Abstract

The Bragg reflection waveguide (BRW), or one-dimensional photonic crystal waveguide, has recently been proposed for a wide spectrum of applications ranging from particle acceleration to nonlinear frequency conversion. Here, we conduct a thorough analytical investigation of the quarter-wave BRW, in which the layers of the resonant cladding have a thickness corresponding to one quarter of the transverse wavelength of a desired guided mode. An analytical solution to the mode dispersion equation is derived, and it is shown that the quarter-wave BRW is polarization degenerate, although the TE and TM mode profiles differ significantly as the external Brewster’s angle condition in the cladding is approached. Analytical expressions for waveguide properties such as the modal normalization constants, propagation loss, and overlap factors between the mode and each waveguide layer are derived, as are dispersion and tuning curves.

© 2006 Optical Society of America

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  1. P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
    [CrossRef]
  2. P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
    [CrossRef]
  3. A. Mizrahi and L. Schachter, "Optical Bragg accelerators," Phys. Rev. E 70, 016505 (2004).
    [CrossRef]
  4. Z. Zhang, S. G. Tantawi, and R. D. Ruth, "Distributed grating-assisted coupler for optical all-dielectric electron accelerator," Phys. Rev. ST Accel. Beams 8, 071302 (2005).
    [CrossRef]
  5. A. S. Helmy and B. R. West, "Phase matching using Bragg reflector waveguides," in Proceedings of the 18th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2005), pp. 424-425.
  6. Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. Part 1 43, L631-L633 (2004).
    [CrossRef]
  7. E. Simova and I. Golub, "Polarization splitter/combiner in high index contrast Bragg reflector waveguides," Opt. Express 11, 3425-3430 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).
  18. P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978).
    [CrossRef]
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    [CrossRef]
  20. Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, "Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length," Opt. Express 12, 1090-1096 (2004).
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2005 (1)

Z. Zhang, S. G. Tantawi, and R. D. Ruth, "Distributed grating-assisted coupler for optical all-dielectric electron accelerator," Phys. Rev. ST Accel. Beams 8, 071302 (2005).
[CrossRef]

2004 (3)

Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. Part 1 43, L631-L633 (2004).
[CrossRef]

A. Mizrahi and L. Schachter, "Optical Bragg accelerators," Phys. Rev. E 70, 016505 (2004).
[CrossRef]

Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, "Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length," Opt. Express 12, 1090-1096 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

2001 (1)

S. Esposito, "Universal photonic tunneling time," Phys. Rev. E 64, 026609 (2001).
[CrossRef]

1995 (1)

P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

1992 (1)

C. Wächter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, "Nonlinear Bragg reflection waveguide," J. Appl. Phys. 71, 3688-3692 (1992).
[CrossRef]

1991 (1)

P. M. Lambkin and K. A. Shore, "Nonlinear semiconductor Bragg reflection waveguide structures," IEEE J. Quantum Electron. 27, 824-829 (1991).
[CrossRef]

1989 (1)

C. M. Kim and R. V. Ramaswamy, "Modeling of graded-index channel waveguides using nonuniform finite-difference method," J. Lightwave Technol. 7, 1581-1589 (1989).
[CrossRef]

1985 (1)

S. Adachi, "GaAs, AlAs, and AlxGa1−xAs material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
[CrossRef]

1984 (1)

1978 (1)

1977 (1)

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

1976 (1)

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Adachi, S.

S. Adachi, "GaAs, AlAs, and AlxGa1−xAs material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
[CrossRef]

Argyros, A.

Asakawa, K.

Chilwell, J.

Deif, A. S.

A. S. Deif, Advanced Matrix Theory for Scientists and Engineers (Routledge, 1987).

Esposito, S.

S. Esposito, "Universal photonic tunneling time," Phys. Rev. E 64, 026609 (2001).
[CrossRef]

Golub, I.

Helmy, A. S.

A. S. Helmy and B. R. West, "Phase matching using Bragg reflector waveguides," in Proceedings of the 18th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2005), pp. 424-425.

