1. Introduction

One of the fascinating recent developments in microscopy consists of achieving spatial resolutions well below the size of an Airy pattern, by the use of suitable masks imposed on the instrument pupil. The mask induces a point-spread function with a narrow central spot and large, low-level wings, whose detrimental effects on imaging may be suppressed by means of non-linear detection methods [1]. This “Optical Superresolution” technique is giving rise to an increasing number of applications [1,2], while new studies are going on to gauge the various amplitude or phase masks [3,4], explore the fundamental limits[5], and introduce further innovative techniques [6,7].

The very concept of superresolution might have broader applications. For instance, there is a well known analogy between the spatial-domain propagation of a light beam, subject to diffraction, and the temporal-domain propagation of a light pulse in a dispersive medium [8,9]. In line with this analogy, it is tempting to explore whether the notion of superresolution can be applied in the temporal domain. Some attempts have already been performed in this direction: amplitude spectral masks have been shown to reduce the effect of gain narrowing in laser amplifiers [10], and it was shown that very specific grating designs may shorten the apparent duration of diffracted pulses [11]. More generally, is it possible to control and shape spectrally ultrashort laser pulses, in such a way as to yield pulses whose full width at half maximum duration is shorter than the temporal “Fourier limit” of Gaussian pulses?

In this paper, we demonstrate this concept of temporal superresolution. A proof of principle experiment is described, that uses a high intensity ultrashort laser system. On the specific example of a steep non linear phenomenon occurring only for few-cycle pulses, we show the large potential of this temporal superresolution method when applied to strongly non linear laser/matter interaction processes.

2. Theoretical approach

We first show theoretically how temporal superresolution can result in pulses shorter than the Fourier limit imposed by the spectral width. The following calculations should not be considered as a survey of all possible schemes to achieve superresolution, but merely as examples demonstrating the possibility of the method. Let us consider a laser pulse, of spectral amplitude E_in(ω), corresponding for instance to a Gaussian spectrum with a flat phase:

where ω₀ is the central frequency and ∆ω the spectral Full Width at Half Maximum (FWHM). In the temporal domain, this pulse presents a FWHM duration of ∆t = 4π ln2 / ∆ω, usually assumed to correspond to the shortest pulse achievable using a ∆ω bandwidth for Gaussian pulses.

By analogy with spatial masks [5,7], we introduce a spectral amplitude or phase mask to modify the temporal profile of laser pulse. The spectral filter transmittance S(ω) relays the optical output with the optical input E(ω)_in by :

In the case of unitary complex S(ω) transmittance one deals with a spectral phase mask; if S(ω) is a purely real positive function, it describes an amplitude filter. Such amplitude filters are similar to those introduced to pre-compensate gain narrowing in amplification chains before amplification [10]. Shorter pulses are obtained in the latter case by attenuating the central components in the spectrum, thus compensating for the reduced amplification of the lateral components, to achieve a broader spectrum at the laser output.

 

Fig. 1. (a) Argument of phase mask (solid line) and profile of an amplitude mask (dashed line).

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An illustrative and simple example of an amplitude mask is a modulation given by S_amp(ω) = (1±αcosτ (ω-ω₀))/(1+α²)^1/2, as displayed in Fig. 1(a) (right-hand scale). This amplitude mask creates two satellite pulses, one advanced and one retarded by τ from the main pulse that interfere destructively with the latter on its edges. This results in a narrower central pulse, flanked by two satellite pulses of smaller intensity.

Phase masks can also be easily designed for temporal superresolution. Several recent works on spatial superresolution have shown definite advantages of spatial phase masks, in terms of Strehl ratios [3]. We present in Fig. 1(a) a suitable phase mask (dashed line, left hand scale), that will be used experimentally in this study :

Figure 1(b) displays the temporal profile of the laser pulse resulting from these amplitude and phase masks, applied to the initial field of Eq. (1) with a 40 nm bandwidth. A FWHM of 20 fs is obtained, while the Fourier limited pulse, shown in dotted line in Fig. 1(b), has a FWHM of 30 fs. A pulse duration reduction of 0.65 is obtained; while the parameters for the mask values can be widely varied, this reduction factor is typical of what one obtains, as long as the intensity of the satellite pulses remains low with respect to that of the main pulse.

 

Fig. 1. b) Normalized intensity profile of superresolved pulses obtained from the amplitude mask (solid line) (S_T = 0.2), from the phase mask (dashed line) (S_T=0.35), applied to a Gaussian, Fourier limited pulse of 40 nm FWHM bandwidth (dotted line).

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3. Superresolution criteria

In order to quantify the quality of temporal superresolution, we introduce numerical criteria by analogy with those used for spatial superresolution [5]. First we define a normalized superresolved pulse duration T, as the ratio between the FWHM using the spectral mask, and the FWHM of the Fourier Transform Limited (FTL) pulse. Second, we introduce a temporal Strehl ratio S_T, as the ratio between the peak intensity of the superresolved pulse and that of the FTL one; this criterion is related to the amount of spectral energy strongly delayed from the peak of the pulse. Finally, one very important figure is the wing intensity ratio R_T between the peak of the superresolved pulse, and that of its most intense satellite in time.

