Abstract

Optical parametric amplifiers using chirped quasi-phase-matching gratings offer gain over wide bandwidths, making them promising candidates for use in ultra-short-pulse laser systems. We discuss the space-time evolution of the amplified light pulses. In the case of a linear phase-matching profile, this problem is exactly solvable. The Green’s functions of the system are investigated in detail and used to calculate the pulse shapes in the long- and short-pulse regimes.

© 2008 Optical Society of America

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References

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  1. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings. I. Practical design formulas,” J. Opt. Soc. Am. B 25, 463-480 (2008).
    [CrossRef]
  2. L. D. Landau, “Zur Theorie der Energieübertragung,” Phys. Z. Sowjetunion 2, 46-51 (1932).
  3. C. Zener, “Non-adiabatic crossing of energy levels,” Proc. R. Soc. London, Ser. A 37, 696-702 (1932).
  4. E. C. G. Stueckelberg, “Theorie der un elastischen Stösse zwischen Atomen,” Helv. Phys. Acta 5, 369-422 (1932).
    [CrossRef]
  5. M. N. Rosenbluth, R. B. White, and C. S. Liu, “Temporal evolution of a three-wave parametric instability,” Phys. Rev. Lett. 31, 1190-1193 (1973).
    [CrossRef]
  6. F. W. Chambers, Ph.D. dissertation (Massachusetts Institute of Technology, 1975).
  7. R. W. Short and A. Simon, “Theory of three-wave parametric instabilities in inhomogeneous plasmas revisited,” Phys. Plasmas 11, 5335-5340 (2004).
    [CrossRef]
  8. B. Afeyan and M. M. Fejer, “Short pulse optical parametric processes with group velocity mismatch and quasi-phase matching,” presented at Nonlinear Optics 2002, Maui, Hawaii, July 29-August 2 (2002).
  9. D. L. Bobroff and H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390-403 (1967).
    [CrossRef]
  10. R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064-1068 (1968).
    [CrossRef]
  11. J. A. Armstrong, S. S. Jha, and N. S. Shiren, “Some effects of group-velocity dispersion on parametric interactions,” IEEE J. Quantum Electron. 6, 123-129 (1970).
    [CrossRef]
  12. S. E. Harris, “Proposed backward wave oscillator in the infrared,” Appl. Phys. Lett. 9, 114-116 (1966).
    [CrossRef]
  13. R. J. Briggs, Electron-Stream Interaction with Plasmas (MIT Press, 1964).
  14. E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Butterworth-Heinemann Ltd., 1981).
  15. A. Bers, Basic Plasma Physics I, A.A.Galeev and R.N.Sudan, eds. (North-Holland, 1983).
  16. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
    [CrossRef]
  17. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).
  18. M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. 29, 565-567 (1972).
    [CrossRef]
  19. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  20. G. Picard and T. W. Johnston, “Absolute parametric instabilities in inhomogeneous plasmas,” Phys. Rev. Lett. 51, 574-577 (1983).
    [CrossRef]
  21. E. A. Williams, J. R. Albritton, and M. N. Rosenbluth, “Effect of spatial turbulence on parametric instabilities,” Phys. Fluids 22, 139-149 (1979).
    [CrossRef]
  22. D. F. DuBois, D. W. Forslund, and E. A. Williams, “Parametric instabilities in finite inhomogeneous media,” Phys. Rev. Lett. 33, 1013-1016 (1974).
    [CrossRef]
  23. F. W. Chambers and A. Bers, “Parametric interactions in an inhomogeneous medium of finite extent with abrupt boundaries,” Phys. Fluids 20, 466-468 (1977).
    [CrossRef]

2008

2004

R. W. Short and A. Simon, “Theory of three-wave parametric instabilities in inhomogeneous plasmas revisited,” Phys. Plasmas 11, 5335-5340 (2004).
[CrossRef]

1992

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

1983

G. Picard and T. W. Johnston, “Absolute parametric instabilities in inhomogeneous plasmas,” Phys. Rev. Lett. 51, 574-577 (1983).
[CrossRef]

1979

E. A. Williams, J. R. Albritton, and M. N. Rosenbluth, “Effect of spatial turbulence on parametric instabilities,” Phys. Fluids 22, 139-149 (1979).
[CrossRef]

