Abstract

The general form for the ABCD matrix in an arbitrarily tapered quadratic-index waveguide is given in terms of the two independent solutions of the second-order differential ray-tracing equation. A procedure for finding taper functions which admit analytical solutions in terms of known functions is presented with several examples.

© 1986 Optical Society of America

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References

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  1. A. Yariv, Optical Electronics (Holt, Rinehart, & Winston, New York, 1985), Chap. 2.
  2. L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Waveguides: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
    [Crossref]
  3. L. W. Casperson, “Beam Propagation in Tapered Quadratic-Index Waveguides: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
    [Crossref]
  4. S. Yamamoto, T. Makimoto, “On the Ray Transfer Matrix of a Tapered Lenslike Medium,” Proc. IEEE 34, 1254 (1971).
    [Crossref]
  5. S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Appl. Opt. 10, 1160 (1971).
    [Crossref] [PubMed]
  6. J. Matthews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965), Chap. 7.
  7. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1970), p. 329.
  8. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (NBS, Washington, DC, 1964), Chap. 8.

1985 (2)

L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Waveguides: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
[Crossref]

L. W. Casperson, “Beam Propagation in Tapered Quadratic-Index Waveguides: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
[Crossref]

1971 (2)

S. Yamamoto, T. Makimoto, “On the Ray Transfer Matrix of a Tapered Lenslike Medium,” Proc. IEEE 34, 1254 (1971).
[Crossref]

S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Appl. Opt. 10, 1160 (1971).
[Crossref] [PubMed]

Casperson, L. W.

L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Waveguides: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
[Crossref]

L. W. Casperson, “Beam Propagation in Tapered Quadratic-Index Waveguides: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
[Crossref]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1970), p. 329.

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1970), p. 329.

Kirkwood, J. L.

L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Waveguides: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
[Crossref]

Makimoto, T.

S. Yamamoto, T. Makimoto, “On the Ray Transfer Matrix of a Tapered Lenslike Medium,” Proc. IEEE 34, 1254 (1971).
[Crossref]

S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Appl. Opt. 10, 1160 (1971).
[Crossref] [PubMed]

Matthews, J.

J. Matthews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965), Chap. 7.

Walker, R. L.

J. Matthews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965), Chap. 7.

Yamamoto, S.

S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Appl. Opt. 10, 1160 (1971).
[Crossref] [PubMed]

S. Yamamoto, T. Makimoto, “On the Ray Transfer Matrix of a Tapered Lenslike Medium,” Proc. IEEE 34, 1254 (1971).
[Crossref]

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rinehart, & Winston, New York, 1985), Chap. 2.

Appl. Opt. (1)

IEEE/OSA J. Lightwave Technol. (2)

L. W. Casperson, J. L. Kirkwood, “Beam Propagation in Tapered Quadratic Index Waveguides: Numerical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).
[Crossref]

L. W. Casperson, “Beam Propagation in Tapered Quadratic-Index Waveguides: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
[Crossref]

Proc. IEEE (1)

S. Yamamoto, T. Makimoto, “On the Ray Transfer Matrix of a Tapered Lenslike Medium,” Proc. IEEE 34, 1254 (1971).
[Crossref]

Other (4)

A. Yariv, Optical Electronics (Holt, Rinehart, & Winston, New York, 1985), Chap. 2.

J. Matthews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965), Chap. 7.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1970), p. 329.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (NBS, Washington, DC, 1964), Chap. 8.

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Figures (1)

Fig. 1
Fig. 1

Taper Profiles for n = 2 (see text). The solid straight line running from corner to corner is for the simple exponentially varying taper radius a(z) = 25 exp(−z/250) and the other curves are tapers which have the same radius and slope at z = 0 and which admit analytical ABCD matrices as described in Sec. III of the text.

