Hostname: page-component-74656f9884-6grmk Total loading time: 0 Render date: 2026-11-06T09:43:03.342Z Has data issue: false hasContentIssue false
  • English
  • Français

Interaction of high power laser beam with magnetized plasma and THz generation

Published online by Cambridge University Press:  19 October 2010

R.P. Sharma ,
A. Monika ,
P. Sharma ,
P. Chauhan  and
A. Ji
R.P. Sharma*
Affiliation:
Centre for Energy Studies, Indian Institute of Technology Delhi, New Delhi, India
A. Monika
Affiliation:
Centre for Energy Studies, Indian Institute of Technology Delhi, New Delhi, India
P. Sharma
Affiliation:
Centre for Energy Studies, Indian Institute of Technology Delhi, New Delhi, India
P. Chauhan
Affiliation:
Grupo de Laser e Plasmas, Instituto Superior Tecnico, Av. Rovisco Pais, Lisbon, Portugal
A. Ji
Affiliation:
Centre for Energy Studies, Indian Institute of Technology Delhi, New Delhi, India
*
Address correspondence and reprint requests to: R.P. Sharma, Centre for Energy Studies, Indian Institute of Technology Delhi, New Delhi 110016, India. E-mail: rpsharma@ces.iitd.ernet.in

Abstract

This paper presents an investigation of the excitation of a Tera hertz (THz) radiation by nonlinear interaction of a circularly polarized high power laser beam and density ripple in collisionless magneto plasma. The ponderomotive force due to the nonlinear interaction between the laser and density ripple generates a nonlinear current at a difference frequency. If the appropriate phase matching conditions are satisfied and the frequency of the ripple is appropriate, then this difference frequency can be brought in the THz range. Filamentation (self focusing) of a circularly polarized beam propagating along the direction of ambient magnetic field in plasma is first investigated within paraxial ray approximation. The beam gets focused when the initial power of the laser beam is greater than its critical power. Resulting localized beam couples with the pre-existing density ripple to produce a nonlinear current driving the THz radiation. Analytical expressions for the beam width of the laser beam, electric vector of the THz wave have been obtained. By changing the strength of the magnetic field, one can enhance or suppress the THz emission. For typical laser beam and plasma parameters with the incident laser power flux = 1014 W/cm2, laser beam radius (r0) = 40 µm, laser frequency (ω0) = 1014 rad/s and plasma density (n0) = 3 × 1018 cm−3, normalized ripple density amplitude (μ) = 0.3, the produced THz emission can be at the level of Giga watt in power.

Keywords

Electron plasma waveFilamentationPonderomotive nonlinearityTHz radiation

Information

Type
Research Article
Information
Laser and Particle Beams , Volume 28 , Issue 4 , December 2010 , pp. 531 - 537
Copyright
Copyright © Cambridge University Press 2010

1. INTRODUCTION

The increasing use of high power laser beams in various applications has worldwide interest in laser-plasma interaction (Sharma & Sharma, Reference Sharma and Sharma2009) in the nonlinear regime. In collisionless plasmas, the nonlinearity arises through the ponderomotive force-induced redistribution of plasma. It is well known that various laser-plasma instabilities like, filamentation (Kaw et al., Reference Kaw, Schmidt and Wilcox1973; Young et al., Reference Young, Baldis, Drake, Campbell and Estrabrook1988), harmonic generation (Ozaki et al., Reference Ozaki, Bom Elouga, Ganeev, Kieffer, Suzuki and Kuroda2007; Baeva et al., Reference Baeva, Gordienko and Pukhov2007; Dombi et al., Reference Dombi, Racz and Bodi2009), stimulated Raman scattering (SRS) (Kline et al., Reference Kline, Montgomery, Rousseaux, Baton, Tassin, Hardin, Flippo, Johnson, Shimada, Yin, Albright, Rose and Amiranoff2009), and stimulated Brillouin scattering (SBS) (Kappe et al., Reference Kappe, Strasser and Ostermeyer2007) might be generated from the interaction of laser pulses with plasmas. Advances in ultra short laser pulse technology have made it possible to exploit the Terahertz (THz) frequency range of the electromagnetic spectrum between the microwave and infrared. The generation of THz radiation is of great practical and theoretical interest because of its non-ionizing character and penetrating power comparable to the microwaves and has been extensively studied by laser-plasma interaction experiments (Dong et al., Reference Dong, Sheng, Wu, Wang and Zhang2009; Schroeder et al., Reference Schroeder, Esarey, Tilborg and Leemans2004; Abo-Bakr et al., Reference Abo-Bakr, Feikes, Holldack, Kuske, Peatman, Schade, Wustefeld and Hubers2003; Yoshii et al., Reference Yoshii, Lai, Katsouleas, Joshi and Mori1997; Yugami et al., 2009; Hamster et al., Reference Hamster, Sullivan, Gordon, White and & Falcone1993), and formation of filaments by intense laser propagating in air or gas (Sprangle et al., Reference Sprangle, Penano, Hafizi and Kapetanakos2004; Xie et al., Reference Xie, Dai and Zhang2006; Yugami et al., Reference Yugami, Ninomiya, Kobayashi and Noda2006; D'Amico et al., Reference D'amico, Houard, Franco, Prade, Mysyrowicz, Couairon and Tikhonchuk2007). This high power radiation has many potential applications in biological imaging, material characterization, remote sensing, and tomography. Some of the specialized reviews for applications of THz are also available in Feruguson and Zhang Ghang (Reference Feruguson and Zhan Ghang2002).

