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Abstract
Single-shot supercontinuum spectral interferometry (SSSI) is an optical technique that can measure ultrafast transients in the complex index of refraction. This method uses chirped supercontinuum reference/probe pulses that need to be pre-characterized prior to use. Conventionally, the spectral phase (or chirp) of those pulses can be determined from a series of phase or spectral measurements taken at various time delays with respect to a pump-induced modulation. Here we propose a novel method to simplify this process and characterize reference/probe pulses up to the third order dispersion from a minimum of 2 snapshots taken at different pump-probe delays. Alternatively, without any pre-characterization, our method can retrieve both unperturbed and perturbed reference/probe phases, including the pump-induced modulation, from 2 time-delayed snapshots. From numerical simulations, we show that our retrieval algorithm is robust and can achieve high accuracy even with 2 snapshots. Without any apparatus modification, our method can be easily applied to any experiment that uses SSSI.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Single-shot spectral interferometry (SSI) is an ultrafast optical method that can measure ultra-rapid refractive index transients induced by ultrashort laser pulses [1, 2]. This measurement can provide a direct view of how a laser-induced perturbation evolves in time and space in a single shot. In this technique, a pump pulse induces a refractive index transient in a medium, and a chirped reference and a time-delayed replica (probe) pulse, upon which the pump-pulse-induced phase shift has been imposed, interfere in an imaging spectrometer, producing a spectral interferogram. The corresponding spatiotemporal (time and 1-dimensional space) evolution of refractive index transient is then reconstructed with frequency-to-time mapping or full Fourier transform methods [2]. In particular, supercontinuum (SC) light has been used for the reference and probe pulses to provide a temporal resolution better than ∼10 fs [2]. This single-shot supercontinuum spectral interferometry (SSSI) and SSI techniques have been successfully applied to capture laser-induced double step ionization of helium [3], laser-heated cluster explosion dynamics [4], laser wakefields [5], optical nonlinearity near the ionization threshold [6], and electronic and inertial nonlinear responses in molecular gases [7–10]. SSI has been also used to capture terahertz waveforms in snapshots without pump-probe scanning [11,12].
Unlike self-referencing nonlinear diagnostics such as FROG [13, 14] and SPIDER [15, 16], linear spectral interferometry including SSSI requires pre-characterized reference/probe pulses prior to its use. One method to pre-characterize a chirped probe in SSSI is to scan the delay between the pump-probe pulses while tracking a characteristic central extremum in the modulated probe phase or spectrum [2]. This method can determine the spectral phase of SC probe light to arbitrary order [17]. However, it relies on stationary-phase and small-perturbation approximations, which can be problematic when the phase modulation is too large or too asymmetric. In addition, this requires repetitive measurements over the entire chirp window for an accurate extraction of higher order dispersion coefficients. In particular, for a SC probe that has an extremely large bandwidth, a large number of pump-probe scans are necessary. This scanning method is impractical for use with low-repetition-rate laser sources.
For these reasons, we aim to develop a method that can simplify or avoid the pre-characterization process if possible and potentially to characterize both SC reference/probe pulses and pump-induced transients with a minimal number of repetitions.
2. Background theory
In SSSI, the probe SC pulse, E(t), is perturbed by a pump-induced modulation, ΔΦ(t − τ), applied at a time delay τ with respect to the probe pulse (see Fig. 1). The reference SC pulse, Er(t), precedes the pump in time and thus remains unaffected.
Fig. 1 Schematic of SSSI consisting of a chirped supercontinuum (SC) pulse (reference, Er(t)) and a time-delayed replica pulse (probe, E(t)) upon which a pump-induced ultrafast modulation ΔΦ(t − τ) has been imparted at a time delay τ with respect to the probe.