Hodgkinson, I.

Hong, C.-S.

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

Ikeda, N.

Inoue, K.

Kim, C. M.

C. M. Kim and R. V. Ramaswamy, "Modeling of graded-index channel waveguides using nonuniform finite-difference method," J. Lightwave Technol. 7, 1581-1589 (1989).
[CrossRef]

Koyama, F.

Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. Part 1 43, L631-L633 (2004).
[CrossRef]

Kwiat, P.

P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Lambkin, P. M.

P. M. Lambkin and K. A. Shore, "Nonlinear semiconductor Bragg reflection waveguide structures," IEEE J. Quantum Electron. 27, 824-829 (1991).
[CrossRef]

Lederer, F.

C. Wächter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, "Nonlinear Bragg reflection waveguide," J. Appl. Phys. 71, 3688-3692 (1992).
[CrossRef]

Leine, L.

C. Wächter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, "Nonlinear Bragg reflection waveguide," J. Appl. Phys. 71, 3688-3692 (1992).
[CrossRef]

Mann, M.

C. Wächter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, "Nonlinear Bragg reflection waveguide," J. Appl. Phys. 71, 3688-3692 (1992).
[CrossRef]

Marom, E.

Mattle, K.

P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Mizrahi, A.

A. Mizrahi and L. Schachter, "Optical Bragg accelerators," Phys. Rev. E 70, 016505 (2004).
[CrossRef]

Nakamura, Y.

Ramaswamy, R. V.

C. M. Kim and R. V. Ramaswamy, "Modeling of graded-index channel waveguides using nonuniform finite-difference method," J. Lightwave Technol. 7, 1581-1589 (1989).
[CrossRef]

Ruth, R. D.

Z. Zhang, S. G. Tantawi, and R. D. Ruth, "Distributed grating-assisted coupler for optical all-dielectric electron accelerator," Phys. Rev. ST Accel. Beams 8, 071302 (2005).
[CrossRef]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).

Sakurai, Y.

Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. Part 1 43, L631-L633 (2004).
[CrossRef]

Schachter, L.

A. Mizrahi and L. Schachter, "Optical Bragg accelerators," Phys. Rev. E 70, 016505 (2004).
[CrossRef]

Sergienko, A.

P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Shih, Y.

P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Shore, K. A.

P. M. Lambkin and K. A. Shore, "Nonlinear semiconductor Bragg reflection waveguide structures," IEEE J. Quantum Electron. 27, 824-829 (1991).
[CrossRef]

Simova, E.

Sugimoto, Y.

Tanaka, Y.

Tantawi, S. G.

Z. Zhang, S. G. Tantawi, and R. D. Ruth, "Distributed grating-assisted coupler for optical all-dielectric electron accelerator," Phys. Rev. ST Accel. Beams 8, 071302 (2005).
[CrossRef]

Trutschel, U.

C. Wächter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, "Nonlinear Bragg reflection waveguide," J. Appl. Phys. 71, 3688-3692 (1992).
[CrossRef]

Wächter, C.

C. Wächter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, "Nonlinear Bragg reflection waveguide," J. Appl. Phys. 71, 3688-3692 (1992).
[CrossRef]

Weinfurter, H.

P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

West, B. R.

A. S. Helmy and B. R. West, "Phase matching using Bragg reflector waveguides," in Proceedings of the 18th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2005), pp. 424-425.

Yariv, A.

P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978).
[CrossRef]

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Yeh, P.

P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978).
[CrossRef]

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

Zeilinger, A.

P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Zhang, Z.

