Polarization properties of E-FISH signals and optimization of simultaneous measurement of electric field vectors

Electric field measurements based on the electric field induced second harmonic (E-FISH) method have been employed in a wide range of studies. Most studies typically measure two components of the electric field separately. Although there have been proposals for the simultaneous measurement of electric field vectors, the imbalance of the two corresponding E-FISH signals has limited its application. Furthermore, the relationship between the polarization of the E-FISH signal and the direction of external electric field remains unclear. In this paper, the general expressions for the polarization and power of both components of E-FISH signals are derived, assuming arbitrary probe beam polarization and external electric field direction. The theoretical results indicate that the polarization of E-FISH signals varies along the interaction length. The final signal’s polarization is elliptically polarized for arbitrary electric field distribution and is correlated with the polarization of the probe beam, which deviates from what is commonly assumed to be consistent with the external electric field. If the polarization of the probe beam is not parallel to the axes, the power of each signal component is determined by both components of the external electric field, which lays the foundation for the simultaneous measurement of electric field vectors. This theoretical prediction is subsequently validated by experimental results. Finally, the power maps suggest that the optimal polarization angle of the probe beam is 45° or 135° to achieve a balanced signal power when measuring an unknown electric field vector. Both components of the electric field can be simultaneously obtained according to the theoretical relationship.