Abstract

We establish a theoretical model of the Doppler effect in absolute distance measurements using frequency scanning interferometry (FSI) and propose a novel FSI absolute distance measurement system. This system incorporates a basic FSI system and a laser Doppler velocimeter (LDV). The LDV results are used to correct for the Doppler effect in the absolute distance measurement signal obtained by the basic FSI system. In the measurement of a target located at 16 m, a measurement resolution of 65.5 μm is obtained, which is close to the theoretical resolution, and a standard deviation of 3.15 μm is obtained. The theoretical measurement uncertainty is 8.6 μm + 0.16 μm/m Rm (k = 2) within a distance range of 1 m to 24 m neglecting the influence of air refractive index, which has been verified with experiments.

© 2016 Optical Society of America

1. Introduction

Prior to the development of swept-wavelength optical sources, frequency scanning interferometry (FSI) was developed in the radio-frequency region to perform free-space range measurements using frequency-modulated continuous-wave (FMCW) radar [1,2]. At present, FSI is 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 extended FMCW radar to the optical spectrum to produce FMCW laser radar systems [10–13], which was aimed at measuring the absolute distance of hard targets and wind speed. Finally, FSI 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 this paper, we focus on absolute distance measurements.

Absolute distance measurement systems, which can be used for distance measurements of several tens of meters with uncertainties on the order of microns, are of significant interest in the field of metrology, especially in the manufacturing industry and large-scale science projects. Measurement systems with these capabilities could directly improve the manufacturing efficiency and accuracy of large assemblies. For large airplane wings, this could lead to reduced reworking through interchangeable rivet hole patterns requiring 40 μm accuracies on hole positions across the wing and less fuel consumption by enabling the production of natural laminar flow wings, which are expected to require profile tolerances of approximately 0.5 mm across 20 m to 30 m long wings [19]. The implementation of tooling able to achieve this tolerance for a dense set of points across the wings would require micron-order accuracy in absolute distance measurements. Other applications lie in particle accelerators and colliders, enabling tighter alignment tolerances and, in turn, more complex and advanced designs.

For basic frequency scanning interferometry, obtaining an uncertainty on the order of microns in the laboratory is easy. However, when the measurement is used in industrial environments, wherein the optical path difference (OPD) of measurement interferometry varies, measurement errors will be hundreds of times larger than the actual changes of the measured OPD. This is caused when the Doppler effect is introduced by the measurement OPD variation. To account for these dynamical effects, a setup with two laser diodes that are simultaneously tuned upward and downward in frequency was developed by Richard Schneider in 2001 [20]. This setup used an auxiliary interferometer and digital signal processing to correct the influence of laser frequency nonlinear tuning. In 2006, Nikon also developed a setup with two laser diodes to solve this problem that was different from the setup of Richard Schneider; this setup used a phase-locked loop to correct for the influence of laser frequency nonlinear tuning [21]. In 2010, Matthew Warden et al. developed a setup similar to that of Richard Schneider using two external cavity lasers. The external cavity laser has a larger frequency sweep range, which could improve the precision of FSI [19,22].

In our work, we revisit the problem of the Doppler effect introduced by dynamical measurement. To solve this problem, we propose a new FSI absolute distance measurement setup. This setup uses an external cavity laser to measure the absolute distance information. The frequency sampling method is used to overcome the problem of laser frequency nonlinear tuning. An added single-frequency laser and two acoustic optical modulators are used to measure the variation in the OPD of the measurement interferometer. The variation in the measurement interferometer OPD is used to correct the FSI signal. Experiments demonstrate the effectiveness of our setup.

