Abstract

A new kind of polarization bandpass filters (P-BPFs), which fulfill the functions of high reflectivity for one polarization and high transmittance in the passband for the other, are proposed. The P-BPFs consist of a stack of two one-dimensional (1D) photonic crystals (PCs) that are referred to as 1D PC heterostructures. The essential function of P-BPFs is realized by making the first transmitted peak near the photonic bandgap edges of one PC coincide with that of the other PC. Based on this method, three kinds of filters have been designed and discussed in detail: an isotropic 1D PC single-polarization bandpass filter (SP-BPF) at oblique incidence with a narrow passband for TM polarization and a broadened stop band for TE polarization, an anisotropic 1D PC SP-BPF at normal incidence with a narrow passband for TE polarization and a broadened stop band for TM polarization, and an anisotropic 1D PC double-polarization bandpass filter (DP-BPF) at oblique incidence with a spectral separated narrow passband for both polarizations, respectively. The filters can be fabricated by a lot of coating methods and integrated easily with other photonic components.

© 2009 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Mead, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
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    [CrossRef] [PubMed]
  3. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  4. Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
    [CrossRef] [PubMed]
  5. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415-419 (2000).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  9. X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (1)

D. Mao, Z. Ouyang, J. Wang, C. Liu, and C. Wu, “A photonic-crystal polarizer integrated with the functions of narrow bandpass and narrow transmission-angle filtering,” Appl. Phys. B 90, 127-131 (2008).
[CrossRef]

2005 (4)

H. Qi, R. Hong, K. Yi, J. Shao, and Z. Fan, “Nonpolarizing and polarizing filter design,” Appl. Opt. 44, 2343-2348 (2005).
[CrossRef] [PubMed]

X. Qin, P. Gu, and B. Guo, “Utilizing one-dimensional dual-periodical thin-film photonic crystals to design the polarization bandpass filters,” Proc. SPIE 5644, 647-653 (2005).
[CrossRef]

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

P. Han and H. Wang, “Criterion of omnidirectional reflection in a one-dimensional photonic heterostructure,” J. Opt. Soc. Am. B 22, 1571-1575 (2005).
[CrossRef]

2004 (2)

2002 (2)

H. Liu, Z. Ouyang, J. Li, and Q. Wang, “Photonic crystal WDM filters,” J. Optoelectron. Laser 13, 145-149 (2002) (in Chinese).

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

2000 (1)

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415-419 (2000).
[CrossRef] [PubMed]

1999 (1)

I. J. Hodgkinson and Q. H. Wu, “Birefringent thin film polarizers for use at normal incidence and with planar technologies,” Appl. Phys. Lett. 74, 1794-1796 (1999).
[CrossRef]

1998 (1)

Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

1997 (1)

I. J. Hodgkinson, S. Kassam, and Q. H. Wu, “Eigenequations and compact algorithms for bulk and layered anisotropic optical media: reflection and refraction at a crystal-crystal interface,” J. Comput. Phys. 133, 75-83 (1997).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

1982 (1)

1972 (1)

Berreman, D. W.

Chen, C.

Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

De La Rue, R. M.

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

de Ridder, R. M.

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

Driessen, A.

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

Fan, S.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Fan, Z.

Fink, Y.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Gu, P.

X. Qin, P. Gu, and B. Guo, “Utilizing one-dimensional dual-periodical thin-film photonic crystals to design the polarization bandpass filters,” Proc. SPIE 5644, 647-653 (2005).
[CrossRef]

Guo, B.

X. Qin, P. Gu, and B. Guo, “Utilizing one-dimensional dual-periodical thin-film photonic crystals to design the polarization bandpass filters,” Proc. SPIE 5644, 647-653 (2005).
[CrossRef]

Han, P.

Hodgkinson, I. J.

I. J. Hodgkinson and Q. H. Wu, “Birefringent thin film polarizers for use at normal incidence and with planar technologies,” Appl. Phys. Lett. 74, 1794-1796 (1999).
[CrossRef]

I. J. Hodgkinson, S. Kassam, and Q. H. Wu, “Eigenequations and compact algorithms for bulk and layered anisotropic optical media: reflection and refraction at a crystal-crystal interface,” J. Comput. Phys. 133, 75-83 (1997).
[CrossRef]

I. J. Hodgkinson and Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, 1998).

Hoekstra, H. J. W. M.

