Abstract

We present an analysis of the phase and amplitude responses of guided resonances in a photonic crystal slab. Through this analysis, we obtain the general rules and conditions under which a photonic crystal slab can be employed as a general elliptical polarization beam splitter, separating an incoming beam equally into its two orthogonal constituents, so that half the power is reflected in one polarization state, and half the power is transmitted in the other state. We show that at normal incidence a photonic crystal slab acts as a dual quarter-wave retarder in which the fast and slow axes are switched for reflection and transmission. We also analyze the case where such a structure operates at oblique incidences. As a result we show that the effective dielectric constant of the photonic crystal slab imposes the Brewster angle as a boundary, separating two ranges of angles with different mechanisms of polarization beam splitting. We show that the diattenuation can be tuned from zero to one to make the structure a circular or linear polarization beam splitter. We verify our analytical analysis through finite-difference time-domain simulations and experimental measurements at infrared wavelengths.

© 2008 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Frequency domain analysis of guided resonances and polarization selectivity in photonic crystal membranes

Jan Kupec, Uğur Akçakoca, and Bernd Witzigmann
J. Opt. Soc. Am. B 28(1) 69-78 (2011)

Circular polarization beam splitter that uses frustrated total internal reflection by an embedded symmetric achiral multilayer coating

R. M. A. Azzam and A. De
Opt. Lett. 28(5) 355-357 (2003)

Angular and polarization properties of a photonic crystal slab mirror

Virginie Lousse, Wonjoo Suh, Onur Kilic, Sora Kim, Olav Solgaard, and Shanhui Fan
Opt. Express 12(8) 1575-1582 (2004)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
    [Crossref]
  2. P. Kok, C. P. Williams, and J. P. Dowling, “Construction of a quantum repeater with linear optics,” Phys. Rev. A 68, 022301 (2003).
    [Crossref]
  3. J. L. Pezzaniti and R. A. Chipman, “Angular dependence of polarizing beam splitter cubes,” Appl. Opt. 33, 1916-1929 (1994).
    [Crossref]
  4. D. Yi, Y. Yan, H. Liu, S. Lu, and G. Jin, “Broadband polarizing beam splitter based on the form birefringence of a subwavelength grating in the quasi-static domain,” Opt. Lett. 29, 754-756 (2004).
    [Crossref] [PubMed]
  5. D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
    [Crossref]
  6. M. C. Gupta and S. T. Peng, “Multifunction grating for signal detection of optical disk,” Proc. SPIE 1499, 303-306 (1991).
    [Crossref]
  7. A. G. Lopez and H. G. Craighead, “Wave-plate polarizing beam splitter based on a form-birefringent multilayer grating,” Opt. Lett. 23, 1627-1629 (1998).
    [Crossref]
  8. M. Schmitz, R. Brauer, and O. Bryngdahl, “Gratings in the resonance domain as polarizing beam splitters,” Opt. Lett. 20, 1830-1831 (1995).
    [Crossref] [PubMed]
  9. R. C. Tyan, P. C. Sun, and Y. Fainman, “Polarizing beam splitters constructed of form-birefringent multilayer gratings,” Proc. SPIE 2689, 82-89 (1996).
    [Crossref]
  10. R. C. Tyan, P. C. Sun, A. Scherer, and Y. Fainman, “Polarizing beam splitter based on anisotropic spectral reflectivity characteristics of form-birefringent multilayer gratings,” Opt. Lett. 21, 761-763 (1996).
    [Crossref] [PubMed]
  11. G. R. Bird and M. Parrish, Jr., “The wire grid as a near-infrared polarizer,” J. Opt. Soc. Am. 50, 886-891 (1960).
    [Crossref]
  12. R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022-1024 (1992).
    [Crossref]
  13. J. R. Wendt, G. A. Vawter, R. E. Smith, and M. E. Warren, “Subwavelength, binary lenses at infrared wavelengths,” J. Vac. Sci. Technol. B 15, 2946-2949 (1997).
    [Crossref]
  14. R. C. Enger and S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220-3228 (1983).
    [Crossref] [PubMed]
  15. L. H. Cescato, E. Gluch, and N. Streibl, “Holographic quarter-wave plate,” Appl. Opt. 29, 3286-3290 (1990).
    [Crossref] [PubMed]
  16. D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492-494 (1983).
    [Crossref]
  17. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566-1572 (1997).
    [Crossref] [PubMed]
  18. G. P. Nordin and P. C. Deguzman, “Broadband form birefringent quarter-wave plate for the mid-infrared wavelength region,” Opt. Express 5, 163-168 (1999).
    [Crossref] [PubMed]
  19. S. D. Jacobs, K. A. Cerqua, K. L. Marshall, A. Schmid, M. J. Guardalben, and K. J. Skerrett, “Liquid-crystal laser optics: design, fabrication, and performance,” J. Opt. Soc. Am. B 5, 1962-1979 (1988).
