Abstract

Equivalence properties of lenslike media with parabolic index profiles are analyzed on the basis of geometrical optics. By equivalent media we mean here those with the same ray-transfer matrix representation. An equivalence theorem is presented that provides a simple method of equivalent transformation of parabolic media. The practical importance of the equivalent transformation is that it leads to flexibility in the design of distributed optical structures, such as gas lenses and inhomogeneous optical fibers.

© 1971 Optical Society of America

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References

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  1. J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).
  2. D. Marcuse, IEEE Trans. MTT–13734 (1965).
  3. P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
    [CrossRef]
  4. K. Nishida, Y. Nannichi, T. Uchida, I. Kitano, Proc. IEEE (Lett.) 58, 790 (1970).
    [CrossRef]
  5. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  6. D. W. Berreman, Bell Syst. Tech. J. 43, 1469 (1964).
  7. H. G. Unger, Arch. Elek. Übertrag. 19, 189 (1965).
  8. H. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  9. Y. Suematsu, T. Kitano, IECE Natl. Conv. Rec. (Japan), No. 535 (1967).
  10. S. E. Miller, Bell Syst. Tech. J. 43, 1741 (1964).

1970

K. Nishida, Y. Nannichi, T. Uchida, I. Kitano, Proc. IEEE (Lett.) 58, 790 (1970).
[CrossRef]

1967

Y. Suematsu, T. Kitano, IECE Natl. Conv. Rec. (Japan), No. 535 (1967).

1966

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

1965

D. Marcuse, IEEE Trans. MTT–13734 (1965).

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

H. G. Unger, Arch. Elek. Übertrag. 19, 189 (1965).

H. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

1964

J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).

S. E. Miller, Bell Syst. Tech. J. 43, 1741 (1964).

D. W. Berreman, Bell Syst. Tech. J. 43, 1469 (1964).

Berreman, D. W.

D. W. Berreman, Bell Syst. Tech. J. 43, 1469 (1964).

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).

Kitano, I.

K. Nishida, Y. Nannichi, T. Uchida, I. Kitano, Proc. IEEE (Lett.) 58, 790 (1970).
[CrossRef]

Kitano, T.

Y. Suematsu, T. Kitano, IECE Natl. Conv. Rec. (Japan), No. 535 (1967).

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

H. Kogelnik, Appl. Opt. 4, 1562 (1965).
[CrossRef]

J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Marcuse, D.

D. Marcuse, IEEE Trans. MTT–13734 (1965).

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 43, 1741 (1964).

Nannichi, Y.

K. Nishida, Y. Nannichi, T. Uchida, I. Kitano, Proc. IEEE (Lett.) 58, 790 (1970).
[CrossRef]

Nishida, K.

K. Nishida, Y. Nannichi, T. Uchida, I. Kitano, Proc. IEEE (Lett.) 58, 790 (1970).
[CrossRef]

Suematsu, Y.

Y. Suematsu, T. Kitano, IECE Natl. Conv. Rec. (Japan), No. 535 (1967).

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Uchida, T.

K. Nishida, Y. Nannichi, T. Uchida, I. Kitano, Proc. IEEE (Lett.) 58, 790 (1970).
[CrossRef]

Unger, H. G.

H. G. Unger, Arch. Elek. Übertrag. 19, 189 (1965).

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Appl. Opt.

Arch. Elek. Übertrag.

H. G. Unger, Arch. Elek. Übertrag. 19, 189 (1965).

Bell Syst. Tech. J.

D. W. Berreman, Bell Syst. Tech. J. 43, 1469 (1964).

J. P. Gordon, H. Kogelnik, Bell Syst. Tech. J. 43, 2873 (1964).

S. E. Miller, Bell Syst. Tech. J. 43, 1741 (1964).

IECE Natl. Conv. Rec. (Japan)

Y. Suematsu, T. Kitano, IECE Natl. Conv. Rec. (Japan), No. 535 (1967).

IEEE Trans.

D. Marcuse, IEEE Trans. MTT–13734 (1965).

Proc. IEEE

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Proc. IEEE (Lett.)

