Abstract

In this paper, the coupled mode theory (CMT) is used to derive the corresponding stochastic differential equations (SDEs) for the modal amplitude evolution inside optical waveguides with random refractive index variations. Based on the SDEs, the ordinary differential equations (ODEs) are derived to analyze the statistics of the modal amplitudes, such as the optical power and power variations as well as the power correlation coefficients between the different modal powers. These ODEs can be solved analytically and therefore, it greatly simplifies the analysis. It is demonstrated that the ODEs for the power evolution of the modes are in excellent agreement with the Marcuse' coupled power model. The higher order statistics, such as the power variations and power correlation coefficients, which are not exactly analyzed in the Marcuse' model, are discussed afterwards. Monte-Carlo simulations are performed to demonstrate the validity of the analytical model.

© 2016 Optical Society of America

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References

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  1. A. Di Donato, M. Farina, D. Mencarelli, A. Lucesoli, S. Fabiani, T. Rozzi, G. M. Di Gregorio, and G. Angeloni, “Stationary Mode Distribution and Sidewall Roughness Effects in Overmoded Optical Waveguides,” J. Lightwave Technol. 28(10), 1510–1520 (2010).
    [Crossref]
  2. T. Barwicz and H. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness in micro-photonic waveguides,” J. Lightwave Technol. 23(9), 2719–2732 (2005).
    [Crossref]
  3. A. W. Snyder and J. Love, Optical Waveguide Theory, 1st ed., Springer, 1983.
  4. D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48(10), 3187–3215 (1969).
    [Crossref]
  5. D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
    [Crossref]
  6. D. Marcuse, “Power distribution and radiation losses in multimode dielectric waveguides,” Bell Syst. Tech. J. 51(2), 429–454 (1972).
    [Crossref]
  7. J. D. Love, T. J. Senden, and F. Ladouceur, “Effect of side wall roughness in buried channel waveguides,” IEE Proc., Optoelectron. 141(4), 242–248 (1994).
    [Crossref]
  8. J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. 137, 282–288, 1990.
    [Crossref]
  9. D. Lenz, D. Erni, and W. Bächtold, “Modal power loss coefficients for highly overmoded rectangular dielectric waveguides based on free space modes,” Opt. Express 12(6), 1150–1156 (2004).
    [Crossref] [PubMed]
  10. F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16(7), 1661–1663 (2004).
    [Crossref]
  11. A. D. Simard, N. Ayotte, Y. Painchaud, S. Bédard, and S. LaRochelle, “Impact of Sidewall Roughness on Integrated Bragg Gratings,” J. Lightwave Technol. 29(24), 3693–3704 (2011).
    [Crossref]
  12. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1974).
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  14. K.-P. Ho and J. M. Kahn, “Frequency diversity in mode-division multiplexing systems,” J. Lightwave Technol. 29(24), 3719–3726 (2011).
    [Crossref]
  15. S. O. Arik, D. Askarov, and J. M. Kahn, “Effect of mode coupling on signal processing complexity in mode-division multiplexing,” J. Lightwave Technol. 31(3), 423–431 (2013).
    [Crossref]
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  18. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” Appl. Opt. 46(26), 6561–6571 (2007).
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    [Crossref]
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    [Crossref]
  21. D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54(6), 985–995 (1975).
    [Crossref]
  22. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
    [Crossref]
  23. A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, and C. R. Menyuk, “An analytical formula for the mean differential group delay of randomly-birefringent spun fibers,” J. Lightwave Technol. 21(7), 1635–1643 (2003).
    [Crossref]
  24. A. Mecozzi, “A Theory of Polarization-Mode Dispersion of Spun Fibers,” J. Lightwave Technol. 27(7), 938–943 (2009).
    [Crossref]
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    [Crossref]

2013 (1)

2011 (3)

2010 (1)

2009 (1)

2007 (1)

2005 (1)

2004 (2)

D. Lenz, D. Erni, and W. Bächtold, “Modal power loss coefficients for highly overmoded rectangular dielectric waveguides based on free space modes,” Opt. Express 12(6), 1150–1156 (2004).
[Crossref] [PubMed]

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

2003 (3)

1996 (1)

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

1994 (2)

J. D. Love, T. J. Senden, and F. Ladouceur, “Effect of side wall roughness in buried channel waveguides,” IEE Proc., Optoelectron. 141(4), 242–248 (1994).
[Crossref]

W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994).
[Crossref]

1988 (1)

J. Seligson, “The orthogonality relation for TE- and TM-modes in guided-wave optics,” J. Lightwave Technol. 6(8), 1260–1264 (1988).
[Crossref]

1975 (1)

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54(6), 985–995 (1975).
[Crossref]

1972 (2)

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
[Crossref]

D. Marcuse, “Power distribution and radiation losses in multimode dielectric waveguides,” Bell Syst. Tech. J. 51(2), 429–454 (1972).
[Crossref]

1969 (1)

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48(10), 3187–3215 (1969).
[Crossref]

Agrawal, G. P.

