Abstract

In this paper we study grating-induced plasmon–plasmon coupling in photorefractive layered media using a weak-coupling approximation. The method used is applicable to general layered structures that support both plasmonic and optical modes, such as photorefractive liquid crystal cells. The approximate equations are accurate when compared to S matrix approaches and capture the plasmon propagation at the surface of the device along with the optical modes guided by the layered geometry underneath. Analysis of the resulting model provides insight into the effect of the control parameters in this device and the means to optimize the diffraction efficiency. For example, by considering the case in which the plasmon is spectrally separated from the guided modes it is possible to determine the optimum gold thickness and grating strength required to obtain the strongest possible diffraction.

© 2013 Optical Society of America

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2012 (3)

2011 (3)

K. R. Daly, C. Holmes, J. C. Gates, P. G. R. Smith, and G. D’Alessandro, “Complete mode structure analysis of tilted Bragg grating refractometers in planar waveguides toward absolute index measurement,” IEEE Photon. J. 3, 861–871 (2011).
[CrossRef]

K. R. Daly, S. B. Abbott, G. D’Alessandro, D. C. Smith, and M. Kaczmarek, “Theory of hybrid photorefractive plasmonic liquid crystal cells,” J. Opt. Soc. Am. B 28, 1874–1881 (2011).
[CrossRef]

C. Holmes, K. R. Daly, I. J. G. Sparrow, J. C. Gates, G. D’Alessandro, and P. G. R. Smith, “Excitation of surface plasmons using tilted planar-waveguide Bragg gratings,” IEEE Photon. J. 3, 777–788 (2011).
[CrossRef]

2005 (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

2004 (1)

M. Kaczmarek, A. Dyadyusha, S. Slussarenko, and I. C. Khoo, “The role of surface charge field in two-beam coupling in liquid crystal cells with photoconducting polymer layers,” J. Appl. Phys. 96, 2616–2623 (2004).
[CrossRef]

2003 (2)

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

2002 (1)

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597–R624 (2002).
[CrossRef]

1999 (2)

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

F. Yang and J. R. Sambles, “Optical fully leaky mode characterization of a standard liquid-crystal cell,” J. Opt. Soc. Am. B 16, 488–497 (1999).
[CrossRef]

1996 (1)

1995 (3)

1994 (1)

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical wave guiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

1993 (1)

M. Malmqvist, “Biospecific interaction analysis using biosensor technology,” Nature 361, 186–187 (1993).
[CrossRef]

1992 (1)

1990 (1)

1988 (1)

1987 (2)

K. R. Welford and J. R. Sambles, “Detection of surface director reorientation in a nematic liquid crystal,” Appl. Phys. Lett. 50, 871–873 (1987).
[CrossRef]

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91–105 (1987).
[CrossRef]

1972 (1)

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. B 62, 502–510 (1972).
[CrossRef]

1968 (1)

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).

1967 (1)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Abbott, S. B.

Anemogiannis, E.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical wave guiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[CrossRef]

Auslender, M.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Berreman, D. W.

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. B 62, 502–510 (1972).
[CrossRef]

Billingham, J.

A. C. King, J. Billingham, and S. R. Otto, Differential Equations: Linear, Nonlinear, Ordinary, Partial (Cambridge University, 2003).

Buchnev, O.

Caldwell, M. E.

Clark, M. G.

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91–105 (1987).
[CrossRef]

Cotter, N. P. K.

D’Alessandro, G.

Daly, K. R.

Dasari, R. R.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597–R624 (2002).
[CrossRef]

Delves, L. M.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Dyadyusha, A.

M. Kaczmarek, A. Dyadyusha, S. Slussarenko, and I. C. Khoo, “The role of surface charge field in two-beam coupling in liquid crystal cells with photoconducting polymer layers,” J. Appl. Phys. 96, 2616–2623 (2004).
[CrossRef]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Feld, M. S.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597–R624 (2002).
[CrossRef]

Garcia Quirino, G. S.

