Analytical approach for the use of different gauges in bubble wakefield acceleration

الموضوعات : Journal of Theoretical and Applied Physics

Hitendra Malik ¹ (Indian Institute of Technology Delhi, India)
Sonu Kumar ² (Department of Physics, Indian Institute of Technology Delhi, New Delhi - 110016, India)
Vidushi Dhaka ³ (Springdale College, Surbhi Colony, Madhotanda Road, Pilibhit, Uttar Pradesh – 262001, India)
Dhananjay Singh ⁴ (Department of Physics, PKRM College, Dhanbad - 826004, Jharkhand, India)

تاريخ الإرسال : 16 الأحد , ربيع الثاني, 1443 تاريخ التأكيد : 20 الجمعة , جمادى الأولى, 1443 تاريخ الإصدار : 17 الجمعة , جمادى الأولى, 1442

الکلمات المفتاحية: Wakefield acceleration, d’Alembert differential equations, Geometrical parameter, Wakefield, Dephasing length,

ملخص المقالة :

Laser and plasma interaction leads to several fascinating nonlinear phenomena, out of which bubble wakefield excitation is one of the recent interests. This field is used for the particle acceleration, which is very useful for high energy physics, betatron radiation emission, cancer therapy, etc. In the present work, an electromagnetic field is evaluated in bubble wakefield regime and the shape of the bubble is shown to be controlled by d’Alembert differential equations and different Gauge conditions. Wakefield potential is calculated in different bubble regimes such as spherical, longitudinal ellipsoid, transverse ellipsoid bubble regimes. A geometrical parameter is found to decide the size of the bubble. A detailed study of the same is conducted under the effect of different electron residual density. A comparative study of different Gauge conditions shows that the accelerator gradient is maximum in transverse ellipsoid bubble case. Also, energy gain in dephasing length is evaluated that shows maximum energy when bubble assumes transverse ellipsoid shape.

نص كامل:



Analytical approach for the use of different Gauges in bubble wakefield acceleration

 

Abstract

Laser and plasma interaction leads to several fascinating nonlinear phenomena, out of which bubble wakefield excitation is one of the recent interests. This field is used for the particle acceleration, which is very useful for high energy physics, betatron radiation emission, cancer therapy, etc. In the present work, an electromagnetic field is evaluated in bubble wakefield regime and the shape of the bubble is shown to be controlled by d’Alembert differential equations and different Gauge conditions. Wakefield potential is calculated in different bubble regimes such as spherical, longitudinal ellipsoid, transverse ellipsoid bubble regimes. A geometrical parameter is found to decide the size of the bubble. A detailed study of the same is conducted under the effect of different electron residual density. A comparative study of different Gauge conditions shows that the accelerator gradient is maximum in transverse ellipsoid bubble case. Also, energy gain in dephasing length is evaluated that shows maximum energy when bubble assumes transverse ellipsoid shape. 

 

Keywords: Bubble regime, bubble wakefield acceleration, dephasing length, wakefield, geometrical parameter, d’Alembert differential equations.

 

 

1. Introduction

Bubble wakefield acceleration is a new concept for acceleration of charged particles in laser wakefield acceleration first given by Tajima and Dawson in 1979 [1]. In bubble wakefield acceleration, an intense short laser pulse excites all the plasma electrons and expells them, forming an ion cavity behind the laser pulse [2–6]. Expelled electrons create an electron sheath around bubble boundary, but these electrons move back to their position due to ions and overshoot their initial positions due to their momentum and create an electron plasma wave called wake. The electrons can be self-trapped, and accelerated otherwise externally injected electrons are accelerate to high energy [7, 8]. In laser Wakefield acceleration, the observation of formation of bubble has been made by Hakimi et al. [9]. Advantage of this bubble formation [10] and self – injection [11][12] of trapped electrons is that there is no need of external bunch of electrons as witness bunch and self -production is possible [13]. In sub-terawatt laser Wakefield acceleration, ionization-induced injection's simulation investigation has been made by Lin et al. [14]. In a sub-mm nitrogen gas jet, Lin et al. [15] have analyzed a few-terawatt laser pulse driven laser Wakefield acceleration. For mid-IR laser drivers in the bubble regime, laser Wakefield acceleration has been studied by Woodbury et al. [16]. Through angular distributions of betatron x-rays in laser Wakefield acceleration, diagnosis of bubble evolution has been done by Ma et al. [17].This acceleration technique is very useful for high energy physics, betatron radiation emission,  cancer therapy and renewable clean energy and many more [18–20].

