Measuring the temporal intensity of ultrashort laser pulses by triple correlation

Transcription

1 Appl. Phys. B 66, (1998) Applied Physics B Lasers and Optics Springer-Verlag 1998 Measuring the temporal intensity of ultrashort laser pulses by triple correlation T. Feurer, S. Niedermeier, R. Sauerbrey Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, Max-Wien-Platz 1, D Jena, Germany (Fax: /636278, feurer@qe.physik.uni-jena.de) Received: 13 May 1997/Revised version: 1 August 1997 Abstract. By measuring the triple correlation of an ultrashort laser pulse, it is possible to determine the temporal envelope of the pulse intensity. No assumptions on the analytic form of the pulse shape and no iterative algorithms are necessary. The method is extremely stable even for noisy data, and is a powerful tool for pulses that have a strong amplitude modulation. A multi-shot and a single-shot arrangement of the triple correlator has been realized. PACS: By; k; Fm Laser pulses as short as several femtoseconds can now be generated in many laboratories all over the world [1]. Since electronic devices, and even the fastest streak cameras, are too slow to measure the temporal evolution of these pulses, there has been an extensive search for novel measurement techniques over the past few years. All of these techniques rely on autocorrelation or crosscorrelation methods of a pulsed electric field, using various nonlinear effects such as second harmonic generation [2] or the optical Kerr effect [3]. In order to extract the temporal intensity, either an analytic pulse shape has to be assumed or iterative algorithms have to be used [4]. With a new experimental setup that uses a cascaded thirdorder nonlinear effect and introduces two independent delay times, background-free triple correlation of a femtosecond laser pulse has been measured in similar fashion to the technique proposed by Weber and Dändliker [5]. The method is different from the scheme proposed in [6], which relies on a single-step third-order nonlinear effect and on counterpropagating beams. The triple-correlation theory has been successfully applied in astronomy in order to increase the resolution of imaging systems [7, 8] and to measure the temporal correlation properties of lasers operating near their threshold [9]. In the late 1960s, it was shown that the third-order intensity correlation is sufficient to determine the time-dependent intensity of a laser pulse [10]. Following a scheme proposed in the mid-1980s [11], the intensity of a short laser pulse can be extracted without any assumptions about the pulse shape and using a recursive algorithm only. The measurement contains redundant information, and therefore the pulse intensity can be calculated by various paths, leading to the correct pulse shape even for very noisy data. Since the pulse intensity is obtained by a direct mathematical operation, this method is well suited for pulses with strong amplitude modulations. It has been proposed that such tailored pulses can, for example, be used in quantum control experiments [12]. 1 Theory In the first part of this section (para 1.1), definitions used throughout the text will be introduced. The second part (paras ) describes the procedure to recover the pulse intensity from the measured triple-correlation trace [11]. Since the intensity of a laser pulse is real and of finite temporal extent, it can be reconstructed from the triple correlation except for an arbitrary time shift t 0. Equivalently, the Fourier transform of the intensity can be determined except for a linear phase factor of exp ( iωt 0 ). 1.1 Definitions The triple correlation G (3) (τ 1,τ 2 ) of a laser pulse that is described by the temporal intensity I(t) is a generalization of a third-order background-free intensity autocorrelation: G (3) (τ 1,τ 2 )= dti(t)i(t τ 1 )I(t τ 2 ). (1) The Fourier transform of the triple correlation is also known as a bispectrum. Thus: G (3) (ω 1,ω 2 )= dτ 1 dτ 2 G (3) (τ 1,τ 2 ) exp ( iω 1 τ 1 iω 2 τ 2 ) = Ĩ(ω 1 + ω 2 )Ĩ( ω 1 )Ĩ( ω 2 ). (2)