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

Hong, R.

Hopman, W. C. L.

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

Hu, X.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Ibanescu, M.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Jia, W.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Joannopoulos, J. D.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

J. D. Joannopoulos, R. D. Mead, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Kaminska, K.

Kassam, S.

I. J. Hodgkinson, S. Kassam, and Q. H. Wu, “Eigenequations and compact algorithms for bulk and layered anisotropic optical media: reflection and refraction at a crystal-crystal interface,” J. Comput. Phys. 133, 75-83 (1997).
[CrossRef]

Lambeck, P. V.

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

Li, J.

H. Liu, Z. Ouyang, J. Li, and Q. Wang, “Photonic crystal WDM filters,” J. Optoelectron. Laser 13, 145-149 (2002) (in Chinese).

Li, Y.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Liang, G. Q.

Liu, C.

D. Mao, Z. Ouyang, J. Wang, C. Liu, and C. Wu, “A photonic-crystal polarizer integrated with the functions of narrow bandpass and narrow transmission-angle filtering,” Appl. Phys. B 90, 127-131 (2008).
[CrossRef]

Liu, H.

H. Liu, Z. Ouyang, J. Li, and Q. Wang, “Photonic crystal WDM filters,” J. Optoelectron. Laser 13, 145-149 (2002) (in Chinese).

Liu, X.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (IoP, Bristol, 2001).
[CrossRef]

Mao, D.

D. Mao, Z. Ouyang, J. Wang, C. Liu, and C. Wu, “A photonic-crystal polarizer integrated with the functions of narrow bandpass and narrow transmission-angle filtering,” Appl. Phys. B 90, 127-131 (2008).
[CrossRef]

Mead, R. D.

J. D. Joannopoulos, R. D. Mead, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Michel, J.

Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Ouyang, Z.

D. Mao, Z. Ouyang, J. Wang, C. Liu, and C. Wu, “A photonic-crystal polarizer integrated with the functions of narrow bandpass and narrow transmission-angle filtering,” Appl. Phys. B 90, 127-131 (2008).
[CrossRef]

H. Liu, Z. Ouyang, J. Li, and Q. Wang, “Photonic crystal WDM filters,” J. Optoelectron. Laser 13, 145-149 (2002) (in Chinese).

Pottier, P.

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

Qi, H.

Qin, X.

X. Qin, P. Gu, and B. Guo, “Utilizing one-dimensional dual-periodical thin-film photonic crystals to design the polarization bandpass filters,” Proc. SPIE 5644, 647-653 (2005).
[CrossRef]

Robbie, K.

Shao, J.

Thomas, E. L.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

van Lith, J.

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

Wang, H.

Wang, H. Z.

Wang, J.

D. Mao, Z. Ouyang, J. Wang, C. Liu, and C. Wu, “A photonic-crystal polarizer integrated with the functions of narrow bandpass and narrow transmission-angle filtering,” Appl. Phys. B 90, 127-131 (2008).
[CrossRef]

Wang, Q.

H. Liu, Z. Ouyang, J. Li, and Q. Wang, “Photonic crystal WDM filters,” J. Optoelectron. Laser 13, 145-149 (2002) (in Chinese).

Wang, X.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

J. D. Joannopoulos, R. D. Mead, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wu, C.

D. Mao, Z. Ouyang, J. Wang, C. Liu, and C. Wu, “A photonic-crystal polarizer integrated with the functions of narrow bandpass and narrow transmission-angle filtering,” Appl. Phys. B 90, 127-131 (2008).
[CrossRef]

Wu, Q. H.

I. J. Hodgkinson and Q. H. Wu, “Birefringent thin film polarizers for use at normal incidence and with planar technologies,” Appl. Phys. Lett. 74, 1794-1796 (1999).
[CrossRef]

I. J. Hodgkinson, S. Kassam, and Q. H. Wu, “Eigenequations and compact algorithms for bulk and layered anisotropic optical media: reflection and refraction at a crystal-crystal interface,” J. Comput. Phys. 133, 75-83 (1997).
[CrossRef]

I. J. Hodgkinson and Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, 1998).

Xu, C.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yeh, P.

