An asymmetrically loaded photonic crystal based polarization rotator has been introduced, designed and simulated. The polarization rotator structure consists of a single defect line photonic crystal slab waveguide with asymmetrically etched upper layer. To continue the rotation from a given input polarization to the desired output polarization the upper layer is alternated on either side of the defect line, periodically. Coupled mode theory based on semi-vectorial modes and plane wave expansion methods are employed to design the polarization rotator structure around a particular frequency band of interest. The 3D-FDTD simulation results agree with the coupled mode analysis around the region of interest specified during the design. Complete polarization rotation is achieved over the propagation length of 12λ. For this length, the coupling efficiency higher than 90% is achieved within the normalized frequency band of 0.258–0.262.
© 2009 Optical Society of America
Photonic crystals (PC) prohibit propagation of electromagnetic wave within frequency band of photonic band gap (PBG) . Compact devices such as waveguides, resonators, delay lines and phase shifters can be realized –.
Polarization rotator is a crucial element of integrated photonic circuits that controls and manipulates polarization of the propagating wave. Polarization rotation devices in optical frequency band are mostly composed of electro-optic material, which utilize the anisotropic property of the materials [4–5]. On the other hand, passive polarization rotators were realized relying on the asymmetry of the structure such as slanted waveguide, periodic loaded rib waveguide based polarization rotator structures and mode-evolution-based polarization rotator [6–8]. The fabrication process of slanted waveguide structure is complicated in the sense that both dry and wet etchings are required, which is a bottleneck for implementation of the device in an integrated circuit. Periodic asymmetric loaded rib waveguide based polarization rotator was demonstrated experimentally by Shani . The total length of the device was several millimeter. Haung and Mao employed coupled mode theory based on scalar modes to analyze the structure theoretically . Obayya, et al. employed a numerical 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 . Watts and Haus proposed a mode-evolution-based polarization rotator structure . The proposed structure was formed by twisting a waveguide which causes the rotation of optical axes. The structure consists of two layers that are tapered oppositely; in other word, the large aspect ratio waveguide (TM guiding) is tapered to small aspect ratio waveguide (TE guiding). The structure was designed using coupled mode theory and simulated by 3D finite-difference time domain (FDTD) method. The simulation results indicated that polarization rotation was achieved at propagation distance less than 100 μm.
Here, we take advantage of the compact guiding structure of PC slab waveguide and introduce a new compact PC based polarization rotator structures. The proposed structure 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 (loaded layer) is the same material as the defect line that has been etched asymmetrically with 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, at half beat lengths, is alternated on either side of the z-axis. Figure 1(a) gives a sketch of the proposed structure. The proposed structure is described as periodic asymmetric loaded PC slab waveguide. Periodic asymmetric loaded structure is the best option for integrated PC circuit in a sense that its fabrication process is compatible with planar integrated printed circuit technology. Moreover, due to the large birefringence of PC structures, the 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. The main obstacles for employing mode-evolution based structure in PC based polarization rotators are the large radiation loss due to tapering and incompatibility of their fabrication process with planar integrated printed circuit technology. Tapering in PC accompanies with it significant radiation losses .
Due to the compactness of the proposed structure, a rigorous numerical method, 3D-FDTD can be employed to analyze and simulate the final designed structure. However, for the quick preliminary design, an analytical method that provides good approximate values of the structural parameters is preferred. Coupled-mode theory is a robust and well-known method for such analyses of perturbed waveguide structures. Thus, a coupled-mode theory based on semi-vectorial modes is developed here for propagation modeling on PC structures. In essence, we wish to develop a simple yet closed form method to carry out the initial design of the device of interest. In the next step, we refine the design by using rigorous but numerically expensive 3D-FDTD simulations. We believe this approach leads to optimization of device parameters easily, if desired. There are other possible formulations such as expansion in terms of Bloch modes, etc [12–13]. However, these methods would add more complexity to the design process and may only lead to slightly more accurate results.
