Geometrical distortions and correction algorithm in single-shot pulse ...

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Geometrical distortions and correction algorithm in single-shot pulse measurements: application to frequency-resolved optical gating An-Chun Tien and Steve Kane Center for Ultrafast Optical Science, University of Michigan, 2200 Bonisteel, Room 1006, Institute of Science and Technology Building, Ann Arbor, Michigan 48109–2099

Jeff Squier Department of Electrical and Computer Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0339

Bern Kohler and Kent Wilson Department of Chemistry, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0339 Received December 14, 1994; revised manuscript received February 9, 1996 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

1.

INTRODUCTION

Characterization of femtosecond optical pulses is an important aspect of ultrafast optics, and many techniques have been developed to measure the temporal profile of pulses.1 – 8 Some techniques of pulse measurement, such as interferometric or scanning background-free autocorrelation, are convenient for high-repetition-rate lasers; however, low-repetition-rate (subkilohertz) high-energy lasers are best characterized by use of single-shot techniques.3 – 5 The geometry of a single-shot second-harmonicgeneration autocorrelator is straightforward3 : Two incoming beams are overlapped in a x s2d material, and the autocorrelation signal propagates along the bisector of the incoming beams. As long as certain conditions are satisfied (large beams, etc.),3 the signal trace measured by the detector is the true autocorrelation of the pulse. In other single-shot measurements, however, the signal measured by the detector is not at all what might be expected. If the signal generated by the nonlinear medium does not propagate along the bisector of the incoming beams, the finite thickness of the nonlinear medium will distort the signal trace and cause the temporal resolution to deteriorate. Frequency-resolved optical gating5 – 8 (FROG) is a practical technique for characterizing the full time-dependent electric field of an individual arbitrary ultrashort pulse, and it can be operated on a single-shot basis.5 FROG measures the spectrum of the signal generated by the incoming pulse under test (probe) gated by its replica (pump) in an instantaneously responding nonlinearoptical medium versus the relative time delay between 0740-3224/96/061160-06$10.00

the pump and the probe; FROG measurements can be performed simply by spectral resolution of the signal in any autocorrelation-type setup. Like all spectrograms, the FROG trace graphically displays a complex waveform in an intuitive way. Furthermore, the deconvolution of a FROG trace is a two-dimensional phase-retrieval problem7 and requires no prior assumptions about the pulse intensity and phase. Single-shot polarization-gate (PG) FROG suffers from the problem of geometrical distortion because the signal propagates in the direction of one of the beams rather than along the bisector. This can result in a measured pulse width that is artificially broadened. For instance, in a PG FROG, when the thickness of the medium is 400 mm and the intersection angle is 8±, the temporal full width at half-maximum of the signal trace is at least 19.5 fs; this means pulses shorter than 20 fs will be unresolvable under these conditions. The temporal resolution can be improved by use of a thinner medium and a smaller intersection angle. This, however, is not always the optimal solution; the signal level will decrease, and the signal-to-noise ratio may approach unacceptable limits. In addition, a shallow angle between the beams limits the time window. In this paper we first analyze the geometrical distortion in a single-shot measurement. Second, we propose algorithms for correcting the errors that are due to the finite thickness of the nonlinear medium. Then, we show computer-simulation results, which can be used for determining the geometrical parameter such that the errors that are due to temporal smearing are below an appropriate level. Finally, we discuss the calibration of time  1996 Optical Society of America

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delay in a single-shot geometry. The analysis is accomplished for the most general two-beam single-shot geometry, so the results are applicable to all types of two-beam single-shot measurement.