Angeloni, G.

Anguita, J. A.

Arik, S. O.

Askarov, D.

Ayotte, N.

Bächtold, W.

Barwicz, T.

Bédard, S.

Cassan, E.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Di Donato, A.

Di Gregorio, G. M.

Erni, D.

Fabiani, S.

Farina, M.

Foschini, G. J.

Galtarossa, A.

Grillot, F.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Haus, H.

Ho, K.-P.

Huang, M.

Huang, W. P.

Kahn, J. M.

Ladouceur, F.

J. D. Love, T. J. Senden, and F. Ladouceur, “Effect of side wall roughness in buried channel waveguides,” IEE Proc., Optoelectron. 141(4), 242–248 (1994).
[Crossref]

LaRochelle, S.

Laval, S.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Lenz, D.

Lin, Q.

Love, J. D.

J. D. Love, T. J. Senden, and F. Ladouceur, “Effect of side wall roughness in buried channel waveguides,” IEE Proc., Optoelectron. 141(4), 242–248 (1994).
[Crossref]

Lucesoli, A.

Marcuse, D.

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54(6), 985–995 (1975).
[Crossref]

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
[Crossref]

D. Marcuse, “Power distribution and radiation losses in multimode dielectric waveguides,” Bell Syst. Tech. J. 51(2), 429–454 (1972).
[Crossref]

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48(10), 3187–3215 (1969).
[Crossref]

Marks, B. S.

Mecozzi, A.

Mencarelli, D.

Menyuk, C. R.

A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, and C. R. Menyuk, “An analytical formula for the mean differential group delay of randomly-birefringent spun fibers,” J. Lightwave Technol. 21(7), 1635–1643 (2003).
[Crossref]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Neifeld, M. A.

Painchaud, Y.

Palmieri, L.

Pascal, D.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Pizzinat, A.

Rozzi, T.

Seligson, J.

J. Seligson, “The orthogonality relation for TE- and TM-modes in guided-wave optics,” J. Lightwave Technol. 6(8), 1260–1264 (1988).
[Crossref]

Senden, T. J.

J. D. Love, T. J. Senden, and F. Ladouceur, “Effect of side wall roughness in buried channel waveguides,” IEE Proc., Optoelectron. 141(4), 242–248 (1994).
[Crossref]

Simard, A. D.

Vasic, B. V.

Vivien, L.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

Wai, P. K. A.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Winzer, P. J.

Yan, X.

Appl. Opt. (1)

Bell Syst. Tech. J. (4)

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48(10), 3187–3215 (1969).
[Crossref]

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
[Crossref]

D. Marcuse, “Power distribution and radiation losses in multimode dielectric waveguides,” Bell Syst. Tech. J. 51(2), 429–454 (1972).
[Crossref]

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54(6), 985–995 (1975).
[Crossref]

IEE Proc., Optoelectron. (1)

J. D. Love, T. J. Senden, and F. Ladouceur, “Effect of side wall roughness in buried channel waveguides,” IEE Proc., Optoelectron. 141(4), 242–248 (1994).
[Crossref]

IEEE Photonics Technol. Lett. (1)

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16(7), 1661–1663 (2004).
[Crossref]

J. Lightwave Technol. (9)

A. D. Simard, N. Ayotte, Y. Painchaud, S. Bédard, and S. LaRochelle, “Impact of Sidewall Roughness on Integrated Bragg Gratings,” J. Lightwave Technol. 29(24), 3693–3704 (2011).
[Crossref]

A. Di Donato, M. Farina, D. Mencarelli, A. Lucesoli, S. Fabiani, T. Rozzi, G. M. Di Gregorio, and G. Angeloni, “Stationary Mode Distribution and Sidewall Roughness Effects in Overmoded Optical Waveguides,” J. Lightwave Technol. 28(10), 1510–1520 (2010).
[Crossref]

T. Barwicz and H. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness in micro-photonic waveguides,” J. Lightwave Technol. 23(9), 2719–2732 (2005).
[Crossref]

K.-P. Ho and J. M. Kahn, “Frequency diversity in mode-division multiplexing systems,” J. Lightwave Technol. 29(24), 3719–3726 (2011).
[Crossref]

