## Reflection minimization at two-dimensional photonic crystal interfaces

Optics Express, Vol. 16, Issue 6, pp. 4270-4277 (2008)

http://dx.doi.org/10.1364/OE.16.004270

Acrobat PDF (987 KB)

### Abstract

We propose a method to design antireflection structures to minimize the reflection of light beams at the interfaces between a two-dimensional photonic crystal and a homogeneous dielectric. The design parameters of the optimal structure to give zero reflection can be obtained from the one-dimensional antireflection coating theory and the finite-difference time-domain simulations. We examine the performance of a Mach-Zehnder interferometer utilizing the self-collimated beams in two-dimensional photonic crystals with and without the optimal antireflection structure introduced. It is shown that the optimal antireflection structure significantly improves the performance of the device.

© 2008 Optical Society of America

## 1. Introduction

1. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. **74**, 1212–1214 (1999). [CrossRef]

2. J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. **8**, 1246–1257 (2002). [CrossRef]

3. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. **90**, 107402 (2003). [CrossRef] [PubMed]

4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B **58**, 10096–10099 (1998). [CrossRef]

**v**

*=∇*

_{g}_{k}

*ω*(

**k**). Thus, the equifrequency contours (EFCs), the cross sections of the dispersion surfaces of the Bloch modes in momentum space, are essential to investigate the propagation properties of lights in PCs and to design the dispersion based PC optical devices such as non-channel waveguides [6

6. D. Chigrin, S. Enoch, C. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express **11**, 1203–1211 (2003). [CrossRef] [PubMed]

7. D.W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. **29**, 50–52 (2004). [CrossRef] [PubMed]

8. P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačiv́, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and Erich P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. **5**, 93–96 (2006). [CrossRef] [PubMed]

9. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **83**, 3251–3253 (2003). [CrossRef]

10. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. **87**, 181106 (2005). [CrossRef]

11. M.-W. Kim, S.-G. Lee, T.-T. Kim, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Experimental demonstration of bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. **90**, 113121 (2007). [CrossRef]

12. S. Shi, A. Sharkawy, C. Chen, D. Pustai, and D. Prather, “Dispersion-based beam splitter in photonic crystals,” Opt. Lett. **29**, 617–619 (2004). [CrossRef] [PubMed]

13. V. Zabelin, L. A. Dunbar, N. Le Thomas, R. Houdré, M. V. Kotlyar, L. O’Faolain, and T. F. Krauss, “Self-collimating photonic crystal polarization beam splitter,” Opt. Lett. **32**, 530–532 (2007). [CrossRef] [PubMed]

14. Z. Y. Li and L. L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B **68**, 245110 (2003). [CrossRef]

15. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. **91**, 207401 (2003). [CrossRef] [PubMed]

16. V. P. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals - imaging by flat lens using negative refraction,” Nature **426**, 404 (2003). [CrossRef] [PubMed]

17. T. Matsumoto, S. Fujita, and T. Baba, “Wavelength demultiplexer consisting of Photonic crystal superprism and superlens,” Opt. Express **13**, 10768–10776 (2005). [CrossRef] [PubMed]

18. K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the superprism phenomena in photonic crystals,” Appl. Phys. Lett. **81**, 1549–1551 (2002). [CrossRef]

*et al.*elongated holes in the first layer [19

19. T. Baba and D. Ohsaki, “Interfaces of photonic crystals for high efficiency light transmission,” Jpn. J. Appl. Phys. **40**, 5920–5924 (2001). [CrossRef]

*et al.*added multilayered diffraction grating at the end of a PC [20

20. J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, “Mode matching interface for efficient coupling of light into planar photonic crystals,” Phys. Rev. E **69**, 046609 (2004). [CrossRef]

21. B. Momeni and A. Adibi, “Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystal,” Appl. Phys. Lett. **87**, 171104 (2005). [CrossRef]

22. H. A. Macleod, *Thin-film optical filters*, (Adam Hilger Ltd, Bristol, 1986). [CrossRef]

*et al.*showed that conventional ARCs can be applied to one-dimensional (1D) PC interfaces [23