    [Crossref]
  20. J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26, 587-589 (2001).
    [Crossref]
  21. R. M. A. Azzam and F. A. Mahmoud, “Symmetrically coated pellicle beam splitters for dual quarter-wave retardation in reflection and transmission,” Appl. Opt. 41, 235-238 (2002).
    [Crossref] [PubMed]
  22. R. M. A. Azzam and A. De, “Circular polarization beam splitter that uses frustrated total internal reflection by an embedded symmetric achiral multilayer coating,” Opt. Lett. 28, 355-357 (2003).
    [Crossref] [PubMed]
  23. D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Photonic crystal polarizers and polarizing beam splitters,” J. Appl. Phys. 93, 9429-9431 (2003).
    [Crossref]
  24. D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Birefringence in two-dimensional bulk photonic crystals applied to the construction of quarter waveplates,” Opt. Express 11, 125-133 (2003).
    [Crossref] [PubMed]
  25. V. Lousse, W. Suh, O. Kilic, S. Kim, O. Solgaard, and S. Fan, “Angular and polarization properties of a photonic crystal slab mirror,” Opt. Express 12, 1575-1582 (2004).
    [Crossref] [PubMed]
  26. V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, and R. M. D. L. Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050-2057 (1999).
    [Crossref]
  27. M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17, 2096-2112 (1999).
    [Crossref]
  28. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
    [Crossref]
  29. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
    [Crossref]
  30. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569-572 (2003).
    [Crossref]
  31. M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
    [Crossref]
  32. M. Meier, A. Mekis, A. Dodabalapur, A. A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7-9 (1999).
    [Crossref]
  33. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123-1125 (2001).
    [Crossref] [PubMed]
  34. H. Y. Ryu, Y. H. Lee, R. L. Sellin, and D. Bimberg, “Over 30-fold enhancement of light extraction from free-standing photonic crystal slabs with InGaAs quantum dots at low temperature,” Appl. Phys. Lett. 79, 3573-3575 (2001).
    [Crossref]
  35. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  36. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).
  37. For a complex number z=∣z∣ejΦ, the phase will be Φ=arg(z). We restrict the phase to the principal value, so that −π<arg(z)⩽π. The argument of a complex number has the properties arg(z1)+arg(z2)=arg(z1z2), and arg(z1)−arg(z2)=arg(z1/z2). The argument of a real number arg(R) will be 0 and π for R⩾0 and R<0, respectively.
  38. E. Hecht, Optics (Addison-Wesley, 1998).
  39. F. Gires and P. Tournois, “Interféromètre utilisable pour la compression d'impulsions lumineuses modulées en fréquence,” C. R. Acad. Sci. Paris 258, 6112-6115 (1964).

2004 (2)

2003 (5)

2002 (2)

2001 (6)

J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. 26, 587-589 (2001).
[Crossref]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
[Crossref]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123-1125 (2001).
[Crossref] [PubMed]

H. Y. Ryu, Y. H. Lee, R. L. Sellin, and D. Bimberg, “Over 30-fold enhancement of light extraction from free-standing photonic crystal slabs with InGaAs quantum dots at low temperature,” Appl. Phys. Lett. 79, 3573-3575 (2001).
[Crossref]

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
[Crossref]

1999 (4)

1998 (1)

1997 (3)

H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. 36, 1566-1572 (1997).
[Crossref] [PubMed]

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

J. R. Wendt, G. A. Vawter, R. E. Smith, and M. E. Warren, “Subwavelength, binary lenses at infrared wavelengths,” J. Vac. Sci. Technol. B 15, 2946-2949 (1997).
[Crossref]

1996 (2)

1995 (1)

1994 (1)

1992 (1)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022-1024 (1992).
[Crossref]

1991 (1)

M. C. Gupta and S. T. Peng, “Multifunction grating for signal detection of optical disk,” Proc. SPIE 1499, 303-306 (1991).
[Crossref]

1990 (1)

1988 (1)

1983 (2)

R. C. Enger and S. K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. 22, 3220-3228 (1983).
[Crossref] [PubMed]

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492-494 (1983).
[Crossref]

1964 (1)

F. Gires and P. Tournois, “Interféromètre utilisable pour la compression d'impulsions lumineuses modulées en fréquence,” C. R. Acad. Sci. Paris 258, 6112-6115 (1964).

