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Terahertz radiation emission from plasma beat-wave interactions with a relativistic electron beam
Optics Communications 401 (2017) 71–74
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Terahertz radiation emission from plasma beat-wave interactions with a relativistic electron beam D.N. Gupta a, *, V.V. Kulagin b , H. Suk c a b c
Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India Sternberg Astronomical Institute, Moscow State University, Moscow, 119 992, Russia Department of Physics and Photon Science, Gwangju Institute of Science and Technology, Gwangju, 61005, South Korea
a r t i c l e
i n f o
Keywords: Electromagnetic radiation Frequency downshift Laser-plasma
a b s t r a c t We present a mechanism to generate terahertz radiation from laser-driven plasma beat-wave interacting with an electron beam. The theory of the energy transfer between the plasma beat-wave and terahertz radiation is elaborated through nonlinear coupling in the presence of a negative-energy relativistic electron beam. An expression of terahertz radiation field is obtained to find out the efficiency of the process. Our results show that the efficiency of terahertz radiation emission is strongly sensitive to the electron beam energy. Emitted field strength of the terahertz radiation is calculated as a function of electron beam velocity. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Terahertz (THz) radiation is the portion of the electromagnetic spectrum at the boundary between the microwaves and the infrared. There are several well-known sources for coherent THz radiation generation such as solid state oscillators, quantum cascade lasers, optically pumped solid state devices and novel free electron devices, which have n turn stimulate a wide range of applications from material science to telecommunications, from biology to biomedicine [1]. Today most widely used sources of pulsed THz radiation are laser driven THz emitters based on frequency down conversion from the optical region [2,3]. In free-electron based sources, to overcome the necessity of reducing the size of the components, several methods have been proposed to let the electrons exchange momentum and allow photon emission. The most familiar method is the use of magnetic wiggler to generate the electromagnetic radiation by the electron beam [4]. In this scheme, an electron beam within a distance comparable to the wavelength of the radiation can emits the coherent radiation. Moreover, a radiofrequency modulated electron beam at wavelengths comparable to the electron bunch length can generates the short pulses of coherent THz radiation. There also exits laser based THz sources, which can fit on or scale of the size of a table-top. A terahertz field greater than 400 kV/cm generated using short 25 fs pulses has been reported by Bartel et al. [5]. Karpowicz et al. [6] reported the generation and detection of broadband terahertz pulse in air covering the spectral range of 0.3 to 10 THz with * Corresponding author.
E-mail address: [email protected] (D.N. Gupta). http://dx.doi.org/10.1016/j.optcom.2017.05.043 Received 13 January 2017; Received in revised form 10 May 2017; Accepted 16 May 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
10% or larger of the maximum electric field. Terahertz wave generation mechanism in air was originally attributed to the four-wave rectification process through the third-order optical nonlinearity of air [7]. However, it has been suggested that the plasma formation plays an important role in the terahertz wave generation process [8]. The four-wave rectification mechanism was tested by Xie et al. [9] by controlling the relative phase, amplitude and polarization of the fundamental and second harmonic pulses. Kim et al. [10] proposed a transient photo current model based on the rapid ionization by femtosecond laser pulses by following the driven motions of electrons by an asymmetric electric field. Laser-plasma based accelerators [11] such as the laser wakefield accelerators [12–14] and the laser beat-wave accelerators [15–17] are widely being researched as possible alternate sources for high energy electron beam. These systems could be used to generate THz radiation. There are several theoretical mechanisms to generate THz from laserplasma interactions [18–20]. Bystrov et al. [18] have proposed the Terahertz radiation of plasma oscillations excited upon the optical breakdown of a gas. Chen [19] has studied the THz radiation using a tailored laser pulse in a plasmas. Such radiation source has a broad tunability range in frequency spectra. Antonsen et al. [20] have a theoretical model for THz radiation generation in a nonuniform plasma. Miniature corrugated channels has been proposed for THz radiation generation by bunched electron beams in plasmas. In our work, we utilize the plasma beat-wave generated in laser beatwave accelerator for THz radiation generation process. In a beat-wave
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Optics Communications 401 (2017) 71–74
the plasma and beam electrons. Solving the equation of motion and continuity, the velocity and density perturbation of the beam electrons can be written as
accelerator, two co-propagating lasers, having frequency difference equal to the plasma electron frequency, can excite a large amplitude plasma wave. We use this plasma beat-wave to generate THz radiation while interacting with a negative energy relativistic electron beam. The negative energy of the beam means that the electron beam propagates opposite to the plasma wave in this scheme. An electron beam propagating through a plasma causes the onset of plasma instability due to the free energy available in the relativistic motion between the beam and the plasma [21]. The plasma beat-waves at moderate laser intensities acquire levels large enough to causes the onset of parametric instabilities. Beam space charge mode can be existed if an electron beam is present. The plasma beat-wave nonlinearly couples with the THz electromagnetic mode to drive the negative-energy beam space-charge mode. Both decayed waves grow in the expense of the beam and the plasma beat-wave. The oscillatory velocity combines with an existing negative-energy beam mode to produce ponderomotive force that drives the electromagnetic sideband. The sidebands couple to the pump wave to produce a nonlinear current that drives the instability. In a result, the plasma beat-wave and the electron beam both lose their energy during the nonlinear interactions and generate THz radiation. In Section 2, we present the theoretical model used to find out the THz radiation field strength. In Section 3, we explain the numerical results and discuss the physics behind the suggested mechanism and we make final remark in the last section.
