Abstract

We apply the boundary element method to the analysis of optical waveguides. After summarizing constant and linear element algorithms for both two- and three-dimensional simulations, we introduce a new recursive series procedure for constructing the diagonal matrix elements. We then demonstrate that our method can be employed to minimize the reflectivity of optical waveguide antireflection coatings with both straight and angled facets.

© 2002 Optical Society of America

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References

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  1. J. C. F. Telles, C. A. Brebbia, L. C. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering (Springer-Verlag, New York, 1984).
  2. K. Tanaka, M. Tanaka, “Computer-aided design of dielectric optical waveguide bends by the boundary-element method based on guided-mode extracted integral equations,” J. Opt. Soc. Am. A 13, 1362–1368 (1996).
    [CrossRef]
  3. S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. 32, 455–461 (1994).
    [CrossRef]
  4. T. Lu, “The boundary element analysis of dielectric waveguides,” M.S. thesis (Queen’s University, Kingston, Ontario, Canada, 1998).
  5. Wei Yang, A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photon. Technol. Lett. 7, 777–779 (1995).
    [CrossRef]
  6. E. Kreyszig, Advanced Engineering Mathematics, 4th ed. (Wiley, New York, 1984).
  7. S. Aoki, J. Yamauchi, M. Mita, H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239–241 (1996).
    [CrossRef]
  8. D. Marcuse, “Reflection loss of laser mode from tilted end mirror,” J. Lightwave Technol. 7, 336–339 (1989).
    [CrossRef]
  9. C. C. Su, “A surface integral equations method for homogeneous optical fibers and coupled image lines of arbitrary cross sections,” IEEE Trans. Microwave Theory Tech. MTT-33, 1114–1119 (1985).
  10. J. S. Gu, P. A. Besse, H. Melchior, “Reflectivity minimization of semiconductor laser amplifiers with coated and angled facets considering two-dimensional beam profiles,” IEEE J. Quantum Electron. 27, 1830–1836 (1991).
    [CrossRef]

1996

K. Tanaka, M. Tanaka, “Computer-aided design of dielectric optical waveguide bends by the boundary-element method based on guided-mode extracted integral equations,” J. Opt. Soc. Am. A 13, 1362–1368 (1996).
[CrossRef]

S. Aoki, J. Yamauchi, M. Mita, H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239–241 (1996).
[CrossRef]

1995

Wei Yang, A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photon. Technol. Lett. 7, 777–779 (1995).
[CrossRef]

1994

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. 32, 455–461 (1994).
[CrossRef]

1991

J. S. Gu, P. A. Besse, H. Melchior, “Reflectivity minimization of semiconductor laser amplifiers with coated and angled facets considering two-dimensional beam profiles,” IEEE J. Quantum Electron. 27, 1830–1836 (1991).
[CrossRef]

1989

D. Marcuse, “Reflection loss of laser mode from tilted end mirror,” J. Lightwave Technol. 7, 336–339 (1989).
[CrossRef]

1985

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

Aoki, S.

S. Aoki, J. Yamauchi, M. Mita, H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239–241 (1996).
[CrossRef]

Besse, P. A.

J. S. Gu, P. A. Besse, H. Melchior, “Reflectivity minimization of semiconductor laser amplifiers with coated and angled facets considering two-dimensional beam profiles,” IEEE J. Quantum Electron. 27, 1830–1836 (1991).
[CrossRef]

Brebbia, C. A.

J. C. F. Telles, C. A. Brebbia, L. C. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering (Springer-Verlag, New York, 1984).

Fukai, I.

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. 32, 455–461 (1994).
[CrossRef]

Gopinath, A.

Wei Yang, A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photon. Technol. Lett. 7, 777–779 (1995).
[CrossRef]

Gu, J. S.

J. S. Gu, P. A. Besse, H. Melchior, “Reflectivity minimization of semiconductor laser amplifiers with coated and angled facets considering two-dimensional beam profiles,” IEEE J. Quantum Electron. 27, 1830–1836 (1991).
[CrossRef]

Kagami, S.

