## Abstract

The physical mechanism for sensing broadband terahertz (THz) wave via using femtosecond (fs) laser induced gas plasma without any local accessory near the plasma, i.e. THz air breakdown coherent detection, is systemically investigated by utilizing the transient photocurrent model. Previous observed results, such as conversion from incoherent to coherent detection, can be numerically obtained. Further calculations and analysis show that it is through modification of the gas ionization process, and not acceleration of freed electrons or through a four-wave-mixing (FWM) process, that the THz waveforms can be encoded into the detected second harmonic (SH) signals.

© 2012 OSA

## 1. Introduction

Laser-induced gas plasma can be used to generate strong (E-field~MV/cm), broadband (0.1-40THz), and coherent THz waves through a complex nonlinear physical process [1–3]. On the other hand, plasma in ambient air or selected gases can also be used as THz wave sensors [3–6]. Unlike solid state materials commonly used in THz time domain spectroscopy (THz-TDS) system, such as electro-optical crystals and photon-conductive antennas [7], gas medium have no phonon resonance or echoes due to THz waves or optical reflection. Moreover, some gaseous medium, from the point of use-cost, is a better choice than the solid state materials mentioned above since there is no concern about their damage.

For the generation of broadband and strong THz pulses from two-color (${\omega}_{0}\text{-}2{\omega}_{0}$) laser induced gas plasma, its physical mechanism was initially treated as a FWM process, which, however, failed to explain some subsequent measured results, e.g. saturation of THz output [2]. To solve these problems, Kim et al developed a so-called transient photocurrent (PC) model, the results of which showed that coherent THz waves had originated from a net electron current surge generated by an asymmetric two-color laser field, indicating that plasma plays a key role for generation [8, 9]. Note that many other models have been also built to explain this complex process [10–12].

Broadband THz detection via using fs-laser induced gas plasma, i.e. THz Air Breakdown Coherent Detection (THz-ABCD), was firstly reported in 2006 [4], the authors of which used a phenomenological FWM model to explain their measured results by introducing the SH component of white light from the plasma as a local oscillator that is depending on probe laser intensity (Note that THz-field induced SH generation effects in liquid medium has been reported in 1999 [13]). However, a detailed microscopic physical picture of their observed phenomena, especially how the THz information is encoded into measured SH emission, was missing up to now since the THz detection with gas plasma, at first glance, was generally treated as an inverse process of THz generation according to the FWM model [4]. Moreover, a subsequent altered sensing method, by adding an AC bias voltage at the location of plasma, namely THz Air Biased Coherent Detection (also abbreviated as THz-ABCD) [5], attracted more attention and was widely used in labs due to its lower laser intensity. Nevertheless for application outside the labs, especially urgent need of remote THz sensing in ambient atmosphere [1], the former THz-ABCD could have more advantages since it doesn’t need any local accessory at the location of plasma, i.e. AC bias. Thus it is very necessary to elucidate the microscopic process of the former THz-ABCD.

In this paper, we theoretically investigate the missed microscopic process of the THz Air Breakdown Coherent Detection by using the well-developed PC model [8, 9]. The observed results in experiments, such as transition of detection mode and SH intensity clamping [4], can be numerically reproduced. Further analysis shows that it is mainly through modification of the gas ionization process that THz waveforms’ information can be encoded into the measured SH signals. Moreover, our results demonstrate that FWM model used in Ref. 4 does not appear to account for the THz Air Breakdown Coherent Detection.

It should be noticed that the followed “THz-ABCD” in this paper means the former one mentioned above.

