Abstract

In the weakly guiding approximation, we present an analysis describing the propagation in bent waveguides characterized by a separable profile (which may be graded) in the transverse direction. We then show how the recently developed matrix method can be used to calculate accurately the bending loss in such waveguides. We then suggest a novel refractive-index profile for which the bending losses can be made extremely small.

© 1987 Optical Society of America

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Corrections

K. Thyagarajan, M. R. Shenoy, and A. K. Ghatak, "Accurate numerical method for the calculation of bending loss in optical waveguides using a matrix approach: erratum," Opt. Lett. 14, 338-338 (1989)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-14-6-338

References

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  1. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
  2. L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
    [CrossRef]
  3. E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
    [CrossRef]
  4. H. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-1, 75 (1975).
    [CrossRef]
  5. K. Petermann, Arch. Elektr. Ubertr. 30, 337 (1976).
  6. H. G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977), p. 157.
  7. D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), Chap. 9.
  8. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.
  9. A. Kumar, K. Thyagarajan, A. K. Ghatak, Opt. Lett. 8, 63 (1983).
    [CrossRef] [PubMed]
  10. A. Kumar, R. K. Varshney, K. Thyagarajan, Electron. Lett. 20, 112 (1984).
    [CrossRef]
  11. R. K. Varshney, A. Kumar, Opt. Lett. 11, 45 (1986).
    [CrossRef] [PubMed]
  12. A. K. Ghatak, Opt. Quantum Electron. 17, 311 (1985).
    [CrossRef]
  13. A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. (to be published).
  14. S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, Appl. Phys. Lett. 48, 92 (1986).
    [CrossRef]

1986 (2)

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, Appl. Phys. Lett. 48, 92 (1986).
[CrossRef]

R. K. Varshney, A. Kumar, Opt. Lett. 11, 45 (1986).
[CrossRef] [PubMed]

1985 (1)

A. K. Ghatak, Opt. Quantum Electron. 17, 311 (1985).
[CrossRef]

1984 (1)

A. Kumar, R. K. Varshney, K. Thyagarajan, Electron. Lett. 20, 112 (1984).
[CrossRef]

1983 (1)

1976 (1)

K. Petermann, Arch. Elektr. Ubertr. 30, 337 (1976).

1975 (2)

E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

H. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-1, 75 (1975).
[CrossRef]

1974 (1)

L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
[CrossRef]

1969 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Bosworth, R. H.

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, Appl. Phys. Lett. 48, 92 (1986).
[CrossRef]

Chang, D. C.

E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

Ghatak, A. K.

A. K. Ghatak, Opt. Quantum Electron. 17, 311 (1985).
[CrossRef]

A. Kumar, K. Thyagarajan, A. K. Ghatak, Opt. Lett. 8, 63 (1983).
[CrossRef] [PubMed]

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. (to be published).

Harris, J. H.

H. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-1, 75 (1975).
[CrossRef]

Heiblum, H.

H. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-1, 75 (1975).
[CrossRef]

Korotky, S. K.

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, Appl. Phys. Lett. 48, 92 (1986).
[CrossRef]

Kuester, E. F.

E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

Kumar, A.

Lewin, L.

L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.

Marcatili, E. A. J.

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, Appl. Phys. Lett. 48, 92 (1986).
[CrossRef]

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), Chap. 9.

Petermann, K.

K. Petermann, Arch. Elektr. Ubertr. 30, 337 (1976).

Shenoy, M. R.

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. (to be published).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.

Thyagarajan, K.

A. Kumar, R. K. Varshney, K. Thyagarajan, Electron. Lett. 20, 112 (1984).
[CrossRef]

A. Kumar, K. Thyagarajan, A. K. Ghatak, Opt. Lett. 8, 63 (1983).
[CrossRef] [PubMed]

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. (to be published).

Unger, H. G.

H. G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977), p. 157.

Varshney, R. K.

