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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 9 — Apr. 27, 2009
  • pp: 7145–7158
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Ultra-compact photonic crystal based polarization rotator

Khadijeh Bayat, Sujeet K. Chaudhuri, and Safieddin Safavi-Naeini  »View Author Affiliations


Optics Express, Vol. 17, Issue 9, pp. 7145-7158 (2009)
http://dx.doi.org/10.1364/OE.17.007145


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Abstract

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

1. Introduction

Photonic crystals (PC) prohibit propagation of electromagnetic wave within frequency band of photonic band gap (PBG) [1

1. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10, 283–295 (1993). [CrossRef]

]. Compact devices such as waveguides, resonators, delay lines and phase shifters can be realized [2

2. W. Bogaerts, R. R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. V. Campnhout, P. Bienstman, and D. V. Thourhout., “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23, 401–421 (2005). [CrossRef]

]–[3

3. W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. V. Campenhout, P. Bienstman, D. V. Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in Silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004). [CrossRef] [PubMed]

].

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

4. R. C. Alferness, “Electrooptic guided-wave devices for general polarization transformations,” IEEE J. Quantum Electron. 17, 965–969 (1981). [CrossRef]

]. 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

6. B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, “Design and Characterization of Compact Single-Section Passive Polarization Rotator,” J. Lightwave Technol. 19, 512–519 (2001). [CrossRef]

]. 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 [7

7. Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991). [CrossRef]

]. 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 [9

9. W. Huang and Z. M. Mao, “Polarization rotation in periodic loaded rib waveguides,” IEEE J. Lightwave Technol. 10, 1825–1831 (1992). [CrossRef]

]. 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 [10

10. S. S. A. Obayya, B. M. A. Rahman, and H. A. El-Mikati, “Vector beam propagation analysis of polarization conversion in periodically loaded waveguides,” IEEE Photon. Tech. Lett. 12, 1346–1348 (2000). [CrossRef]

]. Watts and Haus proposed a mode-evolution-based polarization rotator structure [8

8. M. R. Watts and H. A. Haus, “Integrated mode-evolution-based polarization rotators,” Opt. Lett. 30, 138–140 (2005). [CrossRef] [PubMed]

]. 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 [11

11. M. Palamaru and Ph. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. 78, 1466–1468 (2001). [CrossRef]

].

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.

2. Theory

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 [15

15. W. P. Haung, S. T. Chu, and S. K. chaudhuri, “Scalar coupled-mode theory with vector correction,” IEEE J. Quantum Electron. 28, 184–193 (1992). [CrossRef]

]:

2Exz2+t2Ex+n2k2Ex=x(Ex1n2n2x)x(Ey1n2n2y)
(1.a)
2Eyz2+t2Ey+n2k2Ey=y(Ex1n2n2x)y(Ey1n2n2y)
(1.b)

where, n is the refractive index distribution of the waveguide and ∇t 2 is the transverse differential operator defined as:

t2=2x2+2y2

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 [9

9. W. Huang and Z. M. Mao, “Polarization rotation in periodic loaded rib waveguides,” IEEE J. Lightwave Technol. 10, 1825–1831 (1992). [CrossRef]

].

Fig. 1. The sketch of (a) periodic asymmetric loaded triangular PC slab waveguide (b) asymmetric loaded PC slab waveguide.

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:

E=Exx̂+Eyŷ=ax(z)ex(x,y)ejβxz+ay(z)ey(x,y)ejβyz,
(2)

Where ex(x, y)e -xz and ey(x, y)e -yz 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 xz and e yz, respectively, and assuming that the amplitude of the field are slowly varying along z-direction (propagation direction); the following equation has been obtained:

j2βxexdax(z)dz+ax(z)t2ex+n2k2ax(z)exβx2ax(z)ex=
ax(z)x(ex1n2n2x)ay(z)ejΔzx(ey1n2n2y)
(3.a)
j2βyeyday(z)dz+ay(z)t2ey+n2k2ay(z)eyβy2ay(z)ey=
ay(z)y(ey1n2n2y)ax(z)ejΔzy(ex1n2n2x)
(3.b)

Where: ∆ = βy - βx,

By invoking the following assumption:

t2ex+(n2k2βave2)ex=0
t2ey+(n2k2βave2)ey=0,
(4)

where, βave=βx+βy2,

a simplified form of Eq. (3) is obtained:

j2βxexdax(z)dz+(βave2βx2)ax(z)ex=
ax(z)x(ex1n2n2x)ay(z)ejΔzx(ey1n2n2y)
(5.a)
j2βyeyday(z)dz+(βave2βx2)ay(z)ey=
ay(z)y(ey1n2n2y)ax(z)ejΔzy(ex1n2n2x)
(5.b)

Multiplying both sides of Eq. (5.a), and Eq. (5.b) by ex * and ey * (*- conjugate), respectively and integrating over the cross-section, the following coupled mode equations are obtained:

dax(z)dz=jκxxax(z)jκxyay(z)
day(z)dz=jκyyay(z)jκyxax(z)
(6)

Where:

κxx=(βave2βx2)∫∫ex*·exdxdy+∫∫ex*·y(ey1n2n2y)dxdy2βx∫∫ex*·exdxdy
(7. a)
κxy=ejΔz∫∫ex*·x(ey1n2n2y)dxdy2βx∫∫ex*·exdxdy
(7.b)
κyy=(βave2βy2)∫∫ey*·eydxdy+∫∫ey*·y(ey1n2n2y)dxdy2βy∫∫ey*·eydxdy
(7.c)
κyx=ejΔz∫∫ey*·y(ex1n2n2x)dxdy2βy∫∫ey*·eydxdy
(7.d)

