## Abstract

We establish a theoretical model of dispersion mismatch in absolute distance measurements using swept-wavelength interferometry (SWI) and propose a novel dispersion mismatch compensation method called chirp decomposition. This method separates the dispersion coefficient and distance under test, which ensures dispersion mismatch compensation without introducing additional random errors. In the measurement of a target located at 3.9 m, a measurement resolution of 45.9 μm is obtained, which is close to the theoretical resolution, and a standard deviation of 0.74 μm is obtained, which is better than the traditional method. The measurement results are compared to a single-frequency laser interferometer. The target moves from 1 m to 3.7 m, and the measurement precision using the new method is less than 0.81 μm .

© 2015 Optical Society of America

## 1. Introduction

Prior to the appearance of swept-wavelength optical sources, SWI was developed in the radio-frequency region to perform free-space ranging measurements using frequency modulated continuous-wave (FMCW) radar [1,2]. At present, SWI forms the basis for a variety of measurement techniques used in a diverse array of applications. The first application was targeted at the measurement of reflections in optical fibers, which was termed optical frequency domain reflectometry (OFDR) [3–9]. Another application was the extension of the FMCW radar to the optical spectrum to produce FMCW laser radar systems [10–13], which was aimed at measuring the ranges of hard targets and wind speed. Finally, SWI has been used to perform depth-resolved imaging in two and three dimensions, e.g., optical frequency domain imaging (OFDI) [14,15] and swept-source optical coherence tomography (SS-OCT) [16–18], respectively.

In the practical application of SWI, it is typically not possible to ensure that the frequency-swept characteristics of a laser are entirely linear. Historically, the problem of tuning nonlinearity has been addressed in three ways: the first approach is focused on the design and execution of a tunable laser source with a tuning curve that is linear in time [19–21], but this method is difficult and less convenient than the other methods; the second approach uses an auxiliary interferometer to measure the laser tuning rate to resample the sampling data [22,23]; and the last approach uses the output of an auxiliary interferometer working as a sampling clock to trigger data acquisition, which is called the frequency-sampling method [24–26]. The frequency-sampling method is widely considered to be the best method because it is convenient and avoids the potentially large number of interpolations required in the second method.

Because an optical path typically includes optical fiber, dispersion compensation is important to achieve high resolution. The traditional methods rely on placing the appropriate amount of dispersion balancing material in one interferometer arm [18,27–29], but these methods require bulky components and make it difficult to achieve a higher resolution. Numerical dispersion compensation techniques have also been proposed, and in theory, they can be optimized for any amount of dispersion. The traditional numerical dispersion compensation methods typically measure the phase errors caused by dispersion in advance, and then, the phase errors are used for dispersion compensation. For example, the first numerical dispersion compensation technique optimizes the chirp rate, which is the slope of the phase errors, and then uses it to create a phase term for dispersion compensation [16,30]; ultrahigh-resolution OCT images are demonstrated. Another numerical compensation technique for dispersion mismatch was subsequently proposed by Donghak Choi et al. [31]. This method also measures the phase errors introduced by dispersion mismatch, using them to create a signal for dispersion compensation. Then, Norman Lippok et al. proposed a dispersion compensation technique based on the use of fractional Fourier transform (FrFT). This technique measures the chirp rate to confirm the FrFT order *α* first, and then, the FrFT is used for dispersion compensation, providing a new fundamental perspective of the nature and role of group-velocity dispersion [17].

In our work, an absolute distance measurement system based on SWI is established. The frequency-sampling method is used to correct the tuning nonlinearity. The dispersion mismatch between the auxiliary interferometer and measurement interferometer is analyzed, and a novel numerical dispersion compensation method that decomposes the measuring signal into the sum of a series of chirp signals is proposed. Experiments demonstrate the increased precision obtained with the proposed approach.