1977

F. W. Chambers and A. Bers, “Parametric interactions in an inhomogeneous medium of finite extent with abrupt boundaries,” Phys. Fluids 20, 466-468 (1977).
[CrossRef]

1974

D. F. DuBois, D. W. Forslund, and E. A. Williams, “Parametric instabilities in finite inhomogeneous media,” Phys. Rev. Lett. 33, 1013-1016 (1974).
[CrossRef]

1973

M. N. Rosenbluth, R. B. White, and C. S. Liu, “Temporal evolution of a three-wave parametric instability,” Phys. Rev. Lett. 31, 1190-1193 (1973).
[CrossRef]

1972

M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. 29, 565-567 (1972).
[CrossRef]

1970

J. A. Armstrong, S. S. Jha, and N. S. Shiren, “Some effects of group-velocity dispersion on parametric interactions,” IEEE J. Quantum Electron. 6, 123-129 (1970).
[CrossRef]

1968

R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064-1068 (1968).
[CrossRef]

1967

D. L. Bobroff and H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390-403 (1967).
[CrossRef]

1966

S. E. Harris, “Proposed backward wave oscillator in the infrared,” Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

1932

L. D. Landau, “Zur Theorie der Energieübertragung,” Phys. Z. Sowjetunion 2, 46-51 (1932).

C. Zener, “Non-adiabatic crossing of energy levels,” Proc. R. Soc. London, Ser. A 37, 696-702 (1932).

E. C. G. Stueckelberg, “Theorie der un elastischen Stösse zwischen Atomen,” Helv. Phys. Acta 5, 369-422 (1932).
[CrossRef]

Appl. Phys. Lett.

S. E. Harris, “Proposed backward wave oscillator in the infrared,” Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

Helv. Phys. Acta

E. C. G. Stueckelberg, “Theorie der un elastischen Stösse zwischen Atomen,” Helv. Phys. Acta 5, 369-422 (1932).
[CrossRef]

IEEE J. Quantum Electron.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

J. A. Armstrong, S. S. Jha, and N. S. Shiren, “Some effects of group-velocity dispersion on parametric interactions,” IEEE J. Quantum Electron. 6, 123-129 (1970).
[CrossRef]

J. Appl. Phys.

D. L. Bobroff and H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390-403 (1967).
[CrossRef]

J. Opt. Soc. Am. B

Phys. Fluids

F. W. Chambers and A. Bers, “Parametric interactions in an inhomogeneous medium of finite extent with abrupt boundaries,” Phys. Fluids 20, 466-468 (1977).
[CrossRef]

E. A. Williams, J. R. Albritton, and M. N. Rosenbluth, “Effect of spatial turbulence on parametric instabilities,” Phys. Fluids 22, 139-149 (1979).
[CrossRef]

Phys. Plasmas

R. W. Short and A. Simon, “Theory of three-wave parametric instabilities in inhomogeneous plasmas revisited,” Phys. Plasmas 11, 5335-5340 (2004).
[CrossRef]

Phys. Rev.

R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064-1068 (1968).
[CrossRef]

Phys. Rev. Lett.

M. N. Rosenbluth, R. B. White, and C. S. Liu, “Temporal evolution of a three-wave parametric instability,” Phys. Rev. Lett. 31, 1190-1193 (1973).
[CrossRef]

M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. 29, 565-567 (1972).
[CrossRef]

D. F. DuBois, D. W. Forslund, and E. A. Williams, “Parametric instabilities in finite inhomogeneous media,” Phys. Rev. Lett. 33, 1013-1016 (1974).
[CrossRef]

G. Picard and T. W. Johnston, “Absolute parametric instabilities in inhomogeneous plasmas,” Phys. Rev. Lett. 51, 574-577 (1983).
[CrossRef]

Phys. Z. Sowjetunion

L. D. Landau, “Zur Theorie der Energieübertragung,” Phys. Z. Sowjetunion 2, 46-51 (1932).

Proc. R. Soc. London, Ser. A

C. Zener, “Non-adiabatic crossing of energy levels,” Proc. R. Soc. London, Ser. A 37, 696-702 (1932).

Other

F. W. Chambers, Ph.D. dissertation (Massachusetts Institute of Technology, 1975).

B. Afeyan and M. M. Fejer, “Short pulse optical parametric processes with group velocity mismatch and quasi-phase matching,” presented at Nonlinear Optics 2002, Maui, Hawaii, July 29-August 2 (2002).