G. S. Garcia Quirino, J. J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571–1577 (1995).
[CrossRef]

Gates, J. C.

K. R. Daly, C. Holmes, J. C. Gates, P. G. R. Smith, and G. D’Alessandro, “Complete mode structure analysis of tilted Bragg grating refractometers in planar waveguides toward absolute index measurement,” IEEE Photon. J. 3, 861–871 (2011).
[CrossRef]

C. Holmes, K. R. Daly, I. J. G. Sparrow, J. C. Gates, G. D’Alessandro, and P. G. R. Smith, “Excitation of surface plasmons using tilted planar-waveguide Bragg gratings,” IEEE Photon. J. 3, 777–788 (2011).
[CrossRef]

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

Gaylord, T. K.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical wave guiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

E. N. Glytsis and T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1399–1420 (1990).
[CrossRef]

Glytsis, E. N.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical wave guiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

E. N. Glytsis and T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1399–1420 (1990).
[CrossRef]

Harel, E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[CrossRef]

Hava, S.

Holmes, C.

C. Holmes, K. R. Daly, I. J. G. Sparrow, J. C. Gates, G. D’Alessandro, and P. G. R. Smith, “Excitation of surface plasmons using tilted planar-waveguide Bragg gratings,” IEEE Photon. J. 3, 777–788 (2011).
[CrossRef]

K. R. Daly, C. Holmes, J. C. Gates, P. G. R. Smith, and G. D’Alessandro, “Complete mode structure analysis of tilted Bragg grating refractometers in planar waveguides toward absolute index measurement,” IEEE Photon. J. 3, 861–871 (2011).
[CrossRef]

Homola, J.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

Itzkan, I.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597–R624 (2002).
[CrossRef]

Kaczmarek, M.

Khoo, I. C.

M. Kaczmarek, A. Dyadyusha, S. Slussarenko, and I. C. Khoo, “The role of surface charge field in two-beam coupling in liquid crystal cells with photoconducting polymer layers,” J. Appl. Phys. 96, 2616–2623 (2004).
[CrossRef]

Kik, P. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[CrossRef]

King, A. C.

A. C. King, J. Billingham, and S. R. Otto, Differential Equations: Linear, Nonlinear, Ordinary, Partial (Cambridge University, 2003).

Kneipp, H.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597–R624 (2002).
[CrossRef]

Kneipp, K.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597–R624 (2002).
[CrossRef]

Ko, D. Y. K.

Koel, B. E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[CrossRef]

Kretschmann, E.

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).

Lyness, J. N.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Maier, S. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[CrossRef]

Malmqvist, M.

M. Malmqvist, “Biospecific interaction analysis using biosensor technology,” Nature 361, 186–187 (1993).
[CrossRef]

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Meltzer, S.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[CrossRef]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

Otto, S. R.

A. C. King, J. Billingham, and S. R. Otto, Differential Equations: Linear, Nonlinear, Ordinary, Partial (Cambridge University, 2003).

Palik, E.D.

E.D. Palik, Handbook of Optical Constants of Solids, Volumes I, II, and III: Subject Index and Contributor Index (Academic, 1985).

Podoliak, N.

Preist, T. W.

Quinn, J. J.

J. J. Quinn and K. S Yi, Solid State Physics (Springer, 2009).

Raether, H.

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).

Requicha, A. A. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[CrossRef]

Sambles, J. R.

Sanchez-Mondragon, J. J.

G. S. Garcia Quirino, J. J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571–1577 (1995).
[CrossRef]

Slussarenko, S.

M. Kaczmarek, A. Dyadyusha, S. Slussarenko, and I. C. Khoo, “The role of surface charge field in two-beam coupling in liquid crystal cells with photoconducting polymer layers,” J. Appl. Phys. 96, 2616–2623 (2004).
[CrossRef]

Smith, D. C.

