Abstract

Light passing through a photonic crystal can undergo a negative or a positive refraction. The two refraction states can be functions of the contrast index, the incident angle and the slab thickness. By suitably using these properties it is possible to realize very simple and very efficient optical components to route the light. As an example we present a passive device acting as a polarizing beam splitter where TM polarization is refracted in positive direction whereas TE component is negatively refracted.

© 2005 Optical Society of America

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References

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  1. J.D. Joannopulos, R.D. Mead, and J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals, Springer Verlag (2001).
  3. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10 096–099 (1998).
    [Crossref]
  4. 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. F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001).
    [Crossref]
  6. Lijun Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. 86, 211106, (2005).
    [Crossref]
  7. T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Design of a Compact Photonic-Crystal-Based Polarizing Beam Splitter,” IEEE Photonics Technol. Lett.,  17, 1435–1437 (2005).
    [Crossref]
  8. C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
    [Crossref]
  9. E. Cubukcu and K. Aydin, et al. “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003).
    [Crossref] [PubMed]
  10. J. B. Pendry and D. R. Smith. “Reversing light with negative refraction,” Physics Today 57, 37–43 (2004).
    [Crossref]
  11. S Anantha Ramakrishna , “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
    [Crossref]
  12. V. Mocella, ”Negative refraction in Photonic Crystals: thickness dependence and Pendellösung phenomenon.,“ Opt. Express 13, 1361–1367 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1361.
    [Crossref] [PubMed]
  13. B. W. Battermann and H. Cole, ”Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
    [Crossref]
  14. P.P. Ewald, “Crystal optics for visible light and X rays,” Rev. Mod. Physics 37, 46–56 (1965).
    [Crossref]
  15. G.S. Agarwal, D. N. Pattanyak, and E. Wolf, “Electromagnetic field in spatially dispersive media,” Phys. Rev. B,  10, 1447–1475 (1974).
    [Crossref]
  16. K. Henneberger, “Additional Boundary Condition: an historical mistake,” Phys. Rev. Lett. 80, 2889–2892, (1998).
    [Crossref]
  17. J.J. Hopefield and D.G. Thomas, “Theoretical and Experimental Effects of Spatial Dispersion on the Optical Properties of Crystals,” Phys. Rev. 123, 563–572 (1963).
    [Crossref]

2005 (4)

Lijun Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. 86, 211106, (2005).
[Crossref]

T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Design of a Compact Photonic-Crystal-Based Polarizing Beam Splitter,” IEEE Photonics Technol. Lett.,  17, 1435–1437 (2005).
[Crossref]

S Anantha Ramakrishna , “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
[Crossref]

V. Mocella, ”Negative refraction in Photonic Crystals: thickness dependence and Pendellösung phenomenon.,“ Opt. Express 13, 1361–1367 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1361.
[Crossref] [PubMed]

2004 (1)

J. B. Pendry and D. R. Smith. “Reversing light with negative refraction,” Physics Today 57, 37–43 (2004).
[Crossref]

2003 (1)

E. Cubukcu and K. Aydin, et al. “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003).
[Crossref] [PubMed]

2002 (1)

C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

2001 (1)

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001).
[Crossref]

2000 (1)

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]

1998 (2)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10 096–099 (1998).
[Crossref]

K. Henneberger, “Additional Boundary Condition: an historical mistake,” Phys. Rev. Lett. 80, 2889–2892, (1998).
[Crossref]

1974 (1)

G.S. Agarwal, D. N. Pattanyak, and E. Wolf, “Electromagnetic field in spatially dispersive media,” Phys. Rev. B,  10, 1447–1475 (1974).
[Crossref]

1965 (1)

P.P. Ewald, “Crystal optics for visible light and X rays,” Rev. Mod. Physics 37, 46–56 (1965).
[Crossref]

1964 (1)

B. W. Battermann and H. Cole, ”Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[Crossref]

1963 (1)

J.J. Hopefield and D.G. Thomas, “Theoretical and Experimental Effects of Spatial Dispersion on the Optical Properties of Crystals,” Phys. Rev. 123, 563–572 (1963).
[Crossref]

Agarwal, G.S.

G.S. Agarwal, D. N. Pattanyak, and E. Wolf, “Electromagnetic field in spatially dispersive media,” Phys. Rev. B,  10, 1447–1475 (1974).
[Crossref]

Aydin, K.

E. Cubukcu and K. Aydin, et al. “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003).
[Crossref] [PubMed]

Battermann, B. W.

