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Abstract
Using a streak camera, we directly measure time- and space-resolved dynamics of ${{\mathrm {N}}_{2}}^{+}$ emission from a self-seeded filament. Fluorescence emission does not start with ionization, but with a delay in the tenth of ps range.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser filamentation is a self-guiding phenomenon that occurs in transparent media in which the index of refraction is intensity dependant [1]. Laser filaments are stabilized by the balance between self focusing and defocusing in air leading to the creation of a stable waveguide that can exist for distances exceeding the Rayleigh range [2–4]. Their location can be controlled using temporal focusing [5–7]. Light filaments host enriched spectral contents such as supercontinuum generation and ion emission. Air lasing phenomenon based on the laser filamentation has attracted much attention in the past decade [8,9] and it can be implemented for creation of a high brightness source in atmosphere which has many potential applications including remote detection of gases and atmospheric studies [10–14].
The transitions of the nitrogen cation ${{\mathrm {N}}_{2}}^{+}$ are of particular interest for this application [8] as their emission has been observed in a wide range of pressures and gas mixtures. High gain and fast decay is observed [15–17], with a timescale orders of magnitude shorter than the 67 nanoseconds natural lifetime of the excited state [18]. The mechanism of this gain is still under debate. Some argue that an inversion is created by the ultrashort pulse [19]; others propose a transient gain driven by rotational coherence [20] or lasing without inversion [21] involving coherence in a three level system (the A$^2\Pi _u$ state coupled to the ${{\mathrm {B ^2} \Sigma }_{u}}^{+} \rightarrow {{\mathrm {X ^2} \Sigma }_{g}}^{+}$ transition), and finally some propose that the gain is achieved via superradiance [17,22].
In most experiments, the dynamics of the emission is studied by varying the time delay between an ultrashort “pump” pulse at 800 nm, followed by an ultrashort “probe” pulse which resonantly seeds the single photon transition [15–17,22–25]. This induces measurable changes in plasma radiation depending on the pump-probe delay, from which the temporal dynamics are inferred. However, this inevitably conflates unknown dynamics induced by the probe with the dynamics of interest. In addition, at each delay the reported measurement is integrated over the life time of the emission. In a typical pump probe study the emission spectrum is integrated over the life time of the emission. The probe is known to seed the emission after a certain delay from the ionizing pulse which usually does not create any ion emission by itself. In most cases the time delay scan of probe provides an insight on the wavepacket and plasma dynamics and not the time dependent emission. In some pump-probe studies a cross correlation with a third reference pulse provides a time dependent emission at a particular seed delay. Our streak camera study (which is spatially integrated) can only be compared with the time dependent emission at zero probe delay using a third pulse. A streak camera provides spatial imaging that is not available in pump probe techniques.
A streak camera is used to time-resolve the fluorescence emitted from the side of the plasma, originating from the ${{\mathrm {B ^2} \Sigma }_{u}}^{+} (\nu =0) \rightarrow {{\mathrm {X ^2} \Sigma }_{g}}^{+} (\nu =0,~\nu =1)$ transitions of ${{\mathrm {N}}_{2}}^{+}$. Particular care was taken to achieve an accurate reference for the excitation instant through an original technique detailed in Section 4. After deconvolution for the instrument response, these measurements performed at atmospheric pressure show a delay between excitation and emission of tens of ps for both transitions, in contrast with what is generally assumed. For instance, some suggest an ‘instantaneous’ population inversion mechanism that may be achieved in molecular nitrogen ions at an ultrafast time scale comparable to the 800 nm pump pulse (abstract of Ref. [16]). The interpretation is that “ the population inversion can occur instantly with the ionization of inner-valence electrons” [26].
The measurement technique is described in Section 2. The jitter intrinsic to the streak camera electronic trigger is eliminated by using an optical reference point as detailed in Section 3. Subpicosecond resolution is achieved through a deconvolution technique presented in Section 5. The absolute timing or the cation emission, as well as the fluorescence decay time, are discussed in Section 4.
