## Abstract

We present an all optical approach to measure the value of the carrier-envelope phase (CEP) of a short intense laser pulse. This method relies on photo-ionization of gases with a guided laser beam. This approach that provides the absolute value of the CEP, is compatible with single shot characterization, is scalable in wavelength, does not suffer from bandwidth limitation and is largely intensity independent. It has also the potential to provide a full characterization of the pulse profile via high order autocorrelation on a single shot basis.

© 2014 Optical Society of America

## 1. Introduction

With the appearance of attoscience and the development of strong field physics based on few cycle pulses, the carrier-envelope phase (CEP) [1] of a pulse is often a crucial parameter. Most of the CEP measurement techniques are based on the second harmonic generation [2–6] or photoelectron spectra detection [7–9]. They usually provide relative CEP values. One of the approaches to measure the absolute CEP value was suggested and studied theoretically in [10]; it is based on measuring the asymmetry in the left-right photoelectron spectra. However, in practice highly sophisticated setups are required to provide the exact value of the CEP [11,12].

In this paper we present and numerically investigate a new method to extract the CEP of a few-cycle laser pulse. The method is based on the extreme sensitivity of the ionization rate on the instantaneous field of a short pulse and relies on propagation in a guided geometry (capillary) to change the CEP of the pulse and consecutively the laser field. For a given intensity the peak field strength depends on the CEP: it is maximal for a cos-like pulse and minimal for a sin-like one. The difference is small even for a few-cycle laser pulse, but the extreme sensitivity of the tunneling ionization rate on the field strength leads to a measurable dependence of the ionization probability on the CEP (even though for the cos-like pulse there is one (dominating) ionization event, while for the sin-like there are two). Guided propagation of a pulse in a capillary provides periodic CEP variation, which leads to the ionization degree modulation when a low pressure gas is injected in the capillary. Measuring the spatial evolution of this modulation allows measuring the incident pulse CEP. When the laser is guided in a transparent glass capillary, this ionization rate can be optically observed via the fluorescence emitted by the ionized gas as it recombines. This approach is well adapted to few cycle pulses and provides a direct measurement of their CEP (modulo π). This measure provides the real value of the CEP under the only assumption (justified in the tunneling regime) that the ionization rate is maximum for a cosine pulse (CEP = 0) while most of the other technique provide only relative CEP values. The sensitivity of this approach can even be improved by using the laser pulse in conjunction with its second harmonic. This allows CEP measurement for longer pulses.

In our studies the laser pulse propagation in the EH_{11} mode of a capillary [13] is simulated and ionization probability is calculated as a function of the propagation distance. We show how to extract the pulse CEP value and find maximal pulse durations for which this method is valid.

The experimental realization of this approach would require to guide the ultrashort pulse in a glass capillary filled with low pressure Helium (or other gases depending on the laser intensity and wavelength) and to observe transversally the emitted fluorescence with spatial resolution (by imaging the fluorescence channel onto a CCD camera for instance). In a guided geometry, intensity variation that occurs because of propagation losses, can be directly characterized by measuring the fluorescence after averaging over all the CEP values.

## 2. Theoretical methods

#### 2.1 Pulse propagation

Simulating the pulse propagation we present the field at the entrance to the capillary as:

where*r*is the radial coordinate as here a cylindrical symmetry is assumed.

The temporal structure of this field is defined as:

We assume that the spatial structure of the field in the capillary is described by EH_{11} mode. Note that even if initially there are high order modes in this distribution (say, due to weak coupling of the beam in the capillary), they can disappear after some propagation distance because they have higher attenuation constant than EH_{11} mode [13]. Thus the transverse distribution of the field in Eq. (1) is given by

Propagation induces a dispersion on the pulse. This negative dispersion leads to a pulse broadening and, since phase and group velocities are different, CEP continuously shifts during propagation. Having in mind the broadbandness of the few-cycle laser pulse we describe the field modification while propagation taking into account all dispersion orders. To do this we calculate the field spectrum $S(\omega ,z=0)$ at the capillary entrance and write the spectrum after propagation as

Finally, we calculate the inverse Fourier transform $E(t,z)$ of the propagated spectrum $S(\omega ,z)$ and find the full laser field after propagation as

Note, that for the parameters considered in the paper the attenuation is relatively low, i.e. the ionization signal disappears mainly due to the pulse temporal broadening.

