Abstract

Laser-induced thermal acoustics (LITA) is a four-wave mixing technique that may be employed to measure sound speeds, transport properties, velocities, and susceptibilities of fluids. It is particularly effective in high-pressure gases (>1 bar). An analytical expression for LITA signals is derived by the use of linearized equations of hydrodynamics and light scattering. This analysis, which includes full finite-beam-size effects and the optoacoustic effects of thermalization and electrostriction, predicts the amplitude and the time history of narrow-band time-resolved LITA and broadband spectrally resolved (mulitplex) LITA signals. The time behavior of the detected LITA signal depends significantly on the detection solid angle, with implications for the measurement of diffusivities by the use of LITA and the proper physical picture of LITA scattering. This and other elements of the physics of LITA that emerge from the analysis are discussed. Theoretical signals are compared with experimental LITA data.

© 1995 Optical Society of America

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References

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  1. E. B. Cummings, “Laser-induced thermal acoustics: simple accurate gas measurements,” Opt. Lett. 19, 1361–1363 (1994).
    [CrossRef] [PubMed]
  2. D. E. Govoni, J. A. Booze, A. Sinha, F. F. Crim, “The non-resonant signal in laser-induced grating spectroscopy,” Chem. Phys. Lett. 216, 525–529 (1993).
    [CrossRef]
  3. S. Williams, L. A. Rahn, P. H. Paul, J. W. Forsman, “Laser-induced thermal grating effects in flames,” Opt. Lett. 19, 1361–1363 (1994).
    [CrossRef]
  4. H. J. Eichler, P. Günter, D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986), pp. 84–91.
  5. A. C. Eckbreth, Laser Diagnostics for Combustion Species and Temperature (Abacus, Cambridge, Mass., 1988), pp. 272–277.
  6. E. B. Cummings, “Techniques of single-shot thermometry by degenerate four-wave mixing,” GALCIT Rep. number FM 92-2 (California Institute of Technology, Pasadena, Calif., 1992).
  7. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), pp. 233–240.
  8. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 126–143, 327–331.
  9. L. D. Landau, E. M. Liftshitz, Electrodynamics of Continuous Media (Addison-Wesley, Cambridge, Mass., 1960), 377–383.
  10. D. R. Meacher, P. G. R. Smith, P. Ewart, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
    [CrossRef] [PubMed]
  11. E. B. Cummings, “Laser induced thermal acoustics,” M. S. thesis (California Institute of Technology, Pasadena, Calif., 1995), pp. 89–92.

1994 (2)

1993 (1)

D. E. Govoni, J. A. Booze, A. Sinha, F. F. Crim, “The non-resonant signal in laser-induced grating spectroscopy,” Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

1992 (1)

D. R. Meacher, P. G. R. Smith, P. Ewart, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), pp. 233–240.

Booze, J. A.

D. E. Govoni, J. A. Booze, A. Sinha, F. F. Crim, “The non-resonant signal in laser-induced grating spectroscopy,” Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 126–143, 327–331.

Crim, F. F.

D. E. Govoni, J. A. Booze, A. Sinha, F. F. Crim, “The non-resonant signal in laser-induced grating spectroscopy,” Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

Cummings, E. B.

E. B. Cummings, “Laser-induced thermal acoustics: simple accurate gas measurements,” Opt. Lett. 19, 1361–1363 (1994).
[CrossRef] [PubMed]

E. B. Cummings, “Techniques of single-shot thermometry by degenerate four-wave mixing,” GALCIT Rep. number FM 92-2 (California Institute of Technology, Pasadena, Calif., 1992).

E. B. Cummings, “Laser induced thermal acoustics,” M. S. thesis (California Institute of Technology, Pasadena, Calif., 1995), pp. 89–92.

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Species and Temperature (Abacus, Cambridge, Mass., 1988), pp. 272–277.

Eichler, H. J.

H. J. Eichler, P. Günter, D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986), pp. 84–91.

Ewart, P.

D. R. Meacher, P. G. R. Smith, P. Ewart, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Forsman, J. W.

Govoni, D. E.

