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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 20 — Oct. 1, 2007
  • pp: 13149–13154
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On the relationship between Bloch modes and phase-related refractive index of photonic crystals

Guilin Sun and Andrew G. Kirk  »View Author Affiliations


Optics Express, Vol. 15, Issue 20, pp. 13149-13154 (2007)
http://dx.doi.org/10.1364/OE.15.013149


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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

As a result of their rich band structures and unique dispersion relations, photonic crystals (PhCs) have been shown to display a variety of behaviors such as the superprism [1

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]

], self-collimation [2

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]

], anomalous and negative refraction [3–6

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

]. In particular, the negative refraction of dielectric PhCs can be used to form images using a flat slab leading to subwavelength transversal resolution in the partial bandgap of the first band [6

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]

] and the second band [5

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]

].

Although the dielectric photonic crystals can display negative refraction [5

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

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]

], the mechanisms differ from those of left-handed metamaterials. For negative refraction at the second band and higher, an increase in the normalized frequency causes shrinkage of the equifrequency contours [5

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]

]. The resulting effective refractive index n eff calculated from band structure is negative due to the inwardly directed group velocity [5

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]

]. Using plane wave decomposition from sampled field values [11

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]

], it has been shown that the fundamental wavevector of the Bloch modes in the first Brillouin zone is opposed to the energy flow. Because of this, it was expected that such a photonic crystal would exhibit left-hand behaviors. By applying a theoretical analysis of the Bloch modes using Fourier decomposition [12

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]

], it has been shown that the unfolded band structure cannot give a negative effective index of refraction. This contradiction raises a question about the fundamental wavevector of the Bloch modes.

For a plane wave in an isotropic medium, its phase velocity is associated with its wavevector, and the plane wave is a solution of Maxwell Equations [13

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

]. In photonic crystals, the solution of Maxwell Equations is expressed as a sum of Bloch modes. Each of the modes may be assigned a phase velocity associated with its wavevector [14

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

], which in general means that one may not define a global phase velocity [12

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]

]. However, each of these individual modes (spatial harmonics) is not itself a solution of Maxwell Equations [12

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]

]. When excited, the Bloch modes travel collectively in a definite direction governed by the group velocity [15

15. K. Sakoda, “Optical properties of photonic crystals,”Springer-Verlag, New York,2nd ed., 2005

]. We cannot distinguish the individual phases of each mode, but can only identify the overall phase. We define this to be the measurable phase.

2. Bloch modes, dominant wave and its refractive index

For conciseness, we directly use some known results. The discussion is limited to the two-dimensional (2D) photonic crystals. The wave amplitude u can be expressed as a sum of Bloch modes as [5

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]

]:

u(r)=um,n(r)ej(k'+mG1+nG2)r
(1)

where k⃗′ 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, um,n is the amplitude of the (m, n)th mode, G1 and G2 are the reciprocal lattice vectors, and r⃗ is the position vector.

Our hypothesis is that, although the Bloch modes have an infinite number of spatial harmonics, the measurable phase is dominated by the wavevector that has the largest amplitude, which we define to be the dominant wave. The dominant wave can be determined from the solution of Maxwell Equations with approximate homogenous permittivity and the band structure. It can be shown (by adapting the 1-D model presented in [15

15. K. Sakoda, “Optical properties of photonic crystals,”Springer-Verlag, New York,2nd ed., 2005

]) that the amplitude of each Bloch mode um,n ∝ 1/q , where the quantity q can be written as

q=k'+(mG1+nG2)2ω2με0ε¯
(2)

np=kMk0
(3)

Fig. 1. The reciprocal lattice of the hexagonal photonic crystals. First Brillouin zone is inside the blue hexagon.
Fig. 2. Fourier decomposition of Bloch modes at the first band of the PhC with r/a=0.4 and host material n=3.6 for TE polarization

3. Numerical Fourier decomposition of Bloch modes

To test validity of the above hypothesis, we numerically decompose the Bloch modes from the simulated complex filed (real and imaginary) inside the hexagonal photonic crystal by applying fast Fourier transform. The simulations were performed with the finite-difference time-domain (FDTD) method. To remove the possible effect of the second interface, we use a semi-infinite photonic crystal. The plane wave is incident on it normally from free space. Periodic boundary conditions are applied for the vertical ends perpendicular to the direction of light propagation, and perfectly-matched layers (PML) along the direction of light. The normal to the interface is along ΓM. The simulation stopped before the light reached the second surface.

For the 2nd band, we have also simulated oblique incidence cases using a Gaussian beam with PML conditions and obtained similar results. The value of ∣k⃗'/k 0∣ is almost constant at the normalized frequency 0.30, confirming that the equifrequency contour is almost circular.

However, if the photonic crystal is highly anisotropic, the method can fail to correctly predict the dominant wave. For example, at a normalized frequency 0.4112 of the same PhC (r=0.35a) for TM polarization, the dominant wavevectors as determined by Fourier decomposition are (0.72 x⃗ ± 2.43 y⃗) k 0, which correspond to k⃗'+G1 and k⃗'+G2, both of which differ from the predicted vector k⃗'+G1 +G2. 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 -0.76k 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.

Fig. 3. Fourier decomposition of Bloch modes of the same photonic crystal as in Fig. 2 but at the second band at normalized frequency 0.30.
Fig. 4. Fourier decomposition of Bloch modes of the PhC (r/a=0.35) at the 3rd band for TM-polarization at normalized frequency 0.428.

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 ,,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]

4.

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals, “ J. Opt. Soc. Am, A , 17, 1012–1020, (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]

7.

V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Soviet Phys.Uspekhi , 10, 509–514, (1968). (in Russian,1964) [CrossRef]

8.

J.B. Pentry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. , 85, 3966–3969, (2000). [CrossRef]

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. 84, 4184–4187 (2000). [CrossRef] [PubMed]

10.

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]

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]

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 , 33, 3232–3242, (1986) [CrossRef]

15.

K. Sakoda, “Optical properties of photonic crystals,”Springer-Verlag, New York,2nd ed., 2005

16.

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]

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 14, 9805–9814 (2006) [CrossRef] [PubMed]

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 72, 165112 (2005). [CrossRef]

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

  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]
  4. B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals, " J. Opt. Soc. Am, A,  17, 1012-1020, (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]
  7. V. Veselago, "The electrodynamics of substances with simultaneously negative values of and ," Soviet Phys.Uspekhi, 10, 509-514, (1968). (in Russian,1964) [CrossRef]
  8. J.B. Pentry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett.,  85, 3966-3969, (2000). [CrossRef]
  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. 84, 4184-4187 (2000). [CrossRef] [PubMed]
  10. 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]
  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]
  13. M.  Born, E.  Wolf, Principles of Optics, Pergamon, Oxford, 1989.
  14. P. St.J. Russel, "Interference of integrated Floquet-Bloch waves," Phys. Rev. A,  33, 3232-3242, (1986) [CrossRef]
  15. K. Sakoda, "Optical properties of photonic crystals,"Springer-Verlag, New York,2nd ed., 2005
  16. 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]
  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 14, 9805-9814 (2006) [CrossRef] [PubMed]
  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 72, 165112 (2005). [CrossRef]
  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.

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