Abstract

This paper discusses a numerical method for computing the electromagnetic modes supported by multilayer planar optical waveguides constructed from lossy or active media, having in general a diagonal permittivity tensor. The method solves the dispersion equations in the complex plane via the Cauchy integration method. It is applicable to lossless, lossy and active waveguides, and to AntiResonant Reflecting Optical Waveguides (ARROW’s). Analytical derivatives for the dispersion equations are derived and presented for what is believed to be the first time, and a new algorithm that significantly reduces the time required to compute the derivatives is given. This has a double impact: improved accuracy and reduced computation time compared to the standard approach. A different integration contour, which is suitable for leaky modes is also presented. Comparisons are made with results found in the literature; excellent agreement is noted for all comparisons made.

© 2000 Optical Society of America

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References

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  1. J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguide,” J. Opt. Soc. Am. A 1, 742–753, (1984)
    [Crossref]
  2. L. M. Walpita, “Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix,” J. Opt. Soc. Am. A 2, 595–602, (1985)
    [Crossref]
  3. K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron.,  26, 627–630, (1990)
    [Crossref]
  4. L. Sun and E. Marhic, “Numerical study of attenuation in multilayer infrared waveguides by the circle-chain convergence method”, J. Opt. Soc. Am. B 8, 478–483, (1991)
    [Crossref]
  5. L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp.,  21, 543–560, (1967)
    [Crossref]
  6. L. C. Botten and M. S. Craig, “Complex zeros of analytic functions”, Comput. Phys. Commun.,  29, 245–259, (1983)
    [Crossref]
  7. E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures”, J. Lightwave Tech.,  10, 1344–1351, (1992)
    [Crossref]
  8. R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron.,  28, 1520–1526, (1992)
    [Crossref]
  9. Hermann A. Haus, “Waves and Fields in Optoelectronics,” (New Jersey, Prentice-Hall Inc., 1984). Ch. 11
  10. J. W. Brown and R. V. Churchill, “Complex Variables and Applications,” (Sixth Edition, New York: McGraw-Hill, 1996)
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in C,” (Second Edition, Cambridge, 1994)
  12. J. R. Rice, “Numerical Methods, Software, and Analysis,” (IMSL Reference Edition. New York: McGraw-Hill, 1983)
  13. A. S. Kronrod, “Nodes and Weights of Quadrature Formulas,” (NewYork: Consultants Bureau, 1965)
  14. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures”, J. Lightwave Tech.,  12, 2080–2084, (1994)
    [Crossref]
  15. T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonsnt reflecting optical waveguides-Numerical Results and Analytical Expressions”, IEEE J. Quantum Electron.,  28, 1689–1700, (1992)
    [Crossref]
  16. W. Huang, R. M. Shubiar, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures”, J. Lightwave Tech.,  10, 1015–1022. (1992)
    [Crossref]
  17. J. Deng and Y. Huang,“A novel hybrid coupler based on antiresonant reflecting optical waveguides,” J. Lightwave Tech.,  16, 1062–1069, (1998)
    [Crossref]
  18. B. Ray and G W. Hanson, “Some effects of anisotropy on planar antiresonant reflecting optical waveguides”, J. Lightwave Tech.,  14, 202–208, (1996)
    [Crossref]
  19. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multiplayer optical waveguides: reflection pole method and wavevector density method”, J. Lightwave Tech.,  17, 929–941, (1999)
    [Crossref]

1999 (1)

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multiplayer optical waveguides: reflection pole method and wavevector density method”, J. Lightwave Tech.,  17, 929–941, (1999)
[Crossref]

1998 (1)

J. Deng and Y. Huang,“A novel hybrid coupler based on antiresonant reflecting optical waveguides,” J. Lightwave Tech.,  16, 1062–1069, (1998)
[Crossref]

1996 (1)

B. Ray and G W. Hanson, “Some effects of anisotropy on planar antiresonant reflecting optical waveguides”, J. Lightwave Tech.,  14, 202–208, (1996)
[Crossref]

1994 (1)

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures”, J. Lightwave Tech.,  12, 2080–2084, (1994)
[Crossref]

