Abstract

A new method is proposed for making sharp bends with low radiation losses in dielectric optical waveguides. By modifying the transverse refractive-index profile at curved sections both the pure bend and the transition losses can be minimized. The optimum gradient-index profile requires an inhomogeneous medium. But in practice this can be replaced by a layered medium. By using four homogeneous layers the permitted radius of curvature of a slab waveguide can be reduced, e.g., from 6400 to only 100 wavelengths.

© 1983 Optical Society of America

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References

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  1. L. D. Hutcheson, I. A. White, J. J. Burke, Opt. Lett. 5, 276 (1980).
    [CrossRef] [PubMed]
  2. S. Sheem, J. R. Whinnery, Wave Electron. 1, 61, 105 (1974/1975).
  3. M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (1975).
    [CrossRef]
  4. M. Desai, R. Mittra, in Proceedings, IEEE International Microwave Theory Techniques Symposium (1980), p. 211.
  5. M. Geshiro, Sh. Sawa, IEEE Trans. Microwave Theory Tech. MIT-29, 1182 (1981).
    [CrossRef]
  6. E.-G. Neumann, “Low loss dielectric optical waveguide bends,” to be published in Fibers Integr. Opt.4, No. 2 (1982).
  7. E.-G. Neumann, IEE Proc. 129, 278 (1982).
  8. E.-G. Neumann, German patent pending.
  9. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 1.3.
  10. H.-G. Unger, Optische Nachrichtentechnik (Elitera, Berlin, 1976), p. 28.
  11. E.-G. Neumann, Electron. Lett. 17, 369 (1981).
    [CrossRef]
  12. E. W. Marchand, Appl. Opt. 21, 983 (1982).
    [CrossRef] [PubMed]
  13. K. Simonyi, Theoretische Elektrotechnik (VEB Deutscher Verlag der Wissenschaften, Berlin, 1980), p. 730.
  14. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Sec. 1.4.
  15. H.-G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977), Sec. 2.3.
  16. R. Baets, P. E. Lagasse, “Loss calculation and design of arbitrarily curved integrated optic waveguides,” submitted to J. Opt. Soc. Am.
  17. F. W. J. Olver, Philos. Trans. R. Soc. London Ser. A 247, 328 (1954).
    [CrossRef]
  18. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

1982

E.-G. Neumann, IEE Proc. 129, 278 (1982).

E. W. Marchand, Appl. Opt. 21, 983 (1982).
[CrossRef] [PubMed]

1981

M. Geshiro, Sh. Sawa, IEEE Trans. Microwave Theory Tech. MIT-29, 1182 (1981).
[CrossRef]

E.-G. Neumann, Electron. Lett. 17, 369 (1981).
[CrossRef]

1980

1975

M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (1975).
[CrossRef]

1954

F. W. J. Olver, Philos. Trans. R. Soc. London Ser. A 247, 328 (1954).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Baets, R.

R. Baets, P. E. Lagasse, “Loss calculation and design of arbitrarily curved integrated optic waveguides,” submitted to J. Opt. Soc. Am.

Burke, J. J.

Desai, M.

M. Desai, R. Mittra, in Proceedings, IEEE International Microwave Theory Techniques Symposium (1980), p. 211.

Geshiro, M.

M. Geshiro, Sh. Sawa, IEEE Trans. Microwave Theory Tech. MIT-29, 1182 (1981).
[CrossRef]

Harris, J. H.

M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (1975).
[CrossRef]

Heiblum, M.

M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (1975).
[CrossRef]

Hutcheson, L. D.

Lagasse, P. E.

R. Baets, P. E. Lagasse, “Loss calculation and design of arbitrarily curved integrated optic waveguides,” submitted to J. Opt. Soc. Am.

Marchand, E. W.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Sec. 1.4.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 1.3.

Mittra, R.

M. Desai, R. Mittra, in Proceedings, IEEE International Microwave Theory Techniques Symposium (1980), p. 211.

Neumann, E.-G.

E.-G. Neumann, IEE Proc. 129, 278 (1982).

E.-G. Neumann, Electron. Lett. 17, 369 (1981).
[CrossRef]

E.-G. Neumann, “Low loss dielectric optical waveguide bends,” to be published in Fibers Integr. Opt.4, No. 2 (1982).

E.-G. Neumann, German patent pending.

Olver, F. W. J.

F. W. J. Olver, Philos. Trans. R. Soc. London Ser. A 247, 328 (1954).
[CrossRef]

Sawa, Sh.

M. Geshiro, Sh. Sawa, IEEE Trans. Microwave Theory Tech. MIT-29, 1182 (1981).
[CrossRef]

Sheem, S.

S. Sheem, J. R. Whinnery, Wave Electron. 1, 61, 105 (1974/1975).

Simonyi, K.

K. Simonyi, Theoretische Elektrotechnik (VEB Deutscher Verlag der Wissenschaften, Berlin, 1980), p. 730.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Unger, H.-G.

H.-G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977), Sec. 2.3.

H.-G. Unger, Optische Nachrichtentechnik (Elitera, Berlin, 1976), p. 28.

Whinnery, J. R.

S. Sheem, J. R. Whinnery, Wave Electron. 1, 61, 105 (1974/1975).

White, I. A.

Appl. Opt.

Electron. Lett.

E.-G. Neumann, Electron. Lett. 17, 369 (1981).
[CrossRef]

IEE Proc.

E.-G. Neumann, IEE Proc. 129, 278 (1982).

IEEE J. Quantum Electron.

