We theoretically and experimentally investigate the accuracy with which the magnitude and direction of the electric field (E-field) vector of electromagnetic waves can be determined using the crystal angle dependence of the electro-optic (EO) effect. The mathematical treatment in this paper is a large extension of our previous work to determine the E-field direction of terahertz electromagnetic waves by the spinning EO sensor method [Rev. Sci. Instrum.83, 023104 (2012)]. Here we include misadjustments of the wave plate and polarizer in the experimental setup as well as the effect of the residual birefringence of the EO crystal due to uniform and local strains. The main results are as follows: (1) When there is no residual birefringence in the EO crystal, misadjustments of the wave plate and polarizer do not affect the experimentally determined direction of the E-field vector. This is true even when the term proportional to the square of the E-field magnitude of the EO signal becomes important. (2) The error due to residual birefringence can be effectively eliminated by a signal subtraction algorithm and it is roughly the product of the misadjustment angle of the wave plate and the degree of residual birefringence, which is very small. (3) The error does not depend on the magnitude of the E-field; thus, we can apply the technique when the E-field is weak and the polarization rotation of the probe pulse caused by the EO effect is much smaller than that induced by residual birefringence. These results give a mathematical basis for the accuracy and reliability of the spinning EO sensor method, which is robust, and will be useful for ultrabroadband E-field vector sensing at far-infrared to mid-infrared frequencies.
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