## 2. Principle

The principles of SWI-based absolute distance measurement technology were systematically discussed previously. Although the implementation may differ depending on the application, the defining characteristic of the SWI system is the utilization of a wavelength-tunable source coupled with an interferometer. In practical application, it is often not feasible to ensure that the frequency-tuning characteristics of a laser source are completely linear. In this paper, the frequency-sampling method is used to remove the impact of nonlinear tuning. The schematic diagram of our set-up using the frequency-sampling method is shown in Fig. 1. In this method, the output of an external cavity laser is split into an auxiliary interferometer and measurement interferometer. The output of the auxiliary interferometer can be expressed as

where*A*

_{0}is the amplitude of the auxiliary interferometer output,

*f*is the instantaneous frequency of the laser, and

*τ*

_{0}is the difference between the group delays of the auxiliary interferometer.

The auxiliary interferometer is typically constructed using single-mode fibers (SMFs). To express the influence of chromatic dispersion in the fiber, Eq. (1) can be transformed into

*R*

_{0}is the difference in the lengths of two paths of the auxiliary interferometer,

*n*is the group refraction index of the SMFs,

_{g}*d*is the dispersion coefficient of the SMFs, and

_{f}*n*is the group refraction index at the tuning start frequency

_{g0}*f*

_{0}. Here only the first order dispersion is considered, as the second order dispersion is very small. To correct for the influence of nonlinear tuning, the auxiliary interferometer is used as an external clock, yielding Eq. (3).where

*m*is the sampling point index,

*m*= 1,2,…,

*N*, and

*N*is the number of sampling points. Thus, when the data are sampled, the instantaneous frequency of the laser is

If the measurement light also spreads in the SMFs, the measurement signal can be denoted as

*R*is the difference in the lengths of the measurement path and reference path. Using Eqs. (4) and (5) yields

_{end}#### 2.1 Influence of dispersion mismatch

For the measurement of a target in free space, the measurement light is reflected from the target and returns to the fiber. When it is recombined with the reference light, the interference results in a detector intensity given by

*R*is the distance in air between the fiber end face and target under testing. Because the dispersion is small in air, it is ignored in Eq. (7). With Eqs. (7) and (4), we obtain

_{m}*d*is much less than ${n}_{g0}^{2}{R}_{0}^{2}$, $\sqrt{{n}_{g0}^{2}{R}_{0}^{2}+2{d}_{f}{R}_{0}cm}$ can be expanded as a Taylor series in which higher-order terms can be dropped as

_{f}R_{0}cmThen, Eq. (8) can be simplified as

#### 2. 2 Method to correct dispersion mismatch

In Eq. (10), the first term is a linear phase, and the slope is fixed. The second term is also a linear phase, and the slope is linear to the pending distance. The third term is a quadratic term, which makes the measurement signal a chirp signal, and the chirp rate ${d}_{f}c{R}_{m}/{n}_{g0}^{3}{R}_{0}^{2}$ is linear to the distance under testing. The fourth term is a constant. A traditional Fourier transform is unable to obtain the distance information from this signal, and thus, a dispersion mismatch compensation is necessary.

We propose a novel arithmetic for dispersion compensation to correct the dispersion mismatch without additional errors. In contrast to the Fourier transform, which decomposes the signal into the sum of a series of complex exponential signals, this arithmetic decomposes the signal into the sum of a series of chirp signals, and the transformed result is used as the distance spectrum to confirm the distance under testing. The transform is expressed as

*R*

_{1}~

*R*

_{2}can be obtained using FFT. (

*R*

_{2}-

*R*

_{1})/

*M*is the value of distance spectrum grid, which is related to the signal-to-noise ratio (SNR) of the system. Contrast Eq. (10) with Eq. (11), it is clear that

*P*(

*k*) will be the max when (

*R*

_{2}-

*R*

_{1})

*k*/

*M*+

*R*

_{1}=

*R*. So if

_{m}*P*(k) is used as the distance spectrum, the index

*k*of max in distance spectrum can be used to obtain the distance under testing as

_{max}Before the transform, some parameters should be known in advance. The detailed procedure of the method for dispersion mismatch compensation could be described as follows;

