Laser heterodyne interferometry is widely used in applications such as vibration measurement, velocity measurement, and displacement measurement. Traditional methods for eliminating the effect of laser frequency noise on measurements have mainly focused on reducing the laser noise itself. To suppress the effect of laser frequency noise on heterodyne interferometric signal, we study the model of laser frequency noise’s effect on the heterodyne interferometric signal, simulate the resulting tendencies and suppression extent with compensation fibers, and finally validate the suppression method through experimentation.

A typical laser heterodyne interferometric structure is introduced, and the transitive relation from laser frequency noise to displacement noise is derived mathematically. A simulation system is set up, with input parameters including laser wavelength, laser emission power, laser frequency noise, coupler ratio, modulator frequency, photodetector gain, sampling rate, sampling time, and the passband and stopband frequencies of the low-pass filter. The variable is the measuring optical lengths, which are successively set to 0, 10, 20, and 30 m. The outputs are the power spectral densities of displacement. Three conditions are set: shot noise only, laser frequency noise only, and both shot noise and laser frequency noise considered. Root mean squares (RMSs) of power spectral densities in certain frequency ranges are calculated for the three conditions. The results are presented in a graph to show the trend. Similar work is conducted to show the graph of RMS-laser frequency noise at 0, 10, 20, and 30 m optical lengths. Compensation for the 30 m optical length fiber is simulated. The experiment is conducted using fibers of different optical lengths. The delay fibers with optical lengths of 30 and 154.1 m are used to simulate the real measuring optical length in air. Compensation results are recorded for comparison.

The model analysis shows that in the non-compensation situation, the measuring optical length causes a significant time lag in light propagation. Mathematical derivation indicates that the power spectral density of displacement is directly proportional to the power spectral density of laser frequency and the square of the time lag. The results imply that differences in optical length, causing propagation time lag, may be a significant error source in the laser heterodyne interferometer. Thus, by compensating for the optical length difference, the error could be suppressed. Simulation of the power spectral density of displacement at different measuring optical lengths shows that the total noise level increases as the optical length increases. The shape of the power spectral density changes distinctly, which indicates that the effect of optical length change differs across frequency ranges. In the case of long optical lengths, the power spectral density graph consists of a main lobe and several side lobes. The 0.9?1.1 MHz results of RMS show that shot noise does not change significantly as the optical length increases. Meanwhile, in the case of long optical lengths, the system is mainly affected by laser frequency noise, and shot noise becomes negligible. When laser frequency noise is considered, RMS increases by $\mathrm{0.01}\mathrm{p}\mathrm{m}?\mathrm{H}{\mathrm{z}}^{-\mathrm{1}/\mathrm{2}}$ as the optical length increases by 1 m. The 9?11 MHz results of RMS show that noise increases first and then decreases as the optical length increases from 0 to 30 m. 15 m has the largest value of $\mathrm{0.08}\mathrm{p}\mathrm{m}?\mathrm{H}{\mathrm{z}}^{-\mathrm{1}/\mathrm{2}}$. The 2?30 MHz results of RMS show that the RMS speed decreases as the optical length increases. The RMS-laser frequency noise graph indicates that laser frequency noise has little effect, and displacement noise does not increase as laser frequency noise increases at 0 m optical length. When the optical length is greater than 0 m, displacement noise increases linearly as the laser frequency noise increases. The slope of the line depends on the optical length and increases as the optical length increases. Simulation of compensation with a 30-m optical length fiber shows that displacement noise is efficiently suppressed, and the total noise is reduced to the shot noise level. The experimental results are in accordance with a deduction from derivation and simulation. In the 30-m experiment, the RMSs of 1?5 MHz are respectively 0.031, 0.096, and $\mathrm{0.028}\mathrm{p}\mathrm{m}?\mathrm{H}{\mathrm{z}}^{-\mathrm{1}/\mathrm{2}}$ in the conditions of no delay fiber, with delay fiber, and with delay fiber and compensation fiber. In the 154.1-m experiment, they are 0.027, 0.106, $\mathrm{0.024}\mathrm{p}\mathrm{m}?\mathrm{H}{\mathrm{z}}^{-\mathrm{1}/\mathrm{2}}$. The results validate the suppression method.

We demonstrate the analysis of the effect of laser frequency noise on the heterodyne interferometric signal and the method for its suppression. Theoretical derivation is conducted to obtain the relationship between the power spectral density of laser frequency noise and that of displacement. The power spectral density of displacement is used to represent the magnitude of the noise effect. A numerical simulation is conducted to present the trend of the effect. The results show that displacement noise increases as laser frequency noise increases, and similarly, displacement noise increases as the detecting optical length increases. In the simulation, detecting optical length is set to be 15 m and laser frequency noise is set to be $\mathrm{3}\mathrm{H}\mathrm{z}?\mathrm{H}{\mathrm{z}}^{-\mathrm{1}/\mathrm{2}}$. After adding the compensation fiber of 30-m optical length, the RMS of power spectral density of displacement between 2 and 30 MHz decrease from 0.08 to $\mathrm{0.02}\mathrm{p}\mathrm{m}?\mathrm{H}{\mathrm{z}}^{-\mathrm{1}/\mathrm{2}}$ in the simulation. The experiment using 154.1 m compensation fiber is done for compensating the effect of delay fiber. The RMS of power spectral density of displacement decreases from 0.106 to $\mathrm{0.024}\mathrm{p}\mathrm{m}?\mathrm{H}{\mathrm{z}}^{-\mathrm{1}/\mathrm{2}}$. The experiment validates the method for suppression. It is implied that inserting delay fiber can sufficiently suppress the impact of laser frequency noise in heterodyne interferometry.