Laser propagation basic theory in plasma with periodical density ripple

The laser wave equation governing the electronic field in the plasma can be written as

(1)$$\nabla ^2\vec{E}-\nabla \lpar {\nabla \cdot \vec{E}} \rpar = \displaystyle{{\rm \epsilon} \over {c^2}}\displaystyle{{\partial ^2\vec{E}} \over {\partial t^2}}$$

where ε is the dielectric constant, c is the light speed, and $\vec{E}$ is the electric field.

Applying WKB approximation and paraxial theory (Sodha et al., Reference Sodha, Faisal and Verma2009), the variation in the electric field can be represented as

(2)$$\vec{E} = \vec{A}_0\lpar {r,z,t} \rpar {\rm exp}\lsqb {i\lpar {{\rm \omega} t-k_0z} \rpar } \rsqb $$

where ω is the wave frequency and z is the axial distance along propagation. For further approximation, the electron plasma frequency ω_p is introduced, since ${\rm \omega} _{\rm p}^2 /{\rm \omega} ^2\ll {\rm \epsilon} {\rm In}{\rm \epsilon}$, one can ignore the term $\nabla \lpar {\nabla \cdot \vec{E}} \rpar$ and then substitute the electric field function into Eq. (1) with a combination of the approximation, the following can be obtained:

(3)$$-k_0^2 \vec{A}_0-2ik_0\displaystyle{{\partial {\vec{A}}_0} \over {\partial z}} + \nabla ^2\vec{A}_0 + \displaystyle{1 \over {V_{\rm g}}}\cdot \displaystyle{{\partial {\vec{A}}_0} \over {\partial t}} = \displaystyle{{\rm \omega ^2} \over {c^2}}{\rm \epsilon} \vec{A}_0$$

where $k_0 = {\rm \omega} {\rm \epsilon} _0^{1/2} /c$ is the wave number, ε₀ is the initial value of dielectric constant, V _g = k ₀c ²/ωε is the group velocity, and $\overrightarrow{A}_0$ is a complex function of space which can be written as

(4)$$\vec{A}_0 = \vec{A}_0\lpar {r,z,t} \rpar {\rm exp}\lsqb {-ik_0S_0\lpar {r,z,t} \rpar } \rsqb $$

where S ₀ is the real function in the space which can be written as

(5)$$S_0 = S_{00} + \displaystyle{{r^2} \over {r_0^2}} S_{02}$$

(6)$$S_{02} = \displaystyle{{r_0^2} \over {2f}}\displaystyle{{df} \over {dz}}$$

where r is the radial coordinate of the cylindrical coordinate system and r ₀ is the initial beam width.

Introducing the retarded time τ, we assume the dimensionless retarded time τ′ = t/τ, and substituting $\vec{A}_0$ into Eq. (3), the laser electronic field amplitude ${A^{\prime}}_0^2$ can be obtained:

(7)$${A^{\prime}}_0^2 = \displaystyle{{A_{00}^2} \over {\,f^2}}{\rm exp}\left( {-\displaystyle{{r^2} \over {r_0^2 f^2}}} \right)F\lpar {{\rm {\tau}^{\prime}}} \rpar $$

where $A_{00}^2$ is the initial value, f is the dimensionless beam width parameter in the paraxial region, and τ′ is the dimensionless retarded time. Considering the real part term in the solution of Eq. (3) can be shown in the following form:

(8)$$2\displaystyle{{\partial S_0} \over {\partial z}} + \left( {\displaystyle{{\partial S_0} \over {\partial r}}} \right)^2 = \displaystyle{{{\rm \omega} ^2} \over {k_0^2 c^2}}\epsilon + \displaystyle{1 \over {k_0^2 {{\vec{A}^{\prime}_0}}}}\left( {\displaystyle{1 \over r}\displaystyle{{\partial {{\vec{A}^{\prime}_0}}} \over {\partial r}} + \displaystyle{{\partial^2{{\vec{A}^{\prime}_0}}} \over {\partial r^2}}} \right)$$

Using the partial approximation along with the direction of propagation, the dielectric function ε can be introduced as follows:

(9)$${\rm \epsilon} = {\rm \epsilon}_0-\displaystyle{{r^2} \over {r_0^2}} {\rm \epsilon}_2$$

where ε₀ is the linear part and ε₂ is the nonlinear part.

