Terahertz radiation generation by lasers with remarkable efficiency in electron–positron plasma

JID:PLA AID:23414 /SCO Doctopic: Nonlinear science

[m5G; v1.159; Prn:22/09/2015; 14:52] P.1 (1-4)

Physics Letters A ••• (••••) •••–•••

Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

Terahertz radiation generation by lasers with remarkable efficiency in electron–positron plasma Hitendra K. Malik Plasma Waves and Particle Acceleration Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi-110 016, India

a r t i c l e

i n f o

Article history: Received 19 April 2015 Received in revised form 13 August 2015 Accepted 4 September 2015 Available online xxxx Communicated by F. Porcelli

a b s t r a c t Photo-mixing of spatial-super-Gaussian lasers and electron–positron plasma are proposed for realizing a large amplitude nonlinear current in order to generate an efficient terahertz radiation. An external magnetic field together with a proper index of the lasers helps achieving controllable current and hence, the focused radiation of tunable frequency and power along with a remarkable efficiency of the scheme as ∼6%. © 2015 Elsevier B.V. All rights reserved.

The development of new and improved terahertz (THz) sources and detectors has become a fascinating subject of research due to their enormous applications in medicine, explosive and concealed weapon detection, material characterization, remote sensing, etc. [1–3]. In order to generate the THz radiation, researchers have employed air [4], plasmas [5–8], metamaterials [9,10], semiconductors and electro-optic crystals [11–14] as different media for their exploitation by lasers in different types of schemes. Electromagnetic radiations have been generated by the interaction of ultra-short lasers with gas-jet plasmas [15,16] and water vapor [17]. Corrugated plasma channel has been used to produce an intense THz radiation [6], whereas coherent Cerenkov radiation has been quite useful to generate multimode THz waves [7]. The interaction of electron beam and an electromagnetic wave in a single and multilayered open dielectric waveguide [18,19] has also led to the emission of THz radiation. A two-dimensional array of holes perforated on a semiconductor film has been explored using FDTD methods for the tunability of resonance THz frequencies by the application of external magnetic field [20]. Using crossed static magnetic field and alternating electric field semiconductors have been explored for obtaining the THz radiations [13]. In some of the schemes, the magnetic field alone has effectively enhanced the field of emitted radiations [21–24]. Despite the use of different media and different mechanisms, it has been quite convoluted to achieve tunable and intense THz radiation with its optimal focusing and collimation at the same time. However, in the present Letter, we show that all these properties can be easily attained simultaneously if we generate such radiation through photo-mixing of spatial-super-Gaussian (SSG) lasers and employ an external magnetic field in an electron–positron (e–p)

E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.physleta.2015.09.015 0375-9601/© 2015 Elsevier B.V. All rights reserved.

plasma. The electron–positron pair creation occurs in relativistic plasmas at high temperatures [24–29], and e–p pairs/plasmas have been generated in the laboratory as well [30–34]. The proposal of e–p plasma is due to the fact that we can realize large amplitude nonlinear current due to the contribution of both the species. The SSG lasers are selected due to the steepness in their intensity variation so that a stronger transverse ponderomotive force is realized for driving a very significant current. The magnetic field contributes as an additional controlling parameter for the current. Hence, we get the THz radiation tunable with respect to its frequency, power and focus. We consider photo-mixing of two lasers (beam width b w and profile index p) with frequencies ω1 and ω2 and wave vectors k1 and k2 , linearly polarized along the y-direction (fields  j = E 0L exp[−( y ) p ] exp[i (k j z − ω j t )] yˆ together with j = 1, 2), E bw and co-propagating along the z-direction in the corrugated plasma having density N = N 0 + N α e i α z , together with N α as the amplitude and α as the wave number of the corrugation, under the  applied along the x-direction. Under effect of magnetic field B this situation, a strong transverse nonlinear current is driven by the nonlinear ponderomotive force F pNL = y ik zˆ ] exp[−2( b ) p ] exp[i (kz w

