Abstract

A systematic analytical approach to simulate the propagation of electromagnetic plane waves in multilayer anisotropic structures, where the layers can have arbitrary oriented optical axis, is presented. The explicit expressions for the vector polarizations of electric and magnetic fields inside a randomly oriented anisotropic medium are derived. The developed algorithm operates with analytic 4×4 matrices to calculate the transmission and reflection coefficients. This algorithm is suitable to investigate the near-field/far-field electromagnetic wave interaction at any angle of incidence for numerous intriguing applications. The procedure is applied to design anisotropic single and multilayer lenses for subwavelength imaging.

© 2014 Optical Society of America

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  1. J. Diener, D. I. Kovalev, N. Kuenzner, E. Gross, G. Polisski, F. Koch, and M. Fujii, “Spatially nanostructured silicon for optical applications,” in Proceeding of Sixth International Conference on Material Science and Material Properties for Infrared Optoelectronics, Kiev, Ukraine (2003).
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  3. H. W. Park, T. Kim, J. Huh, M. Kang, J. E. Lee, and H. Yoon, “Anisotropic growth control of polyaniline nanostructures and their morphology-dependent electrochemical characteristics,” ACS Nano 6, 7624–7633 (2012).
    [CrossRef]
  4. Y. Hanazaki, J. Masumoto, S. Sato, K. Furusawa, A. Fukui, and N. Sasaki, “Multiscale analysis of changes in an anisotropic collagen gel structure by culturing osteoblasts,” ACS Appl. Mater. Interfaces 5, 5937–5946 (2013).
    [CrossRef]
  5. K. D. Costa, E. J. Lee, and J. W. Holmes, “Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system,” Tissue Eng. 9, 567–577 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013 (2)

Y. Hanazaki, J. Masumoto, S. Sato, K. Furusawa, A. Fukui, and N. Sasaki, “Multiscale analysis of changes in an anisotropic collagen gel structure by culturing osteoblasts,” ACS Appl. Mater. Interfaces 5, 5937–5946 (2013).
[CrossRef]

P. Ginzburg, F. J. Fortuño, G. A. Wurtz, and W. Dickson, “Manipulating polarization of light with ultrathin epsilon-near-zero metamaterials,” Opt. Express 21, 14907–14917 (2013).
[CrossRef]

2012 (2)

H. W. Park, T. Kim, J. Huh, M. Kang, J. E. Lee, and H. Yoon, “Anisotropic growth control of polyaniline nanostructures and their morphology-dependent electrochemical characteristics,” ACS Nano 6, 7624–7633 (2012).
[CrossRef]

D. Sun, J. Rioux, J. E. Sipe, Y. Zou, M. T. Mihnev, C. Berger, and T. B. Noris, “Evidence for interlayer electronic coupling in multilayer epitaxial graphene from polarization-dependent coherently controlled photocurrent generation,” Phys. Rev. B 85, 165427 (2012).
[CrossRef]

2011 (1)

S. Zhang, Y. Xiong, G. Bartal, X. Yin, and X. Zhang, “Magnetized plasma for reconfigurable subdiffraction imaging,” Phys. Rev. Lett. 106, 243901 (2011).
[CrossRef]

2010 (1)

2008 (2)

2007 (1)

2006 (1)

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

2005 (3)

2003 (1)

K. D. Costa, E. J. Lee, and J. W. Holmes, “Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system,” Tissue Eng. 9, 567–577 (2003).
[CrossRef]

2001 (1)

1998 (1)

I. V. Lindell and F. Olyslager, “Generalized decomposition of electromagnetic media,” IEEE Trans. Antennas Propag. 46, 1584–1585 (1998).
[CrossRef]

1996 (1)

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

1992 (1)

A. N. Parikh and D. L. Allara, “Quantitative determination of molecular structure in multilayered thin films of biaxial and lower symmetry from photon spectroscopies. I. Reflection infrared vibrational spectroscopy,” J. Chem. Phys. 96, 927–945 (1992).
[CrossRef]

1979 (1)

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. A 69, 742–755 (1979).
[CrossRef]

1977 (1)

1972 (2)

Allara, D. L.

