Abstract

A method for obtaining total field solutions from the Helmholtz equation for optical waveguiding structures is presented. The method is based on a conversion of the Helmholtz equation into a matrix total differential equation and direct solution of the resulting equation by means of standard numerical methods. The principle of the method is discussed in detail, and numerical results are presented for planar guiding structures. The method has been extended to three-dimensional guiding structures. Comparisons with the propagating-beam method show that the present method yields better accuracy and is more efficient numerically for planar as well as for three-dimensional waveguiding structures. Further, unlike the propagating-beam method, the present method needs no inherent approximation and hence can be used to obtain any desired accuracy, if the increase in computation time is not a consideration.

© 1989 Optical Society of America

Full Article  |  PDF Article

Corrections

Swagata Banerjee and Anurag Sharma, "Propagation characteristics of optical waveguiding structures by direct solution of the Helmholtz equation for total fields: errata," J. Opt. Soc. Am. A 7, 2156-2156 (1990)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-7-11-2156

References

  • View by:
  • |
  • |
  • |

  1. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–158 (1976).
    [CrossRef]
  2. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  3. M. D. Feit, J. A. Fleck, “Calculation of dispersion in graded-index multimode fibers by a propagating-beam method,” Appl. Opt. 18, 2843–2851 (1979).
    [CrossRef] [PubMed]
  4. D. Yevick, P. Danielson, “Propagating beam analysis of bent optical waveguides,” Opt. Commun. 4, 94–97 (1983).
  5. J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam propagation method: analysis and assessment,”J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  6. L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
    [CrossRef]
  7. D. Yevick, P. Danielson, “Numerical investigation of mode coupling in sinusoidally modulated GRIN planar waveguides,” Appl. Opt. 21, 2727–2733 (1982).
    [CrossRef] [PubMed]
  8. D. Yevick, B. Hermansson, “New approach to perturbed optical waveguides,” Opt. Lett. 11, 103–105 (1986).
    [CrossRef] [PubMed]
  9. A. Sharma, S. Banerjee, “Method for propagation of total fields or beams through optical waveguides,” Opt. Lett. 14, 96–98 (1989).
    [CrossRef] [PubMed]
  10. L. A. Pipes, L. R. Harvill, Applied Mathematics for Engineers and Physicists (McGraw-Hill, New York, 1970).
  11. J. V. Villadsen, W. E. Stewart, “Solution of boundary-value problems by orthogonal collection,” Chem. Eng. Sci. 22, 1483–1501 (1967).
    [CrossRef]
  12. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  13. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  14. R. A. Frazer, W. P. Jones, S. W. Skan, “Approximations to functions and to the solutions of differential equations,” (Great Britain Aeronautical Research Council, London, 1937); reprinted in G. B. Air Minist. Aero. Res. Commun. Tech. Rep. 1, 517–549 (1937).
  15. C. Lanczos, “Trigonometric interpolation of empirical and analytical functions,”J. Math. Phys. 17, 123–199(1938).
  16. C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1956).
  17. B. A. Finlayson, L. E. Scriven, “The method of weighted residuals—a review,” Appl. Mech. Rev. 19, 735–748 (1966).
  18. B. A. Finlayson, Method of Weighted Residuals and Variational Principles with Application to Fluid Mechanics, Heat and Mass Transfer (Academic, New York, 1972).
  19. C. A. J. Fletcher, Computational Galerkin Methods (Springer-Verlag, Berlin, 1984).
    [CrossRef]
  20. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  21. A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, N.J., 1966).
  22. J. B. Scarborough, Numerical Mathematical Analysis (Oxford U. Press, London, 1966).
  23. J. P. Meunier, J. Pigeon, J. N. Massot, “A numerical technique for the determination of propagation characteristics of inhomogeneous planar optical waveguides,” Opt. Quantum Electron. 15, 77–85 (1983).
    [CrossRef]
  24. T. Tamir, “Theory of dielectric waveguides,” in Integrated Optics, Vol. 7 of Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975).
  25. M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, UK, 1981), p. 132.
  26. We used Numerical Algorithm Group (UK) library FFT subroutines for the numerical implementation of the PBM.
  27. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971).
  28. G. Hall, J. M. Watt, Modern Numerical Methods for Ordinary Differential Equations (Clarendon, Oxford, 1976).