1992 (4)

T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonsnt reflecting optical waveguides-Numerical Results and Analytical Expressions”, IEEE J. Quantum Electron.,  28, 1689–1700, (1992)
[Crossref]

W. Huang, R. M. Shubiar, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures”, J. Lightwave Tech.,  10, 1015–1022. (1992)
[Crossref]

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures”, J. Lightwave Tech.,  10, 1344–1351, (1992)
[Crossref]

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron.,  28, 1520–1526, (1992)
[Crossref]

1991 (1)

1990 (1)

K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron.,  26, 627–630, (1990)
[Crossref]

1985 (1)

1984 (1)

1983 (1)

L. C. Botten and M. S. Craig, “Complex zeros of analytic functions”, Comput. Phys. Commun.,  29, 245–259, (1983)
[Crossref]

1967 (1)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp.,  21, 543–560, (1967)
[Crossref]

Anemogiannis, E.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multiplayer optical waveguides: reflection pole method and wavevector density method”, J. Lightwave Tech.,  17, 929–941, (1999)
[Crossref]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures”, J. Lightwave Tech.,  12, 2080–2084, (1994)
[Crossref]

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures”, J. Lightwave Tech.,  10, 1344–1351, (1992)
[Crossref]

Baba, T.

T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonsnt reflecting optical waveguides-Numerical Results and Analytical Expressions”, IEEE J. Quantum Electron.,  28, 1689–1700, (1992)
[Crossref]

Botten, L. C.

L. C. Botten and M. S. Craig, “Complex zeros of analytic functions”, Comput. Phys. Commun.,  29, 245–259, (1983)
[Crossref]

Brown, J. W.

J. W. Brown and R. V. Churchill, “Complex Variables and Applications,” (Sixth Edition, New York: McGraw-Hill, 1996)

Chilwell, J.

Chow, Y. L.

W. Huang, R. M. Shubiar, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures”, J. Lightwave Tech.,  10, 1015–1022. (1992)
[Crossref]

Churchill, R. V.

J. W. Brown and R. V. Churchill, “Complex Variables and Applications,” (Sixth Edition, New York: McGraw-Hill, 1996)

Craig, M. S.

L. C. Botten and M. S. Craig, “Complex zeros of analytic functions”, Comput. Phys. Commun.,  29, 245–259, (1983)
[Crossref]

Delves, L. M.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp.,  21, 543–560, (1967)
[Crossref]

Deng, J.

J. Deng and Y. Huang,“A novel hybrid coupler based on antiresonant reflecting optical waveguides,” J. Lightwave Tech.,  16, 1062–1069, (1998)
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in C,” (Second Edition, Cambridge, 1994)

Forbes, G. W.

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron.,  28, 1520–1526, (1992)
[Crossref]

Gaylord, T. K.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multiplayer optical waveguides: reflection pole method and wavevector density method”, J. Lightwave Tech.,  17, 929–941, (1999)
[Crossref]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures”, J. Lightwave Tech.,  12, 2080–2084, (1994)
[Crossref]

Glytsis, E. N.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multiplayer optical waveguides: reflection pole method and wavevector density method”, J. Lightwave Tech.,  17, 929–941, (1999)
[Crossref]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures”, J. Lightwave Tech.,  12, 2080–2084, (1994)
[Crossref]

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures”, J. Lightwave Tech.,  10, 1344–1351, (1992)
[Crossref]

Hanson, G W.

B. Ray and G W. Hanson, “Some effects of anisotropy on planar antiresonant reflecting optical waveguides”, J. Lightwave Tech.,  14, 202–208, (1996)
[Crossref]

Haus, Hermann A.

Hermann A. Haus, “Waves and Fields in Optoelectronics,” (New Jersey, Prentice-Hall Inc., 1984). Ch. 11

Hodgkinson, I.

Houde-Walter, S. N.

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron.,  28, 1520–1526, (1992)
[Crossref]

Huang, W.

W. Huang, R. M. Shubiar, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures”, J. Lightwave Tech.,  10, 1015–1022. (1992)
[Crossref]

Huang, Y.