𝜐𝟎𝑏 = 𝑛10𝑏 =
𝜐11𝑏 =
𝐄𝟎2 = Re 𝑥𝐴 ̂ 02 exp[−𝑖(𝜔02 𝑡 − 𝑘02 𝑧)],
(2)
𝑛𝑏 =
𝑛=
(6)
.
𝑒𝐄1 3 (𝜔 − 𝑘 𝜐 ) 𝑚𝑖𝛾0𝑏 1 1 0𝑏
(7)
.
𝑘2𝑏 4𝜋𝑒
(8)
𝜒𝑏 (𝜙 + 𝜙𝑝 ),
𝑘2𝑏 4𝜋𝑒
(9)
𝜒𝑒 (𝜙 + 𝜙𝑝 ),
where 𝜒𝑒 = −𝜔2𝑝 ∕𝜔2𝑏 . Using the density perturbation in Poisson’s equation, we obtain 𝜀𝜙 = −𝜒𝑏 𝜙𝑝𝑏 − 𝜒𝑒 𝜙𝑝 , where 𝜀 = 1 + 𝜒𝑒 + 𝜒𝑏 . The nonlinear current density and density perturbation at the THz electromagnetic wave can be written as 𝐉𝐍𝐋 = −𝑛𝑏 𝑒𝝊𝟏0𝑏 ∕2. The wave 1 equation governing the terahertz field can be written as − ∇2 𝐄1 + ∇(∇ ⋅ 𝐄𝟏 ) = −
4𝜋𝑖𝜔1 𝑐2
𝐉𝐍𝐋 + 1
𝜔21 𝑐2
(10)
𝜀1 𝐄1 ,
where 𝜀1 = 1 − 𝜔2𝑝 ∕𝜔21 . Assuming the radial variation of the electrostatic pump mode to be small over the width of the beam and taking the divergence of Eq. (10), we obtain 𝐄1 = −4𝜋𝑖𝜔1 𝐉𝐍𝐋 ∕(𝜔21 − 𝜔2𝑝 ), which 𝟏 gives
(3) |𝐄𝟏 | = −
where 𝜔𝑝 = (4𝜋𝑛0 𝑒2 ∕𝑚)1∕2 is the unperturbed plasma frequency, 𝜐2𝑡ℎ = 3𝑇𝑒 ∕𝑚, 𝑇𝑒 is the electron temperature, 𝛥𝑘 = 𝑘0 and 𝛥𝜔 = 𝜔0 ≈ 𝜔𝑝 are the frequency and wave number differences of the two laser pumps. Under cold plasma approximation, we take 𝜐𝑡ℎ ≈ 0 and the solution of Eq. (3) can be in the form [ ( )] 𝐄𝑝 = Re 𝑧𝐸 ̂ 𝑝0 exp −𝑖 𝜔0 𝑡 − 𝑘0 𝑧 .
− 𝑘0 𝜐0𝑏 )2
where 𝜒𝑏 = −𝜔2𝑝𝑏 ∕𝛾03 (𝜔𝑏 − 𝑘𝑏 𝜐0𝑏 )2 and 𝜔𝑝𝑏 = (4𝜋𝑛0𝑏 𝑒2 ∕𝑚)1∕2 . Similarly, the plasma electron density perturbation is
where |𝐴01 |2𝑥=0 = 𝐴001 exp(−𝑥2 ∕𝑟201 ), |𝐴02 |2𝑥=0 = 𝐴002 exp(−𝑥2 ∕𝑟202 ), 𝜔01 − 𝜔02 ≈ 𝜔𝑝 , 𝜔01 ≥ 𝜔𝑝 , 𝜔02 ≥ 𝜔𝑝 , 𝜔01 and 𝜔02 are the laser frequencies, 𝐤𝟎1 and 𝐤𝟎2 are the wave vectors of the laser beams, and 𝑟01 and 𝑟02 are the laser spot sizes. They produce oscillatory velocities 𝜐𝟎𝑗 = 𝑒𝐄𝟎𝑗 ∕𝑚𝑖𝜔0𝑗 (𝑗 = 1, 2) and exert a ponderomotive force 𝐅𝐩 = −(𝑒∕2𝑐)(𝜐𝟎1 ×𝐁∗𝟎2 +𝜐𝟎2 ×𝐁∗𝟎1 ) = 𝑧𝑖𝑒𝑘 ̂ 0 𝜙𝑝0 exp[−𝑖(𝜔0 𝑡 − 𝑘0 𝑧)] on them, where 𝜙𝑝0 = 𝑒𝐴01 𝐴02 ∕2𝑚𝜔01 𝜔02 , 𝜔0 = 𝜔01 − 𝜔02 , 𝐤𝟎 = 𝐤𝟎1 − 𝐤𝟎2 , 𝐁𝟎𝑗 = 𝑐𝐤𝟎𝑗 × 𝐄𝟎𝑗 ∕𝜔0𝑗 , and −𝑒 and 𝑚 are the electron charge and mass, respectively. Because of the large mass and the slow response of the ions, we