Abstract

A rigorous and efficient computational method is developed to calculate transmission and reflection spectra for a finite number of air-hole arrays in a slab, where the incident waves are propagating modes of the slab. The method is a three-dimensional extension of the Dirichlet-to-Neumann (DtN) map method previously developed for ideal two-dimensional photonic crystals which are infinite and invariant in one spatial direction. The method relies on the DtN maps of the unit cells to avoid repeated calculations in identical unit cells. The DtN map of a unit cell is constructed using eigenmode expansions in the vertical direction (perpendicular to the slab) and cylindrical wave expansions in the horizontal directions.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  21. Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
    [CrossRef]
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    [CrossRef]
  31. Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Comm. Comp. Phys. 1, 1056–1075 (2006).
  32. L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967–1969 (2006).
    [CrossRef]
  33. Y. P. Chiou, Y. C. Chiang, and H. C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices,” J. Lightwave Technol. 18, 243–251 (2000).
    [CrossRef]
  34. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagon photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (2)

2008 (4)

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
[CrossRef] [PubMed]

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466–1473 (2008).
[CrossRef]

2007 (3)

2006 (6)

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222 (2006).
[CrossRef]

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006).
[CrossRef]

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, 2006).
[CrossRef]

Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Comm. Comp. Phys. 1, 1056–1075 (2006).

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967–1969 (2006).
[CrossRef]

Y. Y. Lu, “Minimizing the discrete reflectivity of perfectly matched layers,” IEEE Photon. Technol. Lett. 18, 487–489 (2006).
[CrossRef]

2005 (3)

2004 (3)

2003 (1)

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

2002 (2)

M. Qiu, “Effective index method for heterostructure-slab-waveguide based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
[CrossRef]

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

2001 (1)

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagon photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[CrossRef]

2000 (4)

Y. P. Chiou, Y. C. Chiang, and H. C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices,” J. Lightwave Technol. 18, 243–251 (2000).
[CrossRef]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A 17, 320–327 (2000).
[CrossRef]

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

1996 (2)

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446 (1996).
[CrossRef]

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

1995 (2)

1994 (2)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

1991 (1)

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Armenise, M. N.

Bao, G.

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Boscolo, S.

Botten, L. C.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Centeno, E.

Chang, H. C.

Chen, C.

Chen, Z. M.

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Chiang, Y. C.

Chiou, Y. P.

Ciminelli, C.

Cotter, N. P. K.

Dems, M.

Felbacq, D.

Gaylord, T. K.

Ginste, D. V.

Grann, E. B.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Haider, M. A.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

Hu, Z.

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
[CrossRef] [PubMed]

Huang, Y.

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337–349 (2007).

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006).
[CrossRef]

Jin, J. M.

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Kotynski, R.

Kushta, T.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Lalanne, P.

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

Li, L.

Li, Y. J.

Lu, Y. Y.

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing periodic arrays of cylinders with oblique incident waves,” J. Opt. Soc. Am. B 26, 1442–1449 (2009).
[CrossRef]

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing crossed arrays of circular cylinders,” J. Opt. Soc. Am. B 26, 1984–1993 (2009).
[CrossRef]

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
[CrossRef] [PubMed]

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466–1473 (2008).
[CrossRef]

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337–349 (2007).

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967–1969 (2006).
[CrossRef]

Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Comm. Comp. Phys. 1, 1056–1075 (2006).

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222 (2006).
[CrossRef]

Y. Y. Lu, “Minimizing the discrete reflectivity of perfectly matched layers,” IEEE Photon. Technol. Lett. 18, 487–489 (2006).
[CrossRef]

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446 (1996).
[CrossRef]

Martin, P. A.

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, 2006).
[CrossRef]

McLaughlin, J. R.

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446 (1996).
[CrossRef]

McPhedran, R. C.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Michielssen, E.

Midrio, M.

Moharam, M. G.

Nicorovici, N. A.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Ochiai, T.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagon photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[CrossRef]

Olyslager, F.

Panajotov, K.

Papanicolaou, V.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

Peluso, F.

Pissoort, D.

Pommet, D. A.

Prather, D. W.

Preist, T. W.

