1. Introduction

Harmonic generation is a typical phenomena of extreme nonlinear optics when the strong laser field interacts with the matters. The high harmonic generation (HHG) delivers the extreme ultraviolet ultrashort pulse with high photon energy, which has been widely investigated for its intriguing nonperturbative quantum dynamics and enormous applications as emerging innovative light sources. Comparatively, the low harmonic generation (LHG) refers to the radiation whose photon energy is much lower than the ionization energy. Recently, the LHG has attracted tremendous research interests due to both application and theory meanings. From the application aspect, the LHG from both solid- and gas-phase medium is a reliable approach to generate the ultrashort supercontinuum pulse ranging from terahertz to ultraviolet and beyond [1–4], providing an advanced light source for ultrafast spectroscopy [5–7], subcycle pulse synthesis [8] and frequency-comb technology [9–11]. From the theory aspect, the LHG from solid-state medium is associated with interband and intraband electron dynamics, which is essential for understanding the strong-field electron dynamics and extreme nonlinear optics in solids [12–16]. Thus, the systematical investigation of the LHG in simple atoms and molecules will provide a benchmark for a further understanding of the LHG in complex solid systems.

The physical mechanism behind the HHG can be well explained by the three-step model [17]. Firstly, the strong incident laser field ionizes the atom from the ground state to the continuum state; Then, the continuum electron is accelerated by the the laser field; Finally, the electron recombines with the parent ion, simultaneously emitting the HHG. Nevertheless, the LHG is only related to the first and second steps in three-step model and not necessarily associated with the recombination of electron with the parent ion. In the presence of sufficiently strong fields, the density of free electrons increases step-wise at the each laser half cycle due to tunnelling ionization, and the LHG comes from the acceleration of the transient photocurrent induced by the strong laser fields. This mechanism is also named Brunel model [18].

In the past decades, many efforts have been made to understand the LHG from gas-phase medium [16,19–23]. Previous works exhibit that the LHG has been attribute to different mechanisms according to various laser parameters. In the regime of high-intensity laser field, the nonperturbative nonlinear response of free electrons dominates the LHG process, which can be described by Brunel model. Simultaneously, the terahertz radiation, considered as the 0th order harmonics in LHG, can also be successfully explained with Brunel model [24–28]. In low-intensity regime, the nonlinear response of bound electron is responsible for the LHG which can be described with the perturbation theory [29–31]. In general, it is challenging to accurately distinguish the two mechanisms [32]. Numerical simulation based on time-dependent Schr$\ddot {\mathrm {o}}$dinger equation (TDSE) indicates that the nonlinear response of continuum-state electron is superior to bound electron, dominating the LHG in high-intensity low-frequency fields [33].

In this letter, the third harmonic generation (THG) driven by the bi-chromatic laser field is theoretically and experimentally investigated as an exemplar of the LHG, and the continuum-continuum transition in strong field approximation (SFA) is used to explain the THG. Although the SFA model has well predicted terahertz radiation[34–36], yet it is rarely applied to investigate the THG and LHG. Unlike the THG under the linearly-polarized bi-chromatic field [35], the SFA and photocurrent model are fairly distinguishable by introducing the ellipticity as a new experimental dimension. The SFA model on the basis of the quantum theory gives better predictions, indicating that the photocurrent model has limitation for the THG explanation.