By substituting Eqs (5)–(7) and (9) into Eq. (8), the equation governing the beam width parameter f in the paraxial region can be obtained as

(10)$$\displaystyle{{d^2f} \over {dz^2}} = \displaystyle{1 \over {k_0^2}} \left( {\displaystyle{1 \over {r_0^4 f^3}}-\displaystyle{{\rm \omega^2} \over {c^2}}\displaystyle{\,f \over {r_0^2}} {\rm \epsilon}_2} \right)$$

For the formation of collision plasma in intense laser and matter interaction, the electrons movement function could be written as follows:

(11)$$m\displaystyle{{d{\vec{V}}_{\rm e}} \over {dt}} = -e\vec{E}-mv_{\rm e}\vec{V}_{\rm e}$$

where m is the electronic mass, v _e is the effective frequency of electron collisions, and $\vec{V}_{\rm e}$ is the electrons drift velocity in the plasma. When substituting $\vec{V}_{\rm e}$ into uniform alternating electric field $\vec{E} = \vec{E}_0\,{\rm exp}\lpar {i{\rm \omega} t} \rpar$ with frequency ω at t = 0, and we assuming t ≫ 1/v _e, the expression of $\vec{V}_{\rm e}$ could be written as

(12)$$\vec{V}_{\rm e} = \displaystyle{{-e{\vec{E}}_0e^{i{\rm \omega} t}} \over {m\lpar {v_{\rm e} + i{\rm \omega}} \rpar }}$$

According to thermal energy conservation, the system of plasmas including all the electrons is supported by the following:

(13)$$\displaystyle{d \over {dt}}\left( {\displaystyle{3 \over 2}k_{\rm B}T_{\rm e}} \right) = -e{\rm Re}\lpar {\vec{E}\cdot {\vec{V}}_{\rm e}} \rpar -\displaystyle{3 \over 2}k_{\rm B}{\rm \delta} v_{\rm e}\lpar {T_{\rm e}-T_0} \rpar $$

where $-e{\rm Re}\lpar {\vec{E}\cdot {\vec{V}}_{\rm e}} \rpar$ refers to the ohmic heating rate for one electron; and the latter one − 3k _Bδv _e(T _e − T ₀)/2 refers to the rate of thermal energy loss in one electron during the process of collisions with other heavy particles, T _e is the electron temperature in the field, T ₀ is the temperature of the plasma without the pulse, and δ is the fraction of losing excess energy during the process of collisions.

Substituting Eq. (12) into (13), one obtains

(14)$$\displaystyle{{dT_{\rm e}} \over {dt}} + {\rm \delta} v_{\rm e}\lpar {T_{\rm e}-T_0} \rpar = \displaystyle{{e^2v_{\rm e}EE^{^\ast}} \over {3m{\rm \omega} ^2k_{\rm B}}}$$

The wave frequency ω ≫ v, where v is the electron collision frequency, then the profile of the Gaussian function might by expressed as

(15)$$EE^{^\ast} = {A^{\prime}}_0^2 \,{\rm exp}\left( {-\displaystyle{{r^2} \over {r_0^2}}} \right)F\lpar {{\rm {\tau}^{\prime}}} \rpar $$

where r is the radial coordinate of the cylindrical coordinate system and r ₀ is the initial beam width.

