Abstract

Using Gaussian Constants, a self-contained paraxial theory for rotationally distributed-index media has been formulated. The differential equations governing a ray transfer in inhomogeneous media have been derived together with an analytical solution and a numerical calculation method. The Lagrange-Helmholtz invariant for inhomogeneous media is presented, and a homogeneous model equivalent to ray transfer in inhomogeneous media is proposed. A numerical example is added to confirm the analysis.

© 1984 Optical Society of America

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Corrections

Kazuo Tanaka, "Paraxial theory of rotationally distributed-index media by means of Gaussian Constants: erratum," Appl. Opt. 23, 3265-3265 (1984)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-23-19-3265

References

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  1. L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, Appl. Opt. 21, 993 (1982).
    [CrossRef] [PubMed]
  2. M. Toyama, M. Takami, Appl. Opt. 21, 1013 (1982).
    [CrossRef] [PubMed]
  3. M. Oikawa, K. Iga, Appl. Opt. 21, 1052 (1982).
    [CrossRef] [PubMed]
  4. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), Sec. 209.
  5. P. J. Sands, J. Opt. Soc. Am. 61, 879 (1971).
    [CrossRef]
  6. D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
    [CrossRef]
  7. See, for examaple, E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), p. 32.
  8. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 128.
  9. See, for example, K. Yano, S. Ishihara, Introduction to Analysis (Shokabo, Tokyo, 1968), p. 65.
  10. See, for example, Mathematical Society of Japan, Ed., Encyclopedic Dictionary of Mathematics (Iwanami, Tokyo, 1972), p.618.
  11. See, for example, “Hitac Mathematical Subprogram Library,” 8080-7-005-01 (Hitachi, Yokohama, 1977), p. 54.
  12. K. Tanaka, Optik (Stuttgart) 58, 351 (1981).
  13. K. Tanaka, Optik (Stuttgart) 62, 211 (1982).
  14. K. Tanaka, Optik (Stuttgart) 64, 13 (1983).
  15. M. Herzberger, J. Opt. Soc. Am. 33, 651 (1943).
    [CrossRef]
  16. M. Herzberger, J. Opt. Soc. Am. 42, 637 (1952).
    [CrossRef]
  17. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 457.
  18. G. A. Deschamps, P. E. Mast, Quasi-Optics (Polytechnic Press, Brooklyn, 1964), p. 379.
  19. See, for example, The Institute of Electric Engineers of Japan, Ed., The Theory of Electrical Networks (Gakkensha, Tokyo, 1970), p. 183.
  20. K. Tanaka, Optik (Stuttgart) 64, 89 (1983).

1983 (2)

K. Tanaka, Optik (Stuttgart) 64, 13 (1983).

K. Tanaka, Optik (Stuttgart) 64, 89 (1983).

1982 (4)

1981 (1)

K. Tanaka, Optik (Stuttgart) 58, 351 (1981).

1971 (2)

1952 (1)

1943 (1)

Atkinson, L. G.

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 128.

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), Sec. 209.

Deschamps, G. A.

G. A. Deschamps, P. E. Mast, Quasi-Optics (Polytechnic Press, Brooklyn, 1964), p. 379.

Herzberger, M.

M. Herzberger, J. Opt. Soc. Am. 42, 637 (1952).
[CrossRef]

M. Herzberger, J. Opt. Soc. Am. 33, 651 (1943).
[CrossRef]

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 457.

Houde-Walter, S. N.

Iga, K.

Ishihara, S.

See, for example, K. Yano, S. Ishihara, Introduction to Analysis (Shokabo, Tokyo, 1968), p. 65.

Mast, P. E.

G. A. Deschamps, P. E. Mast, Quasi-Optics (Polytechnic Press, Brooklyn, 1964), p. 379.

Moore, D. T.

O’Neill, E. L.

See, for examaple, E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), p. 32.

Oikawa, M.

Ryan, D. P.

Sands, P. J.

Stagaman, J. M.

Takami, M.

Tanaka, K.

K. Tanaka, Optik (Stuttgart) 64, 89 (1983).

K. Tanaka, Optik (Stuttgart) 64, 13 (1983).

K. Tanaka, Optik (Stuttgart) 62, 211 (1982).

K. Tanaka, Optik (Stuttgart) 58, 351 (1981).

Toyama, M.

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 128.

Yano, K.

See, for example, K. Yano, S. Ishihara, Introduction to Analysis (Shokabo, Tokyo, 1968), p. 65.

Appl. Opt. (3)

J. Opt. Soc. Am. (4)

Optik (Stuttgart) (4)

K. Tanaka, Optik (Stuttgart) 64, 89 (1983).

K. Tanaka, Optik (Stuttgart) 58, 351 (1981).

K. Tanaka, Optik (Stuttgart) 62, 211 (1982).

K. Tanaka, Optik (Stuttgart) 64, 13 (1983).

Other (9)

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), Sec. 209.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 457.

