Abstract

Splitting scalar fields into a sum of counterpropagating waves leads to formulation of the Helmholtz equation as two coupled parabolic partial differential equations. This rigorous formalism naturally leads to the beam-propagation method (BPM) equation when the backward-propagating field is neglected, but it also permits discussion of various corrections. Two problems specifically are analyzed: wide-angle propagation in near-uniform systems and propagation through lenslike systems, with a strongly discontinuous refractive index. Finally, a BPM modeling is proposed for fiber microlens operation.

© 1993 Optical Society of America

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  1. 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]
  2. 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]
  3. D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semi-conductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
    [CrossRef]
  4. H. H. Lin, A. Korpel, “Heuristic scalar paraxial beam-propagation method taking into account continuous reflections,” J. Opt. Soc. Am. B 8, 849–857 (1991).
    [CrossRef]
  5. P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
    [CrossRef]
  6. D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,”IEEE Photon. Lett. 3, 527–529 (1991).
    [CrossRef]
  7. C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).
  8. C. Vassallo, “Pseudo-modes et guides optiques,” Ann. Télécom. 43, 48–65 (1988).
  9. D. Yevick, C. Rolland, B. Hermansson, “Fresnel equation studies of longitudinally varying semiconductor waveguides: reference wavevector dependence,” Electron. Lett. 25, 1254–1256 (1989).
    [CrossRef]
  10. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).
  11. H. M. Presby, “Near 100% efficient fiber microlenses,” in Optical Fiber Communication, Vol. 5 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 408–411.
  12. D. Yevick, B. Hermansson, M. Glasner, “Fresnel and wide-angle equation analyses of microlenses,”IEEE Photon. Technol. Lett. 2, 412–414 (1990).
    [CrossRef]
  13. C. Vassallo, M. J. van der Keur, “Ultimate coupling performances for microlensed fibres,” Electron. Lett. 28, 1913–1915 (1992).
    [CrossRef]
  14. P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

1992 (1)

C. Vassallo, M. J. van der Keur, “Ultimate coupling performances for microlensed fibres,” Electron. Lett. 28, 1913–1915 (1992).
[CrossRef]

1991 (2)

H. H. Lin, A. Korpel, “Heuristic scalar paraxial beam-propagation method taking into account continuous reflections,” J. Opt. Soc. Am. B 8, 849–857 (1991).
[CrossRef]

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,”IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

1990 (2)

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]

D. Yevick, B. Hermansson, M. Glasner, “Fresnel and wide-angle equation analyses of microlenses,”IEEE Photon. Technol. Lett. 2, 412–414 (1990).
[CrossRef]

1989 (2)

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semi-conductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

D. Yevick, C. Rolland, B. Hermansson, “Fresnel equation studies of longitudinally varying semiconductor waveguides: reference wavevector dependence,” Electron. Lett. 25, 1254–1256 (1989).
[CrossRef]

1988 (2)

C. Vassallo, “Pseudo-modes et guides optiques,” Ann. Télécom. 43, 48–65 (1988).

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
[CrossRef]

1981 (1)

Bardyszewski, W.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,”IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

Colson, V.

P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

Doussière, P.

P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

Feit, M. D.

Fernier, B.

P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

Fleck, J. A.

Garabedian, P.

P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

Glasner, M.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,”IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

D. Yevick, B. Hermansson, M. Glasner, “Fresnel and wide-angle equation analyses of microlenses,”IEEE Photon. Technol. Lett. 2, 412–414 (1990).
[CrossRef]

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semi-conductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

Hermansson, B.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,”IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

D. Yevick, B. Hermansson, M. Glasner, “Fresnel and wide-angle equation analyses of microlenses,”IEEE Photon. Technol. Lett. 2, 412–414 (1990).
[CrossRef]

D. Yevick, C. Rolland, B. Hermansson, “Fresnel equation studies of longitudinally varying semiconductor waveguides: reference wavevector dependence,” Electron. Lett. 25, 1254–1256 (1989).
[CrossRef]

Kaczmarski, P.

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
[CrossRef]

Korpel, A.

Lafragette, J. L.

P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

Lagasse, P. E.

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
[CrossRef]

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]

Leblond, F.

P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

Legouezigou, O.

P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

Lin, H. H.

Monnot, M.

P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

Presby, H. M.

H. M. Presby, “Near 100% efficient fiber microlenses,” in Optical Fiber Communication, Vol. 5 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 408–411.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

Rolland, C.

D. Yevick, C. Rolland, B. Hermansson, “Fresnel equation studies of longitudinally varying semiconductor waveguides: reference wavevector dependence,” Electron. Lett. 25, 1254–1256 (1989).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

van der Donk, J.

van der Keur, M. J.

C. Vassallo, M. J. van der Keur, “Ultimate coupling performances for microlensed fibres,” Electron. Lett. 28, 1913–1915 (1992).
[CrossRef]

Van Roey, J.

Vassallo, C.

