Abstract

Transfer matrices for one-dimensional (1-D) multi-layered magneto-optical (MO) waveguides are formulated to analytically calculate the nonreciprocal phase shifts (NRPS). The Cauchy contour integration (CCI) method is introduced in detail to calculate the two complex roots of the transcendental equation corresponding to backward and forward waves. By virtue of perturbation theory and the variational principle, we also present the general upper limit of NRPSs in 1-D MO waveguides. These analytical results are applied to compare silicon-on-insulator (SOI) based MO waveguides. First, a three-layered waveguide system with MO medium is briefly examined and discussed to check the validity and efficiency of the above theory. Then we revisited the reported low-index-gap-enhanced NRPSs in MO waveguides and obtained substantially different results. Finally, the potential of common plasmonic waveguides to enhance the nonreciprocal effect is investigated by studying different waveguides composed of Metal, MO medium and dielectrics. Our study shows that the reasonable NRPSs can be optimized to some extent but not as much as claimed in previous publications.

© 2012 OSA

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2012

P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon.6, 16–24 (2012).
[CrossRef]

2011

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett.98, 171109 (2011).
[CrossRef]

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon.5, 758–762 (2011).
[CrossRef]

M. C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. Bowers, “Silicon ring resonators with bonded nonreciprocal magneto-optic garnets,” Opt. Express19, 11740–11745 (2011), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-639 .
[CrossRef] [PubMed]

2010

2009

2007

2006

2005

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72, 075405 (2005).
[CrossRef]

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B22, 240–253 (2005).
[CrossRef]

2004

1999

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett.74, 2918–2920 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol.17, 929–941 (1999).
[CrossRef]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron.35, 250–253 (1999).
[CrossRef]

1998

A. F. Popkov, “Nonreciprocal TE-mode phase shift by domain walls in magnetooptic rib waveguides,” Appl. Phys. Lett.72, 2508–2510 (1998).
[CrossRef]

1994

Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl.2, 88–92 (1994).

1992

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn.28, 2979–2984 (1992).
[CrossRef]

1981

Z. K. Wang, “An implementation of Kuhn’s rootfinding algorithm for polynomials and related discussion,” J. Numer. Methods Comput. Appl.3, 175–181 (1981).

1974

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys.45, 882–888 (1974).
[CrossRef]

1965

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J.7, 308–313 (1965).

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, 1980).

Anemogiannis, E.

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72, 075405 (2005).
[CrossRef]

Avrutsky, I.

Ayache, M.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

Bahlmann, N.

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B22, 240–253 (2005).
[CrossRef]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron.35, 250–253 (1999).
[CrossRef]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett.74, 2918–2920 (1999).
[CrossRef]

Berini, P.

P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon.6, 16–24 (2012).
[CrossRef]

Bi, L.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon.5, 758–762 (2011).
[CrossRef]

Bowers, J.

Buchwald, W.

Chen, R. Y.

Chen, Y. F.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

Dionne, G. F.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon.5, 758–762 (2011).
[CrossRef]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72, 075405 (2005).
[CrossRef]

Dotsch, H.

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B22, 240–253 (2005).
[CrossRef]

R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood, and H. Dotsch, “Magneto-optical noreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett.29, 941–943 (2004).
[CrossRef] [PubMed]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron.35, 250–253 (1999).
[CrossRef]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett.74, 2918–2920 (1999).
[CrossRef]

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn.28, 2979–2984 (1992).
[CrossRef]

Espinola, R. L.

Fainman, Y.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

Fan, S. H.

Z. F. Yu and S. H. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
[CrossRef]

Fehndrich, M.

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett.74, 2918–2920 (1999).
[CrossRef]

Feng, L.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

Gaylord, T. K.

Gerhardt, R.

Glytsis, E. N.

Hammer, M.

Hao, Y. L.

Hertel, P.

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B22, 240–253 (2005).
[CrossRef]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron.35, 250–253 (1999).
[CrossRef]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett.74, 2918–2920 (1999).
[CrossRef]

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn.28, 2979–2984 (1992).
[CrossRef]

Hu, J.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon.5, 758–762 (2011).
[CrossRef]

Huang, J. Q.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

Huang, W. P.

W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett.98, 171109 (2011).
[CrossRef]

Izuhara, T.

Jiang, G. M.

Jiang, P.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon.5, 758–762 (2011).
[CrossRef]

Jiang, X. Q.

Josef, A.

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett.74, 2918–2920 (1999).
[CrossRef]

Kakihara, K.

