Optical beaming of electrical discharges

Igniting and guiding electrical discharges to desired targets in the ambient atmosphere have been a subject of intense research efforts for decades. Ability to control discharge and its propagation can pave the way to a broad range of applications from nanofabrication and plasma medicine to monitoring of atmospheric pollution and, ultimately, taming lightning strikes. Numerous experiments utilizing powerful pulsed lasers with peak-intensity above air photoionization and photo-dissociation have demonstrated excitation and confinement of plasma tracks in the wakes of laser field. Here, we propose and demonstrate an efficient approach for triggering, trapping and guiding electrical discharges in air. It is based on the use of a low-power continuous-wave vortex beam that traps and transports light-absorbing particles in mid-air. We demonstrate a 30% decrease in discharge threshold mediated by optically trapped graphene microparticles with the use of a laser beam of a few hundred milliwatts of power. Our demonstration may pave the way to guiding electrical discharges along arbitrary paths.


Supplementary
. Calculated breakdown voltage for a single particle heated up to 2000K. These results extend Fig.2(e) of the main manuscript and demonstrate that the breakdown voltage can be reduced by more than 3 times. The particle surface temperature changes from room up to 2000K, corresponding to the thermal ionization threshold.

Supplementary Note 1. THEORETICAL MODEL FOR CALCULATION OF THE MODIFIED BREAKDOWN VOLTAGE
The discharge between two electrodes under normal conditions is can be described by Paschen's law that was determined empirically and later explained by Townsend. Specifically, Townsend studied exponential current increase due to electron avalanche and showed that this current can be self-sustained in the channel (i.e., a breakdown condition) if the following certain criteria is met: where is the so-called first Townsend coefficient that describes electron avalanche (i.e., exponential current increase, = 0 ), is the distance between electrodes, is the so-called second Townsend coefficient that relates physically describes probability of secondary electron emission from the electrodes. Since usually ≫ 1 the Townsend breakdown condition can be simplified to ≃ 1 (2) By definition the first Townsend coefficient, , describes number of ionisations per length of path. It can be presented as a product of number of collisions per unit length and the ionization probability: where 1 e corresponds to the number of collisions per unit length and is related to the electron mean free path λ e ; and −E i /E e is the probability of ionization, i.e., the probability of an electron energy E e = e ℰ exceeds energy of ionization E i ; here -elementary charge, and ℰ = V/ is the electric field between the electrodes, V is the applied voltage. It allows to express the first Townsend coefficient in the following form By considering the first Townsend coefficient as a function of the mean free path, ( e ), and expanding it around a give value e It allows to estimate the strong dependence on mean free path variation. In particular, by substituting all parameters for our experimental setup, =1cm, E i =13eV, e 0 = 90nm, and V=34kV, the values of the first two coefficients are 1 ≈ 10 6 and 2 ≈ 10 13 . This result explains strong dependence of the coefficient on the mean free path variation, shown in Fig.  1c of the main manuscript.
In these expressions, ionization energy, E i , and second Townsend coefficient, , are constants related to the properties of gas and electrodes. Mean free path however depends on local thermodynamics parameters -pressure, p, and temperature, T. Therefore, by controlling it one can control first Townsend coefficient, , and the breakdown voltage, V app . Assuming that at thermodynamic equilibrium pressure and temperature variations occur at distances that are significantly larger than the man free path (which is a natural assumption as pressure and temperature are macroscopic quantities), we may find an expression for the mean free path assuming that locally (i.e., in a small but macroscopic volume) an ideal gas law is satisfied (p = n T, where n is the gas density). By definition: where is the scattering cross-section and is the local coordinate. In our case pressure is constant (p=1atm) and only temperature varies.
Using Supplementary Equations (4) and (6) we can find an expression for the first Townsend coefficient as a function of temperature: By using the axial symmetry of the problem, for a nonuniform Townsend coefficient, ( ), the Townsend breakdown condition may be found as: where we ∫ corresponds to the integration along the path that maximizes breakdown probability, i.e., along the temperature gradient. By knowing the temperature distribution T( ) and second Townsend coefficient, , we solve Supplementary Equation (8) for the applied voltage V, that gives a breakdown voltage, V = V b . To find the value of the second Townsend coefficient we study breakdown condition for a regular parallel plate capacitor under normal conditions. In this case, from Supplementary Equations (2) and (7) we obtain: where and are constant for a given gas (A=112.5 (kPa·cm) −1 and B=2737.5 V (kPa·cm) −1 at room temperature T=300K). V b0 is the breakdown voltage under normal conditions for a given distance between electrodes . In our experiments we apply voltage V = 32 and find distance at which breakdown occurs. We compare these values with table data for air to ensure that our experiment yields a correct result. We then use Supplementary Equation 9 to find coefficient .
For completeness, we extended the calculations for a case of a single particle up to 2000K, which roughly corresponds to the thermal ionization threshold. The results shown in Supplementary Figure 3 indicate that voltage breakdown can be reduced up 3.5 times for various electrodes separations.

Supplementary Note 2. TEMPERATURE DISTRIBUTION OF AIR AROUND A HOT PARTICLE
In general, we are dealing with a non-uniform temperature distribution of air around the hot particle. For simplicity, we consider a single spherical particle with the uniform surface temperature. We also assume that the particle is trapped in the vortex beam for sufficiently long to be in thermodynamic equilibrium. Then the distance-dependent temperature profile outside the particle can be readily approximated as where R is the particle radius, is the surface temperature, determined experimentally, and 0 is the ambient temperature. The temperature distribution for several particles was simulated numerically by using Lumerical HEAT solver.

Supplementary Note 3. ELECTRICAL DISCHARGE GUIDING WITH MULTIPLE TRAPPED PARTICLES
The ability to trap multiple particles between the electrodes creates additional conditions for a discharge guidance along substantially longer paths and channelling the electrical discharge along the particle positions. Supplementary Figures 2 a-c detail the mechanism for discharge channel formation with multiple particles trapped between electrodes. If the particles size is smaller than the diameter of the channel in a tractor beam, multiple particles trapped in the beam are heated to approximately the same temperature, Ts. The heated particles create a hot channel -a virtual "tube" in mid-air -with an elevated average temperature and longer electron mean free path. The uniformity of the channel depends on the interparticle spacing (Supplementary Figure 2 a and b), and the size the channel is approximately comparable with the tractor beam diameter and is defined by the mean lateral temperature profile. This channel creates a favourable path along which the discharge is guided. Supplementary Figure 2 c shows the calculated maximum distance at which breakdown is observed for multiple trapped particles with various temperatures and interparticle spacing values. Clearly, for a given voltage as the number of particles injected between the electrodes increases the maximum distance of the discharge also grows substantially.