In this paper we apply a broadband fiber optic coupler interferometer to the measurement of few-cycle laser pulses. Sub-8-fs pulses delivered by an ultrafast oscillator were characterized spatiotemporally using STARFISH, which is based on spatially resolved spectral interferometry. The reference pulse was measured with the d-scan technique. The pulses were focused by an off-axis parabolic mirror and were characterized at different transverse planes along the focusing region. The evolution of the retrieved pulses is analyzed, exhibiting small variations in the temporal (and spectral) amplitude and phase during propagation. Finally, the peak irradiance evolution is estimated from the integration of the spatiotemporal intensity.
©2012 Optical Society of America
CorrectionsBenjamín Alonso, Miguel Miranda, Íñigo J. Sola, and Helder Crespo, "Spatiotemporal characterization of few-cycle laser pulses: erratum," Opt. Express 21, 5582-5584 (2013)
The field of ultrashort laser pulses is rapidly evolving both from the point of view of the development of laser sources and from their applications. As a consequence, the techniques for the characterization of such pulses are also in constant development. The temporal characterization of ultrashort pulses is a well-established field, with several techniques allowing for the retrieval of pulse amplitude and phase in the temporal domain . Among these techniques, FROG (Frequency-Resolved Optical Gating)  and SPIDER (Spectral phase interferometry for direct electric-field reconstruction)  are the most widely used today. The spatial phase of the pulses can also be obtained with standard methods, as for example using the Hartmann-Shack wavefront sensor .
During the last few years, a huge progress has been done to extend these techniques to characterize the spatiotemporal amplitude and phase (or equivalently, the spatiospectral amplitude and phase) of the pulses. This full characterization is often required in applications where the spatial and temporal dependence of the pulses is coupled and of extreme importance. For this purpose, the following techniques can be underlined: Spatially Encoded Arrangement for Temporal Analysis by Dispersing a Pair Of Light E-fields (SEA TADPOLE) , Hartmann-Shack combined with FROG (Shackled-FROG) , and SpatioTemporal Amplitude-and-phase Reconstruction by Fourier-transform of Interference Spectra of High-complex-beams (STARFISH) .
Regarding the duration of the pulses, currently available technology routinely provides few-cycle near-infrared ultrafast laser pulses (with durations well below 10 fs) either using post-compression schemes or directly from broadband and octave-spanning Ti:sapphire laser oscillators (see, e.g., [8,9]). Octave-spanning oscillators have many applications, for example, in optical frequency metrology and high-precision optical spectroscopy . Moreover, the production of high-energy few-cycle laser pulses has also been recently achieved through direct chirped pulse amplification  as well as in post-compression schemes based on hollow-core fibers  and filamentation [13,14].
The temporal characterization of few-cycle pulses, which have ultra-broad bandwidths, is very demanding for common techniques. Many efforts have been done in adapting these techniques to this regime, where very thin nonlinear crystals and second-harmonic-generation (SHG) spectral signal calibration are required. For example, spatially encoded arrangement of SPIDER (SEA-SPIDER)  and two-dimensional spectral shearing interferometry  have been applied to the measurement of few-cycle pulses. Also, careful calibration of SHG-FROG [17–19] and the use of interferometric FROG  have also been demonstrated in the few-cycle regime. A comparison of the experimental results for these techniques is given in . Recently a new technique known as d-scan (dispersion-scan) has been introduced [22,23]. It achieves the simultaneous compression and characterization of the pulses by tracking the spectrum of the SHG signal during a continuous insertion of dispersion (from negative to positive chirp) and applying an iterative retrieval procedure. The SHG signal can be self-calibrated and the up-converted bandwidth requirements are more relaxed compared to other techniques.
The adaptation of techniques for pulse characterization in the spatiotemporal domain is also a challenge. There is a strong interest in the full spatiotemporal characterization of ultrashort pulses to be used in applications, for example in filamentation . SEA-SPIDER has been recently shown to provide space-time information (excluding the pulse-front tilt) of 10.2 fs pulses . In this paper, we used the technique STARFISH  to measure few-cycle pulses delivered by an oscillator. The phase of the test pulse can be extracted by combining it with a known reference pulse in a spectral interferometer (SI). Here, we used the d-scan technique  to measure the spectral phase of the reference pulse. Due to the large bandwidth of the pulses, we had to calibrate the spectral response (in amplitude and phase) of the fiber optic coupler used in the SI setup, as well as the spectrometer’s response, in order to correct the spectral amplitude and phase retrieved with STARFISH.
