Abstract

The basic relations of geometrical optics for the case of an isotropic, time-dependent medium are derived from Fermat’s principle. The time-dependent theory is applied by discussing the Debye-Sears effect and the frequency fluctuations in a plane light wave induced by atmospheric turbulence and a steady cross wind. In the former case it is shown that the Brillouin scattering relation Δω = VΔk holds in the geometrical optics limit where V is the sound velocity, while in the latter case we find, using a method due to Tatarski, that the fluctuations in frequency are of the order of a few kilohertz under the most extreme conditions of turbulence, wind speed, and range. The intensity law of geometrical optics, = constant, is generalized to read /ν2 = constant, where ν is the frequency of the light wave.

© 1976 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 3.
  2. S. Weinberg, Phys. Rev. 126, 1899 (1962).
    [Crossref]
  3. L. A. Pars, A Treatise on Analytical Dynamics (Wiley, New York, 1968), Chap. XVI.
  4. A. Cohen, An Elementary Treatise on Differential Equations, 2nd. ed. (Heath, Boston, 1933), p. 261.
  5. M. V. Berry, The Diffraction of Light by Ultrasound (Academic, New York, 1966), Chap. 1.
  6. V. I. Tatarski, Propagation of Waves in Turbulent Atmosphere (Nauka, Moscow, 1967).
  7. J. W. Strohbehn, Optical Propagation Through the Turbulent Atmosphere in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1971), Vol. IX, p. 81.

1962 (1)

S. Weinberg, Phys. Rev. 126, 1899 (1962).
[Crossref]

Berry, M. V.

M. V. Berry, The Diffraction of Light by Ultrasound (Academic, New York, 1966), Chap. 1.

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 3.

Cohen, A.

A. Cohen, An Elementary Treatise on Differential Equations, 2nd. ed. (Heath, Boston, 1933), p. 261.

Pars, L. A.

L. A. Pars, A Treatise on Analytical Dynamics (Wiley, New York, 1968), Chap. XVI.

Strohbehn, J. W.

J. W. Strohbehn, Optical Propagation Through the Turbulent Atmosphere in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1971), Vol. IX, p. 81.

Tatarski, V. I.

V. I. Tatarski, Propagation of Waves in Turbulent Atmosphere (Nauka, Moscow, 1967).

Weinberg, S.

S. Weinberg, Phys. Rev. 126, 1899 (1962).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 3.

Phys. Rev. (1)

S. Weinberg, Phys. Rev. 126, 1899 (1962).
[Crossref]

Other (6)

L. A. Pars, A Treatise on Analytical Dynamics (Wiley, New York, 1968), Chap. XVI.

A. Cohen, An Elementary Treatise on Differential Equations, 2nd. ed. (Heath, Boston, 1933), p. 261.

M. V. Berry, The Diffraction of Light by Ultrasound (Academic, New York, 1966), Chap. 1.

V. I. Tatarski, Propagation of Waves in Turbulent Atmosphere (Nauka, Moscow, 1967).

J. W. Strohbehn, Optical Propagation Through the Turbulent Atmosphere in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1971), Vol. IX, p. 81.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 3.

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Equations (73)