consider them immobile. The ponderomotive force drives a large amplitude beat-wave in a plasma. We write the Eulerian equation for the plasma-wave electric field driven by the ponderomotive force of the beating laser pumps as [22] 1 𝐸̈ 𝑝 + 𝜔2𝑝 𝐸𝑝 + 𝜐2𝑡ℎ 𝐸𝑝′′ = 𝐴01 𝐴02 𝜔20 sin(𝛥𝑘𝑧 − 𝛥𝜔𝑡), 2
𝑒𝑘𝑝 𝐄𝐩 3 (𝜔 𝑚𝑖𝛾0𝑏 0
(5)
,
The THz radiation (𝜔1 , 𝐤1 ) couples with the pump wave (𝜔0 , 𝐤𝟎 ) to produce a ponderomotive force (𝐅𝐩 = 𝑚𝛾𝜐 ⋅ ∇𝜐) at frequency of (𝜔𝑏 , 𝐤𝑏 ). The longitudinal ponderomotive force on the beam electrons can be 5 (𝜔 − explicitly written as 𝐅𝐩𝑏 = 𝑧𝑒𝑖𝑘 ̂ 𝑏 𝜙𝑝𝑏 , where 𝜙𝑝𝑏 = −𝑒𝐄∗𝐩 𝐄𝟏 ∕2𝑚𝛾0𝑏 1 𝑘1 𝜐0𝑏 )(𝜔0 − 𝑘0 𝜐0𝑏 ). Ponderomotive potential 𝜙𝑝 for plasma electrons can be deduced by taking 𝜐0𝑏 = 0 in the same expression. 𝜙𝑝 and 𝜙 at (𝜔𝑏 , 𝐤𝑏 ) produce electron beam density perturbation
We consider a plasma of the density 𝑛0 and the electron temperature 𝑇𝑒 . Two collinear laser beams of large amplitude propagate in a plasma with electric fields (1)
− 𝑘0 𝜐0𝑏 )
Here, we perturb the plasma equilibrium as 𝝊𝑏 = 𝝊0𝑏 + 𝝊10𝑏 , 𝑛𝑏 = 1 , where 𝛾 2 2 −1∕2 and 𝛾 1 = 𝑛0𝑏 + 𝑛10𝑏 , 𝛾𝑏 = 𝛾0𝑏 + 𝛾0𝑏 0𝑏 = (1 − 𝜐0𝑏 ∕𝑐 ) 0𝑏 3 2 2 𝛾0𝑏 (𝜐0𝑏 ∕𝑐 ). Corresponding quantities for plasma electrons, 𝝊0𝑒 and 𝑛0𝑒 , can be deduced from Eqs. (5) and (6) by dropping the subscript 𝑏 and taking 𝝊0𝑏 = 0. The plasma beat-wave decays into the negative energy beam mode with potential of 𝜙 = Φ exp[−𝑖(𝜔𝑏 𝑡 + 𝑘𝑏 𝑧)] and a THz electromagnetic wave with electric field of 𝐄𝟏 = 𝐴1 exp[−𝑖(𝜔1 𝑡 + 𝑘1 𝑧)]. The linear response of the beam electrons at (𝜔1 , 𝐤1 ) is
2. Theoretical model
𝐄𝟎1 = Re 𝑥𝐴 ̂ 01 exp[−𝑖(𝜔01 𝑡 − 𝑘01 𝑧)],
𝑒𝐄𝐩 3 (𝜔 𝑚𝑖𝛾0𝑏 0
𝑒𝑘2𝑏 𝜔2𝑝𝑏 |𝐄𝐩 |𝜙𝑝 2𝑚𝛾06 𝜔1 (1 − 𝜔2𝑝 ∕𝜔21 )(𝜔0 − 𝑘0 𝜐0𝑏 )(𝜔𝑏 − 𝑘𝑏 𝜐0𝑏 )2
,
(11)
where we have considered that, in the case of when beam density (or the beam current) is small, 𝜒𝑏 ≪ 1, the self-consistent potential of the beam can be neglected as compared to the ponderomotive potential (𝜙 ≪ 𝜙𝑝𝑏 ).