2 164 For further treatment, all complex quantities will be split into an amplitude and a phase factor, i.e. Ĩ(ω) = j(ω) exp [iϕ(ω)], (3) G (3) (ω 1,ω 2 )= g(ω 1,ω 2 )exp [iβ(ω 1,ω 2 )]. (4) In order to avoid confusion with other phase sensitive methods, we need to stress that ϕ(ω) is not the phase of the field amplitude but the phase associated with the intensity in the spectral domain. Equation (2) implies a number of symmetry relations: first, exchanging the two frequencies ω 1 and ω 2, and, secondly, replacing ω 2 by ω 1 ω 2, leaves the bispectrum unchanged, and this leads to two axes of symmetry, namely ω 1 = ω 2 and ω 1 = 2ω 2. Since the laser pulse intensity is a real quantity, the equality Ĩ(ω) = Ĩ( ω) holds and consequently G (3) (ω 1,ω 2 )=[ G (3) ( ω 1, ω 2 )]. The essence of the symmetry relations is that the bispectrum is completely determined by a single octant in the (ω 1,ω 2 ) space. Therefore, in the case of a noisy signal, the eight octants may be used to calculate an average signal for further processing. The symmetry relations necessarily imply some constraints for the amplitude and the phase of the Fourier transform of the intensity, thus: j(ω) = j( ω), and (5) ϕ(ω) = ϕ( ω). (6) From (6), it immediately follows that ϕ(0) = 0, and a reconstruction procedure alone can determine the phase function for positive frequencies. In our experiment, the two time delays τ 1 and τ 2 are varied by a sufficiently small increment τ, and therefore the measurement yields a discrete set of data. The Fourier transform may then be written as: G (3) (ω 1,ω 2 )= G (3) (p ω, q ω) = Ĩ(p ω + q ω)ĩ( p ω)ĩ( q ω) = Ĩ p+q Ĩ p Ĩ q, (7) where p and q are natural numbers. From this, the corresponding equations for the amplitude and phase of the bispectrum can be obtained: g p,q = j p+q j p j q = j p+q j p j q, and (8) β p,q = ϕ p+q + ϕ p + ϕ q = ϕ p+q ϕ p ϕ q, (9) so that we now can focus on the reconstruction of the amplitude and phase values of the Fourier transform of the laser pulse intensity. 1.2 Reconstruction of the amplitude Setting p and q equal to zero in (8), we have that the amplitude at both delays of zero can be obtained, thus: j 0 = 3 g 0,0. (10) All positive values of p may be obtained by setting q = 0, yielding: g p,0 j p =. (11) j 0 As well as this simple path, there exist a number of different other routes to obtain the amplitude. 1.3 Reconstruction of the phase From (6), it is clear that the phase at both delay times of zero must itself be zero. Thus ϕ 1 may be arbitrarily set to zero, since the triple correlation is not sensitive for a linear phase, which therefore cannot be extracted from the measured data. All other phase samples may be obtained by different paths, which is clear from inspecting (6). One possible path is as follows: ϕ 2 = β 1,1 + 2ϕ 1 p ϕ p+1 = β k,1 +(p+1)ϕ 1. (12) k=1 1.4 Reconstruction of the pulse intensity After extraction of the complex Fourier transform of the pulse intensity from the measured data, the temporal intensity is simply obtained by the inverse Fourier transformation of Ĩ(ω). I(t) = 1 dωĩ(ω) exp (iωt). (13) 2π Figure 1 shows some of the individual steps of the reconstruction procedure described above. The temporal intensity of the laser pulse (Fig. 1e) has a strong amplitude modulation that is simulated by the superposition of four Gaussian pulse envelopes with different maxima and temporal widths. Starting with this intensity, the triple correlation is calculated (see (1) and Fig. 1a), although it would normally be obtained by an appropriate measurement. By (2), the 2D Fourier transform (the bispectrum) of the triple correlation is calculated (see Fig. 1b). From this point, there exist several ways to extract the amplitude and phase of the Fourier transform of the intensity. By use of (10) (12), the amplitude and phase shown in Figs. 1c and 1d, respectively, are obtained. It is clear that the amplitude is reconstructed exactly, whereas the phase shows a linear shift (as discussed above) and a number of 2π phase jumps. After inverse Fourier transformation of Ĩ(ω), the original intensity is obtained, and this is clear by comparison of the two curves in Fig. 1e. The observed time shift is a direct consequence of the linear phase shift. 2 Experiments and discussion For verification of this concept both multi- and single-shot triple-correlation experiments were performed, and Fig. 2 shows the experimental setup. The femtosecond laser pulses stemmed from a TW Ti:sapphire system with a temporal duration of about 150 fs and an energy of 250 mj. The

3 165 Fig. 1a e. Example of a triple correlation (a) and subsequent reconstruction of the temporal intensity of a laser pulse having a strong amplitude modulation. b shows the two-dimensional Fourier transform of the triple correlation, also known as a bispectrum. In c and d, the amplitude and phase, respectively, of the Fourier transform of the temporal intensity are shown. The straight lines are obtained directly by Fourier-transforming the input pulse, and the dots are the result of the reconstruction procedure. e shows the input intensity (straight line) and the reconstructed pulse intensity (dots) Ti:sapphire oscillator was operated at 795 nm and the repetition rate of the amplifier system was 10 Hz. Intensity fluctuations were mainly due to energy fluctuations, and were below 5%. The incoming laser pulse was split into three pulses I 1, I 2,andI 3 by two subsequent beam splitters. The last two pulses, having the same intensity, were directed onto a BBO crystal in order to produce second harmonic radiation I 23.In the case of the multi-shot geometry, the delay τ 2 between the two pulses I 2 and I 3 was adjusted by varying the position of a translation stage. The second harmonic I 23 was then combined with the fundamental field I 1 in a second BBO crystal. Phase matching was optimized for sum-frequency generation,

