All-optical characterization of the two-dimensional waveform and the Gouy phase of an infrared pulse based on plasma fluorescence of gas

Nariyuki Saito; Nobuhisa Ishii; Teruto Kanai; Jiro Itatani

doi:10.1364/OE.26.024591

1. Introduction

Waveform-controlled laser pulses and the complete temporal characterization of these pulses have played central roles in the recent progress of attosecond science, where the waveforms of few-cycle pulses rather than their envelopes determine the electron dynamics in gases and solids [1,2]. Various waveform characterization techniques have been developed, including attosecond streaking [3,4], petahertz optical oscilloscope [5], attosecond resolved interferometric electric-field sampling (ARIES) [6], extreme ultraviolet (XUV) spatial interferometry [7], and frequency-resolved optical gating capable of carrier-envelope phase determination (FROG-CEP) [8]. However, these techniques require either XUV optics or numerical retrieval procedures, which complicate the measurements of waveforms. Recently, a technique called TIPTOE (tunneling ionization with a perturbation for the time-domain observation of an electric field) has been demonstrated to solve this problem [9]. In TIPTOE, a waveform including an absolute CEP is measured using an ion current from tunnel-ionized gas atoms or molecules. Because a strong-field-induced electron excitation and the resulting transient current intrinsically occur on the sub-cycle time scales [10,11], they can be utilized as probes to measure femtosecond oscillation of an optical field. In this approach, the ion current can be collected without a vacuum setup and the waveform can be directly obtained without a numerical retrieval procedure. Although a similar waveform measurement technique based on the third-order nonlinearity has been demonstrated, it cannot measure an absolute CEP [12].

In this paper, we extend the TIPTOE scheme to resolve more degrees of freedom in optical waveforms: two-dimensional polarization and the Gouy phase. In the previous study of TIPTOE [9], linearly polarized waveforms are measured at a specific point in space. However, laser pulses can have arbitrary polarization and their waveforms may be modified by the Gouy phase during propagation. These aspects have attracted significant attention in attosecond science. For example, optical pulses with two-dimensional polarization have been used for high-harmonic generation (HHG) [13–15] and above-threshold ionization (ATI) [16]. The Gouy phase also plays critical roles in ATI [17], photoemission from solids [18], phase matching in HHG [19], and attosecond pump-probe experiments [20,21]. In this context, there is a high demand for methods to characterize two-dimensional optical waveforms and the Gouy phase. Although characterization of two-dimensional waveforms and the Gouy phase for few-cycle pulses has been demonstrated [7,17,18,20,21], our method is based on an all-optical technique with a relatively simple setup.

In our experiment, instead of collecting the ion current [9], we measure plasma fluorescence, which is found to be proportional to the ionization probability. This all-optical technique is used to reveal the spatial distribution of the ionization probability, and thus extract the Gouy phase. Moreover, there is no need to place any apparatus near the interaction region because the plasma fluorescence can be observed from outside the chamber in which the experiment is conducted. Thus, the proposed technique will facilitate powerful and convenient waveform diagnostics for various experiments in strong-field physics and attosecond science.

2. Working principle

2.1 Generalized TIPTOE method and reconstruction of two-dimensional waveforms

The basic principle underlying the measurements in this study is the same as that in [9]. Here, we review the measurement scheme and generalize it to the two-dimensional case. In TIPTOE, a strong fundamental pulse ionizes an atom or molecule while a weak signal pulse slightly modulates the ionization probability, which can be measured using the ion current. Let $E_{F} (t)$ and $E_{S} (t)$ be the electric field vectors of the strong fundamental pulse and the weak signal pulse, respectively, and $w (E)$ be the tunnel ionization rate at a static field amplitude E. Note that the fundamental pulse is linearly polarized and its envelope has a peak at $t = 0$ ; also note that the signal pulse has an arbitrary polarization. The ionization probability $P (Δ t)$ under irradiation by the fundamental and signal pulses with a delay of $Δ t$ can be written as