G. A. Deschamps, P. E. Mast, Quasi-Optics (Polytechnic Press, Brooklyn, 1964), p. 379.

See, for example, The Institute of Electric Engineers of Japan, Ed., The Theory of Electrical Networks (Gakkensha, Tokyo, 1970), p. 183.

See, for examaple, E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), p. 32.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 128.

See, for example, K. Yano, S. Ishihara, Introduction to Analysis (Shokabo, Tokyo, 1968), p. 65.

See, for example, Mathematical Society of Japan, Ed., Encyclopedic Dictionary of Mathematics (Iwanami, Tokyo, 1972), p.618.

See, for example, “Hitac Mathematical Subprogram Library,” 8080-7-005-01 (Hitachi, Yokohama, 1977), p. 54.

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Figures (4)

Fig. 1
Fig. 1

Ray transfer in a rotationally distributed index medium.

Fig. 2
Fig. 2

Equivalent models in the realm of paraxial theory: (a) inhomogeneous configuration; (b) homogeneous configuration. The powers of the i*th and (i + 2)*th refractive element in (b) are, respectively, equal to the powers of the ith and (i + 1)th surface in (a); the power of the (i + 1)*th element in (b) is introduced due to the bending properties caused by inhomogeneity.

Fig. 3
Fig. 3

Inhomogeneously configured doublet as a numerical example: (a) actual configuration; (b) equivalent homogeneous configuration. F′ is the focal point in the image space. P and P′ are the principal points in the object and image space, respectively.

Fig. 4
Fig. 4

Index distribution of the example shown in Fig. 3.

Equations (73)

