Abstract

When light is reflected or refracted at a moving Gaussian surface, the observer sees a number of moving images of the source, which appear or disappear generally in pairs; such an event is called a “twinkle.” In the present paper the number of twinkles per unit time is evaluated in terms of the frequency spectrum of the surface and the distance of the source O and observer Q, on the assumption that the surface is Gaussian and that OQ is perpendicular to the mean surface level.

A solution is found first for a single system of long-crested (or two-dimensional) waves, and then extended to the case of two such systems intersecting at right angles.

The rate of twinkling is found to depend, apart from a scale factor, on two parameters of the surface, one of which, α, increases steadily with the distance of O or Q from the surface; the other, d, discriminates between waves of standing type and waves of progressive type. Over a considerable range of α, the rate of twinkling is almost independent of d, but for large values of α the rate is much greater for standing waves than for progressive waves; waves of intermediate type are included in the analysis.

© 1960 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 838 (1960), paper I of this series.
    [Crossref]
  2. M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 845 (1960), paper II of this series.
    [Crossref]
  3. J. D. Whitehead, J. Terrest. Atm. Phys. 9, 269 (1956).
    [Crossref]
  4. nx has been evaluated previously in the case when the source and observer are at infinite distance. See M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. 56, 234 (1956).
    [Crossref]
  5. A. N. Lowan, “Tables of normal probability functions,” Natl. Bur. Standards, Appl. Math. Ser. 23 (1953).

1960 (2)

1956 (2)

J. D. Whitehead, J. Terrest. Atm. Phys. 9, 269 (1956).
[Crossref]

nx has been evaluated previously in the case when the source and observer are at infinite distance. See M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. 56, 234 (1956).
[Crossref]

1953 (1)

A. N. Lowan, “Tables of normal probability functions,” Natl. Bur. Standards, Appl. Math. Ser. 23 (1953).

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 838 (1960), paper I of this series.
[Crossref]

M. S. Longuet-Higgins, J. Opt. Soc. Am. 50, 845 (1960), paper II of this series.
[Crossref]

nx has been evaluated previously in the case when the source and observer are at infinite distance. See M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. 56, 234 (1956).
[Crossref]

Lowan, A. N.

A. N. Lowan, “Tables of normal probability functions,” Natl. Bur. Standards, Appl. Math. Ser. 23 (1953).

Whitehead, J. D.

J. D. Whitehead, J. Terrest. Atm. Phys. 9, 269 (1956).
[Crossref]

J. Opt. Soc. Am. (2)

J. Terrest. Atm. Phys. (1)

J. D. Whitehead, J. Terrest. Atm. Phys. 9, 269 (1956).
[Crossref]

Natl. Bur. Standards, Appl. Math. Ser. 23 (1)

A. N. Lowan, “Tables of normal probability functions,” Natl. Bur. Standards, Appl. Math. Ser. 23 (1953).

Proc. Cambridge Phil. Soc. (1)

nx has been evaluated previously in the case when the source and observer are at infinite distance. See M. S. Longuet-Higgins, Proc. Cambridge Phil. Soc. 56, 234 (1956).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Graphs of f(α,d) showing the rate of twinkling as a function of α (proportional to distance from surface) for various values of d.

Fig. 2
Fig. 2

Graphs of g(A,d), showing the rate of twinkling for two intersecting systems as a function of A (proportional to the square of the distance from the surface) for various values of d.

Equations (53)

Equations on this page are rendered with MathJax. Learn more.