## 2. THz-ABCD and PC model

Firstly, we briefly describe THz-ABCD process and transient PC model. A probe fs-laser beam (${\omega}_{0}$), as schematically illustrated in Fig. 1 , is focused into gas to ignite gas plasma, then in which the freed electrons will be accelerated by this incident field and thereby form an oscillating current$J$. Thus electromagnetic (EM) waves with all frequencies will radiate from such plasma. When THz signal with a delay time $\Delta t$ is also focused on the plasma, some fluorescence components in the total emission ${E}_{out}(t)$ [14, 15], such as SH emission ($2{\omega}_{0}$), will be significantly influenced by this incident detected waves through a complex interaction that will be investigated in this paper. So by observing such modulated $2{\omega}_{0}$components, one can indirectly get the information of THz waves. Note that the attachment and recombination of such electrons can be ignored since the time of these processes is much longer than that of radiation [11]. The plasma, in addition, can be approximately treated as a uniform spheroid with radius ${w}_{0}$ since its size, compared with the THz wavelength, is sufficiently small [8, 9].

Using PC model, one can write the radiation field from plasma center as [8, 9, 12]:

where, $m$and$e$are the mass and charge of the electron, respectively. The plasma density${N}_{e}(t)$, determined by total incident optical field${E}_{in}(t)$, species and density of gas medium, can be calculated via Ammosov-Delone-Krainov (ADK) tunneling [16], static tunneling [17] or other ionization models [18]. Note that we use the static tunneling model in our simulation of ionization process, and the gas medium, corresponding to the original experiments [4], is the nitrogen.Actually, Eq. (1) represents a net radiated field with all frequencies from the plasma. In order to obtain the SH signal${E}_{2{\omega}_{0}}(t)$from Eq. (1), one can use the Fourier transformation, i.e.${E}_{2{\omega}_{0}}(t)\propto {\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+\infty}{E}_{out}({t}^{\prime}){e}^{-i\omega {t}^{\prime}}d{t}^{\prime}}{f}_{2{\omega}_{0}}(\omega ){e}^{i\omega t}d\omega}$, in which${f}_{2{\omega}_{0}}(\omega )$is a narrow band pass filter function with center frequency at$2{\omega}_{0}$. Then the SH intensity can be calculated according to the expression${I}_{2{\omega}_{0}}\propto {\displaystyle {\int}_{-\tau /2}^{+\tau /2}{[{E}_{2{\omega}_{0}}(t)]}^{2}dt}$in the duration time$\tau $of a probe laser pulse. By neglecting the absorption, total incident field can be written as${E}_{in}(t)={E}_{{\omega}_{0}}(t)+{E}_{\text{THz}}(t+\Delta t)$. Thus treating the delay time$\Delta t$ as independent variable, one can calculate${I}_{2{\omega}_{0}}$containing the information of detected THz waveform with different probe laser energies, as shown in Fig. 2 .

## 3. Simulation and discussion

In our simulation, the density of gas molecules and radius are assumed to be $5\times {10}^{19}{\text{cm}}^{-3}$ and$10\text{\mu m}$, respectively. The probe laser pulse has a Gaussian formation with center wavelength $\lambda =800\text{nm}$and full width at half maximum (FWHM)${T}_{\text{FWHM}}=30\text{fs}$. Note that the THz field is an invariant with a $100\text{kV}/\text{cm}$peak field when laser pulse energy $W$is changed, where$W\propto {w}_{0}^{2}{T}_{\text{FWHM}}{I}_{0}$ and${I}_{0}$ is the peak intensity of probe laser.

The calculated SH intensity signals containing THz information, as shown in Fig. 2(b), have a unipolar feature when$W$is$30\text{\mu J}$, corresponding to the incoherent detection obtained in experiments. The calculated waveforms, with the increasing of probe laser energy, begin to show some bipolar characteristics, and a complete bipolar waveforms appears nearly when$W=150\text{\mu J}$. When probe pulse energy is$200\text{\mu J}$, one can obtain a waveform nearly identical to that of the real incident THz field (shown in Fig. 2(a)), referred to as coherent detection of the THz waves. Such results, i.e. transition of detection mode, have good agreements with that of experiments reported in Refs. 3 and 4.