2. Theory

2.1 The principle of frequency scanning interferometry

The principles of FSI-based absolute distance measurement technology have been systematically discussed in the literature. Although its implementation may differ depending on the application, the defining characteristic of FSI systems is the utilization of a wavelength-tunable source coupled with an interferometer. In practical applications, ensuring that the frequency-tuning characteristics of a laser source are completely linear is difficult. In this paper, the frequency-sampling method is used to remove the impact of nonlinear tuning. A schematic diagram of our setup using the frequency-sampling method is shown in Fig. 1. An external cavity laser is used with a tuning range of 1510-1610 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 in the dashed box in Fig. 1. The OPD of the auxiliary interferometer is 225.7788 m at a wavelength of 1510 nm, which is obtained by comparing to a single-frequency laser interferometer [19]. The output signal of photodetector (PD) in the auxiliary interferometer is used as the external clock of the data acquisition (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. The output of the auxiliary interferometer can be expressed as

I0(f)=A0cos(2πfτ0).
wWhere A0 is the amplitude of the auxiliary interferometer output, f is the instantaneous frequency of the laser, and τ0 is the group delay of the auxiliary interferometer. To correct for the influence of nonlinear tuning, the auxiliary interferometer is used as an external clock, yielding Eq. (2).
2πΔf(k)τ0=2πk,
where Δf is the variation in the laser instantaneous frequency, and k is the sampling point index, where k = 1, 2, …, N, and N is the number of sampling points. Thus, when the data are sampled, the instantaneous frequency of the laser is

 

Fig. 1 Schematic diagram of the FSI-based absolute distance measurement system.

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Δf(k)=kτ0.

The output of the measurement interferometer can also be expressed as

I1(f)=A1cos(2πfτm)=A1cos(2πΔfτm+2πf0τm),
where f0 is the tuning start frequency. 𝜏m is the measured group delay. After frequency sampling method, the measurement signal becomes a digital signal. So substituting Eq. (3) into Eq. (4) yields
I1(k)=A1cos(2πτmτ0k+2πf0τm).
From Eq. (5), we could find the signal obtained using a frequency-sampling method is not affected by laser frequency nonlinear tuning.

2.2 The influence of the Doppler effect

The model introduced above assumes that the measurement OPD is constant. However, in industrial environments, with the influence of vibration and air turbulence, a constant measurement OPD cannot be assumed. Thus, an analysis of dynamical FSI measurements is necessary.

When the measured group delay varies with time, Eq. (5) turns into

I1(k)=A1cos[2πτm(k)τ0k+2πf0τm(k)]=A1cos[2πτm0τ0k+2πf0Δτm(k)+2πf0τm0+2πkτ0Δτm(k)],
where the measured group delay τm(k) = τm0 + Δτm(k), τm0 is the group delay for the first sampling point, and Δτm(k) is the group delay variation for the kth sampling point. The first term of Eq. (6) contains information about the absolute distance. The second term is introduced by the Doppler effect. The third term is a constant. The fourth term is so small it can be neglected. The frequency of the laser f0 is approximately 200 THz, which is much more than the frequency variation, k/τ0. Therefore, even if the group delay variation is very small, the second term of Eq. (6) cannot be neglected.

Here we show the simulation result of Doppler Effect. When the target is located at 10m and the OPD variation of measurement interferometer related to Δτm(k) in Eq. (6) is shown in Fig. 2. The chirp-z transform of Eq. (6) with a Δτm(k) shown as Fig. 2 is shown in Fig. 3. The correct distance of the target cannot be obtained from Fig. 3. At this time, the measurement error is 651 μm, which is much larger than the OPD variation of 1.2 µm. This indicates that basic FSI cannot measure targets in industrial environments.

 

Fig. 2 The OPD variation simulation of measurement interferometer.

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Fig. 3 The distance spectrum of a target located at 10 m with an OPD variation as in Fig. 2.

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2.3 Method to correct the Doppler effect

To correct the Doppler effect in FSI, we proposed a new FSI absolute distance measurement system. This system uses an additional laser Doppler velocimeter (LDV) to monitor the OPD variation of the measurement interferometer. A schematic diagram of this new FSI absolute distance measurement system is shown in Fig. 4. This system contains two parts: one part is a basic FSI system, which is used to measure the absolute distance; the other part is a LDV, which includes a single frequency laser and two acoustic optical modulators (AOM1 and AOM2). In our new FSI absolute distance measurement system, the circulator is replaced with a polarizing beam splitter (PBS) and a quarter wave plate. In this way, the reflected light of fiber end face and focusing system which is needless in our system could be reduced. A visible laser is brought in our system by a wavelength division multiplexer (WDM), which is used as the indicator light.

