Abstract

We analyze the time-resolution constraints for two single-shot, time-resolved femtosecond spectroscopic techniques, an excite–probe method, and a transient-grating method. For pulse lengths of tens of femtoseconds, the former technique is shown to provide pulse-length-limited resolution for samples as much as hundreds of micrometers thick. The latter technique is shown to provide nearly pulse-length-limited resolution under a variety of conditions, independent of the sample thickness.

© 1995 Optical Society of America

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  1. M. M. Malley and P. M. Rentzepis, “Picosecond molecular relaxation displayed with crossed laser beams,” Chem. Phys. Lett. 3, 534–536 (1969).
    [Crossref]
  2. M. M. Malley and P. M. Rentzepis, “Picosecond time-resolved stimulated light emission,” Chem. Phys. Lett. 7, 57–60 (1970).
    [Crossref]
  3. M. R. Topp, P. M. Rentzepis, and R. P. Jones, “Time resolved picosecond emission spectroscopy of organic laser dyes,” Chem. Phys. Lett. 9, 1–5 (1971).
    [Crossref]
  4. M. Malley, “Picosecond laser techniques,” in Creation and Detection of the Excited State, W. R. Ware, ed. (Dekker, New York, 1974), Vol. 2.
  5. R. N. Gyuzalian, S. B. Sogomonian, and Z. Gy. Horvath, “Background-free measurement of time behavior of an individual picosecond laser pulse,” Opt. Commun. 29, 239–242 (1979).
    [Crossref]
  6. A. Brun, P. Georges, G. Le Saux, and F. Salin, “Single-shot characterization of ultrashort light pulses,” J. Phys. D 24, 1225–1233 (1991).
    [Crossref]
  7. R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
    [Crossref]
  8. D. J. Kane and R. Trebino, “Measurement of the intensity and phase of femtosecond pulses using frequency-solved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
    [Crossref]
  9. D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
    [Crossref] [PubMed]
  10. G. Szabo, Z. Bar, and A. Muller, “Phase-sensitive single-pulse autocorrelator for ultrashort laser pulses,” Opt. Lett. 13, 746–748 (1988).
    [Crossref] [PubMed]
  11. L. Dhar, J. T. Fourkas, and K. A. Nelson, “Pure-length-limited ultrafast pump/probe spectroscopy in a single laser shot,” Opt. Lett. 19, 643 (1994).
    [Crossref] [PubMed]
  12. J. T. Fourkas, W. Wang, K. A. Nelson, and R. Trebino, “Ultrafast transient-grating spectroscopy in a single shot,” in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper QWH 22.
  13. H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
    [Crossref]
  14. Y.-X. Yan and K. A. Nelson, “Impulsive stimulated light scattering. I. General theory,” J. Chem. Phys. 87, 6240–6256 (1987).
    [Crossref]
  15. F. W. Wise, M. J. Rosker, and C. L. Tang, “Oscillatory femtosecond relaxation of photoexcited organic molecules,” J. Chem. Phys. 86, 2827–2832 (1987).
    [Crossref]
  16. R. Trebino, K. W. DeLong, and D. J. Kane, “Measurement of the intensity and phase of a single ultrashort pulse: a simple and general technique,” in Lasers and Ultrafast Processes, A. Piskarskas, ed. (Vilnius U. Press, Vilnius, Lithuania, 1993), Vol. 5.

1994 (1)

1993 (3)

1991 (1)

A. Brun, P. Georges, G. Le Saux, and F. Salin, “Single-shot characterization of ultrashort light pulses,” J. Phys. D 24, 1225–1233 (1991).
[Crossref]

1988 (1)

1987 (2)

Y.-X. Yan and K. A. Nelson, “Impulsive stimulated light scattering. I. General theory,” J. Chem. Phys. 87, 6240–6256 (1987).
[Crossref]

F. W. Wise, M. J. Rosker, and C. L. Tang, “Oscillatory femtosecond relaxation of photoexcited organic molecules,” J. Chem. Phys. 86, 2827–2832 (1987).
[Crossref]

1979 (1)

R. N. Gyuzalian, S. B. Sogomonian, and Z. Gy. Horvath, “Background-free measurement of time behavior of an individual picosecond laser pulse,” Opt. Commun. 29, 239–242 (1979).
[Crossref]

1971 (1)

M. R. Topp, P. M. Rentzepis, and R. P. Jones, “Time resolved picosecond emission spectroscopy of organic laser dyes,” Chem. Phys. Lett. 9, 1–5 (1971).
[Crossref]

1970 (1)

M. M. Malley and P. M. Rentzepis, “Picosecond time-resolved stimulated light emission,” Chem. Phys. Lett. 7, 57–60 (1970).
[Crossref]

1969 (1)

M. M. Malley and P. M. Rentzepis, “Picosecond molecular relaxation displayed with crossed laser beams,” Chem. Phys. Lett. 3, 534–536 (1969).
[Crossref]

Bar, Z.

