Abstract

Numerical experiments using the paraxial finite-difference beam propagation method have been performed with the following boundary conditions: perfectly matched layer, Higdon absorbing boundary conditions, complementary operators method, and extended complementary operators method. We have shown that Higdon operators must be modified for the paraxial wave equation to take into account the spectrum of incident rays on the boundaries of the computational domain. Reflection coefficients, accuracy, numerical dissipation/gain, memory requirements, and time computation are compared and discussed for these absorbing techniques.

© 2001 Optical Society of America

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References

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  1. J-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  2. R. L. Higdon, “Absorbing boundary condition for difference approximations to the multi-dimensional wave equation,” Math. Comput. 31, 629–651 (1987).
  3. O. M. Ramahi, “Complementary operators: a method to annihilate artificial reflections arising from the truncation of the computational domain in the solution of partial differential equations,” IEEE Trans. Antennas Propag. 43, 697–704 (1995).
    [CrossRef]
  4. O. M. Ramahi, “Complementary boundary operators for wave propagation problems,” J. Comput. Phys. 133, 113–128 (1997).
    [CrossRef]
  5. W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
    [CrossRef]
  6. D. Jiménez, F. Pérez-Murano, C. Ramírez, A. Guzmán, “Implementation of Bérenger layers as boundary conditions for the beam propagation method: applications to integrated waveguides,” Opt. Commun. 159, 43–48 (1999).
    [CrossRef]
  7. Y-P. Chiou, H-C. Chang, “Complementary operators method as the absorbing boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 10, 976–978 (1998).
    [CrossRef]
  8. D. Jiménez, F. Pérez-Murano, “Improved boundary conditions for the beam propagation method,” IEEE Photon. Technol. Lett. 11, 1000–1002 (1999).
    [CrossRef]
  9. J. Fang, “Absorbing boundary conditions applied to model wave propagation in microwave integrated-circuits,” IEEE Trans. Microwave Theory Tech. 42, 1506–1513 (1994).
    [CrossRef]
  10. R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,” J. Opt. Soc. Am. A 9, 265–273 (1992).
    [CrossRef]
  11. C. Vassallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
    [CrossRef]
  12. T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodical structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [CrossRef]

1999 (2)

D. Jiménez, F. Pérez-Murano, C. Ramírez, A. Guzmán, “Implementation of Bérenger layers as boundary conditions for the beam propagation method: applications to integrated waveguides,” Opt. Commun. 159, 43–48 (1999).
[CrossRef]

D. Jiménez, F. Pérez-Murano, “Improved boundary conditions for the beam propagation method,” IEEE Photon. Technol. Lett. 11, 1000–1002 (1999).
[CrossRef]

1998 (1)

Y-P. Chiou, H-C. Chang, “Complementary operators method as the absorbing boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 10, 976–978 (1998).
[CrossRef]

1997 (1)

O. M. Ramahi, “Complementary boundary operators for wave propagation problems,” J. Comput. Phys. 133, 113–128 (1997).
[CrossRef]

1996 (2)

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

C. Vassallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

1995 (1)

O. M. Ramahi, “Complementary operators: a method to annihilate artificial reflections arising from the truncation of the computational domain in the solution of partial differential equations,” IEEE Trans. Antennas Propag. 43, 697–704 (1995).
[CrossRef]

1994 (2)

J-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Fang, “Absorbing boundary conditions applied to model wave propagation in microwave integrated-circuits,” IEEE Trans. Microwave Theory Tech. 42, 1506–1513 (1994).
[CrossRef]

1992 (1)

1987 (1)

R. L. Higdon, “Absorbing boundary condition for difference approximations to the multi-dimensional wave equation,” Math. Comput. 31, 629–651 (1987).

1971 (1)

Bérenger, J-P.

J-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bertoni, H. L.

Chang, H-C.

Y-P. Chiou, H-C. Chang, “Complementary operators method as the absorbing boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 10, 976–978 (1998).
[CrossRef]

Chiou, Y-P.

Y-P. Chiou, H-C. Chang, “Complementary operators method as the absorbing boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 10, 976–978 (1998).
[CrossRef]

Collino, F.

C. Vassallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

Fang, J.

J. Fang, “Absorbing boundary conditions applied to model wave propagation in microwave integrated-circuits,” IEEE Trans. Microwave Theory Tech. 42, 1506–1513 (1994).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Guzmán, A.

D. Jiménez, F. Pérez-Murano, C. Ramírez, A. Guzmán, “Implementation of Bérenger layers as boundary conditions for the beam propagation method: applications to integrated waveguides,” Opt. Commun. 159, 43–48 (1999).
[CrossRef]

Higdon, R. L.

R. L. Higdon, “Absorbing boundary condition for difference approximations to the multi-dimensional wave equation,” Math. Comput. 31, 629–651 (1987).

Huang, W. P.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

Jiménez, D.

