1. INTRODUCTION
Recent continuous advances in ultra-intense short-pulse lasers and their various applications motivated the research activities in this field such as mono-energetic electron generation, ion block acceleration, and inertial confinement fusion with the fast igniter scheme (Sari et al., Reference Sari, Osman, Doolan, Ghoranneviss, Hora, Höpfl and Hantehzadeh2005; Koyama et al. Reference Koyama, Adachi, Miura, Kato, Masuda, Watanabe and Tanimoto2006; Badziak et al., Reference Badziak, Glowacz, Hora, Jablonski and Wolowski2006). When an intense laser beam propagates in the plasma, due to the induced quivering motion of electrons, the plasma refractive index changes (Hora, Reference Hora1975; Sadighi-Bonabi et al., Reference Sadighi-Bonabi, Habibi and Yazdani2009). In this condition, the plasma behaves initially similar to a positive lens that decreases the laser spot size and continues its focusing and defocusing through plasma (Hora, Reference Hora1985; Faure et al., Reference Faure, Malka, Marquès, David, Amiranoff, Ta Phuoc and Rousse2002; Pukhov, Reference Pukhov2003). In order to achieve a better interaction of laser with plasma, deeper penetration of high-intensity beams in plasma is required. The self-focusing effect plays an important role in the recent advances of the laser–plasma interaction and particularly in fast ignition systems. It is noticed that this effect enables the laser beam to propagate over several Rayleigh lengths in plasma (Schlenvoigt et al., Reference Schlenvoigt, Haupt, Debus, Budde, Jäckel, Pfotenhauer and Brunetti2007; Boyd et al., Reference Boyd, Lukishova and Shen2008).
The laser self-focusing has been studied in the interaction of the laser beam with both homogeneous and inhomogeneous plasmas (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). Prakash studied the propagation of a Gaussian laser beam in a radial inhomogeneous medium with multi-photon absorption (Prakash, Reference Prakash2005). Furthermore, in the classical regime, the propagation of intense Gaussian laser beam in collisional and collisionless plasmas has been studied by many researchers (Upadhyay et al., Reference Upadhyay, Tripathi, Sharma and Pant2002; Sharma et al., Reference Sharma, Prakash, Verma and Sodha2003; Sharma & Kourakis, Reference Sharma and Kourakis2010; Prakash, Reference Prakash2005; Sodha & Sharma, Reference Sodha and Sharma2006; Varshney et al., Reference Varshney, Qureshi and Varshney2006; Kaur & Sharma, Reference Kaur and Sharma2009; Etehadi Abari & Shokri, Reference Etehadi Abari and Shokri2012; Gupta et al., Reference Gupta, Islam, Jang, Suk and Jaroszynski2013; Jafari Milani et al., Reference Jafari Milani, Niknam and Farahbod2014). In principle, classical plasma is introduced by high temperature and low density, while quantum plasma is characterized by high density and low temperature (Shukla, Reference Shukla2009; Chandra et al., Reference Chandra, Paul and Ghosh2012). Distinction between the classical and the quantum models for plasma is determined by the parameter χ = T F/T, where T F and T represent the Fermi temperature and the plasma temperature, respectively. If the plasma temperature is equal to or less than the electron Fermi temperature (χ ≥ 1), then the quantum effects are dominant and the relevant statistical distribution changes from Maxwell–Boltzmann to Fermi–Dirac. The Fermi temperature could be defined as follows (Landau & Lifshitz, Reference Landau and Lifshitz1980):
$$k_{\rm B} T_{\rm F} = E_{\rm F} = \displaystyle{{\hbar ^2} \over {2m_{\rm e}}} (3{\rm \pi} ^2 n_{\rm e} )^{{2 / 3}}$$
$${\rm \chi} = \displaystyle{{T_{\rm F}} \over T} = \displaystyle{1 \over 2}(3{\rm \pi} ^2 )^{{2 / 3}} (n_{\rm e} {\rm \lambda} _{\rm B}^3 )^{{2 / 3}}$$n e represents the electron plasma density. On the other hand, the quantum effects could be measured by the thermal de Broglie wavelength, λB = ħ/(m ek BT)1/2 (ħ, m e, and k B are the rationalized Planck's constant, the electron mass, and the Boltzmann constant, respectively). λB explains roughly the spatial extension of the particle wave function by considering the quantum uncertainty. Therefore, the quantum effects are important when the de Broglie wavelength of the electron is equal to or greater than the average inter-electron distance, 
