Abstract

Ten dispersion relations due to dielectric fluctuations are derived for electromagnetic radiation whose two-point correlation is written as a product of two functions, one in space and the other in time. The time part is assumed to be a Markov process, and five different spatial-correlation functions are considered. The dispersion relations derived are applied to calculate frequency and wave-number dispersions for the irradiance of a monochromatic beam that is used to excite selectively a host species in a solid.

© 1972 Optical Society of America

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References

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  1. V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
    [Crossref] [PubMed]
  2. V. Twersky, J. Math. Phys. 3, 700 (1962).
    [Crossref]
  3. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  4. J. B. Keller, Proc. Symp. Appl. Math. 13, 227 (1962).
    [Crossref]
  5. G. Ross, Opt. Acta 15, 451 (1968).
    [Crossref]
  6. G. Ross, Opt. Acta 16, 95 (1969).
    [Crossref]
  7. G. Ross, Opt. Acta 16, 611 (1969).
    [Crossref]
  8. D. Mintzer, J. Acoust. Soc. Am. 26, 186 (1953).
    [Crossref]
  9. F. C. Karal and J. B. Keller, J. Math. Phys. 5, (1964).
    [Crossref]
  10. V. I. Tatarski, Zh. Eksperim. i Teor. Fiz. 42, 1386 (1962) [Sov. Phys. JETP 15, 961 (1962)].

1969 (2)

G. Ross, Opt. Acta 16, 95 (1969).
[Crossref]

G. Ross, Opt. Acta 16, 611 (1969).
[Crossref]

1968 (1)

G. Ross, Opt. Acta 15, 451 (1968).
[Crossref]

1964 (1)

F. C. Karal and J. B. Keller, J. Math. Phys. 5, (1964).
[Crossref]

1962 (4)

V. I. Tatarski, Zh. Eksperim. i Teor. Fiz. 42, 1386 (1962) [Sov. Phys. JETP 15, 961 (1962)].

V. Twersky, J. Opt. Soc. Am. 52, 145 (1962).
[Crossref] [PubMed]

V. Twersky, J. Math. Phys. 3, 700 (1962).
[Crossref]

J. B. Keller, Proc. Symp. Appl. Math. 13, 227 (1962).
[Crossref]

1953 (1)

D. Mintzer, J. Acoust. Soc. Am. 26, 186 (1953).
[Crossref]

Karal, F. C.

F. C. Karal and J. B. Keller, J. Math. Phys. 5, (1964).
[Crossref]

Keller, J. B.

F. C. Karal and J. B. Keller, J. Math. Phys. 5, (1964).
[Crossref]

J. B. Keller, Proc. Symp. Appl. Math. 13, 227 (1962).
[Crossref]

Mintzer, D.

D. Mintzer, J. Acoust. Soc. Am. 26, 186 (1953).
[Crossref]

Ross, G.

G. Ross, Opt. Acta 16, 95 (1969).
[Crossref]

G. Ross, Opt. Acta 16, 611 (1969).
[Crossref]

G. Ross, Opt. Acta 15, 451 (1968).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Zh. Eksperim. i Teor. Fiz. 42, 1386 (1962) [Sov. Phys. JETP 15, 961 (1962)].

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Twersky, V.

J. Acoust. Soc. Am. (1)

D. Mintzer, J. Acoust. Soc. Am. 26, 186 (1953).
[Crossref]

J. Math. Phys. (2)

F. C. Karal and J. B. Keller, J. Math. Phys. 5, (1964).
[Crossref]

V. Twersky, J. Math. Phys. 3, 700 (1962).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Acta (3)

G. Ross, Opt. Acta 15, 451 (1968).
[Crossref]

G. Ross, Opt. Acta 16, 95 (1969).
[Crossref]

G. Ross, Opt. Acta 16, 611 (1969).
[Crossref]

Proc. Symp. Appl. Math. (1)

J. B. Keller, Proc. Symp. Appl. Math. 13, 227 (1962).
[Crossref]

Zh. Eksperim. i Teor. Fiz. (1)

V. I. Tatarski, Zh. Eksperim. i Teor. Fiz. 42, 1386 (1962) [Sov. Phys. JETP 15, 961 (1962)].

Other (1)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

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Tables (5)

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Table I Dispersion relations for the frequency spectra.

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Table II Dispersion relations for the wavelength spectra.

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Table III Some numerical values for the frequency dispersion Δω(s−1) for the five correlation functions (λ = 1010s−1).

Tables Icon

Table IV Some numerical values for the frequency dispersion Δω(s−1) for the five correlation functions (λ = 5.0×1012 s−1).

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Table V Some numerical values for the wave-number dispersion Δk(cm−1) for four correlation functions.