B. W. Battermann and H. Cole, ”Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[Crossref]

Birner, A.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001).
[Crossref]

Cole, H.

B. W. Battermann and H. Cole, ”Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[Crossref]

Cubukcu, E.

E. Cubukcu and K. Aydin, et al. “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003).
[Crossref] [PubMed]

Ewald, P.P.

P.P. Ewald, “Crystal optics for visible light and X rays,” Rev. Mod. Physics 37, 46–56 (1965).
[Crossref]

Fallahi, M.

T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Design of a Compact Photonic-Crystal-Based Polarizing Beam Splitter,” IEEE Photonics Technol. Lett.,  17, 1435–1437 (2005).
[Crossref]

Gallet, J.-F.

Lijun Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. 86, 211106, (2005).
[Crossref]

Genereux, F.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001).
[Crossref]

Gösele, U.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001).
[Crossref]

Henneberger, K.

K. Henneberger, “Additional Boundary Condition: an historical mistake,” Phys. Rev. Lett. 80, 2889–2892, (1998).
[Crossref]

Hopefield, J.J.

J.J. Hopefield and D.G. Thomas, “Theoretical and Experimental Effects of Spatial Dispersion on the Optical Properties of Crystals,” Phys. Rev. 123, 563–572 (1963).
[Crossref]

Joannopoulos, J. D.

C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

Joannopulos, J.D.

J.D. Joannopulos, R.D. Mead, and J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).

Johnson, S. G.

C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10 096–099 (1998).
[Crossref]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10 096–099 (1998).
[Crossref]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10 096–099 (1998).
[Crossref]

Krauss, T. F.

Lijun Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. 86, 211106, (2005).
[Crossref]

Leonard, S. W.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001).
[Crossref]

Liu, T.

T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Design of a Compact Photonic-Crystal-Based Polarizing Beam Splitter,” IEEE Photonics Technol. Lett.,  17, 1435–1437 (2005).
[Crossref]

Luo, C. Y.

C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

Mansuripur, M.

T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Design of a Compact Photonic-Crystal-Based Polarizing Beam Splitter,” IEEE Photonics Technol. Lett.,  17, 1435–1437 (2005).
[Crossref]

Mazilu, M.

Lijun Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. 86, 211106, (2005).
[Crossref]

Mead, R.D.

J.D. Joannopulos, R.D. Mead, and J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).

Mocella, V.

Moloney, J. V.

T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Design of a Compact Photonic-Crystal-Based Polarizing Beam Splitter,” IEEE Photonics Technol. Lett.,  17, 1435–1437 (2005).
[Crossref]

Notomi, M.

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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10 096–099 (1998).
[Crossref]

Pattanyak, D. N.

G.S. Agarwal, D. N. Pattanyak, and E. Wolf, “Electromagnetic field in spatially dispersive media,” Phys. Rev. B,  10, 1447–1475 (1974).
[Crossref]

Pendry, J. B.

J. B. Pendry and D. R. Smith. “Reversing light with negative refraction,” Physics Today 57, 37–43 (2004).
[Crossref]

Ramakrishna, S Anantha

S Anantha Ramakrishna , “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
[Crossref]

Sakoda, K.

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

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10 096–099 (1998).
[Crossref]

Smith, D. R.

J. B. Pendry and D. R. Smith. “Reversing light with negative refraction,” Physics Today 57, 37–43 (2004).
[Crossref]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10 096–099 (1998).
[Crossref]

Thomas, D.G.

J.J. Hopefield and D.G. Thomas, “Theoretical and Experimental Effects of Spatial Dispersion on the Optical Properties of Crystals,” Phys. Rev. 123, 563–572 (1963).
[Crossref]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10 096–099 (1998).
[Crossref]

van Driel, H. M.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001).
[Crossref]

Winn, J.N.

J.D. Joannopulos, R.D. Mead, and J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).

Wolf, E.

G.S. Agarwal, D. N. Pattanyak, and E. Wolf, “Electromagnetic field in spatially dispersive media,” Phys. Rev. B,  10, 1447–1475 (1974).
[Crossref]

Wu, Lijun

Lijun Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. 86, 211106, (2005).
[Crossref]

Zakharian, A. R.

