Abstract

A new approach is proposed that makes clear the basic assumptions in the beam propagation method (BPM) and that leads to an improved formulation free of difficulties with power conservation wide-angle propagation. The new approach allows one to understand the limitations of the wide-angle BPM. Theoretically, arbitrary accuracy could be achieved for weakly guiding systems, even for angles θ ~ 40° from the z axis, provided that a small enough sampling step is used, together with a BPM solver of sufficient order in nonparaxiality. On the contrary, inevitable errors occur with strongly guiding systems because the local-mode expansion of the physical field rapidly involves evanescent local modes, both forward and backward propagating, that cannot be handled in the BPM propagation algorithms. In this case the new formulation is more accurate than the classical BPM for moderate angles only.

© 1996 Optical Society of America

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References

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  1. D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quant. Electron. 26, S185–S197 (1994).
    [CrossRef]
  2. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
    [CrossRef] [PubMed]
  3. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17, 1743–1745 (1992).
    [CrossRef] [PubMed]
  4. D. Yevick, B. Hermansson, “Convergence properties of wide-angle techniques,” IEEE Photon. Technol. Lett. 6, 1457–1459 (1994).
    [CrossRef]
  5. C. Vassallo, “Wide-angle BPM and power conservation,” Electron. Lett. 31, 130–131 (1995).
    [CrossRef]
  6. C. Vassallo, “Reformulation for the beam-propagation method,” J. Opt. Soc. Am A 10, 2208–2216 (1993).
    [CrossRef]
  7. C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).
  8. J. R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Accurate description of ultra wide-angle beam propagation in homogeneous media by Lanczos orthogonalization,” Opt. Lett. 19, 1284–1286 (1994).
    [CrossRef] [PubMed]
  9. J. 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]
  10. B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
    [CrossRef]
  11. W. H. Press, S. A. Teukolsky, W. T. Veterling, B. P. Flannery, Numerical Recipes, the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).
  12. Y. Chung, N. Dagli, “A wide angle propagation technique using an explicit finite difference scheme”, IEEE Photon. Technol. Lett. 9, 540–542 (1994).
    [CrossRef]
  13. M. Abramovitz, I. A. Stegun, eds. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C.1970).

1995 (1)

C. Vassallo, “Wide-angle BPM and power conservation,” Electron. Lett. 31, 130–131 (1995).
[CrossRef]

1994 (4)

D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quant. Electron. 26, S185–S197 (1994).
[CrossRef]

J. R. P. Ratowsky, J. A. Fleck, M. D. Feit, “Accurate description of ultra wide-angle beam propagation in homogeneous media by Lanczos orthogonalization,” Opt. Lett. 19, 1284–1286 (1994).
[CrossRef] [PubMed]

D. Yevick, B. Hermansson, “Convergence properties of wide-angle techniques,” IEEE Photon. Technol. Lett. 6, 1457–1459 (1994).
[CrossRef]

Y. Chung, N. Dagli, “A wide angle propagation technique using an explicit finite difference scheme”, IEEE Photon. Technol. Lett. 9, 540–542 (1994).
[CrossRef]

1993 (1)

C. Vassallo, “Reformulation for the beam-propagation method,” J. Opt. Soc. Am A 10, 2208–2216 (1993).
[CrossRef]

1992 (4)

Bardyszewski, W.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

Chung, Y.

Y. Chung, N. Dagli, “A wide angle propagation technique using an explicit finite difference scheme”, IEEE Photon. Technol. Lett. 9, 540–542 (1994).
[CrossRef]

Dagli, N.

Y. Chung, N. Dagli, “A wide angle propagation technique using an explicit finite difference scheme”, IEEE Photon. Technol. Lett. 9, 540–542 (1994).
[CrossRef]

Feit, M. D.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Veterling, B. P. Flannery, Numerical Recipes, the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

Fleck, J. A.

Glasner, M.

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

Hadley, G. R.

Hermansson, B.

D. Yevick, B. Hermansson, “Convergence properties of wide-angle techniques,” IEEE Photon. Technol. Lett. 6, 1457–1459 (1994).
[CrossRef]

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Veterling, B. P. Flannery, Numerical Recipes, the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

Ratowsky, J. R. P.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Veterling, B. P. Flannery, Numerical Recipes, the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

Vassallo, C.

C. Vassallo, “Wide-angle BPM and power conservation,” Electron. Lett. 31, 130–131 (1995).
[CrossRef]

C. Vassallo, “Reformulation for the beam-propagation method,” J. Opt. Soc. Am A 10, 2208–2216 (1993).
[CrossRef]

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

Veterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Veterling, B. P. Flannery, Numerical Recipes, the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

Yevick, D.