 Kostyukov et al. [21] and Wu et al. [22] have developed models for spherical bubble regime for bubble wakefield acceleration and, these do not talk about the modification of bubble shape but Sadighi-Bonabi and Rahmatollapur [23] and Li et al. [24] have used such concept and found different bubble shapes called longitudinal ellipsoid and transverse ellipsoid bubbles. They have evaluated the corresponding electromagnetic fields.

To the best of our knowledge, no investigation has been so far done for the calculations of accelerator gradient and maximum energy gain in different shaped bubble regimes. We have developed theoretical model for different shaped bubble regimes and energy gain in dephasing length. In addition, we have considered different gauge conditions for evaluating the acceleration gradient and maximum energy.

1.1 Dephasing length in bubble regime:

When accelerated electrons outrun the plasma wave and decelerate, then the length of acceleration to deceleration region is called as dephasing length [25]. Dephasing length is different in different regimes. It is used to control energy gain in laser wakefield acceleration [26]. Here acceleration distance is limited by dephasing length by [27].

In 3D nonlinear or bubble regime [28], the dephasing length is defined as,

                                                                                                                           (1)

where,  is the plasma wavelength and  is the phase velocity of the wakefield which is same as group velocity of the laser pulse. But, in bubble regime, group velocity of the laser pulse decreases due to pump depletion. In this depletion, laser pulse losses its energy due to etching velocity of the photons owing to its decrement being . Therefore, the group velocity of the laser pulse which is equal to the phase velocity  of bubble.

                                                                                                         (2)

For an underdense plasma, , Hence

                                                                                                               (3)

                                                                                                           (4)

together with  as the radius of bubble wake. This is obtained by equating laser pulse ponderomotive force and electrostatic force of ion cavity on the wakefield electrons being [11].

 

2. ANALYTICAL INVESTIGATION

In laser-plasma interaction when the intensity of the laser is sufficiently high, laser pulse expels all the plasma electrons, creating an ion cavity generally spherical in shape. In the present work, we consider different geometrical bubble shapes with the help of d’Alembert differential equations and find energy gain in the bubble regime.

The d’Alembert differential equations are taken in terms of the electromagnetic field generated by the laser pulse inside plasma with the help of Maxwell’s equations and scalar and vector potentials.

Here we use Gaussian system of units. The dimensionless quantities are obtained by normalizing the length with , velocity with , with , with , with . and  being the scalar and vector potentials of the electromagnetic field,  and  are current and charge densities together with , where ,  is  the phase velocity of bubble,  is the laser frequency [26] and  is the residual electron density normalized by ambient plasma electrons density  .According to Wu et al. [22]  and . The laser pulse is considered to propagate in the x-direction creating an ion cavity. Here we follow Li et al. [24] and consider more Gauge conditions.  is the normalized energy gain, normalized with  [26] and bubble radius  , which is obtained by equating laser pulse ponderomotive force and electrostatic force normalized with . The plasma frequency and dephasing length  [25].

We consider different Gauge conditions and wakefield potentials, defined as follows:

I.   and generated wakefield potential   

II.  and    

III.   and generated wakefield potential

  is vector potential of electromagnetic field. The transverse component

  .

Maxwell’s equations in normalized form can be written as

                                                                                          (5)

                                                                                                  (6)

                                                                                                (7)

                                                                                            (8)

                                                                                              (9)

                                                                                                                               (10)

From (5) and (9),

                                                                           (11)

and from (8) and (10),

                                                                        (12)

The above equations can be written as

                                                                                     (13)

                                                                                 (14)

2.1   and generated wakefield potential   

Under the quasi-static approximation, all the quantities depend on. Hence, L.H.S. of (12) reads

 for x-direction,  for y and z direction.

Combining L.H.S. and R.H.S. of (12), we get

  for x -direction and

                                      (15)

Here  and

 

After solving (15) and (11), we find

                                     for x -direction

For perpendicular directions

                                   

From (15) and (12), we find

                                  

 

Finally the Maxwell’s equations and d’Alembert equations read

                                                                                     (16)

                                                                               (17)

under the quasi-static approximation, all the quantities depend on.