The key issue with plasma is to find appropriate ways of producing THz emission with high conversion efficiencies. The first experimental observation of laser-plasma-produced THz emission was reported by Hamster et al. (Reference Hamster, Sullivan, Gordon, White and & Falcone1993). Their experiment was based on the acceleration of the charged electrons and ions by the ponderomotive forces exerted by the intense light pulse. Ponderomotive force and nonlinear process in laser plasma interaction are the main mechanisms of THz wave generation. Laser plasma interaction also offers the promise for generating coherent radiation (Nuss & Orenstein, Reference Nuss and Orenstein1997; Yampolsky & Frainman, Reference Yampolsky and Frainman2006; Cheng et al., Reference Cheng, Wright and Moloney2001; Antonsen et al., Reference Antonsen, Palastro and Milchberg2007; Gildenburg & Vvedenskii, Reference Gildenburg and Vvedenskii2007). For example, coherent pulses of THz radiation are especially of interest for time domain spectroscopy (Nuss & Orenstein, Reference Nuss and Orenstein1997). It has been observed by Sheng et al. (2005) in their particle in cell simulations that coherent THz radiation can be generated around plasma frequency from vacuum-plasma interface through laser wake filed excitation. Wu et al. (Reference Wu, Sheng, Dong, Xu and Zhang2007) studied the effect of transverse magnetic field on this process and found substantial enhancement in the efficiency of THz generation, which is one of the key challenges in this domain. Yampolsky and Frainman (Reference Yampolsky and Frainman2006) proposed a scheme to produce or amplify THz radiation based on four-wave coupling (two lasers, plasma wave, and terahertz wave) in a plasma filled capillary. Two counter propagating laser pulses interfere to exert an electrostatic ponderomotive force on electrons with frequency close to plasma frequency. The nonlinear current generated in the beating process interacts with the parallel component of the electric field of THz wave and amplifies it. They have also investigated the effect of backward Raman scattering on the growth rate of THz wave. Cheng et al. (Reference Cheng, Wright and Moloney2001) have reported the generation of THz pulse radiation from plasma channels induced by femtosecond light strings, which are generated due to the establishment of a dipole moment in the channel by radiation pressure force of the light string, which oscillates at the plasma frequency and radiates. Antonsen et al. (Reference Antonsen, Palastro and Milchberg2007) proposed the employment of a corrugated plasma channel for phase matched THz radiation generation by the ponderomotive force of a laser pulse.

One of the main motivations of this manuscript is to present a more general analytical approach together with numerical results to investigate the excitation of electromagnetic wave at difference frequency ω2 = ω0 − ω1 in the THz frequency range, by laser and low density electron plasma waves of frequencies ω0, ω1, respectively in the paraxial approximation. We consider a circularly polarized high power laser beam propagating through magnetized plasma with propagation along the static magnetic field. Ponderomotive force at the difference frequency generates a nonlinear current due to the interaction between low density rippled plasma and laser beam and leads to THz generation.

The organization of this paper is as follows. In Section 2, we have given the theory of filamentation of high power laser beam in magnetized plasma when ponderomotive nonlinearity is operative using the paraxial ray approximation. Section 3 investigates the THz generation due to filamentation and resulting localization. We continue in Section 4 with numerical results and discussion. Section 5 summarizes the main conclusion in the manuscript.

2. FILAMENTATION OF GAUSSIAN LASER BEAM IN NONLINEAR MAGNETO PLASMA

Consider the propagation of a circularly polarized laser beam of angular frequency “ω0” in a homogeneous magnetoplasma with electron density η0, along the direction of static magnetic field $B_0 \hat z$. In linear approximation, we have taken the laser beam wave to be propagating in extraordinary mode (Ginzburg, Reference Ginzburg1964). The electric vector of laser field $\vec E_{0+} $ can be written as

(1)
\vec E_{0+}=\vec A_{0+} \left({x\comma \; y\comma \; z} \right)\exp \, \left({i\omega _0 t - ik_{0+} z} \right)\comma \;

where $k_{0+}=\displaystyle{{\omega _0 } / c}\left({\varepsilon _{0+} } \right)^{1/2} \, $is the propagation constant of the pump wave, and is given as

(2)
k_{0+}=\displaystyle{{\omega _0 } \over c}\left[{1 - \displaystyle{{\omega _p^2 } \over {\omega _0^2 \left({1 - \displaystyle{{\omega _{ce} } \over {\omega _0 }}} \right)}}} \right]^{1/2}\comma \;

where $\varepsilon _{0+}=1 - \displaystyle{{\omega _p^2 } \over {\omega _0^2 }}\displaystyle{1 \over {\left({1 - \displaystyle{{\omega _{ce} } \over {\omega _0 }}} \right)}}$ is the dielectric constant, $\omega _p=\left({\displaystyle{{4\pi n_0 e^2 } \over {m_e }}} \right)^{1/2} \, \, $is the plasma frequency, $\omega _{ce}=\displaystyle{{eB_0 } \over {m_e c}}$ is the electron cyclotron frequency, e and m e are the charge and rest mass of electron. The laser imparts oscillatory velocity ${\rm \vec v}_{0+} $ to electrons given by the expression,

(3)
{\rm \vec v}_{0+}=\displaystyle{{ - e\vec E_{0+} } \over {im_e \left({\omega _0 - \omega _{ce} } \right)}}.

The extraordinary mode amplitude is A 0+ = E x + iE y. We consider here the propagation of a three dimensional intense Gaussian electromagnetic beam along the direction of magnetic field. The intensity distribution of this mode at z = 0 in the direction transverse to the direction of propagation is given by

(4)
A_{0+} A_{0+}^{\ast} =E_{00}^2 \exp \left({ - \displaystyle{{r^2 } \over {r_0^2 }}} \right)\comma \;

where r 2 = x 2 + y 2, E 00, and r 0 are the axial amplitude and initial width of main beam, respectively. When a laser beam propagates through the plasma, the transverse intensity gradient generates a ponderomotive force, which modifies the plasma density profile in the transverse direction as in Sodha et al. (Reference Sodha, Ghatak and Tripathi1974).