C. Vassallo, M. J. van der Keur, “Ultimate coupling performances for microlensed fibres,” Electron. Lett. 28, 1913–1915 (1992).
[CrossRef]

C. Vassallo, “Pseudo-modes et guides optiques,” Ann. Télécom. 43, 48–65 (1988).

C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

Yevick, D.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,”IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

D. Yevick, B. Hermansson, M. Glasner, “Fresnel and wide-angle equation analyses of microlenses,”IEEE Photon. Technol. Lett. 2, 412–414 (1990).
[CrossRef]

D. Yevick, C. Rolland, B. Hermansson, “Fresnel equation studies of longitudinally varying semiconductor waveguides: reference wavevector dependence,” Electron. Lett. 25, 1254–1256 (1989).
[CrossRef]

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semi-conductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

Ann. Télécom. (1)

C. Vassallo, “Pseudo-modes et guides optiques,” Ann. Télécom. 43, 48–65 (1988).

Electron. Lett. (4)

D. Yevick, C. Rolland, B. Hermansson, “Fresnel equation studies of longitudinally varying semiconductor waveguides: reference wavevector dependence,” Electron. Lett. 25, 1254–1256 (1989).
[CrossRef]

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semi-conductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
[CrossRef]

C. Vassallo, M. J. van der Keur, “Ultimate coupling performances for microlensed fibres,” Electron. Lett. 28, 1913–1915 (1992).
[CrossRef]

IEEE Photon. Lett. (1)

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,”IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

D. Yevick, B. Hermansson, M. Glasner, “Fresnel and wide-angle equation analyses of microlenses,”IEEE Photon. Technol. Lett. 2, 412–414 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

Other (4)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes—the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987).

H. M. Presby, “Near 100% efficient fiber microlenses,” in Optical Fiber Communication, Vol. 5 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 408–411.

C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).

P. Doussière, P. Garabedian, V. Colson, O. Legouezigou, F. Leblond, J. L. Lafragette, M. Monnot, B. Fernier, “Polarization-insensitive semiconductor optical amplifier with buried laterally tapered active waveguide,” in Optical Amplifiers and Their Applications, Vol. 17 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 140–143.

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Figures (8)

Fig. 1
Fig. 1

Schematics of the slab waveguides considered in Section 4, with nclad = n and ncore = n(1 + Δ). We considered a single normalized frequency V = 1.519 at wavelength λ = 1.55 μm, nclad = 3.17, and ncore = 3.18, 3.23, and 3.52 (Δ = 0.003%, 1.9%, and 11%, respectively).

Fig. 2
Fig. 2

Plots of quadratic error on the field magnitude versus the propagation distance, up to 35 μm, for the three slab waveguides detailed in Fig. 1, for four tilting angles and for the four calculation techniques. Open squares correspond to the classical BPM (na = n), filled squares to BPM + neff, circles to the Yevick BPM (na = n), and triangles to the Yevick BPM + neff. Missing symbols correspond to errors beyond the scale of the figure, except for Δ = 11%, θ = 20°, where filled and open squares are superimposed. All the results were obtained with the finite-difference techniques described in the text, with δx = 0.01 μm and δz = 0.05 μm.

Fig. 3
Fig. 3

Plots of quadratic error on the field magnitude versus the propagation distance, up to 20 μm, for a Gaussian beam at λ = 1.55 μm, with a half-width w = 2 μm (at the left) and w = 1 μm (at the right). The indicated θ values are the angular half-widths, where the field falls at 1/e of its value on the axis. The four kinds of symbol have the same meanings as in Fig. 2.

Fig. 4
Fig. 4

Refraction of a plane wave propagating along the z axis, across a plane interface tilted at angle θ.

Fig. 5
Fig. 5

Propagation of a low-divergence Gaussian beam across a plane interface, from a glass of refractive index n = 1.45 into the air, under 40° incidence; λ = 1 μm and w = 5 μm (half-width). The view at the left shows the contour map for the field intensity superimposed upon the physical system; one can see the slight change in the propagation direction when the beam is passing from the transition domain [na = (n + 1)/2] to the final free-propagation domain (na = neff). This change is enhanced in the orthoscopic view at the right, where the intensity versus x is plotted for z values spaced 1 μm apart.

Fig. 6
Fig. 6

Perfect focusing in the air of a parallel beam propagating along the z axis in a homogeneous medium of refractive index n, across a hyperbolic interface of curvature radius ρ on the axis.

Fig. 7
Fig. 7

Comparison of a BPM analysis of the system depicted in Fig. 6 with geometric theories. The input field is a Gaussian beam of half-width w = 10λ, with λ = 1.55 μm and n = 1.466. The minimum half-width of the refracted field (top panel) and the position zF of the focus (bottom panel) are plotted versus the top curvature radius of the interface. All lengths are in micrometers. Squares are BPM results. Solid curves are geometric estimations, as explained in the text and in Appendix B, and dashed curves are paraxial estimations.