2. Experimental setup
In our experiment, bandwidth-limited ultrashort pulses of 50 fs duration, 1 mJ energy, at 1 kHz repetition rate centered at 800 nm are focused in air. In a tight focusing geometry (numerical aperture of 0.1), the peak intensity reaches $\approx 10^{14}$ ${{\mathrm {Watt/cm^2}}}$ [27–29] generating a plasma that glows over one centimeter. The emission is self-seeded with the supercontinuum generated in the focused short pulse. Indeed, we observed the supercontinuum ring consisting of colorful rings covering the main beam in the far field.
The streak camera (Optronis 10C) captures emission from the side of the filament, as opposed to integrating the emission along its length [15–17,22–24]. Thus, by making point by point measurements along the propagation direction, we access the “longitudinal emission profile” and can therefore monitor the effects of propagation. It should be mentioned that all the measurements presented in this paper have been taken with a sweep speed of 100 ps and the spacing between pixels corresponds to 2.88 ps.
In this configuration, our measurement is the signal collected from the cross section of the plasma and provides a different aspect of the emission, as opposed to integrating the radiation along the light propagation. Transient gain measurements have typically been performed at low pressures, while our observations are in air at atmospheric pressure (630 torr in Albuquerque). The emission of the plasma is very weak, corresponding to a plasma density of the order of 10$^{17}$ cm$^{-3}$. In addition to being accumulated over a very short distance, the total signal is further decreased by the fact that we attempt to measure a number of photons emitted per picosecond time gate with spatial resolution over the length of the plasma. One of the most difficult challenges to address is to collect as efficiently as possible all the light that is emitted radially. Various focusing geometries have been tested. Best results were obtained with a 10 cm focal distance lens, and by using an objective lens and an achromatic telephoto lens to image the transverse emission onto the slit of the streak camera.
3. Optical synchronization
There are three elements that determine the time resolution of the system:
- 1. The accuracy of positioning the frames with respect to each other.
- 2. The intrinsic resolution of the streak camera.
- 3. The reproducibility of the event under observation.
3.1 Delta function reference point
The streak camera is triggered electronically through the master clock of the laser system, thereby inheriting jitters of the laser electronics. In order to be unaffected by this jitter, each image is accurately timed by using a reference optical beam selected from the same pulse that creates the plasma, reference sent directly to the camera via a fixed path (Fig. 1). The path of the reference beam is such that it always illuminates the same point of the photo-cathode in the time frame of the camera streak, providing a temporal reference for every streak camera image. Mechanical and electronic jitters between frames are corrected by using the timing of the reference pulse in the Matlab reconstruction code [30]. The reference beam was sent directly through an optical port on top of the streak camera, which resulted in an order of magnitude better resolution than when using the fiber provided by the streak camera manufacturer.
Fig. 1. Sketch of the streak camera fluorescence detection. The side emission of the plasma created by the focused main beam (thick red line) is imaged onto the streak camera. A weak beam split off from the main beam (dashed red line) provides a temporal reference spot on each frame of the streak camera. The streak camera slit is parallel to the main beam ($z$ direction) and is used at its minimum possible opening for best time resolution. A typical image taken with infrared filter is presented in the inset. The abscissa is the propagation axis, and the ordinate is time. The image is integrated over the transverse dimension as a function of $z$ (coordinate along the beam). The trace of the plasma radiation is recorded. The reference light (on the left) is used for timing reference and correction of spatial jitter. BP represents the band-pass filter. A nebulizer is used to measure the Rayleigh scattering from the focused beam for a weak non-ionizing pulse.
3.2 Imaging the laser focal spot; time and space reference
The response to a delta function light pulse, obtained by scattering the filament off a ground glass diffuser, is extremely sensitive to the slit opening. In fact, it is only with the slit opened to its minimum (i.e. nearly closed) that the resolution quoted by the manufacturer is achieved. The signal is then so weak that it is reduced to a few scattered dots. In order to achieve the best resolution, it is necessary to accumulate and average between 1000 and 2500 frames. In doing so, the frames have to be synchronized as discussed in the previous paragraph. The intensity of the synchronization pulse should also be such that only 10 to 20 pixels are irradiated per frame.
The algorithm used takes the average of all the synchronization dots, excluding those that are outside of an area equal to 4 times the mean square deviation. This average is taken as reference for all frames. Figure 2 shows an image of a filament scattered off ground glass (1000 frames are averaged). Both images of the reference and that of the scatterer have the same width of 38 ps. This picture indicates that we can define the centroid and the rise time of the temporal sweep with a precision of a few ps.