- (1) The sampled signal is processed by band pass filter, after which there is only the interference signal
*I*(_{end}*m*) left, which is introduced by the light reflected from the fiber end face. This signal*I*(_{end}*m*) is not influenced by dispersion as described above, so*R*can be obtained using chirp-z transform (CZT)._{end} - (2) Next the sampled signal is processed by another bandpass filter, after which there is only the interference signal
*I*(_{m}*m*) left, which is introduced by the light reflected from the target. In order to calibrate the ratio between the dispersion coefficient and group refraction index*d*/_{f}*n*, the beat frequencies are measured at different center frequencies using CZT as_{g0}*d*/_{f}*n*is linear to the ratio between the beat frequencies and the center frequencies [32]._{g0} - (3) With the
*I*(_{end}*m*) and*d*/_{f}*n*obtained above, the signal_{g0}*I*(_{m}*m*) could be brought in Eq. (11), and the*P*(*k*) is used as the distance spectrum. - (4) Finally the index
*k*of max in_{max}*P*(*k*) is brought in Eq. (12) to obtain the distance under testing*R*._{m}

Generally the *I _{end}*(

*m*) and

*d*/

_{f}*n*is constant, so the step (1) and (2) are just necessary for the first measurement.

_{g0}## 3. Experiments and results

Our experiment set-up is shown in Fig. 1. An external cavity laser is used with a tuning range of 1,510-1,610 nm and a tuning rate of approximately 100 nm/s or 10 THz/s, defined as the slope of the swept instantaneous frequency as a function of time. The optical power of the external cavity laser can reach 10 mW. After frequency-swept light from the external cavity laser is split using a 99:1 optical coupler, 1% of the optical power goes into an auxiliary Mach-Zehnder-type fiber interferometer, as shown by the dashed box in Fig. 1. The optical path difference (OPD) of the auxiliary interferometer is 225.7788 m at a 1510 nm wavelength, which is obtained by comparing to a single-frequency laser interferometer. The output signal of the auxiliary interferometer is used as the external clock of the DAQ card to correct the nonlinearity of the frequency sweep rate. Then, 99% of the optical power goes into another 99:1 optical coupler, and 1% of the optical power is used as the reference light. Next, 99% of the optical power passes through a circulator, which is focused on the target in free space. The back scattering light is recombined with the reference light at a 3 dB coupler, and the 3 dB coupler output is detected using a balanced detector.

#### 3.1 Measurement of OPD in fiber

In Eq. (11), *R _{end}* should be measured before using chirp composition. To measure

*R*, the laser frequency is swept at 2.73 THz, and

_{end}*R*can be obtained using FFT because it is not affected by dispersion. The FFT result is shown in Fig. 2, and the CZT result is shown in Fig. 3. Fig. 3(a) shows the reflected light signal of the CZT of the fiber end face, from which we can obtain

_{end}*R*= 7.105256 m, and Fig. 3(b) shows the reflected light signal of the CZT of the target in free space. The right side of Fig. 3 demonstrates the influence of dispersion mismatch, which broadens the distance spectrum.

_{end}#### 3.2 Calibration of the dispersion coefficient and group refraction index

Due to the influence of dispersion mismatch, different results will be obtained at different center frequencies using Fourier transform. From Eq. (10), we can obtain the ratio between the dispersion coefficient and group refraction index as

where*R*is the measurement result using Fourier transform at different center frequencies.

_{mcal}In our work, a target located at approximately 2.6 m is measured. The sampling number *N* is 4 million, which corresponds to a wavelength swept from 1,510 to 1,553 nm or a frequency swept from 198 to 193 THz. The measurement results obtained in different sampling sections, which can be approximately considered as functions of the laser frequency, are shown in Fig. 4. Then, using Eq. (13), we can obtain *d _{f}* /

*n*

_{g}_{0}= 2.73 × 10

^{−17}/Hz.

