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Abstract
In this paper, we theoretically and experimentally analyze the frequency-comb interferometry at 518 nm in the underwater environment, which we use to measure the underwater distance with high accuracy and precision. In the time domain, we analyze the principle of pulse cross correlation. The interferograms can be obtained in the vicinity of N∙lpp, where N is an integer and lpp is the pulse-to-pulse length. Due to the strong dispersion of water, the pulse can be broadened as the distance increases. The distance can be measured via the peak position of the interferograms. The experimental results show a difference within 100 μm at 8 m range, compared with the reference values. In the frequency domain, we analyze the principle of dispersive interferometry. The spectrograms can be observed near the location of N∙lpp, due to the low resolution of the optical spectrum analyzer. Because of the strong dispersion of water, the modulation frequency of the spectrogram is not constant. A balanced wavelength will exist with the widest fringe, at which the group optical path difference between the reference and measurement arm is equal to N∙lpp. The position of the widest fringe can be used to measure the distance. Compared with the reference values, the experimental results indicate a difference within 100 μm at 8 m range.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Frequency combs, which are honored by Nobel Prize in 2005, have advanced a wealth of scientific fields since their invention [1,2]. In the time domain, frequency comb emits an ultra-short pulse train with a uniform and stable time spacing. In the optical domain, frequency comb is composed of several hundred thousand equally spaced modes with narrow linewidth over a broad spectral band. These coherent spectral lines can be simply expressed as m × frep + fceo, where m is an integer, frep is the repetition frequency, and fceo is the carrier-envelope-offset frequency. If the two degrees of freedom, repetition frequency and carrier-envelope-offset frequency, are stabilized to an external RF reference, the huge gap between the optical frequency and the RF frequency is bridged in a single step, and all the optical frequency markers will share the same frequency stability as the external reference [3]. This inherent advantage has promoted great progress in many applications, such as precision spectroscopy [4], absolute frequency measurement [5], time/frequency transfer [6], and absolute distance measurement [7], etc.
Technique of absolute distance measurement is of significant importance in both science and technology. During the past two decades, scientists all over the world have proposed various methods based on frequency comb, which can measure the distance with high precision and accuracy [8]. In the time domain, optical sampling has been investigated in great depth, where the time delay between two pulses is continuously tuned, so that the interferograms can be generated. Meanwhile, the optical frequency can be down converted into the RF region. Consequently, many schemes of data processing can be carried out to extract the distance information, e.g., peak finding [9], peak shift [10], intensity evaluation [11], the phase slope of the unwrapped phase [12], and stationary phase point [13,14], etc. In general, a mechanical moving stage is required to shift one pulse with respect to the other. Therefore, the system involves the moving device with lower measurement speed, and the beam alignment is not very easy in the case of long-stroke stage. In the configuration of optical sampling by cavity tuning [15], the mechanical stage is not needed. The pulses can be temporally shifted by continuously changing the repetition frequency. Although the moving part still exists, the small length change of the laser cavity can be greatly enlarged with excellent mechanical stability and high measurement speed. However, large tuning range of the repetition frequency and large path difference between the reference and measurement beam are needed to realize the arbitrary distance measurement [16]. Asynchronous optical sampling is one powerful method [17], which relies on two mode-locked lasers with slightly different repetition frequencies. The pulses can automatically scan each other to generate the interferograms with ultra-high measurement speed, and there is no dead zone in the total measurement path. The limitation could be that two mode-locked lasers are required, making the system expensive and bulky. Please note that, the method of triple-comb interferometry has found applications recently [18–20]. In addition, some incoherent strategies are also well performed, e.g., time-of-flight [21], pulse-to-pulse alignment [22]. In the frequency domain, the distance can be determined based on dispersive interferometry via the slope of the unwrapped phase [23]. When using a low-resolution optical spectrum analyzer, the measurable distance is limited to only several millimeters [24]. The mode-resolved technique, e.g., VIPA based spectrometer [25], and Vernier spectrometer [26], can realize arbitrary distance measurement. It is necessary to mention that the fringe of the dispersive interferometry will be different if the pulse propagates through the strongly dispersive media. In this case, the so-called dispersive reference interferometry [27], or chirped pulse interferometry [28], will turn out. One widest fringe will appear, corresponding to a balanced wavelength (i.e., stationary phase wavelength), where the group optical path difference between the reference and measurement beams is equal to the integer multiple of the pulse-to-pulse length (N∙lpp). The measured distance is related to the position of the balanced wavelength. Additionally, frequency comb can also work as the calibrating source to measure the distance [29]. To date, frequency comb based absolute distance measurement has been used in a wide variety of applications, such as the mission of multi targets [30], angle measurement [31], and refractive index measurement [32], etc. However, there is few report about the underwater distance measurement of frequency comb. We try to use this powerful laser source of frequency comb to carry out underwater distance measurement.