For our studies, an important value is the distance, ${L}_{CEP}$, over which propagation provides a π CEP variation. Taking into account only the first order dispersion of the waveguide (and neglecting the low pressure gas dispersion), this distance can be approximately found from the difference of the group and phase velocity:

where ${\lambda}_{0}$ is the central wavelength of the laser pulse. To choose the appropriate capillary radius we express it in terms of ${L}_{CEP}$:The distance of the pulse spreading is (see, for instance the textbook [14]):

where ${v}_{g}$ is a group velocity. For the ratio of the spreading distance and the $2\pi $ CEP variation distance we find:Note, that $\frac{{\omega}_{0}\tau}{2\pi}$ is a number of the cycles in the pulse. Thus the latter equation shows that even for a few-cycle pulse there are several CEP oscillations at the spreading distance.

#### 2.2 Photoionization

The suggested method of the CEP measurement demands strong and monotonic dependence of the ionization rate on the field strength. The tunnel ionization satisfies this requirement very well, whereas multiphoton ionization does not provide monotonic dependence (see, for instance, numerical studies [15]). So below in this paper the parameters are chosen to provide tunnel ionization of the gas in the capillary.

The instantaneous ionization rate is calculated as an ionization rate in a static field which value is equal to the instantaneous laser field strength. To find the total ionization probability this rate is integrated over the laser pulse. Such quasistatic approximation is applicable in the tunneling limit. Obviously, the ionization probability found with this approach is CEP dependent. The static field tunneling ionization rate for a gas with the ionization potential I_{p} is calculated using expression from the textbook [16] (in atomic units):

Note, that this formula agrees very well with numerical TDSE solution results [15,17] in the intensity range used in our paper (about 10^{14} - 10^{15} W/cm^{2}).

To mimic the ionization throughout the capillary, it is also necessary to consider the radial intensity profile of the guided beam. Below in this paper we present the ionization degree averaged over the capillary radius to take into account this radial intensity profile:

## 3. Calculation results

The ionization rate given by Eq. (11) saturates with the normalized field strength, so we have to use relatively low values of this parameter to provide strong dependence of the ionization rate on the CEP. On the other hand, the ionization should occur in the tunneling regime, i.e. the Keldysh parameter

should be sufficiently less than unity (here ${E}_{0}$ is the laser pulse peak field). Both requirements (low*F*and low $\gamma $) can be satisfied for low ${\omega}_{0}$ and high

*I*. Below we consider cases of low ${\omega}_{0}$ (few-micron radiation ionizing Ar atoms) and high

_{p}*I*(800 nm radiation ionizing He

_{p}^{+}ions), correspondingly. In both cases, we consider the peak laser intensity corresponding to $\gamma =0.5$. The pressure of the medium filling the capillary is supposed to be low enough in such a way linear and non-linear refraction of the medium does not influence the results. The capillary radius is chosen to provide ${L}_{CEP}=2$mm as this length is easy to observe experimentally by imaging onto a CCD.

In Fig. 1 we show the ionization degree calculated using
Eq. (13) for pulses with Gaussian envelope and
different initial CEPs and durations (FWHM durations of intensity envelope are presented in the
figure) for a 3 µm pulse central wavelength. We observe clear spatial modulations of the medium ionization. Comparing results for sin-
and cos-like incident fields, we see that they provide antiphase ionization spatial modulation.
We observe that the cos pulse leads to a maximum ionization at the entrance of the capillary,
while the sin pulse provides a maximum only after a propagation over L_{cep}/2 that
changes the incoming pulse in a cosine pulse. Thus, the incident CEP of the pulse can be found
from the phase of ionization degree modulation. Note, that the phase of the oscillations does
not depend on the incident pulse intensity and only the amplitude is affected by this intensity
(because Eq. (11) provides monotonic dependence
of the ionization probability on the field strength). Thus, the method is applicable for
experimental measurement of the CEP of sequential laser pulses even in presence of intensity
fluctuations.

Figure 1 also shows that the depth of the ionization modulation decreases with the pulse duration increase. When the pulse duration varies from 15 fs to 20 fs (from 1.5 to 2.0 cycles at 3 µm), the ionization modulation depth decreases from 26% to 2.7% but both values can easily be observed experimentally.

