Abstract

The boundaryless beam propagation method uses a mapping function to transform the infinite real space into a finite-size computational domain [Opt. Lett. 21, 4 (1996)]. This leads to a bounded field that avoids the artificial reflections produced by the computational window. However, the method suffers from frequency aliasing problems, limiting the physical region to be sampled. We propose an adaptive boundaryless method that concentrates the higher density of sampling points in the region of interest. The method is implemented in Cartesian and cylindrical coordinate systems. It keeps the same advantages of the original method but increases accuracy and is not affected by frequency aliasing.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Vassallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570–1577 (1996).
    [CrossRef]
  2. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16, 624–626 (1991).
    [CrossRef]
  3. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  4. F. Ladouceur, “Boundaryless beam propagation,” Opt. Lett. 21, 4–5 (1996).
    [CrossRef]
  5. J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, and H. Nakano, “Efficient nonuniform schemes for paraxial and wide-angle finite-difference beam propagation methods,” J. Lightwave Technol. 17, 677–683 (1999).
    [CrossRef]
  6. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Boundaryless finite-difference method for three-dimensional beam propagation,” J. Opt. Soc. Am. A 23, 866–871 (2006).
    [CrossRef]
  7. J. P. Hugonin, P. Lalanne, I. del Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
    [CrossRef]
  8. G. C. des Francs, J. P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long-range plasmonic waveguides,” Opt. Quantum Electron. 42, 557–570 (2011).
    [CrossRef]
  9. C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford, 1998).
  10. A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2005).
  11. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005).
    [CrossRef]
  12. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
    [CrossRef]
  13. I. Ilić, R. Scarmozzino, and R. M. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
    [CrossRef]
  14. A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 18, 944–946 (2006).
    [CrossRef]

2011 (1)

G. C. des Francs, J. P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long-range plasmonic waveguides,” Opt. Quantum Electron. 42, 557–570 (2011).
[CrossRef]

2006 (2)

A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 18, 944–946 (2006).
[CrossRef]

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Boundaryless finite-difference method for three-dimensional beam propagation,” J. Opt. Soc. Am. A 23, 866–871 (2006).
[CrossRef]

2005 (2)

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005).
[CrossRef]

J. P. Hugonin, P. Lalanne, I. del Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

1999 (1)

1996 (3)

F. Ladouceur, “Boundaryless beam propagation,” Opt. Lett. 21, 4–5 (1996).
[CrossRef]

I. Ilić, R. Scarmozzino, and R. M. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

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

1994 (1)

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

1992 (1)

1991 (1)

Agrawal, A.

A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 18, 944–946 (2006).
[CrossRef]

Bandres, M. A.

Berenger, J. P.

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

Collino, F.

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

Ctyroký, J.

G. C. des Francs, J. P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long-range plasmonic waveguides,” Opt. Quantum Electron. 42, 557–570 (2011).
[CrossRef]

del Villar, I.

J. P. Hugonin, P. Lalanne, I. del Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

des Francs, G. C.

G. C. des Francs, J. P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long-range plasmonic waveguides,” Opt. Quantum Electron. 42, 557–570 (2011).
[CrossRef]

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Hadley, G. R.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2005).

Hugonin, J. P.

G. C. des Francs, J. P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long-range plasmonic waveguides,” Opt. Quantum Electron. 42, 557–570 (2011).
[CrossRef]

J. P. Hugonin, P. Lalanne, I. del Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

Ilic, I.

I. Ilić, R. Scarmozzino, and R. M. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

Ladouceur, F.

Lalanne, P.

J. P. Hugonin, P. Lalanne, I. del Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

Matias, I. R.

J. P. Hugonin, P. Lalanne, I. del Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

Matsubara, K.

Nakano, H.

Osgood, R. M.

I. Ilić, R. Scarmozzino, and R. M. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

Pozrikidis, C.

C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford, 1998).

Scarmozzino, R.

I. Ilić, R. Scarmozzino, and R. M. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

Sekiguchi, M.

Sharma, A.

A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 18, 944–946 (2006).
[CrossRef]

Shibayama, J.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2005).