Fig. 8
Fig. 8

Coupling loss between a Gaussian beam of 1-μm half-width at the waist and a typical single-mode fiber at λ = 1.55 μm (nclad = 1.46, ncore = 1.466, core radius = 4 μm), versus the distance between the fiber and the beam waist. The left-hand panel corresponds to hyperbolic microlenses, with ρ = 3.5 μm and half-angles between asymptotes of 47° (solid curve) and 41° (dashed curve). The right-hand panel corresponds to spherical lenses of radius 5.9 μm on the top of cones of half-angles 48° and 51°.

Equations (35)

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Φ ( x , y , z ) = ϕ ( x , y , z ) exp ( - i k n a z ) ,
ϕ z = - i Δ T + k 2 ( n 2 - n a 2 ) 2 k n a ϕ ( x , y , z ) ,
Φ ( x , y , z ) = ϕ f ( x , y , z ) exp [ - i k 0 z n a ( s ) d s ] + ϕ b ( x , y , z ) × exp [ i k 0 z n a ( s ) d s ] .
ϕ f = 1 2 ( 1 + β k n a ) ϕ ( x , y ) exp [ i ( k n a - β ) z ] ,
ϕ b = 1 2 ( 1 - β k n a ) ϕ ( x , y ) exp [ - i ( k n a + β ) z ] .
Φ z = - i k n a ϕ f exp [ - i k 0 z n a ( s ) d s ] + i k n a ϕ b × exp [ i k 0 z n a ( s ) d s ] ,
ϕ f z exp [ - i k 0 z n a ( s ) d s ] + ϕ b z exp [ i k 0 z n a ( s ) d s ] = 0.
( n a ϕ f ) z = - i Δ T + k 2 ( n 2 - n a 2 ) 2 k n a n a ϕ f + exp ( 2 i k n a d z ) [ - i Δ T + k 2 ( n 2 - n a 2 ) 2 k n a × n a ϕ b + d ( log n a ) 2 d z n a ϕ b ] ,
( n a ϕ b ) z = + i Δ T + k 2 ( n 2 - n a 2 ) 2 k n a n a ϕ b + exp ( - 2 i k n a d z ) [ + i Δ T + k 2 ( n 2 - n a 2 ) 2 k n a × n a ϕ f + d ( log n a ) 2 d z n a ϕ b ] .
( n a ϕ f ) z = - i Δ T + k 2 ( n 2 - n a 2 ) 2 k n a ( n a ϕ f ) .
ϕ b 1 2 i k n a ϕ f z exp ( - 2 i k n a d z ) ,
Φ ( x , y , z ) exp ( - i k n a d z ) ( ϕ f + 1 2 i k n a ϕ f z ) .
k 2 n a 2 = ϕ f * ( Δ T + k 2 n 2 ) ϕ f d x d y ϕ f 2 d x d y .
[ α , k n a + k 2 ( n 2 - n a 2 ) - α 2 2 k n a ] ,
( 1 + M 4 k 2 n a 2 ) ( n a ϕ f ) z = - i M 2 k n a ( n a ϕ f ) ,
Ψ p + 1 - Ψ p δ z = - i ( M 2 k n a Ψ ) p + 1 / 2 = - i M p + 1 / 2 4 k n a ( Ψ p + Ψ p + 1 ) + O ( δ z 2 ) .
( M - 4 i k n a δ z ) Ψ p + 1 = - ( M + 4 i k n a δ z ) Ψ p ,
( 1 + M p + 1 / 2 4 k 2 n a 2 ) Ψ p + 1 - Ψ p δ z = - i M p + 1 / 2 4 k n a ( Ψ p + Ψ p + 1 ) ,
( M + 4 k 2 n a 2 1 + i k n a δ z ) Ψ p + 1 = 1 - i k n a δ z 1 + i k n a δ z × ( M + 4 k 2 n a 2 1 - i k n a δ z ) Ψ p .
E = [ ( ψ computed - ψ exact ) 2 d x ψ exact 2 d x ] 1 / 2 .
ψ exact ( r , z ) = 1 2 w 0 2 0 + exp ( - ν 2 w 0 2 / 4 ) J 0 ( ν r ) exp ( - i γ z ) ν d ν .
sin ( θ + β ) = n sin θ ,
2 k n a α cos θ = [ - k 2 ( n 2 - 1 ) + α 2 ] sin θ .
n a = tan θ sin 2 β + n 2 - 1 2 sin β ,
n a = ( n + 1 ) / 2 + O ( θ 2 ) .
n a ϕ f / z = ( d log n a / 2 d z ) n a ϕ b ,
n a ϕ b / z = ( d log n a / 2 d z ) n a ϕ f .
ϕ f ( x , y , z ) = A ( x , y ) + B ( x , y ) / n a ( z ) ,
ϕ b ( x , y , z ) = A ( x , y ) - B ( x , y ) / n a ( z ) ,
n M 0 M + M F C ,
- n M 0 M + M F = 0.
z 2 tan 2 δ - 2 ρ z - r 2 = 0.
tan 2 δ = n 2 - 1
ϕ ( θ ) = ( r d r sin θ d θ ) 1 / 2 exp ( - r 2 w 2 ) .
ϕ G ( θ ) exp [ - ( k 2 w 2 / 4 ) sin 2 θ ] cos θ ,

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