23. J. Ushida, M. Tokushima, M. Shirane, and H. Yamada, “Systematic design of antireflection coating for semi-infinite one-dimensional photonic crystals using Bloch wave expansion,” Appl. Phys. Lett. **82**, 7–9 (2003). [CrossRef]

## 2. Model and Method

*β*is the phase change occurred during the time the light goes across region 2 and

*r*is the reflection coefficient of light propagating from region

_{ij}*i*to

*j*[25]. The reflectance of the incident light, the square of the amplitude of the reflection coefficient

*r*given by Eq. (1), becomes zero when the following two conditions are satisfied simultaneously:

*r*| and

_{ij}*δ*correspond to the amplitude and the phase factor of the reflection coefficient

_{ij}*r*, respectively. In the simple case shown in Fig. 1(a), the optimal ARC parameters, the refractive index

_{ij}*h*=

*λ*/4, are easily obtained from Eqs. (2) and (3) by using the reflection coefficients given by the Fresnel equations. When region 3 is replaced by a 1D PC, the ARC parameters can also be optimized by using the

*r*

_{12}given by the Fresnel equations and

*r*

_{23}given by the numerical calculations as described in Ref. [23

23. J. Ushida, M. Tokushima, M. Shirane, and H. Yamada, “Systematic design of antireflection coating for semi-infinite one-dimensional photonic crystals using Bloch wave expansion,” Appl. Phys. Lett. **82**, 7–9 (2003). [CrossRef]

*R*of the rod (hole) and the distance

_{arc}*d*between the ARC and the PC truncation are chosen as the design parameters of the ARC structure as depicted in Fig. 1(b) that shows the case for a 2D square lattice PC composed of dielectric rods in air. The ARC structure becomes a part of the host PC when

_{arc}*R*=

_{arc}*R*and

*d*=

_{arc}*a*/√2. The ARC parameters for this configuration can also be optimized from Eqs. (2) and (3), provided that

*r*are properly modified. Note that, in this analysis,

_{ij}*r*

_{12}is the reflection coefficient of the ARC structure embedded in air and

*r*

_{23}is that of the semi-infinite PC when the light is incident upon it from the air. In the conventional ARC approach the perfect transmission of incident light is resulted from the resonance in the region 2 of Fig. 1(a), whereas in the 2D PC case this is taking place in the air region located between the end of PC and the ARC structure (see Fig. 1(b)). Therefore, the reflection coefficient

*r*

_{23}in the classical ARC is replaced by the reflection coefficient of the PC starting from the air. We can optimize the parameters for the specific light frequency of interest by using the FDTD simulations. First, the value of

*R*is found to satisfy the condition given by Eq. (2) and then the value of

_{arc}*d*to satisfy Eq. (3) at the optimized value of

_{arc}*R*.

_{arc}26. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. **90**, 231114 (2007). [CrossRef]

27. Y. Zhang, Y. Zhang, and B. Li, “Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals,” Opt. Express **15**, 9287–9292 (2007). [CrossRef] [PubMed]

## 3. Results and Discussion

*ε*=12.0 and the radius

*r*=0.35

*a*, where

*a*is the lattice constant. In our previous work, it was shown that the

*E*-polarized lights, which have the electric field parallel to the rod axis, of frequencies around

*f*=0.194

*c*/

*a*, where

*c*is the speed of light in vacuum, exhibit the self-collimation phenomenon when they propagate along the ΓM-direction in the PC structure considered in this study [10

10. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. **87**, 181106 (2005). [CrossRef]

*r*

_{23}|, and then |

*r*

_{12}| as a function of

*R*for the light of frequency

_{arc}*f*=0.194

*c*/

*a*by using the FDTD simulations. It is found that |

*r*

_{12}|=|

*r*

_{23}|=0.573 at

*R*=0.2064

_{arc}*a*and 0.4347

*a*, as drawn in Fig. 2(a). In order to find the optimal value of

*d*, the total reflectance is calculated as a function of

_{arc}*d*when

_{arc}*R*=0.2064

_{arc}*a*and 0.4347

*a*. As can be seen in Fig. 2(b), there are values of

*d*at which the reflectance becomes zero and the reflectance curves exhibit periodic oscillations with

_{arc}*d*. The period of oscillation is about a half wavelength of the incident beam. This result shows that it is possible to control the total phase of Eq. (3), (2

_{arc}*β*+

*δ*

_{23}-

*δ*

_{12}), by varying the value of

*d*, even though we do not know the specific values of

_{arc}*δ*.

