Jun 1, 2017 - Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Stra·e 2a, 12489 Berlin, Germany. *Corresponding author: ...
Letter
Vol. 42, No. 11 / June 1 2017 / Optics Letters
2185
Interferometric time-domain ptychography for ultrafast pulse characterization JANNE HYYTI, ESMERANDO ESCOTO, GÜNTER STEINMEYER,
AND
TOBIAS WITTING*
Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Straße 2a, 12489 Berlin, Germany *Corresponding author: tobias.witting@mbi‑berlin.de Received 21 March 2017; accepted 9 May 2017; posted 15 May 2017 (Doc. ID 291133); published 31 May 2017
A novel pulse characterization method is presented, favorably combining interferometric frequency-resolved optical gating (FROG) and time-domain ptychography. This new variant is named ptychographic-interferometric frequency-resolved optical gating (πFROG). The measurement device is simple, bearing similarity to standard second-harmonic FROG, yet with a collinear beam geometry and an added bandpass filter in one of the correlator arms. The collinear beam geometry allows tight focusing and circumvents possible geometrical distortion effects of noncollinear methods, making πFROG especially suitable for the characterization of unamplified few-cycle pulses. Moreover, the direction-of-time ambiguity afflicting most second-order FROG variants is eliminated. Possible group delay dispersion of pulses leads to a characteristic tilt in the πFROG traces, allowing the detection of uncompensated dispersion without a retrieval. Using nanojoule, three-cycle pulses at 800 nm, the πFROG method is tested, and the results are compared with spectral phase interferometry for direct electric field reconstruction measurements. Measured pulse durations agree within a fraction of a femtosecond. As a further test, the πFROG measurements are repeated with added group delay dispersion, and found to accurately reproduce the dispersion computed with Sellmeier equations. © 2017 Optical Society of America OCIS codes: (320.0320) Ultrafast optics; (320.7100) Ultrafast measurements; (320.7160) Ultrafast technology. https://doi.org/10.1364/OL.42.002185
Recent years have seen rapid progress in the generation of ultrashort laser pulses, with pulse durations reaching the single-cycle [1,2] and even sub-cycle regimes [3,4]. Characterization methods suitable for these challenging regimes have likewise advanced considerably. Both of perhaps the most popular complete characterization techniques, frequency-resolved optical gating (FROG) and spectral phase interferometry for direct electric field reconstruction (SPIDER), have been used to measure pulses shorter than two optical cycles [5,6]. More recently, the dispersion scan (d-scan) method was as well used to characterize sub-two-cycle pulses from hollow-core fiber compressors [7,8]. While FROG is nearly a quarter-of–a-century-old 0146-9592/17/112185-04 Journal © 2017 Optical Society of America
technique [9], the pulse retrieval algorithms suitable for FROG measurements continue to develop. One of the most promising recent developments in ultrashort pulse characterization is the discovery of time-domain ptychography (TDP), a powerful phase retrieval method suitable for a wide range of measurement techniques [10]. Its spatial domain predecessor, ptychography, was originally developed for lensless x-ray diffraction imaging of objects with atomic-scale features [11]. Analogously to the reconstruction of spatial features from a series of far-field measurements of diffraction patterns, TDP resolves the temporal features of the electric field of an ultrashort laser pulse from a series of far-field spectral measurements. Already in these first few years following the advent of TDP, several pulse characterization modalities based on the technique have surfaced. Some of these modalities simply adapt a ptychographic algorithm for retrieving the temporal pulse shape from previously introduced measurement schemes, such as attosecond streaking [12], and FROG variants including the classic noncollinear FROG [13] and cross-correlation FROG (XFROG) [14]. Other examples bear resemblance to blind FROG [15] in the sense that the pulse under investigation is split into two replicas, one of which is modified, and the ptychographic algorithm retrieves both the modified (gate) and the unmodified (probe) pulse [10,13,16–18]. In these latter modalities, the gate pulse is generated via a plethora of means: spatial light modulators [10,16], a grating combined with spatial filtering optics [17], a glass window [13], or a bandpass filter [18]. While the XFROG implementation assumes a known gate pulse [14], both the gate and the probe pulses are unknown for the blind-FROG-like characterization modalities. While Refs. [10,13,16,17] employed an additional measurement of the fundamental spectrum of the gate pulse, no additional information of either pulse was required for a complete characterization of the unknown electric fields in Ref. [18]. In this Letter, we describe and demonstrate, to the best of our knowledge, the first collinear adaptation of TDP. We call our method ptychographic-interferometric frequency-resolved optical gating (πFROG). Our experimental setup is similar to that of second-harmonic generation interferometric FROG (IFROG) [19], but, as with the above examples, our method is also similar to blind FROG, as both the probe and the gate pulse are retrieved. We use a transmission bandpass filter (BPF) to create the gate pulse, as this optical element introduces a
Vol. 42, No. 11 / June 1 2017 / Optics Letters
Fig. 1. Experimental setup. Abbreviations for the optical elements: DCM, double-chirped mirror; W, glass wedges; DE, dispersive element; BS, beam splitter; DS, delay stage; BPF, bandpass filter; FM, flip mirror; OAPM, off-axis parabolic mirror; ACL, aspherical collimating lens; CGF, color glass filters, P, polarizer; FL, focusing lens; FOC, fiber-optic cable.
