Simulation of THz emission from laser interaction with plasmas

Optik 123 (2012) 2183–2186

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Simulation of THz emission from laser interaction with plasmas Bai Xu a , Yongda Li a , Lijun Song a,b,∗ a b

Department of Physics, Changchun University of Science and Technology, Changchun 130022, PR China Institute of Applied Physics, Changchun University, Changchun 130022, PR China

a r t i c l e

i n f o

Article history: Received 14 May 2011 Accepted 13 October 2011

PACS numbers: 52.38.−r 52.65.−y Keywords: Particle-in-cell simulation Laser-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 plasma. The spectra of THz radiation are discussed under different laser intensity, pulse width, incident angle and density scale length. High-amplitude electron plasma wave driven by a laser wakefield 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 conversion efficiency is several ten thousandths, which is higher then the efficiency of other THz source and suitable for the studies of THz nonlinear physics. © 2011 Elsevier GmbH. All rights reserved.

Strong dynamics, nonlinear effect and a variety of physical processes are involved with ultrashort intense laser pulse incidence into plasma. During the ultrashort interaction, thermal equilibrium cannot be established through particle collision and the problem cannot be treated by hydrodynamics. Particle-in-cell (PIC) simulation is quite adept in dealing with this kind of problems. PIC simulation is a standard simulation technique used to solve a variety of problems in science and industry, particularly in plasma physics. It uses computer to track the movement of huge number of charged particle in external and self-consistent electromagnetic field. In the PIC code, the real particles are replaced by macroparticles with finite size and the charge to mass ratio remains the same as real ones. Replacement with much smaller number of macro-particles makes the simulation more efficient, and keeps the real kinetic features. In this paper, we present a PIC simulation of the terahertz radiation system from the vacuum–plasma interface through linear mode conversion. In Gaussian unit system, Maxwell’s equations are written as:

∇ · E = 4 1 ∂B c ∂t 1 ∂E 1 ∇ ×B= + 4J c ∂t c ∇ ·B = 0

∇ ×E =−

∗ Corresponding author. E-mail address: [email protected] (L. Song). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.10.021

With preassumed initial position and speed of charged particles, the electronic and magnetic field of each grid can be solved, the Lorentz force on each particle can then be computed. According to the equation of motion:



dP V ×B =q E+ c dt



(1)

The position and speed of each particle in next time step can be obtained. Cycling successionally like this, the macroscopic process of plasma can be acquired. It is well know that an intense laser pulse incident into an underdense plasma, i.e. 2 /2P << 1, the ponderomotive force associated with the laser envelope, FP ∼ ∇ a2 , repel electrons from the region of laser pulse. If the length scale Lx is approximately equal to the plasma wavelength P , the ponderomotive force excites large amplitude plasma wave with the phase velocity approximately equal to the laser pulse group velocity [1]. The plasma  oscilla4ne2 /me , tion period is TP = 2/ωP and plasma frequency ωP = where −e is the electron charge, me is the electron mass and n is the electron density. The plasma can sustain a wakefield with amplitude as large as comparable to n1/2 V/cm [1], if the relativistic effect is taken into account, the maximum amplitude can be even larger. Therefore, the laser wakefield can be excited at amplitudes exceeding 1 GV/cm at the plasma density with the plasma frequency ωP /2 = 9 THz. Even though the electrostatic plasma wave usually does not radiate, electrostatic and electromagnetic wave can convert to each other under certain conditions. As shown in Fig. 1, in the inhomogeneous plasma, electromagnetic wave can convert into electrostatic wave through the

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Fig. 2. THz emission from a wakefield generated by a laser pulse incidenced obliquely to the plasma with inhomogeneous density.

Fig. 1. The mode conversion between electrostatic and electromagnetic wave in the plasma with linearly increasing density.

resonance absorption, which names linear mode conversion [2]. Means et al. studied the inverse problem first and found that there is a reversal symmetry in the electromagnetic–electrostatic mode conversion [3], and it was further confirmed by Hinkel-Lipsker et al. [4]. One can get the electromagnetic emission generated through mode conversion from a wakefield that excited with a laser pulse incidence into a plasma. The dispersion relation of plasma wave is ω2 = ωP2 + 3k2 v2e and electromagnetic wave propagating through plasma is ω2 = ωP2 + k2 c 2 . Only if the local wave vector k = 0, their dispersion curves can cross, and the mode conversion would be possible. In the homogeneous plasma, the wave vector of excited plasma wave k = ωP /vg = / 0, where ωP is the plasma frequency, vg is the laser pulse group velocity in plasma[1], and the linear mode conversion does not occur. In the inhomogeneous plasma, the oscillation frequency varies with positions; this leads to a plasma wave number variation with time and space. The local oscillation amplitude of excited plasma wave can be written as ı = ı0 cos [ (x, t)], where 4n(x)e2 /me . (x, t) = ωP (x)(t − x/vg ) cos  and ωP (x) = Assume that the plasma density increases linearly along the direction of laser incidence, n(x) = n0 (x/L), and the plasma wave vector k = − ∂ /∂ x, so

