).

Fig. 12. The contour plot of the cross section of (a) Ey, (b) Hx, (c) Ex and (d) Hy components at

At normalized frequencies higher than 0.267, *Ey* starts leaking energy to the TE-like PC slab modes as it crosses the TE-like PC slab modes, Fig. 6(c). For example, FDTD simulation of the power exchange between x-polarized and y-polarized waves for a/λ=0.275 is graphed in Fig. 16. It is seen that for a/λ=0.275, the slope of the drop of *ay*

much faster than that of a/λ=0.265, Fig. 14. This observation can be interpreted as if *Ey* is dissipating and leaking energy into the TE-like slab modes. Thus, a sudden drop on power exchange rate is observed at normalized frequencies higher than 0.267. Semivectorial BPM analysis utilized for modal analysis is not capable of including the PC modes; thus, in power exchange graph calculated by coupled-mode analysis for normalized frequency of a/λ=0.275 (Fig. 11), no power dissipation is observed as opposed

*2(z)*

*2(z)* is dropping

#### **3.4 3D-FDTD simulation of polarization rotator structure**

Both Design methodologies suggest that high power exchange rate is expected to be observed within the frequency band of 0.258-0.267, the overlap frequency band between the fast and slow mode guiding. To verify the aforementioned results, 3D-FDTD was employed to simulate the polarization rotator. The simulated structure (Fig. 3(a)) consists of 70 rows of holes along the propagation direction (z-direction) and 11 rows of holes (including the defect row) in x-direction. The mesh sizes along the x, y and z-directions (∆x, ∆y and ∆z) are ∆x=∆z=0.0331λ and ∆y=0.0172λ. The perfectly matched layer (PML) boundary condition was applied for all three directions. Time waveforms in 3D\_FDTD were chosen as a single frequency sinusoid. The spatial distribution of the incident field was Guassian. The frequency of the input signal lies within the frequency band of the defect mode (0.258-0.267 corresponding to 586-601 GHz). As the wave proceeds, the polarization of the input signal starts rotating. The power exchange between (*Ex*, *Ey*) and (*Hx*, *Hy*) components was observed. To achieve the maximum power conversion, the size of the last top silicon brick was 15a instead of 10a. Fig. 12 shows the contour plot of transverse field components, *Ex*, *Ey*, *Hx* and *Hy* at the input for a/λ=0.265. The input excitation is TE-like; *Ey* and *Hx* are the dominant components and have even parity as opposed to the non-dominant components *Ex* and *Hy* that have odd symmetry with respect to y=0 plane.

As the wave proceeds, the power exchange is observed between (*Ey*, *Ex*) and (*Hx*, *Hy*) components. The contour plot of Ex and Hy at a point close to the output are plotted in Fig. 13. It is seen that the parity of the *Ex* and *Hy* components have changed and become the dominant component. The amplitudes of *Ey* and *Hx* have been decreased more than an order of magnitude and reached to zero at the output plane. Thus, 90o rotation of polarization is realized at the output.

To show the power exchange between the two polarizations, the z-varying square amplitudes of *Ex* and *Ey* components were graphed. Fig. 14 shows *ax 2(z)* and *ay 2(z)* along the propagation direction for the normalized frequency of 0.265 corresponding to the free space wavelength of 500 μm. The two main elements contributing to the numerical noise are local reflections and imperfections of absorbing layer. Dots in the figure are the actual values of 3D-FDTD analysis. To have a smooth picture of *ax 2(z)* and *ay 2(z)* variations along the propagation direction, a polynomial fit to the data using least square method is also shown in the figure. Each plot consists of more than 100 data points. The FDTD "turn-on" transition of the input wave has also been included in the graph (first 0.5 mm). This portion is obviously a numerical artifact of the FDTD scheme. After almost 6 mm (12λ), the complete exchange, as can be seen in the Fig. 14, has taken place. Comparing this graph with coupled-mode (counterpart plot in Fig. 11), it is seen that the power exchange between the two polarizations takes place at smaller propagation distance; 6 mm in comparison with 7.2 mm. Moreover, the value of P.C.E obtained by 3D-FDTD is close to 100 %; whereas, P.C.E for the same wavelength for coupledmode analysis is 96%. On the other hand, normal mode analysis method predicted that 100% polarization conversion could take place at less than 4.5Lπ, 6 mm. Therefore, normal mode analysis design methodology provides more accurate results.