1989 (1)

1986 (1)

1983 (3)

L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[CrossRef]

D. Yevick, P. Danielson, “Propagating beam analysis of bent optical waveguides,” Opt. Commun. 4, 94–97 (1983).

J. P. Meunier, J. Pigeon, J. N. Massot, “A numerical technique for the determination of propagation characteristics of inhomogeneous planar optical waveguides,” Opt. Quantum Electron. 15, 77–85 (1983).
[CrossRef]

1982 (1)

1981 (1)

1979 (1)

1978 (1)

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–158 (1976).
[CrossRef]

1967 (1)

J. V. Villadsen, W. E. Stewart, “Solution of boundary-value problems by orthogonal collection,” Chem. Eng. Sci. 22, 1483–1501 (1967).
[CrossRef]

1966 (1)

B. A. Finlayson, L. E. Scriven, “The method of weighted residuals—a review,” Appl. Mech. Rev. 19, 735–748 (1966).

1938 (1)

C. Lanczos, “Trigonometric interpolation of empirical and analytical functions,”J. Math. Phys. 17, 123–199(1938).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, UK, 1981), p. 132.

Banerjee, S.

Danielson, P.

D. Yevick, P. Danielson, “Propagating beam analysis of bent optical waveguides,” Opt. Commun. 4, 94–97 (1983).

D. Yevick, P. Danielson, “Numerical investigation of mode coupling in sinusoidally modulated GRIN planar waveguides,” Appl. Opt. 21, 2727–2733 (1982).
[CrossRef] [PubMed]

Feit, M. D.

Finlayson, B. A.

B. A. Finlayson, L. E. Scriven, “The method of weighted residuals—a review,” Appl. Mech. Rev. 19, 735–748 (1966).

B. A. Finlayson, Method of Weighted Residuals and Variational Principles with Application to Fluid Mechanics, Heat and Mass Transfer (Academic, New York, 1972).

Fleck, J. A.

Fletcher, C. A. J.

C. A. J. Fletcher, Computational Galerkin Methods (Springer-Verlag, Berlin, 1984).
[CrossRef]

Frazer, R. A.

R. A. Frazer, W. P. Jones, S. W. Skan, “Approximations to functions and to the solutions of differential equations,” (Great Britain Aeronautical Research Council, London, 1937); reprinted in G. B. Air Minist. Aero. Res. Commun. Tech. Rep. 1, 517–549 (1937).

Gear, C. W.

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971).

Hall, G.

G. Hall, J. M. Watt, Modern Numerical Methods for Ordinary Differential Equations (Clarendon, Oxford, 1976).

Harvill, L. R.

L. A. Pipes, L. R. Harvill, Applied Mathematics for Engineers and Physicists (McGraw-Hill, New York, 1970).

Hermansson, B.

Jones, W. P.

R. A. Frazer, W. P. Jones, S. W. Skan, “Approximations to functions and to the solutions of differential equations,” (Great Britain Aeronautical Research Council, London, 1937); reprinted in G. B. Air Minist. Aero. Res. Commun. Tech. Rep. 1, 517–549 (1937).

Lagasse, P. E.

Lanczos, C.

C. Lanczos, “Trigonometric interpolation of empirical and analytical functions,”J. Math. Phys. 17, 123–199(1938).

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

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Marcuse, D.

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

Massot, J. N.

J. P. Meunier, J. Pigeon, J. N. Massot, “A numerical technique for the determination of propagation characteristics of inhomogeneous planar optical waveguides,” Opt. Quantum Electron. 15, 77–85 (1983).
[CrossRef]

Meunier, J. P.

J. P. Meunier, J. Pigeon, J. N. Massot, “A numerical technique for the determination of propagation characteristics of inhomogeneous planar optical waveguides,” Opt. Quantum Electron. 15, 77–85 (1983).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–158 (1976).
[CrossRef]

Pigeon, J.

J. P. Meunier, J. Pigeon, J. N. Massot, “A numerical technique for the determination of propagation characteristics of inhomogeneous planar optical waveguides,” Opt. Quantum Electron. 15, 77–85 (1983).
[CrossRef]

Pipes, L. A.

L. A. Pipes, L. R. Harvill, Applied Mathematics for Engineers and Physicists (McGraw-Hill, New York, 1970).

Scarborough, J. B.

