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Emission of strong Terahertz pulses from laser wakefields in weakly coupled plasma
Nuclear Instruments and Methods in Physics Research A ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Emission of strong Terahertz pulses from laser wakefields in weakly coupled plasma Divya Singh a,b,n, Hitendra K. Malik a a b
PWAPA Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110 016, India Department of Physics & Electronics, Rajdhani College, University of Delhi, Raja Garden, Ring Road, New Delhi 110015, India
art ic l e i nf o
a b s t r a c t
Article history: Received 19 October 2015 Received in revised form 10 March 2016 Accepted 31 March 2016
The present paper discusses the laser plasma interaction for the wakefield excitation and the role of external magnetic field for the emission of Terahertz radiation in a collisional plasma. Flat top lasers are shown to be more appropriate than the conventional Gaussian lasers for the effective excitation of wakefields and hence, the generation of strong Terahertz radiation through the transverse component of wakefield. & 2016 Elsevier B.V. All rights reserved.
Keywords: Magnetic field Plasma Collisions Wakefield Terahertz radiation
1. Introduction Interaction of intense lasers with plasma is a very enriched field of research, giving rise to very interesting physical phenomena like laser induced fusion, higher harmonic generation, laser-plasma channeling, laser plasma acceleration and plasma based radiation sources. These radiation sources range from X-ray to THz frequency domain. Now a days Terahertz radiation sources have several applications and are being used extensively for imaging, material characterization, topography, tomography, communication, etc. [1]. THz radiation can be produced from plasma and electron beam based THz emitters, such as coherent radiation from plasma oscillations driven by ultrashort laser pulses [2], transition radiation of electron beams [3], synchrotron radiation from accelerator electrons [4], Cherenkov wake radiation in magnetized plasmas [5] and emission from laser plasma channels in air. Plasma is found to be more suitable than the solid targets due to longer sustainability at higher powers of the laser. Theoretical and simulation studies of extremely powerful Terahertz emission by the interaction of chirped [6] and few cycle laser pulses [7] with tenuous plasma have been reported by Wang et al. Ostermayr et al. have used super-Gaussian pulse for the laser plasma acceleration [8]. A wakefield is an electrostatic wave driven by a laser pulse in n Corresponding author at: Department of Physics & Electronics, Rajdhani College, University of Delhi, Raja Garden, Ring Road, New Delhi 110015, India. E-mail address: [email protected] (D. Singh).
plasmas. There have been a number of studies on the excitation of the large amplitude wakefield because of its extensive application prospects, such as in high-gradient electron acceleration [9], proton acceleration [10] and X-ray radiation [11]. Sheng et al. have proposed that high efficiency THz emissions can be produced from a laser driven wakefield in an inhomogeneous plasma through linear mode conversion [12]. Wu et al. [13] have studied the effect of the transverse magnetic field on this process and have found significant enhancement in the efficiency of THz generation. Since the typical plasma oscillation frequency for these applications is in the Gigahertz to Terahertz range, the wakefield can potentially serve as a powerful THz emitter. Above mentioned, all schemes talk about the wakefield excitation but none of them have discussed collisional plasma. In theories as well as in experiments, collisional effects are generally ignored but our calculations and results show that the even small fraction of collisions in the range (ν ¼0.05ωp to 0.5ωp) reduce the longitudinal wakefield and therefore amplitude of emitted THz field is also reduced greatly. The collision frequency depends on the electron temperature as per the relation
νe = ν0 ( Te/T0 )
s /2
.
where, s is a parameter characterizing the type of collision in plasma, where T0 is the equilibrium temperature [14]. For experimentation purpose it is promising to convert a considerable fraction of the energy of plasma oscillations exciting longitudinal wakefield in the plasma is reemitted with their frequencies controlled over a terahertz range radiation by varying the parameters
http://dx.doi.org/10.1016/j.nima.2016.03.108 0168-9002/& 2016 Elsevier B.V. All rights reserved.
Please cite this article as: D. Singh, H.K. Malik, Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.03.108i
D. Singh, H.K. Malik / Nuclear Instruments and Methods in Physics Research A ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
of the ionized gas, i.e. the pressure and temperature. In particular, Golubev et al. [15] have experimentally obtained THz radiation from the optical breakdown of a gas and investigated the effect of gas pressure and temperature on collisions. They found the collision frequency of the order of 1.5 1013 s 1 and plasma frequency 3.0 1014 s 1 at the atmospheric pressure and when the temperature Te ¼15 eV. Therefore, the inequality ν o ωp may be fulfilled at the normal pressure for better amplitude of the THz radiation. However, at the higher pressure of the order of 400 atm, both the frequencies become equal (fp E ν E 6 1015 s 1). Hence, the corresponding radiation frequency falls into the optical range, which, is of little interest. Hence, it is important to consider collisions for normal pressure and temperature in laser plasma interaction experimentally. On the other hand, Chen [16] has studied experimental behavior of Neon plasma with respect to electron collision frequency by varying plasma density for different temperatures and concluded that the electron collisions in uniform density plasma are sensitive to electron temperature at a given gas temperature and show weak dependence on the gas pressure. Thus it become imperative to control collisions in the laser plasma interaction mechanism. Therefore, we justify the discussion on emission of THz radiation pulses from wakefields in collisional plasma when Gaussian and flat top lasers are used and we also justify that the collision frequency that we have considered are experimentally viable. In the present article, we present a theoretical analysis for Terahertz radiation generation based on the scheme of the excitation of wakefield produced by the propagation of highly intense ultra-short flat top laser through collisional magnetized plasma of uniform density, where ions are supposed to be immobile and electron neutral collisions prevail.