 

Fig. 4 Schematic diagram of the novel FSI absolute distance measurement system

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The signal of PD 1 used to correct the impact of nonlinear tuning. The signal of the balanced detector contains two pieces of information; one is the absolute distance measurement signal, which can be expressed as Eq. (6), and the other one is the LDV signal.

The frequency of single frequency laser is f1, the output light frequencies of AOM1 and AOM2 are f1 + fAOM1 and f1 + fAOM2 respectively, so the reference light and measurement light of LDV could be shown as

EDr(t)=ADrexp[j2π(f1+fAOM2)t+j2π(f1+fAOM2)τDr+jφ0],
EDm(t)=ADmexp[j2π(f1+fAOM1)t+j2π(f1+fAOM1)τDm(t)+jφ0].
where the ADr and ADm are amplitudes of reference light and measurement light respectively. ΔfAOM = fAOM1 - fAOM2. τDr and τDm are group delays of reference light path and measurement light path respectively. φ0 is the phase constant. So the interference signal of LDV could be shown as:
I2(t)=A2cos[2πΔfAOMt+2π(f1+fAOM1)τDm(t)2π(f1+fAOM2)τDr]=A2cos[2πΔfAOMt+2π(f1+fAOM1)Δτm(t)+φD].
where A2 is the amplitude of LDV signal. Δτm = τDm - τDm0, τDm0 is the group delay for the first time. φD = 2π(f1 + fAOM1)τDm0 - 2π(f1 + fAOM2)τDr. As the sampling clock is the signal of the auxiliary interferometer1, the sampled data of Eq. (9) should be shown as
I2(k)=A2cos[2πΔfAOMt(k)+2π(f1+fAOM1)Δτm(k)+φD].
In Eq. (10), the time of the sampling points is not recorded, which means that t(k) in Eq. (10)is unknown. In this case, the group delay variation Δτm(k) could not be obtained with Eq. (10). So the auxiliary interferometer2 is essential. Same to Eq. (10), the signal of auxiliary interferometer2 could be expressed as
I3(k)=A3cos[2πΔfAOMt(k)+φD2].
where A3 is the amplitude of auxiliary interferometer2 signal. φD2 is the phase constant related to the OPD of auxiliary interferometer2, as the OPD of auxiliary interferometer2 is invariable. The phase variations at the kth sampling of Eq. (6), (10), and (11) are expressed as
Δφ1(k)=2πτm0τ0k+2πf0Δτm(k)+2πkτ0Δτm(k),
Δφ2(k)=2πΔfAOMt(k)+2π(f1+fAOM1)Δτm(k),
Δφ3(k)=2πΔfAOMt(k).
Δφ1(k), Δφ2(k), and Δφ3(k) could be obtained by Hilbert transform or some ways else. Therefore, the group delay variation Δτm(k) could be obtained by Eq. (13) and (14) as
Δτm(k)=[Δφ2(k)Δφ3(k)]2π(f1+fAOM1).
The measured distance Rm of kth sampling point could be obtained with Eq. (12) and (15) as
Rm(k)=cτm(k)2n=c[τm0+Δτm(k)]2n=c2nτ02πk[Δφ1(k)2πf0Δτm(k)].
where n is the air refractive index. When k = N, the precision is best.