2. ANALYSIS OF SINGLE-SHOT GEOMETRY In a single-shot geometry the detector measures the accumulation of the signal from the entire thickness of the medium. Figure 1 shows the coordinates for a general two-beam single-shot geometry in the nonlinear medium. 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 f, and the intersection angle of zE and zg is 2Q. The bisectors of zE and zg , x and y, are also shown in Fig. 1. The angle uy2 between v and y is the deviation of the signal from the bisector. The angles f, u, and Q are measured in the nonlinear optical medium. In the following analysis we assume that (1) both input beams are plane waves and nondepleted, (2) the response of medium is instantaneous, and (3) the broadening for input fields and the signal field owing to material dispersion is negligible. The signal generated by a probe field Estd and a gate pulse gstd in a nonlinear medium is Esig st, td ­ Estdgst 2 td ,

(1)

where t is the time delay between the two input beams. Because of symmetry, the field distribution of the signal Esigst, td in the bisector frame sx, yd is a function of x only, and the conversion from x to the time delay t is ct ­ 22x sin Q, or Esig st, td ­ Esig st, 22x sin Qycd, where c is the speed of light in the medium. Therefore the intensity distribution in the bisector frame is simply the nth-order intensity correlation function, where n is the order of the optical nonlinearity. In addition, because the nonlinear medium has a finite thickness to accumulate the signal strength and because the signal propagates in the direction of v, the signal measured by an intensity detector is proportional to *É

Z

vmax vmin

x ­ u cos

u . u 1 v sin 2 2

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(4)

By using Eq. (4) and changing variable v to j ; sv 2 u tan fdyslycos fd, we can rewrite Eq. (2) as *É É 2+ Z 1/2 , (5) Esig ft, tsud 2 tG jgdj 21/2

where tsud ­ 2

√ ! 2u u u sin Q cos 1 tan f sin c 2 2

(6)

defines the encoded time per unit length and tG ;

2l sin Q sinsuy2d c cos f

(7)

is a factor dependent on the geometry only. From Eq. (5), it can be seen that the distortion of the signal is simply a function of a single parameter tG . Equation (5) can be used for estimating the temporal resolution in a two-beam single-shot intensity correlation setup with an arbitrary geometry. Another application of Eq. (5) is to analyze the smearing problem in a single-shot FROG. The signal measured by the detector in a single-shot FROG setup can be found by replacement of the timeaveraging-intensity operation kj j2 l with a spectrometer R` operator j 2` exps2ivtddtj2 in Eq. (5):

É 2+ Esig st, tddv *É ­

Z

vmax

vmin

É 2+ Esig st, 22x sin Qycddv

, (2)

where k l denotes a time integral over t, and vmin and vmax are the integration limits defined by the thickness l and the orientation f of the nonlinear material: vmin sud ­ u tan f 2

ly2 , cos f

vmax sud ­ u tan f 1

ly2 . cos f

(3)

The temporal information will be smeared if the signal propagation direction v is different from y since the representation of Esig in frame su, vd is dependent on both u and v, which is a direct consequence from the coordinate transformation

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 f, and the intersection angle of zE and zg is 2Q. The bisectors of zE and zg are x and y. The angle uy2 between v and y is the deviation of the signal from the bisector. The angles f, u, and Q are measured in the nonlinear optical medium.

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É

Z

É2

1/2

21/2

Tien et al.

Esig fv, tsud 2 tG jgdj

(8)

,

where Esig sv, td is the spectrally resolved signal field defined as7 Z ` Esig sv, td ­ Estdgst 2 tdexps2ivtddt . (9)

√ p ! 2 ≠2 p tG2 p ≠ E 1 2 ≠Esig ≠Esig 1 Esig Esig 2 Esig sig 48 ≠t ≠t 2 ≠t ≠t ­

tG2 ≠2 sE p Esig d , 48 ≠t 2 sig

(15)

an approximated expression of Eq. (14) is

2`

tG2 ≠2 sE p Esig d 48 ≠t 2 sig p 2 2 stG y 2d ≠ sE p Esigd ­ jEsig sv, tdj2 1 24 ≠t 2 sig ! √ Z 1/2 tG ø IFROG v, t 2 p j dj . (16) 2 21/2

An alternative approach for analyzing the single-shot geometry is described in Appendix A.

Ssv, td ø jEsig sv, tdj2 1

3. CORRECTION OF THE TEMPORAL SMEARING Equation (8) is equivalent to É ( ) É2 Z 1/2 Z ` Estdgft 2 tsud 1 tG jgexps2ivtddt dj 21/2

É

­

2`

Z

`

(

Z

2`

)

1/2

É2

Estdgft 2 tsud 1 tG jgdj exps2ivtddt

.