k=

ωP (x0 ) cos  3x − vg t · 2x vg

(2)

k is zero only for x = vg t/3. Under this condition, the mode conversion would occur for the same dispersion relations of electromagnetic and electrostatic wave. If the plasma density increases with other profile, the possibility of conversion still exists. Assume further that the plasma density decreases linearly, such as in the interval 0 ≤ x ≤ L, n(x) = n0 (1 − x/L). The same substitution gives the wave vector k=

ωP (x0 ) cos  2L + vg t − 3x · vg 2(L − x)

(3)

Under this condition k is always greater then 0, so the linear mode conversion does not occur. Discussions above show that only when the plasma density increases along the laser propagation direction, electrostatic wave can convert into electromagnetic wave though linear mode conversion. If the plasma density decreases or keeps uniform, the mode conversion does not occur. but density increase is not the sufficient condition for linear mode conversion. As shown in Fig. 2, we consider the laser wakefield excitation when a laser pulse propagates at an angle of  with the density gradient of an inhomogeneous plasma slab, and the plasma is assumed to be cold electrons with static background of ions. We make a Lorentz transformation from the laboratory frame L to the moving frame M. Moving in y-direction parallel to the surface with V = c sin ey so that the pulse is normally incident in frame M: In this frame, one can get a set of equations for the electron momentum P, electron density n, scalar and vector potentials for the wakefield [5]. Here we consider only the equation for the vector potential a associated with the wakefield emission as follows:



2

∂x −

ω2 n 1 2 ∂t − 2 P 2 c c  cos 



a = tan 

 ω 2  P

c

(x) −



n ey  cos  (4)

where a is normalized by mc/e,  is the relativistic factor, (x) is the initial distribution function of plasma density. Eq. (4) suggests that / 0. And the there exists emission from the wakefield only when  = emission is always P polarized no matter whether the incident laser is P or S polarized. In order to distinguish the incident light from the generated emission, we set the incident laser pulse to S polarized. To verify the validity of the above analysis, we conduct one-dimensional PIC simulation. Refer to Fig. 2, a laser pulse incident into plasma with an angle  = 15◦ . The laser wavelength 0 = 1/cos  ␮m, the corresponding critical density of the plasma nc = mω02 /4e2 = 1.1 × 1021 · cos  cm−3 . The electric component of the incident laser have a time profile E(t) = a0 sin (ωt) exp [− (t − )2 /d], where the pulse width = 5 0 ,

0 is the oscillation period of the laser. According to the calculation, d = 2.5 × 10−29 . The normalized field a0 = 1 and the relationship between a0 and the incident laser intensity is I = a20 · 1.37 × 1018 (␮m/0 )2 W/cm2 . The simulation range is 250 , left part is vacuum (between x = 0 and 50 ) and right part is plasma (between x = 50 and 250 ). The plasma density increases linearly from left boundary n(x = 50 ) = 0.3nc to right boundary n(x = 250 ) = 0.9nc . Fig. 3.

B. Xu et al. / Optik 123 (2012) 2183–2186

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Fig. 3. Lorentz transformation from laboratory frame L to moving frame M.

Fig. 6. The time evolution of electric field component Ey at the plasma–vacuum interface.

conversion can ensure greater conversion efficiency, the emission is sharp, the center frequency of radiation is evident, and the generated emissions show good monochromaticity. Next we study the conversion efficiency, which depends upon the plasma density gradient, the incident angle, and the laser parameters such as intensity and pulse duration. According to the theory of resonance absorption, the absorption coefficient of cold plasma is:

 ϕ( ) ≈ 2.3 exp Fig. 4. The evolution of electric field Ey with time and space, dotted line is the plasma–vacuum boundary.

