Abstract

We study resonances of guided-mode resonant gratings in conical mounting. By developing 2D time-dependent coupled-mode theory we obtain simple approximations of the transmission and reflection coefficients. Being functions of the incident light’s frequency and in-plane wave vector components, the obtained approximations can be considered as multi-variable generalizations of the Fano line shape. We show that the approximations are in good agreement with the rigorously calculated transmission and reflection spectra. We use the developed theory to investigate angular tolerances of the considered structures and to obtain mode excitation conditions. In particular, we obtain the cross-polarization mode excitation conditions in the case of conical mounting.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  5. N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  10. H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010).
    [Crossref]
  11. S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
    [Crossref]
  12. D. A. Bykov and L. L. Doskolovich, “ω − kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92, 013845 (2015).
    [Crossref]
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  14. E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, C. Dupuis, S. Collin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Freestanding guided-mode resonance band-pass filters: from 1D to 2D structures,” Opt. Express 20, 13082–13090 (2012).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  24. D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightwave Technol. 31, 793–801 (2013).
    [Crossref]

2016 (2)

2015 (3)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Experimental Theoretical Phys. 121, 785–792 (2015).
[Crossref]

D. A. Bykov and L. L. Doskolovich, “ω − kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92, 013845 (2015).
[Crossref]

D. A. Bykov and L. L. Doskolovich, “Spatiotemporal coupled-mode theory of guided-mode resonant gratings,” Opt. Express 23, 19234–19241 (2015).
[Crossref] [PubMed]

2014 (1)

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

2013 (3)

S. P. Shipman and A. T. Welters, “Resonant electromagnetic scattering in anisotropic layered media,” J. Math. Phys. 54, 103511 (2013).
[Crossref]

D. W. Peters, R. R. Boye, and S. A. Kemme, “Angular sensitivity of guided mode resonant filters in classical and conical mounts,” Proc. SPIE 8633, 86330W (2013).
[Crossref]

D. A. Bykov and L. L. Doskolovich, “Numerical methods for calculating poles of the scattering matrix with applications in grating theory,” J. Lightwave Technol. 31, 793–801 (2013).
[Crossref]

2012 (1)

2010 (1)

2008 (1)

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

2005 (2)

S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
[Crossref]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
[Crossref]

2003 (1)

2002 (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

1997 (2)

A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[Crossref]

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

1995 (2)

1993 (1)

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[Crossref]

Bardou, N.

Boye, R. R.

D. W. Peters, R. R. Boye, and S. A. Kemme, “Angular sensitivity of guided mode resonant filters in classical and conical mounts,” Proc. SPIE 8633, 86330W (2013).
[Crossref]

Bykov, D. A.

Chadha, A.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Chuwongin, S.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Collin, S.

Doskolovich, L. L.

Dupuis, C.

Fan, S.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

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]

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[Crossref]

Friesem, A. A.

A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[Crossref]

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

Gaylord, T. K.

Ghenuche, P.

Gippius, N. A.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Glasberg, S.

Golovastikov, N. V.

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Analytical description of 3D optical pulse diffraction by a phase-shifted Bragg grating,” Opt. Express 24, 18828 (2016).
[Crossref] [PubMed]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Experimental Theoretical Phys. 121, 785–792 (2015).
[Crossref]

Grann, E. B.

Haïdar, R.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Ishihara, T.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Joannopoulos, J. D.

Kemme, S. A.

D. W. Peters, R. R. Boye, and S. A. Kemme, “Angular sensitivity of guided mode resonant filters in classical and conical mounts,” Proc. SPIE 8633, 86330W (2013).
[Crossref]

Ko, Y. H.

Lalanne, P.

H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010).
[Crossref]

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

Liu, H.

H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010).
[Crossref]

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

Liu, V.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Ma, Z.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Magnusson, R.

Moharam, M. G.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Nevière, M.

Niraula, M.

Pardo, F.

Pelouard, J.-L.

Peters, D. W.

D. W. Peters, R. R. Boye, and S. A. Kemme, “Angular sensitivity of guided mode resonant filters in classical and conical mounts,” Proc. SPIE 8633, 86330W (2013).
[Crossref]

Pommet, D. A.

Popov, E.

Reinisch, R.

Rosenblatt, D.

A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[Crossref]

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

Sakat, E.

Seo, J.-H.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Sharon, A.

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

A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[Crossref]

Shipman, S. P.

S. P. Shipman and A. T. Welters, “Resonant electromagnetic scattering in anisotropic layered media,” J. Math. Phys. 54, 103511 (2013).
[Crossref]

S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
[Crossref]

Shuai, Y.-C.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Soifer, V. A.

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Analytical description of 3D optical pulse diffraction by a phase-shifted Bragg grating,” Opt. Express 24, 18828 (2016).
[Crossref] [PubMed]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Experimental Theoretical Phys. 121, 785–792 (2015).
[Crossref]

Suh, W.

Tikhodeev, S. G.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Träger, F.

F. Träger, Springer Handbook of Lasers and Optics (Springer, 2007).
[Crossref]

Venakides, S.