2.1   and generated wakefield potential   

The simplified d’Alembert equations can be written as

                                                                (18)

                                                               (19)

                                                                              (20)

 From (18) and (19), we can find

                                                     (21)

General solutions of this equation can be written as

                                                                 (22)

                                                                 (23)

Here , and  are the coefficients that control the geometry of the bubble. Using (18) - (21) we find

                                                                          (24)

 

                                                     (25)

2.1.1 Bubble geometry:

 For bubble geometry, following parameter is defined

                                              (26)

Here,  is transverse parameter and  is longitudinal parameter and for  then , bubble transforms into transverse ellipsoid bubble and  then , bubble transforms into longitudinal ellipsoid bubble and for , , spherical bubble is formed.

For

                                                                                                                               (27)

 

 

Fig. 1: Variation of geometrical parameter  along y-direction related to shape of bubble having residual electrons density  along x-direction with a change of transverse geometrical coefficient .

Bubble shape transformation depends upon two parameters, one is geometrical coefficient and other is variation of residual electron density. Here, residual electrons play important role for changing bubble shape. If we consider positive value of geometrical coefficient , bubble initially is transverse ellipsoid and if we increase the value of residual electron density, bubble shape changes to longitudinal ellipsoid.

If we change the value of geometrical coefficient , then the shape of the bubble is changed. For example,  gives spherical bubble (Fig.1). The deviation from the spherical shape starts with the finite values of . For the positive values of , the shape changes from spherical to longitudinal (please see graphs for  and ). On the other hand, transverse ellipsoid bubble is obtained for lower value of  (please see graphs for  and ).

2.1.2: Wakefield potential in different shaped bubble regime:

From (23), wakefield potential is         and we have used  for                        

 

For ,

                                              ,                                                                     (28)

                                                                                                             (29)

 

Fig. 2: Variation of wakefield potential  along  direction for different values of .

 

Wakepotential profile is shown in Fig.2 for different values of . Consistent to the variation of parameters , the wakepotential also turns from spherical to longitudinal ellipsoid or transverse ellipsoid in accordance with the positive and negative values of , respectively. Wakefield potential changes from spherical at to longitudinal ellipsoid potential at and at and transverse ellipsoid wake-potential at and at  as depicted in Fig.2.

2.1.3 Electromagnetic field in the bubble regime:

From the above equations, we can find all the components of the electromagnetic field

                                                                 (30)

                            (31)

                            (32)

                                            (33)

These are for the realistic situation of residual electrons. However, for electron free cavity i.e. when  and , we have

,,,,,                                                                                                                                  (34)

2.1.4 Accelerator gradient:

From equation (34), we analyze longitudinal and transverse electric and magnetic fields, but acceleration of electrons takes place by longitudinal electric field. The transverse fields produce radiation due to transverse oscillation of the electrons. Here we focus on the longitudinal electric field in view of the particle acceleration. 

Since acceleration occurs at the front side of the bubble, then,  and the longitudinal electric field

                                                                            (35)

A clear cut variation of field  or the acceleration gradient within the bubble is shown in Fig.3 through the variation of bubble radius. The field increases linearly with the longitudinal direction and bigger bubble is supposed to create larger field and hence, the larger accelerator gradient. Corresponding to Figs.1 and 2, i.e. values of , we notice that the field or gradient in transverse ellipsoid bubble regime for   and  are more than spherical at  and longitudinal ellipsoid bubble regime at  and .

 

 

Fig.3: Variation of accelerating field or wakefield   with bubble radius  along the propagation direction with different values of transverse geometrical coefficient . In unnormalized form, the value of  for  and .

2.1.5 Maximum energy gain:

Based on the expressions of the electric field  and dephasing length , we calculate the maximum energy gain, as

                                                                                                                         (36)

                                                                               (37) 

 

 

Fig.4: Variation of maximum energy gain   with relativistic Lorentz factor  along the x-direction for different values of geometrical coefficient C.

The dependence of maximum energy gain on the Lorentz factor and shape of bubble is shown in Fig. 4 through the variation of  and . As expected maximum energy gain becomes higher and higher with the increase in the Lorentz factor and attains value . In this region, the maximum energy increases almost exponentially with . On the other hand, the maximum energy gain is more in transverse ellipsoid bubble at and  than that of spherical bubble at  and longitudinal ellipsoid bubble regime at  and  for the laser strength parameter . If we consider higher value of  then wakefield is not uniform and for higher value of , density of plasma electrons should be higher but this case is for overdense plasma. With present value of electron density, bubble or nonlinear regime has been made but for higher value of , this regime does not occur with this situation.

