Abstract

An analysis of the butt-joining coupling between two inhomogeneous planar optical waveguides with hyperbolic secant refractive-index profiles is made from the point of view of geometrical optics. Furthermore, the design of a new device to contract or expand a collimated beam is presented, and a relationship between sizes of the input and output beams is obtained.

© 1996 Optical Society of America

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References

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  1. C. Gómez-Reino, Int. J. Optoelectron. 7, 607 (1992).
  2. D. Bertoline, A. Ahkiewicz, C. Pask, Appl. Opt. 26, 2213 (1987).
    [CrossRef]
  3. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 5 and 8.
  4. C. Gómez-Reino, J. Liñares, J. Opt. Soc. Am. A 4, 1337 (1987).
    [CrossRef]
  5. M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977).
  6. A. Fletcher, T. Murphy, A. Young, Proc. R. Soc. London Ser. A 233, 216 (1954).
  7. J. Liñares, C. Gómez-Reino, Appl. Opt. 33, 3427 (1994).
    [CrossRef]
  8. C. Bao, C. Gómez-Reino, Opt. Quantum Electron. 27, 897 (1995).
    [CrossRef]
  9. D. W. Hewak, J. W. Y. Lit, Appl. Opt. 28, 4190 (1989).
    [CrossRef] [PubMed]
  10. R. V. Ramaswamy, R. Srivastava, J. Lightwave Technol. 6, 984 (1988).
    [CrossRef]
  11. N. Takato, K. Jinguyi, M. Yasu, H. Toba, M. Kawachi, J. Lightwave Technol. 6, 1003 (1988).
    [CrossRef]
  12. J. Liñares, C. Gómez-Reino, in Proceedings of the Tenth Topical Meeting on Gradient-Index Optical Systems (University of Santiago de Compostela, Galicia, Spain, 1992), p. 9.
  13. E. Acosta, J. R. Flores, C. Gómez-Reino, J. Liñares, Opt. Eng. 28, 1168 (1989).
  14. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Chap. 4, p. 282.

1995 (1)

C. Bao, C. Gómez-Reino, Opt. Quantum Electron. 27, 897 (1995).
[CrossRef]

1994 (1)

1992 (1)

C. Gómez-Reino, Int. J. Optoelectron. 7, 607 (1992).

1989 (2)

E. Acosta, J. R. Flores, C. Gómez-Reino, J. Liñares, Opt. Eng. 28, 1168 (1989).

D. W. Hewak, J. W. Y. Lit, Appl. Opt. 28, 4190 (1989).
[CrossRef] [PubMed]

1988 (2)

R. V. Ramaswamy, R. Srivastava, J. Lightwave Technol. 6, 984 (1988).
[CrossRef]

N. Takato, K. Jinguyi, M. Yasu, H. Toba, M. Kawachi, J. Lightwave Technol. 6, 1003 (1988).
[CrossRef]

1987 (2)

1954 (1)

A. Fletcher, T. Murphy, A. Young, Proc. R. Soc. London Ser. A 233, 216 (1954).

Acosta, E.

E. Acosta, J. R. Flores, C. Gómez-Reino, J. Liñares, Opt. Eng. 28, 1168 (1989).

Ahkiewicz, A.

Bao, C.

C. Bao, C. Gómez-Reino, Opt. Quantum Electron. 27, 897 (1995).
[CrossRef]

Bertoline, D.

Fletcher, A.

A. Fletcher, T. Murphy, A. Young, Proc. R. Soc. London Ser. A 233, 216 (1954).

Flores, J. R.

E. Acosta, J. R. Flores, C. Gómez-Reino, J. Liñares, Opt. Eng. 28, 1168 (1989).

Ghatak, A. K.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977).

Gómez-Reino, C.