2. Experimental setup
The experiments were performed with a Ti:sapphire ultrafast oscillator (Femtolasers Rainbow CEP) at a repetition rate of 80 MHz, with a central wavelength around 800 nm, a Fourier-transform limit of fs and an energy per pulse of 2.5 nJ. The experimental setup for the full characterization of these pulses is divided into two main parts, corresponding to the combination of the d-scan technique  for measurement of the reference pulse, and the STARFISH technique  for the spatiotemporal characterization of the test pulse (see Fig. 1 ).
The d-scan setup is based on a glass wedge pair and a set of chirped mirrors and allows us to obtain dispersion scans of the pulses around their optimum compression point. We used BK7 wedges (Femtolasers GmbH) with antireflection-coating and an angle of 8°. The chirped mirrors are two pairs of double-chirped mirrors (DCM, Venteon GmbH), with each pair composed of two types of mirrors, named ‘blue’ and ‘green’, designed to compensate for the phase ringing. The group delay dispersion (GDD) introduced by the DCMs is approximately per two bounces at 800 nm. Simultaneously to the d-scan, the pulse is focused in a nonlinear crystal (BBO, type I, 20 μm thick) by an off-axis parabolic (OAP) mirror with a focal length of 5 cm. The SHG signal produced in the BBO crystal is collimated by a lens and a blue filter is used to remove residual infrared radiation before detection with a calibrated fiber-coupled spectrometer (HR4000, Ocean Optics Inc.).
Regarding the STARFISH setup, a broadband beam splitter (BS, Venteon GmbH) is used to produce a replica of the oscillator pulses, to be used as reference pulse in the SI. A flip mirror is used to obtain the calibration of the reference pulses using d-scan. The test pulse is focused by a 5-cm focal length OAP after an iris that selects the most energetic part of the pulse’s profile (diameter of 5 mm). The pulses focused by the OAP are spatiotemporally characterized by scanning their spatial profile along the x-axis with the test pulse fiber. This is performed for different propagation distances around the focus by scanning the z-axis with the fiber. The position of the fiber that collects the reference pulse allows for adjusting the delay in the SI. The single-mode fiber coupler acts as an interferometer, effectively combining the test and reference pulses; it is directly connected to the spectrometer (HR4000), where the resulting spectral interferences are detected. The small diameter of the fiber core () provides high spatial resolution in the measurements.
The spectral phase of the reference pulse is retrieved by d-scan . For this purpose, the SHG trace in a dispersion scan with the glass wedges around the shortest compressed pulse is measured. The dispersion introduced by the wedges is known from the Sellmeier equations of the material (in this case BK7). The fundamental spectrum is separately measured. The SHG trace is simulated for a seed phase and is compared with the experimental d-scan. An iterative algorithm is used to optimize the retrieved phase until the simulated trace converges to the experimental one. This way, the spectral phase of the pulse is univocally determined. Since the dispersion of the wedges is known, the glass insertion that gives the shortest pulse can be obtained simultaneously to the pulse characterization.
The spatiotemporal amplitude and phase characterization is obtained by the STARFISH technique . For each propagation distance, the fiber scans the pulse profile to measure the spatially resolved spectral interferometry with a reference pulse, previously characterized using d-scan. The spectral interferences are analyzed with a Fourier-transform algorithm  to retrieve the spectral phase of the test pulse. The spatially-resolved spectrum (amplitude) is measured in another scan with the fiber. After inverse Fourier-transforming in the frequency coordinate, the spatiotemporal amplitude and phase of the pulses is characterized.
The few-cycle laser pulses delivered by the oscillator have an ultra-broadband spectrum extending from 630 to 980 nm. Since fiber optic couplers are designed to transmit light in a finite bandwidth, checking the transmission of the fiber coupler, as described in the next section, is vital for broadband applications.
