Abstract

The method of assigning irreducible representations to modes in three-dimensional photonic structures is applied to the two-dimensional triangular air-silica lattice with out-of-plane wave propagation. In particular prediction of spatial symmetries of the crystal modes is addressed. We show how the photonic bands are affected by different rod radii and out-of-plane components from a group-theoretical point of view. One particular defect mode is analyzed and the structure which is optimal for air-guidance is found.

©2004 Optical Society of America

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References

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  1. S. G. Johnson and J. Joannopoulus, “Block-iterative frequency domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173 (2001).
    [Crossref] [PubMed]
  2. D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134 (1996).
    [Crossref]
  3. T. Ochiai and K. Sakoda, “Nearly free-photon approximation for two-dimensional photonic crystal slabs,” Phys. Rev. B 64, 045108 (2001).
    [Crossref]
  4. M. Plihal and A. Maradudin, “Photonic band structures of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565 (1991).
    [Crossref]
  5. K. Ohtaka and Y. Tanabe, “Photonic Bands Using Vector Spherical Waves. III Group-Theoretical Treatment,” J. Phys. Soc. Japan 65, 2670 (1996)
    [Crossref]
  6. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
  7. K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55, 15345 (1997).
    [Crossref]
  8. F. Bassani and G. P. Parravicini, Electronic States and Optical Transitions in Solids (Pergamon Press, Oxford, 1975), 8th ed..
  9. R. Meade, K. Brommer, A. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495 (1992).
    [Crossref]
  10. J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25, 96 (2000).
    [Crossref]
  11. Following the procedure in Ref.[7] we can, for incident waves in certain directions, also use the information about the irreducible representations to identify uncoupled modes which behave as band gaps.
  12. M. Løkke, “Triangular photonic band gap crystals - implementation in air-guiding optical fibers,” Thesis, University of Aarhus 2003 (unpublished).
  13. M. M. Sigalas, R Biswas, K. M. Ho, and C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B 58, 6791 (1998).
    [Crossref]
  14. M. L. Povinelli, Steven G. Johnson, Shanhui Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B 64, 075313 (2001).
    [Crossref]

2001 (3)

S. G. Johnson and J. Joannopoulus, “Block-iterative frequency domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173 (2001).
[Crossref] [PubMed]

T. Ochiai and K. Sakoda, “Nearly free-photon approximation for two-dimensional photonic crystal slabs,” Phys. Rev. B 64, 045108 (2001).
[Crossref]

M. L. Povinelli, Steven G. Johnson, Shanhui Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B 64, 075313 (2001).
[Crossref]

2000 (1)

1998 (1)

M. M. Sigalas, R Biswas, K. M. Ho, and C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B 58, 6791 (1998).
[Crossref]

1997 (1)

K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55, 15345 (1997).
[Crossref]

1996 (2)

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134 (1996).
[Crossref]

K. Ohtaka and Y. Tanabe, “Photonic Bands Using Vector Spherical Waves. III Group-Theoretical Treatment,” J. Phys. Soc. Japan 65, 2670 (1996)
[Crossref]

1992 (1)

R. Meade, K. Brommer, A. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

1991 (1)

M. Plihal and A. Maradudin, “Photonic band structures of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565 (1991).
[Crossref]

Barkou, S. E.

Bassani, F.

F. Bassani and G. P. Parravicini, Electronic States and Optical Transitions in Solids (Pergamon Press, Oxford, 1975), 8th ed..

Bertho, D.

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134 (1996).
[Crossref]

Biswas, R

M. M. Sigalas, R Biswas, K. M. Ho, and C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B 58, 6791 (1998).
[Crossref]

Bjarklev, A.

Broeng, J.

Brommer, K.

R. Meade, K. Brommer, A. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

Cassagne, D.

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134 (1996).
[Crossref]

Fan, Shanhui

M. L. Povinelli, Steven G. Johnson, Shanhui Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B 64, 075313 (2001).
[Crossref]

Ho, K. M.

M. M. Sigalas, R Biswas, K. M. Ho, and C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B 58, 6791 (1998).
[Crossref]

Joannopoulos, J.