using equation of continuity where n0 is taken to be the ambient density of the plasma. The components of the nonlinear plasma density are obtained as n(1) =
y
In the present work, we consider a flat top laser (FTL) of frequency ω and wave number k, propagating in the plasma of uniform density n0 along the z-axis. The field of the laser is given as ⎡ ⎛ y ⎞6⎤ ⎢−⎜ ⎟ ⎥ e⎢⎣ ⎝ bw ⎠ ⎥⎦ e[i (kz − ωt )] y^ ,
→ E = E0 where E0 is the field amplitude and bw is the beamwidth of the laser. The laser electric field is taken to be polarized along the y-axis whereas an external magnetic field is taken to be applied along the x-axis. In the presence of the external magnetic field (B), the laser field impart oscillatory velocity to the electrons whereas plasma ions remain immobile. The plasma is weakly coupled, so the electron neutral collisions are taken to be present. We make use of the perturbation approach. Hence, physical quantities like electron velocity, plasma density and plasma currents are expanded to their first order in orders of laser strength eE parameter (a = mc0ω «1 approximated). The electron dynamics in the plasma is completely expressed by the force equation, where collisional force (frequency ν ) acts as a damping cause. The plasma electron velocity is computed as, (0) (1) → υ =→ υ + a→ υ = eE0
⎡ ⎛ y ⎞6⎤ ⎢−⎜ ⎟ ⎥⎡ e⎢⎣ ⎝ bw ⎠ ⎥⎦ ⎢
ei (kz − ωt ).
1 y^ ⎣ im (ω + iν )
+a
{
(ω + iν ) im [(ω + iν )2 − ωc2 ]
y^ +
ωc m [(ω + iν )2 − ωc2 ]
⎤ z^ ⎥ ⎦
}
No component of the velocity is found to arise in the direction of the external magnetic field. The cyclotron frequency is represented eB as ωc = mc x^ . Under the influence of collisional forces and external magnetic field, the electron oscillations become nonlinear. Therefore, nonlinear density perturbations are generated. Different order components of the nonlinear density are computed
+
(1) kn0 → υz ^ z. ω
Due to the
density perturbations in the plasma, coupling of plasma oscillations with density fluctuations take place that drive nonlinear plasma currents. Thus magnitude of these nonlinear plasma currents are obtained as ⎡ ⎛ ⎤ y ⎞6
⎢−⎜ ⎟ ⎥ → ⎢ ⎝b ⎠ ⎥⎡ 1 iω ( ω + iν ) J = n0 e2E0 e⎣ w ⎦ ⎢ y^ + 2 (0) 2 υ y ) m [(ω + iν ) − ωc ] ⎣ (ω − k→
ωc m [(ω + iν )2 − ωc2 ]
⎤ z^⎥ ei (kz − ωt ) . ⎦
Now we will discuss how nonlinear plasma current densities cause the excitation of various components of the wakefield and thereby the emission of THz radiation in collisional plasma. The components of electric and magnetic wakefield are coupled with → → time dependent Maxwell's equations by using Ew and Bw as the components concerning the wakefield. In general, the magnetic wakefield is not considered further, the reason being that they have very low magnitudes as compared to the electric wakefields. Using quasistatic approximation, we consider that the laser pulse does not evolve significantly and therefore, the field variations are assumed to be time independent. Maxwell's equations are expressed in terms of the transformed coordinate ξ = z − υg t as laser propagates along z axes with group velocity vg. Further, the plasma electron motion is analyzed under the excited plasma wakefield using force equation. On combining the Maxwell's equations with the force equation, we obtain differential equations for the wakefields as follows
→ → ⎡→ ⎛ ⎞→ ⎤ ∂ 2E y ∂ 2Ez ω ⎟⎟ Ey ⎥ + + k p2 ⎢ Ez + ⎜⎜ 2 2 0 ( ) ⎢⎣ ∂ξ ∂ξ ⎝ ω − kυy ⎠ ⎥⎦ = −
2. Excitation of plasma currents, wakefield and THz radiation
(1) kn0 → υy y^ (0) → (ω − k υ )
k p2 m e
× eE0
⎡ ⎛ 6⎤ ⎢ −⎜ y ⎞⎟ ⎥ ⎢⎣ ⎝ b w ⎠ ⎥⎦ e
⎡⎛ ⎞ ω ν (ω + iν ) ⎢⎜ ⎟ ⎜ 0 ( ) ⎢⎣ ⎝ ω − kυy ⎟⎠ im [(ω + iν )2 − ωc2 ] ⎡ ⎤⎤ ωc νωc ⎥ ⎥, +⎢ − 2 2 ⎣ im (ω + iν ) m [(ω + iν ) − ωc ] ⎦ ⎥⎦ where k p2 =
4πn0 e
2
mc 2
(1)
is the plasma wave number. Eq. (1) is solved
numerically for the y-and z-components of the wakefield using Runge–Kutta method for collisional magnetized plasma with → boundary condition that Ez = 0 at ξ = 0 and ξ = L/2. Here L is the length of the laser pulse. Numerical solution of Eq. (1) and hence, the wakefields are plotted in the next section. It is observed that the magnitudes strictly depends on the relative amplitude of the laser field, electron neutral collision and cyclotron frequencies. We further calculate the transverse component of the wakefield in terms of the horizontal wakefield analytically. This transverse component of the oscillating wakefield is emitted as THz pulses. The analytical expression of the transverse THz field amplitude is obtained as
