Abstract

A simple method for calculating the transmittance of three-dimensional photonic crystals is proposed. The crystals are divided into multilayer thin films, and each film is divided into rectangles with a minute width to calculate the effective permittivity of the film by the effective medium theory. Transmittance of the multilayer thin films is calculated with the matrix method. As the number of atomic layers increases, remarkable stop bands appear. When the refractive index of photonic atoms increases, the stop band shifts to a lower frequency, the band widens, and the number of bands increases. Polarization and incident angle dependences are also analyzed. The limit of application for this calculation method is also discussed.

© 2006 Optical Society of America

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  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, R10096-R10099 (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. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
    [CrossRef]
  4. H. Kosaka, A. Tomita, T. Kawashima, T. Sato, and S. Kawakami, "Splitting of triply degenerate refractive indices by photonic crystals," Phys. Rev. B 62, 1477-1480 (2000).
    [CrossRef]
  5. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals--Molding the Flow of Light (Princeton University Press, Princeton, N.J., 1995).
  6. J. Maddox, "Photonic band-gaps bite the dust," Nature 348, 481-481 (1990).
    [CrossRef]
  7. S. Satpathy, Z. Zhang, and M. R. Salehpour, "Theory of photon bands in three-dimensional periodic dielectric structures," Phys. Rev. Lett. 64, 1239-1242 (1990).
    [CrossRef] [PubMed]
  8. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).
  9. Y. Ono, Y. Kimura, Y. Ohta, and N. Nishida, "Antireflection effects in ultrahigh spatial-frequency holographic relief gratings," Appl. Opt. 26, 1142-1146 (1987).
    [CrossRef] [PubMed]
  10. P. Lalanne, "Effective medium theory applied to photonic crystals composed of cubic or square cylinders," Appl. Opt. 35, 5369-5380 (1996).
    [CrossRef] [PubMed]
  11. Y. Ono and M. Shinzo, "Transmission spectrum analysis of three-dimensional photonic crystals by the effective medium theory," in Digests of Diffractive Optics 2003, European Optical Society Topical Meeting Digests Series (European Optical Society, Oxford, UK, 2003), pp. 20-21.
  12. D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," Appl. Opt. 32, 1154-1167 (1993).
    [CrossRef] [PubMed]
  13. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, Cambridge, 1999), Chap. 15, p. 837.
  14. R. Bräuer and O. Bryngdahl, "Design of antireflection gratings with approximate and rigorous methods," Appl. Opt. 33, 7875-7882 (1994).
    [CrossRef] [PubMed]
  15. M. Kamiyama, ed., Handbook of Thin Film Engineering (Ohmusha, Tokyo, 1964), Chap. II-7.
  16. Y. Ono and K. Ikemoto, "Fabrication of three-dimensional photonic crystals by holographic lithography," in Technical Digest of Diffractive Optics and Micro-Optics, Vol. 5 of 2002 OSA Techical Digest Series (Optical Society of America, Washington, D. C., 2002), pp. 205-207.

2000 (1)

H. Kosaka, A. Tomita, T. Kawashima, T. Sato, and S. Kawakami, "Splitting of triply degenerate refractive indices by photonic crystals," Phys. Rev. B 62, 1477-1480 (2000).
[CrossRef]

1999 (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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
[CrossRef]

1998 (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, R10096-R10099 (1998).
[CrossRef]

1996 (1)

1994 (1)

1993 (1)

D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," Appl. Opt. 32, 1154-1167 (1993).
[CrossRef] [PubMed]

1990 (2)

J. Maddox, "Photonic band-gaps bite the dust," Nature 348, 481-481 (1990).
[CrossRef]

S. Satpathy, Z. Zhang, and M. R. Salehpour, "Theory of photon bands in three-dimensional periodic dielectric structures," Phys. Rev. Lett. 64, 1239-1242 (1990).
[CrossRef] [PubMed]

1987 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, Cambridge, 1999), Chap. 15, p. 837.

Bräuer, R.

Bryngdahl, O.

Ikemoto, K.

Y. Ono and K. Ikemoto, "Fabrication of three-dimensional photonic crystals by holographic lithography," in Technical Digest of Diffractive Optics and Micro-Optics, Vol. 5 of 2002 OSA Techical Digest Series (Optical Society of America, Washington, D. C., 2002), pp. 205-207.

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals--Molding the Flow of Light (Princeton University Press, Princeton, N.J., 1995).

Kamiyama, M.

M. Kamiyama, ed., Handbook of Thin Film Engineering (Ohmusha, Tokyo, 1964), Chap. II-7.

Kawakami, S.