D. Jiménez, F. Pérez-Murano, “Improved boundary conditions for the beam propagation method,” IEEE Photon. Technol. Lett. 11, 1000–1002 (1999).
[CrossRef]

D. Jiménez, F. Pérez-Murano, C. Ramírez, A. Guzmán, “Implementation of Bérenger layers as boundary conditions for the beam propagation method: applications to integrated waveguides,” Opt. Commun. 159, 43–48 (1999).
[CrossRef]

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

Pérez-Murano, F.

D. Jiménez, F. Pérez-Murano, C. Ramírez, A. Guzmán, “Implementation of Bérenger layers as boundary conditions for the beam propagation method: applications to integrated waveguides,” Opt. Commun. 159, 43–48 (1999).
[CrossRef]

D. Jiménez, F. Pérez-Murano, “Improved boundary conditions for the beam propagation method,” IEEE Photon. Technol. Lett. 11, 1000–1002 (1999).
[CrossRef]

Ramahi, O. M.

O. M. Ramahi, “Complementary boundary operators for wave propagation problems,” J. Comput. Phys. 133, 113–128 (1997).
[CrossRef]

O. M. Ramahi, “Complementary operators: a method to annihilate artificial reflections arising from the truncation of the computational domain in the solution of partial differential equations,” IEEE Trans. Antennas Propag. 43, 697–704 (1995).
[CrossRef]

Ramírez, C.

D. Jiménez, F. Pérez-Murano, C. Ramírez, A. Guzmán, “Implementation of Bérenger layers as boundary conditions for the beam propagation method: applications to integrated waveguides,” Opt. Commun. 159, 43–48 (1999).
[CrossRef]

Ratowsky, R. P.

Tamir, T.

Vassallo, C.

C. Vassallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

Xu, C. L.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

Yokoyama, K.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

Y-P. Chiou, H-C. Chang, “Complementary operators method as the absorbing boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 10, 976–978 (1998).
[CrossRef]

D. Jiménez, F. Pérez-Murano, “Improved boundary conditions for the beam propagation method,” IEEE Photon. Technol. Lett. 11, 1000–1002 (1999).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

O. M. Ramahi, “Complementary operators: a method to annihilate artificial reflections arising from the truncation of the computational domain in the solution of partial differential equations,” IEEE Trans. Antennas Propag. 43, 697–704 (1995).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. Fang, “Absorbing boundary conditions applied to model wave propagation in microwave integrated-circuits,” IEEE Trans. Microwave Theory Tech. 42, 1506–1513 (1994).
[CrossRef]

J. Comput. Phys. (2)

O. M. Ramahi, “Complementary boundary operators for wave propagation problems,” J. Comput. Phys. 133, 113–128 (1997).
[CrossRef]

J-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Lightwave Technol. (1)

C. Vassallo, F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Math. Comput. (1)

R. L. Higdon, “Absorbing boundary condition for difference approximations to the multi-dimensional wave equation,” Math. Comput. 31, 629–651 (1987).

Opt. Commun. (1)

D. Jiménez, F. Pérez-Murano, C. Ramírez, A. Guzmán, “Implementation of Bérenger layers as boundary conditions for the beam propagation method: applications to integrated waveguides,” Opt. Commun. 159, 43–48 (1999).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Numerical reflection coefficient for several ABCs for the experiment of propagating a Gaussian beam in vacuum: HABC-n (solid curves) and B-n (dashed curves).

Fig. 2
Fig. 2

Simulation error of a Gaussian beam propagating in vacuum with use of the HABC, the COM, the ECOM, and the PML.

Fig. 3
Fig. 3

Field distributions at z=500 μm, showing the steps 1–4 in which the operation of the ECOM is divided.

Fig. 4
Fig. 4

Simulation error of a Gaussian beam propagating in vacuum for the COM-n case with computational domain size W=12 μm (RF = 1) and for the ECOM-n case with W=6 μm (RF= 1/2) and W=4 μm (RF = 1/3): (a) n=3 and (b) n=5.

Fig. 5
Fig. 5

Evolution of the Gaussian beam propagating in vacuum with the use of a ten-point Bérenger layer (B-10). The exact solution (dashed curve and lines) is compared with the BPM simulation at different distances.

Fig. 6
Fig. 6

Fractional power of the Gaussian beam propagating in vacuum with the use of various absorbing techniques: (a) HABC, (b) PML, (c) COM, and (d) ECOM. For comparison, the exact fractional power is shown (dashed curves).

Fig. 7
Fig. 7

BPM simulation of a Gaussian beam propagating in a monomode slab waveguide, where an ECOM-3 has been used as the boundary condition. The Gaussian beam has a divergence angle of 45°, and 1.3 μm has been used as the simulation wavelength. The figure shows the modulus of the field. The curve in the inset shows the fraction of the incident power that remains in the computational window.