Smith, P. G. R.

K. R. Daly, C. Holmes, J. C. Gates, P. G. R. Smith, and G. D’Alessandro, “Complete mode structure analysis of tilted Bragg grating refractometers in planar waveguides toward absolute index measurement,” IEEE Photon. J. 3, 861–871 (2011).
[CrossRef]

C. Holmes, K. R. Daly, I. J. G. Sparrow, J. C. Gates, G. D’Alessandro, and P. G. R. Smith, “Excitation of surface plasmons using tilted planar-waveguide Bragg gratings,” IEEE Photon. J. 3, 777–788 (2011).
[CrossRef]

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Sparrow, I. J. G.

C. Holmes, K. R. Daly, I. J. G. Sparrow, J. C. Gates, G. D’Alessandro, and P. G. R. Smith, “Excitation of surface plasmons using tilted planar-waveguide Bragg gratings,” IEEE Photon. J. 3, 777–788 (2011).
[CrossRef]

Stepanov, S.

G. S. Garcia Quirino, J. J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571–1577 (1995).
[CrossRef]

Welford, K. R.

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91–105 (1987).
[CrossRef]

K. R. Welford and J. R. Sambles, “Detection of surface director reorientation in a nematic liquid crystal,” Appl. Phys. Lett. 50, 871–873 (1987).
[CrossRef]

Yang, F.

Yeatman, E. M.

Yee, S. S.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

Yi, K. S

J. J. Quinn and K. S Yi, Solid State Physics (Springer, 2009).

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. R. Welford and J. R. Sambles, “Detection of surface director reorientation in a nematic liquid crystal,” Appl. Phys. Lett. 50, 871–873 (1987).
[CrossRef]

IEEE Photon. J. (2)

C. Holmes, K. R. Daly, I. J. G. Sparrow, J. C. Gates, G. D’Alessandro, and P. G. R. Smith, “Excitation of surface plasmons using tilted planar-waveguide Bragg gratings,” IEEE Photon. J. 3, 777–788 (2011).
[CrossRef]

K. R. Daly, C. Holmes, J. C. Gates, P. G. R. Smith, and G. D’Alessandro, “Complete mode structure analysis of tilted Bragg grating refractometers in planar waveguides toward absolute index measurement,” IEEE Photon. J. 3, 861–871 (2011).
[CrossRef]

J. Appl. Phys. (1)

M. Kaczmarek, A. Dyadyusha, S. Slussarenko, and I. C. Khoo, “The role of surface charge field in two-beam coupling in liquid crystal cells with photoconducting polymer layers,” J. Appl. Phys. 96, 2616–2623 (2004).
[CrossRef]

J. Lightwave Technol. (1)

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical wave guiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

J. Phys. Condens. Matter (1)

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597–R624 (2002).
[CrossRef]

Liq. Cryst. (1)

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91–105 (1987).
[CrossRef]

Math. Comput. (1)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
[CrossRef]

Nat. Mater. (1)

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[CrossRef]

Nature (2)

M. Malmqvist, “Biospecific interaction analysis using biosensor technology,” Nature 361, 186–187 (1993).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Device schematic (bottom) showing its decoupled components (top) as specified in Eqs. (2). The ellipses represent the LC orientation in a photorefractive LC cell. They illustrate how the idealised dielectric layer geometry approximates the LC device.

Fig. 2.
Fig. 2.

Phase matching diagram showing the possible couplings between an incident field, with tangential wave vector nfe^z, and all possible guided and plasmonic modes. The grating vector kg is shown in red and makes and angle θg with the y axis. The region in which the guided modes exist is shaded in blue.

Fig. 3.
Fig. 3.

Plot of the maximum of the relative intensity of the first diffracted orders and of the minimum of the reflected intensity as a function of the gold layer thickness. The curves are calculated using Eqs. (30) and (31) for the fundamental and first diffracted orders, respectively. The cladding is semi-infinite (no guided modes). All other parameters are as in Table 1.