H. Kosaka, A. Tomita, T. Kawashima, T. Sato, and S. Kawakami, "Splitting of triply degenerate refractive indices by photonic crystals," Phys. Rev. B 62, 1477-1480 (2000).
[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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, A. Tomita, T. Kawashima, T. Sato, and S. Kawakami, "Splitting of triply degenerate refractive indices by photonic crystals," Phys. Rev. B 62, 1477-1480 (2000).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
[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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Kimura, Y.

Kosaka, H.

H. Kosaka, A. Tomita, T. Kawashima, T. Sato, and S. Kawakami, "Splitting of triply degenerate refractive indices by photonic crystals," Phys. Rev. B 62, 1477-1480 (2000).
[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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Lalanne, P.

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Maddox, J.

J. Maddox, "Photonic band-gaps bite the dust," Nature 348, 481-481 (1990).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals--Molding the Flow of Light (Princeton University Press, Princeton, N.J., 1995).

Morris, G. M.

D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," Appl. Opt. 32, 1154-1167 (1993).
[CrossRef] [PubMed]

Nishida, N.

Notomi, M.

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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Ohta, Y.

Ono, Y.

Y. Ono, Y. Kimura, Y. Ohta, and N. Nishida, "Antireflection effects in ultrahigh spatial-frequency holographic relief gratings," Appl. Opt. 26, 1142-1146 (1987).
[CrossRef] [PubMed]

Y. Ono and M. Shinzo, "Transmission spectrum analysis of three-dimensional photonic crystals by the effective medium theory," in Digests of Diffractive Optics 2003, European Optical Society Topical Meeting Digests Series (European Optical Society, Oxford, UK, 2003), pp. 20-21.

Y. Ono and K. Ikemoto, "Fabrication of three-dimensional photonic crystals by holographic lithography," in Technical Digest of Diffractive Optics and Micro-Optics, Vol. 5 of 2002 OSA Techical Digest Series (Optical Society of America, Washington, D. C., 2002), pp. 205-207.

Raguin, D. H.

D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," Appl. Opt. 32, 1154-1167 (1993).
[CrossRef] [PubMed]

Salehpour, M. R.

S. Satpathy, Z. Zhang, and M. R. Salehpour, "Theory of photon bands in three-dimensional periodic dielectric structures," Phys. Rev. Lett. 64, 1239-1242 (1990).
[CrossRef] [PubMed]

Sato, T.

H. Kosaka, A. Tomita, T. Kawashima, T. Sato, and S. Kawakami, "Splitting of triply degenerate refractive indices by photonic crystals," Phys. Rev. B 62, 1477-1480 (2000).
[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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Satpathy, S.

S. Satpathy, Z. Zhang, and M. R. Salehpour, "Theory of photon bands in three-dimensional periodic dielectric structures," Phys. Rev. Lett. 64, 1239-1242 (1990).
[CrossRef] [PubMed]

Shinzo, M.

Y. Ono and M. Shinzo, "Transmission spectrum analysis of three-dimensional photonic crystals by the effective medium theory," in Digests of Diffractive Optics 2003, European Optical Society Topical Meeting Digests Series (European Optical Society, Oxford, UK, 2003), pp. 20-21.

Tamamura, T.

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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Tomita, A.

H. Kosaka, A. Tomita, T. Kawashima, T. Sato, and S. Kawakami, "Splitting of triply degenerate refractive indices by photonic crystals," Phys. Rev. B 62, 1477-1480 (2000).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
[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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals--Molding the Flow of Light (Princeton University Press, Princeton, N.J., 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, Cambridge, 1999), Chap. 15, p. 837.

Zhang, Z.

S. Satpathy, Z. Zhang, and M. R. Salehpour, "Theory of photon bands in three-dimensional periodic dielectric structures," Phys. Rev. Lett. 64, 1239-1242 (1990).
[CrossRef] [PubMed]

Appl. Opt. (1)

D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," Appl. Opt. 32, 1154-1167 (1993).
[CrossRef] [PubMed]

Appl. Opt. (3)

Appl. Phys. Lett. (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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Photonic crystals for micro lightwave circuits using wavelength-dependent angular beam steering," Appl. Phys. Lett. 74, 1370-1372 (1999).
[CrossRef]

Nature (1)

J. Maddox, "Photonic band-gaps bite the dust," Nature 348, 481-481 (1990).
[CrossRef]

Phys. Rev. B (1)

H. Kosaka, A. Tomita, T. Kawashima, T. Sato, and S. Kawakami, "Splitting of triply degenerate refractive indices by photonic crystals," Phys. Rev. B 62, 1477-1480 (2000).
[CrossRef]

Phys. Rev. B (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, R10096-R10099 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

S. Satpathy, Z. Zhang, and M. R. Salehpour, "Theory of photon bands in three-dimensional periodic dielectric structures," Phys. Rev. Lett. 64, 1239-1242 (1990).
[CrossRef] [PubMed]

Other (6)

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Y. Ono and M. Shinzo, "Transmission spectrum analysis of three-dimensional photonic crystals by the effective medium theory," in Digests of Diffractive Optics 2003, European Optical Society Topical Meeting Digests Series (European Optical Society, Oxford, UK, 2003), pp. 20-21.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals--Molding the Flow of Light (Princeton University Press, Princeton, N.J., 1995).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, Cambridge, 1999), Chap. 15, p. 837.