21/2

(10) This is a two-dimensional phase-retrieval problem; therefore the solution for Z 1/2 F st, td ­ Estdgst 2 t 1 tG jddj (11) 21/2

is unique. Also, it is easy to show that the integral of F st, td over t is proportional to Estd: " # Z ` Z ` Z 1/2 Estdgst 2 t 1 tG jddj dt F st, tddt ­ 2`

2`

"

­ Estd

21/2

Z

1/2

21/2

Z

#

`

gst 2 t 1 tG jddtdj

2`

/ Estd.

(12)

In principle, the field Estd can be derived when Eqs. (11) and (12) are employed as two constraints in the phaseretrieval iteration algorithm.9 However, it is a nontrivial task to implement these two conditions into an existing retrieval routine. By use of the formula Z a a3 00 f s0d 1 Osa5 d , f sxddx ­ 2af s0d 1 (13) 3 2a the measured FROG signal Ssv, td can be expanded in terms of tG : É É2 Z 1/2 Ssv, td ; Esig sv, t 2 tG jddj É ­

21/2

t 2 ≠2 Esig sv, td 1 G Esig sv, td 1 OstG4 d 24 ≠t 2

É2

­ jEsig sv, tdj2 ! √ ≠2 p ≠2 tG2 p Esig 2 Esig 1 Esig 2 Esig 1 OstG4 d . 1 24 ≠t ≠t (14) The first term in Eq. (14) is the true FROG function IFROG sv, td, and the second term is the lowest-order deviation owing to the finite thickness. If we assume that the second term is approximately equal to

Therefore the partial derivative of Ssv, td with respect to t can be approximated as √ ! Z 1/2 ≠ tG ≠Ssv, td ø IFROG v, t 2 p j dj ≠t 2 21/2 ≠t √ ! p Z 1/2 tG 2 ≠ IFROG v, t 2 p j dj ­2 tG 21/2 ≠j 2 ! √ p " tG 2 IFROG v, t 2 p ­2 tG 2 2 !# √ tG . (17) 2 IFROG v, t 1 p 2 2 This means that the true FROG function can be approximately calculated by use of the following formula: ! √ tG IFROG v, t 1 p 2 ! √ tG tG ≠ ø IFROG sv, td 1 p . (18) S v, t 1 p 2 ≠t 2 2

4.

NUMERICAL EXAMPLE

In this section a single-shot polarization-gate FROG5 is used for the simulation. In this particular type of FROG, the gate function gst 2 td is the intensity of the input pulse jEst 2 tdj2 , and the signal propagates along the same direction as the probe pulse sv ­ zE , uy2 ­ Qd. The array size is 128 3 128 in all simulations in this section, and the best retrieved result is determined by minimization of the FROG-trace error7 in 100 iterations of the algorithm proposed by Trebino et al.7 Figure 2(a) shows the calculated electric field from a severely smeared FROG trace generated by a transformlimited Gaussian pulse from a thick medium. The temporal smearing parameter tG is twice as long as the sGaussiand sGaussiand ­ 2 (l ­ 1 mm, pulse duration tFWHM ; i.e., tGytFWHM sGaussiand ± ± 8 2Q ­ 8 , f ­ 0 , c ­ 2 3 10 mys, and tFWHM ­ 25 fs, for instance). The deconvolved intensity envelope does not match the test pulse well, and the deconvolved phase is not as flat as it is supposed to be. Actually, the FROGtrace error is larger than 1%, which implies that the deconvolution algorithm does not converge well in this case. Figure 2(b) shows the deconvolution result for the FROG trace after being numerically corrected from the smeared case shown in Fig. 2(a) by application of Eq. (18).

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(a)

(b) 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 sGaussiand smearing factor tG ytFWHM ­ 2 in a single-shot PG FROG. The array size used in this simulation is 128 3 128, 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.

