Abstract

A cavity-resonator-integrated guided-mode resonance filter (CRIGF) consists of a grating coupler inside a pair of distributed Bragg reflectors. A combination of a CRIGF with a high-reflection substrate can provide a new type of a band-stop reflector with a small aperture for a vertically incident wave from air. A narrow stopband was theoretically predicted and experimentally demonstrated. It was quantitatively shown that reflection spectra depended on optical-buffer-layer thickness. The reflector of 10-μm aperture was fabricated and characterized. The extinction ratio in reflectance was measured to be lower than –20 dB at a resonance wavelength. The bandwidth at –3 dB was 0.15 nm.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]

2015 (3)

2013 (2)

J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013).
[Crossref]

K. Kintaka, K. Hatanaka, J. Inoue, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter with nonuniform grating coupler for efficient coupling with Gaussian beam,” Appl. Phys. Express 6(10), 102203 (2013).
[Crossref]

2012 (5)

1998 (1)

1997 (2)

S. M. Norton, T. Erdogan, and G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14(3), 629–639 (1997).
[Crossref]

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

1993 (1)

1985 (2)

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985).
[Crossref]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15(7), 886–887 (1985).
[Crossref]

Awatsuji, Y.

J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013).
[Crossref]

K. Hatanaka, T. Majima, K. Kintaka, J. Inoue, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter consisting of curved gratings,” Electron. Lett. 48(12), 717–718 (2012).
[Crossref]

J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of guided-mode resonance filter by cavity-resonator integration,” Appl. Phys. Express 5(2), 022201 (2012).
[Crossref]

S. Ura, J. Inoue, T. Ogura, K. Nishio, Y. Awatsuji, and K. Kintaka, “Reflection-phase variation of cavity-resonator-integrated guided-mode-resonance reflector for guided-mode-exciting surface laser mirror,” in Proceedings of the 63th Electronics Components and Technology Conference (2013), pp. 1874–1879.
[Crossref]

Belharet, D.

Buet, X.

Daran, E.

Erdogan, T.

Fehrembach, A.-L.

Friesem, A. A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

Gauthier-Lafaye, O.

Golubenko, G. A.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15(7), 886–887 (1985).
[Crossref]

Hatanaka, K.

J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013).
[Crossref]

K. Kintaka, K. Hatanaka, J. Inoue, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter with nonuniform grating coupler for efficient coupling with Gaussian beam,” Appl. Phys. Express 6(10), 102203 (2013).
[Crossref]

J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of guided-mode resonance filter by cavity-resonator integration,” Appl. Phys. Express 5(2), 022201 (2012).
[Crossref]

K. Hatanaka, T. Majima, K. Kintaka, J. Inoue, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter consisting of curved gratings,” Electron. Lett. 48(12), 717–718 (2012).
[Crossref]

K. Kintaka, T. Majima, K. Hatanaka, J. Inoue, and S. Ura, “Polarization-independent guided-mode resonance filter with cross-integrated waveguide resonators,” Opt. Lett. 37(15), 3264–3266 (2012).
[Crossref] [PubMed]

K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20(2), 1444–1449 (2012).
[PubMed]

Inoue, J.

J. Inoue, T. Kondo, K. Kintaka, K. Nishio, and S. Ura, “Determination of cavity length of cavity-resonator-integrated guided-mode resonance filter,” Opt. Express 23(3), 3020–3026 (2015).
[Crossref] [PubMed]

K. Kintaka, K. Hatanaka, J. Inoue, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter with nonuniform grating coupler for efficient coupling with Gaussian beam,” Appl. Phys. Express 6(10), 102203 (2013).
[Crossref]

J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013).
[Crossref]

K. Hatanaka, T. Majima, K. Kintaka, J. Inoue, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter consisting of curved gratings,” Electron. Lett. 48(12), 717–718 (2012).
[Crossref]

J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of guided-mode resonance filter by cavity-resonator integration,” Appl. Phys. Express 5(2), 022201 (2012).
[Crossref]

K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20(2), 1444–1449 (2012).
[PubMed]

K. Kintaka, T. Majima, K. Hatanaka, J. Inoue, and S. Ura, “Polarization-independent guided-mode resonance filter with cross-integrated waveguide resonators,” Opt. Lett. 37(15), 3264–3266 (2012).
[Crossref] [PubMed]

S. Ura, J. Inoue, T. Ogura, K. Nishio, Y. Awatsuji, and K. Kintaka, “Reflection-phase variation of cavity-resonator-integrated guided-mode-resonance reflector for guided-mode-exciting surface laser mirror,” in Proceedings of the 63th Electronics Components and Technology Conference (2013), pp. 1874–1879.
[Crossref]

M. Nakata, T. Kondo, K. Kintaka, J. Inoue, and S. Ura, “Reflection optical notch filtering behavior of cavity-resonator-integrated guided-mode resonance mirror,” in Proceedings of International Conference on Electronics Packaging 2015 (2015), paper 652–655.
[Crossref]

Kintaka, K.

