1. INTRODUCTION
The laser driven electron acceleration is an important area of research due to its wide ranging applications, from fast ignition fusion to X-ray and magnetic field generation (Bagchi et al., Reference Bagchi, Kiran, Yang, Rao, Bhuyan, Krishnamurthy and Kumar2011; Hora, Reference Hora2009; Joshi, Reference Joshi2007; Liu et al., Reference Liu, Sheng, Zheng, Li, Xu, Lu, Mori, Liu and Zhang2012; Tajima & Dawson, Reference Tajima and Dawson1979). Improvements in various key factors such as electron bunch energy, number of particles accelerated, and low energy spread are essential for making it attractive for various applications (Faure et al., Reference Faure, Glinec, Pukhov, Kiselev, Gordienko, Lefebvre, Rousseau, Burgy and Malka2004; Miura et al., Reference Miura, Koyama, Kato, Saito, Adachi, Kawada, Nakamura and Tanimoto2005; Kalmykov et al., Reference Kalmykov, Yi, Khudik and Shvets2009; Mirzanejhad et al., Reference Mirzanejhad, Sohbatzadeh, Asri and Ghanbari2010; Kostyukov et al., Reference Kostyukov, Pukhov and Kiselev2004; Yi et al., Reference Yi, Khudik, Siemon and Shvets2013, Upadhyay et al., Reference Upadhyay, Samant and Krishnagopal2013; Jokar & Eslami, Reference Jokar and Eslami2012). In the last two decades, many experiments and simulations have been performed in order to understand the physics of the electron acceleration in underdense plasma. One of the schemes, laser-wakefield acceleration (LWFA) has become a practical reality with numerous recent experimental results (Mangles et al., Reference Mangles, Murphy, Thomas, Dangor, Divall, Foster, Gallacher, Hooker, Jaroszynski, Langley, Mori, Norreys, Tsung, Viskup, Walton and Krushelnick2004; Geddes et al., Reference Geddes, Toth, Tilborg, Esarey, Schroeder, Bruhwiler, Nieter, Cary and Leemans2004; Faure et al., Reference Faure, Glinec, Pukhov, Kiselev, Gordienko, Lefebvre, Rousseau, Burgy and Malka2004; Leemans et al., Reference Leemans, Nagler, Gonsalves, Toth, Nakamura, Geddes, Esarey, Schroeder and Hooker2006; Hafz et al., Reference Hafz, Jeong, Choi, Lee, Pae, Kulagin, Sung, Yu, Hong, Hosokai, Vary, Ko and Lee2008; Lu et al., Reference Lu, Huang, Zhou, Mori and Katsouleas2006; Jha et al. Reference Jha, Saroch and Mishra2013). A short laser pulse of duration equal to the plasma period excites a large amplitude plasma wave in its wake with phase velocity equal to the group velocity of the laser. Electrons trapped in the proper phase of the plasma wave are accelerated to very high energies by the axial electric field of the wave. Several methods have been proposed for better electron trapping and enhancement in the electron bunch energy and charge (Xu et al., Reference Xu, Shen, Zhang, Wen, Ji, Wang, Yu and Nakajima2010; Dahiya et al., Reference Dahiya, Sajal and Tripathi2010; Faure et al., Reference Faure, Rechatin, Norlin, Lifschitz, Glinec and Malik2006; Hidding et al., Reference Hidding, Konigstein, Osterholz, Karsch, Willi and Pretzler2010; Vieira et al., Reference Vieira, Martins, Pathak, Fonseca, Mori and Silwa2011; Reference Vieira, Martins, Pathak, Fonseca, Mori and Silva2012; Esarey et al., Reference Esarey, Hubbard, Leemans, Ting and Sprangle1997; Umstadter et al., Reference Umstadter, Kim and Dodd1996; Hemker et al., Reference Hemker, Tzeng, Mori, Clayton and Katsouleas1998).