3. Calibration of the fiber coupler for ultra-broadband experiments
3.1 Transmission as a function of the wavelength
We measured the spectral transmission of the coupler with a white-light calibration lamp (300-1050 nm, LS1-CAL, Ocean Optics Inc.). The transmission function, , was obtained by comparing the power spectral density with and without the fiber, measured with a calibrated broadband spectrometer (HR4000, Ocean Optics). The result is given in Fig. 2 , where the experimental data is plotted in blue. We fitted an exponential function to the measured data,, where we determined five free parameters (in order to avoid restrictions in the fit of the experimental curve) by least squares optimization. Only the gray shaded area was considered for the fit in order to avoid the noise in the tails. The red curve is the resulting transmission that will be taken into account to correct the amplitude response of the fiber coupler. From this curve, we see that wavelengths below 500 nm and above 1000 nm will not be coupled. Nevertheless, the broadband transmission of the coupler is still adequate for measuring few-cycle, near infrared pulses.
The two interferometer arms are of almost equal length (difference ), and so the dispersion introduced in the reference and test pulses is practically balanced. The slight difference in length gives rise to an actual phase difference but this is calibrated in the SI and is taken into account when retrieving the correct phase from STARFISH. The dispersion of additional optics, e.g. the beam splitter, was also calibrated with SI and introduced in the phase retrieval. Our fiber with almost equal length arms allows their phase difference to be calibrated and the application to SI. If the arms length were not well-balanced, the relative dispersion between the two arms of the interferometer would be arbitrarily large to the point that it caused a large broadening of the side-peaks after calculating the Fourier-transform of the interference . These two peaks may overlap and this fact may enter in conflict with the longer measurable pulses due to the limitations imposed by the experimental spectral resolution. In our case, the signal broadening is well below this limit.
3.2 Transmission as a function of the angle of incidence and the wavelength
We will use STARFISH for the measurement of non-collimated pulses, such as those focused by an off-axis parabola (see Section 4). In these cases, k-vectors with different directions will be coupled with different efficiency into the optical fiber due to its limited numerical aperture (). In general, the transmission of the fiber is a function of the angle of incidence of the light, with the highest efficiency occurring for normal incidence with respect to the fiber axis. The transmission decreases as the angle of incidence increases, which determines a cone of coupled light related to the .
Since the of single-mode fibers also depends on the refractive indices of the fiber core and cladding, and we are dealing with ultra-broadband spectra, the is expected to show a dependence on material dispersion. In this case, the transmission of the fiber depends not only on the angle of incidence , but also on the wavelength . Such situation would imply a spatiospectral distortion in the measurement of focused pulses.
For this reason, we calibrated the coupling efficiency of the fiber as a function of the angle of incidence and the wavelength. We used a white-light source (LS1-CAL, Ocean Optics) and a rotation stage to vary the angle of the fiber with respect to the source while keeping the fiber input in the axis of rotation. The signal was detected with a fiber-coupled spectrometer (AvaSpec-2048, Avantes). The transmission function , where the signal has been normalized for each wavelength, is represented in Fig. 3(a) . The angular dependence of is roughly constant with wavelength. In Fig. 3(b) we plot the integral of in the wavelength axis , obtaining a full-width at half-maximum (FWHM) of the acceptance cone . To study the dispersion of the acceptance angle (see Fig. 3(c)), we calculated the angle for which the signal falls to half the maximum from the center, , as a function of wavelength (blue dots), and compared it to (red curve). If the noisy regions in the extremes of the spectrum are discarded, it can be concluded that the angular response is not dispersive in a broad spectral bandwidth. This means that the angular dependence of the light coupling in the fiber does not depend on the wavelength, so . We therefore conclude that spatiospectral distortions do not occur when measuring few-cycle focused pulses with the fiber. In Fig. 3(d) the numerical aperture for the half-maximum of the cone, , is given. In the case of measuring pulses with higher numerical apertures, the detection modifies the measured pulse with the function , so the signal coming from the peripheral part of the profile of the focused pulse will be detected with less efficiency, as verified experimentally in the near-field of a zone plate . In this work, we measured the oscillator pulses focused with a focal length , so the radius corresponding to is given by . This radius is and the condition in the input profile is fulfilled within the 2.5-mm radius iris used in the experiments. To prove this, we simulated the focus of an ideal plane wave using a 5-mm diameter diaphragm, with and without the effect of the , obtaining a focal spot size (FWHM) of and , respectively. We conclude that the effect is almost negligible in our case, even more taking into account that the outer part of the spatial profile before focusing (the region more afflicted) is the less intense part of the pulse profile.