J. Inoue, T. Kondo, K. Kintaka, K. Nishio, and S. Ura, “Determination of cavity length of cavity-resonator-integrated guided-mode resonance filter,” Opt. Express 23(3), 3020–3026 (2015).
[Crossref] [PubMed]

J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013).
[Crossref]

K. Kintaka, K. Hatanaka, J. Inoue, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter with nonuniform grating coupler for efficient coupling with Gaussian beam,” Appl. Phys. Express 6(10), 102203 (2013).
[Crossref]

K. Hatanaka, T. Majima, K. Kintaka, J. Inoue, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter consisting of curved gratings,” Electron. Lett. 48(12), 717–718 (2012).
[Crossref]

J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of guided-mode resonance filter by cavity-resonator integration,” Appl. Phys. Express 5(2), 022201 (2012).
[Crossref]

K. Kintaka, T. Majima, K. Hatanaka, J. Inoue, and S. Ura, “Polarization-independent guided-mode resonance filter with cross-integrated waveguide resonators,” Opt. Lett. 37(15), 3264–3266 (2012).
[Crossref] [PubMed]

K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20(2), 1444–1449 (2012).
[PubMed]

S. Ura, J. Inoue, T. Ogura, K. Nishio, Y. Awatsuji, and K. Kintaka, “Reflection-phase variation of cavity-resonator-integrated guided-mode-resonance reflector for guided-mode-exciting surface laser mirror,” in Proceedings of the 63th Electronics Components and Technology Conference (2013), pp. 1874–1879.
[Crossref]

M. Nakata, T. Kondo, K. Kintaka, J. Inoue, and S. Ura, “Reflection optical notch filtering behavior of cavity-resonator-integrated guided-mode resonance mirror,” in Proceedings of International Conference on Electronics Packaging 2015 (2015), paper 652–655.
[Crossref]

Kondo, T.

J. Inoue, T. Kondo, K. Kintaka, K. Nishio, and S. Ura, “Determination of cavity length of cavity-resonator-integrated guided-mode resonance filter,” Opt. Express 23(3), 3020–3026 (2015).
[Crossref] [PubMed]

T. Kondo, S. Ura, and R. Magnusson, “Design of guided-mode resonance mirrors for short laser cavities,” J. Opt. Soc. Am. A 32(8), 1454–1458 (2015).
[Crossref] [PubMed]

M. Nakata, T. Kondo, K. Kintaka, J. Inoue, and S. Ura, “Reflection optical notch filtering behavior of cavity-resonator-integrated guided-mode resonance mirror,” in Proceedings of International Conference on Electronics Packaging 2015 (2015), paper 652–655.
[Crossref]

Liu, Z. S.

Lozes-Dupuy, F.

Magnusson, R.

Majima, T.

K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20(2), 1444–1449 (2012).
[PubMed]

K. Kintaka, T. Majima, K. Hatanaka, J. Inoue, and S. Ura, “Polarization-independent guided-mode resonance filter with cross-integrated waveguide resonators,” Opt. Lett. 37(15), 3264–3266 (2012).
[Crossref] [PubMed]

J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of guided-mode resonance filter by cavity-resonator integration,” Appl. Phys. Express 5(2), 022201 (2012).
[Crossref]

K. Hatanaka, T. Majima, K. Kintaka, J. Inoue, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter consisting of curved gratings,” Electron. Lett. 48(12), 717–718 (2012).
[Crossref]

Mashev, L.

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985).
[Crossref]

Monmayrant, A.

Morris, G. M.

Nakata, M.

M. Nakata, T. Kondo, K. Kintaka, J. Inoue, and S. Ura, “Reflection optical notch filtering behavior of cavity-resonator-integrated guided-mode resonance mirror,” in Proceedings of International Conference on Electronics Packaging 2015 (2015), paper 652–655.
[Crossref]

Nishii, J.

Nishio, K.

J. Inoue, T. Kondo, K. Kintaka, K. Nishio, and S. Ura, “Determination of cavity length of cavity-resonator-integrated guided-mode resonance filter,” Opt. Express 23(3), 3020–3026 (2015).
[Crossref] [PubMed]

J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013).
[Crossref]

K. Hatanaka, T. Majima, K. Kintaka, J. Inoue, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter consisting of curved gratings,” Electron. Lett. 48(12), 717–718 (2012).
[Crossref]

J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of guided-mode resonance filter by cavity-resonator integration,” Appl. Phys. Express 5(2), 022201 (2012).
[Crossref]

S. Ura, J. Inoue, T. Ogura, K. Nishio, Y. Awatsuji, and K. Kintaka, “Reflection-phase variation of cavity-resonator-integrated guided-mode-resonance reflector for guided-mode-exciting surface laser mirror,” in Proceedings of the 63th Electronics Components and Technology Conference (2013), pp. 1874–1879.
[Crossref]

Norton, S. M.

