Basic formulation

The present model is set up in an unmagnetized and collisionless TQP considering the relativistic nonlinearity. Assuming wave propagation in the z − direction, the electric vector of laser beam E satisfies the following wave equation:

(1)$$\displaystyle{{\partial ^2E} \over {\partial z^2}} + \nabla _ \bot ^2 E + \displaystyle{{\omega ^2} \over {c^2}}\varepsilon E = 0$$

which may be derived directly from Maxwell's equations by neglecting the term $\nabla (\nabla \times E)$. The effective dielectric constant of homogeneous gaseous plasma can be expressed as:

(2)$${\rm \epsilon} = {\rm \epsilon} _0 + \Phi (EE^ {^\ast} )$$

where ${\rm \epsilon} _0 = 1 - (\omega _{\rm p}^2 /\omega ^2)$ and Φ are the linear and nonlinear parts of the dielectric constant respectively. $\omega _{\rm p}^2 = 4\pi n_0e^2/m_0$ is the plasma frequency and e and m ₀ are charge and rest mass of electron, respectively.

Starting by considering the dielectric constant in an unmagnetized and collisionless electron quantum plasmas including the Fermi gas pressure term as well as the Bohm potential effect caused by the collective interactions (Ali and Shukla, Reference Ali and Shukla2006; Na and Jung, Reference Na and Jung2009):

(3)$${\rm \epsilon} = 1 - \displaystyle{{\omega _{\rm p}^2} \over {\gamma \omega ^2}}\left( {1 - \displaystyle{\delta \over \gamma} - \beta} \right)^{ - 1}$$

where $\beta = k^2v_{{\rm Fe}}^2 /\omega ^2$, $v_{{\rm Fe}} = \surd (2k_{\rm B}T_{{\rm Fe}}/m_0)$ is the Fermi speed, $\delta = 4\pi ^4h^2/m_0^2 \omega ^2\lambda ^4$, $\gamma = \surd 1 + \alpha EE^ * $ is the relativistic factor with $\alpha = e^2/m_0^2 \omega ^2c^2$, and λ is the wavelength of the laser used. It may be mentioned that simple expressions can be found only in the limiting cases: T ≫ T _Fe (corresponding to the classical regime) and T ≪ T _Fe (corresponding to the fully degenerate quantum plasma case). A smooth transition cannot be achieved in a straightforward manner (Manassah et al., Reference Manassah, Baldeck and Alfano1988) using dimensional arguments. However, the thermal speed becomes meaningless in very low temperature limit and should be replaced by the Fermi velocity. The nonlinear part of the dielectric constant can be written as:

(4)$$\Phi (EE^ {^\ast} ) = \displaystyle{{\omega _{\rm p}^2} \over {\omega ^2}}\left[ {1 - \displaystyle{1 \over \gamma} {\left( {1 - \displaystyle{\delta \over \gamma} - \beta} \right)}^{ - 1}} \right]$$

The amplitude of the electric field of laser beam is given by:

$$E = \psi (r,z)\hbox{exp}[\iota (\omega t - kz)]$$

where ψ(r, z) is a complex function of its argument. Substituting the above expression for E into Eq. (1), one may neglect ∂²ψ/∂z ² and obtain the envelop equation governing the propagation of the beam in nonlinear media as shown below:

(5)$$\eqalign{\left[ { - 2\iota k\displaystyle{\partial \over {\partial z}} + \nabla_ \bot^2 + \displaystyle{{\omega^2} \over {c^2}}\left( {\displaystyle{{\omega_{\rm p}^2} \over {\omega^2}}\left( {1 - \displaystyle{1 \over \gamma} {\left( {1 - \displaystyle{\delta \over \gamma} - \beta} \right)}^{ - 1}} \right)} \right)} \right]\psi (r,z) = 0$$

where $\nabla _ \bot ^2 = \displaystyle{{\partial ^2} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{\partial \over {\partial r}}$.

Equation (5) is a nonlinear parabolic partial differential equation. The variational approach, which has rigorous basis, is employed here to investigate nonlinear wave propagation. The exact solution to Eq. (5) is not available and we therefore seek numerical or approximate analytical methods. Further, we choose the latter using a powerful variational method that has been used in several similar investigations (Anderson and Bonnedal, Reference Anderson1979; Gill et al., Reference Gill, Kaur and Mahajan2010a, Reference Gill, Mahajan and Kaur2010b, Reference Gill, Mahajan and Kaur2011; Kaur et al., Reference Kaur, Gill and Mahajan2010, Reference Kaur, Gill and Mahajan2011; Mahajan et al., Reference Mahajan, Gill and Kaur2010). Following the procedure of Anderson and Bonnedal (Reference Anderson1979), we reformulate Eq. (5) into a variational problem corresponding to a Lagrangian L to make (δL/δz) = 0. Solving Eq. (5) is equivalent to making a certain function an extremum. Lagrangian L corresponding to Eq. (5) is given by:

(6)$$\eqalign{L & = \iota k\left( {\psi \displaystyle{{\partial \psi^ {^\ast}} \over {\partial z}} - \psi^ {^\ast} \displaystyle{{\partial \psi} \over {\partial z}}} \right) - \left[ {\left \vert {\displaystyle{{\partial \psi} \over {\partial r}}} \right \vert} \right]^2 \cr & \quad\, + \displaystyle{{\omega _{\rm p}^2} \over {c^2}}\left[ { - (\delta + \beta )\alpha \vert \psi \vert^2 + 0.5(\alpha \vert \psi \vert^2)^2\left( {\delta - \displaystyle{1 \over 2}} \right)} \right. \cr & \quad\,\, \left. { + \displaystyle{\beta \over 4}{(\alpha \vert \psi \vert^2)}^2 - \displaystyle{1 \over {12}}\delta {(\alpha \vert \psi \vert^2)}^3} \right]} $$