D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quant. Electron. 26, S185–S197 (1994).
[CrossRef]

D. Yevick, B. Hermansson, “Convergence properties of wide-angle techniques,” IEEE Photon. Technol. Lett. 6, 1457–1459 (1994).
[CrossRef]

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

Electron. Lett. (1)

C. Vassallo, “Wide-angle BPM and power conservation,” Electron. Lett. 31, 130–131 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

D. Yevick, B. Hermansson, “Convergence properties of wide-angle techniques,” IEEE Photon. Technol. Lett. 6, 1457–1459 (1994).
[CrossRef]

Y. Chung, N. Dagli, “A wide angle propagation technique using an explicit finite difference scheme”, IEEE Photon. Technol. Lett. 9, 540–542 (1994).
[CrossRef]

J. Lightwave Technol. (1)

B. Hermansson, D. Yevick, W. Bardyszewski, M. Glasner, “A comparison of Lanczos electric field propagation methods,” J. Lightwave Technol. 10, 772–776 (1992).
[CrossRef]

J. Opt. Soc. Am A (1)

C. Vassallo, “Reformulation for the beam-propagation method,” J. Opt. Soc. Am A 10, 2208–2216 (1993).
[CrossRef]

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

Opt. Lett. (3)

Opt. Quant. Electron. (1)

D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quant. Electron. 26, S185–S197 (1994).
[CrossRef]

Other (3)

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

W. H. Press, S. A. Teukolsky, W. T. Veterling, B. P. Flannery, Numerical Recipes, the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1992).

M. Abramovitz, I. A. Stegun, eds. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C.1970).

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Figures (13)

Fig. 1
Fig. 1

Schematics of the investigated system, namely, a slab waveguide at angle θ from the z axis, with nclad = 3.17 and ncore = 3.17(1 + Δ). The guidance parameter Δ is either 0.01 or 0.10, with the same normalized frequency V = 1.519. All the lengths are normalized to the half-width, i.e., a = 1. A null field condition is imposed on the lateral sides of the computational domain.

Fig. 2
Fig. 2

Power spectrum of local modes in the weak-guidance case Δ = 0.01. Curves are labeled with the value of the tilting angle (in degrees). Each dot corresponds to a local mode. Curves are alternatively solid and dashed simply for better legibility.

Fig. 3
Fig. 3

Same as Fig. 2 but for the strong-guidance case Δ = 0.10.

Fig. 4
Fig. 4

Classical BPM: variation of the local shape error along the propagation, expressed in terms of the lateral displacement of the waveguide axis, for various sampling steps δx in the weak-guidance case Δ = 0.01. When δx is decreased, the error first decreases (curves with solid symbols), then increases (empty symbols).

Fig. 5
Fig. 5

Output shape error in the weak-guidance case Δ = 0.01 versus the sampling mesh δx in the classical BPM (solid circles) and in the new theory (empty circles). The dashed line is a quadratic fit to the four rightmost points.

Fig. 6
Fig. 6

Same as Fig. 4 but for the strong-guidance case Δ = 0.10 with both the classical BPM (upper curves) and the new theory (lower curves).

Fig. 7
Fig. 7

Variations of the local shape error in a modified BPM based on the propagation of Φ and ∂zΦ in the strong-guidance case Δ = 0.10, without the evanescent local modes (solid symbols) and with a partial account of these modes, as explained in the main text (empty symbols).

Fig. 8
Fig. 8

Variations of the final shape error with θ in the weak-guidance case Δ = 0.01 for the classical BPM (empty symbols) and for the new theory with modal Φ–Ψ conversion (solid symbols). The propagation equation is integrated by means of various Padé approximants, as indicated in the inset. For better legibility the curves for the higher-order Hadley approximants [3, 3] to [5, 5] are plotted as dashed lines without symbols; they always remain close to one another. Computations were made with δx = 0.02 for the classical BPM and with δx = 0.01 for the new theory.

Fig. 9
Fig. 9

Same as Fig. 8 but for the total output error in the new theory—approximately twice the shape error of Fig. 8, indicating the same order for the phase error and the shape error.

Fig. 10
Fig. 10

Same as Fig. 8 (shape error) but for the strong-guidance case Δ = 0.10. These results were obtained with δx = 0.02, but, as shown in Figs. 6 and 7, larger δx lead to the same values.