S. O. Arik, D. Askarov, and J. M. Kahn, “Effect of mode coupling on signal processing complexity in mode-division multiplexing,” J. Lightwave Technol. 31(3), 423–431 (2013).
[Crossref]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, and C. R. Menyuk, “An analytical formula for the mean differential group delay of randomly-birefringent spun fibers,” J. Lightwave Technol. 21(7), 1635–1643 (2003).
[Crossref]

A. Mecozzi, “A Theory of Polarization-Mode Dispersion of Spun Fibers,” J. Lightwave Technol. 27(7), 938–943 (2009).
[Crossref]

J. Seligson, “The orthogonality relation for TE- and TM-modes in guided-wave optics,” J. Lightwave Technol. 6(8), 1260–1264 (1988).
[Crossref]

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

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

Opt. Express (2)

Other (4)

A. W. Snyder and J. Love, Optical Waveguide Theory, 1st ed., Springer, 1983.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1974).

K.-P. Ho and J. M. Kahn, “Mode coupling and its impact on spatially multiplexed systems,” in Optical Fiber Telecommunications (OFC) VI, B, I. P. Kaminow, T. Li, and A. E. Willner, eds. (2013), pp. 491–568.

J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. 137, 282–288, 1990.
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

The radiation loss of E x 11 mode versus correlation length in the x/y direction, the z directional correlation length is 0.75μm.

Fig. 2
Fig. 2

The radiation loss of E x 11 mode versus correlation length in the x/y direction, the z directional correlation length is 50 nm.

Fig. 3
Fig. 3

The power evolution of the fundamental mode (mode 1).

Fig. 4
Fig. 4

The power evolution of mode 2 to mode 4.

Fig. 5
Fig. 5

The power standard deviation evolution of the fundamental mode.

Fig. 6
Fig. 6

The power standard deviation evolution of mode 2 to mode 4.

Fig. 7
Fig. 7

The power correlation coefficients between mode 1 and mode 2 to mode 4 along the propagation distance.

Tables (2)

Tables Icon

Table 1 Effective Indexes of the Ten Modes of the Waveguide

Tables Icon

Table 2 Output Power and Output Power Standard Deviations at the End of the Propagation

Equations (63)