D. E. Govoni, J. A. Booze, A. Sinha, F. F. Crim, “The non-resonant signal in laser-induced grating spectroscopy,” Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

Günter, P.

H. J. Eichler, P. Günter, D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986), pp. 84–91.

Landau, L. D.

L. D. Landau, E. M. Liftshitz, Electrodynamics of Continuous Media (Addison-Wesley, Cambridge, Mass., 1960), 377–383.

Liftshitz, E. M.

L. D. Landau, E. M. Liftshitz, Electrodynamics of Continuous Media (Addison-Wesley, Cambridge, Mass., 1960), 377–383.

Meacher, D. R.

D. R. Meacher, P. G. R. Smith, P. Ewart, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Paul, P. H.

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), pp. 233–240.

Pohl, D. W.

H. J. Eichler, P. Günter, D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986), pp. 84–91.

Rahn, L. A.

Sinha, A.

D. E. Govoni, J. A. Booze, A. Sinha, F. F. Crim, “The non-resonant signal in laser-induced grating spectroscopy,” Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

Smith, P. G. R.

D. R. Meacher, P. G. R. Smith, P. Ewart, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Williams, S.

Chem. Phys. Lett. (1)

D. E. Govoni, J. A. Booze, A. Sinha, F. F. Crim, “The non-resonant signal in laser-induced grating spectroscopy,” Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

D. R. Meacher, P. G. R. Smith, P. Ewart, “Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers,” Phys. Rev. A 46, 2718–2725 (1992).
[CrossRef] [PubMed]

Other (7)

E. B. Cummings, “Laser induced thermal acoustics,” M. S. thesis (California Institute of Technology, Pasadena, Calif., 1995), pp. 89–92.

H. J. Eichler, P. Günter, D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986), pp. 84–91.

A. C. Eckbreth, Laser Diagnostics for Combustion Species and Temperature (Abacus, Cambridge, Mass., 1988), pp. 272–277.

E. B. Cummings, “Techniques of single-shot thermometry by degenerate four-wave mixing,” GALCIT Rep. number FM 92-2 (California Institute of Technology, Pasadena, Calif., 1992).

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976), pp. 233–240.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 126–143, 327–331.

L. D. Landau, E. M. Liftshitz, Electrodynamics of Continuous Media (Addison-Wesley, Cambridge, Mass., 1960), 377–383.

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Figures (6)

Fig. 1
Fig. 1

Optoacoustic generation of phonons and thermons. The curves are field profiles along the cross section A–A of the grating, shown at left. Laser driving and thermalization are assumed to be instantaneous. At time t = 0 the density fields of the phonons and the thermon sum to 0. Afterward, motion of the phonons and damping modulate the density grating amplitude.

Fig. 2
Fig. 2

LITA signal versus driver-laser tuning. Electrostriction, proportional to Re{χ(ω d )}, produces a signal modulated at twice the Brillouin frequency. Thermalization, proportional to −Im{χ(ω d )}, produces a signal modulated at the Brillouin frequency.

Fig. 3
Fig. 3

Experimental LITA signal and theoretical fit by the use of Eqs. (A1) with published calculated properties of atmospheric air, w = 162 μm, σ = 173 μm, q ϕ = 2π/25.3 μm. The driver-laser frequency is tuned to an absorption peak of NO2, present in concentrations of <50 parts in 109.

Fig. 4
Fig. 4

Experimental LITA signal and theoretical fit. The driver-laser frequency is 36 GHz higher than that for the data in Fig. 3. All other parameters are the same as in Fig. 3. The only fitting parameters are U θ , U e and γθ. The signal is normalized with respect to the signal in Fig. 3.

Fig. 5
Fig. 5

Experimental LITA signal and theoretical fit. The driver-laser frequency is 72 GHz higher than that for the data in Fig. 3. Otherwise, the description is the same as for Fig. 4.

Fig. 6
Fig. 6

Experimental LITA signal and theoretical fit. The driver-laser frequency is 120 GHz higher than that for the data in Fig. 3. Otherwise, the description is the same as for Fig. 4.

