Abstract

A multilayer approach (MA) and modified boundary conditions (MBC) are proposed as fast and efficient numerical methods for the design of 1D photonic structures with rough interfaces. These methods are applicable for the structures, composed of materials with an arbitrary permittivity tensor. MA and MBC are numerically validated on different types of interface roughness and permittivities of the constituent materials. The proposed methods can be combined with the 4x4 scattering matrix method as a field solver and an evolutionary strategy as an optimizer. The resulted optimization procedure is fast, accurate, numerically stable and can be used to design structures for various applications.

© 2011 Optical Society of America

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References

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  1. J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic crystals: molding the flow of light (Princeton Univ Pr, 2008).
  2. A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
    [CrossRef] [PubMed]
  3. V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. 5, 862 (2008).
  4. D. Berreman, “Optics in stratified and anisotropic media: 4× 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  5. P. Yeh, Optical waves in layered media (Wiley New York, 1988).
  6. I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646 (1999).
    [CrossRef]
  7. F. K¨artner, N. Matuschek, T. Schibli, U. Keller, H. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, and T. Tschudi, “Design and fabrication of double-chirped mirrors,” Opt. Lett. 22, 831–833 (1997).
    [CrossRef] [PubMed]
  8. T. Yonte, J. Monz’on, A. Felipe, and L. S’anchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A, Pure Appl. Opt. 6, 127–131 (2004).
    [CrossRef]
  9. M. del Rio and G. Pareschi, “Global optimization and reflectivity data fitting for x-ray multilayer mirrors by means of genetic algorithms,” in “Proceedings of SPIE,”, vol. 4145 (2001), vol. 4145, p. 88.
    [CrossRef]
  10. S. Martin, J. Rivory, and M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34, 2247–2254 (1995).
    [CrossRef] [PubMed]
  11. D. Bose, E. McCorkle, C. Thompson, D. Bogdanoff, D. Prabhu, G. Allen, and J. Grinstead, “Analysis and model validation of shock layer radiation in air,” VKI LS Course on hypersonic entry and cruise vehicles, Palo Alto, California, USA (2008).
  12. D. Gerace and L. Andreani, “Low-loss guided modes in photonic crystal waveguides,” Opt. Express 13, 4939–4951 (2005).
    [CrossRef] [PubMed]
  13. P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  14. I. Ohlídal, “Approximate formulas for the reflectance, transmittance, and scattering losses of nonabsorbing multilayer systems with randomly rough boundaries,” J. Opt. Soc. Am. A 10, 158–171 (1993).
    [CrossRef]
  15. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. 13, 1024–1035 (1996).
    [CrossRef]
  16. D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
    [CrossRef]
  17. S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
    [CrossRef]
  18. A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. 56 (2008).
    [CrossRef]
  19. C. Hafner, C. Xudong, J. Smajic, and R. Vahldieck, “Efficient procedures for the optimization of defects in photonic crystal structures,” J. Opt. Soc. Am. A 24, 1177–1188 (2007).
    [CrossRef]
  20. J. Fröhlich, “Evolutionary optimization for computational electromagnetics,” Ph.D. thesis, ETH Zurich, IFH Laboratory (1997).
  21. O. Wiener, “Die Théorie desMischkörpers für das Feld des stationären Strömung. Abh.Math,” Physichen Klasse Königl. Säcsh. GeselWissen 32, 509–604 (1912).
    [PubMed]
  22. M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. 7, 1623–1630 (2010).
    [CrossRef]

2010 (1)

M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. 7, 1623–1630 (2010).
[CrossRef]

2008 (2)

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. 56 (2008).
[CrossRef]

V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. 5, 862 (2008).

2007 (1)

2005 (1)

2004 (1)

T. Yonte, J. Monz’on, A. Felipe, and L. S’anchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A, Pure Appl. Opt. 6, 127–131 (2004).
[CrossRef]

2002 (1)

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

1999 (2)

D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646 (1999).
[CrossRef]

1997 (1)

1996 (2)

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. 13, 1024–1035 (1996).
[CrossRef]

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

1995 (1)

1993 (1)

1981 (1)

1972 (1)

1912 (1)

O. Wiener, “Die Théorie desMischkörpers für das Feld des stationären Strömung. Abh.Math,” Physichen Klasse Königl. Säcsh. GeselWissen 32, 509–604 (1912).
[PubMed]

Abdulhalim, I.