(5)
N_{0e}=N_{00} \exp \left({ - \alpha _+A_{0+} A_{0+}^{\ast} } \right).

where $\alpha _ +=\displaystyle{{e^2 \left({1 - \displaystyle{{\omega _{ce} } \over {2\omega _0 }}} \right)} \over {16k_\beta m_e \omega _0 T\left({1 - \displaystyle{{\omega _{ce} } \over {\omega _0 }}} \right)^2 }}$ is the nonlinearity parameter, e and m e are the electronic charge and mass, and T is the equilibrium temperature of the plasma. N 0e is the time independent component of electron density as modified by main laser beam and N 00 is the electron density in the absence of the beam, k β is the Boltzman's constant.

The wave equation in its general form for a field varying as exp(iωt) is written as

(6)
\nabla ^2 \vec E - \nabla \left({\nabla .\vec E} \right)=- \displaystyle{{\omega ^2 } \over {c^2 }}\underline{\underline \varepsilon } .\vec E\comma \;

where $\underline{\underline \varepsilon } $ is the dielectric tensor having components:

\varepsilon _{xx}=\varepsilon _{yy}=1 - {{\omega _p^2 } / {\left({\omega _0^2 - \omega _c^2 } \right)}}\comma \;
\varepsilon _{xy}=- \varepsilon _{yx}=i\left({{{\omega _c } / {\omega _0 }}} \right){{\omega _p^2 } / {\left({\omega _0^2 - \omega _c^2 } \right)}}\comma
\varepsilon _{xz}=\varepsilon _{zx}=\varepsilon _{yz}=\varepsilon _{zy}=0\comma
\varepsilon _ +=\varepsilon _{xx} - i\varepsilon _{xy}\comma

and

\varepsilon _ -=\varepsilon _{xx}+i\varepsilon _{xy} .

Here the components of the dielectric constant have their usual meaning (Sodha et al., Reference Sodha, Ghatak and Tripathi1974). We assume that the field varies in the z-direction (i.e., along the static magnetic field) with variations being much larger than in the x − y plane so that waves can be treated as transverse in the zeroth order approximation, and hence no space charge is generated in the plasma. Therefore, using

\eqalign{\displaystyle{{\partial E_z } \over {\partial z}}&=- \left({\displaystyle{1 \over {\varepsilon _{zz} }}} \right)\left[{\left({\displaystyle{\partial \over {\partial x}}} \right)\left({\varepsilon _{xx} E_x+\varepsilon _{xy} E_y } \right)}\right. \cr & \qquad \left.{+\left({\displaystyle{\partial \over {\partial y}}} \right)\left({ - \varepsilon _{xy} E_x+\varepsilon _{xx} E_y } \right)} \right],}

and following Sodha et al. (Reference Sodha, Maheshwari, Sharma and Kaushik1980), the wave equation governing the propagation of the circularly polarized wave can be written as

(7)
\displaystyle{{\partial ^2 A_{0+} } \over {\partial z^2 }}+\displaystyle{1 \over 2}\left({\displaystyle{{\partial ^2 } \over {\partial x^2 }}+\displaystyle{{\partial ^2 } \over {\partial y^2 }}} \right)\left({1+\displaystyle{{\varepsilon _{0+} } \over {\varepsilon _{zz} }}} \right)A_{0+}+\displaystyle{{\omega _0^2 } \over {c^2 }}\varepsilon _+A_{0+}=0.

The appropriate components of the dielectric tensor of the magneto plasma can be written as in Sodha et al. (Reference Sodha, Maheshwari, Sharma and Kaushik1980).

\varepsilon _ +=\varepsilon _{0+}+\varepsilon _{2+} \lpar A_{0+} A_{0+}^{\ast} \rpar \comma \;

where ɛ0+ and $\varepsilon _{2+} \lpar A_{0+} A_{0+}^{\ast} \rpar $ are the linear and nonlinear parts of the dielectric constant, ε+, respectively. In Eq. (7), $\varepsilon _{zz}=1 - \displaystyle{{\omega _p^2 } \over {\omega _0^2 }}$ and

(8)
\varepsilon _{2+}=\displaystyle{{\omega _p ^2 } \over {\omega _0 ^2 }}\displaystyle{{\left\{{1 - \exp \left({ - \alpha _+} \right)A_{0+} A_{0+}^{\ast} } \right\}} \over {\left({1 - \displaystyle{{\omega _{ce} } \over {\omega _0 }}} \right)}}\comma \;

where ωce is the cyclotron frequency. Following Sodha et al. (Reference Sodha, Ghatak and Tripathi1974) and Akhamanov et al. (Reference Akhmanov, Sukhorukov and Khokhlov1968), the solution of Eq. (7) for A 0+ may be written in the following form

(9)
A_{0+}=A_{0+}^0 \exp \left\{{ - ik_{0+} \left({z+S_{0+} } \right)} \right\}\comma \;

where

(10)
\left({A_{0+}^0 } \right)^2=\displaystyle{{E_{00}^2 } \over {\,f_{0+}^2 }}\exp \left({ - \displaystyle{{r^2 } \over {r_0^2 \,f_{0+}^2 }}} \right)\comma \;
(11)
S_{0+}=\displaystyle{{r^2 } \over 2}\beta \left(z \right)+\phi _{0+} \left(z \right)\, \comma \;
(12)
\beta \left(z \right)\, =\, \displaystyle{1 \over {\,f_{0+} }}\displaystyle{{df_{0+} } \over {dz}}\displaystyle{{2k_{0+} } \over {\left({1+\displaystyle{{\varepsilon _{0+} } \over {\varepsilon _{zz} }}} \right)}}.