J. B. Scarborough, Numerical Mathematical Analysis (Oxford U. Press, London, 1966).

Scriven, L. E.

B. A. Finlayson, L. E. Scriven, “The method of weighted residuals—a review,” Appl. Mech. Rev. 19, 735–748 (1966).

Secrest, D.

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, N.J., 1966).

Sharma, A.

Skan, S. W.

R. A. Frazer, W. P. Jones, S. W. Skan, “Approximations to functions and to the solutions of differential equations,” (Great Britain Aeronautical Research Council, London, 1937); reprinted in G. B. Air Minist. Aero. Res. Commun. Tech. Rep. 1, 517–549 (1937).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Stewart, W. E.

J. V. Villadsen, W. E. Stewart, “Solution of boundary-value problems by orthogonal collection,” Chem. Eng. Sci. 22, 1483–1501 (1967).
[CrossRef]

Stroud, A. H.

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, N.J., 1966).

Tamir, T.

T. Tamir, “Theory of dielectric waveguides,” in Integrated Optics, Vol. 7 of Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975).

Thylen, L.

L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[CrossRef]

van der Donk, J.

Van Roey, J.

Villadsen, J. V.

J. V. Villadsen, W. E. Stewart, “Solution of boundary-value problems by orthogonal collection,” Chem. Eng. Sci. 22, 1483–1501 (1967).
[CrossRef]

Watt, J. M.

G. Hall, J. M. Watt, Modern Numerical Methods for Ordinary Differential Equations (Clarendon, Oxford, 1976).

Yevick, D.

Appl. Mech. Rev. (1)

B. A. Finlayson, L. E. Scriven, “The method of weighted residuals—a review,” Appl. Mech. Rev. 19, 735–748 (1966).

Appl. Opt. (3)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high-energy laser beams through the atmosphere,” Appl. Phys. 10, 129–158 (1976).
[CrossRef]

Chem. Eng. Sci. (1)

J. V. Villadsen, W. E. Stewart, “Solution of boundary-value problems by orthogonal collection,” Chem. Eng. Sci. 22, 1483–1501 (1967).
[CrossRef]

J. Math. Phys. (1)

C. Lanczos, “Trigonometric interpolation of empirical and analytical functions,”J. Math. Phys. 17, 123–199(1938).

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

D. Yevick, P. Danielson, “Propagating beam analysis of bent optical waveguides,” Opt. Commun. 4, 94–97 (1983).

Opt. Lett. (2)

Opt. Quantum Electron. (2)

L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[CrossRef]

J. P. Meunier, J. Pigeon, J. N. Massot, “A numerical technique for the determination of propagation characteristics of inhomogeneous planar optical waveguides,” Opt. Quantum Electron. 15, 77–85 (1983).
[CrossRef]

Other (15)

T. Tamir, “Theory of dielectric waveguides,” in Integrated Optics, Vol. 7 of Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, UK, 1981), p. 132.

We used Numerical Algorithm Group (UK) library FFT subroutines for the numerical implementation of the PBM.

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971).

G. Hall, J. M. Watt, Modern Numerical Methods for Ordinary Differential Equations (Clarendon, Oxford, 1976).

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

L. A. Pipes, L. R. Harvill, Applied Mathematics for Engineers and Physicists (McGraw-Hill, New York, 1970).

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

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

R. A. Frazer, W. P. Jones, S. W. Skan, “Approximations to functions and to the solutions of differential equations,” (Great Britain Aeronautical Research Council, London, 1937); reprinted in G. B. Air Minist. Aero. Res. Commun. Tech. Rep. 1, 517–549 (1937).

B. A. Finlayson, Method of Weighted Residuals and Variational Principles with Application to Fluid Mechanics, Heat and Mass Transfer (Academic, New York, 1972).

C. A. J. Fletcher, Computational Galerkin Methods (Springer-Verlag, Berlin, 1984).
[CrossRef]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, N.J., 1966).

J. B. Scarborough, Numerical Mathematical Analysis (Oxford U. Press, London, 1966).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Error in the correlation factor as a function of the number of collocation points for a uniform waveguide (s = 0) with index distribution RIP V defined in Table 1. The value of Δz used is 2.5 μm. The dashed curve shows the relative computation time required for these calculations. Also indicated in the figure are the accuracy obtained and the time required for PBM calculations with 128 points.

