Nonlinear error correction for FMCW ladar by the amplitude modulation method

Tong Zhang; Xinghua Qu; Fumin Zhang

doi:10.1364/OE.26.011519

1. Introduction

In contrast with pulsed time of flight system, FMCW enables excellent spatial resolution alongside with high SNR. This is possible since in FMCW the resolution is determined solely by the frequency sweep range, which is independent of the duration of the interrogation pulse of time of flight system. Thus, using longer and more energetic pulses, it is possible to increase the SNR without sacrificing spatial resolution of FMCW. These benefits make FMCW technology has been used in the long range lidar system [1]. In the field of fiber measurement, FMCW technology has been used in optical frequency dominant reflectometry(OFDR) for the measurement of different parameters in optical fiber, such as group velocity dispersion [2], grating characterization [3] and distributed temperature sensing [4], FMCW has also been combined with a raster sweep system for medical high-resolution 3D imaging which is known as OCT technology [5].

The FMCW range measurement system is mainly comprised of a frequency sweep continuous wavelength laser device and an interferometer with a fixed reference path and a measurement path. The frequency swept laser is split into two paths through the interferometer and then combined on a photo-detector to generate an interference signal, while the instantaneous frequency of laser varies linearly with time during at least some fraction of the interrogation period. The interferometer signal is a sinusoidal signal, whose frequency is $f_{F M C W} = α τ = 2 α n R / c$ , where α is the laser’s frequency sweep rate, τ is the relative time delay between the measurement path and reference path, c is the speed of light in vacuum, n is the refractive index in the air. R is the measurement distance corresponding directly to the time delay τ, which can be found by Fourier transforming. The spatial resolution is determined directly by the frequency resolution of FFT algorithm, $δ f_{F M C W} = \frac{1}{T}$ , where T is the sweep duration.. Thus the spatial resolution can be described into $δ R = \frac{c}{2 n α T} = \frac{c}{2 n B}$ , where B is the frequency sweep bandwidth.

FMCW laser range system has a trade-off between frequency sweep bandwidth and laser chirp linearity. Frequency sweep bandwidth determines the maximum achievable range resolution, and chirp linearity is directly related to the achievable range profile of FFT. In theory, the higher spatial resolution of FMCW system can be achieved from the wider frequency sweep scope. However, in practice it’s difficult to design the available lasers that can stabilize their optical frequency sweep rate, due to the mechanical vibration of laser external cavity and the changing of environmental parameters. The increasing of frequency sweep bandwidth will severely distorts the range profile, the peak point of the range profile is overlapped with other noise profiles from the sweep nonlinearity, so that the accurate range is hard to indentify.

Historically, the laser source nonlinear frequency sweeping has been dealt with in hardware and software approaches. The first kind approaches are mainly to focus on the research and design of a tunable laser source with a linear modulation curve [6,7]. The tuning mechanism is often intricate and non-trivially depends on the driving signal to generate a linear frequency modulation output. In many lasers, for example, the tuning mechanism is based on a PZT. In these cases the driving signal must change linearly in time and the use of a linear driving signal is undesired due to the long transient oscillations that follow a jump in the signal. The second kind approaches pay attention on the improvement of optical structure and signal processing algorithm. Some researchers use a reference interferometer actively or passively to correct the nonlinear error result from the fluctuation of laser tuning rate. Active linearization makes use of a reference interferometer to transmit the frequency tuning rate error back to laser drive, forcing the laser optical frequency to be swept linearly [8–12], this method is focused on the design of feedback circuit which is challenged at extremely wide bandwidth and sweep rate because of the rising of laser noise. Passive linearization uses the reference interferometer as a clock signal to resample range signal from time domain to frequency domain [4,13–16], the system with this method is called as resample FMCW system in the following for convenience. The resample FMCW system has two parallel interferemetories, one is used for range measurement, the other one is used to provide auxiliary interference fringe to sample the measurement signal in the post signal processing. The frequency intervals of the sample points are equal with each other. The measurement signal and auxiliary signal are received respectively by two photo-electronic detectors. This system has three comparative deficiencies. First, the light has to be split into two parts to the two parallel interferemetories, weaken the power of light in each of interferometry. Second, the two signals have to be received by two detectors, increasing the complexity of the system, in addition, the two interferometries and detectors will lead to the mismatch of time delay between clock signal and measurement signal, the resample system exists a phase error which distorts the profile of range. Third, the two signals will take up enormous data storage space and decrease the computation efficiency immensely.