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pulse that is broad in time will produce a signal that is spatially broadened. It is necessary to calibrate singleshot measuring devices in order to establish the correspondence between temporal delay and spatial position. The method used for calibration of autocorrelators (where u ­ 0) is fairly straightforward: Introduction of a known delay T into one of the arms of an autocorrelator (either by translation of a delay rail or by insertion of a piece of glass into the beam) causes the signal to shift spatially by an amount du; therefore, Tydu is the encoded time per unit distance. Because of the symmetry of the single-shot autocorrelator, the shift du is the same regardless of which arm of the autocorrelator is delayed. In other single-shot measurements where u fi 0 (such as PG-FROG) the finite thickness of the nonlinear medium leads to a problem of calibration of the spatial translation of the signal trace with respect to the time delay of pulses. Mathematically, this is described in Eq. (6). When the signal propagation direction v is different from y su fi 0d, the spatial translations of the signal trace that are due to the same time delay are different for both arms. Figure 4 shows the pulse wave fronts at a time t and at a later time t 1 T in a PG FROG, in which the signal propagates in the direction of one of the beams, so uy2 is equal to Q. Therefore ≠ty≠u is ssin 2Q 1 2 tan f sin 2 Qdyc in a PG FROG. However, a time delay T added to the signal arm shifts the signal trace by cTytan 2Q, whereas it is cTysin 2Q if the same time delay is applied to the gating arm. As a result, the encoded time per unit distance for a PG FROG must be calibrated by introduction of a delay to the gating arm if the medium is perpendicular to the signal arm sf ­ 0d or by use of the signal arm if the medium is perpendicular to the gating arm sf ­ 2Qd.

6.

CONCLUSION

We have shown a general analytic procedure for temporal information measurements in a single-shot geometry. The geometrical smearing depends on a single parameter tG , defined in Eq. (7). Two different types

The best FROG-trace error in 100 iterations is 0.28%. As shown in Fig. 2(b), the correction algorithm yields quite satisfactory results even with smearing as much as sGaussiand tG ytFWHM ­ 2. The best FROG-trace errors in 100 iterations for other values are plotted in Fig. 3. We found that for the array size used here, roughly a 0.5% or greater FROG-trace error results in poor phase retrieval. When the geometrical smearing factor tG is less than the pulse width t (l ­ 0.4 mm, 2Q ­ 8±, f ­ 0±, c ­ 2 3 108 mys, sGaussiand and tFWHM ­ 20 fs, for example), the smearing causes insignificant FROG-trace errors (,0.25%).

5. CALIBRATION OF SINGLE-SHOT MEASUREMENTS In single-shot pulse measurements, temporal information about the pulse is translated into spatial information; a

Fig. 3. Best FROG-trace errors versus the geometrical smearsGaussiand ing factor tG ytFWHM in a single-shot FROG. The test pulse shape is transform-limited Gaussian. The array size used in this simulation is 128 3 128, and the best deconvolution result is determined by minimization of the FROG-trace error in 100 iterations of the algorithm proposed in Ref. 7.

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! √ ! zg zE g t2 Esig st, u, vd ­ E t 2 c c " √ ! v u ­E t2 cos Q 2 c 2 √ !# u u 2 sin Q 2 c 2 " √ ! v u 3g t2 cos Q 1 c 2 √ !# u u sin Q 1 . 1 c 2 √

(21)

Since the signal is traveling in space in the direction v and the phase-matching condition requires a propagating speed cycossQ 2 uy2d, the total signal accumulated over the thickness of the medium is " √ ! # Z vmax v u , u, v dv (22) Esig t 1 cos Q 2 c 2 vmin 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 1 T . The shortest full width at half-maximum of the signal trace is given by 2W sin u ­ 2l sin2 QycossQ 2 fd in this geometry.

of numerical-correction algorithm for extracting the temporal information from smeared measurements by elimination of the errors caused by the finite thickness of the nonlinear medium have been derived, which can be used in all types of two-beam single-shot measurement. We have presented a graph that can be used to determine how much error can be expected in a given setup. The shortest pulse width that a single-shot setup can measure without serious distortion is tG . We have also pointed out that the time-delay calibration might be different for different arms owing to the asymmetry of the geometry.