 

2.2  , 

The d’Alembert equations in this case can be written as

                              (38)

                                          (39)

                                          (40)

    Solutions of (36), (37) and (38) are (22), (23) are

                                                                                         

                                 (41)

together with                                                          (42)

                                                                                            (43)

                         (44)

                                        (45)

2.2.1 Bubble geometry:

For , the geometrical parameter is obtained as

                                                                                                                  (46)

 

 

Fig.5: Variation of geoemtrical parameter  as a function of residual electrons density  for different transverse coefficient .

         With the consideration of  ,  it is seen that the geometrical shape of bubble changes only in transverse ellipsoid shape for different values of geometrical coefficient  (Fig.6). Wakefield potential changes from spherical at to longitudinal ellipsoid potential at and at and transverse ellipsoid wake-potential at and at .

 

2.2.3 Wakefield potential:

For

                                 (47)

 

Fig. 6: Variation of wakefield potential  as a function of  direction with changing of transverse coefficient .

2.2.4 Accelerating gradient:

                                                                               (48)

 

For

 

                                                     (49)

For

 

 

Fig.7: Variation of accelerating field or wakefield  as a function of bubble radius  along propagation direction x, for different values of transverse geometrical coefficient .

Figure 7 shows the variation of field   or the acceleration gradient with the radius of bubble for different values of geometrical parameter . Clearly larger field or gradient is realized in a bigger size bubble. Moreover, acceleration field transverse ellipsoid bubble regime for and  stays larger than the case of spherical bubble at and longitudinal ellipsoid bubble at and . For all the cases, the field stays larger than the previous case of different Gauge conditions.

2.2.5 Maximum energy gain:

Using the same concept as discussed in previous section, we find the maximum energy gain as

                                                                     (50)

For , it reads

                                                                              (51)

 

 

Fig.8: Variation of maximum energy gain  along y-direction with relativistic Lorentz factor  along x-direction with the change of transverse geometrical coefficient C.

Figure 8 makes a comparative study of the maximum energy gain for different values of geometrical coefficient and Lorentz factor. However, the gain is much larger in the present case (Please see Fig.4 for comparison). With regard to the importance of shape of the bubble for maximum energy gain, it can be observed that the energy is more in transverse ellipsoid bubble at  and at  than that of spherical bubble at  and longitudinal ellipsoid bubble regime at  and  with laser pulse strength .

2.3 Special Case: In order to uncover the role of Gauge selection for realizing appreciable acceleration of electrons, we also compare our results with the ones obtained by Li et al. [10] for the Gauge and wakefield potential

                                                                                                             (52)

The shape of this wakefield potential is shown in Fig. 9. Here also we see that the change in potential profile in view of different values of .

 

 

Fig. 9: Variation of wakefield potential  along  direction with changing of transverse coefficient .

2.3.1 Electromagnetic field in bubble regime:

In addition, we calculate the longitudinal electric field  that is important for the electron acceleration. This is given as

 

                                 (53)

For ,, this reads

                                       (54)

This field (or the acceleration gradient) is plotted in Fig.10 for different size/radius of bubble and geometrical parameter. The behaviour of  remains the same as discussed for the previous cases of Gauge conditions. However, the magnitude remains different. Similar is the case with maximum energy gain,  as shown in Fig 11. A comparative study of Figs.3, 7 and 10 shows that the electric field is the largest for the Gauge condition  and . This Gauge condition also stays the best with respect to the maximum energy gain (please see Figs. 4, 8 and 11)

 

                                        

Fig.10: Variation of accelerating field or wakefield  along y – direction with bubble radius  along propagation direction x- axis with the change of transverse geometrical coefficient .

 

Fig.11: Variation of maximum energy gain  along y-direction with relativistic Lorentz factor  along x-direction with the change of transverse geometrical coefficient C.

 

In order to solve the present problem of bubble wakefield acceleration, we used an analytical approach. However, such a system of equations can also be solved numerically. The numerical and simulation approaches have proved to be very effective in different areas [29–32]. The other interesting part of the present work could be see the role of laser pulse shapes which have proved to be efficient for the THz radiation generation [33–35] and interesting nonlinear phenomena in optics [36, 37].