For a complex structure such as PC slab waveguide where the propagation characteristics is extremely frequency dependent, the frequency band over which both x-polarized and y-polarized waves are guided must be calculated prior to the coupled mode analysis so that the design could be optimized for this frequency band. Therefore, plane wave expansion method (PWEM) was employed for full-vectorial modal analysis of the asymmetric loaded PC slab waveguide . For coupled mode analysis, the semi-vectorial modes of the asymmetric loaded PC slab waveguide, Fig. 1 (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 y-polarized 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 can be expected. To verify the results obtained using the above combination of PWEM and coupled mode theory, a rigorous three-dimensional finite difference time domain (3D-FDTD) simulation was employed on the designed structure. The 3D-FDTD simulation results agree with coupled-mode theory and PWEM method results well, in the frequency band specified in the design.
In section II, the coupled mode theory used here is described briefly. In section III, the design and simulation results are presented and discussed. Finally, we conclude the paper with a summary of the key achievements reported in this paper.
The schematic of the square-hole PC slab polarization rotator is shown in Fig. 1(a). In this structure, the unit cell, the width of the square holes, the thickness of the silicon PC and the thickness of the upper 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 :
where, n is the refractive index distribution of the waveguide and ∇t 2 is the transverse differential operator defined as:
The vector properties are manifested on the right hand side of Eq. (1.a) and Eq. (1.b); which indicates that the two orthogonal polarizations may be coupled to each other as a result of geometrical asymmetry. Huang and Mao employed similar coupled mode theory based on the scalar modes to analyze the 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 . However, scalar modal analysis completely ignores the polarization dependence of the wave propagation; thus, scalar modal analysis 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 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 PC structure, the cross-section varies along the propagation direction within one unit cell. Employing square holes instead of circle holes simplifies the problem of modeling such structures. The PC lattice is triangular. According to Fig. 2, 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. 1(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 y-polarized waves. Assuming that the profile of the total transverse field in the asymmetric loaded PC slab waveguide is represented as following:
Where ex(x, y)e -jβxz and ey(x, y)e -jβyz are x- and y-components of electric field of the semi-vectorial solution of wave equation for x-polarized and y-polarized waves, respectively. βx and βy are propagation constants along x and y directions, respectively. Substituting Eq. (2) into Eq. (1) and multiplying both side of Eq. (1.a), and Eq. (1.b) by e jβxz and e jβyz, respectively, and assuming that the amplitude of the field are slowly varying along z-direction (propagation direction); the following equation has been obtained:
Where: ∆ = βy - βx,
By invoking the following assumption:
a simplified form of Eq. (3) is obtained:
κ xx and κ yy are the self-coupling coefficients; whereas, κ xy and κ yx refer to cross-coupling coefficients. In Eq. (7.a), and Eq. (7.c), in our case, we have noted that the second terms are negligible in comparison with the first terms. The coupling coefficients must be solved for both regions 1 and 2 (see Fig. 2), using Eq. (7). The distribution of the electric fields in both regions are the same; where as, the refractive index profile is different as depicted in Fig. 2 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 (kxy=kyx=k), the coupled-mode equations could be solved analytically as presented in Eq. (8) 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:
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. 2. Assuming that w is the width of a square hole, z1 and z2 are determined as following:
Having set M1 and M2 as the transfer matrix of regions 1 and 2, the transfer matrix for one unit cell is obtained:
the loading period can be approximated as follows:
Thus, the length of one top silicon brick is Λ and the top cladding layer alternates periodically throughout the propagation length. The simulation results revealed that for our structure, ∣kxyȣ ≈ ∣kyx∣ and kxy ≈ kyx *; where the imaginary parts were very small. Numerical, and analytical solutions of the coupled mode theory, Eq. (6.a) and (6.b), give us almost the same results. From Eq. (12) we calculated the preliminary value of the loading period before employing the numerical method to solve the coupled mode equation, Eq. (6).
The propagation characteristics in a PC slab waveguide are strongly polarization dependent and for polarization rotator structure, it is essential to design the PC slab waveguide so that maximum overlap between the TE-like and TM-like guiding could be achieved. We have previously shown that by optimizing the thickness of the PC slab waveguide, maximum overlap between the guiding frequency bands of two polarizations could be realized . Here, we employed the same design methodology introduced in our previous publication  to achieve a thickness that provides maximum overlap between TE-like and TM-like guiding. Then, the coupled mode analysis was carried out in aforementioned frequency band. Finally, to verify the validity of coupled mode analysis, the structure was simulated by 3D-FDTD method, as well.