#### 3.3 Resolution measurement

The dispersion mismatch will broaden the distance spectrum, which will influence the measurement resolution. To verify the effectiveness of chirp decomposition, a target located at 3.9 m is measured. The distance spectrums are shown in Fig. 5, where the solid red curves are the distance spectrums obtained using the traditional CZT and the dotted blue curves are the distance spectrums obtained using the chirp decomposition mentioned above. The frequency-swept ranges of (a)-(f) in Fig. 5 are 1.07, 1.6, 2.13, 2.66, 3.19 and 3.72 THz, respectively. Figure 5 illustrates that the influence of dispersion mismatch becomes increasingly severe with an increase in the frequency-swept range. The chirp decomposition can effectively correct the dispersion mismatch.

When the frequency-swept range is 4.26 THz, the theoretical distance resolution is 35.2 μm. However, because the shape of the spectrum is not perfectly Gaussian, the effective resolution is less than this optimum value. An ideal signal without nonlinearity is created to find the theoretical value of the resolution. Fourier transformation of this signal yields a theoretical resolution of 41.5 μm, as shown in Fig. 6 (dotted blue curve). The experimentally measured distance resolution using chirp decomposition is 45.9 μm, as shown in Fig. 6 (solid red curve).

#### 3.4 Precision using chirp decomposition

In absolute distance measurements, both the resolution and precision are important performances. In theory, the precision will be higher as the frequency-swept range increases. However, the precision will decrease as the frequency-swept range increases because of the influence of dispersion mismatch.

To verify the effectiveness of chirp decomposition, we have carried out 20 repetitive experiments of a target located at 3.9 m. The precision of measurements with different frequency-swept ranges are shown in Table 1, and the signals are processed with CZT, the traditional dispersion compensation arithmetic (“multiplication of a complex phase in the time domain”) [16, 30] and chirp decomposition. If the dispersion mismatch is not corrected, the precision will decrease considerably when the frequency-swept range is greater than 2.4 THz because at this time, the distance spectrum is multimodal. The traditional dispersion compensation arithmetic could improve the precision when the frequency-swept range is greater than 2.13 THz. However, when the frequency-swept range is less than 2.13 THz, the precision is worse than the precision processed without dispersion compensation because the traditional dispersion compensation arithmetic needs to measure the chirp rate ${d}_{f}c{R}_{m}/{n}_{g0}^{3}{R}_{0}^{2}$ first, which is used to the correct measurement signal. However, the chirp rate obtained by using the iterative optimization method or some other methods will also be different due to the influence of environment disturbances, even in repetitive measurements. The error in the measurement of the chirp rate will lead to a random error in the measurement of absolute distance.

This chirp decomposition separates the dispersion coefficient and distance under testing. Because the dispersion coefficient is constant, the dispersion compensation will not introduce an additional random error. When chirp decomposition is used, the precision is better than the traditional dispersion compensation arithmetic, as shown in Table 1. With a frequency-swept range of 3.99 THz, the standard deviation can reach 0.84 μm. However, when the frequency-swept range is bigger than 3.99THz, the standard deviation fluctuate around 0.84 μm, which is because of the influence of environmental disturbance.

The measurement results are compared to a single-frequency laser interferometer (Renishaw ML10). The target moves from 1 to 3.7 m, and the distance residual is shown in Fig. 7. The red data are distance residuals obtained using the traditional compensation arithmetic, and the blue data are distance residuals obtained using chirp decomposition. From Fig. 7, the measurement precision is less than 0.81 μm with chirp decomposition, which is better than the traditional method.

## 4. Conclusion

We analyze the dispersion mismatch between an auxiliary interferometer and measurement interferometer and present a novel, powerful, dispersion mismatch correction method for SWI-based absolute distance measurement. The method decomposes the measuring signal into the sum of a series of chirp signals; in this manner, the dispersion coefficient and distance under test are separated, which enables dispersion mismatch compensation without introducing additional random errors. The distance resolution of measurement can reach 45.9 μm, which is close to the theoretical value. A precision of 0.74 μm has been obtained when a target located at 3.9 m is measured with a frequency-swept range of 4.26 THz, and the precision is less than 0.81 μm with a distance range of 1-3.7 m.

## Acknowledgments

This work was supported by the project supported by the National Natural Science Foundation of China (NSFC) (NO. 51275120 and 61275096). The authors would like to thank the Professors Pu Shaobang for the guidance in experiment.

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