A 0+0(r, z) is a real function of its argument, S 0+ is the eikonal for the main beam, and ωp is the electron plasma frequency. The parameter β−1 may be interpreted as the radius of the curvature of the main beam. In Eq. (11), ø0+(z) is a constant that is not required explicitly in further analysis, f 0+ is the dimensionless beam width parameter satisfying the boundary conditions $f_{0+} \vert _{z=0}=1$, and $\displaystyle{{df_{0+} } \over {dz}}\vert _{z=0}=0$ is governed by the following differential equation

(13)
\displaystyle{{d^2 f_{0+} } \over {dz^2 }}=\displaystyle{{\left({1+\displaystyle{{\varepsilon _{0+} } \over {\varepsilon _{zz} }}} \right)^2 } \over {4R_{d0}^2 \,f_{0+}^3 }} - \displaystyle{{\varepsilon _{2+}^{^{\prime}} } \over {2\varepsilon _{0+} }}\displaystyle{{\left({\displaystyle{{E_{00}^2 } \over {\,f_{0+} }}} \right)\left({1+\displaystyle{{\varepsilon _{0+} } \over {\varepsilon _{zz} }}} \right)} \over {r_0^2 \,f_{0+}^2 }}.

R d0 = k 0+r 02 is the diffraction length and $\varepsilon _{2+}^{^{\prime}} $ is the derivative of ɛ2+ with respect to its argument. The first term on the right-hand side of Eq. (13) is related to the diffraction and the second term is related to the focusing effect due to the ponderomotive force. The density of the plasma varies through the channel due to the ponderomotive force. As a result, the refractive index increases and the laser beam is focused in the plasma. Eq. (10) gives the intensity profile of the laser beam in the plasma and is solved numerically.

3. THZ WAVE DYNAMICS

Now we consider the nonlinear interaction between the high power laser beam (ω0, k 0+) and the density ripple (ω1, k 1) in order to generate the THz radiation. We setup the model equations as follows. For circularly polarized wave (ω2, k 2), we introduce the following complex representation for THz electric field, $\vec E_{T+} $ and the variation of electric field, $\vec E_{T+} $ as

(14)
\vec E_{T+}=\hat rE_{T+} \exp \left({i\omega _2 t - ik_2 z} \right).

For the right-(left) handed wave, we have $\hat r=\hat x+i\hat y\left({\hat r=\hat x - i\hat y} \right)$, with $\hat x$ and $\hat y$ being the unit vectors along the x and y directions, respectively. The wave equation for electric vector E T+ with the magneto plasma may be written as (Shukla & Sharma, Reference Shukla and Sharma1982)

(15)
\nabla ^2 \vec E_{T+}=\displaystyle{1 \over {c^2 }}\displaystyle{{\partial ^2 \vec E_{T+} } \over {\partial t^2 }}+\displaystyle{{4\pi } \over {c^2 }}\displaystyle{{\partial \vec J_{T+} } \over {\partial t}}\comma \;

where $\vec J_{T+} $ is the total current density vector in the presence of low frequency electric field $\vec E_{T+} $ and it is given by

(16)
\vec J_{T+}=\vec J_{1+}+\vec J_{10+},

where J 1+ and J 10+ are the linear and nonlinear current densities, respectively.

Nonlinear interaction of a finite-amplitude plasma wave with high-and low frequency circularly polarized waves is governed by continuity equation,

(17)
\displaystyle{{\partial n_j } \over {\partial t}}+\nabla \cdot \left({n_j {\rm \vec v}_j } \right)=0\comma \;

and momentum transfer equation,

(18)
m_j \left[{\displaystyle{{\partial {\rm \vec v}_j } \over {\partial t}}+\left({{\rm \vec v}_j \cdot \nabla } \right){\rm \vec v}_j } \right]=- e\left[{\vec E_{T+}+\displaystyle{1 \over c}{\rm \vec v}_j \times \left({\vec B_0+\vec B} \right)} \right]\comma \;

where n j, m j, and v j are the particle density, the mass and the fluid velocity of species j = e, i; $\vec E$ and $\vec B$ are the self-consistent electric and magnetic fields associated with the wave.

In the field of the plasma wave (ω1, k 1) with the electric field

(19)
\vec E_1=\hat zE_1 \exp \left({i\omega _1 t - ik_1 z} \right)\comma \;

the electron quiver velocity ${\rm v}_1 \hat z$, and the electron density perturbation $\tilde n_p $ are related by

(20)
\displaystyle{{\tilde n_p } \over {n_0 }}=\displaystyle{{k_1 } \over {\omega _1 }}{\rm v}_1=\mu\comma \;

where ${\rm v}_1=\displaystyle{{eE_1 } \over {im_e \omega _1 }}$ and μ are the normalized ripple density amplitude. Nonlinear interaction of an electron-plasma mode with a high-frequency circularly polarized electromagnetic wave (ω0, k 0+) gives rise to a low frequency circularly polarized wave (ω2, k 2). The phase matching conditions are ω2 = ω0 − ω1 and k 2 = k 0+ − k 1. We write,

n_p=n_0+\tilde n_p \exp i\lpar \omega _1 t - k_1 z\rpar \comma \;
\, n_i=n_0 \comma \;
(21)
\vec J_{1+}=- en_0 {\rm \vec v}_{1+}^{\,e}+en_0 {\rm \vec v}_{1+}^{\,i} \comma \;

and

(22)
\vec J_{10+}=- e\tilde n_p^{\ast} {\rm \vec v}_{0+} - en_0 {\rm \vec v}_{10+}^{\, e} .

