Electric field measurements in plasmas: How focusing strongly distorts the E-FISH signal

Electric field induced second harmonic generation (E-FISH) has recently demonstrated significant potential as a method for making absolute electric field measurements in non-equilibrium plasmas and gas discharges. Previous studies have relied on the plane-wave approximation in quantifying these measurements, while in reality, focused laser beams are almost always used. In this work, we perform a theoretical and experimental study using focused Gaussian beams, and examine the consequent effects on the E-FISH signal. We show that in addition to important parameters such as the external electric field strength, wave vector mismatch and Rayleigh range, the signal is strongly influenced by the full length and shape of this external field profile. We attribute this to the Gouy phase shift associated with focused beams, and note that analogous effects have been previously observed in second and third harmonic generation microscopy. This dependence of the E-FISH signal on the spatial profile of the external field is worth highlighting since it is often not easily determined a priori in a plasma, and neglecting its influence could lead to an incorrect electric field measurement. To minimize any inaccuracies associated with this issue, we propose several recommendations to consider when using the E-FISH diagnostic with focused beams.