Abstract

The optical field in a weakly guiding homogeneous waveguide satisfies scalar Helmholtz equations in both the core and the cladding and the transmission conditions on the boundary. Two different systems of boundary integral equations are derived for numerical solutions of the discrete propagation constants of the optical field; one of them is in the form of Fredholm integral equations of the second kind, and the other is a mixed first and second kind. The Nyström method is used to solve the boundary integral equations numerically. The numerical results show that the two boundary integral formulations are both very efficient in the numerical simulations of homogeneous waveguides but that the second kind is superior because it controls spurious modes better.

© 1998 Optical Society of America

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References

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  1. D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983).
  2. P. K. Banerjee, R. Butterfield, Boundary Element Methods in Engineering Science (McGraw-Hill, London, 1981).
  3. C. C. Su, “A surface integral equations method for homogeneous optical fibers and coupled cross sections,” IEEE Trans. Microwave Theory Tech. MTT-33, 1114–1119 (1985).
  4. E. Yamashita, ed., Analysis Methods for Electromagnetic Wave Problems (Artech, Boston, Mass., 1990).
  5. J. Charles, H. Baudrand, D. Bajon, “A full-wave analysis of an arbitrarily shaped dielectric waveguides using Green’s scalar identity,” IEEE Trans. Microwave Theory Tech. 39, 1029–1034 (1991).
    [CrossRef]
  6. R. Kress, Linear Integral Equations (Springer-Verlag, New York, 1989).
  7. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, New York, 1992).
  8. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).
  9. O. D. Kellog, Foundation of Potential Theory (Dover, New York, 1953).
  10. A. H. Juffer, E. F. F. Botta, B. A. M. van Keulen, A. van der Ploeg, H. J. C. Berendsen, “The electric potential of a macromolecule in a solvent: a fundamental approach,” J. Comput. Phys. 97, 144–171 (1991).
    [CrossRef]
  11. G. Chen, J. Zhou, Boundary Elements Methods (Academic, San Diego, Calif., 1992).
  12. J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguide,” Bell Syst. Tech. J., 2133–2160 (1969).
    [CrossRef]
  13. A. Kumar, K. Thyagarajan, A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation method,” Opt. Lett. 8, 63–65 (1988).
    [CrossRef]
  14. C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact transcendental equation for scalar modes of rectangular dielectric waveguides,” Opt. Quantum Electron. 26, 641–644 (1994).
    [CrossRef]

1994 (1)

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact transcendental equation for scalar modes of rectangular dielectric waveguides,” Opt. Quantum Electron. 26, 641–644 (1994).
[CrossRef]

1991 (2)

J. Charles, H. Baudrand, D. Bajon, “A full-wave analysis of an arbitrarily shaped dielectric waveguides using Green’s scalar identity,” IEEE Trans. Microwave Theory Tech. 39, 1029–1034 (1991).
[CrossRef]

A. H. Juffer, E. F. F. Botta, B. A. M. van Keulen, A. van der Ploeg, H. J. C. Berendsen, “The electric potential of a macromolecule in a solvent: a fundamental approach,” J. Comput. Phys. 97, 144–171 (1991).
[CrossRef]

1988 (1)

1985 (1)

C. C. Su, “A surface integral equations method for homogeneous optical fibers and coupled cross sections,” IEEE Trans. Microwave Theory Tech. MTT-33, 1114–1119 (1985).

1969 (1)

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguide,” Bell Syst. Tech. J., 2133–2160 (1969).
[CrossRef]

Bajon, D.

J. Charles, H. Baudrand, D. Bajon, “A full-wave analysis of an arbitrarily shaped dielectric waveguides using Green’s scalar identity,” IEEE Trans. Microwave Theory Tech. 39, 1029–1034 (1991).
[CrossRef]

Banerjee, P. K.

P. K. Banerjee, R. Butterfield, Boundary Element Methods in Engineering Science (McGraw-Hill, London, 1981).

Baudrand, H.

J. Charles, H. Baudrand, D. Bajon, “A full-wave analysis of an arbitrarily shaped dielectric waveguides using Green’s scalar identity,” IEEE Trans. Microwave Theory Tech. 39, 1029–1034 (1991).
[CrossRef]

Benson, T. M.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact transcendental equation for scalar modes of rectangular dielectric waveguides,” Opt. Quantum Electron. 26, 641–644 (1994).
[CrossRef]

Berendsen, H. J. C.