Equations (52)

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ρ 1 t + ρ ψ 1 = 0 ,
ψ 1 t + c s 2 2 ρ 1 γ ρ + α c s 2 γ 2 T 1 - D V 2 ψ 1 = ψ ˙ ( r , t ; u ) ,
T 1 t - γ - 1 α ρ ρ 1 t - γ D T 2 T 1 = T ˙ ( r , t ; u ) ,
ρ 1 ( r , t = 0 ) = 0 ,             ψ 1 ( r , t = 0 ) = 0 ,             T 1 ( r , t = 0 ) = 0 , ρ 1 ( r , t ) 0 ,             ψ 1 ( r , t ) 0 ,             T 1 ( r , t ) 0 ,             as r .
ρ 1 ( q , s ) ρ = M ( q , s ) - 1 × [ - c s q ( s + γ D T q 2 ) ψ ˙ c s q - ( c s 2 q 2 γ ) T ˙ T ] ,
ψ 1 ( q , s ) c s q = M ( q , s ) - 1 [ s ( s + γ D T q 2 ) ψ ˙ c s q + ( s c s q γ ) T ˙ T ] ,
T 1 ( q , s ) T = M ( q , s ) - 1 × ( - ( γ - 1 ) s ψ ˙ + [ s ( s + q 2 D V ) + c s 2 q 2 γ ] T ˙ T ) ,
M ( q , s ) = s 3 + ( D V q 2 + γ D T q 2 ) s 2 + ( c s 2 q 2 + γ D T q 2 D V q 2 ) s + c s 2 q 2 D T q 2 .
s 1 = - D T q 2 ,             s 2 = - Γ q 2 + i c s q , s 3 = - Γ q 2 - i c s q ,
ρ 1 ρ = [ T ˙ γ T + ( s + γ D T q 2 ) c s q ψ ˙ c s q ] { 1 [ 1 + ( Δ - G ) 2 ] } × [ 1 + i ( Δ - G ) 2 ( s + Γ q 2 - i c s q ) + 1 - i ( Δ - G ) 2 ( s + Γ q 2 + i c s q ) - 1 ( s + D T q 2 ] ,
E d 1 = E 1 + E 1 * ,             E d 2 = E 2 + E 2 * ,
E 1 ( x , y , z , t ) = E ( t ) 2 ( 2 π w 2 ) 1 / 2 × exp [ - ( y cos θ - x sin θ ) 2 + z 2 w 2 ] × exp i [ ω d t - k d ( x cos θ + y sin θ ] , E 2 ( x , y , z , t ) = E ( t ) 2 ( 2 π w 2 ) 1 / 2 × exp [ - ( y cos θ + x sin θ ) 2 + z 2 w 2 ] × exp i [ ω d t - k d ( x cos θ - y sin θ ) ] ,
E 1 * ( r , t ) E 2 ( r , t ) + E 2 * ( r , t ) E 1 ( r , t ) = E d I d ( r ) P d ( t ) ,
I d ( r ) = 2 π w 2 exp [ - 2 ( x 2 I 2 + y 2 h 2 + z 2 w 2 ) ] cos ( q ϕ y ) .
I d ( q ) = 2 π l h 4 w exp ( - l 2 q x 2 8 - w 2 q z 2 8 ) × ( exp [ - h 2 ( q y - q ϕ ) 2 8 ] + exp [ - h 2 ( q y + q ϕ ) 2 8 ] ) .
U e ( t ) = - Re { χ m ( ω ) } 0 ( E 2 / 2 ) ,
ψ ˙ e ( r , t ) = - Re { χ ( ω d ) } ρ 2 [ E d I d ( r ) P d ( t ) ] ,