Z. Zhang, S. G. Tantawi, and R. D. Ruth, "Distributed grating-assisted coupler for optical all-dielectric electron accelerator," Phys. Rev. ST Accel. Beams 8, 071302 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. M. Lambkin and K. A. Shore, "Nonlinear semiconductor Bragg reflection waveguide structures," IEEE J. Quantum Electron. 27, 824-829 (1991).
[CrossRef]

J. Appl. Phys. (2)

S. Adachi, "GaAs, AlAs, and AlxGa1−xAs material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
[CrossRef]

C. Wächter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, "Nonlinear Bragg reflection waveguide," J. Appl. Phys. 71, 3688-3692 (1992).
[CrossRef]

J. Lightwave Technol. (1)

C. M. Kim and R. V. Ramaswamy, "Modeling of graded-index channel waveguides using nonuniform finite-difference method," J. Lightwave Technol. 7, 1581-1589 (1989).
[CrossRef]

J. Opt. Soc. Am. (2)

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978).
[CrossRef]

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

Jpn. J. Appl. Phys. Part 1 (1)

Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. Part 1 43, L631-L633 (2004).
[CrossRef]

Opt. Commun. (1)

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Opt. Express (3)

Phys. Rev. E (2)

A. Mizrahi and L. Schachter, "Optical Bragg accelerators," Phys. Rev. E 70, 016505 (2004).
[CrossRef]

S. Esposito, "Universal photonic tunneling time," Phys. Rev. E 64, 026609 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Phys. Rev. ST Accel. Beams (1)

Z. Zhang, S. G. Tantawi, and R. D. Ruth, "Distributed grating-assisted coupler for optical all-dielectric electron accelerator," Phys. Rev. ST Accel. Beams 8, 071302 (2005).
[CrossRef]

Other (5)

A. S. Helmy and B. R. West, "Phase matching using Bragg reflector waveguides," in Proceedings of the 18th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2005), pp. 424-425.

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

A. S. Deif, Advanced Matrix Theory for Scientists and Engineers (Routledge, 1987).

Mode Solutions version 1.0, Lumerical Solutions, Inc., www.lumerical.com.

J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).

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Figures (8)

Fig. 1
Fig. 1

Index profile of a one-dimensional BRW.

Fig. 2
Fig. 2

Field profiles for two different QtW-BRWs. Fields are normalized to show the stronger localization of the TE mode. Waveguide design parameters are listed in Table 1. Solid curve, E y (TE mode); dotted curve, E x (TM mode). (a) n 1 2 k 2 n 2 2 k 1 = 0.8975 , (b) n 1 2 k 2 n 2 2 k 1 = 1.0768 .

Fig. 3
Fig. 3

Propagation loss in a QtW-BRW. Filled symbols, n 1 n 2 = 0.5 ; open symbols, n 1 n 2 = 0.4 . Circles, TE; squares, TM.

Fig. 4
Fig. 4

Chromatic dispersion of the BRW mode, effective index versus wavelength (nominally quarter wave at λ = 775 nm ). Solid curve, numerical dispersion curve; dashed curve, analytical (linear) model from Eq. (72); dotted curve, error in analytical model.

Fig. 5
Fig. 5

Effective index versus core index for a BRW waveguide (nominally quarter wave at n c = 3.25 ). Solid curve, numerical tuning curve; dashed curve, analytical (linear) model from Eq. (76); dotted curve, error in analytical model.

Fig. 6
Fig. 6

Channel waveguide, utilizing Bragg waveguiding in the lateral direction and total internal reflection in the transverse direction.

Fig. 7
Fig. 7

Effective index versus waveguide width: QtW-BRW #1. Solid curve, TE; dashed curve, TM. The horizontal line at n eff = 2.8566 represents the slab waveguide.

Fig. 8
Fig. 8

Mode intensity, QtW-BRW #1, 3 μ m width. (a) TE 00 , (b) TE 10 , (c) TM 00 , (d) TM 10 . The waveguide geometry is superimposed on the figure.

