Abstract

The Lanczos–Fourier series expansion is employed to analyze the guided-mode field in an asymmetrical slab waveguide, the core of which has an anisotropic and inhomogeneous dielectric permittivity. A system of linear homogeneous equations is derived by the collocation technique with consideration of the wave equation and the appropriate boundary conditions at the interfaces between the core and cladding media. The propagation constants are found from a determinant equation that ensures the existence of a nontrivial solution of the system. Numerical results are presented for several cases of dielectric permittivity, including the constant, parabolic, linear, and anisotropic cases. This approach is found to converge reasonably fast, and Richardson’s extrapolation technique is applied to accelerate the convergence further. The approach can be easily generalized from the scalar to the vector equation, and, as an example, we consider the guided modes of a circular fiber.

© 2002 Optical Society of America

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References

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  1. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981) Chaps. 1–5.
  2. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, London, 1991) Chaps. 1 and 9.
  3. F. Sporleder, H. G. Unger, Waveguide Tapers, Transitions and Couplers (IEE Electromagnetic Waves Series 6, London, 1979).
  4. M. S. Stern, “Semivectorial polarised H-field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc.-J: Optoelectron. 135, 333–338 (1988).
  5. D. Gómez Pedreira, P. Joly, “Mathematical analysis of a method to compute guided waves in integrated optics,” (Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, 2000).
  6. A. Bermudez, D. Gómez Pedreira, P. Joly, “A hybrid approach for the computation of guided modes in integrated optics,” (Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, 2001).
  7. S. Sujecki, T. M. Benson, P. Sewell, P. C. Kendall, “Novel vectorial analysis of optical waveguides,” J. Lightwave Technol. 16, 1329–1335 (1998).
    [CrossRef]
  8. M. N. Armenise, M. De Sario, “Investigation of the guided modes in anisotropic diffused slab waveguide with embedded metal layer,” Fiber Integr. Opt. 3, 197–219 (1980).
    [CrossRef]
  9. M. N. Armenise, A. G. Perri, “A new technique for analysing light propagation in lithium niobate slab waveguides,” J. Mod. Opt. 35, 947–958 (1988).
    [CrossRef]
  10. D. Rafizadeh, S.-T. Ho, “Numerical analysis of vectorial wave propagation in waveguides with arbitrary refractive index profiles,” Opt. Commun. 141, 17–20 (1997).
    [CrossRef]
  11. Z. Cao, J. Yi, Y. Chen, “Analytical investigations of planar optical waveguides with arbitrary index profile,” Opt. Quantum Electron. 31, 637–644 (1999).
    [CrossRef]
  12. S. S. Patrick, K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 40, 692–698 (1992).
    [CrossRef]
  13. A. I. Kleyev, A. B. Manenkov, A. G. Rozhnev, “Numerical methods of calculating dielectric waveguides (or fiber lightguides): special methods (A review),” J. Commun. Technol. Electron. 38, 1–17 (1993).
  14. Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
    [CrossRef]
  15. M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 1992).
  16. F. Fernandez, Y. Lu, Microwave and Optical Waveguide Analysis by the Finite Element Method (Wiley, New York, 1996).
  17. N. E. Nikolaev, V. V. Shevchenko, “The methods of shift formulas applied to embedded planar optical waveguides,” J. Commun. Technol. Electron. 42, 837–839 (1997).
  18. V. Lancellotti, R. Orta, “Modes of open layered anisotropic waveguides: a numerical method with exponential convergence,” Opt. Quantum Electron. 31, 781–796 (1999).
    [CrossRef]
  19. C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1956).
  20. C. A. J. Fletcher, Computational Galerkin Methods (Springer-Verlag, New York, 1984).
  21. I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
    [CrossRef]
  22. V. L. Goncharov, Theory of Function Interpolation and Extrapolation [Gostekhizdat, Moscow, 1954 (in Russian)].
  23. G. E. Forshyte, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  24. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).
  25. G. N. Watson, A Treatise on the Theory of Bessel Functions (MacMillan, New York, 1944).