2. Experimental setup

To control the THG and explore the mechanism of the THG, we prepared the bi-chromatic laser field with elliptical polarization and controllable relative phase. Our set-up is presented in Fig. 1(a). The Ti:sapphire laser with 1 kHz repetition rate delivers linearly polarized ($p$-polarized) 35 fs, 2.5 mJ per pulse, centred around 800 nm with a bandwidth of 40 nm. This fundamental wave (FW) passes through a 200 µm type-I $\beta$-barium borate (BBO) crystal, where a second-harmonic (SH) field of 400 nm is generated (conversion efficiency $\sim$32%). The co-propagating two-color driving fields are separated by a dichroic mirror (DM-2) into the two arms of a Michelson interferometer. The polarization of the two-color laser can be arbitrarily controlled by a pair of half- and quarter-wavelength retardation plates. For instance, the co-rotating or counter-rotating bi-chromatic field when the FW and SH are both circular polarization are shown in Fig. 1(b). The FW (1.2 mJ/pulse) and SH (0.56 mJ/pulse) driving lasers are combined with a DM-3 and focused into atmospheric air by using a lens with focal length of 10 cm. The THG beam emerges from the air plasma then passing through a optical low pass filter to block the pump laser beams. The intensity of the THG can be measured with a grating spectrometer, and the polarization is tested by a glan-laser polarizer before the grating spectrometer.

 

Fig. 1. (a) Schematics of the experimental set-up. DM: dichroic mirror; BBO: $\beta$-phase barium borate crystal; L: lens; GLP: glan-laser polarizer; Piezo: piezo-stage; (b) The co-rotating (green) or counter-rotating (purple) bi-chromatic field combined with circularly polarized FW (red) and SH (blue) beams.

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In order to suppress the sub-femtosecond time jitter of the two-color laser induced by the opto-mechanical components, the actively stabilized Michelson interferometer is adopted in the set-up. To stabilize the relative phase of the two-color laser, another continuous green laser (532 nm) co-propagates with the two-color lasers and interferes before the CCD camera [37]. The interference fringes are monitored by a CCD camera as a feedback signal. The real-time control of a mirror with a piezo actuator is employed to keep the interference fringes stable. The active feedback can efficiently reduce the influence from the airflow and the mechanical noise from the mirror, and minimize the relative phase fluctuations of two-color laser. After stabilization, the relative phase fluctuation in our system is smaller than $\pm 0.02\pi$ during data acquisition.

The time delay can be controlled by two approaches: An optical delay stage roughly compensates the relative time delay between the two-color laser pulses with femtosecond accuracy, and the sub-femtosecond phase delay is precisely tuned by moving the BBO crystal along the propagation direction. The relative phase delay is introduced by the difference between the refractive indices of the two-color fields in the air. When the distance $\Delta L$ between the BBO crystal and air plasma is changed, the relative time delay $\tau = \frac {(n_{ {\operatorname {SH}}} - n_{ {\operatorname {FW}}}) \cdot \Delta L}{c}$ is introduced with the sub-femtosecond accuracy, and the corresponding relative phase can be expressed as $\phi = 2 \phi _{ {\operatorname {SH}}} - \phi _{ {\operatorname {FW}}} = (2 k_{ {\operatorname {SH}}} - k_{ {\operatorname {FW}}}) \cdot \Delta L$, where $n$ is the refractive index of air and $k=2\pi /\lambda$.

3. Theoretical methods

In the photocurrent (PC) model, the THG $ {\boldsymbol {E}}_{ {\operatorname {THG}}} (t)$ comes from the time-variant photocurrent $ {\boldsymbol {j}}(t)$ induced by an external strong field $ {\boldsymbol {E}} (t)$ is,

The transient electron density $N (t)$ originates from accumulated electrons from tunneling ionization, satisfying

Therefore, Eq. (4) is referred to as the single-atom photocurrent (SPC) model.

Compared with the semi-classical SPC model, the THG can also be described under quantum mechanics treatment. The radiation is induced by the time variant dipole moment $ {\boldsymbol {d}} (t) = \langle \Psi (t) | \hat { {\boldsymbol {r}}} | \Psi (t) \rangle = \langle \Psi _0 (t) | \hat {U} (t_0, t) \hat {\mathbf {r}} \hat {U} (t, t_0) | \Psi _0 (t) \rangle$ with the time evolution operator $\hat {U}$ and the initial wave function $| \Psi _0 \rangle$. Using the Dyson series for $\hat {U}$, it is shown that the dipole moment includes three components, $ {\boldsymbol {d}} (t) = {\boldsymbol {d}}^{(0)} (t) + {\boldsymbol {d}}^{(1)} (t) + {\boldsymbol {d}}^{(2)} (t)$. The first term $ {\boldsymbol {d}}^{(0)} (t)$ vanishes in the spherically symmetric system. The second term $ {\boldsymbol {d}}^{(1)} (t) = i \int _{}^t {\operatorname {dt}}' \langle \Psi _0 (t) | \hat {\mathbf {r}} \hat {U} (t, t_{}') \hat {W} (t') | \Psi _0 (t') \rangle + c.c.$ describes the whole process of electron ionization, acceleration in two-color field and recollision to its parent core, corresponding to the CB transition, which are used to explain the HHG. The last term,

