Formation of axially modulated plasma strings by filamentation of interfering femtosecond Bessel beams

TL;DR

Two collinear femtosecond Bessel beams with different central spot sizes form axially modulated plasma strings upon filamentation in air. The field is modeled as a pi-shifted Bessel superposition with a Gaussian envelope and evolved with the standard ultrashort-pulse filamentation equations coupled to a rate equation for electron density, solved numerically by a spectral extended Crank-Nicolson method. The nonlinear modulation period $\Lambda_{NL}$ matches the linear $\Lambda$ and can be tuned by the sub-beam sizes; higher input energy extends the filamentation length but can erode modulation depth, while peak density remains around $10^{17}$ cm$^{-3}$ with intensity clamped near $2\times10^{13}$ W/cm$^2$. An axicon-based interferometer is proposed to realize the two-Bessel-beam superposition without an SLM, enabling tunable axial modulation for THz generation, harmonic generation, and potential microfabrication in transparent media.

Abstract

We numerically investigate the formation of axially modulated plasma strings through the filamentation of two interfering femtosecond Bessel beams. The constituent Bessel beams have different central spot sizes and propagate collinearly. Our results show that filamentation of these beams in air leads to the formation of intense axially modulated light filaments, and corrugated plasma strings with high modulation depths and tunable modulation periods. We find that the modulation periods of the intensity and the plasma density are identical and comparable to the modulation period of the intensity in the linear propagation regime. Furthermore, we show that, in the nonlinear propagation regime, the modulation period is independent of the pulse energy and can be tuned by selecting appropriate central spot sizes for the interfering Bessel beams. Finally, we propose a simple interferometric arrangement using a single axicon to generate two interfering Bessel beams with adjustable spot sizes enabling the creation of axially-modulated plasma strings with tunable periodicity at high input powers.

Figures (9)

• Figure 1: (a) Transverse intensity profile of Beam 1 at $z=0$; (b) Longitudinal intensity profile of Beam 1 in the linear propagation from $z=-1\ \mathrm{m}$ to $z=1\ \mathrm{m}$;(c) Axial intensity profiles of Beam 1 (solid curve), and Beam 2 (dashed curve). Parameters of the beams are as the following: Beam 1: $s_{1}=60\ \mu\mathrm{m}$, $s_2=12\ \mu \mathrm{m}$, and Beam 2: $s_1=90\ \mu \mathrm{m}$, $s_2=18\ \mu\mathrm{m}$; both beams have the pulse energy of 1 nJ, and the same spatiotemporal envelope with $w=1.5\ \mathrm{mm}$, and $t_0=50\ \mathrm{fs}$.
• Figure 2: Evolution of the time-averaged intensity during the nonlinear propagation for (a) Beam 1 with an input power of $6.45\times P_{\text{cr}}$, (b) Beam 2 with an input power of $6.45\times P_{\text{cr}}$, (c) Beam 2 with an input power of $9.65\times P_{\text{cr}}$, and (d) Beam 3 with an input power of $6.45\times P_{\text{cr}}$. Here, $\rho$ denotes the radial distance from the propagation axis, $z$.
• Figure 3: Evolution of the time-integrated plasma density for (a) Beam 1 with an input power of $6.45\times P_{\text{cr}}$, (b) Beam 2 with an input power of $6.45\times P_{\text{cr}}$, (c) Beam 2 with an input power of $9.65\times P_{\text{cr}}$, and (d) Beam 3 with an input power of $6.45\times P_{\text{cr}}$. Here, $\rho$ denotes the radial distance from the propagation axis, $z$.
• Figure 4: Axial intensity and electron density along propagation for (a) Beam 1 with an input power of $6.45\times P_{\text{cr}}$ (b) Beam 2 with an input power of $6.45\times P_{\text{cr}}$, (c) Beam 2 with an input power of $9.65\times P_{\text{cr}}$, and (d) Beam 3 with an input power of $6.45\times P_{\text{cr}}$.
• Figure 5: An interferometric configuration proposed for generating two superposed Bessel beams. An original Bessel beam is generated by an axicon. A converging lens in each arm of the interferometer (i.e., L$_{1}$, and L$_{2}$) relays the original beam to an axial distance beyond the second beam splitter, BS$_{2}$. The parameters of the relayed beams can be tuned by choosing different focal lengths for the two relay lenses or by changing the distances of the relay lenses to the first beam splitter, BS$_{1}$. Superposition of the two relayed modified Bessel beams occurs after the second beam splitter, BS$_2$. M$_{1}$, and M$_{2}$ are flat mirrors.