## Ultra-compact photonic crystal based polarization rotator

Optics Express, Vol. 17, Issue 9, pp. 7145-7158 (2009)

http://dx.doi.org/10.1364/OE.17.007145

Acrobat PDF (688 KB)

### 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 12*λ*. 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

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]

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]

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]

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

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]

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]

## 2. Theory

_{up}, 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.

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]

_{t}

^{2}is the transverse differential operator defined as:

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

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

*e*(

_{x}*x*,

*y*)

*e*

^{-jβxz}and

*e*(

_{y}*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. β

_{x}and β

_{y}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:

*β*-

_{y}*β*,

_{x}*e*

_{x}^{*}and

*e*

_{y}^{*}(

^{*}- conjugate), respectively and integrating over the cross-section, the following coupled mode equations are 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 (

*k*=

_{xy}*k*=

_{yx}*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]

*A*(

*z*)=

*MA*(

*0*); where A is a column vector for coefficients

*a*and

_{x}*a*; the transfer matrix (

_{y}*M*) is expressed as following:

*z*and

_{1}*z*are the length of regions 1 and 2 shown in Fig. 2. Assuming that

_{2}*w*is the width of a square hole,

*z*and

_{1}*z*are determined as following:

_{2}*M*and

_{1}*M*as the transfer matrix of regions 1 and 2, the transfer matrix for one unit cell is obtained:

_{2}*k*ȣ ≈ ∣

_{xy}*k*∣ and

_{yx}*k*≈

_{xy}*k*

_{yx}^{*}; 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).

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]

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]

## 3. Results and discussion

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]

_{up}was chosen to be 0.2a. The refractive index of silicon (n

_{si}) is 3.48.

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]

_{y}, E

_{z}, E

_{x}) and the non-dominant components (E

_{y}, H

_{z}, H

_{x}) have even and odd symmetry w.r.t. y=0 plane, respectively. Similarly, the dominant components of TE-like mode (E

_{y}, H

_{z}, H

_{x}) and the non-dominant components (H

_{y}, E

_{z}, E

_{x}) 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 H

_{y}and H

_{x}components of the defect mode at the normalized frequency of 0.245, respectively. Since the non-dominant components of TM-like PC slab mode (E

_{y}, H

_{x}and H

_{z}) are weak in comparison with the corresponding components of the defect mode, they are not detected in Fig. 4(b). However, H

_{y}component of TM-like PC slab mode is more pronounced in Fig. 4(a). H

_{y}and H

_{x}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 H

_{x}, 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.

_{x}and E

_{y}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.

*λ*). 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.

*λ*=0.275, 0.265 and 0.255,

*λ*=500 μm (600GHz).

*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).

*λ*and ∆y=0.0172

*λ*. 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.

_{x}and E

_{y}and H

_{x}and H

_{y}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, E

_{x}, E

_{y}, H

_{x}and H

_{y}at the input for a/

*λ*=0.265. The input excitation is TE-like ;E

_{y}and H

_{x}are the dominant components and have even parity as opposed to the non-dominant components E

_{x}and H

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

_{y}, E

_{x}) and (H

_{x}, H

_{y}) components. The contour plot at a point close to the output reveals that the parity of the E

_{x}and H

_{y}components have changed and become the dominant component. The amplitudes of E

_{y}and H

_{x}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.

_{x}and E

_{y}components were graphed. Fig. 8 shows

*a*(

_{x}^{2}*z*) and

*a*(

_{y}^{2}*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

*a*(

_{x}^{2}*z*) and

*a*(

_{y}^{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 (12

*λ*), 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%.

*a*(

_{x}^{2}*z*) and

*a*(

_{y}^{2}*z*) are varying slowly with z; thus, expansion of electric field in the from of Eq. (2) is validated for this design case.

_{y}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

*a*(

_{y}^{2}*z*) is much sharper than the slope of the rise of

*a*(

_{x}^{2}*z*). More importantly,

*a*(

_{y}^{2}*z*) is dropping much faster than that for a/

*λ*=0.265. This observation can be interpreted as if E

_{y}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).

*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.

## 4. Conclusion

*λ*=0.265 corresponds to 600 GHz) within the propagation length of 12

*λ*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.

## References and links

1. | E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B |

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. |

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 |

4. | R. C. Alferness, “Electrooptic guided-wave devices for general polarization transformations,” IEEE J. Quantum Electron. |

5. | F. K. Reinhart, R. A. Logan, and W. R. Sinclair, “Electrooptic polarization modulation multielectrode Al |

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. |

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. |

8. | M. R. Watts and H. A. Haus, “Integrated mode-evolution-based polarization rotators,” Opt. Lett. |

9. | W. Huang and Z. M. Mao, “Polarization rotation in periodic loaded rib waveguides,” IEEE J. Lightwave Technol. |

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. |

11. | M. Palamaru and Ph. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. |

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. |

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 |

14. | S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

15. | W. P. Haung, S. T. Chu, and S. K. chaudhuri, “Scalar coupled-mode theory with vector correction,” IEEE J. Quantum Electron. |

16. | D. R. Solli and J. M. Hickmann, “Periodic crystal based polarization control devices,” J. Phys. D |

17. | K. Bayat, S. K. Chaudhuri, and S. Safavi-Naeini, “Polarization and thickness dependent guiding in the photonic crystal slab waveguide,” Opt. Express |

**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

- E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993). [CrossRef]
- 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]
- 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]
- R. C. Alferness, "Electrooptic guided-wave devices for general polarization transformations," IEEE J. Quantum Electron. 17, 965-969 (1981). [CrossRef]
- F. K. Reinhart, R. A. Logan, and W. R. Sinclair, "Electrooptic polarization modulation multielectrode AlxGa1-xAs rib waveguides," IEEE J. Quantum Electron. 18, 763-766 (1982). [CrossRef]
- 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]
- 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]
- M. R. Watts and H. A. Haus, "Integrated mode-evolution-based polarization rotators," Opt. Lett. 30, 138-140 (2005). [CrossRef] [PubMed]
- W. Huang and Z. M. Mao, "Polarization rotation in periodic loaded rib waveguides," IEEE J. Lightwave Technol. 10, 1825-1831 (1992). [CrossRef]
- 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]
- M. Palamaru and Ph. Lalanne, "Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion," Appl. Phys. Lett. 78, 1466-1468 (2001). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- D. R. Solli and J. M. Hickmann, "Periodic crystal based polarization control devices," J. Phys. D 37, R263-R268 (2004). [CrossRef]
- 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]

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