Abstract

We present an analytical method for TE and TM modes in weakly guiding inhomogeneous single-mode slab waveguides. Based on our results, the modal behavior or propagation constants depend on index profiles of the waveguides. It is important to know how the modal behavior depends on the index profile in single-mode waveguides, because it determines wave-front characteristics of propagating modes.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed., Vol. 7 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979), pp. 13–81.
  2. W. Streifer, C. N. Kurtz, “Scalar analysis of radially inhomogeneous guiding media,” J. Opt. Soc. Am. 57, 779–786 (1967).
    [CrossRef]
  3. A. Kumar, K. Thyagarajan, A. K. Ghatak, “Modes in inhomogeneous slab waveguides,” IEEE J. Quantum Electron. QE-10, 902–904 (1974).
    [CrossRef]
  4. A. Weisshaar, “Impedance boundary method of moments for accurate and efficient analysis of planar graded-index optical waveguides,” J. Lightwave Technol. 12, 1943–1951 (1994).
    [CrossRef]
  5. J. Li, A. Weisshaar, “Generalized impedance boundary method of moments for multilayer graded-index dielectric waveguide structures,” IEE Proc. Optoelectron. 143, 167–172 (1996).
    [CrossRef]
  6. D. M. Adams, J. G. Simmons, “Efficient analysis of strongly absorbing inhomogeneous planar waveguides using exact transfer matrices for linearly, exponentially, and parabolically graded media,” Opt. Commun. 110, 293–297 (1994).
    [CrossRef]

1996 (1)

J. Li, A. Weisshaar, “Generalized impedance boundary method of moments for multilayer graded-index dielectric waveguide structures,” IEE Proc. Optoelectron. 143, 167–172 (1996).
[CrossRef]

1994 (2)

D. M. Adams, J. G. Simmons, “Efficient analysis of strongly absorbing inhomogeneous planar waveguides using exact transfer matrices for linearly, exponentially, and parabolically graded media,” Opt. Commun. 110, 293–297 (1994).
[CrossRef]

A. Weisshaar, “Impedance boundary method of moments for accurate and efficient analysis of planar graded-index optical waveguides,” J. Lightwave Technol. 12, 1943–1951 (1994).
[CrossRef]

1974 (1)

A. Kumar, K. Thyagarajan, A. K. Ghatak, “Modes in inhomogeneous slab waveguides,” IEEE J. Quantum Electron. QE-10, 902–904 (1974).
[CrossRef]

1967 (1)

Adams, D. M.

D. M. Adams, J. G. Simmons, “Efficient analysis of strongly absorbing inhomogeneous planar waveguides using exact transfer matrices for linearly, exponentially, and parabolically graded media,” Opt. Commun. 110, 293–297 (1994).
[CrossRef]

Ghatak, A. K.

A. Kumar, K. Thyagarajan, A. K. Ghatak, “Modes in inhomogeneous slab waveguides,” IEEE J. Quantum Electron. QE-10, 902–904 (1974).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed., Vol. 7 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979), pp. 13–81.

Kumar, A.

A. Kumar, K. Thyagarajan, A. K. Ghatak, “Modes in inhomogeneous slab waveguides,” IEEE J. Quantum Electron. QE-10, 902–904 (1974).
[CrossRef]

Kurtz, C. N.

Li, J.

J. Li, A. Weisshaar, “Generalized impedance boundary method of moments for multilayer graded-index dielectric waveguide structures,” IEE Proc. Optoelectron. 143, 167–172 (1996).
[CrossRef]

Simmons, J. G.

D. M. Adams, J. G. Simmons, “Efficient analysis of strongly absorbing inhomogeneous planar waveguides using exact transfer matrices for linearly, exponentially, and parabolically graded media,” Opt. Commun. 110, 293–297 (1994).
[CrossRef]

Streifer, W.

Thyagarajan, K.

A. Kumar, K. Thyagarajan, A. K. Ghatak, “Modes in inhomogeneous slab waveguides,” IEEE J. Quantum Electron. QE-10, 902–904 (1974).
[CrossRef]

Weisshaar, A.

J. Li, A. Weisshaar, “Generalized impedance boundary method of moments for multilayer graded-index dielectric waveguide structures,” IEE Proc. Optoelectron. 143, 167–172 (1996).
[CrossRef]

A. Weisshaar, “Impedance boundary method of moments for accurate and efficient analysis of planar graded-index optical waveguides,” J. Lightwave Technol. 12, 1943–1951 (1994).
[CrossRef]

IEE Proc. Optoelectron. (1)

J. Li, A. Weisshaar, “Generalized impedance boundary method of moments for multilayer graded-index dielectric waveguide structures,” IEE Proc. Optoelectron. 143, 167–172 (1996).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Kumar, K. Thyagarajan, A. K. Ghatak, “Modes in inhomogeneous slab waveguides,” IEEE J. Quantum Electron. QE-10, 902–904 (1974).
[CrossRef]

J. Lightwave Technol. (1)

A. Weisshaar, “Impedance boundary method of moments for accurate and efficient analysis of planar graded-index optical waveguides,” J. Lightwave Technol. 12, 1943–1951 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

D. M. Adams, J. G. Simmons, “Efficient analysis of strongly absorbing inhomogeneous planar waveguides using exact transfer matrices for linearly, exponentially, and parabolically graded media,” Opt. Commun. 110, 293–297 (1994).
[CrossRef]

Other (1)

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed., Vol. 7 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979), pp. 13–81.