_{ij}*d*=0.74

_{arc}*a*and

*R*=0.2064

_{arc}*a*. The minimum value of

*d*is chosen here to minimize the spreading of light beam which may occur during the propagation through the air layer between the ARC structure and the PC and thereby to improve the coupling efficiency. The FDTD simulations are performed for two different PC samples of sizes 12√2

_{arc}*a*and 16√2

*a*in the ΓM-direction. The ARC structures of the same parameters are introduced at both the input and output PC interfaces. The computational geometry is shown in Fig. 3(a). In the

*x*-direction the Bloch periodic boundary condition is applied and the perfectly matched layer (PML) absorbing boundary condition [28

28. J. -P. Berenger, “A perfectly matched layer for the absorption of electomagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

*y*-direction. Fig. 3(b) shows the calculated transmission spectra of the PC without the ARC structure and one can compare them with those shown in Fig. 3(c) which are obtained with the ARC structure applied. The transmission of light beam through the PC without the ARC structure not only strongly oscillates but also depends on the size of the PC; the period of oscillation becomes shortened as the size of PC increases because the optical path of light is increased. The variations in the transmitted power of light result from the constructive or destructive interferences of multiple beams which are reflected and transmitted at the PC interfaces. Hence, it is reasonably expected that the oscillations in the transmission spectra will disappear if the reflection totally vanishes. One can clearly see that the light beams of frequencies around

*f*=0.194

*c*/

*a*show almost perfect transmission, irrespective of the PC size. More than 99% of the incident power is transmitted through the PC samples for the the lights in the frequency range from 0.188 to 0.202

*c*/

*a*.

*r*=0.35

*a*in a high index material of

*ε*=12.0. According to Ref. [9

9. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **83**, 3251–3253 (2003). [CrossRef]

*H*-polarized lights, which have the magnetic field parallel to the hole axis, of frequencies around

*f*=0.190

*c*/

*a*propagate with almost no beam spreading along the ΓM-direction in the PC of the same structure. The optimal values of the ARC parameters are found to be

*R*=0.2565

_{arc}*a*and

*d*=0.58

_{arc}*a*for the light of frequency

*f*=0.190

*c*/

*a*. The transmission spectra are calculated for the cases with and without the optimal ARC structure when the sample size is 16√2

*a*. Figure 4(b) demonstrates that the reflection at the 2D PC interfaces can be efficiently eliminated by the application of the optimal ARC structure. Comparing Fig. 4(b) with Fig. 3(c), one can notice that the frequency range from 0.170 to 0.205

*c*/

*a*which exhibit over 99% transmission in the hole-type PC is much wider than that of the rod-type PC. Because the reflection of light beam at the hole-type PC interface is smaller than that at the rod-type PC interface, the frequency range in which the incident light exhibits high transmission gets wider.

*et al.*theoretically demonstrated that the phase difference of the two split self-collimated beams at the line-defect beam splitter is

*π*/2 in the PC with the same structural parameters considered in this study [26

26. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. **90**, 231114 (2007). [CrossRef]

*I*

_{1}/

*I*

_{0}measured at the port 1 is ideally unity and

*I*

_{2}/

*I*

_{0}at the port 2 is zero due to the constructive and destructive interferences, respectively. We calculate the transmission spectra at the ports 1 and 2 around the frequency

*f*=0.194

*c*/

*a*when the ARC structure is (Fig. 5(b)) and is not (Fig. 5(c)) introduced. In the simulations, a Gaussian pulse with a waist of

*w*=3

*a*is launched into the input interface of the MZI. Figure 5(b) shows that the transmission of light at the port 1 exceeds 92% with almost no fluctuation for the lights in the frequency range between 0.190 and 0.198

*c*/

*a*. On the other hand, Fig. 5(c) shows that the transmitted power at the port 1 strongly fluctuates from 23% to 94% due to the reflection at the input and output interfaces of the device. These results reveal that the performance of the PC MZI is significantly improved by the introduction of the ARC structure into the PC.