$n_{\rm e}^{{{ - 1} / 3}} $, that is, 
$n_{\rm e} {\rm \lambda} _{\rm B}^3 \ge 1$ (Manfredi, Reference Manfredi2005; Shukla & Eliasson, Reference Shukla and Eliasson2010). In the classical regime, the de Broglie wavelength is small enough to ignore overlapping of the wave functions and quantum interferences and consider the particles as points. It is realized that the quantum effects get more effective with the increase in the plasma density or the decrease in the plasma temperature (Patil & Takale, Reference Patil and Takale2013). Another important parameter in quantum plasma, characterized as the ratio of the interaction energy, E int, to the Fermi energy, E F, is the quantum coupling parameter, g Q,
$$g_{\rm Q} = \displaystyle{{E_{{\mathop{\rm int}}}} \over {E_{\rm F}}} = \displaystyle{2 \over {(3{\rm \pi} ^2 )^{{2 / 3}}}} \displaystyle{{e^2 m_{\rm e}} \over {\hbar ^2 {\rm \varepsilon} _0 n_{\rm e} ^{{1 / 3}}}} $$For g Q ≥ 1, the quantum plasma is collisional and for g Q < 1, it is collisionless and the mean-field effects are dominant (Manfredi, Reference Manfredi2005). Hence, for large plasma densities, the quantum plasma is collisionless. By considering the Pauli's exclusion principle, one could find, with increasing of plasma density, the average kinetic energy of the plasma also increases and causes to decrease the quantum coupling parameter (Manfredi, Reference Manfredi2005).
Quantum plasmas is strongly sound in many environments, that is, in astrophysical systems (Opher et al., Reference Opher, Silva, Dauger, Decyk and Dawson2001), biophotonics (Barnes et al., Reference Barnes, Dereux and Ebbesen2003), neutron stars (Chabrier et al., Reference Chabrier, Douchin and Potekhin2002), ultra-cold plasmas (Killian, Reference Killian2006), ultra-small electronic devices (Markowich et al. Reference Markowich, Ringhofer and Schmeiser1990), laser-produced plasmas (Kremp et al., Reference Kremp, Bornath, Bonitz and Schlanges1999; Andreev, Reference Andreev2000; Kremp et al., Reference Kremp, Schlanges and Kraft2005; Marklund, Reference Marklund2005; Becker et al., Reference Becker, Schoenbach and Eden2006), fast ignition (Hu & Keitel, Reference Hu and Keitel1999; Andreev, Reference Andreev2000; Azechi, Reference Azechi2006; Marklund & Shukla, Reference Marklund and Shukla2006; Shukla & Stenflo, Reference Shukla and Stenflo2006; Glenzer & Redmer, Reference Glenzer and Redmer2009), micro plasmas (Becker et al., Reference Becker, Koutsospyros, Yin, Christodoulatos, Abramzon, Joaquin and Brelles-Mariño2005), quantum well, and quantum diodes (Ang et al., Reference Ang, Koh, Lau and Kwan2006; Ang & Zhang, Reference Ang and Zhang2007). The distribution of electrons in quantum plasma is explained by the Wigner function (Wigner, Reference Wigner1932; Hillery et al., Reference Hillery, O'Connell, Scully and Wigner1984; Kozlov & Smolyanov, Reference Kozlov and Smolyanov2007). In recent years, instabilities in plasma, propagation of magneto-acoustic soliton and ion-acoustic solitary Fermi temperature have been studied in quantum plasma physics (Hussain & Mahmood, Reference Hussain and Mahmood2011; Ghosh et al., Reference Ghosh, Chandra and Paul2012; Chandra & Ghosh, Reference Chandra and Ghosh2012). Eliasson & Shukla (Reference Eliasson and Shukla2008) has presented the fluid equations of quantum plasma and the dielectric function of an unmagnetized collisionless quantum plasma has been also introduced (Ali & Shukla, Reference Ali and Shukla2006). In 1970, Mermin derived the dielectric permittivity for collisional quantum plasma (Mermin, Reference Mermin1970). Moreover, Latyshev derived the dielectric permittivity using a kinetic equation in the momentum space in the relaxation approximation (Latyshev & Yushkanov, Reference Latyshev and Yushkanov2014).
