Abstract

The geometry of a single-shot pulse measurement can lead to errors in the pulse characterization. An analysis of this problem is presented, and an algorithm for extracting the correct pulse information is proposed.

© 1996 Optical Society of America

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References

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  1. E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
    [Crossref]
  2. J.-C. M. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond pulses,” Appl. Opt. 24, 1270–1282 (1985).
    [Crossref]
  3. F. Salin, P. Georges, G. Roger, and A. Brun, “Single-shot measurement of a 52-fs pulse,” Appl. Opt. 26, 4528–4531 (1987).
    [Crossref] [PubMed]
  4. G. Szabó, Z. Bor, and A. Müller, “Phase-sensitive single-pulse autocorrelator for ultrashort laser pulses,” Opt. Lett. 9, 746–748 (1988).
    [Crossref]
  5. 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]
  6. J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993).
    [Crossref] [PubMed]
  7. R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
    [Crossref]
  8. K. W. DeLong, R. Trebino, and D. J. Kane, “Comparison of ultrashort-pulse frequency-resolved optical gating traces for three common beam geometries,” J. Opt. Soc. Am. B 11, 1595–1608 (1994).
    [Crossref]
  9. K. W. DeLong and R. Trebino, “Improved ultrashort pulse-retrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11, 2429–2437 (1994).
    [Crossref]

1994 (2)

1993 (3)

1988 (1)

G. Szabó, Z. Bor, and A. Müller, “Phase-sensitive single-pulse autocorrelator for ultrashort laser pulses,” Opt. Lett. 9, 746–748 (1988).
[Crossref]

1987 (1)

1985 (1)

Bor, Z.

G. Szabó, Z. Bor, and A. Müller, “Phase-sensitive single-pulse autocorrelator for ultrashort laser pulses,” Opt. Lett. 9, 746–748 (1988).
[Crossref]

Brun, A.

DeLong, K. W.

Diels, J.-C. M.

Fontaine, J. J.

Fujimoto, J. G.

Georges, P.

Ippen, E. P.

J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993).
[Crossref] [PubMed]

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
[Crossref]

Kane, D. J.

McMichael, I. C.

Müller, A.

G. Szabó, Z. Bor, and A. Müller, “Phase-sensitive single-pulse autocorrelator for ultrashort laser pulses,” Opt. Lett. 9, 746–748 (1988).
[Crossref]

Paye, J.

Ramaswamy, M.

Roger, G.

Salin, F.

Shank, C. V.

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
[Crossref]

Simoni, F.

Szabó, G.

G. Szabó, Z. Bor, and A. Müller, “Phase-sensitive single-pulse autocorrelator for ultrashort laser pulses,” Opt. Lett. 9, 746–748 (1988).
[Crossref]

Trebino, R.

Appl. Opt. (2)

J. Opt. Soc. Am. A (2)

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

Opt. Lett. (3)

Other (1)

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses—Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), p. 83.
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Coordinates in a two-beam single-shot geometry: The propagation vectors of the beams are zE and zg, the propagation direction of the signal is v, and the normal of the nonlinear medium is b; the angle from v to b is ϕ, and the intersection angle of zE and zg is 2Θ. The bisectors of zE and zg are x and y. The angle θ/2 between v and y is the deviation of the signal from the bisector. The angles ϕ, θ, and Θ are measured in the nonlinear optical medium.

Fig. 2
Fig. 2

Deconvolution result (a) without correction and (b) with correction [Eq. (18)] of a FROG image generated by a transform-limited Gaussian pulse in a medium with the geometrical smearing factor τ G / τ FWHM ( Gaussian ) = 2 in a single-shot PG FROG. The array size used in this simulation is 128 × 128, and the best deconvolution result is determined by minimization of the FROG-trace error in 100 iterations of the algorithm proposed in Ref. 7. The correction algorithm yields quite satisfactory results.

Fig. 3
Fig. 3

Best FROG-trace errors versus the geometrical smearing factor τ G / τ FWHM ( Gaussian ) in a single-shot FROG. The test pulse shape is transform-limited Gaussian. The array size used in this simulation is 128 × 128, and the best deconvolution result is determined by minimization of the FROG-trace error in 100 iterations of the algorithm proposed in Ref. 7.

Fig. 4
Fig. 4

Single-shot geometry in a PG FROG. The nonlinear medium is of thickness l, and the pulse wave fronts arrive at a time t and at a later time t + T . The shortest full width at half-maximum of the signal trace is given by 2W sin θ = 2l sin2 Θ/cos(Θ − ϕ) in this geometry.

