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Abstract
We theoretically present the waveform controls of terahertz (THz) radiations generated from homogeneous and rippled plasma within inhomogeneous external electrostatic field. The Particle-in-cell (PIC) simulations is implemented to demonstrate generation and controllability of three types of THz pulses: single frequency THz pulse in homogeneous plasma, broadband THz pulse and dual frequency THz pulse in rippled plasma. The single frequency THz pulse can be tuned via shifting the knob of electron density of homogeneous plasma. Waveform of broadband THz pulse can be regulated into an envelope-like shape by varying amplitude of electron density of rippled plasma. The two center frequencies’ interval of dual frequency THz pulse can be controlled by wave numbers of density distribution of rippled plasma. This work provides a potential means to generate the dual frequency THz pulses with two harmonic frequencies (ω+Ωω, Ω=2) or incommensurate frequencies (ω+Ωω, Ω=1.7,1.8, 2.2…).
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Terahertz (THz) generation is of great importance in time-domain spectroscopy field, such as the Rydberg states of silicon [1], spin waves [2], molecular orientation and alignment [3], the lattice vibrations of SrTiO3 [4], and quantum rotational coherence [5]. For producing ultra-short THz pulses, ultra-short laser pulse driving is a common method. Based on optical rectification, the femtosecond (fs) laser pulse driving organic crystals is used to generate narrowband THz pulses with strength ∼ GV/m [6]. Utilizing transient photocurrent induced by mixing the ultra-short fundamental frequency and second harmonic laser fields in gas, the super-broadband (sub-hundred THz) THz pulses with output energy > 5µJ are generated [7]. Under the photocurrent mechanism, transient quasi-dc current is dependent on ionization rates and drift velocity, and the conversion efficiency can be effectively improved by optimizing parameters of gas target [7] and laser waveform [8]. Based on the Cherenkov emission from charges oscillation of laser driving plasma, when a transverse electrostatic field is applied to plasma of laser filament, the conversion efficiency is enhanced effectively because plasma electrons acquire a biased acceleration in transverse direction [9,10]. So far, besides efforts of enhancing conversion efficiency, waveform control of THz pulse is another challenge. For the narrowband THz pulse, the waveform controls rely on photoconductive switching and optical rectification, containing applying the orientation-inverted nonlinear crystal with non-periodic structures [11], manipulating the temporal or spatial shape of laser pulse [12,13], and spatially manipulating dispersed multi-frequency components generated in a fanned-out periodically poled crystal [14]. Because the radiation based on photocurrent mechanism covers in a wide frequency range consisting of THz, infrared, and extreme ultraviolet [7], the control of waveform only for THz portion of emission remains largely difficult. Under the Cherenkov emission of plasma, the bias field is applied to the preformed plasma, the non-uniformly distributed response field caused by the accelerated electrons can be converted into THz radiation [15]. According to Maxwell's classical electromagnetic theory, since the distribution of response field depends on distributions of external electrostatic field and plasma electron density, THz waveforms of plasma emission can be controlled by presetting the distributions of external electrostatic field and plasma electron density.
In this paper, we exploit an inhomogeneous electrostatic field, which is set as a tip-shaped field with symmetrical distribution, to drive the homogeneous and rippled plasma separately, and generate the controllable waveform THz pulses. The preformed plasma with homogeneously distributional electron density is adopted to generate the single frequency THz pulses, and this THz pulse can be tuned via varying electron density of plasma. The rippled plasma with periodically distributional electron density is adopted to generate the broadband THz pulses and the dual frequency THz pulses. Waveform of broadband THz pulse can be drastically regulated by varying amplitude of electron density of rippled plasma. In the generations of dual frequency THz pulses, its two center frequencies’ interval can be controlled by adjusting the wave numbers of electron density distribution of rippled plasma. The dual frequency THz pulses with two harmonic frequencies or incommensurate frequencies are also produced. The generational mechanisms of the three types of THz pulses are discussed by approximately solving the d’Alembert equation of response field excited by external electrostatic field.