2 e 2 E 0L [− b2pw 2mω1 ω2

( byw ) p −1 yˆ +

− ωt )] acting on the electrons and positrons both (unlike the usual case of electron–ion plasmas) at the frequency ω = ω1 − ω2 and wave number k = k1 − k2 . The nonlinear current J NL (≡ − 12 N α e v NL e i α z ) comprises the contribution from nonlinear perturbations in the density N NL = N0   NL   NL  c )] and linear density pertur2 [i ω ∇ · F p + ∇ · ( F p × ω 2 mi ω(ω −ωc )

bation N L =

 ∇φ  N 0 e ∇· m(ω2 −ωc2 )

due to the potential φ produced by N NL .

Unlike the usual plasma, the velocity v NL in the expression of J NL is the net nonlinear velocity of the electrons and positrons under

JID:PLA

AID:23414 /SCO Doctopic: Nonlinear science

[m5G; v1.159; Prn:22/09/2015; 14:52] P.2 (1-4)

H.K. Malik / Physics Letters A ••• (••••) •••–•••

2

 , and Lorentz force exerted by the the forces F pNL and F L (≡ −∇φ)  This would be just double of the electron velocmagnetic field B. ity. Hence, the current density J NL reads

J NL =





2

2 c

ωc ω − ω + ω

2 p









p ωc2 ω2p bw + kω ω − ω + zˆ ω2 py p −1 2

2 c

 p   bw + i ωc k ω2 − ωc2 + ω2p p −1 py



 ω ω 2 2 + i ω ω − ωc + yˆ 2 ×

py

2 c

2 p

ω ω

p −1

2 N α e 3 E 0L ( 2 − c2 )−1 e i ((k+α )z−ωt ) . y p p 2( ) b w m2 1 2 ( 2 − h2 )e b w

ω

ω ω ω

(1)

ω

In the presence of magnetic field, the electric permittivity of



the plasma evolves into a tensor quantity

εxx εxy εxz

ε =

ε yx ε y y ε yz

together εzx εzy εzz

2ω2p

2ω 2

εxx = 1 − ω2 , ε y y = εzz = 1 − (ω2 −ωp 2 ) c and εzy = −ε yz = εxz = εzx = εxy = ε yx = 0. Hence, the Maxwell’s with its components as

equations along with the current J NL yield the following wave equation, which governs the THz radiation emission 2  ∇  · E THz ) + 4π i ω J NL − ω ε E THz = 0. −∇ 2 E THz + ∇( 2 2

c

c

(2)

 THz , we put E THz = For the fast phase variation of the field E  0 THz ( y , z)e i (kz−ωt ) and separate out the components of the wave E equation. The transverse component gives the THz field E 0 THz as





   p −1 

E 0 THz

−2 ( y ) p

= λ1 zE 0L kλ2 + λ3 ω 2p y

e b w .

E



b b 0L

Here λ1 = −

w

(3)

w

ω N α e ω2p 4kc 2 mN 0 ω1 ω2 (ω2 −ωh2 )(ω2 −ωc2 )

, λ2 = ωc (ω2 − ωc2 + ω2p ) and

ω2 ω2

λ3 = ω2 − ωc2 + ωc 2 p . In order to obtain the above solution, we 2 have used the condition k2 − ωc 2 ε y y = 0, which is known as the

dispersion relation. Based on this relation we find that the frequency of emitted radiation is in the range of THz. Hence, Eq. (2) is called to govern the emission of THz radiation only. This relation  2

α c ) = ω [ (1 − 2ω p ) − 1], inferring that the resoalso yields ( ω ωp p ω2 −ωc2



nance occurs close to the frequency

Fig. 1. (Color online.) Electric field profile ( y /b w ) of THz radiation under the effect of magnetic field (B) and index of laser beams (p), when b w = 0.05 mm, z = 100λcorru , ω1 = 2.4 × 1014 rad/s, ω p = 2 × 1013 rad/s, N α / N 0 = 0.1, and ω/ω p = 1.45.