Qiu, M.

M. Qiu, “Effective index method for heterostructure-slab-waveguide based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
[CrossRef]

Sakoda, K.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagon photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[CrossRef]

Sambles, J. R.

Shi, S.

Smith, G. H.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Toyama, H.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Vassallo, C.

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Venakides, S.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Wu, H. J.

Wu, Y.

Yasumoto, K.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Yuan, J.

Yuan, L.

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967–1969 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

M. Qiu, “Effective index method for heterostructure-slab-waveguide based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
[CrossRef]

Comm. Comp. Phys. (1)

Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Comm. Comp. Phys. 1, 1056–1075 (2006).

IEEE J. Quantum Electron. (1)

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

Y. Y. Lu, “Minimizing the discrete reflectivity of perfectly matched layers,” IEEE Photon. Technol. Lett. 18, 487–489 (2006).
[CrossRef]

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967–1969 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

J. Acoust. Soc. Am. (1)

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446 (1996).
[CrossRef]

J. Comput. Math. (1)

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337–349 (2007).

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Lightwave Technol. (5)

J. Opt. Soc. Am. A (7)

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

Microwave Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Opt. Express (2)

Opt. Quantum Electron. (1)

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

Phys. Rev. B (1)

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagon photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[CrossRef]

Phys. Rev. E (1)

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

SIAM J. Appl. Math. (1)

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

Other (4)

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, 2006).
[CrossRef]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

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Figures (8)

Fig. 1
Fig. 1

Dielectric slab with five arrays of holes on a triangular lattice. Left: 3D view; right: top view.

Fig. 2
Fig. 2

Possible cross sections of the sub-domains Ω l for a triangular lattice of air-holes on a slab.

Fig. 3
Fig. 3

Shifted unit cell Ω l containing two half air-holes and the nearby regular unit cell Ω l .

Fig. 4
Fig. 4

Cross section of a regular unit cell Ω l containing one air-hole. The vertical axis z is terminated by PMLs.

Fig. 5
Fig. 5

Transmission spectrum and out-of-plane radiation loss of ten arrays of air-holes on a slab for a fundamental TE mode incident wave at normal incidence.

Fig. 6
Fig. 6

Transmission spectrum of seven arrays of air-holes for a fundamental TE mode incident wave with an incident angle π / 6 .

Fig. 7
Fig. 7

Top view of six arrays of air-holes on a slab separated by a line defect.

Fig. 8
Fig. 8

Transmission spectrum of six arrays of air-holes on a slab separated by a line defect for a fundamental TE mode incident wave with incident angle π / 6 .

Equations (84)