Brun, A.

A. Brun, P. Georges, G. Le Saux, and F. Salin, “Single-shot characterization of ultrashort light pulses,” J. Phys. D 24, 1225–1233 (1991).
[Crossref]

DeLong, K. W.

R. Trebino, K. W. DeLong, and D. J. Kane, “Measurement of the intensity and phase of a single ultrashort pulse: a simple and general technique,” in Lasers and Ultrafast Processes, A. Piskarskas, ed. (Vilnius U. Press, Vilnius, Lithuania, 1993), Vol. 5.

Dhar, L.

Eichler, H. J.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
[Crossref]

Fourkas, J. T.

L. Dhar, J. T. Fourkas, and K. A. Nelson, “Pure-length-limited ultrafast pump/probe spectroscopy in a single laser shot,” Opt. Lett. 19, 643 (1994).
[Crossref] [PubMed]

J. T. Fourkas, W. Wang, K. A. Nelson, and R. Trebino, “Ultrafast transient-grating spectroscopy in a single shot,” in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper QWH 22.

Georges, P.

A. Brun, P. Georges, G. Le Saux, and F. Salin, “Single-shot characterization of ultrashort light pulses,” J. Phys. D 24, 1225–1233 (1991).
[Crossref]

Günter, P.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
[Crossref]

Gyuzalian, R. N.

R. N. Gyuzalian, S. B. Sogomonian, and Z. Gy. Horvath, “Background-free measurement of time behavior of an individual picosecond laser pulse,” Opt. Commun. 29, 239–242 (1979).
[Crossref]

Horvath, Z. Gy.

R. N. Gyuzalian, S. B. Sogomonian, and Z. Gy. Horvath, “Background-free measurement of time behavior of an individual picosecond laser pulse,” Opt. Commun. 29, 239–242 (1979).
[Crossref]

Jones, R. P.

M. R. Topp, P. M. Rentzepis, and R. P. Jones, “Time resolved picosecond emission spectroscopy of organic laser dyes,” Chem. Phys. Lett. 9, 1–5 (1971).
[Crossref]

Kane, D. J.

R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
[Crossref]

D. J. Kane and R. Trebino, “Measurement of the intensity and phase of femtosecond pulses using frequency-solved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[Crossref]

D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
[Crossref] [PubMed]

R. Trebino, K. W. DeLong, and D. J. Kane, “Measurement of the intensity and phase of a single ultrashort pulse: a simple and general technique,” in Lasers and Ultrafast Processes, A. Piskarskas, ed. (Vilnius U. Press, Vilnius, Lithuania, 1993), Vol. 5.

Le Saux, G.

A. Brun, P. Georges, G. Le Saux, and F. Salin, “Single-shot characterization of ultrashort light pulses,” J. Phys. D 24, 1225–1233 (1991).
[Crossref]

Malley, M.

M. Malley, “Picosecond laser techniques,” in Creation and Detection of the Excited State, W. R. Ware, ed. (Dekker, New York, 1974), Vol. 2.

Malley, M. M.

M. M. Malley and P. M. Rentzepis, “Picosecond time-resolved stimulated light emission,” Chem. Phys. Lett. 7, 57–60 (1970).
[Crossref]

M. M. Malley and P. M. Rentzepis, “Picosecond molecular relaxation displayed with crossed laser beams,” Chem. Phys. Lett. 3, 534–536 (1969).
[Crossref]

Muller, A.

Nelson, K. A.