(4)
3. Numerical results and discussion
The plasma beat-wave acquires a large amplitude to onset of the parametric instabilities. We also consider a relativistic electron beam of density 𝑛0𝑏 passes through it in opposite direction with initial velocity of −𝜐𝑜𝑏 𝑧. ̂ Thus the plasma beat wave is traveling in opposite direction to the electron beam. The fundamental beat-wave (𝜔0 , 𝐤0 ) parametrically couples to the electron beam and excites a negative energy beam mode (𝜔𝑏 , 𝐤𝐛 ) and a THz electromagnetic wave (𝜔1 , 𝐤1 ), where 𝜔1 = 𝜔𝑏 + 𝜔0 , and 𝐤𝟏 = 𝐤𝐛 +𝐤𝟎 . The slow-beam space charge mode has negative energy (𝜔𝑏 = −𝐤𝑏 ⋅ 𝜐0𝑏 ). The plasma beat-wave imparts oscillatory velocity to
Using Eq. (11), we depict the THz field strength with respect to the THz frequency. The used numerical parameter are as follows: 𝑎01 = 𝑒𝐴01 ∕𝑚𝜔01 𝑐 = 1 and 𝑎02 = 𝑒𝐴02 ∕𝑚𝜔02 𝑐 = 1 (corresponding to peak laser intensity 𝐼01 ≈ 𝐼02 ≈ 1.37 × 1018 W∕cm2 ), the considered wavelength of first laser is 𝜆01 ∼ 1 µm and the wavelength of second laser has been chosen according to the resonant condition for beat-wave excitation, and the initial waist size is 𝑟01 ≈ 𝑟02 ∼ 10 µm. The plasma electron density considered for this calculation is 𝑛0 ≈ 1016 cm−3 . We consider that an 72
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Fig. 1. Normalized field strength of THz radiation with frequency (in Hz) for different electron beam velocities of 𝜐0𝑏 ≈ 0.2𝑐 (red curve), 𝜐0𝑏 ≈ 0.4𝑐 (green curve), and 𝜐0𝑏 ≈ 0.6𝑐 (black curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Efficiency of THz radiation generation with frequency (in Hz) for different electron beam velocities of 𝜐0𝑏 ≈ 0.2𝑐 (red curve), 𝜐0𝑏 ≈ 0.4𝑐 (green curve), and 𝜐0𝑏 ≈ 0.6𝑐 (black curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
electron beam of density of the order of 𝑛0𝑏 ≈ 1011 cm−3 is propagates through a plasma with different velocities ranging from 𝜐0𝑏 ≈ 0.1𝑐 to 0.6𝑐. Fig. 1 shows the normalized THz radiation field (𝐴1 ∕𝐴01 ) with the radiation frequency (in Hz) for different electron beam velocities of 𝜐0𝑏 ≈ 0.2𝑐 (red curve), 𝜐0𝑏 ≈ 0.4𝑐 (green curve), and 𝜐0𝑏 ≈ 0.6𝑐 (black curve). The field strength of THz radiation is maximized at the resonance (𝜔𝑝 ≈ 𝜔1 ) due to the maximum energy transfer from the beat-wave coupling. The efficient radiation can be generated if a proper plasma density is chosen. Our results show that the field strength decrease as 𝜔1 decreases due to the frequency mismatch. The decreasing field strength is attributed in reduction to the nonlinear current during this parametric process. Thus the strength of THz field decreases significantly for higher values of THz frequencies. Our results also show that the electron beam velocity significantly affects the radiated field strength. The radiation field strength is increased with the electron beam velocity. For higher beam velocity, the plasma beat-wave may amplify to attain large amplitude. Also, the larger electron velocity makes the space charge mode to have more negative energy density. Thus these both aspects contribute to enhance the growth rate of the energy transferring in this process. By increasing the beam velocity from 0.2𝑐 to 0.6𝑐, the maximum field strength of THz radiation rises almost two fold. The efficiency (𝜂) of THz radiation generation process can be defined by the ratio of the average energy densities of the emitted 2 radiation (𝑊𝑇 𝐻𝑧 = 𝜔1 (𝜕𝜀0 ∕𝜕𝜔1 )|𝐸⃗1 | ∕8𝜋) to the incident lasers (𝑊𝐿 = 2 𝜔01 (𝜕𝜀0 ∕𝜕𝜔01 )|𝐸⃗01 | ∕8𝜋), i.e., 𝜂 = 𝑊𝑇 𝐻𝑧 ∕𝑊𝐿 . Fig. 2 shows the efficiency
plasma beat-wave and may radiate THz via nonlinear current generation. In this process, a pump (plasma beat-wave) produces oscillatory velocities to the plasma electrons and the beam electrons. The oscillatory velocity combines with an existing counter propagating beam mode to produce a ponderomotive force that drives the electromagnetic sideband via nonlinear current generation. The sidebands couple to the pump wave to produce a nonlinear current driving the instability. In a result, the plasma beat-wave and the electron beam loss their energy during the nonlinear interaction to generate the THz radiation. Our calculation shows that, using currently available experimental parameters, THz radiation can be excited via this scheme by supplying of power to the radiation by the electron beam through the plasma beat-wave. The efficiency of the process strongly depends on the electron beam velocity. Higher beam velocity is seen to be good to generate THz radiation with higher efficiency. The launching of the electrostatic wave requires the existence of a plasma which is also favorable medium for the propagation of high-current beams, much above the vacuum current limit. Acknowledgments This work was supported by the Department of Science and Technology, Govt. of India under DST-RFBR joint research proposal (Grant No. INT/RUS/RFBR/P-186) and RFBR (Grant No. 15-52-45119-Inda). This research was also financially supported by the National Research Foundation of Korea (Grant No. 2014M1A7A1A01030173 and 2017R1A2B3010765).
of THz radiation emission for different electron beam velocities of 𝜐0𝑏 ≈ 0.2𝑐 (red curve), 𝜐0𝑏 ≈ 0.4𝑐 (green curve), and 𝜐0𝑏 ≈ 0.6𝑐 (black curve). The efficiency of THz radiation generation strongly sensitive to the electron beam velocity. At resonance (𝜔𝑝 ≈ 𝜔1 ), the efficiency is maximum and decreases rapidly as the frequency mismatch occurs. From our results, a significant enhancement in the efficiency of THz radiation generation can be seen for higher electron beam velocity. Basically the energy to the radiation is fed by the beam through the nonlinear current generation. Thus decreasing electron beam velocity reduces the growth rate of instability and hence the efficiency of the coupling process.