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

× E = i k 0 μ H ,     × H = i k 0 ε E ,
μ d d z ( 1 μ d ϕ ( 1 ) d z ) + k 0 2 ε μ ϕ ( 1 ) = [ η ( 1 ) ] 2 ϕ ( 1 ) ,     < z < ,
ε d d z ( 1 ε d ϕ ( 2 ) d z ) + k 0 2 ε μ ϕ ( 2 ) = [ η ( 2 ) ] 2 ϕ ( 2 ) ,     < z < ,
ϕ ( p ) ( z ) 0     as   | z | ,     for   p = 1 , 2.
H z ( i ) = 1 μ ( z ) ϕ 1 ( 1 ) ( z ) exp [ i ( α 0 x β 1 , 0 ( 1 ) y ) ] ,     E z ( i ) = 0 ,
E z ( i ) = 1 ε ( z ) ϕ 1 ( 2 ) ( z ) exp [ i ( α 0 x β 1 , 0 ( 2 ) y ) ] ,     H z ( i ) = 0 ,
α k = α 0 + 2 π k / L ,     β j k ( p ) = [ η j ( p ) ] 2 α k 2 ,     p = 1 , 2.
H z ( r ) = 1 μ ( z ) j k = R j k ( 1 ) ϕ j ( 1 ) ( z ) exp [ i ( α k x + β j k ( 1 ) y ) ] ,
E z ( r ) = 1 ε ( z ) j k = R j k ( 2 ) ϕ j ( 2 ) ( z ) exp [ i ( α k x + β j k ( 2 ) y ) ] ,
H z ( t ) = 1 μ ( z ) j k = T j k ( 1 ) ϕ j ( 1 ) ( z ) exp [ i ( α k x β j k ( 1 ) y ) ] ,
E z ( t ) = 1 ε ( z ) j k = T j k ( 2 ) ϕ j ( 2 ) ( z ) exp [ i ( α k x β j k ( 2 ) y ) ] ,
TP j k ( 1 ) = | T j k ( 1 ) | 2 β j k ( 1 ) [ η j ( 1 ) ] 2 1 μ ( z ) | ϕ j ( 1 ) ( z ) | 2 d z .
TP j k ( 2 ) = | T j k ( 2 ) | 2 β j k ( 2 ) [ η j ( 2 ) ] 2 1 ε ( z ) | ϕ j ( 2 ) ( z ) | 2 d z .
TP = j , k [ TP j k ( 1 ) + TP j k ( 2 ) ] ,
Ω = { ( x , y , z ) : 0 < x < L , 0 < y < D , < z < } .
w ( x + L , y , z ) = ρ w ( x , y , z )     for   ρ = e i α 0 L .
w ( L , y , z ) = ρ w ( 0 , y , z ) ,     w x ( L , y , z ) = ρ w x ( 0 , y , z ) .
μ ( z ) f ( x , z ) = j = 1 f ̃ j ( x ) ϕ j ( 1 ) ( z ) .
f ( x , z ) = 1 μ ( z ) j = 1 k = f ̂ j k ϕ j ( 1 ) ( z ) exp ( i α k x ) ,
S ( 1 ) [ 1 μ ( z ) ϕ j ( 1 ) ( z ) exp ( i α k x ) ] = β j k ( 1 ) [ 1 μ ( z ) ϕ j ( 1 ) ( z ) exp ( i α k x ) ] ,
[ S ( 1 ) f ] ( x , z ) = 1 μ ( z ) j = 1 k = β j k ( 1 ) f ̂ j k ϕ j ( 1 ) ( z ) exp ( i α k x ) .
H z ( r ) y = i S ( 1 ) H z ( r ) ,     H z ( t ) y = i S ( 1 ) H z ( t ) .
S ( 2 ) [ 1 ε ( z ) ϕ j ( 2 ) ( z ) exp ( i α k x ) ] = β j k ( 2 ) [ 1 ε ( z ) ϕ j ( 2 ) ( z ) exp ( i α k x ) ] ,
E z ( r ) y = i S ( 2 ) E z ( r ) ,     E z ( t ) y = i S ( 2 ) H z ( t ) .
w ( r ) y = i S w ( r ) ,     w ( t ) y = i S w ( t ) .
w y = i S w ,     y = 0.
w ( i ) y = i S w ( i ) .
w y = i S w 2 i S w ( i ) ,     y = D .
Q l w l = ν w l ,     Y l w l = w 0 ,
Q 0 = i S ,     Y 0 = I ,