Fig. 4 shows that the P polarized electromagnetic emission is generated from the interaction between short-pulse laser and underdense plasma whose density increases linearly. Furthermore, the electromagnetic radiation generated from linear mode conversion is stable. Fig. 5 shows the evolution of electrostatic field Ex. It is clear that Ex does not exist in the vacuum region, which is in agreement with the previous theory[1]. Fig. 6 shows the time evolution of Ey component of the THz radiation from the vacuum–plasma interface where x = 50 . It is obvious that the radiation tends to stabilize quickly. Although the emission from plasma–vacuum boundary covers a broad radiation spectrum, only those frequencies satisfying the conditions of linear

 (5)

where = (ωLω ˜ P0 /c)1/3 sin , ω ˜ = ω/ωP0 , L is plasma scale length and  is incident angel. As mentioned above, the mode conversion from electrostatic to electromagnetic waves is symmetric with its inverse problem[3,4]. Therefore, the mode conversion coefficient is ϕ( ) too. According to the theory of mode conversion, the radiation spectrum can be written as: 2 ˜ L, , dL ) = ϕ( )Em S(ω, (ω) ˜

(6)

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: Em (ωP ) =

Fig. 5. The evolution of electrostatic field Ex with time and space, dotted line is the plasma–vacuum boundary.

−2 3 3

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

(7)

Fig. 7. Emission spectrum as a function of the incident angle  for dL = 50 and L = 200 .

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Fig. 8. Emission spectrum as a function of the plasma density scale length L for  = 15◦ and dL = 50 .

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

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

(8)

Eqs. (6) and (8) suggest that the emission power is proportional to a40 , P ∝ I02 ∝ a40

(9)

If other parameters keep fixed, the emission power will be increased by 3 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. 7 shows the radiation spectrum S for different incident angles  at the fixed dL = 50 and L = 200 .Note that there is no emission at  = 0◦ , and after the spectrum peak, peaks are shifted to lower frequencies and the spectrum widths reduce with the incident angle increases. For the linear mode conversion from the wakefield to electromagnetic radiation, there is an optimized incident angle for high conversion efficiency at particular frequency. As shown in Fig. 7, we set ω ˜ = 0.3, i.e. ω = 0.3ωP0 , the emission peak occurs under the condition:

 ωL 1/3 c

sin  ≈ 0.8

(10)

The optimized incident angle is calculated to be 13.8◦ . Fig. 8 shows the radiation spectrum for different scale length L at the fixed  = 15◦ and dL = 50 . It indicates that the radiation power varies with L increases, while the center frequency and the spectrum width of emission depend on the density scale length weakly. So the scale length L can be used to adjust the emission

Fig. 9. Emission spectrum as a function of the pulse width dL for  = 15◦ and L = 200 .

power. Similar to the optimized incident angle, there is an optimized scale length for the emission spectrum peak. For ω = 0.3ωP0 , the optimized scale length is 15.70 calculated with Eq. (10). Fig. 9 shows the radiation spectrum as a function of the pulse width dL at the fixed  = 15◦ 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 changing P0 (or critical density nc ) and dL . If dL /P0 is fixed, the spectrum profile is fixed for given L and . Through theoretical analysis and one-dimensional PIC simulation, the THz emissions generated with short laser pulse incidence obliquely into inhomogeneous plasma have been investigated, power and spectrum of THz radiation are discussed under different laser intensity, pulse width, incident angle and plasma density scale length. The P-polarized electromagnetic radiation generated in a region enclosed by the normal to the interface and the direction of reflection. The simulation results are well consistent with the mode conversion theory, and it is important to understand the progress of the THz radiation generated through linear mode conversion References [1] E. Esarey, P. Sprangle, J. Krall, A. Ting, Overview of plasma-based accelerator concepts, IEEE Trans. Plasma Sci. 24 (1996) 252–288. [2] W.L. Kruer, The Physics of Laser Plasma Interaction, Addison-Wesley, New York, 1988. [3] R.W. Means, L. Muschietti, M.Q. Tran, J. Vaclavik, Electromagnetic radiation from an inhomogeneous plasma: theory and experiment, Phys. Fluids 24 (1981) 2197–2207. [4] D.E. Hinkel-Lipsker, B.D. Frid, G.J. Morales, Analytical expression for mode conversion of Langmuir and electromagnetic waves, Phys. Rev. Lett. 62 (1989) 2680–2683. [5] R. Lichters, J. Meyer-ter-Vehn, A. Pukhov, Short-pulse laser harmonics from oscillating plasma surface driven at relativistic intensity, Phys. Plasma 3 (1996) 3425–3437.