Phase retrieval of ultrashort laser pulses using a MIIPS algorithm

Alberto Comin; Richard Ciesielski; Nicolás Coca-López; Achim Hartschuh

doi:10.1364/OE.24.002505

1. Introduction

The phase retrieval of structured and broad-band laser pulses continues to be a topic of intense research [1–3]. Numerous powerful techniques have been established with different advantages and experimental challenges [1, 4]. For instance, Multiphoton Intrapulse Interference Phase Scan (MIIPS) is a particularly effective techique for optimizing the typical output of a Titanium-Sapphire laser source [5]. MIIPS yields bi-dimensional datasets with sequences of second harmonic spectra for varying phase modulation parameters, after which a simple peak-finding algorithm can yield an estimate of the group delay dispersion for each frequency [5, 6]. Frequency Resolved Optical Gating (FROG) is an alternative technique [7], and it achieves a broader applicability [8], albeit at the expense of adding complexity in the phase retrieval algorithm [9]. Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER) is another very popular technique [10]. Recently, there have been efforts toward developing chirp (or dispersion) scan methods, which are related to MIIPS, but differ from it in that they rely on modulating the input pulse using a nearly linear spectral chirp rather than a sinusoidal function [11, 12]. The primary advantage of these techniques is that they are relatively simple to implement, because they do not require a pulse shaper. Another particularly powerful technique is the Phase-Resolved Interferometric Spectral Modulation (PRISM), a recently developed pulse-compression method that employs a pulse shaper without requiring the use of a spectrometer [3]. This can be a clear advantage in cases, for instance, when the second harmonic spectrum lies in the UV range and the sensitivity of detection systems is low [3].

Once the pulse has been characterized, it may require optimization by compensating for the temporal chirp accumulated along the optical path. For instance, a microscope objective can introduce a group delay dispersion (GDD) of several thousands fs², as well as higher order phase distortions [3]. The most flexible solution is to use a pulse-shaper capable of producing nearly arbitrary phase modulations. A technique that relies on pulse-shaping for the characterization step thus has an advantage, in that it can potentially measure and compress the pulse in a single step. This is one of the main factors behind the success of MIIPS. The other is that MIIPS data are intuitively represented in the generated trace: A second order phase shifts the offset of the MIIPS trace and a third order phase adjusts its slope [5]. In a previous publication, we demonstrated that the accuracy of a MIIPS iteration can be improved by implementing a scanning amplitude gate [13]. Beyond intra-focus pulse compression, MIIPS-based methods can be employed to investigate the spectral phase response of single plasmonic nanoantennas [14]. In this approach, MIIPS maps are recorded following detection of the second harmonic generation (SHG) of the nanoantennas using a compressed laser pulse.

In this work, we introduce a method that allows for efficient phase retrieval in a single iteration, providing that the laser spectrum at the reference sample is known. This technique is expected to be particularly beneficial for measuring photosensitive samples [14], and it will also be helpful in the case of samples which produce very low second harmonic spectra. We also show that the new method of analysis avoids a type of non-trivial ambiguity that arises for structured amplitude pulse profiles and can provide better feedback on the accuracy of the phase retrieval.

2. Theory

MIIPS is a frequency domain technique based on second harmonic generation (SHG) of a reference sample. The main assumptions for the technique are that the SHG process is instantaneous and that the nonlinear coefficients of the reference sample are spectrally constant. Within these approximations, the SHG is proportional to the auto-convolution of the frequency domain electric field [6]:

E (2 ω) = \int_{- \infty}^{+ \infty} | E (ω - Ω) | | E (ω + Ω) | exp {i [ϕ (ω - Ω) + ϕ (ω + Ω)]} d Ω

The key concept here is that the SHG intensity, |E(2ω)|², is maximized when the argument of the exponential in Eq. (1) is zero. A MIIPS measurement requires modulating the spectral phase of the pulse while simultaneously recording SHG spectra. The total spectral phase can then be written as the sum of the input laser phase, φ(ω), and a modulation term, φ_mod(ω):

ϕ (ω) = φ (ω) + φ_{mod} (ω)

The goal of the technique is to determine the modulation parameters that maximize the SHG intensity for each frequency component.

In the most common implementation of MIIPS, the phase modulation is a sinusoid given by the expression [6]:

φ_{mod} (ω) = Φ_{0} sin (τ (ω - ω_{0}) - ψ)

where τ is the modulation frequency and ψ is a scanning parameter. The SHG spectra are stacked together to form a map, like those shown in Figs. 1(c) and 1(d).