⎤ ⎡ 2 (0) 2 → iν (ω + iν ) 1 6y5 ωc (ω − kυy )(ω − ωc + iνω) ⎥ + ETHz = ⎢ 2 6 2 2 ⎥ ⎢k b ⎡ (ω + iν )2 − ω 2 ⎤ [( ω + ν ) − ω ] i ω ( ω − ν ) i p w c ⎣ ⎦ c ⎦ ⎣ 6
E0 e−(y / b w ) .
(2)
It is evident from the mathematical calculations that the plasma wakefields are excited in the plasma due to propagation of highly intense laser through it. These wakes are longitudinal and transverse in nature. It is well-known that the longitudinal wakefields are used for the acceleration purpose, whereas the transverse component is of less importance. Our calculations
Please cite this article as: D. Singh, H.K. Malik, Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.03.108i
D. Singh, H.K. Malik / Nuclear Instruments and Methods in Physics Research A ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
reveal that the transverse component of the wake is originated by the horizontal wakefield and is responsible for the emission of electromagnetic radiation in THz frequency range.
3. Results and discussion Here we will talk about the evolution of wakefields due to the propagation of flat top laser through the plasma, electric field profile of the longitudinal and transverse wakefields under the impact of collisions. A comparison of the results obtained based on the proposed flat top laser with the ones of Gaussian laser is also made. The evolution of the longitudinal wakefield is shown in Fig. 1 for different collision frequencies between the electrons and neutrals for the case of flat top laser. Here the solid lines correspond to the Gaussian laser (GL) whereas the dotted lines represent the case of flat top laser (FTL). Normalization of the laser propagation distance is made with the laser pulse length L. The magnitude of the excited wakefield is found to decrease with the increased collision frequency. The wake amplitude is sufficiently good for low collisions (ν«ωp) but falls sharply to further increased collisions (ν ¼0.5 ωp to 1.5 ωp). It is also clearly seen that the large amplitude wakefiled is realized when FTL is used. These large amplitude wakes are attributed to the uniform maximum intensity distribution of the FTL transverse to the laser propagation axis in comparison with the GL [17,18]. Therefore, maximum energy of the laser is utilized to excite plasma wakefield Fig. 2 describes the transverse profile of the emitted THz field for different laser frequencies, when the collision frequency is kept fixed as ν ¼0.05 ωp. Here it is evident that the THz field amplitude maximizes when the laser frequency remains closer to the plasma frequency (ω ¼ ωp). This is due to the frequency matching at the resonance which leads to the maximum energy transfer from the laser to the plasma oscillations that causes the large magnitude
Fig. 2. Normalized amplitude of THz field with normalized transverse distance from propagation axis of the laser in plasma for different laser frequency (normalized with plasma frequency), when ωp ¼ 2.0 1013 rad/s, ν ¼ 0.05 ωp, y¼ 0.5bw, E0 ¼ 5.0 108 V/m, bw ¼0.01 cm and L ¼ 0.5λp.
wakefield. As the resonance condition departs, the wake amplitude tends to decay and the peak of the emitted radiation is also shifted. If we compare the results of Fig. 2 (ν ¼0.05 ωp) with Fig. 3 (ν ¼0.5 ωp), we find that in the presence of high collision rate the THz field amplitude decreases but the peak of the radiation is not affected. In the case of higher collisions, a uniform illumination of the emitted radiation is obtained instead of sharply focused radiation. Fig. 3 further explains that the FTL gives higher THz field amplitude than the GL. The collisions in the plasma act as a damping force to the electron oscillations. These reduce the nonlinear currents, the amplitude of the excited wakefield and also affect the peak of the emitted radiation.