At relativistically high laser intensity and small spot size, the laser pulse expels all the electrons from the axial region, through axial and radial ponderomotive force. The scenario known as bubble regime has been extensively studied using Particle-in-Cell (PIC) simulations. PIC has allowed analyzing the complex behavior of the bubble scenario in a systematic way, which led to many refinements in the scheme. In many studies, a preformed plasma channel is employed for the efficient laser guiding much beyond the diffraction length and yielded electron beams with narrow energy spread (Rao et al., Reference Rao, Chakera, Naik, Kumar and Gupta2011; Martins et al., Reference Martins, Fonseca, Lu, Mori and Silva2010; Kumar et al., Reference Kumar, Dahiya and Sharma2011; Gu et al., Reference Gu, Zhu, Kong, Li, Li, Chen and Kawata2011). Zhang et al. (Reference Zhang, Chen, Wang, Yan, Yuan, Mao, Wang, Liu, Shen, Faenov, Pikuz, Li, Li, Dong, Lu, Ma, Wei, Sheng and Zhang2012) have studied electron acceleration in a submicron sized cluster-gas targets. Wang et al. (Reference Wang, Sheng and Zhang2008) have suggested a scheme for better electron injection by using two orthogonally directed laser pulses. Murakami et al. (Reference Murakami and Tanaka2013) have utilized the extraordinary material and mechanical property of CNT to generate quasimonoenergetic proton beams. Shen et al. (Reference Shen, Li, Nemeth, Shang and Chae2007) have studied the triggering wave breaking using a nanowire. Ma et al. (Reference Ma, Sheng, Li, Chang and Yuan2006) conjectured the production of attosecond electron bunches by ponderomotive force acceleration along the surface of nanowire. Davoine et al. (Reference Davoine, Lefebvre, Rechatin, Faure and Malka2009) have reported another injection scheme by modifying the electron's spatial trajectory. Karmakar and Pukhov (Reference Karmakar and Pukkov2007) have obtained GeV electrons using high-Z materials. Uhm et al. (Reference Uhm, Nam, Kur and Suk2013) investigated the electron dynamics trapped in a cavity using a theoretical model. Kumar and Tripathi (Reference Kumar and Tripathi2012) have developed a bubble regime electron acceleration using Lorentz boosted frame and calculate the energy of the trapped electrons.
In this paper, we develop a model of laser wakefield electron acceleration in the blow out regime in a pre-existing non-uniform plasma channel having a nanotube along the axis. An intense short pulse, with pulse duration of the plasma period ωp, spot size r 0 on the order of (c/ωp), where c is the speed of light in free space, and normalized amplitude a 0, exerts axial, and radial ponderomotive force on the plasma electrons creating an ion bubble co-moving with the laser pulse. The model and simulations in 2D creates a cylindrical ion bubble with the axis transverse to the direction of pulse propagation. The presence of a nanotube three-dimensional (3D)/sheet (2D) leads to creation of a very thin charged line/sheet layer (with charge density n tube and thickness Δ□ << λ, where λ is the wavelength of the laser used) which affect little the overall propagation of laser mode. The bubble radius depends on the laser spot size and laser amplitude and eventually decides the electron energy and its energy spread. The presence of a nanotube augments the charge density by Z An T along the axis, Z A being the atomic number of the CNT and n T is the number density. The laser ponderomotive forces expel the CNT electrons, along with the plasma electrons radically outward. In the moving frame, the expelled electrons appear travelling backwards toward the stagnation point which appears as a stagnation line in 2D. The presence of high density ionic sheet modifies the scenario in three ways helpful in the desired goals of electron acceleration. First, the finite ionic sheet along the direction of propagation leads to generation of an additional electric field along the direction of propagation providing an additional pull on the electrons. Second, the inward addition pull toward the axis by the sheet field restricts the momentum spread in the transverse direction leading to bunch directionality. Third, the injection increases the number of accelerated electrons. We calculate addition energy gain of a test electron for a given thickness and density of the sheet and compare the results with PIC simulations.
In Section 2, we provide an analytical background to the CNT aided acceleration. Simulation results are presented in Section 3. The numerical results of the model and simulation results are compared and discussed in Section 4.
2. ANALYSIS
Let us consider a parabolic plasma channel embedded with a nanotube along the propagation axis. In two dimensions, the plasma density can be taken as
$$n = {n_0} \left(1 + \displaystyle{{{y^2}} \over {y_0^2 }} \right)+ {n_{tube}}\comma \;$$n tube = Z AN T for −Δ/2 < y < Δ/2 or 0 otherwise, where Z A is the charge on one carbon atom, N T is the number density of carbon atoms inside the tube and Δ is the tube thickness (Fig. 1). In the wake of the pulse, one gets a plasma ion bubble of radius 2R embedded with thin high density charged sheet of length 2R along the axis. A preformed plasma channel assists favorably to the self guiding of a high intensity pulse. The additional electric field due to the trapped ion sheet inside the bubble can be calculated as follows.
Fig. 1. Schematic of the bubble and embedded CNT. The width of CNT is Δ = 25 nm.