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. 32, 455–461 (1994).
[CrossRef]

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics, 4th ed. (Wiley, New York, 1984).

Lu, T.

T. Lu, “The boundary element analysis of dielectric waveguides,” M.S. thesis (Queen’s University, Kingston, Ontario, Canada, 1998).

Marcuse, D.

D. Marcuse, “Reflection loss of laser mode from tilted end mirror,” J. Lightwave Technol. 7, 336–339 (1989).
[CrossRef]

Melchior, H.

J. S. Gu, P. A. Besse, H. Melchior, “Reflectivity minimization of semiconductor laser amplifiers with coated and angled facets considering two-dimensional beam profiles,” IEEE J. Quantum Electron. 27, 1830–1836 (1991).
[CrossRef]

Mita, M.

S. Aoki, J. Yamauchi, M. Mita, H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239–241 (1996).
[CrossRef]

Nakano, H.

S. Aoki, J. Yamauchi, M. Mita, H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239–241 (1996).
[CrossRef]

Su, C. C.

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

Tanaka, K.

Tanaka, M.

Telles, J. C. F.

J. C. F. Telles, C. A. Brebbia, L. C. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering (Springer-Verlag, New York, 1984).

Wrobel, L. C.

J. C. F. Telles, C. A. Brebbia, L. C. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering (Springer-Verlag, New York, 1984).

Yamauchi, J.

S. Aoki, J. Yamauchi, M. Mita, H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239–241 (1996).
[CrossRef]

Yang, Wei

Wei Yang, A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photon. Technol. Lett. 7, 777–779 (1995).
[CrossRef]

IEEE J. Quantum Electron.

J. S. Gu, P. A. Besse, H. Melchior, “Reflectivity minimization of semiconductor laser amplifiers with coated and angled facets considering two-dimensional beam profiles,” IEEE J. Quantum Electron. 27, 1830–1836 (1991).
[CrossRef]

IEEE Photon. Technol. Lett.

Wei Yang, A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photon. Technol. Lett. 7, 777–779 (1995).
[CrossRef]

S. Aoki, J. Yamauchi, M. Mita, H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239–241 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. 32, 455–461 (1994).
[CrossRef]

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

J. Lightwave Technol.

D. Marcuse, “Reflection loss of laser mode from tilted end mirror,” J. Lightwave Technol. 7, 336–339 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Other

J. C. F. Telles, C. A. Brebbia, L. C. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering (Springer-Verlag, New York, 1984).

T. Lu, “The boundary element analysis of dielectric waveguides,” M.S. thesis (Queen’s University, Kingston, Ontario, Canada, 1998).

E. Kreyszig, Advanced Engineering Mathematics, 4th ed. (Wiley, New York, 1984).

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Figures (13)

Fig. 1
Fig. 1

2-D regions and integration over region boundaries. The input field is incident on boundary Γs.

Fig. 2
Fig. 2

Subdivision of different elements into simple triangles centered at A0. (a) Subdivision of the triangular element into T1, T2, and T3 and (b) subdivision of the quadrilateral element into T1, T2, T3, and T4.

Fig. 3
Fig. 3

Linear element algorithm for the triangular element; any triangular element can be mapped in (η1, η2) coordinates.

Fig. 4
Fig. 4

2-D slab waveguides: (a) single-layer AR coating and (b) double-layer AR coating.

Fig. 5
Fig. 5

Modal reflection versus coating thickness for a single-layer AR-coated slab waveguide as calculated with both the BEM and the FD-TD method incorporating the PML boundary condition.

Fig. 6
Fig. 6

Discretization error: power reflectivity versus grid point spacing dx for the constant BEM and for the linear BEM with uniform and nonuniform grids.

Fig. 7
Fig. 7

Power reflectivity versus tilt angle for a single-layer AR-coated waveguide.

Fig. 8
Fig. 8

Modal reflectivity versus wavelength for single-layer and double-layer AR-coated slab waveguides as calculated with both the BEM and the FD-TD method with the PML boundary condition.

Fig. 9
Fig. 9

3-D view of a single-layer AR-coated rectangular waveguide.

