1 Introduction

Harmonics generation is a fundamental nonlinear optical effect that enables frequency up-conversion of intense laser radiation interacting with a material when phase-matching conditions are fulfilled. The harmonics radiation is generated if the laser electric field becomes comparable to the intra-atomic electric field (i.e., 108–109 V/m or higher), inducing strong perturbation of the atomic electrons motion [1, 2]. Solid materials are commonly used as nonlinear media for generation of odd and even harmonics, though the solids are limited by their finite size and by the damage threshold. By contrast, gaseous media are free of optical damage issues, and can generate high odd-order harmonics due to their high ionization potential and symmetry properties [3,4,5,6,7,8,9,10,11].

Intense harmonics may be used for high-resolution nanoscale processing of materials and for non-perturbing measurements of the atomic and molecular systems, making it possible to assess properties such as the optical susceptibility of the materials [1,2,3, 12, 13]. The coherence of the harmonics enables generation of attosecond bursts which can be used to study the dynamics of the atomic and molecular electrons of the materials at sub-femtosecond timescale [5,6,7,8,9,10].

Two important factors may limit the efficiency of harmonics generation in gases: self-defocusing of the pump radiation and phase mismatch between the harmonics and the pump radiation [14, 15]. Self-defocusing would lower the pump laser intensity and the laser field, whereas phase mismatch would prevent constructive addition of the elementary emissions from the individual atoms within the interaction length. The optimum conditions for generation of intense harmonics require low densities of the gas (1018–1019 cm−3) to weaken the nonlinear effect of self-defocusing, and low ionization of the gas to compensate the neutral and ion dispersions [16, 17].

There are important efforts directed at finding new methods for generation of ever higher-order harmonics and to increase the conversion efficiency of preferred harmonics [7, 18,19,20,21,22,23,24,25,26]. Studies on the efficiency and properties of the harmonics generated in gases employ fs lasers in complex irradiation geometries and gaseous targets [3, 18, 26,27,28,29,30]. Thus, for generation of harmonics radiation, linearly polarized femtosecond pump pulses are focused at intensities of the order of PW/cm2 in gas jets generated in vacuum by specially designed valves; the gas jets have controlled composition, shape, and delay relative to the pump pulse [26, 30]. In other studies, the gas jets emitted by the valves were replaced by laser ablation plasmas produced by delay-controlled laser pre-pulses [18, 31, 32]. It was demonstrated that one can obtain maximum efficiencies of the order of 0.1% for the TH generation process [27, 31, 33], and that filamentation of the femtosecond laser pulses in gases is a source of intense TH radiation in the axial direction [31, 33]. The divergence of the harmonics generated in gas jets or ablation plasmas is several times smaller than the fundamentals when using irradiation geometries that favor small trajectories of the electrons freed from the parent atoms under the intense laser field [34, 35].

Here, we analyze the generation mechanism and the properties of the TH radiation generated by linearly polarized IR nanosecond laser pulses focused in ambient air at intensities of the order of TW/cm2. Our data indicate that the breakdown plasma acts as a self-induced nonlinear medium, controlled by the fundamental (F) pulse intensity, enabling filamentation and generation of intense TH. We determine the efficiency of the TH generation process by analyzing the variation of TH intensity with F intensity. We also analyzed the properties of TH radiation such as beam divergence and spectral bandwidth as compared to F radiation properties.

2 Experiments

The experimental setup that we used for generation and characterization of the TH radiation is presented in Fig. 1. For generation of TH, we used a Brilliant Q-switched Nd–YAG laser system delivering linearly polarized IR radiation at \({\lambda _{\text{L}}}=\) 1064 nm wavelength with 0.7 cm1 (i.e., 0.07 nm) bandwidth, ~ 4.5 ns pulse duration, and 6 mm beam diameter with a Gaussian intensity profile. The fundamental beam diameter was measured with a CCD camera (1500M-GE Thorlabs) placed in the axial position (i.e., along the F beam direction). The energy of the F pulses was varied in the range of 100–300 mJ by increasing the delay between the flash lamp and Q-switch signals and measured with a power meter (S370C Thorlabs). The pulse duration is demonstrated to be approximately constant for the whole range of delays employed here by using a rapid photodetector RP (SV2-FC Thorlabs); a typical temporal profile of the F pulses provided by the photodetector is presented in Fig. 2a.