In the above equation, e and $\tilde n_p $ are charge and density perturbation of the electron, and n 0 is the background density, ${\rm \vec v}_{1+}\comma \; {\rm \vec v}_{10+} $ are linear and nonlinear velocities. Solving Eq. (18) we get the linear velocity, ${\rm \vec v}_{1+}^{\, e} $of electron for low frequency (ω2) right-handed circularly polarized wave:

(23)
{\rm \vec v}_{1+}^e=\displaystyle{{ - e\vec E_{T+} } \over {m_e i\left({\omega _2 - \omega _{ce} } \right)}}\comma \;

and the linear velocity for ion:

(24)
{\rm \vec v}_{1+}^i=\displaystyle{{e\vec E_{T+} } \over {m_i i\left({\omega _2+\omega _{ci} } \right)}}\comma \;

where $\omega _{ci}=\displaystyle{{eB_0 } / {m_i c}}$ is the ion cyclotron frequency with the ion mass m i.

Substituting ${\rm \vec v}_{1+}^{\, e} $and ${\rm \vec v}_{1+}^{\, i} $ from Eqs. (23) and (22) into Eq. (21) we find the linear current density $\vec J_{1+} $ for the right circularly mode, $\vec J_{1+} $ will be:

(25)
\vec J_{1+}=- in_0 e^2 \left({\displaystyle{{\omega _2 } \over {m_e \left({\omega _2 - \omega _{ce} } \right)\left({\omega _2+\omega _{ci} } \right)}}} \right)\vec E_{T+} .

The nonlinear velocity is produced by the beating of electron-induced velocity in density ripple with the laser velocity ${\rm \vec v}_{0+} $, and the corresponding laser-frequency magnetic field $\vec B_{0_+}=\left({ck_{0+} /\omega _0 } \right)\hat z \,\times\, \vec E_{0+} $. In particular, ${\rm \vec v}_{10+} $ is obtained by solving the following equation (Shukla & Sharma, Reference Shukla and Sharma1982),

m_e \left({\displaystyle{\partial \over {\partial t}}{\rm \vec v}_{10+}+\omega _{ce} {\rm \vec v}_{10+} \times \hat z} \right)=- m_e {\rm v}_1^{\ast} \left({\displaystyle{{\partial {\rm \vec v}_{0+} } \over {\partial z}} - \displaystyle{e \over {m_e }}\displaystyle{{k_{0+} } \over {\omega _0 }}\vec E_{0+} } \right)
(26)
\simeq - \displaystyle{{e\vec E_{0+} } \over {\omega _0 }}\displaystyle{{\omega _{ce} } \over {\left({\omega _0 - \omega _{ce} } \right)}}k_{0+} {\rm v}_1^{\ast} .

If we let ${\rm \vec v}_{10+}={\rm v}_{10+} \left({\hat x+i\hat y} \right)$, Fourier transformation of Eq. (26) then gives

(27)
{\rm \vec v}_{10+}=\displaystyle{{ - ie\vec E_{0+} \omega _{ce} k_{0+} {\rm v}_1^{\ast} \over {m_e \omega _0 \left({\omega _0 - \omega _{ce} } \right)\left({\omega _2 - \omega _{ce} } \right)}}}\comma \;

where superscript $ \ast $ implies a complex conjugate of that quantity. The low-frequency wave is taken to be a right-handed polarized wave. Substituting Eq. (27) and laser velocity, ${\rm \vec v}_{0+}=- \displaystyle{{e\vec E_{0+} } \over {im_e \left({\omega _0 - \omega _{ce} } \right)}}$ into Eq. (22), we get

(28)
\vec J_{10+}=- i\displaystyle{{n_0 e^2 } \over {m_e \left({\omega _0 - \omega _{ce} } \right)}}\left[{1 - \displaystyle{{\omega _1\, k_{0+} \omega _{ce} } \over {\omega _0\, k_1 \left({\omega _{ce} - \omega _2 } \right)}}} \right]\mu ^{\ast} \vec E_{0+} .

In the above equation, contribution of the ion term to the nonlinear coupling coefficient is small and is, therefore, neglected. Combining Eqs. (25) and (28) and substituting in (15), we obtain the following wave equation for $\vec E_{T+} $ in terms of the electric fields of the pump and plasma wave (at r = 0).

(29)
\eqalign{& \displaystyle{{\partial ^2 \vec E_{T+} } \over {\partial z^2 }}+\displaystyle{{\omega _2^2 } \over {c^2 }}\left({1 - \displaystyle{{\omega _P^2 } \over {\left({\omega _2+\omega _{ci} } \right)\left({\omega _2 - \omega _{ce} } \right)}}} \right)\vec E_{T+} \cr &=\displaystyle{1 \over 2}\displaystyle{{\omega _p^2 } \over {c^2 }}\displaystyle{{\omega _2 } \over {\left({\omega _0 - \omega _{ce} } \right)}}\mu ^{\ast} \left[{1 - \displaystyle{{\omega _1 } \over {\omega _0 }}\displaystyle{{k_{0+} } \over {k_1 }}\displaystyle{{\omega _{ce} } \over {\left({\omega _{ce} - \omega _2 } \right)}}} \right]\exp \lpar\! -\! ik_2 z\rpar \displaystyle{{E_{00} } \over {\,f_{0+} }}. }

Here f 0+ is the function of z and it is governed by Eq (10). To evaluate E T+ we employ the Green function technique. The solution of Eq. (26) may be written as (Mathews & Walker, Reference Mathews and Walker1965)

