Third harmonic generation by a weak femtosecond probe pulse intersecting a pump laser-induced plasma in air is investigated and a general model is developed to describe such signal, applicable to a wide range of focusing and plasma conditions. The effect of the surrounding air on the generated signal is discussed. The third-order nonlinear susceptibility of an air plasma with electron density Ne is determined to be with γp = 2 ± 1 × 10−49 m5 V−2 and being the third-order susceptibility in air. Lateral scans of the probe through the plasma are used to determine electron density profiles and the effect of focusing and phase mismatch is discussed.
© 2011 OSA
Third harmonic generation (THG) microscopy  has been recognized as a useful nonlinear imaging modality to observe spatial inhomogeneities in both the linear (refractive index) and nonlinear (third-order susceptibility χ (3)) optical properties of materials, including biological species. Since all materials have a non-vanishing χ (3), this technique can be applied to study virtually any material, as long as it is transparent at the wavelengths involved.
THG microscopy was applied to probe laser plasma plumes from solid targets , which are produced in laser induced breakdown spectroscopy for example. The potential of this technique from the standpoint of plasma diagnostics was suggested, but it remained unclear how the observed structures in the TH signal relate to plasma properties. There have been a number of experiments in which THG by a filament producing signal pulse was studied in the presence of a plasma string generated by an energetic pump in gaseous media (see for example [3,4]). This scenario is of potential interest for the generation of ultra-short pulses in the UV wavelength range (e.g. ). The two-order-of-magnitude increase of the THG efficiency in the presence of the pump induced plasma was attributed to a plasma-enhanced χ (3) .
In this paper we investigate the third harmonic (TH) signal generated by a weak probe beam intersecting transversely a pump laser-induced plasma in air and develop a general model for the generation of such a signal, applicable for a wide range of focusing conditions and a radially symmetric distribution of the electron density (Ne). Unlike in previous experiments, the intensity of the probe pulse was chosen small enough such that filamentation was avoided and the TH signal was proportional to the probe intensity raised to the third power. This setup allowed us to study the properties of the pump-induced plasma and THG without interference by probe-induced filamentation and associated nonlinearities. We measure the functional dependence of the TH signal with respect to the electron density, estimate χ (3) of such a medium, and propose a method by which the electron density distribution of a laser induced plasma can be measured with temporal resolution. The conditions under which THG can be used for 3D imaging of plasmas are evaluated and the limitations discussed.
2. Theoretical background
The TH field generated by a focused Gaussian fundamental beam propagating in an infinitely long homogeneous medium (material a) vanishes due to destructive interference of the TH waves generated before and after the focus, provided the material exhibits normal dispersion, see for example 
A convenient way to visualize the value of the integral in Eq. (1) is using a vibrational diagram , as shown in Fig. 1(a), where the TH field E 3ω (z) is plotted in the complex plane, with z as the running parameter. For a particular z position the resultant TH field is represented by the vector from the origin to the corresponding point on the trace. As shown in Fig. 1(a) the TH field created in the negative half-space −∞ < z < 0 is canceled by the TH field created in the positive half-space 0 < z < +∞. As a result the integration over the infinite medium yields no THG.
If a slab of a material of thickness D is introduced with different linear and/or nonlinear optical parameters, see Fig. 2, the TH field E 3 ω will in general no longer vanish. The corresponding integral for the TH field can be split into three components - before the slab, the slab and after the slab. After adding the three contributions, taking into consideration that at the interface of two nonlinear media the phase mismatch between the fundamental and harmonic waves has to be continuous, the TH field takes on the formEq. (2) we assume a homogeneous distribution of both Δkp and across the slab material. Figure 1(b) shows the corresponding vibrational diagram for this scenario. The total TH field at infinity does not vanish, contrary to the case shown in Fig. 1(a) for the infinitely long medium. Instead, the total TH field is a contribution from the TH generated in medium a and in the slab.
Using Eq. (2) we can calculate the TH field produced by a probe beam propagating through a plasma (material p) region in air (material a) centered around the focus of the probe. According to the Drude model the refractive index of the plasma at frequency ω can be written as :
From Eq. (3) the wave-vector mismatch in the plasma can be written as:
The third-order nonlinear susceptibility of air can be estimated by . For the third-order nonlinear susceptibility of the plasma we make the following ansatz:10]. It assumes that the “background” nonlinearity of the neutral and ionic gas molecules is that of air. It is conceivable that for high electron densities terms of order and higher have to be added to account for electron-electron interactions, electron-ion interactions and a change in the background nonlinearity produced by the molecules.