As to the drawback of the active linearization and the resample FMCW system, a novel method of amplitude modulation is presented to correct the frequency sweep nonlinear error, a FMCW ranging experiment is performed to verify the feasibility of this approach.

2. Measurement method and theory

As shown in the Fig. 1. the experimental set up is composed of four parts, the amplitude modulation system is our system(green dotted border).The typical resample FMCW system(blue dotted border) is used to compared with our system for analyzing the difference between the two nonlinear correcting methods. The nonlinear FMCW system is used to demonstrate the nonlinear error correcting ability of our system(purple dotted border), The cw laser interferometer is used to evaluate the uncertainty and standard deviation(red dotted border).

Fig. 1 the schematic diagram of amplitude modulation system and the experiment set ups used to estimate our system

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The measurement system based amplitude modulation method is comprised of two tandem fiber Mach-Zehnder interferometers, the first interferometer near the laser source is auxiliary interferometer used to generate a sinusoidal carrier signal, the frequency interval between the peak points of sinusoidal carrier signal is equal with each other. In the second interferometer, the carrier signal is split into signals and combined into a mix beat signal whose amplitude is modulated by the measurement range.

In the light source of amplitude modulation system, the laser is modulated by saw-tooth sweep way, the instantaneous frequency of the laser source can be given by

f (t) = f_{o} + α t, 0 \leq t \leq T .

Where $f_{0}$ is the initial frequency of modulated laser, $α$ is the laser frequency modulation rate, $T$ is the modulation period.

The phase of laser can be given as:

\begin{matrix} ϕ (t) = 2 π \int_{0}^{t} f (ε) d ε + φ_{0} \\ = 2 π f_{0} t + π α t^{2} + φ_{0}, 0 \leq t \leq T . \end{matrix}

Where $φ_{0}$ is the initial phase of the laser.

The electric field of the laser transmitting signal can be defined as

E (t) = A_{o} e^{i [2 π f_{0} t + π α t^{2} + φ_{0}]}, 0 \leq t \leq T .

Where $A_{0}$ is a constant amplitude of the laser.

In the auxiliary interferometer unit, the input light is split into two parts that have relative optical path length difference and recombined to generate beat carrier signal, the delay time of the fiber a is $τ_{a}$ , the delay time of the fiber b is $τ_{b}$ , the time delay difference between fiber a and b is $τ_{a b} = τ_{a} - τ_{b} = \frac{n L_{a b}}{c}$ , $τ_{a}$ > $τ_{b}$ where $L_{a b}$ is the corresponding optical path difference between fiber a and b

$τ_{a b}$ should be double longer than the time delay corresponding to the measurement distance at least by Nyquist criterion.

The signal from the fiber a can be given as:

E_{a} (t) = ξ_{a} A_{o} e^{i [2 π f_{0} (t - τ_{a}) + π α {(t - τ_{a})}^{2} + φ_{0}]}, τ_{a} \leq t \leq T + τ_{a} .

The signal from the fiber b can be given as:

E_{b} (t) = ξ_{b} A_{o} e^{i [2 π f_{0} (t - τ_{b}) + π α {(t - τ_{b})}^{2} + φ_{0}]}, τ_{b} \leq t \leq T + τ_{b} .

Where $ξ_{a}$ and $ξ_{b}$ are the attenuation coefficient of fiber a and b.

The signals from fiber a and b are mixed to generate a carrier signal at the coupler1, ignoring the higher order term, the carrier signal can be given as:

\begin{matrix} C (t) = ξ_{a}^{2} A_{0}^{2} + ξ_{b}^{2} A_{0}^{2} + 2 ξ_{a} ξ_{b} A_{0}^{2} \cos [2 π α t τ_{a b} + 2 π f_{0} τ_{a b}] \\ = A + B \cos [2 π α t τ_{a b} + 2 π f_{0} τ_{a b}] . \\ τ_{a} \leq t \leq T + τ_{b} \\ A = ξ_{a}^{2} A_{0}^{2} + ξ_{b}^{2} A_{0}^{2} \\ B = 2 ξ_{a} ξ_{b} A_{0}^{2} \end{matrix}

In the measurement interferometer unit, the carrier light is divided into two lights again, the one is called measurement light, which goes through the circulator and is forced on the surface of the target, the reflected light is received by the collimating lens and goes out from the circulator. The measurement light can be given as:

\begin{array}{l} E_{m} (t) = ξ_{m} [A + B \cos (2 π α t τ_{a b} + 2 π f_{0} τ_{a b})] e^{i [2 π f_{0} (t - τ_{d}) + π α {(t - τ_{d})}^{2} + φ_{0}]} \\ τ_{d} \leq t \leq T + τ_{d} . \end{array}

Where $ξ_{m}$ is the attenuation coefficient of measurement light and $τ_{d}$ is the time delay of the path through by the measurement light.