2. Theoretic model
An external electrostatic field (also referred to a dc bias field) is applied to the preformed plasma region along the y direction by placing a needle electrode. The preformed plasma is placed along the x direction. After plasma electrons are driven by this dc bias field, redistribution of electron density causes the new electromagnetic fields, and the interaction between plasma electrons and electromagnetic fields is described by Maxwell’s equations:
In the simulation, the 300µm-long simulation box which is placed on the x-axis is discretized by 32768 grids. The 270µm-long plasma with the homogeneous and inhomogeneous density distribution are set along the x direction. Considering the electron density of preformed plasma is within 1017- 1018 /cm3, an average collision frequency ve ∼1013 Hz is adopted in calculations [18]. 84000 macro particles are placed in the plasma region, resulting in averaged 2.8 initial simulation particles per cell. The observation position of the THz field is placed at vacuum 2.6 µm away from the boundary of plasma. The external electric field is provided by a needle electrode. The pointed profile of external electrostatic field is expressed by the formula: $E(x )= {E_0}({x - {x_l}} )({{x_r} - x} )$, where x goes between left xl and right boundary xr of plasma, E0 = 18 kV/cm.
3. Results and discussion
3.1 Singe frequency THz pulse generation in homogeneous density plasma
We start with the THz wave generations of homogeneous plasma exposed in external electrostatic field. When there is a transversely bias in preformed plasma, plasma electrons are driven to produce the response field. The non-uniformly distributed response field, whose distribution is related to distribution of external electrostatic field, is converted into the THz radiation field. The waveform properties of THz generated in homogeneous plasma are show in Fig. 1. Figure 1(b) show three THz waveforms when plasma density ne = 0.6 × 1018/cm3, 1.2 × 1018/cm3, and 2.4 × 1018/cm3, respectively. With the increscent of plasma density, the THz peaks are 1.17 kV/cm, 0.89 kV/cm, 0.66 kV/cm. The three waveforms can be regarded as cosine waves (see Eq. (10)), and their initial phases are given by fitting: φ = 1.15π, π and 0.8π. Figure 1(c) show the normalized THz spectra. The central frequencies of three THz waves vTHz = 7 THz, 10 THz, and 14 THz, respectively. These three frequencies are close to plasma frequencies ${\nu _p} = \sqrt {4\pi {e^2}{n_e}/{m_e}} /2\pi \approx $6.87THz, 9.86 THz, 13.90THz corresponding to ne = 0.6 × 1018/cm3, 1.2 × 1018/cm3, 2.4 × 1018/cm3. It should be noted that besides the ac component with frequency ωTHz ∼ ωp, the generated THz wave includes a dc component [15]. This dc component is abandoned from THz fields as the THz waveform is unaffected. As the numerical results shown, the narrowband THz wave is generated in plasma with homogeneous density by an inhomogeneous electrostatic field, with the central frequency ωTHz determined by plasma density ne. THz conversion efficiency can be expressed by $\textrm{g} = {I_{THz}}/{I_{in}}$, where ${I_{THz}}$ and ${I_{in}}\; $ are THz output and input powers. Considering power is proportional to square of amplitude of electric field, efficiency is estimated via peak of electric field: $\textrm{g} = E_{THZ0}^2/E_{ex0}^2$, where ${E_{THz0}}$ and ${E_{ex0}}$ are peaks of THz and external electrostatic fields. For fixed external electrostatic field of amplitude ∼18 kV/cm, THz conversion efficiencies are ∼4.22%, ∼2.44%, ∼1.34%. Efficiency decrease with the increscent of plasma density ne. As a whole this scheme presents a low conversion efficiency.
Fig. 1. Singe frequency THz generations in homogeneous plasma with different electron densities: (a) distribution of external electrostatic field, (b) THz temporal waveforms, and (c) the corresponding normalized spectra.