J. Deng and Y. Huang,“A novel hybrid coupler based on antiresonant reflecting optical waveguides,” J. Lightwave Tech.,  16, 1062–1069, (1998)
[Crossref]

Kokubun, Y.

T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonsnt reflecting optical waveguides-Numerical Results and Analytical Expressions”, IEEE J. Quantum Electron.,  28, 1689–1700, (1992)
[Crossref]

Kronrod, A. S.

A. S. Kronrod, “Nodes and Weights of Quadrature Formulas,” (NewYork: Consultants Bureau, 1965)

Lyness, J. N.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp.,  21, 543–560, (1967)
[Crossref]

Marhic, E.

Nathan, A.

W. Huang, R. M. Shubiar, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures”, J. Lightwave Tech.,  10, 1015–1022. (1992)
[Crossref]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in C,” (Second Edition, Cambridge, 1994)

Ray, B.

B. Ray and G W. Hanson, “Some effects of anisotropy on planar antiresonant reflecting optical waveguides”, J. Lightwave Tech.,  14, 202–208, (1996)
[Crossref]

Rice, J. R.

J. R. Rice, “Numerical Methods, Software, and Analysis,” (IMSL Reference Edition. New York: McGraw-Hill, 1983)

Schlereth, K. H.

K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron.,  26, 627–630, (1990)
[Crossref]

Shubiar, R. M.

W. Huang, R. M. Shubiar, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures”, J. Lightwave Tech.,  10, 1015–1022. (1992)
[Crossref]

Smith, R. E.

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron.,  28, 1520–1526, (1992)
[Crossref]

Sun, L.

Tacke, M.

K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron.,  26, 627–630, (1990)
[Crossref]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in C,” (Second Edition, Cambridge, 1994)

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in C,” (Second Edition, Cambridge, 1994)

Walpita, L. M.

Comput. Phys. Commun. (1)

L. C. Botten and M. S. Craig, “Complex zeros of analytic functions”, Comput. Phys. Commun.,  29, 245–259, (1983)
[Crossref]

IEEE J. Quantum Electron. (3)

R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron.,  28, 1520–1526, (1992)
[Crossref]

T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonsnt reflecting optical waveguides-Numerical Results and Analytical Expressions”, IEEE J. Quantum Electron.,  28, 1689–1700, (1992)
[Crossref]

K. H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: An algorithm for a personal computer,” IEEE J. Quantum Electron.,  26, 627–630, (1990)
[Crossref]

J. Lightwave Tech. (6)

W. Huang, R. M. Shubiar, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures”, J. Lightwave Tech.,  10, 1015–1022. (1992)
[Crossref]

J. Deng and Y. Huang,“A novel hybrid coupler based on antiresonant reflecting optical waveguides,” J. Lightwave Tech.,  16, 1062–1069, (1998)
[Crossref]

B. Ray and G W. Hanson, “Some effects of anisotropy on planar antiresonant reflecting optical waveguides”, J. Lightwave Tech.,  14, 202–208, (1996)
[Crossref]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multiplayer optical waveguides: reflection pole method and wavevector density method”, J. Lightwave Tech.,  17, 929–941, (1999)
[Crossref]

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures”, J. Lightwave Tech.,  10, 1344–1351, (1992)
[Crossref]

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures”, J. Lightwave Tech.,  12, 2080–2084, (1994)
[Crossref]

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

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

Math. Comp. (1)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp.,  21, 543–560, (1967)
[Crossref]

Other (5)

Hermann A. Haus, “Waves and Fields in Optoelectronics,” (New Jersey, Prentice-Hall Inc., 1984). Ch. 11

J. W. Brown and R. V. Churchill, “Complex Variables and Applications,” (Sixth Edition, New York: McGraw-Hill, 1996)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in C,” (Second Edition, Cambridge, 1994)

J. R. Rice, “Numerical Methods, Software, and Analysis,” (IMSL Reference Edition. New York: McGraw-Hill, 1983)

A. S. Kronrod, “Nodes and Weights of Quadrature Formulas,” (NewYork: Consultants Bureau, 1965)

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Figures (4)

Figure 1.
Figure 1.