Thus, the solution to the variational problem

(7)$$\delta \int_{ - \infty} ^\infty \int_{ - \infty} ^\infty Ldrdz = 0$$

also solves the nonlinear Schrödinger Eq. (5). We use the cosh-Gaussian field distribution ansatz for the amplitude ψ as trial function:

(8)$$\eqalign{\psi (r,z) & = \displaystyle{{\psi _0(z)} \over 2}\hbox{exp}\left( {\displaystyle{{b^2} \over 4} - 2k_iz} \right) \cr & \quad \times \left( {\hbox{exp}\left[ { - {\left( {\displaystyle{r \over {a(z)}} + \displaystyle{b \over 2}} \right)}^2} \right] + \hbox{exp}\left[ { - {\left( {\displaystyle{r \over {a(z)}} - \displaystyle{b \over 2}} \right)}^2} \right]} \right) \cr & \quad \times \hbox{exp}(\iota {q}^{\prime}(z)r^2 + \iota \phi (z))} $$

where a(z) is the beam width of the Gaussian amplitude distribution, ψ ₀ is the amplitude at the central position, k _i is the absorption coefficient, b the decentered parameter, also termed the normalized modal parameter, q′(z) is the spatial chirp, and ϕ(z) is the phase of laser beam. Using the expression for ψ as trial function into the Lagrangian L of Eq. (6) and after integrating L, we obtain:

(9)$$ \lt L \gt \; = \int_{ - \infty} ^\infty Ldr$$

The reduced variational problem is obtained by solving the integration in Eq. (9) using some standard integrals to get:

(10)$$\eqalign{\lt L \gt & = \displaystyle{{\iota ka\surd \pi } \over {8\surd 2}}e^{ - 4k_iz}(b^2 + 8)\left( {\psi _0\displaystyle{{\partial \psi _0^ * } \over {\partial z}} - \psi _0^ * \displaystyle{{\partial \psi _0} \over {\partial z}}} \right) \cr & \quad + \displaystyle{{k \vert \psi _0 \vert ^2} \over {32\surd 2}}\displaystyle{{d{q}^{\prime}} \over {dz}}a^3\surd \pi e^{ - 4k_iz}(b^2 + 8) \cr & \quad + \displaystyle{{k \vert \psi _0 \vert ^2} \over {4\surd 2}}\displaystyle{{d\phi } \over {dz}}a\surd \pi e^{ - 4k_iz}(b^2 + 8) \cr & \quad - \displaystyle{{ \vert \psi _0 \vert ^2} \over {4\surd 2}}e^{ - 4k_iz}\surd \pi \left[ {\displaystyle{2 \over a} + \displaystyle{{b^4} \over {2a}} + 2{{q}^{\prime}}^2a^3 + \displaystyle{{{{q}^{\prime}}^2b^2a^3} \over 4}} \right.\left. { - \displaystyle{{7b^2} \over {4a}}} \right] \cr & \quad - \displaystyle{{\omega _{\rm p}^2 } \over {c^2}}\displaystyle{{ \vert \alpha ^2\psi _0 \vert ^2} \over {8\surd 2}}(\delta + \beta )e^{ - 4k_iz}\surd \pi (b^2 + 8) \cr & \quad + \displaystyle{{\omega _{\rm p}^2 } \over {64c^2}}a(\alpha ^2 \vert \psi _0 \vert ^2)^2(\delta - 0.5)e^{ - {{b^2} \over 2} - 8k_iz}\surd \pi (b^2 + 8) \cr & \quad + \displaystyle{{\omega _{\rm p}^{\rm 2} } \over {64c^2}}a(\alpha ^2 \vert \psi _0 \vert ^2)^2\beta e^{ - {{b^2} \over 2} - 8k_iz}\surd \pi (b^2 + 8) \cr & \quad - \displaystyle{{\omega _{\rm p}^2 } \over {768c^2}}a\delta (\alpha ^2 \vert \psi _0 \vert ^2)^3e^{ - 12k_iz}\surd \pi \left( {\displaystyle{{10} \over {\surd 6}} + \displaystyle{{1.5b^2} \over {\surd 6}}} \right) \cr & \quad - \displaystyle{{\omega _{\rm p}^2 } \over {768c^2}}a\delta (\alpha ^2 \vert \psi _0 \vert ^2)^3e^{{{b^2} \over 2} - 12k_iz}\surd \pi \left( {\displaystyle{3 \over {\surd 6}} + 6 + \displaystyle{{3b^2} \over 4}} \right)} $$

We differentiate < L > with respect to the variables ψ ₀ and $\psi _0^ * $. Using the Euler–Lagrange's equations, we arrive at two equations. Further, we multiply these two equations with variables ψ ₀ and $\psi _0^ * $. On adding the resultant equations, we arrive at Eq. (11). The variation of < L >, i.e., (δ< L >/δS) = 0 where S denotes $\displaystyle{{\partial \psi _0} / {\partial z}},\displaystyle{{\partial \psi _0 ^{*} } / {\partial z}},a,{q}{\prime},\displaystyle{{d{q}{\prime}} / {dz}}$ etc., and following the procedure of Anderson and Bonnedal (Reference Anderson1979) and Saini and Gill (Reference Saini and Gill2006), we arrive at the following equations:

(11)$$\eqalign{& \displaystyle{{\iota ka} \over {4\surd 2}}\left( {\psi _0\displaystyle{{\partial \psi _0^ * } \over {\partial z}} - \psi _0^ * \displaystyle{{\partial \psi _0} \over {\partial z}}} \right) = - \displaystyle{{k \vert \psi _0 \vert ^2} \over {16\surd 2}}\displaystyle{{d{q}^{\prime}} \over {dz}}a^3 - \displaystyle{{k \vert \psi _0 \vert ^2} \over {2\surd 2}}\displaystyle{{d\phi } \over {dz}}a \cr & + \displaystyle{{ \vert \psi _0 \vert ^2} \over {2\surd 2}}\displaystyle{1 \over {(b^2 + 8)}}\left( {\displaystyle{2 \over a} + \displaystyle{{b^4} \over {2a}} + 2{{q}^{\prime}}^2a^3 + \displaystyle{{{{q}^{\prime}}^2b^2a^3} \over 4} - \displaystyle{{7b^2} \over {4a}}} \right) \cr & + \displaystyle{{\omega _p^2 } \over {c^2}}\displaystyle{{ \vert \alpha ^2\psi _0 \vert ^2} \over {4\surd 2}}(\delta + \beta )e^{ - 4k_iz} - \displaystyle{{\omega _{\rm p}^2 } \over {16c^2}}a(\alpha ^2 \vert \psi _0 \vert ^2)^2(\delta - 0.5)e^{ - {{b^2} \over 2} - 4k_iz} \cr & - \displaystyle{{\omega _{\rm p}^2 } \over {16c^2}}a(\alpha ^2 \vert \psi _0 \vert ^2)^2\beta e^{ -{{b^2} \over 2} - 4k_iz} \cr & + \displaystyle{{\omega _{\rm p}^2 } \over {128c^2}}a\delta (\alpha ^2 \vert \psi _0 \vert ^2)^3e^{ - 8k_iz}\left( {\displaystyle{{10} \over {\surd 6}} + {{1.5b^2} \over {\surd 6}}} \right) \cr & + \displaystyle{{\omega _p^2 } \over {128c^2}}a\delta (\alpha ^2 \vert \psi _0 \vert ^2)^3e^{{{b^2} \over 2} - 8k_iz}\left( {\displaystyle{3 \over {\surd 6}} + 0.75b^2 + 6} \right)} $$

and

(12)$$a^2 \vert \psi _0 \vert ^2 = a_0^2 A_0^2 = \hbox{constant}$$

(δ<L>/δa) = 0 gives,

(13)$$\! \eqalign{&\displaystyle{{\iota k} \over {8\surd 2}}\left( {\psi _0\displaystyle{{\partial \psi _0^ * } \over {\partial z}} - \psi _0^ * \displaystyle{{\partial \psi _0} \over {\partial z}}} \right) = - \displaystyle{3 \over {32}}\displaystyle{{k \vert \psi _0 \vert ^2} \over {\surd 2}}\displaystyle{{d{q}^{\prime}} \over {dz}}a^2 \cr & \! - \displaystyle{{k \vert \psi _0 \vert ^2} \over {2\surd 2}}\displaystyle{{d\phi } \over {dz}} + \displaystyle{{ \vert \psi _0 \vert ^2} \over {4\surd 2}}\displaystyle{1 \over {(b^2 + 8)}}\left( {\displaystyle{{ - 2} \over {a^2}} - \displaystyle{{b^4} \over {2a^2}} + 6{{q}^{\prime}}^2a^2 + \displaystyle{{3{{q}^{\prime}}^2b^2a^2} \over 4} - \displaystyle{{2b^2} \over {4a^2}}} \right) \cr & + \displaystyle{{\omega _{\rm p}^2 } \over {c^2}}\displaystyle{{ \vert \alpha ^2\psi _0 \vert ^2} \over {8\surd 2}}(\delta + \beta ) - \displaystyle{{\omega _{\rm p}^2 } \over {64c^2}}a(\alpha ^2 \vert \psi _0 \vert ^2)^2(\delta - 0.5)e^{ - {{b^2} \over 2} - 4k_iz} \cr & - \displaystyle{{\omega _{\rm p}^2 } \over {64c^2}}a(\alpha ^2 \vert \psi _0 \vert ^2)^2\beta e^{ - {{b^2} \over 2} - 4k_iz} \cr & + \displaystyle{{\omega _{\rm p}^2 } \over {768c^2}}a\delta (\alpha ^2 \vert \psi _0 \vert ^2)^3e^{ - 8k_iz}\left( {\displaystyle{{10} \over {\surd 6}} + 1.5b^2} \right) \cr & + \displaystyle{{\omega _{\rm p}^2 } \over {768c^2}}a\delta (\alpha ^2 \vert \psi _0 \vert ^2)^3e^{{{b^2} \over 2} - 8k_iz}\left( {\displaystyle{3 \over {\surd 6}} + 0.75b^2 + 6} \right)} $$

(δ<L>/δq′) = 0 gives;

(14)$${q}^{\prime} = - \displaystyle{k \over {4a}}\displaystyle{{da} \over {dz}}$$

(15)$$\displaystyle{{d{q}^{\prime}} \over {dz}} = - \displaystyle{k \over {4a}}\displaystyle{{d^2a} \over {dz^2}} + \displaystyle{k \over {4a^2}}\left( {\displaystyle{{da} \over {dz}}} \right)^2$$

Substituting Eq. (11) into Eq. (13) and using Eqs. (14) and (15), we arrive at the following equation for a:

(16)$$\eqalign{\displaystyle{{d^2a} \over {dz^2}} & = \displaystyle{{32} \over {k^2a^3(b^2 + 8)}}\left( {2 - \displaystyle{{7b^2} \over 4} + \displaystyle{{b^4} \over 2}} \right) \cr & \quad + \displaystyle{1 \over {6\surd 3}}\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2a}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz}} \over {(b^2 + 8)}}(1.5b^2 + 10) \cr & \quad + \displaystyle{{\surd 2} \over 6}\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2a}} (\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz + {{b^2} \over 2}}} \over {(b^2 + 8)}}\left( {0.75b^2 + 6 + \displaystyle{3 \over {\surd 6}}} \right) \cr & \quad - \surd 2\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2a}}(\alpha ^2 \vert \psi _0 \vert ^2)(\delta - 0.5)e^{ - 4k_iz + {{b^2} \over 2}} \cr & \quad - \surd 2 \displaystyle{{\omega _{\rm p}^2} \over {k^2c^2a}}(\alpha ^2 \vert \psi _0 \vert ^2)\beta e^{ - 4k_iz - {{b^2} \over 2}}} $$

Finally, the phase ϕ(z) of amplitude ψ ₀(z) is obtained by using $\psi _0(z) = \vert \psi _0 \vert e^{\iota \phi} $ and also using Eq. (16):

(17)$$\eqalign{\displaystyle{{d\phi} \over {dz}} & = \displaystyle{2 \over {ka^2(b^2 + 8)}}\left( {1 - \displaystyle{{7b^2} \over 8} + \displaystyle{{b^4} \over 4}} \right) \cr & \quad + \left( {\displaystyle{1 \over {384\surd 3}} + \displaystyle{1 \over {128}}} \right)\displaystyle{{\omega _{\rm p}^2} \over {kc^2}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\displaystyle{{e^{ - 8k_iz}} \over {(b^2 + 8)}}\,\,(1.5b^2 + 10) \cr & \quad + \displaystyle{{\omega _{\rm p}^2} \over {4kc^2}}(\delta + \beta )e^{ - 4k_iz} + \displaystyle{{\surd 2} \over {96}}\displaystyle{{\omega _{\rm p}^2} \over {kc^2}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz + {{b^2} \over 2}}} \over {(b^2 + 8)}} \cr & \quad \times \! \left( {0.75b^2 + 6 + \displaystyle{3 \over {\surd 6}}} \right) \cr & \quad - \displaystyle{{5\surd 2} \over {64}}\displaystyle{{\omega _{\rm p}^2} \over {kc^2}}(\alpha ^2 \vert \psi _0 \vert ^2)(\delta - 0.5)e^{ - 4k_iz + {{b^2} \over 2}} - \displaystyle{{5\surd 2} \over {64}}\displaystyle{{\omega _{\rm p}^2} \over {kc^2}}(\alpha ^2 \vert \psi _0 \vert ^2) \cr & \quad \times \beta e^{ - 4k_iz - {{b^2} \over 2}}} $$

After normalization using $\eta = cz/\omega a_0^2 $, we arrive at the following equations for a _n and ϕ:

(18)$$\eqalign{\displaystyle{{d^2a_n} \over {d\eta ^2}} & = \displaystyle{{32} \over {a_n^3 (b^2 + 8)}}\left( {2 - \displaystyle{{7b^2} \over 4} + \displaystyle{{b^4} \over 2}} \right) \cr & \quad + \displaystyle{1 \over {6\surd 3}}\displaystyle{{\omega _{\rm p}^2 a_0^2} \over {c^2a_n}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k_i^{\prime}\eta}} \over {(b^2 + 8)}}(1.5b^2 + 10) \cr & \quad + \displaystyle{{\surd 2} \over 6}\displaystyle{{\omega _{\rm p}^2 a_0^2} \over {c^2a_n}}\,(\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k_i^{\prime}\eta + \displaystyle{{b^2} \over 2}}} \over {(b^2 + 8)}}\left( {0.75b^2 + 6 + \displaystyle{3 \over {\surd 6}}} \right) \cr & \quad - \surd 2\displaystyle{{\omega _{\rm p}^2 a_0^2} \over {c^2a_n}}(\alpha ^2 \vert \psi _0 \vert ^2)(\delta - 0.5)e^{ - 4k_i^{\prime}\eta + \displaystyle{{b^2} \over 2}} \cr & \quad - \surd 2 \displaystyle{{\omega _{\rm p}^2 a_0^2} \over {c^2a_n}}(\alpha ^2 \vert \psi _0 \vert ^2)\beta e^{ - 4k_i^{\prime}\eta - \displaystyle{{b^2} \over 2}}} $$