K. Nishida, Y. Nannichi, T. Uchida, I. Kitano, Proc. IEEE (Lett.) 58, 790 (1970).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Refractive-index distributions of exponentially tapered parabolic medium and its equivalents: (a) n0 (z) = const (original), (b) n2 (z) = const, (c) n2 (z)/n0 (z) = const.

Fig. 2
Fig. 2

Comparison of equivalent lengths. (The case of Fig. 1 is indicated.)

Equations (35)

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n ( x , z ) = n 0 ( z ) - ( 1 2 ) n 2 ( z ) · x 2 ,
n 0 ( z ) n 2 ( z ) · x 2 0.
N ( x , Z ) = N 0 ( Z ) - ( 1 2 ) N 2 ( Z ) · x 2 ,
[ 1 Δ z i / n 0 ( z i ) - n 2 ( z i ) Δ z i 1 ] ,
[ 1 Δ Z i / N 0 ( Z i ) - N 2 ( Z i ) Δ Z i 1 ] .
Δ z i / n 0 ( z i ) = Δ Z i / N 0 ( Z i ) ,
n 2 ( z i ) Δ z i = N 2 ( Z i ) Δ Z i .
[ n 2 ( z i ) / n 0 ( z i ) ] 1 2 Δ z i = [ N 2 ( Z i ) / N 0 ( Z i ) ] 1 2 Δ Z i .
N 0 ( Z ) / n 0 ( z ) = n 2 ( z ) / N 2 ( Z ) = d Z / d z .
Z = Z ( z ) ,
z = z ( Z ) .
[ n 0 ( z ) , n 2 ( z ) ] ;             ( 0 z L )
[ N 0 ( Z ) , N 2 ( Z ) ] ;             ( Z 0 Z Z L ) ,
N 0 ( Z ) = n 0 [ z ( Z ) ] · d z ( Z ) / d Z ,
N 2 ( Z ) = n 2 [ z ( Z ) ] · d z ( Z ) / d Z ,
( Z 0 , Z L ) = [ Z ( 0 ) , Z ( L ) ] .
F [ N 0 ( Z ) , N 2 ( Z ) ] = f ( Z ) ,
F [ n 0 ( z ) · d Z / d z , n 2 ( z ) / d Z / d z ] = f ( Z ) ,
G [ n 0 ( z ) , n 2 ( z ) ] - 1 / m d z = f ( Z ) - 1 / m d Z .
Z = N 00 0 z d z / n 0 ( z ) ,
Z = ( 1 / N 02 ) 0 z n 2 ( z ) d z ,
Z = ( N 00 / N 02 ) 1 2 0 z [ n 2 ( z ) / n 0 ( z ) ] 1 2 d z .
J ( p , q ) 0 L [ n 2 ( z ) q / n 0 ( z ) p ] d z ,             ( p + q = 1 )
J ( 1 , 0 ) 0 L d z / n 0 ( z ) ,
J ( 1 2 , 1 2 ) 0 L [ n 2 ( z ) / n 0 ( z ) ] 1 2 d z ,
n ( x , z ) = n 00 - ( 1 2 ) n 02 · x 2 ,             ( 0 z L ) .
N ( x , Z ) = n 00 g ( Z ) - ( 1 2 ) n 02 / g ( Z ) · x 2 ,             ( 0 Z L )
0 L d Z / g ( Z ) = L .
n 1 ( x , z ) = n 01 - ( 1 2 ) n 21 exp ( μ z ) · x 2 ,             ( 0 z L 1 ) ,
n 2 ( x , z ) = n 02 [ 1 + μ ( n 01 / n 02 ) z ] - ( 1 2 ) n 22 · x 2 ,             ( 0 z L 2 ) ,
L 2 = ( n 02 / n 01 ) ( 1 / μ ) [ exp ( μ L 1 ) - 1 ] ,
n 3 ( x , z ) = n 03 [ 1 + ( μ / 2 ) ( n 01 / n 03 ) z ] [ 1 - ( 1 2 ) ( n 23 / n 03 ) · x 2 ] ,             ( 0 z L 3 ) ,
L 3 = ( n 03 / n 01 ) ( 2 / μ ) [ exp ( μ L 1 / 2 ) - 1 ] ,
n 01 n 21 = n 02 n 22 = n 03 n 23 .
n ( x , z ) = n 00 - ( 1 2 ) n 2 ( z ) · x 2 .

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