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).

R. J. Briggs, Electron-Stream Interaction with Plasmas (MIT Press, 1964).

E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Butterworth-Heinemann Ltd., 1981).

A. Bers, Basic Plasma Physics I, A.A.Galeev and R.N.Sudan, eds. (North-Holland, 1983).

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Figures (9)

Fig. 1
Fig. 1

Green’s functions of the uniform medium, Eqs. (19, 21), for (a) γ ( z z 0 ) γ L = 5 and (b) γ L = 10 .

Fig. 2
Fig. 2

Nature of saddle points and local behavior of the Green’s function as a function of position inside the causal region. t ¯ s and t ¯ i are the wavefront arrival times, defined by Eqs. (70, 71). U s and U i are the delays with respect to the wavefronts, defined by Eqs. (72, 73).

Fig. 3
Fig. 3

Trajectory of the saddle points in the complex plane as U s U i goes from 0 to ∞. The saddle points are denoted by crosses, while the branch points are represented by solid dots.

Fig. 4
Fig. 4

Steepest descent contour in the regime U s U i 4 λ . The crosses represent saddle points; the dashed lines, the contour of integration; the zig-zag line, the branch cut; and the solid dots, the branch points.

Fig. 5
Fig. 5

Stationary phase contour in the regime U s U i 4 λ .

Fig. 6
Fig. 6

Contour for numerical integration.

Fig. 7
Fig. 7

Signal (left) and idler (right) Green’s functions calculated numerically (solid curves), and comparison with the asymptotic expressions developed in Subsection 3J (dashed curves). The gain parameter is λ = 2 . The input and output positions are (a), (b) z ¯ 0 = 5 , z ¯ = 5 ; (c), (d) z ¯ 0 = 10 , z ¯ = 10 ; (e), (f) z ¯ 0 = 15 , z ¯ = 15 . The corresponding normalized lengths, L ¯ = z ¯ z ¯ 0 , are written on each plot.

Fig. 8
Fig. 8

Amplified signal pulses of various initial durations Δ t ¯ = 50 , 30, 10, 5, 2, 1, and 0.5. The pulses are incident at z ¯ 0 = 15 , and observed at z ¯ = 15 , for a grating length of L ¯ = z ¯ z ¯ 0 = 30 . The gain parameter is λ = 2 , corresponding to a Rosenbluth gain factor e π λ 535 . The solid curve is the numerical solution while the dashed curve corresponds to the analytical expressions developed in the Appendix.

Fig. 9
Fig. 9

Amplified idler pulses of various initial durations Δ t ¯ = 50 , 30, 10, 5, 2, 1, and 0.5. The pulses are incident at z ¯ 0 = 15 , and observed at z ¯ = 15 , for a grating length of L ¯ = z ¯ z ¯ 0 = 30 . The gain parameter is λ = 2 , corresponding to a Rosenbluth gain factor e π λ 535 . The solid curve is the numerical solution while the dashed curve corresponds to the analytical expressions developed in the Appendix.

Equations (127)