In next section, first the band structure of the asymmetric loaded PC slab waveguide is calculated using PWEM. Combining the coupled mode and PWEM results, the frequency band over which lossless polarization rotation can be realized is estimated. Finally, to verify and fine-tune the design parameters of the polarization rotator structure, 3D-FDTD is employed.
3. Results and discussion
The first step of the design of the polarization rotator structure is to calculate the frequency band over which loss-less propagation takes place and then proceed with the design using coupled mode theory within the aforementioned frequency band. The asymmetric loaded PC slab waveguide shown in Fig. 1(b) was first simulated using PWEM to obtain the band diagram and the frequency band of the defect modes. The thickness and width of the squares were chosen based on the design methodology presented in previous publication in Optics Express . For t=0.8a and w=0.6a, it is expected to reach a wide frequency band over which both polarizations are guided. The larger the thickness of the upper layer is, the stronger the geometrical asymmetry is leading to smaller rotation length. However, to comply with the fabrication constrains the thickness of the upper layer tup was chosen to be 0.2a. The refractive index of silicon (nsi) is 3.48.
In PWEM analysis, it is assumed that the structure is periodic in all direction. However, the PC slab structure has a finite thickness (vertical dimension). The periodicity can be artificially introduced in the vertical (y) direction by introducing a sequence of slabs separated by sufficient amount of air to maintain electromagnetic isolation. This method is called super-cell method . The super-cell becomes the new unit cell and now the periodic boundary condition will apply in all directions. Figure 3(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. In the diagram shown in Fig. 3(b) calculated by PWEM, the PC slab modes are divided into TE-like and TM-like waves and the guided mode inside the defect line is depicted by defect mode in the figure. It is seen that there is a defect mode within the frequency band of 0.242–0.25 that lies inside the frequency band of the TM-like PC slab modes. Thus, this mode leaks energy to the TM-like PC slab modes. Figures 4(a) and (b) show the cross sections of Hy and Hx components of the defect mode at the normalized frequency of 0.245, respectively. Since the non-dominant components of TM-like PC slab mode (Ey, Hx and Hz) are weak in comparison with the corresponding components of the defect mode, they are not detected in Fig. 4(b). However, Hy component of TM-like PC slab mode is more pronounced in Fig. 4(a). Hy and Hx distribution over y=0 plane is plotted in Figs. 4(c) and (d), respectively. The presence of PC slab modes can be clearly seen in Fig. 4(c). Decay of amplitude of Hx, Fig. 4(d), is the indication of detachment of energy by TM-like PC slab modes. Thus, the defect mode is lossy and can not be used for polarization rotation application.
In Fig. 3(b), there is another mode sitting within the frequency band of 0.258–0.267 for which all six components are guided. It lies above the TM-like PC slab modes. It crosses the TE-like PC slab modes at the normalized frequency of 0.267 where Hx and Ey start to couple energy to the TE-like PC slab modes. As a result, the frequency band over which both x-polarized and x-polarized modes are guided and the polarization rotator is expected to function properly is 0.258–0.268.
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). Next, we use the coupled mode analysis discussed earlier 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. 1(b), must be calculated. Semi-vectorial BPM was employed for semi-vectorial modal analysis of the structure. The outputs of semi-vectorial BPM analysis are electric field distribution and effective index of the mode. The normalized electric field for x-polarized (TM-like) and y-polarized (TE-like) waves for the normalized frequency of a/λ=0.265 are shown in Figs. 5(a) and 5(b), respectively.
Figure 5 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 in 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. 1(b)) were calculated using Eq. (7) for both regions of 1 and 2, depicted in Fig. 2. Using Eq. (9), the loading period was calculated to be approximately, 10a. Fig. 6 shows the power exchange between the two polarizations along the propagation distance for a/λ=0.275, 0.265 and 0.255, λ=500 μm (600GHz).
Defining the power conversion efficiency (P.C.E.) as following:
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. 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 (0.258-0.268).