\begin{matrix} P (Δ t) = \int w (| E_{F} (t) + E_{S} (t + Δ t) |) d t \\ ≅ \int {\frac{d w}{d E} |}_{E = | E_{F} (t) |} \frac{E_{F} (t) \cdot E_{S} (t + Δ t)}{| E_{F} (t) |} d t + const . \\ = \int {\frac{d w}{d E} |}_{E = | E_{F} (t) |} E_{S p r o j} (t + Δ t) d t + c o n s t . \end{matrix}

where

E_{S p r o j}

is the projection of the signal electric field along the polarization direction of the fundamental pulse. We assume that the signal pulse is sufficiently weaker than the fundamental pulse and extract only the first-order term of the signal pulse. Furthermore, we ignore the depletion of the ground state.

Tunnel ionization is highly nonlinear and mostly confined to the regions in which $| E_{F} (t) |$ becomes maximal (i.e., the peaks in the fundamental field occurring every half cycle). Therefore, if the carrier wave of the fundamental electric field is proportional to $\cos (ω t)$ , the ionization probability can further be approximated as

P (Δ t) ≅ \sum_{n} a_{n} E_{S p r o j} (Δ t + \frac{n π}{ω}) + const .

where

a_{n}

is a constant representing the derivative of the ionization rate with its sign alternating between adjacent half cycles. If the fundamental pulse is short enough and its waveform is cosine-like, the component at n=0 becomes dominant in the summation in Eq. (2), and

E_{S p r o j} (Δ t)

becomes proportional to the modulation of the ionization probability

P (Δ t)

. Therefore, the signal waveform projected along the direction parallel to the fundamental polarization can be measured by scanning the delay

Δ t

. Then, by measuring the two orthogonal polarization components of the signal waveform, the complete two-dimensional signal waveform can be reconstructed.

To verify the validity of this approach, a simulation of a two-dimensional waveform measurement is performed, as shown in Fig. 1. The fundamental pulse used in the simulation is similar to that used in our experiment: a Gaussian pulse with a central wavelength of 1600 nm, a peak electric field of 310 MV/cm (1.28 x 10¹⁴ W/cm²), and a pulse duration of 11 fs. First, the time evolution of the ionization probability of a nitrogen molecule irradiated by only the fundamental pulse is calculated (Fig. 1(a)). In the calculation, the molecular ADK model [22] for a randomly oriented nitrogen molecule is used. The use of the ADK model is justified by the fact that the Keldysh parameter γ is sufficiently small (γ~0.5). Figure 1(a) shows that the ionization is mostly confined to the central part of the pulse, and thus acts as an attosecond temporal gate. Next, a two-dimensional signal pulse is superimposed on the fundamental pulse and the ionization modulation is simulated. The central wavelength and temporal duration of the signal pulse are the same as those of the fundamental pulse; however, the peak electric field amplitude is reduced to 3 MV/cm (1.20 x 10¹⁰ W/cm²) and linear chirps are added to the two polarization components to create a complex polarization shape. The reconstructed waveform (blue circles) and original waveform (red curve) shown in Fig. 1(b) agree well, confirming that the proposed technique can be used to measure the two-dimensional waveform of few-cycle pulses.

Fig. 1 Simulation of the two-dimensional waveform measurement. (a) Electric field of the fundamental pulse (black dashed curve) and the ionization probability of a nitrogen molecule irradiated by the fundamental pulse (red curve). (b) Reconstructed two-dimensional signal waveform (blue circles) and the original signal waveform (red curve); the projections of the reconstructed and original waveforms in the vertical and horizontal directions are shown as gray circles and curves, respectively.