The nonlinear effects are present more effectively in quantum plasma than in the classical case (Shukla et al., Reference Shukla, Ali, Stenflo and Marklund2006; Shukla & Eliasson, Reference Shukla and Eliasson2010). In the quantum regime, the laser spot size oscillates with greater frequency and less amplitude while propagating deeper in the medium. Therefore, these effects will result in stronger self-focusing compared with the classical regime (Shukla et al., Reference Shukla, Ali, Stenflo and Marklund2006; Shukla & Eliasson, Reference Shukla and Eliasson2010; Marklund & Brodin, Reference Marklund and Brodin2007; Bulanov et al., Reference Bulanov, Esirkepov, Habs, Pegoraro and Tajima2009; Patil et al., Reference Patil, Takale, Navare, Dongare and Fulari2013). Despite the fact that the laser self-focusing in collisionless quantum plasma has been studied over the last decades (Manfredi, Reference Manfredi2005; Ali & Shukla, Reference Ali and Shukla2006; Shukla et al., Reference Shukla, Ali, Stenflo and Marklund2006, Shukla and Eliasson (Reference Shukla and Eliasson2010); Na & Jung, Reference Na and Jung2009; Habibi & Ghamari, Reference Habibi and Ghamari2014), the interaction of relativistic laser intensities with collisional quantum plasma has never been presented.
The current study is devoted to investigate the self-focusing of the relativistic Gaussian X-ray laser beam in collisional quantum plasma. Using the ansatz for the electric field in the wave equation, together with the Wentzel–Kramers–Brillouin (WKB) and the paraxial approximations, a mathematical formulation for the beam-width parameter in collisional quantum plasma is obtained. By considering the dielectric permittivity derived by Latyshev (Latyshev & Yushkanov, Reference Latyshev and Yushkanov2014), the evolution of the beam-width parameter is introduced along the propagation direction. It is noticed that in collisional plasma, the laser beam width initially oscillates along the propagation direction (focusing) and then defocuses due to divergence and energy absorption. Greater collision frequencies result in the higher energy absorption rate, and as a consequence, the laser spot size oscillates with higher amplitude and defocuses earlier. Furthermore, the effect of the collision frequency and the plasma density on the self-focusing conditions is thoroughly explained. It is noticed that in denser plasmas, the laser self-focusing occurs earlier with higher oscillation amplitude, smaller spot size, and more oscillations.
2. THEORY
The cylindrical coordinate system is used to study the propagation of a Gaussian laser beam along the z-axis. In this coordinate, the scalar wave equation is,
$$\displaystyle{{{\rm \partial} ^2 E} \over {{\rm \partial} \,z^2}} + \displaystyle{1 \over r}\displaystyle{{\rm \partial} \over {{\rm \partial} r}}\left( {r\displaystyle{{{\rm \partial} E} \over {{\rm \partial} r}}} \right) + \displaystyle{{{\rm \omega} ^2} \over {c^2}} {\rm \varepsilon} (r,z)E = 0 $$Akhmanov et al. (Reference Akhmanov, Sukhorukov and Khokhlov1968) and Sodha et al. (Reference Sodha, Ghatak and Tripathi1974) suggest the solution of Eq. (3), as the following ansatz:
$$E(r,z) = A(r,z)\,\exp \left( {i{\rm \omega} t - i\int_0^z {k(z)\,dz}} \right), $$
where 
$k = {{\sqrt {{\rm \varepsilon} _{0{\rm r}}} {\rm \omega}} / c}$, and ε0r is the real part of the linear dielectric constant. By considering the WKB approximation for slowly converging and diverging beams, and neglecting ∂2A/∂z 2, the following equation is obtained by substituting Eq. (4) into Eq. (3):
