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Self-focusing up to the incident laser wavelength by an appropriate density ramp

Published online by Cambridge University Press:  15 December 2011

R. Sadighi-Bonabi  and
M. Moshkelgosha
R. Sadighi-Bonabi*
Affiliation:
Department of Physics, Sharif University of Technology, Tehran, Iran
M. Moshkelgosha
Affiliation:
Department of Physics, Sharif University of Technology, Tehran, Iran
*
Address correspondence and reprint requests to: R. Sadighi-Bonabi, Department of Physics, Sharif University of Technology, 11365-9161, Tehran, Iran. E-mail: sadighi@sharif.ir
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Abstract

This work is devoted to improving relativistic self-focusing of intense laser beam in underdense unmagnetized plasma. New density profiles are introduced to achieve beam width parameter up to the wavelength of the propagating laser. By investigating variations of the beam width parameter in presence of different density profiles it is found that the beam width parameter is considerably decreased for the introduced density ramp comparing with uniform density and earlier introduced density ramp profiles. By using this new density profile high intensity laser pulses are guided over several Rayleigh lengths with extremely small beam width parameter.

Keywords

Density rampPropagating laserSelf-focusingUnderdense plasmaUnmagnetized plasma
Type
Research Article
Information
Laser and Particle Beams , Volume 29 , Issue 4 , December 2011 , pp. 453 - 458
Copyright
Copyright © Cambridge University Press 2011

INTRODUCTION

Since the first suggestion of Askarian (Reference Askarian1962), self-focusing is an extensively studied phenomenon in the field of high intensity laser interaction with nonlinear media. When laser intensity exceeds the critical power P cr ≅ 17 (ω/ωp)2 GW, which ω and ωp are the laser frequency and the plasma frequency, respectively (Sun et al., Reference Sun, Ott, Lee and Guzdar1987), quiver motion of the generated relativistic electrons increase the mass of electrons. As a consequence, transverse gradient of the refractive index leads to confinement of the laser beam to the propagation axes and decreases the beam width parameter that gives rise to the relativistic self-focusing (Hora et al., Reference Hora1975; Osman et al., Reference Osman, Castillo and Hora2000). Due to mass increase in relativistic self-focusing, the electron mass replaced by m 0γ where γ = (1 + a 2/2)1/2 is relativistic factor, a = e|E|/(m 0cω) is normalized laser amplitude and E, e, m 0 are the amplitude of the laser electric field, the electron charge, and the electron rest mass, respectively. Therefore, the plasma dielectric function modified as ɛ = 1 − ω2p/γω2, where ωp = (4πn ee 2/m 0γ)1/2 is the plasma frequency (Boyd et al., Reference Boyd, Lukishova and Shen2008). The ponderomotive force also causes nonlinear electron perturbation that exerts a radial force and expels the electrons radialy outward from the intense laser beam axes, which ends to ponderomotive self-focusing (Mori et al., Reference Mori, Joshi, Dawson and Forslund1988; Perkins & Valeo, Reference Perkins and Valeo1974). Ponderomotive self-focusing causes decreasing of electron density and increasing the refractive index.