Khurgin, J. B.

J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett.89, 251115 (2006).
[CrossRef]

Kim, D

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon.5, 758–762 (2011).
[CrossRef]

Kimerling, L. C.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon.5, 758–762 (2011).
[CrossRef]

Kleine-Borger, J.

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett.74, 2918–2920 (1999).
[CrossRef]

Kono, N.

Koshiba, M.

Kromer, H.

Leon, I. D.

P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon.6, 16–24 (2012).
[CrossRef]

Lohmeyer, M.

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett.74, 2918–2920 (1999).
[CrossRef]

N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron.35, 250–253 (1999).
[CrossRef]

Long, Y. L.

Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl.2, 88–92 (1994).

Lu, M. H.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

Luhrmann, B.

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn.28, 2979–2984 (1992).
[CrossRef]

Makimoto, T.

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys.45, 882–888 (1974).
[CrossRef]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J.7, 308–313 (1965).

Mizumoto, T.

Mu, J. W.

W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett.98, 171109 (2011).
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J.7, 308–313 (1965).

Osgood, R. M.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

Pintus, P.

Polman, A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72, 075405 (2005).
[CrossRef]

Popkov, A. F.

Ross, C. A.

L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon.5, 758–762 (2011).
[CrossRef]

Saitoh, K.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer-Verlag, 2004).

Scherer, A.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

Shoji, Y.

Soref, R.

Sure, S.

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn.28, 2979–2984 (1992).
[CrossRef]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72, 075405 (2005).
[CrossRef]

Tien, M. C.

Tsai, M. C.

Wang, M. H.

Wang, Z. K.

Z. K. Wang, “An implementation of Kuhn’s rootfinding algorithm for polynomials and related discussion,” J. Numer. Methods Comput. Appl.3, 175–181 (1981).

Wen, X. L.

Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl.2, 88–92 (1994).

Wilkens, L.

H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B22, 240–253 (2005).
[CrossRef]

M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett.74, 2918–2920 (1999).
[CrossRef]

Winkler, H. P.

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn.28, 2979–2984 (1992).
[CrossRef]

Xie, C. F.

Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl.2, 88–92 (1994).

Xu, Y. L.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

Yamamoto, S.

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys.45, 882–888 (1974).
[CrossRef]

Yang, J. Y.

Ye, M.

H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn.28, 2979–2984 (1992).
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Figures (6)

Fig. 1
Fig. 1

Schematic of a multi-layered slab waveguide system composed of N + 1 isotropic layers, in which magnetooptical medium can be included. The propagation direction z points into the paper and the waveguide is infinite in y.

Fig. 2
Fig. 2

Nonreciprocal phase shifts in three-layered SOI-based waveguides, where the MO medium (Ce:YIG) is placed (a) at one side and (b) in the center, respectively. The calculated Δγ by Eq. (30) is also shown for symmetric MO(+)/Si/MO(−) waveguide in (a), where γ0 is numerically prepared by 1-D finite difference frequency domain (FDFD) method.

Fig. 3
Fig. 3

(a) The dependence of NRPS in a Ce:YIG/Air/Si/SiO2 waveguide on the structural parameters and (b) the field distributions of Ex (in the lower half plane) and Hy (upper) with increasing air gap width and a fixed silicon thickness of 200nm.

Fig. 4
Fig. 4

The comparison of NRPSs. The soild line is for the dependence of analytical NRPSs to the MO layer width in a 1-D Air/180-nm Si/x-nm MO(+)/x-nm MO(−)/180-nm Si/Air waveguide,where the two silicon layer width is 180nm. The green line corresponds to the Fig. 4. in [8], where the waveguide height is 500nm and the width of the slotted MO layer is 2x nm. The red line is the same as Fig.5. in [5], where the rib height h is 200nm and the rib width is 2x nm.

Fig. 5
Fig. 5

The field distributions of the Ex and Hy mode components in different waveguide structures, including (a) 1-D Air/180-nm Si/MO(+)/MO(−)/180-nm Si/Air waveguides with slot widths of 30nm and 1200nm; 2-D Si/MO/Si slotted waveguides with x=15nm in (b) and x=600nm in (c); and (d) a 2-D Ce:YIG/GGG rib waveguide. The left and right columns in (b–d) are for the Ex and Hy components, respectively.

Fig. 6
Fig. 6

The characteristic length for (a) π–NRPS and (b) 1-dB propagation loss and (c) the field distributions, of the dominant modes in several typical 1-D plasmonic waveguides. The field components are for Ex component that was normalized to their maximums. The modes are distinguished in color in all panels. The field distributions are calculated with a sandwiched layer thickness of 10nm.

