Abstract

We analyze wave propagation in coupled planar waveguides, pointing specific attention to modal cross-talk and out-of-plane scattering in quasi-planar photonics. An algorithm capable of accurate numerical computation of wave coupling in arrays of planar structures is developed and illustrated on several examples of plasmonic and volumetric waveguides. An analytical approach to reduce or completely eliminate scattering and modal cross-talk in planar waveguides with anisotropic materials is also presented.

© 2009 Optical Society of America

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  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,” Nature Mater. 2, 229-232 (2003).
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  5. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508-511 (2006).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2009 (1)

2008 (3)

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photon. 2, 242-246 (2008).
[CrossRef]

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100, 066402 (2008).
[CrossRef] [PubMed]

2007 (6)

R. F. Oulton, D. F. P. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 76, 035408 (2007).
[CrossRef]

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41-48 (2007).
[CrossRef]

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young's double-slit experiment,” Nature Nanotech. 2, 426-429 (2007).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Huang, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

2006 (8)

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247-8256 (2006).
[CrossRef] [PubMed]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 171108 (2006).
[CrossRef]

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, “Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media,” J. Opt. Soc. Am. B 23, 498-505 (2006).
[CrossRef]

2005 (3)

T. Sondergaard and S. I. Bozhevolnyi, “Out-of-plane scattering properties of long-range surface plasmon polariton gratings,” Phys. Status Solidi B 242, 3064-3069 (2005).
[CrossRef]

I. I. Smolyaninov, J. Elliot, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005).
[CrossRef] [PubMed]

Y. F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005) and Reply Y. F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef] [PubMed]

Y. F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005) and Reply Y. F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef] [PubMed]

2004 (1)

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

2003 (2)

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

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,” Nature Mater. 2, 229-232 (2003).
[CrossRef]

2002 (1)

S. I. Bozhevolnyi, V. S. Volkov, and K. Leosson, “Localization and waveguiding of surface plasmon polaritons in random nanostructures,” Phys. Rev. Lett. 89, 186801 (2002).
[CrossRef] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1997 (1)

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, “Experimental study of surface-plasmon scattering by individual surface defects,” Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

1981 (1)

1966 (1)

P. J. B. Clarricoats and K. R. Slinn, “Numerical method for the solution of waveguide-discontinuity problems,” Electron. Lett. 2, 226-228 (1966).
[CrossRef]

Alekseyev, L. V.

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,” Nature Mater. 2, 229-232 (2003).
[CrossRef]

Barnes, W. L.

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

Bartal, G.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Boardman, A. D.

A. D. Boardman, Electromagnetic Surface Modes (Wiley, 1982).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

T. Sondergaard and S. I. Bozhevolnyi, “Out-of-plane scattering properties of long-range surface plasmon polariton gratings,” Phys. Status Solidi B 242, 3064-3069 (2005).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, and K. Leosson, “Localization and waveguiding of surface plasmon polaritons in random nanostructures,” Phys. Rev. Lett. 89, 186801 (2002).
[CrossRef] [PubMed]

Brongersma, M. L.

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young's double-slit experiment,” Nature Nanotech. 2, 426-429 (2007).
[CrossRef]

Chen, Y. F.

Y. F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005) and Reply Y. F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef] [PubMed]

Y. F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005) and Reply Y. F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef] [PubMed]

Chichkov, B. N.

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

Clarricoats, P. J. B.

P. J. B. Clarricoats and K. R. Slinn, “Numerical method for the solution of waveguide-discontinuity problems,” Electron. Lett. 2, 226-228 (1966).
[CrossRef]

Davis, C. C.

I. I. Smolyaninov, Y. J. Huang, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, J. Elliot, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005).
[CrossRef] [PubMed]

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, “Experimental study of surface-plasmon scattering by individual surface defects,” Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

Dereux, A.

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

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Dickson, W.

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

Dolling, G.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

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

Elliot, J.

I. I. Smolyaninov, J. Elliot, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005).
[CrossRef] [PubMed]

Elser, J.