 

2.4. Conclusions

Geometrical shape of bubble in bubble wakefield acceleration mechanism changes to transverse, spherical, and longitudinal ellipsoid depending on different Gauges conditions and the wakefield potential assumes spherical to longitudinal and transverse ellipsoid profiles. The calculations show that the energy gain limited by dephasing length yields its maximum value in transverse ellipsoid bubble than that of spherical and longitudinal ellipsoid bubbles in each Gauge condition. In particular energy gain is found to be greater for  ,  than ,  and ,   cases.

 

References

1.         Tajima, T., Dawson, J.M.: Laser electron accelerator. Phys. Rev. Lett. 43, 267 (1979)

2.         Toosi, E.S., Mirzanejhad, S., Dorranian, D.: Bubble structure in laser wake-field acceleration. Laser Part. Beams. 34, 193 (2016)

3.         Malik, H.K., Kumar, S., Nishida, Y.: Electron acceleration by laser produced wake field: Pulse shape effect. Opt. Commun. 280, 417–423 (2007)

4.         Malik, H.K., Kumar, S., Singh, K.P.: Electron acceleration in a rectangular waveguide filled with unmagnetized inhomogeneous cold plasma. Laser Part. Beams. 26, 197 (2008)

5.         Nakajima, K., Enomoto, A., Kobayashi, H., Nakanishi, H., Nishida, Y., Ogata, A., Ohsawa, S., Oogoe, T., Shoji, T., Urano, T.: Plasma wake-field accelerator experiments at KEK. Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers, Detect. Assoc. Equip. 292, 12–20 (1990)

6.         Golian, Y., Dorranian, D.: Proton driven plasma wakefield generation in a parabolic plasma channel. J. Theor. Appl. Phys. 11, 27–35 (2017)

7.         Malik, H.K.: Analytical calculations of wake field generated by microwave pulses in a plasma filled waveguide for electron acceleration. J. Appl. Phys. 104, 53308 (2008)

8.         Singh, D.K., Davies, J.R., Sarri, G., Fiuza, F., Silva, L.O.: Dynamics of intense laser propagation in underdense plasma: Polarization dependence. Phys. Plasmas. 19, 73111 (2012)

9.         Hakimi, S., Beier, N., Ma, Y., Hinojosa, J., Hussein, A., Maksimchuk, A., Nees, J., Tajima, T., Krushelnick, K., Thomas, A.: Observation of bubble formation in laser wakefield acceleration. In: APS Division of Plasma Physics Meeting Abstracts. pp. GP20-012 (2020)

10.         Bloom, M.S., Streeter, M.J. V, Kneip, S., Bendoyro, R.A., Cheklov, O., Cole, J.M., Döpp, A., Hooker, C.J., Holloway, J., Jiang, J.: Bright x-ray radiation from plasma bubbles in an evolving laser wakefield accelerator. Phys. Rev. Accel. Beams. 23, 61301 (2020)

11.         Schlenvoigt, H.P., Haupt, K., Debus, A., Budde, F., Jäckel, O., Pfotenhauer, S., Schwoerer, H., Rohwer, E., Gallacher, J.G., Brunetti, E., Shanks, R.P., Wiggins, S.M., Jaroszynski, D.A.: A compact synchrotron radiation source driven by a laser-plasma wakefield accelerator. Nat. Phys. 4, 130–133 (2008). https://doi.org/10.1038/nphys811

12.         Barzegar, S., Niknam, A.R.: Laser pulse-electron beam synergy effect on electron self-injection and higher energy gain in laser wakefield accelerators. Sci. Rep. 11, 1–8 (2021)

13.         Lu, W., Huang, C., Zhou, M., Tzoufras, M., Tsung, F.S., Mori, W.B., Katsouleas, T.: A nonlinear theory for multidimensional relativistic plasma wave wakefields. Phys. Plasmas. 13, 56709 (2006)

14.         Lin, M.-W., Hsieh, C.-Y., Tran, D.K., Chen, S.-H.: Simulation study of ionization-induced injection in sub-terawatt laser wakefield acceleration. Phys. Plasmas. 27, 13102 (2020)

15.         Lin, M.-W., Chu, T.-Y., Chen, Y.-Z., Tran, D.K., Chu, H.-H., Chen, S.-H., Wang, J.: Laser wakefield acceleration driven by a few-terawatt laser pulse in a sub-mm nitrogen gas jet. Phys. Plasmas. 27, 113102 (2020)

16.         Woodbury, D., Korolov, A., Schwartz, R., Milchberg, H.: Laser wakefield acceleration in the bubble regime for mid-IR laser drivers. In: APS Division of Plasma Physics Meeting Abstracts. pp. YO7-008 (2018)