Fig. 2. The blue area is a typical average of 1000 frames, that captured the reference (turquoise spot in the middle-top of the figure) and the scattering of a diffuser (turquoise spot in the middle left). The ordinate is a spatial coordinate, the abscissa time. The total width of the image is swept in 100 ps. The spacing between pixels corresponds to 2.88 ps. The inserted graphs below reference and scattering spots are space integrated recordings of intensity versus time. For ideal resolution, the reference (top) and the image have the same intensity. Note that they have the same width.
4. Determination of an accurate time origin for temporal measurements
The reference pulse ensures temporal resolution by providing a common origin to the numerous frames being averaged. However, the objective is to measure the temporal profile of the UV emission, with respect to the 800 nm excitation at the same location. The problem is to ascertain with precision that exactly the same spot is being observed in both experiments. Using a diffuser plate — such as was done in the experiment to determine the instrumental resolution — puts one at the mercy of an error in positioning. An accurate time origin can be provided by Rayleigh scattering. However, Rayleigh scattering at 800 nm from air molecules is too weak to be observed (and time resolved) with the streak camera. Since we have achieved accurate timing of each frame with the reference pulse as demonstrated in Fig. 2, we can perform independent measurements to determine the arrival time of the 800 nm pulses.
The solution that we chose is to use Rayleigh scattering enhanced by aerosols. The aerosols are blown through the camera field of view along the path of the focused beam. The challenge here is to create droplets that enhance scattering without creating plasma and/or producing an optical resonance [31] in the droplet. The plasma is avoided by producing droplets of the order of 1 $\mu$m diameter with a nebulizer (mist generator based on Bernoulli principle). These droplets were sufficiently small as not to create any visible plasma or local illumination. The reference point accurately determined by recording Rayleigh scattering of the lowest density mist is shown in Fig. 3.
Fig. 3. (a) In yellow: the reference pulse obtained by sending the low intensity laser beam through a dispersed mist. Green: reference pulse obtained with a solid diffuser surface. The center of gravity of the yellow reference will be used as the time of arrival of the fs pulse. (b) Typical averaged streak camera frame.
5. De-convolution
As it can be seen in Fig. 2 for the selected slit opening of the streak camera the streaks of the fs laser pulse reflected from the diffuser and the reference optical beam extracted from the same fs laser pulse, have the same Full Width at Half Maximum (FWHM) of 40 ps. This width is orders of magnitude larger than the actual duration ($\approx$ 50 fs) of the laser pulse. The 40 ps wide streak in the insets of Fig. 2 can be seen as the $\delta$-function response of the streak camera for the particular slit opening used. The measured time delays and decay times that will be presented in section 6 have the same order of magnitude as this time resolution. The accurate knowledge of the $\delta$-function response enable us to correct the measured response for the instrument response. The measured signal $I_S$ is the convolution of the real signal $S$ and the $\delta$-function response $R$: $I_S=S\circledast R$ [32,33]. By taking the Fourier transform we have: $\mathfrak {F}[I_S]= \mathfrak {F}[S] \times \mathfrak {F}[R]$. Hence, the actual signal can be calculated as: $S= \mathfrak {F}^{-1} [{\mathfrak {F}[I_S]} / {\mathfrak {F}[R]}]$. The Rayleigh scattering pulse of Fig. 3 is used to de-convolute the raw data and extract the true temporal profile of the fluorescence.
6. Time resolved measurements in selected spectral bands
We characterize the emission in three different spectral regions, selected by appropriate combination of filters: $391 \pm 1$ nm (corresponding to the ${{\mathrm {B ^2} \Sigma }_{u}}^{+} (\nu =0) \rightarrow {{\mathrm {X ^2} \Sigma }_{g}}^{+} (\nu =0)$ transition), $428 \pm 1$ nm (corresponding to the ${{\mathrm {B ^2} \Sigma }_{u}}^{+} (\nu =0) \rightarrow {{\mathrm {X ^2} \Sigma }_{g}}^{+} (\nu =1)$ transition), and $1 \mu m >\lambda >750~$nm that covers $X$ to $A$ transition. The time delay associated with each filter is accounted for. Note that the individual rotational transitions cannot be resolved.