Ogura, T.

J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013).
[Crossref]

S. Ura, J. Inoue, T. Ogura, K. Nishio, Y. Awatsuji, and K. Kintaka, “Reflection-phase variation of cavity-resonator-integrated guided-mode-resonance reflector for guided-mode-exciting surface laser mirror,” in Proceedings of the 63th Electronics Components and Technology Conference (2013), pp. 1874–1879.
[Crossref]

Popov, E.

Rassem, N.

Rosenblatt, D.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

Sharon, A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

Shin, D.

Svakhin, A. S.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15(7), 886–887 (1985).
[Crossref]

Sychugov, V. A.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15(7), 886–887 (1985).
[Crossref]

Tibuleac, S.

Tishchenko, A. V.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15(7), 886–887 (1985).
[Crossref]

Ura, S.

T. Kondo, S. Ura, and R. Magnusson, “Design of guided-mode resonance mirrors for short laser cavities,” J. Opt. Soc. Am. A 32(8), 1454–1458 (2015).
[Crossref] [PubMed]

J. Inoue, T. Kondo, K. Kintaka, K. Nishio, and S. Ura, “Determination of cavity length of cavity-resonator-integrated guided-mode resonance filter,” Opt. Express 23(3), 3020–3026 (2015).
[Crossref] [PubMed]

J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013).
[Crossref]

K. Kintaka, K. Hatanaka, J. Inoue, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter with nonuniform grating coupler for efficient coupling with Gaussian beam,” Appl. Phys. Express 6(10), 102203 (2013).
[Crossref]

K. Hatanaka, T. Majima, K. Kintaka, J. Inoue, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter consisting of curved gratings,” Electron. Lett. 48(12), 717–718 (2012).
[Crossref]

J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of guided-mode resonance filter by cavity-resonator integration,” Appl. Phys. Express 5(2), 022201 (2012).
[Crossref]

K. Kintaka, T. Majima, J. Inoue, K. Hatanaka, J. Nishii, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter for aperture miniaturization,” Opt. Express 20(2), 1444–1449 (2012).
[PubMed]

K. Kintaka, T. Majima, K. Hatanaka, J. Inoue, and S. Ura, “Polarization-independent guided-mode resonance filter with cross-integrated waveguide resonators,” Opt. Lett. 37(15), 3264–3266 (2012).
[Crossref] [PubMed]

M. Nakata, T. Kondo, K. Kintaka, J. Inoue, and S. Ura, “Reflection optical notch filtering behavior of cavity-resonator-integrated guided-mode resonance mirror,” in Proceedings of International Conference on Electronics Packaging 2015 (2015), paper 652–655.
[Crossref]

S. Ura, J. Inoue, T. Ogura, K. Nishio, Y. Awatsuji, and K. Kintaka, “Reflection-phase variation of cavity-resonator-integrated guided-mode-resonance reflector for guided-mode-exciting surface laser mirror,” in Proceedings of the 63th Electronics Components and Technology Conference (2013), pp. 1874–1879.
[Crossref]

Wang, S. S.

Young, P. P.

Appl. Opt. (1)

Appl. Phys. Express (2)

J. Inoue, T. Majima, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Aperture miniaturization of guided-mode resonance filter by cavity-resonator integration,” Appl. Phys. Express 5(2), 022201 (2012).
[Crossref]

K. Kintaka, K. Hatanaka, J. Inoue, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter with nonuniform grating coupler for efficient coupling with Gaussian beam,” Appl. Phys. Express 6(10), 102203 (2013).
[Crossref]

Electron. Lett. (1)

K. Hatanaka, T. Majima, K. Kintaka, J. Inoue, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter consisting of curved gratings,” Electron. Lett. 48(12), 717–718 (2012).
[Crossref]

IEEE J. Quantum Electron. (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).
[Crossref]

IEICE Electron. Express (1)

J. Inoue, T. Ogura, K. Hatanaka, K. Kintaka, K. Nishio, Y. Awatsuji, and S. Ura, “Cavity-resonator-integrated guided-mode resonance filter in channel waveguide,” IEICE Electron. Express 10(15), 20130444 (2013).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Sov. J. Quantum Electron. (1)

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15(7), 886–887 (1985).
[Crossref]

Other (5)

S. Ura, J. Inoue, K. Kintaka, and Y. Awatsuji, “Proposal of small-aperture guided-mode-resonance filter,” in Proceedings of the 13th International Conference on Transparent Optical Networks (2011), paper Th.A4.4.
[Crossref]

K. Kintaka, K. Asai, T. Kondo, J. Inoue, and S. Ura, “Reflectance change by grating-position shifting in cavity-resonator-integrated guided-mode resonance filter,” in Technical Digest of Integrated Photonics Research, Silicon and Nano Photonics 2015 (2015), paper IM2A.3.