ψ ˙ e ( q , s ) = - Re { χ ( ω d ) } ρ q 2 E d I d ( q ) P d ( s ) .
U ex t + γ θ U ex + γ n θ U ex - D s 2 U ex = - 2 k d Im { χ ( ω d ) } [ E d P d ( t ) I d ( r ) ] , U ex ( r , t = 0 ) = 0.
U θ ( q , s ) = - 2 γ θ k d Im { χ ( ω d ) } E d I d ( q ) P ( s ) s ( s + γ n θ + γ θ + D s q 2 ) .
T ˙ ( q , s ) = - 2 γ θ k d Im { χ ( ω d ) } E d I d ( q ) P ( s ) ρ c v ( s + γ n θ + γ θ + D s q 2 ) .
T ˙ ( q , s ) = - 2 k d Im { χ ( ω d ) } E d I d ( q ) P d ( s ) Θ ( q , s ) ρ c v ,
ρ 1 ( q , s ) ρ = I d ( q ) P d ( s ) E d [ - 2 k d γ θ Im { χ ( ω d ) } ρ c p T × 1 ( s + γ n θ + γ θ + D s q 2 ) - Re { χ ( ω d ) } ρ c s 2 × ( s + γ D T q 2 ) ] ( 1 [ 1 + ( Δ - G ) 2 ] ) × [ 1 + i ( Δ - G ) 2 ( s + Γ q 2 - i c s q ) + 1 + i ( Δ - G ) 2 ( s + Γ q 2 + i c s q ) - 1 ( s + D T q 2 ) ] .
U θ = - 2 k d Im { χ ( ω d ) } E d w 2 ρ c p T γ θ γ n θ + γ θ + D s q 2 , U e = - q Re { χ ( ω d ) } E d w 2 ρ c s 2 .
f ( t )     g ( t ) 0 f ( τ ) g ( t - τ ) d τ .
ρ 1 ( q , t ) ρ = - w 2 I d ( q ) P d ( t )     [ H θ ( q , t ) U θ + H e ( q , t ) U e ] .
H θ ( q , t ) H θ P ( q ) Φ P ( q , t ) + H θ P * ( q ) Φ P * ( q , t ) + H θ T ( q ) Φ T ( q , t ) + H θ D ( q ) Φ D ( q , t ) ,
H e ( q , t ) H e P ( q ) Φ P ( q , t ) + H e P * ( q ) Φ P * ( q , t ) + H e T ( q ) Φ T ( q , t ) ,
Φ P ( q , t ) exp ( - Γ q 2 t + i c s q t ) , Φ T ( q , t ) exp ( - D T q 2 t ) , Φ D ( q , t ) exp [ - ( γ θ + γ n θ ) t - D s q 2 t ] ,
H θ P [ 1 + i ( Δ - G ) ] ( 1 - G Π - i Π ) 2 [ 1 + ( Δ - G ) 2 ] [ ( 1 - G Π ) 2 + Π 2 ] , H θ T - 1 [ 1 + ( Δ - G ) 2 ] ( 1 - Δ Π ) , H θ D Π 2 [ ( 1 - G Π ) 2 + Π 2 ] ( 1 - Δ Π ) ,
H e P i [ 1 - i ( γ Δ - G ) ] [ 1 + i ( Δ - G ) ] 2 [ 1 + ( Δ - G ) 2 ] , H e T ( γ - 1 ) Δ [ 1 + ( Δ - G ) 2 ] ,
Π c s q / ( γ θ + γ n θ + D s q 2 ) .
Φ P ( d ) ( r ) = 2 l h π w Λ P Ω P H P × exp [ - 2 x 2 Λ P 2 - 2 z 2 Ω P 2 - 2 ( y + c s t ) 2 H P s Γ q ϕ 2 t η P 2 ] × cos [ ( y + c s t ) q ϕ η P 2 ] ,