Equation (2) holds for sufficiently weakly focused beams (numerical aperture NA≲ 0.4, corresponding to w 0 ≳ 1 μm), which is a consequence of E 3 ω from Eq. (2) being the solution to the paraxial wave equation, and radially symmetric electron density distributions. Equation (2) can be simplified assuming D ≪ z 0 to take on the formEq. (6), with as a prefactor, represents the contribution from air, and the second term the contribution from the plasma. For low electron densities (Ne ≪ Nc) as is typical for weakly ionized plasmas, and sufficiently small plasma thickness D, the phase mismatch |ΔkpD| ≪ 1. This allows us to expand the trigonometric functions and the exponential prefactor in Eq. (6), using Eq. (4), to first order in Ne. We obtain
Figures 3(a) and 3(b) depict the calculated TH signal as a function of the electron density of a pump induced plasma (D = 100 μm) for weak and strong focusing conditions of the probe beam, respectively. The solid blue lines in Figures 3(a) and 3(b) show the exact results using Eq. (2), while the solid red lines neglect the contribution from air to the generated TH signal. For weak focusing conditions, Fig. 3(a), the contribution from air cannot be neglected. The TH generated in air and in the plasma are of the same order of magnitude but opposite signs. For tight focusing conditions, Fig. 3(b), the contribution from air is important for the low electron densities. Only for high electron densities this contribution becomes negligible, since the effect of the “quasi-free” electrons on the signal becomes dominant, cf. Eq. (5). In general, the air contribution can be approximately neglected if , where the factors (γpNeD) and are an approximate measure of the magnitude of the signal generated in the plasma and air, respectively.
The dotted line in Fig. 3(a) shows a parabolic fit, confirming that in the low electron density limit. For electron densities Ne > 1024 m−3 and D = 100 μm, |ΔkpD| ≳ 1 and the approximations leading to Eq. (7) are not valid. The oscillations of the TH signal are a consequence of the phase mismatch approaching and exceeding π.
Figure 3(c) shows the TH signal as a function of the confocal parameter 2z 0 for constant probe energy. If focused tightly, the probe will generate TH in a small volume, representing a small fraction of the plasma, with the limiting case where the TH signal becomes negligible for a bulk sample. For large z 0 the small probe intensity prevents efficient THG. Optimal focusing conditions are found for 2z 0 ≈ D, for the case where |ΔkpD| ≪ 1.
In typical experimental conditions the electron density distribution is not constant along z, Ne(z) = Ne f (z). Such an electron density distribution produces a gradient in the plasma non-linearity , cf. Eq. (5), and in the wave-vector mismatch Δkp(z), cf. Eq. (4). The total TH field can be calculated using a relation similar to Eq. (2). The integral corresponding to the plasma region is split into smaller regions of thickness Δz within which Ne(z) can be regarded constant. As before, when adding these contributions, the accumulated phase in one region has to be correctly added to the phase in the following region. The TH field takes on the formEq. (8) to first order in Ne.
3. Experimental setup
The experiments were carried out using a Ti:saphire chirped-pulse-amplification laser system providing 40-fs, 800-nm pulses, with energies of 2 mJ, at a 1-kHz repetition rate. The pulse to pulse energy variations of the laser were ∼ 2%. A schematic diagram of the experimental setup is shown in Fig. 4. The output from the amplifier was split into a pump and a probe pulse. The pump pulse with an energy of up to 1.5 mJ was focused using a lens L1, generating a plasma in air. The beam waist of the probe pulse after passing through lens L2 intersected the plasma transversely. The beam waist of the probe was determined through knife-edge measurements. The time delay between the pump and probe pulses was adjusted using an optical delay line. The TH signal was filtered from the fundamental using dichroic mirrors, a prism, and an interference filter, and was detected by a photomultiplier tube (PMT). The background (measured when blocking the pump beam) was subtracted from the measured TH signal. For a time delay of 30 ps the signal-to-background ratio was ∼ 100 : 1.
4. Results and discussion
Figure 5 shows the TH signal as a function of the probe pulse energy with a pump induced plasma present. This 10-cm long plasma of diameter D ≈ 140 μm was produced by 1.5 mJ pulses using a lens L1 with f = 1 m. The probe was focused with an f = 30-cm lens (L2) producing a beam waist of w 0 ≈ 17 μm at zw = 0. For small probe energies (≲ 5 μJ) the TH signal shows the expected cubic dependence. For larger energies the slope is less than three, which was also observed in  and is likely a consequence of the generation of a plasma by the probe and subsequent probe defocusing. From experimentally obtained ionization rates of N2 and O2 as a function of laser intensity [11,12], we can estimate Ne ≈ 4 × 1024 m−3 at probe energies of about 20 μJ.