The reference light from the fiber c can be given as:

\begin{array}{l} E_{r} (t) = ξ_{r} [A + B \cos (2 π α t τ_{a b} + 2 π f_{0} τ_{a b})] e^{i [2 π f_{0} (t - τ_{c}) + π α {(t - τ_{c})}^{2} + φ_{0}]} . \\ τ_{c} \leq t \leq T + τ_{c} \end{array}

Where $ξ_{r}$ is the attenuation coefficient of the reference light and $τ_{c}$ is the time delay of fiber c. $τ_{d} > τ_{c}$ .

The measurement and reference lights combined with each other to generate a frequency mix signal at the photo-detector. Ignoring the constant term, the mixing signal could be written by:

\begin{matrix} M_{t} (t) = [C + D \cos (2 π α t τ_{c d} + 2 π f_{0} τ_{c d})] [A + B \cos (2 π α t τ_{a b} + 2 π f_{0} τ_{a b})] \\ τ_{a} + τ_{d} \leq t \leq T + τ_{b} + τ_{c} . \\ C = ξ_{m}^{2} A_{0}^{2} + ξ_{r}^{2} A_{0}^{2} \\ D = 2 ξ_{m} ξ_{r} A_{0}^{2} \end{matrix}

Where $τ_{c d}$ is the time delay difference of measurement distance $L_{c d}$ . According to the formula (6), the first term $C + D \cos (2 π α t τ_{c d} + 2 π f_{0} τ_{c d})$ is the modulated amplitude of the carrier signal, the second term $A + B \cos (2 π α t τ_{a b} + 2 π f_{0} τ_{a b})$ is the carrier signal, the frequency of the modulated amplitude signal is related to the time delay of the measurement distance.

The laser instantaneous frequency can be given as $υ = f_{0} + α t$ , the modulated amplitude signal can be extracted at the peak ( $υ = \frac{n}{τ_{a b}}$ ) of the carrier signal:

\begin{matrix} U_{m} (t) = (A + B) (C + D \cos 2 π \frac{τ_{c d}}{τ_{a b}} n) \\ n = 0, 1, \dots, N - 1 . \end{matrix}

Where N is the number of peak point, $\frac{1}{τ_{a b}}$ is the frequency interval between two peak points, $\frac{N}{τ_{a b}}$ is equal to the frequency modulation bandwidth of the tunable laser. The frequency intervals of the discrete points of measurement signal are equal, satisfying the requirement of fast Fourier transform algorithm.

In the case of nonlinear sweep, the instantaneous frequency is given as $υ = f_{0} + α_{0} t + α_{e} t^{2} = f_{0} + α t$ , $α_{0}$ is the constant sweep rate, $α_{e}$ is the first order nonlinear coefficient, at the peak ( $υ = \frac{n}{τ_{a b}}$ ) of the carrier signal, the modulated amplitude signal still can be given as formula(10), it can be observed that the sweep rate is not related to the modulated amplitude signal, this method can avoid the influence of sweep nonlinearity.

At the frequency domain, the frequency of measurement signal is $f_{A} = \frac{τ_{c d}}{τ_{a b}}$ , which is independent of the laser sweep rate $α$ , avoiding the influence of sweep nonlinearity.

The measured range $L_{c d}$ can be given as:

L_{c d} = \frac{n_{f} L_{a b}}{2 n_{a}} f_{A} .

Where $n_{f}$ and $n_{a}$ are respectively corresponding to the refractive index in the fiber and air. $f_{A}$ is the frequency of modulated amplitude signal, which can be given as $f_{A} = \frac{τ_{c d}}{τ_{a b}}$ . And then, the frequency is acquired through the fast Fourier transform algorithm.

According to the formula (10) and (11), the range resolution of the amplitude modulation system can be given as:

δ L = \frac{c}{2 n_{a} N \frac{1}{τ_{a b}}} .