L. Dhar, J. T. Fourkas, and K. A. Nelson, “Pure-length-limited ultrafast pump/probe spectroscopy in a single laser shot,” Opt. Lett. 19, 643 (1994).
[Crossref] [PubMed]

Y.-X. Yan and K. A. Nelson, “Impulsive stimulated light scattering. I. General theory,” J. Chem. Phys. 87, 6240–6256 (1987).
[Crossref]

J. T. Fourkas, W. Wang, K. A. Nelson, and R. Trebino, “Ultrafast transient-grating spectroscopy in a single shot,” in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper QWH 22.

Pohl, D. W.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
[Crossref]

Rentzepis, P. M.

M. R. Topp, P. M. Rentzepis, and R. P. Jones, “Time resolved picosecond emission spectroscopy of organic laser dyes,” Chem. Phys. Lett. 9, 1–5 (1971).
[Crossref]

M. M. Malley and P. M. Rentzepis, “Picosecond time-resolved stimulated light emission,” Chem. Phys. Lett. 7, 57–60 (1970).
[Crossref]

M. M. Malley and P. M. Rentzepis, “Picosecond molecular relaxation displayed with crossed laser beams,” Chem. Phys. Lett. 3, 534–536 (1969).
[Crossref]

Rosker, M. J.

F. W. Wise, M. J. Rosker, and C. L. Tang, “Oscillatory femtosecond relaxation of photoexcited organic molecules,” J. Chem. Phys. 86, 2827–2832 (1987).
[Crossref]

Salin, F.

A. Brun, P. Georges, G. Le Saux, and F. Salin, “Single-shot characterization of ultrashort light pulses,” J. Phys. D 24, 1225–1233 (1991).
[Crossref]

Sogomonian, S. B.

R. N. Gyuzalian, S. B. Sogomonian, and Z. Gy. Horvath, “Background-free measurement of time behavior of an individual picosecond laser pulse,” Opt. Commun. 29, 239–242 (1979).
[Crossref]

Szabo, G.

Tang, C. L.

F. W. Wise, M. J. Rosker, and C. L. Tang, “Oscillatory femtosecond relaxation of photoexcited organic molecules,” J. Chem. Phys. 86, 2827–2832 (1987).
[Crossref]

Topp, M. R.

M. R. Topp, P. M. Rentzepis, and R. P. Jones, “Time resolved picosecond emission spectroscopy of organic laser dyes,” Chem. Phys. Lett. 9, 1–5 (1971).
[Crossref]

Trebino, R.

R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
[Crossref]

D. J. Kane and R. Trebino, “Measurement of the intensity and phase of femtosecond pulses using frequency-solved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[Crossref]

D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
[Crossref] [PubMed]

J. T. Fourkas, W. Wang, K. A. Nelson, and R. Trebino, “Ultrafast transient-grating spectroscopy in a single shot,” in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper QWH 22.

R. Trebino, K. W. DeLong, and D. J. Kane, “Measurement of the intensity and phase of a single ultrashort pulse: a simple and general technique,” in Lasers and Ultrafast Processes, A. Piskarskas, ed. (Vilnius U. Press, Vilnius, Lithuania, 1993), Vol. 5.

Wang, W.

J. T. Fourkas, W. Wang, K. A. Nelson, and R. Trebino, “Ultrafast transient-grating spectroscopy in a single shot,” in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper QWH 22.

Wise, F. W.

F. W. Wise, M. J. Rosker, and C. L. Tang, “Oscillatory femtosecond relaxation of photoexcited organic molecules,” J. Chem. Phys. 86, 2827–2832 (1987).
[Crossref]

Yan, Y.-X.

Y.-X. Yan and K. A. Nelson, “Impulsive stimulated light scattering. I. General theory,” J. Chem. Phys. 87, 6240–6256 (1987).
[Crossref]

Chem. Phys. Lett. (3)

M. M. Malley and P. M. Rentzepis, “Picosecond molecular relaxation displayed with crossed laser beams,” Chem. Phys. Lett. 3, 534–536 (1969).
[Crossref]

M. M. Malley and P. M. Rentzepis, “Picosecond time-resolved stimulated light emission,” Chem. Phys. Lett. 7, 57–60 (1970).
[Crossref]

M. R. Topp, P. M. Rentzepis, and R. P. Jones, “Time resolved picosecond emission spectroscopy of organic laser dyes,” Chem. Phys. Lett. 9, 1–5 (1971).
[Crossref]

IEEE J. Quantum Electron. (1)

