1. INTRODUCTION

Short laser pulses are widely used in microscopy, spectroscopy, and micro-machining [1]. They allow the study of elementary processes in real time and, thanks to their high peak power, they are ideal for investigating nonlinear phenomena. However, they are difficult to handle, since as they propagate through any medium, including air, they collect phase distortions. This is particularly critical in the case of focusing through high numerical aperture objectives, since the amount of glass contained in such devices is enough to broaden the pulse duration of a femtosecond pulse by several orders of magnitude [2,3]. There are several techniques that can in principle be used to compensate phase distortions and deliver transform-limited pulses to the focal plane of a microscope. The most straightforward ones are based on prism and grating compressors, which can be used to minimize the quadratic and cubic terms in the phase expansion [1].

Pulse shaping has, in principle, the ability to compensate arbitrary phase distortions [4], but how to determine which phase to program in the pulse shaper is less straightforward. Genetic algorithms, for instance, require substantial measurement time and computational effort and do not always yield the desired pulse phase profile in a reproducible way. Frequency-resolved optical gating (FROG) is perhaps one of the most popular tools to retrieve the phase of femtosecond pulses [5], and there are several other ingenious techniques [6,7]. A significant step toward reliable and reproducible compression of ultrashort laser pulses was achieved by the group of Dantus, with the development of MIIPS [8]. The authors demonstrated the possibility to compensate the phase distortion introduced by a 0.60 NA objective by only using MIIPS [9]. Most recent implementations of MIIPS allow real-time pulse compression [10]. It was also recently shown that single plasmonic nanocrystals can be used as nonlinear material [3]. This could provide a way to perform subdiffraction-limited coherent control with ultrashort pulses [3]. However, the compensation of higher NA objectives benefits from external compression (for instance, by a prism pair) to retain the accessible phase range of the shaper [9].

In this paper, we show that MIIPS can be improved by complementing it with an amplitude gate that is scanned across the spectrum. We demonstrate that this gated version of MIIPS, which we called G-MIIPS (Fig. 1), estimates more accurately the spectral phase, without the need to increase the measurement time. It is also less affected by systematic errors originating from higher-order phase distortions and shaper artifacts.

 

Fig. 1. Multiphoton intrapulse interference phase scans. (a) Simulated MIIPS trace of a femtosecond laser pulse with large cubic phase distortion. (b) Equivalent G-MIIPS scan (c) from the top: phase modulation used in the MIIPS scan; real part of the phase-amplitude modulation used in G-MIIPS; normalized SHG spectrum after phase compensation using either MIIPS (blue curve) or G-MIIPS (green curve). (d) Group delay dispersion obtained geometrically by skewing and rescaling the G-MIIPS trace.

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The work is organized as follows: in Section 2 we give a brief introduction to the MIIPS technique; in Section 3 we discuss the limitations of the standard implementation of MIIPS; in Section 4 we introduce G-MIIPS; in Section 5 we derive analytical models for MIIPS and G-MIIPS; in Section 6 we illustrate, by numerical simulation, the advantages of G-MIIPS and, in Section 7, we show experimental data indicating that G-MIIPS is more suitable than standard MIIPS in estimating the phase of a femtosecond laser pulse.

2. MULTIPHOTON INTRAPULSE INTERFERENCE PHASE SCAN

MIIPS was designed as a tool to control multiphoton processes and to characterize and compress ultrashort laser pulses [11]. It employs a pulse-shaper setup in order to estimate the group delay dispersion (GDD) and to compensate arbitrary complex spectral phase distortions. The main concept is that the GDD is correlated with the intensity of second harmonic generation (SHG). At any given frequency, the SHG is maximized if the GDD is null [12]. It follows that a transform-limited (flat phase) pulse can be obtained by applying suitable phase masks and maximizing the SHG for the whole spectrum [12]. This can be seen from the equation for the SHG at the angular frequency $\omega$ in the ideal case (assuming ${\chi }_2=1$) as

Table 1. List of Symbols Used in this Article

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Expanding the phase in series and retaining the second order, the approximated equation for the SHG becomes

It follows that the SHG is maximized when the $\mathrm{GDD}=\ddot{\varphi }(\omega )$ is equal to zero. This condition is at the basis of MIIPS, and it is valid as long as the phase can be locally approximated by a second-order polynomial. The standard implementation of MIIPS involves adding a sinusoidal modulation to the phase of the laser pulse, typically using a pulse-shaper:

3. CHALLENGES OF MIIPS

A known limitation of MIIPS is that it only considers the second-order correction to the phase at a given frequency, neglecting higher orders in the Taylor expansion [9,15]. Although this can lead to tolerable errors for not overly distorted pulses, in other cases it can significantly reduce the accuracy of MIIPS as discussed below. Xu et al. showed that the systematic error is of the order of ${(\tau /\mathrm{\Delta }t)}^2/12$, where $\tau$ is the MIIPS modulation frequency and $\mathrm{\Delta }t$ is the transform limited pulse duration [9]. From the above estimate, it might seem that accurate results could always be obtained for short pulse duration just by choosing a small modulation frequency, and indeed an ideal MIIPS simulation, which does not take into account any of the shaper artifacts, would confirm this conclusion.

In Fig. 2, we report several simulated, single MIIPS scans performed on a 10 fs laser pulse centered at ${\omega }_0=2.4\mathrm{rad}/\mathrm{fs}$ (785 nm), after propagation through 10 cm of BK7 glass, which mimic the optical path length of a microscope objective. For each scan we report the residual spectral phase and the SHG spectrum, after a single scan. In order to compare the results obtained with different modulation parameters, we kept the maximum GDD correction constant [see Eq. (5)]. As expected, the best performance in terms of maximum SHG and residual phase are obtained for the smaller values of the modulation frequency $\tau$.

 

Fig. 2. Result of single MIIPS iterations, simulated for various phase modulation frequencies while keeping the maximum correction constant ${\mathrm{\Phi }}_0{\tau }^2=2.5·{10}^4{\mathrm{fs}}^2$. The figure refers to a $\mathrm{\Delta }t=10\mathrm{fs}$ laser pulse after propagation through 10 cm of glass. (a) MIIPS-corrected SHG spectra for different phase-modulation frequencies; the ideal SHG spectrum (black-dashed line) is also shown for reference. (b) Residual phase after MIIPS correction using different phase-modulation frequencies.

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In practice the adjustment of the modulation parameters is subtler because, for highly distorted ultrashort pulses, it requires setting the modulation amplitude to very high values, which in turn deteriorates the spectrum by introducing shaper artifacts, like for instance diffraction caused by phase grating and ripples in transmitted spectra. The former can be minimized by employing a double-pass $4f$ pulse shaper setup [16]. The latter can be explained as follows.

By definition the spectral phase is given by multiples of $2\pi$ radians. However, due to cross talk between the SLM pixels, phase and amplitude distortions can be observed in the spectral regions where the phase crosses $2\pi$, a phenomenon which is known as wrapping [17]. Therefore, if the amount of wrapping is very high, the spectrum transmitted through the SLM can be too distorted to be accurately characterized. This poses a limitation to the total phase, which can be compensated, and it is particularly severe for very short pulses ($<10\mathrm{fs}$) in combination with high-numerical aperture objectives.

For the above reasons, a common choice is to set the phase-modulation frequency $\tau$ equal to the transform-limited pulse duration $\mathrm{\Delta }t$ [9]. This is a good compromise because a lower modulation frequency would result in having a too high modulation amplitude, which would cause phase-wrapping artifacts, as we previously discussed. Higher modulation frequencies, on the other hand, are not optimal because the second-order polynomial expansion in Eq. (2) loses validity. Indeed, since MIIPS uses a sinusoidal phase mask, it introduces both second (GDD) and fourth-order (FOD) corrections, linked by the relation: $\mathrm{FOD}=-\mathrm{SOD}·{\tau }^2$. In a 2D map, which represents the SHG intensity as a function of GDD and FOD values (Fig. 3), MIIPS maximizes the signal along a linear trajectory, and from Fig. 3 it can be seen that this maximum does not necessarily correspond to a zero GDD. More generally, a sinusoidal phase modulation introduces an infinite number of higher-order phase distortions, each of them proportional to ${\tau }^n$, where $n$ is the order of the term in the Taylor expansion. Therefore maps similar to the one reported in Fig. 3 could be drawn to illustrate the systematic errors that affect MIIPS in case of higher-order phase distortions.