4. Results

The electric components of the two-color fields take the form,

The field amplitudes $E_\omega = 5.2\times 10^{8}$ $\rm {V/cm}$, $E_{2\omega } = 2.6\times 10^{8}$ $\rm {V/cm}$. Figure 2(a) shows that the SPC and SFA-CC models have very similar prediction, since the both of them come from the same physical origin, depicting the radiation induced by continuum electron. The intensity of the LHG has rapidly decreased with the increasing harmonic orders, and the plateau in high energy photon region is not found. It is easy to understand that the “hard” recollision predominantly contributes to high energy photons, while the CC transition mainly produces the low energy photons. The LHG yield in the counter-rotating bi-chromatic field is several orders of magnitude larger than that in the co-rotating field, which has been not experimentally validated yet. The 3$n$ order harmonics are suppressed in the counter-rotating field, and only the 3$n \pm$1 order harmonics can exist. The suppression of the 3$n$ order harmonics in the LHG is analogous to that in the HHG [40–43], due to the triple rotational symmetry of two-color field and the selection rules [44,45]. The suppression of the 3$n$ order harmonics in the LHG has been experimentally validated in the terahertz region, i.e. 0th order harmonics of the LHG, and our measurement proves that the THG follows the same rule as well. The intensity ratio of the THG in co-rotating field to that in counter-rotating field is about 100:1, as shown in Fig. 2(b). The THG in co-rotating field is analyzed by a polarizer, Fig. 2(c) shows that the intensity of THG is almost unchanged with the rotation of glan-laser polarizer, which is consistent with the theoretical simulation.

 

Fig. 2. (a) The spectrum of low-order harmonics. The red and blue curves represent the harmonic spectra with the SFA-CC and SPC, respectively. The solid and dashed line represent the co-rotating and counter-rotating field, respectively. (b) Measured intensities of the THG in co-rotating and counter-rotating field. The THG yield in counter-rotating field is much lower than that in co-rotating field. (c) The intensity of the THG in co-rotating field along arbitrary polarization directions in the experiment (red dots) and theory (green line).

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In order to further study the mechanism of the THG, the THG yields $I$ as a function of the ellipticity of the SH $\varepsilon _{2 \omega }$ and relative time delay $\tau$ are measured when the FW is circular polarized ($\varepsilon _{\omega } = 1$), as shown in Fig. 3(a). The THG is normalized to the maximum value of its the $p$ polarization component. The $\varepsilon _{2\omega }$-dependent THG provides a new dimension of information, which is not accessible only with the $\tau$-dependent measurement. Firstly, it can be found that only at special $\varepsilon$ the intensity modulation along $\tau$ is clearly visible. The modulation of $I_p (\tau )$ is clearly observed at $\varepsilon _{2 \omega } \approx 0.25$, while the modulation of $I_s (\tau )$ is more clear at $\varepsilon _{2 \omega } \approx -0.4$. Although Our laser system is freely running, yet the phase of 2$\omega$ beam is passively locked to the phase of $\omega$ beam. In multi-cycle regime, the THG is not sensitive to the CEP of driving laser, whereas only the relative phase has critical impact on low harmonics. Secondly, the difference between the SPC and SFA-CC model is more distinct when introducing the $\varepsilon$ dimension, as shown in Fig. 3(b) and (c). For instance, the ratios between the maximum value of $I_s$ and $I_p$, $\max (I_s)/\max (I_p)$, predicted by the SPC and SFA-CC model are different: The SFA-CC predicts $\max (I_s)/\max (I_p)\approx 0.37$, while the prediction of the SPC is $\sim 0.13$. The measurement shows that the SFA-CC gives better prediction of $\max (I_s)/\max (I_p)$.