In the experiments to be described below where we used THG as a diagnostic tool the probe intensity was kept sufficiently small. This ensured that the TH signal generated by the probe alone was negligible compared to the case were a pump induced plasma was present and that the cubic power relationship applied. For example, for a probe energy of 1.7 μJ under the focusing conditions of Fig. 5 and no plasma present, the TH signal was below our detection limit. From experimentally obtained ionization rates of N2 and O2 as a function of laser intensity [11, 12], we estimate Ne ≈ 3 × 1017 m−3 for such a probe pulse.
In order to investigate the relationship of TH conversion and the electron density of the plasma, we measured the TH signal as a function of the time delay τ between pump and probe pulses and determined the time-dependent electron density in the plasma Ne(τ) by means of a diffraction experiment . This experiment used the fact that the plasma induced diffraction of the zero-spatial frequency component of a probe beam is proportional to Ne(τ). We used an f = 30-cm focusing lens (L2) for the probe producing a beam waist of ≈ 17 μm at zw = 0. The laser amplifier repetition rate was reduced to 250 Hz to avoid thermal effects (see below). Figure 6 shows the so obtained TH signal from the probe as a function of the electron density in the plasma. The solid line illustrates the good agreement between the experiment and the predictions from Eq. (8). The simulations also took into account the change of the electron density spatial profile . Ne(z) changes from Gaussian at large Ne(0) (small delays) to a more flat-top profile for low Ne(0) (large delays) because of the bimolecular nature of the electron-ion recombination.
To estimate the plasma nonlinearity we compared the measured TH conversion efficiency as a function of electron density (Fig. 6) with the model predictions, according to Eq. (8). From Eq. (5) we obtained γp = 2 ± 1 × 10−49 m5 V−2, where the main sources of error came from the uncertainties on the beam waist and M 2 value of the probe, and on . For electron densities of fs pulse induced filaments in air (Ne ≈ 1.5 × 1023 m−3 ) the third-order nonlinear susceptibility is estimated to be , a value that is approximately 1 5 times greater than the third-order nonlinear susceptibility of air, . This result means that the effective second order hyperpolarizability of a single “quasi-free” electron in the plasma is about 70 times larger than that of an average air molecule ( where N air is the total number of air molecules ≈ 2 × 1025 m−3). In reference  an estimate for of 4.5 ± 3 × 10−24 m2/V2 was reported for Ne ≈ 2 × 1025 m−3, in agreement with what we obtain using our value for γp and Eq. (5).
In order to determine the spatial distribution of the electron density in a laser produced plasma, the TH signal was recorded while the plasma was scanned laterally (y direction) across the probe beam, which was realized by moving the focusing lens of the pump beam (L1) along the y direction using a translation stage. This resulted in the filament moving by the same amount Δy with respect to the probe beam, which is only true in the vicinity of the focal plane of lens L1, in which case aberrations and a change in the time delay between pump and probe beams can be neglected. The results of the scan are shown in Fig. 7. We used an f = 30-cm focusing lens (L2) for the probe producing a beam waist of ≈ 17 μm at zw = 0. The solid line in Fig. 7 was obtained from Eq. (8), and shows a distribution with a width of ≈ 100 μm. For each y position we assumed a constant Ne over the probe beam with respect to y and integrated over z. Since the spatial distribution shown in Fig. 7 corresponds to . From this we obtain that the spatial distribution of Ne(y) has a width of ≈ 140 μm.
It should be mentioned that for repetition rates ≳ 500 Hz the TH scan profiles exhibit pedestals extending over about 1 mm. We attribute this to an increase in the temperature of the air surrounding the plasma. Our THG model, Eq. (8), predicts a TH contribution from an air channel with a refractive index and nonlinear susceptibility gradient expected from a temperature and resulting particle density gradient. To avoid this, the laser amplifier repetition rate was set to 250 Hz (Figures 6 and 7).
To study THG for higher plasma densities we generated an air plasma of a few mm length by focusing the pump pulse with a 10-cm-focal-length lens L1, cf. Fig. 4. The probe beam was focused using a 6-cm-focal-length lens (L2) producing a waist of ≈ 6 μm. The laser amplifier repetition rate was set to 1 kHz, since the thermal effects were negligible under these experimental conditions. Figure 8 shows the TH signal while the plasma was scanned laterally (y direction) across the probe beam for several pump pulse energies (left column), and simulation results (right column).