 

Fig. 3. Phase dependence of the SHG intensity at the central frequency of a 100 nm broad laser pulse centered at 800 nm. (a) Map of the SHG in function of the second- and fourth-order phase terms. The dashed line represents the points explored by a typical MIIPS trace. (b) The maximum SHG intensity along the MIIPS trajectory does not correspond to zero GDD.

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A. MIIPS Resolution

For a defined set of modulation parameters, the maximum GDD, which can be compensated by a single MIIPS iteration, is ${\mathrm{GDD}}_{\max }={\mathrm{\Phi }}_0{\tau }^2$ [see Eq. (5)]. On the other hand, the minimum GDD which can be compensated depends on the size of the phase steps and on the maximum relative error, which can be accepted for the GDD. If $\mathrm{\Delta }\psi$ is the increment of the argument of the sinusoidal modulation, and $ϵ=\mathrm{\Delta }\mathrm{GDD}/\mathrm{GDD}$ is the maximum relative error on the GDD, then it follows that

4. G-MIIPS

We developed G-MIIPS to avoid the limitations of MIIPS when compensating large phase distortions caused by broad spectra and high numerical aperture objectives. The idea, illustrated in Fig. 1, is to enable using higher modulation frequencies for the sinusoidal phase, while avoiding systematic errors due to higher-order phase terms. In the next sections, however, we will show that G-MIIPS is more accurate than MIIPS and also with the same choice of modulation parameters.

G-MIIPS can be readily implemented using a pulse shaper, which provides both phase and amplitude modulation. The amplitude modulation is exploited to gate the spectrum around a specific frequency [Fig. 1(c)], therefore improving the validity of the second-order polynomial expansion of the phase, even in the case of significant higher-order contributions, as will be shown below. It then becomes possible to use higher values of the modulation frequency $\tau$ and lower values of the modulation amplitude ${\mathrm{\Phi }}_0$ minimizing the phase wrapping. In the case of a Gaussian gate, the modulation that is applied by the pulse shaper can be written as a function of the phase terms:

As can be seen from Eq. (8), in G-MIIPS a Gaussian amplitude mask is translated alongside the phase modulation, using the scanning parameter $\psi$. In the following, we will refer to the dimensional parameter $\sigma$ as the gate width. To improve the accuracy of the GDD values closer to the gate boundaries, the G-MIIPS signal $\mathcal{G}(2\omega )$ is then obtained by dividing the SHG spectrum by the square of the Gaussian amplitude mask:

By combining Eq. (1) with Eq. (8), in analogy with what we previously discussed for the standard MIIPS case, it can be shown that

On a related topic, G-MIIPS can be particularly useful when working with highly structured pulses. Fast spectral modulations pose a challenge for standard MIIPS. Additionally, given that MIIPS assumes the spectral phase to be $C^2$ continuous, it is straightforward to imagine examples of spectral phases that cannot be compensated by MIIPS, even iteratively. Perhaps the most evident case is that of a $\pi$ phase step at the central frequency ($\varphi (\omega )=\pi \times \mathrm{Heaviside}(\omega -{\omega }_0)$). Remarkably, thanks to the reduced spectral bandwidth, G-MIIPS works nicely even in those cases. Fast phase oscillations do not create particular difficulties as long as the gate width is narrow enough and the spectrum can be considered locally smooth [see Eq. (18)]. Of course a limitation is given by the signal-noise ratio: the SHG detection must be carefully optimized because narrowing the gate means reducing the SHG intensity. Phase discontinuities are still challenging but, at least in simple cases like the $\pi$ phase step discussed above, we could maximize the SHG by iterating the G-MIIPS a few times.

In analogy with standard MIIPS [14], the G-MIIPS signal can also be expressed analytically for a few simple pulse shapes. For instance, analytical expressions for the SHG signal of a transform limited Gaussian pulse, subject to sinusoidal phase modulation, have been given by Hacker et al. [13] and reproposed by Lozovoy and Dantus [14]. However, to the best of our knowledge, an analytical expression of the MIIPS signal in the case of a chirped Gaussian pulse has not been shown yet. In the next section, we will derive it for the cases of both MIIPS and G-MIIPS. Such an expression can be useful for fitting an MIIPS trace without relying on a peak finding algorithm, as it is usually done [9]. Additionally, we will show that an approximated analytic expression for G-MIIPS signal can be also be written for a generic pulse shape, provided that the field amplitude varies slowly with respect to the Gaussian gate [Eq. (8)].