(30)
E_{T+}=- \displaystyle{1 \over {4\pi }}\vint\limits_0^z {G\lpar z\comma \; z^{\prime}\rpar } F\left({z^{\prime}} \right)dz^{\prime}\comma \;

where

\eqalign{F\left({z^{\prime}} \right)&=\displaystyle{1 \over 2}\displaystyle{{\omega _p^2 } \over {c^2 }}\displaystyle{{\omega _2 } \over {\left({\omega _0 - \omega _{ce} } \right)}}\mu ^{\ast} \left[{1 - \displaystyle{{\omega _1 } \over {\omega _0 }}\displaystyle{{k_{0+} } \over {k_1 }}\displaystyle{{\omega _{ce} } \over {\left({\omega _2 - \omega _{ce} } \right)}}} \right]\cr & \quad \times \exp \lpar \!\!-\! i\lpar k_2 \rpar z^{\prime}\rpar \displaystyle{{E_{00} } \over {\,f_{0+} }}\comma \; }

and $G\lpar z\comma \; z^{\prime}\rpar $ is the Green's function. Using the appropriate value of the Green's function,

G\lpar z\comma \; z^{\prime}\rpar =\displaystyle{{\exp \lpar - ik_{T1} \left\vert {z - z^{\prime}} \right\vert \rpar } \over {\left\vert {z - z^{\prime}} \right\vert }}.

Eq. (30) may be written as

(31)
E_{T+}=- \displaystyle{1 \over {4\pi }}\mathop\vint\limits_0^z {\displaystyle{{\exp \lpar - ik_{T1} \left\vert {z - z^{\prime}} \right\vert \rpar } \over {\left\vert {z - z^{\prime}} \right\vert }}} F\lpar z^{\prime}\rpar dz^{\prime}\comma \;

where $k_{T1}^2=\displaystyle{{\omega _2^2 } \over {c^2 }}\left({1 - \displaystyle{{\omega _P^2 } \over {\left({\omega _2+\omega _{ci} } \right)\left({\omega _2 - \omega _{ce} } \right)}}} \right)$. The amplitude of emitted electromagnetic wave may be given as

(32)
E_{T+}=- \displaystyle{{\exp \left({ - ik_{T1} z} \right)} \over {4\pi }}\, \mathop\vint\limits_0^z {\displaystyle{{\exp \lpar ik_{T1} z^{\prime}\rpar F\left({z^{\prime}} \right)} \over {\left\vert {z - z^{\prime}} \right\vert }}\, } dz^{\prime}.

One can obtain the THz field intensity with Eq. (32) where the laser beam intensity is changing with the distance of propagation as given by Eq. (10). The THz field generated by this mechanism can also be obtained numerically straight from Eq. (32) with the appropriate boundary conditions. The laser imparts oscillatory velocity to plasma electrons and exerts a ponderomotive force on them. The ponderomotive force has a transverse component that drives nonlinear current, producing THz radiation.

4. RESULTS AND DISCUSSION

This section deals with the results obtained by numerically solving Eqs. (10), (29), and (32), respectively. The following set of parameters has been used in the numerical calculations: we have used laser beam intensity $\alpha _+E_{00}^2 $ = 3 (laser power flux $ \sim $1014 W/cm2), beam radius r 0 = 40 µm, laser frequency (ω0) = 1014 rad/s, plasma density n 0 = 3 × 1018 cm−3, applied magnetic field B 0 = 60 and 77 and 105 kG for T = 1 KeV and ripple density amplitude μ = 0.3. For initial wave front of the beam, the initial conditions for f 0+ are $f_{0+} \vert _{z=0}=1 \ {\rm and}\ df_{0+} /dz\vert _{z=0}=0$. When laser beam propagate through the plasma, then density of plasma will be varying through the channel due to ponderomotive force. But the ponderomotive nonlinearity introduced in the plasma depends upon the intensity of laser beam. We solve Eq. (10) for initial field of the laser that is greater than the critical value. In that case, f 0+ is obtained from the numerical solution. Figure 1 illustrates the behavior of the laser beam (variation in laser beam intensity versus normalized distance $\left({\xi=\displaystyle{z \over {R_{d0} }}} \right)$ and radial distance), when the ponderomotive nonlinearity and paraxial approximation are taken into consideration for n 0 = 3 × 1018 cm−3, B 0 = 60 kG, and α+E 002 = 3. The laser intensity increases as f 0+ decreases, attains a maximum at an optimum distance and then falls.

Fig. 1.

(Color online) Variation of laser beam intensity with normalized distance $\left({\xi=\displaystyle{z \over {R_{d0} }}} \right)$ and radial distance ( r/r 0) (r) in paraxial case by using ponderomotive nonlinearity.

Here we have also developed the theory of THz wave generation. We solve Eqs. (26) and (29) numerically by using the data of the solution of Eq. (10). Figure 2 depicts the variation of the THz field with the normalized distance of propagation when laser beam exhibit oscillatory focusing at different values of the static magnetic field. In Figure 2, for ωce/ω0 = .01, .016, and .019, i.e., by increasing the magnetic field, the normalized THz amplitude rises at ω2 = around 1 THz. These parameters, for plasma frequency n 0 = 3 × 1018 cm−3, laser frequency (ω0) 1014 rad/s (CO2 laser), correspond toB 0 = 60 kG, 77 kG, and 105 kG. It is quite apparent that the THz amplitude exhibits maxima and minima. We find that the radiated electromagnetic wave has power level in the GW range as shown in Figure 2. We report the generation of THz radiation with maximum conversion efficiencies 20%, 27%, and 35% for ωce/ω0 = .01, .016, and .019. It should be mentioned here that if the laser beam is not self focused (f 0+ = 1), the conversion efficiency is constant at the level of 0.6%. Therefore, self focusing of the laser beam plays a very important role in this conversion mechanism. This is on account of the fact that the self focusing of the laser beam leads to enhanced intensity of the laser therefore the conversion process which depends on the laser intensity gets enhanced.