1960 (1)

Adachi, J.

Astratov, V. N.

Azzam, R. M. A.

Bhat, R.

Bimberg, D.

H. Y. Ryu, Y. H. Lee, R. L. Sellin, and D. Bimberg, “Over 30-fold enhancement of light extraction from free-standing photonic crystal slabs with InGaAs quantum dots at low temperature,” Appl. Phys. Lett. 79, 3573-3575 (2001).
[Crossref]

Bird, G. R.

Boroditsky, M.

Brauer, R.

Bryngdahl, O.

Busch, A.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

Case, S. K.

Cerqua, K. A.

Cescato, L. H.

Chiao, R. Y.

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Birefringence in two-dimensional bulk photonic crystals applied to the construction of quarter waveplates,” Opt. Express 11, 125-133 (2003).
[Crossref] [PubMed]

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Photonic crystal polarizers and polarizing beam splitters,” J. Appl. Phys. 93, 9429-9431 (2003).
[Crossref]

Chipman, R. A.

Chutinan, A.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123-1125 (2001).
[Crossref] [PubMed]

Coccioli, R.

Craighead, H. G.

Culshaw, I. S.

Davis, J. A.

De, A.

Deguzman, P. C.

Dias, D.

D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
[Crossref]

Dodabalapur, A.

M. Meier, A. Mekis, A. Dodabalapur, A. A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7-9 (1999).
[Crossref]

Dowling, J. P.

P. Kok, C. P. Williams, and J. P. Dowling, “Construction of a quantum repeater with linear optics,” Phys. Rev. A 68, 022301 (2003).
[Crossref]

Enger, R. C.

Erchak, A.

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

Fainman, Y.

Fan, S.

V. Lousse, W. Suh, O. Kilic, S. Kim, O. Solgaard, and S. Fan, “Angular and polarization properties of a photonic crystal slab mirror,” Opt. Express 12, 1575-1582 (2004).
[Crossref] [PubMed]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569-572 (2003).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

Fernández-Pousa, C. R.

Ferstl, M.

D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
[Crossref]

Flanders, D. C.

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492-494 (1983).
[Crossref]

Franson, J. D.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
[Crossref]

Gires, F.

F. Gires and P. Tournois, “Interféromètre utilisable pour la compression d'impulsions lumineuses modulées en fréquence,” C. R. Acad. Sci. Paris 258, 6112-6115 (1964).

Gluch, E.

Guardalben, M. J.

Gupta, M. C.

M. C. Gupta and S. T. Peng, “Multifunction grating for signal detection of optical disk,” Proc. SPIE 1499, 303-306 (1991).
[Crossref]

Haidner, H.

D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
[Crossref]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1998).

Hickmann, J. M.

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Birefringence in two-dimensional bulk photonic crystals applied to the construction of quarter waveplates,” Opt. Express 11, 125-133 (2003).
[Crossref] [PubMed]

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Photonic crystal polarizers and polarizing beam splitters,” J. Appl. Phys. 93, 9429-9431 (2003).
[Crossref]

Imada, M.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123-1125 (2001).
[Crossref] [PubMed]

Ippen, E. P.

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

Iwata, K.

Jacobs, B. C.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
[Crossref]

Jacobs, S. D.

Jin, G.

Joannopoulos, J. D.

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569-572 (2003).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

M. Meier, A. Mekis, A. Dodabalapur, A. A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7-9 (1999).
[Crossref]

Johnson, S. R.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

Kanskar, M.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

Kikuta, H.

Kilic, O.

Kim, S.

Kok, P.

P. Kok, C. P. Williams, and J. P. Dowling, “Construction of a quantum repeater with linear optics,” Phys. Rev. A 68, 022301 (2003).
[Crossref]

Kolodziejski, L. A.

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

Krauss, T. F.

Lee, Y. H.