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2 φ+ k 2 φ=0
2 φ+δ k 2 φ+ k 2 φ=0 δ k 2 = k 2 k 2
φ= a m ϕ m e j β m z
2 ϕ m +( k 2 β m 2 )φ=0
ϕ m 2j β m a m ( z ) z e j β m z +δ k 2 a m' ( z ) ϕ m' e j β m z =0
d a m ( z ) dz = C mm' ( z ) e j( β m β m' )z a m' ( z )
C mm' ( z )= 1 2j β m xy ϕ m * ϕ m' δ k 2 dS xy | ϕ m | 2 dS
xy ϕ m ϕ n * dS = 2ωμ β m δ mn
C mm' ( z )= k 0 2 j4ωμ xy ϕ m * ϕ m' δ n 2 dS δ k 2 =δ n 2 k 0 2 δ n 2 = n 2 n 2
δ n 2 ( x,y,z )δ n 2 ( x',y',z' ) = R xy ( xx',yy' )η( zz' ) C mm' ( z ) C mm' * ( z' ) = κ mm' η( zz' )
κ mm' = k 0 4 16 ω 2 μ 2 xy x'y' ϕ m * ( x,y ) ϕ m ( x',y' ) ϕ m' ( x,y ) ϕ m' * ( x',y' )R( xx',yy' )dSdS'
C mm' ( z )= ς mm' f( z ) f( z )f( z' ) =η( zz' )
ς mm' = k 0 2 j4ωμ ϕ m * ( a,y ) ϕ m' ( a,y )dy
d a m ( z )= a m ( z+dz ) a m ( z ) = m' a m' ( z ) z z+dz C mm' e j( β m β m' )z' dz' + m' d a m' ( z ) 2 z z+dz C mm' e j( β m' β m )z' dz' = m' a m' ( z ) z z+dz C mm' e j( β m β m' )z' dz' + 1 2 ( m' z z+dz C mm' e j( β m β m' )z' dz' z z+dz C mm' * e j( β m β m' )z' dz' ) a m ( z )+O( d z 3/2 )
P t = a H a
d P t dz =0
P m ( z )= a m ( z ) a m * ( z )
d( a m ( z ) a m * ( z ) )=d( a m ( z ) ) a m * ( z )+ a m ( z )d( a m * ( z ) )+d a m ( z )d a m * ( z )
d P m ( z )= P m' ( z ) z z+dz C mm' e j( β m β m' )z' dz' z z+dz C mm' * e j( β m β m' )z'' dz'' P m ( z ) z z+dz C mm' e j( β m β m' )z' dz' z z+dz C mm' * e j( β m β m' )z'' dz''
z z+dz C mm' e j( β m β m' )z' dz' z z+dz C mm' * e j( β m β m' )z'' dz'' =dz κ mm' dz dz η( z' ) e j( β m β m' )z' dz'
κ mm' = k 0 4 16 ω 2 μ 2 xy x'y' ϕ m * ( x,y ) ϕ m' ( x',y' )R( xx',yy' )dSdS'
dz dz η( z' ) e j( β m β m' )z' dz' FT( η )( β m β m' )
d P m dz = κ mm' FT( η )( β m β m' )( P m' ( z ) P m ( z ) )
P m 2 ( z ) = a m 2 ( z ) a m * 2 ( z )
d P m 2 ( z )=2 a m ( z ) a m * 2 ( z )d a m ( z )+2 a m * ( z ) a m 2 ( z )d a m * ( z ) +2 a m ( z )2 a m * ( z )d a m ( z )d a m * ( z )+ a m * 2 ( z )d a m ( z )d a m ( z ) + a m 2 ( z )d a m * ( z )d a m * ( z )
d P m 2 ( z ) dz =2 P m ( z ) 2 m' κ mm' FT( η )( β m β m' ) +4 m' P m ( z ) P m' ( z ) κ mm' FT( η )( β m β m' ) 2 κ mm FT( η )( 0 ) P m ( z ) 2
d P m ( z ) P n ( z )=d a m ( z ) a m * ( z ) a n ( z ) a n * ( z )+ a m ( z )d a m * ( z ) a n ( z ) a n * ( z ) + a m ( z ) a m * ( z )d a n ( z ) a n * ( z )+ a m ( z ) a m * ( z ) a n ( z )d a n * ( z ) +d a m ( z )d a m * ( z ) a n ( z ) a n * ( z )+ a m ( z ) a m * ( z )d a n ( z )d a n * ( z ) +d a m ( z ) a m * ( z )d a n ( z ) a n * ( z )+ a m ( z )d a m * ( z ) a n ( z )d a n * ( z ) +d a m ( z ) a m * ( z ) a n ( z )d a n * ( z )+ a m ( z )d a m * ( z )d a n ( z ) a n * ( z )
d P m P n dz = P m P n m' κ mm' FT( η )( β m β m' ) P m P n n' κ nn' FT( η )( β n β n' ) + m' P m ( z ) P m' ( z ) κ mm' FT( η )( β m β m' ) + n' P n ( z ) P n' ( z ) κ nn' FT( η )( β n β n' ) 2 κ mn FT( η )( 0 ) P m P n
δ P m 2 = P m 2 P m 2 C( P m , P n )= P m P n P m P n
dP dz =KP
P( z )=exp( Kz )P( 0 )
dQ( z ) dz =MQ( z )
Q=( P 1 P 1 P 1 P n P 2 P 1 P 2 P n P n P 1 P n P n )
Q( z )=exp( Mz )Q( 0 )
C mm' ( z )= jω ε 0 xy e m * e m' δ n 2 dS ( ( e m × h m * )+( e m * × h m ) ) z dS
