Abstract

The Brillouin line shape is evaluated under the assumption of space- and time-limited coherence of the acoustic field. The coherence limitation, regarded as a means of taking into account the phonon–phonon interaction, is studied by introducing stochastic phase jumps in the acoustic waves. Under this assumption, any acoustic wave contributes to the spectral distribution of the Brillouin line intensity. The main effect is due to the reduced spatial coherence and consists of an asymmetry of the Brillouin line and of a shift of the intensity maximum towards the frequency of the incident light.

© 1971 Optical Society of America

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References

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  1. L. Brillouin, Am. Phys. 17, 88 (1922).
  2. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1959), p. 391.
  3. A preliminary attempt to consider the spatial incoherence of the acoustic field was made by R. De Micheli and L. Giulotto, Nuovo Cimento 46, 277 (1966).
    [Crossref]
  4. See, for example, R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966). For a description taking into account the not complete coherence of the incident field see L. Mandel, Phys. Rev. 181, 75 (1969).
    [Crossref]
  5. C. J. Montrose, V. A. Solovyev, and T. A. Litovitz, J. Acoust. Soc. Am. 43, 117 (1968).
    [Crossref]
  6. W. H. Nichols and E. F. Carome, J. Chem. Phys. 49, 1000 (1968).
  7. E. F. Carome, W. H. Nichols, C. R. Kinsitis-Swyt, and S. P. Singal, J. Chem. Phys. 49, 1013 (1968).
    [Crossref]

1968 (3)

C. J. Montrose, V. A. Solovyev, and T. A. Litovitz, J. Acoust. Soc. Am. 43, 117 (1968).
[Crossref]

W. H. Nichols and E. F. Carome, J. Chem. Phys. 49, 1000 (1968).

E. F. Carome, W. H. Nichols, C. R. Kinsitis-Swyt, and S. P. Singal, J. Chem. Phys. 49, 1013 (1968).
[Crossref]

1966 (2)

A preliminary attempt to consider the spatial incoherence of the acoustic field was made by R. De Micheli and L. Giulotto, Nuovo Cimento 46, 277 (1966).
[Crossref]

See, for example, R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966). For a description taking into account the not complete coherence of the incident field see L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

1922 (1)

L. Brillouin, Am. Phys. 17, 88 (1922).

Brillouin, L.

L. Brillouin, Am. Phys. 17, 88 (1922).

Carome, E. F.

W. H. Nichols and E. F. Carome, J. Chem. Phys. 49, 1000 (1968).

E. F. Carome, W. H. Nichols, C. R. Kinsitis-Swyt, and S. P. Singal, J. Chem. Phys. 49, 1013 (1968).
[Crossref]

De Micheli, R.

A preliminary attempt to consider the spatial incoherence of the acoustic field was made by R. De Micheli and L. Giulotto, Nuovo Cimento 46, 277 (1966).
[Crossref]

Giulotto, L.

A preliminary attempt to consider the spatial incoherence of the acoustic field was made by R. De Micheli and L. Giulotto, Nuovo Cimento 46, 277 (1966).
[Crossref]

Kinsitis-Swyt, C. R.

E. F. Carome, W. H. Nichols, C. R. Kinsitis-Swyt, and S. P. Singal, J. Chem. Phys. 49, 1013 (1968).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1959), p. 391.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1959), p. 391.

Litovitz, T. A.

C. J. Montrose, V. A. Solovyev, and T. A. Litovitz, J. Acoust. Soc. Am. 43, 117 (1968).
[Crossref]

Montrose, C. J.

C. J. Montrose, V. A. Solovyev, and T. A. Litovitz, J. Acoust. Soc. Am. 43, 117 (1968).
[Crossref]

Mountain, R. D.

See, for example, R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966). For a description taking into account the not complete coherence of the incident field see L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

Nichols, W. H.

W. H. Nichols and E. F. Carome, J. Chem. Phys. 49, 1000 (1968).

E. F. Carome, W. H. Nichols, C. R. Kinsitis-Swyt, and S. P. Singal, J. Chem. Phys. 49, 1013 (1968).
[Crossref]

Singal, S. P.