Resent advances in ultra-intense short-pulse lasers and their numerous applications stimulated the research activities in this field such as generation of high-energy electron and ion beams and their acceleration (Leemans et al., Reference Leemans, Nagler, Gonsalves, Toth, Nakamura, Geddes, Esarey, Schroeder and Hooker2006; Láska et al., Reference Láska, Jungwirth, Krása, Krouský, Pfeifer, Rohlena, Ullschmied, Badziak, Parys, Wolowski, Gammino, Torrisi and Boody2006; Lihua et al., Reference Cao, Yu, Xu, Zheng, Liu and Li2004; Geddes, Reference Geddes2005; Hoffmann et al., Reference Hoffmann, Blazevic, Ni, Rosmej, Roth, Tahir, Tauschwitz, Udera, Vanentsov, Weyrich and Maron2005; Schlenvoigt et al., Reference Schlenvoigt, Haupt, Debus, Budde, Ja Ckel, Pfotenhauer, Schwoerer, Rohwer, Gallacher, Brunetti., Shanks, Wiggins and Jaroszynski2008; Xie et al., Reference Xie, Aimidula, Niu, Liu and Yu2009; Zhou et al., Reference Zhou, Yu and He2007), monoenergetic electron beam (Fature et al., Reference Fature, Gline, Pukhov, Kiselev, Gordienko, Lefebvre, Rousseau, Burgy and Malka2004; Singh et al., Reference Singh, Sajal and Gupta2008; Sadighi-Bonabi et al., Reference Sadighi-Bonabi, Navid and Zobdeh2009a, Reference Sadighi-Bonabi, Rahmatallahpor, Navid, Lotfi, Zobdeh, Reiazi, Nik and Mohamadian2009b, Reference Sadighi-Bonabia and Rahmatollahpur2010a, Reference Sadighi-Bonabi and Rahmatollahpur2010b), monoenergetic ion beam generation (Hegelich et al., Reference Hegelich, Albright, Cobble, Flippo, Letzring, Paffett, Ruhl, Schreiber, Schulze and Fernandez2006), X-ray emission and X-ray lasers (Zhang et al., Reference Zhang, He, Chen, Li, Zhang, Lang, Li, Feng, Zhang, Tang and Zhang1998), harmonic generation (Butylkin & Fedorova, Reference Butylkin and Fedorova1994), fusion with the fast ignition scheme (Lalousis & Hora, Reference Lalousis and Hora1983; Hora, Reference Hora2004, Reference Hora2009; Hora et al., Reference Hora, Miley, Azizi, Malkynia, Ghoranneviss and He2009; Ghoranneviss et al., Reference Ghoranneviss, Malekynia, Hora, Miley and He2008; Yazdani et al., Reference Yazdani, Cang, Sadighi-Bonabi, Hora and Osman2009; Sadighi-Bonabi et al., Reference Sadighi-Bonabi, Hora, Riazi, Yazdani and Sadighi2010c, Reference Sadighi-Bonabi, Yazdani, Cang and Hora2010d). These lasers are also used in the transmutation of hazardous radioactive wastes to valuable nuclear medicine (Sadighi-Bonabi & Kokabi, Reference Sadighi-Bonabi and Kokabi2006; Sadighi-Bonabi et al., Reference Sadighi-Bonabi, Irani, Safaie, Imani, Silatani and Zare2010e; Sadighi & Sadighi-Bonabi, Reference Sadighi and Sadighi-Bonabi2010).

In the entire above mentioned ultra-intense laser interactions, self-focusing has an important role and it should be carefully studied. In order to guide high intensity laser pulses over several Rayleigh lengths, it is important to employ self-focusing by a defined density profile. This achievement can have very fundamental impact in the recent advances of laser-plasma interaction and fast ignition systems. Self-focusing effects have been observed in laser-plasma interaction that enables the laser beam to propagate over several Rayleigh lengths (Boyd et al., Reference Boyd, Lukishova and Shen2008; Schlenvoigt et al., Reference Schlenvoigt, Haupt, Debus, Budde, Ja Ckel, Pfotenhauer, Schwoerer, Rohwer, Gallacher, Brunetti., Shanks, Wiggins and Jaroszynski2008). Self-focusing has been investigated in the interaction of laser beam with homogenous and inhomogeneous plasma (Upadhyay et al., Reference Upadhyay, Tripathi, Sharma and Pant2002; Varshney et al., Reference Varshney, Qureshi and Varshney2006; Kaur & Sharma, Reference Kaur and Sharma2009; Sharma & Kourakis, Reference Sharma and Kourakis2010).

In this work, a complete study of density ramp profiles is presented and the self-focusing of laser beam along the propagation axis in axially inhomogeneous plasma is simulated. It is shown that with the introduced density ramp profile and by optimizing the laser and plasma parameters, the laser beam width parameter is reduced up to the laser wavelength.