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A s z + 1 v s A s t = i γ A i * ,
A i * z + 1 v i A i * t = i γ A s .
A s ˜ z i ω v s A s ˜ = i γ A i * ˜ + 1 v s A s ( z , 0 ) ,
A i * ˜ z i ω v i A i * ˜ = i γ A s ˜ ,
A s ˜ = y s exp [ i 2 ( 1 v s + 1 v i ) ω ( z z 0 ) ] ,
A i * ˜ = y i * exp [ i 2 ( 1 v s + 1 v i ) ω ( z z 0 ) ] .
d 2 y s d z 2 + { [ 1 2 ( 1 v s 1 v i ) ω ] 2 γ 2 } y s = 1 v s [ i ω v i A s ( z , 0 ) + d d z A s ( z , 0 ) ] e i ( 1 v s + 1 v i ) ω ( z z 0 ) 2 .
d 2 y i d z 2 + { [ 1 2 ( 1 v s 1 v i ) ω ] 2 γ 2 } y i = i γ v s A s * ( z , 0 ) e i ( 1 v s + 1 v i ) ω ( z z 0 ) 2 .
1 v ¯ = 1 2 ( 1 v s + 1 v i ) ,
1 δ v = 1 2 ( 1 v s 1 v i ) .
Ψ 1 , 2 = e ± i z ( ω δ v ) 2 γ 2 .
y s ̂ ( k , ω ) = i ( ω δ v k ) e i k z 0 ( ω δ v ) 2 γ 2 k 2 ,
y i ̂ ( k , ω ) = i γ e i k z 0 ( ω δ v ) 2 γ 2 k 2 .
G s ( z , t ) = 1 4 π 2 d ω C k d k y s ̂ ( k , ω ) e i k z i [ t ( z z 0 ) v ¯ ] ω .
G s = 1 4 π i ( z + 1 v i t ) { + i ϵ + i ϵ e i ( ω δ v ) 2 γ 2 ( z z 0 ) i [ t ( z z 0 ) v ¯ ] ω ( ω δ v ) 2 γ 2 d ω + i ϵ + i ϵ e i ( ω δ v ) 2 γ 2 ( z z 0 ) i [ t ( z z 0 ) v ¯ ] ω ( ω δ v ) 2 γ 2 d ω } .
V s = { t s t , if v s < v i t t s , if v s > v i ,
V i = { t t i , if v s < v i t i t , if v s > v i .
G s = δ v 4 π i ( z + 1 v i t ) θ ( V s V i ) × e ( i 2 ) δ v γ V s V i ( u 1 u ) d u u ,
G s = δ ( t t s ) + 1 2 δ v γ V i V s θ ( V s V i ) I 1 ( δ v γ V s V i ) .
A i * = 1 i γ A s V i .
G i = i 2 δ v γ θ ( V s V i ) I 0 ( δ v γ V s V i ) .
A s z + 1 v s A s t + ν s A s = i γ A i * ,
A i * z + 1 v i A i * t + ν i A i * = i γ A s ,
G s ( z , t ) = i 4 π 2 e ν ¯ ( z z 0 ) C k ω δ v + i δ ν k ( ω δ v + i δ ν ) 2 γ 2 k 2 × e i k ( z z 0 ) i [ t ( z z 0 ) v ¯ ] ω d k d ω ,
ν ¯ = 1 2 ( ν s + ν i ) ,
δ ν = 1 2 ( ν s ν i ) .
G s = e δ v ( ν s V i + ν i V s ) 2 [ δ ( t t s ) + 1 2 δ v γ V i V s θ ( V s V i ) I 1 ( δ v γ V s V i ) ] .
A s z + 1 v s A s t + ν s A s = i γ A i * ,
A i * z 1 v i A i * t ν i A i * = i γ A s .
G s ( z , t ) = i 4 π 2 e δ ν ( z z 0 ) C k ω v ¯ + i ν ¯ k ( ω v ¯ + i ν ¯ ) 2 + γ 2 k 2 × e i k ( z z 0 ) i [ t ( z z 0 ) δ v ] ω d k d ω ,