*f*=0.160

*c*/

*a*at which the light does not exhibit self-collimated propagation in the larger size rod-type PC structure employed in this study. The obtained results showed more than 99% transmission for the lights in the frequency range from 0.153 to 0.170

*c*/

*a*with the optimal ARC structure applied.

## 4. Conclusion

*R*and

_{arc}*d*. We also show that the performance of a PC MZI based on the self-collimated beams can be significantly improved by the introduction of the optimal ARC structure. The proposed method can be important for implementing other PC devices.

_{arc}## Acknowledgments

## References and links

1. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. |

2. | J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. |

3. | S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. |

4. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

5. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B |

6. | D. Chigrin, S. Enoch, C. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express |

7. | D.W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. |

8. | P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačiv́, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and Erich P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. |

9. | X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. |

10. | S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. |

11. | M.-W. Kim, S.-G. Lee, T.-T. Kim, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Experimental demonstration of bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. |

12. | S. Shi, A. Sharkawy, C. Chen, D. Pustai, and D. Prather, “Dispersion-based beam splitter in photonic crystals,” Opt. Lett. |

13. | V. Zabelin, L. A. Dunbar, N. Le Thomas, R. Houdré, M. V. Kotlyar, L. O’Faolain, and T. F. Krauss, “Self-collimating photonic crystal polarization beam splitter,” Opt. Lett. |

14. | Z. Y. Li and L. L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B |

15. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. |

16. | V. P. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals - imaging by flat lens using negative refraction,” Nature |

17. | T. Matsumoto, S. Fujita, and T. Baba, “Wavelength demultiplexer consisting of Photonic crystal superprism and superlens,” Opt. Express |

18. | K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the superprism phenomena in photonic crystals,” Appl. Phys. Lett. |

19. | T. Baba and D. Ohsaki, “Interfaces of photonic crystals for high efficiency light transmission,” Jpn. J. Appl. Phys. |

20. | J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, “Mode matching interface for efficient coupling of light into planar photonic crystals,” Phys. Rev. E |

21. | B. Momeni and A. Adibi, “Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystal,” Appl. Phys. Lett. |

22. | H. A. Macleod, |

23. | J. Ushida, M. Tokushima, M. Shirane, and H. Yamada, “Systematic design of antireflection coating for semi-infinite one-dimensional photonic crystals using Bloch wave expansion,” Appl. Phys. Lett. |

24. | A. Taflove, |

25. | M. Born and E. Wolf, |

26. | D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. |

27. | Y. Zhang, Y. Zhang, and B. Li, “Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals,” Opt. Express |

28. | J. -P. Berenger, “A perfectly matched layer for the absorption of electomagnetic waves,” J. Comput. Phys. |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(310.1210) Thin films : Antireflection coatings

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: January 28, 2008

Revised Manuscript: February 29, 2008

Manuscript Accepted: March 2, 2008

Published: March 13, 2008

**Citation**

Sun-Goo Lee, Jin-sun Choi, Jae-Eun Kim, Hae-Yong Park, and Chul-Sik Kee, "Reflection minimization at two-dimensional photonic crystal interfaces," Opt. Express **16**, 4270-4277 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-4270