1999 (3)

Z. Cao, J. Yi, Y. Chen, “Analytical investigations of planar optical waveguides with arbitrary index profile,” Opt. Quantum Electron. 31, 637–644 (1999).
[CrossRef]

V. Lancellotti, R. Orta, “Modes of open layered anisotropic waveguides: a numerical method with exponential convergence,” Opt. Quantum Electron. 31, 781–796 (1999).
[CrossRef]

I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
[CrossRef]

1998 (1)

1997 (2)

D. Rafizadeh, S.-T. Ho, “Numerical analysis of vectorial wave propagation in waveguides with arbitrary refractive index profiles,” Opt. Commun. 141, 17–20 (1997).
[CrossRef]

N. E. Nikolaev, V. V. Shevchenko, “The methods of shift formulas applied to embedded planar optical waveguides,” J. Commun. Technol. Electron. 42, 837–839 (1997).

1993 (1)

A. I. Kleyev, A. B. Manenkov, A. G. Rozhnev, “Numerical methods of calculating dielectric waveguides (or fiber lightguides): special methods (A review),” J. Commun. Technol. Electron. 38, 1–17 (1993).

1992 (1)

S. S. Patrick, K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 40, 692–698 (1992).
[CrossRef]

1990 (1)

Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

1988 (2)

M. N. Armenise, A. G. Perri, “A new technique for analysing light propagation in lithium niobate slab waveguides,” J. Mod. Opt. 35, 947–958 (1988).
[CrossRef]

M. S. Stern, “Semivectorial polarised H-field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc.-J: Optoelectron. 135, 333–338 (1988).

1980 (1)

M. N. Armenise, M. De Sario, “Investigation of the guided modes in anisotropic diffused slab waveguide with embedded metal layer,” Fiber Integr. Opt. 3, 197–219 (1980).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981) Chaps. 1–5.

Armenise, M. N.

M. N. Armenise, A. G. Perri, “A new technique for analysing light propagation in lithium niobate slab waveguides,” J. Mod. Opt. 35, 947–958 (1988).
[CrossRef]

M. N. Armenise, M. De Sario, “Investigation of the guided modes in anisotropic diffused slab waveguide with embedded metal layer,” Fiber Integr. Opt. 3, 197–219 (1980).
[CrossRef]

Benson, T. M.

Bermudez, A.

A. Bermudez, D. Gómez Pedreira, P. Joly, “A hybrid approach for the computation of guided modes in integrated optics,” (Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, 2001).

Cao, Z.

Z. Cao, J. Yi, Y. Chen, “Analytical investigations of planar optical waveguides with arbitrary index profile,” Opt. Quantum Electron. 31, 637–644 (1999).
[CrossRef]

Chen, Y.

Z. Cao, J. Yi, Y. Chen, “Analytical investigations of planar optical waveguides with arbitrary index profile,” Opt. Quantum Electron. 31, 637–644 (1999).
[CrossRef]

Chung, Y.

Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

Dagli, N.

Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

De Sario, M.

M. N. Armenise, M. De Sario, “Investigation of the guided modes in anisotropic diffused slab waveguide with embedded metal layer,” Fiber Integr. Opt. 3, 197–219 (1980).
[CrossRef]

Fernandez, F.

F. Fernandez, Y. Lu, Microwave and Optical Waveguide Analysis by the Finite Element Method (Wiley, New York, 1996).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

Fletcher, C. A. J.

C. A. J. Fletcher, Computational Galerkin Methods (Springer-Verlag, New York, 1984).

Forshyte, G. E.

G. E. Forshyte, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Gómez Pedreira, D.

D. Gómez Pedreira, P. Joly, “Mathematical analysis of a method to compute guided waves in integrated optics,” (Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, 2000).

A. Bermudez, D. Gómez Pedreira, P. Joly, “A hybrid approach for the computation of guided modes in integrated optics,” (Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, 2001).

Goncharov, V. L.