Fig. 2.

(Color online) Variation of normalized THz field amplitude (normalized with laser field amplitude) with normalized distance $\left({\xi=\displaystyle{z \over {R_{d0} }}} \right)$ in paraxial case by using ponderomotive nonlinearity. Dotted blue line, semi-dotted red line and black solid line, are respectively for ωce/ω0 = .01,ωce/ω0 = 0.016, andωce/ω0 = 0.019.

5. CONCLUSION

We have studied the dynamics of the generation of THz wave by self focusing of high power laser beam in magnetized plasma that couples with low density ripples, when ponderomotive nonlinearity is operative. The nonlinear mechanism which generates the THz radiation can be understood as follows: The nonlinear interaction of laser beam and density ripple generates a nonlinear current. It is obvious from Eq. (23) that the nonlinear current is generated due to the ponderomotive force by the beating of pump (laser) wave and density ripple electric field. The ponderomotive force has two components: Due to nonlinear terms $\left({{\rm \vec v}_{\rm 1}^ {\,\ast} \cdot \vec \nabla } \right){\rm \vec v}_{\rm 0} $ and $\left({{\rm \vec v}_{\rm 1}^ {\, \ast} \times \vec B_{0+} } \right)$. By the difference in pump wave frequency (ω0) and density ripple frequency(ω1), we get a frequency (ω2) in the range of THz frequency. Thus the THz radiation is resonantly excited if the matching conditions are satisfied. THz amplitude can be controlled by applied magnetic field, laser amplitude, and rippled density amplitude. The dependence on the ripple density amplitude (μ) is obvious from Eq. (26) of THz wave. But the dependence on laser amplitude and applied magnetic field is quite complex. Let us discuss first the dependence on laser amplitude. From THz expression Eq. (26) THz amplitude depends directly on laser amplitude. But in addition to this the self focusing of the laser beam also affect the THz generation. (Please confirm from Eq. (26)).The self focusing of laser beam gets affected if initial laser beam intensity is changed. Therefore on account of the self focusing, the THz generation is having a complex dependence on laser amplitude. Now we discuss the effect of magnetic field on THz generation. The magnetic field affects the THz generation in three ways. Firstly the laser beam propagation in our geometry is a right circularly polarized mode and its phase velocity and group velocity depends on the background magnetic field strength. Secondly, self focusing of the laser beam gets affected by changing the strength of the magnetic field. THz wave geometry also depends on the background magnetic field. Acutely THz wave in our geometry is basically an Alfven wave which gets affected by change in background magnetic field. Hence magnetic field plays an important role in the generation of THz radiation.

The effect of magnetic field on the THz wave amplitude is shown in the fig. (2).The emission frequency and phase can also be controlled by this process. The related simulation being complicated and beyond the scope of the present work will be done in future. Numerical Simulation shows that for the incident laser power flux = 1014 W/cm2, laser beam radius $\left({r_0 } \right)=40\, \rm \mu m$, laser frequency $\left({\omega _0 } \right)=10^{14} \rm rad/s$ and plasma density $\left({n_0 } \right)=3 \times 10^{18} {\rm cm}^{{\rm - 3}} $, one can obtain GW power at around 1 THz frequency. The radiation that is emitted in this process can be a powerful THz source at the GW level. Self focusing of the laser beam plays a very important role in this conversion mechanism. This technique could be a source of generating high-power THz radiation in future.

ACKNOWLEDGMENTS

This work was partially supported by DST, Government of India. One of the authors (Monika) is grateful to Dr. Karuna Batra for useful discussions.