A. N. Parikh and D. L. Allara, “Quantitative determination of molecular structure in multilayered thin films of biaxial and lower symmetry from photon spectroscopies. I. Reflection infrared vibrational spectroscopy,” J. Chem. Phys. 96, 927–945 (1992).
[CrossRef]

Babilotte, P.

Bartal, G.

S. Zhang, Y. Xiong, G. Bartal, X. Yin, and X. Zhang, “Magnetized plasma for reconfigurable subdiffraction imaging,” Phys. Rev. Lett. 106, 243901 (2011).
[CrossRef]

Berger, C.

D. Sun, J. Rioux, J. E. Sipe, Y. Zou, M. T. Mihnev, C. Berger, and T. B. Noris, “Evidence for interlayer electronic coupling in multilayer epitaxial graphene from polarization-dependent coherently controlled photocurrent generation,” Phys. Rev. B 85, 165427 (2012).
[CrossRef]

Berreman, D. W.

Braat, J. J.

Bria, D.

Costa, K. D.

K. D. Costa, E. J. Lee, and J. W. Holmes, “Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system,” Tissue Eng. 9, 567–577 (2003).
[CrossRef]

de Boer, D. K.

Dickson, W.

Diener, J.

J. Diener, D. I. Kovalev, N. Kuenzner, E. Gross, G. Polisski, F. Koch, and M. Fujii, “Spatially nanostructured silicon for optical applications,” in Proceeding of Sixth International Conference on Material Science and Material Properties for Infrared Optoelectronics, Kiev, Ukraine (2003).

Djafari-Rouhani, B.

Eichmann, G.

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]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub–diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

Farrell, R. A.

Fortuño, F. J.

Fujii, M.

J. Diener, D. I. Kovalev, N. Kuenzner, E. Gross, G. Polisski, F. Koch, and M. Fujii, “Spatially nanostructured silicon for optical applications,” in Proceeding of Sixth International Conference on Material Science and Material Properties for Infrared Optoelectronics, Kiev, Ukraine (2003).

Fukui, A.

Y. Hanazaki, J. Masumoto, S. Sato, K. Furusawa, A. Fukui, and N. Sasaki, “Multiscale analysis of changes in an anisotropic collagen gel structure by culturing osteoblasts,” ACS Appl. Mater. Interfaces 5, 5937–5946 (2013).
[CrossRef]

Furusawa, K.

Y. Hanazaki, J. Masumoto, S. Sato, K. Furusawa, A. Fukui, and N. Sasaki, “Multiscale analysis of changes in an anisotropic collagen gel structure by culturing osteoblasts,” ACS Appl. Mater. Interfaces 5, 5937–5946 (2013).
[CrossRef]

Ginzburg, P.

Gross, E.

J. Diener, D. I. Kovalev, N. Kuenzner, E. Gross, G. Polisski, F. Koch, and M. Fujii, “Spatially nanostructured silicon for optical applications,” in Proceeding of Sixth International Conference on Material Science and Material Properties for Infrared Optoelectronics, Kiev, Ukraine (2003).

Hanazaki, Y.

Y. Hanazaki, J. Masumoto, S. Sato, K. Furusawa, A. Fukui, and N. Sasaki, “Multiscale analysis of changes in an anisotropic collagen gel structure by culturing osteoblasts,” ACS Appl. Mater. Interfaces 5, 5937–5946 (2013).
[CrossRef]

Holmes, J. W.

K. D. Costa, E. J. Lee, and J. W. Holmes, “Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system,” Tissue Eng. 9, 567–577 (2003).
[CrossRef]

Huh, J.

H. W. Park, T. Kim, J. Huh, M. Kang, J. E. Lee, and H. Yoon, “Anisotropic growth control of polyaniline nanostructures and their morphology-dependent electrochemical characteristics,” ACS Nano 6, 7624–7633 (2012).
[CrossRef]

Jia, Y.

Kang, M.