ω ≈ (2ω2p + ωc2 ) for maxi-

mum momentum transfer from the lasers to the plasma species. This means the THz radiation shall emit with higher frequency in the e–p plasma as compared to the case of usual √ion–electron plasma, being the plasma frequency ω p raised by 2 times due to the motion of the electrons and positrons. Since the plasma species are taken to oscillate in the same phase in the calculations, monochromatic THz radiations are expected to be emitted in the present mechanism. However, in the real experiments, all the species do not possess the same phase during oscillations and hence, the THz emission is realized in a broad frequency band due to the different currents owing to the phase difference in the oscillations. This can be seen from the expressions of the current J NL (source of THz radiation) and the ponderomotive force F NL p (driver) that the tuning/resonance between J NL and F pNL is achieved with the help of corrugation (wave number α ) in the plasma density. Such corrugation in density may be produced using various techniques, which involve transmissive ring grating and a patterned mask where the control of ripple parameters might be possible by changing the groove structure, groove period, and duty cycle in such a grating and by adjusting the period and size of the

masks [35–39]. The periodicity (repetition) of the corrugation can be obtained as |λcorru | = | 2απ |. The dependence of λcorru on α and hence ωc reveals that the magnetic field plays a vital role in realthe resonant excitation of the THz radiation at the frequency izing

ω≈

(2ω2p + ωc2 ) by photo-mixing of the SSG lasers. The steep

corrugations in the density that are constituted at closer distances are suggested for obtaining strong THz radiation [6]. Moreover, Eq. (3) shows that the radiation with larger field is obtained for the case of higher amplitude corrugation (N α ), which is attributed to the more number of electrons and positrons that take part in exciting the nonlinear current J NL . The most striking feature of the proposed scheme in the e–p plasma is the tuning of power and focus (transverse profile: y /b w ) of the emitted THz radiation with the help of magnetic field and index of the lasers. In this regard, it is clear from Fig. 1 that the THz field amplitude attains a maximum value at a particular value of y /b w . This peak moves away from the z-axis (direction of propagation) if the lasers of higher index are used. The transverse profile also becomes more symmetrical about the y-axis for the case of higher index lasers, which can be understood as follows. If we look at the profiles of the lasers for p = 2 (Gaussian profile) and p > 2 (Super-Gaussian: SSG profile), it is found that there is a gradient in the intensity that changes its sign for the first and second halves of the laser pulses. For Gaussian profile, this change takes place suddenly. However, in SSG profile of the lasers, this change in the intensity is not a sudden change due to the flat top distribution (flatness becomes larger for larger value of p). Since the ponderomotive force changes its sign according to the gradient in the intensity, the direction of motion of the electrons and positrons will also change as per the direction of ponderomotive force. In view of the flatness (zero gradient) in case of SSG lasers, these species get sufficient time to revert back their motion, resulting in a symmetrical profile of the emitted radiation. On the other hand in the case of Gaussian lasers, these species change their motion rapidly, which results in an unsymmetrical profile of the radiation. The peak value of the THz field amplitude in Fig. 1 is also increased when the lasers of higher indices are used. Actually for the higher index lasers, a stronger nonlinear current J NL is realized due to the stronger force F pNL in the presence of sharp gradient in the SSG lasers’ intensity. This gradient is larger in the case of lasers of smaller beam width (b w ) also and hence, the THz field of higher amplitude is expected to be further achieved when the lasers of smaller b w are used. Here a point of observation is that in the present scheme the THz field is larger than the field obtained in the other schemes [4–13,40–44]. The larger field in the present

JID:PLA AID:23414 /SCO Doctopic: Nonlinear science

[m5G; v1.159; Prn:22/09/2015; 14:52] P.3 (1-4)

