Abstract

We present a simple and general approach for analysis of absorbing and leaky waveguides. The real and imaginary parts of the propagation constant β of a planar optical waveguide are obtained by evaluating a function, defined through the eigenvalue equation, in real β. The applicability of the method is demonstrated for simple structures. To apply the method for arbitrarily graded inhomogeneous or multilayered structures we use a simple matrix approach to obtain the eigenvalue equation. The method is straightforward, accurate, and requires no iteration in the complex β plane.

© 1989 Optical Society of America

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References

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  1. P. K. Tien, R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  2. E. M. Garmire, H. Stoll, IEEE J. Quantum Electron. QE-8, 206 (1972).
  3. W. W. Anderson, IEEE J. Quantum Electron. QE-1, 229 (1965).
  4. A. Reisinger, Appl. Opt. 12, 1015 (1973).
    [CrossRef] [PubMed]
  5. I. P. Kaminow, W. L. Mammel, H. P. Weber, Appl. Opt. 13, 396 (1974).
    [CrossRef] [PubMed]
  6. T. Findakly, C.-L. Chen, Appl. Opt. 17, 469 (1978).
    [CrossRef] [PubMed]
  7. A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
    [CrossRef]
  8. It is possible to obtain a Lorentzian even when the loss is zero. This would be a Lorentzian having a HWHM of zero (not a delta function7) and would be of the form 1/(βr − βm,r)2, which may be easily plotted. Thus the method is also applicable for cases in which resonances are sharp. However, one may solve Eq. (13) directly if the waveguide is lossless.
  9. J. T. Chilwell, I. J. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984).
    [CrossRef]
  10. L. M. Walpita, J. Opt. Soc. Am. A 2, 595 (1985).
    [CrossRef]
  11. M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, UK, 1981).
  12. M. R. Ramadas, R. K. Varshney, K. Thyagarajan, A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” IEEE J. Lightwave Technol. (to be published).

1987 (1)

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[CrossRef]

1985 (1)

1984 (1)

1978 (1)

1974 (1)

1973 (1)

1972 (1)

E. M. Garmire, H. Stoll, IEEE J. Quantum Electron. QE-8, 206 (1972).

1970 (1)

1965 (1)

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 229 (1965).

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, UK, 1981).

Anderson, W. W.

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 229 (1965).

Chen, C.-L.

Chilwell, J. T.

Findakly, T.

Garmire, E. M.

E. M. Garmire, H. Stoll, IEEE J. Quantum Electron. QE-8, 206 (1972).

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[CrossRef]

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” IEEE J. Lightwave Technol. (to be published).

Hodgkinson, I. J.

Kaminow, I. P.

Mammel, W. L.

Ramadas, M. R.

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” IEEE J. Lightwave Technol. (to be published).

Reisinger, A.

Shenoy, M. R.

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[CrossRef]

Stoll, H.

E. M. Garmire, H. Stoll, IEEE J. Quantum Electron. QE-8, 206 (1972).

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[CrossRef]

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” IEEE J. Lightwave Technol. (to be published).

Tien, P. K.

Ulrich, R.

Varshney, R. K.

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” IEEE J. Lightwave Technol. (to be published).

Walpita, L. M.

Weber, H. P.

Appl. Opt. (3)

IEEE J. Lightwave Technol. (1)

A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, IEEE J. Lightwave Technol. LT-5, 660 (1987).
[CrossRef]

IEEE J. Quantum Electron. (2)

E. M. Garmire, H. Stoll, IEEE J. Quantum Electron. QE-8, 206 (1972).

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 229 (1965).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Other (3)

It is possible to obtain a Lorentzian even when the loss is zero. This would be a Lorentzian having a HWHM of zero (not a delta function7) and would be of the form 1/(βr − βm,r)2, which may be easily plotted. Thus the method is also applicable for cases in which resonances are sharp. However, one may solve Eq. (13) directly if the waveguide is lossless.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, Chichester, UK, 1981).

M. R. Ramadas, R. K. Varshney, K. Thyagarajan, A. K. Ghatak, “A matrix approach to study the propagation characteristics of a general nonlinear planar waveguide,” IEEE J. Lightwave Technol. (to be published).

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Figures (2)

Fig. 1
Fig. 1

Variation of |1/F(βr)|2 as a function of βr for a worst-case symmetric dielectric film (nf = 1.5884) with metal (ns = 0.1620–i3.2103) sidewalls and d = 0.2 μm.

Fig. 2
Fig. 2

Transverse cross section of a multilayer planar waveguide having N layers sandwiched between the two semi-infinite media—the substrate and the cover.

Equations (14)

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F ( β m ) = 0.
β m = β m , r + i β m , i .
| 1 F ( β r ) | 2 = A [ ( β r - β m , r ) 2 + β m , i 2 ] .
F ( β ) = F ( β m ) + ( β - β m ) d F d β + .
F ( β ) = ( β - β m ) C .
F ( β r ) = ( β r - β m , r + i β m , i ) C .
F ( β r ) = H tan ( H ) - P ,
| 1 F ( β r ) | 2 = 1 ( H tan H - P r ) 2 + P i 2 ,
E i = A i cos k i ( x - d i ) + B i ξ i sin k i ( x - d i ) ,
[ A i + 1 B i + 1 ] = S i S i - 1 S 2 S 1 [ A 1 B 1 ] ,
S i = [ cos Δ i ξ i sin Δ i - ( 1 / ξ i ) sin Δ i cos Δ i ]
S i = [ cosh Δ i γ i sinh Δ i ( 1 / γ i ) sinh Δ i cosh Δ i ] ,
E s = A 1 cosh ( p s x ) + B 1 γ s sinh ( p s x ) = [ ( A 1 + γ s B 1 ) / 2 ] exp ( p s x ) + [ ( A 1 - γ s B 1 ) / 2 ] exp ( - p s x ) ,
A c ( β ) + γ c B c ( β ) = 0 ,

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