Fig. 2
Fig. 2

Similar to Fig. 1 except that the index distribution now is RIP IV with s = 0.

Fig. 3
Fig. 3

Fractional power lost from the fundamental mode in a taper of length 5 μm with index distribution RIP V and s = 0.25, as a function of relative computation time [with respect to PBM 128 (WRT PBM 128)], using our method as well as the PBM.

Fig. 4
Fig. 4

Similar to Fig. 3 except that here the index distribution is RIP IV, s = 0.1, and the taper length is 5 μm.

Fig. 5
Fig. 5

Variation of error in the modal field after propagation over a distance of 10 μm as a function of the value of the parameter α. All other parameters are the same as those in Fig. 1.

Fig. 6
Fig. 6

Error in the correlation factor and computation time (in arbitrary units) as a function of the number of collocation points after propagation by a distance of 10 μm through a uniform (s = 0) fiber with a refractive-index distribution given by Eq. (31).

Fig. 7
Fig. 7

Fractional power lost from a tapered fiber after propagation through a distance of 10 μm as a function of computation time. The index distribution is given by Eq. (31), with s = 0.05.

Tables (6)

Tables Icon

Table 1 RIP’s for Numerical Examples

Tables Icon

Table 2 Convergence of Numerical Results for β/k0 for a Cladded Parabolic Profile Waveguidea

Tables Icon

Table 3 Convergence of Numerical Results for β/k0 for a Gaussian Profile Waveguidea

Tables Icon

Table 4 Propagation Constant β/k0 for Profiles with Discontinuity

Tables Icon

Table 5 Fractional Power Loss from the Propagating Fundamental Mode in a Tapered Waveguide

Tables Icon

Table 6 Percent Power Loss in a Taper with RIP V, s = 0.1, and zf = 10 μm, Using Two Different Methods to Solve the Differential Equation

Equations (52)

