Abstract

We present a theoretical model of light-beam propagation in anisotropic and inhomogeneous dielectric structures obtained as a direct extension of the scalar fast-Fourier-transform beam-propagation method. We solve Maxwell’s equations in a generalized geometrical optics approximation, in which the reflected fields are neglected. This is a full-vectorial model because it accounts for the polarization effects that are due to both the anisotropy and the inhomogeneity of the medium.

© 1999 Optical Society of America

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  1. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  2. J. Van Roey, J. Van der Donk, P. E. Lagasse, “Beam propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  3. M. D. Feit, J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1158 (1980).
    [CrossRef] [PubMed]
  4. M. D. Feit, J. A. Fleck, “Calculation of dispersion in graded-index multimode fibers by a propagating beam-method,” Appl. Opt. 8, 2843–2851 (1979).
    [CrossRef]
  5. M. D. Feit, J. A. Fleck, “Analysis of rib waveguides and couplers by the propagating beam method,” J. Opt. Soc. Am. A 7, 73–79 (1990).
    [CrossRef]
  6. M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. 132, 336–342 (1985).
  7. D. Yevick, L. Thilén, “Analysis of gratings by the beam-propagation method,” J. Opt. Soc. Am. 72, 1084–1089 (1982).
    [CrossRef]
  8. R. Baets, P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junctions,” Appl. Opt. 21, 1972–1979 (1982).
    [CrossRef] [PubMed]
  9. R. Baets, P. E. Lagasse, “Loss calculation and design of arbitrarily curved integrated-optic waveguides,” J. Opt. Soc. Am. 73, 177–182 (1983).
    [CrossRef]
  10. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  11. D. Yevick, B. Hermansson, “New formulation of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
    [CrossRef]
  12. J. Van Roey, P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin film gratings,” Appl. Opt. 20, 423–429 (1981).
    [CrossRef] [PubMed]
  13. J. A. Hermann, “Beam propagation and optical power limiting with nonlinear media,” J. Opt. Soc. Am. B 1, 729–736 (1984).
    [CrossRef]
  14. L. Thylen, D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21, 2751–2754 (1982).
    [CrossRef] [PubMed]
  15. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920–926 (1983).
    [CrossRef]
  16. J. M. Liu, L. Gomelsky, “Vectorial beam propagation method,” J. Opt. Soc. Am. A 9, 1574–1585 (1992).
    [CrossRef]
  17. W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photon. Technol. Lett. 3, 910–913 (1991).
    [CrossRef]
  18. K. Hayata, M. Koshiba, “Full-vectorial analysis of nonlinear-optical waveguides,” J. Opt. Soc. Am. B 5, 2494–2501 (1988).
    [CrossRef]
  19. S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
    [CrossRef]
  20. C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
    [CrossRef] [PubMed]
  21. E. Santamato, G. Abbate, P. Maddalena, L. Marrucci, “Lagrangian approach to light propagation in liquid crystals,” Phys. Rev. E 52, 5053–5059 (1995).
    [CrossRef]
  22. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4 matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]

1995 (2)

S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
[CrossRef]

E. Santamato, G. Abbate, P. Maddalena, L. Marrucci, “Lagrangian approach to light propagation in liquid crystals,” Phys. Rev. E 52, 5053–5059 (1995).
[CrossRef]

1992 (1)

1991 (1)

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

1990 (1)

1989 (2)

D. Yevick, B. Hermansson, “New formulation of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[CrossRef] [PubMed]

1988 (1)

1985 (1)

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. 132, 336–342 (1985).

1984 (1)

1983 (2)

1982 (3)

1981 (2)

1980 (1)

1979 (1)

M. D. Feit, J. A. Fleck, “Calculation of dispersion in graded-index multimode fibers by a propagating beam-method,” Appl. Opt. 8, 2843–2851 (1979).
[CrossRef]

1978 (1)

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1972 (1)

Abbate, G.

E. Santamato, G. Abbate, P. Maddalena, L. Marrucci, “Lagrangian approach to light propagation in liquid crystals,” Phys. Rev. E 52, 5053–5059 (1995).
[CrossRef]

Baets, R.

Berreman, D. W.

Chaudhuri, S. K.

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

Chu, S. T.

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

Dayan, P.

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. 132, 336–342 (1985).

Feit, M. D.

Fleck, J. A.

Gomelsky, L.

Hayata, K.

Hermann, J. A.

Hermansson, B.

D. Yevick, B. Hermansson, “New formulation of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

Huang, W. P.