Φ P * ( d ) ( r ) = 2 l h π w Λ P * Ω P * H P × exp [ - 2 x 2 Λ P * 2 - 2 z 2 Ω P * 2 - 2 ( y - c s t ) 2 H P 2 Γ q ϕ 2 t η P 2 ] × cos [ ( y - c s t ) q ϕ η P 2 ] ,
Φ T ( d ) ( r ) = 2 l h π w Λ T Ω T H T × exp ( - 2 x 2 Λ T 2 - 2 y 2 H T 2 - 2 z 2 Ω T 2 - D T q Φ 2 t η T 2 ) × cos ( y q ϕ η T 2 ) ,
Φ D ( d ) ( r ) = 2 l h π w Λ D Ω D H D × exp [ - 2 x 2 Λ D 2 - 2 y 2 H D 2 - 2 z 2 Ω D 2 - D s q ϕ 2 t η D 2 - ( γ θ + γ n θ ) t ] cos ( y q ϕ η D 2 ) ,
Λ P 2 l 2 + 8 Γ t - 4 i c s t / q ϕ ,             Λ T 2 l 2 + 8 D T t , Λ D 2 l 2 + 8 D s t ,             H P 2 h 2 + 8 Γ t ,             H T 2 h 2 + 8 D T t , H D 2 h 2 + 8 D s t ,             Ω P 2 w 2 + 8 Γ t - 4 i c s t / q ϕ , Ω T 2 w 2 + 8 D T t ,             Ω D 2 w 2 + 8 D s t , η P h / H P ,             η T h / H T ,             η D h / H D .
E s ( R , t ; q ) = - k s 2 4 π R cos ( k s · R - ω o t ) μ 1 ( q , t ) ,
μ 1 ( r , t ) = χ 1 ( r , t ; ω o ) E o ( r , t ) .
I o = ( 2 π σ 2 ) 1 / 2 exp [ - ( y σ y - x σ x ) 2 - z σ ] ;
Φ P ( d , o ) ( q ) = N P Ψ P ( q ) Σ P ( t ) ,             Φ T ( d , o ) = N T Ψ T ( q ) Σ T ( t ) , Φ D ( d , o ) = N D Ψ D ( q ) Σ D ( t ) ,
N P ( π ξ P ζ P ) 1 / 2 l h w σ H P Ω P ,             N T ( π ξ T ζ T ) 1 / 2 l h w σ H T Ω T , N D ( π ξ D ζ D ) 1 / 2 l h w σ H D Ω D ,
Ψ P ( ξ P ζ P 4 π ) 1 / 2 exp ( - ξ P 2 q y 2 8 - ζ P 2 q z 2 8 + ξ P 2 H P 2 i c s q y t ) , Ψ T ( ξ T ζ T 8 π ) 1 / 2 exp ( - ξ T 2 q y 2 8 - ζ T 2 q z 2 8 ) , Ψ D ( ξ D ζ D 8 π ) 1 / 2 exp ( - ξ D 2 q y 2 8 - ζ D 2 q z 2 8 ) ,
Σ P exp ( - 2 c s 2 t 2 H P 2 + 2 σ y 2 - Γ q ϕ 2 t η P 2 + i q ϕ c s t η P 2 ) , Σ T exp ( - D T q ϕ 2 t η T 2 ) , Σ D exp [ - D s q ϕ 2 t η D 2 - ( γ θ + γ n θ ) t ] ,
ξ { P , T , D } ( 1 H { P , T , D } 2 + 1 2 σ y 2 ) - 1 / 2 , ζ { P , T , D } ( 1 Ω { P , T , D } 2 + 1 2 σ 2 ) - 1 / 2