G s = e v ¯ ( ν s V i + ν i V s ) 2 [ δ ( t t s ) + 1 2 v ¯ γ V i V s θ ( V s V i ) I 1 ( v ¯ γ V s V i ) ] ,
G i = i 2 v ¯ γ θ ( V s V i ) e v ¯ ( ν s V i + ν i V s ) 2 I 0 ( v ¯ γ V s V i ) ,
V s = t t s ,
V i = t + t i ,
G s = g ( k , ω ) Δ ( k , ω ) e i [ k ( z z 0 ) + i ω t ] d k d ω ,
Δ ( k , ω ) = ( k + ω v s + i ν s ) ( k ω v i i ν i ) γ 2 .
Δ ( k , ω ) = ( k + ω v s + i ν s ) ( k + ω v i + i ν i ) + γ 2 .
k ± = ( ω δ v + i δ ν ) ± ( ω v ¯ + i ν ¯ ) 2 + γ 2 .
k ± = ( ω v ¯ + i ν ¯ ) ± ( ω δ v + i δ ν ) 2 γ 2 .
A s z + 1 v s A s t = i γ A i * e i φ ( z ) ,
A i * z + 1 v i A i * t = i γ A s e i φ ( z ) ,
φ ( z ) = z 0 z κ ( z ) d z ,
A s = a s e i φ ( z ) 2 ,
A i * = a i * e i φ ( z ) 2 .
a s ˜ z + i [ 1 2 κ ( z ) ω v s ] a s ˜ = i γ a i * ˜ + 1 v s a s ( z , 0 ) ,
a i * ˜ z i [ 1 2 κ ( z ) + ω v i ] a i * ˜ = i γ a s ˜ ,
a s ˜ = y s exp [ i 2 ( 1 v s + 1 v i ) ω ( z z 0 ) ] ,
a i * ˜ = y i * exp [ i 2 ( 1 v s + 1 v i ) ω ( z z 0 ) ] .
d 2 y s d z 2 + { 1 4 [ κ ( 1 v s 1 v i ) ω ] 2 + i 2 d κ d z γ 2 } y s = 1 v s [ i ( 1 2 κ ( z ) + ω v i ) a s ( z , 0 ) + d d z a s ( z , 0 ) ] e i ( 1 v s + 1 v i ) ω ( z z 0 ) 2
d 2 y i d z 2 + { 1 4 [ κ ( 1 v s 1 v i ) ω ] 2 + i 2 d κ d z γ 2 } y i = i γ v s a s * ( z , 0 ) e i ( 1 v s + 1 v i ) ω ( z z 0 ) 2 .
κ ( z ) = κ ( z z p m 0 ) .
z ¯ = κ ( z z p m 0 ) ,
ω ¯ = 2 δ v ω κ ,
t ¯ = δ v κ 2 t ,
λ = γ 2 κ .
ζ = z ¯ σ ω ¯ ,
d 2 y d ζ 2 + ( i 2 λ + 1 4 ζ 2 ) y = 0 ,
y s ( ζ 0 + ) = i ζ 0 2 ,
y s ( ζ 0 + ) = 1 .
Ψ 1 = D i λ ( ζ e i π 4 ) ,
Ψ 2 = D 1 i λ ( ζ e i π 4 ) ,
G s ˜ ( z ¯ , ω ¯ ) = e i π 4 W θ ( z ¯ z ¯ 0 ) exp { i [ 1 4 ( z ¯ 2 z ¯ 0 2 ) + 1 2 v s + v i v s v i σ ω ¯ ( z ¯ z ¯ 0 ) ] } × [ D i λ ( ζ 0 e i π 4 ) D i λ ( ζ e i π 4 ) + λ D i λ 1 ( ζ 0 e i π 4 ) D i λ 1 ( ζ e i π 4 ) ] .
G i ˜ ( z ¯ , ω ¯ ) = i λ 1 2 W θ ( z ¯ z ¯ 0 ) exp { i [ 1 4 ( z ¯ 2 z ¯ 0 2 ) 1 2 v s + v i v s v i σ ω ¯ ( z ¯ z ¯ 0 ) ] } × [ D i λ 1 ( ζ 0 e i π 4 ) D i λ ( ζ e i π 4 ) D i λ ( ζ 0 e i π 4 ) D i λ 1 ( ζ e i π 4 ) ] .
G s ˜ e i ( δ v 2 v s ) ω ¯ ( ζ ζ 0 ) e i λ ln ζ ζ 0 × [ 1 λ ζ 0 ζ e 2 i λ ln ζ ζ 0 e i ( ζ 2 ζ 0 2 ) 2 ] ,