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100, 066402 (2008).
[CrossRef] [PubMed]

R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, “Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media,” J. Opt. Soc. Am. B 23, 498-505 (2006).
[CrossRef]

V. A. Podolskiy and J. Elser, “Electroplasmonics: dynamical plasmonic circuits with minimized parasitic scattering (QTuJ2),” presented at the Conference on Lasers and Electro-Optics (CLEO) and the International Quantum Electronics Conference (IQEC), Washington, D.C., May 4-8, 2008.

Engheta, N.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Evans, P.

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

Fan, S.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

Fischer, P.

Y. F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005) and Reply Y. F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef] [PubMed]

Y. F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005) and Reply Y. F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, W. T. Wetterling, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge Univ. Press, 1992).

Gaylord, T. K.

Govyadinov, A. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

Green, W. M. J.

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photon. 2, 242-246 (2008).
[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,” Nature Mater. 2, 229-232 (2003).
[CrossRef]

Huang, Y. J.

I. I. Smolyaninov, Y. J. Huang, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

Jacob, Z.

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,” Nature Mater. 2, 229-232 (2003).
[CrossRef]

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,” Nature Mater. 2, 229-232 (2003).
[CrossRef]

Laluet, J. Y.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 171108 (2006).
[CrossRef]

Leosson, K.

S. I. Bozhevolnyi, V. S. Volkov, and K. Leosson, “Localization and waveguiding of surface plasmon polaritons in random nanostructures,” Phys. Rev. Lett. 89, 186801 (2002).
[CrossRef] [PubMed]

Linden, S.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Liu, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

R. F. Oulton, D. F. P. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 76, 035408 (2007).
[CrossRef]

Liu, Z.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 171108 (2006).
[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,” Nature Mater. 2, 229-232 (2003).
[CrossRef]

Mait, J.

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, “Experimental study of surface-plasmon scattering by individual surface defects,” Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

Mazzoni, D. L.

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, “Experimental study of surface-plasmon scattering by individual surface defects,” Phys. Rev. B 56, 1601-1611 (1997).
[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,” Nature Mater. 2, 229-232 (2003).
[CrossRef]

Moharam, M. G.

Mongiardo, M.

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides (Inspec/IEE, 1997).
[CrossRef]

Narimanov, E.

Narimanov, E. E.

Noginov, M. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

Oulton, R. F.

R. F. Oulton, D. F. P. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 76, 035408 (2007).
[CrossRef]

Passinger, S.

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Pile, D. F. P.

R. F. Oulton, D. F. P. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 76, 035408 (2007).
[CrossRef]

Podolskiy, V. A.

S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34, 890-892 (2009).
[CrossRef] [PubMed]

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100, 066402 (2008).
[CrossRef] [PubMed]

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, “Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media,” J. Opt. Soc. Am. B 23, 498-505 (2006).
[CrossRef]

V. A. Podolskiy and J. Elser, “Electroplasmonics: dynamical plasmonic circuits with minimized parasitic scattering (QTuJ2),” presented at the Conference on Lasers and Electro-Optics (CLEO) and the International Quantum Electronics Conference (IQEC), Washington, D.C., May 4-8, 2008.

Pollard, R.

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

Press, W. H.

W. H. Press, W. T. Wetterling, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge Univ. Press, 1992).

Reinhardt, C.

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

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,” Nature Mater. 2, 229-232 (2003).
[CrossRef]

Rozzi, T.

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides (Inspec/IEE, 1997).
[CrossRef]

Salandrino, A.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Schevchenko, V. V.

V. V. Schevchenko, Continuous Transitions in Open Waveguides (Golem, 1971).

Shalaev, V. M.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41-48 (2007).
[CrossRef]

Shin, H.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

Slinn, K. R.

P. J. B. Clarricoats and K. R. Slinn, “Numerical method for the solution of waveguide-discontinuity problems,” Electron. Lett. 2, 226-228 (1966).
[CrossRef]

Smolyaninov, I. I.

I. I. Smolyaninov, Y. J. Huang, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, J. Elliot, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005).
[CrossRef] [PubMed]

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, “Experimental study of surface-plasmon scattering by individual surface defects,” Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

Sondergaard, T.

T. Sondergaard and S. I. Bozhevolnyi, “Out-of-plane scattering properties of long-range surface plasmon polariton gratings,” Phys. Status Solidi B 242, 3064-3069 (2005).
[CrossRef]

Soukoulis, C. M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Stacy, A. M.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Steele, J. M.