H.K. Malik / Physics Letters A ••• (••••) •••–•••

Fig. 2. (Color online.) Efficiency of THz radiation scheme as a function of magnetic field and index of laser beams for the same parameters as used in Fig. 1.

mechanism is also attributed to the larger Coulomb force between the electrons and positrons that influences the linear current den . sity N L via F L (≡ −e ∇φ) Based on the distribution of electric field in Fig. 1, we can examine the collimation of emitted THz radiation. From this figure it is evident that most of the field is situated near the peak for the case of larger value of p and there is a faster fall in the field when we move away from the peak. Such a distribution of the field is called to correspond to the collimated radiation. The peak field and also the collimation are found to be enhanced in the presence of stronger magnetic field, though the effect of magnetic field is a bit less significant for the case of higher index lasers in the present scheme. However, the radiation of highest intensity at a desired position can be obtained by changing the index and magnetic field. Finally, we examine the efficiency (say η ) of the present scheme for proving its supremacy over the schemes proposed by the other investigators [4–13,30–44]. For this, we calculate the average energy densities of the input lasers  W LE and that of the emitted W THz radiation  W THzE , and define the efficiency as η =  WTHzE . LE The average energy density is calculated [41,45] using  W = 1 ∂ [εω]| E |2 . Finally, taking as the gamma function the ef8π ∂ ω ficiency η of the THz radiation is obtained as

η=

1 p

2 2 λ21 z2 ω E 0L



1 p

 λ2 k 2  1  2

p

1 p

4 pω

+

p ωλ23

4

 2p −1 

1− 1p

p

b2w

+

k λ2 λ3 bw

 .

(4)

Equation (4) shows that the present scheme is very effective, where a remarkable efficiency of about 6% can be obtained (Fig. 2) with the use of the lasers of higher index and stronger magnetic field. The efficiency can be further enhanced if the plasma with higher density N 0 is employed. Then the magnetic field also has very significant effect. A comparison of Fig. 2 with Fig. 1 infers that in the situation of higher efficiency the THz radiation also emits with higher field and is more focused. This is evident from the figures that unlike the other investigators [13,40–44] we can get the THz radiation at a desired position along with its tunable power and frequency with the application of an external magnetic field and proper choice of index of the SSG lasers. In the presence of collisions, the lasers field shall impart lower oscillatory velocity to the electrons and positrons in the plasma. This shall lead to a reduction in the magnitude of the ponderomotive force and nonlinear current as well. Since the nonlinear current contributes to the THz radiation generation, the radiation with a lower field is expected in the presence of collisions. We can estimate the impact of collisions on the emitted radiation based

3

on the collision frequency. In this respect, this can be seen from the expressions of ponderomotive force and the current densities that the square of collision frequency (say υ ) is expected to be added to the square of frequency ω . It means the impact of collision frequency will be trivial when υ /ω < 0.1. However, the larger collisions (υ /ω > 0.1) will have much significant influence on the THz radiation. Hence, there is a need of calculating the THz field in the presence of collisions in order to realize the exact amount of THz field/intensity. In conclusion, the proposed photo-mixing of the SSG lasers in e–p plasma under the effect of magnetic field is very significant technique for getting efficient THz radiation, where the resonance is achieved by creating corrugation in the plasma density and tuning of the THz frequency and power is done with the help of magnetic field and index of the lasers. The magnetic field also helps getting more collimated radiation and enhanced efficiency of the scheme, which can be further increased by employing the higher amplitude of the density corrugation. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

[24] [25] [26] [27]

[28] [29] [30] [31]

[32]