Fig. 8
Fig. 8

Error in the shape of the field for the different absorbing techniques (solid curves): (a) HABC, (b) GHABC-4 with several attenuation rates, (c) PML, (d) COM, and (e) ECOM. For (a), (c), (d), and (e), the dashed curves correspond to the error in both the shape and the phase of the field, as defined in Eq. (26).

Fig. 9
Fig. 9

Numerical reflection coefficient for several Bérenger layers and HABCs for the experiment of propagation of a Gaussian beam in a slab waveguide.

Fig. 10
Fig. 10

Field distribution at the steady state for the HABC and PML techniques. Owing to the reflection of the evanescent waves on the boundaries, the field in the cladding and in the substrate is distorted with respect to the exact solution (solid curve, thin trace).

Fig. 11
Fig. 11

Error in both the shape and the phase of the field for the HABC-2. An oscillating behavior is observed with angular frequency equal to the numerical propagation β^.

Tables (5)

Tables Icon

Table 1 Ranges of Propagation Constants and the Associated Ranges of Transverse Propagation Constants in the Substrate, Corresponding to the Various Mode Types of the Helmholtz and Paraxial Equations, for the Slab Waveguide

Tables Icon

Table 2 Reflection Coefficients Corresponding to the Outer Boundaries of the Computational Domain for the ECOM

Tables Icon

Table 3 Angles of Zero Reflection for the HABC, the COM, and the ECOM

Tables Icon

Table 4 Elapsed Time to Simulate the Propagation of a Gaussian Beam in Vacuum at a Distance of 500 μm for Each Absorbing Technique

Tables Icon

Table 5 Numerical Dissipation/Gain (dB/cm)

Equations (43)

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

[x2+z2+k2n2(x)]Ψ=0.
Ψ(x, z)=exp(-jkn0z)ϕ(x, z).
-12kn0z2ϕ+jzϕ=12kn0x2ϕ+kn02n2(x)n02-1ϕ.
jzϕ=12kn0x2ϕ+kn02n2(x)n02-1ϕ.
kx2+kz2=k2n2(x),
kx2+2kn0kz=k2[n2(x)+n02],
ϕ(x,z)=ϕ0(x)exp(-jβz),
ϕ(x,z)=ϕ0(x)exp(-jβz),
ϕ0(x)=ϕ0(x),
β=-kn01±1+2βkn01/2.
i=1n(x+jknrsin θi+αi)ϕ|(x=0,z)=0,
ϕ=exp(-jkxx-jkzz)+Rn(θ)exp(jkxx-jkzz),
Rn(θ)=-i=1n-jkx+jknrsin θi+αijkx+jknrsin θi+αi,
i=1n(x+jkx,i+αi)ϕ|(x=0,z)=0,
kx,i=-kn0+k[n02+(n02+nr2)tan2 θi]1/2tan θi
ifθi=(0, π/2),
kx,i=0, ifθi=0,
kx,i=kn02+nr2,ifθi=π/2.
Iϕi=ϕi,
Sϕi=ϕi+1.
i=1nI-S-1Δx+(jkx,i+αi)I+S-12ϕN=0,
i=1n(I+aiS-1)ϕN=0,
ai=-1+(jkx,i+αi)Δx/21+(jkx,i+αi)Δx/2.
ϕ=exp(-jkxiΔx-jkzmΔz)+Rnd(θ)exp(jkxiΔx-jkzmΔz).
Rnd(θ)=-i=1n1+aiexp(jkxΔx)1+aiexp(-jkxΔx)exp(-2jkxNΔx),
Rn-=Rn-1-jkx+j0+0jkx+j0+0=-Rn-1,
Rn+=Rn-1-jkx+jX+0jkx+jX+0Rn-1.
Rnd(θ)
=(-1)1-exp(jkxΔx)1-exp(-jkxΔx)i=1n-11+aiexp(jkxΔx)1+aiexp(-jkxΔx)×exp(-2jkxNΔx),
an=-1+j0Δx/21+j0Δx/2=-1,
Rnd,C(θ)
=(-1)1+exp(jkxΔx)1+exp(-jkxΔx)i=1n-11+aiexp(jkxΔx)1+aiexp(-jkxΔx)×exp(-2jkxNΔx),
an=-1+jXΔx/21+jXΔx/21.
1±exp(jkxΔx)1±exp(-jkxΔx)=±exp(jkxΔx).
ϕ=14i=14ϕi,
ϕ=12i=12ϕi,
ϕ(x, z)=(w0/Wz)exp(-x2/Wz2),Wz2=w02+2jz/k,
(z)=k|ϕkBPM-ϕkexact|2/k|ϕkexact|2.
PBPM(z)=k|ϕkBPM|2.
Pexact(z)=k|ϕkexact|2.
(z)=k||ϕkBPM|-|ϕksteady||2/k|ϕksteady|2.
β=kn02Neffn02-1.
(z)=1+PBPMPsteady-2Psteadykϕ^0,kϕ0,kcos(β^z),

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