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 171108 (2006).
[CrossRef]

Stockman, M. I.

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

Sun, C.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, W. T. Wetterling, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge Univ. Press, 1992).

Thongrattanasiri, S.

Vlasov, Y.

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photon. 2, 242-246 (2008).
[CrossRef]

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, and K. Leosson, “Localization and waveguiding of surface plasmon polaritons in random nanostructures,” Phys. Rev. Lett. 89, 186801 (2002).
[CrossRef] [PubMed]

Wang, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Wangberg, R.

Wegener, M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Wetterling, W. T.

W. H. Press, W. T. Wetterling, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge Univ. Press, 1992).

Wise, F. W.

Y. F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005) and Reply Y. F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef] [PubMed]

Y. F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005) and Reply Y. F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

Wurtz, G. A.

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

Xia, F.

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photon. 2, 242-246 (2008).
[CrossRef]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

Yao, J.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Zayats, A. V.

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

I. I. Smolyaninov, J. Elliot, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005).
[CrossRef] [PubMed]

Zhang, X.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

R. F. Oulton, D. F. P. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 76, 035408 (2007).
[CrossRef]

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 171108 (2006).
[CrossRef]

Zia, R.

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young's double-slit experiment,” Nature Nanotech. 2, 426-429 (2007).
[CrossRef]

Appl. Phys. Lett. (3)

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 171108 (2006).
[CrossRef]

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007).
[CrossRef]

C. Reinhardt, S. Passinger, B. N. Chichkov, W. Dickson, G. A. Wurtz, P. Evans, R. Pollard, and A. V. Zayats, “Restructuring and modification of metallic nanorod arrays using femtosecond laser direct writing,” Appl. Phys. Lett. 89, 231117 (2006).
[CrossRef]

Electron. Lett. (1)

P. J. B. Clarricoats and K. R. Slinn, “Numerical method for the solution of waveguide-discontinuity problems,” Electron. Lett. 2, 226-228 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nat. Photon. (2)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41-48 (2007).
[CrossRef]

Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photon. 2, 242-246 (2008).
[CrossRef]

Nature (2)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

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

Nature 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,” Nature Mater. 2, 229-232 (2003).
[CrossRef]

Nature Nanotech. (1)

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young's double-slit experiment,” Nature Nanotech. 2, 426-429 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (3)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103 (2006).
[CrossRef]

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, “Experimental study of surface-plasmon scattering by individual surface defects,” Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

R. F. Oulton, D. F. P. Pile, Y. Liu, and X. Zhang, “Scattering of surface plasmon polaritons at abrupt surface interfaces: implications for nanoscale cavities,” Phys. Rev. B 76, 035408 (2007).
[CrossRef]

Phys. Rev. Lett. (7)

J. Elser and V. A. Podolskiy, “Scattering-free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. 100, 066402 (2008).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, and K. Leosson, “Localization and waveguiding of surface plasmon polaritons in random nanostructures,” Phys. Rev. Lett. 89, 186801 (2002).
[CrossRef] [PubMed]

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef] [PubMed]

I. I. Smolyaninov, J. Elliot, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005).
[CrossRef] [PubMed]

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Y. F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005) and Reply Y. F. Chen, P. Fischer, and F. W. Wise, Phys. Rev. Lett. 98, 059702 (2007).
[CrossRef] [PubMed]

Phys. Status Solidi B (1)

T. Sondergaard and S. I. Bozhevolnyi, “Out-of-plane scattering properties of long-range surface plasmon polariton gratings,” Phys. Status Solidi B 242, 3064-3069 (2005).
[CrossRef]

Science (4)

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-894 (2006).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef] [PubMed]

I. I. Smolyaninov, Y. J. Huang, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Other (8)

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

W. H. Press, W. T. Wetterling, S. A. Teukolsky, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge Univ. Press, 1992).

E.Palik, ed., The Handbook of Optical Constants of Solids (Academic, 1997).

For details see COMSOL Multiphysics User's Guide and RF Module User's Guide; COMSOL (1994-2009); www.comsol.com.

