Electro-optic modulators can change the phase and polarization state of incident light, therefore having a wide range of applications in many fields, such as optical communication, integrated optics, and super-resolution microscopy. They feature fast response and reliability. However, due to the differences among electro-optic modulator devices, the relationship between the applied voltage and the phase change in actual operation is not consistent with the corresponding relationship in the technical manuals. Meanwhile, when modulators are working at different wavelengths, applying the same voltage to the modulator results in different phase changes. Therefore, before utilization, it is necessary to calibrate the relationship between the phase change and voltage change of the electro-optic modulator. Since its modulation is linear, the slope of the function only needs to be obtained for subsequent experiments.

Common calibration methods include the contour method and the Michelson interferometry method. The contour method employs two optical paths for interference, one of which passes through an electro-optic modulator to change the phase. Interference fringes are taken at a certain voltage interval. The displacement between two fringes is divided by the fringe period and then multiplied by 2π to obtain the phase difference which is combined with the voltage interval to get the half-wave voltage. This method is simple but requires repeated adjustment and correction, which is time-consuming and laborious. Additionally, the limitation of camera pixels reduces the accuracy. The Michelson interferometry method passes one of the interferometer arms through an electro-optic modulator, applies a voltage to produce a phase shift, and then moves this arm to change the optical path difference and make the interference fringes disappear to determine the phase difference. This method has high accuracy but requires optical path rebuilding with too much consumed time.

To quickly and accurately calibrate the half-wave voltage of an electro-optic modulator without rebuilding an optical path, we study a method using cross-correlation in the frequency domain. The complex conjugate of the high-order spectrum of the previous interference fringe is multiplied by the high-order spectrum of the subsequent fringe to calculate the phase difference for calibrating half-wave voltage.

We adopt the phase angle of the cross-correlation function between multiple fringe images to determine the phase difference. After converting the fringe illumination light to the frequency domain, the high-order spectrum is extracted. In the spectra of multiple fringe patterns, the conjugate of the high-order spectrum of the previous one is multiplied by the next one to obtain the cross-correlation function of adjacent images. The angle of this function is the phase difference between two fringe images. The feasibility of this method is verified by generating fringe images with the same phase difference in Matlab. The optical path for experimental calibration is part of a structured illumination super-resolution microscopy system. This system achieves high-speed imaging based on electro-optic modulators and galvanometers. One optical path passes through a phase electro-optic modulator, while the other does not. Finally, the two beams interfere at the camera through a mirror to form fringes.

The experiment applies voltage to the EOM through a computer-controlled acquisition card, with a voltage interval of 1 V and a voltage range of -10 V to 9 V. When the voltage is changed each time, the camera is controlled to acquire an image and save it. A total of 20 interference fringe images with equal interval displacement are collected. The 640 nm and 561 nm lasers are utilized for calibration, and the calibration results of the Michelson interferometry method serve as the correct results for accuracy consideration. To further eliminate the errors caused by interference, we take nine sets of fringe images for each laser wavelength, calculate the average value of the nine sets of results, and then compare this value with the accurate calibration value. The half-wave voltage obtained by calibrating the 640 nm using the Michelson interferometry method is 6.6 V, and the result obtained using this method is 6.57 V, with a difference of 0.03 V and an error of 0.45%. The half-wave voltage obtained by calibrating the 561 nm using the Michelson interferometry method is 5.87 V, and the result obtained by this method is 5.84 V, with a difference of 0.03 V and an error of 0.51%. After converting to phase difference, the phase difference calculated for 640 nm is 0.478 rad, the standard phase difference is 0.476 rad, and the difference is 0.002 rad. The phase difference calculated for 561 nm is 0.538 rad, the standard phase difference is 0.535 rad, and the difference is 0.003 rad. By employing the contour method to process the 640 nm image, the obtained half-wave voltage is 6 V, which has a larger error than the result obtained by this method. The half-wave voltage obtained by our method is close to that obtained by the Michelson interferometry method, with the same accuracy and faster speed.

Before adopting an electro-optic modulator, it is often necessary to calibrate the half-wave voltage. Previous methods such as Michelson interferometry require additional optical path construction, and the calibration process is slow and easily interfered by noise and jitter. Thus, the contour method is not accurate enough. Therefore, a method based on the high-order cross-correlation of the interference fringe frequency domain is proposed to calculate the phase difference between fringe images and calibrate the half-wave voltage of the electro-optic modulator. This method employs a specific mask to remove the 0th-order spectrum and extract the high-order spectrum. The complex conjugate of the high-order spectrum of the previous image is multiplied by the high-order spectrum of the next image to obtain the angle and then solve for the phase difference. The phase error in actual detection reaches 0.002 rad and the half-wave voltage error is 0.03 V, which meet the calibration requirements of electro-optic modulators. Since the proposed method has a large calibration speed and does not require optical path rebuilding, it can check whether the electro-optic modulator drifts at any time and whether corrections are needed or not.