Fig. 1. Normalized amplitude of the plasma wake field with normalized distance of propagation of the laser in plasma for different collision frequencies for Gaussian (GL) and flat top (FTL) lasers, when ω¼ 2.4 1014 rad/s, ωp ¼ 2.0 1013 rad/s, y¼ 0.8bw, E0 ¼ 5.0 108 V/m, bw ¼ 0.01 cm, B ¼1 T and L ¼0.5λp.
Please cite this article as: D. Singh, H.K. Malik, Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.03.108i
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Fig. 3. Normalized amplitude of THz field with normalized distance of propagation of laser in plasma for Gaussian (GL) and flat top (FTL) lasers for different laser frequency (normalized with plasma frequency) when ν ¼1.5 ωp, ν ¼0.5 ωp, ωp ¼ 2.0 1013 rad/s, y¼ 0.5bw, E0 ¼ 5.0 108 V/m, bw ¼ 0.01 cm and L ¼ 0.5λp.
The effect of magnetic field on the radiated field amplitude can be seen in Fig. 4, where the normalized field amplitude of the THz radiation is plotted with normalized laser frequency for different collision frequency and pulse length. The applied magnetic field is found to enhance the THz amplitude even in the presence of significant collisions. This is due to the additional cyclotron motion of the plasma electrons that enhances the nonlinear plasma currents and therefore, the THz amplitude. Fig. 4a and b shows an enhancement in the amplitude for a laser of same pulse length L ¼0.5 λp whereas Fig. 4c shows the same for L¼1.5λp. This reflects that the emitted field amplitude is independent of the laser pulse length in the present approximation. However, the magnetic field serves the purpose even in the presence of high collisions. A further enhancement in the field amplitude may be realized with use of flat top laser (FTL) than the Gaussian laser (GL), as depicted in
Fig. 5. Normalized amplitude of THz field with normalized frequency of laser in plasma for Gaussian and flat top laser in plasma for different collision rate when ωp ¼2.0 1013 rad/s, y¼0.5bw, E0 ¼5.0 108 V/m and L ¼0.5λp.
Fig. 5, and this increment is very significant in the presence of low collisions. Therefore, the FTLs are concluded to be more useful than the GLs in the presence of transverse magnetic field for achieving large amplitude THz radiation generation.
4. Conclusions The numerical study of analytically derived Eq. (1) shows that the large amplitude Langmuir waves can be excited in the plasma, when the laser frequency matches with the plasma frequency and the flat top lasers (FTL) are used. The role of collisions in the
Fig. 4. Normalized amplitude of THz field with normalized frequency of laser in plasma for different collision frequencies and pulse length, when ωp ¼2.0 1013 rad/s, y¼ 0.5 bw, E0 ¼ 5.0 108 V/m, bw ¼0.01 cm and L ¼ 0.5λp.
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plasma is to act as a damping force, which reduces the plasma wake field. The present study made on the THz radiation generation in collisional plasma with the application of external magnetic field clearly depicts that the longitudinal component of the wakefield gives rise to the transverse fields, which are responsible for the THz radiation. The negative impact of the collisions may be overcome by the application of external transverse magnetic field and the use of FTLs.
Acknowledgments DST and DRDO, Government of India are gratefully acknowledged for the financial support.