H. W. Park, T. Kim, J. Huh, M. Kang, J. E. Lee, and H. Yoon, “Anisotropic growth control of polyaniline nanostructures and their morphology-dependent electrochemical characteristics,” ACS Nano 6, 7624–7633 (2012).
[CrossRef]

Kim, T.

H. W. Park, T. Kim, J. Huh, M. Kang, J. E. Lee, and H. Yoon, “Anisotropic growth control of polyaniline nanostructures and their morphology-dependent electrochemical characteristics,” ACS Nano 6, 7624–7633 (2012).
[CrossRef]

Koch, F.

J. Diener, D. I. Kovalev, N. Kuenzner, E. Gross, G. Polisski, F. Koch, and M. Fujii, “Spatially nanostructured silicon for optical applications,” in Proceeding of Sixth International Conference on Material Science and Material Properties for Infrared Optoelectronics, Kiev, Ukraine (2003).

Kovalev, D. I.

J. Diener, D. I. Kovalev, N. Kuenzner, E. Gross, G. Polisski, F. Koch, and M. Fujii, “Spatially nanostructured silicon for optical applications,” in Proceeding of Sixth International Conference on Material Science and Material Properties for Infrared Optoelectronics, Kiev, Ukraine (2003).

Kuenzner, N.

J. Diener, D. I. Kovalev, N. Kuenzner, E. Gross, G. Polisski, F. Koch, and M. Fujii, “Spatially nanostructured silicon for optical applications,” in Proceeding of Sixth International Conference on Material Science and Material Properties for Infrared Optoelectronics, Kiev, Ukraine (2003).

Lee, E. J.

K. D. Costa, E. J. Lee, and J. W. Holmes, “Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system,” Tissue Eng. 9, 567–577 (2003).
[CrossRef]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub–diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

Lee, J. E.

H. W. Park, T. Kim, J. Huh, M. Kang, J. E. Lee, and H. Yoon, “Anisotropic growth control of polyaniline nanostructures and their morphology-dependent electrochemical characteristics,” ACS Nano 6, 7624–7633 (2012).
[CrossRef]

Lindell, I. V.

I. V. Lindell and F. Olyslager, “Generalized decomposition of electromagnetic media,” IEEE Trans. Antennas Propag. 46, 1584–1585 (1998).
[CrossRef]

Liu, H.

Masumoto, J.

Y. Hanazaki, J. Masumoto, S. Sato, K. Furusawa, A. Fukui, and N. Sasaki, “Multiscale analysis of changes in an anisotropic collagen gel structure by culturing osteoblasts,” ACS Appl. Mater. Interfaces 5, 5937–5946 (2013).
[CrossRef]

McCally, R. L.

Mihnev, M. T.

D. Sun, J. Rioux, J. E. Sipe, Y. Zou, M. T. Mihnev, C. Berger, and T. B. Noris, “Evidence for interlayer electronic coupling in multilayer epitaxial graphene from polarization-dependent coherently controlled photocurrent generation,” Phys. Rev. B 85, 165427 (2012).
[CrossRef]

Mounier, D.

Nguyen, Q. D.

Q. D. Nguyen, “Electrochemistry in anisotropic etching of silicon in alkaline solutions: a kinetic wave analysis,” Ph.D. thesis (University of Twente, 2007).

Noris, T. B.

D. Sun, J. Rioux, J. E. Sipe, Y. Zou, M. T. Mihnev, C. Berger, and T. B. Noris, “Evidence for interlayer electronic coupling in multilayer epitaxial graphene from polarization-dependent coherently controlled photocurrent generation,” Phys. Rev. B 85, 165427 (2012).
[CrossRef]

Nougaoui, A.

Olyslager, F.

I. V. Lindell and F. Olyslager, “Generalized decomposition of electromagnetic media,” IEEE Trans. Antennas Propag. 46, 1584–1585 (1998).
[CrossRef]

Ouchani, N.

Parikh, A. N.

A. N. Parikh and D. L. Allara, “Quantitative determination of molecular structure in multilayered thin films of biaxial and lower symmetry from photon spectroscopies. I. Reflection infrared vibrational spectroscopy,” J. Chem. Phys. 96, 927–945 (1992).
[CrossRef]

Park, H. W.