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 (3)

Fig. 1
Fig. 1

Coordinates of a slab waveguide. The wave propagation is in the z direction and the refractive index varies in the x direction.

Fig. 2
Fig. 2

Solutions of Eq. (9) for m = 1, 2, 3,⋯ , and 9 are given by the intersecting solid curves from right to left. Each intersection represents a different propagation constant β m for TE modes. The intersection of a dotted curve and a solid curve represents the propagation constant of β m→.

Fig. 3
Fig. 3

Solutions of Eq. (17) for m = 1, 2, 3,⋯ , and 9 are given by the intersecting solid curves from right to left. Each intersection represents a different propagation constant β m for TM modes. The intersection of a dotted curve and a solid curve represents the propagation constant of β m→.

Equations (19)

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

2 E y m x ,   z + n x 2 ω 2 ε 0 μ 0 E y m x ,   z = 0 ,
d 2 Z y m z / d z 2 = - β m 2 Z y m z , d 2 X y m ξ / d ξ 2 + a / 2 2 k 1 2 - β m 2 - 2 Δ | ξ | m k 1 2 + Δ 2 ξ 2 m k 1 2 X y m ξ = 0 ,
X y m ξ = e 0 m ξ + Δ e 1 m ξ + Δ 2 e 2 m ξ + .
d 2 e 0 m ξ / d ξ 2 + θ m 2 e 0 m = 0 ,
d 2 e 1 m ξ / d ξ 2 + θ m 2 e 1 m ξ - 2 a / 2 2 k 1 2 | ξ | m e 0 m ξ = 0 ,
e 0 m ξ = cos θ m ξ
d 2 e 1 m ξ / d ξ 2 + θ m 2 e 1 m ξ - 2 ak 1 / 2 2 | ξ | m cos θ m ξ = 0 .
e 1 m ξ = f m ξ cos θ m ξ + g m ξ sin θ m ξ ,
f m ξ = 2 ak 1 / 2 2 1 / 2 θ m 2 m ! m + 0 !   ξ m + 0 - 1 / 2 θ m 4 m ! m - 2 !   ξ m - 2 + ,   g m ξ = 2 ak 1 / 2 2 1 / 2 θ m m ! m + 1 !   ξ m + 1 - 1 / 2 θ m 3 m ! m - 1 !   ξ m - 1 + for   ξ     0 .
- Δ d f m ξ / d ξ + Δ θ m g m ξ + Δ d g m ξ / d ξ - θ m 1 + Δ f m ξ tan θ m ξ 1 + Δ f m ξ + Δ g m ξ tan θ m ξ ξ = 1 = V 2 - θ m 2 ,
d 2 X y m ξ / d ξ 2 + θ m 2 X y m ξ - 2 Δ - m | ξ | m - 1 d X y m ξ / d ξ + ak 1 / 2 2 | ξ | m X y m ξ = 0 .
X y m ξ = h 0 m ξ + Δ h 1 m ξ + Δ 2 h 2 m ξ + .
d 2 h 0 m ξ / d ξ 2 + θ m 2 h 0 m ξ = 0 ,
d 2 h 1 m ξ / d ξ 2 + θ m 2 h 1 m ξ - 2 - m | ξ | m - 1 d h 0 m ξ / d ξ + ak 1 / 2 2 | ξ | m h 0 m ξ = 0 .
h 0 m ξ = cos θ m ξ
d 2 h 1 m ξ / d ξ 2 + θ m 2 h 1 m ξ - 2 m | ξ | m - 1 θ m sin θ m ξ + ak 1 2 / 2 2 | ξ | m cos θ m ξ = 0 .
h 1 m ξ = F m ξ cos θ m ξ + G m ξ sin θ m ξ ,
F m ξ = 2 ak 1 / 2 2 - 2 θ m 2 1 / 2 θ m 2 ξ m - 1 / 2 θ m 4 m ! m - 2 !   ξ m - 2 + 1 / 2 θ m 6 m ! m - 4 !   ξ m - 4 - G m ξ = 2 ak 1 / 2 2 1 2 θ m m ! m + 1 !   ξ m + 1 - 2 ak 1 2 2 1 2 θ m 3 - 1 2 θ m m ! m - 1 !   ξ m - 1 + 2 ak 1 2 2 1 2 θ m 5 - 1 2 θ m 3 m ! m - 3 !   ξ m - 3 - .
- Δ d F m ξ / d ξ + Δ θ m G m ξ + Δ d G m ξ / d ξ - θ m 1 + Δ F m ξ tan θ m ξ 1 + Δ F m ξ + Δ G m ξ tan θ m ξ ξ = 1 = V 2 - θ m 2 .

Metrics