4. Experimental demonstration

The experimental demonstration of the feasibility of temporal superresolution is performed using a 1 kHz Ti:sapphire multistage chirped pulse amplified laser system, emitting 30fs duration pulses, centred at a wavelength of 814 nm, with an average output power of 9W. This laser is mostly used for extreme non linear optics studies, such as high harmonic generation, or other non linear phenomena depending strongly on the central pulse duration. The superresolving mask is obtained by means of an acousto-optic programmable filter (Dazzler™, [12]), inserted between the stretcher and the regenerative amplifier. This position makes it difficult to apply amplitude masks on the spectrum, as the shape of the latter is modified by gain narrowing later in the amplifying chain. In contrast, phase masks can be applied easily, with a direct transcription of the imposed phase to the output pulse phase, thanks to the low value of the non linear phase B. The output pulses are fully characterized by the SPIDER method [13,14].

A preliminary experiment is required to correct the laser spectral phase. We apply on the Dazzler™ the opposite of the phase measured on the freely operating laser, therefore constraining the final phase to an almost flat level, to within 0.05 rad. We then apply to the Dazzler™ a Gaussian-shaped spectral phase (Figure 2, thin black line). From the SPIDER characterization, we obtain an output phase shown as a thin red line in Fig. 2. The imposed and obtained phases are extremely similar, although not strictly identical in this case. Figure 3 presents the corresponding temporal profiles: the thick dashed line is the FTL pulse, to be compared with the experimental profile in red solid line. The central peak duration is very clearly reduced down to only 19fs FWHM, yielding a normalized duration ratio T = 0.63. This reduction induces the appearance of time satellites, with a temporal Strehl ratio of S_T=0.35 , and a wing intensity ratio of R_T=0.4.

 

Fig. 2. Experimental laser spectral profile (FWHM = 40 nm) (thick line), imposed theoretical spectral phase (black thin line), and experimentally measured spectral phase (red line).

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The experimental phase displayed here was obtained by a manual try-and-error procedure, playing with only three parameters on the imposed Gaussian phase: central position, width, height. Increasing the latter turns out to improve the wing intensity ratio; however, we were experimentally limited by the onset of phase-amplitude coupling effects in the Dazzler™ . We also show as a thin black line in Fig. 3 the profile of a pulse calculated with the experimental spectrum, and the Gaussian phase displayed in Fig. 2, culminating at 4.8 rad. The intensity of the satellites is strongly reduced to only 0.2, even though the temporal Strehl ratio is not noticeably modified. This gives the actual Strehl ratio obtainable with our target phase mask, if artefacts from the phase modulator could be removed.

This improved wing intensity ratio may play an instrumental role to make the temporal superresolution method widely applicable. As for the applications of spatial superresolution to multiphotonic confocal microscopy, non linear effects may be used to reduce dramatically the importance of the satellite peaks. As a first example, let us consider the applicability to high harmonic generation. High harmonics, being generated by a highly non linear effect, have a very steep intensity behaviour; harmonics within the plateau may exhibit an increase with typically the 5^th power of intensity [15]. Moreover, the limiting phenomenon for high harmonics generation is the progressive ionization of the gas medium, which has a severe detrimental effect on phase-matching as soon as the ionization level exceeds a few percent. For fundamental reasons due to the process itself, high harmonics generation and ionization in the tunnel regime have a similarly steep intensity behavior. Such a I⁵ power law applied to satellites with an intensity ratio of 0.2 will reduce the net effect of tunnel ionization rate by the pedestal by more than 3000, to an almost negligible level.

 

Fig. 3. Temporal profiles of the pulse measured with Spider (red line, FWHM = 19 fs) , of the corresponding Fourier limited 30 fs pulse (thick dashed line), and of a pulse calculated from experimental spectra and theoretical phase displayed in Fig. 2 as a black thin line (black thin line).

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Temporal superresolution might be particularly interesting when applied to few cycle pulses. The time duration is so short that the position of the peak of the pulse with respect to the oscillating electric field, characterized by the Carrier-Envelope Offset (CEO) phase, may play an important role for strongly non linear phenomena. Paulus et al. [16] have used such pulses to ionize noble gas atoms, and investigated the emission direction of photoelectrons. While a “sine” pulse (electric field crossing null value at the peak of the pulse) results in an equal distribution of electrons in the polarization plane, a “cosine” pulse (maximal field at the peak of the pulse) results in an asymmetrical distribution: electrons are preferentially emitted along the electric field direction at the peak of the pulse. Paulus et al. have shown experimentally that this asymmetry varies very strongly with the pulse duration. The effect was clear with 6fs pulses, while lengthening the pulse by only 2 femtoseconds (8fs) was sufficient to bring the asymmetry below detection thresholds. Such an effective pulse reduction from 8 to 6 fs is well within reach of temporally superresolved pulses. To evaluate the enhancement of ionization asymmetry with superresolved pulses, one may compute the asymmetry factor R, defined as the ratio between the differential (left minus right) ionization probability, normalized by the total ionization probability. We performed this calculation using standard instantaneous Ammosov-Delone-Krainov rates [17] applied to the ionization of argon atoms, subjected to short cosine laser pulses at a normalized intensity of 3∙10¹³ W/cm².

