Abstract

A numerical method for the determination of the bound modes of multilayer planar dielectric waveguides is presented. It is based on the idea of tracing the evolutions of the eigenvalues (effective indices) of the waveguide as the physical parameters of the waveguide change. The eigenvalues are obtained by numerical integration of an initial-value problem. A mathematical proof is given that shows that the method guarantees that all bound modes of a lossless multilayer waveguide will be found. For a lossy waveguide, a similar mathematical proof is not available; however, numerical evidence shows that the method is still capable of finding all bound modes whose real parts of the effective indices are greater than those of the substrate and cover. The method is conceptually and computationally simple, but it is limited to searching for bound modes only.

© 1994 Optical Society of America

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References

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  1. J. W. Y. Lit, Y-F Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
    [CrossRef]
  2. E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
    [CrossRef]
  3. R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Numerical determination of planar waveguide modes using the analyticity of the dispersion relation,” Opt. Lett. 16, 1316–1318 (1991).
    [CrossRef] [PubMed]
  4. R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron. 28, 1520–1526 (1992).
    [CrossRef]
  5. L. M. Delves, J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560 (1967).
    [CrossRef]
  6. E. Marantonio, R. E. Zich, I. Montrosset, “Alternative expression of the dispersion equation in multilayered structures,” Proc. Inst. Electr. Eng. Part J 137, 357–360 (1990).
  7. G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
    [CrossRef]
  8. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956), Chap. 2, Sec. 13.
  9. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Chap. 1.
  10. J. Chilwell, I. Hodgkinson, “Thin-film field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984).
    [CrossRef]
  11. A. I. Markushevich, Theory of Functions of a Complex Variable, 2nd ed. (Chelsea, New York, 1977), Vol. 2, Pt. 1, Chap. 2, Sec. 15.
  12. The NAG FORTRAN Library–Mark 12(Numerical Algorithm Group, Inc., Downers Grove, Ill., 1987).
  13. R. J. Hawkins, R. J. Deri, O. Wada, “Optical power transfer in vertically integrated impedance-matched waveguide/ photodetectors: physics and implications for diode-length reduction,” Opt. Lett. 16, 470–472 (1991).
    [CrossRef] [PubMed]

1992 (2)

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

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

1991 (2)

1990 (1)

E. Marantonio, R. E. Zich, I. Montrosset, “Alternative expression of the dispersion equation in multilayered structures,” Proc. Inst. Electr. Eng. Part J 137, 357–360 (1990).

1988 (1)

J. W. Y. Lit, Y-F Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

1984 (2)

1967 (1)

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

Anemogiannis, E.

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

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Chap. 1.

Chilwell, J.

Delves, L. M.

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

Deri, R. J.

Forbes, G. W.

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

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Numerical determination of planar waveguide modes using the analyticity of the dispersion relation,” Opt. Lett. 16, 1316–1318 (1991).
[CrossRef] [PubMed]

Glytsis, E. N.

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

Hawkins, R. J.

Hewak, D. W.

J. W. Y. Lit, Y-F Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Hodgkinson, I.

Houde-Walter, S. N.

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

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Numerical determination of planar waveguide modes using the analyticity of the dispersion relation,” Opt. Lett. 16, 1316–1318 (1991).
[CrossRef] [PubMed]

Ince, E. L.

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956), Chap. 2, Sec. 13.

Li, Y-F

J. W. Y. Lit, Y-F Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Lit, J. W. Y.

J. W. Y. Lit, Y-F Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Lyness, J. N.

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

Marantonio, E.

E. Marantonio, R. E. Zich, I. Montrosset, “Alternative expression of the dispersion equation in multilayered structures,” Proc. Inst. Electr. Eng. Part J 137, 357–360 (1990).

Markushevich, A. I.

A. I. Markushevich, Theory of Functions of a Complex Variable, 2nd ed. (Chelsea, New York, 1977), Vol. 2, Pt. 1, Chap. 2, Sec. 15.

Montrosset, I.

E. Marantonio, R. E. Zich, I. Montrosset, “Alternative expression of the dispersion equation in multilayered structures,” Proc. Inst. Electr. Eng. Part J 137, 357–360 (1990).

Petit, R.

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Smith, R. E.

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

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Numerical determination of planar waveguide modes using the analyticity of the dispersion relation,” Opt. Lett. 16, 1316–1318 (1991).
[CrossRef] [PubMed]

Tayeb, G.

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Wada, O.

Zich, R. E.

E. Marantonio, R. E. Zich, I. Montrosset, “Alternative expression of the dispersion equation in multilayered structures,” Proc. Inst. Electr. Eng. Part J 137, 357–360 (1990).

Can. J. Phys. (1)

J. W. Y. Lit, Y-F Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

J. Lightwave Technol. (1)

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

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

Math. Comput. (1)

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

Opt. Acta (1)

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Opt. Lett. (2)

Proc. Inst. Electr. Eng. Part J (1)

E. Marantonio, R. E. Zich, I. Montrosset, “Alternative expression of the dispersion equation in multilayered structures,” Proc. Inst. Electr. Eng. Part J 137, 357–360 (1990).

Other (4)

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956), Chap. 2, Sec. 13.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Chap. 1.