Equations (35)

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× E = j 2 π ν H
× H = j 2 π ν ε r E ,
ε r = ( ε x x j ε x y j ε x z j ε x y ε y y j ε y z j ε x z j ε y z ε z z ) .
z E x x E z = j 2 π ν H y
( z H y x H y ) = j 2 π ν ( ε x x j ε x z j ε x z ε z z ) ( E x E z )
( E x E z ) = 1 j 2 π ν 1 ε x x ε z z ε x z 2 ( ε z z j ε x z j ε x z ε x x ) ( z H y x H y )
x 2 H y ( 2 π ν ) 2 [ ε z z ε x x γ 2 ( ε z z ε x z 2 ε x x ) ] H y = 0.
κ = + ε z z ε x x γ 2 ε e .
H y ( n ) = { A n exp [ 2 π κ n ( x x n ) ] + B n exp [ 2 π κ n ( x x n ) ] } e i 2 π ν γ z
E z = 1 j ε e ( 1 2 π ν x H y γ ε x z ε x x H y )
f ( n ) ( d n ) = f ( n + 1 ) ( 0 ) ,
f ( n ) ( x ) = ( 1 1 1 ε e ( κ ε x z γ ε x x ) 1 ε e ( κ ε x z γ ε x x ) ) ( A exp ( 2 π ν κ x ) B exp ( + 2 π ν κ x ) ) ( n ) ,
( A ( n + 1 ) B ( n + 1 ) ) = S n ( A ( n ) B ( n ) )
S n = 1 2 ( ( 1 + a b ) exp ( ϕ n ) ( 1 a b ) exp ( ϕ n ) ( 1 a + b ) exp ( ϕ n ) ( 1 + a + b ) exp ( ϕ n ) )
a n , n + 1 = ε e ( n + 1 ) κ n ε e ( n ) κ n + 1
b n , n + 1 = ( ε x z ( n + 1 ) ε x x ( n + 1 ) ε e ( n + 1 ) ε e ( n ) ε x z ( n ) ε x x ( n ) ) γ κ n + 1 ,
( A N 0 ) = S T ( 1 B 0 ) .
N = 1 2 π i C f ( z ) f ( z ) d z
n = 1 N z n k = 1 2 π i C z k f ( z ) f ( z ) d z ,
Δ γ = ν 2 E * Δ ε r E d x d y ,
S = 1 4 ( E × H * + E * × H ) z d x d y .
Δ γ 2 γ 2 Re ε x z E x * 1 ε e x ε e E x d x d y [ | E x 2 | 1 2 π ν γ E x x ( 1 ε e x ε e E x * ) ] d x d y
Δ γ 2 2 π ν ε x z ε x x 2 H y * x H y d x d y
| Δ γ | 1 π ν max | ε x z ε x x 2 | M O | H y x H y d x |
γ 2 = [ H y 2 ε ( x ) ( 1 2 π ν x H y ) 2 ] d x
H y x H y d x 2 π ν H y 2 ε ( x ) d x γ 2
| Δ γ | 2 max ( ε x z ε x x 2 ) ε m n 2 2
exp ( 4 π ν κ 1 d 1 ) = 1 ( a 01 + b 01 ) 1 + ( a 01 + b 01 ) 1 ( a 21 b 21 ) 1 + ( a 21 b 21 )
2 π ν i κ 1 d 1 = atan [ i ( a 01 + b 01 ) ] + atan [ i ( a 21 b 21 ) ] .
Δ γ 2 | c 01 b 01 + c 21 b 21 c 01 γ a 01 + c 21 γ a 21 + 2 π ν γ 0 d 1 κ 1 |
d 1 = 1 2 π ν ε 1 ε 2 atan ( ε 1 ε 2 ε 2 ε 0 ε 1 ε 2 ) .
Δ γ h [ ε x z ( x h , y ) ε x x 2 ( x h , y ) | H y ( x h , y ) | 2 ε x z ( x h + , y ) ε x x 2 ( x h + , y ) | H y ( x h + , y ) | 2 ] d y ,
20 log 10 exp [ 2 π ν Im ( γ ) L 1 d B ] = 1
ε z z ε x x γ 2 ε e + ε x z ε x x γ = ε e ε m γ 2 ε m ,
Δ γ ε x z ε e ε m 2 ε m 2 ε e 2 γ spp 2 ε e .

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