The perturbed probe pulse Ē(t) can be written as
where E(t) is the unperturbed probe pulse. Then ΔΦ(t), the same pump-induced modulation but applied at τ = 0, can be extracted from the interference between the reference and probe pulses in the frequency domain as where F denotes the Fourier transform, |Ē(ω)| and |E(ω)| is the spectral amplitude of the perturbed and unperturbed probe pulses, respectively, that can be directly measured by a spectrometer; Δφτ(ω) is the phase difference between the modulated probe and reference pulses at a given τ which can be obtained from an interferometer; and φ(ω) is the spectral phase of the unperturbed probe (or reference) pulse. Here to retrieve ΔΦ(t), the spectral phase φ(ω) needs to be characterized. In general, the spectral phase of a chirped pulse can be expressed in a Taylor expansion around the central frequency ωc as where φ0 is the absolute spectral phase; b1 is the first order dispersion coefficient related to a pulse shift in time; b2 and b3 are the second and third order dispersion coefficients, respectively. Here the first two terms are not required in retrieving ΔΦ(t), but b2 and b3 need to be characterized for SSSI operation.In Eq. (2), it is important to note that the modulation ΔΦ(t) remains unchanged even if the time delay τ changes. This is because the term e−iωτ shifts the modulation occurring at t − τ back to t. In other words, ΔΦ(t) must be uniquely retrieved from many different τ delayed shots if the spectral phase φ is correctly characterized.
For illustration, we consider a Gaussian-type phase modulation given by
where we choose A = 0.4 and σ = 50 fs. Here the probe pulse is also a Gaussian pulse centered at 800 nm with a full-width-at-half-maximum (FWHM) bandwidth of 170 nm and chirped with b2 = 1000 fs2 and b3 = 400 fs3. Figure 2(a) shows a series of differential probe power spectrum [17], ΔI(ω), as a function of the pump-probe delay τ. Figure 2(b) shows two spectral line-outs at τ = 0 fs and 400 fs. One prominent feature is that the position of the central minimum ω0 shifts with respect to the pump-probe delay τ. Here the central minimum is defined as the point where ΔI(ω) oscillates most slowly; mathematically, it is given by the condition Therefore, by tracing ω0 at each τ, one can determine b2 and b3 with a polynomial fit [17]. This method, however, is limited by the validity of the stationary phase approximation. Moreover, it is inefficient as only the central minimum/maximum point or at most some adjacent extrema are used in each shot for characterization.Fig. 2 (a) Simulated differential probe power spectrum, ΔI(ω, τ), modulated by a phase transient given by Eq. (4) as a function of pump-probe delay. The probe is chirped with b2 = 1000 fs2 and b3 = 400 fs3. (b) Differential probe spectral line-outs at τ = 0 fs and τ = 400 fs
It is obvious that one needs the correct values of the second and third dispersion coefficients to properly characterize the phase modulation, but what is the consequence if the known values differ by Δb2 and Δb3 from the true ones? Our simulation shows that nonzero Δb2 or Δb3 leads to ambiguity in the retrieved modulation ΔΦ(t). For illustration, we show the retrieved ΔΦ(t)’s from two different time delays τ = 0 fs and τ = 400 fs with Δb2 = 0 fs2, Δb3 = −40 fs3 in Fig. 3(a) and Δb2 = −60 fs2, Δb3 = 40 fs3 in Fig. 3(b). Those two retrieved ΔΦ(t)’s are different, and furthermore neither is identical to the true modulation. For a wider range of Δb2 and Δb3, we quantify the difference in shape of the retrieved modulations obtained from multiple time-delayed shots by
where is the average of ΔΦτ retrieved from all different time delays τ. This ΔS2 strongly depends on how well the probe phase is characterized. For example, the dependence of ΔS2 on both Δb2 and Δb3 is computed and shown in Fig. 3(c). It clearly shows a deep global minimum located at (0, 0), which corresponds to the initially assigned probe phase (b2 = 1000 fs2 and b3 = 400 fs3). This shows that the modulations obtained from all different time delays converge only when the pulse’s phase used for retrieval matches the true form. At the same time, the converged function represents the real form of modulation. A mathematical proof of this observation is provided in Appendix.Fig. 3 (a), (b) Extracted modulations ΔΦ(t) obtained from two different time-delayed shots at τ = 0 fs (green dashed line) and τ = 400 fs (red dotted line) using intentionally incorrect spectral phase coefficients, along with the correct modulation (black line). The modulations ΔΦ(t) obtained from two shots of different time delay are nonidentical when Δb2 or Δb3 is non-zero. (c) The dependence of ΔS2 on Δb2 and Δb3 shows a deep global minimum located at (0, 0). The square and the triangle correspond to (Δb2, Δb3) = (0, −40) and (−60, 40) as illustrated in (a), (b) respectively.