A. H. Juffer, E. F. F. Botta, B. A. M. van Keulen, A. van der Ploeg, H. J. C. Berendsen, “The electric potential of a macromolecule in a solvent: a fundamental approach,” J. Comput. Phys. 97, 144–171 (1991).
[CrossRef]

Botta, E. F. F.

A. H. Juffer, E. F. F. Botta, B. A. M. van Keulen, A. van der Ploeg, H. J. C. Berendsen, “The electric potential of a macromolecule in a solvent: a fundamental approach,” J. Comput. Phys. 97, 144–171 (1991).
[CrossRef]

Butterfield, R.

P. K. Banerjee, R. Butterfield, Boundary Element Methods in Engineering Science (McGraw-Hill, London, 1981).

Charles, J.

J. Charles, H. Baudrand, D. Bajon, “A full-wave analysis of an arbitrarily shaped dielectric waveguides using Green’s scalar identity,” IEEE Trans. Microwave Theory Tech. 39, 1029–1034 (1991).
[CrossRef]

Chen, G.

G. Chen, J. Zhou, Boundary Elements Methods (Academic, San Diego, Calif., 1992).

Colton, D.

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983).

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, New York, 1992).

Ghatak, A. K.

Goell, J. E.

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguide,” Bell Syst. Tech. J., 2133–2160 (1969).
[CrossRef]

Juffer, A. H.

A. H. Juffer, E. F. F. Botta, B. A. M. van Keulen, A. van der Ploeg, H. J. C. Berendsen, “The electric potential of a macromolecule in a solvent: a fundamental approach,” J. Comput. Phys. 97, 144–171 (1991).
[CrossRef]

Kellog, O. D.

O. D. Kellog, Foundation of Potential Theory (Dover, New York, 1953).

Kendall, P. C.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact transcendental equation for scalar modes of rectangular dielectric waveguides,” Opt. Quantum Electron. 26, 641–644 (1994).
[CrossRef]

Kress, R.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, New York, 1992).

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983).

R. Kress, Linear Integral Equations (Springer-Verlag, New York, 1989).

Kumar, A.

Love, J. D.

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

Smartt, C. J.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact transcendental equation for scalar modes of rectangular dielectric waveguides,” Opt. Quantum Electron. 26, 641–644 (1994).
[CrossRef]

Snyder, A. W.

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

Su, C. C.

C. C. Su, “A surface integral equations method for homogeneous optical fibers and coupled cross sections,” IEEE Trans. Microwave Theory Tech. MTT-33, 1114–1119 (1985).

Thyagarajan, K.

van der Ploeg, A.

A. H. Juffer, E. F. F. Botta, B. A. M. van Keulen, A. van der Ploeg, H. J. C. Berendsen, “The electric potential of a macromolecule in a solvent: a fundamental approach,” J. Comput. Phys. 97, 144–171 (1991).
[CrossRef]

van Keulen, B. A. M.

A. H. Juffer, E. F. F. Botta, B. A. M. van Keulen, A. van der Ploeg, H. J. C. Berendsen, “The electric potential of a macromolecule in a solvent: a fundamental approach,” J. Comput. Phys. 97, 144–171 (1991).
[CrossRef]

Zhou, J.

G. Chen, J. Zhou, Boundary Elements Methods (Academic, San Diego, Calif., 1992).

Bell Syst. Tech. J. (1)

J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguide,” Bell Syst. Tech. J., 2133–2160 (1969).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

C. C. Su, “A surface integral equations method for homogeneous optical fibers and coupled cross sections,” IEEE Trans. Microwave Theory Tech. MTT-33, 1114–1119 (1985).

J. Charles, H. Baudrand, D. Bajon, “A full-wave analysis of an arbitrarily shaped dielectric waveguides using Green’s scalar identity,” IEEE Trans. Microwave Theory Tech. 39, 1029–1034 (1991).
[CrossRef]

J. Comput. Phys. (1)

A. H. Juffer, E. F. F. Botta, B. A. M. van Keulen, A. van der Ploeg, H. J. C. Berendsen, “The electric potential of a macromolecule in a solvent: a fundamental approach,” J. Comput. Phys. 97, 144–171 (1991).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact transcendental equation for scalar modes of rectangular dielectric waveguides,” Opt. Quantum Electron. 26, 641–644 (1994).
[CrossRef]

Other (8)

G. Chen, J. Zhou, Boundary Elements Methods (Academic, San Diego, Calif., 1992).

R. Kress, Linear Integral Equations (Springer-Verlag, New York, 1989).

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, New York, 1992).