D. J. Kane and R. Trebino, “Measurement of the intensity and phase of femtosecond pulses using frequency-solved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[Crossref]

J. Chem. Phys. (2)

Y.-X. Yan and K. A. Nelson, “Impulsive stimulated light scattering. I. General theory,” J. Chem. Phys. 87, 6240–6256 (1987).
[Crossref]

F. W. Wise, M. J. Rosker, and C. L. Tang, “Oscillatory femtosecond relaxation of photoexcited organic molecules,” J. Chem. Phys. 86, 2827–2832 (1987).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

A. Brun, P. Georges, G. Le Saux, and F. Salin, “Single-shot characterization of ultrashort light pulses,” J. Phys. D 24, 1225–1233 (1991).
[Crossref]

Opt. Commun. (1)

R. N. Gyuzalian, S. B. Sogomonian, and Z. Gy. Horvath, “Background-free measurement of time behavior of an individual picosecond laser pulse,” Opt. Commun. 29, 239–242 (1979).
[Crossref]

Opt. Lett. (3)

Other (4)

J. T. Fourkas, W. Wang, K. A. Nelson, and R. Trebino, “Ultrafast transient-grating spectroscopy in a single shot,” in Quantum Electronics and Laser Science Conference, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper QWH 22.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
[Crossref]

M. Malley, “Picosecond laser techniques,” in Creation and Detection of the Excited State, W. R. Ware, ed. (Dekker, New York, 1974), Vol. 2.

R. Trebino, K. W. DeLong, and D. J. Kane, “Measurement of the intensity and phase of a single ultrashort pulse: a simple and general technique,” in Lasers and Ultrafast Processes, A. Piskarskas, ed. (Vilnius U. Press, Vilnius, Lithuania, 1993), Vol. 5.

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Figures (15)

Fig. 1
Fig. 1

Single-shot techniques used in early ultrafast experiments. (a) In the perpendicular pump–probe geometry, a relatively narrow pump beam passes through a cell at a right angle to a wide probe beam. The delay is encoded in the spatial intensity profile of the probe pulse, which is detected by an array detector. (b) In the transmission echelon technique, the pump and the probe beams are more nearly copropagating, improving the time resolution. The spatially encoded decay arises because different portions of the probe beam pass through different thicknesses of the echelon.

Fig. 2
Fig. 2

In the pump–probe geometry, both the excitation and the probe beams are cylindrically focused on the same face of a thin sample, allowing the beams to be nearly copropagating. The delay is encoded in the spatial profile of the probe beam, which is monitored with an array detector.

Fig. 3
Fig. 3

Single-shot pump–probe data taken with 70-fs laser pulses on a 100-μm sample of Ethyl Violet in methanol; 155-fs oscillations can be seen clearly. The time resolution is pulse-length limited.

Fig. 4
Fig. 4

In the grating geometry the excitation beams are cylindrically focused onto opposite faces of a sample, creating a grating that spatially encodes the delay time along the length of the sample. The probe beam is brought in at 90° along the Bragg cone, such that all delay times are interrogated simultaneously. The diffracted signal is incident upon an array detector.

Fig. 5
Fig. 5

Coordinate system used in calculations for the pump probe geometry.

Fig. 6
Fig. 6

Instrument response functions for the pump–probe technique for various sample thicknesses, given a 70-fs pulse duration and an interbeam angle of 25°.

Fig. 7
Fig. 7

Pump–probe resolution as a function of sample thickness at several interbeam angles, given a 70-fs pulse duration.

Fig. 8
Fig. 8

Pump–probe resolution versus interbeam angle for several sample thicknesses, given a 70-fs pulse duration. Note the break in the scale.

Fig. 9
Fig. 9

Log–log plot of pump–probe resolution versus pulse duration for several sample thicknesses, given a 25° interbeam angle.

Fig. 10
Fig. 10

Log–log plot of pump–probe resolution versus pulse length at several interbeam angles, given a sample thickness of 100 μm.

Fig. 11
Fig. 11

Coordinate systems used in grating calculations. (a) System used for excitation portion of the calculation. (b) Coordinates used for probe portion of the calculation. Note the different perspectives in (a) and (b).

Fig. 12
Fig. 12

Grating resolution as a function of interbeam angle for various pulse lengths, given excitation and probe waists of 20 μm.