Yi, K.

Yudistira, D.

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

Zi, J.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

D. Mao, Z. Ouyang, J. Wang, C. Liu, and C. Wu, “A photonic-crystal polarizer integrated with the functions of narrow bandpass and narrow transmission-angle filtering,” Appl. Phys. B 90, 127-131 (2008).
[CrossRef]

Appl. Phys. Lett. (2)

I. J. Hodgkinson and Q. H. Wu, “Birefringent thin film polarizers for use at normal incidence and with planar technologies,” Appl. Phys. Lett. 74, 1794-1796 (1999).
[CrossRef]

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

W. C. L. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. W. M. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11-16 (2005).
[CrossRef]

J. Comput. Phys. (1)

I. J. Hodgkinson, S. Kassam, and Q. H. Wu, “Eigenequations and compact algorithms for bulk and layered anisotropic optical media: reflection and refraction at a crystal-crystal interface,” J. Comput. Phys. 133, 75-83 (1997).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Optoelectron. Laser (1)

H. Liu, Z. Ouyang, J. Li, and Q. Wang, “Photonic crystal WDM filters,” J. Optoelectron. Laser 13, 145-149 (2002) (in Chinese).

Opt. Lett. (1)

Phys. Rev. Lett. (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Proc. SPIE (1)

X. Qin, P. Gu, and B. Guo, “Utilizing one-dimensional dual-periodical thin-film photonic crystals to design the polarization bandpass filters,” Proc. SPIE 5644, 647-653 (2005).
[CrossRef]

Science (2)

Y. Fink, J. N. Winn, S. Fan, J. Michel, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Other (3)

J. D. Joannopoulos, R. D. Mead, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (IoP, Bristol, 2001).
[CrossRef]

I. J. Hodgkinson and Q. H. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, 1998).

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Figures (8)

Fig. 1
Fig. 1

Transmission spectra of perfect 1D PC at (a) normal and (b) 45° incidence. p u ( s u ) and p d ( s d ) are the first transmission peaks of TM (TE) polarization at the upper and lower band edge, respectively.

Fig. 2
Fig. 2

Schematic diagram of P-BPFs. (a) Upper band edge of PC1. (b) Lower band edge of PC2. (c) Transmission spectrum near the passband of heterostructure ( PC 1 + PC 2 ) .

Fig. 3
Fig. 3

(a) Variations of normalized wavelength of p u 1 and p d 2 with θ and κ. (b) 2D plot of the intersection curve shown in (a). (c), (d), (e) Transmission spectra of PC1, PC2, and the heterostructure ( PC 1 + PC 2 ) , respectively. (f) Local zoom-in of passband in (e). Spectra in (c), (d), (e), and (f) are calculated in the condition of κ = 0.68325 and θ = π 4 .

Fig. 4
Fig. 4

Physical structures of 1D PC heterostructures ( ( H L ) m ( κ H κ L ) n ) . (a) H and L are both isotropic materials. (b) H and L are isotropic and uniaxial anisotropic materials, respectively.

Fig. 5
Fig. 5

Electric and magnetic fields in uniaxial birefringent layer sandwiched by isotropic layers. Y - Z plane is the incidence plane, and Y is the optic axis; θ in is the angle of incidence; l k is the wave vector of light that propagated in uniaxial birefringent layer; l so and l se are ray vectors of ordinary and extraordinary waves respectively; E o ( E e ) , D o ( D e ) , and H o ( H e ) are the electric field, displacement vector, and magnetic field of ordinary (extraordinary) waves, respectively; α and β are refraction angles of ordinary and extraordinary waves, respectively; γ is the angle between E e and D e .