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n ( x , h ) = n 0 ( x ) + n 1 ( x ) h 2 + n 2 ( x ) h 4 + n 3 ( x ) h 6 ,
2 n 1 ( x ) h ( x ) - d d x [ n 0 ( x ) d d x h ( x ) ] = 0 ,
a ( x ) = n 0 ( x ) u ( x ) = - n 0 ( x ) d d x h ( x ) ,
[ h ( x ) a ( x ) ] = [ A ( x ) B ( x ) C ( x ) D ( x ) ] [ h ( 0 ) a ( 0 ) ] ,
A ( 0 ) = 1 , B ( 0 ) = 0 , C ( 0 ) = 0 , D ( 0 ) = 1.
d 2 d x 2 A ( x ) + 1 n 0 ( x ) [ d d x n 0 ( x ) ] d d x A ( x ) - 2 n 1 ( x ) n 0 ( x ) A ( x ) = 0 ,
A ( 0 ) = 1 , d d x A ( x ) | x = 0 = 0 ,
d 2 d x 2 B ( x ) + 1 n 0 ( x ) [ d d x n 0 ( x ) ] d d x B ( x ) - 2 n 1 ( x ) n 0 ( x ) B ( x ) = 0 ,
B ( 0 ) = 0 , d d x B ( x ) | x = 0 = - 1 n 0 ( 0 ) .
d 2 d x 2 X ( x ) + 1 n 0 ( x ) [ d d x n 0 ( x ) ] d d x X ( x ) - 2 n 1 ( x ) n 0 ( x ) X ( x ) = 0.
C ( x ) + n 0 ( x ) d d x A ( x ) = 0 ,
D ( x ) + n 0 ( x ) d d x B ( x ) = 0 ,
d d x C ( x ) + 2 n 1 ( x ) A ( x ) = 0 ,
C ( 0 ) = 0 ,
d d x D ( x ) + 2 n 1 ( x ) B ( x ) = 0 ,
D ( 0 ) = 1.
Y i ( x ) + n 0 ( x ) d d x X i ( x ) = 0 ,
d d x Y i ( x ) + 2 n 1 ( x ) X i ( x ) = 0.
n ( x , h ) = n 00 ( α x + β ) 2 + n 10 h 2 .
2 n 10 α 2 n 00 > - 0.25 ,
A ( x ) = ( t - s ) - 1 [ t β - s ( α x + β ) s - s β - t ( α x + β ) t ] , B ( x ) = n 00 - 1 ( t - s ) - 1 ( α β ) - 1 [ β - s ( α x + β ) s - β - t ( α x + β ) t ] , C ( x ) = - ( t - s ) - 1 s t α n 0 ( x ) [ β - s ( α x + β ) s - 1 - β - t ( α x + β ) t - 1 ] , D ( x ) = - n 00 - 1 ( t - s ) - 1 β - 1 n 0 ( x ) × [ β - s s ( α x + β ) s - 1 - β - t t ( α x + β ) t - 1 ] ,
y 2 + y - ( 2 n 10 ) / ( α 2 n 00 ) = 0.
2 n 10 α 2 n 00 = - 0.25
A ( x ) = 0.5 β 0.5 [ 2 - log β + log ( α x + β ) ] ( α x + β ) - 0.5 , B ( x ) = n 00 - 1 α - 1 β - 0.5 [ log β - log ( α x + β ) ] ( α x + β ) - 0.5 , C ( x ) = - 0.25 α β 0.5 n 0 ( x ) [ log β - log ( α x + β ) ] ( α x + β ) - 1.5 , D ( x ) = - 0.5 n 00 - 1 β - 0.5 n 0 ( x ) × [ - 2 - log β + log ( α x + β ) ] ( α x + β ) - 1.5 .
2 n 10 α 2 n 00 < - 0.25 ,
A ( x ) = β 0.5 t - 1 ( α x + β ) - 0.5 { [ t sin ( t log β ) + 0.5 cos ( t log β ) ] × sin [ t log ( α x + β ) ] + { - 0.5 sin ( t log β ) + t cos ( t log β ) ] × cos [ t log ( α x + β ) ] } , B ( x ) = n 00 1 α - 1 β - 0.5 t - 1 ( α x + β ) - 0.5 { - cos ( t log β ) × sin [ t log ( α x + β ) ] + sin ( t log β ) cos [ t log ( α x + β ) ] } , C ( x ) = - α β 0.5 ( 0.25 t - 1 + t ) n 0 ( x ) ( α x + β ) - 1.5 × { - cos ( t log β ) sin [ t log ( α x + β ) ] + sin ( t log β ) cos [ t log ( α x + β ) ] } , D ( x ) = - n 00 - 1 β - 0.5 t - 1 n 0 ( x ) ( α x + β ) - 1.5 { [ 0.5 cos ( t log β ) - t sin ( t log β ) ] sin [ t log ( α x + β ) ] - [ t cos ( t log β ) + 0.5 sin ( t log β ) ] cos [ t log ( α x + β ) ] } ,
d Z 3 d x = - 1 n 0 ( x ) [ d d x n 0 ( x ) ] Z 3 + 2 n 1 ( x ) n 0 ( x ) Z 2 , d Z 2 d x = Z 3 , d Z 1 d x = - 2 n 1 ( x ) Z 2 .
Z 3 ( 0 ) = 0 , Z 2 ( 0 ) = 1 , Z 1 ( 0 ) = 0 ,
Z 2 ( x ) = A ( x ) , Z 1 ( x ) = C ( x ) .
Z 3 ( 0 ) = - 1 n 0 ( 0 ) , Z 2 ( 0 ) = 0 , Z 1 ( 0 ) = 1 ,
Z 2 ( x ) = B ( x ) , Z 1 ( x ) = D ( x ) .
a ( x ) h ¯ ( x ) - a ¯ ( x ) h ( x ) = a ( 0 ) h ¯ ( 0 ) - a ¯ ( 0 ) h ( 0 )
a ( x ) h ¯ ( x ) - a ¯ ( x ) h ( x ) a ( 0 ) h ¯ ( 0 ) - a ¯ ( 0 ) h ( 0 ) = A ( x ) D ( x ) - B ( x ) C ( x ) .