C. Bao, C. Gómez-Reino, Opt. Quantum Electron. 27, 897 (1995).
[CrossRef]

J. Liñares, C. Gómez-Reino, Appl. Opt. 33, 3427 (1994).
[CrossRef]

C. Gómez-Reino, Int. J. Optoelectron. 7, 607 (1992).

E. Acosta, J. R. Flores, C. Gómez-Reino, J. Liñares, Opt. Eng. 28, 1168 (1989).

C. Gómez-Reino, J. Liñares, J. Opt. Soc. Am. A 4, 1337 (1987).
[CrossRef]

J. Liñares, C. Gómez-Reino, in Proceedings of the Tenth Topical Meeting on Gradient-Index Optical Systems (University of Santiago de Compostela, Galicia, Spain, 1992), p. 9.

Hewak, D. W.

Jinguyi, K.

N. Takato, K. Jinguyi, M. Yasu, H. Toba, M. Kawachi, J. Lightwave Technol. 6, 1003 (1988).
[CrossRef]

Kawachi, M.

N. Takato, K. Jinguyi, M. Yasu, H. Toba, M. Kawachi, J. Lightwave Technol. 6, 1003 (1988).
[CrossRef]

Liñares, J.

J. Liñares, C. Gómez-Reino, Appl. Opt. 33, 3427 (1994).
[CrossRef]

E. Acosta, J. R. Flores, C. Gómez-Reino, J. Liñares, Opt. Eng. 28, 1168 (1989).

C. Gómez-Reino, J. Liñares, J. Opt. Soc. Am. A 4, 1337 (1987).
[CrossRef]

J. Liñares, C. Gómez-Reino, in Proceedings of the Tenth Topical Meeting on Gradient-Index Optical Systems (University of Santiago de Compostela, Galicia, Spain, 1992), p. 9.

Lit, J. W. Y.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Chap. 4, p. 282.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 5 and 8.

Murphy, T.

A. Fletcher, T. Murphy, A. Young, Proc. R. Soc. London Ser. A 233, 216 (1954).

Pask, C.

Ramaswamy, R. V.

R. V. Ramaswamy, R. Srivastava, J. Lightwave Technol. 6, 984 (1988).
[CrossRef]

Sodha, M. S.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977).

Srivastava, R.

R. V. Ramaswamy, R. Srivastava, J. Lightwave Technol. 6, 984 (1988).
[CrossRef]

Takato, N.

N. Takato, K. Jinguyi, M. Yasu, H. Toba, M. Kawachi, J. Lightwave Technol. 6, 1003 (1988).
[CrossRef]

Toba, H.

N. Takato, K. Jinguyi, M. Yasu, H. Toba, M. Kawachi, J. Lightwave Technol. 6, 1003 (1988).
[CrossRef]

Yasu, M.

N. Takato, K. Jinguyi, M. Yasu, H. Toba, M. Kawachi, J. Lightwave Technol. 6, 1003 (1988).
[CrossRef]

Young, A.

A. Fletcher, T. Murphy, A. Young, Proc. R. Soc. London Ser. A 233, 216 (1954).

Appl. Opt. (3)

Int. J. Optoelectron. (1)

C. Gómez-Reino, Int. J. Optoelectron. 7, 607 (1992).

J. Lightwave Technol. (2)

R. V. Ramaswamy, R. Srivastava, J. Lightwave Technol. 6, 984 (1988).
[CrossRef]

N. Takato, K. Jinguyi, M. Yasu, H. Toba, M. Kawachi, J. Lightwave Technol. 6, 1003 (1988).
[CrossRef]

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

Opt. Eng. (1)

E. Acosta, J. R. Flores, C. Gómez-Reino, J. Liñares, Opt. Eng. 28, 1168 (1989).

Opt. Quantum Electron. (1)

C. Bao, C. Gómez-Reino, Opt. Quantum Electron. 27, 897 (1995).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

A. Fletcher, T. Murphy, A. Young, Proc. R. Soc. London Ser. A 233, 216 (1954).

Other (4)

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 5 and 8.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Chap. 4, p. 282.