E. F. Carome, W. H. Nichols, C. R. Kinsitis-Swyt, and S. P. Singal, J. Chem. Phys. 49, 1013 (1968).
[Crossref]

Solovyev, V. A.

C. J. Montrose, V. A. Solovyev, and T. A. Litovitz, J. Acoust. Soc. Am. 43, 117 (1968).
[Crossref]

Am. Phys. (1)

L. Brillouin, Am. Phys. 17, 88 (1922).

J. Acoust. Soc. Am. (1)

C. J. Montrose, V. A. Solovyev, and T. A. Litovitz, J. Acoust. Soc. Am. 43, 117 (1968).
[Crossref]

J. Chem. Phys. (2)

W. H. Nichols and E. F. Carome, J. Chem. Phys. 49, 1000 (1968).

E. F. Carome, W. H. Nichols, C. R. Kinsitis-Swyt, and S. P. Singal, J. Chem. Phys. 49, 1013 (1968).
[Crossref]

Nuovo Cimento (1)

A preliminary attempt to consider the spatial incoherence of the acoustic field was made by R. De Micheli and L. Giulotto, Nuovo Cimento 46, 277 (1966).
[Crossref]

Rev. Mod. Phys. (1)

See, for example, R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966). For a description taking into account the not complete coherence of the incident field see L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

Other (1)

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1959), p. 391.

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Figures (3)

Fig. 1
Fig. 1

Power spectrum of the Brillouin line according to the stochastic model for different values of the parameters of the spatial coherence. (a) α′ = β′ = 0.080, (b) α′ = β′ = 0.056, and (c) α′ = β′ = 0.040 (the broadening of the lines by the temporal incoherence is not considered).

Fig. 2
Fig. 2

Power spectrum of the Brillouin line for α′ = 0.080 in the case of unlimited transversal coherence.

Fig. 3
Fig. 3

Power spectrum in the Brillouin line for β′ = 0.080 in the case of unlimited longitudinal coherence.

Equations (31)