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[4] M. Abo-Bakr, J. Feikes, K. Holldack, P. Kuske, W.B. Peatman, U. Schade, G. Wustefeld, H.W. Hübers, Phys. Rev. Lett. 90 (2003) 094801. [5] (a) J. Yoshii, C.H. Lai, T. Katsouleas, C. Joshi, W.B. Mori, Phys. Rev. Lett. 79 (1997) 4194; (b) N. Yugami, T. Higashiguchi, H. Gao, S. Sakai, K. Takahashi, H. Ito, Y. Nishida, T. Katsouleas, Phys. Rev. Lett. 89 (2002) 065003. [6] W.M. Wang, Z.M. Sheng, H.-C. Wu, M. Chen, C. Li, J. Zhang, K. Mima, Opt. Express 16 (2008) 16999. [7] W.M. Wang, S. Kawata, Z.M. Sheng, Y.T. Li, J. Zhang, Phys. Plasmas 18 (2011) 073108. [8] T. Ostermayr, S. Petrovics, K. Iqbal, C. Klier, H. Ruhl, A.K. Nakajima, A. Deng, X. Zhang, B. Shen, J. Liu, R. Li, Z. Xu, T. Tajima, J. Plasma Phys. 78 (2012) 447. [9] J. Faure, C. Rechatin, A. Norlin, et al., Nature 444 (2006) 737. [10] A. Pukhov, Phys. Rev. Lett. 86 (2001) 3562. [11] A. Rousse A, K.T. Phuoc, R. Shah, et al., Phys. Rev. Lett. 93 (2004) 135005. [12] (a) Z.M. Sheng, K. Mima, J. Zhang, H. Sanuki, Phys. Rev. Lett. 94 (2005) 095003; (b) Z.M. Sheng, K. Mima, J. Zhang, Phys. Plasmas 12 (2005) 123103. [13] H.C. Wu, Z.M. Sheng, Q.L. Dong, H. Xu, J. Zhang, Phys. Rev. E 75 (2007) 016407. [14] M.S. Sodha, M. Faisal, J. Plasma Phys. 79 (2009) 563. [15] S.V. Golubev, E.V. Suvorov, A.G. Shalashov, JETP Lett. 79 (2004) 361, Pis’ma Zh. Éksp. Teor. Fiz. 79 443. [16] C.L. Chen, Phys. Rev. 135 (3A) (1964). [17] D. Singh, H.K. Malik, Phys. Plasmas 21 (2014) 083105. [18] D. Singh, H.K. Malik, Plasma Sources Sci. Technol. 24 (2015) 045001.
Please cite this article as: D. Singh, H.K. Malik, Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.03.108i
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Emission of strong Terahertz pulses from laser wakefields in weakly coupled plasma Divya Singh a,b,n, Hitendra K. Malik a a b
PWAPA Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110 016, India Department of Physics & Electronics, Rajdhani College, University of Delhi, Raja Garden, Ring Road, New Delhi 110015, India
art ic l e i nf o
a b s t r a c t
Article history: Received 19 October 2015 Received in revised form 10 March 2016 Accepted 31 March 2016
The present paper discusses the laser plasma interaction for the wakefield excitation and the role of external magnetic field for the emission of Terahertz radiation in a collisional plasma. Flat top lasers are shown to be more appropriate than the conventional Gaussian lasers for the effective excitation of wakefields and hence, the generation of strong Terahertz radiation through the transverse component of wakefield. & 2016 Elsevier B.V. All rights reserved.
Keywords: Magnetic field Plasma Collisions Wakefield Terahertz radiation
1. Introduction Interaction of intense lasers with plasma is a very enriched field of research, giving rise to very interesting physical phenomena like laser induced fusion, higher harmonic generation, laser-plasma channeling, laser plasma acceleration and plasma based radiation sources. These radiation sources range from X-ray to THz frequency domain. Now a days Terahertz radiation sources have several applications and are being used extensively for imaging, material characterization, topography, tomography, communication, etc. [1]. THz radiation can be produced from plasma and electron beam based THz emitters, such as coherent radiation from plasma oscillations driven by ultrashort laser pulses [2], transition radiation of electron beams [3], synchrotron radiation from accelerator electrons [4], Cherenkov wake radiation in magnetized plasmas [5] and emission from laser plasma channels in air. Plasma is found to be more suitable than the solid targets due to longer sustainability at higher powers of the laser. Theoretical and simulation studies of extremely powerful Terahertz emission by the interaction of chirped [6] and few cycle laser pulses [7] with tenuous plasma have been reported by Wang et al. Ostermayr et al. have used super-Gaussian pulse for the laser plasma acceleration [8]. A wakefield is an electrostatic wave driven by a laser pulse in n Corresponding author at: Department of Physics & Electronics, Rajdhani College, University of Delhi, Raja Garden, Ring Road, New Delhi 110015, India. E-mail address: [email protected] (D. Singh).
plasmas. There have been a number of studies on the excitation of the large amplitude wakefield because of its extensive application prospects, such as in high-gradient electron acceleration [9], proton acceleration [10] and X-ray radiation [11]. Sheng et al. have proposed that high efficiency THz emissions can be produced from a laser driven wakefield in an inhomogeneous plasma through linear mode conversion [12]. Wu et al. [13] have studied the effect of the transverse magnetic field on this process and have found significant enhancement in the efficiency of THz generation. Since the typical plasma oscillation frequency for these applications is in the Gigahertz to Terahertz range, the wakefield can potentially serve as a powerful THz emitter. Above mentioned, all schemes talk about the wakefield excitation but none of them have discussed collisional plasma. In theories as well as in experiments, collisional effects are generally ignored but our calculations and results show that the even small fraction of collisions in the range (ν ¼0.05ωp to 0.5ωp) reduce the longitudinal wakefield and therefore amplitude of emitted THz field is also reduced greatly. The collision frequency depends on the electron temperature as per the relation
νe = ν0 ( Te/T0 )
s /2
.
where, s is a parameter characterizing the type of collision in plasma, where T0 is the equilibrium temperature [14]. For experimentation purpose it is promising to convert a considerable fraction of the energy of plasma oscillations exciting longitudinal wakefield in the plasma is reemitted with their frequencies controlled over a terahertz range radiation by varying the parameters
http://dx.doi.org/10.1016/j.nima.2016.03.108 0168-9002/& 2016 Elsevier B.V. All rights reserved.