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minimal complication to the already simple IFROG system. Previously, in the context of pulse characterization, a BPF inserted into an interferometer arm has been used for noncollinear single-shot measurements [20]. Akin to IFROG, the use of a collinear geometry allows tight focusing, and consequently the characterization of unamplified few-cycle pulses directly from an oscillator. Moreover, detrimental geometrical smearing effects [21] of noncollinear geometries due to a finite crossing angle of the focused beams in the nonlinear medium are avoided. In contrast to IFROG, the πFROG can readily distinguish pre- and post-pulses as the direction-of-time ambiguity afflicting most FROG modalities [21] is lifted. Furthermore, TDP algorithms are highly robust, capable of retrieving a pulse even from incomplete spectrograms consisting of but a few recorded spectra [13]. The πFROG experimental setup is depicted in Fig. 1. A variably delayed pulse replica is provided by the Michelson interferometer comprising a pair of beam splitters and a piezoelectric delay stage. The BPF (50-nm-wide passband centered at 800 nm) is inserted into one of the two interferometer arms, a marked difference to the interferometric FROG setup [19]. The width of the passband is not critical, as long as it is less than half the fundamental spectrum width. Choosing a broader band pass improves signal strength at the expense of convergence speed of the algorithm [16]. Several different BPFs were tested, and the passband of 50 nm was found to be the best compromise. The collinearly propagating pulses are focused by an f 2.54 cm off-axis parabolic mirror to a 20-μm-thick beta-barium borate (BBO) crystal for sum-frequency generation (SFG) in a type–I phase-matching configuration. The transmitted beam is collimated with an aspherical lens, and the fundamental field is suppressed by a pair of BG39 color glass filters and a polarizer. The filtered second-harmonic beam is guided to a spectrograph with a fiber-optic cable, and the spectrogram is recorded by a charge-coupled device (CCD) camera; see Figs. 2(a) and 3(a)–3(c). The recorded spectra are calibrated with the phase-matching efficiency of the BBO, the spectral response of the optical elements after SFG, as well as the measured sensitivity of the spectrometer. Reference measurements with a commercial SPIDER [6,22] apparatus are executed by the insertion of a flip mirror (FM), and by blocking one of the two interferometer arms. Pulses from the Ti:sapphire oscillator were precompensated for dispersion by pairs of double-chirped mirrors and glass wedges. In order to test the accuracy of the retrieval, additional dispersion can be provided by a dispersive element (DE). Windows of fused
Letter
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SPIDER πFROG
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Fig. 2. Experimental results for a sub-10-fs pulse. (a) Measured πFROG trace, comprising SFG spectra as a function of delay. (b) DFT of the data along the delay axis. The peak centered at ω is isolated with a super-Gaussian window. (c) The inverse DFT of the filtered ω 0 band is our input trace for the retrieval. (d) Retrieved trace with 0.24% rms error. (e) Reconstructed pulse in the spectral domain. Spectral intensity (solid red line), spectral phase (dashed red line), and comparison with SPIDER (gray). (f ) Temporal intensity.