PWEM and coupled mode analysis of the structure suggest that high power exchange rate is expected within the frequency band of the defect mode. To verify the aforementioned results, 3D-FDTD was employed to simulate the structure numerically. The simulated structure (Fig. 1(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 and Ey and Hx and Hy components was observed. To achieve the maximum power conversion, the size of the last top silicon brick was 15a instead of 10a. Figure 7 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 at a point close to the output reveals 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, 90° 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. 8 shows ax2(z) and ay2(z) along the propagation direction for the normalized frequency of 0.265 corresponding to the free space wavelength of 500 μm. The numerical noise mainly consists of two elements including imperfection in absorbing boundaries (PML) and local reflections from PC walls. Dots in the figure are the actual values of 3D-FDTD analysis. To have a smooth picture of ax2(z) and ay2(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.8, has taken place. Comparing this graph with coupled-mode (counterpart plot in Fig. 6), 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 coupled-mode analysis is 96%.
In coupled mode theory, it is assumed that the amplitude of electric field is varying slowly along the propagation direction [see Eq. (2)]. The polynomials extrapolated to 3D-FDTD data presents a good approximation of the variation of the square of the amplitude of electric field. Both ax2(z) and ay2(z) are varying slowly with z; thus, expansion of electric field in the from of Eq. (2) is validated for this design case.
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. 9. Coupled-mode analysis shows that P.C.E of higher than 80% is achieved within the normalized frequency band of 0.255–0.27. The 3D-FDTD simulation results show that the P.C.E higher than 90% is realized within the frequency band of 0.259–0.267; over which the defect mode lies. 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. 3(b). For example, the FDTD simulation of the power exchange between x-polarized and y-polarized waves for a/λ=0.275 is graphed in Fig. 10. It is seen that for a/λ=0.275, the slope of the drop of ay2(z) is much sharper than the slope of the rise of ax2(z). More importantly, ay2(z) is dropping much faster than that for a/λ=0.265. This observation can be interpreted as if Ey is dissipating and coupling energy to the TE-like slab modes. Thus, a sudden drop on power exchange rate is observed at normalized frequencies higher than 0.267. Semi-vectorial 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. 6), no power dissipation is observed as opposed to 3D-FDTD simulation (Fig. 10).
The behavior of the wave propagation at lower frequencies, a/λ<0.255, is more complicated in a sense that another mode, the defect mode sitting within the frequency band of 0.245–0.255, is involved and our recommendation is to avoid this region for the design of the polarization rotator. Having compared FDTD and coupled-mode analyses, the effectiveness of our approach described in Sec. 2 has been verified. Thus, within the frequency band of the defect mode the coupled mode theory can be employed for preliminary design. The design can be fine-tuned by 3D-FDTD simulation.
A novel and compact PC based periodic asymmetric loaded polarization rotator was proposed and analyzed. The polarization rotator structure consists of single defect line photonic crystal slab waveguide for which the upper layer is asymmetrically etched with respect to the z-axis, propagation direction. To avoid power conversion reversal and synchronize the coupling, the upper layer is alternated on either side of the defect line periodically. Full-vectorial modal analysis was carried out using PWEM to obtain the frequency band over which loss less propagation for both TE-like and TM-like waves could be achieved. Coupled mode theory was developed for triangular PC slab waveguide and employed to analyze the coupling between the x-polarized and y-polarized waves. By applying coupled mode analysis based on semi-vectorial modes within the aforementioned frequency band, the polarization rotator structure was designed in an efficient fashion. The structure was simulated using 3D-FDTD method, as well. The rigorous simulations verified that power conversion efficiency higher than 90% was achieved over the frequency band of 586.4-604.5 GHz corresponding to the normalized frequency of 0.259–0.267 (assuming that a/λ=0.265 corresponds to 600 GHz) within the propagation length of 12λ suggesting that the device is quite compact. Moreover, due to the scalability of Maxwell equations the aforementioned normalized frequency band can be scaled up to optical frequency band. Our approach of combining “analytical” formulations like coupled mode theory and PWEM allows us to follow a “synthesis” route rather than a “brute force analysis” route for the design. This approach, we believe can be generalized for other pc slab waveguide based devices in the future. It also facilitates optimization and integration of these devices for future applications. We are currently fabricating polarization rotating devices on SOI wafers for potential applications in the THz (500GHz- 3THz) region with novel usage in the nano-sensors and biological fields.
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