3. Experiment and result

As shown in the layout of Fig. 1. the experimental system of amplitude modulate system was set up using single-mode polarization-maintain fiber for avoiding polarization fading of the interference fringes, the optical source was external cavity tunable laser with the wavelength of 1550nm for using in the C telecommunications band. The optical frequency was set to sweep as saw-tooth way at sweep rate of 20nm/s over sweep frequency bandwidth of 1540nm-1560nm. The light was split by a 50/50 splitter and combined on the coupler1 with 90/10 splitting ratio, the 90% of the light from the coupler1 was measurement light to the target, the rest 10% was reference light through the fiber c, the measurement light was focused to a spot with diameter of 60μm by a collimating lens. The measurement and reference light was combined on the coupler2 with 50/50 splitter ratio. The light signal was transformed into electrical signal by a balanced electronic detector for decreasing the interference of common-mode signal.

In order to demonstrate the nonlinear error correcting ability of the amplitude modulation method, the nonlinear FMCW system and the amplitude modulation system are simultaneously used to measure the same target. The frequency modulation of the nonlinear FMCW system hasn’t using any nonlinear error correcting methods. The measurement data sets were multiplied by a wavelet filter prior to the application of the transform to suppress sidebands.

Figure 2 is the mixing signal from the amplitude modulation system, the amplitude values of the peak point are modulated by the distance to the target, the frequency of the amplitude is corresponded to the measurement distance. It can be observed that the amplitude at the valley point is also varied by the same frequency. By choosing a matched attenuation coefficient of fiber a, b and r, the contrast ratio of the peak point amplitude signal can be increased, the valley point amplitude also can be eliminated.

Fig. 2 the mixing signal of the amplitude modulation system

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Figure 3 shows the range profile of nonlinear FMCW system and amplitude modulation system, the rate of the frequency modulation at the center of the sweep bandwidth was used to transform the beat frequency axis into a distance axis. The range profile (blue line) of nonlinear FMCW system is come from the FFT of the interference signal produced directly by the interferometry with a frequency sweep laser. The range profile(red line) of the amplitude modulation system is from the FFT of the modulated amplitude signal that is abstracted from the mixing signal produced by the amplitude modulation system. The suppression of sweep nonlinearity can be clearly observed between the two measurement systems. Part a is the range profile of nonlinear FMCW system (blue line), the distance of the targets can’t be distinguished due to the overlapping of the main lobe and side lobe. Part b is the range profile of amplitude modulation system (red line), the main lobe amplitude of the target is much higher than the side lobes, the FWHM (full width at half maximum) of the main lobe is about 69μm, close to the theoretical range resolution (61μm), which is attributed to two factors. The first is frequency bandwidth, The signal frequency bandwidth between the peak point is slightly less than the mechanical sweep frequency bandwidth, which will decrease the practical range resolution according to the formula(12). The second is the refraction index $n_{a}$ , which is uncertain in the actual environment, also decreases the practical range resolution slightly.

Fig. 3 Part a represents the nonlinear range profile, part b represents the range profile by amplitude modulation

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In order to analyze the measurement performance of the amplitude modulation system, the typical resample system [13–16] was used to compare with the amplitude modulation system by measuring the same target. Figure 4 shows the range profile of amplitude modulation(dark blue) and resample system with different time delay(green, red and light blue). It can be observed that the profiles of the resample system have distortion compared to the amplitude modulation, the ratio between the main lobe and side lobe also has some decrease, in addition to the distortion, the peak of the resample system has a slight shift, which is attributed to the different time delay between the auxiliary and the measurement interferometer of the typical resample system. The delay introduces a random phase error to the frequency interval between two adjacent peak points. The amplitude modulation system has just one signal sampled by one balance electronic detector, completely avoiding the error from the different time delay from optical and electrical path.

Fig. 4 the range profile of resample system and amplitude modulation

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Besides the character that immune to the phase error from the time delay mismatch, the data size of amplitude modulation system is half of the resample system, effectively saving the computer storage space and enhancing the speed of data processing, the real mean data processing time is about 1 second by the amplitude modulation method, which is about half of the time by resample algorithm. And then, the power of output signal(5.56mW) from amplitude modulation system is higher than the resample system(2.34mW) with the same input laser power(8mW).

In order to evaluate the uncertainty index of the amplitude modulation system, this system is compared with a resample FMCW system by measuring a fixed target.A cw laser interferometry is used as the standard range of the two systems. The lights from the two systems and the cw laser interferometry have been modulated to the same direction along the linear guide. An arbitrary range offset because of the different initial optical paths between two measurement systems has been removed.