W ( x ) = X 1 d d x X 2 - X 2 d d x X 1 .
W ( x ) = A ( x ) d d x B ( x ) - B ( x ) d d x A ( x ) = - 1 n 0 ( x ) [ A ( x ) D ( x ) - B ( x ) C ( x ) ] .
W ( x ) = W ( 0 ) exp { - 0 x 1 n 0 ( x ) [ d d x n 0 ( x ) ] d x } = - 1 n 0 ( x ) .
A ( x ) D ( x ) - B ( x ) C ( x ) = 1.
0 x n 1 ( x ) A ( x ) B ( x ) d x ,
0 x 1 n 0 ( x ) C ( x ) D ( x ) d x .
n i ( x , h ) = n 0 , i ( x ) + n 1 , i ( x ) h 2 + n 2 , i ( x ) h 4 + ,
( h i a i ) = ( 1 0 ϕ i 1 ) ( h i a i ) ,
ϕ i = [ n 0 , i ( 0 ) - n 0 , i - 1 ( d i - 1 ) ] / r i .
( h i + 1 a i + 1 ) = [ A i ( d i ) B i ( d i ) C i ( d i ) D i ( d i ) ] ( h i a i ) ,
( h i + 2 * a i + 2 * ) = ( A i + 1 i + 2 * B i i + 2 * C i + 1 i + 1 * D i i + 1 * ) ( h i * a i * ) ,
A i + 1 i + 2 * = [ ϕ i + 1 * , - e i + 1 * ] , B i i + 2 * = [ - e i * , ϕ i + 1 * , - e i + 1 * ] , C i + 1 i + 1 * = [ ϕ i + 1 * ] , D i i + 1 * = [ - e i * , ϕ i + 1 * ] .
ϕ i + 1 * = C i ( d i ) ,
e i * = [ 1 - D i ( d i ) ] / C i ( d i ) ,
e i + 1 * = [ 1 - A i ( d i ) ] / C i ( d i ) .
[ A i ( d i ) B i ( d i ) C i ( d i ) D i ( d i ) ] = ( 1 - e i + 1 * 0 1 ) ( 1 0 ϕ i + 1 * 1 ) ( 1 - e i * 0 1 ) .
ϕ i + 1 * = C i ( d i ) = 0 ,
e i * = - B i ( d i ) = 0 d 1 1 n 0 ( x ) d x ,
e i + 1 * = 0.
r 1 = 90 , r 2 = 40 , r 3 = - 100 , d 1 = 20 , d 2 = 5 , n 0 = 1.00 , n 1 ( x , h ) = 1.55 × ( 0.0025 x + 1 ) 2 - 0.00015 h 2 , n 2 ( x ) = 1.75 / ( - 0.0054 x + 1 ) , n 3 = 1.33.
A 1 ( 20 ) = 0.96395469 , B 1 ( 20 ) = - 12.13836366 , C 1 ( 20 ) = 0.00592656 , D 1 ( 20 ) = 0.96276445.
A 1 ( 20 ) D 1 ( 20 ) - B 1 ( 20 ) C 1 ( 20 ) = 1.
A 2 ( 5 ) = 1.0 , B 2 ( 5 ) = - 2.81857143 , C 2 ( 5 ) = 0.0 , D 2 ( 5 ) = 1.0.
A 2 ( 5 ) D 2 ( 5 ) - B 2 ( 5 ) C 2 ( 5 ) = 1.
ϕ 1 * = [ n 1 ( 0 , 0 ) - n 0 ] / r 1 = 0.00611111 , ϕ 3 * = [ n 2 ( 0 ) - n 1 ( 20 , 0 ) ] / r 2 = - 0.00102813 , ϕ 5 * = [ n 3 - n 2 ( 5 ) ] / r 3 = 0.00468561.
ϕ 2 * = C 1 ( 20 ) = 0.00592656.
e 1 * = [ 1 - D 1 ( 20 ) ] / C 1 ( 20 ) = 6.28283034 , e 2 * = [ 1 - A 1 ( 20 ) ] / C 1 ( 20 ) = 6.08199990.
ϕ 4 * = C 2 ( 5 ) = 0.0 ,
e 3 * = - B 2 ( 5 ) = 2.81857143 , e 4 * = 0.0.
A 1 5 * = [ ϕ 1 * , - e 1 * , ϕ 2 * , - e 2 * , ϕ 3 * , - e 3 * , ϕ 4 * , - e 4 * ] = 0.85906657 , B 1 5 * = [ - e 1 * , ϕ 2 * , - e 2 * , ϕ 3 * , - e 3 * , Q 4 * , - e 4 * ] = - 14.88715912 , C 1 5 * = [ ϕ 1 * , - e 1 * , ϕ 2 * , - e 2 * , ϕ 3 * , - e 3 * , Q 4 * , - e 4 * , ϕ 5 * ] = 0.01492057 , D 1 5 * = [ - e 1 * , ϕ 2 * , - e 2 * , ϕ 3 * , - e 3 * , ϕ 4 * , - e 4 * , ϕ 5 * ] = 0.90548876.
A 1 5 * D 1 5 * - B 1 5 * C 1 5 * = 1.
s P = ( 1 - D 1 5 * ) / C 1 5 * = 6.33429201 ,
s P = ( A 1 5 * - 1 ) / C 1 5 * = - 9.44558060.
S F = A 1 5 * / C 1 5 * = 57.57599623.
f = 1 / C 1 5 * = 67.02157683.
I 1 0 x n 1 ( x ) A ( x ) B ( x ) d x = - ½ 0 x 3333 b k d d x C ( x ) ] B ( x ) d x .
I 1 = - ½ [ C ( x ) B ( x ) | 0 x + 0 x C ( x ) D ( x ) n 0 ( x ) d x ] .
I 2 0 x n 1 ( x ) A ( x ) B ( x ) d x = - ½ 0 x [ d d x D ( x ) ] A ( x ) d x .
I 2 = - ½ [ A ( x ) D ( x ) | 0 x + 0 x D ( x ) C ( x ) n 0 ( x ) d x ] .
A ( x ) D ( x ) - B ( x ) C ( x ) = A ( 0 ) D ( 0 ) - B ( 0 ) C ( 0 ) = 1.

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