Tables (1)

Tables Icon

Table 1 Waveguide Design Parameters Used in Fig. 2

Equations (100)

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E ( x , y , z , t ) = E ( x ) exp [ i ( ω t β z ) ] ,
2 E ( x ) x 2 + k 2 ( x ) E ( x ) = 0 .
k i = k 0 n i 2 n eff 2 .
E y ( x ) = { C 1 TE cos ( k c x ) , x t c 2 ( core ) C 2 TE E K , TE ( x t c 2 ) exp [ i K TE ( x t c 2 ) ] x > t c 2 ( cladding ) } ,
E K , TE ( x + Λ ) = E K , TE ( x ) .
E clad ( x ) = { a n , TE exp [ i k 1 ( x t c 2 n Λ ) ] + b n , TE exp [ i k 1 ( x t c 2 n Λ ) ] , n Λ ( x t c 2 ) n Λ + a c n , TE exp [ i k 2 ( x t c 2 n Λ a ) ] + d n , TE exp [ i k 2 ( x t c 2 n Λ a ) ] , n Λ + a ( x t c 2 ) ( n + 1 ) Λ } ,
( a n 1 b n 1 ) TE = [ A B C D ] TE ( a n b n ) TE .
A TE = exp ( i k 1 a ) [ cos k 2 b + i 2 ( k 2 k 1 + k 1 k 2 ) sin k 2 b ] ,
B TE = exp ( i k 1 a ) [ i 2 ( k 2 k 1 k 1 k 2 ) sin k 2 b ] ,
C TE = B TE * , D TE = A TE * .
( a n 1 b n 1 ) TE = exp ( i K TE Λ ) ( a n b n ) TE ,
[ A B C D ] TE ( a n b n ) TE = exp ( i K TE Λ ) ( a n b n ) TE .
exp ( i K TE Λ ) = Re ( A TE ) ± [ Re ( A TE ) ] 2 1 ,
( a n b n ) TE = [ B TE exp ( i K TE Λ ) A TE ] = exp ( i n K TE Λ ) ( a 0 b 0 ) TE
K TE = 1 Λ cos 1 [ Re ( A TE ) ] .
( c n d n ) TE = 1 2 [ exp ( i k 1 a ) ( 1 + k 1 k 2 ) exp ( i k 1 a ) ( 1 k 1 k 2 ) exp ( i k 1 a ) ( 1 k 1 k 2 ) exp ( i k 1 a ) ( 1 + k 1 k 2 ) ] ( a n b n ) TE .
E y ( x ) = { C 1 TE cos ( k c x ) , 0 x t c 2 { + b 0 , TE exp [ i k 1 ( x t c 2 n Λ ) ] } exp ( i n K TE Λ ) , C 2 TE { a 0 , TE exp [ i k 1 ( x t c 2 n Λ ) ] } n Λ ( x t c 2 ) n Λ + a { + d 0 , TE exp [ i k 2 ( x t c 2 n Λ a ) ] } exp ( i n K TE Λ ) , C 2 TE { c 0 , TE exp [ i k 2 ( x t c 2 n Λ a ) ] } n Λ + a ( x t c 2 ) ( n + 1 ) Λ } ,
E y ( x ) = E y ( x ) .
1 k c cot ( k c t c 2 ) = i k 1 exp ( i K TE Λ ) A TE + B TE exp ( i K TE Λ ) A TE B TE .
E x ( x ) = β ω ϵ H y ( x )
θ B = tan 1 ( n 2 n 1 ) .
n 1 2 k 2 = n 2 2 k 1 .