G i ˜ λ 1 2 e i ( z ¯ 2 z ¯ 0 2 ) 2 e i ( δ v 2 v s ) ω ¯ ( ζ ζ 0 ) × [ 1 ζ ζ ζ 0 i λ 1 ζ 0 ζ ζ 0 i λ e i ( ζ 2 ζ 0 2 ) 2 ] .
G s ˜ e π λ e i ( δ v 2 v s ) ω ¯ ( ζ ζ 0 ) e i λ ln ζ ζ 0 [ 1 + i λ 1 2 ζ 0 e i λ [ ln ( λ ζ 0 2 ) 1 ] e i ζ 0 2 2 ] [ 1 i λ 1 2 ζ e i λ [ ln ( λ ζ 2 ) 1 ] e i ζ 2 2 ] ,
G i ˜ i e π λ e i ω ¯ 2 2 e i ( δ v 2 ) ω ¯ ( ζ v i ζ 0 v s ) e i ζ 0 2 2 e i λ ln ( ζ ζ 0 λ ) [ 1 i λ 1 2 ζ 0 e i λ [ ln ( λ ζ 0 2 ) 1 ] e i ζ 0 2 2 ] [ 1 i λ 1 2 ζ e i λ [ ln ( λ ζ 2 ) 1 ] e i ζ 2 2 ] .
G s ( z ¯ , t ¯ ) = 1 4 π δ v κ + i ϵ + + i ϵ G s ˜ ( z ¯ , ω ¯ ) e i ω ¯ t ¯ d ω ¯ ,
D ν ( x ) = 1 i 2 π e x 2 4 i i e x p + p 2 2 p ν d p ,
t ¯ s = δ v κ 2 ( z z 0 v s ) ,
t ¯ i = δ v κ 2 ( z z 0 v i ) .
U s = { t ¯ s t ¯ , if v s < v i t ¯ t ¯ s , if v s > v i ,
U i = { t ¯ t ¯ i , if v s < v i t ¯ i t ¯ , if v s > v i .
G s ( z ¯ , t ¯ ) = δ v κ 2 [ δ ( U s ) λ 2 π θ ( U s U i ) U s e i U s ( z ¯ + z ¯ 0 ) 2 × e i U s U i p ( p 1 2 p + 1 2 ) i λ d p ( p 1 2 ) ( p + 1 2 ) ] ,
G s ( z ¯ , t ¯ ) = δ v κ 2 [ δ ( U s ) + 1 2 π θ ( U s U i ) U i e i U s ( z ¯ + z ¯ 0 ) 2 × e i U s U i p ( p 1 2 p + 1 2 ) i λ d p ] .
G i ( z ¯ , t ¯ ) = δ v κ 2 λ 2 π θ ( U s U i ) e i U i ( z ¯ + z ¯ 0 ) 2 × e i U s U i p ( p 1 2 p + 1 2 ) i λ d p p + 1 2 .
e ( 1 2 ) δ v γ V s V i ( u + 1 u ) d u u 2 ,
G s ( z ¯ , t ¯ ) = δ v κ 2 e ν ¯ s U i ν ¯ i U s [ δ ( U s ) + 1 2 π θ ( U s U i ) U i e i U s ( z ¯ + z ¯ 0 ) 2 e i U s U i p ( p 1 2 p + 1 2 ) i λ d p ] .
G s ( z ¯ , t ¯ ) = v ¯ κ 2 e ( ν ¯ s U i + ν ¯ i U s ) [ δ ( U s ) + 1 2 π θ ( U s U i ) U i e i U s ( z ¯ + z ¯ 0 ) 2 e i U s U i p ( p 1 2 p + 1 2 ) i λ d p ] .
G i ( z ¯ , t ¯ ) = v ¯ κ 2 e ( ν ¯ s U i + ν ¯ i U s ) λ 2 π θ ( U s U i ) e i U i ( z ¯ + z ¯ 0 ) 2 × e i U s U i p ( p 1 2 p + 1 2 ) i λ d p p 1 2 ,
U s = t ¯ t ¯ s ,
U i = t ¯ + t ¯ i ,
t ¯ = v ¯ κ 2 t .
G s = δ v κ 2 [ δ ( U s ) + sinh π λ π θ ( U s U i ) U i e i U s ( z ¯ + z ¯ 0 ) 2 e i U s U i 2 × 0 1 e i U s U i p ( 1 p p ) i λ d p ] ,
G i = δ v κ 2 λ sinh π λ π θ ( U s U i ) e i U i ( z ¯ + z ¯ 0 ) 2 e i U s U i 2 × 0 1 e i U s U i p ( 1 p p ) i λ d p p .