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4. Conclusion In conclusion, we have proposed a mechanism for generation of THz radiation through nonlinear coupling between a plasma beat-wave and a negative-energy electron beam in a plasma. Two co-propagating lasers with frequency difference equal to the plasma frequency can excite a large amplitude plasma beat-wave in underdense plasma. A negative energy electron beam propagating through a plasma destabilizes the 73
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Terahertz radiation emission from plasma beat-wave interactions with a relativistic electron beam D.N. Gupta a, *, V.V. Kulagin b , H. Suk c a b c
Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India Sternberg Astronomical Institute, Moscow State University, Moscow, 119 992, Russia Department of Physics and Photon Science, Gwangju Institute of Science and Technology, Gwangju, 61005, South Korea
a r t i c l e
i n f o
Keywords: Electromagnetic radiation Frequency downshift Laser-plasma
a b s t r a c t We present a mechanism to generate terahertz radiation from laser-driven plasma beat-wave interacting with an electron beam. The theory of the energy transfer between the plasma beat-wave and terahertz radiation is elaborated through nonlinear coupling in the presence of a negative-energy relativistic electron beam. An expression of terahertz radiation field is obtained to find out the efficiency of the process. Our results show that the efficiency of terahertz radiation emission is strongly sensitive to the electron beam energy. Emitted field strength of the terahertz radiation is calculated as a function of electron beam velocity. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Terahertz (THz) radiation is the portion of the electromagnetic spectrum at the boundary between the microwaves and the infrared. There are several well-known sources for coherent THz radiation generation such as solid state oscillators, quantum cascade lasers, optically pumped solid state devices and novel free electron devices, which have n turn stimulate a wide range of applications from material science to telecommunications, from biology to biomedicine [1]. Today most widely used sources of pulsed THz radiation are laser driven THz emitters based on frequency down conversion from the optical region [2,3]. In free-electron based sources, to overcome the necessity of reducing the size of the components, several methods have been proposed to let the electrons exchange momentum and allow photon emission. The most familiar method is the use of magnetic wiggler to generate the electromagnetic radiation by the electron beam [4]. In this scheme, an electron beam within a distance comparable to the wavelength of the radiation can emits the coherent radiation. Moreover, a radiofrequency modulated electron beam at wavelengths comparable to the electron bunch length can generates the short pulses of coherent THz radiation. There also exits laser based THz sources, which can fit on or scale of the size of a table-top. A terahertz field greater than 400 kV/cm generated using short 25 fs pulses has been reported by Bartel et al. [5]. Karpowicz et al. [6] reported the generation and detection of broadband terahertz pulse in air covering the spectral range of 0.3 to 10 THz with * Corresponding author.
E-mail address: [email protected] (D.N. Gupta). http://dx.doi.org/10.1016/j.optcom.2017.05.043 Received 13 January 2017; Received in revised form 10 May 2017; Accepted 16 May 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
10% or larger of the maximum electric field. Terahertz wave generation mechanism in air was originally attributed to the four-wave rectification process through the third-order optical nonlinearity of air [7]. However, it has been suggested that the plasma formation plays an important role in the terahertz wave generation process [8]. The four-wave rectification mechanism was tested by Xie et al. [9] by controlling the relative phase, amplitude and polarization of the fundamental and second harmonic pulses. Kim et al. [10] proposed a transient photo current model based on the rapid ionization by femtosecond laser pulses by following the driven motions of electrons by an asymmetric electric field. Laser-plasma based accelerators [11] such as the laser wakefield accelerators [12–14] and the laser beat-wave accelerators [15–17] are widely being researched as possible alternate sources for high energy electron beam. These systems could be used to generate THz radiation. There are several theoretical mechanisms to generate THz from laserplasma interactions [18–20]. Bystrov et al. [18] have proposed the Terahertz radiation of plasma oscillations excited upon the optical breakdown of a gas. Chen [19] has studied the THz radiation using a tailored laser pulse in a plasmas. Such radiation source has a broad tunability range in frequency spectra. Antonsen et al. [20] have a theoretical model for THz radiation generation in a nonuniform plasma. Miniature corrugated channels has been proposed for THz radiation generation by bunched electron beams in plasmas. In our work, we utilize the plasma beat-wave generated in laser beatwave accelerator for THz radiation generation process. In a beat-wave
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the plasma and beam electrons. Solving the equation of motion and continuity, the velocity and density perturbation of the beam electrons can be written as
accelerator, two co-propagating lasers, having frequency difference equal to the plasma electron frequency, can excite a large amplitude plasma wave. We use this plasma beat-wave to generate THz radiation while interacting with a negative energy relativistic electron beam. The negative energy of the beam means that the electron beam propagates opposite to the plasma wave in this scheme. An electron beam propagating through a plasma causes the onset of plasma instability due to the free energy available in the relativistic motion between the beam and the plasma [21]. The plasma beat-waves at moderate laser intensities acquire levels large enough to causes the onset of parametric instabilities. Beam space charge mode can be existed if an electron beam is present. The plasma beat-wave nonlinearly couples with the THz electromagnetic mode to drive the negative-energy beam space-charge mode. Both decayed waves grow in the expense of the beam and the plasma beat-wave. The oscillatory velocity combines with an existing negative-energy beam mode to produce ponderomotive force that drives the electromagnetic sideband. The sidebands couple to the pump wave to produce a nonlinear current that drives the instability. In a result, the plasma beat-wave and the electron beam both lose their energy during the nonlinear interactions and generate THz radiation. In Section 2, we present the theoretical model used to find out the THz radiation field strength. In Section 3, we explain the numerical results and discuss the physics behind the suggested mechanism and we make final remark in the last section.