(19)$$\openup6pt\eqalign{\displaystyle{{d\phi} \over {d\eta}} & = \displaystyle{2 \over {a_n^2 (b^2 + 8)}}\left( {1 - \displaystyle{{7b^2} \over 8} + \displaystyle{{b^4} \over 4}} \right) \cr &\quad + \left( {\displaystyle{1 \over {384\surd 3}} + \displaystyle{1 \over {128}}} \right)\displaystyle{{\omega _{\rm p}^2 a_0^2} \over {c^2}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\displaystyle{{e^{ - 8k{\rm ^{\prime}}_i\eta}} \over {(b^2 + 8)}}\,(1.5b^2 + 10) \cr &\quad + \displaystyle{{\omega _{\rm p}^2 a_0^2} \over {4c^2}}\delta e^{ - 4k{\rm ^{\prime}}_i\eta} + \displaystyle{{\omega _{\rm p}^2 a_0^2} \over {4c^2}}\beta e^{ - 4k{\rm ^{\prime}}_i\eta} \cr &\quad + \displaystyle{{\surd 2} \over {96}}\displaystyle{{\omega _{\rm p}^2 a_0^2} \over {c^2}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k{\rm ^{\prime}}_i\eta + {{b^2} \over 2}}} \over {(b^2 + 8)}}\,\left( {0.75b^2 + 6 + \displaystyle{3 \over {\surd 6}}} \right) \cr &\quad - \displaystyle{{5\surd 2} \over {64}}\displaystyle{{\omega _{\rm p}^2 a_0^2} \over {c^2}}(\alpha ^2 \vert \psi _0 \vert ^2)(\delta - 0.5)e^{ - 4k{\rm ^{\prime}}_iz + {{b^2} \over 2}} \cr &\quad - \displaystyle{{5\surd 2} \over {64}}\displaystyle{{\omega _{\rm p}^2 a_0^2} \over {c^2}}(\alpha ^2 \vert \psi _0 \vert ^2)\beta e^{ - 4k{\rm ^{\prime}}_i\eta - {{b^2} \over 2}}} $$

with ${k}^{\prime}_i = k_iR_d$, ${k}^{\prime}_i$ is the normalized absorption coefficient.

Self-trapping

For an initially plane wave front, a = a ₀ = a _e, da/dz = 0, and a = 1 at z = 0, the condition d ²a/dz ² = 0 leads to propagation of a cosh-Gaussian laser beam without any change in its beam width. This is known as the uniform waveguide propagation. If we put $\displaystyle{{d^2a} \over {dz^2}} = 0$ in Eq. (16), we obtain a relation between the dimensionless initial beam width parameter (ρ₀) and critical values of the intensity parameter Π (= α²|ψ ₀|²). The following general expression is obtained for determination of critical threshold for various values of b:

(20)$$\openup6pt\eqalign{&\displaystyle{{32} \over {k^2a^2(b^2 + 8)}}\left( {2 - \displaystyle{{7b^2} \over 4} + \displaystyle{{b^4} \over 2}} \right) \cr & = - \displaystyle{1 \over {6\surd 3}}\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz}} \over {(b^2 + 8)}}(1.5b^2 + 10) \cr & \quad - \displaystyle{{\surd 2} \over 6}\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz + {{b^2} \over 2}}} \over {(b^2 + 8)}}\left( {0.75b^2 + 6 + \displaystyle{3 \over {\surd 6}}} \right) \cr & \quad + \surd 2\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2}}(\alpha ^2 \vert \psi _0 \vert ^2)(\delta - 0.5)e^{ - 4k_iz + {{b^2} \over 2}} \cr & \quad + \surd 2\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2}}(\alpha ^2 \vert \psi _0 \vert ^2)\beta e^{ - 4k_iz - {{b^2} \over 2}}} $$

where ρ₀ = a _eω/c, is the initial dimensionless beam width parameter. The results are shown in the form of graphs. Such graphs are called the critical curves and these curves characterize the self-focusing region in the $\rho _0^2 - \Pi $ plane. The region below the critical curve corresponds to propagation of a cosh-Gaussian beam with self-focusing whereas the region above the critical curve corresponds to either oscillatory or steady state self-focusing of a cosh-Gaussian beam.

Numerical results and discussion

Equations (18) and (19) are nonlinearly coupled ordinary second order differential equations governing the normalized beam width parameter a _n and the phase ϕ as a function of distance of propagation η. These equations cannot be solved analytically. Therefore, we employ numerical computational techniques to study the dynamics of the beam. The first term on right hand side (R.H.S.) of Eq. (18) has its origin in the Laplacian $(\nabla _ \bot ^2 )$ appearing in the evolution Eq. (5) and it leads to the diffractional divergence of laser beam. The other terms in Eq. (18) arise due to the relativistic nonlinear effect when laser beam of high intensity is used. This effect arises due to the relativistic mass correction and it depends on various factors such as the intensity parameter (Π = α²|ψ ₀|²), relative plasma density, etc. The second, third, and the fourth terms have dependence on the cold quantum contribution δ as well as on the decentered parameter, b. However, the last term (fifth) is dominated by the TQP whose contribution comes from the term proportional to β which contains quantum effects via the Fermi temperature. A similar explanation can be given for the terms appearing in Eq. (19). The first term leads to the diffractional divergence of laser beam. The other terms in Eq. (19) appear because of the relativistic effects. The third term has dependence on the cold quantum contribution δ, the fifth and the sixth terms depend on δ as well as on the decentered parameter, b. The contribution of the fourth term comes from the term proportional to β which consists of the quantum effects. The last term is proportional to β, the decentered parameter, b and Π. The self-focusing/defocusing of laser beam is determined by the competing mechanisms on the R.H.S. of Eq. (18). The normalized beam width parameter, a _n < 1 corresponds to the self-focusing and a _n > 1 is the result of diffractional dominance over all the other terms leading to the defocusing of laser beam. If we set δ and β to zero, i.e., ignoring quantum effects, one obtains a differential equation for the normalized beam width parameter in the CR plasma. In Figure 1, the plot of normalized beam width parameter, a _n as a function of the dimensionless distance of propagation, η is displayed for b = 0 at different values of absorption levels ${k}^{\prime}_i$ considering the relativistic warm quantum plasma. In absence of the decentered parameter, a large value of ${k}^{\prime}_i$ is observed to weak the self-focusing effect. An analysis of evolution of the normalized beam width parameter as a function of distance of propagation, η is performed at b = 0, 1, 2 for three values of $k_i^{\prime}$ with other parameters chosen as follows: a ₀ = 0.002 cm, k = 0.53 × 10⁴ cm⁻¹, ω_p = 2 × 10⁻⁶ × ω, ω = 1.778 × 10²⁰(rad/sec), T _Fe = 10⁹ K, and λ = 0.0106 nm. Figure 2 plots a _n as a function of η for b = 1 with different values of absorption levels ${k}^{\prime}_i$. The beam propagates oscillatory over several number of Rayleigh lengths. However, the situation has changed significantly in Figure 3, where we have plotted a _n as a function of η for b = 2 with varying ${k}^{\prime}_i$. All curves are seen to exhibit sharp self-focusing effects.