APPENDIX A: ALTERNATIVE APPROACH FOR ANALYZING SINGLE-SHOT GEOMETRY In a laboratory frame the field distribution for the probe pulse Estd moving in the direction of zE can be described as Est 2 zE ycd, where c is the phase velocity of Estd in the medium. Similarly, the field distribution for the gate pulse gstd moving in the direction of zg can be described as gst 2 zgycd. Here, we assume that both the probe pulse and the gate pulse propagates at the same speed in the medium. From Fig. 1 the coordinate transformations from zE and zg to su, vd are ! √ u zE ­ v cos Q 2 1 u sin Q 2 2 √ ! √ u zg ­ v cos Q 1 2 u sin Q 1 2 √

u 2 u 2

! ,

(19)

.

(20)

!

Therefore the signal field generated at a certain time t and at a certain point su, vd in space is

or Z

"

vmax

Esig vmin

"

√ !# u u t 2 sin Q 2 c 2

√ !# v u u u 3 g t 1 2 sin Q sin 1 sin Q 1 dv , (23) c 2 c 2 where vmin and vmax are defined in Eq. (3). For an infinitesimally thin medium the integral is proportional to the integrand evaluated at v ­ u tan f. This yields " √ !# u u E t 2 sin Q 2 c 2 " √ !# v u u u 3 g t 1 2 tan u sin Q sin 1 sin Q 1 . c 2 c 2 (24) The encoded time per unit distance can be found by the delay of g by a time td . Then, by equating the arguments of E and g, one finds the relation between td and u: √ ! u u . u td ­ 2 sin Q tan u sin 1 cos (25) c 2 2 Since a time delay td in the gate function g introduces a shift 2td to the signal trace, the encoded time per unit distance is √ ! u u , u tsud ­ 2td ­ 22 sin Q tan u sin 1 cos (26) c 2 2 which is identical to what we found in Eq. (6). By a proper change of variable, Eq. (23) can be rewritten as √ !# " Z 1/2 u u E t 2 sin Q 2 c 2 21/2 " √ ! # u u 3g t2 sin Q 2 2 tsud 1 tG j dj , (27) c 2

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where tG is defined in Eq. (7). The signal measured by a FROG is then É " √ !# Z ` Z 1/2 u u E t 2 sin Q 2 c 2 2` 21/2 √ ! # " u u 2 tsud 1 tG j 3 g t 2 sin Q 2 c 2 É2 3 dj exps2ivtddt

(28)

.

By a change in the order of integration, this becomes É

(

√ !# u u E t 2 sin Q 2 c 2 21/2 2` " # √ ! u u 3 g t 2 sin Q 2 2 tsud 1 tG j c 2 ) É2

Z

1/2

Z

"

`

3 exps2ivtddt dj É ­ É ­

Z

1/2

21/2

Z

(

Z

)

`

Estdgft 2 tsud 1 tG jgexps2ivtddt dj

2`

1/2

21/2

É2

Esig fv, tsud 2 tG jgdj

É2 .

(29)

This is exactly the same as Eq. (8).

ACKNOWLEDGMENTS This study is partially supported by the National Science Foundation through the Center for Ultrafast Optical

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Science under STC PHY 8920108. This study was also made possible through the support of IMRA America. Steve Kane acknowledges the support of the Torrey Foundation.

REFERENCES 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. 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). 3. F. Salin, P. Georges, G. Roger, and A. Brun, “Single-shot measurement of a 52-fs pulse,” Appl. Opt. 26, 4528 – 4531 (1987). 4. G. Szab´o, Z. Bor, and A. Muller, ¨ “Phase-sensitive singlepulse autocorrelator for ultrashort laser pulses,” Opt. Lett. 9, 746 – 748 (1988). 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). 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). 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). 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). 9. K. W. DeLong and R. Trebino, “Improved ultrashort pulseretrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11, 2429 – 2437 (1994).