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2.2 Retrieval of the Gouy phase

Next, we show that the Gouy phase of a laser beam can be retrieved from the spatial distribution of the ionization probability. We assume that the fundamental and the signal pulses have the same central frequency $ω$ , and are linearly polarized along the same direction; they are expressed as $E_{F} (z, t) = E_{F 0} (z, t) \cos (ω t + φ_{F} (z))$ and $E_{S} (z, t) = E_{S 0} (z, t) \cos (ω t + φ_{S} (z))$ , respectively, where z is the position along the laser propagation axis, $E_{F 0} (z, t)$ and $E_{S 0} (z, t)$ are the respective pulse envelopes, and $φ_{F} (z)$ and $φ_{S} (z)$ are the respective CEPs. Note that the CEPs depend on z due to the Gouy phase. In this case, the ionization probability $P (z, Δ t)$ can be written as

\begin{matrix} P (z, Δ t) ≅ \sum_{n} a_{n} E_{S 0} (z, Δ t + \frac{n π - φ_{F} (z)}{ω}) \cos (ω (Δ t + \frac{n π - φ_{F} (z)}{ω}) + φ_{S} (z)) + const . \\ = \cos (ω Δ t + φ_{S} (z) - φ_{F} (z)) \sum_{n} | a_{n} | E_{S 0} (z, Δ t + \frac{n π - φ_{F} (z)}{ω}) + c o n s t . \end{matrix}

Because the summation on the right-hand side of Eq. (3) varies slowly with

Δ t

, the oscillation phase of the ionization modulation is determined by

\cos (ω Δ t + φ_{S} (z) - φ_{F} (z)) .

Thus, by recording the ionization probability at each position along the laser propagation axis and extracting its oscillation phase, the relative Gouy phase

φ_{S} (z) - φ_{F} (z)

can be retrieved. Furthermore, if the signal pulse is collimated (i.e.,

φ_{S} (z) = const .

), the Gouy phase of the fundamental pulse alone can be extracted. Different from the waveform measurement, the Gouy phase can be retrieved even if the fundamental pulse is not a cosine-like pulse (see Eq. (3)).

To confirm that the oscillation phase of the ionization modulation reflects the relative phase between the fundamental and signal fields, a numerical simulation is performed. The intensity and the envelope shape of the fundamental electric field used in the simulation is the same as that used in the previous section, and its CEP $φ_{F}$ is varied from 0 to 2π rad. The signal electric field has a wavelength of 1600 nm, a pulse duration of 10 fs, and is linearly chirped as shown by the black curve in Fig. 2(a). Its CEP is fixed to cosine-like (i.e., $φ_{S} (z) = 0$ ). The peak electric field amplitudes of the fundamental and signal pulses are 310 MV/cm (1.28 x 10¹⁴ W/cm²) and 3 MV/cm (1.20 x 10¹⁰ W/cm²), respectively. As in the previous simulation, the molecular ADK model for a randomly oriented nitrogen molecule is employed. The simulated ionization modulation for various fundamental pulse’s CEPs are shown as red, green, yellow, and blue curves in Fig. 2(a). A clear phase shift can be seen in the ionization modulations. Fourier transform is applied to them to extract the phase at the central frequency, and thus extract the oscillation phase. The result is shown in Fig. 2(b) (blue circles). The phase shift and the fundamental pulse’s CEP coincide perfectly, indicating that the relative phase change between the fundamental and signal waveforms can be retrieved by the proposed technique.

Fig. 2 Simulation of the retrieval of the relative phase shift between the fundamental and signal fields. (a) Original electric field of the signal pulse (black curve) and the ionization modulations with various fundamental pulse’s CEPs (i.e., different $φ_{F}$ values) (red, green, yellow and blue curves). (b) Phase shift retrieved from the ionization modulation as a function of the fundamental pulse’s CEP $φ_{F}$ (blue circles). The black dashed line serves as a guide for the eye, by connecting (0, 0) and (2, 2).

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3. Experimental setup

In the experiment, we employ infrared (IR) pulses from a BiB₃O₆-based optical parametric chirped-pulse amplifier (1600 nm, 11 fs, 1.5 mJ, 1 kHz) [23]. Figure 3(a) shows a schematic of the setup. The IR pulses are split into two arms by a hole-drilled mirror. The reflected beam serves as the fundamental beam. It passes through a half-wave plate (HWP, Thorlabs AHWP10M-1600), variable neutral density filter (VND), and 2-mm-thick fused silica plate for dispersion compensation. The transmitted beam is the signal beam. It is passed through a pair of fused silica wedges for CEP control, VND, quarter-wave plate (QWP, Thorlabs AQWP10M-1600), and iris. Time delay between the two beams is controlled by a delay stage in the fundamental arm. Then, the two beams are recombined by another hole-drilled mirror and focused into a nitrogen gas cell by a concave mirror (f = 37.5 cm). The pressure of the nitrogen is set to 50 mbar to suppress nonlinear propagation.

