Holographic Plasma Lenses

A hologram fully encodes a three-dimensional light field by imprinting the interference between the field and a reference beam in a recording medium. Here we show that two collinear pump lasers with different foci overlapped in a gas jet produce a holographic plasma lens capable of focusing or collimating a probe laser at intensities several orders-of-magnitude higher than the limits of a nonionized optic. We outline the theory of these diffractive plasma lenses and present simulations for two plasma mechanisms that allow their construction: spatially varying ionization and ponderomotively driven ion-density fluctuations. Damage-resistant plasma optics are necessary for manipulating high-intensity light, and divergence control of high-intensity pulses—provided by holographic plasma lenses—will be a critical component of high-power plasma-based lasers.

Figure 1

Schematic of a holographic plasma lens. (a) Two pump lasers overlap in a gas (extent along $z$ is $D$), arranged so that their interference pattern is a sinusoidal zone plate. (b) At a delayed time, a probe laser passes through the resulting structure and is diffracted into one or more orders. (c) The intensity profile of the overlapped pumps and the resultant index modulation. (d) The peak index of refraction modulation as a function of time, showing the formation of the structure with the arrival of the pumps, followed by a decay and possible modification by the probe. The amplitude and timescale both depend on the chosen nonlinear mechanism.

Figure 2

Calculations (3D PPS) for a thick plasma lens, showing collimation (a) and focusing (b) depending on the initial probe focus position. (a) Probe (blue) near-collimated to focus at $z=6\mathrm{mm}$ by lens formed in $D=1\mathrm{mm}$ initial gas (orange), with $\mathrm{\Delta }n=1.3\times {10}^{-4}$, ${\lambda }_p={\lambda }_0=400\mathrm{nm}$, $f_A=1\mathrm{mm}$, and $f_B=6\mathrm{mm}$; 81% of energy is within the focal spot at $z=6\mathrm{mm}$. (b) Focusing lens with $D=200\mu \mathrm{m}$ and $\mathrm{\Delta }n=3.3\times {10}^{-3}$ for ${\lambda }_0=800\mathrm{nm}$ probe. Focal spot full-width-half-maximum is $2.2\mu \mathrm{m}$. Pumps have ${\lambda }_p=800\mathrm{nm}$, $f_A=0.5\mathrm{mm}$, and $f_B=3\mathrm{mm}$. (c) Efficiency (${\eta }_1$) for lens shown in (b), defined as probe energy a within $10\text{−}\mu \mathrm{m}$ radius at $z=0.5\mathrm{mm}$ as $\mathrm{\Delta }n$ is varied between $1\times {10}^{-5}$ and $9\times {10}^{-3}$ for fixed $D$.

Figure 3

Simulations of focusing by thick plasma lenses formed by SVI [(a),(b)] and ion fluctuations [(c),(d)] using a 3D nonlinear envelope equation solver [(a),(b)] and 3D PIC simulations [(c),(d)]. (a) The density of plasma after passage of the pumps (upper half) and the probe (lower half) through a $D=200\mu \mathrm{m}$ column of molecular nitrogen. (b) The probe intensity at the $z=3\mathrm{mm}$ focal spot. In (a),(b), the pumps are 10 fs, 800 nm, 0.65 mJ pulses focused at $z=3\mathrm{mm}$ and $z=0.5\mathrm{mm}$, and the probe is 1 ps, 800 nm, and 60 mJ. In (c),(d), the pumps are 15 TW, $1\mu \mathrm{m}$ pulses with 500 fs duration, the probe is a 100 TW, $1\mu \mathrm{m}$ pulse with 40 fs duration, and the hydrogen plasma has $D=40\mu \mathrm{m}$ with an initial density density $n_e/n_c=0.045$ and electron temperature $T_e=100\mathrm{eV}$. (c) The ion density after both the pump and the probe have passed, and (d) the probe intensity near the focal spot.

Figure 4

Plasma lens efficacy (peak focal intensity normalized by incident probe power) at varied incident probe energy for (a) SVI calculated with the 3D nonlinear envelope equation and (b) the ponderomotive ion mechanism using 2D PIC. (a) Parameters apart from probe energy are the same as in Figs. 3 and 3. (b) Physical parameters are the same as in Figs. 3 and 3, although these simulations are 2D and the resolution is $20\mathrm{cells}/\lambda$ and 10 particles per cell.