A. I. Markushevich, Theory of Functions of a Complex Variable, 2nd ed. (Chelsea, New York, 1977), Vol. 2, Pt. 1, Chap. 2, Sec. 15.

The NAG FORTRAN Library–Mark 12(Numerical Algorithm Group, Inc., Downers Grove, Ill., 1987).

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Figures (9)

Fig. 1
Fig. 1

Coordinate system and notation for a general multilayer waveguide.

Fig. 2
Fig. 2

Schematic refractive-index distributions of a parameterized waveguide W(t) (a) at t = 1.0 and (b) at t = 0.

Fig. 3
Fig. 3

Schematic diagram illustrating the relationships among the modes of waveguides W(0) and W(1).

Fig. 4
Fig. 4

Complex planes of (a) β2 and (b) β. In (a) the center dot indicates the branch point, and the heavy line indicates the branch cut. The upper and lower Riemann sheets in (a) are mapped onto the upper and lower half-planes in (b), respectively. In (b) the upper sheet is indicated by the closely spaced dots, and the lower sheet is indicated by the loosely spaced dots.

Fig. 5
Fig. 5

Evolution of the eigenvalues for the bound modes of the lossless waveguide described in Table 1. The solid curves are for the TE modes, and the dashed curves are for the TM modes. The dotted–dashed line indicates the cutoff position on the vertical axis.

Fig. 6
Fig. 6

Trajectories of the first four bound modes of W(0) for a low-loss waveguide of the following physical parameters: ns = 1.5, nc = 1.0, n1 = 1.66, n2 = 1.6, n3 = 1.53 + i1.53 × 10−4, n4 = 1.66 + i1.66 × 10−4, h1 = h2 = h3 = h4 = 0.5 μm, λ = 0.6328 μm.

Fig. 7
Fig. 7

Trajectory of the fifth bound mode of W(0) for the waveguide described in Fig. 6. The two insets, one within another, show in two different magnifications the trajectory in the vicinity of the branch point. In the smallest inset 1.0E-5 is 1.0 × 10−5, for example.

Fig. 8
Fig. 8

Trajectories of the bound modes of W(0) for the lossy waveguide described in Table 2.

Fig. 9
Fig. 9

Trajectories of the highest seven bound modes of W(0) for the lossy waveguide described in Table 2. The waveguide is parameterized with n* = 5.0.

Tables (2)

Tables Icon

Table 1 Effective Indices of the Bound Modes of a Lossless Waveguidea

Tables Icon

Table 2 Effective Indices of the TE Polarized Bound Modes of a Lossy Waveguidea

Equations (28)

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

I * = exp ( x 2 ) 1 + x 2 d x .
I ( t ) = exp ( t x 2 ) 1 + x 2 d x , t 0 .
d I d t = I π t .
I ( t ) = exp ( t ) [ C π 0 t exp ( x ) x d x ] ,
I ( t ) = π exp ( t ) erfc ( t ) ,
I * = I ( 1 ) = ( π e ) erfc ( 1 ) ,
n * = max { Re [ n j ] : j = 1 , 2 , . . . , N } .
Re [ n c ] < Re [ n s ] < n * .
j ( t ) = { n * 2 + ( j n * 2 ) t , 1 j N Re [ j ] + i Im [ j ] t , j = 0 , N + 1 ,
ρ = ( k n eff ) 2 ,
δ ρ ψ 2 d y = k 2 δ ψ 2 d y
δ ρ 1 ψ 2 d y = δ ( 1 ) [ ρ ψ 2 + ( d ψ d y ) 2 ] d y ,
R ( 0 ) = φ R ( 1 ) R * ( 0 ) .
R ( 1 ) R * ( 1 ) = φ 1 R * ( 0 ) .
h ( κ f 2 ρ ) 1 / 2 arctan [ τ c ( ρ κ c 2 κ f 2 ρ ) 1 / 2 ] arctan [ τ s ( ρ κ s 2 κ f 2 ρ ) 1 / 2 ] m π = 0 ,
F j ( x , y ; t ) = A j exp ( i α x + i β j y ) + B j exp ( i α x i β j y ) ,
α = ρ , β j = [ k j 2 ( t ) ρ ] 1 / 2 ,
k j 2 ( t ) = k 2 j ( t ) μ j , 0 j N + 1 .
G ( ρ , t ) = 0 .
d ρ d t = G / t G / ρ ,
G ρ 0 .
ρ ( 1 ) = 0 1 G / t G / ρ d t + ρ ( 0 ) ,
σ ( y ) = { μ ( y ) for TE ( y ) for TM .
L ψ = 1 σ ρ ψ , ψ ( | y | ) = 0 ,
L = d d y ( 1 σ d d y ) + k 2 μ σ .
( δ L ) ψ + L ( δ ψ ) = ρ ψ δ ( 1 σ ) + 1 σ ψ δ ρ + 1 σ ρ δ ψ ,
δ L = d d y [ δ ( 1 σ ) d d y ] + k 2 δ ( μ σ ) .
δ ρ 1 σ ψ 2 d y = [ k 2 δ ( μ σ ) ρ δ ( 1 σ ) ] ψ 2 d y δ ( 1 σ ) ( d ψ d y ) 2 d y .

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