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 [9

9. W. Huang and Z. M. Mao, “Polarization rotation in periodic loaded rib waveguides,” IEEE J. Lightwave Technol. 10, 1825–1831 (1992). [CrossRef]

]. 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:

Mi±=(cos(Ωizi)jcos(φi/2)sin(Ωizi)jsin(φi/2)sin(Ωizi)jsin(φi/2)sin(Ωizi)cos(Ωizi)+jcos(φi/2)sin(Ωizi))i=1,2
(8)
Ωi=δi2+κi2
δi=κxxiκyyi2
tan(φi/2)=κiδi
(9)

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:

z1=aw
(10.a)
z2=w
(10.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±=M1±M2±,
(11)

the loading period can be approximated as follows:

ΛπΩ1+Ω2,
(12)

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 kxykyx *; 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).

Fig. 2. 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. Results and discussion

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 [14

14. S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]

]. 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.

Fig. 3. (a) The supercell of the asymmetric loaded PC slab waveguide for PWEM analysis. (b) The band diagram for the asymmetric loaded PC slab waveguide obtained by PWEM.

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.

Fig. 4. The distribution of (a) Hy and (b) Hx at (x-y) plane and (c) Hy and (d) Hx at y=0 plane in asymmetric loaded PC slab waveguide with w=0.6a, t=0.8a, tup=0.2a, a=132.5 μm and a/λ=0.245; input has launched at z=0.

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:

P.C.E=PTMPTM+PTE×100=ax2ax2+ay2×100,
(13)

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).

Fig. 5. The profile of (a) Ex and (b)Ey components of x-polarized and y-polarized modes of the structure shown in Fig. 1(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).
Fig. 6. 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.

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.

Fig. 7. The contour plot of the cross section of (a) Ey, (b) Hx, (c) Ex and (d) Hy components of electromagnetic field propagating in asymmetric loaded PC slab waveguide at the input plane and (e) Ex and (f) Hy at z=5.5 mm (w=0.6a, t=0.8a, tup=0.2a, a/λ=0.265 and λ=500 μm).

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.

Fig. 8. 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).
Fig. 9. 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).

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.

Fig. 10. 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).

4. Conclusion

References and links

1.

E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10, 283–295 (1993). [CrossRef]

2.

W. Bogaerts, R. R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. V. Campnhout, P. Bienstman, and D. V. Thourhout., “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23, 401–421 (2005). [CrossRef]

3.

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. V. Campenhout, P. Bienstman, D. V. Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in Silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004). [CrossRef] [PubMed]

4.

R. C. Alferness, “Electrooptic guided-wave devices for general polarization transformations,” IEEE J. Quantum Electron. 17, 965–969 (1981). [CrossRef]

5.

F. K. Reinhart, R. A. Logan, and W. R. Sinclair, “Electrooptic polarization modulation multielectrode Alx-Ga1-xAs rib waveguides,” IEEE J. Quantum Electron. 18, 763–766 (1982). [CrossRef]

6.

B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, “Design and Characterization of Compact Single-Section Passive Polarization Rotator,” J. Lightwave Technol. 19, 512–519 (2001). [CrossRef]

7.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991). [CrossRef]

8.

M. R. Watts and H. A. Haus, “Integrated mode-evolution-based polarization rotators,” Opt. Lett. 30, 138–140 (2005). [CrossRef] [PubMed]

9.

W. Huang and Z. M. Mao, “Polarization rotation in periodic loaded rib waveguides,” IEEE J. Lightwave Technol. 10, 1825–1831 (1992). [CrossRef]

10.

S. S. A. Obayya, B. M. A. Rahman, and H. A. El-Mikati, “Vector beam propagation analysis of polarization conversion in periodically loaded waveguides,” IEEE Photon. Tech. Lett. 12, 1346–1348 (2000). [CrossRef]

11.

M. Palamaru and Ph. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. 78, 1466–1468 (2001). [CrossRef]

12.

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljacic, “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys.Lett. 84, 3639–3641 (2004). [CrossRef]

13.

S. G. Johnson, P. Bienstman, M. A. Skorobogati, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002). [CrossRef]

14.

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]

15.

W. P. Haung, S. T. Chu, and S. K. chaudhuri, “Scalar coupled-mode theory with vector correction,” IEEE J. Quantum Electron. 28, 184–193 (1992). [CrossRef]

16.

D. R. Solli and J. M. Hickmann, “Periodic crystal based polarization control devices,” J. Phys. D 37, R263–R268 (2004). [CrossRef]

17.

K. Bayat, S. K. Chaudhuri, and S. Safavi-Naeini, “Polarization and thickness dependent guiding in the photonic crystal slab waveguide,” Opt. Express 15, 8391–8400 (2007). [CrossRef] [PubMed]

OCIS Codes
(220.0220) Optical design and fabrication : Optical design and fabrication
(230.0230) Optical devices : Optical devices
(250.0250) Optoelectronics : Optoelectronics

ToC Category:
Optical Devices

History
Original Manuscript: November 19, 2008
Revised Manuscript: January 24, 2009
Manuscript Accepted: March 30, 2009
Published: April 15, 2009

Citation
Khadijeh Bayat, Sujeet K. Chaudhuri, and Safieddin Safavi-Naeini, "Ultra-compact photonic crystal based polarization rotator," Opt. Express 17, 7145-7158 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7145


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References

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