Abstract

There has been increasing interest in making photonic devices more and more compact in the integrated photonics industry, and one of the important questions for manufacturers and design engineers is how to quantify the effect of the finite cladding thickness on the modal confinement loss of photonic waveguides. This requires at least six to seven digits of accuracy for the computation of propagation constant β since the modal confinement loss is proportional to the imaginary part of β that is six to seven orders of magnitude smaller than its real part by the industrial standard. In this paper, we present an accurate and efficient method to compute the propagation constant of the electromagnetic modes of photonic waveguides with an arbitrary number of (nonsmooth) inclusions in a layered media. The method combines a well-conditioned boundary integral equation formulation for photonic waveguides which requires the discretization of the material interface only, and efficient Sommerfeld integral representations to treat the effect of the layered medium. Our scheme is capable of calculating the propagation loss of the electromagnetic modes with high fidelity, even for waveguides with corners embedded in a cladding material of finite thickness. The numerical results, with a more than 10-digit accuracy, show quantitatively that the modal confinement loss of the rectangular waveguide increases exponentially fast as the cladding thickness decreases.

© 2016 Optical Society of America

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References

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2014 (1)

J. Lai, M. Kobayashi, and L. Greengard, “A fast solver for multi-particle scattering in a layered medium,” Opt. Express 14, 302–307 (2014).

2012 (1)

W. Lu and Y. Y. Lu, “Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations,” J. Comput. Phys. 231, 1360–1371 (2012).
[Crossref]

2009 (1)

2007 (2)

2004 (1)

2002 (3)

2001 (1)

S. Selleri, L. Vincetti L, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quant. Electron. 33, 359–371 (2001).
[Crossref]

1999 (1)

1956 (1)

D. E. Müller, “A method for solving algebraic equations using an automatic computer,” Math. Tables Aids Comput. 10, 208–215 (1956).
[Crossref]

Andres, M.V.

Andres, P.

Benson, T. M.

N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2007).
[Crossref]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Boriskina, S. V.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Botten, L. C.

Chang, H. C.

Cheng, H.

Chiang, Y. C.

Chiou, Y. P.

Crutchfield, W. Y.

Cucinotta, A.

S. Selleri, L. Vincetti L, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quant. Electron. 33, 359–371 (2001).
[Crossref]

Doery, M.

Du, C. H.

El-Mikati, H. A.

Ferrando, A.

Grattan, K. T. V.

Greengard, L.

J. Lai, M. Kobayashi, and L. Greengard, “A fast solver for multi-particle scattering in a layered medium,” Opt. Express 14, 302–307 (2014).

H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express 12(16), 3791–3805 (2004).
[Crossref] [PubMed]

Hassani, A.

Kabashin, A.

Kobayashi, M.

J. Lai, M. Kobayashi, and L. Greengard, “A fast solver for multi-particle scattering in a layered medium,” Opt. Express 14, 302–307 (2014).

Kress, R.

R. Kress, Linear Integral Equations (Springer–Verlag, 1989).
[Crossref]

Kuhlmey, B. T.

Lacroix, S.

Lai, C. H.

Lai, J.

J. Lai, M. Kobayashi, and L. Greengard, “A fast solver for multi-particle scattering in a layered medium,” Opt. Express 14, 302–307 (2014).

Lu, W.

W. Lu and Y. Y. Lu, “Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations,” J. Comput. Phys. 231, 1360–1371 (2012).
[Crossref]

Lu, Y. Y.

W. Lu and Y. Y. Lu, “Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations,” J. Comput. Phys. 231, 1360–1371 (2012).
[Crossref]

Martijn de Sterke, C.

Maystre, D.

McPhedran, R. C.

Miret, J.J.

Müller, D. E.

D. E. Müller, “A method for solving algebraic equations using an automatic computer,” Math. Tables Aids Comput. 10, 208–215 (1956).
[Crossref]

Nosich, A. I.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Obayya, S. S. A.

Pone, E.

Rahman, B. M. A.