Fig. 1
figure 1

Experimental setup: FL focusing lens, BP breakdown plasma acting as a nonlinear medium, DM dichroic mirror, F fundamental pump radiation, TH third harmonic radiation, FT/CCD fiber tip of the spectrometer/CCD camera, RP rapid photodiode

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Fig. 2
figure 2

a Temporal profile of the F pulses. Upper graph: F profile recorded in front of the focus. Bottom graph: F profile recorded after the focus. b Side view of the filamentary breakdown plasma. The F pulse propagates from right to left, and we highlight the lens focus position

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The laser pulses were focused in open air (i.e., in ordinary laboratory conditions) by a convergent lens of 10 cm focal length (F/16), giving a beam diameter in focus of ~ 22 µm and peak intensities of 4–12 TW/cm2. The breakdown plasma plume (BP) ignited near the focus was imaged from lateral with the CCD camera equipped with an objective; the camera was placed in radial position (i.e., perpendicular to the F beam direction) and was triggered by the Q-switch signal of the laser system. We illustrate in Fig. 2b a typical 2D image of the plasma plume along the axial direction. The image was obtained for 7.6 TW/cm2 pump intensity, and the length L of the filamentary plasma plume evaluated from this image is ~ 4 mm. Similar images recorded at different F intensities indicate that the length L of the filamentary plasma plume increases from 2 to 5 mm when the F intensity increases between 4 and 12 TW/cm2.

The radiation emerging from the breakdown plasma was spectrally analyzed in the axial and radial directions with a fiber-coupled spectrometer (Ocean Optics HR2000+, 0.1 nm FWHM spectral resolution). The spectrometer was triggered by the Q-switch signal of the laser system, and the integration time was set to 100 ms which is the time interval between two consecutive laser pulses. The collimating lens of the spectrometer fiber tip (FT) was mounted on a 1D translation stage which enables translation of the fiber tip with 10 µm resolution across the analyzed beam. The fiber tip was set at different axial and radial positions at distances of centimeters to tens of centimeters away from the lens focus. For analysis of TH in the axial direction with the spectrometer and CCD camera, the driving F beam was reflected out of the radiation by using a dichroic mirror (DM) placed at 45° angle in front of FT/CCD.

Typical spectral signals in the 200–400 nm domain recorded in these conditions are presented in Fig. 3. Each spectrum presented in Fig. 3 represents an average over ten spectra recorded during ten consecutive pulses. The upper part of Fig. 3a presents the 200–300 nm spectrum recorded in the axial and radial directions. We note that the axial and radial spectra are very different: the radial signal is a quasi-continuum spectrum, whereas the axial one exhibits strong molecular oxygen emission. The bottom graph of Fig. 3a presents in more detail the emission spectrum of molecular oxygen. We highlight on this graph the most important absorption bands characteristic of high-pressure molecular oxygen [36, Table 8]. They are superimposed on the A–X band emission [36, Table 11], resulting from the \({A^3}\Sigma _{u}^{+} - {X^3}\Sigma _{g}^{ - }\) transitions of neutral molecular oxygen. The presence of strong absorption lines of molecular oxygen indicate that the pressure of the breakdown plasma is very high (at least 60 atm. [36]), due to heating induced by inverse Bremsstrahlung in the field of neutral and ionized species.

Fig. 3
figure 3

Typical signals from the spectrometer recorded via a USB cable to a computer. a Upper plot: axial and radial spectra between 200 and 300 nm. Bottom plot: close look at the diffusive high-pressure absorption spectrum of molecular oxygen. b Upper plot: axial and radial spectra between 300 and 390 nm. Bottom plot: close look at the TH line, with the FWHM calculated from the Gaussian fitting curve

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The upper part of Fig. 3b presents the 300–400 nm spectrum recorded in the axial and radial directions. TH radiation is not detected in the radial direction, while low emission of ionized molecular nitrogen due to \({B^2}\Sigma _{u}^{+} - {X^2}\Sigma _{g}^{+}\) transitions [37, Table 42] is detected at several centimeters away from the plasma in the radial direction. Figure 3b reveals instead very strong emission at the TH wavelength (i.e., 354.7 nm) in the axial direction, the TH line being the most important emission in the whole 300–400 spectral range. To estimate the TH line intensity and bandwidth FWHM, we fitted the TH line with a Gaussian function (bottom plot of Fig. 3b). We found that, for all F intensities employed here, the TH line bandwidth is 0.24–0.25 nm, which is approximately three times larger than that of the F pulse.