f / x = 0 ,             f / y = 0 ,
f ( x , y , t ) = ζ ( x , y , t ) + 1 2 κ ( x 2 + y 2 )
κ = 1 2 [ ( 1 / h 1 ) + ( 1 / h 2 ) ] .
κ = ( μ 1 h 1 + μ 2 h 2 ) / [ ( μ 2 - μ 1 ) h 1 h 2 ] .
( 2 f / x 2 ) ( 2 f / y 2 ) - ( 2 f / x y ) 2 = 0
f / x = 0 ,             2 f / x 2 = 0 ,
f ( x , t ) = ζ ( x , t ) + 1 2 κ x 2 .
ζ ( x , t ) = n = 1 n 0 c n cos ( k n x + σ n t + n ) ,
n 1 2 c n 2 = E ( k , σ ) d k d σ ,
ζ / x ,             2 ζ / x 2 ,             2 ζ / x t , 3 ζ / x 3 = ξ 1 , ξ 2 , ξ 3 , ξ 4 ,
( ξ i ξ j av ) = ( m 2 0 0 - m 4 0 m 4 m 3 0 0 m 3 m 2 0 - m 4 0 0 m 6 ) ,
m r = - 0 E ( k , σ ) k r d k d σ , m r = - 0 E ( k , σ ) k r σ d k d σ , m r = - 0 E ( k , σ ) k r σ 2 d k d σ .
p ( ξ 1 , ξ 2 , ξ 3 , ξ 4 ) = ( M i j ) 1 2 4 π 2 exp [ - 1 2 i , j M i j ξ i ξ j ] ,
p ( ξ 1 , ξ 2 , ξ 3 , ξ 4 ) = p ( ξ 1 , ξ 4 ) p ( ξ 2 , ξ 3 ) ,
p ( ξ 1 , ξ 4 ) = 1 2 π ( m 2 m 6 - m 4 2 ) 1 2 × exp [ - m 6 ξ 1 2 + 2 m 4 ξ 1 ξ 4 + m 2 ξ 4 2 2 ( m 2 m 6 - m 4 2 ) ] , p ( ξ 2 , ξ 3 ) = 1 2 π ( m 4 m 2 - m 3 2 ) 1 2 × exp [ - m 2 ξ 2 2 - 2 m 3 ξ 2 ξ 3 + m 4 ξ 3 2 2 ( m 4 m 2 - m 3 2 ) ] .
η 1 , η 2 , η 3 , η 4 = f / x , 2 f / x 2 , 2 f / x t , 3 f / x 3 = ( ξ 1 + κ x ) , ( ξ 2 + κ ) , ξ 3 , ξ 4 ,
p ( η 1 , η 2 , η 3 , η 4 ) = p ( ξ 1 , ξ 2 , ξ 3 , ξ 4 ) ,
ξ 1 , ξ 2 , ξ 3 , ξ 4 = ( η 1 - κ x ) , ( η 2 - κ ) , η 3 , η 4 .
η 1 = 0 ,             η 2 = 0
n = - n x d x .
| ( η 1 , η 2 ) ( x , t ) | = 2 f / x 2 3 f / x 3 2 f / x t 3 f / x 2 t d x d t ,
n x d x d t = - - p ( η 1 , η 2 , η 3 , η 4 ) η 3 η 4 d x d t d η 3 d η 4 ,
n = - - - p ( η 1 , η 2 , η 3 , η 4 ) η 3 η 4 d x d η 3 d η 4 .
n = 1 ( 2 π ) 1 2 κ m 6 1 2 - exp ( - η 4 2 / 2 m 6 ) η 4 d η 4 × 1 2 π ( m 4 m 2 - m 3 2 ) 1 2 × - exp [ - m 2 κ 2 + 2 m 3 κ η 3 + m 4 η 3 2 2 ( m 4 m 2 - m 3 2 ) ] .
n = 2 1 2 π 3 2 m 6 1 2 ( m 4 m 2 - m 3 2 ) 1 2 κ m 4 exp ( - κ 2 / 2 m 4 ) × [ exp ( - 1 2 ϕ 2 ) + ϕ 0 ϕ exp ( - 1 2 z 2 ) d z ] ,
ϕ = κ m 3 / [ m 4 1 2 ( m 4 m 2 - m 3 2 ) 1 2 ] .
α = m 4 1 2 / κ ,
d = [ ( m 4 m 2 - m 3 2 ) / m 4 m 2 ] 1 2 ,
n = [ ( m 6 m 2 ) 1 2 / m 4 ] f ( α , d ) ,
f ( α , d ) = 2 1 2 π 3 2 α d exp ( - 2 α 2 ) - 1 × [ exp ( - 1 2 ϕ 2 ) + ϕ 0 ϕ exp ( - 1 2 z 2 ) d z ]
ϕ = ( 1 - d 2 ) 1 2 / α d .
( m 6 m 2 ) 1 2 / m 4 = 2 π / τ ,
2 ( m 4 m 2 - m 3 2 ) = - 0 - 0 E ( k 1 , σ 1 ) E ( k 2 , σ 2 ) × ( k 1 4 k 2 2 σ 2 2 + k 2 4 k 1 2 σ 1 2 - 2 k 1 3 k 2 3 σ 1 σ 2 ) × d k 1 d σ 1 d k 2 d σ 2 ,
m 4 m 2 - m 3 2 0 ,
0 d 1.
f ( α , d ) ( 1 / π ) exp ( - 2 α 2 ) - 1 ,
f ( α , d ) ( 2 1 2 / π 3 2 ) α exp ( - 2 α 2 ) - 1 .
f ( α , d ) ~ ( 2 1 2 / π 3 2 ) α d ,
ζ ( x , y , t ) = ζ 1 ( x , t ) + ζ 2 ( y , t ) ,
f 1 / x = 0 ,             f 2 / x = 0 ,             ( 2 f 1 / x 2 ) ( 2 f 2 / y 2 ) = 0 ,
f 1 ( x , t ) = ζ 1 ( x , t ) + 1 2 κ x 2 , f 2 ( x , t ) = ζ 2 ( x , t ) + 1 2 κ y 2 .
n = n ( 1 ) N ( 2 ) + n ( 2 ) N ( 1 ) ,
N ( i ) = ( 2 π ) 1 2 [ α exp ( - 2 α 2 ) - 1 + 0 1 / α exp ( - 1 2 z 2 ) d z ] ,
N ~ ( 2 / π ) 1 2 α
n = 2 n ( 1 ) N ( 1 ) .
A = 1 2 α 2 = 1 4 κ - 2 D ,
n = [ ( m 6 m 2 ) 1 2 / m 4 ] g ( A , d ) .
g ( A , d ) ( 8 / π 2 ) A d .
L = N / n ,
L ( 1 ) = N ( 1 ) / n ( 1 ) ,             L ( 2 ) = N ( 2 ) / n ( 2 ) .
N = N ( 1 ) N ( 2 ) .
1 / L = ( 1 / L ( 1 ) ) + ( 1 / L ( 2 ) ) .
1 / L = 2 / L ( 1 ) ,