E s ( q , R , t ) P o ( t ) = - k s 2 w 2 4 π R χ ( ω o ) exp i ( k s · R - ω o t ) P d ( t ) [ 2 Re { Φ P ( d , o ) ( q , t ) ( U θ H θ P + U e H e P ) } + Φ T ( d , o ) ( q , t ) ( U θ H θ T + U e H e T ) + Φ D ( d , o ) ( q , t ) U θ H θ D ] .
L het P o ( t ) = π l h w σ χ ( ω o ) k o 2 w 2 4 π × exp ( - i ω o t ) P d ( t ) ( Re { ( ξ P ζ P ) 1 / 2 H P Ω P × exp ( - 2 c s 2 t 2 H P 2 - Γ q ϕ 2 t η P 2 + i q ϕ c s t η P 2 ) ) × ( U θ H θ P + U e H e P ) } + ( ξ T ζ T ) 1 / 2 H T Ω T × exp ( - D T q ϕ 2 t η T 2 ) ( U θ H θ T + U e H e T ) + ( ξ D ζ D ) 1 / 2 H D Ω D × exp [ - D s q ϕ 2 t η D 2 - ( γ θ + γ n θ ) t ] U θ H θ D ) ,
A P U θ H θ P + U e H e P ,             A T U θ H θ T + U e H e T , A D U θ H θ D .
L hom P o 2 ( t ) = π 4 l 2 λ o 2 χ ( ω o ) 2 ξ ζ σ 2 cos 2 ϕ cos 2 θ ( Re { 2 A P 2 ( w 2 ζ P Ω P 2 ζ ) × exp ( - 4 c s 2 t 2 h 2 + 2 σ y 2 - 2 Γ q ϕ 2 t + 2 i q ϕ c s t ) × exp ( - 4 ξ 2 c s 2 t 2 h 4 ) + 4 A P A T [ ( 2 ζ P 2 ζ P 2 + ζ 2 ) ] 1 / 2 w Ω P × exp [ - 2 c s 2 t 2 h 2 + 2 σ y 2 - ( D T + Γ ) q θ 2 t + i q ϕ c s t ] × exp ( - ξ 2 c s 2 t 2 h 4 ) + 4 A P A D [ ( 2 ζ P 2 ζ P 2 + ζ 2 ) ] 1 / 2 w Ω P × exp [ - 2 c s 2 t 2 h 2 + 2 σ y 2 - ( D s + Γ ) q ϕ 2 t + i q ϕ c s t - ( γ θ + γ n θ ) t ] exp ( - ξ 2 c s 2 t 2 h 4 ) } + 2 A P A P * ( w 2 ζ P Ω P 2 ζ ) × exp ( - 4 c s 2 t 2 h 2 + 2 σ y 2 - 2 Γ q ϕ 2 t ) + 2 A T A D × exp [ - ( D T + D s ) q ϕ 2 t - ( γ θ + γ n θ ) t ] + A T 2 exp ( - 2 D T q ϕ 2 t ) + A D 2 exp [ - 2 D s q ϕ 2 t - 2 ( γ θ + γ n θ ) t ] ) ,
ξ ( 1 h 2 + 1 2 σ y 2 ) - 1 / 2 ,             ζ ( 1 w 2 + 1 2 σ 2 ) - 1 / 2 .
L hom P o 2 ( t ) = π 4 ( l λ o ) 2 χ ( ω o ) 2 ( w σ ) 2 × ( Re { 2 A P 2 exp ( - 2 Γ q ϕ 2 t + 2 i q ϕ c s t - 4 c s 2 t 2 w 2 ) + 4 A P A T exp [ - ( D T + Γ ) q ϕ 2 t + i q ϕ c s t - c s 2 t 2 w 2 ] + 4 A P A D exp [ - ( D s + Γ ) q ϕ 2 t + i q ϕ c s t - ( γ θ + γ n θ ) t - ξ 2 c s 2 t 2 h 4 ] } + 2 A P A P * × exp ( - 2 Γ q ϕ 2 t ) + 2 A T A D × exp [ - ( D T + D s ) q ϕ 2 t - ( γ θ + γ n θ ) t ] + A T 2 exp ( - 2 D T q ϕ 2 t ) + A D 2 × exp [ - 2 D s q ϕ 2 t - 2 ( γ θ + γ n θ ) t ] ) .