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides (Inspec/IEE, 1997).
[CrossRef]

V. V. Schevchenko, Continuous Transitions in Open Waveguides (Golem, 1971).

A. D. Boardman, Electromagnetic Surface Modes (Wiley, 1982).

V. A. Podolskiy and J. Elser, “Electroplasmonics: dynamical plasmonic circuits with minimized parasitic scattering (QTuJ2),” presented at the Conference on Lasers and Electro-Optics (CLEO) and the International Quantum Electronics Conference (IQEC), Washington, D.C., May 4-8, 2008.

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Figures (5)

Fig. 1
Fig. 1

Schematic geometry of the multilayered structures and electromagnetic mode types used in the manuscript. Geometry of the single multilayer stack is shown in (a); panel (b) explains the composition of top, bottom, and guided modes; field profiles in the outside layers of the multilayer are shown; the interface between two multilayer stacks is shown in (c).

Fig. 2
Fig. 2

(a) Light reflection in the planar air–Si–air waveguide coupled to a homogeneous dielectric; λ 0 = 1.5 μ m , waveguide thickness d = 0.6 μ m ; the geometry and profiles of the waveguide modes supported by the system are shown in the inset; excitation by TM 2 mode is assumed; the graph shows the comparison between the technique presented here (lines) and FEM simulations (dots); R 2 = 0 since the symmetry of TM 1 mode is different from that of TM 0 and TM 2 modes. (b) Reflection in plasmonic gap (Au–Air–Au) waveguide; λ 0 = d = 0.6 μ m ; thickness of lines illustrates the convergence of the computations. Panels (c) and (d) illustrate the field distributions in the system (a) with ε D = 12.12 (c) and ε D x x = 3.6 + 0.05 i , ε D y z = 12.2 + 1.36 i (d).

Fig. 3
Fig. 3

Scattering of the SPP propagating across the Au–Air step; panels (a) and (b) show transmission, reflection, and scattering ( S = 1 R T ) of an SPP that is incident on the step; geometry of the structures is shown in insets; lines correspond to the formalism developed in this work (line thickness represents data variation due to changes in spectra of bulk modes); dots represent results from FEM simulations; panels (c) and (d) illustrate the field distributions obtained from scattering-matrix (c) and FEM (d) simulations for d = 0.7 μ m .

Fig. 4
Fig. 4

(a,c) An interface between an air–Si–air waveguide and isotropic air– ( ε = 6.06 ) –air guide leads to substantial modal cross-talk, polarization mixing, and out-of-plane scattering; while the interface between air–Si–air waveguide and its anisotropic truly planar optics analog allows for ideal mode matching with light steering capabilities (b,d); guided modes in (c,d) correspond to mode number > 2000 ; the system is excited by a TM 2 guided mode propagating at the angle 30° to the z = 0 interface; the amplitudes of modes in panels (c,d) are normalized to the amplitude of the incident mode.

Fig. 5
Fig. 5

Anisotropic coatings (b, d) can significantly reduce (and almost eliminate) the scattering losses in plasmonic circuits; the figure shows reflection, transmission, and scattering in conventional plasmonic circuit (a) and in a plasmonic system where the space x > 0 , z > 0 filled with material ε y z = 1 , ε x x = ε D (b); field distribution for ε D = 4 (c) clearly shows that only a fraction of the energy of incident SPP is transferred into SPP on the right-hand side of the interface; on the other hand, the interface between the anisotropic system with ε y z = 1 , ε x x = 4 (d) allows for substantial modulation of SPP index, and results in almost perfect SPP–SPP coupling across a z = 0 interface.

Equations (34)