S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
[Crossref]

Vincent, G.

Wang, K. X.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Wang, S. S.

Welters, A. T.

S. P. Shipman and A. T. Welters, “Resonant electromagnetic scattering in anisotropic layered media,” J. Math. Phys. 54, 103511 (2013).
[Crossref]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Yang, H.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Zhao, D.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Zhou, W.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

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

J. Experimental Theoretical Phys. (1)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Experimental Theoretical Phys. 121, 785–792 (2015).
[Crossref]

J. Lightwave Technol. (1)

J. Math. Phys. (1)

S. P. Shipman and A. T. Welters, “Resonant electromagnetic scattering in anisotropic layered media,” J. Math. Phys. 54, 103511 (2013).
[Crossref]

J. Opt. Soc. Am. A (5)

Nature (1)

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[Crossref]

Phys. Rev. A (1)

D. A. Bykov and L. L. Doskolovich, “ω − kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92, 013845 (2015).
[Crossref]

Phys. Rev. B (2)

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Phys. Rev. E (1)

S. P. Shipman and S. Venakides, “Resonant transmission near nonrobust periodic slab modes,” Phys. Rev. E 71, 026611 (2005).
[Crossref]

Proc. SPIE (1)

D. W. Peters, R. R. Boye, and S. A. Kemme, “Angular sensitivity of guided mode resonant filters in classical and conical mounts,” Proc. SPIE 8633, 86330W (2013).
[Crossref]

Prog. Quantum Electron. (1)

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Other (2)

F. Träger, Springer Handbook of Lasers and Optics (Springer, 2007).
[Crossref]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

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Figures (3)

Fig. 1
Fig. 1 Pulse diffraction by guided-mode resonant grating
Fig. 2
Fig. 2 TFs describing coupling between the modes and the incident, reflected, and transmitted fields
Fig. 3
Fig. 3 The transmission coefficient of the grating: (a) |T (kx, ω)|2 at ky = 0, TM-polarized incident wave; (b) |T (kx, ω)|2 at ky = 0, TE-polarized incident wave; (c) |T (ky, ω)|2 at kx = 0, TM-polarized incident wave; (d) |T (ky, ω)|2 at kx = 0, TE-polarized incident wave

Tables (2)

Tables Icon

Table 1 Parameters of the modes

Tables Icon

Table 2 Parameters of the resonances

Equations (43)