H. W. Park, T. Kim, J. Huh, M. Kang, J. E. Lee, and H. Yoon, “Anisotropic growth control of polyaniline nanostructures and their morphology-dependent electrochemical characteristics,” ACS Nano 6, 7624–7633 (2012).
[CrossRef]

Picart, P.

Polisski, G.

J. Diener, D. I. Kovalev, N. Kuenzner, E. Gross, G. Polisski, F. Koch, and M. Fujii, “Spatially nanostructured silicon for optical applications,” in Proceeding of Sixth International Conference on Material Science and Material Properties for Infrared Optoelectronics, Kiev, Ukraine (2003).

Rioux, J.

D. Sun, J. Rioux, J. E. Sipe, Y. Zou, M. T. Mihnev, C. Berger, and T. B. Noris, “Evidence for interlayer electronic coupling in multilayer epitaxial graphene from polarization-dependent coherently controlled photocurrent generation,” Phys. Rev. B 85, 165427 (2012).
[CrossRef]

Rouseff, D.

Ruello, P.

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]

Sasaki, N.

Y. Hanazaki, J. Masumoto, S. Sato, K. Furusawa, A. Fukui, and N. Sasaki, “Multiscale analysis of changes in an anisotropic collagen gel structure by culturing osteoblasts,” ACS Appl. Mater. Interfaces 5, 5937–5946 (2013).
[CrossRef]

Sato, S.

Y. Hanazaki, J. Masumoto, S. Sato, K. Furusawa, A. Fukui, and N. Sasaki, “Multiscale analysis of changes in an anisotropic collagen gel structure by culturing osteoblasts,” ACS Appl. Mater. Interfaces 5, 5937–5946 (2013).
[CrossRef]

Schesser, J.

Schubert, M.

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

Sherman, G. C.

Sipe, J. E.

D. Sun, J. Rioux, J. E. Sipe, Y. Zou, M. T. Mihnev, C. Berger, and T. B. Noris, “Evidence for interlayer electronic coupling in multilayer epitaxial graphene from polarization-dependent coherently controlled photocurrent generation,” Phys. Rev. B 85, 165427 (2012).
[CrossRef]

Sithambaranathan, G. S.

Sluijter, M.

Stamnes, J.

Stamnes, J. J.

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub–diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

Sun, D.

D. Sun, J. Rioux, J. E. Sipe, Y. Zou, M. T. Mihnev, C. Berger, and T. B. Noris, “Evidence for interlayer electronic coupling in multilayer epitaxial graphene from polarization-dependent coherently controlled photocurrent generation,” Phys. Rev. B 85, 165427 (2012).
[CrossRef]

Webb, K. J.

Wurtz, G. A.

Xiong, Y.

S. Zhang, Y. Xiong, G. Bartal, X. Yin, and X. Zhang, “Magnetized plasma for reconfigurable subdiffraction imaging,” Phys. Rev. Lett. 106, 243901 (2011).
[CrossRef]

Yang, X.

Yeh, P.

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. A 69, 742–755 (1979).
[CrossRef]

Yin, X.

S. Zhang, Y. Xiong, G. Bartal, X. Yin, and X. Zhang, “Magnetized plasma for reconfigurable subdiffraction imaging,” Phys. Rev. Lett. 106, 243901 (2011).
[CrossRef]

Yoon, H.

H. W. Park, T. Kim, J. Huh, M. Kang, J. E. Lee, and H. Yoon, “Anisotropic growth control of polyaniline nanostructures and their morphology-dependent electrochemical characteristics,” ACS Nano 6, 7624–7633 (2012).
[CrossRef]

Zhang, S.

S. Zhang, Y. Xiong, G. Bartal, X. Yin, and X. Zhang, “Magnetized plasma for reconfigurable subdiffraction imaging,” Phys. Rev. Lett. 106, 243901 (2011).
[CrossRef]

Zhang, T.

