## On the relationship between Bloch modes and phase-related refractive index of photonic crystals

Optics Express, Vol. 15, Issue 20, pp. 13149-13154 (2007)

http://dx.doi.org/10.1364/OE.15.013149

Acrobat PDF (224 KB)

### Abstract

It has previously been shown that the phase-related refractive index is positive in photonic crystals that display negative refraction at higher bands. We hypothesize that the phase velocity is governed by a wave that can be related to the dominant Bloch mode. This dominant wave can be identified from an approximate solution of Maxwell Equations using a homogeneously averaged dielectric constant and the dominant wavevector is related to the fundamental wavevector and the reciprocal lattice vectors. We validate this hypothesis by numerical Fourier decomposition of the field in the entire simulation domain. It confirms that for negative refraction at higher bands, the phase-related refractive index is indeed positive and differs significantly from the negative value of effective refractive index calculated from the band structure.

© 2007 Optical Society of America

## 1. Introduction

1. 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]

2. 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]

3. J.P. Dowling and C.M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. , **41**, 345–351, (1994). [CrossRef]

6. C. Luo, S.G. Johnson, J.D. Joannopoulos, and J.B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev., B **65**, 201104, (2002). [CrossRef]

5. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev., B , **62**, 10696–10705, (2000). [CrossRef]

5. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev., B , **62**, 10696–10705, (2000). [CrossRef]

6. C. Luo, S.G. Johnson, J.D. Joannopoulos, and J.B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev., B **65**, 201104, (2002). [CrossRef]

5. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev., B , **62**, 10696–10705, (2000). [CrossRef]

*n*

_{eff}calculated from band structure is negative due to the inwardly directed group velocity [5

**62**, 10696–10705, (2000). [CrossRef]

11. A. Martínez, H. Míguez, J. Sánchez-Dehesa, and J. Martí, “Analysis of wave propagation in a two-dimensional photonic crystal with negative index of refraction: plane wave decomposition of the Bloch modes,” Opt. Express **13**, 4160–4174 (2005). [CrossRef] [PubMed]

12. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B **22**, 1179–1190 (2005) [CrossRef]

14. P. St.J. Russel, “Interference of integrated Floquet-Bloch waves,” Phys. Rev. A , **33**, 3232–3242, (1986) [CrossRef]

12. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B **22**, 1179–1190 (2005) [CrossRef]

12. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B **22**, 1179–1190 (2005) [CrossRef]

## 2. Bloch modes, dominant wave and its refractive index

*u*can be expressed as a sum of Bloch modes as [5

**62**, 10696–10705, (2000). [CrossRef]

*k*⃗′ is the fundamental wavevector [21

21. S. Foteinopoulou and C.M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects,” Phys. Rev. B **72**, 165112 (2005). [CrossRef]

*m*and

*n*are integers,

*u*is the amplitude of the (

_{m,n}*m*,

*n*)

_{th}mode,

*G*⃗

_{1}and

*G*⃗

_{2}are the reciprocal lattice vectors, and

*r*⃗ is the position vector.

*u*∝ 1/

_{m,n}*q*, where the quantity

*q*can be written as

## 3. Numerical Fourier decomposition of Bloch modes

*k*(ΓM) and

_{x}*k*(ΓK) are the wavenumbers along axes. The wavevector with the largest amplitude has a refractive index of 2.564, as defined in (3). The pseudo-interference technique gives a value of 2.59. From the band structure, the effective refractive index is calculated to be

_{y}*n*

_{eff}=2.562. Therefore, it is recognized that the dominant wavevector is the fundamental wavevector since no other wavevectors have a refractive index close to 2.56. To verify this result, some other simulations at different wavelengths were also done and similar results are obtained. Therefore we conclude that the effective index of refraction calculated from the band structure is the same as that of the fundamental wavevector of the Bloch modes at the 1

^{st}band. This confirms our previous analysis. The wavevector sampling interval is 0.04

*k*

_{0}in this paper, thus the precision of the calculated refractive index (as extracted from Fourier decomposition) is ± 0.04.

^{nd}band, we have also simulated oblique incidence cases using a Gaussian beam with PML conditions and obtained similar results. The value of ∣

*k*⃗'/

*k*

_{0}∣ is almost constant at the normalized frequency 0.30, confirming that the equifrequency contour is almost circular.

*r*=0.35

*a*) for TM polarization, the dominant wavevectors as determined by Fourier decomposition are (0.72

*x*⃗ ± 2.43

*y*⃗)

*k*

_{0}, which correspond to

*k*⃗'+

*G*⃗

_{1}and

*k*⃗'+

*G*⃗

_{2}, both of which differ from the predicted vector

*k*⃗'+

*G*⃗

_{1}+

*G*⃗

_{2}. Even though the hypothesis fails to predict the correct dominant wave in this situation, the numerical Fourier decomposition correctly give the wavevectors. For example, Fourier decomposition gives the fundamental wavevector of -0.76

*k*

_{0}which is close to -0.74

*k*

_{0}calculated from band structure. Most importantly, it shows that the fundamental wavevector is indeed negative, and the phase velocity is positive.