Fig. 13
Fig. 13

Grating resolution versus pulse length for a variety of beam waists (the excitation and probe waists are taken to be the same), given an interbeam angle of 45°.

Fig. 14
Fig. 14

Grating resolution as a function of excitation waist for a variety of pulse lengths. The interbeam angle is 45°, and the probe waist is 20 μm.

Fig. 15
Fig. 15

Grating resolution versus probe waist for a variety of pulse lengths. The interbeam angle is 45°, and the excitation waist is 10 μm.

Equations (83)

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x e = x cos θ z sin θ ,
y e = y ,
z e = x sin θ + z cos θ ,
x s = x cos ( θ / 2 ) z sin ( θ / 2 ) ,
y s = y ,
z s = x sin ( θ / 2 ) + z cos ( θ / 2 ) .
( 2 z 2 1 c 2 2 t 2 ) E ( t , z ) = 4 π c 2 2 t 2 P ( t , z ) .
E ( t , z ) = Ê ( t , z ) exp [ i ω ( t z / c ) ] + c . c . ,
P ( t , z ) = P ̂ ( t , z ) exp [ i ω ( t z / c ) ] + c . c .
k Ê t 2 Ê t 2 , k Ê z 2 Ê z 2 , k 2 P ̂ k P ̂ t 2 P ̂ t 2 ,
( z + c t ) Ê ( t , z ) = 2 π i k P ̂ ( t , z ) .
z Ê ( t 0 + z c , z ) = 2 π i k P ̂ ( t 0 + z c , z ) .
q 1 = [ d / 2 + x sin ( θ / 2 ) ] / cos ( θ / 2 ) ,
q 2 = [ d / 2 x sin ( θ / 2 ) ] / cos ( θ / 2 ) .
̂ s ( t 0 , z ) = ̂ p ( t 0 , z ) 2 π i k q 1 q 2 P ̂ ( t 0 , z ) d z .
P ̂ ( t 0 , z ) = P ̂ ( 1 ) ( t 0 , z ) + P ̂ ( 3 ) ( t 0 , z ) + .
S pp = 4 π Re d t 0 [ i ̂ p * ( t 0 , q 1 ) q 1 q 2 P ̂ ( 3 ) ( t 0 , z ) d z ] .
P ̂ ( 3 ) ( t ) = ( i ) 3 N t d t 3 t 3 d t 2 t 2 d t 1 ̂ p ( t 3 ) ̂ e * ( t 2 ) × ̂ e ( t 1 ) R ( t t 3 , t 3 t 2 , t 2 t 1 ) ,
̂ e ( t ) = E e exp [ x e 2 4 ω e x 2 y e 2 4 ω e y 2 ( z e c t ) 2 4 ω e z 2 ] ,
̂ p ( t ) = E p exp [ x 2 4 ω p x 2 y 2 4 ω p y 2 ( z c t ) 2 4 ω p z 2 ] .
P ̂ i ( 3 ) ( t 0 , x , y , z ) = ( i ) 3 N | E e | 2 E p × exp [ ( α x + 1 4 ω p x 2 ) x 2 ( α y 1 4 ω p y 2 ) y 2 ( α t 1 4 ω p z 2 ) ( c t 0 ) 2 ] × exp [ α z z 2 + z ( β x x + β t c t 0 ) + γ x c t 0 x ] ,
α x = cos 2 θ 2 ω e x 2 + sin 2 θ 2 ω e x 2 ,
α y = 1 2 ω e y 2 + 1 2 ω p y 2 ,
α z = sin 2 θ 2 ω e x 2 + ( 1 cos θ ) 2 2 ω e x 2 ,
α z = sin 2 θ 2 ω e x 2 + ( 1 cos θ ) 2 2 ω e x 2 ,
β x = sin 2 θ 2 ω e x 2 + ( 1 cos θ ) sin θ ω e z 2 ,
β t = ( 1 cos θ ) ω e z 2 ,
γ x = sin θ ω e z 2 .
