Objective The synthetic wavelength method based on a femtosecond optical frequency comb has been widely used in high-precision long distance ranging systems owing to the capability to measure absolute distances, traceability to the length standard and simple setup. However, optical power variations will cause phase shift variations in photodetection. This phenomenon, which is frequently referred to as the power-to-phase conversion (PPC) effect, will eventually lead to distance measurement errors, thus deteriorating the precision of the ranging system. The conventional methods, such as phase bridge measurement and impulse response measurement, usually focus on reducing the PPC effect and generating ultralow phase noise, ultrahigh stable microwave signals by intermode beating from a femtosecond frequency comb. However, it still lacks a comprehensive research about the influence of PPC on the ranging errors of a femtosecond laser ranging system and the corresponding correction technique. In the present study, a polynomial fitting correction method is proposed to improve the precision of the ranging system. Combined with the phase ranging method, a correction look up table (LUT) is formed by referencing to a length standard and adopting the least-square based polynomial fitting. We believe that our work can extend the femtosecond laser based high-precision ranging technique to be applied to outdoors, complicated industrial environments or even non-cooperative targets, which significantly broadens its application area.

Methods In this study, a repetition-rate-locked femtosecond laser is used as the laser source. First, the Michelson-like interferometer is established, and optimal experimental parameters are determined by studying the PPC effect of different synthetic radiofrequency (RF) signals consisting of the repetition rate and its high-order harmonics via intermode beating of the femtosecond laser after photodetection. Then, using the fourth order harmonic as the fine ruler, phase shifts are extracted and investigated as a function of the optical power based on fast Fourier transform (FFT). Combined with the phase ranging method, a correction LUT is formed by adopting the least-square based polynomial fitting with different degrees. Finally, comparisons of corrected and uncorrected ranging results are made to verify the effectiveness of the proposed correction method. By comparing the distance errors after corrections with polynomial fitting of different degrees, we have determined that using the 4th degree correction method can obtain the best correction performance. In addition, by comparing corresponding residual errors of corrected and uncorrected distances versus linear stage displacements after applying linear fit in the measurement range, the proposed correction method is further proven to be very effective in improving the precision of the femtosecond laser ranging system.

Results and Discussions A custom-made, 200MHz repetition-rate-locked all-polarization-maintaining femtosecond fiber laser referenced to a highly stable frequency standard is used in the ranging system. With the increase of incident optical power, we have investigated the RF power of different beat notes and identified three operation zones for the applied photodiodes (FGA015, Thorlabs), i.e. the linear regime for low optical power, the saturation regime and above saturation regime for high optical power (Fig.2). The latter two are usually classified as the non-linear regime. The results demonstrate all the beat notes are under the linear regime if the optical power is lower than 2.3mW. Besides, we have also investigated PPC coefficients of different beat notes in detail (Fig.3). The overall results show that the PPC effect remains at a relatively low level for all beat notes when the optical power is less than 2mW. Considering the low PPC effect and high signal-to-noise ratio for high precision distance measurement, the incident optical power is chosen to be 2mW. Then, using the 800MHz (fourth order harmonic) RF signal as the fine ruler, we have formed a correction LUT under different RF power levels (from -19.55dBm to -10.87dBm) using polynomial fitting of 2nd degree, 3rd degree, and 4th degree by referencing the calculated distance results to a length standard of 10mm (Fig.4). By comparing the distance errors after corrections with polynomial fitting of different degrees, we have found the 4th degree correction method can achieve a higher precision. Specifically, the errors are reduced to ±0.05mm for the 4th degree polynomial correction method while the range error correction results of the 2nd and 3rd degree polynomial correction are both -0.15 mm to 0.1 mm (Fig.5). In addition, we have tested the effectiveness and feasibility of our correction method by comparing corrected and uncorrected distance results at different incremental displacements of a high precision linear stage. After linear fit of corrected and uncorrected distances versus linear stage displacement in the range of 110mm, the residual errors can be significantly reduced from ±0.25mm to ±0.08mm (Fig.6), and the linear correlation coefficient is increased from 0.999989 (uncorrected) to 0.999998 (corrected).

Conclusions Aiming at the error caused by the power-to-phase conversion (PPC) in high-precision absolute distance measurements using the femtosecond laser synthetic wavelength method, a 4th degree polynomial fitting correction method is proposed to improve the precision of the ranging system. In this study, a Michelson-like interferometer is established, and synthetic radiofrequency beat signals are achieved via intermode beating of the femtosecond laser after photodetection. Then, phase differences are extracted and investigated as a function of the optical power based on FFT. Combined with the phase ranging method, a correction LUT is formed under different optical power levels by adopting the least-square based polynomial fitting method. Experimental results show that using the fourth order harmonic, the measurement distance error has a slope of 2.7mm/mW with the optical power ranging from 1mW to 3mW without correction, while the residual error range can be significantly reduced from ±0.25mm to ±0.08mm in 110mm measurement range. This verifies the effectiveness of the proposed correction method. Meanwhile, we have qualitatively investigated the transport of electron-hole carriers in the intrinsic part of the junction, indicating the generality of our correction method for other semiconductor photodiodes such as avalanche photodiodes (APDs) and positive-intrinsic-negative (PIN) photodiodes. Consequently, our present work demonstrates that the femtosecond laser based high-precision ranging technique can be potentially extended to outdoors, complicated industrial environments where the optical power variations are large, which will significantly promote its real industrial applications, such as advanced large-volume manufacturing, assembly and precise shape, and dimensional measurement of workpieces.