Abstract

Exact solutions can be obtained for electromagnetic wave propagation in a medium with a simple uniform refractive-index distribution. For more-complex distributions, approximate or numerical methods have to be utilized. We describe an elegant approximation scheme called the decomposition method for nonlinear differential equations, which was introduced by Adomian [Non-linear Stochastic Systems Theory and Applications to Physics (Kluwer, Dordrecht, The Netherlands, 1989)]. The method is described and applied to waveguide problems (planar waveguides with step and parabolic refractive-index profiles), and the results are compared with those obtained by JWKB and modified Airy function methods.

© 1998 Optical Society of America

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  1. For a discussion of variational method see, for example, H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley Series in Physics (Addison-Wesley, Reading, Mass., 1980), Chap. 2. For applications to waveguides see A. Sharma, P. Bindal, “Analysis of diffused planar and channel waveguide,” IEEE J. Quantum Electron. 29, 150–153 (1993); “An accurate variational analysis of single mode diffused channel waveguides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
    [CrossRef]
  2. N. Froman, P. O. Froman, JWKB Approximation: Contributions to the Theory (North-Holland, Amsterdam, 1965).
  3. C. M. Bender, S. A. Orzsag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  4. A. K. Ghatak, R. L. Gallawa, I. C. Goyal, Modified Airy Function and WKB Solutions to the Wave Equation, National Institute of Standards and Technology Monogr. 176 (U.S. GPO, Washington, D.C., 1991).
  5. G. Baym, Lectures on Quantum Mechanics (Benjamin, New York, 1969).
  6. I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “An approximate solution to the wave equation—revisited,” J. Electromagn. Waves Appl. 5, 623–636 (1991).
    [CrossRef]
  7. I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “Approximate solutions to the scalar wave equations for optical waveguides,” Appl. Opt. 29, 2985–2990 (1991).
    [CrossRef]
  8. A. K. Ghatak, R. L. Gallawa, I. C. Goyal, “Accurate solutions to Schrodinger’s equation using modified Airy functions,” IEEE J. Quantum Electron. 28, 400–403 (1992).
    [CrossRef]
  9. M. L. Calvo, V. Lakshminarayanan, “Light propagation in optical waveguides: a dynamic programming approach,” J. Opt. Soc. Am. A 14, 872–881 (1997).
    [CrossRef]
  10. G. Adomian, “Convergent series solution of non-linear equations,” J. Comput. Appl. Math. 11, 225–230 (1984).
    [CrossRef]
  11. G. Adomian, “A new approach to the heat equation—an application of the decomposition method,” J. Math. Anal. Appl. 113, 202–209 (1986).
    [CrossRef]
  12. G. Adomian, “An investigation of asymptotic decomposition method for non-linear equations in physics,” Appl. Math. Comput. 24, 1–17 (1987).
    [CrossRef]
  13. G. Adomian, “Non-linear oscillations in physical systems,” Math Comput. Sim. 29, 275–284 (1987).
    [CrossRef]
  14. G. Adomian, Non-linear Stochastic Systems Theory and Applications to Physics (Kluwer, Dordrecht, The Netherlands, 1989).
  15. G. Adomian, “Non-linear stochastic differential equations,” in Selected Topics in Mathematical Physics, R. Sridhar, K. Srinivasa Rao, V. Lakshminarayanan, eds. (Allied, New Delhi, 1995), pp. 47–57.
  16. R. Rach, “On the Adomian decomposition method and comparisons with Picard’s method,” J. Math. Anal. Appl. 128, 480–483 (1987).
    [CrossRef]
  17. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).
  18. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1972).
  19. All the computations were done with the commercial software Maple V Release 4, is a product of Waterloo Maple, Inc., Waterloo, Ontario, Canada.
  20. C. Yu, D. Yevick, “Application of the bidirectional parabolic equations to optical waveguide facets,” J. Opt. Soc. Am. A 14, 1448–1450 (1997).
    [CrossRef]
  21. I. R. Bellman, G. Adomian, Partial Differential Equations: New Methods for Their Treatment and Solution (Reidel, Dordrecht, The Netherlands, 1985).

