Self-focusing

We begin by considering the propagation of finite AiG laser beams along the z-direction. The initial electric field distribution of the beams is expressed as follows (Ouahid et al., Reference Ouahid, Dalil-Essakali and Belafhal2018a):

(1)$$\eqalign{&E\lpar {x\comma \;y\comma \;0} \rpar = \cr& E_0{\rm Ai}\left({\displaystyle{x \over {r_0}}} \right){\rm exp}\left({\displaystyle{{a_{0}x} \over {r_0}}} \right){\rm exp}\left({\displaystyle{{-x^2} \over {r_0^2 }}} \right){\rm Ai}\left({\displaystyle{y \over {r_0}}} \right){\rm exp}\left({\displaystyle{{a_{0}y} \over {r_0}}} \right){\rm exp}\left({\displaystyle{{-y^2} \over {r_0^2 }}} \right)}$$

where E ₀ is the constant amplitude of electric field, Ai( ⋅ ) is the Airy function of the first kind, r ₀ is the initial beam-width, and a₀ is the modulation parameter also called the aperture coefficient.

The propagation of finite AiG laser beam in plasma is characterized by the dielectric function which can, in general, be expressed as follows (Sodha et al., Reference Sodha, Ghatak and Tripathi1976):

(2)$${\rm \varepsilon } = {\rm \varepsilon }_0 + {\rm \Phi }\lpar {{\rm E}{\rm E}^\ast } \rpar -i{\rm \varepsilon }_i$$

where ${\rm \varepsilon }_0 = 1-{\rm \omega }_{\rm p}^2 /{\rm \omega }^2$ is the linear part of the dielectric function. In the relativistic regime, the usual expression for nonlinear part ${\rm \Phi }\lpar {{\rm E}{\rm E}^{\rm \ast }} \rpar$ of the dielectric function for the plasma is written as follows (Sharma and Kourakis, Reference Sharma and Kourakis2010):

(3)$${\rm \Phi }\lpar {{\rm E}{\rm E}^\ast } \rpar = \displaystyle{{{\rm \omega }_{\rm p}^2 } \over {{\rm \gamma }{\rm \omega }^2}}\lpar {{\rm \gamma }-1} \rpar $$

where ω is the angular frequency of laser beam, ω_p is the plasma frequency given by ω_p = (4πne ²/m)^1/2, n ₀ is the density of plasma electrons in the absence of the beam, γ is the relativistic factor expressed as γ = (1 + αEE*)^1/2. Here, α = e ²/m ²ω²c ² with e is the charge on electron and m is the rest mass of electron. We limit ourselves to the case when ɛ_i is field independent and ɛ_i ≪ ɛ₀.

The wave equation governing the electric vector of the beam in plasma with the dielectric function given by Eq. (2) can be written as follows:

(4)$$\nabla ^2E-\displaystyle{{\rm \varepsilon } \over {c^2}}\displaystyle{{\partial ^2E} \over {\partial t^2}} = 0$$

In writing Eq. (4), the term $\nabla \lpar {\nabla \cdot E} \rpar$ has been neglected which has been justified when $\lpar {c^2/{\rm \omega }^2} \rpar \vert {\nabla \lpar {\ln {\rm \varepsilon }} \rpar /{\rm \varepsilon }} \vert \ll 1$. For the convenience of field distribution given by Eq. (1), we have adopted the Cartesian coordinate system. Within the framework of WKB approximation, the solution of Eq. (4) can be written as follows:

(5)$$E = A\lpar {x\comma \;y\comma \;z} \rpar {\rm exp}\lsqb {i\lpar {{\rm \omega }t-kz} \rpar } \rsqb $$

Substituting for E and ɛ from Eqs. (5) and (2) in Eq. (4), one obtains

(6)$$2ik\displaystyle{{\partial A} \over {\partial z}} = \displaystyle{{\partial ^2A} \over {\partial x^2}} + \displaystyle{{\partial ^2A} \over {\partial y^2}} + \displaystyle{{k^2{\rm \Phi }\lpar {{\rm A}{\rm A}^\ast } \rpar } \over {\rm \varepsilon }}A$$

Equation (6) is known as the parabolic wave equation that describes the evolution of beam envelope in plasma.