In order to reveal mechanisms of THz radiation, based on classical electromagnetic field theory, the theoretical analysis for electromagnetic radiation is presented. As introduced above, the electromagnetic wave emitted in the preformed plasma with the external electrostatic field is caused by the response field. The response field originates from electric current formed by electrons accelerated under the external field. The vector potential $\vec{A}$ of plasma response field satisfies d’Alembert equation (Coulomb gauge)
If the homogeneous plasma is exposed in a homogeneous external electric field, same response field is generated inside plasma at any position except boundaries. When an inhomogeneous external electric field is applied, the spatial dependence of response field need been considered. The external bias field Eb(x) driving response field satisfies Eq. (7):
Then electromagnetic radiation fields are calculated by ${\vec{B}_{rad}} = \nabla \times \vec{A}$ and ${\vec{E}_{rad}} ={-} \nabla \varphi - \frac{{\partial \vec{A}}}{{\partial t}}$, where ∇φ and $\frac{{\partial \vec{A}}}{{\partial t}}$ are the longitudinal and transversal components. If external electric field ${E_b}(x )$ is homogeneous, since vector potential Ay distributes uniformly inside plasma, radiation magnetic field ${\vec{B}_{rad}} = \nabla \times \vec{A} = 0$. Thus, the radiation is generated only at the plasma-vacuum boundary. When ${E_b}(x )$ is inhomogeneous, the total THz field is compounded from radiated fields generated at different positions:
The constant of the derivative of Ay, which represents dc component, is ignored. ${t_i}$ is time for THz field propagating from the generated position xi to observational position. The features of numerical THz fields can be described by Eq. (10). Since the THz frequency is dependent on oscillation frequency of plasmas, the THz pulse can be tuned via varying electron density of homogeneous plasma. In addition, Eq. (10) shows that THz wave is born with zero phase, hence phase of the observed THz wave originates from propagation phase $\phi = \eta {k_0}r$, where refractive index η and wave vector in vacuum k0 satisfy dispersion relation: $\omega _{THz}^2 = \omega _{pe}^2 + {({c\eta {k_0}} )^2}$. In generations of singe frequency THz, because the geometric size (i.e propagation distance r) is fixed, the THz phase is proportional to $\eta {k_0} = \frac{1}{c}\sqrt {\omega _{THz}^2 - \omega _{pe}^2} $. With the increscent of plasma density, wave vector ηk0 decreases, and THz phase decreases.
3.2 Broadband THz pulse generation in plasma of ripple with few wave numbers
Now we show generations of THz wave in rippled plasma. The density distribution of rippled plasma is set as ${n_e}(x )= {n_0} - {n_0}\mathrm{\delta cos}({{{\rm k}_{\rm q}}{\rm x}} )$, there n0 is uniform density of plasma, δ is ratio of ripple amplitude to n0, and kq is angular wave number of ripple. According to implement scheme of density ripple in plasma [19,20,21], the rippled wave number depends on periodic distribution of the interference pattern which is induced by two oppositely propagating lasers and can be regulated by focal system of laser. Considering that gas ionization is proportional to total laser intensity, the rippled amplitude could be controlled by the intensities of two lasers. In order to display the dependence of angular wave number kq on THz wave properties, the THz generational cases of ripple are discussed by varying wave numbers of ripple.
The angular wave number of rippled plasma k1 = 2π/270 µm-1 is set, i.e., there is only one rippled wave number (q = 1) in plasma (see in Fig. 2(a)). With fixing angular wave number and changing the rippled amplitude as δ = 0.1, 0.3, 0.5, 0.7, the generated THz waves are shown in Fig. 2(b). Under the same external electrostatic field, the amplitude of generated THz wave increases with the increscent of rippled amplitude. THz amplitude is approximately 0.95 kV/cm in the case of rippled amplitude δ = 0.1. The amplitude increases to ∼ 8 kV/cm when rippled amplitude is 70%. For external electrostatic field of amplitude ∼18 kV/cm, the former presents the conversion efficiency of 2.5%, while the latter is close to 19%, the conversion efficiency is improved by a factor of 7.6. Comparing with the case of homogeneous density, THz generation in rippled plasma presents higher conversion efficiency. Furthermore, the THz waveform is sensitively dependent on the rippled amplitude. THz wave at case of the rippled amplitude δ = 0.1 is approximately a sinusoidal wave. With the increasing rippled amplitude, THz waveforms are significantly modulated in time domain of 200∼1000fs. At case of the rippled amplitude δ=0.7, THz pulse presents waveform like an envelope.
Fig. 2. Broadband THz generations in rippled plasma (q = 1) with different ripple amplitudes: (a) density distributions of rippled plasma with different ripple amplitudes, (b) THz temporal waveforms, and (c) the corresponding normalized spectra.