 

Fig. 3. The distribution of the THG with the phase delay $\tau$ and ellipticity of the SH $\varepsilon _{2\omega }$ when the FW is circular polarized ($\varepsilon _{\omega } = 1$). Upper row: The $p$-polarized $I_p (\tau , \varepsilon _{2\omega })$; Lower row: The $s$-polarized $I_s (\tau , \varepsilon _{2\omega })$. The $I_p$ and $I_s$ obtained from the measurement (a) and the SFA-CC model (b), the SPC model (c).

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Considering the THG distribution via the phase delay $\tau$ and the SH ellipticity $\varepsilon _{\omega }$ when the SH is circular polarized ($\varepsilon _{2 \omega } = 1$), as shown in Fig. 4, the $I_s (\tau , \varepsilon _{2 \omega })$ predicted by the SFA-CC [Fig. 4(b2)] and SPC [Fig. 4(c2)] show totally different patterns. Compared with the measurement [Fig. 4(a2)], the SFA-CC gives better agreement with the $\varepsilon _{\omega }$-dependent yield of the THG. The contrast of the THG modulation along the $\tau$ axis can not be fully reproduced by the theoretical models. There are several possible reasons as follows: Firstly, the grating spectrometer is an incoherence detection method, the incoherence component of the radiation may decrease the contrast of the modulation. Secondly, the Coulomb potential of the ionic core may have a nonnegligible impact on the THG [46]. Thirdly, other mechanisms, for instance the bound-bound transition [29,30,32,33], may contribute to the THG, which results in the background of the measurement.

 

Fig. 4. The distribution of the THG with the phase delay $\tau$ and ellipticity fo the FW $\varepsilon _{\omega }$ when the SH is circular polarization ($\varepsilon _{2 \omega } = 1$). Upper row: The $p$-polarized $I_p (\tau , \varepsilon _{2 \omega })$; Lower row: The $s$-polarized $I_s (\tau , \varepsilon _{2 \omega })$. The $I_p$ and $I_s$ obtained from the measurement(a) and theoretical models of the SFA-CC (b), the SPC (c).

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The Fig. 5 shows the THG intensity versus the intensity ratio of two color fields $I_{\omega } / I_{2 \omega }$. The intensity of the SH was fixed at 0.56 mJ/pulse, and the intensity of the FW was changed in the experiment. The experimental result shows the intensity of the THG increases rapidly with the increasing $I_{\omega } / I_{2 \omega }$. Comparatively, the SFA-CC well reproduces the experimental results, while the THG predicted by the SPC saturates at $I_{\omega } / I_{2 \omega } = 1.5$.

 

Fig. 5. The THG yield as a function of the ratio between the FW and SH intensities in co-rotating bi-chromatic field. The experiment (black scatter), the SFA-CC (red curve) and the SPC (blue curve) are presented.

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5. Discussions

Comparing the experiment and theory, we find that both the SPC and SFA-CC model can roughly replicate the measurement. Nevertheless, there are some discrepancies between the SPC and SFA-CC model in details, especially the dependence of ellipticity. In the following, comparing between the analytic formulas, the similarity and difference of two models are discussed.

In PC model, the depletion of the neutral atoms can be neglected, and the PC model can be simplified into the form of Eq. (4), referred to as the SPC model. As shown in Fig. 6, the approximation in SPC model does not significantly modify the THG yields $I_p(\varepsilon _{2\omega }=1,\tau = 0)$ and $I_s(\varepsilon _{2\omega }=1,\tau = 0)$.

 

Fig. 6. The THG yields $I_p(\varepsilon _{2\omega }=1,\tau = 0)$ and $I_s(\varepsilon _{2\omega }=1,\tau = 0)$ predicted by PC model, SPC model, $\boldsymbol {a}_1(t)$ and $\boldsymbol {a}_1(t) + \boldsymbol {a}_2(t)$ in SFA-CC model.