V. L. Goncharov, Theory of Function Interpolation and Extrapolation [Gostekhizdat, Moscow, 1954 (in Russian)].

Ho, S.-T.

D. Rafizadeh, S.-T. Ho, “Numerical analysis of vectorial wave propagation in waveguides with arbitrary refractive index profiles,” Opt. Commun. 141, 17–20 (1997).
[CrossRef]

Joly, P.

A. Bermudez, D. Gómez Pedreira, P. Joly, “A hybrid approach for the computation of guided modes in integrated optics,” (Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, 2001).

D. Gómez Pedreira, P. Joly, “Mathematical analysis of a method to compute guided waves in integrated optics,” (Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, 2000).

Kendall, P. C.

Kleyev, A. I.

A. I. Kleyev, A. B. Manenkov, A. G. Rozhnev, “Numerical methods of calculating dielectric waveguides (or fiber lightguides): special methods (A review),” J. Commun. Technol. Electron. 38, 1–17 (1993).

Lancellotti, V.

V. Lancellotti, R. Orta, “Modes of open layered anisotropic waveguides: a numerical method with exponential convergence,” Opt. Quantum Electron. 31, 781–796 (1999).
[CrossRef]

Lanczos, C.

C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1956).

Lu, Y.

F. Fernandez, Y. Lu, Microwave and Optical Waveguide Analysis by the Finite Element Method (Wiley, New York, 1996).

Malcolm, M. A.

G. E. Forshyte, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Manenkov, A. B.

I. G. Tigelis, A. B. Manenkov, “Scattering from an abruptly terminated asymmetrical slab waveguide,” J. Opt. Soc. Am. A 16, 523–532 (1999).
[CrossRef]

A. I. Kleyev, A. B. Manenkov, A. G. Rozhnev, “Numerical methods of calculating dielectric waveguides (or fiber lightguides): special methods (A review),” J. Commun. Technol. Electron. 38, 1–17 (1993).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, London, 1991) Chaps. 1 and 9.

Moler, C. B.

G. E. Forshyte, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Nikolaev, N. E.

N. E. Nikolaev, V. V. Shevchenko, “The methods of shift formulas applied to embedded planar optical waveguides,” J. Commun. Technol. Electron. 42, 837–839 (1997).

Orta, R.

V. Lancellotti, R. Orta, “Modes of open layered anisotropic waveguides: a numerical method with exponential convergence,” Opt. Quantum Electron. 31, 781–796 (1999).
[CrossRef]

Patrick, S. S.

S. S. Patrick, K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 40, 692–698 (1992).
[CrossRef]

Perri, A. G.

M. N. Armenise, A. G. Perri, “A new technique for analysing light propagation in lithium niobate slab waveguides,” J. Mod. Opt. 35, 947–958 (1988).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

Rafizadeh, D.

D. Rafizadeh, S.-T. Ho, “Numerical analysis of vectorial wave propagation in waveguides with arbitrary refractive index profiles,” Opt. Commun. 141, 17–20 (1997).
[CrossRef]

Rozhnev, A. G.

A. I. Kleyev, A. B. Manenkov, A. G. Rozhnev, “Numerical methods of calculating dielectric waveguides (or fiber lightguides): special methods (A review),” J. Commun. Technol. Electron. 38, 1–17 (1993).

Sadiku, M. N. O.

M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 1992).

Sewell, P.

Shevchenko, V. V.

N. E. Nikolaev, V. V. Shevchenko, “The methods of shift formulas applied to embedded planar optical waveguides,” J. Commun. Technol. Electron. 42, 837–839 (1997).

Sporleder, F.

F. Sporleder, H. G. Unger, Waveguide Tapers, Transitions and Couplers (IEE Electromagnetic Waves Series 6, London, 1979).

Stern, M. S.

M. S. Stern, “Semivectorial polarised H-field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc.-J: Optoelectron. 135, 333–338 (1988).

Sujecki, S.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

Tigelis, I. G.

Unger, H. G.