1997 (2)

1992 (1)

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, “Accurate solutions to Schrodinger’s equation using modified Airy functions,” IEEE J. Quantum Electron. 28, 400–403 (1992).
[CrossRef]

1991 (2)

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “An approximate solution to the wave equation—revisited,” J. Electromagn. Waves Appl. 5, 623–636 (1991).
[CrossRef]

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “Approximate solutions to the scalar wave equations for optical waveguides,” Appl. Opt. 29, 2985–2990 (1991).
[CrossRef]

1987 (3)

G. Adomian, “An investigation of asymptotic decomposition method for non-linear equations in physics,” Appl. Math. Comput. 24, 1–17 (1987).
[CrossRef]

G. Adomian, “Non-linear oscillations in physical systems,” Math Comput. Sim. 29, 275–284 (1987).
[CrossRef]

R. Rach, “On the Adomian decomposition method and comparisons with Picard’s method,” J. Math. Anal. Appl. 128, 480–483 (1987).
[CrossRef]

1986 (1)

G. Adomian, “A new approach to the heat equation—an application of the decomposition method,” J. Math. Anal. Appl. 113, 202–209 (1986).
[CrossRef]

1984 (1)

G. Adomian, “Convergent series solution of non-linear equations,” J. Comput. Appl. Math. 11, 225–230 (1984).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1972).

Adomian, G.

G. Adomian, “An investigation of asymptotic decomposition method for non-linear equations in physics,” Appl. Math. Comput. 24, 1–17 (1987).
[CrossRef]

G. Adomian, “Non-linear oscillations in physical systems,” Math Comput. Sim. 29, 275–284 (1987).
[CrossRef]

G. Adomian, “A new approach to the heat equation—an application of the decomposition method,” J. Math. Anal. Appl. 113, 202–209 (1986).
[CrossRef]

G. Adomian, “Convergent series solution of non-linear equations,” J. Comput. Appl. Math. 11, 225–230 (1984).
[CrossRef]

G. Adomian, Non-linear Stochastic Systems Theory and Applications to Physics (Kluwer, Dordrecht, The Netherlands, 1989).

G. Adomian, “Non-linear stochastic differential equations,” in Selected Topics in Mathematical Physics, R. Sridhar, K. Srinivasa Rao, V. Lakshminarayanan, eds. (Allied, New Delhi, 1995), pp. 47–57.

I. R. Bellman, G. Adomian, Partial Differential Equations: New Methods for Their Treatment and Solution (Reidel, Dordrecht, The Netherlands, 1985).

Baym, G.

G. Baym, Lectures on Quantum Mechanics (Benjamin, New York, 1969).

Bellman, I. R.

I. R. Bellman, G. Adomian, Partial Differential Equations: New Methods for Their Treatment and Solution (Reidel, Dordrecht, The Netherlands, 1985).

Bender, C. M.

C. M. Bender, S. A. Orzsag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Calvo, M. L.

Froman, N.

N. Froman, P. O. Froman, JWKB Approximation: Contributions to the Theory (North-Holland, Amsterdam, 1965).

Froman, P. O.

N. Froman, P. O. Froman, JWKB Approximation: Contributions to the Theory (North-Holland, Amsterdam, 1965).

Gallawa, R. L.

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, “Accurate solutions to Schrodinger’s equation using modified Airy functions,” IEEE J. Quantum Electron. 28, 400–403 (1992).
[CrossRef]

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “Approximate solutions to the scalar wave equations for optical waveguides,” Appl. Opt. 29, 2985–2990 (1991).
[CrossRef]

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “An approximate solution to the wave equation—revisited,” J. Electromagn. Waves Appl. 5, 623–636 (1991).
[CrossRef]

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, Modified Airy Function and WKB Solutions to the Wave Equation, National Institute of Standards and Technology Monogr. 176 (U.S. GPO, Washington, D.C., 1991).

Ghatak, A. K.

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, “Accurate solutions to Schrodinger’s equation using modified Airy functions,” IEEE J. Quantum Electron. 28, 400–403 (1992).
[CrossRef]

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “An approximate solution to the wave equation—revisited,” J. Electromagn. Waves Appl. 5, 623–636 (1991).
[CrossRef]

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “Approximate solutions to the scalar wave equations for optical waveguides,” Appl. Opt. 29, 2985–2990 (1991).
[CrossRef]

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, Modified Airy Function and WKB Solutions to the Wave Equation, National Institute of Standards and Technology Monogr. 176 (U.S. GPO, Washington, D.C., 1991).

Goldstein, H.

For a discussion of variational method see, for example, H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley Series in Physics (Addison-Wesley, Reading, Mass., 1980), Chap. 2. For applications to waveguides see A. Sharma, P. Bindal, “Analysis of diffused planar and channel waveguide,” IEEE J. Quantum Electron. 29, 150–153 (1993); “An accurate variational analysis of single mode diffused channel waveguides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

Goyal, I. C.

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, “Accurate solutions to Schrodinger’s equation using modified Airy functions,” IEEE J. Quantum Electron. 28, 400–403 (1992).
[CrossRef]

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “Approximate solutions to the scalar wave equations for optical waveguides,” Appl. Opt. 29, 2985–2990 (1991).
[CrossRef]

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “An approximate solution to the wave equation—revisited,” J. Electromagn. Waves Appl. 5, 623–636 (1991).
[CrossRef]

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, Modified Airy Function and WKB Solutions to the Wave Equation, National Institute of Standards and Technology Monogr. 176 (U.S. GPO, Washington, D.C., 1991).