R. Meade, K. Brommer, A. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

Joannopoulos, J. D.

M. L. Povinelli, Steven G. Johnson, Shanhui Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B 64, 075313 (2001).
[Crossref]

Joannopoulus, J.

Johnson, S. G.

Johnson, Steven G.

M. L. Povinelli, Steven G. Johnson, Shanhui Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B 64, 075313 (2001).
[Crossref]

Jouanin, C.

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134 (1996).
[Crossref]

Løkke, M.

M. Løkke, “Triangular photonic band gap crystals - implementation in air-guiding optical fibers,” Thesis, University of Aarhus 2003 (unpublished).

Maradudin, A.

M. Plihal and A. Maradudin, “Photonic band structures of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565 (1991).
[Crossref]

Meade, R.

R. Meade, K. Brommer, A. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

Ochiai, T.

T. Ochiai and K. Sakoda, “Nearly free-photon approximation for two-dimensional photonic crystal slabs,” Phys. Rev. B 64, 045108 (2001).
[Crossref]

Ohtaka, K.

K. Ohtaka and Y. Tanabe, “Photonic Bands Using Vector Spherical Waves. III Group-Theoretical Treatment,” J. Phys. Soc. Japan 65, 2670 (1996)
[Crossref]

Parravicini, G. P.

F. Bassani and G. P. Parravicini, Electronic States and Optical Transitions in Solids (Pergamon Press, Oxford, 1975), 8th ed..

Plihal, M.

M. Plihal and A. Maradudin, “Photonic band structures of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565 (1991).
[Crossref]

Povinelli, M. L.

M. L. Povinelli, Steven G. Johnson, Shanhui Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B 64, 075313 (2001).
[Crossref]

Rappe, A.

R. Meade, K. Brommer, A. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

Sakoda, K.

T. Ochiai and K. Sakoda, “Nearly free-photon approximation for two-dimensional photonic crystal slabs,” Phys. Rev. B 64, 045108 (2001).
[Crossref]

K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55, 15345 (1997).
[Crossref]

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

Sigalas, M. M.

M. M. Sigalas, R Biswas, K. M. Ho, and C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B 58, 6791 (1998).
[Crossref]

Søndergaard, T.

Soukoulis, C. M.

M. M. Sigalas, R Biswas, K. M. Ho, and C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B 58, 6791 (1998).
[Crossref]

Tanabe, Y.

K. Ohtaka and Y. Tanabe, “Photonic Bands Using Vector Spherical Waves. III Group-Theoretical Treatment,” J. Phys. Soc. Japan 65, 2670 (1996)
[Crossref]

Appl. Phys. Lett. (1)

R. Meade, K. Brommer, A. Rappe, and J. Joannopoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

J. Phys. Soc. Japan (1)

K. Ohtaka and Y. Tanabe, “Photonic Bands Using Vector Spherical Waves. III Group-Theoretical Treatment,” J. Phys. Soc. Japan 65, 2670 (1996)
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (6)

M. M. Sigalas, R Biswas, K. M. Ho, and C. M. Soukoulis, “Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,” Phys. Rev. B 58, 6791 (1998).
[Crossref]

M. L. Povinelli, Steven G. Johnson, Shanhui Fan, and J. D. Joannopoulos, “Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap,” Phys. Rev. B 64, 075313 (2001).
[Crossref]

D. Cassagne, C. Jouanin, and D. Bertho, “Hexagonal photonic-band-gap structures,” Phys. Rev. B 53, 7134 (1996).
[Crossref]

T. Ochiai and K. Sakoda, “Nearly free-photon approximation for two-dimensional photonic crystal slabs,” Phys. Rev. B 64, 045108 (2001).
[Crossref]

M. Plihal and A. Maradudin, “Photonic band structures of two-dimensional systems: The triangular lattice,” Phys. Rev. B 44, 8565 (1991).
[Crossref]

K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55, 15345 (1997).
[Crossref]

Other (4)

F. Bassani and G. P. Parravicini, Electronic States and Optical Transitions in Solids (Pergamon Press, Oxford, 1975), 8th ed..