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

Koshiba, M.

Lagasse, P. E.

Liu, J. M.

Maddalena, P.

E. Santamato, G. Abbate, P. Maddalena, L. Marrucci, “Lagrangian approach to light propagation in liquid crystals,” Phys. Rev. E 52, 5053–5059 (1995).
[CrossRef]

Marrucci, L.

E. Santamato, G. Abbate, P. Maddalena, L. Marrucci, “Lagrangian approach to light propagation in liquid crystals,” Phys. Rev. E 52, 5053–5059 (1995).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Oldano, C.

C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[CrossRef] [PubMed]

Ritchie, S.

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. 132, 336–342 (1985).

Robertson, M. J.

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. 132, 336–342 (1985).

Santamato, E.

E. Santamato, G. Abbate, P. Maddalena, L. Marrucci, “Lagrangian approach to light propagation in liquid crystals,” Phys. Rev. E 52, 5053–5059 (1995).
[CrossRef]

Selleri, S.

S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
[CrossRef]

Thilén, L.

Thylen, L.

Van der Donk, J.

Van Roey, J.

Xu, C. L.

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

Yevick, D.

Zoboli, M.

S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Yevick, B. Hermansson, “New formulation of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. Selleri, M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic waveguides,” IEEE Trans. Microwave Theory Tech. 43, 887–892 (1995).
[CrossRef]

J. Opt. Soc. Am. (5)

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

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

Phys. Rev. A (1)

C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (1)

E. Santamato, G. Abbate, P. Maddalena, L. Marrucci, “Lagrangian approach to light propagation in liquid crystals,” Phys. Rev. E 52, 5053–5059 (1995).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. 132, 336–342 (1985).

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Figures (1)

Fig. 1
Fig. 1

Schematic picture of the BPM procedure for propagating an electromagnetic beam from the z = 0 plane to the z = L plane. Every BPM step consists of one propagation half-step in a transversally homogeneous medium, one correction step represented as a thin lens in the figure, and a second propagation half-step. The two propagation half-steps between two successive lenses can be joined together.

Equations (40)