To determine the divergence of the TH radiation beam, we measured the beam diameter 10 cm away from the focus, in the axial direction. Thus, we scanned the fiber tip of the spectrometer across the beam using the 10 µm resolution translation stage and measured the TH intensity. The dependence of the TH intensity on the fiber tip position is described by the sech function (Fig. 4a).

Fig. 4
figure 4

a TH line intensity vs fiber tip position. The continuous line is the sech fitting curve. b 1D transverse intensity profile of the TH beam, determined from the CCD image of the TH intensity profile (inset). The continuous line is the sech fitting curve

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The transverse intensity profile of the TH beam was further analyzed by direct projection of the beam on the 1500M-GE Thorlabs CCD camera placed after the dichroic mirror, 10 cm behind the lens focus in the axial direction. The IR-filtered CCD camera, with 1098 × 1390 pixels (6.45 × 6.45 µm2), was triggered by the Q-switch signal of the laser system. The 1D transverse profile of the TH intensity was obtained using the 1:1 image from the CCD camera (the inset plot of Fig. 4b), indicating the sech spatial profile of the TH intensity (continuous curve Fig. 4b).

3 Results and discussion

The spectroscopic study of the radiation from breakdown plasma indicates that intense TH radiation may be detected at large distances (i.e., up to 50 cm) away from the lens focus along the axial direction, but not in the radial direction (Fig. 3). We characterized the directionality of the TH radiation in the axial direction by measuring the TH beam diameter (from the sech intensity profile) at 10 cm after the lens focus. The data presented in Fig. 4 indicate that the diameter of the TH beam is ~ 2 mm, which is three times smaller than the F beam diameter at the same position. Thereby, the harmonic beam divergence is directly related to the wavelength, and the divergence of the TH radiation beam generated within the breakdown plasma is ~ 10 mrad.

The high intensity of TH in the axial direction indicates that this radiation results from nonlinear optical effects in the breakdown plasma involving the nitrogen molecules [32, 37]. Thus, the nitrogen molecules are ionized via impact with energetic electrons absorbing F energy through inverse Bremsstrahlung effect. Then, the intense quasi-resonant transition at 354.9 nm, from the \({B^2}\Sigma _{u}^{+} - {X^2}\Sigma _{g}^{+}\) system of the ionic molecular nitrogen [37], facilitates emission of intense radiation at 354.7 nm wavelength corresponding to TH.

The duration of TH pulses is estimated to be approximately three times smaller than that of the F pulses, as indicated by the “chopped” temporal profile of the F pulse recorded after the breakdown plasma (bottom plot in Fig. 2a) as compared to the temporal profile recorded in front of the breakdown plasma (upper plot of Fig. 2a). The chopping of the F pulse after the breakdown plasma indicates strong heating and ionization of the plasma during the leading part of an F pulse. The consequent increase of electron density \({n_{\text{e}}}\) leads to increase of plasma frequency \({\omega _{\text{p}}}\sim \sqrt {{n_{\text{e}}}}\) above the F frequency, so that the trailing part of the F pulse could not penetrate the plasma in the axial direction. Thereby, the breakdown plasma could support generation of TH radiation in the axial direction only during the leading part of a pump pulse.

To determine the optimum irradiation conditions for TH generation, we analyzed the variation of the TH intensity (\({I_{3\omega }}\)) as a function of F pump intensity (\({I_\omega }\)). The data presented in Fig. 5 reveal two important characteristics of the TH signal. First, the maximum TH signal is obtained when \({I_\omega } \approx\)11 TW/cm2. Second, there is a periodic variation, with equidistant minima, of \({I_{3\omega }}\) with the square root of the F intensity.