3. Experiments and results

The novel FSI absolute distance system is shown in Fig. 4. The OPD of auxiliary interferometer 1 is 217.9425 m. The number of samplings is 2 × 106, which is equivalent to a frequency sweep range of 2.75 THz. The frequency modulation laser is an external cavity laser with a mode-hop-free tuning range of 1510 nm to 1610 nm at a tuning speed of 2 nm s−1 to 100 nm s−1, providing up to 10 mW of fiber-coupled power. The external cavity laser is connected to two interferometers. One is auxiliary interferometer 1, whose signal is used as the sampling clock to correct the impact of nonlinear tuning. The other is the measurement interferometer, whose signal contains the measured distance. The single-frequency laser is also connected to two interferometers. One is auxiliary interferometer 2, whose signal is used to ensure the time of sampling points, t(k). The other is the measurement interferometer, whose signal is used to measure the group delay variation Δτm(k) in the measurement interferometer. The frequencies of AOM1 and AOM2 are 80 MHz and 81 MHz, respectively.

3.1 Measurement summary

The Fourier transform of the measurement interferometer signal with a target located at 16 m is shown in Fig. 5. The signal in the red rectangle is used for the target distance measurement, which could be extracted used a filter. The signal in the green rectangle is used to measure the OPD variation of the measurement interferometer. As the target is not set on a vibration isolation platform, the measurement interferometer OPD has a variation of approximately 1.5 μm. If the Doppler effect is not corrected, the distance spectrum is broadened (Fig. 6). The blue line is the chirp-z transform (CZT) of the distance measurement signal, and the green dotted line is the simulation of an ideal signal. The exact distance information cannot be obtained from this distance spectrum. In 20 repeated experiments, the measurement standard deviation is 645 μm, which is much larger than the practical variation in the OPD of the measurement interferometer.

 

Fig. 5 The Fourier transform of the measurement interferometer signal with a target located at 16 m. The signal in the red rectangle is used for target distance measurement. The signal in the green rectangle is used to measure the OPD variation of the measurement interferometer.

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Fig. 6 The distance spectrum of a target located at 16 m. The blue line is the distance spectrum without the Doppler effect correction. The green line is the simulation result of an ideal signal.

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To correct for the Doppler effect, a filter should first be used to obtain the OPD variation measurement signal in Fig. 5. As the sampling clock is the signal of auxiliary interferometer 1, the time of the sampling points t(k) is unknown. To obtain the OPD variation of the measurement interferometer, the phase difference between the OPD variation measurement signal and the signal of auxiliary interferometer 2 should be obtained, as shown in Eq. (15). The result of the OPD variation measurement is shown in Fig. 7.

 

Fig. 7 The OPD variation measurement results of our novel system, which is used to correct the Doppler effect in our distance measurement signal.

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This OPD vibration measurement result can be used to correct the influence of the Doppler effect, as shown in Eq. (16). The distance spectrum after the Doppler effect correction is shown in Fig. 8, where the blue line is the CZT of the measurement signal after the Doppler effect correction and the green line is the simulation of an ideal signal. The influence of the dispersion mismatch is corrected by a chirp decomposition method [23]. The distance resolution in our system is 65.5 μm, which is equal to the theoretical distance resolution. In 20 repeated experiments, the measurement standard deviation is 3.15 μm, which is a much better result than that without the Doppler effect correction.

 

Fig. 8 The distance spectrum with the Doppler effect correction of a target located at 16 m. The blue line is the distance spectrum of our novel system, which is corrected with the OPD variation measurement result shown in Fig. 7. The green dotted line is the simulation result of an ideal signal.

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3.2 Uncertainty

In absolute distance measurements, the uncertainty is the most importance performance. To verify the uncertainty of our system, an experiment is set up in a 25 m linear guideway, in which the target can be moved through a range of distances (0.2 to 25 m) while being measured by our novel FSI system. The temperature is observed to be stable over the time of the experiment. The slider on the linear guideway has two targets that are relatively dormant. The moving distance of the slider is simultaneously measured by our FSI system and a laser interferometer (Renishaw ML10), as shown in Fig. 9.

 

Fig. 9 The comparison setup to verify the uncertainty of our system.

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The slider is moved from 1 m to 24 m, and ten FSI measurements are obtained at each position. The displacement of the slider is both measured by our FSI system and the laser interferometer. Our FSI system and the laser interferometer use the same air refractive index. The measured results of the laser interferometer could be used as the truth value of the slider displacement. So our FSI system measurement residual could be expressed by the difference between the measured results of our novel FSI system and the laser interferometer, which is shown in Fig. 10. The empirical uncertainty limits are estimated graphically from the scattering of the experimental data as ± 8.3 μm.