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

O. D. Kellog, Foundation of Potential Theory (Dover, New York, 1953).

E. Yamashita, ed., Analysis Methods for Electromagnetic Wave Problems (Artech, Boston, Mass., 1990).

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983).

P. K. Banerjee, R. Butterfield, Boundary Element Methods in Engineering Science (McGraw-Hill, London, 1981).

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Figures (6)

Fig. 1
Fig. 1

Homogeneous waveguide.

Fig. 2
Fig. 2

Rectangular waveguide.

Fig. 3
Fig. 3

Function w(s), here [α]=j1π/N.

Fig. 4
Fig. 4

Eigenvalues of the eigenmodes that are symmetric with respect to the x axis and the y axis of a rectangular waveguide with V=10 and a/b=2.

Fig. 5
Fig. 5

Differences between the numerical and perturbation methods.

Fig. 6
Fig. 6

Relative differences between the numerical and perturbation methods.

Tables (3)

Tables Icon

Table 1 Numerical Results for LP02 Mode (Circular)

Tables Icon

Table 2 Numerical Results for E42 Mode (Rectangular)

Tables Icon

Table 3 Normalized Propagation Constant (P) As a Function of Waveguide Frequency (V) for the Fundamental Mode of a Rectangular Waveguide with a/b=2

Equations (69)