Let the laser propagation be in x-direction with origin at the back of the bubble. The charged sheet lies in the x-z plane. Considering an element of charged sheet of length dx′ at x′ (with x′ = 0 coinciding with the origin at x = 0) the electric field due to this sheet element inside the bubble at distance r can be written as
$${E_r} = \left(\displaystyle{{{n_{tube}}e} \over {2{{\rm \epsilon} _0}}} \right)r\comma \; \quad r \lt \Delta \semicolon \;$$
$$ \hskip 36pt= \left(\displaystyle{{{n_{tube}}e} \over {2{{\rm \epsilon} _0}{\rm \pi} }} \right)\left(\displaystyle{{\Delta d{x^{\prime}}} \over r} \right)\comma \; \quad r \gt \Delta .$$The axial and transverse electric field at a point P(x,y) is
$${E_{\vert \square \vert }} = \int_0^{2R} {\displaystyle{{{Z_A}{N_T}e\Delta } \over {2{{\rm \epsilon} _0}{\rm \pi} }}\displaystyle{{\lpar x - {x^{\prime}}\rpar } \over {{{\lcub {{\lpar x - {x^{\prime}}\rpar }^2} + {y^2}\rcub }^{1/2}}}}d{x^{\prime}}\semicolon \; }$$
$${E_ \bot } = \int_0^{2R} {\displaystyle{{{Z_A}{N_T}e\Delta } \over {2{{\rm \epsilon} _0}{\rm \pi} }}\displaystyle{y \over {{{\lcub {{\lpar x - {x^{\prime}}\rpar }^2} + {y^2}\rcub }^{1/2}}}}d{x^{\prime}}.}$$The additional energy gained by an electron due to the axial field by the sheet depends on its transverse distance from the laser axis and is given by
$$K{E_{add}} = \displaystyle{{{Z_A}{N_T}{e^2}\Delta } \over {2{{\rm \epsilon} _0}{\rm \pi} }} \int_{\rm \alpha} ^R {\int_0^{2R} {\displaystyle{{\lpar x - {x^{\prime}}\rpar d{x^{\prime}}} \over {{{[{{\lpar x - {x^{\prime}}\rpar }^2} + {y^2}] }^{1/2}}}}dx\comma \; } }$$where α is the starting point close to the stagnation point.
3. SIMULATION RESULTS
We simulate the described model using PIC code VORPAL and compare it with the results of a non CNT embedded case of bubble acceleration using standard parameters. A linearly polarized Gaussian laser pulse is launched from the left boundary in the pre-ionized plasma. Moving window is used to simulate 300 µm long plasma systems which contain two bubbles. Physical parameters used in the simulations are as follows: laser wavelength λ = 1µm, normalized peak laser intensity a 0 = 5, initial spot size r 0 = 7µm, pulse duration τL = 40fs (FWHM), uniform background plasma density n 0 = 1.0 × 1019 cm −3, and 
$\displaystyle{{{n_0}} \over {{n_c}}} = 0.01$, number of grids in 2D simulation box N xN y = 1200, 800 with grid size 
${N_x} = \displaystyle{{\rm \lambda} \over {40}}$ and 
${N_y} = \displaystyle{{\rm \lambda} \over {80}}$ resulting in physical simulation size of 300 × 20 µm2. Number of macro-particles per cell is 16 in 2D simulations. The presence of a preformed plasma channel keeps the laser pulse self-focused over a sufficient diffraction length. The nanotube of thickness (25 nm) appears as a thin dark line along the laser axis (Fig. 2).
Fig. 2. (Color online) Moving Simulation window showing CNT at the axis. The dimensions are (30 µm × 20 µm).
We run a reference simulation without a CNT, calling it case-0, for a typical standard parameters of LWFA and obtain two accelerated electron bunches in the first two bubbles after 600 fs. The energy of the first bunch ranges from 89 MeV to 111 MeV peaking at 100 MeV (Fig. 3a-1). The electron bunch in the second bubble has an energy range from 27 MeV to 46 MeV (Fig. 3a-2). The two bunches are well separated in energy and their energy gap is approximately 43 MeV. The minimum cut-off energy of the second bunch (27 MeV) is well separated from the maximum cut-off energy of the bulk electrons (17 MeV) (Fig. 4.). Hence, we obtain two distinct bunches of accelerated electrons in the first two buckets with their characteristic energy gap. We run two more simulations, called case-1 and case-2, using the same set of parameters but introducing a CNT along the laser axis. The thickness of the tube is 25 nm. In case-1, the density inside tube is considered 30 times the background plasma density where as it is 90 times in case-2. The choices restrict the change in the average plasma density to less than 3–4%.
Fig. 3. (Color online) Phase-space diagram for the kinetic energy of electrons in three distinct cases: (a) Case-0, Without CNT (b) Case-1, n tube = 30n 0 (c) Case-2, n tube = 90n 0. The kinetic energy is given in eV.
Fig. 4. (Color online) Energy spectrum for three cases: (a) Case-1 without CNT (b) Case-2 with CNT at n tube = 30n 0 (c) Case-3 n tube = 90n 0.