H y ( x ) = { C 1 TM cos ( k c x ) , 0 x t c 2 , n 1 2 k 2 < n 2 2 k 1 C 1 TM sin ( k c x ) , 0 x t c 2 , n 1 2 k 2 > n 2 2 k 1 C 2 TM H K , TM ( x t c 2 ) exp [ i K TM ( x t c 2 ) ] , x > t c 2 } ,
H y ( x ) = { H y ( x ) , n 1 2 k 2 < n 2 2 k 1 H y ( x ) , n 1 2 k 2 > n 2 2 k 1 } ,
A TM = exp ( i k 1 a ) [ cos k 2 b + i 2 ( n 2 2 k 1 n 1 2 k 2 + n 1 2 k 2 n 2 2 k 1 ) sin k 2 b ] ,
B TM = exp ( i k 1 a ) [ i 2 ( n 2 2 k 1 n 1 2 k 2 n 1 2 k 2 n 2 2 k 1 ) sin k 2 b ] ,
C TM = B TM * D TM = A TM * ,
( c n d n ) TM = 1 2 ( n 2 k 1 n 1 k 2 ) [ exp ( i k 1 a ) ( 1 + n 1 2 k 2 n 2 2 k 1 ) exp ( i k 1 a ) ( 1 n 1 2 k 2 n 2 2 k 1 ) exp ( i k 1 a ) ( 1 n 1 2 k 2 n 2 2 k 1 ) exp ( i k 1 a ) ( 1 + n 1 2 k 2 n 2 2 k 1 ) ] ( a n b n ) TM .
1 k c cot ( k c t c 2 ) = i k 1 ( n 1 n c ) 2 exp ( i K TM Λ ) A TM + B TM exp ( i K TM Λ ) A TM B TM , n 1 2 k 2 < n 2 2 k 1 ,
k c cot ( k c t c 2 ) = i k 1 ( n c n 1 ) 2 exp ( i K TM Λ ) A TM B TM exp ( i K TM Λ ) A TM + B TM , n 1 2 k 2 > n 2 2 k 1 .
k 1 a = k 2 b = π 2 .
A TE = D TE = 1 2 ( k 2 k 1 + k 1 k 2 ) ,
B TE = C TE = 1 2 ( k 2 k 1 k 1 k 2 ) ,
A TM = D TM = 1 2 ( n 2 2 k 1 n 1 2 k 2 + n 1 2 k 2 n 2 2 k 1 ) ,
B TM = C TM = 1 2 ( n 2 2 k 1 n 1 2 k 2 n 1 2 k 2 n 2 2 k 1 )
K Λ = { ( 2 m 1 ) π ± i ln ( k 2 k 1 ) ( TE ) ( 2 m 1 ) π ± i ln ( n 2 2 k 1 n 1 2 k 2 ) ( TM ) } ,
exp ( i K Λ ) = { ( k 2 k 1 ) ( TE ) ( n 1 2 k 2 n 2 2 k 1 ) ( TM , n 1 2 k 2 < n 2 2 k 1 ) ( n 2 2 k 1 n 1 2 k 2 ) ( TM , n 1 2 k 2 > n 2 2 k 1 ) } .
k c t c 2 = ( p + 1 ) π 2 , p = 0 , 2 , 4 , ,
k c = 2 π λ n c 2 n eff 2 = π t c
n eff = [ n c 2 ( λ 2 t c ) 2 ] 1 2 ,
λ 2 n c < t c < λ 2 n c 2 n 2 2 ( n c > n 2 ) ,
λ 2 n c < t c ( n c < n 2 ) .
a 0 = b 0 ( TE , TM [ n 1 2 k 2 < n 2 2 k 1 ] ) ,
a 0 = b 0 ( TM [ n 1 2 k 2 > n 2 2 k 1 ] ) .
c 0 = d 0 = i a 0 ( TE , TM [ n 1 2 k 2 < n 2 2 k 1 ] ) ,
c 0 = d 0 = i a 0 ( TM [ n 1 2 k 2 > n 2 2 k 1 ] ) .