J.A. Zeitler, L.F. Gladden, Eur. J. Pharm. Biopharm. 71 (2009) 2. P.H. Siegel, IEEE Trans. Microw. Theory Tech. 52 (2004) 2438. B. Ferguson, X.C. Zhang, Nat. Mater. 1 (2002) 26. T. Loffler, F. Jacob, H.G. Roskos, Appl. Phys. Lett. 77 (2000) 453. Y. Chen, M. Yamaguchi, M. Wang, X.C. Zhang, Appl. Phys. Lett. 91 (2007) 251116. T.M. Antonsen, J. Palastra Jr., H.M. Milchberg, Phys. Plasmas 14 (2007) 033107. K. Kan, J. Yang, A. Ogata, T. Kondor, K. Norizawa, Y. Yoshida, Appl. Phys. Lett. 99 (2011) 251503. W.M. Wang, S. Kawata, Z.M. Sheng, Y.T. Li, J. Zhang, Phys. Plasmas 18 (2011) 073108. X. Ropagnol, F. Blanchard, T. Ozaki, M. Reid, Appl. Phys. Lett. 103 (2013) 161108; X. Ropagnol, F. Blanchard, T. Ozaki, M. Reid, IEEE Photonics J. 3 (2011) 174. I. Al-Naib, G. Sharma, M. Dignam, H. Hafez, A. Ibrahim, D.G. Cooke, T. Ozaki, R. Morandotti, Phys. Rev. B 88 (2013) 195203. P. Zhao, S. Ragam, Y.J. Ding, I.B. Zotova, Opt. Lett. 35 (2010) 3979. Y. Jiang, D. Li, Y.J. Ding, I.B. Zotova, Opt. Lett. 36 (2011) 1608. V.A. Kukushkin, Europhys. Lett. 84 (2008) 60002. A.N. Bugay, S.V. Sazonov, Phys. Lett. A 374 (2010) 1093. D. Dorranian, M. Starodubtsev, H. Kawakami, H. Ito, N. Yugami, Y. Nishida, Phys. Rev. E 68 (2003) 026409. D. Dorranian, M. Ghoranneviss, M. Starodubtsev, H. Ito, N. Yugami, Y. Nishida, Phys. Lett. A 331 (2004) 77. K. Johnson, M. Price-Gallagher, O. Mamerc, A. Lesimple, C. Fletcherb, Y. Chend, X. Lu, M. Yamaguchi, X.-C. Zhang, Phys. Lett. A 372 (2008) 6037. V.A. Namiot, L.Yu. Shchurova, Phys. Lett. A 375 (2011) 2759. L.Yu. Shchurovaa, V.A. Namiot, Phys. Lett. A 377 (2013) 2440. L. Luo, W. Jia, C. Lei, J. Liu, H. Huang, Phys. Lett. A 375 (2011) 3213. H.C. Wu, Z.-M. Sheng, Q.L. Dong, H. Xu, J. Zhang, Phys. Rev. E 75 (2007) 016407. H.K. Malik, A.K. Malik, Appl. Phys. Lett. 99 (2011) 251101. R. McLaughlin, A. Corchia, M.B. Johnston, Q. Chen, C.M. Ciesla, D.D. Arnone, G.A.C. Jones, E.H. Linfield, A.G. Davies, M. Pepper, Appl. Phys. Lett. 76 (2000) 2038. C. Weiss, R. Wallenstein, R. Beigang, Appl. Phys. Lett. 77 (2000) 4160. H.R. Miller, P.J. Witta, Active Galactic Nuclei, Springer-Verlag, Berlin, 1987. F.C. Michel, Rev. Mod. Phys. 54 (1982) 1. O. Adriani, G.C. Barbarino, G.A. Bazilevskaya, R. Bellotti, M. Boezio, E.A. Bogomolov, L. Bonechi, M. Bongi, V. Bonvicini, S. Bottai, A. Bruno, F. Cafagna, D. Campana, P. Carlson, M. Casolino, G. Castellini, M.P. De Pascale, G. De Rosa, N. De Simone, V. Di Felice, A.M. Galper, L. Grishantseva, P. Hofverberg, S.V. Koldashov, S.Y. Krutkov, A.N. Kvashnin, A. Leonov, V. Malvezzi, L. Marcelli, W. Menn, V.V. Mikhailov, E. Mocchiutti, S. Orsi, G. Osteria, P. Papini, M. Pearce, P. Picozza, M. Ricci, S.B. Ricciarini, M. Simon, R. Sparvoli, P. Spillantini, Y.I. Stozhkov, A. Vacchi, E. Vannuccini, G. Vasilyev, S.A. Voronov, Y.T. Yurkin, G. Zampa, N. Zampa, V.G. Zverev, Nature (London) 458 (2009) 607. N.C. Lee, Phys. Plasmas 18 (2011) 062310. E. Heidari, M. Aslaninejad, H. Eshraghi, L. Rajaee, Phys. Plasmas 21 (2014) 032305. R.G. Greaves, C.M. Surko, Phys. Rev. Lett. 75 (1995) 3846. S. Hatchett, C. Brown, T. Cowan, E. Henry, J. Johnson, M. Key, J. Koch, A. Langdon, B. Lasinsky, R. Lee, A. Mackinnon, D. Pennington, M. Perry, T. Phillips, M. Roth, T. Sangster, M. Singh, R. Snavely, M. Stoyer, S. Wilks, K. Yasuike, Phys. Plasmas 7 (2000) 2076. E.P. Liang, S.C. Wilks, M. Tabak, Phys. Rev. Lett. 81 (1998) 4887.