3. Algorithm details
Experimentally, it is possible that modulations from different shots are similar in shape but slightly different in magnitude due to pump pulse power fluctuation. Therefore, the modulation extracted from each shot is normalized prior to comparison. We also emphasize that the retrieved modulation often exhibits smooth variations in the vicinity of the central extremum, but it is very noisy in the far away region. Therefore, in practice, only a region of interest is used for an input. This should cover as much meaningful features of modulation as possible but be narrow enough to avoid too much noise.
One feature needs to be discussed is how to choose different time delays τ to optimize the operation of our algorithm. In a stationary phase approximation, the perturbed probe pulse can be expressed as [17]
where C is a constant, ΔΦ(ω) is the Fourier transform of ΔΦ(t) in the frequency domain, b2′ = b2 + 3b3(ω0 − ωc), and ω0 is given by φ′(ω0) = τ. In the case of a small τ, ω0 becomes close to ωc and b2′ ≈ b2, and the dominant part containing the third-order dispersion 3b3(ωc − ω0)(ω − ω0)2 becomes insignificant. In that case, the third order dispersion is hard to be determined. Therefore, for an effective operation, we want the change caused by the third-order dispersion to be greater than its measurement error ε where ω0 − ωc ≈ τ/(2b2). Therefore, the time delay separation between two shots should be Equation (9) establishes the relation between the time delay and experimental conditions. Furthermore, the upper limit of the time delay is fundamentally set by the probe pulse duration, i.e. the pump pulse should lie within the probe pulse. As a result, a low-chirp probe requires shorter shot-to-shot separations Δτ.As illustrated in Fig. 3, the probe spectral phase can be found by minimizing ΔS2. This process can be performed by a genetic algorithm (GA). Figure 4 shows a diagram of our algorithm routine to characterize both probe and modulation simultaneously. First, the spectral modulations of probe (Δφs1, Δφs2, .., ΔφsN) are measured at multiple pump-probe time delays τi. Along with an initial population of b2’s and b3’s, the corresponding temporal modulation functions (ΔΦ1, ΔΦ2, .., ΔΦN) are constructed within Eq. (2). Then the GA is used to minimize ΔS2 defined by Eq. (6). Finally, those b2, b3 that provides the global minimum of ΔS2 will be selected for the best fitting parameters. Simultaneously, the calculated from these optimized values is deemed to be the correct form of the applied modulation.
Fig. 4 Algorithm routine for simultaneous characterization of both probe chirp (b2 and b3) and modulation (ΔΦ(t)). It uses a genetic algorithm (GA) to minimize ΔS2 such that the modulation functions obtained from all pump-probe delays can converge to equality.
4. Performance test
In this section, we test the reliability of our algorithm with numerical simulations. Here we simulate two types of modulations. The first one mimics a Kerr-induced refractive index modulation, where the modulation is proportional to the intensity of a co-propagating pump pulse. The second one simulates a femtosecond photo-ionization process, where the modulation asymptotically approaches zero at t → −∞ and a non-zero value at t → ∞. In both cases, the probe pulse is set to be the same as in the previous sections with b2 = 1000 fs2 and b3 = 400 fs3.