Equations (43)

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n ( r , z ) = n 0 [ 1 - k 2 ( z ) 2 k 0 r 2 ] ,
d 2 r d z 2 + k 2 k 0 r = 0.
r ( z ) = α f ( z ) + β g ( z ) ,
W ( z ) = f ( z ) g ( z ) - g ( z ) f ( z ) = W 0 ,
[ r ( z ) n 0 r ( z ) ] = [ f ( z ) g ( z ) n 0 f ( z ) n 0 g ( z ) ] [ α β ] ,
[ α β ] = [ f 0 g 0 n 0 f n 0 g 0 ] - 1 [ r 0 n 0 r 0 ] = 1 W 0 [ g 0 - g 0 / n 0 - f 0 f 0 / n 0 ] [ r 0 n 0 r 0 ] .
[ r n 0 r ] = [ A B C D ] [ r 0 n 0 r 0 ] ,
[ A B C D ] = 1 W 0 [ f ( z ) g ( z ) n 0 f ( z ) n 0 g ( z ) ] [ g 0 - g 0 / n 0 - f 0 f 0 / n 0 ] .
A = W 0 - 1 [ g 0 f ( z ) - f 0 g ( z ) ] ,
B = W 0 - 1 [ g 0 f ( z ) - f 0 g ( z ) ] / n 0 ,
C = n 0 W 0 - 1 [ g 0 f ( z ) - f 0 g ( z ) ] ,
D = - W 0 - 1 [ g 0 f ( z ) - f 0 g ( z ) ] .
d A d z = g 0 f ( z ) - f 0 g ( z ) = C ( z ) / n 0 .
d B d z = D ( z ) / n 0 ,
d C d z = - n 0 k 2 k 0 A ( z ) ,
d D d z = - n 0 k 2 k 0 B ( z ) .
d 2 F d t 2 + P ( t ) d F d t + Q ( t ) F ( t ) = 0.
f ( z ) = F [ t ( z ) ] ,
d 2 F d t 2 ( d t d z ) 2 + d F d t ( d 2 t d z 2 ) + k 2 k 0 F = 0.
( d 2 t d z 2 ) ( d t d z ) - 2 = P ( t ) ,
k 2 ( z ) k 0 = ( d t d z ) - 2 Q [ t ( z ) ] .
P ( t ) = d d t ln [ R ( t ) ] ,
d t d z = γ R ( t )
t 0 t ( z ) d t R ( t ) = γ z ,
k 2 ( z ) k 0 = γ 2 R 2 [ t ( z ) ] Q [ t ( z ) ] .
d 2 F d t 2 + 1 t d F d t + ( 1 - n 2 t 2 ) F = 0.
t ( z ) = t 0 exp ( γ z ) .
k 2 ( z ) k 0 = γ 2 [ t 2 ( z ) - n 2 ] = γ 2 [ t 0 2 exp ( 2 γ z ) - n 2 ] ,
R ( t ) = t 2 , Q ( t ) = [ 1 - n ( n + 1 ) / t 2 ] , t ( z ) = t 0 ( 1 - γ t 0 z ) - 1 ,
k 2 / k 0 = γ 2 [ t 2 ( z ) - n ( n + 1 ) ] t 2 ( z ) .
R ( t ) = t , Q ( t ) = ( n + 1 / 2 - t / 4 ) / t , t ( z ) = t 0 exp ( γ z ) ,
k 2 ( z ) / k 0 = γ 2 [ n + 1 / 2 - t ( z ) / 4 ] t ( z ) .
R ( t ) = 1 - t 2 , Q ( t ) = n ( n + 1 ) / ( 1 - t 2 ) , t ( z ) = tanh [ γ z + tanh - 1 ( t 0 ) ] ,
k 2 ( z ) / k 0 = γ 2 n ( n + 1 ) / cosh 2 [ γ z + tanh - 1 ( t 0 ) ] .
R ( t ) = t c / 2 ( 1 - t ) a + b + 1 - c / 2 , Q ( t ) = - a b / [ t ( 1 - t ) ] , t t ( z ) d t ( t ) c / 2 ( 1 - t ) a + b + 1 - c / 2 = γ z ,
k 2 ( z ) / k 0 = - a b γ 2 t c - 1 ( 1 - t ) 2 ( a + b ) - c + 1
t ( z ) = t 0 exp ( γ z ) ,
k 2 ( z ) / k 0 = n 2 γ 2 t ( z ) / [ 1 - t ( z ) ] .
t ( z ) = t 0 exp ( γ z ) / [ ( 1 - t 0 ) + t 0 exp ( γ z ) ] ,
k 2 ( z ) / k 0 = ( b - 1 ) b γ 2 t ( z ) [ 1 - t ( z ) ] .
n ( r , z ) = n 0 [ 1 - Δ r 2 / a 2 ( z ) ] ,
k 2 ( z ) / k 0 = 2 Δ / a 2 ( z ) ,
a ( z ) = 25.0 exp ( - z / 250 )

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