Numerical simulations of THz emission from the laser interaction with magnetized plasmas

Optik 124 (2013) 1372–1375

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Numerical simulations of THz emission from the laser interaction with magnetized plasmas Bai Xu ∗ , Yongda Li Department of Physics, Changchun University of Science and Technology, Changchun 130022, PR China

a r t i c l e

i n f o

Article history: Received 8 November 2011 Accepted 20 March 2012

PACS: 52.38 .−r 52.65 .−y Keywords: Particle-in-cell simulation Magnetized plasma Linear mode conversion THz emission

a b s t r a c t One-dimensional particle-in-cell (PIC) program is used to simulate the generation of high power terahertz (THz) emission from the interaction of an ultrashort intense laser pulse with underdense magnetized plasma. The spectra of THz radiation are discussed with different laser intensity, pulse width, density scale length and external magnetic field. A high-amplitude electron plasma wave driven by a laser pulse can produce powerful THz emission through linear mode conversion under certain conditions. With incident laser intensity of 1018 W/cm2 , the generated emission is computed to be of the order of several MV/cm field and tens of MW level power. The corresponding energy converrsion efficiency is several ten thousandths, which is higher than the efficiency of other THz source and suitable for the studies of high-field and nonlinear physics in the THz regime. © 2012 Elsevier GmbH. All rights reserved.

1. Introduction Recent progress in making ultrashort (10–100 fs) laser pulses has opened new possibilities for investigating relativistic laser–plasma interaction and corresponding generation of THz radiation. Currently the THz radiation from laser irradiated plasma is attracting increasing interest. A laser wakefield is an electron plasma wave driven by the ponderomotive force of a laser pulse [1], which can be converted into an electromagnetic wave under certain condition. In the past ten years, Hamster et al. found that the interaction between ultrashort laser pulses and plasmas leads to the emission of coherent, short-pulse radiation at THz frequencies [2], Loffler and Roskos introduced that THz radiations can be generated by photoionization of electrically biased gases with amplified laser pulses, and studied the efficiency of generation process [3]. Sheng et al. proposed that THz radiation can be generated from a laser wakefield through linear mode conversion, which is confirmed by particle-in-cell (PIC) simulations [4]. More recently, this method has been used under inhomogeneous magnetized plasmas [5]. In this paper, we use one-dimensional PIC program to simulate the generation of high power THz emission from the interaction of an ultrashort intense laser pulse with an underdense magnetized plasma, and discuss the spectra of THz radiation with different laser

∗ Corresponding author. E-mail address: [email protected] (B. Xu). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.03.066

intensity, pulse width, density scale length and external magnetic field. 2. Cold relativistic plasma equations As shown in Fig. 1, we consider the laser wakefield excitation when a laser pulse irradiates normally into the magnetized inhomogeneous plasma slab. The plasma is underdense, i.e. 2 /2P  1, where  is laser wavelength and P is plasma wave-



4ne2 /me is plasma frequency. The length, P = 2c/ωP and ωP = plasma density increases linearly with distance, i.e. n(x) = n0 x/L, where L = nc /n is plasma density scale length and nc is the critical density. The external magnetic field B0 is along the z direction perpendicular to the incident plane. Maxwell’s equations then read 2

(∂x −

1 2 1 ∂t )A(x, t) = − J⊥ (x, t) c2 ε0 c 2

(1)

E⊥ (x, t) = −∂t A(x, t)

(2)

Ex (x, t) = −∂x ␾(x, t)

(3)

B(x, t) = ∇ × A(x, t)

(4)

The motion equation is: dt P = dt (mV) = q(E + V × B)

(5)

The plasma is assumed to be a cold electron slab with a static background of ions, where m and q are electron mass and charge respectively. In the fluid picture, the total time derivatives dt

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Fig. 1. Laser pulse irradiates normally into the magnetized inhomogeneous plasma slab.