T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Design of a Compact Photonic-Crystal-Based Polarizing Beam Splitter,” IEEE Photonics Technol. Lett.,  17, 1435–1437 (2005).
[Crossref]

Appl. Phys. Lett. (2)

C. Y. Luo, S. G. Johnson, and J. D. Joannopoulos. “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

Lijun Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. 86, 211106, (2005).
[Crossref]

IEEE Photonics Technol. Lett. (1)

T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Design of a Compact Photonic-Crystal-Based Polarizing Beam Splitter,” IEEE Photonics Technol. Lett.,  17, 1435–1437 (2005).
[Crossref]

Nature (1)

E. Cubukcu and K. Aydin, et al. “Negative refraction by photonic crystals,” Nature 423, 604–605 (2003).
[Crossref] [PubMed]

Opt. Express (1)

Phys. Rev. (1)

J.J. Hopefield and D.G. Thomas, “Theoretical and Experimental Effects of Spatial Dispersion on the Optical Properties of Crystals,” Phys. Rev. 123, 563–572 (1963).
[Crossref]

Phys. Rev. B (4)

G.S. Agarwal, D. N. Pattanyak, and E. Wolf, “Electromagnetic field in spatially dispersive media,” Phys. Rev. B,  10, 1447–1475 (1974).
[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, R10 096–099 (1998).
[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]

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001).
[Crossref]

Phys. Rev. Lett. (1)

K. Henneberger, “Additional Boundary Condition: an historical mistake,” Phys. Rev. Lett. 80, 2889–2892, (1998).
[Crossref]

Physics Today (1)

J. B. Pendry and D. R. Smith. “Reversing light with negative refraction,” Physics Today 57, 37–43 (2004).
[Crossref]

Rep. Prog. Phys. (1)

S Anantha Ramakrishna , “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
[Crossref]

Rev. Mod. Phys. (1)

B. W. Battermann and H. Cole, ”Dynamical diffraction theory of X rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[Crossref]

Rev. Mod. Physics (1)

P.P. Ewald, “Crystal optics for visible light and X rays,” Rev. Mod. Physics 37, 46–56 (1965).
[Crossref]

Other (2)

J.D. Joannopulos, R.D. Mead, and J.N. Winn, Photonic crystal: Molding the flow of light, Princeton University Press (Princeton, 1995).

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

Supplementary Material (1)

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Figures (6)

Fig. 1.
Fig. 1.

When the incoming wavelength is larger than the lattice period of the PhC the medium can be considered homogeneous and the dispersion surfaces are spheres (a). When the wavelength decreases the spheres approach one another and a Bragg gap appears (b). The conservation of the tangential component of the wavevector in the external medium, k i , determines the wavevectors in the PhC: P 1 O , P 2 O , P 1 H , P 2 H (c).

Fig. 2.
Fig. 2.

Dispersion (equi-frequency) surface for an air hole square lattice PhC in silicon (ε=11.9), r/a=0.195, in two adjacent Brillouin zones along the ΓX direction for TM polarization (E along the holes’ axis). By increasing the normalized frequency from 0.06 to 0.1848, the dispersion surfaces intersect over the lower 0-band (a). In such a case the upper 1-band has to be considered as well to get the complete dispersion surface, where the Bragg gap 2π/λ0 occurs (b).

Fig. 3.
Fig. 3.

Band diagram of the PhC of Fig. 2 for TM polarization (a). The regions along the XM direction, where many wavevectors are allowed for a given frequency, are highlighted in gray and labeled as α-β-γ. The dispersion surfaces corresponding to a frequency in region β (b) and region γ (c) show the overlap of different bands.

Fig. 4.
Fig. 4.

Band diagram for TM and TE polarization referring to the same PhC as in Fig. 2, along the XM direction (a). Λ0 /a as a function of ωn (b). The difference between the two polarization states is apparent.

Fig. 5
Fig. 5

(2.42 Mb) Movie versus time of FDTD simulation for TM (a) and TE polarization (b). The incident wave (λ=1.55 μm) has a Gaussian profile with 4 μm FWHM and impinges at an angle 20.57° over a PhC square lattice of air holes in silicon with r/a=0.195, a=0.64 μm. The grid size in calculation is 15 nm, in x and z direction.

Fig. 6.
Fig. 6.

The power flows of the refracted beams and of the reflected beam as a function of the wavelength for TE (a) and TM (b) polarization.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

k 2 = k 2 k = ( ω c ) 2 ε ˜ ( k , ω )
I + max t = 2 m Λ 0 2
I max t = ( 2 m i ) Λ 0 2 m = 1,2
t = 2 m Λ 0 TE 2 = ( 2 m 1 ) Λ 0 TM 2

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