 

Fig. 10 The measurement length residual between our absolute distance measurement system and the laser interferometer (Renishaw ML10). The dotted lines indicate a linear estimate of the error of the experiments based on a graphic analysis. The dashed line represents the theoretical measurement uncertainty.

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From Eq. (16), we could find the measurement uncertainty is related to the uncertainties of air refractive index n, group delay of the auxiliary interferometer1 τ0, and phase variations (Δφ1(k), Δφ2(k), and Δφ3(k)). In our comparison setup, the FSI system and the laser interferometer use the same air refractive index, so the uncertainty of air refractive index will not introduce a measurement residual between our system and the interferometer. The group delay of the auxiliary interferometer1 τ0 will be influenced by the temperature. In our system it will introduce a measurement standard uncertainty of 0.08 × 10−6Rm. The phase variations will introduce a measurement standard uncertainty of 4.3 μm. So the measurement uncertainty of our system is 8.6 μm + 0.16 μm/m Rm (k = 2), as shown in Fig. 10.

4. Conclusion

In the basic FSI absolute distance measurement system, the OPD of the measurement interferometer should be constant, which is impossible in many industrial environments. When there is a variation in the measurement interferometer OPD, the Doppler effect will influence the measurement results. To overcome this problem, we propose a novel FSI absolute distance measurement system, which combines an additional LDV to measure the OPD variation in the measurement interferometer and to correct the influence of the Doppler effect. When a target is located at 16 m, the standard deviation with the Doppler effect correction is 3.15 μm, but when the Doppler effect correction is not applied, the standard deviation is 645 μm, which verifies the effectiveness of our system. Compared with tests using a laser interferometer (Renishaw ML10), we find the empirical uncertainty is ± 8.3 μm within a distance range of 1 m to 24 m. The theoretical measurement uncertainty is 8.6 μm + 0.16 μm/m Rm (k = 2) neglecting the influence of air refractive index. Another advantage of this system is the ability to measure a target without a cooperative target, which will be very convenient for most industrial measurements.

Funding

National Natural Science Foundation of China (NSFC) (51275120)

References and links

1. D. N. Keep, “Frequency-modulation radar for use in the mercantile marine,” Proc. IEEE: Radio Electron. Eng. 103, 519–523 (1956). [CrossRef]  

2. B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982). [CrossRef]  

3. W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981). [CrossRef]  

4. S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985). [CrossRef]  

5. K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991). [CrossRef]  

6. R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995). [CrossRef]  

7. D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005). [CrossRef]   [PubMed]  

8. A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007). [CrossRef]  

9. M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008). [CrossRef]  

10. E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990). [CrossRef]  

11. A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994). [CrossRef]  

12. E. D. Moore and R. R. McLeod, “Phase-sensitive swept-source interferometry for absolute ranging with application to measurements of group refractive index and thickness,” Opt. Express 19(9), 8117–8126 (2011). [CrossRef]   [PubMed]  

13. M. S. Warden, “Precision of frequency scanning interferometry distance measurements in the presence of noise,” Appl. Opt. 53(25), 5800–5806 (2014). [CrossRef]   [PubMed]  

14. S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003). [CrossRef]   [PubMed]  

15. I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000). [CrossRef]  

16. M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004). [CrossRef]   [PubMed]  

17. N. Lippok, S. Coen, P. Nielsen, and F. Vanholsbeeck, “Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform,” Opt. Express 20(21), 23398–23413 (2012). [CrossRef]   [PubMed]  

18. K. Asaka and K. Ohbayashi, “Dispersion matching of sample and reference arms in optical frequency domain reflectometry-optical coherence tomography using a dispersion-shifted fiber,” Opt. Express 15(8), 5030–5042 (2007). [CrossRef]   [PubMed]  