P m = 1 4 ( ( e m × h m * )+( e m * × h m ) ) z dS=1
C mm' ( z )= k 0 2 j4ω μ 0 xy e m * e m' δ n 2 dS
κ mm' = ω 2 ε 0 2 16 xy x'y' e m * ( x,y )· e m' ( x,y ) e m ( x',y' )· e m' * ( x',y' )R( xx',yy' )dSdS'
κ mm' = ω 2 ε 0 2 16 x x' e m * ( x,b ) e m' ( x,b ) e m ( x',b ) e m' * ( x',b ) R x ( xx' )dxdx'
e p ( k x , k y )= ρ p ( k x , k y )exp( j k x x+j k y y ) ρ p ( k x , k y )= 1 2π 2ω μ 0 β p ( k x , k y ) β p ( k x , k y )= n 2 2 k 0 2 k x 2 k y 2
xy e p ( k x , k y ) e p ( k x ', k y ' )dxdy = 2ω μ 0 β p ( k x , k y ) δ( k x k x ' )δ( k y k y ' )
α m = k x k y κ mp FT( η )( β m β p )d k x d k y κ mp = ω 2 ε 0 2 | ρ p | 2 16 x x' e m * ( x,b ) e m ( x',b )exp( j k x ( xx' ) ) R x ( xx' )dxdx'
α m = ω 2 ε 0 2 ω μ 0 16π ( n 1 2 n 2 2 ) 2 e m * ( b ) e m ( b ) k y FT( η )( β m β p ) 1 n 2 2 k 0 2 k y 2 d k y
k y = n 2 k 0 sinθ
α m = ω 2 ε 0 2 ω μ 0 16π ( n 1 2 n 2 2 ) 2 e m * ( b ) e m ( b ) 0 π FT( η )( β m n 2 k 0 cosθ )dθ
α m = k x k y ω 2 ε 0 2 | ρ p | 2 ( n 1 2 n 2 2 ) 2 16 | x e m * ( x,b )exp( j k x x )dx | 2 FT( η )( β m β p )d k x d k y
C mm' ( z )= ( n 1 2 n 2 2 ) k 0 2 ϕ m ϕ m' γ m γ m' 2j β m β m' ( 1+ γ m d )( 1+ γ m' d ) ( f( z ) ( 1 ) m+m' h( z ) ) ϕ m ={ cos( κ m d )m=0,2,4 sin( κ m d )m=1,3,5 γ m = β m 2 n 2 2 k 0 2 κ m = n 1 2 k 0 2 β m 2
f( z )f( z' ) = h( z )h( z' ) =η( zz' )= σ ¯ 2 e ( zz' ) 2 D 2
FT( σ ¯ 2 e ( zz' ) 2 D 2 )= σ ¯ 2 π D e D 2 ω 2 4
C mm' ( z ) C mm' * ( z' ) = ( n 1 2 n 2 2 ) 2 k 0 4 ϕ m 2 ϕ m' 2 γ m γ m' 2 β m β m' ( 1+ γ m d )( 1+ γ m' d ) σ ¯ 2 e ( zz' ) 2 D 2
α m = n 2 k 0 n 2 k 0 FT( η )( β m β ) I m ( β ) dβ I m ( β )= ( n 1 2 n 2 2 ) k 0 3 n 1 sin 2 θ m 2πdcos θ m ( 1+ γ m d ) ( ρ cos 2 σd ρ 2 cos 2 σd+ σ 2 sin 2 σd + ρ sin 2 σd ρ 2 sin 2 σd+ σ 2 cos 2 σd ) ρ= n 2 2 k 0 2 β 2 σ= n 1 2 k 0 2 β 2 sin θ m = ( m+1 )π 2 n 1 k 0 d
α m = π ( n 1 2 n 2 2 ) k 0 3 n 1 sin 2 θ m 2cos θ m ( 1+ 1 γ m d ) σ ¯ 2 D d
α m = σ ¯ 2 e D 2 4 ( β m n 2 k 0 ) 2 n 1 k 0 3 sin 2 θ m 2d n 2 k 0 ( β m n 2 k 0 ) ( 1+ 1 γ m d )cos θ m ·{ 2 n 2 k 0 D 2 ( β m n 2 k 0 ) ( cot σ 2 d+ tan 2 σd )and tanσd0 cotσd0 ( n 1 2 n 2 2 )or tanσd=0 cotσd=0
z σ ¯ 2 k 0 2 d 2
d a m ( z )= m' a m' ( z+ dz 2 ) z z+dz C mm' e j( β m β m' )z' dz'
F T 1 ( FT( C mm' ( z ) C mm' * ( z' ) ) )
C mm' ( z ) C mm' * ( z' ) = 1 2α exp( α| zz' | )
d C mm' dz +α C mm' =n( z ) n( z )n( z' ) =δ( zz' )
a m' ( z+ dz 2 )= a m' ( z )+ d a m' ( z ) dz dz 2 + = a m' ( z )+ d a m' ( z ) 2 +
d a m ( z )= m' a m' ( z ) z z+dz C mm' e j( β m β m' )z' dz' + m' d a m' ( z ) 2 z z+dz C mm' e j( β m β m' )z' dz'
d a m ( z )= m' a m' ( z ) z z+dz C mm' e j( β m β m' )z' dz' + m' n a n ( z ) z z+dz C m'n e j( β m' β n )z' dz' 2 z z+dz C mm' e j( β m β m' )z' dz' +O( d z 3/2 )
d a m ( z )= m' a m' ( z ) z z+dz C mm' e j( β m β m' )z' dz' + 1 2 ( m' z z+dz C m'm e j( β m' β m )z' dz' z z+dz C mm' e j( β m β m' )z' dz' ) a m ( z )+O( d z 3/2 )
d a m ( z )= m' a m' ( z ) z z+dz C mm' e j( β m β m' )z' dz' + 1 2 ( m' z z+dz C mm' e j( β m β m' )z' dz' z z+dz C mm' * e j( β m β m' )z' dz' ) a m ( z )+O( d z 3/2 )

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