Abstract

Properties of the wide-angle equation (WAEQ), a nonparaxial scalar wave equation used to propagate light through media characterized by inhomogeneous refractive-index profiles, are studied. In particular, it is shown that the WAEQ is not equivalent to the more complicated but more fundamental Helmholtz equation (HEQ) when the index of refraction profile depends on the position along the propagation axis. This includes all nonstraight waveguides. To study the quality of the WAEQ approximation, we present a novel method for computing solutions to the WAEQ. This method, based on a short-time iterative Lanczos (SIL) algorithm, can be applied directly to the full three-dimensional case, i.e., systems consisting of the propagation axis coordinate and two transverse coordinates. Furthermore, the SIL method avoids series-expansion procedures (e.g., Padé approximants) and thus convergence problems associated with such procedures. Detailed comparisons of solutions to the HEQ, WAEQ, and the paraxial equation (PEQ) are presented for two cases in which numerically exact solutions to the HEQ can be obtained by independent analysis, namely, (i) propagation in a uniform dielectric medium and (ii) propagation along a straight waveguide that has been tilted at an angle to the propagation axis. The quality of WAEQ and PEQ, compared with exact HEQ results, is investigated. Cases are found for which the WAEQ actually performs worse than the PEQ.

© 2003 Optical Society of America

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References

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  1. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  2. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).
  3. N. Peyghambarian, S. W. Koch, A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  4. M. D. Feit, J. A. Fleck, A. Steiger, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).
    [CrossRef] [PubMed]
  5. C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
    [CrossRef]
  6. D. K. Pant, R. D. Coalson, M. I. Hernández, J. Campos-Martínez, “Optimal control theory for the design of optical waveguides,” J. Lightwave Technol. 16, 292–300 (1998).
    [CrossRef]
  7. D. K. Pant, R. D. Coalson, M. I. Hernández, J. Campos-Martínez, “Optimal control theory for optical waveguide design: application to Y-branch structures,” Appl. Opt. 38, 3917–3923 (1999).
    [CrossRef]
  8. R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000), and references therein.
    [CrossRef]
  9. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
    [CrossRef]
  10. I. Ilić, R. Scarmozzino, R. M. Osgood, “Investigation of the Padé approximants-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813–2822 (1996).
    [CrossRef]
  11. See, for example, D. S. Saxon, Elementary Quantum Mechanics (Holden-Day, San Francisco, Calif., 1968).
  12. See, for example, S. E. Koonin, D. C. Meredith, Computational Physics: Fortran Version (Addison-Wesley, Reading, Mass., 1990).
  13. J. W. Cooley, J. W. Tukey, “An algorithm for machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  14. T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,” J. Chem. Phys. 85, 5870–5876 (1986).
    [CrossRef]
  15. C. Iung, C. Leforestier, “Exact time evolution methods for large bound systems,” Comput. Phys. Commun. 63, 135–153 (1991).
    [CrossRef]
  16. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  17. M. D. Feit, J. A. Fleck, A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
    [CrossRef]
  18. S. Banerjee, A. Sharma, “Propagation characteristics of optical waveguiding structures by direct solution of the Helmholtz equation for total fields,” J. Opt. Soc. Am. A 6, 1884–1894 (1989).
    [CrossRef]
  19. R. D. Coalson, D. K. Pant, A. Ali, D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
    [CrossRef]
  20. See, for example, R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics (Wiley, New York, 1988).

2000 (1)

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000), and references therein.
[CrossRef]

1999 (1)

1998 (1)

1996 (1)

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

1994 (1)

R. D. Coalson, D. K. Pant, A. Ali, D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
[CrossRef]

1992 (1)

1991 (2)

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

C. Iung, C. Leforestier, “Exact time evolution methods for large bound systems,” Comput. Phys. Commun. 63, 135–153 (1991).
[CrossRef]

1989 (1)

1986 (1)

T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,” J. Chem. Phys. 85, 5870–5876 (1986).
[CrossRef]

1982 (1)

M. D. Feit, J. A. Fleck, A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

1980 (1)

1978 (1)

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Ali, A.

R. D. Coalson, D. K. Pant, A. Ali, D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
[CrossRef]

Banerjee, S.

Bisseling, R.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Campos-Martínez, J.

Cerjan, C.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Coalson, R. D.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Feit, M. D.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

M. D. Feit, J. A. Fleck, A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

M. D. Feit, J. A. Fleck, A. Steiger, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).
[CrossRef] [PubMed]

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

Fleck, J. A.

Friesner, R.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000), and references therein.
[CrossRef]

Guldberg, A.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Hadley, G. R.

Hammerich, A.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000), and references therein.
[CrossRef]

Hernández, M. I.

Ilic, I.

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

Iung, C.