Fig. 1. Variation of normalized beam width parameter a _n(η) as a function of dimensionless distance of propagation η considering relativistic nonlinearity for b = 0 with the following set of parameters for the various values of ${k}^{\prime}_i$: a ₀ = 0.002 cm, ω_p = 2 × 10⁻⁶ × ω, ω = 1.778 × 10²⁰(rad/sec), k = 0.53 × 10⁴ cm⁻¹, and intensity parameter, Π = 0.1. Solid curve corresponds to $k_i^{\prime} = 0.01$, dashed to ${k}^{\prime}_i = 0.02$, and dotted to ${k}^{\prime}_i = 0.03$.

Fig. 2. Plot of a _n(η) as a function of dimensionless distance of propagation η for b = 1 for the same set of parameters as in the caption of Figure 1.

Fig. 3. Variation of normalized beam width a _n(η) as a function of dimensionless distance of propagation η for b = 2 for the same set of parameters as in the caption of Figure 1.

The variation of the normalized beam width parameter, a _n as a function of the dimensionless distance of propagation, η with various intensity parameters (Π = α²|ψ ₀|²), keeping b and ${k}^{\prime}_i$ fixed is displayed in Figure 4. It is observed from Figure 4 that the self-focusing takes place at lower values of Π in comparison with the earlier investigations (Gill et al., Reference Gill, Mahajan and Kaur2011). However, there is a substantial increase in the self-focusing with increase in Π. Figure 5 highlights the variation of the beam width parameter with η for the three cases of plasma. If we consider all the terms in Eq. (18), we obtain a plot between a _n and η for the case of relativistic TQP (dotted curve). If we put β → 0, a curve for the RCQ plasma is obtained (dashed curve). Further if β → 0 and δ → 0, i.e., ignoring the quantum effects, one obtains a curve for the CR plasma (solid curve). The results show much higher oscillations and better self-focusing in the TQP than the RCQ and the CR cases. Another aspect of this phenomenon observed here is that there is a decrease in the focusing length for the TQP in comparison with the earlier RCQ and the CR cases. The stronger pinching effect offered by the quantum effects enhances laser self-focusing. The largest self-focusing length is observed in the CR regime. Further, Eq. (19) describes the evolution of the longitudinal phase, ϕ(η, α²|ψ ₀|²) with the dimensionless distance of propagation (η) in Figure 6. The R.H.S. of Eq. (19) is a complicated function of b, ${k}^{\prime}_i$, the normalized beam width a _n and the intensity parameter, α²|ψ ₀|². Though we can fix b, ${k}^{\prime}_i$, and α²|ψ ₀|², the evolution of the normalized beam width parameter, a _n significantly affects the longitudinal phase with η. The longitudinal phase may be positive or negative depending upon the value of α²|ψ ₀|² and ${k}^{\prime}_i$. It is positive for the present model and does not show an oscillatory character as the R.H.S. of Eq. (19) for b = 0 depends only on α²|ψ ₀|² and ${k}^{\prime}_i$ maintaining a nonlinear relationship. The regularized phase, ϕ _reg, is defined as:

(21)$$\phi _{{\rm reg}} = \phi (\eta ) - \phi (\eta ) \vert _{\alpha ^2 \vert \psi _0 \vert ^2 = 0}.$$

Fig. 4. Graph of a _n(η) as a function of dimensionless distance of propagation η for b = 1 and ${k}^{\prime}_i = 0.01$ at varying values of intensity parameter for the same set of parameters as in the caption of Figure 1. Solid curve corresponds to Π = 0.09, dashed to Π = 0.1, and dotted to Π = 0.3.

Fig. 5. Variation of a _n(η) as a function of dimensionless distance of propagation η for b = 1, ${k}^{\prime}_i = 0.01$, and Π = 0.1 for different plasma regimes using same set of parameters as in the caption of Figure 1. Solid curve corresponds to CR region, dashed to RCQ, and dotted to TQP.

Fig. 6. Plot of longitudinal phase ϕ(η) versus dimensionless distance of propagation η for b = 0 for the same set of parameters as mentioned in the caption of Figure 1. Solid curve corresponds to ${k}^{\prime}_i = 0.01$, dashed to ${k}^{\prime}_i = 0.02$, and dotted to ${k}^{\prime}_i = 0.03$.

In the linear limit (α²|ψ ₀|² = 0), the phase will have a more rapid growth during the propagation. But the presence of nonlinear forces counteracts the diffraction and tends to keep the pulse intensity higher in the plasma medium. However, the regularized phase always has a negative value. This is a consequence of frequency chirp as the time derivative of the phase will have same sign as in the case of a conventional plane wave. Thus, the red will always lead blue in the super continuum. This aspect of the phase change is well presented using the variational approach in the present investigation. Figure 7 displays the plot of the regularized phase versus η. This feature confirms the finding of Karlsson et al. (Reference Karlsson, Anderson and Desaix1992) and is contrary to the results of Manassah et al. (Reference Manassah, Baldeck and Alfano1988). The latter predicted that the regularized phase may be positive or negative during the beam propagation. This inconsistent result stems from the fact that the PRA used by Manasaah et al., does not correctly predict the ϕ _reg.