To simulate the TH signal as a function of the scan coordinate we first estimated the electron density distribution from Keldysh theory  assuming a Gaussian pump intensity of waist w 0 ≈ 50 μm, a value larger than the predicted waist generated by a 10-cm-focal-length lens. For the high intensities used, however, the generation of a plasma and subsequent defocusing prevents the pump from reaching such small waists. The value assumed for the simulations was chosen based on the best agreement found between the simulations and the experimental results. It should be noted that the pump intensities involved lead to tunneling ionization and are higher than intensities for which rates have been determined experimentally . The simulated electron densities are shown in the right column (dashed lines). These electron density distributions Ne f (y, z) were then used in Eq. (8) to obtain the TH signal, which is shown as a solid line. We find good overall qualitative agreement between the model predictions and the experimental data. For the higher pump energies Ne(y, z) broadens and its peak value increases. The model predicts local minima of the TH produced in the plasma center even though this is where Ne(y, z) peaks, which is a consequence of the phase mismatch exceeding π, cf. Fig. 3(a). Therefore, structure found in the TH signal of high-density plasmas does not necessarily point to similar structures in the electron density itself.
In the low density regime, cf. Fig. 7, and assuming cylindrical symmetry of the plasma, the lateral scans can be used to measure plasma density distributions with temporal resolution by measuring and using Eq. (5) as a function of pump-probe delay. This approach is advantageous over fluorescence techniques, for example, where the response time is given by the shorter of the integration time of the detector and the fluorescence lifetime.
The question arises whether microscopic 3D imaging of plasmas using THG is possible. To resolve feature sizes Δz one would need focusing conditions of the probe for which the confocal parameter 2z 0 ≲ Δz. At the same time, to ensure for unambiguous interpretation, ΔkpΔz ≪ 1 is required. On the other hand, if 2z 0 ≪ Δz the TH field would vanish due to destructive interference of the TH waves generated before and after the focus, adding another source of ambiguity in interpreting the structure found on the TH signal. As an example, boundaries of relatively uniform regions of thickness D ≫ 2z 0 and individual structures with dimensions ≪ 2z 0 separated by D could give rise to similar TH signals. In certain cases this ambiguity could be addressed by performing series of scans with different focusing conditions and scan directions. The detection limit of the experiment will dictate how strong the probe beam can be focused while keeping the intensity small enough to ensure a cubic power law for the TH conversion and to avoid significant contributions to the TH signal in the absence of the pump. As an example, for an air plasma with an electron density Ne = 1025 m−3, the upper limit on the dimension D which can be probed without ambiguities using a single lateral scan is D ≈ 1/Δkp ≈ 16 μm.
In conclusion, 3D mapping of plasma densities using THG can be done for certain plasma symmetries, such as those found in air-filaments, where single scans can provide this information, along with modeling when necessary.
It is worth mentioning that the methods and results described in this work are not restricted to femtosecond laser-induced plasmas. This methodology can be applied, for example, to study plasmas generated from a nanosecond pulsed laser using a femtosecond probe, so long as both sources are synchronized.
A model of THG by a probe in the presence of a pump induced plasma was developed that can be applied to a wide range of focusing and plasma conditions. Contributions to the TH signal from the plasma and ambient gas can be distinguished and their relative strength depends on probe beam parameters and plasma size and density. When the probe is weakly focused the contribution from ambient gas molecules cannot be neglected. For higher density plasmas and when the probe is tightly focused the contribution from ambient gas molecules becomes negligible. In general, we conclude that the air contribution can be approximately neglected if . To use THG as plasma probe, care must be taken to keep the probe pulse fluence small enough to ensure a cubic power law for the TH conversion and to avoid significant contributions to the TH signal by the focused probe in the absence of the pump. Using this probe regime, with γp = 2 ± 1 × 10−49 m5 V−2 was determined for the third-order susceptibility of the plasma. Under certain conditions (cylindrically symmetric plasma, phase mismatch ≪ 1, and a confocal parameter of the probe approximately equal to the size of the plasma), lateral scans of the probe beam through the plasma can be used to determine 2D plasma density profiles with sub picosecond time resolution. In the general case, phase mismatch critically affects the scan profiles and must be taken into account to derive electron density profiles. 3D mapping of plasma densities using THG seems possible when additional information is available, such as the peak densities and approximate dimension of the structures. The latter could be obtained via modeling or confocal fluorescence imaging combined with up-conversion if temporal resolution is needed.
We are grateful to L. Emmert, D. Cremers, and R. Multari for helpful discussions and we thank J. Chen for support with the experiments. This work was supported by NSF ( PHY-0722622) and ONR ( N000140810350).
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