E TE ( x ) = { C 1 TE cos ( k c x ) , x t c 2 C 2 TE ( k 2 k 1 ) n sin [ k 1 ( x t c 2 n Λ ) ] , n Λ ( x t c 2 ) n Λ + a C 2 TE ( k 2 k 1 ) n cos [ k 2 ( x t c 2 n Λ a ) ] , n Λ + a ( x t c 2 ) ( n + 1 ) Λ } ,
E TM ( x ) = { C 1 TM n c 2 cos ( k c x ) , x t c 2 C 2 TM n 1 2 ( n 1 2 k 2 n 2 2 k 1 ) n sin [ k 1 ( x t c 2 n Λ ) ] , n Λ ( x t c 2 ) n Λ + a C 2 TM n 2 2 ( n 1 2 k 2 n 2 2 k 1 ) n cos [ k 2 ( x t c 2 n Λ a ) ] , n Λ + a ( x t c 2 ) ( n + 1 ) Λ } ,
E TM ( x ) = { C 1 TM n c 2 sin ( k c x ) , x t c 2 C 2 TM n 1 2 ( n 2 2 k 1 n 1 2 k 2 ) n cos [ k 1 ( x t c 2 n Λ ) ] , n Λ ( x t c 2 ) n Λ + a C 2 TM n 2 2 ( n 2 2 k 1 n 1 2 k 2 ) n sin [ k 2 ( x t c 2 n Λ a ) ] , n Λ + a ( x t c 2 ) ( n + 1 ) Λ } ,
C 2 = { k c k 1 C 1 ( TE , TM [ n 1 2 k 2 < n 2 2 k 1 ] ) C 1 , ( TM [ n 1 2 k 2 > n 2 2 k 1 ] ) } .
E ( x ) 2 d x = 2 0 E ( x ) 2 d x = 1 ,
I c + I 1 + I 2 = 1 2 ,
I c = 0 t c 2 C 1 TE 2 cos 2 ( k c x ) d x = C 1 TE 2 t c 4 .
I 1 = C 2 TE 2 n = 0 t c 2 + n Λ t c 2 + n Λ + a ( k 2 k 1 ) 2 n sin 2 [ k 1 ( x t c 2 n Λ ) ] d x = C 2 TE 2 0 a sin 2 ( k 1 x ) d x n = 0 ( k 2 k 1 ) 2 n = C 2 TE 2 a σ TE 2 ,
σ n = 0 exp ( i 2 K Λ ) = { k 1 2 k 1 2 k 2 2 ( TE ) n 2 4 k 1 2 n 2 4 k 1 2 n 1 4 k 2 2 ( TM , n 1 2 k 2 < n 2 2 k 1 ) n 1 4 k 2 2 n 1 4 k 2 2 n 2 4 k 1 2 ( TM , n 1 2 k 2 > n 2 2 k 1 ) } ,
I 2 = C 2 TE 2 b σ TE 2 .
C 1 TE = [ t c 2 + π k c 2 σ TE ( k 1 + k 2 ) 2 k 1 3 k 2 ] 1 2 ,
C 2 TE = k c k 1 [ t c 2 + π k c 2 σ TE ( k 1 + k 2 ) 2 k 1 3 k 2 ] 1 2 .
C 1 TM = [ t c 2 n c 4 + π k c 2 σ TM ( n 1 4 k 1 + n 2 4 k 2 ) 2 n 1 4 n 2 4 k 1 3 k 2 ] 1 2 ,
C 2 TM = k c k 1 [ t c 2 n c 4 + π k c 2 σ TM ( n 1 4 k 1 + n 2 4 k 2 ) 2 n 1 4 n 2 4 k 1 3 k 2 ] 1 2 ,
C 1 TM = C 2 TM = [ t c 2 n c 4 + π σ TM ( n 1 4 k 1 + n 2 4 k 2 ) 2 n 1 4 n 2 4 k 1 k 2 ] 1 2 ,
Γ c TE = ( C 1 TE ) 2 t c 2 ,
Γ 1 TE = ( C 1 TE ) 2 π k c 2 σ 2 k 1 3 ,
Γ 2 TE = ( C 1 TE ) 2 π k c 2 σ 2 k 1 2 k 2 ,
Γ c TM = ( C 1 TM ) 2 t c 2 n c 4 ,
Γ 1 TM = { ( C 1 TM ) 2 π k c 2 σ 2 n 1 4 k 1 3 , n 1 2 k 2 < n 2 2 k 1 ( C 1 TM ) 2 π σ 2 n 1 4 k 1 , n 1 2 k 2 > n 2 2 k 1 } ,