Zhang, X.

S. Zhang, Y. Xiong, G. Bartal, X. Yin, and X. Zhang, “Magnetized plasma for reconfigurable subdiffraction imaging,” Phys. Rev. Lett. 106, 243901 (2011).
[CrossRef]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub–diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

Zhao, Y.

Zhu, X.

Zou, Y.

D. Sun, J. Rioux, J. E. Sipe, Y. Zou, M. T. Mihnev, C. Berger, and T. B. Noris, “Evidence for interlayer electronic coupling in multilayer epitaxial graphene from polarization-dependent coherently controlled photocurrent generation,” Phys. Rev. B 85, 165427 (2012).
[CrossRef]

ACS Appl. Mater. Interfaces (1)

Y. Hanazaki, J. Masumoto, S. Sato, K. Furusawa, A. Fukui, and N. Sasaki, “Multiscale analysis of changes in an anisotropic collagen gel structure by culturing osteoblasts,” ACS Appl. Mater. Interfaces 5, 5937–5946 (2013).
[CrossRef]

ACS Nano (1)

H. W. Park, T. Kim, J. Huh, M. Kang, J. E. Lee, and H. Yoon, “Anisotropic growth control of polyaniline nanostructures and their morphology-dependent electrochemical characteristics,” ACS Nano 6, 7624–7633 (2012).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

I. V. Lindell and F. Olyslager, “Generalized decomposition of electromagnetic media,” IEEE Trans. Antennas Propag. 46, 1584–1585 (1998).
[CrossRef]

J. Chem. Phys. (1)

A. N. Parikh and D. L. Allara, “Quantitative determination of molecular structure in multilayered thin films of biaxial and lower symmetry from photon spectroscopies. I. Reflection infrared vibrational spectroscopy,” J. Chem. Phys. 96, 927–945 (1992).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (6)

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (3)

D. Sun, J. Rioux, J. E. Sipe, Y. Zou, M. T. Mihnev, C. Berger, and T. B. Noris, “Evidence for interlayer electronic coupling in multilayer epitaxial graphene from polarization-dependent coherently controlled photocurrent generation,” Phys. Rev. B 85, 165427 (2012).
[CrossRef]

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

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

Phys. Rev. Lett. (1)

S. Zhang, Y. Xiong, G. Bartal, X. Yin, and X. Zhang, “Magnetized plasma for reconfigurable subdiffraction imaging,” Phys. Rev. Lett. 106, 243901 (2011).
[CrossRef]

Science (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub–diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

Tissue Eng. (1)

K. D. Costa, E. J. Lee, and J. W. Holmes, “Creating alignment and anisotropy in engineered heart tissue: role of boundary conditions in a model three-dimensional culture system,” Tissue Eng. 9, 567–577 (2003).
[CrossRef]

Other (2)

J. Diener, D. I. Kovalev, N. Kuenzner, E. Gross, G. Polisski, F. Koch, and M. Fujii, “Spatially nanostructured silicon for optical applications,” in Proceeding of Sixth International Conference on Material Science and Material Properties for Infrared Optoelectronics, Kiev, Ukraine (2003).

Q. D. Nguyen, “Electrochemistry in anisotropic etching of silicon in alkaline solutions: a kinetic wave analysis,” Ph.D. thesis (University of Twente, 2007).

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Figures (10)

Fig. 1.
Fig. 1.

Incidence, reflectance, and transmittance of a plane wave.

Fig. 2.
Fig. 2.

Flowchart of transfer matrix development for the lth layer.

Fig. 3.
Fig. 3.

Transfer functions for lossy and optically active anisotropic slab.

Fig. 4.
Fig. 4.

Subwavelength imaging with an anisotropic slab εxx=εyy=1j·02εzz=1, d=400nm, λ=700nm.

Fig. 5.
Fig. 5.

Transfer function for layered anisotropic–isotropic lens, the specifications of the layered strictures are provided in Table 1.

Fig. 6.
Fig. 6.

Subwavelength imaging with layered anisotropic–isotropic lens.