Fig. 7. Plot of regularized phase ϕ _reg versus dimensionless distance of propagation η for b = 1 for the same set of parameters as mentioned in the caption of Figure 1. Solid curve corresponds to ${k}^{\prime}_i = 0.01$, dashed to ${k}^{\prime}_i = 0.02$, and dotted to ${k}^{\prime}_i = 0.03$.

To further elucidate the results for delineating the underlying physics for the propagation of a cosh-Gaussian laser beam in the TQP, we numerically analyze the dimensionless initial beam width parameter (ρ₀) as a function of critical values of beam power Π (= α²|ψ ₀|²) for different values of b when the relativistic nonlinearity is considered. The results are depicted in the form of graphs. The critical curves for a cosh-Gaussian laser beam characterize the self-focusing region in the $\rho _0^2 - \Pi $ space. In Figure 8, $\rho _0^2 $ versus Π is plotted for two values of b, i.e., b = 0 and b = 1. The solid curve corresponds to b = 0 and the dashed curve to b = 1. The solid curve depicts that the dependence of $\rho _0^2 $ for b = 0 is much weaker on the high intensity, a result consistent with earlier calculations based on the variational approach (Anderson, Reference Anderson and Bonnedal1978). The initial beam width is much higher than the earlier investigations. Higher the value of b, the faster is the initial decrease in $\rho _0^2 $ with Π.

Fig. 8. Dependence of dimensionless initial beam width (ρ₀) as a function of Π with b = 0 and b = 1. Solid curve corresponds to b = 0 and dashed curve to b = 1.

Lastly, Figure 9 depicts the variation of a _n with η in the relativistic TQP for three different values of the Fermi temperature. It is observed that the self-focusing length decreases with an increase in the Fermi electron temperature. A propagation to several Rayleigh lengths is observed. Also, it occurs at much lower value of the intensity parameter in comparison with the earlier investigations. This is apparently a consequence of the variational approach where the contribution of the whole wave front is considered in the averaging process. On the other hand, the PRA takes into account only those rays which are very close to the beam-axis.

Fig. 9. Comparison of a _n(η) with dimensionless distance of propagation η at different values of electron temperature (T _Fe). Other parameters are same as in the caption of Figure 1. Solid curve corresponds to T _Fe = 10⁷ K, dashed to T _Fe = 10⁸ K, and dotted to T _Fe = 10⁹ K.

Stability criterion of beam dynamics

Variational method is used in several branches of physics and mathematics, can also be applied to study the stability characteristics of the evolution of cosh-Gaussian laser beam in TQP when relativistic effects are taken into account. The methods of nonlinear dynamics applied to dissipative solitons (Skarka et al., Reference Skarka, Berezhiani and Miklaszewski1997, Reference Skarka, Berezhiani and Miklaszewski1999; Skarka and Aleksic, Reference Skarka and Aleksic2006) can also be used to study stability properties in the present investigation. The Euler–Lagrange equations are the starting point to establish stability criterion. The dependent variables are disturbed about their equilibrium values and method of Lyapunov's exponents (Lakshman and Rajasekar, Reference Lakshman and Rajasekar2003; Skarka and Aleksic, Reference Skarka and Aleksic2006) is used. Thus, for stability characteristics of the system, the following Jacobi determinant is constructed from derivatives with respect to amplitude, width and curvature in terms of S, F, and G where

(22)$$S = \displaystyle{{d\psi _0} \over {dz}} = \displaystyle{{ - 4{q}^{\prime} \vert \psi _0 \vert} \over k}$$

(23)$$F = \displaystyle{{da} \over {dz}} = \displaystyle{{ - 4a{q}^{\prime}} \over k}$$

(24)$$\eqalign{G & = \displaystyle{{d{q}^{\prime}} \over {dz}} = \displaystyle{{4{{q}^{\prime}}^2} \over k} - \displaystyle{8 \over {ka^4(b^2 + 8)}}\left( {2 - \displaystyle{{7b^2} \over 4} + \displaystyle{{b^4} \over 2}} \right) \cr &\quad - \displaystyle{1 \over {24\surd 3}}\displaystyle{{\omega _{\rm p}^2} \over {kc^2a^2}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz}} \over {(b^2 + 8)}}(1.5b^2 + 10) \cr &\quad - \displaystyle{{\surd 2} \over {24}}\displaystyle{{\omega _{\rm p}^2} \over {kc^2a^2}}(\alpha ^2 \vert \psi _0 \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz + {{b^2} \over 2}}} \over {(b^2 + 8)}}\left( {0.75b^2 + 6 + \displaystyle{3 \over {\surd 6}}} \right) \cr &\quad + \displaystyle{{\surd 2} \over 4}\displaystyle{{\omega _{\rm p}^2} \over {kc^2a^2}}(\alpha ^2 \vert \psi _0 \vert ^2)e^{ - 4k_iz + {{b^2} \over 2}}(\delta - 0.5) \cr &\quad + \displaystyle{{\surd 2} \over 4}\displaystyle{{\omega _{\rm p}^2 \beta} \over {kc^2a^2}}(\alpha ^2 \vert \psi _0 \vert ^2)e^{ - 4k_iz - {{b^2} \over 2}}} $$

