Diagnostics of the electric field vector is crucial for the investigation of surface streamer discharges and their transition mechanisms. The electric-field-induced secondary harmonic method has the advantages of high spatial and temporal resolution

non-intrusiveness and a wide range of applications. In this paper

the method of simultaneous measurement of electric field vectors is proposed to satisfy the strong demand for the electric field diagnostics of surface streamer discharge. Firstly

the general expressions for the polarization and power of E-FISH signals are derived

assuming arbitrary probe beam polarization and external electric field direction. The theoretical relationships between the power of both components of E-FISH signals and the external electric field vector are analyzed. This theoretical prediction is validated by experimental results. Furthermore

the polarization properties of E-FISH signals are analyzed based on the expressions and the method of simultaneous measurement is proposed. Finally

based on the measurement method and platform

the spatial and temporal evolutions of surface streamer discharge are obtained. Comparison with the results measured separately shows the accuracy and applicability of this method. The method has the potential to be applied in implementation of the measuring electric field vectors in unstable streamer discharges along surfaces.