H. Y. Ryu, Y. H. Lee, R. L. Sellin, and D. Bimberg, “Over 30-fold enhancement of light extraction from free-standing photonic crystal slabs with InGaAs quantum dots at low temperature,” Appl. Phys. Lett. 79, 3573-3575 (2001).
[Crossref]

Liu, H.

Lopez, A. G.

Lousse, V.

Lu, S.

Mackenzie, J.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

Magnusson, R.

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022-1024 (1992).
[Crossref]

Mahmoud, F. A.

Marshall, K. L.

McCormick, C. F.

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Birefringence in two-dimensional bulk photonic crystals applied to the construction of quarter waveplates,” Opt. Express 11, 125-133 (2003).
[Crossref] [PubMed]

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Photonic crystal polarizers and polarizing beam splitters,” J. Appl. Phys. 93, 9429-9431 (2003).
[Crossref]

Meier, M.

M. Meier, A. Mekis, A. Dodabalapur, A. A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7-9 (1999).
[Crossref]

Mekis, A.

M. Meier, A. Mekis, A. Dodabalapur, A. A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7-9 (1999).
[Crossref]

Mochizuki, M.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123-1125 (2001).
[Crossref] [PubMed]

Moreno, I.

Morin, R.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

Nalamasu, O.

M. Meier, A. Mekis, A. Dodabalapur, A. A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7-9 (1999).
[Crossref]

Noda, S.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123-1125 (2001).
[Crossref] [PubMed]

Nordin, G. P.

Ohira, Y.

Pacradouni, V.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

Paddon, P.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

Parrish, M.

Peng, S. T.

M. C. Gupta and S. T. Peng, “Multifunction grating for signal detection of optical disk,” Proc. SPIE 1499, 303-306 (1991).
[Crossref]

Petrich, G. S.

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

Pezzaniti, J. L.

Pittman, T. B.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
[Crossref]

Rakich, P.

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

Ripin, D. J.

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

Rue, R. M. D. L.

Ryu, H. Y.

H. Y. Ryu, Y. H. Lee, R. L. Sellin, and D. Bimberg, “Over 30-fold enhancement of light extraction from free-standing photonic crystal slabs with InGaAs quantum dots at low temperature,” Appl. Phys. Lett. 79, 3573-3575 (2001).
[Crossref]

Sakoda, K.

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

Scherer, A.

Schmid, A.

Schmitz, M.

Sellin, R. L.

H. Y. Ryu, Y. H. Lee, R. L. Sellin, and D. Bimberg, “Over 30-fold enhancement of light extraction from free-standing photonic crystal slabs with InGaAs quantum dots at low temperature,” Appl. Phys. Lett. 79, 3573-3575 (2001).
[Crossref]

Skerrett, K. J.

Skolnick, M. S.

Slusher, R. E.

M. Meier, A. Mekis, A. Dodabalapur, A. A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7-9 (1999).
[Crossref]

Smith, R. E.

J. R. Wendt, G. A. Vawter, R. E. Smith, and M. E. Warren, “Subwavelength, binary lenses at infrared wavelengths,” J. Vac. Sci. Technol. B 15, 2946-2949 (1997).
[Crossref]

Solgaard, O.

Solli, D. R.

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Birefringence in two-dimensional bulk photonic crystals applied to the construction of quarter waveplates,” Opt. Express 11, 125-133 (2003).
[Crossref] [PubMed]

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Photonic crystal polarizers and polarizing beam splitters,” J. Appl. Phys. 93, 9429-9431 (2003).
[Crossref]

Stankovic, S.

D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
[Crossref]

Steingrüber, R.

D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
[Crossref]

Stevenson, R. M.

Streibl, N.

Suh, W.

Sun, P. C.

Tiedje, T.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

Timko, A. A.

M. Meier, A. Mekis, A. Dodabalapur, A. A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7-9 (1999).
[Crossref]

Tournois, P.

F. Gires and P. Tournois, “Interféromètre utilisable pour la compression d'impulsions lumineuses modulées en fréquence,” C. R. Acad. Sci. Paris 258, 6112-6115 (1964).

Tschudi, T.

D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
[Crossref]

Tyan, R. C.

Vawter, G. A.

J. R. Wendt, G. A. Vawter, R. E. Smith, and M. E. Warren, “Subwavelength, binary lenses at infrared wavelengths,” J. Vac. Sci. Technol. B 15, 2946-2949 (1997).
[Crossref]

Vrijen, R.