Figure 5 shows the statistical uncertainty of amplitude modulation system is 2.9μm over 20 times measurement, the statistical uncertainty of resample FMCW system is 9.6μm over 20 times measurement, it can be observed that the statistical uncertainty of amplitude modulation system is better than the resample FMCW system because the amplitude modulation system eliminate the phase error from the sample time delay mismatch between the clock signal and the measurement signal., According to the formula (11), the fractional contributions to the uncertainty of amplitude modulation system include the refractive index $n_{a}$ in the air, the refractive index $n_{f}$ , the optical path difference $L_{a b}$ of the auxiliary fiber interferometer and the frequency $f_{A}$ of the amplitude.

Fig. 5 the statistical uncertainty of the amplitude modulation measurement system and resample FMCW system. The blue points are the measurement result of the amplitude modulation system for a fixed target, the red points are the residual error of the amplitude modulation system to the cw laser interferometry. The green points are the measurement result of the resample FMCW system, the purple points are the residual error of the resample FMCW system to the cw laser interferometry.

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The first term $n_{a}$ is the main contribution to the total uncertainty in the case of absolute distance measurement in the air. This index is limited by the experimental environment parameters include the temperature, pressure and humidity of the air. In our experiments, with consideration of the inherent uncertainty of the sensors, the environmental stability, and the environmental inhomogeneity, the temperature, air pressure, and humidity can be stabilized at 34 mK, 11 Pa, and 2% respectively, correspond to 3.2 × 10⁻⁸, 3 × 10⁻⁸, and 1.4 × 10⁻⁸ uncertainty of air refractive index. The second tern $n_{f}$ is mainly dominated by the temperature and the mechanical deformation of the fiber, the third term $L_{a b}$ (range difference of the fiber) is affected by the fiber mechanical deformation, which is mostly attributed to the thermal expansion effect of the fiber material and mechanical vibration. With a well control of the air parameters and vibration insulation, the combine uncertainty of the optical path difference $n_{f} L_{a b}$ can be close to 3μm for the optical path difference of 30m, corresponding to the relative precision of 10⁻⁷. The uncertainty of frequency $f_{A}$ is mainly controlled by the sample frequency interval error of and the noise on the path from the target. The interval error can be limited by well control of the optical path difference $n_{f} L_{a b}$ of auxiliary interferometer. The noise effect can be suppressed by greater return powers from the target and longer integration times.

In order to evaluate the accuracy index of the amplitude modulation system, the amplitude modulation system and the resample FMCW system are used to measure the range along a line guide. Figure 6 shows the max relative error of amplitude modulation system (4.3μm) is better than the max relative error of resample FMCW system (16.6μm). The relative error of amplitude modulation system is mainly attributed to the Abbe error and optical dispersion error. The Abbe error is from that the cw laser interferometer and our amplitude modulation system haven’t share the same optical path, which can be limited in 1μm along 1m measurement range. The optical dispersion error is resulted from the optical frequency sweep of FMCW system. Because the refractive index of fiber is related with the optical frequency, the frequency interval between two peak points will has a slightly linear increase during the process of optical frequency modulation, making the main lobe of the range profile has been distorted and the measurement range has a little shift. Since the measurement range is less than 1m, the dispersion error has little effect to the real measurement result.

Fig. 6 The accuracy comparison of amplitude modulation system and resample FMCW system. The blue points are the measurement range by amplitude modulation system, The red points are the measurement range by resample FMCW system, the green points are the relative error of the amplitude modulation to the cw laser interferometry. the purple points are the relative error of the resample FMCW system to the cw laser interferometry.

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4. Conclusion

In this article, a method for improving the measuring resolution of FMCW laser ranging is presented. This system comprises a measurement interferometer and an auxiliary interferometer, which is all fiber Mach–Zehnder configuration. The auxiliary interferometer is used to produce the sinusoidal carrier signal which is used to load the measurement signal from measurement interferometer, and the range information can be sampled at the peak points of carrier signal. It has been demonstrated that this method is able to suppress the nonlinear scanning of the laser source and eliminate the error of sample time delay mismatch. In the experiment, The range resolution of the method can be achieved to 69μm, the statistical uncertainty is approach to 2.9μm and the standard deviation is achieve to 4.3μm with sweep bandwidth of 1540nm-1560nm along a range of 1m compared to the cw laser interferometer.

Funding

National Natural Science Foundation of China (NSFC) (51675380); Aeronautical Science Foundation of China (20160948001).

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Nonlinear error correction for FMCW ladar by the amplitude modulation method

Abstract

1. Introduction

2. Measurement method and theory

3. Experiment and result

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (12)

Optics Express