$$2ik(z)\displaystyle{{{\rm \partial} A(r,z)} \over {{\rm \partial} z}} = \displaystyle{1 \over r}\displaystyle{{\rm \partial} \over {{\rm \partial} r}}\left( {r\displaystyle{{{\rm \partial} A(r,z)} \over {{\rm \partial} r}}} \right) + \displaystyle{{{\rm \omega} ^2} \over {c^2}} \left( {{\rm \varepsilon} (r,z) - {\rm \varepsilon} _{0{\rm r}}} \right)\ A(r,z) $$The complex amplitude, A(r,z), is expressed below:
$$A(r,z) = A_0 (r,z)\,\exp \left( { - ik(z)S(r,z)} \right), $$S(r, z) is the eikonal function which is complex for collisional plasmas (Wang et al., Reference Wang, Yuan, Zhou, Li and Du2011),
$$S(r,z) = S_{\rm r} (r,z) + i\,S_{\rm i} (r,z), $$In this relation, S r and S i are real functions, where S i represents the decay of the laser intensity during propagation in an absorbing plasma. In the paraxial approximation, the dielectric constant of absorbing plasma can be written similar to the case of non-absorbing plasma (Wang et al., Reference Wang, Yuan, Zhou, Li and Du2011),
$${\rm \varepsilon} (r,z) = {\rm \varepsilon} _0 (z) + {\rm \varepsilon} _2 (r,z) $$For collisional plasma, the dielectric constant is a complex function. Therefore,
$${\rm \varepsilon} _0 (z) = {\rm \varepsilon} _{0{\rm r}} (z) + i{\rm \varepsilon} _{0i} (z)$$
$${\rm \varepsilon} _2 (r,z) = {\rm \varepsilon} _{2{\rm r}} (r,z) + i{\rm \varepsilon} _{2{\rm i}} (r,z)$$Substituting A(r,z) and S(r,z) from Eqs. (6) and (7) into Eq. (5), and separately equating the real and imaginary parts to zero, Eqs. (10a) and (10b) are resulted,
$$\eqalign{2\displaystyle{{{\rm \partial} S_{\rm r}} \over {{\rm \partial} z}} + \left( {\displaystyle{{{\rm \partial} S_{\rm r}} \over {{\rm \partial} r}}} \right)^2 - \left( {\displaystyle{{{\rm \partial} S_{\rm i}} \over {{\rm \partial} r}}} \right)^2 &= \displaystyle{1 \over {k^2 A_0}} \left( {\displaystyle{{{\rm \partial} ^2 A_0} \over {{\rm \partial} r^2}} + \displaystyle{1 \over r}\displaystyle{{{\rm \partial} A_0} \over {{\rm \partial} r}}} \right) + \displaystyle{2 \over {kA_0}} \displaystyle{{{\rm \partial} A_0} \over {{\rm \partial} r}}\displaystyle{{{\rm \partial} S_{\rm i}} \over {{\rm \partial} r}}\cr &\quad + \displaystyle{1 \over k}\left( {\displaystyle{{{\rm \partial} {}^2S_{\rm i}} \over {{\rm \partial} r^2}} + \displaystyle{1 \over r}\displaystyle{{{\rm \partial} S_{\rm i}} \over {{\rm \partial} r}}} \right) + \displaystyle{{{\rm \varepsilon} _{2{\rm r}}} \over {{\rm \varepsilon} _{0{\rm r}}}}}$$
$$\eqalign{& \displaystyle{{{\rm \partial} A_0^2} \over {{\rm \partial} z}} + \displaystyle{{{\rm \partial} A_0^2} \over {{\rm \partial} r}}\displaystyle{{{\rm \partial} S_{\rm r}} \over {{\rm \partial} r}} + A_0^2 \left( {\displaystyle{{{\rm \partial} {}^2S_{\rm r}} \over {{\rm \partial} r^2}} + \displaystyle{1 \over r}\displaystyle{{{\rm \partial} S_{\rm r}} \over {{\rm \partial} r}}} \right) + 2A_0^2 k\left( {\displaystyle{{{\rm \partial} S_{\rm i}} \over {{\rm \partial} z}} + \displaystyle{{{\rm \partial} S_{\rm i}} \over {{\rm \partial} r}}\displaystyle{{{\rm \partial} S_{\rm r}} \over {{\rm \partial} r}}} \right) \cr & \quad = A_0^2 k\displaystyle{{{\rm \varepsilon} _{0i} + {\rm \varepsilon} _{2{\rm i}}} \over {{\rm \varepsilon} _{0{\rm r}}}}}$$In the next step, S r and S i are defined (Sodha et al., Reference Sodha, Ghatak and Tripathi1974; Prakash, Reference Prakash2005),
$$S_{\rm r} (r,z) = {\rm \varphi} _{\rm r} (z) + \displaystyle{{r^2} \over {2f}}\displaystyle{{df} \over {dz}}$$