𝜐𝟎𝑏 = 𝑛10𝑏 =
𝜐11𝑏 =
𝐄𝟎2 = Re 𝑥𝐴 ̂ 02 exp[−𝑖(𝜔02 𝑡 − 𝑘02 𝑧)],
(2)
𝑛𝑏 =
𝑛=
(6)
.
𝑒𝐄1 3 (𝜔 − 𝑘 𝜐 ) 𝑚𝑖𝛾0𝑏 1 1 0𝑏
(7)
.
𝑘2𝑏 4𝜋𝑒
(8)
𝜒𝑏 (𝜙 + 𝜙𝑝 ),
𝑘2𝑏 4𝜋𝑒
(9)
𝜒𝑒 (𝜙 + 𝜙𝑝 ),
where 𝜒𝑒 = −𝜔2𝑝 ∕𝜔2𝑏 . Using the density perturbation in Poisson’s equation, we obtain 𝜀𝜙 = −𝜒𝑏 𝜙𝑝𝑏 − 𝜒𝑒 𝜙𝑝 , where 𝜀 = 1 + 𝜒𝑒 + 𝜒𝑏 . The nonlinear current density and density perturbation at the THz electromagnetic wave can be written as 𝐉𝐍𝐋 = −𝑛𝑏 𝑒𝝊𝟏0𝑏 ∕2. The wave 1 equation governing the terahertz field can be written as − ∇2 𝐄1 + ∇(∇ ⋅ 𝐄𝟏 ) = −
4𝜋𝑖𝜔1 𝑐2
𝐉𝐍𝐋 + 1
𝜔21 𝑐2
(10)
𝜀1 𝐄1 ,
where 𝜀1 = 1 − 𝜔2𝑝 ∕𝜔21 . Assuming the radial variation of the electrostatic pump mode to be small over the width of the beam and taking the divergence of Eq. (10), we obtain 𝐄1 = −4𝜋𝑖𝜔1 𝐉𝐍𝐋 ∕(𝜔21 − 𝜔2𝑝 ), which 𝟏 gives
(3) |𝐄𝟏 | = −
where 𝜔𝑝 = (4𝜋𝑛0 𝑒2 ∕𝑚)1∕2 is the unperturbed plasma frequency, 𝜐2𝑡ℎ = 3𝑇𝑒 ∕𝑚, 𝑇𝑒 is the electron temperature, 𝛥𝑘 = 𝑘0 and 𝛥𝜔 = 𝜔0 ≈ 𝜔𝑝 are the frequency and wave number differences of the two laser pumps. Under cold plasma approximation, we take 𝜐𝑡ℎ ≈ 0 and the solution of Eq. (3) can be in the form [ ( )] 𝐄𝑝 = Re 𝑧𝐸 ̂ 𝑝0 exp −𝑖 𝜔0 𝑡 − 𝑘0 𝑧 .
− 𝑘0 𝜐0𝑏 )2
where 𝜒𝑏 = −𝜔2𝑝𝑏 ∕𝛾03 (𝜔𝑏 − 𝑘𝑏 𝜐0𝑏 )2 and 𝜔𝑝𝑏 = (4𝜋𝑛0𝑏 𝑒2 ∕𝑚)1∕2 . Similarly, the plasma electron density perturbation is
where |𝐴01 |2𝑥=0 = 𝐴001 exp(−𝑥2 ∕𝑟201 ), |𝐴02 |2𝑥=0 = 𝐴002 exp(−𝑥2 ∕𝑟202 ), 𝜔01 − 𝜔02 ≈ 𝜔𝑝 , 𝜔01 ≥ 𝜔𝑝 , 𝜔02 ≥ 𝜔𝑝 , 𝜔01 and 𝜔02 are the laser frequencies, 𝐤𝟎1 and 𝐤𝟎2 are the wave vectors of the laser beams, and 𝑟01 and 𝑟02 are the laser spot sizes. They produce oscillatory velocities 𝜐𝟎𝑗 = 𝑒𝐄𝟎𝑗 ∕𝑚𝑖𝜔0𝑗 (𝑗 = 1, 2) and exert a ponderomotive force 𝐅𝐩 = −(𝑒∕2𝑐)(𝜐𝟎1 ×𝐁∗𝟎2 +𝜐𝟎2 ×𝐁∗𝟎1 ) = 𝑧𝑖𝑒𝑘 ̂ 0 𝜙𝑝0 exp[−𝑖(𝜔0 𝑡 − 𝑘0 𝑧)] on them, where 𝜙𝑝0 = 𝑒𝐴01 𝐴02 ∕2𝑚𝜔01 𝜔02 , 𝜔0 = 𝜔01 − 𝜔02 , 𝐤𝟎 = 𝐤𝟎1 − 𝐤𝟎2 , 𝐁𝟎𝑗 = 𝑐𝐤𝟎𝑗 × 𝐄𝟎𝑗 ∕𝜔0𝑗 , and −𝑒 and 𝑚 are the electron charge and mass, respectively. Because of the large mass and the slow response of the ions, we