We may express A as

(7)$$A = A_0\lpar {x\comma \;y\comma \;z} \rpar {\rm exp}\lsqb {-ikS\lpar {x\comma \;y\comma \;z} \rpar } \rsqb $$

where A ₀ and S are real functions of x, y, and z. Here, S is the eikonal of the beam which determines convergence or divergence of the beam. Equation (7) is valid when the polarization of the beam does not change with propagation. Substituting for A from Eq. (7) in Eq. (6) and separating real and imaginary parts, we get

(8a)$$2\displaystyle{{{\rm \;}\partial S} \over {\partial z}} + \left({\displaystyle{{\partial S} \over {\partial x}}} \right)^2 + \left({\displaystyle{{\partial S} \over {\partial y}}} \right)^2 = \displaystyle{1 \over {k^2A_0}}\left({\displaystyle{{\partial^2A_0} \over {\partial x^2}} + \displaystyle{{\partial^2A_0} \over {\partial y^2}}} \right) + \displaystyle{{\Phi \lpar {{\rm A}_0{\rm A}_0^{\rm \ast } } \rpar } \over {{\rm \varepsilon }_0}}$$

and

(8b)$$\eqalign{\displaystyle{{\partial A_0^2 } \over {\partial z}} + \left({\displaystyle{{\partial S} \over {\partial x}}} \right)\left({\displaystyle{{\partial A_0^2 } \over {\partial x}}} \right) + \left({\displaystyle{{\partial S} \over {\partial y}}} \right)\left({\displaystyle{{\partial A_0^2 } \over {\partial y}}} \right) \cr + \left({\displaystyle{{\partial^2{\rm S}} \over {\partial x^2}} + \displaystyle{{\partial^2{\rm S}} \over {\partial y^2}}} \right)A_0^2 -\left({\displaystyle{{k{\rm \varepsilon }_i} \over {{\rm \varepsilon }_0}}} \right)A_0^2 = 0}$$

Following the approach given by Akhmanov et al. (Reference Akhmanov, Sukhorov and Khokhlov1968) and Sodha et al. (Reference Sodha, Ghatak and Tripathi1976), the solutions corresponding to Eq. (8) can be written as follows:

(9a)$$S = {\rm \beta }_1\lpar z \rpar \displaystyle{{x^2} \over 2} + {\rm \beta }_2\lpar z \rpar \displaystyle{{y^2} \over 2} + {\rm \varphi }\lpar z \rpar $$

and

(9b)$$\eqalign{A_0^2 = &\displaystyle{{E_0^2 } \over {\,f_1f_2}}\left({{\rm Ai}\left({\displaystyle{x \over {r_0f_1}}} \right){\rm Ai}\left({\displaystyle{y \over {r_0f_2}}} \right)} \right)^2 \cr& {\rm exp}\left({\displaystyle{{2a_{0}x} \over {r_0f_1}} + \displaystyle{{2a_{0}y} \over {r_0f_2}}-\displaystyle{{2x^2} \over {r_0^2 f_1^2 }}-\displaystyle{{2y^2} \over {r_0^2 f_2^2 }}-k_iz} \right)}$$

where β₁(z) = (1/f ₁(z))(∂f ₁(z)/∂z) and β₂(z) = (1/f ₂(z))(∂f ₂(z)/∂z) are the inverse of radius of curvatures of the beam along the x and y directions, respectively, φ(z) is the axial phase, f ₁(z) and f ₂(z) are the dimensionless beam-width parameters along the x and y directions, respectively, and k _i is the absorption coefficient.

Substituting for S and $A_0^2$ from Eq. (9) in Eq. (8a) and using the paraxial approximation, we find

(10a)$$\displaystyle{{d^2f_1\lpar z \rpar } \over {d{\rm \xi }^2}} = \displaystyle{A \over {\,f_1{\lpar z \rpar }^3}} + \displaystyle{{BCe^{{-}2{\rm \xi }k_i^{\prime} }\;{\left({1 + \displaystyle{{{\rm \alpha }E_0^2 g_2^4 e^{-{\rm \xi }k_i^{\prime} }} \over {\,f_1\lpar z \rpar f_2\lpar z \rpar }}} \right)}^{{-}3/2}} \over {\,f_1{\lpar z \rpar }^3f_2{\lpar z \rpar }^2}}$$

(10b)$$\displaystyle{{d^2f_2\lpar z \rpar } \over {d{\rm \xi }^2}} = \displaystyle{A \over {\,f_2{\lpar z \rpar }^3}} + \displaystyle{{BCe^{{-}2{\rm \xi }k_{i}^{^{\prime}} }{\left({1 + \displaystyle{{{\rm \alpha }E_0^2 g_2^4 e^{-{\rm \xi }k_i^{\prime} }} \over {\,f_1\lpar z \rpar f_2\lpar z \rpar }}} \right)}^{{-}3/2}} \over {\,f_2{\lpar z \rpar }^3f_1{\lpar z \rpar }^2}}$$