Wang, L. L.

D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
[Crossref]

Wang, S. S.

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022-1024 (1992).
[Crossref]

Warren, M. E.

J. R. Wendt, G. A. Vawter, R. E. Smith, and M. E. Warren, “Subwavelength, binary lenses at infrared wavelengths,” J. Vac. Sci. Technol. B 15, 2946-2949 (1997).
[Crossref]

Wendt, J. R.

J. R. Wendt, G. A. Vawter, R. E. Smith, and M. E. Warren, “Subwavelength, binary lenses at infrared wavelengths,” J. Vac. Sci. Technol. B 15, 2946-2949 (1997).
[Crossref]

Whittaker, D. M.

Williams, C. P.

P. Kok, C. P. Williams, and J. P. Dowling, “Construction of a quantum repeater with linear optics,” Phys. Rev. A 68, 022301 (2003).
[Crossref]

Yablonovitch, E.

Yan, Y.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Yi, D.

Yokoyama, M.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123-1125 (2001).
[Crossref] [PubMed]

Young, J. F.

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

Appl. Opt. (5)

Appl. Phys. Lett. (6)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022-1024 (1992).
[Crossref]

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492-494 (1983).
[Crossref]

A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a 2D photonic crystal in a semiconductor LED,” Appl. Phys. Lett. 78, 563-565 (2001).
[Crossref]

M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438-1440 (1997).
[Crossref]

M. Meier, A. Mekis, A. Dodabalapur, A. A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7-9 (1999).
[Crossref]

H. Y. Ryu, Y. H. Lee, R. L. Sellin, and D. Bimberg, “Over 30-fold enhancement of light extraction from free-standing photonic crystal slabs with InGaAs quantum dots at low temperature,” Appl. Phys. Lett. 79, 3573-3575 (2001).
[Crossref]

C. R. Acad. Sci. Paris (1)

F. Gires and P. Tournois, “Interféromètre utilisable pour la compression d'impulsions lumineuses modulées en fréquence,” C. R. Acad. Sci. Paris 258, 6112-6115 (1964).

J. Appl. Phys. (1)

D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Photonic crystal polarizers and polarizing beam splitters,” J. Appl. Phys. 93, 9429-9431 (2003).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. A, Pure Appl. Opt. (1)

D. Dias, S. Stankovic, H. Haidner, L. L. Wang, T. Tschudi, M. Ferstl, and R. Steingrüber, “High-frequency gratings for applications to DVD pickup systems,” J. Opt. A, Pure Appl. Opt. 3, 164-173 (2001).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Vac. Sci. Technol. B (1)

J. R. Wendt, G. A. Vawter, R. E. Smith, and M. E. Warren, “Subwavelength, binary lenses at infrared wavelengths,” J. Vac. Sci. Technol. B 15, 2946-2949 (1997).
[Crossref]

Opt. Express (3)

Opt. Lett. (6)

Phys. Rev. A (2)

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
[Crossref]

P. Kok, C. P. Williams, and J. P. Dowling, “Construction of a quantum repeater with linear optics,” Phys. Rev. A 68, 022301 (2003).
[Crossref]

Phys. Rev. B (1)

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

Proc. SPIE (2)

M. C. Gupta and S. T. Peng, “Multifunction grating for signal detection of optical disk,” Proc. SPIE 1499, 303-306 (1991).
[Crossref]

R. C. Tyan, P. C. Sun, and Y. Fainman, “Polarizing beam splitters constructed of form-birefringent multilayer gratings,” Proc. SPIE 2689, 82-89 (1996).
[Crossref]

Science (1)

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123-1125 (2001).
[Crossref] [PubMed]

Other (4)

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

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

For a complex number z=∣z∣ejΦ, the phase will be Φ=arg(z). We restrict the phase to the principal value, so that −π<arg(z)⩽π. The argument of a complex number has the properties arg(z1)+arg(z2)=arg(z1z2), and arg(z1)−arg(z2)=arg(z1/z2). The argument of a real number arg(R) will be 0 and π for R⩾0 and R<0, respectively.

E. Hecht, Optics (Addison-Wesley, 1998).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1 Illustration showing light normally incident on a PCS.