Lakshminarayanan, V.

Love, J. D.

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

Orzsag, S. A.

C. M. Bender, S. A. Orzsag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Rach, R.

R. Rach, “On the Adomian decomposition method and comparisons with Picard’s method,” J. Math. Anal. Appl. 128, 480–483 (1987).
[CrossRef]

Snyder, A. W.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1972).

Yevick, D.

Yu, C.

Appl. Math. Comput. (1)

G. Adomian, “An investigation of asymptotic decomposition method for non-linear equations in physics,” Appl. Math. Comput. 24, 1–17 (1987).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, “Accurate solutions to Schrodinger’s equation using modified Airy functions,” IEEE J. Quantum Electron. 28, 400–403 (1992).
[CrossRef]

J. Comput. Appl. Math. (1)

G. Adomian, “Convergent series solution of non-linear equations,” J. Comput. Appl. Math. 11, 225–230 (1984).
[CrossRef]

J. Electromagn. Waves Appl. (1)

I. C. Goyal, R. L. Gallawa, A. K. Ghatak, “An approximate solution to the wave equation—revisited,” J. Electromagn. Waves Appl. 5, 623–636 (1991).
[CrossRef]

J. Math. Anal. Appl. (2)

G. Adomian, “A new approach to the heat equation—an application of the decomposition method,” J. Math. Anal. Appl. 113, 202–209 (1986).
[CrossRef]

R. Rach, “On the Adomian decomposition method and comparisons with Picard’s method,” J. Math. Anal. Appl. 128, 480–483 (1987).
[CrossRef]

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

Math Comput. Sim. (1)

G. Adomian, “Non-linear oscillations in physical systems,” Math Comput. Sim. 29, 275–284 (1987).
[CrossRef]

Other (11)

G. Adomian, Non-linear Stochastic Systems Theory and Applications to Physics (Kluwer, Dordrecht, The Netherlands, 1989).

G. Adomian, “Non-linear stochastic differential equations,” in Selected Topics in Mathematical Physics, R. Sridhar, K. Srinivasa Rao, V. Lakshminarayanan, eds. (Allied, New Delhi, 1995), pp. 47–57.

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

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1972).

All the computations were done with the commercial software Maple V Release 4, is a product of Waterloo Maple, Inc., Waterloo, Ontario, Canada.

For a discussion of variational method see, for example, H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley Series in Physics (Addison-Wesley, Reading, Mass., 1980), Chap. 2. For applications to waveguides see A. Sharma, P. Bindal, “Analysis of diffused planar and channel waveguide,” IEEE J. Quantum Electron. 29, 150–153 (1993); “An accurate variational analysis of single mode diffused channel waveguides,” Opt. Quantum Electron. 24, 1359–1371 (1992).
[CrossRef]

N. Froman, P. O. Froman, JWKB Approximation: Contributions to the Theory (North-Holland, Amsterdam, 1965).

C. M. Bender, S. A. Orzsag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, Modified Airy Function and WKB Solutions to the Wave Equation, National Institute of Standards and Technology Monogr. 176 (U.S. GPO, Washington, D.C., 1991).

G. Baym, Lectures on Quantum Mechanics (Benjamin, New York, 1969).

I. R. Bellman, G. Adomian, Partial Differential Equations: New Methods for Their Treatment and Solution (Reidel, Dordrecht, The Netherlands, 1985).

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Figures (6)

Fig. 1
Fig. 1

Exact (solid curve), 10-term decomposition (dotted curve) and 20-term decomposition (dashed curve) solutions to the wave equation in a planar waveguide with step-profile refractive-index distribution. The third (j=2) mode is considered here.

Fig. 2
Fig. 2

Percentage difference between the exact and the 10-term decomposition solutions (solid curve) and between the exact and the 20-term decomposition solutions to Eq. (14). The percentage difference between the exact and the 20-term decomposition solutions is zero, and hence its graph coincides with the x axis.

Fig. 3
Fig. 3

Exact, JWKB, and MAF solutions to the wave equation for a planar waveguide with a parabolic-index profile. The second antisymmetric mode (n=3) is considered. The solid curve is the exact solution. The MAF solution coincides with the exact solution. The JWKB solution (dashed curve) is seen to diverge at X=7.

Fig. 4
Fig. 4

Percentage difference between the exact solution and the JWKB (solid curve) and the MAF (dotted curve) solutions. The MAF solution has zero percentage error at all values of X except near the origin and the zero crossing (see text).