with

$$\;A = \left({4 + a_0-\displaystyle{{2a_0g_1^3 } \over {g_2^3 }} + \displaystyle{{4g_1^2 } \over {g_2^2 }}} \right)\;$$

$$B = \lpar {g_1^2 g_2^2 -4a_0g_1g_2^3 -2g_2^4 + 2a_0^2 g_2^4 } \rpar \;$$

$$C = \displaystyle{{{\rm \alpha }E_0^2 g_2^4 r_0^2 {\rm \omega }_{\rm p}^2 } \over {2c^2}}$$

$$g_1 = \displaystyle{1 \over {3^{1/3}{\rm \Gamma }\lpar {1/3} \rpar }}$$

and

$$g_2 = \displaystyle{1 \over {3^{2/3}{\rm \Gamma }\lpar {2/3} \rpar }}$$

where ξ = z/R _d the dimensionless distance of propagation, $R_d = kr_0^2$ is the Rayleigh diffraction length, and $k_i^{\prime} = k_iR_d$ is the normalized absorption coefficient.

Equations (10a) and (10b) are the nonlinear coupled differential equations governing the variation of the beam-width parameters f ₁ and f ₂ with the distance of propagation ξ. The first term on the right-hand side corresponds to the diffraction divergence of the beam and the second term corresponds to convergence due to the nonlinearity. We now devote our efforts for obtaining and analyzing the domain of propagation of finite AiG beams through plasma.

Numerical results and discussion

For an initial plane wavefront of the beam, the initial conditions on f ₁ and f ₂ are f ₁(ξ = 0) = f ₂(ξ = 0) = 1 and df ₁/dξ = df ₂/dξ = 0. When the two terms on the right-hand side of Eqs. (10a) and (10b) cancel each other at ξ = 0, d ²f ₁/dξ² = d ²f ₂/dξ² = 0 since df ₁/dξ and df ₂/dξ are also zero and f ₁ = f ₂ = 1 for all values of ξ. In other words, the beam propagates without convergence or divergence. The condition for self-trapping is therefore

(11)$${\rm \rho} \lpar {a_0\comma \;p} \rpar = \sqrt {\displaystyle{{-2\left({4 + a_0-\displaystyle{{2a_0g_1^3 } \over {g_2^3 }} + \displaystyle{{4g_1^2 } \over {g_2^2 }}} \right) {\lpar {1 + pg_2^4 } \rpar }^{3/2}} \over {\lpar {g_1^2 g_2^2 -4a_0g_1g_2^3 -2g_2^4 + 2a_0^2 g_2^4 } \rpar pg_2^4 }}} $$

where ρ = r ₀ω_p/c is the dimensionless initial beam radius and $p = {\rm \alpha }E_0^2$ is the initial intensity parameter. Equation (11) is definitely more amenable to mathematical manipulations. With the help of this equation, the critical values of a ₀ and p pertaining to uniform propagation of finite AiG beams can be easily determined. We have determined the minimum value of ρ by minimizing it with respect to a ₀ and p using the theorem on extremum values in two variable cases. This gives respective critical values of modulation and intensity parameters as a ₀ = a _0c = 0.70635 and p = p _c = 125.88654. At these two values, one can get the minimum value of the critical beam radius as ρ = ρ_c = 28.50137. Now one may investigate the response of ρ against p around a _0c by using Eq. (11).

In Figure 1, we have plotted the dimensionless initial beam radius ρ as a function of the initial intensity parameter p for different values of a ₀. Such variation of ρ against p is regarded as critical curves for finite AiG beams propagation in plasma. Each of these curves divides (p, ρ) plane into two regions. Initial points (p, ρ) lying above and below the each curve corresponds to self-focusing and divergence of finite AiG beams, respectively, which accords with earlier investigation (Sharma et al., Reference Sharma, Prakash, Verma and Sodha2003). One should note from this figure that as a ₀ increases, the critical curve shifts downward till a ₀ takes a critical value a _0c = 0.70635. However, with further increase in a ₀, the curve shifts upward. Figure 2 shows the variation of beam-width parameters f ₁ and f ₂ with dimensionless distance of propagation ξ for same values of a ₀ as varied in Figure 1 and ${k}^{\prime}_i = 0$. We have chosen a representative point (p, ρ) which lie below the all critical curves of Figure 1, i.e., ρ = ρ_c = 28.50137 and p = p _c = 125.88654. It is evident from Figure 2 that as a ₀ increases, AiG beams suffers with defocusing character of f ₁ and f ₂ with ξ up to a _0c. At a _0c = 0.70635, beam shows a stationary self-trapped mode. With further increase in a ₀ above a _0c (a ₀ > a _0c) causes defocusing of finite AiG beams. It is to be noted that for ρ < ρ_c, AiG beams always get defocused, although it has p > p _c.