19. J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014). [CrossRef]   [PubMed]  

20. R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001). [CrossRef]  

21. M. Rezk and A. Slotwinski, “Compact fiber optic geometry for a counter-chirp FMCW coherent laser radar,” U.S. Patent No. 8,687,173 (2014

22. M. Warden, “Absolute distance metrology using frequency swept lasers,” Ph.D. thesis, University of Oxford (2011).

23. C. Lu, G. Liu, B. Liu, F. Chen, T. Hu, Z. Zhuang, X. Xu, and Y. Gan, “Method based on chirp decomposition for dispersion mismatch compensation in precision absolute distance measurement using swept-wavelength interferometry,” Opt. Express 23(25), 31662–31671 (2015). [CrossRef]   [PubMed]  

References

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  • |

  1. D. N. Keep, “Frequency-modulation radar for use in the mercantile marine,” Proc. IEEE: Radio Electron. Eng. 103, 519–523 (1956).
    [Crossref]
  2. B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
    [Crossref]
  3. W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
    [Crossref]
  4. S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
    [Crossref]
  5. K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
    [Crossref]
  6. R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
    [Crossref]
  7. D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
    [Crossref] [PubMed]
  8. A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
    [Crossref]
  9. M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
    [Crossref]
  10. E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
    [Crossref]
  11. A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
    [Crossref]
  12. E. D. Moore and R. R. McLeod, “Phase-sensitive swept-source interferometry for absolute ranging with application to measurements of group refractive index and thickness,” Opt. Express 19(9), 8117–8126 (2011).
    [Crossref] [PubMed]
  13. M. S. Warden, “Precision of frequency scanning interferometry distance measurements in the presence of noise,” Appl. Opt. 53(25), 5800–5806 (2014).
    [Crossref] [PubMed]
  14. S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003).
    [Crossref] [PubMed]
  15. I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
    [Crossref]
  16. M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004).
    [Crossref] [PubMed]
  17. N. Lippok, S. Coen, P. Nielsen, and F. Vanholsbeeck, “Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform,” Opt. Express 20(21), 23398–23413 (2012).
    [Crossref] [PubMed]
  18. K. Asaka and K. Ohbayashi, “Dispersion matching of sample and reference arms in optical frequency domain reflectometry-optical coherence tomography using a dispersion-shifted fiber,” Opt. Express 15(8), 5030–5042 (2007).
    [Crossref] [PubMed]
  19. J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
    [Crossref] [PubMed]
  20. R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
    [Crossref]
  21. M. Rezk and A. Slotwinski, “Compact fiber optic geometry for a counter-chirp FMCW coherent laser radar,” U.S. Patent No. 8,687,173 (2014
  22. M. Warden, “Absolute distance metrology using frequency swept lasers,” Ph.D. thesis, University of Oxford (2011).
  23. C. Lu, G. Liu, B. Liu, F. Chen, T. Hu, Z. Zhuang, X. Xu, and Y. Gan, “Method based on chirp decomposition for dispersion mismatch compensation in precision absolute distance measurement using swept-wavelength interferometry,” Opt. Express 23(25), 31662–31671 (2015).
    [Crossref] [PubMed]

2015 (1)

2014 (2)

2012 (1)

2011 (1)

2008 (1)

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

2007 (2)

K. Asaka and K. Ohbayashi, “Dispersion matching of sample and reference arms in optical frequency domain reflectometry-optical coherence tomography using a dispersion-shifted fiber,” Opt. Express 15(8), 5030–5042 (2007).
[Crossref] [PubMed]

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

2005 (1)

2004 (1)

2003 (1)

2001 (1)

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

2000 (1)

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

1995 (1)

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

1994 (1)

A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
[Crossref]

1991 (1)

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

1990 (1)

E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
[Crossref]

1985 (1)

S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
[Crossref]

1982 (1)

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
[Crossref]

1981 (1)

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
[Crossref]

Asaka, K.

Bouma, B.

Burrows, E. C.

E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
[Crossref]

Chen, D.

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

Chen, F.