$$S_{\rm i} (r,z) = {\rm \varphi}_{\rm i} (z) + \displaystyle{{r^2} \over 2}{\rm \beta}_{\rm i} (z)$$where ϕr and ϕi represent the axial phase and attenuation functions, independently; r and f are the radial coordinate of the cylindrical system and the dimensionless beam-width parameter, respectively. Considering Eq. (8) and the paraxial approximation, ε2r and ε2i of Eq. (9b) are represented,
$${\rm \varepsilon} _{2{\rm r}} (r,z) = r^2 {\rm \theta} _{\rm r} (z)$$
$${\rm \varepsilon} _{2{\rm i}} (r,z) = r^2 {\rm \theta} _{\rm i} (z)$$Besides, for the laser beam with the Gaussian intensity distribution, the real amplitude function, A 0, is given as the following:
$$A_0 (r,z) = \displaystyle{{A_{00}} \over {\,f\,(z)}}\exp \,\left( {\displaystyle{{ - r^2} \over {2r_0^2 \ f^2 (z)}}} \right) $$A 00 and r 0 are the initial electric field amplitude and the initial laser beam width, respectively. Expanding A 0 in a power series of r2 and substituting Eqs. (11a), (12a) and (13) into Eq. (10a), and by separately equating the r independent and r 2 terms to zero, the two following relations are obtained:
$$\displaystyle{{d{\rm \phi} _{\rm r} (z)} \over {dz}} = \displaystyle{{{\rm \beta} _{\rm i} (z)} \over {k(z)}} - \displaystyle{1 \over {k^2 (z)\,r_0^2\ f^2 (z)}}$$
$$\displaystyle{1 \over {\,f\,(z)}}\displaystyle{{d^2 f\,(z)} \over {d^2 z}} = \left( {\displaystyle{1 \over {r_0^2 k(z)\ f^2 (z)}} - {\rm \beta} _i (z)} \right)^2 + \displaystyle{{{\rm \theta} _{\rm r} (z)} \over {{\rm \varepsilon} _{{\rm 0r}} (z)}}$$Similarly Eqs. (11b), (12b), and (13) are substituted into Eq. (10b), and these relations are resulted,
$$2\displaystyle{{d{\rm \phi} _{\rm i} (z)} \over {dz}} = \displaystyle{{{\rm \varepsilon} _{0{\rm i}} (z)} \over {{\rm \varepsilon} _{{\rm 0r}} (z)}},$$
$$\displaystyle{{d{\rm \beta} _{\rm i} (z)} \over {dz}} + \displaystyle{2 \over {\,f\,(z)}}\displaystyle{{df\,(z)} \over {dz}}{\rm \beta} _{\rm i} (z) = \displaystyle{{{\rm \theta} _{\rm i} (z)} \over {{\rm \varepsilon} _{0{\rm r}} (z)}}$$More to the point, in quantum plasma physics, by using the quantum kinetic Wigner–Vlasov–Boltzmann (WVB) equation and the Bhatnagar, Gross, and Krook (BGK) collision integral in the coordinate space, the plasma dielectric function can be derived (Latyshev & Yushkanov, Reference Latyshev and Yushkanov2014),
$${\rm \varepsilon} = 1 + \displaystyle{{\hbar \,e^2} \over {\,{\rm \pi} ^2 {\rm \gamma} \,m_{\rm e} {\rm \omega} \,q^2}} \int {R(k,q,{\rm \omega}, v)kq\,dk} $$
$$R(k,q,{\rm \omega}, v) = \displaystyle{{\,f_{k \,+\, {q / 2}} - f_{k - {q / 2}}} \over {E_{k \, + \, {q / 2}} - E_{k - {q / 2}}}} \left( {1 - \displaystyle{{\hbar ({\rm \omega}\, + \,iv)(1 - {\rm \alpha} (q,{\rm \omega}, v))} \over {E_{k \,+\, {q / 2}} - E_{k - {q / 2}} + \hbar \,({\rm \omega}\, +\, iv)}}} \right) $$
$$1 - {\rm \alpha} (q,{\rm \omega}, v) = \displaystyle{{{\rm \omega} \,(B(q,0) - B(q,{\rm \omega} + iv)} \over {\omega \,B(q,0) + ivB(q,{\rm \omega} + iv)}}$$
$$B(q,{\rm \omega} + iv) = \displaystyle{1 \over {4{\rm \pi} ^3}} \int {\displaystyle{{\,f_{k + {q / 2}} - f_{k - {q / 2}}} \over {E_{k + {q / 2}} - E_{k - {q / 2}} + \hbar \,({\rm \omega} + iv)}}} \,dk $$
$$E_k = \displaystyle{{\hbar^2} {k^2} \over {2m}},\quad f_k = (\exp ({{E_k} / {k_{\rm B} T}}) + 1)^{ - 1}$$
where f k and q represent the electron distribution and the wave vector, respectively. Also, ω and v are the laser frequency and the collision frequency of the plasma electrons, seperately. By considering Eqs. (4), (6), (7), and (13), the relativistic factor, 