Fig. 4.
Fig. 4.

Plot of the maximum of the intensity of the first diffracted orders scaled to that of the input beam as a function of the grating spacing Λ. The curves are computed numerically (N) or using the analytical model (A) given by Eq. (31). The cladding is semi-infinite (no guided modes). All other parameters are as in Table 1.

Fig. 5.
Fig. 5.

Plot of the maximum of the intensity of the first diffracted orders, scaled to that of the input beam, as a function of the coupling strength ϵg,cl. The curves are computed numerically (N) or using the analytical model (A) given by Eq. (31). The cladding is semi-infinite (no guided modes). All other parameters are as in Table 1.

Fig. 6.
Fig. 6.

Plot of the relative intensity of the reflected fundamental beam and diffracted orders as a function of the angle of incidence of the beam predicted by Eqs. (30) and (31), for ϵg,cl={0.01,0.03,0.06,0.12} left to right, top to bottom, respectively, and Λ=1μm. The cladding is semi-infinite (no guided modes). All other parameters areas in Table 1.

Fig. 7.
Fig. 7.

Effect of guided modes for thick cladding, no grating in the core. Reflection spectrum computed using (top panel) Eqs. (30) and (31), i.e., no guided modes; (middle panel) Eqs. (17), (23), and (24); and (bottom panel) an S-matrix code. Left cladding thickness, Lcl=600nm; core width, Lco=2.5μm; semi-infinite right cladding, Λ=1μm, ϵg,co=0. All other parameters are as in Table 1.

Fig. 8.
Fig. 8.

Effect of guided modes for thick cladding, no grating in the cladding. Reflection spectrum computed using (top panel) Eqs. (30) and (31), i.e., no guided modes; (middle panel) Eqs. (17), (23), and (24); and (bottom panel) an S-matrix code. Left cladding thickness, Lcl=600nm; core width, Lco=2.5μm; semi-infinite right cladding, Λ=1μm, ϵg,cl=0. All other parameters are as in Table 1.

Fig. 9.
Fig. 9.

Effect of guided modes for thin cladding, no grating in the cladding. Reflection spectrum is computed using (top panel) Eqs. (30) and (31), i.e., no guided modes; (middle panel) Eqs. (17), (23), and (24); and (bottom panel) an S-matrix code. Left cladding thickness, Lcl=100nm; core width, Lco=2.5μm; semi-infinite right cladding, Λ=1μm, ϵg,cl=0. All other parameters are as in Table 1.

Tables (1)

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Table 1. Default Parameter Values Used in All Figuresa,b

Equations (42)