Φ ( β , γ , x ) = Γ ( γ ) Γ ( β ) Γ ( γ β ) 0 1 ( 1 s ) γ β 1 s β 1 e s x d s .
G s = δ v κ 2 [ δ ( U s ) + λ θ ( U s U i ) U i e i U s ( z ¯ + z ¯ 0 ) 2 e i U s U i 2 Φ ( 1 i λ , 2 , i U s U i ) ] ,
G i = δ v κ 2 i λ θ ( U s U i ) e i U i ( z ¯ + z ¯ 0 ) 2 e i U s U i 2 Φ ( i λ , 1 , i U s U i ) .
G s = δ v κ 2 [ δ ( U s ) i θ ( U s U i ) U i e i U s ( z ¯ + z ¯ 0 ) 2 e i U s U i 2 L i λ 1 1 ( i U s U i ) ] ,
G i = δ v κ 2 i λ θ ( U s U i ) e i U i ( z ¯ + z ¯ 0 ) 2 e i U s U i 2 L i λ ( i U s U i ) .
ρ ( p ) = i U s U i p + i λ ln ( p 1 2 p + 1 2 ) .
p ± = ± 1 2 1 4 λ U s U i .
G s δ v κ 2 1 2 π ( λ U i U s 3 ) 1 4 e i U s ( z ¯ + z ¯ 0 ) 2 e 2 λ U s U i , ( 0 U s U i 4 λ ) ,
G i δ v κ 2 i 2 π ( λ U s U i ) 1 4 e i U i ( z ¯ + z ¯ 0 ) 2 e 2 λ U s U i , ( 0 U s U i 4 λ ) .
G s δ v κ 2 2 λ π 1 U s e π λ e i U s ( z ¯ + z ¯ 0 ) 2 × cos { U s U i 2 λ [ ln ( U s U i λ ) + 1 ] π 4 } , ( U s U i 4 λ ) .
G i δ v κ 2 1 2 π e π λ e i U i ( z ¯ + z ¯ 0 ) 2 × exp { i U s U i 2 + i λ [ ln ( U s U i λ ) + 1 ] + i π 4 } × ( 1 i λ U s U i exp { i U s U i 2 i λ [ ln ( U s U i λ ) + 1 ] } ) , ( U s U i 4 λ ) .
exp [ i ( U s U i 4 λ ) p 16 3 i λ p 3 ] d p ,
G s δ v κ 2 1 2 ( 2 λ ) 1 3 e π λ U i e i U s ( z ¯ + z ¯ 0 ) 2 Ai [ 4 λ U s U i 2 ( 2 λ ) 1 3 ] , ( U s U i 4 λ ) .
A s ( z ¯ , t ¯ ) = 2 δ v κ A s o ( t ¯ t ¯ ) G s ( z ¯ , z ¯ 0 , t ¯ ) d t ¯ ,
A i ( z ¯ , t ¯ ) = 2 δ v κ A s o * ( t ¯ t ¯ ) G i ( z ¯ , z ¯ 0 , t ¯ ) d t ¯ .
A s 0 ( t ¯ ) = B s 0 ( t ¯ ) e i δ ω ¯ t ¯ ,
A s A s 0 ( t ¯ t ¯ s ) e π λ e i λ ln ζ ζ 0 ,
A i i A s 0 * ( t ¯ t ¯ s i ) e π λ e i λ [ ln ( ζ ζ 0 λ ) + 1 ] e i ζ 0 2 2 .
Δ ω B W = δ v 2 κ L ,
A s 0 ( t ¯ ) = e ( t ¯ τ ¯ ) 2 ,
A s e π λ A s 0 ( t ¯ t ¯ s + 4 λ L ¯ ) .
A i τ ¯ 2 e π λ e [ τ ¯ ( t ¯ t ¯ s i ) 2 ] 2 e i ( t ¯ t ¯ s i ) 2 2 .
G ̃ s { e π λ , δ ω < Δ ω B W 2 0 , δ ω > Δ ω B W 2 ,
G ̃ i { i e π λ e i δ ω ¯ 2 2 , δ ω < Δ ω B W 2 0 , δ ω > Δ ω B W 2 .
A s 0 ( t ¯ ) = B s 0 ( t ¯ ) e i δ ω ¯ t ¯ .
A s = 1 2 π ( p + 1 2 p 1 2 ) i λ t ¯ i t ¯ s A s 0 ( t ¯ t ¯ ) ( t ¯ t ¯ i ) e i ( t ¯ t ¯ s ) ( z ¯ + z ¯ 0 ) 2 + i ( t ¯ t ¯ s ) ( t ¯ t ¯ i ) p d t ¯ d p .