Fig. 11
Fig. 11

Same as Fig. 8 (Δ = 0.01) but with a Φ–Ψ conversion by means of Padé approximants.

Fig. 12
Fig. 12

Same as Fig. 10 (Δ = 0.10) but with a Φ–Ψ conversion by means of Padé approximants.

Fig. 13
Fig. 13

Analysis of the double dielectric mirror shown in the inset, with nclad = 3.377, ncore = 3.38, core width 4 μm, transverse displacement 15 μm, middle part tilted by 30° (mirrors at 15°), and λ = 1.15 μm; the total device length is 86 μm. The sampling step is taken to be δx = 0.02 μm, so as to produce enough points in the decay of the field in the air. The figure displays the global power transmission for the dominant mode obtained with various integration steps (in micrometers) and with various Padé solvers, as indicated at the top. The various solvers converge poorly and not toward the same value. Taking δx = 0.01 μm does not improve the results.

Tables (4)

Tables Icon

Table 1 Relative Magnitude of Backward-Propagating and Evanescent Components

Tables Icon

Table 2 Coefficients for [ηn, δzn] Propagators

Tables Icon

Table 3 δz Dependence of Integration Error

Tables Icon

Table 4 Comparison Between Modal and Padé Φ–Ψ Conversion

Equations (27)

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( z 2 + H 2 ) · Φ ( x , z ) = 0
z Φ = - i H · Φ
Φ = Φ a + Φ b ,
z Φ = - i H · ( Φ a - Φ b ) .
z Φ a = - i H · Φ a - 1 2 H - 1 · z H · ( Φ a - Φ b ) ,
z Φ b = + i H · Φ b + 1 2 H - 1 · z H · ( Φ a - Φ b ) .
Ψ a = H 1 / 2 · Φ a ,
P = Re ( i Φ * z Φ d x ) = ( Ψ a 2 - Ψ b 2 ) d x .
z Ψ a = - i H · Ψ a + p · Ψ a + q · Ψ b ,
z Ψ b = + i H · Ψ b + p · Ψ b + q · Ψ a ,
p = 1 2 ( z H 1 / 2 · H - 1 / 2 - H - 1 / 2 · z H 1 / 2 ) ,
q = 1 2 H - 1 / 2 · z H · H - 1 / 2 .
H 2 = k 2 n 0 2 ( 1 + η ) ,
p = ( η · z η - z η · η ) / 32 + .
H 2 · ϕ j = β j 2 ϕ j ( x ) ,             j = 1 , 2 , .
H · Φ = j = 1 N β j Φ ϕ j ϕ j ( x ) ,
H 1 / 2 · Φ = j = 1 N β j 1 / 2 Φ ϕ j ϕ j ( x ) .
2 Φ a = j = 1 N ( Φ ϕ j + i β j - 1 z Φ · ϕ j ) ϕ j ( x ) ,
2 Φ b = j = 1 N ( Φ ϕ j - i β j - 1 z Φ · ϕ j ) ϕ j ( x ) .
( 1 + η ) 1 / 4 const . × j = 1 n ( η - c j ) / ( η - d j ) ,
( 1 + η ) 1 / 4 const . × ( 1 + i s ) 1 / 4 j = 1 n ( η - c ˜ j ) / ( η - d ˜ j ) ,
z ψ = - i h · ψ ,
ψ ( z + δ z ) = P ( δ z ) · ψ ( z ) = exp ( - i h δ z ) · ψ ( z ) = j = 1 N exp [ - i ( β j - k n 0 ) δ z ] ψ ϕ j ϕ j ,
exp ( - i h δ z ) = ( 1 + i δ z 2 h ) - 1 · ( 1 - i δ z 2 h ) + O ( δ z 3 ) ,
P = 1 - i u 2 η + ( i u 8 - u 2 8 ) η 2 + ( - i u 16 + u 2 16 + i u 3 48 ) η 3 + ( 5 i u 128 - 5 u 2 128 - i u 3 64 + u 4 384 ) η 4 + ,
z 0 z 1 f ( z ) d z = δ z 2 ( f 0 + f 1 ) - δ z 2 12 ( f 0 - f 1 ) + O ( δ z 5 )
( 1 + i δ z 2 h - δ z 2 12 h 2 ) · ψ 1 = ( 1 - i δ z 2 h - δ z 2 12 h 2 ) · ψ 0 + δ z 3 24 ( h · z h - z h · h - i 2 z 2 h ) × ( ψ 0 + ψ 1 ) + O ( δ z 5 ) .

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