Considering 800 nm radiation, we use He^{+} as an ionizing medium, 1.8 10^{15}
W/cm^{2} peak intensity (providing $\gamma =0.5$) and 22 µm capillary radius (providing
${L}_{CEP}=2$mm). We find the behavior of the ionization degree very similar to
the one presented in Fig. 1. When the pulse duration
increases from 4 fs (1.5 cycles) to 5.0 fs (1.9 cycles), the ionization modulation depth
decreases from 16% to 3%. Thus, in this case the upper limit of the pulse duration (measured in
optical period) for which the method is applicable is slightly lower than for the few-micron
radiation. The lower efficiency of the method for 800 nm wavelength is due to higher field
values used in this case to provide tunneling ionization, leading to lower sensitivity of
ionization on the field strength. Assuming that 1% modulation depth is a limit of its
experimental measurement we find the maximal pulse durations up to which our method is
applicable. The pulse durations calculated for the wavelengths 3 µm, 2 µm, 800nm
are presented in Table 1. Moreover, using a high dynamics (16 bit) camera and post processing by Fourier
transform to extract the oscillatory part, one can assume the 0.1% modulation depth as a
reasonably conservative limit of this technique; reducing the observable modulation depth
increases of the maximum pulse duration (see Table 1)
that can be characterized with this approach.

In the following calculations, we consider the 3 µm radiation propagating in 43 µm radius capillary filled with Argon; its incident intensity is 3.7 10^{13} W/cm^{2} corresponding to $\gamma =0.5$.

As mentioned in the previous section, the number of oscillations of the ionization degree observed before ionization vanishes because of the pulse spreads characterizes the minimum pulse duration. For instance, in Fig. 1, one sees approximately 4.5 oscillations before the ionization degree achieves its half level for a 35 fs pulse and 6.5 oscillations for a 40 fs pulse. Thus our method allows accessing the pulse duration measurement in conjunction with the CEP measurement.

This approach, sensitive to the electric field, can be compared to a new autocorrelation approach based on chirp control [6] that provides the pulse intensity profile by observing the impact of chirp on its doubling. Here we use a higher non linear order and CEP oscillations provide a self calibration of the chirp and the capillary characteristics. Moreover, it also provides access to the chirp of the pulse as shown in the following.

We write the spectrum of the chirped pulse at the capillary input (*z* = 0) using
the spectrum of the Fourier-limited one ${S}^{F-l}(\omega ,0)$ as

This approach can be used to characterize pulses with a limited number of optical cycles but gets less efficient for longer pulses as the integrated ionization rate gets insensitive to CEP when the pulse has too many cycles. To reduce artificially the number of optical cycles contributing to ionization, it is also possible to combine the fundamental pulse with its second harmonic (SH). Combining a laser field and its relatively weak second harmonic with identical polarization and a proper relative phase allows suppression of the ionization at every second laser half-cycle (when the fundamental and the SH instantaneous fields are antiparallel) and enhancement of the ionization at every next half-cycle (when they are parallel). Thus addition of the SH field could enhance the dependence of the ionization probability on the fundamental CEP. However, in the presence of the SH the ionization probability depends also on the relative phase between the fundamental and the second harmonic. This phase varies with propagation and the behavior of the ionization degree gets more complicated; generally, this behavior reflects the competition between influences of these two phases. If both fields propagate in the capillary in the EH_{11} mode, the distance of the $2\pi $ relative phase variation is:

In Fig. 3 we present ionization degree calculated for the joint propagation of the fundamental and the SH. One can see that in this case the ionization degree does not oscillate in antiphase for the sin- and cos-like pulses. This is because the oscillation is mainly due to the relative phase variation (although we optimize the peak SH intensity to 0.6% of the fundamental intensity to emphasize the oscillation due to the CEP variation). Moreover, the lengths ${L}_{rel}^{11-11}$ and ${L}_{CEP}$ are close to each other (see Eq. (15)), which results in the beatings for the upper curves in Fig. 3. Note however that the asymmetry of the beatings is an heterodyne way to get the CEP and not only its absolute value and therefore to measure the exact value of the pulse CEP modulo 2π (and not this value modulo π).

If the SH propagates in the EH_{12} mode, the distance of the $2\pi $ relative phase variation is:

For both cases of propagation of the SH, the difference between sin- and cos-like fundamental pulses is nevertheless nicely seen in the spatial spectra of the ionization degree presented in Figs. 4(a) and 4(b) after Fourier transforming the spatial evolution of the signal . Indeed, in these figures the strong right peak almost does not depend on the CEP (it is just due to the dephasing between the two fields), but the phase of the weaker left peak is directly linked to the CEP of the input pulse. This phase can be extracted by filtering out the main peak and back Fourier transforming. Assuming that the phase variation of the latter peak can be measured experimentally until the modulation depth at this frequency is 1% we find the upper limits of the pulse duration, which CEP can be measured with this method presented in Table 1.