Fig. 3 (a) Experimental setup (CCD, charge-coupled device camera; PMT, photomultiplier; BPF, band-pass filter; FS, fused silica; VND, variable neutral density filter; HWP, half-wave plate; QWP, quarter-wave plate; CM, chirped mirror.) (b) Spectrum of the plasma fluorescence from nitrogen gas (blue curve) and the transmission of the BPF (red curve). (c) Dependence of the plasma fluorescence yield on the laser intensity (red circles) and the ion yield calculated using the molecular ADK model (blue curve).

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We employ two methods to detect the generated plasma fluorescence. In the first method, the fluorescence is collected by a BK7 lens, filtered through a band-pass filter (BPF, Edmund Optics #65-189) and iris, and detected by a photomultiplier tube (PMT, Hamamatsu H10720-210). The BPF transmits only the 337-nm line (2P(0, 0)) in the fluorescence spectrum (Fig. 3(b)), which corresponds to the transition from the lowest vibration level of the C³Π_u state to the lowest vibration level of the B³Π_g state in a neutral nitrogen molecule [24]. The electric current from the PMT is measured by a lock-in amplifier. For lock-in detection, the repetition rate of the signal pulse is reduced to 500 Hz by a chopper. This method cannot spatially resolve the fluorescence but has a good signal-to-noise ratio. In the second method, a charge-coupled device (CCD) camera (BITRAN BU-52LN-F) with a camera lens is used to resolve the spatial distribution of the plasma fluorescence. In this method, lock-in detection is not employed. Also, the CCD detects unfiltered fluorescence which contains multiple emission lines.

For accurate waveform measurements, it is critical to set the fundamental pulse to be a cosine-like pulse as we mention in the section 2.1. To detect the CEP of the fundamental pulse, the second harmonic (SH) of the IR pulse is introduced as a signal pulse and the intensity of the plasma fluorescence is measured as a function of the delay between the fundamental and the SH pulses as described in [9]. In our setup, the signal arm can be replaced with the SH arm that comprises a BBO crystal to generate the SH and a chirped mirror pair for pulse compression. The CEP of the fundamental pulse can be fixed to cosine-like with an error of ~0.04π rad.

Unlike measuring the ion current [9], it is not trivial to utilize the plasma fluorescence to measure optical waveforms. In fact, the mechanism of the fluorescence is still under debate. There are two possible mechanisms proposed for the formation of a nitrogen molecule in the C³Π_u state. In the first mechanism, N₂⁺, which is generated by ionization of N₂, collides into another N₂ to form N₄⁺. Subsequently, the N₄⁺ captures an electron, and splits into N₂ in the C³Π_u state and N₂ in the ground state [25]. In the second mechanism, N₂ is excited to a higher-lying singlet excited state by an intense laser field, and collides with another N₂. As a result, N₂ in the C³Π_u state and N₂ in the ground state are formed [26]. To verify that the generation of the plasma fluorescence is strongly related to the tunnel ionization, the laser intensity dependence of the 2P(0, 0) fluorescence yield is measured under irradiation with only the fundamental beam (Fig. 3(c), red circles). Note that the absolute intensity in Fig. 3(c) is calibrated to match the calculated N₂⁺ ion yield using the ADK model for a randomly-oriented nitrogen molecule (Fig. 3(c), blue curve). The calibrated laser intensity agrees with the intensity estimated based on the pulse energy, pulse duration, and spot size within a factor of two. The agreement between the measured intensity dependence and the ADK calculation indicates that the plasma fluorescence is almost proportional to the ionization yield, and thus can be utilized to measure optical waveforms.