Γ 2 TM = { ( C 1 TM ) 2 π k c 2 σ 2 n 2 4 k 1 2 k 2 , n 1 2 k 2 < n 2 2 k 1 ( C 1 TM ) 2 π σ 2 n 2 4 k 2 , n 1 2 k 2 > n 2 2 k 1 } .
Γ i TE , n = Γ i TE 2 σ ( k 2 k 1 ) 2 n = Γ i TE 2 [ ( k 2 k 1 ) 2 n ( k 2 k 1 ) 2 n + 2 ] ,
Γ i TM , n = { Γ i TM 2 σ ( n 1 2 k 2 n 2 2 k 1 ) 2 n = Γ i TM 2 [ ( n 1 2 k 2 n 2 2 k 1 ) 2 n ( n 1 2 k 2 n 2 2 k 1 ) 2 n + 2 ] ( n 1 2 k 2 < n 2 2 k 1 ) Γ i TM 2 σ ( n 2 2 k 1 n 1 2 k 2 ) 2 n = Γ i TM 2 [ ( n 2 2 k 1 n 1 2 k 2 ) 2 n ( n 2 2 k 1 n 1 2 k 2 ) 2 n + 2 ] ( n 1 2 k 2 > n 2 2 k 1 ) } .
Γ TE , N = ( Γ 1 TE + Γ 2 TE ) 2 σ n = 0 N 1 ( k 2 k 1 ) 2 n = ( Γ 1 TE + Γ 2 TE ) 2 [ 1 ( k 2 k 1 ) 2 N ] ,
Γ TM , N = { ( Γ 1 TM + Γ 2 TM ) 2 [ 1 ( n 1 2 k 2 n 2 2 k 1 ) 2 N ] ( n 1 2 k 2 < n 2 2 k 1 ) ( Γ 1 TM + Γ 2 TM ) 2 [ 1 ( n 2 2 k 1 n 1 2 k 2 ) 2 N ] ( n 1 2 k 2 > n 2 2 k 1 ) } .
θ i = sin 1 ( n eff n c ) = sin 1 { [ 1 ( λ 2 n c t c ) 2 ] 1 2 } ,
M j = [ cos k j t j ( i γ j ) sin k j t j i γ j sin k j t j cos k j t j ] ,
γ j = 1 ( n j ) 2 ρ n j 2 n eff 2 , ρ = { 0 ( TE ) 1 ( TM ) }
M = j M j = [ m 11 m 12 m 21 m 22 ] ,
r = γ c m 11 + γ c γ s m 12 m 21 γ s m 22 γ c m 11 + γ c γ s m 12 + m 21 + γ s m 22 ,
M = ( M 1 M 2 ) N = ( [ 0 i γ 1 i γ 1 0 ] [ 0 i γ 2 i γ 2 0 ] ) N = ( 1 ) N [ ( γ 2 γ 1 ) N 0 0 ( γ 1 γ 2 ) N ] ,
r = γ c ( γ 2 γ 1 ) N γ s ( γ 1 γ 2 ) N γ c ( γ 2 γ 1 ) N + γ s ( γ 1 γ 2 ) N .
Loss ( decibels   per   centimeter ) = λ ln r 10 n c t c 2 [ 1 ( λ 2 n c t c ) 2 ] 1 2 .
n ̃ i = n i + Δ n i .
k ̃ i = k i + Δ k i = ( 2 π λ + Δ λ ) ( n i + Δ n i ) 2 ( n eff + Δ n eff ) 2 2 π λ ( 1 Δ λ λ ) n i 2 n eff 2 ( 1 + n i Δ n i n eff Δ n eff n i 2 n eff 2 ) ,
Δ k i k i ( n i Δ n i n eff Δ n eff n i 2 n eff 2 Δ λ λ ) = F i Δ λ k i ( n eff Δ n eff n i 2 n eff 2 + Δ λ λ ) ,
F i k 0 2 n i n i λ k i
Δ n eff Δ λ = π k c t c ( k 2 2 k 1 2 ) [ F 1 + F 2 ( k 1 + k 2 ) λ 1 ] F c + k c λ 1 k 0 2 n eff [ π k c t c k 1 k 2 ( k 2 k 1 ) 1 k c ] .