Fig. 6
Fig. 6

(a) Variations of normalized wavelength of s d 2 and p d 2 with κ at normal incidence. Normalized wavelengths of s u 1 and p u 1 are λ 0 λ = 1.1939 (shown as s u 1 line) and 1.2492 (shown as p u 1 line), respectively. The s u 1 line and s d 2 line are intersecting at C s 1 with κ = 0.6758 , under which the normalized wavelengths of p d 2 and p u 1 are 1.2026 (shown as C p 3 ) and 1.2492 (shown as C p 2 ), respectively. p d 2 is at the lower side of p u 1 , and the heterostructure is a P-BPF [see Fig. 2] for TE polarization. The p u 1 line and the p d 2 line are intersecting at C p 1 with κ = 0.651 , under which the normalized wavelengths of s d 2 and s u 1 are 1.2390 (shown as C s 3 ) and 1.1939 (shown as C s 2 ), respectively. Then the heterostructure is not a P-BPF for TM polarization because s d 2 is at the upper side of s u 1 and it can not absolutely cut off the TE polarization [see Fig. 2]. (b), (c), (d) Transmission spectra of PC1, PC2, and the heterostructure ( PC 1 + PC 2 ) , respectively. (e) Local zoom-in of passband in (d). Spectra in (b), (c), (d), and (e) are calculated in the condition of κ = 0.6758 and θ = 0 .

Fig. 7
Fig. 7

(a) Variations of normalized wavelength of s u 1 and s d 2 with θ and κ. (b) Variations of normalized wavelength of p u 1 and p d 2 with θ and κ. (c) 2D plot of intersection curves shown in (a) and (b). The two curves intersect at ( θ = 0.2732 , κ = 0.6639 ), which means that s u 1 , s d 2 , p u 1 , and p d 2 coincide at this situation. (d), (e), (f) Transmission spectra of PC1, PC2, and the heterostructure ( PC 1 + PC 2 ) , respectively. (g) Local zoom-in of passband in (f). Spectra in (d), (e), (f), and (g) are calculated in the condition of κ = 0.6639 and θ = 0.2732 rad .

Fig. 8
Fig. 8

Transmission spectra of 1D PC SP-BPF based on isotropic materials at 45° incidence with different κ. (a) κ = 0.6828 , (b) κ = 0.6831 , (c) κ = 0.6835 , (d) κ = 0.6838 .

Equations (8)

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M = ( m 11 m 12 m 21 m 22 ) = m = 1 N M m ,
M m = A B ,
A = [ 1 0 exp ( i δ o m ) 0 0 cos ( α m + γ m ) 0 cos ( α m + γ m ) exp ( i δ e m ) 0 n e m ζ 0 cos γ m 0 n e m ζ 0 cos γ m exp ( i δ e m ) n o m ζ 0 cos β m 0 n o m ζ 0 cos β m exp ( i δ o m ) 0 ] ,
B = [ exp ( i δ o m ) 0 1 0 0 cos ( α m + γ m ) exp ( i δ e m ) 0 cos ( α m + γ m ) 0 n e m ζ 0 cos γ m exp ( i δ e m ) 0 n e m ζ 0 cos γ m n o m ζ 0 cos β m exp ( i δ o m ) 0 n o m ζ 0 cos β m 0 ] 1 ,
[ 1 0 1 0 0 cos θ in 0 cos θ in 0 η p ζ 0 0 η p ζ 0 η s ζ 0 cos θ in 0 η s ζ 0 cos θ in 0 ] [ i s i p r s r p ] = M [ 1 0 0 cos θ out 0 η p ζ 0 η s ζ 0 cos θ out ] [ t s t p ] ,
[ r s r p t s t p ] = [ T 2 , M T 3 ] 1 T 1 [ i s i p ] ,
T 1 = [ 1 0 0 cos θ in 0 η p ζ 0 η s ζ 0 cos θ in 0 ] , T 2 = [ 1 0 0 cos θ in 0 η p ζ 0 η s ζ 0 cos θ in 0 ] , T 3 = [ 1 0 0 cos θ out 0 η p ζ 0 η s ζ 0 cos θ out 0 ] .
T = | t s | 2 + | t p | 2 | i s | 2 + | i p | 2 , R = | r s | 2 + | r p | 2 | i s | 2 + | i p | 2 .

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