Fig. 10
Fig. 10

Power reflectivity versus coating thickness in the rectangular and square waveguides.

Fig. 11
Fig. 11

Reflected field in a square waveguide (0.4 μm×0.4 μm). The refractive index of the AR coating has been set to nAC=1.92, and the semivectorial formalism is used.

Fig. 12
Fig. 12

Power reflectivity versus tilt angle for rectangular waveguides with different geometries.

Fig. 13
Fig. 13

Modal reflectivity spectrum of a 0.15-μm×3.0-μm rectangular waveguide (TE mode).

Equations (72)

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n=f(x, z)xi+f(x, z)zk
γU(r)+Γ-ΓU(r) G(r,r)ndl
=Γ-ΓG(r, r) U(r)ndl+S(r).
G(r, r)=G(ρ)=14j H0(2)(kρ),
G(r, r)n=G(ρ)n=jk4 H1(2)(kρ)cos (ρˆ, nˆ).
S(r)=-ΓsUs(r) G(r, r)ndl+ΓsG(r, r) Us(r)ndl.
γrϕ(ri)=-ji0ΓjGr(rj-ri)njr ϕ(rj)dl-jΓjGr(rj-ri)αFrψ(rj)dl+SR(ri),
γlϕ(ri)=-jiΓjGl(rj-ri)njl ϕ(rj)dl+jΓjGl(rj-ri)αFlψ(rj)dl+SL(ri).
αTMl,r=kl,r2,αTEl,r=αSCl,r=1.
ϕ(r)=Eyl(r)=Eyr(r),
ψ(r)=Eyl(r)n=-Eyr(r)nr,
ϕ(r)=Hyl(r)=Hyr(r),
ψ(r)=1kl2Hyl(r)nl=-1kr2Hyr(r)nr.
12 ϕi=-jiϕjΓjGr(rj-ri)njdl-jψjαFrΓjGr(rj-ri)dl+SRi,
12 ϕi=-jiϕjΓjGl(rj-ri)njdl+jψjαFlΓjGl(rj-ri)dl+SLi.
V=ϕ1ϕ2ϕNψ1ψ2ψN,S=SR1SR2SRNSL1SL2SLN,
HV=S.
H=HrGrHl-Gl.
gijl,r=αFl,r-1+114j H0(2)(kρ)dξ lj2,
hijl,r=-1+114 jkH1(2)(kρ)[cos (ρˆ, nˆ)]dξ lj2.
giil,r=12jkk=1(-1)k(2k+1)22k(k!)2kli22k+1×1-i 2πlnkli4+γE-12k+1-m=1k1m+1-i 2πlnkli4-1+γEkli2,
ϕ(ξ)=1-ξ2 ϕ1+1+ξ2 ϕ2,
ψ(ξ)=1-ξ2 ψ1+1+ξ2 ψ2.
γrϕi+e=1N(h1e,rh2e,r)ϕ1eϕ2e
=-e=1N(g1e,rg2e,r)ψ1eψ2e+SRi,
γlϕi+e=1N(h1e,lh2e,l)ϕ1eϕ2e
=e=1N(g1e,lg2e,l)ψ1eψ2e+SLi.
h1eh2e=-1112 (1±ξ) jk4 H1(2)(kρ)[cos (ρˆ, nˆ)]dξ le2,
g1eg2e=-1112 (1±ξ) 14j H0(2)(kρ)dξ le2.
g1e=k=1(-1)k(2k+1)22k(k!)2 (kle)2k+11-i 2πlnkle2+γ-12k+1+2iπm=1k1m+1-i 2πlnkle2-1+γ×(kle)-kle4kj H1(2)(kle)+12kπ,
g2e=14kj H1(2)(kle)-12k2leπ.
G(ρ)=14πρexp(-jkρ),