5. ANALYTICAL MODELS FOR MIIPS AND G-MIIPS

A. MIIPS on a Chirped Gaussian Pulse

Let us assume a Gaussian pulse given by

After some algebraic manipulation, one obtains

B. Analytical Expression for G-MIIPS

For the specific case of a Gaussian pulse, the analytical expression for the G-MIIPS signal is similar to the one we just derived for MIIPS. By substituting Eq. (11) into Eq. (10), after some algebraic manipulation, one arrives at a formula similar to Eq. (12), except for the substitution $\mathrm{\Delta }{\omega }^2\to L^2=(\mathrm{\Delta }{\omega }^2{\sigma }^2)/({\sigma }^2+\mathrm{\Delta }{\omega }^2{\tau }^2)$ inside the integration sign:

6. NUMERICAL COMPARISON

In order to test the efficacy of the G-MIIPS, we performed a series of simulated measurements [see Figs. 4(a) and 4(b)], in which we compared the ability of MIIPS and G-MIIPS in correctly estimating the phase introduced by 10 cm of glass on a 10 fs laser pulse centered at ${\omega }_0=2.4\mathrm{rad}/\mathrm{fs}$. The modulation parameters were ${\mathrm{\Phi }}_0=100\mathrm{rad}$ and $\tau =10\mathrm{fs}$, and the gate width was $\sigma =0.5\mathrm{rad}$.

 

Fig. 4. Simulation of the compensation of a 10 fs laser pulse centered at $2.4\mathrm{rad}/\mathrm{fs}$ after propagation through 10 cm of BK7 glass. Standard MIIPS (a) and G-MIIPS (b) maps, obtained with ${\mathrm{\Phi }}_0=100\mathrm{rad}$, $\tau =10\mathrm{fs}$ and $\sigma =0.5\mathrm{rad}$. The black-dashed lines indicate the GDD range, which can be measured with this set of parameters; the white-dashed line represents both the center of the gate and the points where the GDD is zero. (c) GDD measured by a single iteration of MIIPS (red line) and G-MIIPS (blue line), together with actual GDD value (black line). (d) Residual phase after a single iteration of MIIPS (red line) and G-MIIPS (blue line). (e) SHG signal of the ideal SHG (black line) and after compensation with a single iteration of MIIPS (red line) and G-MIIPS (blue line).

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In Figs. 4(c) and 4(d), we report, for the two cases, the actual and estimated values of GDD, the SHG spectrum before and after phase correction obtained by a single iteration, the estimated phase and the residual phase. From the traces reported in Fig. 4, it can be seen that the gated version of MIIPS is superior in both estimating the GDD and yielding greater SHG after one iteration. The difference could be even greater if we consider that, to compensate such a considerable GDD, we had to employ a modulation amplitude ${\mathrm{\Phi }}_0$ of 100 rad, which could cause significant phase-wrapping distortions. In order to reduce phase wrapping, we would have to decrease ${\mathrm{\Phi }}_0$ and correspondingly increase $\tau$ [see Eq. (5)]. This would reduce the performance of MIIPS because of the increased contribution from higher-order phase terms, as discussed above. G-MIIPS, however, would be relatively unaffected.

One important parameter that needs to be considered when comparing MIIPS with G-MIIPS is the choice of the gate width $\sigma$ with respect to the pulse bandwidth $\mathrm{\Delta }\mathrm{\Omega }$ and the modulation frequency $\tau$. To some extent, narrower gate values yield better results, as shown in Fig. 5. However, reducing the gate also means decreasing the intensity of the SHG spectra recorded during the G-MIIPS scan. Therefore the integration time needs to be adjusted accordingly. The ideal choice of the gate width is then a compromise between phase accuracy and measurement time.