VARIATION OF BEAM WIDTH PARAMETER

For investigating of electron density variation of unmagnetized cold plasm, the propagating of a Gaussian laser beam $E = \hat{x}A \, e^{-i\lpar {\rm \omega} t - k_{0}z\rpar }$ in a cylindrical coordinate system is considered, where k 0(z) = (ω0/c)ω0/c)ɛ01/2 and ɛ0 is the plasma dielectric constant. Following the Tripathi et al. (Reference Tripathi, Taguchi and Liu2005) approach, at z > 0 one have a 2 = (a 02/f 2)exp(−r 2/r 20f 2), where f is the laser beam width parameter and a 0 = eA 0/(mωc) is the laser intensity parameter.

As the Gaussian beam have an intensity gradient along its cross-section, the radial ponderomotive force pushes the electrons outward of propagation axis, on the time scale of a plasma period ωp−1 and creates a radial space charge field of E s = −∇φs, therefore, considering Poisson's equation, electron density modified as:

(1)
n_{e} = n_{e0}+\lpar 1/4{\rm \pi} e\rpar \nabla^{2}_{\perp}{\rm \varphi} _{s}.

Where φs is the electric potential, which is due to the radial space charge. In quasi-steady state one can have F p = −∇φp = eE s, where φp is the radial ponderomotive force and it is defined as

(2)
{\rm \varphi} _{\,p} = - \lpar mc^{2}/e\rpar \lpar \lpar 1 + a^{2}/2\rpar ^{1/2} - 1\rpar.

Therefore, the modified electron density is obtained similar to the works of Tripathi et al. (Reference Tripathi, Taguchi and Liu2005), Gupta et al. (Reference Gupta, Hur, Hwang and Suk2007a), and Sadighi-Bonabi et al. (Reference Sadighi-Bonabi, Yazdani, Habibi and Lotfi2010f, Reference Sadighi-Bonabi, Habibi and Yazdani2010g):

(3)
\eqalign{n_e & = n_{e0} \left(z \right)\left\{1 - \displaystyle{{c^2 } \over {{\rm \omega} _p^2 \,r_0^2 \,f^2 }}\displaystyle{{a^2 } \over {\left({1 + a^2 /2} \right)^{0.5} }}\right.\cr &\quad \left. \times \ 1 - \left({\displaystyle{{r^2 } \over {r_0^2 \,f^2 }}\displaystyle{{1 + a^2 /4} \over {1 + a^2 /2}}} \right)\right\}.}.

And the dielectric constant of plasma is obtained as

(4)
\eqalignb{{\rm \varepsilon} &= 1 - {\rm \omega} _p^2 /{\rm \gamma} {\rm \omega} ^2 \cr &= 1 - \left(\displaystyle{\matrix{4{\rm \pi} n_0 \left(z \right)\left\{1 - \displaystyle{{c^2 } \over {{\rm \omega} _p^2 r_0^2 f^2 }}\displaystyle{{a^2 } \over {\left({1 + a^2 /2} \right)^{0.5} }}\right.\cr \left. \times \ 1 - \left({\displaystyle{{r^2 } \over {r_0^2 f^2 }}\displaystyle{{1 + a^2 /4} \over {1 + a^2 /2}}} \right)\right\}e^2 }} \over {m_0 \sqrt {1 + \displaystyle{{a^2 } \over 2}{\rm \omega} ^2 }} \right)^{1/2} \comma \; }

in paraxial approximation (r 2 « r 02f 2), the dielectric constant of plasma can be expanded as ɛ = ɛ0 – r 2/r 02. As a result expanding of in this approximation, leads to:

(5)
\eqalign{{\rm \varepsilon} &\cong \left({1 - \displaystyle{{{\rm \omega} _p^2 } \over {{\rm \omega} ^2 }}\displaystyle{{\sqrt 2 } \over {\sqrt {2 + a^2 } }} + \displaystyle{{c^2 a^2 } \over {{\rm \omega} ^2 \left({1 + \displaystyle{{a^2 } \over 2}} \right)r_0^2 \,f^2 }}} \right)\cr &\quad - \left\{\vphantom{\vscale 400%(}\left(\displaystyle{{\sqrt 2 } \over 2}{}\displaystyle{{{\rm \omega} _p^2 } \over {{\rm \omega} ^2 }}\displaystyle{{a^2 } \over {\root 3 \of {2 + a^2 } }} + \displaystyle{{c^2 a4} \over {{\rm \omega} ^2 }}\displaystyle{1 \over {\,f^4 r_0^2 \left({2 + a^2 } \right)^2 }} \hskip58pt(5)\right.\right. \cr &\quad \left. \left. + \displaystyle{{c^2 a4} \over {{\rm \omega} ^2 \,f^4 }}\displaystyle{1 \over {\left({2 + a^2 } \right)\left({1 + \displaystyle{{a^2 } \over 2}} \right)}} + \displaystyle{{4c^2 a^2 } \over {{\rm \omega} ^2 }}\displaystyle{1 \over {\,f^8 r_0^2 \left({2 + a^2 } \right)^2 }} \right)\right\}\times \displaystyle{{r^2 } \over {r_0^{\,\,2} }}.}

Then expansion coefficients are obtained as:

(6)
\eqalign{&{\rm \varepsilon} _0 = 1 - \displaystyle{{{\rm \omega} _p^2 } \over {{\rm \omega} ^2 }}\left({\displaystyle{1 \over {\left({1 + a^2 /2} \right)^{1/2} }}} \right)+ \displaystyle{{c^2 a^2 } \over {{\rm \omega} ^2 \,r_0^2 \,f^2 \left({1 + a^2 /2} \right)}}\comma \; \cr &\quad {\rm \phi} = \displaystyle{{{\rm \omega} _p^2 } \over {4{\rm \omega} ^2 }}\displaystyle{{a^2 } \over {\,f^2 \left({1 + a^2 /2} \right)^{3/2} }}\left({1 + \displaystyle{{c^2 \left({8 + a^2 } \right)} \over {r_0^2 {\rm \omega} _p^2 \,f^2 \left({1 + a^2 /2} \right)^{1/2} }}} \right).}

Regarding the wave equation approaches (Gupta et al., Reference Gupta, Hur, Hwang and Suk2007a); the second order boundary equation for the laser beam width parameter is obtained as:

(7)
\displaystyle{{\partial ^2 f} \over {\partial {\rm \xi} ^2 }} = \displaystyle{1 \over {\,f^3 }} - \displaystyle{1 \over {2{\rm \varepsilon} _0 }}\displaystyle{{\partial f} \over {\partial {\rm \xi} }}\displaystyle{{\partial {\rm \varepsilon} _0 } \over {\partial {\rm \xi} }} - \displaystyle{{R^{2}_{\,\,d} } \over {r_0^2 \,{\rm \varepsilon} _0 }}{\rm \phi} \ f.

Where ξ = z/R d is dimensionless propagation length, with R d = ωr 02/c as Rayleigh length. The first term on the right-hand side of Eq. (7) is due to the diffraction effect, the second term is due to the plasma inhomogeneities, and the last term is the nonlinear term that is responsible for relativistic self-focusing. Using initial boundary condition at z = 0 as f = 1 and initial plane front wave (df/dξ = 0), one can solve this equation and investigate changing of the beam width parameter along the laser propagation in plasma.

The critical laser relativistic intensity of I = 1.21 × 1018 (w/cm2) is assumed for Nd:Glass laser wavelength λ = 1.06 µm and initial laser spot size chosen as r 0 = 10λ = 10.6 µm. For propagation of laser in plasma without reflecting, electron density must be regarded less than the critical density n cr = (ω2m/4πe 2)≃ 1021(cm−3). Therefore, laser frequency must be larger than the plasma frequency.