A s e i π 4 2 π 1 2 e i ( ζ 2 ζ 0 2 ) 4 e i δ ω ¯ ( t ¯ t ¯ s ) × B s 0 ( t ¯ t ¯ 0 ) ( t ¯ 0 t ¯ i ) ( p + 1 2 p 1 2 ) i λ × e i p ( ζ ζ 0 ) 2 4 i ( ζ + ζ 0 ) 2 16 p d p p 1 2 ,
t ¯ 0 = δ ν 2 ν ¯ ( ζ ζ 0 ) + ( ζ + ζ 0 ) 4 p .
ρ ( p ) = 1 4 p ( ζ ζ 0 ) 2 1 16 p ( ζ + ζ 0 ) 2 + λ ln ( p + 1 2 p 1 2 ) .
p 1 , 2 ± ζ + ζ 0 2 ( ζ ζ 0 ) ,
p 3 , 4 ± 1 2 1 + 4 λ ζ ζ 0 .
A s A s 0 ( t ¯ t ¯ s ) e π λ e i λ ln ζ ζ 0 × { 1 + i λ 1 2 ζ 0 e i ζ 0 2 2 + i λ [ ln ( ζ 0 2 λ ) + 1 ] i λ 1 2 ζ e i ζ 2 2 i λ [ ln ( ζ 2 λ ) + 1 ] } .
A i i A s 0 * ( t ¯ t ¯ s i ) e π λ e i λ [ ln ( ζ ζ 0 λ ) + 1 ] e i ζ 0 2 2 { 1 i λ 1 2 ζ 0 e i ζ 0 2 2 i λ [ ln ( ζ 0 2 λ ) + 1 ] i λ 1 2 ζ e i ζ 2 2 i λ [ ln ( ζ 2 λ ) + 1 ] } ,
A s 0 ( t ¯ ) = e ( t ¯ τ ¯ ) 2 ,
A s = 1 2 π ( p + 1 2 p 1 2 ) i λ t ¯ i t ¯ s e [ ( t ¯ t ¯ ) τ ¯ ] 2 ( t ¯ t ¯ i ) e i ( t ¯ t ¯ s ) ( z ¯ + z ¯ 0 ) 2 + i ( t ¯ t ¯ s ) ( t ¯ t ¯ i ) p d t ¯ d p .
x = 1 τ ¯ 2 i p ( t ¯ t ¯ ) + i [ ( U s U i ) p + ( z ¯ + z ¯ 0 ) 2 ] 2 1 τ ¯ 2 i p .
A s 1 4 π 1 2 e i U s ( z ¯ + z ¯ 0 ) 2 x i [ erf ( x s ) erf ( x i ) ] 1 τ ¯ 2 i p exp { [ ( U s U i ) p + ( z ¯ + z ¯ 0 ) 2 ] 2 4 ( 1 τ ¯ 2 i p ) } e i U s U i p ( p + 1 2 p 1 2 ) i λ d p .
A s 2 π τ ¯ δ v κ G s .
A s e π λ { A s 0 ( t ¯ t ¯ s + 4 λ L ¯ ) + 4 λ L ¯ 2 A s 0 ( t ¯ t ¯ i 4 λ L ¯ ) e i ( z ¯ z ¯ 0 ) 2 2 } + 1 U s λ 2 τ ¯ ( 1 + τ ¯ 4 4 ) 1 4 e π λ e i U s ( z ¯ + z ¯ 0 ) 2 ( exp { τ ¯ 2 4 ( t ¯ t ¯ i s ) 2 ( 1 + τ ¯ 4 4 ) + i 2 U s U i i λ [ ln ( U s U i λ ) + 1 ] i π 4 i 2 arctan ( τ ¯ 2 2 ) } + exp { τ ¯ 2 4 ( t ¯ t ¯ s i ) 2 ( 1 + τ ¯ 4 4 ) i 2 U s U i + i λ [ ln ( U s U i λ ) + 1 ] + i π 4 + i 2 arctan ( τ ¯ 2 2 ) } ) ,
A s e π λ A s 0 ( t ¯ t ¯ s ) .
A i λ L ¯ e π λ [ A s 0 * ( t ¯ t ¯ i 4 λ L ¯ ) + A s 0 * ( t ¯ t ¯ s + 4 λ L ¯ ) e i ( z ¯ z ¯ 0 ) 2 2 ] + τ ¯ 2 ( 1 + τ ¯ 4 4 ) 1 4 e π λ e [ τ ¯ ( t ¯ t ¯ s i ) 2 ] 2 e i U i ( z ¯ + z ¯ 0 ) 2 × exp { i [ 1 2 U s U i + λ [ ln ( U s U i λ ) + 1 ] + π 4 + 1 2 arctan ( τ ¯ 2 2 ) ] } ,
A i τ ¯ 2 e π λ e [ τ ¯ ( t ¯ t ¯ s i ) 2 ] 2 e i ( t ¯ t ¯ s i ) 2 2 .

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