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### References

- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999). [CrossRef]
- J. Witzens, M. Loncar, and A. Scherer, "Self-collimation in planar photonic crystals," IEEE J. Sel. Top. Quantum Electron. 8, 1246-1257 (2002). [CrossRef]
- S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction in media with a negative refractive index," Phys. Rev. Lett. 90, 107402 (2003). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, 10096-10099 (1998). [CrossRef]
- D. Chigrin, S. Enoch, C. Sotomayor Torres, and G. Tayeb, "Self-guiding in two-dimensional photonic crystals," Opt. Express 11, 1203-1211 (2003). [CrossRef] [PubMed]
- D.W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Schneider, and J. Murakowski, "Dispersion-based optical routing in photonic crystals," Opt. Lett. 29, 50-52 (2004). [CrossRef] [PubMed]
- P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljaciv, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and Erich P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5, 93-96 (2006). [CrossRef] [PubMed]
- X. Yu and S. Fan, "Bends and splitters for self-collimated beams in photonic crystals," Appl. Phys. Lett. 83, 3251-3253 (2003). [CrossRef]
- S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, "Line-defect-induced bending and splitting of selfcollimated beams in two-dimensional photonic crystals," Appl. Phys. Lett. 87, 181106 (2005). [CrossRef]
- M.-W. Kim, S.-G. Lee, T.-T. Kim, J.-E. Kim, H. Y. Park, and C.-S. Kee, "Experimental demonstration of bending and splitting of self-collimated beams in two-dimensional photonic crystals," Appl. Phys. Lett. 90, 113121 (2007). [CrossRef]
- S. Shi, A. Sharkawy, C. Chen, D. Pustai, and D. Prather, "Dispersion-based beam splitter in photonic crystals," Opt. Lett. 29, 617-619 (2004). [CrossRef] [PubMed]
- V. Zabelin, L. A. Dunbar, N. Le Thomas, R. Houdr’e, M. V. Kotlyar, L. O’Faolain, and T. F. Krauss, "Selfcollimating photonic crystal polarization beam splitter," Opt. Lett. 32, 530-532 (2007). [CrossRef] [PubMed]
- Z. Y. Li and L. L. Lin, "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction," Phys. Rev. B 68, 245110 (2003). [CrossRef]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopou, C. M. Soukoulis, "Subwavelength resolution in a twodimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003). [CrossRef] [PubMed]
- V. P. Parimi,W. T. Lu, P. Vodo, and S. Sridhar, "Photonic crystals - imaging by flat lens using negative refraction," Nature 426, 404 (2003). [CrossRef] [PubMed]
- T. Matsumoto, S. Fujita, and T. Baba, "Wavelength demultiplexer consisting of Photonic crystal superprism and superlens," Opt. Express 13, 10768-10776 (2005). [CrossRef] [PubMed]
- K. B. Chung and S. W. Hong, "Wavelength demultiplexers based on the superprism phenomena in photonic crystals," Appl. Phys. Lett. 81, 1549-1551 (2002). [CrossRef]
- T. Baba and D. Ohsaki, "Interfaces of photonic crystals for high efficiency light transmission," Jpn. J. Appl. Phys. 40, 5920-5924 (2001). [CrossRef]
- J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, "Mode matching interface for efficient coupling of light into planar photonic crystals," Phys. Rev. E 69, 046609 (2004). [CrossRef]
- B. Momeni and A. Adibi, "Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystal," Appl. Phys. Lett. 87, 171104 (2005). [CrossRef]
- H. A. Macleod, Thin-film Optical Filters, (Adam Hilger Ltd, Bristol, 1986). [CrossRef]
- J. Ushida, M. Tokushima, M. Shirane, and H. Yamada, "Systematic design of antireflection coating for semiinfinite one-dimensional photonic crystals using Bloch wave expansion," Appl. Phys. Lett. 82, 7-9 (2003). [CrossRef]
- A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Boston, 1995).
- M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, Cambridge, 2002).
- D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, "Photonic crystal Mach-Zehnder interferometer based on self-collimation," Appl. Phys. Lett. 90, 231114 (2007). [CrossRef]
- Y. Zhang, Y. Zhang, and B. Li, "Optical switches and logic gates based on self-collimated beams in twodimensional photonic crystals," Opt. Express 15, 9287-9292 (2007). [CrossRef] [PubMed]
- J. -P. Berenger, "A perfectly matched layer for the absorption of electomagnetic waves," J. Comput. Phys. 114, 185-200 (1994). [CrossRef]

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