3D-FDTD simulations were repeated for other frequencies to obtain the frequency dependence of polarization conversion. The power exchange rate for both coupled-mode analysis and 3D-FDTD are graphed versus the normalized frequency in Fig. 15. Coupledmode analysis shows that P.C.E of higher than 90% is achieved within the normalized

Both Design methodologies suggest that high power exchange rate is expected to be observed within the frequency band of 0.258-0.267, the overlap frequency band between the fast and slow mode guiding. To verify the aforementioned results, 3D-FDTD was employed to simulate the polarization rotator. The simulated structure (Fig. 3(a)) consists of 70 rows of holes along the propagation direction (z-direction) and 11 rows of holes (including the defect row) in x-direction. The mesh sizes along the x, y and z-directions (∆x, ∆y and ∆z) are ∆x=∆z=0.0331λ and ∆y=0.0172λ. The perfectly matched layer (PML) boundary condition was applied for all three directions. Time waveforms in 3D\_FDTD were chosen as a single frequency sinusoid. The spatial distribution of the incident field was Guassian. The frequency of the input signal lies within the frequency band of the defect mode (0.258-0.267 corresponding to 586-601 GHz). As the wave proceeds, the polarization of the input signal starts rotating. The power exchange between (*Ex*, *Ey*) and (*Hx*, *Hy*) components was observed. To achieve the maximum power conversion, the size of the last top silicon brick was 15a instead of 10a. Fig. 12 shows the contour plot of transverse field components, *Ex*, *Ey*, *Hx* and *Hy* at the input for a/λ=0.265. The input excitation is TE-like; *Ey* and *Hx* are the dominant components and have even parity as opposed to the non-dominant components *Ex* and *Hy*

As the wave proceeds, the power exchange is observed between (*Ey*, *Ex*) and (*Hx*, *Hy*) components. The contour plot of Ex and Hy at a point close to the output are plotted in Fig. 13. It is seen that the parity of the *Ex* and *Hy* components have changed and become the dominant component. The amplitudes of *Ey* and *Hx* have been decreased more than an order of magnitude and reached to zero at the output plane. Thus, 90o rotation of polarization is

To show the power exchange between the two polarizations, the z-varying square amplitudes

direction for the normalized frequency of 0.265 corresponding to the free space wavelength of 500 μm. The two main elements contributing to the numerical noise are local reflections and imperfections of absorbing layer. Dots in the figure are the actual values of 3D-FDTD analysis.

polynomial fit to the data using least square method is also shown in the figure. Each plot consists of more than 100 data points. The FDTD "turn-on" transition of the input wave has also been included in the graph (first 0.5 mm). This portion is obviously a numerical artifact of the FDTD scheme. After almost 6 mm (12λ), the complete exchange, as can be seen in the Fig. 14, has taken place. Comparing this graph with coupled-mode (counterpart plot in Fig. 11), it is seen that the power exchange between the two polarizations takes place at smaller propagation distance; 6 mm in comparison with 7.2 mm. Moreover, the value of P.C.E obtained by 3D-FDTD is close to 100 %; whereas, P.C.E for the same wavelength for coupledmode analysis is 96%. On the other hand, normal mode analysis method predicted that 100% polarization conversion could take place at less than 4.5Lπ, 6 mm. Therefore, normal mode

3D-FDTD simulations were repeated for other frequencies to obtain the frequency dependence of polarization conversion. The power exchange rate for both coupled-mode analysis and 3D-FDTD are graphed versus the normalized frequency in Fig. 15. Coupledmode analysis shows that P.C.E of higher than 90% is achieved within the normalized

*2(z)* and *ay*

*2(z)* and *ay*

*2(z)* variations along the propagation direction, a

*2(z)* along the propagation

**3.4 3D-FDTD simulation of polarization rotator structure** 

that have odd symmetry with respect to y=0 plane.

of *Ex* and *Ey* components were graphed. Fig. 14 shows *ax*

analysis design methodology provides more accurate results.

realized at the output.

To have a smooth picture of *ax*

frequency band of 0.258-0.267. 3D-FDTD simulation results show that the P.C.E higher than 90% is realized within the frequency band of 0.258-0.267; over which the defect mode lies.

Fig. 12. The contour plot of the cross section of (a) Ey, (b) Hx, (c) Ex and (d) Hy components at the input plane (t=0.8a, tup=0.2a, nsi=3.48, a/λ=0.265, λ=500 µm).

At normalized frequencies higher than 0.267, *Ey* starts leaking energy to the TE-like PC slab modes as it crosses the TE-like PC slab modes, Fig. 6(c). For example, FDTD simulation of the power exchange between x-polarized and y-polarized waves for a/λ=0.275 is graphed in Fig. 16. It is seen that for a/λ=0.275, the slope of the drop of *ay 2(z)* is much sharper than the slope of the rise of *ax2(z)*. More importantly, *ay 2(z)* is dropping much faster than that of a/λ=0.265, Fig. 14. This observation can be interpreted as if *Ey* is dissipating and leaking energy into the TE-like slab modes. Thus, a sudden drop on power exchange rate is observed at normalized frequencies higher than 0.267. Semivectorial BPM analysis utilized for modal analysis is not capable of including the PC modes; thus, in power exchange graph calculated by coupled-mode analysis for normalized frequency of a/λ=0.275 (Fig. 11), no power dissipation is observed as opposed to 3D-FDTD simulation (Fig. 16).