C. Iung, C. Leforestier, “Exact time evolution methods for large bound systems,” Comput. Phys. Commun. 63, 135–153 (1991).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Jolicard, G.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Karrlein, W.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Koch, S. W.

N. Peyghambarian, S. W. Koch, A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Koonin, S. E.

See, for example, S. E. Koonin, D. C. Meredith, Computational Physics: Fortran Version (Addison-Wesley, Reading, Mass., 1990).

Kosloff, R.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Langer, D. W.

R. D. Coalson, D. K. Pant, A. Ali, D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
[CrossRef]

Leforestier, C.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

C. Iung, C. Leforestier, “Exact time evolution methods for large bound systems,” Comput. Phys. Commun. 63, 135–153 (1991).
[CrossRef]

Light, J. C.

T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,” J. Chem. Phys. 85, 5870–5876 (1986).
[CrossRef]

Lipkin, N.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).

Meredith, D. C.

See, for example, S. E. Koonin, D. C. Meredith, Computational Physics: Fortran Version (Addison-Wesley, Reading, Mass., 1990).

Meyer, H.-D.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Mysyrowicz, A.

N. Peyghambarian, S. W. Koch, A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Osgood, R. M.

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

Pant, D. K.

Park, T. J.

T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,” J. Chem. Phys. 85, 5870–5876 (1986).
[CrossRef]

Peyghambarian, N.

N. Peyghambarian, S. W. Koch, A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000), and references therein.
[CrossRef]

Roncero, O.

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Saxon, D. S.

See, for example, D. S. Saxon, Elementary Quantum Mechanics (Holden-Day, San Francisco, Calif., 1968).

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000), and references therein.
[CrossRef]

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

Sharma, A.

Steiger, A.

Stieger, A.

M. D. Feit, J. A. Fleck, A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Zare, R. N.

See, for example, R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics (Wiley, New York, 1988).

Appl. Opt. (3)

Comput. Phys. Commun. (1)

C. Iung, C. Leforestier, “Exact time evolution methods for large bound systems,” Comput. Phys. Commun. 63, 135–153 (1991).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000), and references therein.
[CrossRef]

J. Chem. Phys. (1)

T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,” J. Chem. Phys. 85, 5870–5876 (1986).
[CrossRef]

J. Comput. Phys. (2)

M. D. Feit, J. A. Fleck, A. Stieger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982).
[CrossRef]

C. Leforestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.-D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,” J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

J. Lightwave Technol. (3)

D. K. Pant, R. D. Coalson, M. I. Hernández, J. Campos-Martínez, “Optimal control theory for the design of optical waveguides,” J. Lightwave Technol. 16, 292–300 (1998).
[CrossRef]

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

R. D. Coalson, D. K. Pant, A. Ali, D. W. Langer, “Computing the eigenmodes of lossy field-induced optical waveguides,” J. Lightwave Technol. 12, 1015–1022 (1994).
[CrossRef]

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

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, “An algorithm for machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Opt. Lett. (1)

Other (6)

See, for example, D. S. Saxon, Elementary Quantum Mechanics (Holden-Day, San Francisco, Calif., 1968).

See, for example, S. E. Koonin, D. C. Meredith, Computational Physics: Fortran Version (Addison-Wesley, Reading, Mass., 1990).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).

N. Peyghambarian, S. W. Koch, A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice-Hall, Englewood Cliffs, N.J., 1993).

See, for example, R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics (Wiley, New York, 1988).

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Figures (10)

Fig. 1
Fig. 1

Propagation in a homogeneous medium of a 2-D Gaussian field profile as calculated via several wave equations and numerical techniques. All curves depict the optical power density |ψ(x, z)|2 at a particular value z along the propagation axis. The power density at z = 0 μm (localized around x = 0 μm) is shown by the thin dashed curve. The other three curves depict the power density at z = 16 μm. In particular, SIL WAEQ denotes the short-time iterative Lanczos used to propagate the wide-angle equation, SPO PEQ denotes the split operator used to propagate the paraxial equation, and Padé WAEQ denotes the result of propagation using a (1, 1) Padé approximant to the wide-angle equation. The SPO PEQ curve is indistinguishable from that obtained when we propagate according to the PEQ using SIL, while the Fourier integral profile is indistinguishable from the SIL WAEQ result.

Fig. 2
Fig. 2

As in Fig. 1 but for the 3-D case. Here contour plots of |ψ(x, y, z = 16 μm)|2 are plotted. Again, PEQ corresponds to the paraxial equation propagated with SPO, and WAEQ corresponds to the wide-angle equation propagated with SIL. The Fourier integral profile is indistinguishable from the SIL WAEQ result. The power density |ψ(x, y, z = 0 μm)|2 associated with the initial field, labeled initial, is also plotted.