Fig. 1 Simulated MIIPS measurement of two 10 fs laser pulses, centered at 400 THz. a) Spectral amplitude (gray line) and GDD (blue line) of a Gaussian pulse with GDD oscillating between 970 fs² and 1030 fs². b) Spectral amplitude (gray line) and GDD (black line) of a pulse with constant GDD of 1000 fs² and modulated spectral amplitude. c,d) MIIPS maps of the pulses shown in (a) and (b), with two lines indicating the retrieved (solid red) and the correct (dashed white) GDD. For both maps the modulation parameters were: τ = 10 fs and Φ₀ = 15 rad. e) Comparison of the GDD retrieved from the MIIPS maps in panels (c) and (d): The two curves are overlapped, showing that MIIPS does not distinguish between amplitude and phase variations. f) A simulated MIIPS map obtained using, as input, the GDD retrieved from the map in panel (e), together with the derived GDD (solid red line) and correct GDD (dashed white line).

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The standard analysis of the MIIPS map is based on the Taylor expansion of the spectral phase term in Eq. (1), which permits the second harmonic field to be expressed as a function of the GDD:

E (2 ω) \approx e^{2 i ϕ (ω)} \int_{- \infty}^{+ \infty} | E (ω - Ω) | | E (ω + Ω) | exp (i \frac{d^{2} ϕ (ω)}{d ω^{2}} Ω^{2}) d Ω

It follows that the SHG intensity, |E(2ω)|², is maximized when:

GDD (ω) \equiv \frac{d^{2} φ (ω)}{d ω^{2}} = - τ^{2} Φ_{0} sin (τ (ω - ω_{0}) - ψ)

This equation allows for the calculation of the GDD from the values of ψ that maximize the MIIPS map and which can be determined using a peak finding algorithm [6]. The exponential prefactor in Eq. (4) contains information about the carrier-envelope phase offset. It does not contribute to the SHG intensity and it is not measured by MIIPS, meaning that it is valid only for pulses longer than several optical oscillations. However, disregarding the frequency dependence of the amplitude, as in Eq. (4), may lead to incorrect results for a single MIIPS iteration, especially if the laser pulse has a modulated amplitude profile, as is typically the case for ultrafast laser pulses.

In order to illustrate this point, we simulated MIIPS for two different pulses, both with the same central frequency (400 THz ≡ 750 nm) and temporal duration (10 fs). The first pulse, shown in Fig. 1(a), had a Gaussian spectrum and a GDD oscillating between 970 fs² and 1030 fs². The second pulse had a constant GDD of 1000 fs² but a modulated spectral amplitude, as shown in Fig. 1(b). The MIIPS modulation parameters, chosen to reflect typical values used in experiments, were in both cases τ = 10 fs and Φ₀ = 15 rad (see Eq. (3)). A more accurate simulation could be obtained using a smaller modulation frequency and a correspondingly larger modulation amplitude (e.g. τ = 5 fs and Φ₀ = 60 rad), but in practice, phase wrapping at the spatial light modulator (SLM) would lead to substantial shaper artifacts [15].

The resultant phase-frequency MIIPS maps are shown in Figs. 1(c) and 1(d). For convenience, we also plotted two pairs of lines in the same graphs showing the GDD retrieved by MIIPS (solid red lines) and its correct value (dashed white lines). The red lines follow the positions of the maxima of the SHG as a function of the scanning parameter ψ, while the white line indicates the actual position of the flat spectral phase for the chosen pulse characteristics. In addition to the shift between the retrieved and actual phases, we note that the retrieved phase trace is curved. According to the usual interpretation of MIIPS data, the trace curvature would indicate the presence of higher order phase terms, which are not actually present in the pulse.

From the above discussion, it follows that there is a systematic error for both of the pulses shown in Fig. 1. This error is particularly significant for the pulse with modulated amplitude, reflected by the curvature of the red line in Fig. 1(d), which depends solely on the pulse amplitude, rather than the phase.

The phase-only MIIPS algorithm seems not to distinguish between amplitude and phase variations. This conclusion is supported by the observation that, despite the two pulses having different spectra and phase, the red lines in panels (c) and (d) of Fig. 1 are in the same position, meaning that the corresponding maps are indistinguishable by a single MIIPS iteration. This fact can be clearly observed by plotting the retrieved GDDs in the same graph, as shown in Fig. 1(e), and noting that the two curves are perfectly overlapped. As already discussed, the missing point is that the main equation of MIIPS, reported in Eq. (4), does not consider the frequency dependence of the spectral amplitude.