Following the procedure in Ref.[7] we can, for incident waves in certain directions, also use the information about the irreducible representations to identify uncoupled modes which behave as band gaps.

M. Løkke, “Triangular photonic band gap crystals - implementation in air-guiding optical fibers,” Thesis, University of Aarhus 2003 (unpublished).

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

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Figures (8)

Fig. 1.
Fig. 1. Two-dimensional Brillouin zone for the hexagonal lattice. The irreducible Brillouin zone is bounded by ΓQP in-plane. The highly symmetric lattice points with a non-zero kz -component are denoted by ZLH, respectively (out-of-plane). The three equivalent P points, which only differ by a linear combination of reciprocal lattice vectors are also shown.
Fig. 2.
Fig. 2. Free-photon hexagonal bands with kz =1.4. The dispersion relation is slightly convex due to the non-zero value of kz .
Fig. 3.
Fig. 3. The spatial symmetry of the z-component of the modes on each symmetry point in the irreducible Brillouin zone belonging to the different one-dimensional irreducible representations. The black and white colors correspond to opposite signs of the field amplitude.
Fig. 4.
Fig. 4. The triangular air-silica band structure as kz and ra are varied. The error bars show the predicted positions and widths of the band splits at the high symmetry points calculated in Table 5. Notice the different frequency scales. (a) (kz, ra )=(0.1,0.1) the near-uniform crystal. (b) (kz, ra )=(1.4,0.1), small variation in the crystal structure (c) (kz, ra )=(0.1,0.44), long wavelength limit. (d) (kz, ra )=(1.4,0.44), no error bars have been added because this case is beyond the approximation. The irreducible representations for the electrical field have been assigned according to Table 1 and Table 4. A band gap above the air line appears for the normalized frequency in [1.405;1.434].
Fig. 5.
Fig. 5. The Ez field distribution for the lowest 8 bands at the different symmetry points. The lattice considered has a rod radius of ra =0.44 and kz =1.4. The blue asnd red colors correspond to opposite signs of the field amplitude.
Fig. 6.
Fig. 6. The largest complete band gaps for the lowest 20 bands in the triangular air-silica structure with ra =0.44 (The white areas in the inset are silica). The dashed line is the air-line [10].
Fig. 7.
Fig. 7. The fraction of electromagnetic field energy localized inside the air defect as the defect radius, rd , is varied. The dashed line shows the frequency of the corresponding defect mode and the horizontal lines represent the boundaries of the photonic band gap for the triangular air-silica crystal with (kz, ra )=(1.4,0.44).
Fig. 8.
Fig. 8. The field distribution of the Γ2+ defect mode at two different defect radii. (a) rd =1.42 with an energy fraction of 92.5% confined in the air defect. (b) rd =1.50 where the confined energy fraction is 76.1%.

Tables (5)

Tables Icon

Table 1. Compatibility relations for the hexagonal lattice.

Tables Icon

Table 2. Characters for the rotations and mirror reflections.

Tables Icon

Table 3. The characters of the two lowest representations at the Z-point in the extended zone scheme of the hexagonal lattice.

Tables Icon

Table 4. The irreducible representations for out-of-plane electromagnetic waves in free space, whose wave vectors (in units of 2π/a) are reduced in the Brillouin zone of the hexagonal lattice.

Tables Icon

Table 5. Upper and lower boundaries for the normalized frequency bands in triangular crystals with (kz, ra ) equal to (0.1,0.1), (1.4,0.1) and (0.1,0.44), respectively.

Equations (7)

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1 ε 2 X i ( r ) = ω 2 c 2 X i ( r ) , i { x , y , z }
G k + G η ( G G ) k + G X ˜ i ( G ) = ω 2 c 2 X ˜ i ( G ) ,
a i = 1 h H R H χ ( i ) ( R ) χ ( R ) .
k 1 = e ̂ 1 e i k · r ,
k 2 = e ̂ 2 e i k · r .
P ̂ R f ( r ) = R f ( R 1 r ) .
ω 2 ( k z , r a ) = k z 2 η 0 ( r a ) .

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