In the paraxial region, the electron temperature in the field T _e can be expressed as

(16)$$T_{\rm e} = T_0\lpar {1 + T_{\rm p}} \rpar $$

where T _p is the dimensionless temperature, thermal energy conservation can be gained as

(17)$$\eqalign{\displaystyle{{dT_{\rm p}} \over {d{\rm {\tau} ^{\prime}}}} +& {\rm \delta} v_0{\rm \tau} \lpar {1 + T_{\rm p}} \rpar ^{1/2}T_{\rm p} \cr= &\displaystyle{{\tau v_0} \over {T_0}}\displaystyle{{e^2} \over {3m{\rm \omega} ^2k_{\rm B}}} \times \lpar {1 + T_{\rm p}} \rpar ^{1/2}\displaystyle{{A_{00}^2} \over {\,f^2}}e^{-(r^2/r_0^2 f^2)}F\lpar {{\rm {\tau}^{\prime}}} \rpar } $$

The dimensionless temperature T _p can be expanded as the second-order formula approximation along the axial direction of propagation distance r:

(18)$$T_{\rm P}\approx T_{{\rm P}0}-\displaystyle{{r^2} \over {r_0^2}} T_{{\rm P}2}$$

Submitting T _p above into Eq. (17), and equating the coefficients of r ² and $r_0^2$ on both sides of the equation, the following can be obtained:

(19)$$\eqalign{\displaystyle{{dT_{{\rm p}0}} \over {d{\rm {\tau} ^{\prime}}}} + &{\rm \delta} v_0{\rm \tau} \left( {T_{{\rm p}0} + \displaystyle{1 \over 2}T_{{\rm p}0}^2} \right) \cr = &{{{\rm \tau} v_0} \over {T_0}}\displaystyle{{e^2} \over {3m{\rm \omega} ^2k_{\rm B}}}\displaystyle{{A_{00}^2} \over {\,f^2}} \,{\times}{\left( {1 + \displaystyle{{T_{{\rm p}0}} \over 2}} \right)F\lpar {{\rm {\tau}^{\prime}}} \rpar}}$$

(20)$$\eqalign{\displaystyle{{dT_{{\rm p}2}} \over {d{\rm {\tau} ^{\prime}}}} + & {\rm \delta} v_0{\rm \tau} \lpar {T_{{\rm p}0}T_{{\rm p}2} + T_{{\rm p}2}} \rpar \cr {=}& \displaystyle{{{\rm \alpha} A_{00}^2} \over {\,f^2}}\left[ {\displaystyle{{T_{{\rm p}0}} \over 2} + \displaystyle{1 \over {\,f^2}}\left( {1 + \displaystyle{{T_{{\rm p}0}} \over 2}} \right)} \right]F\lpar {{\rm {\tau}^{\prime}}} \rpar} $$

(21)$${\rm \alpha} = e^2{\rm \tau} v_0/3m{\rm \omega} ^2k_{\rm B}T_0$$

The two second-order non-homogeneous differential equations have introduced the dimensionless variable of temperature T _p in plasma with periodical density ripple, and with which the extensions T _p0 and T _p2, the temperatures particularly depend on the dimensionless retarded time τ′.

By using the coordinate conversion, we introduce the dimensionless distance of propagation

(22)$${\rm \xi} = zc\lpar {{\rm \omega} r_0^2} \rpar $$

and the dimensionless initial beam width

(23)$${\rm \beta} = r_0{\rm \omega} /c$$

The dielectric function ε as shown in Eq. (9) has the linear part ε₀ and the nonlinear part ε₂ which can be expanded and the electron plasma frequency and the electron temperature can be applied:

(24)$${\rm \epsilon} _0 = 1-\displaystyle{{{\rm \omega} _{\rm p}^2} \over {{\rm \omega} ^2}}\displaystyle{2 \over {2 + T_{{\rm p}0}}}$$

(25)$${\rm \epsilon} _2 = \displaystyle{{{\rm \omega} _{\rm p}^2} \over {{\rm \omega} ^2}}\displaystyle{{2T_{{\rm p}2}} \over {{(2 + T_{{\rm p}0})}^2}}$$

Substituting Eqs (22)–(25) with the wavenumber $k_0 = {\rm \omega} {\rm \epsilon} _0^{1/2} /c$ into Eq. (10), the modified equation of the beam width in the paraxial region could be obtained:

(26)$$\eqalign{\displaystyle{{d^2f} \over {d{\rm \xi} ^2}} = &\displaystyle{{2 + T_{{\rm p}0}} \over {2 + T_{{\rm p}0}-2{\rm \omega} _{\rm p}^2 /{\rm \omega} ^2}}\displaystyle{1 \over {\,f^3}} \cr &-\displaystyle{{2T_{{\rm p}2}{\rm \omega} _{\rm p}^2 /{\rm \omega} ^2} \over {\lpar {2 + T_{{\rm p}0}} \rpar \lpar {2 + T_{{\rm p}0}-2{\rm \omega}_{\rm p}^2 /{\rm \omega}^2} \rpar }}{\rm \beta} ^2f} $$

The equation of the beam width mainly depends on the temperature of the electron and spatial–temporal coordinate parameters. On the right part, the first term refers to the beam diffraction and the later one represents the nonlinear part of the laser–plasma interaction system.

According to previous researches (Xia, Reference Xia2014), the relations between electron density and electron frequency can be expressed as $n_{\rm e}/n_{{\rm e}0} = {\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm p}0}^2$, where ω_p is the electron plasma frequency for the interactions with the laser beam. So Eq. (26) in inhomogeneous collisional plasmas can be changed into the following form:

(27)$$\eqalign{\displaystyle{{d^2f} \over {d{\rm \xi} ^2}} = &\displaystyle{{2 + T_{{\rm p}0}} \over {2 + T_{{\rm p}0}-2({\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm p}0}^2 ) \times ({\rm \omega} _{{\rm p}0}^2 /{\rm \omega} ^2)}}\displaystyle{1 \over {\,f^3}} \cr &-\displaystyle{{2T_{{\rm p}2}({\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm p}0}^2 ) \times ({\rm \omega} _{{\rm p}0}^2 /{\rm \omega} ^2)} \over {\lpar {2 + T_{{\rm p}0}} \rpar \lpar {2 + T_{{\rm p}0}-2({\rm \omega}_{\rm p}^2 /{\rm \omega}_{{\rm p}0}^2 ) \times ({\rm \omega}_{{\rm p}0}^2 /{\rm \omega}^2)} \rpar }}{\rm \beta} ^2f} $$

Collisional plasma with periodic density ripple have been found according to the experiment. In this paper, the collisional plasmas are assumed as n _e/n _e0 = A + B cos (Cz), where A refers to the rate of electron density and initial electron density, B is the rippled parameter of collisional plasma, respectively, and c is the wave number.

Numerical results and discussions

According to the above analysis, during the interaction in laser and plasma with periodical density ripple, numerical calculations for Eqs (19), (20), (24), (25), and (27) yielded a relationship between electron temperature, dielectric constants, and beam width. The relevant parameters are specified in Figures 1–6.

Fig. 1. Variation of axial electron temperatures T _p0 and T _p2 with the variations of dimensionless retarded time τ′ = t/τ for the parameters of δv ₀ in the Gaussian pulse. ${\rm \alpha} A_{00}^2 = 10$, f = 1, ${\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm p}0}^2 = 1 + 0.1\,{\rm cos}\lpar {2{\rm \pi} z/3} \rpar$, ${\rm \omega} _{{\rm p}0}^2 /{\rm \omega} ^2 = 0.8$, δv ₀ = 0.5 (I), 1.5 (III), and 2.5 (V) for T _p0; and δv ₀ = 0.5 (II), 1.5 (IV), and 2.5 (VI) for T _p2.

Fig. 2. Variation of axial dielectric constant (ε₀ and ε₂) with dimensionless retarded time τ′ at z = 0 for the parameters δv ₀ in Gaussian pulse. ${\rm \alpha} A_{00}^2 = 10$, f = 1, ${\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm p}0}^2 = 1 + 0.1\,{\rm cos}\lpar {2{\rm \pi} z/3} \rpar$, ${\rm \omega} _{{\rm p}0}^2 /{\rm \omega} ^2 = 0.8$, δv ₀ = 0.5 (I), 1.5 (III), and 2.5 (V) for ε₀, and δv ₀ = 0.5 (II), 1.5 (IV), and 2.5 (VI) for ε₂.