J. Liñares, C. Gómez-Reino, in Proceedings of the Tenth Topical Meeting on Gradient-Index Optical Systems (University of Santiago de Compostela, Galicia, Spain, 1992), p. 9.

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Figures (5)

Fig. 1
Fig. 1

Geometry, coordinate system, and coupling conditions for HS planar waveguides.

Fig. 2
Fig. 2

Beam-size contraction by butt joining of HS optical waveguides. Ray-tracing calculations have been made for n01 = n02 and α2 = 2α1.

Fig. 3
Fig. 3

Variation of the effective input aperture versus n02. Calculations have been made for n01 = 1.6, α1 = 0.1 mm−1, and α2 = 2α1.

Fig. 4
Fig. 4

Variation of the input and output beam aperture relationship versus n02 for different highs of the input marginal ray. Calculations have been made for n01 = 1.5, α1 = 0.1 mm−1, and α2 = 2α1.

Fig. 5
Fig. 5

Block diagram for obtaining focuser and collimator devices by two coupled HS waveguides. For instance, coupling of two HS waveguides with π/1 and π/α2 lengths can operate as a collimator for point source illumination or as a focuser for collimated beam illumination (dashed lines).

Equations (23)

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n ( x ) = n 0 sech ( α x ) ,
u = sinh ( α x ) .
( u u ˙ ) = [ H 2 ( z ) H 1 ( z ) H ˙ 2 ( z ) H ˙ 1 ( z ) ] ( u 0 u ˙ 0 ) ,
H 1 ( z ) = sin ( α z ) / α , H 2 ( z ) = cos ( α z ) .
x i ( z ) = 1 α i sinh 1 ( u i ) ,
x ˙ i ( z ) = u ˙ i α i cosh ( α i x i ) , i = 1 , 2 ,
u 02 = sinh { α 2 α 1 sinh 1 [ u 1 ( d 1 ) ] } ,
u ˙ 02 = l 01 α 2 cosh [ α 2 x 1 ( d 1 ) ] l 02 α 1 cosh [ α 1 x 1 ( d 1 ) ] [ u ˙ 01 H ˙ 1 1 ( d 1 ) + u 01 H ˙ 2 1 ( d 1 ) ] ,
l 01 x ˙ 1 ( z ) = l 02 x ˙ 2 ( z )
u 2 = u ˙ 02 H 1 2 ( z ) + u 02 H 2 2 ( z ) ,
u ˙ 2 = u ˙ 02 H ˙ 1 2 ( z ) + u 02 H ˙ 2 2 ( z ) ,
u 2 ( d 2 ) = ( 1 ) p 1 + p 2 + 1 l 01 l 02 u 01 ,
u ˙ 2 ( d 2 ) = 0 ,
u 1 ( z = 0 ) = u 01 , u ˙ 1 ( z = 0 ) = u ˙ 01 = 0
A 0 A i = α 1 α 2 sinh 1 ( l 01 l 02 u 0 M 1 ) sinh 1 ( u 0 M 1 ) ,
l 02 2 l 01 2 = ( n 02 2 n 01 2 1 ) cosh 2 ( α 1 x 0 M 1 ) + 1 ,
A 0 / A i = α 1 / α 2 ,
x 0 M 1 max = 1 α 1 cosh 1 [ n 01 ( n 01 2 n 02 2 ) 1 / 2 ] ,
n 1 2 ( x 0 M 1 max ) = n 01 2 n 02 2 .
A i eff = 2 x 0 M 1 max ( A i eff < A wg ) .
A 0 A i eff = α 1 α 2 sinh 1 ( l 01 l 02 u 0 M 1 max ) sinh 1 ( u 0 M 1 max ) ,
u 0 M 1 max = sinh ( α x 0 M 1 max ) .
A i eff = A i = A wg ,

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