Photonic Crystal for Polarization Rotation 317

At frequencies lower than a/λ<0.255, only y-polarized wave is guided; thus, no power exchange between the two polarization takes place. Our recommendation is to avoid this region for the design of the polarization rotator. Having compared FDTD and coupled-mode analyses, coupled-mode theory approach is effective within the frequency band where xpolarized and y-polarized guiding overlap. The other approach described in 3.3.1.2 can predict the frequency response of the PC based polarization rotator. In polarization rotation angel graph versus frequency obtained using normal mode analysis, a sudden jump in the polarization rotation angel at normalized frequencies larger than 0.268 and smaller than 0.257 is observed that is a sign of changing the behavior of the modes; hence the polarization rotator.

Fig. 15. Power exchange between the x-polarized and y-polarized waves versus frequency for both coupled-mode analysis and 3D-FDTD simulations ((t=0.8a, tup=0.2a, w=0.6a,

Fig. 16. Power exchange between the x-polarized and y-polarized wave versus the propagation length for a/λ=0.275 obtained by 3D-FDTD simulation (t=0.8a, tup=0.2a,

a=132.5 μm, nsi=3.48).

w=0.6a, a=132.5 μm, nsi=3.48, λ=500 μm).

(b)

Fig. 13. Contour plot of the cross section of (a) Ex and (b) Hy at z=5.5 mm (t=0.8a, tup=0.2a, nsi=3.48, a/λ=0.265, λ=500 µm).

Fig. 14. Power exchange between the x-polarized and y-polarized wave versus the propagation length for a/λ=0.265 obtained by 3D-FDTD simulation (t=0.8a, tup=0.2a, w=0.6a, a=132.5 μm, nsi=3.48, λ=500 μm).

(a)

 (b) Fig. 13. Contour plot of the cross section of (a) Ex and (b) Hy at z=5.5 mm (t=0.8a, tup=0.2a,

Fig. 14. Power exchange between the x-polarized and y-polarized wave versus the propagation length for a/λ=0.265 obtained by 3D-FDTD simulation (t=0.8a, tup=0.2a,

nsi=3.48, a/λ=0.265, λ=500 µm).

w=0.6a, a=132.5 μm, nsi=3.48, λ=500 μm).

At frequencies lower than a/λ<0.255, only y-polarized wave is guided; thus, no power exchange between the two polarization takes place. Our recommendation is to avoid this region for the design of the polarization rotator. Having compared FDTD and coupled-mode analyses, coupled-mode theory approach is effective within the frequency band where xpolarized and y-polarized guiding overlap. The other approach described in 3.3.1.2 can predict the frequency response of the PC based polarization rotator. In polarization rotation angel graph versus frequency obtained using normal mode analysis, a sudden jump in the polarization rotation angel at normalized frequencies larger than 0.268 and smaller than 0.257 is observed that is a sign of changing the behavior of the modes; hence the polarization rotator.

Fig. 15. Power exchange between the x-polarized and y-polarized waves versus frequency for both coupled-mode analysis and 3D-FDTD simulations ((t=0.8a, tup=0.2a, w=0.6a, a=132.5 μm, nsi=3.48).

Fig. 16. Power exchange between the x-polarized and y-polarized wave versus the propagation length for a/λ=0.275 obtained by 3D-FDTD simulation (t=0.8a, tup=0.2a, w=0.6a, a=132.5 μm, nsi=3.48, λ=500 μm).

Photonic Crystal for Polarization Rotation 319

shows the SEM picture of the backside. It can be seen that the back is etched uniformly; the

(a)

Polarization rotation devices for potential applications in the THz frequency band (200 GHz – 1THz) were fabricated. The fabrication of this PC based polarization rotator is more complex in a sense that the front side processing requires two sets of masks. The first mask is employed to create the periodic loading layers. The second mask is for patterning of the PC slab waveguide. The third mask is used to open window at the back side of the structure. Fig. 18(a) shows the SEM picture of the periodic asymmetric loaded PC slab waveguide with square holes. The SEM picture shows that the walls are very sharp. In Fig. 18(b), the SEM picture of the periodic asymmetric loaded PC slab waveguide for circular air

Fig. 17. SEM picture of fabricated (a) PC membrane slab waveguide (b) front side and

(b) (c)

backside of the PC structure.

holes pattern is presented.

oxide at the back can be easily removed by buffered HF (BHF).