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

E ( k y , k z ) = { a 1 + E 1 ( k x ; 1 ( k y , k z ) , k y , k z ) + a 1 E 1 ( k x ; 1 ( k y , k z ) , k y , k z ) , x < x 1 a j + E j ( k x ; j ( k y , k z ) , k y , k z ) + a j E j ( k x ; j ( k y , k z ) , k y , k z ) , x j 1 < x < x j a N + E N ( k x ; N ( k y , k z ) , k y , k z ) + a N E N ( k x ; N ( k y , k z ) , k y , k z ) , x N 1 < x }
H ( k y , k z ) = { a 1 + H 1 ( k x ; 1 ( k y , k z ) , k y , k z ) + a 1 H 1 ( k x ; 1 ( k y , k z ) , k y , k z ) , x < x 1 a j + H j ( k x ; j ( k y , k z ) , k y , k z ) + a j H j ( k x ; j ( k y , k z ) , k y , k z ) , x j 1 < x < x j a N + H N ( k x ; N ( k y , k z ) , k y , k z ) + a N H N ( k x ; N ( k y , k z ) , k y , k z ) , x N 1 < x }
TE polarized waves : { E j = e i ω t + i k r k y 2 + k z 2 { 0 , k z , k y } H j = e i ω t + i k r c ω k y 2 + k z 2 { ( k y 2 + k z 2 ) , k x ; j k y , k x ; j k z } k x ; j 2 + k y 2 + k z 2 ε j y z = ω 2 c 2 }
TM polarized waves : { E j = e i ω t + i k r k y 2 + k z 2 { k y 2 + k z 2 , ε j x x k x ; j k y ε j y z , ε j x x k x ; j k z ε j y z } H j = e i ω t + i k r ω ( k y 2 + k z 2 ) c { 0 , ε j x x k z , ε j x x k y } k x ; j 2 ε j y z + k y 2 + k z 2 ε j x x = ω 2 c 2 } .
( a j + 1 a j + 1 + ) = α j [ ( 1 + K j ) φ j ( 1 K j ) φ j + ( 1 K j ) φ j + ( 1 + K j ) φ j ] ( a j a j + ) ,
E = q [ A ( q ) + E ( k y , k z ( q ) ) + A ( q ) E ( k y , k z ( q ) ) ] + 0 [ A top + ( k x ) E ( k y , k z ( k x ; N ) ) + A top ( k x ) E ( k y , k z ( k x ; N ) ) + A btm + ( k x ) E ( k y , k z ( k x ; 1 ) ) + A btm ( k x ) E ( k y , k z ( k x ; 1 ) ) ] d k x ,
H = q [ A ( q ) + H ( k y , k z ( q ) ) + A ( q ) H ( k y , k z ( q ) ) ] + 0 [ A top + ( k x ) H ( k y , k z ( k x ; N ) ) + A top ( k x ) H ( k y , k z ( k x ; N ) ) + A btm + ( k x ) H ( k y , k z ( k x ; 1 ) ) + A btm ( k x ) H ( k y , k z ( k x ; 1 ) ) ] d k x .
E 1 | H 2 = ( E 1 × H 2 ) z ̂ d x ,
E L x = E R x ; H L y = H R y ,
E L y = E R y ; H L x = H R x .
E = m [ A ( m ) + E ( m ) + + A ( m ) E ( m ) ] w ( m ) ,
H = m [ A ( m ) + H ( m ) + + A ( m ) H ( m ) ] w ( m ) ,
m = 1 N L [ A L ( m ) + E L x ( m ) + + A L ( m ) E L x ( m ) ] w L ( m ) = m = 1 N R [ A R ( m ) + E R x ( m ) + + A R ( m ) E R x ( m ) ] w R ( m ) ,
m = 1 N L [ A L ( m ) + H L y ( m ) + + A L ( m ) H L y ( m ) ] w L ( m ) = m = 1 N R [ A R ( m ) + H R y ( m ) + + A R ( m ) H R y ( m ) ] w R ( m ) ,
m = 1 N L [ A L ( m ) + E L y ( m ) + + A L ( m ) E L y ( m ) ] w L ( m ) = m = 1 N R [ A R ( m ) + E R y ( m ) + + A R ( m ) E R y ( m ) ] w R ( m ) ,