K. Asai, K. Kintaka, J. Inoue, and S. Ura, “Coupled-mode analysis of grating-position-shifted cavity-resonator-integrated guided-mode resonance filter,” in Technical Digest of 20th Microoptics Conference (2015), paper H-4.
[Crossref]

S. Ura, J. Inoue, T. Ogura, K. Nishio, Y. Awatsuji, and K. Kintaka, “Reflection-phase variation of cavity-resonator-integrated guided-mode-resonance reflector for guided-mode-exciting surface laser mirror,” in Proceedings of the 63th Electronics Components and Technology Conference (2013), pp. 1874–1879.
[Crossref]

M. Nakata, T. Kondo, K. Kintaka, J. Inoue, and S. Ura, “Reflection optical notch filtering behavior of cavity-resonator-integrated guided-mode resonance mirror,” in Proceedings of International Conference on Electronics Packaging 2015 (2015), paper 652–655.
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Basic configuration (a) and wave coupling (b) of a band-stop reflector with a vertical cavity consisting of a CRIGF and a high-reflection substrate.
Fig. 2
Fig. 2 Schematic cross-sectional view of designed CRIGF.
Fig. 3
Fig. 3 Reflection spectra calculated with κLoss of 0.5 dB/mm (a), and 2.0 dB/mm (b) for various buffer layer thicknesses tb.
Fig. 4
Fig. 4 SEM image of a part of the fabricated reflector.
Fig. 5
Fig. 5 Measured reflection spectrum of the fabricated device.

Equations (21)

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dg dt =( j ω 0 1 τ )gj κ ^ * || a IN
| a OUT =| a IN jg| κ ^ ,
g=j κ ^ * || a IN j( ω ω 0 )+1/τ .
d | g | 2 dt =2 1 τ | g | 2 2( 1 τ Rad + 1 τ DBR + 1 τ Loss ) | g | 2 .
1 τ Rad = c N L GC L Cav α , 1 τ DBR = c N ln( r DBR ) L Cav , 1 τ Loss = c N κ Loss ,
E νa ( x )={ E νa 0 { ( t a t s r a r s )exp( j k ax x ) r s exp( j k ax x ) }exp( j β ν z ) ; in air , E νa 0 t a exp( j k sx x )exp( j β ν z ) ; in substrate
E νs ( x )={ E νs 0 t s exp( j k ax x )exp( j β ν z ) ; in air E νs 0 { r a exp( j k sx x )+( t a t s r a r s )exp( j k sx x ) }exp( j β ν z ) ; in substrate ,
k a = k ax 2 + β ν 2 , k s = k sx 2 + β ν 2 ,
π 2ω μ 0 E νa 0 2 { k a ( | t a t s r a r s | 2 + | r s | 2 )+ k s | t a | 2 }=1 ,
π 2ω μ 0 E νs 0 2 { k a | t s | 2 + k s ( | t a t s r a r s | 2 + | r a | 2 ) }=1 .
E Aν (x)={ c a E νa (x)+ c s E νs (x) } E Aν 0 ,
c a = r a t a t s r a r s 1 E νa 0 , c s = t a t a t s r a r s 1 E νs 0
| a IN = ( c a | E INν 0 c s | E INν 0 )=( c a c s )| E INν 0 ,
| a OUT =( d a | E OUTν 0 d s | E OUTν 0 ) =( d a d s )| E OUTν 0 .
| E INν 0 = E IN0 0 (ω)δ( β ν =0),
( d a d s )=( c a c s ) κ ^ 0a * c a + κ ^ 0s * c s j( ω ω 0 )+1/τ ( κ ^ a κ ^ s ) ,
( κ ^ a κ ^ s )= 2πc N L GC L Cav ( κ νa κ νs ) ,
( κ ^ 0a * κ ^ 0s * )=sinc( Δ ω L GC 2 ) 2πc N L GC L Cav ( κ νa * κ νs * ) ,
Δ ω = N c ( ω ω 0 ).
κ νi = ε 0 ω 4 E νi * (x)Δε E g (x)dx ( i=a or s ) ,
r CRIGF = r a sinc( Δ ω L GC 2 ) 2πc N L GC L Cav | κ νa | 2 r a + κ νs * κ νa t a E νa 0 / E νs 0 j( ω ω 0 )+1/τ .

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