${\rm \gamma} = \sqrt {1 + {{e^2 EE^ * } / {(m_{\rm e}^2 {\rm \omega} ^2 c^2}} )} $ , is expressed as a function of the radial coordinate, r, and the beam-width parameter, f,
$${\rm \gamma} = \left[ {1 + \displaystyle{{e^2} \over {m_0^2 {\rm \omega} ^2 c^2}} \displaystyle{{A_{00}^2} \over {\,f^2}} \exp \left( {\displaystyle{{ - r^2} \over {r_0^2\, f^2}}} \right)} \right]^{{1 / 2}} $$This equation can be expanded in a power series of r 2. It should be realized that by using Eqs. (16)–(18) and (19), the dielectric constant is calculated and divided into real and imaginary parts. Then all parts of the dielectric constant in Eqs. (8) and (9), that is, ε0r, ε2r, ε0i, and ε2i are obtained and according to Eq. (12), the θr and θi functions are found. It is better to express Eqs. (14) and (15) in terms of the succeeding dimensionless variables,
$${\rm \xi} = {z / {\left( {r_0^2 k} \right)}},\quad \beta _{\rm i} = {\beta / {\left( {r_0^2 k} \right)}},\quad {\rm \varphi} = k{\rm \varphi} _{\rm i} $$ξ represents the dimensionless distance that is concerned to the Rayleigh length, and c is the velocity of light in vacuum. By utilizing these dimensionless variables, Eqs. (14) and (15) take these forms,
$$f'' = f\left( {{\rm \beta} - \displaystyle{1 \over {\,f^2}}} \right)^2 + \left( {\displaystyle{{r_0 {\rm \omega}} \over c}} \right)^2 \displaystyle{{\,p_0} \over {2f^3}} {\rm \theta} _{\rm r} \left( {1 + \displaystyle{{\,p_0} \over {\,f^2}}} \right)^{{{ - 3} / 2}}$$
$${\rm \beta} ' = \displaystyle{{ - 2f'} \over f}{\rm \beta} + \left( {\displaystyle{{r_0 {\rm \omega}} \over c}} \right)^2 \displaystyle{{\,p_0} \over {2f^4}} {\rm \theta} _{\rm i} \left( {1 + \displaystyle{{\,p_0} \over {\,f^2}}} \right)^{{{ - 3} / 2}}$$
$${\rm \varphi} ' = \displaystyle{{{\rm \varepsilon} _{0i}} \over 2}\left( {\displaystyle{{r_0 {\rm \omega}} \over c}} \right)^2 \left( {1 + \displaystyle{{\,p_0} \over {\,f^2}}} \right)^{{{ - 1} / 2}}$$
where 
${{e^2 A_{00}^2} / {(m_0^2 {\rm \omega} ^2 c^2 )}} = p_0 $ is the dimensionless quantity proportional to the laser beam power. Assuming a Gaussian intensity distribution and the initial plane wave front, the boundary conditions are,
$${{\,f = 1,\quad df} / {d{\rm\xi} = 0,\quad {\rm \beta} = 0,\quad {\rm \varphi} = 0\,\,\,}}{\rm at}\quad {\rm \xi} = 0 $$Additionally, from Eqs. (4), (6), (7), (13) and (20), the irradiation intensity, I(r, ξ) is
$$\eqalign{I(r,{\rm \xi} ) & = EE^{\ast} = \displaystyle{{A_{00}^2} \over {\,f^2}} \exp \left[ { - \displaystyle{{r^2} \over {r_0^2}} \left( {\displaystyle{1 \over {\,f^2}} - {\rm \beta}} \right)} \right]\exp (2{\rm \varphi} ) \cr & = \displaystyle{{A_{00}^2} \over {\,f^2}} \exp \left( { - \displaystyle{{r^2} \over {r_0^2 \,F^2}}} \right)\exp (2{\rm \varphi} )}$$and subsequently it is deduced that the modified beam-width parameter should have the following form:
$$\displaystyle{1 \over {F^2}} = \displaystyle{1 \over {\,f^2}} - {\rm \beta} $$ In Eq. (23), 
${\rm exp}(2{\rm \varphi})$ is related to the energy attenuation. It should be realized when the nonlinear absorption in the plasma is considered, f and F are different parameters. The axial intensity is determined by f 
$\left( { \propto {1 / {\,f^2}}} \right)$ but the radial intensity depends on both f and the modified beam-width parameter, F. Therefore, in this manner F (not f) corresponds to the beam width, that is, r 0F. From now on, for simplicity, “modified beam-width parameter” is meant by “beam-width parameter” through the text.