4. Experimental characterization of few-cycle pulses delivered by an oscillator
4.1 Measurement of the reference pulse: d-scan
The reference pulse required by STARFISH was characterized with the d-scan technique. The compressor was composed of two pairs of chirped mirrors and a pair of BK7 wedges with an angle of 8°. The dispersion scan was done, as usual, by translating one wedge along the direction illustrated in Fig. 1. The total scan corresponds to 59 points with a step of in the direction of the scan. This translates into a total glass insertion of in the propagation direction of the pulse. We measured three independent d-scans of the pulse in order to perform several pulse retrievals.
An experimental d-scan trace is shown in Fig. 4(a) . The corresponding retrieved trace is given in Fig. 4(b) and shows a good matching to the measurement. The spectrum and phase of the retrieved pulse for the best achieved compression are shown in Fig. 4(c). The full width at of the characterized spectrum is . The standard deviation of the phase (gray curve) for the different retrievals shows the small precision error present in the retrieval. In Fig. 4(d) the temporal intensity and phase of the pulse is depicted. The Fourier-limited duration of the measured spectrum is (FWHM) and the duration of the retrieved pulse is (FWHM). The gray curves are the standard deviation of the amplitude and phase calculated from the different traces, showing a small variation between them: for the spectral phase, for the temporal phase and for the normalized temporal intensity.
4.2 Spatiospectral and spatiotemporal characterization: STARFISH
We used STARFISH to characterize the focusing region of the oscillator after the OAP (). The measurements were done for 7 consecutive propagation distances z around the focus, in order to track the evolution of the focused pulses: . The spatiospectral (and spatiotemporal) amplitude and phase were retrieved for each z-plane. The spatial features in the transverse plane were measured in one axis (x-axis), since the system was assumed to have cylindrical symmetry. Similar sets of measurements can be performed in the full x-y plane just by spatially scanning with two perpendicular actuators.
The evolution of the spatially-resolved spectrum is shown in Fig. 5(a) as a function of the longitudinal position z. The x-axis extends from −50 to 50 mm in all cases. The x-scan was done in steps of . During propagation, it is mainly the spatial width (x-axis) that is changing, whereas the spectrum is almost undistorted. Due to the pulses’ broad bandwidth, the effect of smaller focal spot size for the shorter wavelengths is visible in the spatially-resolved spectrum at the focus of the OAP [Fig. 5(a)]. Apart from this effect, the main consequence is a change of the relative amplitude of the spectral components, in particular the fact that shorter wavelengths exhibit slightly larger amplitudes with respect to longer wavelengths. Other small distortions can be attributed to a misalignment of the OAP or to a non-homogeneous spatial profile of the oscillator (before the OAP).
In Fig. 5(b), the spatiospectral phase (or wavefront) is represented for the same set of axial distances. Since STARFISH retrieves the frequency-resolved wavefront  (with a small noise due to phase drifts), we represent the phase for different wavelengths . For clarity, we represent these wavefronts, , using a different color for each wavelength and shifted to in , what is equivalent to remove the spectral phase on-axis. To corroborate the result, we did two independent measurements (-scans) for each position , so we also represent the error (black curves) calculated from their difference. In spite of the presence of some noise, these measurements can be used to determine qualitatively and quantitatively the convergence and divergence of the pulse, thus helping to identify the propagation distance analyzed in each measurement. As observed in the measurements, for the pulses focused by the OAP the reddish wavelengths have smaller curvature than the bluer ones, as given by the dependence of the wavenumber on wavelength, .