Fig. 7.
Fig. 7.

Normal component of k vector as function of kx and εg. (a) Gyrotropic dielectric function and (b) diagonally anisotropic dielectric function.

Fig. 8.
Fig. 8.

Transfer function of the seven-layer thick superlens for different values of the off-diagonal tensor element εg.

Fig. 9.
Fig. 9.

Subwavelength imaging with layered gyrotropic-isotropic lens.

Fig. 10.
Fig. 10.

Imaging with gyrotropic layered superlens varying the diagonal elements of the dielectric tensor.

Tables (2)

Tables Icon

Table 1. Specifications of Five-Layer and Seven-Layer Superlens

Tables Icon

Table 2. Specifications of Layered Superlens

Equations (39)

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

×H=jωD,
×E=jωμH,
{Hyz=jωDxHxzHzx=jωDyHyx=jωDz.
kz(i)Hy=ω(εxxEx+εxyEy+εxzEz),
kxHy=ω(εzxEx+εzyEy+εzzEz).
E⃗=[1αγβλξγλξ]Ex,
βl=εxzkz(i)+εzzkx,
αl=εxxkz(i)+εzxkx,
γl=εxykz(i)+εzykx,
λl=kxεxx+kz(i)εxz(εxyεyz)αlγl,
ξl=kxεzx+kz(i)εzz+(εxyεzy)βlγl,
H⃗=×Ejωμ=[kz(i)ωμ(αγβλξγ)1μω(kz(i)kxλξ)kxωμ(αγβλξγ)]Ex.
[Ai.TMBr,TMAi,TEBr,TE]=TF[At,TM0At,TE0],
TF=BC01(i=1NTl(dl,ε¯¯l)BCN,
[HylExlEylHxl]=BCl[Hl1Hl2Hl3Hl4],
BCl=[A1,lA2,lA3,lA4,l1111B1,lB2,lB3,lB4,lC1,lC2,lC3,lC4,l],
Ai,l=kz(i)ω[εxx+εxy[βlλlγlξlαlγl]λlξlεxz,
Bi,l=(βlλlγlξlαlγl)Ai,l,
Ci,l=kz(i)ωμBi,l.
[Hl1Hl2Hl3Hl4]=Pl[Hl1Hl2H3lHl4],
Pl=[eikzl,1dl0000eikzl,2dl0000eikzl,3dl0000eikzl,4dl].
[Hyl1Exl1Eyl1Hxl1]=Tl[Hyl+1Exl+1Eyl+1Hxl+1].
Tl=BClPlBCl1.
T=l=1NTl.
BC0=[1100kz0ωε0kz0ωε000001100kz0ωμ0kz0ωμ0],
BCN=[1100kzNωεNkzNωεN00001100kzNωμNkzNωμN].
ε¯¯=[εxx000εyy000εzz].
kz,1=kz,2=k02εxxεxxεzzkx2,
kz,3=kz,4=εyyk02kx2.
TTM=Ct,TMAi,TM=T33T11T33T13T31.
ε=[εxxiεg0iεgεxx000εzz],
kz,1=kz,2(εxx+εzz)kx22εxxεzzk02+(εxxkx2εzzkxx2)2+4εzzεg2(εzzk02kx2)2εzz,
kz,3=kz,42εxxεzzk02(εxx+εzz)kx2+(εxxkx2εzzkx2)2+4εzzεg2(εzzk02kx2)2εzz,
[HyExEyHx]=[1kzlωεl00ωεlkzl100001ωμlkzl00kzlωμl1][HyExEyHx].
BCl,TM=[11kzlωεlkzlωεl],
BCl,TE=[11kzlωμlkzlωμl],
Pl,TM=Pl,TE=[eikzldl00eikzldl].
Tl,iso=BCl.isoPl,isoBCl,iso1,
Tl,iso=[cos(kzldl)iωεlkzlsin(kzldl)00ikzlωεlsin(kzldl)cos(kzldl)0000cos(kzldl)iωμlkzlsin(kzldl)00ikzlωμlsin(kzldl)cos(kzldl)].

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