Fig. 3. Variation of width parameter f with dimensionless distance of propagation ξ for the dimensionless retarded time τ′ in Gaussian pulse. ${\rm \alpha} A_{00}^2 = 10$, f = 1, ${\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm p0}}^2 = 1 + 0.1\,{\rm cos}\lpar {2{\rm \pi} z/3} \rpar$, and ${\rm \omega} _{{\rm p}0}^2 /{\rm \omega} ^2 = 0.8$ at τ′ = −2.5 (I), −2 (II), −1.5 (III), −1 (IV), −0.5 (V), and 0 (VI) for (a), and τ′ = 0 (I), 0.5 (II), 1 (III), 1.5 (IV), 2 (V), and 2.5 (VI) for (b).

Fig. 4. Variation of axial dielectric constant (ε₀ and ε₂) with dimensionless retarded time τ′ at z = 0 for the parameters δv ₀ in Gaussian pulse. ${\rm \alpha} A_{00}^2 = 10$, f = 1, ${\rm \omega} _{{\rm p}0}^2 /{\rm \omega} ^2 = 0.8$, δv ₀ = 0.5 (I), 1.5 (III), and 2.5 (V) for ε₀, and δv ₀ = 0.5 (II), 1.5 (IV), and 2.5 (VI) for ε₂ in (a), (b), and (c), $\omega _{\rm p}^2 /\omega _{{\rm p}0}^2 = 1.5 + 0.1\,{\rm cos}\lpar {2{\rm \pi} z/3} \rpar$ for (a), 1.5 + 0.6 cos(2πz/3) for (b), and 1.5 + 0.6 cos(5πz/6) for (c).

Fig. 5. Variation of width parameter f with dimensionless distance of propagation ξ for the dimensionless retarded time τ′ in Gaussian pulse. ${\rm \alpha} A_{00}^2 = 10$, f = 1, ${\rm \omega} _{{\rm p}0}^2 /{\rm \omega} ^2 = 0.8$ at τ′ = −2.5 (I), −2 (II), −1.5 (III), −1 (IV), −0.5 (V), and 0 (VI) for (a), (b), and (c), and τ′ = 0 (I), 0.5 (II), 1 (III), 1.5 (IV), 2 (V), and 2.5 (VI) for (d), (e), and (f), and ${\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm p0}}^2 = 1.5 + 0.1\,{\rm cos}\lpar {2{\rm \pi} z/3} \rpar$ for (a) and (d), 1.5 + 0.6 cos(2πz/3) for (b) and (e), and 1.5 + 0.6 cos(5πz/6) for (c) and (f).

Fig. 6. Variation of width parameter f with dimensionless distance of propagation ξ for the dimensionless retarded time τ′ in Gaussian pulse. ${\rm \alpha} A_{00}^2 = 15$, ${\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm p}0}^2 = 1.5 + 0.6\,{\rm cos}\lpar {2{\rm \pi} z/3} \rpar$, and ${\rm \omega} _{{\rm p}0}^2 /{\rm \omega} ^2 = 0.8$ at τ′ = −2.5 (I), −2 (II), −1.5 (III), −1 (IV), −0.5 (V), and 0 (VI) for (a) and (b), and τ′ = 0 (I), 0.5 (II), 1 (III), 1.5 (IV), 2 (V), and 2.5 (VI) for (c) and (d), f = 1 for (a) and (c), as well as f = 0.8 for (b) and (d).

Figure 1 shows the electron temperatures T _p0 and T _p2 as a function of dimensionless retarded time τ′ = t/τ. On the whole, both the values of T _p0 and T _p2 increase first and then decrease, showing a single peak. In addition, with δv ₀ increasing, the amplitudes of T _p0 and T _p2 decrease. While T _p0 and T _p2 do not reach the peak values at the same time. In addition, with the increase in δv ₀, peak values of both T _p0 and T _p2 have decreased.