Please cite this article as: D. Singh, H.K. Malik, Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.03.108i
D. Singh, H.K. Malik / Nuclear Instruments and Methods in Physics Research A ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
of the ionized gas, i.e. the pressure and temperature. In particular, Golubev et al. [15] have experimentally obtained THz radiation from the optical breakdown of a gas and investigated the effect of gas pressure and temperature on collisions. They found the collision frequency of the order of 1.5 1013 s 1 and plasma frequency 3.0 1014 s 1 at the atmospheric pressure and when the temperature Te ¼15 eV. Therefore, the inequality ν o ωp may be fulfilled at the normal pressure for better amplitude of the THz radiation. However, at the higher pressure of the order of 400 atm, both the frequencies become equal (fp E ν E 6 1015 s 1). Hence, the corresponding radiation frequency falls into the optical range, which, is of little interest. Hence, it is important to consider collisions for normal pressure and temperature in laser plasma interaction experimentally. On the other hand, Chen [16] has studied experimental behavior of Neon plasma with respect to electron collision frequency by varying plasma density for different temperatures and concluded that the electron collisions in uniform density plasma are sensitive to electron temperature at a given gas temperature and show weak dependence on the gas pressure. Thus it become imperative to control collisions in the laser plasma interaction mechanism. Therefore, we justify the discussion on emission of THz radiation pulses from wakefields in collisional plasma when Gaussian and flat top lasers are used and we also justify that the collision frequency that we have considered are experimentally viable. In the present article, we present a theoretical analysis for Terahertz radiation generation based on the scheme of the excitation of wakefield produced by the propagation of highly intense ultra-short flat top laser through collisional magnetized plasma of uniform density, where ions are supposed to be immobile and electron neutral collisions prevail.
using equation of continuity where n0 is taken to be the ambient density of the plasma. The components of the nonlinear plasma density are obtained as n(1) =
y
In the present work, we consider a flat top laser (FTL) of frequency ω and wave number k, propagating in the plasma of uniform density n0 along the z-axis. The field of the laser is given as ⎡ ⎛ y ⎞6⎤ ⎢−⎜ ⎟ ⎥ e⎢⎣ ⎝ bw ⎠ ⎥⎦ e[i (kz − ωt )] y^ ,
→ E = E0 where E0 is the field amplitude and bw is the beamwidth of the laser. The laser electric field is taken to be polarized along the y-axis whereas an external magnetic field is taken to be applied along the x-axis. In the presence of the external magnetic field (B), the laser field impart oscillatory velocity to the electrons whereas plasma ions remain immobile. The plasma is weakly coupled, so the electron neutral collisions are taken to be present. We make use of the perturbation approach. Hence, physical quantities like electron velocity, plasma density and plasma currents are expanded to their first order in orders of laser strength eE parameter (a = mc0ω «1 approximated). The electron dynamics in the plasma is completely expressed by the force equation, where collisional force (frequency ν ) acts as a damping cause. The plasma electron velocity is computed as, (0) (1) → υ =→ υ + a→ υ = eE0
⎡ ⎛ y ⎞6⎤ ⎢−⎜ ⎟ ⎥⎡ e⎢⎣ ⎝ bw ⎠ ⎥⎦ ⎢
ei (kz − ωt ).
