Abstract

The k function is the arbitrary function that arises in the general solution of the eikonal equation, a nonlinear, first-order partial differential equation, and describes completely the geometrical properties of a wave-front train and the associated caustic in a homogeneous, isotropic optical medium. The archetypal wave front in such a train is that unique wave front whose optical path length from the ultimate object point is zero. As an example, the k function of a wave-front train resulting from a plane wave front incident upon a spherical refracting surface is calculated. From it the archetype and the caustic are obtained. The results agree exactly with ray-trace data.

© 1995 Optical Society of America

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References

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  1. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chap. 8.
  2. More properly, this should be referred to as Jacobi’s second method of which an excellent account can be found in A. R. Forsyth, Theory of Differential Equations (Dover, New York, 1959), Vol. 5, Chap. 4.
  3. O. N. Stavroudis, R. C. Fronczek, “Caustic surfaces and the structure of the geometric image,” J. Opt. Soc. Am. 66, 795–800 (1976).
    [Crossref]
  4. O. N. Stavroudis, R. C. Fronczek, R.-S. Chang, “Geometry of the half-symmetric image,” J. Opt. Soc. Am. 68, 739–742 (1978).
    [Crossref]
  5. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), Part 1. See also Ref. 1, pp. 228–235.
  6. O. N. Stavroudis, “Tracing wavefronts: can it be done?,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop,R. E. Fischer, C. Londono, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 18–26 (1987).
    [Crossref]
  7. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966), pp. 36–38.
  8. O. N. Stavroudis, “Refraction of wave fronts: a special case,” J. Opt. Soc. Am. 59, 114–115 (1969).
    [Crossref]
  9. Ref. 1, p. 170–179.
  10. R. T. Farouki, J.-C. A. Chastang, “Exact equations of ‘simple’ wavefronts,” Optik 91, 109–121 (1992).
  11. D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 278–301 (1973).
    [Crossref]
  12. O. N. Stavroudis, “Simpler derivation of the formulas for generalized ray tracing,” J. Opt. Soc. Am. 66, 1330–1333 (1976).
    [Crossref]
  13. A. M. Kassim, D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 27, 516–522 (1988).
    [Crossref] [PubMed]
  14. J. M. Rebordão, M. Grosmann, “Refraction on spherical surfaces. I: an exact algebraic approach,” J. Opt. Soc. Am. A 1, 51–61 (1984).
    [Crossref]

1992 (1)

R. T. Farouki, J.-C. A. Chastang, “Exact equations of ‘simple’ wavefronts,” Optik 91, 109–121 (1992).

1988 (1)

1984 (1)

1978 (1)

1976 (2)

1973 (1)

D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 278–301 (1973).
[Crossref]

1969 (1)

Burkhard, D. G.

D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 278–301 (1973).
[Crossref]

Chang, R.-S.

Chastang, J.-C. A.

R. T. Farouki, J.-C. A. Chastang, “Exact equations of ‘simple’ wavefronts,” Optik 91, 109–121 (1992).

Farouki, R. T.

R. T. Farouki, J.-C. A. Chastang, “Exact equations of ‘simple’ wavefronts,” Optik 91, 109–121 (1992).

Forsyth, A. R.

More properly, this should be referred to as Jacobi’s second method of which an excellent account can be found in A. R. Forsyth, Theory of Differential Equations (Dover, New York, 1959), Vol. 5, Chap. 4.

Fronczek, R. C.

Grosmann, M.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), Part 1. See also Ref. 1, pp. 228–235.

Kassim, A. M.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966), pp. 36–38.

Rebordão, J. M.

Shealy, D. L.

A. M. Kassim, D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 27, 516–522 (1988).
[Crossref] [PubMed]

D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 278–301 (1973).
[Crossref]

Stavroudis, O. N.

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

D. L. Shealy, D. G. Burkhard, “Flux density for ray propagation in discrete index media expressed in terms of the intrinsic geometry of the deflecting surface,” Opt. Acta 20, 278–301 (1973).
[Crossref]

Optik (1)

R. T. Farouki, J.-C. A. Chastang, “Exact equations of ‘simple’ wavefronts,” Optik 91, 109–121 (1992).

Other (6)

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chap. 8.