have to be interpreted as dt = ∂t + vx ∂x . The electron momen1 + (P/mc)2 . The tum P = mV = P⊥ + Px and relativistic factor  = transverse momentum P⊥ is written in the form

dt (P⊥ + qA) = −qV x B0 eˆ y

(6)

And for the longitudinal momentum Px one has dt Px = q(−∂x ␾ + V⊥ ∂x A + Vy B0 )

(7)

It is convenient to get dt (Py + qAy ) = −qV x B0 eˆ y and dt (Pz + qAz ) = 0 from Eq. (6), and transverse current J⊥ = −ene V⊥ = ene (eAz eˆ z − Py eˆ y )/m. Eqs. (1) and (5) can then be written in the form (

∂

2

2 ∂x

−

∂

2

2 ∂t

)a = (

ωp0 2 n(az eˆ z − py eˆ y ) ) c 

dt (py − ay ) = vx B0 eˆ y dt (pz − az ) = 0

(8) (9) (10)

where a = eA/mc, p = P/mc, v = V/c, B0 = B0 e/m, n = ne /n0 . According to Eq. (10), we have az = pz = 0 in Eq. (8) as it is satisfied initially. Eq. (8) determines the occurrence of the radiation from the wakefield. If B0 = 0, Eq. (9) leads to py = py0 + ay , where py0 = 0, thus there is no radiation. However, in magnetized plasmas, the term vx B0 in Eq. (9) leads to a nonzero source and the THz emission is always p polarized. It is well known that an electromagnetic wave can convert into an electrostatic wave near the point where the dielectric function for electrostatic waves vanishes, and anomalous absorption of the electromagnetic wave energy takes place with the generation of large-amplitude electrostatic waves. This absorption progress is named as linear mode conversion. Means et al. studied the inverse problem and found that there is a reversal symmetry in the electromagnetic–electrostatic mode conversion [6], which was confirmed later by Hinkel-Lipsker et al. [7]. One can get an electromagnetic emission generated through mode conversion from a wakefield excited with a laser pulse incidence into a plasma. 3. PIC simulation results In order to simulate the process of THz emission with a normal laser pulse irradiate into the magnetized plasma, one-dimensional PIC program is compiled. The 1-D model is valid as long as the radiation spot size is large compared to the plasma wavelength, i.e., rs  p = 2/kp . According to the derivation above, we set incident laser pulse with s polarization in order to distinguish it from the p polarized THz emission. The laser wavelength 0 = 1 ␮m, the cor2 /4e2 = 1.1 × responding critical density of the plasma nc = mωP0

Fig. 2. Spatial–temporal plots of the transverse electric field Ey (a), longitudinal electric field Ex (b), and the time evolution of Ey (c) at the plasma–vacuum interface.

1021 cm−3 and the plasma frequency is ωP0 /2 = 293 THz. The simulation range is 250 , the left part is vacuum (between x = 0 and 50 ) and the right part is the plasma (between x = 50 and 250 ). The plasma density increases linearly with distance from left boundary n(x = 50 ) = 0.03nc to right boundary n(x = 250 ) = 0.09nc . The external magnetic field B0 = 6.4 T. We select an incident laser pulse width large enough to see 10 cycles and the continuous modulated Gaussian source is E(t) = a0√sin (2t/ 0 ) exp [− (t − 5 0 )2 /2 2 ], where  0 = 0 /c and = 50 /2 2 ln 2 = 7.078 × 10−15 s. The normalized intensity is a0 = 1, corresponding to the laser intensity is I = a20 1.37 × 1018 (␮m/0 )2 W/cm2 = 1.37 × 1018 W/cm2 . Fig. 2a and b shows the evolution of transverse field Ey and longitudinal field Ex with time and space respectively. The laser pulse is s polarized and the electric component of incident wave is Ez , Ex is the electrostatic field generated from the interaction between laser pulse and underdense plasma, and Ey is generated from Ex though linear mode conversion. Fig. 2a shows there are fields both inside plasma (between x = 50 and 250 ) and across the left vacuum–plasma boundary in the vacuum region (x < 50 ). This is just due to the wakefield emission through linear mode conversion. Fig. 2b shows that the pure electrostatic field Ex only exists in the plasma. Fig. 2c shows the time evolution of Ey from the vacuum–plasma interface where x = 50 . Compared with the situation of unmagnetized plasma, the THz emission from magnetized plasma has a time delay, it is because the magnetic field B0