The frequency characteristics of THz wave are shown as the normalized spectra in Fig. 2(c). Comparing with Fig. 1(c), THz spectra at the cases of rippled plasma present more broadband. The bandwidth are 1.8 THz, 2.77 THz, 3.85 THz, and 5.35 THz when δ=0.1, 0.3, 0.5, and 0.7, presenting a positive correlation. Considering the existed density gradient in rippled plasma, according to Eq. (10), oscillation frequency of response field stimulated at any position by external electrostatic field depends on electron density ne (x) of this position. Therefore, frequency of THz wave driven by the response field is determined by electron density distribution ne (x). The small THz frequency vs = 9.86 THz for boundary density n0 = 1.2 × 1018/cm3, and the larger frequency vl is estimated by the density ne(xc), where xc is the center of plasma region. At the case of weakly ripple (δ=0.1), the larger THz frequency is close to the small THz frequency because ne(xc) is close to n0, and the THz spectrum of frequency vs ≈ vl ∼ 10 THz presents a narrowband of 2THz. With the increscent of rippled amplitude, the frequency vl will be separate from vs owing to the increase of difference between ne(xc) and n0. When δ = 0.7, vl is equal to 12.85THz for density ne(xc) = 1.7n0, and its bandwidth is 5.35 THz.
The further modulation of THz waveform is carried out by increasing angular wave number of rippled plasma. Ripple with two wave numbers (q = 2) is set, i.e. k2 = 2π/135 µm-1, as Fig. 3(a) shown. Figure 3(b) shows four THz waveforms generated in the rippled plasma with rippled amplitude δ = 0.1, 0.3, 0.5, and 0.7. Compared with the case of one rippled wave number (Fig. 2(b)), the trailing edges of these THz waveforms are modulated further. With the increasing rippled amplitude, more serious modulation for the trailing edge of THz wave occurs. When rippled amplitude δ = 0.7, the THz pulse becomes an asymmetric envelope in time domain. Corresponding THz spectra are shown in Fig. 3(c). THz spectrum of rippled amplitude δ=0.1 is a narrowband with central frequency ∼ 10 THz, like the case of one rippled wave number. With the increscent of rippled amplitude, bandwidth of THz spectrum is extended.
Fig. 3. Broadband THz generations in rippled plasma (q = 2) with different ripple amplitudes: (a) density distributions of rippled plasma with different ripple amplitudes, (b) THz temporal waveforms, and (c) the corresponding normalized spectra.
3.3 Dual frequency THz pulse generation in plasma of ripple with multi wave numbers
Next, we discuss THz wave generations at the case of ripple with multi wave numbers (q > 2). Figure 4(a) shows THz waveforms of ripple wave numbers q = 7, 9, 11, and the corresponding spectra is shown in Fig. 4(b). The three THz spectra possess two central frequencies whose values are close. When rippled wave numbers q = 7, the two central frequencies v1 = 9THz and v2 = 13THz. With the increment of rippled wave number, the large central frequency value v2 increase, but the small central frequency v1 is almost constant. The frequency difference between two central frequencies is enlarged. When q = 11, two central frequencies are v1 = 9.8 THz and v2 = 16 THz, and the frequency difference Δv is equal to 6.2 THz. The further increment of rippled wave number q = 12, 14, 16, 18, 20 can realize significantly tuning for large central frequency: v2 = 17 THz, 18 THz, 20 THz, 22 THz, 24 THz, and substantially enlarge its frequency difference: Δv = 7 THz, 8 THz, 10 THz, 12 THz. Above all, THz waves with harmonic frequencies or incommensurate frequencies can be generated in rippled plasma by adjusting ripple wave number, and the two frequencies’ interval can be controlled arbitrarily. In generations of dual frequency THz, phases of the two THz signals with frequencies v1 and v2 are extracted by Fourier transform. Figure 5 shows phases of the two THz signals at the cases of wave number q = 12, 14, 16, 18, 20. Black square (Red circle) represents phase of THz with frequency v1 (v2). With the increment of wave number q, phase of THz signal with the large central frequency v2 presents larger increment than THz signal with the v1. It was obvious that THz signal with short wavelength (i. e large frequency v2) obtains more phase shift in plasma due to its lager refractive index $\eta = \sqrt {1 - {{({{\omega_{pe}}/{\omega_{THz}}} )}^2}} $ than THz signal with long wavelength.
