Abstract

A new method for tracing rays through graded-index media is presented. The method essentially consists of transforming the ray equation into a convenient form and solving the resulting equation using a standard numerical technique. A detailed comparison of this method with existing methods has also been made, and it is shown that for obtaining a desired accuracy this method requires much less computational effort.

© 1982 Optical Society of America

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References

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  1. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 4 and 5.
  2. A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), Chap. 1.
    [CrossRef]
  3. H. A. Buchdahl, J. Opt. Soc. Am. 63, 46 (1973).
    [CrossRef]
  4. E. W. Marchand, Appl. Opt. 11, 1104 (1972).
    [CrossRef] [PubMed]
  5. E. W. Marchand, J. Opt. Soc. Am. 60, 1 (1970).
    [CrossRef]
  6. L. Montagnino, J. Opt. Soc. Am. 58, 1667 (1968).
    [CrossRef]
  7. D. T. Moore, J. Opt. Soc. Am. 65, 451 (1975).
    [CrossRef]
  8. W. Streifer, K. B. Paxton, Appl. Opt. 10, 769 (1971).
    [CrossRef] [PubMed]
  9. K. B. Paxton, W. Streifer, Appl. Opt. 10, 1164 (1971).
    [CrossRef] [PubMed]
  10. J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U. P., Baltimore, 1966), Chap. 13, Article 116.
  11. E. W. Marchand, Appl. Opt. 19, 1044 (1980).
    [CrossRef] [PubMed]
  12. E. W. Marchand, J. Opt. Soc. Am. 66, 1326 (1976).
    [CrossRef]

1980

1976

1975

1973

1972

1971

1970

1968

Buchdahl, H. A.

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), Chap. 1.
[CrossRef]

Marchand, E. W.

Montagnino, L.

Moore, D. T.

Paxton, K. B.

Scarborough, J. B.

J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U. P., Baltimore, 1966), Chap. 13, Article 116.

Streifer, W.

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), Chap. 1.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Position error at z = 2 of a ray launched at an angle of 17°27′ with the z axis at z = 0 in a medium characterized by Eq. (15) with α = 0.2 as a function of extrapolation distance Δt; the solid curve corresponds to the present method and the dashed curve to Montagnino’s method.6

Fig. 2
Fig. 2

Spherical aberrations S and meridional coma Cm of a radial singlet characterized by Eqs. (17), (18), and Table II as a function of N2; N2 = 0.0 corresponds to Marchand’s design.11

Tables (5)

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Table I Fractional Error in Position of Typical Skew Rays in Medium Characterized by Eq. (14)

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Table II Radial Singlet Data11

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Table III Aberration of the Radial Singlet

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Table IV Wood Lens Data12

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Table V Aberration of the Wood Lens

Equations (20)

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d d s [ n ( r ) d r d s ] = n ( r ) ,
t = d s n ;             d t = d s n .
d 2 r d t 2 = n n ,
d 2 r d t 2 = ½ n 2 .
T d r d t .
T d r d t n d r d s i ^ n d x d s + j ^ n d y d s + k ^ n d z d s i ^ n cos α + j ^ n cos β + k ^ n cos γ ,
T a = T b + u N ^ ,
u = n a 2 - n b 2 + ϑ - ϑ .
R ( x y z ) ; T ( T x T y T z ) n ( d x / d s d y / d s d z / d s ) ,
D n ( n / x n / y n / z ) ½ ( n 2 / x n 2 / y n 2 / z ) ,
d 2 R d t 2 = D ( R ) .
R n + 1 = R n + Δ t [ T n + 1 6 ( A + 2 B ) ] ,
T n + 1 = T n + 1 6 ( A + 4 B + C ) ,
A = Δ t D ( R n ) , B = Δ tD ( R n + Δ t 2 T n + 1 8 Δ t A ) , C = Δ t D ( R n + Δ t T n + ½ Δ t B ) ,
n 2 = 2.5 - 0.1 ( x 2 + y 2 ) .
n ( z ) = 1.55 - α z
n = 1.55 - 0.005 ( x 2 + y 2 ) + 0.02 z - 0.005 z 2 ,
n 2 = N 0 + N 1 r 2 + N 2 r 4
N 0 = 1.5215 2             N 1 = - 0.0109548.
n = N 0 + N 1 r 2 + N 2 r 4 ,

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