Abstract

The problem of ray tracing in inhomogeneous media has been examined in an attempt to find an accurate yet computationally efficient method. Several numerical techniques have been compared. It is shown that Montagnino’s Taylor-series-expansion method [ J. Opt. Soc. Am. 58, 1667 ( 1968)] is superior to Euler’s method, which is a standard numerical method for solving the differential equation. It is also shown that, by a transformation of the independent coordinate from the physical path length to the optical path length, more-accurate optical path calculations may be obtained.

© 1982 Optical Society of America

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References

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  1. L. Montagnino, “Ray tracing in inhomogeneous media,” J. Opt. Soc. Am. 58, 1667–1668 (1968).
    [Crossref]
  2. D. T. Moore, “Ray tracing in gradient-index media,” J. Opt. Soc. Am. 65, 451–455 (1975).
    [Crossref]
  3. W. H. Southwell, “Inhomogeneous optical waveguide lens analysis,” J. Opt. Soc. Am. 67, 1004–1009 (1977).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 110–122.
  5. G. Dahlquist and A. Bjorck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 338.
  6. W. H. Southwell, “Sine-wave optical paths in gradient-index media,” J. Opt. Soc. Am. 61, 1715 (1971).
    [Crossref]
  7. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966), p. 187.

1977 (1)

1975 (1)

1971 (1)

1968 (1)

Bjorck, A.

G. Dahlquist and A. Bjorck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 338.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 110–122.

Dahlquist, G.

G. Dahlquist and A. Bjorck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 338.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966), p. 187.

Montagnino, L.

Moore, D. T.

Southwell, W. H.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 110–122.

J. Opt. Soc. Am. (4)

Other (3)

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966), p. 187.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 110–122.

G. Dahlquist and A. Bjorck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 338.

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Figures (2)

Fig. 1
Fig. 1

Plot of the ray-height error data of Table 1 as a function of equivalent step size.

Fig. 2
Fig. 2

Plot of the optical-path-difference error data of Table 1 as a function of equivalent step size.

Tables (1)

Tables Icon

Table 1 Comparison of Ray-Tracing Methods*

Equations (27)

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S ( r ) = constant ,
( S ) 2 = n 2 ,
S = n ŝ ,
ŝ = d r d s ,
d d s ( n ŝ ) = d d s ( S ) .
d d s = d r d s · = s ·
d d s = 1 n ( S ) · .
d d s ( n d r d s ) = 1 n ( S ) · ( S ) .
d d s ( n d r d s ) = 1 2 n ( S ) 2 ,
d d s ( n d r d s ) = n .
d r d s = T n , d T d s = n .
d F d s = F ( s + h ) - F ( s ) h ,
r = r 0 + h T 0 / n , T = T 0 + h n , O p = O p 0 + h n ,
t = n d s ,
d t = n d s .
d 2 r d t 2 = n n 3 - 2 n ( T · n ) d r d t ,
d r d t = T , d T d t = n n 3 - 2 n ( T · n ) T .
r = r 0 + h T 0 , T = T 0 + h [ n n 3 - 2 n ( T 0 · n ) T 0 ] , O p = O p 0 + h ,
r = r 1 / 3 - 2 r 2 + 8 r 3 / 3.
r = r 0 + s h + 1 2 K h 2 + 1 6 L h 3 , s = s 0 + K h + 1 2 L h 2 , O p = O p 0 + n h ,
K = 1 n [ n - s ( s · n ) ] ,
L = 1 n { d d s ( n ) - 2 K ( s · n ) - s [ K · n + s · d d s ( n ) ] } ,
r = r 0 + T h + 1 2 d 2 r d t 2 h 2 + 1 6 d 3 r d t 3 h 3 , T = T 0 + d 2 r d t 2 h + 1 2 d 3 r d t 3 h 2 , O p = O p 0 + h ,
d 2 r d t 2 = n n 3 - T ( T · n ) .
d 3 r d t 3 = 1 n 4 { d d s ( n ) - 5 K ( s · n ) - 2 S [ K · n + s · d d s ( n ) ] + 3 s n ( s · n ) 2 } .
h = c g ln | n + ( n 2 - c 2 ) 1 / 2 n 0 + ( n 0 2 - c 2 ) 1 / 2 | ,
y ( z ) = B sin ( π x l + δ ) ,