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(DiiϵD)(EH)=0,
ϵs={ϵrx<ξ1,rΩsϵclx>ξ1,rΩs,
ϵw={ϵclx<ξ2,rΩwϵcoξ2<x<ξ3,rΩwϵclξ3<x,rΩw,
ϵf={ϵglx<0,rΩfϵm0<x,rΩf,
Hs=AsHs(x)einsυ^·r,rΩs,
Hs(r)={υ^eβs1(xξ1)x<ξ1υ^eβs2(xξ1)x>ξ1,
ns=ϵrϵclϵr+ϵcl,βs1=ϵrϵrϵr+ϵcl,βs2=ϵclϵclϵr+ϵcl.
Hw=q=0NAqHq(x)einqυ^·r,rΩw,
Hq={υ^cos[Lcoγq2]e(xξ2)βq,x<ξ2,υ^cos[(xξ3+ξ22)γq],ξ2<x<ξ3,υ^cos[Lcoγq2]e(xξ4)βq,ξ3<x,
Hq={υ^sin[Lcoγq2]e(xξ2)βq,x<ξ2,υ^sin[(xξ3+ξ22)γq],ξ2<x<ξ3,υ^sin[Lcoγq2]e(xξ3)βq,ξ3<x.
Hf=Hfeinfz,
Hf={υ^eiγfx+Arυ^eiγfxx<0,rΩf,Atυ^eβfxx>0,rΩf,
+Hp*·Hqdxδpq.
+Hα*·HβdxO(η),
ϵ¯s={ϵglϵrx<0iϵi0<x<ξ1ϵcoξ2<x<ξ30otherwise,
ϵ¯w={ϵglϵclx<0ϵmϵcl0<x<ξ10otherwise,
ϵ¯f={ϵclϵmξ1<x<ξ2ϵcoϵmξ2<x<ξ3ϵclϵmξ3<x0otherwise.
ϵ¯g={ϵ¯g,clξ1<x<ξ2ϵ¯g,coξ2<x<ξ3.
Hs(m)=As(m)Hs(m)(x)eikf(m)·r,
Hw(m)=q=1NAq(m)Hq(m)(x)eikf(m)·r.
(κs,s+Psδn¯s(m))As(m)+κs,fAt(m)+q=0Nκs,qAq(m)+χs,sAs(m+1)+χs,fAt(m+1)+q=0Nχs,qAq(m+1)+χs,s*As(m1)+χs,f*At(m1)+q=0Nχs,q*Aq(m1)=0,
Ppδn¯p(m)Ap(m)+κp,fAt(m)+κp,sAs(m)+q=0Nκp,qAq(m)+χp,sAs(m+1)+χp,fAt(m+1)+q=0Nχp,qAq(m+1)+χp,s*As(m1)+χp,f*At(m1)+q=0Nχp,q*Aq(m1)=0.
Ps=ns(βs1ϵcl+βs2ϵr)βs1ϵclβs2ϵr,
PpnpLcoϵco.
κα,β=ϵ¯αEα(m)·Eβ(m)dx,
χα,β=ϵ¯gEα(m)·Eβ(m+1)dx,
χs,s=ϵg,cl(ns2+βs22)(1e2βs2Lcl)2βs2ϵcl2ϵg,co(ns2+βs22)(1e2βs2Lco)e2βs2Lcl2βs2ϵcl2,
Hs(m)As(m)+At(m)+q=1NHq(m)=δ0m+Ar(m),
Es(m)As(m)βf(m)ϵ2At(m)+q=1NEq(m)=iγf(m)ϵ1(δ0mAt(m)),
(1111iβ˜fiβ˜s10κs,fiκ˜s,s+δn˜s)(ArAtA˜s)=(110),
Ar=(1+iβ˜f)(iκ˜s,s+δn˜s)(1iβ˜s1)κs,f(1iβ˜f)(iκ˜s,s+δn˜s)(1+iβ˜s1)κs,f.
nfns+κs,feβs1LauPs(βs12ϵgl2+ns2ϵm2ϵglϵm2βs12ϵgl2ns2ϵm2+ϵglϵm2).
(K+1χ+1,00χ0,+1K0χ1,00χ0,1K1)(A+1A0A1)=(0S0),
A0=P1S,
A±1=K±11χ0,±1P1S,
Ar(0)=ξ0[(1+iβ˜f(0))X(1iβ˜s1(0))κs,f(0)],
Ar(±1)=2iζ0χ˜s,s(0,±1)κs,f(0)(β˜f(±1)+β˜s1(±1))(iκ˜s,s+δn˜s(±1))(iβ˜f(±1)1)+κs,f(±1)(iβ˜s1(±1)+1),
ζ0=1(1iβ˜f(0))X(1+iβ˜s1(0))κs,f(0),
X=iκ˜s,s+δn˜s(0)ζ(+1)ζ(1),
ζ(±1)=(iβ˜f(±1)1)χ˜s,s(0,±1)χ˜s,s(±1,0)κs,f(±1)(1+iβ˜s1±1)(δn˜s(±1)+iκ˜s,s)(1+iβ˜f±1),
θm=arcsin(|kg|2ns).
|kg|2=ns2ns22,

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