Fig. 3
Fig. 3

X, Z and X′, Z′ coordinate systems, the latter tilted along with the waveguide by angle θ, for a square guide of width wb - a. In particular, note that b′ - a′w′ = cos θ.

Fig. 4
Fig. 4

Optical power density |ψ(x, z = 160 μm)|2 obtained by propagation according to the WAEQ, the PEQ, and the exact solution of the HEQ for a tilt angle of θ = 4°.

Fig. 5
Fig. 5

Overlaps [defined in the text by Eq. (34)] at different z values for the PEQ solution and the exact solution of the HEQ (open circles) and for the WAEQ and the exact solution of the HEQ (open squares). The tilt angle is θ = 4°.

Fig. 6
Fig. 6

Same as in Fig. 4, but for a tilt of θ = 20°.

Fig. 7
Fig. 7

Same as in Fig. 5, but for a tilt of θ = 20°.

Fig. 8
Fig. 8

(a) Comparison of optical power density |ψ(x, y, z)|2 obtained from an exact solution of the HEQ (dotted curves) versus the PEQ (solid curves) at a propagation distance z = 40 μm in the 3-D case for a tilt angle of θ = 20° and azimuthal angle ϕ = 0°. (b) Same as (a) but a comparison of the exact (dotted curves) with the wide-angle (solid curves) solution.

Fig. 9
Fig. 9

As in Fig. 5, but for a 3-D calculation and tilt angle of θ = 20°.

Fig. 10
Fig. 10

Overlap versus propagation distance for a sequence of index mismatches n g - n 0 between guiding and bulk (substrate) regions; n g - n 0 = (a) 0.309, (b) 0.109, (c) 0.049, (d) 0.009. In (a)–(d) the tilt angle is 8°, and the bulk index is n 0= 3.560. Note that the PEQ performs better than the WAEQ for larger index mismatches, whereas the situation is reversed for sufficiently small index mismatch values.

Tables (2)

Tables Icon

Table 1 Initial Parameters of the Gaussian Wave Packet

Tables Icon

Table 2 Parameters for the 2-D Tilted Waveguide Calculations

Equations (37)

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

2Φx, y, z+n2x, y, zk2Φx, y, z=0,
Φx, y, z=ψx, y, zexpiβ0z.
2z2+2iβ0zψx, y, z=-2x2+2y2+β02-k2n2x, y, zψx, y, z.
i z ψx, y, z=Ĥψx, y, z,
Ĥ=-12β02x2+2y2+12β0β02-k2n2x, y, z.
Ĥ=Tˆx, y+Vx, y, z.
i z ψx, y, z=Ôψx, y, z,
Ô=OĤ=β01-1-2Ĥ/β01/2.
ĥ-2z2-2iβ0zψx, y, z=0,
ĥ-2x2-2y2+β02-k2n2x, y, z=2β0Ĥ.
Dˆ-λˆ+Dˆ-λˆ-=Dˆ2-2β0Dˆ+ĥ,
i ψx, y, zz=λˆ-ψx, y, z,
λˆ+Dˆ+Dˆλˆ-λˆ++λˆ-Dˆ,
ϕ1=1β1Ĥϕ0-α0ϕ0,
ϕ2=1β2Ĥϕ1-α1ϕ1-β1ϕ0,
HNk=α0β100β1α1β2000β2α2β30000αNk-2βNk-1000βNk-1αNk-1,
αjϕj|Ĥ|ϕj; βj+1=ϕj+1|Ĥ|ϕj.
ψx, y, z+Δz=j=0Nk-1 ajz+Δzϕjx, y,
az+Δz=ZdZaz,
β01-1-kx2+ky2/β021/2.
ψx, y, z=-dkx-dkyfkx, kyexpizβ01-1-kx2+ky2/β02]1/2exp(ikxx+ikyy.
kx2+ky22β0.
ψx, z=0=2Imαπ1/4 ×expiαx-x02+p0x-x0.
n2x, z=nb2-nb2-ng2x-x1z-x-x2z,
u=1if u>00if u<0.
x1z=tgθz+α-w/2,
x2z=tgθz+α+w/2,
xz=cos θsin θ-sin θcos θxz,
2x2+2z2Φx, z+k2n2xΦx, z=0,
n2x=nb2-nb2-ng2x-a-x-b.
Φx, z=Axexpiβz
d2Adx2+k2n2x-β2A=0.
Φx, z=ψx, zexpiβ0z;
Φx, z=ψx, zexpiβ0z=Axexpiβz,
ψx, z=Axexpiβzexp-iβ0z.
ψx, 0=Ax cos θexpiβx sin θ,
Ovz=|ψex|ψap|2ψex|ψexψap|ψap,

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