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

¯¯x, y, z=xxxyxzxyyyyzxzyzzz,
x, y, z, t=Ex, y, zexp-iωt,x, y, z, t=Hx, y, zexp-iωt,Dx, y, z, t=Dx, y, zexp-iωt,x, y, z, t=Bx, y, zexp-iωt.
ψz=ik0Aˆx, y, z; x, yψ,
ψ=ExEyHy-Hx=ψ1ψ2,
Hz=1μ0iωExy-Eyx, Ez=1zziωHyx-Hxy-zxEx-zyEy,
Aˆx, y, z; x, yAˆ0z; x, y+ΔAˆx, y, z,
Aˆ0z; x, y=Aˆ0, 0, z; x, y,ΔAˆx, y, z=Aˆx, y, z; 0, 0-Aˆ00, 0, z; 0, 0.
ψz=ik0Aˆ0z; x, y+ΔAˆx, y, zψ.
ψz=ik0Aˆ0z; x, yψ,
ψz=ik0ΔAˆx, y, zψ.
ψx, y, z+Δz=expik0zz+Δz/2dzAˆ0×expik0zz+ΔzdzΔAˆ×expik0z+Δz/2z+ΔzdzAˆ0ψx, y, z.
ψ˜βx, βy, zz=ik0A0z; βx, βyψ˜βx, βy, z,
ψ˜βx, βy, z=12π-dxdyψx, y, z×exp-ik0βxx+βyy,
A0z; βx, βyMz; βx, βy=Mz; βx, βyΛz; βx, βy.
MTGMii=1i=1, 2-1i=3, 4,
G=0010000110000100.
A0TG=GA0
Λ=diagλe+, λo+, λe-, λo-.
ψ¯=M-1ψ˜
ψ¯=ψ¯e+ψ¯o+ψ¯e-ψ¯o-=ψ¯+ψ¯-.
GGOA  ψ¯-=0  ψ˜1=m11ψ¯+ψ˜2=m21ψ¯+  ψ˜2=m21m11-1ψ˜1,
ψ¯+z=ik0Λ11ψ¯+-V11ψ¯+,
ψ˜1z=ik0m11Λ11m11-1ψ˜1+m12V21m11-1ψ˜1=Pψ˜1,
ψ˜1Pβx, βy, z+Δz2=expik0zz+Δz/2dzPβx, βy, z×ψ˜1βx, βy, z;
ψ˜2Pβx, βy, z+Δz2=m21m11-1ψ˜1Pβx, βy, z+Δz2;
ψPx, y, z+Δz2=12π-dβxdβyψ˜βx, βy, z+Δz2×expik0βxx+βyy.
ψz=ik0ΔAˆx, y, zψ
ψx, y, z+Δz/2=expik0zz+Δz ΔAˆx, y, zdz×ψPx, y, z+Δz/2.
ΔAˆx, y, z=Aˆx, y, z; x=0, y=0-Aˆ0z; x=0, y=0,
Aˆ=Aˆ11Aˆ12Aˆ21Aˆ22,
Aˆ11=ik0zxzzx+ik0xzxzzik0zyzzx+ik0xzyzzik0zxzzy+ik0yzxzzik0zyzzy+ik0yzyzz, Aˆ12=μ0c+1ck021zz2x2+1ck02x1zzx1ck021zz2xy+1ck02x1zzy1ck021zz2yx+1ck02y1zzxμ0c+1ck021zz2y2+1ck02y1zzy, Aˆ21=cxx-czxxzzz+1ck022y2cxy-czyxzzz-1ck022yxcyx-czxyzzz-1ck022xycyy-czyyzzz+1ck022x2, Aˆ22=ik0xzzzxik0xzzzyik0yzzzxik0yzzzy ,
Aˆ0=Aˆ011Aˆ012Aˆ021Aˆ022 ,
A^011=ik0zx0zz0x+ik0xzxzz0ik0zy0zz0x+ik0xzyzz0ik0zx0zz0y+ik0yzxzz0ik0zy0zz0y+ik0yzyzz0,A^012=μ0c+1ck021zz02x2+1ck02x1zz0x1ck021zz02xy+1ck02x1zz0y1ck021zz02yx+1ck02y1zz0xμ0c+1ck021zz02y2+1ck02y1zz0y , Aˆ021=cxx0-czx0xz0zz0+1ck022y2cxy0-czy0xz0zz0-1ck022yxcyx0-czx0yz0zz0-1ck022xycyy0-czy0yz0zz0+1ck022x2 , Aˆ022=ik0xz0zz0xik0xz0zz0yik0yz0zz0xik0yz0zz0y ,
ΔAˆ=ik0 Δxzxzzik0 Δxzyzz00ik0 Δyzxzzik0 Δyzyzz00cΔ˜xxcΔ˜xy00cΔ˜yxcΔ˜yy00,
˜xx=xx-zxxzzz, ˜xy=xy-zyxzzz, ˜yx=yx-yzzxzz, ˜yy=yy-zyyzzz, Δ˜ij=˜ijx, y, z-˜ij0, 0, z, i, j  x, y.
ψz=exp0z aˆζdζψ0=1+0z aˆzdz+0z aˆzdz 0z aˆzdz+Oz3ψ=1+0z aˆ1zdz+0z aˆ2zdz+0z aˆ1zdz 0z aˆ1zdz+0z aˆ2zdz 0z aˆ2zdz+0z aˆ1zdz 0z aˆ2zdz+0z aˆ2zdz 0z aˆ1zdz+Oz3+Oz3ψ,
aˆ1=aˆ1z; x, y=ik0Aˆ0z; x, y, aˆ2=aˆ2x, y, z=ik0ΔAˆx, y, z.
ψz=exp0z/2 aˆ1zdzexp0z aˆ2zdz×expz/2z aˆ1zdzψ0=1+0z/2 aˆ1zdz+z/2z aˆ1zdz+0z aˆ2zdz+0z/2 aˆ1zdz 0z/2 aˆ1zdz+0z aˆ2zdz 0z aˆ2zdz+z/2z aˆ1zdz z/2z aˆ1zdz+0z/2 aˆ1zdz 0z aˆ2zdz+0z/2 aˆ1zdz z/2z aˆ1zdz+0z aˆ2zdz z/2z aˆ1zdz+Oz3ψ0.
aˆ1z=n aˆ1nzn,  aˆ2z=n aˆ2nzn,
Δψ=nm aˆ1naˆ1mz2+n+m12n+1n+1m+n+2-122n+m+3n+1m+n+2+12m+n+2m+1m+n+2-122m+n+3m+1m+n+2-12n+1n+1m+1+12n+m+2n+1m+1+n,maˆ1naˆ2m-aˆ2maˆ1nzn+m+2×1m+1n+m+2-12n+1n+1m+1ψ.

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