S PP , i ( x ) = 4 π 2 N | E e | 2 | E p | 2 c α y α t × exp [ ( α x + 1 2 ω p x 2 γ x 2 4 α t ) x 2 ] q 1 q 2 d z × exp [ ( α z β t 2 4 α t ) z 2 + ( β z + γ x β t 2 α t ) x z ] .
S PP , c ( x ) = 16 π 2 N | E e | 2 | E p | 2 ( π α y ) 1 / 2 ω e z 2 ω p z 2 c 4 × exp ( x 2 2 ω p x 2 ) × q 1 q 2 d z exp [ ( z sin θ x cos θ ) 2 2 ω e x 2 ] .
S ¯ PP ( x ) exp { x 2 2 [ ( ω e z 2 + ω p z 2 ) cot 2 ( θ / 2 ) + ω e x 2 ] } × { Φ [ δ z d 2 cos ( θ / 2 ) δ x x ] Φ [ δ z d 2 cos ( θ / 2 ) δ x x ] } / { Φ [ z d 2 cos ( θ / 2 ) x x ] Φ [ z d 2 cos ( θ / 2 ) x x ] } ,
δ z = ( α z β t 2 4 α t ) 1 / 2 ,
δ x = 2 α t β x + γ x β t 4 α t δ z + δ z tan ( θ / 2 ) ,
z = sin θ 2 ω e x ,
x = cos θ 2 ω e x + z tan ( θ / 2 ) ,
Φ ( ζ ) = 0 ζ exp ( ξ 2 ) d ξ .
S ¯ pp ( x , τ ) exp { [ x c τ cot ( θ / 2 ) ] 2 2 [ ( ω e z 2 + ω p z 2 ) cot 2 ( θ / 2 ) + ω e x 2 ] } × { Φ [ δ z d 2 cos ( θ / 2 ) δ x x + δ τ c τ ] Φ [ δ z d 2 cos ( θ / 2 ) δ x x + δ τ c τ ] } / { Φ [ z d 2 cos ( θ / 2 ) x x ] Φ [ z d 2 cos ( θ / 2 ) x x ] } ,
δ τ = 1 cos θ 2 δ z ( ω e z 2 + ω p z 2 ) .
S ¯ pp , d 0 ( x ) exp { [ α x γ x 2 4 α t + ( α z β t 2 4 α t ) tan 2 ( θ / 2 ) + ( β x + γ x β t 2 α t ) tan 2 ( θ / 2 ) ] x 2 } ,
α x = sin 2 θ 2 ω e z 2 ,
α z = ( 1 cos θ ) 2 2 ω e z 2 ,
β x = ( 1 cos θ ) sin θ ω e z 2 .
S ¯ PP , d 0 ( x ) exp { [ 2 x tan ( θ / 2 ) ] 2 2 ( ω e z 2 + ω p z 2 ) } .
σ pp , d 0 = 1 c ω e z 2 + ω p z 2 ,
S ¯ PP , d ( x ) exp { [ α x γ x 2 4 α t ( 2 β x α t + γ x β t ) 2 4 α t ( 4 α z α t β t 2 ] x 2 } ,
S ¯ PP , d ( x ) exp { x 2 2 [ ( ω e z 2 + ω p z 2 ) cot 2 ( θ / 2 ) + ω e x 2 ] } .
σ pp , d = 1 c ( ω e z 2 + ω p z 2 ) + ω e x 2 tan 2 ( θ / 2 ) .
E 1 ( t , z 1 ) = Ê 1 ( t , z 1 ) exp [ i ω ( t z 1 / c ) ] + c . c . ,
E 2 ( t , z 2 ) = Ê 2 ( t , z 2 ) exp [ i ω ( t z 2 / c ) ] + c . c . ,
Ê 1 ( t , x 1 , y 1 , z 1 ) = E 1 exp [ x 1 2 4 ω e x 2 y 1 2 4 ω e y 2 ( z 1 c t ) 2 4 ω e z 2 ] ,
Ê 2 ( t , x 2 , y 2 , z 2 ) = E 2 exp [ x 2 2 4 ω e x 2 y 2 2 4 ω e y 2 ( z 2 c t ) 2 4 ω e z 2 ] .
x 1 = z s cos ( θ / 2 ) x s sin ( θ / 2 ) ,
y 1 = y s ,
z 1 = z s sin ( θ / 2 ) + x s cos ( θ / 2 ) ,
x 2 = z s cos ( θ / 2 ) + x s sin ( θ / 2 ) ,
y 2 = y s ,