Download Full Size | PPT Slide | PDF

Fig. 2
Fig. 2 FDTD simulations showing that the background is not significantly affected by an introduction of a form birefringence. The simulations were done for a dielectric slab with a dielectric constant of 12 and a thickness of 0.55 a , a being the lattice constant of rectangular through holes on a square lattice. A simulation was first done for square-shaped holes of size 0.5 a × 0.5 a . Then, the β value at the frequency ω = 0.3 ( 2 π c ∕ a ) was calculated, where it has a value of 0. Afterward, the width of the rectangular holes was gradually increased (along the direction 2) by keeping the area of the hole ( 0.5 a × 0.5 a ) constant. The horizontal axis corresponds to the hole width. The hole width of 0.5 a corresponds to square-shaped holes, while the hole width of 1.0 a corresponds to a 1-D grating, since the rectangular holes become connected. The rectangular and circular data points correspond to the β values at the frequency ω = 0.3 ( 2 π c ∕ a ) for the two polarizations.

Download Full Size | PPT Slide | PDF

Fig. 3
Fig. 3 FDTD simulations showing that the linewidth does not change significantly when the doubly degenerate resonances split. The data are from the same simulations that are used in Fig. 2. The graph shows how a guided resonance splits through form birefringence. The solid curve corresponds to a doubly degenerate resonance for a hole width of 0.5 a , which is a square hole. By introducing birefringence through making the holes rectangular, the doubly degenerate resonance splits into two. The dashed curves correspond to split resonances for a hole width of 0.55 a , and the dotted curves correspond to split resonances for a hole width of 0.63 a . The guided resonances used in the calculations were isolated from the background spectrum by passing the complex field amplitude through a high-pass filter.

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4 Calculated spectra of the phase and amplitude responses for transmission (top) and reflection (bottom) for three different form-birefringent PCS structures (a), (b), and (c). The circle-marked, square-marked, and solid curves correspond to the normalized powers in polarization 1, polarization 2, and both polarizations, respectively. The dashed line corresponds to the retardance in units of π. The structure has a dielectric constant of 12 with a thickness of 0.55 a , a being the pitch. The guided resonances possess a half-linewidth of γ = 0.064 ( 2 π c ∕ a ) , are separated by Δ = γ , and have ω ( 1 ) = ω ( 2 ) = ω 0 values of 0.47 ( 2 π c ∕ a ) , 0.53 ( 2 π c ∕ a ) , and 0.58 ( 2 π c ∕ a ) for the cases (a), (b), and (c), respectively. (a) corresponds to a linear PBS with β = − 1 , (b) to a circular PBS with β = 0 , and (c) to a linear PBS with β = + 1 , in the same way as it is illustrated in Fig. 5.

Download Full Size | PPT Slide | PDF

Fig. 5
Fig. 5 Illustration showing how the PCS-based PBS operates for several special β values.

Download Full Size | PPT Slide | PDF

Fig. 6
Fig. 6 Illustration of the PCS device on the Poincaré sphere. The arrow with its head just at the center of the sphere denotes the incident polarization, while the two arrows pointing away from the center show the reflected and transmitted polarizations. The circle passing through P + 45 ° , L , P − 45 ° , and R shows the polarizations aligned to the PBS axes, as in Eq. (13). Note that for the case when the incident polarization is aligned to the PBS, the arrows denoting the transmission and reflection polarizations are always antiparallel to each other and orthogonal to the arrow depicting the incidence polarization, showing that the incident polarization is separated into its two orthogonal constituents. The paths show what happens when some of the parameters are changed gradually, summarizing the operation of this type of PBS. (a) β changes from 0 to 1 2 to 1 for the incidence polarization fixed at L . (b) Incidence polarization changes from L to elliptical to P + 45 ° , for β fixed at 1. (c) β changes from 1 to 1 2 to 0 for the incidence polarization fixed at P + 45 ° . (d) Incidence polarization changes from P + 45 ° to elliptical to L for β fixed at 0. (e) The incident polarization is misaligned, rotating between the P + 45 ° and H polarizations, and then between H and L polarizations.

Download Full Size | PPT Slide | PDF

Fig. 7
Fig. 7 Illustration showing light incident on a PCS at an oblique angle. The azimuthal orientation of the incidence plane is irrelevant.