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In the SFA-CC model, the Eq. (6) can be written as $\ddot {\boldsymbol {d}}^{(2)} (t) = \boldsymbol {a}_1(t)+\boldsymbol {a}_2(t)$, where $\boldsymbol {a}_1(t)$ and $\boldsymbol {a}_2(t)$ correspond to the first and second terms in Eq. (6). The $\boldsymbol {a}_2(t)$ only contributes to radiation when the emission time $t$ approaches the ionization time $t'$, which can always be neglected due to the minor contribution [35]. The Fig. 6 shows that the $\boldsymbol {a}_1(t)$ dominates the THG, while $\boldsymbol {a}_2(t)$ can be safely neglected. Hence, the Eq. (6) can be approximated and rewritten as a new form,

Since $w'(t',t)$ in Eq. (9) is roughly independent of $t$ [35], $w'(t',t)$ versus the ionization instant $t'$ can be directly compared with the $w(t')$ in Eq. (4).

As shown in Fig. 7 (a) and (b), $w(t')$ and $w'(t')$ have similar distributions along the ionization time $t'$, manifesting the correspondence between the semi-classical and quantum theories. Nevertheless, there are minute differences, leading to the discrepancies between SPC and FA-CC models. As shown in Fig. 7(c), $w'(t')$ gives apparently larger value than $w(t')$ at $\varepsilon _{\omega }=0$, resulting in different patterns predicted by the SPC and SFA model in Fig. 4 (b) and (c).

 

Fig. 7. The distributions of $w(t')$ in the SPC model (Eq. (4)) and $w'(t')$ in the SFA-CC model (Eq. (9)) when $\varepsilon _{2\omega }=1$ and $\tau = 0$ fs. (a) and (b), $w(t')$ and $w'(t',t\rightarrow \infty )$ versus the ellipticity of $\omega$ beam. (c), $w(t')$ and $w'(t',t\rightarrow \infty )$ at $\varepsilon _{\omega }=0$. Gray line, $|E(t)|=\sqrt {E_p^2+E_s^2}$; Green line, $w'(t', t\rightarrow \infty )$ in the SFA-CC; Red line, $w(t')$ in the SPC model.

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Both $w'(t')$ and $w(t')$ represent the weights of the electron released at ionization instant $t'$ on the radiation at emission time $t$, yet they have different physical meanings in the two models. In the SPC model, $w(t')$ is simply estimated by quasi-static tunneling rate under strong electric field. In contrast, $w'(t')$ in SFA-CC model derives from quantum mechanical treatment, which describes the overlapping between two quantum paths of a single electron released at different ionization instants $t'$ and $t''$, resulting in the coherence between the corresponding continuum states and yielding the radiation. The better agreement to the experimental results based on SFA-CC model indicates that the quantum effect may be significant in the THG.

6. Conclusion

In summary, the THG is experimentally and theoretically investigated by controlling the phase delay and ellipticity of the bi-chromatic laser field. The THG is found to be strong suppressed in the counter-rotating bi-chromatic fields, which can be explained with the selection rules of harmonic emissions. Furthermore, the dependency of the THG on the phase delay and ellipticity of bi-chromatic fields is experimentally acquired. The THG yields are periodically modulated along the phase delay, and highly dependent upon the ellipticity of the laser field. The THG as a function of the phase delay and ellipticity can be be roughly reproduced by both the photocurrent model and SFA-CC model, but the SFA-CC model gives better prediction of the elliptical dependency of the THG. We find that after formula simplification the photocurrent and SFA-CC model have similar analytic forms, but the overlapping between the continuum states via different quantum paths released at distinct ionization instants is not included in quasi-static ionization rate in photocurrent model, which leads to the slightly different prediction of elliptically-dependent THG. Our research provides a new perspective for further understanding of the mechanism of the LHG in gases and solid mediums.