The most common way to improve the accuracy of MIIPS is to iterate the process of measuring and correcting the GDD, until the residual error drops below a certain threshold [6]. Figures 2(a) and 2(b) shows the residual GDD after several phase-only iterations for a Gaussian (Fig. 1(a)) and an amplitude modulated (Fig. 1(b)) pulse. Both pulses were initially linerly chirped, with a GDD of 1000 fs². As discussed previously, phase-only iterations tend to introduce additional features in the GDD of the amplitude modulated pulse; namely, they eventually decrease in intensity and become negligible after several iterations. For comparison, the residual GDD after a single experimental iteration using the new method, introduced in the subsequent section, is also reported as a black line in Figs. 2(a) and 2(b). Figures 2(c) and 2(d) report the same data, but averaged over 2 standard deviations around the central frequency of the laser pulse. These data demostrate that, by iterating a sufficient number of times, the difficulties caused by amplitude modulations can be overcome by phase-only MIIPS. At the same time, including information about the pulse spectrum may allow for faster measurements and reduce the risk of introducing unwanted phase modulations.

Fig. 2 Result of a single experimental iteration using the new method and convergence of phase-only MIIPS alghorithm for a Gaussian and a amplitude modulated pulse. Both pulses where initially linearly chirped, with a GDD of 1000 fs². MIIPS parameters are τ = 10 fs and Φ₀ = 15 rad for all cases. a,b) GDD residual after five successive MIIPS experimental iteration using phase-only MIIPS for a Gaussian (a) and an amplitude modulated pulse (b). Also shown are the GDD after a single experimental iteration using the new method (black curve) and the spectral amplitude (gray shaded area). Data are vertically offset for clarity. c,d) GDD residual for a Gaussian (c) and an amplitude modulated (d) pulse, averaged over an interval of two standard deviations around the central frequency

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Further insight can be obtained by simulating a second MIIPS map using the retrieved GDD as input. Figure 1(f) shows a map obtained using the GDD retrieved from the amplitude modulated pulse, as shown in Fig. 1(e). The maxima of the SHG signal (red solid line) are further shifted to the left, and the distance between the red and the white lines is increased with respect to Fig. 1(d). This is a consequence of the monotonicity of Eq. (5) in the interval [−π/2, +π/2]. In the following section, we will see how this observation can be used to derive an improved method for estimating the GDD.

3. Improved GDD retrieval

We observe that, because we are using a periodic modulation function (Eq. (3)), the MIIPS map is also periodic. We can thus limit our analysis to the central part of the map, delineated by the two lines τ(ω − ω₀) = ±π/2. As previously discussed, the second derivative of the modulation function, Eq. (5), is monotonic within this range.

The primary motivation of the present approach is to determine the spectral phase which, substituted in Eq. (1), produces a MIIPS map identical to the experimental map. This procedure is similar to the method used for analyzing FROG data [9], but it is much simpler, in that it does not involve switching forth and back between frequency and time domains. In contrast to the phase-only MIIPS procedure, we simply require the spectral amplitude of the pulse to use Eq. (1). Using this information, the new algorithm can also be accurate for pulses with complex profiles.

We begin with an input MIIPS map, M_exp(ω, ψ), and use Eq. (5) to obtain a first guess for the GDD, φ̈₀(ω), as illustrated below:

M^{exp} (ω, ψ) \overset{peak finding (Eq . (5))}{\to} {\ddot{φ}}_{0} (ω)

By integrating the GDD twice with respect to the angular frequency, we obtain an initial guess for the spectral phase, φ₀(ω), and, using Eq. (1), a first simulated MIIPS map,

M_{0}^{sim} (ω, ψ)

:

{\ddot{φ}}_{0} (ω) \overset{\iint d ω d ω}{\to} φ_{0} (ω) \overset{MIIPS simulation (Eq . (1))}{\to} M_{0}^{sim} (ω, ψ)

The GDD retrieved from the simulated MIIPS map,

M_{0}^{sim} (ω, ψ)

, provides a feedback term,

{\ddot{φ}}_{0}^{feedback} (ω)

, on the accuracy of the first guess:

M_{0}^{sim} (ω, ψ) \overset{peak finding (Eq . (5))}{\to} {\ddot{φ}}_{0}^{feedback} (ω)