(25)$$\openup6pt\hbox{det} \vert J - \lambda I \vert = \left \vert {\matrix{ {\displaystyle{{\partial S} \over {\partial \psi_0}} - \lambda} & {\displaystyle{{\partial S} \over {\partial a}}} & {\displaystyle{{\partial S} \over {\partial {q}^{\prime}}}} \cr {\displaystyle{{\partial F} \over {\partial \psi_0}}} & {\displaystyle{{\partial F} \over {\partial a}} - \lambda} & {\displaystyle{{\partial F} \over {\partial {q}^{\prime}}}} \cr {\displaystyle{{\partial G} \over {\partial \psi_0}}} & {\displaystyle{{\partial G} \over {\partial a}}} & {\displaystyle{{\partial G} \over {\partial {q}^{\prime}}} - \lambda} \cr}} \right \vert = 0$$

This leads to the following characteristic equation cubic in λ:

(26)$$\lambda ^3 + \alpha _1\lambda ^2 + \alpha _2\lambda + \alpha _3 = 0$$

where

(27)$$\alpha _1 = - \displaystyle{{4{q}^{\prime}} \over k}$$

(28)$$\eqalign{\alpha _2 & = \displaystyle{{ - 32{{q}^{\prime}}^2} \over {k^2}} + \displaystyle{{128} \over {k^2a^4(b^2 + 8)}}\left( {2 - \displaystyle{{7b^2} \over 4} + \displaystyle{{b^4} \over 2}} \right) \cr &\quad + \displaystyle{1 \over {\surd 3}}\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2a^6}}(\alpha ^2A_0^2 \vert \psi _{00} \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz}} \over {(b^2 + 8)}}(1.5b^2 + 10) \cr &\quad + \surd 2\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2a^6}}(\alpha ^2A_0^2 \vert \psi _{00} \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz + {{b^2} \over 2}}} \over {(b^2 + 8)}}\left( {0.75b^2 + 6 + \displaystyle{3 \over {\surd 6}}} \right) \cr &\quad - 4\surd 2\displaystyle{{\omega _{\rm p}^2} \over {k^2c^2a^4}}(\alpha ^2A_0^2 \vert \psi _{00} \vert ^2)e^{ - 4k_iz + {{b^2} \over 2}}(\delta - 0.5) \cr &\quad - 4\surd 2\displaystyle{{\omega _{\rm p}^2 \beta} \over {k^2c^2a^4}}(\alpha ^2A_0^2 \vert \psi _{00} \vert ^2)e^{ - 4k_iz -{{b^2} \over 2}}} $$

(29)$$\eqalign{\alpha _3 & = \displaystyle{{ - 128{{q}^{\prime}}^3} \over {k^3}} + \displaystyle{{512{q}^{\prime}} \over {k^3a^4(b^2 + 8)}}\left( {2 - \displaystyle{{7b^2} \over 4} + \displaystyle{{b^4} \over 2}} \right) \cr &\quad + \displaystyle{4 \over {\surd 3}}\displaystyle{{\omega _{\rm p}^2 {q}^{\prime}} \over {k^3c^2a^6}}(\alpha ^2A_0^2 \vert \psi _{00} \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz}} \over {(b^2 + 8)}}(1.5b^2 + 10) \cr &\quad + 4\surd 2\displaystyle{{\omega _{\rm p}^2 {q}^{\prime}} \over {k^3c^2a^6}}(\alpha ^2A_0^2 \vert \psi _{00} \vert ^2)^2\delta \displaystyle{{e^{ - 8k_iz + {{b^2} \over 2}}} \over {(b^2 + 8)}}\left( {0.75b^2 + 6 + \displaystyle{3 \over {\surd 6}}} \right) \cr &\quad - 16\surd 2{q}^{\prime}\displaystyle{{\omega _{\rm p}^2} \over {k^3c^2a^4}}(\alpha ^2A_0^2 \vert \psi _{00} \vert ^2)e^{ - 8k_iz + {{b^2} \over 2}}(\delta - 0.5) \cr &\quad - 16\surd 2{q}^{\prime}\displaystyle{{\omega _{\rm p}^2 \beta} \over {k^2c^2a^4}}(\alpha ^2A_0^2 \vert \psi _{00} \vert ^2)e^{ - 4k_iz - {{b^2} \over 2}}} $$

In order to have Lyapunov's stability, Hurwitz conditions must be fulfilled, i.e., α₁α₂ − α₃ must be positive. According to the Routh–Hurwitz criterion, a necessary and sufficient condition for the stationary solutions to be stable is:

(30)$$\alpha _1\alpha _2 - \alpha _3 \gt 0$$

Equation (26) has a pair of purely imaginary roots at a critical point (Lugiato and Narducci, Reference Lugiato and Narducci1985):

(31)$$\lambda = \pm \iota v,\quad v \gt 0$$

Substituting Eq. (31) into Eq. (26), we get:

(32)$$v^2 - \alpha _2 = 0$$

and

(33)$$\alpha _1v^2 - \alpha _3 = 0$$

The critical condition of the Hopf-bifurcation is:

(34)$$f = \alpha _1\alpha _2 - \alpha _3 = 0$$

f > 0 is a necessary condition for the stationary solution to be stable, f < 0 is a necessary condition for the Hopf-bifurcation to emerge. It is observed that the condition f ≥ 0 is satisfied for chosen set of parameters in the present analysis and therefore Hopf-bifurcation, resulting from the unstable fixed point does not come into play and it leads to overall stability of the beam dynamics (Wang, Reference Wang1990).