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

k k 1 = ( ω ω 0 ) / v g .
u ( x , y , t ) = G ( k x , k y ) e i ( k x x + k y y ω t ) d k x d k y ,
v g 2 [ 2 u x 2 + 2 u y 2 ] = 2 u t 2 2 i ( v g k 1 ω 0 ) u t ( v g k 1 ω 0 ) 2 u .
u ¯ = u e i k 1 x + i ω 0 t
2 u t 2 2 i ω 0 u t + ω 0 2 u , 2 u x 2 2 i k 1 u x + k 1 2 u .
u t = v g u x + i v g 2 k 1 2 u y 2 + i ( v g k 1 ω 0 ) u .
v t = v g v x + i v g 2 k 1 2 v y 2 + i ( v g k 1 ω 0 ) v .
2 π d m = k 1 .
{ u t = v g u x + i v g 2 k 1 2 u y 2 + i ( v g k 1 ω 0 ) u c 1 u + c 2 e 2 i k 1 x v + c e e i k 1 x f v t = v g v x + i v g 2 k 1 2 v y 2 + i ( v g k 1 ω 0 ) v c 1 v + c 2 e 2 i k 1 x u + c e e i k 1 x f .
{ f R = r 0 f + ( c r e i k 1 x u + c r e i k 1 x v ) , f T = t 0 f + ( c t e i k 1 x u + c t e i k 1 x v ) .
{ u ˜ t = v g u ˜ x + i v g 2 k 1 2 u ˜ y 2 ( c 1 + i ω 0 ) u ˜ + c 2 v ˜ + c e f , v ˜ t = v g v ˜ x + i v g 2 k 1 2 v ˜ y 2 ( c 1 + i ω 0 ) v ˜ + c 2 u ˜ + c e f , f R = r 0 f + c r ( u ˜ + v ˜ ) , f T = t 0 f + c t ( u ˜ + v ˜ ) .
u = u ( x , y , t ) e i ( k x x + k y y ω t ) d k x d k y d ω .
{ i ω U = v g i k x U i v g 2 k 1 k y 2 U ( c 1 + i ω 0 ) U + c 2 V + c e F , i ω V = v g i k x V i v g 2 k 1 k y 2 V ( c 1 + i ω 0 ) V + c 2 U + c e F , F R = r 0 F + c r ( U + V ) , F T = t 0 F + c t ( U + V ) ,
T = F T F = t 0 v g 2 k x 2 ( ω ω z t η k y 2 ) ( ω ω p 2 η k y 2 ) v g 2 k x 2 ( ω ω p 1 η k y 2 ) ( ω ω p 2 η k y 2 ) ,
v g 2 k x 2 = ( ω ω p 1 η k y 2 ) ( ω ω p 2 η k y 2 )
{ u ˜ t = v g u ˜ x + i v g 2 k 1 2 u ˜ y 2 i ω 0 u ˜ c ^ 1 u ˜ + c ^ 2 v ˜ + c ^ e 1 f , v ˜ t = v g v ˜ x + i v g 2 k 1 2 v ˜ y 2 i ω 0 v ˜ c ^ 1 v ˜ + c ^ 2 u ˜ + c ^ e 2 f , f R TE = r ^ 0 TE f + c ^ r 1 TE u ˜ + c ^ r 2 TE v ˜ , f T TE = t ^ 0 TE f + c ^ t 1 TE u ˜ + c ^ t 2 TE v ˜ , f R TM = r ^ 0 TM f + c ^ r 1 TM u ˜ + c ^ r 2 TM v ˜ , f T TM = t ^ 0 TM f + c ^ t 1 TM u ˜ + c t 2 TM v ˜ .
( c ^ e f ) = C e F .
{ i ω U = v g i k x U i v g 2 k 1 k y 2 U ( i ω 0 + C 1 ) U + C 2 V + C e 1 F , i ω V = v g i k x V i v g 2 k 1 k y 2 V ( i ω 0 + C 1 ) V + C 2 U + C e 2 F , F T TE = T 0 TE F + C t 1 TE U + C t 2 TE V , F R TE = R 0 TE F + C r 1 TE U + C r 2 TE V , F T TM = T 0 TM F + C t 1 TM U + C t 2 TM V F R TM = R 0 TM F + C r 1 TM U + C r 2 TM V ,
C e 1 ( k x , k y , ω ) C e 2 ( k x , k y , ω ) c e .
C t 1 TM ( k x , k y , ω ) k y c t TM ,
c ^ t 1 TM i c t TM y .
C t 2 TM ( k x , k y , ω ) k y c t TM .
C r 1 TM ( k x , k y , ω ) C r 2 TM ( k x , k y , ω ) k y c r TM .
R 0 TM ( k x , k y , ω ) k x k y r 0 TM ,
T 0 TM ( k x , k y , ω ) k x k y t 0 TM .
C 1 ( k x , k y , ω ) c 1 , C 2 ( k x , k y , ω ) c 2 .
T same = t 0 same v g 2 k x 2 ( ω ω zt η zt k y 2 ) ( ω ω p 2 η p 2 k y 2 ) v g 2 k x 2 ( ω ω p 1 η p 1 k y 2 ) ( ω ω p 2 η p 2 k y 2 ) ,
T cross = k x k y [ t 0 cross + b v g 2 k x 2 ( ω ω p 1 η p 1 k y 2 ) ( ω ω p 2 η p 2 k y 2 ) ] ,
t 0 same = t 0 TE , t 0 cross = t 0 TM , b = 2 i c e c t TM v g , ω p 1 = ω 0 i ( c 1 c 2 ) , ω p 2 = ω 0 i ( c 1 + c 2 ) , ω zt = ω p 1 2 i c e c t TE / t 0 TE , η p 1 = η p 2 = η zt = v g / ( 2 k 1 ) .
C 1 ( k x , k y , ω ) c 1 + c 1 k y 2 , C 2 ( k x , k y , ω ) c 2 + c 2 k y 2 .
η p 1 = v g / ( 2 k 1 ) i ( c 1 c 2 ) , η p 2 = v g / ( 2 k 1 ) i ( c 1 + c 2 ) .
C e 1 ( k x , k y , ω ) C e 2 ( k x , k y , ω ) k y c e .
C t 1 TE ( k x , k y , ω ) C t 2 TE ( k x , k y , ω ) k y c t TE .
C t 1 TM ( k x , k y , ω ) C t 2 TM ( k x , k y , ω ) c t TM .
T same = t 0 same v g 2 k x 2 ( ω ω p 1 η p 1 k y 2 ) ( ω ω p 2 η zt k y 2 ) v g 2 k x 2 ( ω ω p 1 η p 1 k y 2 ) ( ω ω p 2 η zt k y 2 ) .
ω = ω p 1 + η p 1 k y 2 .
ω = ω p 1 + v g 2 ω p 1 ω p 2 k x 2 + O ( k x 4 ) .
[ E x E y E z H x H y H z ] = [ k 0 k z 0 k 0 k x k x k y k x 2 + k z 2 k y k z ] exp { i ( k x x + k y y + k z z ω t ) } .
[ E x E y E z H x H y H z ] = [ 0 k 0 k z k 0 k y ( k y 2 + k z 2 ) k x k y k x k z ] exp { i ( k x x + k y y + k z z ω t ) } .
A m same ( k x , k y ) = A m same ( k x , k y ) ,
A m cross ( k x , k y ) = A m cross ( k x , k y ) .
A m same ( k x , k y ) = A m same ( k x , k y ) ,
A m cross ( k x , k y ) = A m cross ( k x , k y ) .

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