Fig. 5
figure 5

TH intensity (\({I_{3\omega }}\)) vs square root of the pump intensity (\({I_\omega }\)). The equidistant minima of \({I_{3\omega }}\) are marked as min1–min5

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We demonstrate that the periodic variation of \({I_{3\omega }}\) with \(\sqrt {{I_\omega }}\) is a sinc2 type variation which may be explained by using the theory of harmonic generation phenomenon in a nonlinear medium of length L [2, 38]:

$${I_{3\omega }}\sim \frac{{{L^2}I_{\omega }^{3}}}{{{n_{3\omega }}n_{\omega }^{3}}}{\text{sin}}{{\text{c}}^2}\left( {\frac{{\Delta k~L}}{2}} \right).$$
(1)

Here, \(\Delta k=\left( {3{k_\omega } - {k_{3\omega }}} \right)\) is the wavevector mismatch (WM) between F and TH radiations [39, 40], and \({n_\omega }\) and \({n_{3\omega }}\) are the refractive indexes of the dispersive nonlinear medium corresponding to F and TH radiation, respectively. We note that the length L of the nonlinear medium, which in our case is the self-induced breakdown plasma, depends on the intensity \({I_\omega }\) as described below.

The radial view of the breakdown plasma presented in Fig. 2b and the sech profile of the TH transverse intensity presented in Fig. 4 indicate that laser filamentation occurs in air at the high F intensities employed here. Filamentation is due to two nonlinear counter effects: self-focusing determined by the intensity-dependent refractive index \(n={n_0}+{n_2}{I_\omega }\) as in the optical Kerr effect, and self-defocusing caused by the free electrons in the plasma [1, 2, 41]. The length L of the filament induced by the F beam depends directly on the square root of the pump intensity \({I_\omega }\) [42]:

$$L\sim \frac{{\sqrt {{I_\omega }} }}{{{\lambda _{\text{L}}}R_{0}^{2}}},$$
(2)

where \({R_0}\) is the initial radius of the F beam.

Thus, the sinc2 dependence of \({I_{3\omega }}\) on \(\sqrt {{I_\omega }}\) arises from the fact that the length L of the nonlinear medium (appearing as an argument of sinc2 function in Eq. (1)) represents the length L ~ \(\sqrt {{I_\omega }}\) of the self-induced filament (given by Eq. (2)). Moreover, considering in Eq. (1) that \({L^2}\sim {I_\omega }\) as predicted by Eq. (2), and \({n_\omega }\sim {I_\omega }\) as in the optical Kerr effect [1, 2], we find that the proportionality coefficient \({L^2}I_{\omega }^{3}/n_{\omega }^{3}\) of \({I_{3\omega }}\) in Eq. (1) is directly related to \({I_\omega }\). This explains the good fitting curve of the maxima \({I_{3\omega }}\sim {I_\omega }\) presented in Fig. 5.

To estimate the efficiency of the TH generation phenomenon, we measured the average power of the TH pulses and calculated the ratio of TH power to F power. In optimum phase-matching conditions, we found that the average power of TH pulses is ~ 0.1 mW when using F pulses of 270 mW average power. This gives an overall efficiency of ~ 0.04% for the TH generation process, which is only several times smaller than the maximum efficiency obtainable in expensive and complicated experimental conditions involving powerful fs laser and gas-puffs targets [27, 31, 33].

4 Conclusions

We presented an experimental study of the fundamentals of the TH generation process in air. We analyzed the characteristics and the efficiency of TH generated axially by focusing nanosecond IR laser pulses in air at intensities of the order of TW/cm2. We showed that TH intensity varies as a sinc2 function of the square root of the F intensity, and the maximum TH signal is obtained for ~ 11 TW/cm2 F intensity.

The sinc2 variation of the TH intensity demonstrates that the breakdown plasma acts as a self-induced nonlinear medium with the length controlled by the F pulse intensity. We indicated that the ionized nitrogen molecules from the high-pressure breakdown plasma facilitate emission of intense TH radiation.

The temporal and spatial characteristics of the TH radiation were analyzed and it was found that the TH pulses were three times shorter than the F pulses, and the divergence of the TH beam was three times smaller than that of the F beam.

The maximum efficiency of the TH generation process was measured to be ~ 0.04%, which is only several times smaller than the maximum efficiency obtainable in expensive and complicated experimental conditions involving powerful fs laser and gas-puffs targets.