R. K. Varshney, A. Kumar, Opt. Lett. 11, 45 (1986).
[CrossRef] [PubMed]

A. Kumar, R. K. Varshney, K. Thyagarajan, Electron. Lett. 20, 112 (1984).
[CrossRef]

Veselka, J. J.

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, Appl. Phys. Lett. 48, 92 (1986).
[CrossRef]

Appl. Phys. Lett. (1)

S. K. Korotky, E. A. J. Marcatili, J. J. Veselka, R. H. Bosworth, Appl. Phys. Lett. 48, 92 (1986).
[CrossRef]

Arch. Elektr. Ubertr. (1)

K. Petermann, Arch. Elektr. Ubertr. 30, 337 (1976).

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Electron. Lett. (1)

A. Kumar, R. K. Varshney, K. Thyagarajan, Electron. Lett. 20, 112 (1984).
[CrossRef]

IEEE J. Quantum Electron. (2)

E. F. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

H. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-1, 75 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

L. Lewin, IEEE Trans. Microwave Theory Tech. MTT-22, 718 (1974).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

A. K. Ghatak, Opt. Quantum Electron. 17, 311 (1985).
[CrossRef]

Other (4)

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, “Numerical analysis of planar optical waveguides using matrix approach,” IEEE J. Lightwave Technol. (to be published).

H. G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977), p. 157.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), Chap. 9.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.

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Figures (5)

Fig. 1
Fig. 1

The coordinate system for a bent planar waveguide.

Fig. 2
Fig. 2

(a) The solid line shows the effective refractive-index profile corresponding to a bent-slab waveguide. The dashed line shows the corresponding profile when the waveguide is straight. (b) The effective profile replaced by a large number of homogeneous layers.

Fig. 3
Fig. 3

Typical variation of |Ej+/E1+|2 as a function of β for ρ = 1 cm, corresponding to the waveguide defined by Eq. (9). The peak corresponds to β r ; the full width at half-maximum, 2Γ, gives the loss coefficient of the bent waveguide; and j corresponds to a layer inside the core of the waveguide.

Fig. 4
Fig. 4

The effective refractive-index profile of the planar waveguide defined by Eq. (9) and the corresponding field distribution of the quasi-mode for a bend radius of ρ = 0.5 cm.

Fig. 5
Fig. 5

Variation of bend loss with radius of curvature for a planar waveguide defined by Eq. (9). The solid curve corresponds to the present method, and the dashed curve corresponds to that obtained by using the approximate formula of Marcuse.7

Equations (10)

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n 2 ( x , y ) = n 2 ( x ) + n 2 ( y ) .
ψ ( r , ϕ , z ) = ψ ( r , z ) exp ( - i β ρ ϕ ) ,
1 r r ( r ψ r ) + 2 ψ z 2 - β 2 ρ 2 r 2 ψ + k 0 2 n 2 ( r , z ) ψ = 0.
1 r d d r ( r d R d r ) + [ k 0 2 n 2 ( r ) - β 2 ρ 2 r 2 + β z 2 ] R = 0 ,
d 2 Z d z 2 + [ k 0 2 n 2 ( z ) - β z 2 ] Z = 0.
d 2 u d ξ 2 + [ k 0 2 n ˜ 2 ( ξ ) - β 2 ] u ( ξ ) = 0 ,
n ˜ 2 ( ξ ) = n 2 ( r ) + { β 2 k 0 2 [ 1 - ρ 2 ( ρ + ξ ) 2 ] + 1 4 k 0 2 ( ρ + ξ ) 2 } + β z 2 k 0 2 .
| E j + E 1 + | 2 ~ 1 ( β - β r ) 2 + Γ 2 ,
n ( x ) = { n 1 = 1.503 x < α n 2 = 1.500 x > a , a = 2 μ m ,             λ 0 = 1 μ m ,
n ( x ) = { n 1 - a < x < a n 2 - α 2 x a < x < d n 3 x > d .

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