Another observed result, i.e. the dependence of peak SH signal${I}_{2{\omega}_{0}}$on probe laser energies$W$, based on which detection mode can be categorized as incoherent, hybrid and coherent detection, is reproduced as red line in Fig. 3 . The measured${I}_{2{\omega}_{0}}$, in original report [4], could be phenomenally fitted as a quadratic function at lower probe energy according to FWM model, which, however, failed to explain the situation at high-power probe laser, namely the intensity clamping [4–6]. Our further calculation shows that the blue line shown in Fig. 3, i.e. plasma density${N}_{e}$induced by probe lasers versus$W$, has a similar evolution characteristic with that of${I}_{2{\omega}_{0}}$. Particularly, when the complete coherent detection, corresponding to such intensity clamping, emerges at$W=200\text{\mu J}$,${N}_{e}$begins to be saturated, indicating that plasma density could play a key role in whole detection process. It is worth noted that there exist other additional saturation mechanisms, e.g. plasma defocusing effects. Although such mechanism could change some numerical results, e.g. the lowest probe laser intensity for coherent detection, the main result, i.e. conversion from incoherent to coherent detection as shown in Fig. 3, will not be significantly affected.

To manifest the physical nature of such experimental and our numerical results obtained above, a detailed analysis is performed as followed. Since plasma density${N}_{e}$is determined by total incident field${E}_{in}(t)$, Eq. (1) can be rewritten as${E}_{out}^{{\omega}_{0}+T,{\omega}_{0}+T}(t)\propto {E}_{in}(t){N}_{e}(\left|{E}_{in}(t)\right|)$, from which it is clear that the contribution of THz field${E}_{\text{THz}}$ to plasma emission can be divided into two parts: one is participating the acceleration of freed electrons in plasma, the other is that to plasma density${N}_{e}$. To answer the question that through which process the THz waves can be encoded into the observed SH signals in the whole detection process, we firstly investigate the effect of such waves on plasma emission${E}_{out}^{{\omega}_{0}+T,{\omega}_{0}}(t)\propto {E}_{in}(t){N}_{e}^{{\omega}_{0}}(\left|{E}_{{\omega}_{0}}(t)\right|)$only by means of accelerating freed electrons, i.e. meanwhile neglecting their contribution to the plasma density. Calculated spectrum of ${E}_{out}^{{\omega}_{0}+T,{\omega}_{0}}(t)$is plotted with blue lines in Fig. 4 , from which it is clear that SH signals, with both low and high probe laser intensities, do not show any visible fluctuation at$2{\omega}_{0}$. However, the emission${E}_{out}^{{\omega}_{0},{\omega}_{0}+T}(t)\propto {E}_{{\omega}_{0}}(t){N}_{e}^{{\omega}_{0}+T}(\left|{E}_{in}(t)\right|)$, i.e. only considering THz waves contribute to plasma density, has significant changes at $2{\omega}_{0}$ (red dot lines in Fig. 4), indicating that it is by involving gas ionization process that information of THz waves is encoded into the detected SH signals ($2{\omega}_{0}$). Moreover the coincidence between red and green dot lines (the spectrum of real emission ${E}_{out}^{{\omega}_{0}+T,{\omega}_{0}+T}(t)$from gas plasma in THz-ABCD), especially near$2{\omega}_{0}$, also supports such conclusion.