Vassallo, C.

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

Yamauchi, J.

IEEE Photon. Technol. Lett. (1)

A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 18, 944–946 (2006).
[CrossRef]

J. Comput. Phys. (1)

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

J. Lightwave Technol. (3)

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

I. Ilić, R. Scarmozzino, and R. M. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
[CrossRef]

J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, and H. Nakano, “Efficient nonuniform schemes for paraxial and wide-angle finite-difference beam propagation methods,” J. Lightwave Technol. 17, 677–683 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (3)

Opt. Quantum Electron. (2)

J. P. Hugonin, P. Lalanne, I. del Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37, 107–119 (2005).
[CrossRef]

G. C. des Francs, J. P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long-range plasmonic waveguides,” Opt. Quantum Electron. 42, 557–570 (2011).
[CrossRef]

Other (2)

C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford, 1998).

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

(a) Mapping function x=αtanθ. This mapping densely discretizes the center of the physical domain but scarcely discretizes the exterior. (b) Geometric interpretation of the mapping function, equivalent to a stereographic projection.

Fig. 2.
Fig. 2.

(a) Adaptive function x(θ)=x0+αtan(θθ0), densely discretizes the region in the physical domain where the field is concentrated. (b) Geometric interpretation in terms of a stereographic projection.

Fig. 3.
Fig. 3.

(a) Cosine-Gauss field at z=0. (b) Comparison between the analytical solution (solid black curve) and the numerical solution using our adaptive method (gray dots) at z=zR. (c) Propagation of the cosine-Gauss field in the plane (θ,z). (d) ERR function [Eq. (16)] for the proposed adaptive boundary method and the standard static method.

Fig. 4.
Fig. 4.

(a) Long propagation of the cosine-Gauss beam using the standard static method. The dotted lines show the fictitious boundary produced by frequency aliasing due to the scarcely sampled region. (b) Long propagation using our adaptive method, showing better performance than the standard method.

Equations (26)

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

x=αtanθ,
x(θ)=x0+αtan(θθ0),
2U(x,z)x2+2ikU(x,z)z=0,
U=αsec(|θ|θ0)ψ(x,z),
1α2cos(|θ|θ0)42ψθ2+1α2cos(|θ|θ0)4ψ+2ikψz=0.
θppΔθ,p=0,1,,N1,
ψp(n)ψ(θp,nΔz),n=0,1,2,,
apψp1(n+1)+(bp+i4kΔz)ψp(n+1)+apψp+1(n+1)=apψp1(n)(bpi4kΔz)ψp(n)apψp+1(n)
ap=cos(|θp|θ0)4α2Δθ2,
bp=(Δθ22)ap,
Ψ(n+1)=ML1MRΨ(n),
ML=C+4ikΔzIN,MR=C+4ikΔzIN,
C=[b0a0000a1b1a10000aN3bN3aN300aN2bN2aN2000aN1bN1],Ψ(n)=[ψ0(n)ψ1(n)ψN3(n)ψN2(n)ψN1(n)].
U(x,0)=exp(x2/w02)cos(ktx),
ψp(0)=1αexp(xp2/w02)cos(ktxp)cos(θp),
ERR=1|Uexact*Ucomputeddx|2|Uinput|2dx|Uexact|2dx,
1rr[rψr]+2ikψzl2r2ψ=0,
U(r,ϕ,z)=ψ(r,z)exp(ilϕ).
A(θ)2ψθ2+B(θ)ψθ+C(θ)ψ+2ikψz=0
A(θ)=1α2cos(θθ0)4,
B(θ)=2α2cos(θθ0)3sin(θθ0)+cos(θθ0)2αr(θ),
C(θ)=l2r(θ)2.
apψp1(n+1)+[bp+4ikΔz]ψp(n+1)+cpψp+1(n+1)=apψp1(n)[bp4ikΔz]ψp(n)cpψp+1(n),
ap=A(θp)Δθ2B(θp)2Δθ,
bp=2A(θp)Δθ2+C(θp),
cp=A(θp)Δθ2+B(θp)2Δθ.

Metrics