silica (FS) or calcium fluoride with thicknesses 1.1 (FS), 3.0 (CaF2 ), and 6.4 mm (FS) were used to provide a group delay dispersion of 39, 84, or 232 fs2 at 800 nm, respectively. Analogously to second-harmonic IFROG [19], the πFROG trace can be expressed as the Fourier transform of the squared sum of the probe and the gate pulses, 2 Z ∞ Gt Et − τ2 e −iωt dt : (1) I ω; τ ∝ −∞
Expansion of this expression in a manner similar to what was done for IFROG in Ref. [19] yields multiple terms, pertaining to different modulational bands located at multiples of the carrier frequency ω0 in the delay-frequency domain; see Fig. 2(b). Of these, only the DC band at ω 0 is used by the πFROG algorithm. Defining the analytic electric fields Et ≡ Et expiω0 t and Gt ≡ Gt expiω0 t for the probe and gate pulses, respectively, we write the second-harmonic fields as Z ∞ E 2 te −iωt dt; (2) E SH ω ≡ −∞
Z G SH ω ≡
∞ −∞
G 2 te −iωt dt:
(3)
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+ 1.1 mm SiO
+ 3.0 mm CaF
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We note that at large delays the probe and gate pulses no longer overlap, so that jE X ω; τj2 → 0 and only the delayindependent second harmonic (SH) fields remain. Therefore, the SH intensity can be removed from the trace by subtracting a spectrum at the edge of the trace from the entire trace, leaving only the cross term jE X ω; τj2 . This cross term has exactly the same mathematical form as the one described in Ref. [18], allowing us to use the algorithm therein to retrieve the pulse from this filtered DC band of an πFROG trace. For the ptychographic reconstruction of the unknown laser pulse from the πFROG trace, we make a discrete Fourier transform (DFT) of our data along the delay axis τ; see Figs. 2(a) and 2(b). Apart from the DC peak at ω 0, the DFT has two distinct peaks at ω0 and 2ω0 . We isolate the peak around ω 0 by multiplication with a fourth-order super-Gaussian window function [indicated by the thin light red lines in Fig. 2(b)]. The magnitude of the inverse Fourier transform (after the SH intensity is removed as described above) in Fig. 2(c) represents a ptychographic trace similar to the ones presented in Ref. [18]. Note that we plot the extracted ptychographic trace with wavelength on the horizontal and delay on the vertical axis. We feed this subtrace of the measured πFROG trace into the time-domain extended ptychographic iterative engine (tdePIE) algorithm discussed in depth in Ref. [18]. The algorithm retrieves both the unknown pulse and the gate pulse. While not shown here, it is also possible to compute the complex-valued spectral response function gω of the BPF, defined through Gω gωEω, using the retrieved fields. Taking the Fourier transform of this relation gives a convolution Gt gt Et, from which the response function can be iteratively retrieved via a deconvolution strategy similar to what was used in Ref. [23]. The results of the retrieval for the unchirped pulse (i.e., with DE removed) are shown in Fig. 2. The πFROG measured trace is shown in Fig. 2(a). We scanned the delay in steps of 0.2 fs over a range of 500 fs. To enable visibility of the fringes in print, we only show a 300-fs window around the temporal overlap of the probe and the gate pulses. The Fourier transform of the data is shown in Fig. 2(b). The ptychographic subtrace after Fourier filtering of the DC lobe at ω ≈ 0 is shown in Fig. 2(c). Note that the trace is now resampled onto a delay grid spanning from −300 to 200 fs in 5-fs steps. The choice of delay step size and range is independent of the original delay sampling of the interferometric trace. In the tdePIE algorithm the grid size in frequency and delay directions are not coupled, and every delay slice of the data matrix is treated independently. The reconstructed trace after 50 iterations of the tdePIE algorithm is shown in Fig. 2(d). The agreement to the original trace Fig. 2(c) is excellent; the rms error is only 0.24%. The spectral domain reconstruction is presented in Fig. 2(e). The red solid line is the reconstructed spectrum and the red dashed line the spectral phase. The temporal intensity is shown in Fig. 2(f ). In order to measure the pulse with SPIDER, one of the two arms
0 τ (fs)
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= 0.16 %
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I DC ω; τ ∝ jE SH ωj2 jG SH ωj2 4jE X ω; τj2 :
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Using the above definitions, the DC band can be written as
of our interferometer was blocked and a FM was inserted in the beam path (see FM in Fig. 1). The SPIDER data is shown in gray in Figs. 2(e) and 2(f ). The agreement of the spectrum and the spectral phase between πFROG and SPIDER is good, showing a relatively flat phase with a third-order component. The third-order phase is also apparent in the time domain in Fig. 2(f ), where trailing satellites are found after the main lobe. The full width at half maximum (FWHM) pulse durations for πFROG and SPIDER are virtually identical: 8.8 and 8.9 fs, respectively. We further test the robustness of the πFROG method by measuring the pulse after propagation through three different dispersive elements (see DE in Fig. 1). The corresponding πFROG subtraces and pulse reconstructions are shown in Fig. 3. From left to right, the columns show the results for an increasingly chirped pulse after traversing DEs with progressively higher dispersion. Presented in Figs. 3(a)–3(c) are the interferometric traces from which we extract the ptychography traces Figs. 3(d)–3(f ). Note that the trace tilts with increasing dispersion. In fact, the tilt of the ptychography trace directly corresponds to the group delay dispersion of the unknown pulse, giving a convenient, intuitive way to read the πFROG subtraces. The reconstructed traces after 500 iterations of the tdePIE algorithm are shown in Figs. 3(g)–3(i), with
|E(t)| (arb.u.)