## 4. Discussion and conclusion

## References and links

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

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

3. | J.P. Dowling and C.M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. , |

4. | B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals, “ J. Opt. Soc. Am, A , |

5. | M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev., B , |

6. | C. Luo, S.G. Johnson, J.D. Joannopoulos, and J.B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev., B |

7. | V. Veselago, “The electrodynamics of substances with simultaneously negative values of 10, 509–514, (1968). (in Russian,1964) [CrossRef] |

8. | J.B. Pentry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. , |

9. | D.R. Smith, W.J. Padina, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz, , “A composite medium with simultaneously negative permeability and permittivity,” Phys.Rev.Lett. |

10. | M. Anioniades and G.V. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas & Wireless Propag. Lett. , |

11. | A. Martínez, H. Míguez, J. Sánchez-Dehesa, and J. Martí, “Analysis of wave propagation in a two-dimensional photonic crystal with negative index of refraction: plane wave decomposition of the Bloch modes,” Opt. Express |

12. | B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B |

13. | M. Born and E. Wolf, Principles of Optics, Pergamon, Oxford, 1989. |

14. | P. St.J. Russel, “Interference of integrated Floquet-Bloch waves,” Phys. Rev. A , |

15. | K. Sakoda, “Optical properties of photonic crystals,”Springer-Verlag, New York,2 |

16. | G. Sun, A.S. Jugessur, and A.G. Kirk, “Imaging properties of dielectric photonic crystal slabs for large object distances,” Opt. Express , |

17. | G. Sun and A. G. Kirk, “Pseudo-interference and its application in determining averaged phase refractive index of photonic crystals,” IEEE LEOS 2006 Annual Meeting, 29 October - 2 November 2006, Montreal. |

18. | A. Martínez and J. Martí, “Positive phase evolution of waves propagating along a photonic crystal with negative index of refraction,” Opt. Express |

19. | G. Sun, A. Bakhtazad, A. Jugessur, and A. Kirk, “Open cavities using photonic crystals with negative refraction,” Proc. SPIE, 6343, ed. P. Mathieu, (2006), doi:10.1117/12.708026 |

20. | A. Yariv and P. Yeh, Optical waves in crystals: propagation and control of laser radiation, Wiley, New York, 2003. |

21. | S. Foteinopoulou and C.M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects,” Phys. Rev. B |

22. | G. Sun and A. G. Kirk, “Lattice resonance inside photonic crystal slab with negative refraction,” OSA Frontiers in Optics 2006/Laser Science XXII conferences, October 8–12, 2006, Rochester, New York. |

**OCIS Codes**

(110.0110) Imaging systems : Imaging systems

(350.3950) Other areas of optics : Micro-optics

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: August 1, 2007

Revised Manuscript: September 15, 2007

Manuscript Accepted: September 18, 2007

Published: September 26, 2007

**Citation**

Guilin Sun and Andrew G. Kirk, "On the relationship between Bloch modes and phase-related refractive index of photonic crystals," Opt. Express **15**, 13149-13154 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-13149

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### References

- 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]
- 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.P. Dowling and C.M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt., 41, 345-351, (1994). [CrossRef]
- B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals, " J. Opt. Soc. Am, A, 17, 1012-1020, (2000). [CrossRef]
- M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev., B, 62, 10696-10705, (2000). [CrossRef]
- C. Luo, S.G. Johnson, J.D. Joannopoulos, and J.B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev., B 65, 201104, (2002). [CrossRef]
- V. Veselago, "The electrodynamics of substances with simultaneously negative values of and ," Soviet Phys.Uspekhi, 10, 509-514, (1968). (in Russian,1964) [CrossRef]
- J.B. Pentry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett., 85, 3966-3969, (2000). [CrossRef]
- D.R. Smith, W.J. Padina, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys.Rev.Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
- M. Anioniades and G.V. Eleftheriades, "Compact linear lead/lag metamaterial phase shifters for broadband applications, " IEEE Antennas & Wireless Propag. Lett., 2, 103 - 106, (2003) [CrossRef]
- A. Martínez, H. Míguez, J. Sánchez-Dehesa, and J. Martí, "Analysis of wave propagation in a two-dimensional photonic crystal with negative index of refraction: plane wave decomposition of the Bloch modes," Opt. Express 13, 4160-4174 (2005). [CrossRef] [PubMed]
- B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, "Fourier analysis of Bloch wave propagation in photonic crystals," J. Opt. Soc. Am. B 22, 1179-1190 (2005) [CrossRef]
- M. Born, E. Wolf, Principles of Optics, Pergamon, Oxford, 1989.
- P. St.J. Russel, "Interference of integrated Floquet-Bloch waves," Phys. Rev. A, 33, 3232-3242, (1986) [CrossRef]
- K. Sakoda, "Optical properties of photonic crystals,"Springer-Verlag, New York,2nd ed., 2005
- G. Sun, A.S. Jugessur, and A.G. Kirk, "Imaging properties of dielectric photonic crystal slabs for large object distances, " Opt. Express, 14, 6755-6765 (2006). [CrossRef] [PubMed]
- G. Sun and A. G. Kirk, "Pseudo-interference and its application in determining averaged phase refractive index of photonic crystals," IEEE LEOS 2006 Annual Meeting, 29 October - 2 November 2006, Montreal.
- A. Martínez and J. Martí, "Positive phase evolution of waves propagating along a photonic crystal with negative index of refraction," Opt. Express 14, 9805-9814 (2006) [CrossRef] [PubMed]
- G. Sun, A. Bakhtazad, A. Jugessur, and A. Kirk, "Open cavities using photonic crystals with negative refraction," Proc. SPIE, 6343, ed. P. Mathieu, (2006), doi:10.1117/12.708026
- A. Yariv and P. Yeh, Optical waves in crystals: propagation and control of laser radiation, Wiley, New York, 2003.
- S. Foteinopoulou and C.M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects," Phys. Rev. B 72, 165112 (2005). [CrossRef]
- G. Sun and A. G. Kirk, "Lattice resonance inside photonic crystal slab with negative refraction," OSA Frontiers in Optics 2006/Laser Science XXII conferences, October 8-12, 2006, Rochester, New York.

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