z s = z s sin ( θ / 2 ) + x s cos ( θ / 2 ) .
Ê 1 ( x , y , z , t ) Ê 2 * ( x , y , z , t ) = E 1 E 2 * exp [ η z z s 2 η y y s 2 η x ( x s Qct ) 2 η t ( c t ) 2 ] ,
η z = cos 2 ( θ / 2 ) 2 ω e x 2 + sin 2 ( θ / 2 ) 2 ω e x 2 ,
η y = 1 2 ω e y 2 ,
η x = sin 2 ( θ / 2 ) 2 ω e x 2 + cos 2 ( θ / 2 ) 2 ω e z 2 ,
Q = ω e x 2 cos ( θ / 2 ) ω e z 2 sin 2 ( θ / 2 ) + ω e x 2 cos 2 ( θ / 2 ) ,
η t = sin 2 ( θ / 2 ) 2 [ ω e z 2 sin 2 ( θ / 2 ) + ω e x 2 cos 2 ( θ / 2 ) ] .
x s = x ,
y s = y sin ( θ / 2 ) + z cos ( θ / 2 ) ,
z s = y cos ( θ / 2 ) + z sin ( θ / 2 ) .
Ê p ( t , x p , y p , z p ) = E p exp [ x p 2 4 ω p x 2 y p 2 4 ω p y 2 ( z p c t ) 2 4 ω p z 2 ] ,
x p = x ,
y p = y cos θ + z sin θ ,
z p = y sin θ + z cos θ .
P ̂ i ( 3 ) ( t 0 , x , y , z ) = ( i ) 3 N E 1 E 2 * E p exp [ η z z 2 η y y 2 η x x 2 η t ( c t 0 ) 2 + κ x x c t 0 κ z z c t 0 κ y y c t 0 + κ x x z μ y y z ] ,
η z = sin 2 θ 8 ω e x 2 + 1 + cos θ 4 ω e y 2 + ( 1 cos θ ) 2 + 4 8 ω e z 2 + sin 2 θ 4 ω p y 2 + ( 1 cos θ ) 2 4 ω p z 2 ,
η y = ( 1 + cos θ ) 2 8 ω e x 2 + 1 cos θ 4 ω e y 2 + sin 2 θ 8 ω e z 2 + cos 2 θ 4 ω p y 2 + sin 2 θ 4 ω p z 2 ,
η x = 1 cos θ 4 ω e x 2 + 1 + cos θ 4 ω e x 2 + 1 4 ω p x 2 ,
η t = 1 2 ω e x 2 + 1 4 ω p z 2 ,
κ x = cos ( θ / 2 ) ω e x 2 ,
κ z = 1 cos θ 2 ω p z 2 + 1 2 ω e z 2 ,
κ y = sin θ 2 ω p z 2 ,
μ y = ( 1 + cos θ ) sin θ 4 ω e x 2 sin θ 2 ω e y 2 + ( 1 cos θ ) sin θ 4 ω e z 2 + sin θ cos θ 2 ω p y 2 + ( 1 cos θ ) sin θ 4 ω p z 2 .
S TG , i ( x ) exp [ ( 1 cos θ 2 ω e x 2 + 1 2 ω p x 2 + 1 + cos θ 2 ω e z 2 ) x 2 ] exp [ κ x 2 2 ( 4 η t η y κ y 2 + 4 η y η z μ y 2 + 2 κ y μ y 4 η y κ z 4 η t η z η y η t μ y 2 η y κ z 2 κ y 2 η z + κ y κ z μ y ) x 2 ] .
P ̂ c ( 3 ) exp ( x 1 2 + x 2 2 4 ω e x 2 y 1 2 + y 2 2 w ω e y 2 x p 2 4 ω p x 2 y p 2 4 ω p z 2 ) × t d t 3 exp [ ( z p c t 3 ) 2 4 ω p z 2 ] × d t 2 exp { [ t 2 z s sin ( θ / 2 ) ω e z ] 2 } × t 1 d t 1 exp ( t 1 2 ) .
S TG , c ( x ) exp [ ( 1 cos θ 2 ω e x 2 + 1 2 ω p x 2 ) x 2 ] .
S ¯ TG ( x ) exp ( 1 2 [ 1 + cos θ ω e z 2 κ x 2 ( 4 η t η y κ y 2 + 4 η y η z μ y 2 + 2 κ y μ y 4 η y κ z 4 η t η z η y η t μ y 2 η y κ z 2 κ y 2 η z + κ y κ z μ y ) ] x 2 ) .

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