( Q l i S ) w l = 2 i S w ( i ) .
w 0 = Y m w l .
M [ w l w l 1 ] = [ M 11 M 12 M 21 M 22 ] [ w l w l 1 ] = [ ν w j ν w j 1 ] ,
Z = ( Q l 1 M 22 ) 1 M 21 ,
Q l = M 11 + M 12 Z ,
Y l = Y l 1 Z .
Λ [ w l w l 1 w x = L w x = 0 ] = [ Λ 11 Λ 12 Λ 13 Λ 14 Λ 21 Λ 22 Λ 23 Λ 24 Λ 31 Λ 32 Λ 33 Λ 34 Λ 41 Λ 42 Λ 43 Λ 44 ] [ w l w l 1 w x = L w x = 0 ] = [ ν w l ν w l 1 x w x = L x w x = 0 ] ,
M = [ M 11 M 12 M 21 M 22 ] = [ Λ 11 Λ 12 Λ 21 Λ 22 ] + [ C 1 D 1 C 1 D 2 C 2 D 1 C 2 D 2 ] ,
C 1 = Λ 14 + ρ Λ 13 ,     C 2 = Λ 24 + ρ Λ 23 ,    
D 0 = ρ 2 Λ 43 ρ Λ 33 + ρ Λ 44 Λ 34 ,
D 1 = D 0 1 ( Λ 31 ρ Λ 41 ) ,     D 2 = D 0 1 ( Λ 32 ρ Λ 42 ) .
w l = T w l ,     w l 1 = T w l 1     for   T = [ 0 I ρ I 0 ] ,
M = [ T 0 0 T ] 1 M [ T 0 0 T ] .
μ s d d z ( 1 μ s d ϕ ( 1 ) d z ) + k 0 2 ε μ ϕ ( 1 ) = [ η ( 1 ) ] 2 ϕ ( 1 ) ,     z bot < z < z top ,
ϕ ( 1 ) = 0 ,     z = z top ,     z = z bot ,
ε s d d z ( 1 ε s d ϕ ( 2 ) d z ) + k 0 2 ε μ ϕ ( 2 ) = [ η ( 2 ) ] 2 ϕ ( 2 ) ,     z bot < z < z top ,
d ϕ ( 2 ) d z = 0 ,     z = z bot ,     z = z top .
H z ( x , y , z ) = 1 μ ( z ) j = 1 H ̃ z , j ( x , y ) ϕ j ( 1 ) ( z ) ,    
E z ( x , y , z ) = 1 ε ( z ) j = 1 E ̃ z , j ( x , y ) ϕ j ( 2 ) ( z ) .
f ( x , y , z ) = 1 μ ( z ) j = 1 f ̃ j ( x , y ) ϕ j ( 1 ) ( z ) ,    
g ( x , y , z ) = 1 ε ( z ) j = 1 g ̃ j ( x , y ) ϕ j ( 2 ) ( z ) .
A ̃ w ̃ = [ A ̃ 11 A ̃ 12 A ̃ 21 A ̃ 22 ] w ̃ = [ f ̃ 1 f ̃ 2 g ̃ 1 g ̃ 2 ] ,     for   w ̃ = [ H ̃ z , 1 H ̃ z , 2 E ̃ z , 1 E ̃ z , 2 ] .
S ̃ j j p p   exp ( i α k x ) = β j k ( p )   exp ( i α k x ) ,     k = 0 , ± 1 , ± 2 , ,
w ( r , θ , z ) = m = w ̂ m ( r , z ) e i m θ ,     where   w ̂ m ( r , z ) = [ H ̂ z , m ( r , z ) E ̂ z , m ( r , z ) ] .
H ̂ z , m = 1 μ ( z ) j = 1 [ a j m ( 1 ) J m ( η j ( 1 ) r ) + b j m ( 1 ) H m ( 1 ) ( η j ( 1 ) r ) ] ϕ j ( 1 ) ( z ) ,     r > a ,
E ̂ z , m = 1 ε ( z ) j = 1 [ a j m ( 2 ) J m ( η j ( 2 ) r ) + b j m ( 2 ) H m ( 1 ) ( η j ( 2 ) r ) ] ϕ j ( 2 ) ( z ) ,     r > a ,
H ̂ z , m = 1 μ h ( z ) j = 1 c j m ( 1 ) J m ( η h , j ( 1 ) r ) ϕ h , j ( 1 ) ( z ) ,     r < a ,
E ̂ z , m = 1 ε h ( z ) j = 1 c j m ( 2 ) J m ( η h , j ( 2 ) r ) ϕ h , j ( 2 ) ( z ) ,     r < a .
a m = [ a 1 m ( 1 ) a 2 m ( 1 ) a 1 m ( 2 ) a 2 m ( 2 ) ] ,     b m = [ b 1 m ( 1 ) b 2 m ( 1 ) b 1 m ( 2 ) b 2 m ( 2 ) ] ,     c m = [ c 1 m ( 1 ) c 2 m ( 1 ) c 1 m ( 2 ) c 2 m ( 2 ) ] ,