k ̃ i = k i + Δ k i = ( 2 π λ ) n i 2 ( n eff + Δ n eff ) 2 k i ( 1 n eff Δ n eff n i 2 n eff 2 )
Δ k i = k 0 2 n eff Δ n eff k i .
k ̃ c = ( 2 π λ ) ( n c + Δ n c ) 2 ( n eff + Δ n eff ) 2 ,
Δ k c ( k 0 2 k c ) ( n c Δ n c n eff Δ n eff ) .
Δ n eff Δ n c = n c n eff [ 1 + π k c 2 ( k 2 + k 1 ) t c k 1 k 2 ( k 1 2 k 2 2 ) ] 1 .
Δ ( β 2 ) k 0 2 core Δ n 2 ( x ) E ( x ) 2 d x E ( x ) 2 d x = k 0 2 Δ n c 2 Γ c ,
1 k c + Δ k c cot [ ( k c + Δ k c ) t c 2 ] = i k 1 + Δ k 1 ( { exp ( i K TE Λ ) + Δ [ exp ( i K TE Λ ) ] } ( A TE + Δ A TE ) + ( B TE + Δ B TE ) { exp ( i K TE Λ ) + Δ [ exp ( i K TE Λ ) ] } ( A TE + Δ A TE ) ( B TE + Δ B TE ) ) .
1 k c + Δ k c cot [ ( k c + Δ k c ) t c 2 ] 1 k c ( 1 Δ k c k c ) ( Δ k c t c 2 ) Δ k c t c 2 k c ,
i k 1 ( 1 Δ k 1 k 1 ) [ exp ( i K TE Λ ) A TE + B TE exp ( i K TE Λ ) A TE B TE ] { 1 + Δ [ exp ( i K TE Λ ) ] Δ A TE + Δ B TE exp ( i K TE Λ ) A TE + B TE Δ [ exp ( i K TE Λ ) ] Δ A TE Δ B TE exp ( i K TE Λ ) A TE B TE } i k 1 { Δ [ exp ( i K TE Λ ) ] Δ A TE + Δ B TE exp ( i K TE Λ ) A TE B TE } ,
A ̃ TE = A TE + Δ A TE = exp [ i ( k 1 + Δ k 1 ) a ] { cos [ ( k 2 + Δ k 2 ) b ] + i 2 ( k 2 + Δ k 2 k 1 + Δ k 1 + k 1 + Δ k 1 k 2 + Δ k 2 ) sin [ ( k 2 + Δ k 2 ) b ] } i ( 1 + i Δ k 1 a ) [ Δ k 2 b + i 2 ( k 2 k 1 + k 1 k 2 ) ( 1 + 2 k 2 Δ k 2 + 2 k 1 Δ k 1 k 2 2 + k 1 2 Δ k 1 k 1 Δ k 2 k 2 ) ] ,
Δ A TE = Δ k 1 [ 1 2 k 1 ( k 2 k 1 + k 1 k 2 ) 1 k 2 i ( π 4 k 1 ) ( k 2 k 1 + k 1 k 2 ) ] + Δ k 2 [ 1 2 k 2 ( k 2 k 1 + k 1 k 2 ) 1 k 1 i ( π 2 k 2 ) ] .
Δ B TE = Δ k 1 [ 1 2 k 1 ( k 2 k 1 k 1 k 2 ) 1 k 2 i ( π 4 k 1 ) ( k 2 k 1 k 1 k 2 ) ] + Δ k 2 [ 1 2 k 2 ( k 2 k 1 k 1 k 2 ) + 1 k 1 ] ,
Δ D TE = Δ A TE * , Δ C TE = Δ B TE * .
Ξ TE = [ Δ A TE Δ B TE Δ B TE * Δ A TE * ] .
Δ [ exp ( i K TE Λ ) ] = u TE + , Ξ TE u TE + = 1 2 [ 1 1 ] [ Δ A TE Δ B TE Δ B TE * Δ A TE * ] ( 1 1 ) = Re ( Δ A TE Δ B TE ) ,
Δ k c = π k c ( Δ k 1 + Δ k 2 ) t c ( k 2 2 k 1 2 ) .

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