4.1. Kerr-like modulation
An intense laser pulse can induce a transient in the index of refraction of a medium it propagates through, leading to a phase modulation on the co-propagating probe pulse. We assume the modulation has a form of
with A1 = 0.2, a = 1.6×10−7 fs−4, b = 5×10−2 fs−1, t0 = 30 fs, A2 = 0.1, and c = 2.5×10−3 fs−2. The imaginary part (second term) represents nonzero absorption in the medium. Here we assume the modulation is not symmetric in time. We also introduce a random error of ≤5% to the simulated spectrogram to test the stability of our algorithm and a fluctuation of ≤10% to the magnitude of ΔΦ for each time-delayed shot.We first test the convergence speed of our GA. In this simulation, each generation comprises 80 sets of b2, b3 with the search range of 600 − 1200 fs2 for b2 and 200 − 800 fs3 for b3. We perform the simulation in three cases with different numbers of time-delayed shots, namely two shots at τ = 0 and 400 fs, three shots at τ = 0, 400 and 600 fs, and four shots at τ = 0, 400, 600 and −200 fs. In each case, we use the same initial population that is intentionally chosen to be far away from the converged values. The optimized ΔS2 at each generation is shown in Fig. 5(a). Despite the unfavorable condition we set, ΔS2 converges fast in all three cases after 15 generations.
Fig. 5 (a) Minimal ΔS2 after each generation when using data from two shots at τ = 0 and 400 fs (black solid line), three shots at τ = 0, 400 and 600 fs (red dashed line), and four shots at τ = 0, 400, 600 and −200 fs (green dotted line). ΔS2 is normalized to the final converged value. (b)–(d) The real part of reconstructed average modulation (green solid line) compared to the exact function (red dashed line) given by Eq. (10), with its best fitting parameters ( and ) obtained from 2, 3, and 4 shots, respectively, as defined in (a). (e)–(g) Distribution of retrieved (b2, b3) after 300 trials, corresponding to (b–d) respectively.
In Figs. 5(b)–5(d), we show the optimized set of , , and the average modulation obtained from two, three, and four delayed shots. In this example, the second order dispersion coefficient b2 can be characterized within a 1% error, and the shape of modulation can be reconstructed fairly well even with 2 shots. However, the third order dispersion coefficient b3, less significant compared to b2, suffers from a 5% error when only two shots are used. It is noticeable that when 3 and 4 shots are used, the retrieval errors of b2 and b3 reduce to less than 1% and 2%, respectively.
We emphasize that the GA is so effective that it converges quickly to the almost exact global minimum of ΔS2 regardless of the number of shots used. Note that we also introduce ≤10% fluctuations to the modulation amplitude, but it is neutralized by the normalization step in our algorithm. Therefore, the retrieval error as shown previously is solely due to the random error introduced to the spectrogram. To examine how this error affects the retrieved values, we repeat the same simulation for 300 times and plot the extracted b2 and b3 in Figs. 5(e)–5(g). Firstly, compared to the typical uncertainty in b2 (2%), b3 spans much wider with the standard deviation of ∼10% in the case of 2 shots. This is understandable as the effect of third order dispersion on the spectra is quite small and can be overwhelmed by the random noise. Secondly, the overall certainty is diminished by increasing the number of snapshots. The error margins of retrieved b2 and b3 shrink significantly when the number of shots increase from 2 to 4, specially from ∼10% to ∼2% for b3. This is not surprising as the effect of random noise can be lessened by repetition.
To further investigate the robustness of the algorithm, we introduce more sources of errors, for example, shot-to-shot variation in the reference pulse’s chirp and pump-induced modulation’s width and amplitude. In general, with a moderate level of random errors, i.e. the total variation of less than 10%, the retrieved chirp coefficients converge to acceptable values. We note that our GA can always retrieve the exact b2 and b3, and ΔΦ(t) when no random fluctuation is included in the simulations.