L hom P o 2 ( t ) = k o 2 w 4 16 π 2 cos 2 ϕ χ ( ω o ) 2 0 0 P d ( t - τ 1 ) P d ( t - τ 2 ) × [ Re { 2 A P 2 N P ( τ 1 ) N P ( τ 2 ) × ( 4 ζ P ( τ 1 ) ξ P ( τ 1 ) ζ P ( τ 2 ) ξ P ( τ 2 ) [ ξ P 2 ( τ 1 ) + ξ P 2 ( τ 2 ) ] [ ζ P 2 ( τ 1 ) + ζ P 2 ( τ 2 ) ] ) 1 / 2 × exp ( - 2 c s 2 [ τ 1 2 ξ P 2 ( τ 1 ) H P 4 ( τ 1 ) + τ 2 2 ξ P 2 ( τ 2 ) H P 4 ( τ 2 ) ] ) Σ P ( τ 1 ) Σ P ( τ 2 ) + 4 A P A T N P ( τ 1 ) N T ( τ 2 ) × ( 4 ζ P ( τ 1 ) ξ P ( τ 1 ) ξ T ( τ 2 ) ξ T ( τ 2 ) [ ξ P 2 ( τ 1 ) + ξ T 2 ( τ 2 ) ] [ ζ P 2 ( τ 1 ) + ζ T 2 ( τ 2 ) ] ) 1 / 2 × exp ( - 2 ξ P 4 ( τ 1 ) c s 2 τ 1 2 H P 4 ( τ 1 ) [ ξ P 2 ( τ 1 ) + ξ T 2 ( τ 2 ) ] ) Σ P ( τ 1 ) Σ T ( τ 2 ) + 4 A P A D N P ( τ 1 ) N D ( τ 2 ) × ( 4 ζ P ( τ 1 ) ξ P ( τ 1 ) ζ D ( τ 2 ) ξ D ( τ 2 ) [ ξ P 2 ( τ 1 ) + ξ D 2 ( τ 2 ) ] [ ζ P 2 ( τ 1 ) + ζ D 2 ( τ 2 ) ] ) 1 / 2 × exp ( - 2 ξ P 4 ( τ 1 ) c s 2 τ 1 2 H P 4 ( τ 1 ) [ ξ P 2 ( τ 1 ) + ξ D 2 ( τ 2 ) ] ) Σ P ( τ 1 ) Σ D ( τ 2 ) } + 2 A P A P * N P ( τ 1 ) N P * ( τ 2 ) × ( 4 ζ P ( τ 1 ) ξ P ( τ 1 ) ζ P ( τ 2 ) ξ P ( τ 2 ) [ ξ P 2 ( τ 1 ) + ξ P 2 ( τ 2 ) ] [ ζ P 2 ( τ 1 ) + ζ P 2 ( τ 2 ) ] ) 1 / 2 × Σ P ( τ 1 ) Σ P * ( τ 2 ) + 2 A T A D N T ( τ 1 ) N D ( τ 2 ) × ( 4 ζ T ( τ 1 ) ξ T ( τ 1 ) ζ D ( τ 2 ) ξ D ( τ 2 ) [ ξ T 2 ( τ 1 ) + ξ D 2 ( τ 2 ) ] [ ζ T 2 ( τ 1 ) + ζ D 2 ( τ 2 ) ] ) 1 / 2 × Σ T ( τ 1 ) Σ D ( τ 2 ) + A T 2 N T ( τ 1 ) N T ( τ 2 ) × ( 4 ζ T ( τ 1 ) ξ T ( τ 1 ) ζ T ( τ 2 ) ξ T ( τ 2 ) [ ξ T 2 ( τ 1 ) + ξ T 2 ( τ 2 ) ] [ ζ T 2 ( τ 1 ) + ζ T 2 ( τ 2 ) ] ) 1 / 2 × Σ T ( τ 1 ) Σ T ( τ 2 ) + A D 2 N D ( τ 1 ) N D ( τ 2 ) × ( 4 ζ D ( τ 1 ) ξ D ( τ 1 ) ζ D ( τ 2 ) ξ D ( τ 2 ) [ ξ D 2 ( τ 1 ) + ξ D 2 ( τ 2 ) ] [ ζ D 2 ( τ 1 ) + ζ D 2 ( τ 2 ) ] ) 1 / 2 × Σ D ( τ 1 ) Σ D ( τ 2 ) ] d τ 1 d τ 2 .

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