4 166 Fig. 2. Diagrammatic representation of the experimental arrangement for measuring triple correlation. In the first BBO crystal the second harmonic is produced, and the second BBO crystal is optimized for sum-frequency generation, yielding the third harmonic of the fundamental. Two independent delay times can be inserted and thereby the third harmonic of the fundamental was produced. Again, the delay between the two pulses τ 1 was able to be varied by a second translation stage. By recording the third harmonic signal as a function of the two delay times τ 1 and τ 2 the triple correlation trace defined in (1) was obtained. In Fig. 3a, a measured triple correlation of a pulse with a smooth amplitude modulation is displayed. The delay times τ 1 and τ 2 were varied in our experiments in increments of 67 fs. The total number of data points was Although some of the information in the triple correlation trace was redundant due to symmetry properties, the full triple correlation was recorded and the redundant data used for averaging. Following the pulse reconstruction procedure, the temporal intensity shown in Fig. 3c was extracted from the data. In detail, the symmetry relation G (3) (τ 1,τ 2 )=G (3) (τ 2,τ 1 )was exploited and, by taking the average of the corresponding pixels, the trace was symmetrized. From this averaged triple correlation trace, the two-dimensional Fourier transform was calculated, yielding the bispectrum according to (2). The bispectrum G (3) (ω 1,ω 2 ) was then split into amplitude and phase (see (4)). By application of (10) (12), the amplitude and phase of the Fourier transform of the laser pulse intensity Ĩ(ω) were thus derived. By calculating the inverse Fourier transform of the spectral intensity, the amplitude in the time domain was finally obtained. The FWHM of the pulse was found to be about (150± 15) fs, which is in good agreement with results obtained by the frequency-resolved optical-gating technique. The substructure in the wings of the extracted temporal intensity was mostly due to the limited number of data points. Setting the delay τ 2 equal to zero yields the third-order autocorrelation trace equivalent, for example, to an autocorrelation measurement using the optical Kerr effect [3]. Assuming a Gaussian pulse shape from Fig. 3b, a FWHM of (156 ± 12) fs of the laser pulse can be deduced. Again, this is in good agreement with the FWHM extracted from the triple-correlation measurement. Since two delay times have to be varied, recording the triple correlation of a laser pulse is quite time-consuming. It would therefore be desirable, especially for low repetitionrate systems, to record the whole triple correlation in just one shot. In a second experiment, the setup for which is shown in Fig. 4, such a single-shot arrangement has been achieved. The beam is split into three laser-pulse planes of area 3 3mm 2. The spatial intensity profile along the two axes perpendicular to the direction of propagation is flat. The first two planes are tilted horizontally by 3 with respect to each other, leading to Fig. 3. a Measured multi-shot triple-correlation trace of a femtosecond laser pulse stemming from a CPA Ti:sapphire system. b Setting one delay time equal to zero yields the third-order intensity autocorrelation of the pulse. c Reconstruced temporal intensity of the laser pulse, the FWHM of the pulse being about 150 fs, which is compatible with the result obtained from a FROG measurement a temporal window of about 1ps. This is equivalent to recording the second-order background-free autocorrelation trace in single-shot geometry. The only difference is that two planes