1 y^ ⎣ im (ω + iν )
+a
{
(ω + iν ) im [(ω + iν )2 − ωc2 ]
y^ +
ωc m [(ω + iν )2 − ωc2 ]
⎤ z^ ⎥ ⎦
}
No component of the velocity is found to arise in the direction of the external magnetic field. The cyclotron frequency is represented eB as ωc = mc x^ . Under the influence of collisional forces and external magnetic field, the electron oscillations become nonlinear. Therefore, nonlinear density perturbations are generated. Different order components of the nonlinear density are computed
+
(1) kn0 → υz ^ z. ω
Due to the
density perturbations in the plasma, coupling of plasma oscillations with density fluctuations take place that drive nonlinear plasma currents. Thus magnitude of these nonlinear plasma currents are obtained as ⎡ ⎛ ⎤ y ⎞6
⎢−⎜ ⎟ ⎥ → ⎢ ⎝b ⎠ ⎥⎡ 1 iω ( ω + iν ) J = n0 e2E0 e⎣ w ⎦ ⎢ y^ + 2 (0) 2 υ y ) m [(ω + iν ) − ωc ] ⎣ (ω − k→
ωc m [(ω + iν )2 − ωc2 ]
⎤ z^⎥ ei (kz − ωt ) . ⎦
Now we will discuss how nonlinear plasma current densities cause the excitation of various components of the wakefield and thereby the emission of THz radiation in collisional plasma. The components of electric and magnetic wakefield are coupled with → → time dependent Maxwell's equations by using Ew and Bw as the components concerning the wakefield. In general, the magnetic wakefield is not considered further, the reason being that they have very low magnitudes as compared to the electric wakefields. Using quasistatic approximation, we consider that the laser pulse does not evolve significantly and therefore, the field variations are assumed to be time independent. Maxwell's equations are expressed in terms of the transformed coordinate ξ = z − υg t as laser propagates along z axes with group velocity vg. Further, the plasma electron motion is analyzed under the excited plasma wakefield using force equation. On combining the Maxwell's equations with the force equation, we obtain differential equations for the wakefields as follows
→ → ⎡→ ⎛ ⎞→ ⎤ ∂ 2E y ∂ 2Ez ω ⎟⎟ Ey ⎥ + + k p2 ⎢ Ez + ⎜⎜ 2 2 0 ( ) ⎢⎣ ∂ξ ∂ξ ⎝ ω − kυy ⎠ ⎥⎦ = −
2. Excitation of plasma currents, wakefield and THz radiation
(1) kn0 → υy y^ (0) → (ω − k υ )
k p2 m e
× eE0
⎡ ⎛ 6⎤ ⎢ −⎜ y ⎞⎟ ⎥ ⎢⎣ ⎝ b w ⎠ ⎥⎦ e
⎡⎛ ⎞ ω ν (ω + iν ) ⎢⎜ ⎟ ⎜ 0 ( ) ⎢⎣ ⎝ ω − kυy ⎟⎠ im [(ω + iν )2 − ωc2 ] ⎡ ⎤⎤ ωc νωc ⎥ ⎥, +⎢ − 2 2 ⎣ im (ω + iν ) m [(ω + iν ) − ωc ] ⎦ ⎥⎦ where k p2 =
4πn0 e
2
mc 2
(1)
is the plasma wave number. Eq. (1) is solved
numerically for the y-and z-components of the wakefield using Runge–Kutta method for collisional magnetized plasma with → boundary condition that Ez = 0 at ξ = 0 and ξ = L/2. Here L is the length of the laser pulse. Numerical solution of Eq. (1) and hence, the wakefields are plotted in the next section. It is observed that the magnitudes strictly depends on the relative amplitude of the laser field, electron neutral collision and cyclotron frequencies. We further calculate the transverse component of the wakefield in terms of the horizontal wakefield analytically. This transverse component of the oscillating wakefield is emitted as THz pulses. The analytical expression of the transverse THz field amplitude is obtained as
⎤ ⎡ 2 (0) 2 → iν (ω + iν ) 1 6y5 ωc (ω − kυy )(ω − ωc + iνω) ⎥ + ETHz = ⎢ 2 6 2 2 ⎥ ⎢k b ⎡ (ω + iν )2 − ω 2 ⎤ [( ω + ν ) − ω ] i ω ( ω − ν ) i p w c ⎣ ⎦ c ⎦ ⎣ 6
E0 e−(y / b w ) .
(2)
It is evident from the mathematical calculations that the plasma wakefields are excited in the plasma due to propagation of highly intense laser through it. These wakes are longitudinal and transverse in nature. It is well-known that the longitudinal wakefields are used for the acceleration purpose, whereas the transverse component is of less importance. Our calculations
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reveal that the transverse component of the wake is originated by the horizontal wakefield and is responsible for the emission of electromagnetic radiation in THz frequency range.
3. Results and discussion Here we will talk about the evolution of wakefields due to the propagation of flat top laser through the plasma, electric field profile of the longitudinal and transverse wakefields under the impact of collisions. A comparison of the results obtained based on the proposed flat top laser with the ones of Gaussian laser is also made. The evolution of the longitudinal wakefield is shown in Fig. 1 for different collision frequencies between the electrons and neutrals for the case of flat top laser. Here the solid lines correspond to the Gaussian laser (GL) whereas the dotted lines represent the case of flat top laser (FTL). Normalization of the laser propagation distance is made with the laser pulse length L. The magnitude of the excited wakefield is found to decrease with the increased collision frequency. The wake amplitude is sufficiently good for low collisions (ν«ωp) but falls sharply to further increased collisions (ν ¼0.5 ωp to 1.5 ωp). It is also clearly seen that the large amplitude wakefiled is realized when FTL is used. These large amplitude wakes are attributed to the uniform maximum intensity distribution of the FTL transverse to the laser propagation axis in comparison with the GL [17,18]. Therefore, maximum energy of the laser is utilized to excite plasma wakefield Fig. 2 describes the transverse profile of the emitted THz field for different laser frequencies, when the collision frequency is kept fixed as ν ¼0.05 ωp. Here it is evident that the THz field amplitude maximizes when the laser frequency remains closer to the plasma frequency (ω ¼ ωp). This is due to the frequency matching at the resonance which leads to the maximum energy transfer from the laser to the plasma oscillations that causes the large magnitude
Fig. 2. Normalized amplitude of THz field with normalized transverse distance from propagation axis of the laser in plasma for different laser frequency (normalized with plasma frequency), when ωp ¼ 2.0 1013 rad/s, ν ¼ 0.05 ωp, y¼ 0.5bw, E0 ¼ 5.0 108 V/m, bw ¼0.01 cm and L ¼ 0.5λp.