In the spatiotemporal domain, the intensity of the pulses along the focusing region is given in Fig. 6(a) . Here, the temporal features are also roughly constant both in the x-axis and along z, with the spatial width (x-axis) of the pulses exhibiting the largest variation as the pulse approaches and moves away from the focus. Figure 6(b) depicts slices of the on-axis (x = 0) temporal intensity, with the instantaneous wavelength (calculated from the inverse of the derivative of the temporal phase) shown in different colors that give us information on the temporal chirp of the pulses. Due to the overall positive chirp of the pulses, combined with the relative amplitude decrease of redder wavelengths with respect to the bluer part of the spectrum (as mentioned above), the temporal intensity presents a small decrease in the leading part of the pulse in the planes closer to the focus. The chirp is almost constant along the direction of propagation. These results are consistent with the expected almost constant temporal profile in the focusing region.
4.3 Comparison of the on-axis results
Here, we compare the spectral and temporal retrievals obtained on-axis (x = 0) for the 7 propagation distances considered. In Fig. 7(a) , we represent the mean spectral power (blue curve) and corresponding standard deviation (gray curves), together with the spectral phase (red curve) and standard deviation (gray curves). From the results, it is clear that the spectral amplitude deviation is higher than the phase deviation, indicating that the differences observed in the temporal profiles of Fig. 6(b) are mainly originated by the differences in the spectral amplitude. In Fig. 7(b), the temporal duration (FWHM) of the on-axis pulses is plotted as a function of the propagation distance. The variation of these widths is compared with the FWHM of the Fourier-transform limit (FTL) of the corresponding spectra, exhibiting a correlation between the width of the FTL and the actual width of the pulses. This result again supports the idea that the differences mainly come from the amplitude and not from the phase. The explanation of this result is that the spectral amplitude reshaping due to the focusing flattens the spectrum close to the focus (since the shape of the input spectrum on Fig. 4(c) has lower signal for shorter wavelengths) and this induces a small reduction in the FTL and the duration of the pulse, as seen in Fig. 7(b).
As mentioned above, the variations observed in the spectral domain are also present in the temporal domain. In Fig. 7(c), we show the mean temporal intensity (blue) and its standard deviation (gray). The same applies for the temporal phase (red) and its deviation (gray). Here, it is also observed that the difference in the phase is smaller than the difference in the amplitude. The statistics of the pulse width retrieved on-axis gives a FWHM of , which is consistent with the retrieval of the d-scan. In Fig. 7(d), we also represent the (mean) temporal profile colored with the (mean) instantaneous wavelength to show the good match with the results of Fig. 6(b).
These results can be interpreted as the validation of the current experimental implementation of the d-scan technique (and could also be extrapolated to other techniques, in which the pulses to be characterized are focused in the nonlinear crystal with an OAP), in the sense that it is assumed that the focus of the OAP does not distort the temporal (or equivalently spectral) amplitude and phase of the pulse. Here, we have found that small differences can occur, although they do not hinder proper pulse retrieval.
4.4 Measurement of the peak irradiance of ultrashort laser pulses
The measurement of the peak intensity of ultrashort laser pulses is often difficult to be addressed, many times due to the high intensities involved. We will show how the characterization of the spatiotemporal intensity of the pulses can be used to calculate an estimation of the peak intensity. Although the results presented in Fig. 6(a) are commonly called intensity, they are actually the irradiance of the pulse, measured in units. It is usually represented in arbitrary units (a.u.) whenever the absolute value is unknown or irrelevant.
The integral in the two spatial coordinates and the temporal dimension gives the energy of the pulse , as given in Eq. (1), where represents the experimental normalized spatiotemporal intensity at a certain and is a constant that gives the peak irradiance of the pulse:
Since we are assuming cylindrical symmetry, the characterization was done only in one spatial dimension (x-axis). The scan was done in the full axis, i.e., over the two sides of the beam profile with respect to the center (x = 0). Consequently, we have double information and we can obtain two values of the peak irradiance per measurement, corresponding to the polar radius and , respectively.
Often, this full information is not available and we have to make approximations to obtain the peak intensity. Here, we will do a first rough calculation just for comparison. To simplify, we can consider a focused pulse with a Gaussian profile both in the temporal and in the spatial coordinates, thus with separable dependence in time and space. In this case, the irradiance is given byFig. 7(b). For the spatial width , we consider the full width in the x-axis after integration in wavelengths of the spatiospectral traces shown in Fig. 5(a). The results for are shown in Fig. 8(a) as a function of the propagation distance. After integrating the Gaussian irradiance , we obtain the following relation between the pulse energy, the x- and t-widths, and the peak irradiance:
From Eq. (3) we see that the Gaussian peak irradiance actually corresponds to the pulse energy divided by the spatiotemporal volume above half the peak intensity, that is, the pulse energy divided by the temporal duration, , by the spatial section, , and corrected by a factor of .