Structure of the multilayer planar optical waveguide

Figure 2.
Figure 2.

The integral contours in the complex plane. The ‘ 1 correspond to the poles, ñ max is the complex refractive index with the maximum real part and is enclosed by C1, for TE modes in anisotropic media: ñ max=ñ yymax, ñs =ñyys and ñc =ñyyc . For TM modes in anisotropic media: ñ max=ñ xxmax, ñs =ñxxs and ñc =ñxxc .

Figure 3.
Figure 3.

Comparison of the normalized execution times required for the ADR and the CIM based on a numerical derivative (ND) and an analytical derivative (AD).

Figure 4.
Figure 4.

(a) Field distribution for the symmetric TM mode: TM1. (b) Field distribution for the first anti-symmetric TM mode: TM2.

Tables (5)

Tables Icon

TABLE I. Effective Index of Guided Modes Supported by an Inhomogeneous Waveguide.

Tables Icon

TABLE II. Effective Index of Guided Modes Supported by a 6-layer Quantum Well Active Waveguide.

Tables Icon

TABLE III. Effective Index of Leaky Modes Supported by a 4-layer Lossless Waveguide.

Tables Icon

TABLE IV. Effective Index of ARROW Modes Supported by a 9-layer ARROW Waveguide

Tables Icon

TABLE V. Effective Index of ARROW Modes Supported by a 3-layer Anisotropic ARROW Waveguide

Equations (37)