JID:PLA

4

AID:23414 /SCO Doctopic: Nonlinear science

[m5G; v1.159; Prn:22/09/2015; 14:52] P.4 (1-4)

H.K. Malik / Physics Letters A ••• (••••) •••–•••

[33] C. Gahn, G.D. Tsakiris, G. Pretzler, K.J. Witte, C. Delfin, C.-G. Wahlstrom, D. Habs, Appl. Phys. Lett. 77 (2000) 2662. [34] H. Chen, S.C. Wilks, J.D. Bonlie, E.P. Liang, J. Myatt, D.F. Price, D.D. Meyerhofer, P. Beiersdorfer, Phys. Rev. Lett. 102 (2009) 105001. [35] K.Y. Kim, A.J. Taylor, T.H. Glownia, G. Rodriguez, Nat. Photonics 153 (2008) 1. [36] C.C. Kuo, C.H. Pai, M.W. Lin, K.H. Lee, J.Y. Lin, J. Wang, S.Y. Chen, Phys. Rev. Lett. 98 (2007) 033901. [37] S. Hazra, T.K. Chini, M.K. Sanyal, J. Grenzer, Phys. Rev. B 70 (2004) 121307(R). [38] B.D. Layer, A. York, T.M. Antonson, S. Varma, Y.-H. Chen, Y. Leng, H.M. Milchberg, Phys. Rev. Lett. 99 (2007) 035001.

H.K. Malik, Europhys. Lett. 106 (2014) 55002. K.Y. Kim, A.J. Taylor, T.H. Glownia, G. Rodriguez, Nat. Photonics 153 (2008) 1. A.K. Malik, H.K. Malik, S. Kawata, J. Appl. Phys. 107 (2010) 113105. J. Yoshii, C.H. Lai, T. Katsouleas, C. Joshi, W.B. Mori, Phys. Rev. Lett. 79 (1997) 4194. [43] N. Yugami, T. Higashiguchi, H. Gao, S. Sakai, K. Takahashi, H. Ito, Y. Nishida, T. Katsouleas, Phys. Rev. Lett. 89 (2002) 065003. [44] H.C. Wu, Z.-M. Sheng, J. Zhang, Phys. Rev. E 77 (2008) 046405. [45] E.J. Rothwell, M.J. Cloud, Electromagnetics, second edition, CRC Press, Taylor & Francis Group, 2009, p. 211. [39] [40] [41] [42]