4. Waveform measurement of linearly polarized few-cycle pulses

To confirm the validity of the proposed measurement scheme, we first measure one-dimensional (i.e., linearly polarized) optical waveforms. Figure 4(a) shows a typical waveform measured with a long delay scan. The estimated peak electric fields of the fundamental and signal pulses are 310 MV/cm (1.28 x 10¹⁴ W/cm²) and 3 MV/cm (1.20 x 10¹⁰ W/cm²), respectively. The intensity spectrum and phase obtained from the Fourier-transform of the waveform in Fig. 4(a) are represented by the red and black circles, respectively, in Fig. 4(b). We obtain good agreement between the spectrum transformed from the measured waveform (red circles) and that measured directly by a grating-based spectrometer (blue curve), indicating that the waveform measurement covers the whole spectrum spanning nearly one octave. The CEP-dependent waveforms are also measured (Fig. 4(c)). The fused silica wedge in the signal arm is adjusted to change the CEP of the signal pulse. The red and blue circles in Fig. 4(c) represent two measured waveforms with a CEP-difference of π rad. The solid curves represent the data after smoothing by Fourier filtering and the dashed curves are their envelopes. As expected, the carrier waves are inverted while their envelopes are almost unchanged.

Fig. 4 Measured one-dimensional waveforms. (a) A waveform obtained by a long delay scan and (b) its Fourier transformed spectral intensity (red circles) and phase (black circles). The laser spectrum which is measured using a grating-based spectrometer is shown as a reference (blue curve). (c) CEP-dependent waveforms; the red and blue circles represent the data measured with a CEP difference of π rad. The solid curves are the smoothed data and the dashed curves are their envelopes.

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5. Waveform measurement of circularly polarized few-cycle pulses

Next, we measure two-dimensional waveforms. The polarization of the signal pulse is set to either linear or circular by rotating the QWP in the beam path. The two orthogonal polarization components of a signal waveform are measured by changing the polarization of the fundamental pulse between s- and p-polarization. The peak intensities of the fundamental pulse and the linearly polarized signal pulse are the same as those in the one-dimensional waveform measurements. The peak intensity of the circularly polarized signal pulse is reduced by two while the pulse energy is kept to the same. Figures 5(a) and 5(b) represent the measured linearly and circularly polarized signal waveforms, respectively. Both of the waveforms accurately depict the two-dimensional nature of the signal pulse. The small residual oscillation in the x-component seen in Fig. 5(a) is probably due to misalignment or imperfectness of the HWP in the fundamental arm.

Fig. 5 Measured (a) linearly and (b) circularly polarized waveforms. The red circles and curves are the measured data and the grey curves are their projections in the vertical and horizontal directions.

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6. Space-resolved measurement of the Gouy phase

Next, we perform a space-resolved measurement for the Gouy phase using a CCD camera. As we discuss in the section 2.2, the Gouy phase can be determined by the setup to measure waveforms, but without the need to use cosine-like fundamental pulses. Again, the peak intensities of the signal and fundamental pulses are kept the same as in the previous experiments. As for the focusing condition of the fundamental and signal beams, the fundamental beam is focused from an annular profile with an inner diameter of 4.5 mm and an outer diameter of 9.5 mm at the concave mirror. On the other hand, the signal beam is almost collimated with a beam diameter of ~700 μm. Therefore, based on the discussion in the section 2.2, the Gouy phase of the fundamental beam alone is expected to be retrieved from the measurement. Figure 6(a) shows the measured space-resolved fluorescence. The horizontal axis represents the delay between the two pulses and the vertical axis represents the position along the propagation axis of the IR beam. The data shows that the delay modulates the fluorescence and that the modulation phase is spatially shifted. To show the shift explicitly, the modulated fluorescence signals at different positions (one between −5 and −3 mm and the other between 3 and 5 mm in Fig. 6(a)) are shown in Fig. 6(b). The circles are the measured data points, the solid curves are the smoothed data points and the dashed curves are their envelopes. It can be seen that there is a clear phase shift in the modulation. The phase shift is determined for each position, as shown in Fig. 6(c) (blue curve). The results are compared with the calculated on-axis Gouy phase of the fundamental beam based on the parameter obtained from its near-field beam profile (red curve in Fig. 6(c)). The measured and calculated phases agree well. The linear Gouy phase is characteristic of an annular beam [27]. An arctangent-like Gouy phase of a Gaussian beam is also shown in Fig. 6(c) as a reference (black dashed curve). The parameter of the arctangent phase is adjusted to have the same tilt as the measured phase at the origin. The measured Gouy phase clearly deviates from that of a Gaussian beam, which implies that the proposed measurement technique can be used to correctly determine the Gouy phase.