Fig. 1. Critical curves for different modulation parameters a ₀.

Fig. 2. Dependence of beam-width parameters f ₁ and f ₂ with dimensionless propagation distance ξ for different modulation parameters a ₀. The other numerical parameters are ρ = 28.50137, p = 125.88654, and ${k}^{\prime}_i = 0$.

Figure 3 shows the variation of d ²f ₁/dξ² and d ²f ₂/dξ² with a ₀ for ρ = 40 ( > ρ_c) with p = p _c = 125.88654. Three domains of a ₀ have been observed characterizing the nature of propagation of finite AiG beams as follows:

1. (i) Self-focusing

   d ²f ₁/dξ² < 0 and d ²f ₂/dξ² < 0 for  −0.07179 < a ₀ < 1.50680.

2. (ii) Defocusing

   d ²f ₁/dξ² > 0 and d ²f ₂/dξ² > 0 for  −0.07179 > a ₀ > 1.50680.

3. (iii) Self-trapping

   d ²f ₁/dξ² = 0 and d ²f ₂/dξ² = 0 for  −0.07179 = a ₀ = 1.50680.

In Figure 4, we have displayed beam-width parameters f ₁ and f ₂ as a function of ξ for different values of a ₀ with ρ = 40 and $k_{i} ^{\prime} = 0$. From this figure, we have observed exact propagation behaviors of f ₁ and f ₂ with ξ as per the domains of a ₀ discussed in Figure 3. Consequently, by increasing the modulation parameter a ₀, self-focusing of finite AiG beams becomes better and shifted toward lower values of propagation distance ξ as reported earlier in Ouahid et al. (Reference Ouahid, Dalil-Essakali and Belafhal2018a). However, such early and strong self-focusing trend of f ₁ and f ₂ is observed to be reversed beyond critical modulation parameter a _0c. As such finite AiG beams suffer more defocused character of f ₁ and f ₂. Figure 5 illustrates the three domains of a ₀ for different values of ρ. The most striking feature of this figure is that the range of a ₀ remain unchanged as discussed in Figure 3. However, the self-focusing region enhances with an increase in ρ values. It may be noted that critical beam radius ρ has an infinite value at two values of modulation parameter a ₀. These are a ₀ = −0.39603 ( = a ₀₁) and a ₀ = 1.85406 ( = a ₀₂). Thus, it is of interest to find the range of a ₀, within which the self-trapping of beam is valid which demands the following inequality to be satisfied.

1. (i) For a ₀₁ < a ₀ < a ₀₂, ρ is real.

2. (ii) For a ₀₁ > a ₀ > a ₀₂, ρ is imaginary.

3. (iii) For a ₀₁ = a ₀ = a ₀₂, ρ is undefined.

Fig. 3. Domains of modulation parameter a ₀ with ρ = 40 and p = 125.88654.

Fig. 4. Dependence beam-width parameters f ₁ and f ₂ with dimensionless propagation distance ξ for different modulation parameters a ₀ with ${k}^{\prime}_i = 0$. Other parameters are same as in Figure 3.

Fig. 5. Domains of modulation parameter a ₀ for different ρ with p = 125.88654.

Hence, the range of a ₀, for which self-trapping of finite AiG beams is valid, is −0.39603 < a ₀ < 1.85406. The above three domains are independent on the power (p) of the AiG beam. To further elucidate the results for delineating the propagation of finite AiG beams through plasma, we numerically analyze the dependence of beam-width parameters f ₁ and f ₂ as a function of ξ for different values of normalized absorption coefficient ${k}^{\prime}_i$ when relativistic nonlinearity is taken into account. The results are depicted in the form of a set of graphs in Figure 6. This figure demonstrates that with an increase in a ₀ for given ${k}^{\prime}_i$, the beam exhibits strong ad early self-focusing. But such focusing trends get reversed to sharp defocusing depending on the location of a ₀ in the relevant domain as defined earlier. Further, at a given modulation parameter, an increase in ${k}^{\prime}_i$ causes a substantial reduction in self-focusing. This is because the weakening of self-focusing action takes place due to absorption and thus beam suffers sharp steady divergence for higher ${k}^{\prime}_i$ values.

Fig. 6. Dependence beam-width parameters f ₁ and f ₂ with dimensionless propagation distance ξ for different modulation parameters a ₀ with (a) ${k}^{\prime}_i = 0.00$, (b) ${k}^{\prime}_i = 0.04$, (c) ${k}^{\prime}_i = 0.08$, and (d) ${k}^{\prime}_i = 0.12$. Other parameters are same as in Figure 3.