Download Full Size | PPT Slide | PDF

Fig. 8
Fig. 8 Calculated spectra of the phase and amplitude responses for transmission (top) and reflection (bottom) for an oblique incidence angle larger than the Brewster angle, θ > θ b r ( θ = 80 ° and θ b r ≈ 74 ° ) for the case of β = 0 . The structural parameters of the PCS are the same as the ones used for Fig. 4. The guided resonances possess a half-linewidth of γ = 0.040 ( 2 π c ∕ a ) . The PBS frequencies are ω ( 1 ) = 0.51 ( 2 π c ∕ a ) and ω ( 2 ) = 0.59 ( 2 π c ∕ a ) , respectively.

Download Full Size | PPT Slide | PDF

Fig. 9
Fig. 9 FDTD simulations showing (a) transmission results and (b) reflection results for a circular PBS.

Download Full Size | PPT Slide | PDF

Fig. 10
Fig. 10 SEM image of a fabricated sample with capsule shaped holes on a silicon slab.

Download Full Size | PPT Slide | PDF

Fig. 11
Fig. 11 Capsule-shaped holes. The shape defined by two parameters, the rectangle width w and semicircle radius u, is responsible for the form birefringence in the PCS.

Download Full Size | PPT Slide | PDF

Fig. 12
Fig. 12 (a) Experimental results showing two guided resonances corresponding to two different polarizations that are separated by one linewidth. This corresponds to β ≈ 0 . (b) Experimental results showing the transmission spectrum for the PCS that is illuminated with linearly polarized light at 45° (solid curve). As expected, the spectrum is the average of the two spectra in (a). When we put a quarter-wave plate followed by a linear polarizer at 45° after the PCS, we observe a drop in the transmission (dashed curve). For ideal conditions, the drop in the transmission at the center frequency should be 100%. Here, however, we observe a 60% drop in transmission, suggesting some ellipticity in the polarization. The inset shows the polarization state of the transmitted light calculated from these measurement data. Compared with the ideal case, a circular polarization state (shown as a dotted circle in the inset), we clearly see that there is a significant amount of ellipticity in the transmitted light.

Download Full Size | PPT Slide | PDF

Fig. 13
Fig. 13 Experimental results showing two guided resonances corresponding to two different polarizations that are separated by one linewidth. This corresponds to a linear PBS with β ≈ − 1 .

Download Full Size | PPT Slide | PDF

Fig. 14
Fig. 14 Some applications of the PCS structure. (a) The PCS-PBS structure in a Gires–Tournois configuration. By changing the distance between the mirror and the PCS, it is possible to control the polarization state of the output beam. Such a device can be used as a mirror in a laser cavity enabling dynamic polarization control. (b) The PCS can be designed to be highly chromatic, normally a disadvantage for QWRs. In such a case, a slight change in the PCS background, such as through heat, mechanical stress, etc., will destroy the orthogonality in the polarizations of the transmitted and reflected waves. For the configuration shown, one would then see interference in the output beam for any small effect on the PCS. The nice feature of such a detection scheme is that there is no cumbersome alignment requirement even though it is based on two-beam interferometry.

Download Full Size | PPT Slide | PDF

Tables (1)

Tables Icon

Table 1 Special Values of the Splitting Ratio β and the Corresponding Values of the Amplitudes ∣ t 1 ∣ and ∣ t 2 ∣ a

View Table

Equations (45)

