Optimization of power compression and stability of relativistic and ponderomotive self-channeling of $248\mathrm{nm}$ laser pulses in underdense plasmas

The controlled formation in an underdense plasma of stable multi-PW relativistic micrometer-scale channels, which conduct a confined power at $248\mathrm{nm}$ exceeding ${10}^4$ critical powers and establish a peak channel intensity of $\sim {10}^{23}\mathrm{W}∕{\mathrm{cm}}^2$, can be achieved with the use of an appropriate gradient in the electron density in the initial launching phase of the confined propagation. This mode of channel formation optimizes both the power compression and the stability by smoothing the transition from the incident spatial profile to that associated with the lowest channel eigenmode, the dynamically robust structure that governs the confined propagation. A chief outcome is the ability to stably conduct coherent energy at fluences greater than ${10}^9\mathrm{J}∕{\mathrm{cm}}^2$.

Figure 1

Highly unstable relativistic and ponderomotive self-channeling of an azimuthally perturbed multi-TW laser pulse in an initially uniform underdense plasma. The laser and plasma parameters are $\lambda =248\mathrm{nm}$, $P_0=39\mathrm{TW}$, $r_0=10\mu \mathrm{m}$, $I_0=1.2\times {10}^{19}\mathrm{W}∕{\mathrm{cm}}^2$, and $N_{e,0}=3.8\times {10}^{20}{\mathrm{cm}}^{-3}$. The normalized power and beam radius in this case are, respectively, $\eta =P_0∕P_{\mathrm{cr}}\cong 50$ and ${\rho }_0\cong 37$. The incident azimuthally perturbed transverse amplitude distribution is defined by Eq. (12) with ${\epsilon }_q=0.01$ $\left(q=1–4\right)$. The corresponding level of weak azimuthal perturbations is characterized by ${\delta }_{\mathrm{pert}}$ from Eq. (13) , ${\delta }_{\mathrm{pert}}=0.063$. (a) presents the weakly azimuthally perturbed transverse intensity profile of the incident laser beam. (b) illustrates the transverse structure with the multiple peripheral filaments that have evolved in the process of highly unstable propagation.

Figure 2

Dynamics of stable multi-TW channel formation in the initial stage of the propagation with the use of an exponential longitudinal electron density profile defined by $N_{e,0}\left(z\right)=N_{e,0}\left(0\right)\exp \left(\alpha z\right)$ with $N_{e,0}\left(0\right)=1.9\times {10}^{19}{\mathrm{cm}}^{-3}$, $N_{e,0,\max }=3.8\times {10}^{20}{\mathrm{cm}}^{-3}$, and $z_{\max }=3.16L_R$ $\left(\alpha =0.949\right)$. All other laser pulse parameters are the same as in the previous case depicted in Fig. 1: $\lambda =248\mathrm{nm}$, $P_0=39\mathrm{TW}$, $r_0=10\mu \mathrm{m}$, and $I_0=1.2\times {10}^{19}\mathrm{W}∕{\mathrm{cm}}^2$. The incident azimuthally perturbed transverse intensity profile is the same as the distribution shown in Fig. 1a with ${\epsilon }_q\equiv 0.01$ $\left(q=1–4\right)$ and ${\delta }_{\mathrm{pert}}=0.063$ [see Eqs. (12, 13)]. The development of a stable channel is presented in the $\left(x,z\right)$ plane, where $z$ is the axis of propagation and $x$ is one of the transverse coordinates.

Figure 3

Dynamics of stable multi-PW channel formation in the initial stage of propagation with the use of an exponential longitudinal electron density profile defined by $N_{e,0}\left(z\right)=N_{e,0}\left(0\right)\exp \left(\alpha z\right)$ with $N_{e,0}\left(0\right)=5\times {10}^{19}{\mathrm{cm}}^{-3}$, $N_{e,0,\max }=3.8\times {10}^{20}{\mathrm{cm}}^{-3}$, and $z_{\max }=3.95L_R$ $\left(\alpha =0.514\right)$. The laser pulse parameters are $\lambda =248\mathrm{nm}$, $P_0=7.8\mathrm{PW}$, $r_0=2\mu \mathrm{m}$, and $I_0=6.2\times {10}^{22}\mathrm{W}∕{\mathrm{cm}}^2$. The incident azimuthally perturbed transverse amplitude distribution is defined by Eq. (12) with ${\epsilon }_q\equiv 0.01$ $\left(q=1–4\right)$ and the corresponding level of azimuthal perturbation is ${\delta }_{\mathrm{pert}}=0.063$ [see Eq. (13)]. The power trapped in the stable relativistic multi-PW laser channel is in the range of ${10}^4$ critical powers.

Figure 4

Dynamics of stable multi-PW relativistic channel formation in the initial stage of propagation with the use of a longitudinal electron density profile defined by Eq. (14) with $N_{e,0}\left(0\right)=5\times {10}^{19}{\mathrm{cm}}^{-3}$, $N_{e,0,\max }=3.8\times {10}^{20}{\mathrm{cm}}^{-3}$, and $z_{\exp ,2}=2.96L_R$. The laser pulse parameters are identical to the case illustrated in Fig. 3: $\lambda =248\mathrm{nm}$, $P_0=7.8\mathrm{PW}$, $r_0=2\mu \mathrm{m}$, and $I_0=6.2\times {10}^{22}\mathrm{W}∕{\mathrm{cm}}^2$. The incident azimuthally perturbed transverse amplitude distribution is defined by Eq. (12) with ${\epsilon }_q\equiv 0.01$ $\left(q=1–4\right)$ and the corresponding level of azimuthal perturbation is ${\delta }_{\mathrm{pert}}=0.063$ [see Eq. (13)]. The power trapped in the stable relativistic multi-PW laser channel is in the range of ${10}^4$ critical powers.