All the three features (depicted by 1, 2, 3, respectively, in Fig. 3) of accelerated electrons in a LWFA show a remarkable shift when a CNT is introduced along the laser axis. In case-1, the entire bunch gains energy by 4–5 % as compared to the reference case-0. The new energy range is (93–115 MeV) instead of (89–111 MeV), showing about a 4% enhancement without any additional energy spread (Fig. 4). However, in case-2, the energy spectrum of the first bubble changes drastically. The bunch energy shows a flattened behavior with reduced number density and enhanced energy spread 63–125 MeV expanding on either side of the spectrum (Fig. 4). The number of accelerated electrons in the bunch is enhanced, though at the expense of energy spread, by merely changing the tube density. The bunch of electrons in the second bubble also experiences an enhanced acceleration due to the tube. The maximum energy cut-off of the second bunch in case-0 is 46 MeV. Due to the presence of tube it reaches up to 54 MeV. The total number of accelerated particles is also enhanced as shown by the enhanced peaks. The gap in the energy spectrum of the two bunches from bubble-1 and bubble-2 remains almost unchanged in case-1 around 40 MeV. However, this gap reduces drastically to about 10 MeV for case-2. The tube has a noticeable effect on the energy of the bulk electrons (Fig. 3a-3). In case-0, the maximum energy cut-off is 15 MeV. This cut-off increases to about 25 MeV for case-1 (Fig. 3b-3) but is still much lower than the lower cut-off of the second bunch . This reduced energy gap is totally vanished for case-2 (Fig. 3c-3). The merged energy spectrum has a combined cut-off at 54 MeV.
We have numerically evaluated and plotted the axial and the transverse electric field profile as given by Eqs. (4) and (5) up to the bunch location (1 µm) from the laser axis. The axial field is strong around the stagnation point (at x′ = 0) facilitating the additional pull inside the bubble (Fig. 5). The transverse field is strong closer to the axis and gradually becomes weak as one goes away from the laser axis (Fig. 6). One can calculate the average gain as the accelerated bunch remains confined closer to the laser axis. We numerically evaluate the additional gain in the energy of a test electron due to the axial electric field generated by the charged sheet. The numerical value of the additional kinetic energy, following Eq. (6), can be averaged out up to the transverse extent of the bunch as
$${[ K{E_{add}}] _{\rm avg}} = \displaystyle{{{Z_A}{N_T}{e^2}\Delta } \over {2{{\rm \epsilon} _0}{\rm \pi} }}{I_{\rm avg}},$$where
$${I_{avg}} = \sum\limits_1^n {\displaystyle{1 \over n} \int_{\rm \alpha} ^R {\int_0^{2R} {\displaystyle{{\lpar x - {x^{\prime}}\rpar d{x^{\prime}}} \over {{{[{{\lpar x - {x^{\prime}}\rpar }^2} + y_n^2] }^{1/2}}}}dx} } }$$
and α is the starting point close to the stagnation point but much less than the bubble radius (
$ \lt \displaystyle{R \over {50}}$). The distance from the laser axis y n ranges from 100 nm to 1 µm. The additional gain is maximum for electrons accelerated in the vicinity of the laser axis and gradually decreases away from it. The additional fractional enhancement in energy due to the tube as compared to the bubble is calculated for a range of practical parameters. For a typical bubble radius R = 5 µm, normalized nanotube thickness 
$\displaystyle{{\Delta} \over {\lambda}} = 0.025$, normalized background plasma density 
$\displaystyle{{{n_0}} \over {{n_{cr}}}} = 0.01$, one finds that enhancement is about 4–5% for case-1 and about 12–15% for case-2. This is in agreement with the simulation results described above.
Fig. 5. (Color online) Comparison of axial electrostatic field profile parallel to the laser axis at radial distances: (i) 100 nm, (ii) 300 nm, (iii) 500 nm, (iv) 700 nm, and (v) 900 nm.
Fig. 6. (Color online) Comparison of transverse electrostatic field profile parallel to the laser axis at distances: (i) 100 nm, (ii) 300 nm, (iii) 500 nm, (iv) 700 nm, and (v) 900 nm.
4. DISCUSSION
The bubble regime electron acceleration by a short laser pulse in underdense plasma appears to be significantly influenced by the embedded CNT. The immobile charge of the central region affects the goals of LWFA favorably. In all the cases considered, an enhancement in the bunch energy and charge is indicated. The nanometer dimension of the axial high density region changes the average charge density of the plasma only marginally, but it facilitates significant energy enhancement and improves particle confinement in the transverse direction. An equivalent change in the background plasma density doesn't indicate the above benefits. The CNT modifies the electrostatic field (Fig. 5 and Fig. 6) in the bubble most significantly near the axis hence has stronger influence over the electrons pulled from the stagnation point. The spatial location of the CNT relative to the laser axis and its effect on the injection process can be investigated separately as this structure provides a strong field closer to the bubble rear.
ACKNOWLEDGEMENT
The authors are grateful to Prof. C. S. Liu, University of Maryland for his valuable inputs in the present work. The authors acknowledge the financial support from DST, Government of India.