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leads to the resonant absorption delay. For the case of linear density increasing, i.e. n(x) = nc x/L, the delay distance x = B02 L/4nc mc 2 . 4. Emission spectra with different parameters In a magnetized, inhomogeneous plasma, two new features are introduced into the process of resonant absorption. First, the resonant frequency is now the upper hybrid frequency ωuh = (ωP2 +

ωc2 )1/2 , instead of the plasma frequency ωP and ωc = eB0 /m is the electron cyclotron frequency. Second, the Lorentz force provides coupling between the electromagnetic and electrostatic waves so that, even for normal incidence, mode conversion into an upper hybrid wave takes place and significant absorption of the incident laser energy occurs. When an electromagnetic radiation propagating into a magnetized inhomogeneous plasma, the transmission coefficient is |T| = e−s0 /2 and the reflection coefficient is |R| = 1 − e−s0 , where s0 = k0 x0 , k0 = ω0 /c, ω0 is laser frequency, x0 = Lωc /ω [8]. We set ˜ 0 . We can get the absorption ω ˜ = ω/ω0 , then s0 = 2ωc L/ω0 ω coefficient a = 1 − |T|2 − |R|2 . According to the mention above [6,7], the mode conversion from electrostatic to electromagnetic waves is symmetric with its inverse problem. Therefore, the mode conversion coefficient  = a.  = e−s0 − e−2s0

(11)

Next we study the conversion efficiency, which depends upon the plasma density gradient, the external magnetic field, and the laser parameters such as intensity and pulse duration. According to the theory of mode conversion, the radiation spectrum can be written as: 2 S(ω, ˜ L, ωc , dL ) = (s0 )Em (ω) ˜

(12)

where Em is the wakefield amplitude. If the plasma density is weakly inhomogeneous, the local wakefield amplitude can be approximated by a corresponding value in homogeneous plasma. For a pulse profiled aL = a0 sin 2 [(x − ct)/dL ], the wakefield amplitude is [9]: Em (ωP )

=

(mωP c/e)(3a20 /4) sin(dL /P )/{[1− (dL /P )2 ][4 − (dL /P )2 ]}

Fig. 3. (a) Emission spectrum as a function of the electron cyclotron frequency ωc for L = 200 and dL = 50 . (b) Emission spectrum as a function of the density scale length L for ωc = 7.0 × 10−4 ω0 and dL = 50 . (c) Emission spectrum as a function of the pulse duration dL for ωc = 7.0 × 10−4 ω0 and L = 200 .

(13)

To show the radiation spectrum, we substitute ωP with ω and dL /P = ωdL /2c = ωd ˜ L /P0 , so: Em (ω) ˜

=

(mωcω ˜ P0 /e)(3a20 /4) sin(ωd ˜ L /P0 )/{[1− (ωd ˜ L /P0 )2 ][4 − (ωd ˜ L /P0 )2 ]}

(14)

Eqs. (12) and (14) suggest that the emission power is proportional to a40 , P ∝ I02 ∝ a40

(15)

If other parameters keep fixed, the emission power will be increased by three times with double laser intensity. So it is a very effective way to change the emission power by changing the laser intensity. Note that this result is valid only under weakly relativistic laser condition. Otherwise, the wakefield amplitude Em is not proportional to a20 . Fig. 3a shows the radiation spectrum S for different external magnetic fields B0 at the fixed dL = 50 and L = 200 . It is convenient to represent the external magnetic field B0 by electron cyclotron frequency ωc for ωc = eB0 /mc. Note that there is no emission at ωc = 0, and after the spectrum peak, peaks are shifted to higher frequencies and the spectrum widths reduce with the magnetic field increases. For the linear mode conversion from the wakefield to electromagnetic radiation, there is an optimized external magnetic field for high conversion efficiency at particular frequency. According