 

Fig. 5. G-MIIPS for different settings of the gate width, simulated for a 10 fs laser pulse centered at $2.4\mathrm{rad}/\mathrm{fs}$ after propagation through 10 cm of BK7 glass. For all cases, the modulation parameters were ${\mathrm{\Phi }}_0=200\mathrm{rad}$ and $\tau =10\mathrm{fs}$. Standard MIIPS is also shown for comparison. (a) G-MIIPS map corresponding to $\sigma =1\mathrm{rad}$. (b) G-MIIPS map corresponding to $\sigma =0.2\mathrm{rad}$. The black-dashed lines indicate the GDD range, which can be measured with this set of parameters; the white-dashed line represents both the center of the gate and the points where the GDD is zero. (c) SHG after a single iteration of G-MIIPS for $\sigma$ varying between 1 and 0.2 rad. (d) Residual phase after a single iteration of G-MIIPS for $\sigma$ varying between 1 and 0.2 rad.

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As previously discussed, the reason of the success of G-MIIPS in compensating the phase introduced by propagation through transparent media lies in the ability of dealing with higher-order phase contributions. To highlight this aspect, we performed another simulation in which we assumed the presence of only a fourth-order phase term (${10}^5{\mathrm{fs}}^4$). This situation could mimic the case in which the second-order (GDD) term has been pre-compensated by a prism compressor. The results, for a 10 fs laser pulse centered at ${\omega }_0=2.4\mathrm{rad}/\mathrm{fs}$ are reported in Fig. 6. The modulation parameters where ${\mathrm{\Phi }}_0=20\mathrm{rad}$, $\tau =10\mathrm{fs}$, and the gate width was $\sigma =0.5\mathrm{rad}$. It can be seen that, in this specific situation, G-MIIPS offers a significant advantage over standard MIIPS. Since pre-compensating the GDD before pulse-shaping is common practice, this result underlines the usefulness of G-MIIPS.

 

Fig. 6. Simulation of the compensation of a 10 fs laser pulse centered at $2.4\mathrm{rad}/\mathrm{fs}$ with significant (${10}^5{\mathrm{fs}}^4$) fourth-order phase distortion. Standard MIIPS (a) and G-MIIPS (b) maps, obtained with ${\mathrm{\Phi }}_0=20\mathrm{rad}$, $\tau =10\mathrm{fs}$, and gate width $\sigma =0.5\mathrm{rad}$. The black-dashed lines indicate the GDD range, which can be measured with this set of parameters; the white-dashed line represents both the center of the gate and the points where the GDD is zero. (c) GDD measured by a single iteration of MIIPS (red line) and G-MIIPS (blue line), together with actual GDD value (black line). (d) Residual phase after a single iteration of MIIPS (red line) and G-MIIPS (blue line). (e) SHG signal of the ideal SHG (black line) and compensated with a single iteration of MIIPS (red line) and G-MIIPS (blue line). Standard MIIPS in this case reduces SHG.

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Since MIIPS is commonly iterated multiple times when used for pulse compression, we conclude this section by giving a few examples that demonstrate that G-MIIPS converges faster than standard MIIPS also when used iteratively. In Fig. 7 we considered the two cases discussed before (propagation through glass and significant fourth-order phase distortion), but, instead of stopping after the first iteration, we repeated MIIPS and G-MIIPS until the full SHG spectrum was recovered. In both case, we assumed a 10 fs pulse. In Figs. 7(a) and 7(c), we assumed that the pulse had propagated through 10 cm of BK7 glass, and we used the following modulation parameters: ${\mathrm{\Phi }}_0=100\mathrm{rad}$, $\tau =10\mathrm{fs}$, and $\sigma =0.5\mathrm{rad}$. In Figs. 7(b) and 7(d), we assumed a significant fourth-order phase distortion (${10}^5{\mathrm{fs}}^4$). The modulation parameters in this case were ${\mathrm{\Phi }}_0=20\mathrm{rad}$, $\tau =10\mathrm{fs}$, and $\sigma =0.5\mathrm{rad}$. We employed two different metrics to express the convergence of the iterations: the maximum absolute value of the residual phase on the spectral interval $|\omega -{\omega }_0|<{\sigma }_{\omega }$, where ${\sigma }_{\omega }$ is the standard deviation of the envelope of the electric field, and the intensity of the SHG, normalized to its theoretical transform limit value. For comparison we also reported the starting phase and SHG intensity, labeled as iteration 0. On a more technical side, we note that there are many strategies for tuning the modulation parameters after each iteration, in order to achieve faster convergence. Here, for simplicity, we kept the modulation parameters constant. However, since this choice could lead to instabilities when the phase is already very small compared to the modulation parameters, we added the constrain that the (G-MIIPS) correction is applied only when it leads either to an improvement of the SHG or to a reduction of the residual phase. From Fig. 7 we see that G-MIIPS recovers the full SHG spectrum and an almost flat phase in just two iterations. The performance difference with respect to standard MIIPS is particularly apparent when the fourth-order phase distortion is considered [Fig. 7(b)]. We emphasize here that we limited our analysis to two simple cases, with the aim of mimicking real situations that may occur in the lab. As previously discussed, it is relatively straightforward to come up with more complicated examples; for instance, including fast spectral oscillations or phase discontinuities, in which the advantage of G-MIIPS is much greater.