Next we will show how the THz waves affect the ionization process. Since pulse width of a THz wave (~ps) is much longer than that of a probe laser (~fs), such detected wave can be approximately treated as a constant field in the duration time of laser pulse, leading to a small displacement on the previous symmetric laser waveform (Fig. 5(a) schematically shows such relation between monochromatic light${\omega}_{0}$and THz E-field). Note that this asymmetry is directional due to positive and negative THz E-filed relative to probe laser field. As shown in experiments and our simulation, the bipolar characteristics of a THz waves, i.e. positive and negative fields of THz wave, can be resolved in coherent detection, while not in incoherent detection. Moreover it is by involving gas ionization process, as analyzed above, that information of THz waves is encoded into the detected SH signals. Thus to seek how the THz fields with different polarities affect to ionization process, we calculate the time derivative of plasma density (TDPD) with single probe laser pulse${E}_{{\omega}_{0}}$,${E}_{{\omega}_{0}}+{E}_{\text{THz}}$ and${E}_{{\omega}_{0}}-{E}_{\text{THz}}$, denoted as$d{N}_{e}^{{\omega}_{0}}/dt$, $d{N}_{e}^{{\omega}_{0}+T}/dt$and$d{N}_{e}^{{\omega}_{0}-T}/dt$respectively, as shown in Fig. 5. From the insets of Fig. 5(b)-5(d), the TDPD distribution with both ${E}_{{\omega}_{0}}+{E}_{\text{THz}}$ (red line) and ${E}_{{\omega}_{0}}-{E}_{\text{THz}}$(green line), compared with that induced only by${E}_{{\omega}_{0}}$ (blue line), has a small increment, indicating that THz field can affect the ionization process. When laser pulse is not strong ($W=30\text{\mu J}$), as illustrated in Fig. 5(b), the TDPD distribution has a similar symmetric profile with probe pulse (cyan line), showing that gas ionization process occurs at both leading and trailing edge of a laser pulse. Note that the inset of Fig. 5(b) shows that although red line is different with green line at the probe pulse leading edge or trailing edge, further calculation shows that the area below these two lines is almost equivalent, while larger than that below blue line, indicating that both of these two hybrid incident fields, i.e. ${E}_{{\omega}_{0}}+{E}_{\text{THz}}$ and${E}_{{\omega}_{0}}-{E}_{\text{THz}}$, will produce more electrons than that generated by the single probe laser${E}_{{\omega}_{0}}$. Thus for low probe intensity, THz fields with different polarities has nearly same influence to the generation of gas plasma, indicating that both of positive and negative field in a real THz pulse cannot be resolved, namely detecting a unipolar THz waveform.

With the increasing of probe laser intensity, the TDPD distribution, as show in Fig. 5(c), begins to move to the leading edge of laser pulse and the previous symmetric profile is broken since more plasma is induced at the leading edge, in other words gas depletion appears at the trailing edge of laser pulse, which means that the intensity of SH emissions from gas plasma induced by ${E}_{{\omega}_{0}}+{E}_{\text{THz}}$ and${E}_{{\omega}_{0}}-{E}_{\text{THz}}$ will be different, leading that some bipolar characteristics of THz waveform can be resolved. Furthermore, Fig. 5(d) illustrates that when $W=200\text{\mu J}$the TDPD distribution has nonzero values only at the leading edge of laser pulse, in other words generation of plasma mainly occurs at leading edge, corresponding to the ionization saturation obtained above. Thus from the inset of Fig. 5(d), the TDPD caused by three incident fields${E}_{{\omega}_{0}}$,${E}_{{\omega}_{0}}+{E}_{\text{THz}}$ and${E}_{{\omega}_{0}}-{E}_{\text{THz}}$, respectively, has different profile in details, indicating different generation process of plasma, from which the E-field of THz pulse with different polarities can be resolved .

## 4. Conclusion

In conclusion, by using PC model, we systematically investigate the physical mechanism of THz-ABCD, which was previously understood as a reverse process of THz generation based on FWM model. The measured results, such as the transition of detection mode and SH intensity clamping, can be reproduced in our numerical calculation. Further analysis show that the information of THz waves can be encoded into the SH signals through modification of the gas ionization process, but not acceleration of freed electrons or through a FWM process.

## Acknowledgments

The authors gratefully acknowledge support from the National Natural Science Foundation of China under Grant No.10974063, 60907045 and 61177095, Hubei Natural Science Foundation under grant No. 2010CDA001, Ph.D. Programs Foundation of Ministry of Education of China under grant No. 20100142110042, and the Fundamental Research Funds for the Central Universities, HUST: 2011TS001, 2012QN094 and 2012QN097.

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