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646 (1999).
[CrossRef]

Andreani, L.

Berreman, D.

Bousquet, P.

Braginsky, L.

M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. 7, 1623–1630 (2010).
[CrossRef]

V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. 5, 862 (2008).

Chen, J.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Culshaw, I.

D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

Fallahi, A.

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. 56 (2008).
[CrossRef]

Fan, S.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Felipe, A.

T. Yonte, J. Monz’on, A. Felipe, and L. S’anchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A, Pure Appl. Opt. 6, 127–131 (2004).
[CrossRef]

Flory, F.

Gerace, D.

Gippius, N.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Hafner, C.

M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. 7, 1623–1630 (2010).
[CrossRef]

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. 56 (2008).
[CrossRef]

V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. 5, 862 (2008).

C. Hafner, C. Xudong, J. Smajic, and R. Vahldieck, “Efficient procedures for the optimization of defects in photonic crystal structures,” J. Opt. Soc. Am. A 24, 1177–1188 (2007).
[CrossRef]

Haus, H.

Heine, C.

Ishihara, T.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Joannopoulos, J.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

K¨artner, F.

Keller, U.

Kurland, I.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Li, L.

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. 13, 1024–1035 (1996).
[CrossRef]

Martin, S.

Matuschek, N.

Mekis, A.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Mishrikey, M.

M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. 7, 1623–1630 (2010).
[CrossRef]

V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. 5, 862 (2008).

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. 56 (2008).
[CrossRef]

Monz’on, J.

T. Yonte, J. Monz’on, A. Felipe, and L. S’anchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A, Pure Appl. Opt. 6, 127–131 (2004).
[CrossRef]

Morf, R.

Muljarov, E.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Ohlídal, I.

Rivory, J.

Roche, P.

S’anchez-Soto, L.

T. Yonte, J. Monz’on, A. Felipe, and L. S’anchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A, Pure Appl. Opt. 6, 127–131 (2004).
[CrossRef]

Scheuer, V.

Schibli, T.

Schoenauer, M.

Shklover, V.

V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. 5, 862 (2008).

Smajic, J.

Tikhodeev, S.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Tilsch, M.

Tschudi, T.

Vahldieck, R.

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. 56 (2008).
[CrossRef]

C. Hafner, C. Xudong, J. Smajic, and R. Vahldieck, “Efficient procedures for the optimization of defects in photonic crystal structures,” J. Opt. Soc. Am. A 24, 1177–1188 (2007).
[CrossRef]

Villeneuve, P.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Whittaker, D.

D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

Wiener, O.

O. Wiener, “Die Théorie desMischkörpers für das Feld des stationären Strömung. Abh.Math,” Physichen Klasse Königl. Säcsh. GeselWissen 32, 509–604 (1912).
[PubMed]

Witz, G.

V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. 5, 862 (2008).

Xudong, C.

Yablonskii, A.

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Yonte, T.

T. Yonte, J. Monz’on, A. Felipe, and L. S’anchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A, Pure Appl. Opt. 6, 127–131 (2004).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Ant. Prop. (1)

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Ant. Prop. 56 (2008).
[CrossRef]

J. Comput. Theor. Nanosci. (1)

M. Mishrikey, L. Braginsky, and C. Hafner, “Light propagation in multilayered photonic structures,” J. Comput. Theor. Nanosci. 7, 1623–1630 (2010).
[CrossRef]

J. Comput. Theoret. Nanosci. (1)

V. Shklover, L. Braginsky, G. Witz, M. Mishrikey, and C. Hafner, “High-Temperature Photonic Structures. Thermal Barrier Coatings, Infrared Sources and Other Applications,” J. Comput. Theoret. Nanosci. 5, 862 (2008).