3. RESULTS
By means of the fourth-order Runge–Kutta method, Eqs. (21) are numerically solved under the boundary conditions given by Eq. (22). Equation (23) is plotted with plasma and laser parameters as follows: ω = 1.778 × 1020 s−1, T = 1000 °K, r 0 = 20 μm, n e = 4 × 1022 cm−3, and p 0 = 1. The plasma temperature and density are chosen in a way that the conditions of collisional quantum plasma, that is, g Q ≥ 1 and 
${n_{\rm e} {\rm \lambda} _{\rm B}^3 \ge 1} $, are satisfied. As the laser beam in collisional plasma is focused, the spatial diffraction becomes stronger and it grows until becoming predominant. The laser spot size increases after a minimum value and converges again, showing an oscillatory behavior.
Figure 1 makes a comparison for the laser self-focusing effect between two distinct mediums, a collisional plasma with v = 0.5ωp (dashed curve) and a collisionless plasma (solid curve). It is noticed that initially the beam-width parameter shows similar oscillations to ξ for both cases. Due to energy absorption in the collisional plasma, the laser beam power reduces and when it becomes lower than the critical value of the self-focusing, the diffraction effects overcome the focusing. Accordingly, F initially oscillates with ξ (self-focusing) and then defocuses due to energy absorption. For the collisional plasma, however, the oscillation amplitude of the laser beam width enhances by passing through the plasma, and the laser beam defocuses at a few Rayleigh lengths.
(color online) Comparing the beam-width parameter for the collisionless plasma (solid) and the collisional plasma with v = 0.5ωp (dash) at T = 1000 K, n e = 4 × 1022 cm−3.
Figure 2 shows the dependency of the beam-width parameter on the propagation distance for various collision frequencies, v = 0.2ωp (solid), 0.5ωp (dash), and 0.9ωp (dash-dot). For greater collision frequencies, the laser absorption rate increases. Therefore, the oscillation amplitude of the beam width enhances and the laser beam defocusing occurs sooner.
(color online) Variation of the beam-width parameter with ξ for different collision frequencies, v = 0.2ωp (solid), v = 0.5ωp (dash), and v = 0.9ωp (dash-dot) with the temperature and density similar to the case of Fig. 1
Figure 3 compares the beam-width parameter for different plasma densities, n e = 4 × 1022 cm−3 (solid), n e = 4 × 1021 cm−3 (dash-dot), and n e = 1021 cm−3 (dash). In denser plasmas, the self-focusing length is reduced, and the laser spot size acquires a smaller minimum and higher oscillation amplitude. These results are valid for both collisional and collisionless quantum plasmas (Patil & Takale, Reference Patil and Takale2013). In other words, higher plasma densities induce earlier laser self-focusing. Also, by increasing the plasma density, energy absorption increases subsequently and the laser beam defocuses quicker.
(color online) Comparing the beam-width parameter for various plasma densities, n e = 4 × 1022 cm−3 (solid), n e = 4 × 1021 cm−3 (dash-dot), and n e = 1021 cm−3 (dash) at T = 1000 K, v = 0.5ωp.
4. CONCLUSION
The propagation of relativistic Gaussian X-ray laser beam is studied in collisional quantum plasma. In this scheme based on the obtained equations for the beam-width parameter, the laser power becomes lower than the critical value of the self-focusing effect. Therefore, the divergence effects overcome the self-focusing effects and the laser beam defocuses after a few oscillations in the plasma. On account of energy absorption in the plasma, it is found that in the propagation of the laser beam through collisional plasma, the oscillation amplitude of the laser beam width increases. In addition, it is noticed that for the greater collision frequencies, the laser energy absorption rate enhances and the laser spot size oscillates with higher amplitude and defocuses sooner. Furthermore, for higher densities in collisional plasmas, early and soon self-focusing with smaller spot size could be achieved.