Equations (31)

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2 E ( r , t ) μ c 2 2 E ( r , t ) t 2 = μ δ ( r , t ) c 2 2 E ( r , t ) t 2 .
E ( r , t ) = 1 ( 2 π ) 4 d κ d s e ( i κ r + i s t ) E 1 ( κ , s ) ,
δ ( r , t ) = ( 2 π ) 4 d κ d s e ( i κ r + i s t ) δ ( κ , s ) .
[ κ 2 μ c 2 ] E ( κ , s ) = μ c 2 d κ d s δ ( κ κ , s s ) s 2 E ( κ , s ) .
E ( κ , s ) = E 0 ( κ , s ) + μ c 2 d κ d s δ ( κ κ , s s ) [ κ 2 ( μ / c 2 ) s 2 ] s 2 E 0 ( κ , s ) + μ 2 2 c 4 d κ d s δ ( κ κ , s s ) [ κ 2 ( μ / c 2 ) s 2 ] s 2 × d κ d s δ ( κ κ , s s ) [ κ 2 ( μ / c 2 ) s 2 ] s 2 E 0 ( κ , s ) + .
[ 2 μ c 2 2 t 2 ] G ( r , t | r , t ) = δ ( r r ) δ ( t t ) .
d κ d s ( 4 π 3 ) 1 ( κ 2 μ c 2 s 2 ) 1 e i κ ( r r ) e is ( t t )
G ( r , t | r , t ) = δ [ t t + ( μ / c ) | r r | ] π | r r | .
E ( r , t ) = E t ( r , t ) = E 0 δ ( κ k ) δ ( s ω ) ,
[ k 2 μ ω 2 c 2 ] = μ 2 2 ω 2 c 4 × d r d t 2 δ [ t t + | r r | μ c 1 ] t 2 × δ ( r , t ) δ ( r , t ) e i k | r r | i ω τ 4 π | r r | .
N ( r , t | r , t ) = δ ( r , t ) δ ( r , t ) / δ 2
[ k 2 μ ω 2 c 2 ] = ω 2 δ 2 μ 2 c 4 d R d τ 2 δ [ τ + R μ c 1 ] τ 2 × N 1 ( R ) N 2 ( τ ) e i k R 4 π R e i ω τ .
k 2 μ ω 2 c 2 = δ 2 μ 2 ω 2 c 4 ( λ + i ω ) 2 × 0 d R ( sin k R ) N 1 ( R ) × exp ( i ( ω + i λ ) μ R c ) .
k 2 k 0 2 k 0 4 δ 2 2 k × 0 d R ( sin k R ) N 1 ( R ) e i k 0 R .
I = I 0 ( 2 Δ ω ) 1 exp { [ ( ω ω ¯ ) / 2 Δ ω ] 2 } .
I ω B / I ω A = exp { [ ( ω ω ¯ ) / 2 Δ ω ] }
= 1 ; δ 2 10 3 ; ω 0 = 6.72 × 10 12 s 1
ω 2 c 2 k 2 μ = δ 2 ω 2 γ a 2 k [ ( 1 i a γ ) 2 + a 2 k 2 ]
ω 2 c 2 k 2 μ = ω 2 γ 2 δ 2 a π 1 2 4 k i exp { ( a 2 / 4 ) k 2 } exp { ( a 2 / 4 ) γ } × [ exp { ( a 2 / 2 ) k γ } erfc { i a 2 ( γ + k ) } exp { + ( a 2 / 2 ) k γ } erfc { i a 2 ( γ k ) } ]
ω 2 c 2 k 2 μ = δ 2 ω 2 γ 2 a 2 a 2 k 2 a γ 2 1 [ ( 1 i a γ ) 2 + a 2 k 2 ] 2
( 1 + x 2 a 2 ) 2 e λ t
ω 2 c 2 k 2 μ = δ 2 ω 2 γ 2 a π 1 2 e i 3 π / 2 4 π k { [ a ( γ k ) ] 1 2 K 1 2 [ a ( γ k ) ] [ a ( γ + k ) ] 1 2 K 1 2 [ a ( γ + k ) ] }
x a K 1 ( x a ) e λ t
ω 2 c 2 k 2 μ = δ 2 ω 2 γ 2 2 a k i [ i ( k + γ ) a ( k + γ ) 2 + a 2 + [ ( k + γ ) 2 + a 2 ] 1 2 a log ( a i { ( γ + k ) + [ ( γ + k ) 2 + a 2 ] 1 2 } ) i a ( γ k ) ( γ + k ) 2 + a 2 [ ( γ k ) 2 + a 2 ] 1 2 a log ( a i { ( γ k ) + [ ( γ k ) 2 + a 2 ] 1 2 } ) ]
γ = ( ω + i λ ) [ μ ] 1 / 2 c
k 2 k 0 2 = δ 2 k 0 4 a 2 2 [ ( 1 i a k 0 ) 2 + a 2 k 2 ]
k 2 k 0 2 = k 0 4 δ 2 a π 1 2 2 4 R i exp { ( a 2 / 4 ) k 2 } exp { ( a 2 / 4 ) k 0 2 } × [ exp { ( a 2 / 2 ) k k 0 } erfc { i a 2 ( k 0 + k ) } exp { ( a 2 / 2 ) k k 0 } erfc { i a 2 ( k 0 k ) } ]
( 1 + x 2 a 2 ) 2
k 2 k 0 2 = δ 2 k 0 4 a π 1 2 e i 3 π / 2 4 π k c l { [ a ( k 0 k ) ] 1 2 K 1 2 [ a ( k 0 k ) ] [ a ( k 0 + k ) ] 1 2 K 1 2 [ a ( k 0 + k ) ] }
x a K 1 ( x a )
k 2 k 0 2 = δ 2 k 0 4 2 a k 2 i [ i ( k + k 0 ) a ( k + k 0 ) 2 + a 2 + [ ( k + k 0 ) 2 + a 2 ] 1 2 a log ( a i { ( k 0 + k ) + [ ( k 0 + k ) 2 + a 2 ] 1 2 } ) i ( k 0 k ) a ( k 0 k ) + a 2 [ ( k 0 k ) 2 + a 2 ] 1 2 a log ( a i { ( k 0 k ) + [ ( k 0 k ) 2 + a 2 ] 1 2 } ) ]