J. Opt. A, Pure Appl. Opt. (2)

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646 (1999).
[CrossRef]

T. Yonte, J. Monz’on, A. Felipe, and L. S’anchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A, Pure Appl. Opt. 6, 127–131 (2004).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (2)

D. Whittaker and I. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 45102 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Physichen Klasse Königl. Säcsh. Gesel (1)

O. Wiener, “Die Théorie desMischkörpers für das Feld des stationären Strömung. Abh.Math,” Physichen Klasse Königl. Säcsh. GeselWissen 32, 509–604 (1912).
[PubMed]

Other (5)

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic crystals: molding the flow of light (Princeton Univ Pr, 2008).

P. Yeh, Optical waves in layered media (Wiley New York, 1988).

M. del Rio and G. Pareschi, “Global optimization and reflectivity data fitting for x-ray multilayer mirrors by means of genetic algorithms,” in “Proceedings of SPIE,”, vol. 4145 (2001), vol. 4145, p. 88.
[CrossRef]

J. Fröhlich, “Evolutionary optimization for computational electromagnetics,” Ph.D. thesis, ETH Zurich, IFH Laboratory (1997).

D. Bose, E. McCorkle, C. Thompson, D. Bogdanoff, D. Prabhu, G. Allen, and J. Grinstead, “Analysis and model validation of shock layer radiation in air,” VKI LS Course on hypersonic entry and cruise vehicles, Palo Alto, California, USA (2008).

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Figures (6)

Fig. 1
Fig. 1

(a) Dependence of the total reflectivity of the unpolarized light on the energy of photons and the angle of incidence θ. (b) Thicknesses of individual layers of the optimal structure. Total number of layers N=14.

Fig. 2
Fig. 2

Rough interface between interfaces with permittivities ɛ1 and ɛ2.

Fig. 3
Fig. 3

(a) Sinusoidal grating on top of the slab with thickness L=100 nm. Height and periods of the grating are h = 10 nm, dx = dy = 10 nm. (b) Random hilly roughness on top of the slab with thickness L=100 nm. Periods of the unit cell are dx = dy = 50 nm, maximum height of the hills is hmax = 10 nm. (c) Multilayered structure with rough interfaces 1–3.

Fig. 4
Fig. 4

Reflectivity of the structure shown in Fig. 3(a). (a) and (b) correspond to an isotropic material with ɛ = 5, (c) and (d) correspond to an anisotropic material with ɛ11 = ɛ22 = 7, ɛ33 = 9. Red curve - reference solution (CST Studio), black dash-dotted curve - flat slab with thickness 105 nm, blue curve - multilayer approach (MA), green curve - MBC between air and slab.

Fig. 5
Fig. 5

Reflectivity of the structure shown in Fig. 3(b). (a) and (b) correspond to an isotropic material with ɛ = 5, (c) and (d) correspond to an anisotropic material with ɛ11 = ɛ22 = 7, ɛ33 = 9. Red curve - reference solution (CST Studio), black dash-dotted curve - flat slab with thickness 105 nm, blue curve - multilayer approach (MA), green curve - MBC between air and slab.

Fig. 6
Fig. 6

Reflectivity of the structure shown in Fig. 3(c) for normally incident light. Permittivities of materials are ɛ1 = 2 and ɛ2 = 7 (on top). Red curve - reference solution (CST Studio), black dash-dotted curve - flat interfaces, blue curve - multilayer approach (MA) at interfaces, green curve - MBC at interfaces.

Equations (24)