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× E = j ω μ 0 H
× H = j ω ε 0 ε ¯ ¯ r E
ε ¯ ¯ r = [ ε ˜ r , xx ε ˜ r , yy ε ˜ r , zz ] = [ n ˜ xx 2 n ˜ yy 2 n ˜ zz 2 ]
E i = y ̂ E yi ( x ) exp ( j ω t j γ ˜ z )
H i = [ x ̂ H xi ( x ) + z ̂ H zi ( x ) ] exp ( j ω t j γ ˜ z )
d 2 E yi ( x ) dx + κ ˜ i 2 E yi ( x ) = 0
H zi ( x ) = j ω μ 0 dE yi ( x ) dx
H xi ( x ) = γ ˜ ω μ 0 E yi ( x )
E yi ( x ) = cos [ κ ˜ i ( x x i ) ] E yi ( x i ) + 1 κ ˜ i sin [ κ ˜ i ( x x i ) ] dE yi ( x i ) dx
dE yi ( x i ) dx = κ ˜ i sin [ κ ˜ i ( x x i ) ] E yi ( x i ) + cos [ κ ˜ i ( x x i ) ] dE yi ( x i ) dx
( E yi ( x i ) dE yi ( x i ) dx ) = ( cos [ κ ˜ i ( x x i ) ] 1 κ ˜ i sin [ κ ˜ i ( x x i ) ] κ ˜ i sin [ κ ˜ i ( x x i ) ] cos [ κ ˜ i ( x x i ) ] ) ( E yi ( x ) dE yi ( x ) dx )
( E ys dE ys dx ) = i = 1 r M i ( E yc dE yc dx )
= ( m 11 m 12 m 21 m 22 ) ( E yc dE yc dx )
M i = ( cos ( κ ˜ i d i ) 1 κ ˜ i sin ( κ ˜ i d i ) κ ˜ i sin ( κ ˜ i d i ) cos ( κ ˜ i d i ) ) for i = 1 , 2 , , r
E ys ( x ) = A s exp ( γ ˜ s x ) dE ys ( x ) dx = γ ˜ s A s exp ( γ ˜ s x ) } for x 0
E yc ( x ) = B c exp [ γ ˜ c ( x x r + 1 ) ] dE yc ( x ) dx = γ ˜ c B c exp [ γ ˜ c ( x x r + 1 ) ] } for x x r + 1
F ( γ ˜ ) = γ ˜ s m 11 + γ ˜ c m 22 m 21 γ ˜ s γ ˜ c m 12 = 0
d 2 H yi ( x ) dx + κ ˜ i 2 H yi ( x ) = 0
E zi ( x ) = j ω ε 0 n ˜ zzi 2 dH yi ( x ) dx
E xi ( x ) = γ ˜ ω ε 0 n ˜ xxi 2 H yi ( x )
M i = ( cos ( κ ˜ i d i ) n ˜ zzi 2 κ ˜ i sin ( κ ˜ i d i ) κ ˜ i n ˜ zzi 2 sin ( κ ˜ i d i ) cos ( κ ˜ i d i ) ) for i = 1 , 2 , , r
F ( γ ˜ ) = γ ˜ s n ˜ zzs 2 m 11 + γ ˜ c n ˜ zzc 2 m 22 m 21 γ ˜ s γ ˜ c n ˜ zzs 2 n ˜ zzc 2 m 12 = 0
S 0 = 1 j 2 π c f ( z ) f ( z ) dz = N z N p
S m = 1 j 2 π c z m f ( z ) f ( z ) dz = i = 1 s 0 z i m for m = 1 , 2 , , S 0
p ( z ) = i = 1 s 0 ( z z i ) = k = 0 s 0 C k z k
C k = 1 ( k S 0 ) j = 1 s 0 k S j C k + j for k = S 0 1 , , 0
f ( z 0 ) = 1 j 2 π D f ( z ) ( z z 0 ) 2 dz
R f ( z 0 ) R [ f ( z 0 ) ] m = 1 m l = 1 m f ( z 0 + Re j 2 π m l ) e j 2 π m l
a b F ( z ) = i = 1 K W i F ( z i )
dM i d u ˜ = u ˜ ( k 0 2 d i κ ˜ i sin ( κ ˜ i d i ) k 0 2 d i κ ˜ i 2 cos ( κ ˜ i d i ) k 0 κ ˜ i 3 sin ( κ ˜ i d i ) k 0 2 κ ˜ i sin ( κ ˜ i d i ) k 0 2 d i cos ( κ ˜ i d i ) k 0 2 d i κ ˜ i sin ( κ ˜ i d i ) ) for i = 1 , , r
dM d u ˜ = ( dm 11 d u ˜ dm 12 d u ˜ dm 21 d u ˜ dm 22 d u ˜ ) = j = 1 r ( dM j d u ˜ · i = 1 i j r M i )
dF ( u ˜ ) d u ˜ = k 0 2 u ˜ γ ˜ s m 11 + k 0 2 u ˜ γ ˜ c m 22 + γ ˜ s dm 11 d u ˜ + γ ˜ c dm 22 d u ˜ dm 21 d u ˜ k 0 2 u ˜ γ ˜ c γ ˜ s m 12 k 0 2 u ˜ γ ˜ s γ ˜ c m 12 γ ˜ s γ ˜ c dm 12 d u ˜
dM i d u ˜ = u ˜ ( k 2 d i κ ˜ i sin ( κ ˜ i d i ) k 2 n ˜ zzi 2 d i κ ˜ i 2 cos ( κ ˜ i d i ) k 2 n ˜ zzi 2 κ ˜ i 3 sin ( κ ˜ i d i ) k 2 κ ˜ i n ˜ zzi 2 sin ( κ ˜ i d i ) k 2 d i n ˜ zzi 2 cos ( κ ˜ i d i ) k 2 d i κ ˜ i sin ( κ ˜ i d i ) ) for i = 1 , , r
dF ( u ˜ ) d u ˜ = k 0 2 u ˜ γ ˜ s n ˜ xxs 2 m 11 + k 0 2 u ˜ γ ˜ c n ˜ xxc 2 m 22 + γ ˜ s n ˜ zzs 2 dm 11 d u ˜ + γ ˜ c n ˜ zzc 2 dm 22 d u ˜ dm 21 d u ˜ γ ˜ s γ ˜ c n ˜ zzs 2 n ˜ zzc 2 dm 12 d u ˜
k 0 2 u ˜ γ ˜ c n ˜ xxs 2 n ˜ zzc 2 γ ˜ s m 12 k 0 2 u ˜ γ ˜ s n ˜ xxc 2 n ˜ zzs 2 γ ˜ c m 12
n ( x ) = n s 2 + 2 Δ n s e ( x α ) for x < 0
n ( x ) = n c for x > 0

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