Fig. 5
Fig. 5

Second antisymmetric exact and 10-term decomposition solutions to the wave equation in the planar waveguide with parabolic refractive index. The decomposition solution differs from the exact solution only for large values of X.

Fig. 6
Fig. 6

Percentage difference between the exact and the 10-term decomposition solutions. The difference is less than 1% for most values of X.

Tables (1)

Tables Icon

Table 1 Number of Iterations for 1% Accuracy by Adomian’s Decomposition Methoda

Equations (68)

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

2ψ(x)+Γ2(x)ψ(x)=0.
d2ψdx2+Γ2(x)ψ=0.
Lψ+Nψ=f(x),
ψ=ψ(0)+xψ(0)+L-1[f(x)-Nψ],
ψ(0)=dψdxx=0,
ψ=n=0ψn(x),
ψ0=ψ(0)+xψ(0)-L-1[f(x)].
n=0ψn=ψ0-L-1Nn=0ψn.
ψn+1=-L-1(Nψn).
ψ1=-L-1(Nψ0),
ψ2=-L-1(Nψ1),
ψ3=-L-1(Nψ2),
g(ψ)=Nψ.
g(ψ)=n=0λnAn,
An=1n!dngdλnλ=0.
hn=1n!dngdψnλ=0,
A0=h0(ψ0),
A1=h1(ψ0)ψ1,
A2=(1/2)h2(ψ0)ψ12+h1(ψ0)ψ2,
A3=(1/3!)h3(ψ0)ψ13+h2(ψ0)ψ1ψ2+h1(ψ0)ψ3.
n(x)=n1|x|ρn2|x|>ρ,
d2ψdX2+U2ψ=0,
ψj=cos(UX)cos U,jeven,
ψj=sin(UX)sin U,jodd.
ψ=ψ(0)+Xψ(0)-L-1ψ,
ψ0=-1,
ψ1=4.9348X2,
ψ2=-4.0587X4,
ψ=-1+4.9348X2-4.0587X4+,
ψ=cos(UX)cos U.
n(x)=n01-2Δxρ2,
d2ψdx2+[k2n2(x)-β2]ψ=0,
d2ψdX2+(λ2-X2)ψ=0,
ψn(X)=12nn!π½ exp-X22Hn(X),
d2ψdx2+Γ2(x)ψ(x)=0
ψJWKB(x)=AΓ(x)sinxx0Γ(x)dx+BΓ(x)cosxx0Γ(x)dx.
ψJWKB(x)=B-A2η(x)expx0xη(x)dx+B+A22η(x)exp-x0xη(x)dx,
ψ(x)=F(x)exp[±iu(x)],
d2ψdx2-xψ(x)=0.
Ai(x)=a1f(x)-a2g(x),
Bi(x)=3[a1f(x)+a2g(x)],
f(x)=1+13!x3+(1)(4)6!x6+(1)(4)(7)9!x9+,
g(x)=x+24!x4+(2)(5)7!x7+(2)(5)(8)10!x10+,
ψ(x)=F(x)Ai[ξ(x)],
ξ(x)=-32xx0Γ(x)dx,x<x0,
ξ(x)=32x0xη(x)dx,x>x0,
F(x)=1ξ(x).
ψMAF(x)=Cξ(x)Ai[ξ(x)]+Dξ(x)Bi[ξ(x)].
Γ2(X)=λ2-X2,
η2(X)=X2-λ2,
XX0Γ(X)dX=λ2π4-X2λ2-X2-λ22sin-1Xλ,
X0Xη(X)dX=X2X2-λ2-λ22ln|X+X2-λ2|+λ22ln|λ|,
ψJWKB(0)=ψ(0),ψJWKB(0)=ψ(0)
ψ3(0)=0,ψ3(0)=-1448/π
ψ3(0)=A sinλ2π4+B cosλ2π4λ,
ψ3(0)=-Aλ4 cosλ2π4+Bλ4 sinλ2π4λ7/2.
A=3 cosλ2π4λπ,
B=-3 sinλ2π4λπ.
ψ3(X)=Cξ(X)Ai[ξ(X)]
C=323π7/12711/1227Ai[ξ(0)]-7(3π)Ai[ξ(0)].
ψ(X)=ψ(0)+Xψ(0)-L-1[Γ(X)ψ(X)]
ψ0=ψ(0)+Xψ(0)=-1448π X.
ψ0=-1448π X,
ψ1=L-1[Γ(X)ψ0]=-1448π X520-λ2X36,
ψ2=L-1[Γ(X)ψ1]=-1448π X91440-132520λ2X7+1120λ4X5,
ψ3=L-1[Γ(X)ψ2]=-1448π X13224640-591108800λ2X11+1790720λ4X9-15040λ6X7
····
ψ=ψ0+ψ1+ψ2+ψ3+.

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