Fig. 4. Spatially integrated emission versus time. (a) and (b) show the normalized emission of the nitrogen cation at 391 nm (a) and at 428 nm (b), respectively. The dashed line indicates the center of gravity of the yellow reference, which is the Rayleigh scattering measured separately at low power as shown in Fig. 3(a) The raw data for both (a) and (b) is shown in blue and red respectively, the green dotted curve is the fit to data in proximity of the peak. The de-convoluted fast emission of 391 and 428 is calculated using the instrument response function of the Rayleigh scattering of the femtosecond pulse and is shown with black curve. (c) The 391 nm and 428 nm emission and broadband IR emission ($\lambda >$ 750 nm) are plotted without normalization. There are two time constants for the ion emission known as “fast” and “slow” decay [25,34].
The time dependence of the emission (integrated in space) of the plasma created by a focused 1 mJ pulse, is presented in the plots of Fig. 4. The origin $t=0$ is defined as the time of the Rayleigh scattering. Both raw and de-convoluted temporal emission profiles are plotted. The de-convoluted emission lines exhibit characteristic delays of 33 $\pm 0.3$ ps at 391 nm (Fig. 4(a)) and 25 $\pm 0.3~$ps for 428 nm (Fig. 4(b)). Streak camera measurements performed with different experimental techniques that are made along the filament report a 5 ps delay between excitation and emission, which they attribute to the narrowness of the emission line [35]. In side emission measurements performed with streak camera by others, the emission is seen to start at the same time as the excitation [34,36]. Unlike the Rayleigh scattering the emission profiles are asymmetric and two decay regimes are present in them; a “fast” decay regime, which is followed by a “slow” exponentially decaying tail. They exhibit short temporal widths of 14 $\pm$ 0.5 ps at 391 nm and 10 $\pm$ 0.5 ps at 428 nm for the fast decay regime. De-convolution of time dependent signals requires an accurate knowledge of the instrument response function. Estimating a 5% accuracy in the Rayleigh signal, we calculate that a 5% increase in the instrument response changes the fast decay of the 391 and 428 nm signals by 0.5 ps, leaving the rise time of the emission unchanged. The position of the peak is unchanged, confirming that the de-convolution process does not affect the time delay. To estimate the precision in the delay measurement, we take as error bar for the delay which is the width of the de-convoluted signal at 1% from its peak, or $\pm$ 0.3 ps.
Using a streak camera “fast” and “slow” decays for the 391 nm emissions at much lower than atmospheric pressures have been observed [34]. In [34] the authors claim that the “fast” decay originates from the collision between the ${{\mathrm {N}}_{2}}^{+}$ and free electrons. In this paper we have investigated simultaneously both the delay times between the ionizing laser filament and emissions, and the decay dynamics, which leads to a better understanding of the underlying phenomenon. As it is discussed in details below we believe that our observations suggest the involvement of Dicke superfluorescence [37] rather than attributing the delay to the narrowness of the emission line [35] or invoking the role of electron-cation collisions in fast decay mechanism [34].
Infrared emission at $\lambda >$ 750 nm (green curve) plotted in Fig. 4(c), has zero delay with respect to the Rayleigh scattering and decays exponentially with a time constant of 700 ps. It has a similar decay behavior as seen in both 391 and 428 nm emission profiles, suggesting the involvement of the $A$ state in the dynamics [19]. The population exchange between $X$ and $A$ states affects the gain at the $B$ to $X$ transition wavelength. Such a long exponential tail has to be associated with N$_2^+$, since it is not observed at low intensity as shown in Fig. 3(a). We can estimate this slow decay assuming it arises from population decay due to electron-cation collisions as $1/\rho v\sigma$, where $\rho$ is the density of emitting dipoles, $v$ is the velocity of electrons and $\sigma$ is the collision cross section. $\rho = 0.01 \rho _{air}$ based on an intensity of $\approx 10^{14}$ Watt/cm$^2$ [27–29] which yields an ionization rate [38] of one percent. The average energy of released electrons is 0.1 eV with the total electron cross section [39] of $5 \times 10^{-16}$ cm$^2$. This yields a decay time of 700 picoseconds, in good agreement with the slow decay of the measured ion emission as well as the infrared emission in Fig. 4.