17.         Ma, Y., Seipt, D., Dann, S.J.D., Streeter, M.J. V, Palmer, C.A.J., Willingale, L., Thomas, A.G.R.: Angular streaking of betatron X-rays in a transverse density gradient laser-wakefield accelerator. Phys. Plasmas. 25, 113105 (2018)

18.         Dhawan, R., Malik, H.K.: Sheath characteristics in plasma carrying finite mass negative ions and ionization at low frequency. Chinese J. Phys. 66, 560–572 (2020)

19.         Dhawan, R., Malik, H.K.: Sheath formation criterion in collisional electronegative warm plasma. Vacuum. 177, 109354 (2020)

20.         Dhawan, R., Kumar, M., Malik, H.K.: Influence of ionization on sheath structure in electropositive warm plasma carrying two-temperature electrons with non-extensive distribution. Phys. Plasmas. 27, 63515 (2020)

21.         Kostyukov, I., Pukhov, A., Kiselev, S.: Phenomenological theory of laser-plasma interaction in “bubble” regime. Phys. Plasmas. 11, 5256–5264 (2004)

22.         Wu, H.-C., Xie, B.-S., Zhang, S., Hong, X.-R., Zhao, X.-Y., Liu, M.-P.: Bubble core field modification by residual electrons inside the bubble. Phys. Plasmas. 17, 113103 (2010)

23.         Sadighi-Bonabi, R., Rahmatollahpur, S.H.: A complete accounting of the monoenergetic electron parameters in an ellipsoidal bubble model. Phys. Plasmas. 17, 33105 (2010)

24.         Li, X.F., Yu, Q., Gu, Y.J., Huang, S., Kong, Q., Kawata, S.: Bubble shape and electromagnetic field in the nonlinear regime for laser wakefield acceleration. Phys. Plasmas. 22, 83112 (2015)

25.         Wood, J.: Betatron radiation from laser wakefield accelerators and its applications, (2016)

26.         Esarey, E., Pilloff, M.: Trapping and acceleration in nonlinear plasma waves. Phys. Plasmas. 2, 1432–1436 (1995)

27.         Esarey, E., Schroeder, C.B., Leemans, W.P.: Physics of laser-driven plasma-based electron accelerators. Rev. Mod. Phys. 81, 1229 (2009)

28.         Lu, W., Tzoufras, M., Joshi, C., Tsung, F.S., Mori, W.B., Vieira, J., Fonseca, R.A., Silva, L.O.: Generating multi-GeV electron bunches using single stage laser wakefield acceleration in a 3D nonlinear regime. Phys. Rev. Spec. Top. Beams. 10, 61301 (2007)

29.         Malik, L., Tevatia, A.: Comparative Analysis of Aerodynamic Characteristics of F16 and F22 Combat Aircraft using Computational Fluid Dynamics. Def. Sci. J. 71, (2021)

30.         Malik, L., Rawat, S., Kumar, M., Tevatia, A.: Simulation studies on aerodynamic features of Eurofighter Typhoon and Dassault Rafale combat aircraft. Mater. Today Proc. 38, 191–197 (2021)

31.         Malik, L., Escarguel, A., Kumar, M., Tevatia, A., Sirohi, R.S.: Uncovering the remarkable contribution of lasers peak intensity region in holography. Laser Phys. Lett. 18, 86003 (2021)

32.         Malik, L., Kumar, M., & Singh, I. V.: A Three-Coil Setup for Controlled Divergence in Magnetic Nozzle. IEEE Trans. Plasma Sci. 49, 2227-2237 (2021).

33.         Malik, H.K.: Laser-Matter Interaction for Radiation and Energy. CRC Press (2021)

34.         Malik, H.K.: Generalized treatment of skew-cosh-Gaussian lasers for bifocal terahertz radiation. Phys. Lett. A. 384, 126304 (2020)

35.         Gill, R., Singh, D., Malik, H.K.: Multifocal terahertz radiation by intense lasers in rippled plasma. J. Theor. Appl. Phys. 11, 103–108 (2017)

36.         Malik, L., Escarguel, A.: Role of the temporal profile of femtosecond lasers of two different colours in holography. EPL (Europhysics Lett. 124, 64002 (2019)

37.         Malik, L.: Dark hollow lasers may be better candidates for holography. Opt. Laser Technol. 132, 106485 (2020)

 

 

1

 

---