B 11 a m + B 12 b m = B 13 c m ,
[ H θ E θ ] = m = [ H ̂ θ , m E ̂ θ , m ] e i m θ .
H ̂ θ , m = i m r μ ( z ) j = 1 ϕ j ( 1 ) ( z ) [ η j ( 1 ) ] 2 [ a j m ( 1 ) J m ( η j ( 1 ) r ) + b j m ( 1 ) H m ( 1 ) ( η j ( 1 ) r ) ] + i k 0 j = 1 ϕ j ( 2 ) ( z ) η j ( 2 ) [ a j m ( 2 ) J m ( η j ( 2 ) r ) + b j m ( 2 ) H m ( 1 ) ( η j ( 2 ) r ) ] ,    
r > a ,
E ̂ θ , m = i m r ε ( z ) j = 1 ϕ j ( 2 ) ( z ) [ η j ( 2 ) ] 2 [ a j m ( 2 ) J m ( η j ( 2 ) r ) + b j m ( 2 ) H m ( 1 ) ( η j ( 2 ) r ) ] i k 0 j = 1 ϕ j ( 1 ) ( z ) η j ( 1 ) [ a j m ( 1 ) J m ( η j ( 1 ) r ) + b j m ( 1 ) H m ( 1 ) ( η j ( 1 ) r ) ] ,    
r > a .
H ̂ θ , m = j = 1 [ i m c j m ( 1 ) r μ ( z ) ϕ h , j ( 1 ) ( z ) [ η h , j ( 1 ) ] 2 J m ( η h , j ( 1 ) r ) + i k 0 c j m ( 2 ) η h , j ( 2 ) ϕ h , j ( 2 ) ( z ) J m ( η h , j ( 2 ) r ) ] ,     r < a ,
E ̂ θ , m = j = 1 [ i m c j m ( 2 ) r ε ( z ) ϕ h , j ( 2 ) ( z ) [ η h , j ( 2 ) ] 2 J m ( η h , j ( 2 ) r ) i k 0 c j m ( 1 ) η h , j ( 1 ) ϕ h , j ( 1 ) ( z ) J m ( η h , j ( 1 ) r ) ] ,     r < a .
B 21 a m + B 22 b m = B 23 c m ,
b m = D m a m .
H z = 1 μ ( z ) j = 1 H ̃ z , j ( r , θ ) ϕ j ( 1 ) ( z ) ,     E z = 1 ε ( z ) j = 1 E ̃ z , j ( r , θ ) ϕ j ( 2 ) ( z ) ,
H ̃ z , j ( r , θ ) = m = [ a j m ( 1 ) J m ( η j ( 1 ) r ) + b j m ( 2 ) H m ( 1 ) ( η j ( 1 ) r ) ] e i m θ ,     r > a ,
E ̃ z , j ( r , θ ) = m = [ a j m ( 2 ) J m ( η j ( 2 ) r ) + b j m ( 2 ) H m ( 1 ) ( η j ( 2 ) r ) ] e i m θ ,     r > a .
w ̃ = F a ,
w ̃ ν = G a ,
w ̃ ν = Λ ̃ w ̃ ,     Λ ̃ = G F 1 .
ϕ ( z h ) 2 ϕ ( z ) + ϕ ( z + h ) h 2 = ϕ ( z h ) + 10 ϕ ( z ) + ϕ ( z + h ) 12 + O ( h 4 ) ,
f ( z + h / 2 ) f ( z h / 2 ) h = f ( z h ) + 22 f ( z ) + f ( z + h ) 24 + O ( h 4 ) .
d 1 f ( z + h / 2 ) + d 2 f ( z h / 2 ) d 3 f ( z h ) + d 4 f ( z ) + d 5 f ( z + h ) ,
ϕ ( p ) ( z ) = ϕ ( p ) ( z + ) ,     p = 1 , 2 ,
1 μ ( z ) d ϕ ( 1 ) d z ( z ) = 1 μ ( z + ) d ϕ ( 1 ) d z ( z + ) ,
1 ε ( z ) d ϕ ( 2 ) d z ( z ) = 1 ε ( z + ) d ϕ ( 2 ) d z ( z + ) .
d 2 ϕ ( p ) d z ̂ 2 + k 0 2 ε μ ϕ ( p ) = [ η ( p ) ] 2 ϕ ( p ) ,
z l = z bot + l h ,     1 l J 1 ,
z l + 1 / 2 = z bot + ( l + 1 2 ) h ,     0 l J 1 ,

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