The energy per pulse is calculated from the measured average power and the pulse repetition rate . The power measured after the iris and the OAP was , so the energy per pulse was . We consider that the fraction of this energy that is lost (spatially spread) in the focus is negligible. Also, there are other factors that can affect the result, for example part of the radiation being incoherent (e.g. amplified spontaneous emission) or small pre- or post-pulses that are not measured. Therefore, the peak irradiance obtained by this procedure should be considered as an estimate of the actual value.
The comparison of the results for the peak irradiance as a function of the propagation distance is shown in Fig. 8(b). As expected, the peak irradiance is higher closer to the focus, where the spatial width decreases. The values obtained at the focus () are , and in the Gaussian irradiance approximation, for the measured spatiotemporal irradiance from the set and from the set , respectively. The two values for the full spatiotemporal calculation are overlapped, whereas the Gaussian estimation gives higher values. This occurs because the irradiance is more spread in the temporal dimension than in a Gaussian function with the same and accordingly the actual peak irradiance is lower.
Current techniques for the temporal characterization of laser pulses have already reached the few-cycle regime. In particular, the d-scan technique is very powerful due to its simple and low-demanding experimental implementation. The STARFISH technique for the spatiotemporal characterization of the pulses is based in spectral interferometry, and thus requires a calibrated reference pulse. Firstly, we have shown the capabilities of fiber optic coupler based interferometry for ultra-broadband pulse measurements in terms of the operating spectral bandwidth. Then, we have demonstrated the application of STARFISH to few-cycle pulses, using the d-scan to measure the reference pulse.
We have reconstructed spatiotemporally the pulses delivered by an ultrafast oscillator (6.7 fs FWHM Fourier-transform limit) focused by an OAP. The full retrieval of the amplitude-and-phase in the spatiotemporal and spatiospectral domains gives additional information that is lost with usual temporal characterization techniques. We have measured pulses with durations below 8 fs (FWHM) and have studied the evolution of the pulses along the focusing region. We found that temporal dependence of the pulses is practically preserved around the focus of the OAP, presenting small changes in the spectral and temporal amplitude (due to the dependence of the focal spot size for different wavelengths in ultra-broadband pulses), and almost invariant spectral and temporal phases. OAPs are important devices that find many uses in pulse focusing and characterization (especially of ultra-broadband pulses), precisely because of the absence of dispersion and chromatic aberrations, provided that they are properly aligned. STARFISH allows us to know whether the focusing is being properly performed in both the xy-plane (actually, we measured the x-axis) and the z-axis.
We have calculated the peak irradiance of the pulses from the spatiotemporal reconstruction. We have checked that assuming Gaussian profiles and uncoupled space-time dependence is not enough to estimate the peak irradiance. This will be absolutely relevant in pulses with stronger spatiotemporal coupling.
We expect the availability of spatiotemporal characterization techniques in the few-cycle regime to be extremely helpful to study processes involving ultrafast oscillators, as well as processes employing high-energy pulses such as pulse post-compression and high-order harmonic generation, among others.
We acknowledge support from Spanish Ministerio de Ciencia e Innovación (MICINN) through the Consolider Program SAUUL (CSD2007-00013), Research projects FIS2009-09522, and grant programs Formación de Profesorado Universitario (No. AP2007-00236 for B. Alonso) and Ramón y Cajal (for I. J. Sola); and from the Junta de Castilla y León through the Program for Groups of Excellence (GR27). H. Crespo acknowledges Fundos FEDER, through Programa Operacional Factores de Competitividade – COMPETE, and Fundação para a Ciência e a Tecnologia (FCT) under grants No. PTDC/FIS/115102/2009 and PTDC/FIS/122511/2010. M. Miranda acknowledges FCT and FEDER grant No. SFRH/BD/37100/2007. We also acknowledge support from Centro de Láseres Pulsados (CLPU), Salamanca, Spain.
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