4.2. Femtosecond stepwise modulation
As a second example, we consider an ultrafast transient commonly observed in optical field ionization. In strong laser electric fields, atoms and molecules can be tunnel ionized, producing free electrons in continuum states. Macroscopically, the free electron density rises in time until the intense pulse passes by. The density modulation induced by the pump pulse can be picked up by a co-propagating probe pulse. For simplicity, we consider the following phase modulation caused by tunneling ionization,
Similar to the previous section, we simulate the spectrograms at different time delays and use data from 2, 3 or 4 shots to reconstruct the modulation. The spectrograms are also subject to ≤5% random fluctuations. Our simulation results are presented in Figs. 6(a)–6(c). In the 2-shot case, the optimized and exhibit 1% and 10% errors, respectively. With the FWHM bandwidth of 170 nm, the fastest resolvable phase transient is ∼5.5 fs for a Gaussian temporal modulation. For an abruptly changing function as in this example, its Fourier transform spreads much wider in the frequency domain than that of a Gaussian function (for instance, 1/|ω| for a step function in time). This leads to an even worse temporal resolution. Thus a relatively large uncertainty is expected in the extraction process. However, when using three or four shots, highly accurate characterization is possible.In conclusion, our algorithm works well for two examples of ultrafast modulations even with ≤5% random noise applied in the spectrograms. It works equally well, we believe, for any reasonably shaped modulations. However, depending on the modulation shape, more than 2 shots are needed to obtain very high accuracy, especially when non-negligible random errors are present.
4.3. Extending the number of fitting parameters
In this section, we test the flexibility of the algorithm when more fitting parameters are introduced. In one possible scenario, the probe and reference pulses can be non-identical with different b2 and/or b3. This disparity can occur when the reference and probe pulses pass through a beam splitter different numbers of times, thus leading to unequal dispersion. In that case, Eq. (2) needs to be modified as
where δφ is the phase difference between the probe and reference pulses. We estimate where B2 has an order of 10 fs2. In this example, there are three parameters to be optimized (b2, b3 and B2). Here we choose b2 = 1000 fs2, b3 = 400 fs3, and B2 = 30 fs2, with the same modulation and noise (≤5%) as in Section 4.1 for simulation. The optimal parameters b2, b3 and B2 retrieved from 2, 3 and 4 time-delayed shots are presented in Tab. 1.
Table 1. Best-fit parameters (, , and ) retrieved with 2, 3, and 4 time-delayed shots when the probe and reference pulses are allowed to have second order dispersion coefficients different by B2.
As shown in Tab. 1, b2 and B2 can be determined within 2% regardless of the number of shots. Noticeably, when 3 or 4 shots are used, all three parameters can be obtained within 1% error. Note that b2 is retrieved with a 3% difference, which is comparable to the 5% error obtained when the reference and probe pulses are set to be identical (B2 = 0) in Section 4.1. Therefore, the addition of more chirp parameters does not significantly affect the performance of our algorithm.
5. Conclusion
In summary, we have presented a simple method to determine both probe spectral phases and pump-induced modulations in a conventional SSSI setup. Our GA-based routine is shown to work for typical ultrafast modulations and capable of characterizing the probe phase with high accuracy. Also our algorithm can be easily modified to include more chirp parameters if necessary. With numerical simulations, we show that our algorithm is robust against random errors (≤5%) and can provides satisfactory accuracy even with 2 time-delayed shots. With three or more shots, our algorithm can retrieve nearly the exact modulation and spectral phase. We believe our technique can be readily applied to any SSSI setup to simplify or eliminate its routine chirp characterization process.
Appendix: Proof of ΔS2 minimum uniqueness
This section attempts to prove mathematically that the standard deviation ΔS2 exhibits a zero value when the phase function used in extraction has the correct form. Suppose that φ is slightly deviated as φ → φ + δφ, then we have
where M(ω) = F−1[eiΔΦ(t)]. The Fourier transform of this term is given by where F{eiδφ(ω)} = C(t)eiδφ̃(t) and E(t) = |E(t)|eiφ̃(t). Generally, the modulation varies in a shorter time scale than the probe pule, so φ̃ and δφ̃ vary much faster than ΔΦ. Using the stationary phase approximation, Eq. (15) can be approximated as where g is the point that contributes the most to the integral. Within the stationary phase approximation, this point can be given by As a result, Eq. (2) now gives Because g depends on τ according to Eq. (17), the extraction now produces different results at various time delays. Only when δφ = 0, making C(t)eiδϕ̃ → δ(t) and g → 0, Eq. (18) yields the same function form regardless of τ.Funding
National Science Foundation (NSF) (Award No. 1351455); Air Force Office of Scientific Research (AFOSR) (FA9550-16-1-0163).
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