The method of laser FMCW measurement has many advantages，such as no blind area and low SNR detection^［1-6］. Compared with the current injection semiconductor laser frequency modulation technology，the frequency modulation bandwidth of the external cavity laser is large；thus，it can achieve high-resolution measurement in theory^［7-10］. However，to ensure the stability and debugging convenience of the system，the optical system usually adopts the optical fiber structure. When combined with a broadband tuning light source，the dispersion mismatch effect will occur，which results in distortion of the peak profile of the target spectrum and shift of the peak position.

To eliminate the dispersion mismatch effect of the system，Zeb W. Barber analyzed the influence of optical fiber dispersion of the reference interferometer on ranging in the active chirp linearization method. The dispersion causes the beat frequency of the reference interferometer to change with the frequency modulation bandwidth，which leads to a change in the frequency modulation slope，widens the target spectrum peak and limits further improvement in the extent of the range. To solve the dispersion problem，Barber proposed introducing the chirp frequency into the beat frequency of the reference interferometer to counteract the dispersion effect ^［11-12］. Gan Yu et al. studied the phenomenon of dispersion mismatch in broadband laser frequency sweep interferometry in detail. They constructed a theoretical model of dispersion mismatch and further proposed adaptive dispersion phase compensation and chirp decomposition dispersion correction methods，the full-width at half-maximum（FWHM）of the compensated target spectrum being close to the theoretical resolution^［13-15］. Dispersion mismatch also exists in swept OCT^［16］. Lippok N et al. proposed that the group velocity dispersion coefficient of materials be measured by the fractional Fourier transform（FrFT）^［17］. The group velocity dispersion coefficient and symbols are obtained by optimizing the fractional Fourier order，and then the dispersion compensation is completed. When estimating the fractional order of the FrFT，the relationship between the iteration times of the algorithm and the optimal compensation coefficient must be balanced.

The above method requires calibration of the dispersion coefficient or iterative compensation in advance. In this paper，by studying the dispersion law of the measurement system，the method of dispersion cancellation by forward and reverse tuning is proposed. It can avoid estimation of the dispersion coefficient of the system in advance or the iterative compensation process，which can improve the compensation efficiency.

Ideally，the laser frequency performs linear tuning，and the target distance is calculated by detecting the beat frequency signal generated by the local and transmitted path light. In practice，it is difficult for lasers to achieve perfect linear tuning，which leads to broadening of the target spectrum. To eliminate this effect，an auxiliary interferometer is constructed for tuning the nonlinear correction. A structure diagram of the optical path is shown in Fig. 1.

The external cavity laser is tuned linearly. The output light is divided into two channels after passing through the optical isolator and coupler 1. Part of the light passes through coupler 2. It then returns to form the beat signal with the local path light at balanced detector A. The rest of the light passes through coupler 3. It then passes through the fiber-optic Mach-Zehnder interferometer with an unequal arm length to form the auxiliary interferometer beat signal on balanced detector B. The measured beat signal is synchronously sampled by the zero-amplitude crossing point of the auxiliary interferometer signal. Then，the sampled signal is changed to a sine signal，and the frequency of the sampled signal can be calculated by a Fourier transform. The target distance is calculated by combining the optical path length of the auxiliary interferometer.

The sampled signal^［13］can be expressed as：

where τair and τaux are the time delays of the measurement and auxiliary interferometers，respectively. When the time delay of the auxiliary interferometer is known by calibration，the optical path of the target can be obtained by FFT of the sampled signal.

In practice，due to the use of a broadband tuning light source，it is necessary to consider the influence of optical fiber dispersion^［18］ of the auxiliary interferometer on the measurement. Then，the beat frequency formed by the auxiliary interferometer is as follows^［11］：

where

where Raux denotes the fiber length of the auxiliary interferometer and μ is the slope of the linear tuning. β1=1/υg，and υg denotes the group velocity. The dispersion coefficient is β2=dβ1/dω. Oζn refers to an error of order ζn.

After the measurement signal is resampled by the signal clock of the auxiliary interferometer，the measurement beat frequency is derived as follows.

The measurement optical path is primarily in air，and the dispersion coefficient of air can be ignored. Therefore，the ratio of the time delay by the measurement and auxiliary interferometer can be expressed as Eq. 4：

Because Δτaux/τaux<<1，the higher-order terms in Eq. 4 can be ignored with a negligible loss in accuracy. The higher-order terms of Eq. 4 are denoted by Oζn.

