Abstract

We present a simple, yet accurate, method to obtain the intersection point of a ray in a gradient-index medium and the surface following the medium. The method does not involve additional ray tracing, unlike the commonly used iterative technique, and the coordinates of the intersection point are obtained analytically.

© 1986 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, “Rays in Gradient Index Media: Separable Systems,” J. Opt. Soc. Am. 63, 46 (1973).
    [CrossRef]
  4. A. Sharma, A. K. Ghatak, “A New Method for Tracing Rays Through Graded-Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, DC, 1981), paper MA3.
  5. A. Sharma, D. V. Kumar, A. K. Ghatak, “Tracing Rays Through Graded-Index Media: A New Method,” Appl. Opt. 21, 984 (1982).
    [CrossRef] [PubMed]
  6. L. Montagnino, “Ray Tracing in Inhomogeneous Media,” J. Opt. Soc. Am. 58, 1667 (1968).
    [CrossRef]
  7. D. T. Moore, “Ray Tracing in Gradient-Index Media,” J. Opt. Soc. Am. 65, 451 (1975).
    [CrossRef]
  8. A. Sharma, “Computing Optical Path Length in Gradient-Index Media: A Fast and Accurate Method,” Appl. Opt. 24, 4367 (1985).
    [CrossRef] [PubMed]
  9. See e.g., J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U.P., Baltimore, 1966), Chap. 10.
  10. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 3.

1985

1982

1975

1973

1968

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 3.

Buchdahl, H. A.

Ghatak, A. K.

A. Sharma, D. V. Kumar, A. K. Ghatak, “Tracing Rays Through Graded-Index Media: A New Method,” Appl. Opt. 21, 984 (1982).
[CrossRef] [PubMed]

A. Sharma, A. K. Ghatak, “A New Method for Tracing Rays Through Graded-Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, DC, 1981), paper MA3.

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

Kumar, D. V.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 4 and 5.

Montagnino, L.

Moore, D. T.

Scarborough, J. B.

See e.g., J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U.P., Baltimore, 1966), Chap. 10.

Sharma, A.

A. Sharma, “Computing Optical Path Length in Gradient-Index Media: A Fast and Accurate Method,” Appl. Opt. 24, 4367 (1985).
[CrossRef] [PubMed]

A. Sharma, D. V. Kumar, A. K. Ghatak, “Tracing Rays Through Graded-Index Media: A New Method,” Appl. Opt. 21, 984 (1982).
[CrossRef] [PubMed]

A. Sharma, A. K. Ghatak, “A New Method for Tracing Rays Through Graded-Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, DC, 1981), paper MA3.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 3.

Thyagarajan, K.

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

Appl. Opt.

J. Opt. Soc. Am.

Other

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps. 4 and 5.

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

A. Sharma, A. K. Ghatak, “A New Method for Tracing Rays Through Graded-Index Media,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, DC, 1981), paper MA3.

See e.g., J. B. Scarborough, Numerical Mathematical Analysis (Johns Hopkins U.P., Baltimore, 1966), Chap. 10.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 3.

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Figures (1)

Fig. 1
Fig. 1

Geometry of ray–surface intersection.

Equations (25)

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d 2 R d t 2 = D ( R ) ,
D = 1 2 ( n 2 x , n 2 y , n 2 z ) .
T = d R d t = n d R d s
x = a 0 + a 1 t ˜ + a 2 t ˜ 2 + a 3 t ˜ 3 , y = b 0 + b 1 t ˜ + b 2 t ˜ 2 + b 3 t ˜ 3 , z = c 0 + c 1 t ˜ + c 2 t ˜ 2 + c 3 t ˜ 3 ,
F ( x , y , z ) = 0.
F ( x , y , z ) = F 0 + F 1 t ˜ + F 2 t ˜ 2 + F 3 t ˜ 3 ,
F ( x , y , z ) = F 0 + F 1 t ˜ + F 1 t ˜ 2 + F 3 t ˜ 3 .
F ( x , y , z ) C v [ x 2 + y 2 + ( z - d ) 2 ] - 2 ( z - d ) = 0 ,
d F d t ˜ d F d t = 2 { C v [ x d x d t + y d y d t + ( z - d ) d z d t ] - d z d t } ,
F m = F ( x , y , z ) at P m , F m = d F d t at P m , F m + 1 = F ( x , y , z ) at P m + 1 , F m + 1 = d F d t at P m + 1 , }
F 0 = F m , F 1 = F m , F 2 = [ 3 ( F m + 1 - F m ) - ( 2 F m - F m + 1 ) Δ t ] / ( Δ t ) 2 , F 3 = [ ( F m + 1 + F m ) Δ t - 2 ( F m + 1 - F m ) ] / ( Δ t ) 3 . }
F 3 t ˜ 3 + F 2 t ˜ 2 + F 1 t ˜ + F 0 = 0
t ˜ 2 + A 1 t ˜ + A 0 = - A 3 t ˜ 3 ,
t ˜ = u ( 1 + u 1 + 2 u 2 + 3 u 3 + ) ,
t ˜ 2 + A 1 t ˜ + A 0 = 0 ,
u 1 = - ( A 3 u ) / p ,
u 2 = ( A 3 u ) 2 ( 3 p - 1 ) / p 3 ,
u 3 = - ( A 3 u ) 2 ( 6 p - 1 ) / p 4 .
F 1 t ˜ + F 0 = - ( F 2 t ˜ 2 + F 3 t ˜ 3 ) ,
t ˜ = u ( 1 + u 1 + u 2 + u 3 + ) ,
u = - F 0 / F 1 ,
u 1 = - u ( F 3 u + F 2 ) / F 1 ; u 2 = u 2 [ 3 ( F 3 u ) 2 + 5 F 2 ( F 3 u ) + 2 F 2 2 ] / F 1 2 , u 3 = - u 3 [ 12 ( F 3 u ) 3 + 28 F 2 ( F 3 u ) 2 + 21 F 2 2 ( F 3 u ) + 5 F 2 3 ] / F 1 3 }
n 2 ( x , y ) = 2.5 - 0.5 ( x 2 + y 2 ) .
u = 0.0725936 ;             u 2 = 0.262661 × 10 - 9 ; u 1 = 0.957911 × 10 - 5 ;             u 3 = 0.493870 × 10 - 14 .
u 2 = 0.758286 × 10 - 7 ,             u s = 0.242088 × 10 - 10 ,

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