M. Kamiyama, ed., Handbook of Thin Film Engineering (Ohmusha, Tokyo, 1964), Chap. II-7.

Y. Ono and K. Ikemoto, "Fabrication of three-dimensional photonic crystals by holographic lithography," in Technical Digest of Diffractive Optics and Micro-Optics, Vol. 5 of 2002 OSA Techical Digest Series (Optical Society of America, Washington, D. C., 2002), pp. 205-207.

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Figures (13)

Fig. 1
Fig. 1

Simple cubic lattice that consists of spherical photonic atoms; a is the lattice constant, R, the photonic atom radius.

Fig. 2
Fig. 2

Calculation method of effective permittivity for a thin, two-dimensional photonic crystal that consists of cylinder-shaped atoms sliced from the three-dimensional photonic crystal shown in Fig. 1. Value ε j is the effective permittivity calculated by the EMT for the jth rectangle.

Fig. 3
Fig. 3

Definitions for the parallel polarization wave and normal polarization wave.

Fig. 4
Fig. 4

Multilayer thin-film structure: nj , refractive index; ϕ j , incident angle; d j , thickness; δ j , phase.

Fig. 5
Fig. 5

Transmittance with respect to division number per lattice constant in the depth direction at a normalized frequency of 0.4 for a ten-atomic-layer crystal.

Fig. 6
Fig. 6

Effective refractive index distribution in the depth direction. The vertical axis, height, is normalized by the lattice constant a.

Fig. 7
Fig. 7

Transmittance versus normalized frequency for five-, ten-, twenty-, and thirty-atomic-layer crystals with R / a = 0.5 .

Fig. 8
Fig. 8

Minimum transmittance with respect to the number of atomic layers at the first dip of Fig. 7, that is, at the normalized frequency of around 0.4 for indices n = 1.5 and 2.0 .

Fig. 9
Fig. 9

Cut-off frequency versus refractive index of photonic atoms for thirty-atomic-layer crystals.

Fig. 10
Fig. 10

Photonic atom radius dependence of transmittance for twenty-layer crystals.

Fig. 11
Fig. 11

Transmittance versus normalized frequency when the refractive indices of atom and background materials are exchanged for those of the twenty-layer crystal with R / a = 0.3 .

Fig. 12
Fig. 12

Crystal surface phase dependence of transmittance for thirty-layer crystals with R / a = 0.5 .

Fig. 13
Fig. 13

Incident angle and polarization dependences of transmittance for the twenty-layer crystal with R / a = 0.5 . Bold curve is at θ i = 0 ° for both p and s polarizations. Solid and dotted curves are for p and s polarizations, respectively.

Equations (15)

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ε ( 2 ) = ε ( 0 ) [ 1 + π 2 3 ( n 1 + n 2 ) 2 ( Λ λ ) 2 f 2 ( 1 f ) 2 ( α n 1 ) 2 1 + f ( α n 2 1 ) ] ,
ε ( 2 ) = ε ( 0 ) [ 1 + π 2 3 ( n 1 + n 2 ) 2 ( Λ λ ) 2 f 2 ( 1 f ) 2 ( α n 1 ) 2 1 + f ( α n 2 1 ) [ α n 2 f ( α n 2 1 ) ] 2 ] ,
ε ( 0 ) = ( 1 f ) n 1 2 + f n 2 2 ,
ε ( 0 ) = 1 / [ ( 1 f ) / n 1 2 + f / n 2 2 ] .
D j = ε j E .
D ¯ = E t j ε j / t j = E f j ε j ,
ε = D ¯ / E = f j ε j .
E = D / ε j .
E ¯ = ( D t j / ε j ) / t j = D f j / ε j .
ε = D / E ¯ = 1 / f j / ε j .
T j = [ cos δ j ( i / w j ) sin δ j i w j sin δ j cos δ j ] ,
δ j = ( 2 π / λ ) n j d j cos ϕ j ,
w j = { n j cos ϕ j : s  polarization, n j / cos ϕ j : p  polarization .
II j = 1 N T j = [ A B C D ] .
T = 4 w 0 w s | w 0 A + w 0 w s B + C + w s D | 2 .

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