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2ψ+(n12k2-β2)ψ=0inΩ,
2ψ+(n22k2-β2)ψ=0inΩc,
ψanditsnormalderivativearecontinuousacrossΩ,
ψ(X, Y)0as(X2+Y2)1/2,
2ψ+V2(1-P)ψ=0inΩ,
2ψ-V2Pψ=0inΩc,
ψandψ/narecontinuousacrossΩ,
ψ(x, y)0as(x2+y2)1/2,
ψ(r)=-Ω G1(r, r)nψ(r)dl+ΩG1(r, r)ψ(r)ndlinΩ,
ψ(r)=Ω G2(r, r)nψ(r)dl-ΩG2(r, r)ψ(r)ndlinΩc,
2Gl(r, r)+hl2Gl(r, r)=-δ(|r-r|),l=1, 2,
Gl(r, r)=H0(2)(hl|r-r|)/(4i),l=1, 2.
12ψ(r)+Ω G1(r, r)nψ(r)dl-ΩG1(r, r)ϕ(r)dl=0,rΩ,
12ψ(r)-Ω G2(r, r)nψ(r)dl+ΩG2(r, r)ϕ(r)dl=0,rΩ,
ψ(r)-Ωn(G2-G1)ψ(r)dl+Ω(G2-G1)ϕ(r)dl=0,rΩ.
ϕ(r)-Ω2nn(G2-G1)ψ(r)dl+Ωn(G2-G1)ϕ(r)dl=0,rΩ.
r(t)=[x(t), y(t)],0t2π,
Sl(t, τ)Gldldτ=14if1(τ)H0(2)(hlR),
Dl(t, τ)Glndldτ=ihl4f2(τ, t)H1(2)(hlR),
D(t, τ)(G2-G1)ndldτ=if2(t, τ)f1(τ)4f1(t)×[h2H1(2)(h2R)-h1H1(2)(h1R)],
T(t, τ)2(G2-G1)nndldτ=-i4f1(t){f3(t, τ)[h2H1(2)(h2R)-h1H1(2)(h1R)]+f2(t, τ)f2(τ, t)[h22H2(2)(h2R)-h12H2(2)(h1R)]},
f1(t)={[x(t)]2+[y(t)]2}1/2,
f2(t, τ)={y(t)[x(t)-x(τ)]-x(t)[y(t)-y(τ)]}/R(t, τ),
f3(t, τ)=[y(t)y(τ)+x(t)x(τ)]/R(t, τ),
R=R(t, τ)={[x(t)-x(τ)]2+[y(t)-y(τ)]2}1/2.
Yn(z)=2πlnz2+γJn(z)-1πp=0n-1(n-1-p)!p!2zn-2p-1πp=0(-1)pp!(n+p)!z2n+2p×m=1p+n1m+m=1p1m,
Sl(t, τ)=Sl1(t, τ)ln4 sin2 t-τ2+Sl2(t, τ),
Dl(t, τ)=Dl1(t, τ)ln4 sin2 t-τ2+Dl2(t, τ),
D(t, τ)=D1(t, τ)ln4 sin2 t-τ2+D2(t, τ),
T(t, τ)=T1(t, τ)ln4 sin2 t-τ2+T2(t, τ),
Sl1(t, τ)=-f1(τ)J0(hlR)/(4π),
Dl1(t, τ)=hlf2(τ, t)J1(hlR)/(4π),
D1(t, τ)=f2(t, τ)f1(τ)[h2J1(h2R)-h1J1(h1R)]/[4πf1(t)],
T1(t, τ)=-{f3(t, τ)[h2J1(h2R)-h1J1(h1R)]+f2(t, τ)f2(τ, t)[h22J2(h2R)-h12J2(h1R)]}/[4πf1(t)],
02π ln4 sin2 t-τ2S1l(t, τ)ψ(τ)dτj=02N-1Rj(N)(t)S1l(t, tj)ψ(tj),0t2π,
Rj(N)(t)=-(2π/N)m=1N-1 cos[m(t-tj)]/m-π cos[N(t-tj)]/N2,
02πSl2(t, τ)ψ(τ)dτ(π/N)j=02N-1Sl2(t, tj)ψ(tj),
12I+D1-S112I-D2S2ΨΦ=0,
I-(D2-D1)S2-S1-TI+DΨΦ=0,
Ψ=(ψ(t0), ψ(t1), , ψ(t2N-1)),
Φ=(ϕ(t0), ϕ(t1), , ϕ(t2N-1)),
Sl(i, j)=Sl1(ti, tj)Rj(N)(ti)+πSl2(ti, tj)/N,
l=1, 2,
Dl(i, j)=Dl1(ti, tj)Rj(N)(ti)+πDl2(ti, tj)/N,
l=1, 2,
D(i, j)=D1(ti, tj)Rj(N)(ti)+πD2(ti, tj)/N,
T(i, j)=T1(ti, tj)Rj(N)(ti)+πT2(ti, tj)/N.
|Q(P)|det Q(P)=0.
w(s)=2α2Up(s+α1-c)Up(s+α1-c)+Up(α1+c-s)-α2+c,
U(s)=12-1ps-α1α13+s-α1pα1+1/2,
c=0, π/2, π, 3π/2, 2π,
α1=j1π/N, π/2-j1π/N, j1π/N, π/2-j1π/N, j1π/N,
α2=α, π/2-α, α, π/2-α, α,
Ψ=(ψ¯(slx), , ψ¯(sj1-1), ψ¯(sj1+1), , ψ¯(sN/2-ly)),
Φ=(ϕ¯(slx), , ϕ¯(sj1-1), ϕ¯(sj1+1), , ϕ¯(sN/2-ly)),
Sl(i, j)=w(sj)[Sl1(si, sj)Rj(N)(si)+πSl2(si, sj)/N]+(-1)lyw(sN-j)[Sl1(si, sN-j)RN-j(N)(si)+πSl2(si, sN-j)/N]+(-1)lx+lyw(sN+j)×[Sl1(si, sN+j)RN+j(N)(si)+πSl2(si, sN+j)/N]+(-1)lxw(s2N-j)[Sl1(si, s2N-j)R2N-j(N)(si)+πSl2(si, s2N-j)/N]
Jn(z)exp(iz)=k=0akzk=k=0Lakzk+O(zL+1),
cJn(z, L)=exp(-iz)k=0Lakzk.
cJn(z, L)=exp(-iz)zn×k=0L-nm=0[k/2](iz)km!(m+n)!(k-2m)!2(2m+n).
P(n+1)=P(n)-1/tr[Q-1(P(n))Q(P(n))],
n=0, 1, 2, ,
ψ1(r)=Ω G1(r,r)nu(r)dl+ΩG1(r,r)v(r)dlinΩ,
ψ2(r)=Ω G2(r,r)nu(r)dl+ΩG2(r,r)v(r)dlinΩc.
ψ1=ψ2,ψ1/n=ψ2/n
u(r)+Ωn(G2-G1)u(r)dl+Ω(G2-G1)v(r)dl=0,
v(r)-Ω2nn(G2-G1)u(r)dl-Ωn(G2-G1)v(r)dl=0,
abK(t, τ)f(τ)dτj=02N-1wj(t)f(tj),
u(t)=j=0Nαj cos jt+j=1N-1βj sin jt,
02πK(t, τ)f(τ)dτ02πK(t, τ)u(τ)dτ=j=02N-1wj(t)f(jπ/N).

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