Figure 5

Stable multi-PW relativistic channel formation in the initial stage of propagation with the use of an exponential longitudinal electron density profile defined by $N_{e,0}\left(z\right)=N_{e,0}\left(0\right)\exp \left(\alpha z\right)$ with $N_{e,0}\left(0\right)=1.9\times {10}^{19}{\mathrm{cm}}^{-3}$, $N_{e,0,\max }=3.8\times {10}^{20}{\mathrm{cm}}^{-3}$, and $z_{\max }=2.37L_R$ $\left(\alpha =1.65\right)$. The laser pulse parameters are $\lambda =248\mathrm{nm}$, $P_0=9\mathrm{PW}$, $r_0=5\mu \mathrm{m}$, and $I_0=1.1\times {10}^{22}\mathrm{W}∕{\mathrm{cm}}^2$. The incident azimuthally perturbed transverse amplitude distribution is defined by Eq. (12) with ${\epsilon }_q\equiv 0.01$ $\left(q=1–4\right)$ and the corresponding level of azimuthal perturbation is ${\delta }_{\mathrm{pert}}=0.063$ [see Eq. (13)]. (a) shows the incident azimuthally perturbed transverse intensity profile. The transverse profile of the evolved stable multi-PW relativistic channel is presented in (b). (c) illustrates the dynamics of stable channel formation in the $\left(x,z\right)$ plane, where $z$ is the axis of propagation and $x$ is one of the transverse coordinates. The trajectory (solid curve) of stable channel development in the $\left(\eta ,{\rho }_0\right)$ plane [see Eq. (1)] is depicted in (d). The dotted lines represent the boundary between the zone of stable relativistic and ponderomotive self-channeling resulting in the formation of the main powerful channel and the zone of highly unstable self-channeling, which leads to strong filamentation of the laser beam. The dashed curve illustrates the normalized radius of the channel eigenmode ${\rho }_{e,0}$ defined by Eq. (4) as a function of normalized power $\eta$. The initial point of the self-channeling trajectory $A_{\text{stable}}$ corresponds to the $\left(\eta ,{\rho }_0\right)$ parameters of the incident laser beam, and the final point of the trajectory $B_{\text{stable}}$ illustrates the values of the $\left(\eta ,{\rho }_0\right)$ parameters for the evolved stable ultrapowerful relativistic channel. The power conducted by the stable multi-PW channel exceeds ${10}^4$ critical powers. The peak channel intensity is in the range of ${10}^{23}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 6

Stable relativistic channel formation in the case of a severely azimuthally aberrated incident multi-PW laser beam having a relatively large initial beam radius. The exponential longitudinal electron density profile is defined by $N_{e,0}\left(z\right)=N_{e,0}\left(0\right)\exp \left(\alpha z\right)$ with $N_{e,0}\left(0\right)=9.5\times {10}^{17}{\mathrm{cm}}^{-3}$, $N_{e,0,\max }=3.8\times {10}^{20}{\mathrm{cm}}^{-3}$, and $z_{\max }=4.93L_R$ $\left(\alpha =1.21\right)$. The laser pulse parameters are $\lambda =248\mathrm{nm}$, $P_0=9\mathrm{PW}$, $r_0=5\mu \mathrm{m}$, and $I_0=1.0\times {10}^{22}\mathrm{W}∕{\mathrm{cm}}^2$. The severely distorted incident transverse amplitude distribution is defined by Eq. (12) with ${\epsilon }_q\equiv 0.05$ $\left(q=1–4\right)$ and the corresponding level of azimuthal perturbation is ${\delta }_{\mathrm{pert}}=0.393$ [see Eq. (13)]. The strongly azimuthally perturbed transverse intensity profile of the incident laser beam is presented in (a). The transverse profile of the evolved stable multi-PW relativistic channel is presented in (b). (c) illustrates in the $\left(x,z\right)$ plane the dynamics of the propagation.. The trajectory (solid curve) of stable channel development in the $\left(\eta ,{\rho }_0\right)$ plane [see Eq. (1)] is depicted in (d). The dotted lines represent the boundary between the zone of stable relativistic and ponderomotive self-channeling resulting in the formation of the main powerful channel and the zone of highly unstable self-channeling, which leads to strong filamentation of the laser beam. The dashed curve illustrates the normalized radius of the channel eigenmode ${\rho }_{e,0}$ defined by Eq. (4) as a function of normalized power $\eta$. The initial point of the self-channeling trajectory $A_{\text{stable}}$ corresponds to the $\left(\eta ,{\rho }_0\right)$ parameters of the incident laser beam, and the final point of the trajectory $B_{\text{stable}}$ illustrates the values of the $\left(\eta ,{\rho }_0\right)$ parameters for the evolved stable ultrapowerful relativistic channel. The power conducted by the channel exceeds ${10}^4$ critical powers and the peak channel intensity is in the range of ${10}^{23}\mathrm{W}∕{\mathrm{cm}}^2$.