References

REFERENCES

Abo-Bakr, M., Feikes, J., Holldack, K., Kuske, P., Peatman, W.B., Schade, U., Wustefeld, G. & Hubers, H.-W. (2003). Brilliant, coherent far-infrared (THz) synchrotron radiation. Phys. Rev. Lett. 90, 094801.Google Scholar
Akhmanov, A.S., Sukhorukov, A.P. & Khokhlov, R.V. (1968). Self-focusing and diffraction of light in a nonlinear medium. Soviet. Phys. Usp. 10, 609–636.CrossRefGoogle Scholar
Antonsen, T.M., Palastro, J. Jr. & Milchberg, H.M. (2007). Excitation of terahertz radiation by laser pulses in no uniform plasma channels. Phys. Plasmas 14, 033107.Google Scholar
Baeva, T., Gordienko, S. & Pukhov, A. (2007). Relativistic plasma control for single attosecond pulse generation: Theory, simulations and structure of the pulse. Laser Part. Beams 25, 339–346.Google Scholar
Cheng, Chung-Chieh, Wright, E.M. & Moloney, J.V. (2001). Generation of electromagnetic pulses from plasma channels induced by femtosecond light strings. Phys. Rev. Lett. 87, 213001.CrossRefGoogle ScholarPubMed
D'amico, C., Houard, A., Franco, M., Prade, B., Mysyrowicz, A., Couairon, I.A. & Tikhonchuk, V.T. (2007). Conical forward THz emission from femtosecond-laser-beam filamentation in air. Phys. Rev. Lett. 98, 235002.Google Scholar
Dombi, P., Racz, P. & Bodi, B. (2009). Surface plasmon enhanced electron acceleration with few cycle laser pulses. Laser Part. Beams 27, 291–296.Google Scholar
Dong, X.G., Sheng, Z.M., Wu, H.C., Wang, W.M. & Zhang, J. (2009). Single-cycle strong terahertz pulse generation from a vacuum-plasma interface driven by intense laser pulses. Phys. Rev. E 79, 046411CrossRefGoogle ScholarPubMed
Feruguson, B. & Zhan Ghang, X.-C. (2002). Materials for terahertz science and technology. Nat. Mater. 1, 26.Google Scholar
Gildenburg, V.B. & Vvedenskii, V.V. (2007). Optical-to-THz wave conversion via excitation of plasma oscillations in the tunneling-ionization process. Phys. Rev. Lett. 98, 245002.Google Scholar
Ginzburg, V.L. (1964). The Propagation of Electromagnetic Waves in Plasmas. London: Pergamom.Google Scholar
Hamster, H., Sullivan, A., Gordon, S., White, W.& Falcone, R.W. (1993). Subpicosecond, electromagnetic pulses from intense laser-plasma interaction. Phys. Rev. Lett. 71, 2725.CrossRefGoogle ScholarPubMed
Kappe, P., Strasser, A. & Ostermeyer, M. (2007). Investigation of the impact of SBS-parameters and loss modulation on the mode locking of an SBS-laser oscillator. Laser Part. Beams 25, 107–116.Google Scholar
Kaw, P.K., Schmidt, G. & Wilcox, T. (1973). Filamentation and trapping of electromagnetic radiation in plasmas. Phys. Fluids 16, 1522–1525.Google Scholar
Kline, J.L., Montgomery, D.S., Rousseaux, C., Baton, S.D., Tassin, V., Hardin, R.A., Flippo, K.A., Johnson, R.P., Shimada, T., Yin, L., Albright, B.J., Rose, H.A. & Amiranoff, F. (2009). Investigation of stimulated Raman scattering using a short-pulse diffraction limited laser beam near the instability threshold. Laser Part. Beams 27, 185–190.Google Scholar
Mathews, J. & Walker, R.L. (1965). Mathematical Methods of Physics. New York: Benjamin.Google Scholar
Nuss, M.C. & Orenstein, J. (1997). Millimeter-Wave Spectroscopy of Solids. Heidelberg: Springer-Verlag.Google Scholar
Ozaki, T., Bom Elouga, L.B., Ganeev, R., Kieffer, J.-C., Suzuki, M. & Kuroda, H. (2007). Intense harmonic generation from silver ablation. Laser Part. Beams 25, 321–325.CrossRefGoogle Scholar
Schroeder, C.B., Esarey, E., Tilborg, J. Van, & Leemans, W.P. (2004). Theory of coherent transition radiation generated at a plasma-vacuum interface. Phys. Rev. E 69, 016501.Google Scholar
Sharma, R.P. & Sharma, P. (2009). Effect of laser beam filamentation on second harmonic spectrum in laser plasma interaction. Laser Part. Beams 27, 157–169 .Google Scholar
Shukla, P.K. & Sharma, R.P. (1982). Alfvén-wave generation in a beam-plasma system. Phys. Rev. A 25, 2816–2819.Google Scholar
Sodha, M.S., Ghatak, A.K. & Tripathi, V.K. (1974). Self Focusing of Laser Beams in Dielectrics, Plasma and Semiconductors. Delhi: Tata-McGraw-Hill.Google Scholar
Sodha, M.S., Maheshwari, K.P., Sharma, R.P. & Kaushik, S.C. (1980). Plasma wave and second harmonic generation by a Gaussian EM beam in extraordinary/ordinary mode. Proc. Indian Nat. Sci. Acad. 46, 343–355.Google Scholar
Sprangle, P., Penano, J.R., Hafizi, B. & Kapetanakos, C.A. (2004). Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces. Phys. Rev. E 69, 066415.Google Scholar
Wu, H.C., Sheng, Z.M., Dong, Q.L., Xu, H. & Zhang, J. (2007). Powerful terahertz emission from laser wakefields in inhomogeneous magnetized plasmas. Phys. Rev. E 75, 016407.Google Scholar
Xie, X., Dai, J.M. & Zhang, X.C. (2006). Coherent control of THz wave generation in ambient air. Phys. Rev. Lett. 96, 075005.Google Scholar
Yampolsky, N.A. & Frainman, G.M. (2006). Conversion of laser radiation to terahertz frequency waves in plasma. Phys. Plasmas 13, 113108.CrossRefGoogle Scholar
Yoshii, J., Lai, C.H., Katsouleas, T., Joshi, C. & Mori, W.B. (1997). Radiation from Cerenkov wakes in a magnetized plasma. Phys. Rev. Lett. 79, 4194.Google Scholar
Young, P.E., Baldis, H.A., Drake, R.P., Campbell, E.M. & Estrabrook, K.G. (1988). Direct evidence of ponderomotive Filamentation in laser-produced plasma. Phys. Rev. Lett. 61, 2336–2339.CrossRefGoogle ScholarPubMed
Yugami, N., Ninomiya, K., Kobayashi, K. & Noda, H. (2006). Observation of millimeter range radiation with TM01 mode by laser plasma interaction experiments. Jpn. J. Appl. Phys. 45, L1051.Google Scholar
Figure 0

Fig. 1. (Color online) Variation of laser beam intensity with normalized distance $\left({\xi=\displaystyle{z \over {R_{d0} }}} \right)$ and radial distance ( r/r0) (r) in paraxial case by using ponderomotive nonlinearity.

Figure 1

Fig. 2. (Color online) Variation of normalized THz field amplitude (normalized with laser field amplitude) with normalized distance $\left({\xi=\displaystyle{z \over {R_{d0} }}} \right)$ in paraxial case by using ponderomotive nonlinearity. Dotted blue line, semi-dotted red line and black solid line, are respectively for ωce/ω0 = .01,ωce/ω0 = 0.016, andωce/ω0 = 0.019.