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c T = n * r
c T v = n v * ( r + δ r ) .
c d δ T d α = r δ n * + n * δ r ,
δ T T v ( α , t 0 + δ t 0 ) - T ( α , t 0 ) ,
δ r r · ( δ r ) / r
δ n * ( n ) * · δ r + ( n t ) * ( δ T + δ t 0 ) .
d δ T d α - P δ T = ( Q 1 · δ r ) + Q 2 · δ r + P δ t 0 ,
c Q 1 ( α ) n * r / r , c Q 2 ( α ) r ( n ) * - ( n * r / r ) , c P ( α ) r ( n t ) * ,
δ T = exp ( 0 α P d α ) 0 α [ ( Q 1 · δ r ) + Q 2 · δ r + P δ t 0 ] × exp ( - 0 α P d α ) d α .
δ T ( α ) = ( Q 1 · δ r ) α - ( Q 1 · δ r ) 0 exp ( 0 α P d α ) + δ t 0 [ exp ( 0 α P d α ) - 1 ] + exp ( 0 α P d α ) × [ 0 α ( P Q 1 + Q 2 ) · δ r exp ( - 0 α P d α ) d α ] .
P Q 1 + Q 2 = 0 ,
T = ( 1 / c ) n r .
ω d T = k · d r - k 0 · d r 0 + ( ω 0 - ω ) d t 0
k ω ( n * / c ) k ˆ
ω ω 0 exp ( - 1 c 0 α r n * t d α ) ,
- ω 0 d t 0 = k · d r - k 0 · d r 0 - ω d t .
S = k
S t + ω ( S , r , t ) = 0 ,
r ˙ = ω k ,             k ˙ = - ω r ,
( T ) 2 = ( 1 / c 2 ) n 2 ( r , t 0 + T ) .
t = - ( S / ω 0 ) + T ( r 0 , r , - S / ω 0 ) .
Ω ( r 0 ) = 0 ,
0 S ( r 0 , r , t ) · d r 0 = 0             ( d r 0 lying in Ω )
S ( r 0 , r , t ) = - ω 0 t 0 .
× E + ( 1 / c ) B ˙ = 0 , × H - ( 1 / c ) D ˙ = 0 , · D = 0 , · B = 0 ,
w l + 1 8 π ( ˙ E 2 + μ ˙ H 2 ) + · S = 0.
E = Re [ e 0 ( r , t ) e i S ( r , t ) ] , H = Re [ h 0 ( r , t ) e i S ( r , t ) ] ,
( S t ) 2 = c 2 n 2 ( r , t ) ( S ) 2 ,
k = S
ω = - S / t
1 c ( I n ) t + 1 2 I n c ln ( μ ) t + · ( I k ˆ ) = 0 ,
ln ( I ) t + c n k ˆ · ln ( I ) + c n · k ˆ + 2 ln ( n ) t = 0.
d ln ( I σ ) d t = - 2 ln ( n ) t .
I σ I 0 σ 0 = exp ( - 2 0 t 1 n n t d t ) ,
I σ ω 2 = I 0 σ 0 ω 0 2
n = 0 + 1 cos ( K y - Ω t ) .
ω = ( c / n ) k .
x ˙ = ω k x = c n k x k , y ˙ = ω k y = c n k y k , k ˙ y = - ω x = 0 , k ˙ y = - ω y = c k n 2 n y .
( ω - ω 0 ) = V ( k y - k 0 y ) ,
I 0 ( 1 / f ) ( 1 + p 2 ) 1 / 2 - p = ( 1 / f 0 ) ( 1 + p 0 2 ) 1 / 2 - p 0 ,
f n V / c , p d y d x = k y k x ,
d ϕ d x = K p - Ω x ˙ = K p - Ω n c ( 1 + p 2 ) 1 / 2 ,
1 K d ϕ d x = p - f ( 1 + p 2 ) 1 / 2 .
p = - f 2 I 0 ± ( 1 / W 0 ) ( f 2 - W 0 2 ) 1 / 2 f 2 - 1 ,
W 0 1 / ( 1 + I 0 2 ) 1 / 2 .
- 1 K d ϕ d x = ± 1 W 0 ( f 2 - W 0 2 ) 1 / 2 ,
F ( ψ , κ ) = ± Q x + F ( ψ 0 , κ ) ,
κ = ( 2 1 ( V 2 / c ) ( V 2 / c 2 ) ( 0 + 1 ) - W 0 2 ) 1 / 2 ,
Q = ( K / 2 W 0 ) [ ( V 2 / c 2 ) ( 0 + 1 ) - W 0 2 ] 1 / 2 .
W 0 2 = ( V 2 / c 2 ) ( 0 - 1 )
y = 0 x p d x = ϕ 0 ϕ p d x d ϕ d ϕ ,
lim Ω 0 κ = ( 2 1 - cos ϕ 0 ) 1 / 2 , lim Ω 0 Q = κ 1 1 / 2 sin ( ϕ 0 / 2 ) 2 n 0 .
t - v ρ v ,
ψ = - 1 c 0 S L ( n t ) * d s ,
ψ = v 2 c 0 L ( ρ v d s d x ) * d x ,
ψ = v 2 c 0 L ρ v d x
ρ = - 1 2 0 L M ( L , x , ξ ) ( ξ , 0 ) d ξ ,
σ ψ 2 = v 2 4 c 2 0 L d x 0 L d x ρ v ( x , ρ ( x ) ) ρ v ( x , ρ ( x ) ) ,
σ ψ 2 = [ v 2 / 4 ( 2 π ) 4 c 2 ] 0 L d x 0 L d x d ρ d ρ d k d k e i k · ρ e i k · ρ B ( k , k , ρ , ρ , x , x ) ,
B = e - i k · ρ ( x ) e - i k · ρ ( x ) ρ v ( x , ρ ) ρ v ( x , ρ ) .
B = - ( δ 2 Φ δ a v ( x , ρ ) δ a v ( x , ρ ) ) a = a 0 ,
Φ = exp ( i 0 L d x d ρ a ( x , ρ ) · ( x , ρ ) )
a 0 = 1 2 δ ( ρ ) [ k M ( L , x , x ) + k M ( L , x , x ) ] .
σ ψ 2 = ψ 2 1 + ψ 2 2 ,
ψ 2 1 ( π v 2 / 2 c 2 ) L d q q v 2 ϕ ( q )
ψ 2 2 - ( π v 2 / 4 c 2 ) 0 L d x 0 L d x d q d q Φ 0 ( q , q , x , x ) ϕ ( q ) ϕ ( q ) q v q l q v q l [ q l M ( L , x , x ) + q l M ( L , x , x ) ] × [ q l , M ( L , x , x ) + q l M ( L , x , x ) ] .
Φ 0 = e - i q · ρ ( x ) e - i q · ρ ( x ) .
ϕ ( q ) = 0.33 C 2 q - 11 / 3 exp ( - q 2 / κ m 2 ) ,
σ ψ 2 = 0.33 π 2 4 Γ ( 1 6 ) v 2 c 2 C 2 κ m 1 / 3 L .
6 × 10 - 9 ( v / c ) L 1 / 2 < σ ψ < 3 × 10 - 7 ( v / c ) L 1 / 2 ,
e ψ = exp ( ψ + ( 1 / 2 ) σ ψ 2 ) .
e 2 ψ - e ψ 2 = ( ν 2 - ν 2 ) / ν 0 2 ( 1 + 2 ψ + 2 σ ψ 2 ) - ( 1 + 2 ψ + σ ψ 2 ) .
( ν - ν ) 2 1 / 2 = ν 0 σ ψ ,