Fig. 4. Dual frequency THz generations in rippled plasma with different wave numbers of ripple: (a) THz Temporal waveforms and (b) the corresponding normalized spectra at case of wave numbers q = 7, 9, 11 ; (c) THz temporal waveforms and (d) the corresponding spectra at case of wave numbers q = 12, 14, 16, 18, 20.
Fig. 5. Phases of dual frequency THz at cases of wave number q = 12, 14, 16, 18, 20: Black square and Red circle represent phases of THz with central frequencies v1 and v2.
At the case of ripple with multi wave numbers, the control of dual frequency of generated THz pulse can be explained with the assumption of the vector potential with periodical distribution: ${A_y}(x )= {A_0}\textrm{sin}({{k_q}x} )$, where A0 is amplitude, and ${k_q} = \frac{{2\pi }}{{{\lambda _q}}}$ is angular wave number. λq is the wavelength of periodical density distribution of rippled plasma, and q expresses the total wave number of the whole length of plasma. The variable separation form of vector potential ${A_y}({x,t} )= {A_y}(x ){A_y}(t )$ is inserted into Eq. (7), it is simplified into
According to Eq. (11), the vector potential of response field becomes an oscillatory solution with frequency being equal to $\sqrt {{c^2}k_q^2 + \omega _p^2} $ . The corresponding radiation field is written as
In the rippled plasma, there are coexistent of the THz radiations represented by Eq. (10) and Eq. (12). Hence, the total generated THz field can be expressed as the linear superposition of the two groups of radiation fields:
According to Eq. (13), the total THz field appears two center frequencies. The small frequency (${\omega _p} = \sqrt {4\pi {e^2}{n_e}/{m_e}} $) is determined by the electron density near the position where THz radiates, and the lager frequency ($\omega ^{\prime} = \sqrt {{c^2}{k_q}^2 + {\omega _p}^2} $) depends on both of the electron density ne and the angular wave number of ripple kq. The interval of two frequencies is related to angular wave number kq. If it is a large angular wave number kq, radiation frequency $\omega ^{\prime}$ is far larger than ωp, and the generated THz pulse appears two separate central frequencies. The two center frequencies’ interval can be controlled via varying the angular wave number kq of rippled plasma, as Fig. 4 (c) and (d) shown. At the case of q = 14 in Fig. 4 (d), the small frequency of generated THz is ∼10 THz, and it corresponds plasma frequency ωp ∼ 9.8 THz which is estimated by ${\omega _p} = \sqrt {4\pi {e^2}{n_0}/{m_e}} $, where the electron density n0 = 1.2 × 1018/cm3. Besides, the large frequency of generated THz is ∼18 THz, and it is consistent with the estimated value 18.4 THz by $\omega ^{\prime} = \sqrt {{c^2}{k_q}^2 + {\omega _p}^2} $, where the angular wave number kq = 2π×5.18 × 104/m as the wave numbers q = 14 in 270µm-long plasma.
4. Conclusions
In conclusion, we propose that waveform-controlled THz radiation can be produced from the preformed plasma within an inhomogeneous external electrostatic field. In this scheme, the preformed plasma could be seen as the oscillator of THz source. When the plasma electron density that represents the eigen oscillation frequency of plasma (${\omega _p} = \sqrt {4\pi {e^2}{n_e}/{m_e}} $) is shifted as value of THz frequency range, the external electrostatic field which is used as the pump source can drive plasma electron to excite response fields. These inhomogeneous response fields can be converted into THz radiation. Three types of ultra-short intense THz pulses with duration ∼ ps, i.e. single frequency THz pulse in homogeneous plasma, broadband THz pulse and dual frequency THz pulse in rippled plasma, are generated by this THz source. The maximum conversion efficiency is close to 44%, and the peak of THz output can reach to 8 kV/cm. The single frequency THz pulse can be tuned by shifting electron density of homogeneous plasma. By the modulation of the amplitude of electron density of rippled plasma, the waveform of broadband THz can be regulated into an envelope-like shape. By the modulation of the wave numbers of density distribution of rippled plasma, the two center frequencies’ interval of dual frequency THz pulse can be controlled to generate the dual frequency THz pulse with two harmonic frequencies or incommensurate frequencies.
Funding
National Natural Science Foundation of China (11765018, 11964033, 12164044).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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