Renversez, G.

Selleri, S.

S. Selleri, L. Vincetti L, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quant. Electron. 33, 359–371 (2001).
[Crossref]

Sewell, P.

N. Thomas, P. Sewell, and T. M. Benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2007).
[Crossref]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Silvestre, E.

Skorobogatiy, M.

Thomas, N.

Vincetti L, L.

S. Selleri, L. Vincetti L, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quant. Electron. 33, 359–371 (2001).
[Crossref]

White, T. P.

Zoboli, M.

S. Selleri, L. Vincetti L, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quant. Electron. 33, 359–371 (2001).
[Crossref]

IEEE J. Selected Topics in Quantum Electron. (1)

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Highly efficient full-vectorial integral equations method solution for the bound, leaky, and complex modes of dielectric waveguides,” IEEE J. Selected Topics in Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

J. Comput. Phys. (1)

W. Lu and Y. Y. Lu, “Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations,” J. Comput. Phys. 231, 1360–1371 (2012).
[Crossref]

J. Lightwave Technol. (3)

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

Math. Tables Aids Comput. (1)

D. E. Müller, “A method for solving algebraic equations using an automatic computer,” Math. Tables Aids Comput. 10, 208–215 (1956).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Opt. Quant. Electron. (1)

S. Selleri, L. Vincetti L, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quant. Electron. 33, 359–371 (2001).
[Crossref]

Other (3)

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds. NIST Handbook of Mathematical Functions (Cambridge University Press, 2010).

R. Kress, Linear Integral Equations (Springer–Verlag, 1989).
[Crossref]

J. Lai and S. Jiang, “Second kind integral equation formulation for the mode calculation of optical waveguides,” Appl. Comput. Harmon. Anal. in press, doi: http://dx.doi.org/10.1016/j.acha.2016.06.009 .

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Figures (5)

Fig. 1
Fig. 1 Cross section of a rectangular waveguide in layered medium.
Fig. 2
Fig. 2 The hyperbolic tangent contour used in the Sommerfeld representation
Fig. 3
Fig. 3 Modal confinement loss versus the lower cladding thickness for Example 1. The y-axis uses logarithmic scale (base 10). Dashed lines are the least squares fit to the data.
Fig. 4
Fig. 4 Magnitude of the electromagnetic field Ez and Hz of the first and second modes with hl = 4μm in Example 1. The square represents the boundary of the waveguide. The colorbar uses logarithmic scale.
Fig. 5
Fig. 5 Modal confinement loss versus the lower cladding thickness for Example 2. The y-axis uses logarithmic scale (base 10). Dashed lines are the least squares fit to the data.

Tables (3)

Tables Icon

Table 1 Convergence study for the effective index of the second mode when the lower cladding thickness is 4μm. The first column lists the number of discretization points on each side of the square; the second and third columns list the real and imaginary parts of the effective index of the second mode.

Tables Icon

Table 2 Effective indices versus lower cladding thickness hl for Example 1.

Tables Icon

Table 3 Effective indices of the photonic waveguide in Example 2.

Equations (17)

