Abstract

The perfectly matched layer boundary conditions are incorporated into the R-matrix propagator method to achieve an extended capability for the modeling of integrated optical devices. As examples, with this technique we calculate the free-space coupling of a Gaussian beam into a planar waveguide, both with and without surface roughness on the waveguide surface.

© 1999 Optical Society of America

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References

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  1. J. M. Elson, P. Tran, “Dispersion and diffraction in photonic media: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
    [CrossRef]
  2. J. M. Elson, P. Tran, “Coupled mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 54, 1711–1715 (1996).
    [CrossRef]
  3. J. M. Elson, P. Tran, “Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator,” in Photonic Band Gap Materials, Vol. 315 of NATO Advanced Study Institute Series E: Applied Sciences, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 341–354.
  4. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  5. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 11, 2816–2828 (1995).
  6. J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2773–2775 (1992).
    [CrossRef]
  7. Jean-Pierre Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994);J.-P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
    [CrossRef]
  8. A. Maradudin, T. Michel, A. McGurn, E. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]

1996 (1)

J. M. Elson, P. Tran, “Coupled mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 54, 1711–1715 (1996).
[CrossRef]

1995 (2)

1994 (1)

Jean-Pierre Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994);J.-P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

1993 (1)

1992 (1)

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2773–2775 (1992).
[CrossRef]

1990 (1)

A. Maradudin, T. Michel, A. McGurn, E. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Berenger, Jean-Pierre

Jean-Pierre Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994);J.-P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

Cotter, N. P. K.

Elson, J. M.

J. M. Elson, P. Tran, “Coupled mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 54, 1711–1715 (1996).
[CrossRef]

J. M. Elson, P. Tran, “Dispersion and diffraction in photonic media: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
[CrossRef]

J. M. Elson, P. Tran, “Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator,” in Photonic Band Gap Materials, Vol. 315 of NATO Advanced Study Institute Series E: Applied Sciences, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 341–354.

Li, L.

MacKinnon, A.

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2773–2775 (1992).
[CrossRef]

Maradudin, A.

A. Maradudin, T. Michel, A. McGurn, E. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

McGurn, A.

A. Maradudin, T. Michel, A. McGurn, E. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Mendez, E.

A. Maradudin, T. Michel, A. McGurn, E. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Michel, T.

A. Maradudin, T. Michel, A. McGurn, E. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Pendry, J. B.

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2773–2775 (1992).
[CrossRef]

Preist, T. W.

Sambles, J. R.

Tran, P.

J. M. Elson, P. Tran, “Coupled mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 54, 1711–1715 (1996).
[CrossRef]

J. M. Elson, P. Tran, “Dispersion and diffraction in photonic media: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
[CrossRef]

J. M. Elson, P. Tran, “Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator,” in Photonic Band Gap Materials, Vol. 315 of NATO Advanced Study Institute Series E: Applied Sciences, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 341–354.

Ann. Phys. (1)

A. Maradudin, T. Michel, A. McGurn, E. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

J. Comput. Phys. (1)

Jean-Pierre Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994);J.-P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

J. Opt. Soc. Am. A (3)

Phys. Rev. B (1)

J. M. Elson, P. Tran, “Coupled mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 54, 1711–1715 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2773–2775 (1992).
[CrossRef]

Other (1)

J. M. Elson, P. Tran, “Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator,” in Photonic Band Gap Materials, Vol. 315 of NATO Advanced Study Institute Series E: Applied Sciences, C. M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1996), pp. 341–354.

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Figures (8)

Fig. 1
Fig. 1

Schematic of a Gaussian beam incident endwise on a semi-infinite planar waveguide aperture. The Gaussian beam and the waveguide channel are invariant in the y direction. The shaded substrate regions with permittivity ε1 and ε3 border the planar waveguide channel of width w which has permittivity ε2. The calculation region in the x direction is limited to |x|L/2 with the vicinity of the x calculation boundary regions (darker gray) L/2|x|L/2-δ consisting of perfectly matched absorbing layers. At the z=0 plane, the dotted line symbolizes the x discretization points which have spacing Δx. The waveguide channel in the substrate regions 0z>-d1 and z<-d has smooth boundaries. In the region -d1z-d, the waveguide boundaries have noncorrelated random roughness.

Fig. 2
Fig. 2

Schematic illustration of dividing the roughness region into sublayers such that each sublayer can be approximated as independent of z. In the lower figure the dashed line is the actual roughness shape and the horizontal solid lines are the z-invariant approximation.

Fig. 3
Fig. 3

Intensity of the y component of the electric field versus x coordinate for various depths within the waveguide. From left to right, the material dielectric constants are ε1=(2.25, 0), ε2=(2.5, 0), and ε3=(2.25, 0). For clarity, all the z/λ curves, which represent distance from the z=0 interface, have been displaced in multiples of 0.2 relative to zero intensity. The shaded areas are the waveguide channel and the right-hand portion of the two PML regions.

Fig. 4
Fig. 4

Spectral intensity of the y component of the electric field versus wave number for various depths. The spectral intensity is shown where the transmitted intensity gradually transitions to a single-mode guided wave. The vertical line at kx/(ω/c)=0.5 is the transmitted specular component, and all z/λ values are relative to the z=0 plane.

Fig. 5
Fig. 5

Intensity of the x component of the electric field versus x coordinate for various depths within the waveguide. From left to right, the material dielectric constants are ε1=(2.25, 0), ε2=(4, 0), and ε3=(-16, 0). For clarity, all the z/λ curves, which represent distance from the z=0 interface, have been displaced in multiples of 0.2 relative to zero intensity. The shaded areas are the waveguide channel and the left-hand portion of the two PML regions.