Coen, S.

Culshaw, B.

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
[Crossref]

Dale, J.

Davies, D. E. N.

S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
[Crossref]

de Boer, J.

Dickerson, B. D.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

Dieckmann, A.

A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
[Crossref]

Duker, J.

Eickhoff, W.

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
[Crossref]

Fielder, B. F.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

Froggatt, M. E.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
[Crossref] [PubMed]

Fujimoto, J.

Gan, Y.

Gifford, D. K.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
[Crossref] [PubMed]

Giles, I. P.

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
[Crossref]

Gisin, N.

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

He, S.

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

Horiguchi, T.

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

Hu, T.

Hughes, B.

Iftimia, N.

Jiang, M.

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

Kingsley, S. A.

S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
[Crossref]

Ko, T.

Kowalczyk, A.

Koyamada, Y.

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

Lancaster, A. J.

Lewis, A. J.

Liou, K.-Y.

E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
[Crossref]

Lippok, N.

Liu, B.

Liu, G.

Lu, C.

McLeod, R. R.

Moore, E. D.

Nielsen, P.

Ohbayashi, K.

Passy, R.

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

Reichold, A. J. H.

Sang, A. K.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

Schneider, R.

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

Shimizu, K.

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

Soller, B. J.

Srinivasan, V.

Stockmann, M.

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

Tearney, G.

Thuermel, P.

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

Ulrich, R.

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
[Crossref]

Vanholsbeeck, F.

von der Weid, J. P.

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

Warden, M. S.

Wojtkowski, M.

Wolfe, M. S.

Xu, X.

Yamaguchi, I.

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

Yamamoto, A.

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

Yano, M.

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

Yun, S.

Zhuang, Z.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
[Crossref]

Electron. Lett. (3)

S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
[Crossref]

E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
[Crossref]

A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
[Crossref]

IEEE Photonics Technol. Lett. (3)

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
[Crossref]

Opt. Eng. (2)

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

Opt. Express (7)

C. Lu, G. Liu, B. Liu, F. Chen, T. Hu, Z. Zhuang, X. Xu, and Y. Gan, “Method based on chirp decomposition for dispersion mismatch compensation in precision absolute distance measurement using swept-wavelength interferometry,” Opt. Express 23(25), 31662–31671 (2015).
[Crossref] [PubMed]

M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004).
[Crossref] [PubMed]

N. Lippok, S. Coen, P. Nielsen, and F. Vanholsbeeck, “Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform,” Opt. Express 20(21), 23398–23413 (2012).
[Crossref] [PubMed]

K. Asaka and K. Ohbayashi, “Dispersion matching of sample and reference arms in optical frequency domain reflectometry-optical coherence tomography using a dispersion-shifted fiber,” Opt. Express 15(8), 5030–5042 (2007).
[Crossref] [PubMed]

J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
[Crossref] [PubMed]

E. D. Moore and R. R. McLeod, “Phase-sensitive swept-source interferometry for absolute ranging with application to measurements of group refractive index and thickness,” Opt. Express 19(9), 8117–8126 (2011).
[Crossref] [PubMed]

S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003).
[Crossref] [PubMed]

Proc. SPIE (1)

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

Other (3)

D. N. Keep, “Frequency-modulation radar for use in the mercantile marine,” Proc. IEEE: Radio Electron. Eng. 103, 519–523 (1956).
[Crossref]

M. Rezk and A. Slotwinski, “Compact fiber optic geometry for a counter-chirp FMCW coherent laser radar,” U.S. Patent No. 8,687,173 (2014

M. Warden, “Absolute distance metrology using frequency swept lasers,” Ph.D. thesis, University of Oxford (2011).

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Figures (10)