m = 1 N L [ A L ( m ) + H L x ( m ) + + A L ( m ) H L x ( m ) ] w L ( m ) = m = 1 N R [ A R ( m ) + H R x ( m ) + + A R ( m ) H R x ( m ) ] w R ( m ) .
E R L + ˆ m n A L ( m ) + + E R L ˆ m n A L ( m ) = E R R + ˆ m n A R ( m ) + + E R R ˆ m n A R ( m ) ,
H R L + ˆ m n A L ( m ) + + H R L ˆ m n A L ( m ) = H R R + ˆ m n A R ( m ) + + H R R ˆ m n A R ( m ) ,
E L L + ˆ m n A L ( m ) + + E L L ˆ m n A L ( m ) = E L R + ˆ m n A R ( m ) + + E L R ˆ m n A R ( m ) ,
H L L + ˆ m n A L ( m ) + + H L L ˆ m n A L ( m ) = H L R + ˆ m n A R ( m ) + + H L R ˆ m n A R ( m ) ,
E R { L | R } ± m n = w { L | R } ( m ) { E { L | R } y ( m ) ± H R x ( n ) d x , n N R TE E { L | R } x ( m ) ± H R y ( n ) d x , n > N R TE } ,
H R { L | R } ± m n = w { L | R } ( m ) { H { L | R } x ( m ) ± E R y ( n ) d x , n N R TE H { L | R } y ( m ) ± E R x ( n ) d x , n > N R TE } ,
E L { L | R } ± m n = w { L | R } ( m ) { E { L | R } y ( m ) ± H L x ( n ) d x , n N R TE E { L | R } x ( m ) ± H L y ( n ) d x , n > N R TE } ,
H L { L | R } ± m n = w { L | R } ( m ) { H { L | R } x ( m ) ± E L y ( n ) d x , n N R TE H { L | R } y ( m ) ± E L x ( n ) d x , n > N R TE } . .
{ A R = R 11 ˆ A L + R 12 ˆ A L + A R + = R 21 ˆ A L + R 22 ˆ A L + } ,
{ A L = L 11 ˆ A R + L 12 ˆ A R + A L + = L 21 ˆ A R + L 22 ˆ A R + } ,
{ R 11 ˆ = [ E R R + ˆ 1 E R R ˆ H R R + ˆ 1 H R R ˆ ] 1 ( E R R + ˆ 1 E R L ˆ H R R + ˆ 1 H R L ˆ ) R 12 ˆ = [ E R R + ˆ 1 E R R ˆ H R R + ˆ 1 H R R ˆ ] 1 ( E R R + ˆ 1 E R L + ˆ H R R + ˆ 1 H R L + ˆ ) R 21 ˆ = [ E R R ˆ 1 E R R + ˆ H R R ˆ 1 H R R + ˆ ] 1 ( E R R ˆ 1 E R L ˆ H R R ˆ 1 H R L ˆ ) R 22 ˆ = [ E R R ˆ 1 E R R + ˆ H R R ˆ 1 H R R + ˆ ] 1 ( E R R ˆ 1 E R L + ˆ H R R ˆ 1 H R L + ˆ ) } ,
{ L 11 ˆ = [ E L L + ˆ 1 E L L ˆ H L L + ˆ 1 H L L ˆ ] 1 ( E L L + ˆ 1 E L R ˆ H L L + ˆ 1 H L R ˆ ) L 12 ˆ = [ E L L + ˆ 1 E L L ˆ H L L + ˆ 1 H L L ˆ ] 1 ( E L L + ˆ 1 E L R + ˆ H L L + ˆ 1 H L R + ˆ ) L 21 ˆ = [ E L L ˆ 1 E L L + ˆ H L L ˆ 1 H L L + ˆ ] 1 ( E L L ˆ 1 E L R ˆ H L L ˆ 1 H L R ˆ ) L 22 ˆ = [ E L L ˆ 1 E L L + ˆ H L L ˆ 1 H L L + ˆ ] 1 ( E L L ˆ 1 E L R + ˆ H L L ˆ 1 H L R + ˆ ) } .
{ A L = ( I ̂ L 12 ˆ R 21 ˆ ) 1 L 11 ˆ A R + ( I ̂ L 12 ˆ R 21 ˆ ) 1 L 12 ˆ R 22 ˆ A L + A R + = ( I ̂ R 21 ˆ L 12 ˆ ) 1 R 21 ˆ L 11 ˆ A R + ( I ̂ R 21 ˆ L 12 ˆ ) 1 R 22 ˆ A L + } , ,
ε L j y z = ε R j y z ,
n 2 ε L j x x = ε R j x x ,
sin ( θ i ) = sin ( θ r ) = n sin ( θ t ) ,
A L ( m ) A L ( m ) + = k L z ( m ) k R z ( m ) k L z ( m ) + k R z ( m ) , A R ( m ) + A L ( m ) + = 2 k L z ( m ) k L z ( m ) + k R z ( m ) .
k y 2 + k z 2 ε d x x ( 1 + ε d y z | ε m | ) ω 2 c 2 ,

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