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

2 E x 2 + 2 E y 2 + 2 E z 2 + k 0 2 n 2 ( x , y , z ) E ( x , y , z ) = 0 ,
2 E x 2 + 2 E z 2 + k 0 2 n 2 ( x , z ) E ( x , z ) = 0.
E ( x , z ) = n = 1 N C n ( z ) ϕ n ( x ) ,
ϕ n ( x ) = G n H n - 1 ( α x ) exp [ - ( 1 / 2 ) α 2 x 2 ] ,
H N ( α x j ) = 0.
2 E x 2 | x = x j + d 2 E j d z 2 + k 0 2 n 2 ( x j , z ) E j ( z ) = 0 ,             j = 1 , 2 , , N ,
( d 2 / d z 2 ) E + D + RE = 0 ,
E = [ E 1 E 2 E N ] ,             R = k 0 2 [ n 2 ( x 1 ) 0 n 2 ( x 2 ) 0 n 2 ( x N ) ] , D = [ ( 2 E / x 2 ) x = x N ( 2 E / x 2 ) x = x 2 ( 2 E / x 2 ) x = x N ] .
E ( x j , z ) = n = 1 N C n ( z ) ϕ n ( x j ) ,             j = 1 , 2 , , N ,
E = AC ,
C = [ C 1 C 2 C N ] , A = [ ϕ 1 ( x 1 ) ϕ 2 ( x 1 ) . . . ϕ N ( x 1 ) ϕ 1 ( x 2 ) ϕ 2 ( x 2 ) . . . ϕ N ( x 2 ) . . . . . . . . . . . . . . . . . . ϕ 1 ( x N ) ϕ 2 ( x N ) . . . ϕ N ( x N ) ] .
D = BC ,
B = [ ( 2 ϕ 1 / x 2 ) x 1 ( 2 ϕ 2 / x 2 ) x 1 . . . ( 2 ϕ N / x 2 ) x 1 ( 2 ϕ 1 / x 2 ) x 2 ( 2 ϕ 2 / x 2 ) x 2 . . . ( 2 ϕ N / x 2 ) x 2 . . . . . . . . . . . . . . . . . . ( 2 ϕ 1 / x 2 ) x N ( 2 ϕ 2 / x 2 ) x N . . . ( 2 ϕ N / x 2 ) x N ]
D = BA - 1 E ,
( d 2 / d z 2 ) E + SE ( z ) = 0 ,
S = BA - 1 + R .
E ( z ) = F ( z ) exp ( - i p z ) ,
d 2 F d z 2 - 2 i p d F d z + ( S - p 2 I ) F ( z ) = 0 ,
( d / d z ) F = ( S - p 2 I ) F ( z ) / ( 2 i p ) ,
SE = β 2 E ,
β 2 = - + E 2 k 0 2 n 2 ( x ) d x - - + d E / d x 2 d x - + E 2 d x .
d F d z = Q ( z ) F ( z ) ,
Q ( z ) = ( BA - 1 + R ( z ) - p 2 I ) / 2 i p .
E ( x , z = 0 ) = cosh - W ( x / a ) ,
W = a [ β 2 - k 0 2 n c 1 2 ] 1 / 2 = [ ( 1 + 4 V 2 ) 1 / 2 - 1 ] / 2.
E ( x , y , z ) = m = 1 N n = 1 N K n m ( z ) ϕ n ( x ) ψ m ( y ) ,
( d 2 / d z 2 ) E + XE + EY T + RE = 0 ,
E = [ E ( x 1 , y 1 ) E ( x 1 , y 2 ) . . . E ( x 1 , y N ) E ( x 2 , y 1 ) E ( x 2 , y 2 ) . . . E ( x 2 , y N ) . . . . . . . . . . . . . . . . . . E ( x N , y 1 ) E ( x N , y 2 ) . . . E ( x N , y N ) ] ,
X = BA - 1 ,             Y = H G - 1 , G = [ ψ 1 ( y 1 ) ψ 2 ( y 1 ) . . . ψ N ( y 1 ) ψ 1 ( y 2 ) ψ 2 ( y 2 ) . . . ψ N ( y 2 ) . . . . . . . . . . . . . . . . . . ψ 1 ( y N ) ψ 2 ( y N ) . . . ψ N ( y N ) ] ,
H = [ ( 2 ψ 1 / y 2 ) y 1 ( 2 ψ 2 / y 2 ) y 1 . . . ( 2 ψ N / y 2 ) y 1 ( 2 ψ 1 / y 2 ) y 2 ( 2 ψ 2 / y 2 ) y 2 . . . ( 2 ψ N / y 2 ) y 2 . . . . . . . . . . . . . . . . . . ( 2 ψ 1 / y 2 ) y N ( 2 ψ 2 / y 2 ) y N . . . ( 2 ψ N / y 2 ) y N ] .
d F / d ( z ) = 1 / 2 i k ( XF + F Y T + RF - p 2 F ) .
n 2 ( x , y , z ) = n 1 2 - ( n 1 2 - n 2 2 ) { 1 - exp [ - ( x 2 + y 2 ) / a 2 ] } ,
A = { A j n :             A j n = ϕ n ( u j ) ; n , j = 1 , 2 , , N } ,
ϕ n ( u ) = G n H n - 1 ( u ) exp ( - u / 2 ) ,
H N ( u j ) = 0 ,             j = 1 , 2 , , N .
- + ϕ n ( u ) ϕ m ( u ) d u = j = 1 N W j 2 ϕ m ( u j ) ϕ n ( u j ) = δ m n ,
( WA ) T ( WA ) = I ,
W = diag ( W 1 , W 2 , , W N ) .
AA T = ( W T W ) - 1 ,
A - 1 = A T ( W T W ) .
S = BA - 1 + R ,
B = { B j n :     B j n = α 2 d 2 ϕ n d u 2 | u = u j ; j , n = 1 , 2 , , N } ,
R = k 0 2 × diag { n 2 ( u 1 ) , n 2 ( u 2 ) , , n 2 ( u N ) } ,
d 2 ϕ n d u 2 = [ u 2 - ( 2 n - 1 ) ] ϕ n .
B j n = α 2 [ u j 2 - ( 2 n - 1 ) ] A j n ,
B = D 1 A - AD 2 ,
D 1 = α 2 diag { u 1 2 , u 2 2 , , u N 2 }
D 2 = α 2 diag { 1 , 3 , 5 , , ( 2 N - 1 ) } .
S = D 3 - AD 2 A - 1 ,
S = A - 1 D 3 A - D 2 .
A - 1 D 3 A = ( A - 1 D 3 A ) T = A T D 3 ( A T ) - 1 ,
D 3 = ( AA T ) D 3 ( AA T ) - 1 .

Metrics