Here，

where Δτaux denotes the time delayed variation on the auxiliary interferometer caused by the fiber dispersion effect. After deducing the above process，the sampled signal is as follows^［13］：

If μ<0，

If μ>0，

where，PT，PL and ηH represent the transmitted light，local light power and heterodyne interference efficiency，respectively.

To reduce the dispersion effect，we proposed a method of dispersion cancellation by forward and reverse tuning. The principle is as follows.

The Hilbert transform^［19］ is used to extract the phase of the measured signal under forward and reverse tuning，and phase unwrapping is carried out. Furthermore，the phase of the signal for dispersion elimination can be averaged by summing the phases of the two signals. The formula is as follows：

where δdisp=τairτaux2πβ2β1. Use Eq.（8） to reconstruct the measured signal：

Then，Eq.（9） is the measured signal after eliminating the influence of dispersion，and the target distance can be obtained by further ChirpZ^［20］ transformation of the signal. The ChirpZ transform can be used to subdivide the frequency spectrum. Its effect is equivalent to the zero filling Fourier transform，but the algorithm is more efficient.

The external cavity laser was set to triangle wave tuning mode. It tuned from 1552 nm to 1542 nm（corresponding to forward tuning）and from 1552 nm to 1562 nm（corresponding to reverse tuning）. The output power and tuning speed of the laser were 1.5 mW and 100 nm/s，respectively. The optical path of the auxiliary interferometer was 220 m. The target was placed on an air flotation optical platform. The acquired time domain measured signal diagram is shown in Fig. 2. The spectrum of the measured signal after application of the Fourier transform is shown in Fig. 3. The optical path of the fiber end face was 4.524271 m，and part of the target in air is shown in the diagram.

To study the dispersion effect of the measurement system，the measured signal formed by forward and reverse tuning of the laser was divided into five sections. Then，the ChirpZ transformation was carried out for each section of the signal，and the corresponding distance was calculated. The results are shown in Fig. 4. In the case of forward tuning，the ranging value of the target increased linearly with increasing tuning range. In the case of reverse tuning，the ranging value decreased linearly with increasing tuning range. The above phenomena were caused by dispersion mismatch，which resulted in the spectral peak of the target shifting with increasing tuning range.

A Hilbert transform was used to extract the phase of the measured signal under the conditions of forward and reverse tuning. Then，the phase was unwrapped and fitted linearly. Figure 5 shows that the residual phase was a curved line that opened upward in the case of forward tuning，indicating that the beat frequency was not a constant value and changed with the tuning range. The residual phase after linear fitting of the measured signal phase in the case of reverse tuning is shown in Fig. 6. As shown，the residual phase appeared as a downward curved line. The phase of dispersion cancellation by forward and reverse tuning is shown in Fig. 7. The residual phase fluctuated around the value of zero，indicating that the dispersion effect of the measurement system was eliminated.

The measured signal was reconstructed by the phase of dispersion cancellation and divided into five sections. Then，the target distance of each section was extracted by the ChirpZ transform. The results are shown in Fig. 8. Clearly，the ranging value（green star symbol in the figure）remained basically horizontal with increasing tuning range，indicating that the system dispersion was eliminated.

The whole measured signal after dispersion cancellation was extracted by the ChirpZ transform. The distance profile of the target peak before and after dispersion cancellation is shown in Fig. 9. The target peak of the forward and reverse tuning had some degree of distortion，which was caused by system dispersion. The distortion of the target peak（green solid line）after dispersion cancellation was little.

In this paper，an external cavity tunable laser FMCW system is constructed. The theoretical model of dispersion mismatch is established. The theoretical and experimental results show that the beat frequency of the measured signal changes with time due to dispersion，which leads to the distortion of the target spectrum peak profile and measurement instability. To reduce the influence of dispersion，a dispersion mismatch model of the forward and reverse tuning of the external cavity laser is established. It is found that the dispersion has symmetry under forward and reverse tuning. According to this characteristic，dispersion cancellation is realized by summing and averaging the phases of the forward and reverse tuned signals. The advantage of this method is that it does not require estimation of the dispersion coefficient nor the carrying out of iterative compensation. The measurement is completed by a single compensation，which can improve the efficiency of the dispersion compensation.