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ɛ ^ 1 = ( 2 + 0.01 i 0.02 i 0 0.02 i 2 + 0.01 i 0 0 0 3 + 0.01 i ) ɛ ^ 2 = ( 9 + 0.01 i 0.03 i 0 0.03 i 9 + 0.01 i 0 0 0 11 + 0.01 i )
Fitness = R ( d 1 , d 2 , , d N ) = R 1 + R 2 2 1 ,
R 1 , 2 ( d 1 , d 2 , , d N ) = ω 1 , 2 θ 0.5 ( R ss + R sp + R ps + R pp ) d ω d θ ω 1 , 2 θ d ω d θ 1 .
( f 1 ɛ 1 + f 2 ɛ 2 ) 1 = ɛ min ɛ eff ɛ max = ( f 1 ɛ 1 + f 2 ɛ 2 )
ɛ eff i = ɛ min i + ɛ max i 2 , i = 1 , , M
ɛ = { ɛ 1 if z < δ ɛ ( z ) if | z | < δ ɛ 2 , if z > + δ .
H d l = 1 c t D d S , E d l = 1 c t B d S , ( CGS ) H d l = t D d S , E d l = t B d S . ( SI )
H d l = H τ ( 1 ) ( δ ) l H τ ( 2 ) ( δ ) l + δ δ H n ( l / 2 ) d z δ δ H n ( l / 2 ) d z ,
δ δ H n d z = B n δ δ d z μ ( z ) ,
H d l = [ H τ ( 1 ) ( δ ) H τ ( 2 ) ( δ ) ] l + [ B n ( l / 2 ) B n ( l / 2 ) ] δ δ d z μ ( z )
1 c t D d S = i ω c E τ δ δ ɛ ( z ) d z d l .
H x ( 1 ) H x ( 2 ) δ [ H x ( 1 ) z + H x ( 2 ) z ] + B z x δ δ d z μ ( z ) = i ω c E y δ δ ɛ ( z ) d z H y ( 1 ) H y ( 2 ) δ [ H y ( 1 ) z + H y ( 2 ) z ] + B z y δ δ d z μ ( z ) = i ω c E x δ δ ɛ ( z ) d z E x ( 1 ) E x ( 2 ) δ [ E x ( 1 ) z + E x ( 2 ) z ] + D z x δ δ d z ɛ ( z ) = i ω c H y δ δ μ ( z ) d z E y ( 1 ) E y ( 2 ) δ [ E y ( 1 ) z + E y ( 2 ) z ] + D z y δ δ d z ɛ ( z ) = i ω c H x δ δ μ ( z ) d z .
E = ( 0 , E y , 0 ) H = ( H x , 0 , H z ) .
E y ( 1 ) E y ( 2 ) = 0 E y ( 1 ) z E y ( 2 ) z = ω 2 c 2 t 1 E y ( 1 )
E = ( E x , 0 , E z ) H = ( 0 , H y , 0 )
H y ( 1 ) H y ( 2 ) = t 1 ɛ 1 H y ( 1 ) z 1 ɛ 1 H y ( 1 ) z 1 ɛ 2 H y ( 2 ) z = t 2 2 H y ( 1 ) x 2 .
t 1 = δ ( ɛ 1 + ɛ 2 ) δ δ ɛ ( z ) d z t 2 = δ δ d z ɛ ( z ) δ ɛ 1 δ ɛ 2 .
t 1 = δ ( ɛ 11 ( 1 ) + ɛ 11 ( 2 ) ) δ δ ɛ 11 ( z ) d z t 2 = δ δ d z ɛ 33 ( z ) δ ɛ 33 ( 1 ) δ ɛ 33 ( 2 ) .
2 E y z 2 + 2 E y x 2 + ɛ ( z ) ω 2 c 2 E y = 0 .
( E 2 E 2 ) = ( ( 1 + β ω 2 c 2 ) 0 α ω 2 c 2 ( 1 + β ω 2 c 2 ) 1 ) ( E 1 E 1 )
z [ 1 ɛ 11 ( z ) H y z ] + x [ 1 ɛ 33 ( z ) H y z ] + ω 2 c 2 H y = 0 .
z [ 1 ɛ 11 ( z ) H y z ] + ( ω 2 c 2 k x 2 ɛ 33 ) H y = 0 .
( H 2 H 2 ɛ 11 ( 2 ) ) = ( ( 1 + β ω 2 c 2 ) 1 α γ ω 2 2 c 2 sin 2 θ α ω 2 c 2 γ sin 2 θ ( 1 + β ω 2 c 2 ) α γ ω 2 2 c 2 sin 2 θ ) ( H 1 H 1 ɛ 11 ( 1 ) ) .
ɛ ( x , y , z ) = { ɛ 2 = 5 , if z > h ( 1 2 sin ( π x d x ) sin ( π y d y ) ) ɛ 1 = 1 , if z < h ( 1 2 sin ( π x d x ) sin ( π y d y ) )

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