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ρ K ( r , t ) = ρ ˚ ( K ) exp i ( K · r - Ω t ) ,
A ( k 1 , k 2 ) V ρ ˚ ( K ) e i ( K - k 2 + k 1 ) · r d r .
± K 0 = q 0 .
ρ K ( r , t ) = ρ ˚ ( K ) exp [ - t / τ + i ( K · r - Ω t ) ] .
L ( ω ) ρ ˚ ( K 0 ) 2 1 / τ ( 1 / τ ) 2 + ( ω ± Ω 0 ) 2 ,
ρ K ( r , t ) = ρ 0 exp i ( K · r - Ω t )
ρ K ( r , t ) = ρ 0 exp i [ K · r - Ω t + φ ( r , t ) ] .
Ψ D ( x , y , z , t ) = ρ K ( 1 ) ρ K * ( 2 ) p 1 [ ρ K ( 1 ) ] × p 12 [ ρ K ( 2 ) ρ K ( 1 ) ] d ρ K ( 1 ) d ρ K ( 2 ) .
ρ K ( 1 ) = ρ 0 e i φ ,
ρ K ( 2 ) = ρ 0 exp i [ Ω t - K z + φ + ( m - 2 u ) Δ φ 1 + ( m - 2 u ) Δ φ s + ( n - 2 v ) Δ φ t ] ,
p 12 [ ρ K ( 2 ) ρ K ( 1 ) ] = exp ( - z / Λ ) ( z / Λ ) m m ! 1 2 m C m u exp [ - ( x 2 + y 2 ) 1 2 / σ ] × 1 2 m C m u exp ( - t / τ ) ( t / τ ) n n ! 1 2 n C n v ,
C m u = m ! / [ ( m - u ) ! u ! ] ,             C m u = m ! / [ ( m - u ) ! u ! ] ,
Ψ K ( x , y , z , t ) = ρ 0 2 0 2 π 1 2 π m = 0 m = 0 n = 0 u = 0 m u = 0 m v = 0 n exp ( - z / λ ) × ( z / λ ) m m ! 1 2 m C m u exp [ - ( x 2 + y 2 ) 1 2 / σ ] [ ( x 2 + y 2 ) 1 2 / σ ] m m ! × 1 2 m C m u exp ( - t / τ ) ( t / τ ) n n ! 1 2 n C n v exp ( - i φ ) × exp i [ Ω t - K z + φ + ( m - 2 u ) Δ φ 1 + ( m - 2 u ) Δ φ s + ( n - 2 v ) Δ φ t ] d φ .
Ψ K ( x , y , z , t ) = ρ 0 2 m = 0 u = 0 m exp ( - z / λ ) ( z / λ ) m m 1 2 m C m u × exp i [ ( m - 2 u ) Δ φ 1 ] m = 0 u = 0 m exp [ - ( x 2 + y 2 ) 1 2 / σ ] × [ ( x 2 + y 2 ) 1 2 / σ ] m m ! 1 2 m C m u exp i [ ( m - 2 u ) Δ φ s ] × n = 0 v = 0 n exp ( - t / τ ) ( t / τ ) n n ! 1 2 n C n u × exp i [ ( n - 2 v ) Δ φ t ] exp i ( Ω t - K z ) .
Ψ K ( x , y , z , t ) = ρ 0 2 m = 0 exp ( - z / λ ) ( z / λ ) m m ! cos m Δ φ 1 × m = 0 exp [ - ( x 2 + y 2 ) 1 2 / σ ] [ ( x 2 + y 2 ) 1 2 / σ ] m m ! cos m Δ φ s × n = 0 exp ( - t / τ ) ( t / τ ) n n ! cos n Δ φ t exp i ( Ω t - K z ) .
Ψ K ( x , y , z , t ) = ρ 0 2 exp [ - z ( 1 - cos Δ φ s ) / λ - ( x 2 + y 2 ) 1 2 ( 1 - cos Δ φ s ) / σ - t ( 1 - cos Δ φ t ) / τ ] exp i ( Ω t - K z ) .
α = ( 1 - cos Δ φ 1 ) / λ ,             β = ( 1 - cos Δ φ s ) / σ ,
γ = ( 1 - cos Δ φ , ) / τ .
Ψ K ( x , y , z , t ) = ρ 0 2 exp [ - α z - β ( x 2 + y 2 ) 1 2 - γ t ] × exp i ( Ω t - K z ) .
Ψ ˚ K ( q x , q y , q z ) = - exp ( - α z - β ( x 2 + y 2 ) 1 2 - γ t ) × exp i [ ( K - q x ) z - q x x - q y y ] d x d y d z .
Ψ ˚ K ( q x , q y , q z ) = Ψ ˚ K ( q ζ , q r ) = d ζ 0 2 π d ϕ 0 d r r × exp ( - α ζ - β r ) exp [ - i r q r cos ( ϕ - θ ) ] ,
J 0 ( r q r ) = 1 2 π 0 2 π exp [ - i r q r cos ( ϕ - θ ) ] d ϕ
0 r exp ( - β r ) J 0 ( r q r ) d r = β ( β 2 + q r 2 ) 3 2 .
Ψ ˚ K ( q ζ , q r ) = ρ 0 2 α [ α 2 + ( K - q ζ ) 2 · β ( β 2 + q r 2 ) 3 2 .
Ψ ˚ q 0 ( K , ψ ) = ρ 0 2 α [ α 2 + ( K - q 0 cos ψ ) 2 ] · β [ β 2 + ( q 0 sen ψ ) 2 ] 3 2 ,
S q 0 ( K , ψ , ω ) = Ψ ˚ q 0 ( K , ψ ) L ( K , ω ) ,
L ( ω , K ) = L ( ω - Ω ) = γ / [ γ 2 + ( ω - Ω ) 2 ] .
S q 0 ( K , ψ , ω ) S q 0 ( Ω , ψ , ω ) Ψ ˚ q 0 ( Ω , ψ ) L ( ω - Ω ) .
S q 0 ( Ω , ω ) = I q 0 ( Ω ) L ( ω - Ω ) ,
I q 0 ( Ω ) = I q 0 ( Ω / Ω 0 ) = 0 1 α [ α 2 + ( Ω / Ω 0 - cos ψ ) 2 ] × β ( β 2 + 1 - cos 2 ψ ) 3 2 d cos ψ ,
S q 0 ( ω ) = - I q 0 ( Ω ) L ( ω - Ω ) d Ω ,