Currently another important criterion is the accuracy with which the CEP is measured as it impacts the measurements in metrology. As here the CEP measurement is performed by measuring the position of some fringes, it is possible to extract it with a high accuracy limited by the number of fringes that can be observed. We anticipate that this optical approach can improve the CEP control necessary for metrology and strong field physics.

## 4. Discussion and conclusions

Throughout this paper we do not consider the creation of the excited states of atoms or ions during the laser pulse. The fluorescence from such excited states might provide some background for the measurement. However, this background should be low because the excitation is less probable than the ionization in the tunneling regime. Namely, the excitation of the neutral atoms measured in [18] is typically order of magnitude less probable than the ionization, and the excitation of the ions due to shakeup process is several orders of magnitude less probable [19]. Moreover, this background signal could be further suppressed if we record only the fluorescence after a given delay since fluorescence from the laser created excited states is much faster than the recombining one.

Our technique is based on the measurement of the fluorescence due to recombination of the photoelectrons and fluorescence of the states populated by these recombining electrons. Application of our method requires some limitations for the gas pressure. Its upper limit originates from the assumption that the gas dispersion and the pulse self action are negligible (if this is not the case, the method is still applicable, but the interpretation might be more complicated, see below). In the recent study [20] it was shown that the propagation of a laser pulse (with intensity much higher than the ones considered here) in a capillary with 1 mbar of helium at a distance of 40 cm does barely provides self action by broadening the pulse spectrum. So for the propagation over centimeters required in our method this limit corresponds to pressure of tens of mbars. The lower limit of the pressure comes from the requirement that the fluorescence pattern due to recombination is an image of the ionization pattern. This should imply that electron and ion recombine at the place where they were created and that the recombination distance should be less than the typical size of the fluorescence pattern. In fact recombination distances longer than the typical size of the ionization pattern are still allowable as the population of ions follows this pattern and wherever the electron comes recombination can only occur where ions where created; nevertheless signal coming from electron hitting the wall of the capillary should be avoided (it can be filtered with spectral filters) and to fulfill this criteria a short recombination distance is preferred. Experimentally, focusing a pulse in a cell with a 1 mbar pressure we can clearly see the fluorescence pattern tracing the beam shape near the focus. Note, that ionization probabilities of 10^{−4}-10^{−6} found in our calculations provide enough photoelectrons to detect the signal for such pressures. Thus, there is a gap between the lower and the upper pressure limits, and the gas pressure of about 1-10 mbars appears to be the optimal one.

Thus, in this paper we have described an all optical setup suggested to measure the CEP of short pulses and we have shown that this measurement is possible even by considering a limited set of parameters. Further improvements are however possible by slight modifications of this approach. The number of oscillations of the ionization degree is limited with the pulse spreading which reduces the contrast of the oscillations; this number is given by Eq. (10) if the dispersion originates only from the propagation in a capillary. However, if the gas pressure is higher and the gas dispersion is comparable with the waveguide dispersion, they can provide zero group velocity. Practically this is hardly possible with the usual capillary because it requires too high gas pressure or very large capillary radius which provides too large inter-fringes distance ${L}_{CEP}$. However, such propagation can be achieved in a photonic crystal wave-guide. Note, that in the case of higher gas pressure the pulse propagation can be studied via numerical approach [21].

In conclusion, we have described a new approach that can be used to fully characterize the field of an intense ultrashort pulse. This approach gives the value of the CEP with high accuracy, its sign and also the chirp of the pulse. It is well fitted to characterize few cycle pulses. Using weak second harmonic co-propagating in the capillary allows measuring CEP for almost twice longer pulses. This method is fully optical and can be adapted to several wavelength ranges or even improved by considering other guided geometry

## Acknowledgments

We acknowledge financial supports from the CNRS (Pics n°PICS06038), the European community (Laserlab program Inrex and Eurolite), the ANR (Attowave project), the region Aquitaine (Nasa project), Russian Foundation for Basic Research (12-02-91059-NCNI-a, 12-02-00627-a), and the Dynasty Foundation.

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