The difference between the GDD retrieved from the simulated and experimental maps allows for the estimation of an error,

Δ {\ddot{φ}}_{0}^{err} (ω)

:

Δ {\ddot{φ}}_{0}^{err} (ω) = {\ddot{φ}}_{0}^{feedback} (ω) - {\ddot{φ}}_{0} (ω)

The monotonicity of Eq. (5) also allows us to determine the sign of the error. In other words we can determine if we are over- or under-estimating the GDD. We can then derive a second guess φ̈₁(ω) for the GDD by compensating for the estimated error:

{\ddot{φ}}_{1} (ω) = {\ddot{φ}}_{0} (ω) - k \cdot Δ {\ddot{φ}}_{0}^{err} (ω)

Here, k is a numerical constant (≈ 0.5) that can be tuned to optimize the convergence of the algorithm. The process then starts over from Eq. (7) and concludes when the error of the n^th iteration becomes smaller than a user defined tolerance ε_GDD.

| Δ {\ddot{φ}}_{n}^{err} (ω) | = | {\ddot{φ}}_{n}^{feedback} (ω) - {\ddot{φ}}_{0} (ω) | \leq ε_{GDD}

The tolerance can be set by considering that, according to Eq. (5), the minimum variation of the GDD that can be appreciated using MIIPS is defined by the equation:

min [Δ \ddot{φ} (ω)] = - \frac{τ^{2} \cdot Φ_{0}}{N_{ψ}}

The term N_ψ is the number of phase steps per radian. Therefore, in general, the GDD tolerance should be chosen such that ε_GDD ≥ τ²Φ₀/N_ψ.

The complete process, for the case of the amplitude modulated pulse shown in Fig. 1(b), is illustrated in Fig. 3. Figure 3(a) shows the GDD error for the first four iterations of the algorithm. It can be seen that the third iteration already yields an error close to zero. Figure 3(b) reports the comparison between the true spectral phase and its estimated value after each iteration. Figures 3(c) and 3(d) show, respectively, the input and the retrieved MIIPS map after four iterations. The two maps are identical, supporting the conclusion that the GDD has been correctly retrieved.

Fig. 3 Iterative numerical analysis of the MIIPS map of an amplitude modulated 10 fs laser pulse centered at 400 THz, with constant GDD of 1000 fs². The MIIPS parameters were τ = 10 fs and Φ₀ = 15 rad. a) Error of the retrieved GDD for several iterations of the algorithm. b) Convergence of the retrieved phase (solid lines) to the true spectral phase of the input pulse (dashed black line). c) Initial MIIPS map. d) Retrieved MIIPS map after four numerical iterations. Initial and retrieved MIIPS maps are virtually identical, confirming the accuracy of the phase retrieval.

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We have shown that the new method for phase retrieval based on MIIPS maps has two advantages. First, it avoids the amplitude-phase ambiguities described previuosly, and second, it can be substantially faster by permitting retrieval of the spectral phase from a single experimental MIIPS iteration. However, as in the present implementation, it still does not allow MIIPS to compensate for an arbitrarily shaped pulse. For instance, by introducing steps in the spectral phase, it is possible to synthesize pulses that cannot be retrieved using either the phase-only or the new phase-amplitude analysis methods [8]. Experimentally, such phase steps could, for example, occur in cases of double pulses. To the best of our knowledge, the more general question of whether or not it is always possible to uniquely invert an arbitrary MIIPS map has not yet been explored. We suggest that answering this question would be an essential step for understanding the limits of validity of MIIPS and for developing even more accurate methods.

4. Conclusion

In this work, we presented an improved means for retrieving the spectral phase of an ultrashort laser pulse based on MIIPS. We have shown that the new method has two primary advantages; namely, it avoids amplitude-phase ambiguities that can occur in case of structured laser spectra, and it can be substantially faster through retrieval of the spectral phase from a single experimental MIIPS iteration.

Acknowledgments

We acknowledge support from the Nanosystems Initiative Munich (NIM), the ERC starting Grant NEWNANOSPEC (279494), and the Deutsche Forschungsgemeinschaft through SPP 1391 (HA4405/6-1). We thank Dr.Giovanni Piredda for useful discussions.

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Phase retrieval of ultrashort laser pulses using a MIIPS algorithm

Abstract

1. Introduction

2. Theory

3. Improved GDD retrieval

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (12)

Optics Express