F. Sporleder, H. G. Unger, Waveguide Tapers, Transitions and Couplers (IEE Electromagnetic Waves Series 6, London, 1979).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (MacMillan, New York, 1944).

Webb, K. J.

S. S. Patrick, K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 40, 692–698 (1992).
[CrossRef]

Yi, J.

Z. Cao, J. Yi, Y. Chen, “Analytical investigations of planar optical waveguides with arbitrary index profile,” Opt. Quantum Electron. 31, 637–644 (1999).
[CrossRef]

Fiber Integr. Opt. (1)

M. N. Armenise, M. De Sario, “Investigation of the guided modes in anisotropic diffused slab waveguide with embedded metal layer,” Fiber Integr. Opt. 3, 197–219 (1980).
[CrossRef]

IEE Proc.-J: Optoelectron. (1)

M. S. Stern, “Semivectorial polarised H-field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc.-J: Optoelectron. 135, 333–338 (1988).

IEEE J. Quantum Electron. (1)

Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. S. Patrick, K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 40, 692–698 (1992).
[CrossRef]

J. Commun. Technol. Electron. (2)

A. I. Kleyev, A. B. Manenkov, A. G. Rozhnev, “Numerical methods of calculating dielectric waveguides (or fiber lightguides): special methods (A review),” J. Commun. Technol. Electron. 38, 1–17 (1993).

N. E. Nikolaev, V. V. Shevchenko, “The methods of shift formulas applied to embedded planar optical waveguides,” J. Commun. Technol. Electron. 42, 837–839 (1997).

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

M. N. Armenise, A. G. Perri, “A new technique for analysing light propagation in lithium niobate slab waveguides,” J. Mod. Opt. 35, 947–958 (1988).
[CrossRef]

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

Opt. Commun. (1)

D. Rafizadeh, S.-T. Ho, “Numerical analysis of vectorial wave propagation in waveguides with arbitrary refractive index profiles,” Opt. Commun. 141, 17–20 (1997).
[CrossRef]

Opt. Quantum Electron. (2)

Z. Cao, J. Yi, Y. Chen, “Analytical investigations of planar optical waveguides with arbitrary index profile,” Opt. Quantum Electron. 31, 637–644 (1999).
[CrossRef]

V. Lancellotti, R. Orta, “Modes of open layered anisotropic waveguides: a numerical method with exponential convergence,” Opt. Quantum Electron. 31, 781–796 (1999).
[CrossRef]

Other (13)

C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1956).

C. A. J. Fletcher, Computational Galerkin Methods (Springer-Verlag, New York, 1984).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981) Chaps. 1–5.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, London, 1991) Chaps. 1 and 9.

F. Sporleder, H. G. Unger, Waveguide Tapers, Transitions and Couplers (IEE Electromagnetic Waves Series 6, London, 1979).

D. Gómez Pedreira, P. Joly, “Mathematical analysis of a method to compute guided waves in integrated optics,” (Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, 2000).

A. Bermudez, D. Gómez Pedreira, P. Joly, “A hybrid approach for the computation of guided modes in integrated optics,” (Institut National de Recherche en Informatique et en Automatique, Le Chesnay Cedex, France, 2001).

V. L. Goncharov, Theory of Function Interpolation and Extrapolation [Gostekhizdat, Moscow, 1954 (in Russian)].

G. E. Forshyte, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J., 1977).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

G. N. Watson, A Treatise on the Theory of Bessel Functions (MacMillan, New York, 1944).

M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 1992).

F. Fernandez, Y. Lu, Microwave and Optical Waveguide Analysis by the Finite Element Method (Wiley, New York, 1996).

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Figures (9)

Fig. 1
Fig. 1

Geometry of an asymmetrical slab waveguide with an arbitrary relative dielectric permittivity for the core region.

Fig. 2
Fig. 2

Normalized-mode wave number B=(β2-k02n12)/[k02(n22-n12)] with dimensionless frequency V=k0d(n22-n12)1/2 for the first even TE guided mode for a symmetrical slab waveguide with n1=n3=3.24 and n(x)=n2=3.6.