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

{ × E = μ 0 H t , E = 0 , × H = ϵ 0 n 2 E t , H = 0 ,
[ E ( x , y , z , t ) , H ( x , y , z , t ) ] = [ E ( x , y ) , H ( x , y ) ] e i ( β z ω t ) ,
[ Δ + ( k 2 β 2 ) ] u = 0 ,
[ E z ] = 0 , [ H z ] = 0 , [ E τ ] = 0 , [ H τ ] = 0 ,
G k ( P , Q ) = i 4 H 0 ( 1 ) ( k 2 β 2 P Q ) ,
{ S Γ k [ σ ] ( P ) = Γ G k ( P , Q ) σ ( Q ) d s Q , D Γ k [ σ ] ( P ) = Γ G k ( P , Q ) ν ( Q ) σ ( Q ) d s Q , T Γ k [ σ ] ( P ) = Γ G k ( P , Q ) τ ( Q ) σ ( Q ) d s Q .
G k ( P , Q ) = 1 4 π e λ 2 k 2 + β 2 | y y 0 | λ 2 k 2 + β 2 e i λ ( x x 0 ) d λ .
{ S ^ Γ t k 1 [ σ ^ ] ( P ) = 1 4 π e λ 2 k 1 2 + β 2 | y y t | λ 2 k 1 2 + β 2 e i λ x σ ^ ( λ ) d λ , D ^ Γ t k 1 [ σ ^ ] ( P ) = 1 4 π e λ 2 k 1 2 + β 2 | y y t | e i λ x σ ^ ( λ ) d λ , T ^ Γ t k 1 [ σ ^ ] ( P ) = 1 4 π i λ e λ 2 k 1 2 + β 2 | y y t | λ 2 k 1 2 + β 2 e i λ x σ ^ ( λ ) d λ ,
{ ϕ Γ , x k [ J , M ] = 1 i k ν x Τ Γ k [ J τ ] + β k ν x S Γ k [ J z ] + k 2 i k ν S Γ k [ J τ τ x ] y S Γ k [ M z ] + i β S Γ k [ M τ τ y ] , ϕ Γ , y k [ J , M ] = 1 i k ν y Τ Γ k [ J τ ] + β k ν y S Γ k [ J z ] + k 2 i k ν S Γ k [ J τ τ y ] + x S Γ k [ M z ] i β S Γ k [ M τ τ x ] , ϕ Γ , z k [ J , M ] = β k ν Τ Γ k [ J τ ] + ( k 2 β 2 ) i k ν S Γ k [ J z ] + D Γ k [ M τ ] .
{ ψ Γ , x k [ J , M ] = 1 i k ν x Τ Γ k [ M τ ] β k ν x S Γ k [ M z ] k 2 i k ν S Γ k [ M τ τ x ] k 2 k ν 2 y S Γ k [ J z ] + i β k 2 k ν 2 S Γ k [ J τ τ y ] , ψ Γ , y k [ J , M ] = 1 i k ν y Τ Γ k [ M τ ] β k ν y S Γ k [ M z ] k 2 i k ν S Γ k [ M τ τ y ] + k 2 k ν 2 x S Γ k [ J z ] i β k 2 k ν 2 S Γ k [ J τ τ x ] , ψ Γ , z k [ J , M ] = β k ν Τ Γ k [ M τ ] ( k 2 β 2 ) i k ν S Γ k [ M z ] + k 2 k ν 2 D Γ k [ J τ ] .
{ ( E , H ) = ( ϕ Γ 0 k 0 [ J , M ] , ψ Γ 0 k 0 [ J , M ] ) in Ω 0 , ( E , H ) = ( ϕ ^ Γ t k 1 [ J ^ t , M ^ t ] , ψ ^ Γ t k 1 [ J ^ t , M ^ t ] ) in Ω 1 , ( E , H ) = ( ϕ ^ Γ t k 2 [ J ^ t , M ^ t ] , ψ ^ Γ t k 2 [ J ^ t , M ^ t ] ) + ( ϕ Γ 0 k 2 [ J , M ] , ψ Γ 0 k 2 [ J , M ] ) + ( ϕ ^ Γ b k 2 [ J ^ b , M ^ b ] , ψ ^ Γ b k 2 [ J ^ b , M ^ b ] ) in Ω 2 , ( E , H ) = ( ϕ ^ Γ b k 3 [ J ^ b , M ^ b ] , ψ ^ Γ b k 3 [ J ^ b , M ^ b ] ) in Ω 3 .
λ = t tanh ( t ) 2 i , t .
L = 20 ln ( 10 ) 2 π λ I m ( n e ) 10 9
L 10 K 1 h l .
S 1 = 4 π n 2 2 n e 2 ( log 10 e ) / λ 0.738 μ m 1 ,
L 10 K 2 h l .
S 2 = 4 π n 2 2 n e 2 ( log 10 e ) / λ 0.468 μ m 1 ,

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