Fig. 6 (a) Spatially resolved plasma fluorescence measured by a CCD camera. (b) Modulated plasma fluorescence signals extracted from different z-positions in (a). The red circles are integrated from −5 to −3 mm and the blue circles are integrated from 3 to 5 mm. The solid curves are the smoothed data and the dashed curves are their envelopes. (c) Gouy phase extracted from the measurement (blue curve), the calculated phase from a measured near-field profile of the fundamental beam (red curve), and the arctangent-like Gouy phase of a Gaussian beam (black dashed curve).

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7. Comparison between plasma fluorescence and ion current measurements

Finally, we compare the previously demonstrated TIPTOE measurement based on the ion current [9] with the plasma fluorescence measurement proposed here. A pair of 1-cm-square stainless steel plates with a bias voltage of 10 V is introduced to the setup with a gap of 1 mm between them to sandwich the plasma and allows the ion current to be collected. The waveform is measured by collecting the current from the metal plates with the lock-in amplifier. The same waveform is also measured by the PMT and CCD methods. Figure 7 shows the results. All the methods reproduce almost the same waveforms, indicating that both the plasma fluorescence and ion current can be used to measure waveforms. This finding also indicates that the wavelength of the fluorescence is not important for the measurement because the PMT detects the 2P(0,0) line at 337 nm while the CCD detects multiple emission lines.

Fig. 7 Waveforms measured based on the PMT (blue circles), CCD (green circles), and ion current (red circles). The solid curves are the smoothed data and the dashed curves are their envelopes.

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8. Conclusion

In conclusion, we demonstrate an all-optical method for waveform characterization of few-cycle laser pulses using plasma fluorescence. We show that the proposed technique can be used to resolve a two-dimensional waveform and the Gouy phase. This technique does not require the placement of any apparatus in the vicinity of the interaction volume because the fluorescence signal can be detected from outside the experimental chamber. Based on our results, simple beam diagnostics systems for few-cycle intense pulses can be realized. Moreover, the space-resolved fluorescence measurement can be extended to a single-shot waveform measurement. Using cylindrical focusing, the fundamental pulse can generate a sheet-like plasma distribution. If a large, collimated signal pulse is superimposed on the fundamental pulse with a small tilt, the delay between the two pulses can be mapped in one direction. Thus, by measuring the spatial distribution of the fluorescence with a camera, the waveform can be retrieved from a single image. This method would be highly beneficial for low-repetition-rate and high-power few-cycle lasers.

Funding

Japan Society for the Promotion of Science (JSPS) (KAKENHI JP18H05250 and JP17H04816); Grant-in-Aid for JSPS Research Fellow (DC2); Advanced Leading Graduate Course for Photon Science (ALPS) by Ministry of Education, Culture, Sports, Science and Technology (MEXT) and JSPS.

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All-optical characterization of the two-dimensional waveform and the Gouy phase of an infrared pulse based on plasma fluorescence of gas

Abstract

1. Introduction

2. Working principle

2.1 Generalized TIPTOE method and reconstruction of two-dimensional waveforms

2.2 Retrieval of the Gouy phase

3. Experimental setup

4. Waveform measurement of linearly polarized few-cycle pulses

5. Waveform measurement of circularly polarized few-cycle pulses

6. Space-resolved measurement of the Gouy phase

7. Comparison between plasma fluorescence and ion current measurements

8. Conclusion

Funding

References

Cited By

Figures (7)

Equations (3)

Optics Express