Figure 2 shows the variation of the dielectric constants ε₀ and ε₂ along with the dimensionless retarded time τ′ in different pulse widths at ξ = 0. It is shown that ε₀ and ε₂ have shown the obvious oscillatory during the dimensionless retarded time, besides the linear part ε₀ is always larger than the nonlinear part ε₂, and ε₂ is relatively more complex than ε₀. This is because that the electron temperature and density varied under the influence of collisional nonlinear effect, leading to the nonlinear variations of the dielectric function. Additionally, it shows that the peak values of ε₀ and ε₂ decrease steadily with the increase in losing excess energy factor δv ₀. This is because effective collision decreases with the increase of δv ₀, leading to the peak of dielectric function on decline.

Figure 3 shows the variation of the dimensionless beam width parameter f with the variation of the dimensionless distance of propagation ξ for different values of the dimensionless retarded time τ′. In Figure 3, in plasma with periodical density ripple, laser beam presents three obvious nonlinear phenomena, which are self-defocusing [see Fig. 3a(I–III)], oscillational self-focusing [see Fig. 3a(IV and VI) and 3b(I)], and stable self-focusing [see Fig. 3a(V) and 3b(I–IV and VI)] in different dimensionless retarded time τ′. This is actually the competition results between the convergence term caused by nonlinearity and the discrete term generated by diffraction, such a competitive mechanism leads to stable self-focusing when the convergence term is greater than the discrete term, and self-focusing with oscillational variation when the convergence term has similar degree with the discrete term, and self-defocusing when the convergence term is less than the discrete term. In the initial stage, due to the obvious diffraction effect, self-defocusing and oscillational self-focusing mainly occur. The effect of the collision nonlinearity is significantly strengthened with time, in this case, the convergence term caused by the collision nonlinearity is enhanced, resulting in the significantly enhanced stable self-focusing, while the other two are weakened or even disappear.

Figure 4 reflects the variations in the dielectric function in different plasmas with periodic density distributions. Comparing with Figure 2, with the increase in the rate of electron density and initial electron density A in (a), the amplitude of the dielectric function is significantly larger, based on the profile in (a), when increasing the rippled parameter of collisional plasma B in (b), the whole amplitude will be slightly reduced. And then increasing the wave number c in (c), the amplitude of the plasma dielectric constant is restored to a reduced level. This is because when the density increases, the frequency of electron collision increases, and the dielectric function increases as well. With the strengthening of B and c, the variation of fluctuation increases and the dielectric function decreases.

Figure 5 discusses the effect on beam width in plasmas with different parameters. Specifically, compared with (a) in Figure 3, it can be seen with Figure 5a as well, when increasing A, the maximum amplitude of the curve with a larger amplitude is slightly lower, the other different curves seem to be without any variations. From (a), (b), (d), and (e), when increasing B, there are relatively obvious variations in the amplitude which occur on the curve with a larger amplitude and which become larger again. When increasing c, as is shown in (b) and (c), it could be observed that the curve with a larger amplitude has a slightly larger maximum value, the change of the ratio curve is not obvious, and (d) is similar to the change of (e), and most of the amplitudes are still at the low level, the wavelength is basically stabilized. The possible reason for the above phenomenon is that under the influence of collision nonlinear factors, the parameters of plasma with periodical density ripple have a greater impact on the effect of self-defocusing and self-focusing with oscillational variation than the stable self-focusing.

Figure 6 shows the effects of variations in the beam width parameter from f = 1 to f = 0.8 during the propagation. When f = 1 in (a) and (c), it can be found that there are several phenomena here, forming stable self-focusing [VI in (a), and II, III, IV, and V in (c)], self-focusing with oscillational variations [I and V in (a), and I in (c)], and self-defocusing phenomenon [III, IV, and V in (a)]. Obviously, the parameter f has a significant effect on self-defocusing and oscillational self-focusing. As f decreases, amplitudes of the two nonlinear phenomena increase slightly, while self-focusing becomes faster on the same length.