5 167 Fig. 4a f. Experimental arrangement for measuring the triple correlation using single-shot geometry. The laser pulse is split into three parts denoted by I 1, I 2, and I 3, respectively. Each beam has a flat spatial intensity profile in both dimensions perpendicular to the direction of propagation, with a size of 3 3mm 2. I 2 and I 3 (a) overlap in a BBO crystal (b) and produce the second-order autocorrelation function that is equal along the z-axes, and the delay τ 2 being mapped onto the y-axes. The second harmonic radiation (c) is subsequently combined with the third part I 1 of the laser pulse (d) in a second BBO crystal and the third harmonic radiation at 266 nm is produced (e) where both beams overlap spatially and temporally yielding the triple correlation trace (f). The delay τ 1 is now parallel to the z-axes rather than two line foci overlap in the nonlinear medium. The second harmonic trace subsequently overlaps with the third plane, which is tilted vertically by 3 (see Fig. 4). Therefore, the two delay times are mapped onto two perpendicular spatial coordinates, each having a length of 3mmcorresponding to a temporal width of about 1ps. The third harmonic radiation is then recorded by a CCD camera equipped with an interference bandpass filter with a center wavelength of 266 nm. The calibration of both time axes has been realized by introducing external delays τ 1 and τ 2, respectively, and observing the spatial shift of the triple correlation trace along the two axes. In Figs. 5a and 5c, two single-shot triple correlations are shown. The grating compressor of the Ti:sapphire laser system has been misaligned by increasing the distance between the two gratings leading to a longer pulse width as can be seen in the figure. From both traces the laser pulse intensity has been extracted. The corresponding intensities are shown in the Figs. 5b and 5d. It can be clearly seen that the pulse width increases from (185 ± 15) fs to (315 ± 15) fs with increasing grating separation as it is expected from theory. The substructures in the wings in Figs. 5b and 5d are probably due to the fact that the recorded triple correlation traces (especially in Fig. 5c) deviate somewhat from the symmetry property G (3) (τ 1,τ 2 )=G (3) (τ 2,τ 1 ). 3 Conclusion We have shown that by recording the triple correlation of a short laser pulse, it is possible to reconstruct the temporal envelope of the electric field without any assumptions on the analytic form of the pulse shape. This procedure is necessary, for example, for second-order intensity autocorrelation measurements, where the temporal width derived from the measurements depends on the assumed pulse shape. Also, no iterative algorithms (as used, for example, for frequency-resolved optical gating) are necessary. Therefore, the retrieved laser pulse intensity is unambiguous. For noisy data, the measurement yields good results since the triple-correlation trace contains redundant data that can be used for averaging prior to determining the pulse intensity. In addition, there are various paths to determine amplitude and phase of the Fourier transform of the temporal intensity, and therefore an additional possibility for averaging arises.

6 168 Fig. 5a d. Measured single-shot triple-correlation traces for different compressor alignments. First, the compressor was set to a position where the temporal width was near minimum (a), and then the grating separation was increased, leading to the triple correlation shown in c. The reconstructed temporal intensity for both cases is shown in b and d, respectively It has been demonstrated for the first time that this technique may be used in a multi-shot as well as in a single-shot arrangement. The multi-shot setup is preferable for characterizing low-intensity laser pulses with high repetition rates, whereas the single-shot setup is suitable for high-powered systems with low repetition rates, such as TW-Ti:sapphire laser systems. In an intermediate intensity range, it would also be possible to arrange a hybrid setup of the multi-shot and single-shot experiment. First, the second harmonic would be produced using single-shot geometry, and subsequently it would be correlated with the fundamental in multi-shot geometry. This would reduce the recording time to a value equivalent to a normal multi-shot autocorrelator. Acknowledgements. The authors thank A.W. Lohmann, H. Bartelt, and D. von der Linde for stimulating and fruitful discussion, R. Fernsler for assistance with the experiments, and the Deutsche Forschungsgemeinschaft (grant numbers: SFB 196 and SA 325/3-1) for financial support. References 1. A. Stingl, M. Lenzner, Ch. Spielmann, F. Krausz, R. Szipöcs: Opt. Lett. 20, 602 (1995) 2. F. Salin, P. Georges, G. Roger, A. Brun: Appl. Opt. 26, 4528 (1987) 3. H.S. Albrecht, P. Heist, J. Kleinschmidt, D. van Lap, T. Schröder: Appl. Phys. B 55, 362 (1992) 4. K.W. DeLong, R. Trebino, D.J. Kane: J. Opt. Soc. Am. B 11, 1595 (1994) 5. H.P. Weber, R. Dändliker: Phys. Lett. 28, 77 (1968) 6. J. Jansky, G. Corradi: Opt. Commun. 60, 251 (1986) 7. T. Sato, S. Wadaka, J. Yamamoto, J. Ishii: Appl. Opt. 17, 2047 (1978) 8. A.W. Lohmann, G. Weigelt, B. Wirnitzer: Appl. Opt. 22, 4028 (1983) 9. S. Chopra, L. Mandel: Phys. Rev. Lett. 30, 60 (1973) 10. E.I. Blount, J.R. Klauder: J. Appl. Phys. 40, 2874 (1969) 11. H. Bartelt, A.W. Lohmann, B. Wirnitzer: Appl. Opt. 23, 3121 (1984) 12. R. Kosloff, S.A. Rice, P. Gaspard, S. Tersigni, D.J. Tannor: Chem. Phys. 139, 201 (1989)