Figure 1 shows the plots of Eq. (7) for propagation of the laser beam in plasma with constant density (n e0 = n 0), by considering the self-focusing effect (solid curve) and neglecting the self-focusing (dashed curve). This plot indicates that by neglecting the self-focusing term in Eq. (8), the laser beam diverges during travelling in the plasma medium. However, by considering the self-focusing, laser beam width parameter decreases due to the relativistic effects and ponderomotive force. Later the beam width starts to increase due to the attraction of the electrons with the ions and decreasing of the dielectric constant. As a consequence, the laser beam undergoes an oscillatory focusing/defocusing behavior along the propagation direction.

Fig. 1. Dependence of beam width parameter f on distance of propagation in underdense plasma neglecting self-focusing (dash curve) and for self-focusing with uniform electron density (solid curve).

Figure 2 shows the electron density distribution n e, versus r. As one can notice from the beam width parameter f = 1 that is related to unfocused beam, the electrons distributed uniformly along r (solid curve). However, for the focused beams (f is less than one) that are shown by dashed and point curves, one can see for smaller f the electrons distribution become more inhomogeneous and the electrons distributed in region far from the axis due to the ponderomotive force.

Fig. 2. (Color online) The changes of electron density along r: for f = 1(solid curve), f = 0.6 (dash curve), f = 0.5 (dot curve).

One can also investigate the electron density distribution along the propagation direction of the laser beam. Regarding Eq. (3) and substituting the beam width variation (f), one can have an oscillatory distribution of ne and this is completely agrees with the result of self-focusing of the laser beam. This is shown in Figure 3 and as one can see for regions with n e(z)/n 0 = 1, which is related to the uniform distribution of the electron density, the beam width parameter value is f = 1 and in the regions with n e(z)/n 0 = 0 that is related to the maximum self-focusing, the beam width parameter is minimum. Figures 1 and 3 are in good agreement with the work of Brandi et al. (Reference Brandi, Manus, Mainfray and Lehner1993), which is obtained with a different approach.

Fig. 3. (Color online) The changes of electron density (solid curve) and beam width parameter along laser beam propagation (z).

THE PROPOSED DENSITY RAMP AND THE RESULTS

To overcome the defocusing of laser beam due to attraction of centralized ions at the axis, Gupta et al. (Reference Gupta, Hur, Hwang and Suk2007a) introduced a density ramp varying along laser propagation (z) in the form of n 0 + tan(z/d). Where n 0 is the electron density at z = 0 and d is a constant factor. Regarding dimension correction revised papers are presented and an acceptable form of density ramp is introduced as n 0 × tan(z/d) (Gupta et al., Reference Gupta and Suk2007b; Sadighi-Bonabi et al., Reference Sadighi-Bonabi, Yazdani, Habibi and Lotfi2010f, Reference Sadighi-Bonabi, Habibi and Yazdani2010g). Introducing the new density profile of n e0 = n 0 + n 0tan(z/d), one can investigate the beam width variation for mentioned profile.

In Figure 4, the beam width parameter variation, for two different density ramps are compared at the same conditions. A considerable decrease of beam width parameter for introduced density ramp of n e0 = n 0 + n 0tan(z/d) is obtained in comparison to the earlier profile of n e0 = n 0 × n 0tan(z/d). By using here introduced density ramp, the laser beam width parameter decreased up to 20% of the initial value. The frequency of oscillation also decreased noticeably.

Fig. 4. Dependence of beam width parameter f on distance of propagation ξ in underdense plasma mass for ramp density profile with function as n e0 = n 0 × tan(z/d) (dot curve) and n e0 = n 0 + n 0 tan(z/d) (solid curve).