M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. QE-11, 75 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

M. Geshiro, Sh. Sawa, IEEE Trans. Microwave Theory Tech. MIT-29, 1182 (1981).
[CrossRef]

Opt. Lett.

Philos. Trans. R. Soc. London Ser. A

F. W. J. Olver, Philos. Trans. R. Soc. London Ser. A 247, 328 (1954).
[CrossRef]

Wave Electron.

S. Sheem, J. R. Whinnery, Wave Electron. 1, 61, 105 (1974/1975).

Other

M. Desai, R. Mittra, in Proceedings, IEEE International Microwave Theory Techniques Symposium (1980), p. 211.

E.-G. Neumann, German patent pending.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 1.3.

H.-G. Unger, Optische Nachrichtentechnik (Elitera, Berlin, 1976), p. 28.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

E.-G. Neumann, “Low loss dielectric optical waveguide bends,” to be published in Fibers Integr. Opt.4, No. 2 (1982).

K. Simonyi, Theoretische Elektrotechnik (VEB Deutscher Verlag der Wissenschaften, Berlin, 1980), p. 730.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Sec. 1.4.

H.-G. Unger, Planar Optical Waveguides and Fibers (Clarendon, Oxford, 1977), Sec. 2.3.

R. Baets, P. E. Lagasse, “Loss calculation and design of arbitrarily curved integrated optic waveguides,” submitted to J. Opt. Soc. Am.

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Figures (4)

Fig. 1
Fig. 1

90° bend connecting two straight waveguide sections. Refractive-index profiles at the straight and bent sections.

Fig. 2
Fig. 2

Step approximation of the optimum gradient-index profile by a layered medium (∂/∂z = 0).

Fig. 3
Fig. 3

Optimum values of the refractive indices of the layers and axes offset necessary for obtaining a total attenuation smaller than 0.5 dB.

Fig. 4
Fig. 4

Transverse field distributions at the straight and curved waveguides with the refractive indices and the offset optimized for b = 100λ0.

Equations (35)

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R λ 0 , R ω 0 .
2 E ( ρ , φ , z ) + n 2 ( ρ , z ) k 2 E ( ρ , φ , z ) = 0.
E ( ρ , φ , z ) = E s ( ρ , z ) exp ( j β R φ ) .
n 2 ( ρ , z ) = ( R ρ ) 2 ( β k ) 2 1 k 2 E s ( 1 ρ E s ρ + 2 E s ρ 2 + 2 E s z 2 ) .
n s 2 ( ρ , z ) = ( β s k ) 2 1 k 2 E s ( 2 E s ρ 2 + 2 E s z 2 ) .
n 2 ( ρ , z ) = n s 2 ( ρ , z ) + ( R ρ ) 2 ( β k ) 2 ( β s k ) 2 .
| β / k n s ( ρ , z ) | n s ( ρ , z ) 1 ,
n 2 ( ρ , z ) = ( R ρ ) 2 n s 2 ( ρ , z ) + [ β 2 k 2 n s 2 ( ρ , z ) ] [ ( R ρ ) 2 1 ] ( R ρ ) 2 n s 2 ( ρ , z ) .
n ( ρ , z ) = R ρ n s ( ρ , z )
( R ρ ) = R ρ 2 e ρ ,
| ( R ρ ) | = 1 R .
n e ( ρ , z ) = ρ R · n ( ρ , z ) .
Π = Π i e z ,
E z = k i 2 Π i , E ρ = 0 , E φ = 0 ,
H z = 0 , H ρ = j ω i ρ Π i φ , H φ = j ω i Π i ρ ,
i = n i 2 0 ,
k i = n i k .
Π i = [ A i J ν ( k i ρ ) + B i Y ν ( k i ρ ) ] exp ( j ν φ ) ,
B 1 = 0 and B 4 = j A 4
ν = ν j ν ,
J ν ( k 1 a ) J ν ( k 2 a ) Y ν ( k 2 a ) 0 0 0 0 J ν ( k 2 b ) Y ν ( k 2 b ) J ν ( k 3 b ) Y ν ( k 3 b ) 0 0 0 0 J ν ( k 3 c ) Y ν ( k 3 c ) H ν ( 2 ) ( k 4 c ) n 1 J ν + 1 ( k 1 a ) n 2 J ν + 1 ( k 2 a ) n 2 Y ν + 1 ( k 2 a ) 0 0 0 0 n 2 J ν + 1 ( k 2 b ) ν 2 Y ν + 1 ( k 2 b ) n 3 J ν + 1 ( k 3 b ) n 3 Y ν + 1 ( k 3 b ) 0 0 0 0 n 3 J ν + 1 ( k 3 c ) n 3 Y ν + 1 ( k 3 c ) n 4 H ν + 1 ( 2 ) ( k 4 c ) = 0.
A 4 = 1
η c s = 1 P s P c | A ( E × H * s ) d A | 2 ,
P c = Δ z n 4 4 k 3 | A 4 | 2 exp ( ν π ) Z 0 π ν ,
η s c = η c s = η .
ν n 2 k b = n 2 2 π b / λ 0 1.
J * ν ( x ) = J ν * ( x * )
a 90 ° = ν · 10 · π log e = 13.64 ν .
a T = 10 log η .
a = a T + a 90 ° + a T = a 90 ° + 2 a T .
n 1 = 1.5 ;
Δ = 0.00316 ;
V = π ;
b a = c b = d / 2 = 2.097 λ 0 .
a = 0.2445 dB .

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