 

Fig. 7. Comparison of iterative MIIPS and G-MIIPS. Panels (a) and (c) represent the case of a 10 fs pulse after propagation through 10 cm of glass. Panels (b) and (d) refer to the case of a 10 fs subject to a fourth-order spectral phase of ${10}^5{\mathrm{fs}}^4$. In the panels (a) and (b), it is plotted the residual phase after multiple iterations of either MIIPS (blue dots) or G-MIIPS (green diamonds). In these graphs the first point, labeled with 0, corresponds to the situation before the first iteration. The panels (c) and (d) report the SHG intensity, normalized to its theoretical limit.

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7. EXPERIMENTAL DATA

In this section we experimentally compare the ability of MIIPS and G-MIIPS in estimating the chirp accumulated by a femtosecond laser pulse after traveling through glass. To perform the experiment, we started by compressing a 15 fs laser pulse, centered at 800 nm, and focused through a 1.3 NA microscope objective. Then we introduced additional glass on the optical path by a matched pair of SF10 prisms and performed a single iteration of either MIIPS or G-MIIPS. The two prisms were in contact, with a thin layer of optical oil between them to minimize reflections at the interface. In both cases, the scan parameters were: ${\mathrm{\Phi }}_0=10\mathrm{rad}$ and $\tau =25\mathrm{fs}$, with 401 phase steps and 1 s integration per step. The gate width was set to 0.18  rad. The measurements were performed using a $4f$ pulse shaping setup, based on a 128 pixel spatial light modulator. The SHG was obtained using BBO micro-crystals, deposited on a microscope slide.

The results [Fig. 8(a)], for 23 mm of glass, illustrate that the GDD introduced by the glass is better estimated by G-MIIPS. The contrast is particularly striking if one compares the resulting SHG signal after one iteration [Fig. 8(d)]: G-MIIPS in just one iteration recovers almost all the SHG spectrum of the compressed pulse. This result, together with the numerical simulations reported in the previous section, unambiguously show that G-MIIPS is a significant improvement toward a quick and reliable characterization of ultrashort laser pulses.

 

Fig. 8. Experimental comparison of MIIPS and G-MIIPS regarding the compensation of the GDD introduced by 23 mm of SF10 glass on a 15 fs pulse centered at 800 nm. (a) Standard MIIPS trace obtained with scanning parameters ${\mathrm{\Phi }}_0=10\mathrm{rad}$ and $\tau =25\mathrm{fs}$. (b) G-MIIPS scan obtained with scanning parameters ${\mathrm{\Phi }}_0=10\mathrm{rad}$, $\tau =25\mathrm{fs}$, $\sigma =0.18\mathrm{rad}$. (c) GDD measured by a single MIIPS (black curve) and G-MIIPS (red curve) iteration, compared with the theoretical GDD calculated using the dispersion equation. (d) SHG spectrum after a single iteration of either MIIPS (black curve) or G-MIIPS (red curve), compared with the SHG obtained after full pulse compression (green curve).

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As a note of caution, we would like to remind the reader that reducing the bandwidth of a laser pulse, as done in G-MIIPS, is not safe if the pulse shaper is followed by an amplifier because of the risk of damaging the amplifier optics. For those cases, the standard MIIPS would therefore be a more appropriate choice.

8. CONCLUSION

We introduced an improved scheme for ultrashort laser pulse compression and characterization, which avoids some limitations of MIIPS for severely phase-distorted ultrashort pulses. We demonstrated its effectiveness in giving a more accurate correction to substantial phase distortions, as those caused by propagation through high-NA microscope objectives.