Dicke superfluorescence predicts that these delay and decay times should be inversely proportional to the number of dipole emitters involved in the emission [37], which in turn is proportional to the time integral of the non-normalized emission profiles [Fig. 4(c)]. We find the ratio of the delay time for 391 nm emission to that of 428 nm to be $1.32\pm 0.02$, while the ratio of temporal widths is $1.4\pm 0.09$. Indeed, both these numbers are in close agreement with $1.4\pm 0.5$, the ratio of number of emitters in the 428 nm to 391 nm emission profiles, obtained from their time integral. Note that this collective emission does not require an inverted sample [40]. The higher gain of the 428 nm fluorescence can be due to higher inversion of the (${{\mathrm {B ^2} \Sigma }_{u}}^{+} \nu =0 \rightarrow {{\mathrm {X ^2} \Sigma }_{g}}^{+} \nu =1$) transition due to unequal occupation of the ${{\mathrm {X ^2} \Sigma }_{g}}^{+}$ vibrational states following strong-field ionization [41].
In order to further investigate the collective contribution of emission, we analyze the spatial properties of the measured fluorescence. The emission is divided in three equal slices along the propagation direction as shown in the inset of Fig. 5. The emission is integrated in the radial direction over each section and plotted as a function of streak camera sweep. Figure 5 shows the spectral emission with a long pass filter and the emission at 391 nm is represented in Fig. 6. The integrated signal in both IR and 391 nm is strongest in the central region of the plasma labeled as “B”, followed by “C” and “A”. The normalized emission in the IR shows that all regions have a comparable time dependent profile and no observable delay between regions could be measured in our system. IR emissions from the selected regions exhibit their maximum at the Rayleigh peak. However a measurable difference is recorded for the peak emission at 391 nm. The brightest region of the plasma has the least delay with respect to the Rayleigh peak, with a delay inversely proportional to the strength of the signal. The width of the emission increases as the total signal decreases. The 3 to 4.5 ps relative delay between emission originating from the center and sides of the plasma is much longer than the time it would take for light to travel between the sub-regions. We show that density of the emitters not only affect the strength of the signal but also the temporal profile of the emission. Altogether these observations suggest that it is important to include collective effects in modelling gain and propagation in air-lasing.
Fig. 5. Normalized integrated IR emission from selected spatial regions of the plasma labeled as “A” , “B”, and “C” as shown in the legend. The inset shows the relative intensity along the filament for different slices.
Fig. 6. Comparing fast decay and built up time of the emission at 391 nm from the selected spatial regions of the plasma labeled as “A” , “B”, and “C” as shown in the legend.
7. Conclusion
Our unique capability of sub-picosecond timing resolution along with absolute timing measurement using the Rayleigh scattering enables us to report accurate measurement in timing and temporal profile of the emission from (${{\mathrm {B ^2} \Sigma }_{u}}^{+}$) to (${{\mathrm {X ^2} \Sigma }_{g}}^{+} \nu =0~\&~ \nu =1$) in $N_2^+$. It has been observed that these emissions have a delay with respect to the ionizing laser filament. Our study suggests that the fast decay and high peak gain observed in air lasing is due to collective emission introduced by Dicke in 1954 [37]. The temporal behaviour of the pulse as a function of density agrees with the predictions of superfluorescence [17,42]. The presence of a long tail emission at infrared wavelength with no observable delay with respect to the initial pulse suggests the influence of $X$ to $A$ coupling in air lasing. We emphasize that we observe this superfluorescence in air under ambient conditions, making it suitable for practical applications. Our study suggests that collective emission and spatial distribution of the plasma needs to be considered for applications in remote source design using $N_2^+$ as a gain medium.
Funding
Air Force Research Laboratory (FA9451-15-1-0039); Army Research Office (W911NF-19-1-0272).
Acknowledgments
We would like to acknowledge Andreas Velten, Paul Corkum, Michael Spanner, David Villeneuve, Mathew Britton and Andreas Schmitt-Sody for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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