Equations (29)

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E sig ( t , τ ) = E ( t ) g ( t τ ) ,
| v min v max E sig ( t , τ ) d v | 2 = | v min v max E sig ( t , 2 x sin Θ / c ) d v | 2 ,
v min ( u ) = u tan ϕ l / 2 cos ϕ , v max ( u ) = u tan ϕ + l / 2 cos ϕ .
x = u cos θ 2 + v sin θ 2 .
| 1 / 2 1 / 2 E sig [ t , τ ( u ) τ G ξ ] d ξ | 2 ,
τ ( u ) = 2 u c sin Θ ( cos θ 2 + tan ϕ sin θ 2 )
τ G 2 l sin Θ sin ( θ / 2 ) c cos ϕ
| 1 / 2 1 / 2 E sig [ ω , τ ( u ) τ G ξ ] d ξ | 2 ,
E sig ( ω , τ ) = E ( t ) g ( t τ ) exp ( i ω t ) d t .
| 1 / 2 1 / 2 { E ( t ) g [ t τ ( u ) + τ G ξ ] exp ( i ω t ) d t } d ξ | 2 = | { 1 / 2 1 / 2 E ( t ) g [ t τ ( u ) + τ G ξ ] d ξ } exp ( i ω t ) d t | 2 .
F ( t , τ ) = 1 / 2 1 / 2 E ( t ) g ( t τ + τ G ξ ) d ξ
F ( t , τ ) d τ = [ 1 / 2 1 / 2 E ( t ) g ( t τ + τ G ξ ) d ξ ] d τ = E ( t ) [ 1 / 2 1 / 2 g ( t τ + τ G ξ ) d τ d ξ ] E ( t ) .
a a f ( x ) d x = 2 af ( 0 ) + a 3 3 f ( 0 ) + O ( a 5 ) ,
S ( ω , τ ) | 1 / 2 1 / 2 E sig ( ω , τ τ G ξ ) d ξ | 2 = | E sig ( ω , τ ) + τ G 2 24 2 τ 2 E sig ( ω , τ ) + O ( τ G 4 ) | 2 = | E sig ( ω , τ ) | 2 + τ G 2 24 ( E sig 2 τ 2 E sig * + E sig * 2 τ 2 E sig ) + O ( τ G 4 ) .
τ G 2 48 ( E sig 2 τ 2 E sig * + E sig * 2 τ 2 E sig + 2 E sig τ E sig * τ ) = τ G 2 48 2 τ 2 ( E sig * E sig ) ,
S ( ω , τ ) | E sig ( ω , τ ) | 2 + τ G 2 48 2 τ 2 ( E sig * E sig ) = | E sig ( ω , τ ) | 2 + ( τ G / 2 ) 2 24 2 τ 2 ( E sig * E sig ) 1 / 2 1 / 2 I FROG ( ω , τ τ G 2 ξ ) d ξ .
S ( ω , t ) τ 1 / 2 1 / 2 τ I FROG ( ω , τ τ G 2 ξ ) d ξ = 2 τ G 1 / 2 1 / 2 ξ I FROG ( ω , τ τ G 2 ξ ) d ξ = 2 τ G [ I FROG ( ω , τ τ G 2 2 ) I FROG ( ω , τ + τ G 2 2 ) ] .
I FROG ( ω , τ + τ G 2 ) I FROG ( ω , τ ) + τ G 2 τ S ( ω , τ + τ G 2 2 ) .
z E = v cos ( Θ θ 2 ) + u sin ( Θ θ 2 ) ,
z g = v cos ( Θ + θ 2 ) u sin ( Θ + θ 2 ) .
E sig ( t , u , v ) = E ( t z E c ) g ( t z g c ) = E [ t v c cos ( Θ θ 2 ) u c sin ( Θ θ 2 ) ] × g [ t v c cos ( Θ + θ 2 ) + u c sin ( Θ + θ 2 ) ] .
v min v max E sig [ t + v c cos ( Θ θ 2 ) , u , v ] d v
v min v max E sig [ t u c sin ( Θ θ 2 ) ] × g [ t + 2 v c sin Θ sin θ 2 + u c sin ( Θ + θ 2 ) ] d v ,
E [ t u c sin ( Θ θ 2 ) ] × g [ t + 2 v c tan θ sin Θ sin θ 2 + u c sin ( Θ + θ 2 ) ] .
τ d = 2 u c sin Θ ( tan θ sin θ 2 + cos θ 2 ) .
τ ( u ) = τ d = 2 u c sin Θ ( tan θ sin θ 2 + cos θ 2 ) ,
1 / 2 1 / 2 E [ t u c sin ( Θ θ 2 ) ] × g [ t u c sin ( Θ θ 2 ) τ ( u ) + τ G ξ ] d ξ ,
| 1 / 2 1 / 2 E [ t u c sin ( Θ θ 2 ) ] × g [ t u c sin ( Θ θ 2 ) τ ( u ) + τ G ξ ] × d ξ exp ( i ω t ) d t | 2 .
| 1 / 2 1 / 2 { E [ t u c sin ( Θ θ 2 ) ] × g [ t u c sin ( Θ θ 2 ) τ ( u ) + τ G ξ ] × exp ( i ω t ) d t } d ξ | 2 = | 1 / 2 1 / 2 { E ( t ) g [ t τ ( u ) + τ G ξ ] exp ( i ω t ) d t } d ξ | 2 = | 1 / 2 1 / 2 E sig [ ω , τ ( u ) τ G ξ ] d ξ | 2 .

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