Self-focusing can produce extremely high laser intensity that can have numerous important applications such as ELI (www.extreme-light-infrastructure.eu/eli-home.php), ion acceleration in fusion applications (Ting et al., Reference Ting, Moore, Krushelnick, Manka, Esarey, Sprangle, Hubbard, Burris and Baine1997; Hegelich et al., Reference Hegelich, Albright, Cobble, Flippo, Letzring, Paffett, Ruhl, Schreiber, Schulze and Fernandez2006). The realistic expectation is focusing of high power lasers in dimensions comparable to the laser wavelength. For this purpose, even more useful density ramp is introduced as n e0 = n 0 + n 1tan(z/d). Based on this new ramp profile Figure 5 is produced where self-focusing for different n 1 is plotted.

Fig. 5. (Color online) Dependence of the beam width parameter f on distance of propagation ξ in underdense plasma for ramp density profile with function as n e0 = n 0 + n 1tan(z/d), with different n 1 (n 1 = 0 (dot curve), n 1 = n 0 (solid curve), n 1 = 2 n 0 (dash curve).

Simulating the Nd:Glass laser parameters in the intensities of I = 1017, I = 1.2 × 1018, I = 1019 (W/cm2), one can see that the best self-focusing, occur at the critical relativistic intensity of I cr = 1.2 × 1018 (W/cm2). This is shown in Figure 6.

Fig. 6. Comparing self-focusing of different intensities; I = 1019 W/cm2 (dot curve), I = 1018 W/cm2 (solid curve), I = 1019 W/cm2 (dot-dashed curve).

Regarding the optimized laser and plasma parameters and introducing the above mentioned density ramp, beam width parameter of close to 10% of the initial value is obtained and this is shown in Figure 7. In this condition, the laser beam width parameter reduced up to about 10% of its initial beam width parameter and this is equal to about one laser wavelength.

Fig. 7. Dependence of beam width parameter f on distance of propagation ξ in underdense plasma for ramp density profile with function as n e0 = n 0 × tan(z/d) (dot-dash curve), n e0 = n 0 + n 0 tan(z/d) (dash curve) and n e0 = n 0 + n 1 tan(z/d) (solid curve).

CONCLUSION

In this work, simulation and optimization of the self-focusing of laser beam along the propagation axis is investigated. Focusing of the laser beam up to the laser wavelength by the new density profile is achieved. The effect of different laser intensities is also studied and the best intensity for achieving minimum laser beam width parameter is shown. This achievement can have very important impact in fast ignition processes in which self focusing is very important factor.

ACKNOWLEDGMENTS

The authors want to thank Pars oil and gas company of ministry of oil for their support of this project through contract number PT131. We also acknowledge the research deputy of Sharif University of Technology.

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Figure 0

Fig. 1. Dependence of beam width parameter f on distance of propagation in underdense plasma neglecting self-focusing (dash curve) and for self-focusing with uniform electron density (solid curve).

Figure 1

Fig. 2. (Color online) The changes of electron density along r: for f = 1(solid curve), f = 0.6 (dash curve), f = 0.5 (dot curve).

Figure 2

Fig. 3. (Color online) The changes of electron density (solid curve) and beam width parameter along laser beam propagation (z).

Figure 3

Fig. 4. Dependence of beam width parameter f on distance of propagation ξ in underdense plasma mass for ramp density profile with function as ne0 = n0 × tan(z/d) (dot curve) and ne0 = n0 + n0 tan(z/d) (solid curve).

Figure 4

Fig. 5. (Color online) Dependence of the beam width parameter f on distance of propagation ξ in underdense plasma for ramp density profile with function as ne0 = n0 + n1tan(z/d), with different n1 (n1 = 0 (dot curve), n1 = n0 (solid curve), n1 = 2 n0 (dash curve).

Figure 5

Fig. 6. Comparing self-focusing of different intensities; I = 1019 W/cm2 (dot curve), I = 1018 W/cm2 (solid curve), I = 1019 W/cm2 (dot-dashed curve).

Figure 6

Fig. 7. Dependence of beam width parameter f on distance of propagation ξ in underdense plasma for ramp density profile with function as ne0 = n0 × tan(z/d) (dot-dash curve), ne0 = n0 + n0 tan(z/d) (dash curve) and ne0 = n0 + n1 tan(z/d) (solid curve).