Fig. 6
Fig. 6

Intensity of the x component of the electric field versus x coordinate at depth z/λ=-1000. The waveguide supports two guided-wave modes.

Fig. 7
Fig. 7

Spectral intensity of the x component of the electric field versus wave number at the z=0 plane. The two major spectral peaks denote the specular reflected beam and the excitation of surface plasmons at the air–metal interface.

Fig. 8
Fig. 8

Transmission and reflection of incident energy versus length of the roughness region. The transmitted energy is in the form of a guided wave. The solid and open dots at roughness length 100λ are the transmission values with roughness height h=0. These values are the same as those for a roughness length of 0λ.

Equations (53)

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TM:Binc(x, z)TE:Einc(x, z)
=aλπ -π/2π/2dθ exp[(a/λ)2(θ-θinc)2]×exp[i(ω/c)(x sin θ-z cos θ)].
TM:Ex(x, z)z
=iωc 1+4πiσz*(x, z)ωByz(x, z),
Ez(x, z)x=-iωc 1+4πiσx*(x, z)ωByx(x, z),
x [Byx(x, z)+Byz(x, z)]
=-iωc ε(x, z)+4πiσx(x, z)ωEz(x, z),
z [Byx(x, z)+Byz(x, z)]
=iωc ε(x, z)+4πiσz(x, z)ωEx(x, z),
TE:z [Eyx(x, z)+Eyz(x, z)]
=-iωc 1+4πiσz*(x, z)ωBx(x, z),
x [Eyx(x, z)+Eyz(x, z)]
=iωc 1+4πiσx*(x, z)ωBz(x, z),
Bz(x, z)x=iωc ε(x, z)+4πiσx(x, z)ωEyx(x, z),
Bx(x, z)z=-iωc ε(x, z)+4πiσz(x, z)ωEyz(x, z).
TM:Ex(x, z)z
=iωc [1+4πiσz*(x, z)/ω]By(x, z)+icω 1+4πiσz*(x, z)/ω1+4πiσx*(x, z)/ω×x 1ε(x, z)+4πiσx(x, z)/ω By(x, z)x,
By(x, z)z
=iωc [ε(x, z)+4πiσz(x, z)/ω]Ex(x, z),
TE:Ey(x, z)z
=-iωc [1+4πiσz*(x, z)/ω]Bx(x, z),
Bx(x, z)z
=-iωc (ε(x, z)+4πiσz(x, z)/ω)Ey(x, z)-icω ε(x, z)+4πiσz(x, z)/ωε(x, z)+4πiσx(x, z)/ω×x 11+4πiσx*(x, z)/ω Ey(x, z)x.
TM:Ex(x, z)z
=iωc By(x, z)+ic/ω1+4πiσx*(x)/ω×x 1ε(x)+4πiσx(x)/ω By(x, z)x,
By(x, z)z=iωε(x)c Ex(x, z),
TE:Ey(x, z)z
=-iωc Bx(x, z),
Bx(x, z)z=-iωc ε(x)Ey(x, z)-icε(x)/ωε(x)+4πiσx(x)/ω×x11+4πiσx*(x)/ω Ey(x, z)x.
TM:Ex(x, z)z=iωc By(x, z)+icωΔx2β(x)×By(x+Δx, z)-By(x, z)α(x+Δx/2)+By(x-Δx, z)-By(x, z)α(x-Δx/2),
By(x, z)z=iωε(x)c Ex(x, z),
TE:Ey(x, z)z=-iωc Bx(x, z),
Bx(x, z)z=-iωc ε(x)Ey(x, z)-icε(x)/ωΔx2α(x)×Ey(x+Δx, z)-Ey(x, z)β(x+Δx/2)+Ey(x-Δx, z)-Ey(x, z)β(x-Δx/2),
α(x)=ε(x)+4πiσx(x)/ω,
β(x)=1+4πiσx*(x)/ω.
z A(z)=MA(z),
TM:A(z)=Ex(z)By(z)orTE:A(z)=Ey(z)Bx(z).
A(z)=S11S12S21S22eλz00e-λzC+C-.
E(z=0)E(z=-d)=R11R12R21R22B(z=0)B(z=-d).
Einc(0)+Er(0)Et(-d)=R11R12R21R22Binc(0)+Br(0)Bt(-d).
Et(z)Bt(z)=S11eλzC+S21eλzC+.
Bt(z)=S21[S11]-1Et(z),
Et(z)=S11eλ(z+d)[S11]-1Et(-d),z-d.
TM:Er(x, z)Br(x, z)=12π dk[xˆ-zˆ(k/q)]ETMr(k)yˆBTMr(k)×exp[i(kx+qz)],
TE:Er(x, z)Br(x, z)=12π dkyˆETEr(k)[xˆ-zˆ(k/q)]BTEr(k)×exp[i(kx+qz)],
BTMr(k)=(ω/c)εincq(k) ETMr(k),BTEr(k)=-q(k)ω/c ETEr(k).
BTMr(k)=ZTMETMr(k),BTEr(k)=ZTEETEr(k),
Br(k)=ZEr(k),
Er(x, 0)=F(x, k)Er(k, 0),
Br(x, 0)=F(x, k)Br(k, 0),
Br(0)=FZF-1Er(0),
I-R11FZF-1-R12S21S11-1-R21FZF-1I-R22S21S11-1Er(0)Et(-d)
=R11Binc(0)-Einc(0)R21Binc(0),

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