consider them immobile. The ponderomotive force drives a large amplitude beat-wave in a plasma. We write the Eulerian equation for the plasma-wave electric field driven by the ponderomotive force of the beating laser pumps as [22] 1 𝐸̈ 𝑝 + 𝜔2𝑝 𝐸𝑝 + 𝜐2𝑡ℎ 𝐸𝑝′′ = 𝐴01 𝐴02 𝜔20 sin(𝛥𝑘𝑧 − 𝛥𝜔𝑡), 2
𝑒𝑘𝑝 𝐄𝐩 3 (𝜔 𝑚𝑖𝛾0𝑏 0
(5)
,
The THz radiation (𝜔1 , 𝐤1 ) couples with the pump wave (𝜔0 , 𝐤𝟎 ) to produce a ponderomotive force (𝐅𝐩 = 𝑚𝛾𝜐 ⋅ ∇𝜐) at frequency of (𝜔𝑏 , 𝐤𝑏 ). The longitudinal ponderomotive force on the beam electrons can be 5 (𝜔 − explicitly written as 𝐅𝐩𝑏 = 𝑧𝑒𝑖𝑘 ̂ 𝑏 𝜙𝑝𝑏 , where 𝜙𝑝𝑏 = −𝑒𝐄∗𝐩 𝐄𝟏 ∕2𝑚𝛾0𝑏 1 𝑘1 𝜐0𝑏 )(𝜔0 − 𝑘0 𝜐0𝑏 ). Ponderomotive potential 𝜙𝑝 for plasma electrons can be deduced by taking 𝜐0𝑏 = 0 in the same expression. 𝜙𝑝 and 𝜙 at (𝜔𝑏 , 𝐤𝑏 ) produce electron beam density perturbation
We consider a plasma of the density 𝑛0 and the electron temperature 𝑇𝑒 . Two collinear laser beams of large amplitude propagate in a plasma with electric fields (1)
− 𝑘0 𝜐0𝑏 )
Here, we perturb the plasma equilibrium as 𝝊𝑏 = 𝝊0𝑏 + 𝝊10𝑏 , 𝑛𝑏 = 1 , where 𝛾 2 2 −1∕2 and 𝛾 1 = 𝑛0𝑏 + 𝑛10𝑏 , 𝛾𝑏 = 𝛾0𝑏 + 𝛾0𝑏 0𝑏 = (1 − 𝜐0𝑏 ∕𝑐 ) 0𝑏 3 2 2 𝛾0𝑏 (𝜐0𝑏 ∕𝑐 ). Corresponding quantities for plasma electrons, 𝝊0𝑒 and 𝑛0𝑒 , can be deduced from Eqs. (5) and (6) by dropping the subscript 𝑏 and taking 𝝊0𝑏 = 0. The plasma beat-wave decays into the negative energy beam mode with potential of 𝜙 = Φ exp[−𝑖(𝜔𝑏 𝑡 + 𝑘𝑏 𝑧)] and a THz electromagnetic wave with electric field of 𝐄𝟏 = 𝐴1 exp[−𝑖(𝜔1 𝑡 + 𝑘1 𝑧)]. The linear response of the beam electrons at (𝜔1 , 𝐤1 ) is
2. Theoretical model
𝐄𝟎1 = Re 𝑥𝐴 ̂ 01 exp[−𝑖(𝜔01 𝑡 − 𝑘01 𝑧)],
𝑒𝐄𝐩 3 (𝜔 𝑚𝑖𝛾0𝑏 0
𝑒𝑘2𝑏 𝜔2𝑝𝑏 |𝐄𝐩 |𝜙𝑝 2𝑚𝛾06 𝜔1 (1 − 𝜔2𝑝 ∕𝜔21 )(𝜔0 − 𝑘0 𝜐0𝑏 )(𝜔𝑏 − 𝑘𝑏 𝜐0𝑏 )2
,
(11)
where we have considered that, in the case of when beam density (or the beam current) is small, 𝜒𝑏 ≪ 1, the self-consistent potential of the beam can be neglected as compared to the ponderomotive potential (𝜙 ≪ 𝜙𝑝𝑏 ).