Fig. 1
Fig. 1 Schematic diagram of the FSI-based absolute distance measurement system.
Fig. 2
Fig. 2 The OPD variation simulation of measurement interferometer.
Fig. 3
Fig. 3 The distance spectrum of a target located at 10 m with an OPD variation as in Fig. 2.
Fig. 4
Fig. 4 Schematic diagram of the novel FSI absolute distance measurement system
Fig. 5
Fig. 5 The Fourier transform of the measurement interferometer signal with a target located at 16 m. The signal in the red rectangle is used for target distance measurement. The signal in the green rectangle is used to measure the OPD variation of the measurement interferometer.
Fig. 6
Fig. 6 The distance spectrum of a target located at 16 m. The blue line is the distance spectrum without the Doppler effect correction. The green line is the simulation result of an ideal signal.
Fig. 7
Fig. 7 The OPD variation measurement results of our novel system, which is used to correct the Doppler effect in our distance measurement signal.
Fig. 8
Fig. 8 The distance spectrum with the Doppler effect correction of a target located at 16 m. The blue line is the distance spectrum of our novel system, which is corrected with the OPD variation measurement result shown in Fig. 7. The green dotted line is the simulation result of an ideal signal.
Fig. 9
Fig. 9 The comparison setup to verify the uncertainty of our system.
Fig. 10
Fig. 10 The measurement length residual between our absolute distance measurement system and the laser interferometer (Renishaw ML10). The dotted lines indicate a linear estimate of the error of the experiments based on a graphic analysis. The dashed line represents the theoretical measurement uncertainty.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

I 0 ( f ) = A 0 cos ( 2 π f τ 0 ) .
2 π Δ f ( k ) τ 0 = 2 π k ,
Δ f ( k ) = k τ 0 .
I 1 ( f ) = A 1 cos ( 2 π f τ m ) = A 1 cos ( 2 π Δ f τ m + 2 π f 0 τ m ) ,
I 1 ( k ) = A 1 cos ( 2 π τ m τ 0 k + 2 π f 0 τ m ) .
I 1 ( k ) = A 1 cos [ 2 π τ m ( k ) τ 0 k + 2 π f 0 τ m ( k ) ] = A 1 cos [ 2 π τ m 0 τ 0 k + 2 π f 0 Δ τ m ( k ) + 2 π f 0 τ m 0 +2 π k τ 0 Δ τ m ( k ) ] ,
E D r ( t ) = A D r exp [ j 2 π ( f 1 + f A O M 2 ) t + j 2 π ( f 1 + f A O M 2 ) τ D r + j φ 0 ] ,
E D m ( t ) = A D m exp [ j 2 π ( f 1 + f A O M 1 ) t + j 2 π ( f 1 + f A O M 1 ) τ D m ( t ) + j φ 0 ] .
I 2 ( t ) = A 2 cos [ 2 π Δ f A O M t + 2 π ( f 1 + f A O M 1 ) τ D m ( t ) 2 π ( f 1 + f A O M 2 ) τ D r ] = A 2 cos [ 2 π Δ f A O M t + 2 π ( f 1 + f A O M 1 ) Δ τ m ( t ) + φ D ] .
I 2 ( k ) = A 2 cos [ 2 π Δ f A O M t ( k ) + 2 π ( f 1 + f A O M 1 ) Δ τ m ( k ) + φ D ] .
I 3 ( k ) = A 3 cos [ 2 π Δ f A O M t ( k ) + φ D 2 ] .
Δ φ 1 ( k ) = 2 π τ m 0 τ 0 k + 2 π f 0 Δ τ m ( k ) +2 π k τ 0 Δ τ m ( k ) ,
Δ φ 2 ( k ) = 2 π Δ f A O M t ( k ) + 2 π ( f 1 + f A O M 1 ) Δ τ m ( k ) ,
Δ φ 3 ( k ) = 2 π Δ f A O M t ( k ) .
Δ τ m ( k ) = [ Δ φ 2 ( k ) Δ φ 3 ( k ) ] 2 π ( f 1 + f A O M 1 ) .
R m ( k ) = c τ m ( k ) 2 n = c [ τ m 0 + Δ τ m ( k ) ] 2 n = c 2 n τ 0 2 π k [ Δ φ 1 ( k ) 2 π f 0 Δ τ m ( k ) ] .

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