wakefield. As the resonance condition departs, the wake amplitude tends to decay and the peak of the emitted radiation is also shifted. If we compare the results of Fig. 2 (ν ¼0.05 ωp) with Fig. 3 (ν ¼0.5 ωp), we find that in the presence of high collision rate the THz field amplitude decreases but the peak of the radiation is not affected. In the case of higher collisions, a uniform illumination of the emitted radiation is obtained instead of sharply focused radiation. Fig. 3 further explains that the FTL gives higher THz field amplitude than the GL. The collisions in the plasma act as a damping force to the electron oscillations. These reduce the nonlinear currents, the amplitude of the excited wakefield and also affect the peak of the emitted radiation.
Fig. 1. Normalized amplitude of the plasma wake field with normalized distance of propagation of the laser in plasma for different collision frequencies for Gaussian (GL) and flat top (FTL) lasers, when ω¼ 2.4 1014 rad/s, ωp ¼ 2.0 1013 rad/s, y¼ 0.8bw, E0 ¼ 5.0 108 V/m, bw ¼ 0.01 cm, B ¼1 T and L ¼0.5λp.
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Fig. 3. Normalized amplitude of THz field with normalized distance of propagation of laser in plasma for Gaussian (GL) and flat top (FTL) lasers for different laser frequency (normalized with plasma frequency) when ν ¼1.5 ωp, ν ¼0.5 ωp, ωp ¼ 2.0 1013 rad/s, y¼ 0.5bw, E0 ¼ 5.0 108 V/m, bw ¼ 0.01 cm and L ¼ 0.5λp.
The effect of magnetic field on the radiated field amplitude can be seen in Fig. 4, where the normalized field amplitude of the THz radiation is plotted with normalized laser frequency for different collision frequency and pulse length. The applied magnetic field is found to enhance the THz amplitude even in the presence of significant collisions. This is due to the additional cyclotron motion of the plasma electrons that enhances the nonlinear plasma currents and therefore, the THz amplitude. Fig. 4a and b shows an enhancement in the amplitude for a laser of same pulse length L ¼0.5 λp whereas Fig. 4c shows the same for L¼1.5λp. This reflects that the emitted field amplitude is independent of the laser pulse length in the present approximation. However, the magnetic field serves the purpose even in the presence of high collisions. A further enhancement in the field amplitude may be realized with use of flat top laser (FTL) than the Gaussian laser (GL), as depicted in
Fig. 5. Normalized amplitude of THz field with normalized frequency of laser in plasma for Gaussian and flat top laser in plasma for different collision rate when ωp ¼2.0 1013 rad/s, y¼0.5bw, E0 ¼5.0 108 V/m and L ¼0.5λp.
Fig. 5, and this increment is very significant in the presence of low collisions. Therefore, the FTLs are concluded to be more useful than the GLs in the presence of transverse magnetic field for achieving large amplitude THz radiation generation.
4. Conclusions The numerical study of analytically derived Eq. (1) shows that the large amplitude Langmuir waves can be excited in the plasma, when the laser frequency matches with the plasma frequency and the flat top lasers (FTL) are used. The role of collisions in the
Fig. 4. Normalized amplitude of THz field with normalized frequency of laser in plasma for different collision frequencies and pulse length, when ωp ¼2.0 1013 rad/s, y¼ 0.5 bw, E0 ¼ 5.0 108 V/m, bw ¼0.01 cm and L ¼ 0.5λp.
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plasma is to act as a damping force, which reduces the plasma wake field. The present study made on the THz radiation generation in collisional plasma with the application of external magnetic field clearly depicts that the longitudinal component of the wakefield gives rise to the transverse fields, which are responsible for the THz radiation. The negative impact of the collisions may be overcome by the application of external transverse magnetic field and the use of FTLs.
Acknowledgments DST and DRDO, Government of India are gratefully acknowledged for the financial support.
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Please cite this article as: D. Singh, H.K. Malik, Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.03.108i