 

Fig. 4. Ionization asymmetry coefficient as function of spectral width for cosine superresolved pulses (solid line) , FTL pulses (dashed line) and superresolved pulses with Strehl ratio correction (dotted line).

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Figure 4 presents the asymmetry coefficient R calculated for pulses of various spectral widths, both in Fourier Transform Limited conditions, and using the full time evolution of superresolved pulses analogous to that of Fig. 2, ie, for which the phase mask is scaled proportionally to the full width of the spectrum. For all widths, all FTL pulses are seen to result in a much lower asymmetry, except in the saturation regime when the asymmetry approaches 1. The left-hand curve (short dashes) displays the asymmetry coefficient for a superresolved pulse carrying the same energy as the FTL pulse, and for which the maximum intensity is therefore reduced. In all cases, temporal superresolution does mimic shorter pulses for a given spectral width, at the expense of a higher energy required to reach any given intensity.

5. Discussion and conclusions

We have presented here a new method for temporal control of short laser pulses, based on an generalization of the well-known concept of optical superresolution. We have shown both theoretically and experimentally the possibility to shape pulses into a central very narrow pulse, with a full width duration reduced well below the Fourier limit value of Gaussian pulses, at the expense of the creation of satellite pulses.

A host of applications and method variants can be considered. Using as guidelines all the contributions on spatial superresolution, many studies may be initiated to optimize the masks in terms of improved Strehl and wing ratios, for instance by using complex (phase and amplitude) masks [18], to try to determine any fundamental limits of temporal superresolution [5], and to compare the effects of smooth masks, as required by the Dazzler, to those of binary masks, that can produced by phase plates in zero dispersion lines.

The existence of side lobes is naturally a concern. We have stressed that, for applications based on highly nonlinear processes such as high order harmonics generation, the sidelobes will not induce any problems if their intensities are sufficiently low not to cause any significant ionization. Other applications will require however to reduce the wing ratio : laser/solid interactions often require good temporal contrast, and therefore low pre-pulses; conversely, studies of decay dynamics, as probed in a pump probe experiments, may be hindered by the post pulse of superresolved pulses.

Several possibilities can be explored to reduce strongly the influence of those side lobes.

First, Gundu et al. [18] have shown in the spatial domain that mixed amplitude and phase masks can be used to damp side lobes. In an analogous way, efficiently decreasing the wing ratio in the temporal domain can hence be achieved by using simultaneously amplitude and phase spectral masks, at the expense of a decrease of the intensity of principal superresolved peak with respect to Fourier Transform limited pulse (Strehl ratio). In each specific application, one has to find a compromise between high superresolution Strehl ratio and intensity of sidelobes.

Secondly, the pre-pulse and the post-pulse are not necessarily of equal intensity. Indeed, while masks in spatial superresolution are usually rotationally symmetric, spectral masks for temporal superresolution may easily be made asymmetric, in order to decrease strongly either the leading edge or the trailing edge side pulse in a selective way. The large number of degrees of freedom to design superresolution masks make the method very versatile, allowing one to find the optimized masks for each specific application.

Finally, for experiments demanding very high femtosecond contrast, we suggest to combine temporal superresolution with new powerful methods of contrast enhancement: in particular, the suppression of sidelobes by non-linear methods was demonstrated by Cross-Polarized Wave (XPW) generation in a BaF₂ crystal [19]. This method was shown experimentally to increase the contrast ratio by at least four orders of magnitude (40 dB) and is based on a four-wave mixing process governed by anisotropy of the real part of the crystal third-order nonlinearity tensor. Its application to superresolved laser pulses can therefore be expected to decrease dramatically the intensity of the side lobes, down to a level for which their effect remains of little importance on a femtosecond time scale.

The method of temporal superresolution may thus open new exciting possibilities in laser/matter interaction studies at very short times, especially close to the single cycle limit of optical pulses. Indeed, spatial superresolution turns out to be really important for confocal microscopes getting to work close to the wavelength limit, and therefore in conditions in which it is more and more difficult to reduce further the size of the point spread function. The situation is again analogous in the time domain: shorter and shorter pulses can now be generated, down to only few cycles, with however ever increasing difficulties. The relatively easy reduction in FWHM duration for very broad spectrum pulses, along with the recent advent and spread of spectral modulators like the Dazzler or liquid crystal spatial light modulators, will certainly stimulate many applications of temporal superresolution.