G(ρ)n=-1+jkρ4πρ2exp(-jkρ)cos (ρˆ, nˆ).
γU(r)+S-SU(r) G(r, r)nds
=S-SG(r, r) U(r)nds+S(r).
ϕ=Eyl=Eyr,
ϕ=kl2Eyl=kr2Eyr,
ψ=Eylnl=-Eyrnr.
γrαFr(r)ϕ(r)=-ji0SjGr(ρ)njr αFr(rj)ϕ(rj)ds-jSjGr(ρ)ψ(rj)ds+SR,
γlαFl(r)ϕ(r)=-ji0SjGl(ρ)njl αFl(rj)ϕ(rj)ds+jSjGl(ρ)ψ(rj)ds+SL,
αSCl,r=1,
αTEl,r=1,n(r)yˆ1kl,r2,n(r)|yˆ,
αTMl,r=1,n(r)xˆ1kl,r2,n(r)|xˆ.
12 αFr(ri)ϕi=-jiϕjSjαFr(rj) Gr(ρ)njrds-jψiSjGr(ρ)ds+SRi,
12 αFl(ri)ϕi=-jiϕjSjαFl(rj) Gl(ρ)njlds+jψjSjGl(ρ)ds+SLi.
hijl,r=-αFl,rSj1+jkρ4πρ2exp(-jkρ)[cos(ρˆ, nˆ)]ds
gijl,r=Sj14πρexp(-jkρ)ds.
giil,r=k=13gk=k=13Δk14πρexp(-jkρ)ds.
g1=Δr114πρexp(-jkρ)ds=0θ01dθ0ρ0(θ)14πρexp(-jkρ)ρdρ=-14πjkn=1(-jkr1sin θ11)nn! pn(θ11, θ01),
g2=-14πjkn=1(-jkr2sin θ22)nn! pn(θ22, θ02),
g3=-14πjkn=1(-jkr3sin θ33)nn! pn(θ33, θ03),
pn(θ1, θ2)=0θ2dθsinn(θ+θ1),
p0(θ1, θ2)=θ2,
p1(θ1, θ2)=lntanθ1+θ22-lntanθ12,
pn(θ1, θ2)=cos θ1(n-1)sinn-1θ1-cos(θ1+θ2)(n-1)sinn-1(θ1+θ2)+n-2n-1 pn-2(θ1, θ2).
γrαFrϕi=-eiN(h1e,rh2e,rh3e,r)ϕ1eϕ2eϕ3e-e=1N(g1e,rg2e,rg3e,r)ψ1eψ2eψ3e+SRi,
γlαFlϕi=-eiN(h1e,lh2e,lh3e,l)ϕ1eϕ2eϕ3e+e=1N(1e,lg2e,lg3e,l)ψ1eψ2eψ3e+SLi,
h1eh2eh3e=αF4πΔe-1-jkρρ2exp(-jkρ)[cos (ρˆ, nˆ)]×η1η2η3ds,
g1eg2egee=14πΔe1ρexp(-jkρ)η1η2η3ds.
ρsin θ=η1r13sin(Θ3-θ)=η2r23sin θ.
g3e=14πΔeexp(-jkρ)ρ (1-η1-η2)ds=g0e-g1e-g2e.
g0e=14πΔeexp(-jkρ)ρds
=14π(-jk)n=1(-jkr13sin Θ1)nn! pn(Θ1, Θ3)
=14π(-jk)n=1an(13).
an(lm)=(-jkrlmsin Θl)nn! pn(Θl, Θm).
pn(Θ1, Θ3)=0Θ3dθsinn(0+Θ1),
g1e=14πΔeexp(-jkρ)ρ η1ds
=14π0Θ3dθ0(r13sin Θ1)/sin(Θ1+θ)×exp(-jkρ)ρρ sin(Θ3-0)r13sin Θ3 ρdρ.
g1e=r23cos Θ24πr12(-jk)n=1nn+1 an(13)+sin Θ1exp(-jkr13)-sin Θ2exp(-jkr23)4πr12(-jk)2+sin Θ2-sin Θ14πr12(-jk)2,
g2e=r13cos Θ14πr12(-jk)n=1nn+1 an(13)+sin Θ2exp(-jkr23)-sin Θ1exp(-jkr13)4πr12(-jk)2+sin Θ1-sin Θ24πr12(-jk)2,
g3e=g0e-g1e-g2e
=14π(-jk)n=1an(13)1-nn+1cos(Θ1-Θ2).

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