Fig. 3
Fig. 3

First TE guided-mode wave number β with core thickness 2d for a symmetrical slab waveguide with n1=n3=3.24 and parabolic refractive-index profile n(x) given by Eq. (13).

Fig. 4
Fig. 4

Variation of the electric field distribution of the first TE guided mode with transverse normalized distance x/d for a symmetrical slab waveguide with d=0.15 μm, n1=n3=3.24, and parabolic refractive-index profile n(x) given by Eq. (13).

Fig. 5
Fig. 5

First TE guided-mode wave number β with core thickness 2d for an asymmetrical slab waveguide with n1=3.42, n3=3.24, and linear refractive-index profile n(x) given by Eq. (14).

Fig. 6
Fig. 6

Variation of the electric field distribution of the first TE guided mode with transverse normalized distance x/d for an asymmetrical slab waveguide with d=0.15 μm, n1=3.24, n3=3.42, and linear refractive-index profile n(x) given by Eq. (14).

Fig. 7
Fig. 7

Normalized-mode wave number B=(β2-k02n12)/[k02(x,max-n12)] with dimensionless frequency V=k0d(x,max-n12)1/2 of TM guided modes for a symmetrical slab waveguide with n1=n3=1.46 and x(x)=y(x)=z(x)=3.62.

Fig. 8
Fig. 8

Normalized-mode wave number B=(β2-k02n12)/[k02(x,max-n12)] with dimensionless frequency V=k0d(x,max-n12)1/2 of TM guided modes for a symmetrical slab waveguide with n1=n3=1.46, x(x)=2.7225[1-2Δx(x/d)2], y(x)=z(x)=2.1904[1-2Δz(x/d)2], Δx=5.75%, and Δz=0.675%.

Fig. 9
Fig. 9

Normalized-mode wave number B=(β2-k02n12)/[k02(n22-n12)] of dominant TE guided mode with term number N for a slab waveguide with semithickness d=0.08722466 μm. The other parameters are the same as for Fig. 2.

Tables (2)

Tables Icon

Table 1 Extrapolation of the Normalized-Mode Wave Number B for Planar Geometry

Tables Icon

Table 2 Extrapolation of the Normalized-Mode Wave Number B for Circular Geometry

Equations (21)

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

x(x)z(x)d2Hy(x)dx2-x(x)z2(x)dz(x)dxdHy(x)dx
+[k02x(x)-β2]Hy(x)=0.
Hy(x)=C3exp[-h3(x-d)],x>dC1exp[+h1(x+d)],x<-d,
Hy(x)=A0+B0xd+n=1NBnsinnπ2d (x+d),
x=-dC1=A0-B0,
h1C1n12=1z(-d)B0d+n=1NBnnπ2d;
x=+dC3=A0+B0,
-h3C3n32=1z(+d)B0d+n=1NBnnπ2d(-1)n.
xj=-d+j 2dN+1,j=1, 2 ,, N.
A0Rj+B0Sj+n=1NBnQjn=0,j=1, 2 ,, N,
Rj=k02x(xj)-β2,
Sj=Rjxjd-x(xj)z2(xj)z(xj)d,
Qjn=Rj-x(xj)z(xj)nπ2d2sinnπ2d (xj+d)-x(xj)z2(xj) z(xj)nπ2dcosnπ2d (xj+d).
n2(x)=n22[1-2Δ(x/d)2],-d<x<d.
n(x)=n1+(n2-n1)x+d2d,-d<x<d.
xj=d sgn(tj)|tj|p,tj=(2j-N-1)/(N+1),
β(N)βext+q/N,
β˜(N2)=[N2β(N2)-N1β(N1)](N2-N1).
n=1+sinπn(x+d)2d=π(d-x)4d.
E(ρ, ϕ)=Aρm+n=1NBnJm(vn(m)ρ/a)×cos(mϕ+ϕ0),ρ<a,
f(θ)|1+R0|2-+U(x)exp[ikn0x sin(θ)]dx2,

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