(4)
3. Numerical results and discussion
The plasma beat-wave acquires a large amplitude to onset of the parametric instabilities. We also consider a relativistic electron beam of density 𝑛0𝑏 passes through it in opposite direction with initial velocity of −𝜐𝑜𝑏 𝑧. ̂ Thus the plasma beat wave is traveling in opposite direction to the electron beam. The fundamental beat-wave (𝜔0 , 𝐤0 ) parametrically couples to the electron beam and excites a negative energy beam mode (𝜔𝑏 , 𝐤𝐛 ) and a THz electromagnetic wave (𝜔1 , 𝐤1 ), where 𝜔1 = 𝜔𝑏 + 𝜔0 , and 𝐤𝟏 = 𝐤𝐛 +𝐤𝟎 . The slow-beam space charge mode has negative energy (𝜔𝑏 = −𝐤𝑏 ⋅ 𝜐0𝑏 ). The plasma beat-wave imparts oscillatory velocity to
Using Eq. (11), we depict the THz field strength with respect to the THz frequency. The used numerical parameter are as follows: 𝑎01 = 𝑒𝐴01 ∕𝑚𝜔01 𝑐 = 1 and 𝑎02 = 𝑒𝐴02 ∕𝑚𝜔02 𝑐 = 1 (corresponding to peak laser intensity 𝐼01 ≈ 𝐼02 ≈ 1.37 × 1018 W∕cm2 ), the considered wavelength of first laser is 𝜆01 ∼ 1 µm and the wavelength of second laser has been chosen according to the resonant condition for beat-wave excitation, and the initial waist size is 𝑟01 ≈ 𝑟02 ∼ 10 µm. The plasma electron density considered for this calculation is 𝑛0 ≈ 1016 cm−3 . We consider that an 72
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Fig. 1. Normalized field strength of THz radiation with frequency (in Hz) for different electron beam velocities of 𝜐0𝑏 ≈ 0.2𝑐 (red curve), 𝜐0𝑏 ≈ 0.4𝑐 (green curve), and 𝜐0𝑏 ≈ 0.6𝑐 (black curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. Efficiency of THz radiation generation with frequency (in Hz) for different electron beam velocities of 𝜐0𝑏 ≈ 0.2𝑐 (red curve), 𝜐0𝑏 ≈ 0.4𝑐 (green curve), and 𝜐0𝑏 ≈ 0.6𝑐 (black curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
electron beam of density of the order of 𝑛0𝑏 ≈ 1011 cm−3 is propagates through a plasma with different velocities ranging from 𝜐0𝑏 ≈ 0.1𝑐 to 0.6𝑐. Fig. 1 shows the normalized THz radiation field (𝐴1 ∕𝐴01 ) with the radiation frequency (in Hz) for different electron beam velocities of 𝜐0𝑏 ≈ 0.2𝑐 (red curve), 𝜐0𝑏 ≈ 0.4𝑐 (green curve), and 𝜐0𝑏 ≈ 0.6𝑐 (black curve). The field strength of THz radiation is maximized at the resonance (𝜔𝑝 ≈ 𝜔1 ) due to the maximum energy transfer from the beat-wave coupling. The efficient radiation can be generated if a proper plasma density is chosen. Our results show that the field strength decrease as 𝜔1 decreases due to the frequency mismatch. The decreasing field strength is attributed in reduction to the nonlinear current during this parametric process. Thus the strength of THz field decreases significantly for higher values of THz frequencies. Our results also show that the electron beam velocity significantly affects the radiated field strength. The radiation field strength is increased with the electron beam velocity. For higher beam velocity, the plasma beat-wave may amplify to attain large amplitude. Also, the larger electron velocity makes the space charge mode to have more negative energy density. Thus these both aspects contribute to enhance the growth rate of the energy transferring in this process. By increasing the beam velocity from 0.2𝑐 to 0.6𝑐, the maximum field strength of THz radiation rises almost two fold. The efficiency (𝜂) of THz radiation generation process can be defined by the ratio of the average energy densities of the emitted 2 radiation (𝑊𝑇 𝐻𝑧 = 𝜔1 (𝜕𝜀0 ∕𝜕𝜔1 )|𝐸⃗1 | ∕8𝜋) to the incident lasers (𝑊𝐿 = 2 𝜔01 (𝜕𝜀0 ∕𝜕𝜔01 )|𝐸⃗01 | ∕8𝜋), i.e., 𝜂 = 𝑊𝑇 𝐻𝑧 ∕𝑊𝐿 . Fig. 2 shows the efficiency
plasma beat-wave and may radiate THz via nonlinear current generation. In this process, a pump (plasma beat-wave) produces oscillatory velocities to the plasma electrons and the beam electrons. The oscillatory velocity combines with an existing counter propagating beam mode to produce a ponderomotive force that drives the electromagnetic sideband via nonlinear current generation. The sidebands couple to the pump wave to produce a nonlinear current driving the instability. In a result, the plasma beat-wave and the electron beam loss their energy during the nonlinear interaction to generate the THz radiation. Our calculation shows that, using currently available experimental parameters, THz radiation can be excited via this scheme by supplying of power to the radiation by the electron beam through the plasma beat-wave. The efficiency of the process strongly depends on the electron beam velocity. Higher beam velocity is seen to be good to generate THz radiation with higher efficiency. The launching of the electrostatic wave requires the existence of a plasma which is also favorable medium for the propagation of high-current beams, much above the vacuum current limit. Acknowledgments This work was supported by the Department of Science and Technology, Govt. of India under DST-RFBR joint research proposal (Grant No. INT/RUS/RFBR/P-186) and RFBR (Grant No. 15-52-45119-Inda). This research was also financially supported by the National Research Foundation of Korea (Grant No. 2014M1A7A1A01030173 and 2017R1A2B3010765).
of THz radiation emission for different electron beam velocities of 𝜐0𝑏 ≈ 0.2𝑐 (red curve), 𝜐0𝑏 ≈ 0.4𝑐 (green curve), and 𝜐0𝑏 ≈ 0.6𝑐 (black curve). The efficiency of THz radiation generation strongly sensitive to the electron beam velocity. At resonance (𝜔𝑝 ≈ 𝜔1 ), the efficiency is maximum and decreases rapidly as the frequency mismatch occurs. From our results, a significant enhancement in the efficiency of THz radiation generation can be seen for higher electron beam velocity. Basically the energy to the radiation is fed by the beam through the nonlinear current generation. Thus decreasing electron beam velocity reduces the growth rate of instability and hence the efficiency of the coupling process.
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4. Conclusion In conclusion, we have proposed a mechanism for generation of THz radiation through nonlinear coupling between a plasma beat-wave and a negative-energy electron beam in a plasma. Two co-propagating lasers with frequency difference equal to the plasma frequency can excite a large amplitude plasma beat-wave in underdense plasma. A negative energy electron beam propagating through a plasma destabilizes the 73
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