We further define a cross term similar to a standard FROG trace, but with the second field replaced by the gate, Z ∞ GtEt − τe −iωt dt: (4) E X ω; τ ≡
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Fig. 3. Accuracy test using dispersive elements. Each column shows the πFROG trace (a)–(c), extracted subtrace (d)–(f ), retrieved subtrace (g)–(i), spectral intensity (light blue line) and phase (red line) compared to the calculated phase from Sellmeier coefficients (dashed black line) (j)–(l), and temporal intensity (red line) compared to the unchirped pulse intensity (black line) (m)–(o).
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rms errors of 0.18%, 0.13%, and 0.16%, respectively. The reconstructed pulses are shown in the spectral domain in Figs. 3(j)–3(l), as well as in the temporal domain in Figs. 3(m)– 3(o). The spectral phase is shown as the difference to the spectral phase of pulse reconstruction without a DE [cf. Fig. 2(e)]. For comparison, the dispersion calculated from Sellmeier data is shown as well. The agreement between the Sellmeier and retrieved spectral phases for all three dispersive elements is excellent. While the DC band provided the tools for pulse retrieval, the 2ω0 -peak can be used to detect and correct for measurement errors; see Fig. 2(b). For instance, the nonuniform movement of the delay stage results in the widening of the band [19]. Analogously to second-harmonic IFROG, the 2ω0 band in πFROG is formed primarily due to a modulation by the factor cos2ω0 Δωτ, where Δω is the deviation from the second harmonic of the carrier wave; see Eq. (6) in Ref. [19]. The explicit dependence of the modulation frequency to detection frequency through Δω leads to the tilt of the 2ω0 band. Small deviations from this modulation can occur if the phases of E SH ω and G SH ω are not equal, or if the carrier frequencies of the two pulses differ. Besides these, the nonuniform movement of the delay stage causes the recorded delay steps to deviate from the presumed values of τ, leading to the abovementioned broadening of the 2ω0 band. To quantify this phenomenon, we extract a rms phase jitter of 0.3 rad for the second-harmonic band, corresponding to a timing jitter of 70 as, or about 1/3 of the delay step of 200 as. Furthermore, a systematic error of the delay step size, e.g., due to a finite angle of the delay stage with respect to the beam path, can also be detected and corrected. This is possible, as a properly calibrated measurement yields the 2ω0 peak exactly at a line where the detection frequency equals the delay-frequency. For the same reasons, a miscalibration of the spectrometer would result in deviations from a linear band, and could thus be observed by a simple visual inspection of the Fourier-transformed πFROG trace. Such corrections were, however, not found to be necessary for our measurements. Similar capability to calibrate a FROG variant against systematic delay and wavelength errors has been previously demonstrated for noncollinear beam geometry [24]. In conclusion, we have introduced a new pulse characterization technique based on time-domain ptychography, the first such technique employing a collinear beam geometry to the best of our knowledge. The collinearity facilitates tight focusing conditions, making πFROG especially suitable for characterization of unamplified few-cycle pulses. We demonstrated this capability by successfully characterizing 7-nJ, 8.9-fs few-cycle pulses from a titanium-sapphire oscillator, finding the retrieved pulse shape to be in excellent agreement with a reference measurement using SPIDER. The robustness of our method was further validated by comparing the retrieved spectral phases of chirped pulses with known Sellmeier data. Furthermore, we showed that the fringe structure of an πFROG spectrogram provides additional means for detecting and quantifying experimental uncertainties, such as the positioning error of the translation stage of the Michelson interferometer, which we measured to be as small as 70 as. We believe our method is a viable alternative for the characterization of few-cycle pulses, especially when the pulse energy is low. While we chose SFG as the nonlinear process, πFROG could just as well be applied for
Letter third-harmonic generation, and as such serve as a probe for ultrafast polarization dynamics of both χ 2 and χ 3 processes [23]. The πFROG method also appears to be an ideal tool to measure pulse shapes in the focus of high-numerical-aperture objectives and in nonlinear microscopy [25]. Moreover, for measurement systems where harmonic generation under such focusing conditions occurs, πFROG offers pulse characterization in situ. Funding. 762/11-1).
Deutsche Forschungsgemeinschaft (DFG) (STE
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