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

t = t d − ( t d ± r d ) γ ∕ [ γ + j ( ω − ω 0 ) ] ,
r = r d − ( r d ± t d ) γ ∕ [ γ + j ( ω − ω 0 ) ] .
t d = [ cos ( k z h ) − j k z 0 2 + k z 2 2 k z 0 k z sin ( k z h ) ] − 1 ,
r d = [ j k z 0 2 − k z 2 2 k z 0 k z sin ( k z h ) ] t d ,
α ( ω ) = ( ω − ω 0 ) ∕ γ ,
β ( ω ) = k z 0 2 − k z 2 2 k z 0 k z sin ( k z h ) .
α 1 ( ω ) = [ ω − ( ω 0 − Δ ) ] ∕ γ ,
α 2 ( ω ) = [ ω − ( ω 0 + Δ ) ] ∕ γ ,
t 1 , 2 = t d − ( t d ± r d ) ∕ ( 1 + j α 1 , 2 ) .
t = t d ( j + α ) ( α ∓ β ) ∕ ( 1 + α 2 ) .
Δ Φ t = arg ( t 2 ∕ t 1 ) = arg ( j + α 2 j + α 1 ) + arg ( α 2 ∓ β α 1 ∓ β ) .
Δ Φ t = { Φ 0 ρ t ⩾ 0 Φ 0 + π ρ t < 0 } .
Δ Φ r = { Φ 0 ρ r ⩾ 0 Φ 0 + π ρ r < 0 } .
∣ t 1 , 2 ∣ 2 = ( α 1 , 2 ∓ β ) 2 ∕ [ ( 1 + β 2 ) ( 1 + α 1 , 2 2 ) ] .
∣ r 1 , 2 ∣ 2 = ( 1 ± β α 1 , 2 ) 2 ∕ [ ( 1 + β 2 ) ( 1 + α 1 , 2 2 ) ] .
J t = [ ∣ t 1 ∣ 0 0 ∣ t 2 ∣ e j Δ Φ t ] , J r = [ ∣ r 1 ∣ 0 0 ∣ r 2 ∣ e j Δ Φ r ] .
∣ i ⟩ = 1 2 [ 1 e j δ ] .
Tr ( J t † J t ) = Tr ( J r † J r ) = 1 .
ρ r 2 ρ t 2 = ∣ r 1 ∣ 2 ∣ t 2 ∣ 2 ∣ r 2 ∣ 2 ∣ t 1 ∣ 2 = 1 ,
ρ r = − ρ t , ρ r = + ρ t .
Tr ( J t † J r ) = 0 .
ω ( 1 ) = ω 0 − Δ 2 − γ 2 ,
ω ( 2 ) = ω 0 + Δ 2 − γ 2 .
J t = [ ∣ t 1 ∣ 0 0 j ∣ t 2 ∣ ] , J r = [ j ∣ t 2 ∣ 0 0 ∣ t 1 ∣ ] .
t d 1 = [ cos ( k z h ) − j k z 0 2 + k z 2 2 k z 0 k z sin ( k z h ) ] − 1 ,
t d 2 = [ cos ( k z h ) − j ϵ 2 k z 0 2 + ϵ 0 2 k z 2 2 ϵ ϵ 0 k z 0 k z sin ( k z h ) ] − 1 ,
r d 1 = [ j k z 0 2 − k z 2 2 k z 0 k z sin ( k z h ) ] t d 1 ,
r d 2 = [ j − ϵ 2 k z 0 2 + ϵ 0 2 k z 2 2 ϵ ϵ 0 k z 0 k z sin ( k z h ) ] t d 2 ,
β 2 β 1 = 1 − k ∥ 2 ω 2 ∕ c 2 ϵ + ϵ 0 ϵ ϵ 0 .
β 2 β 1 = 1 − sin 2 θ sin 2 θ br = f .
Δ Φ t = arg ( t d 2 ∕ t d 1 ) + Φ 0 + arg ( ρ t ) .
Δ Φ r = arg ( r d 2 ∕ r d 1 ) + Φ 0 + arg ( ρ r ) .
arg ( f ) = arg ( 1 − sin 2 θ sin 2 θ br ) = { 0 θ ⩽ θ br π θ > θ br } ,
ω ( 1 ) = ω 0 − Δ 2 − γ 2 ∓ 2 β Δ γ 1 − f 1 + f β 2 ,
ω ( 2 ) = ω 0 + Δ 2 − γ 2 ∓ 2 β Δ γ 1 − f 1 + f β 2 .
ω ( 1 ) = ω 0 − Δ 2 − γ 2 ,
ω ( 2 ) = ω 0 + Δ 2 − γ 2 .
Δ min = γ ( β 2 + 1 ± β ) .
ω ( 1 ) = ω 0 ± β γ 1 + f 1 − f β 2 − Δ 2 + γ 2 + ( β γ 1 + f 1 − f β 2 ) 2 ,
ω ( 2 ) = ω 0 ± β γ 1 + f 1 − f β 2 + Δ 2 + γ 2 + ( β γ 1 + f 1 − f β 2 ) 2 .
ω ( 1 ) = ω 0 − Δ 2 + γ 2 ,
ω ( 2 ) = ω 0 + Δ 2 + γ 2 .
ω m = 1 2 π h ϵ [ sin − 1 ( 2 ϵ 1 − ϵ β ) + m π ]
m = … − 2 , − 1 , 0 , 1 , 2 , … ,
ω m = m 2 h ϵ m = 1 , 2 , 3 , … .

Metrics