Fig. 4. Absorption coefficient  for linear absorption.

to Eq. (11), the absorption coefficient  is shown in Fig. 4 as a function of s0 . As shown in Fig. 4, the max absorption coefficient  = 0.25 when s0 = 0.225. For ω = 0.3ωP0 , the optimized electron cyclotron frequency ωc = 5.37 × 10−4 ωP0 , i.e. B0 = 6.389T. Fig. 3b shows the radiation spectrum for different scale length L at the fixed ωc = 7.0 × 10−4 , i.e. B0 = 8.3 T, and dL = 50 . It indicates that the radiation power varies with L increases, and the scale length L dependent variation trends of center frequency shift and spectrum width are similar to those of ωc dependent. Similar to the optimized external magnetic field, there is an optimized scale length for the emission spectrum peak. For ω = 0.3ωP0 , the optimized scale length is 15.70 . Fig. 3c shows the radiation spectrum as a function of the pulse width dL at the fixed ωc = 7.0 × 10−4 and L = 200 . It shows that the radiation center frequencies are downshifted significantly, and the emission spectrum widths are narrowed obviously. This reminds us that the pulse duration dL is a key parameter to control the central frequency and spectrum width. One can change the spectrum by

B. Xu, Y. Li / Optik 124 (2013) 1372–1375

changing P0 (or critical density nc ) and dL . If dL /P0 is fixed, the spectrum profile is fixed for given ωc and L. 5. Conclusion The THz radiation generated with short laser pulse incidence normally into magnetized inhomogeneous plasma is investigated through theoretical analysis and one-dimensional PIC simulation. Power and spectrum of THz radiation are also discussed under different laser intensity, pulse width, external magnetic field and plasma density scale length. The simulation results are well consistent with the mode conversion theory. It is important to understand the progress of the THz radiation generated through the interaction between laser and magnetic plasma. References [1] P. Sprangle, E. Esarey, A. Ting, Nonlinear theory of intense laser–plasma interactions, Phys. Rev. Lett. 64 (1990) 2011.

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[2] H. Hamster, A. Sullivan, S. Gordon, et al., Subpicosecond, electromagnetic pulses from intense laser–plasma interaction, Phys. Rev. Lett. 71 (1993) 2725. [3] T. Loffler, H.G. Roskos, Gas-pressure dependence of terahertz-pulse generation in a laser-generated nitrogen plasma, J. Appl. Phys. 91 (2002) 2611. [4] Z.M. Sheng, K. Mima, J. Zhang, et al., Emission of electromagnetic pulses from laser wakefields through linear mode conversion, Phys. Rev. Lett. 94 (2005) 095003. [5] Z.M. Sheng, H.C. Wu, W.M. Wang, et al., Simulation of high power THz emission from laser interaction with tenuous plasma and gas targets, Commun. Comput. Phys. 4 (2008) 1258. [6] R.W. Means, L. Muschietti, M.Q. Tran, et al., Electromagnetic radiation from an inhomogeneous plasma: theory and experiment, Phys. Fluids 24 (1981) 2197. [7] D.E. Hinkel-Lipsker, B.D. Fried, G.J. Morales, Analytic expressions for mode conversion in a plasma with a linear density profile, Phys. Fluids B 4 (1992) 559. [8] R.B. White, F.F. Chen, Amplification and absorption of electromagnetic waves in overdense plasmas, Plasma Phys. 16 (1974) 565. [9] H.C. Wu, Z.M. Sheng, Q.L. Dong, et al., Powerful terahertz emission from laser wakefields in inhomogeneous magenetized plasmas, Phys. Rev. E 75 (2007) 016407.