More properly, this should be referred to as Jacobi’s second method of which an excellent account can be found in A. R. Forsyth, Theory of Differential Equations (Dover, New York, 1959), Vol. 5, Chap. 4.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), Part 1. See also Ref. 1, pp. 228–235.

O. N. Stavroudis, “Tracing wavefronts: can it be done?,” in Recent Trends in Optical Systems Design and Computer Lens Design Workshop,R. E. Fischer, C. Londono, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 18–26 (1987).
[Crossref]

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966), pp. 36–38.

Ref. 1, p. 170–179.

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Figures (7)

Fig. 1
Fig. 1

Caustic surface associated with the spherical reflecting surface, with the object at infinity.

Fig. 2
Fig. 2

Rays and caustics from a plane wave front refracted by a sphere, from generalized ray tracing. (a) Some incident rays, (b) the caustic. r = 100, n = 1, n′ = 1.9. Data are given in Table 1.

Fig. 3
Fig. 3

Archetype and caustic calculated from the k function. r = 100, n = 1, n = 1.9. Data are given in Table 2.

Fig. 4
Fig. 4

Rays and caustics from a plane wave front refracted by a sphere, from generalized ray tracing. (a) Some incident rays, (b) the caustic. r = 100, n =1.9, n′ = 1.

Fig. 5
Fig. 5

Archetype and caustic calculated from the k function. r = 100, n = 1.9, n′ = 1.

Fig. 6
Fig. 6

Rays and caustics from a plane wave front refracted by a sphere, from generalized ray tracing. (a) Some incident rays, (b) the caustic. r = −100, n = 1.4, n′ = 1.

Fig. 7
Fig. 7

Archetype and caustic calculated from the k function. r = −100, n = 1.4, n′ = 1.

Tables (3)

Tables Icon

Table 1 Curvatures and Caustic Coordinates from Generalized Ray Tracinga

Tables Icon

Table 2 Caustic Coordinates Generated by the k Functiona

Tables Icon

Table 3 Caustic Coordinates from the k Function, with Use of Cosines from Table 1

Equations (27)

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W ( υ , w ; s ) = q n 2 S K ,
S = ( u , υ , w ) , ( S 2 = n 2 ) .
q = ( n s k ) + S · K ,
K = ( 0 , k υ , k w ) .
C ± ( υ , w ) = 1 2 n 2 ( H ± Σ ) S K ,
H = ( n 2 υ 2 ) k υ υ + ( n 2 w 2 ) k w w 2 υ w k υ w , T 2 = k υ υ k w w k υ w 2 , Σ 2 = H 2 4 n 2 u 2 T 2 .
S = n Z ,
y = r sin α , z = r ( 1 cos α ) .
N = ( 0 , sin α , cos α ) .
S = S + γ N ,
γ = n cos α + ( n 2 n 2 sin 2 α ) 1 / 2 .
υ = γ sin α , w = n + γ cos α .
P = ( 0 , y , z ) = r ( 0 , sin α , 1 cos α ) = r [ 0 , υ / γ , 1 ( w n ) / γ ] ,
P = r γ [ ( γ + n ) Z S ] .
γ 2 = n 2 + n 2 2 n w .
z = r γ ( γ + n w ) .
s = n r n γ ( γ + n w ) .
W arch = r γ [ ( γ + n ) Z S ] n r n 2 γ [ γ + n w ] S = r n 2 γ [ n 2 + n 2 n w + n γ ] S + r γ ( γ + n Z .
W ( υ , w ; s ) = W arch ( υ , w ) + ( s / n ) S .
q = k + ( S K ) = r γ ( n 2 + n 2 n w + n γ )
K = r γ ( γ + n ) Z .
k ( υ , w ) = r ( γ + n w ) .
k υ = 0 , k w = r ( 1 + n / γ ) ,
k υ υ = 0 , k υ w = 0 , k w w = n 2 r / γ 3 .
H = n 2 r ( n 2 w 2 ) / γ 3 , T = 0 ,
C + = n 2 r n 2 γ 3 ( n 2 w 2 ) S + r ( 1 + n / γ ) Z , C = r ( 1 + n / γ ) Z .
C + ( 0 , n ) = C ( 0 , n ) = n r n n Z .

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