Traditionally, the underwater distance measurement mostly relies on the single-beam or multi-beam echo sounder since the lower attenuation of the sound. The single-beam echo sounder emits an acoustic pulse train, and the distance can be measured by the flight time of the acoustic pulse [33]. However, due to the large beam size the measurement resolution is lower, and the measurement speed is also lower because each single measurement can only get one distance. The multi-beam echo sounder can simultaneously output the acoustic pulse train at a wide angle range [34]. Therefore, the detected area has been updated to a line, and the measurement speed can be greatly enhanced. The measurement resolution can be also improved with the adjustment of the sound footprint. Despite these advantages, the measurement accuracy and precision (generally ± 0.3% of the measured distance) are not very high if we directly read out the distance through the flight time of the pulsed sound, and the measurement resolution is limited by the beam size and the width of the sound pulse. On the other hand, considerable noises will be involved into the returned signal due to the multi-path effect. Photogrammetry can be also used to measure the distance underwater [35], but an auxiliary light source with high power is required, and the working distance (about 10 m) is not very long. The airborne laser radar can map a sea area with high efficiency and low cost [36], but the incoherent scheme of time-of-flight measurement cannot satisfy the requirement of the high precision (e.g., micrometer level). It is necessary to mention that the incremental measurement is not applicable in the underwater environment any more. It is difficult to fabricate a long rail underwater. Additionally, the continuous movement of the target can cause wave to make the light scattered strongly. Therefore, the ultrafast measurement of frequency comb will play important role in the underwater environment. In this paper, we, for the first time, carry out underwater distance measurement using the laser of frequency comb. We perform the underwater distance measurement in the time and frequency domains, respectively. The measurement principles are analyzed. The distances up to 8 m can be measured, and the measurement uncertainty can achieve well below 100 μm in the lab environment.
2. Underwater propagation of frequency comb
Figure 1 shows the experimental setup. The frequency comb (Menlosystem Orange, 100 MHz repetition frequency, 300 mW maximum output power, 518 nm center wavelength), which is locked to the Rb clock (Microsemi 8040), emits a pulse train into a Michelson interferometer, and split at a beam splitter. One part is reflected by the reference mirror, which is fixed on a scanning platform (PI M521, 200 mm travel range). The other part is guided into the water container after the beam size is expanded to 15 mm by a pair of lens, and reflected by the measurement mirror underwater. The two parts are combined, and finally detected by the photodetector (Thorlabs PDA10) and the optical spectrum analyzer (Thorlabs OSA201C), respectively. The oscilloscope is the LeCroy 640zi. The environmental sensors are placed along the optical path underwater, and the temperature of the water is well controlled with below 20 mK stability. We fabricate an optical board, where a series of positioning holes are designed. The length interval between the positioning holes are measured and calibrated by the coordinate measurement machine (HAXAGON TORO Advantage 120.16.21) with precision of better than 48 μm when measuring large-scale workpiece, which we use as the reference values.
2.1 Pulse cross correlation
In the time domain, frequency comb can be expressed as:
where E(t) is the electric field of the pulse, ωc is the center frequency, φ0 is an initial phase of the carrier pulse, ∆φce is carrier phase slip rate due to the difference between the group and phase velocities, m is an integer, and TR is the time interval between two pulses. TR = 1/frep, where frep is the repetition frequency of the comb. fceo = ∆φcefrep/2π, where fceo is the carrier-envelope-offset frequency.In the Michelson interferometer, the reference pulse reflected by the reference mirror can be expressed as:
where N = floor (2L/lpp), floor rounds the element of 2L/lpp to the nearest integer, less than or equal to 2L/lpp.The measurement pulse reflected by the measurement mirror underwater can be expressed as:
where n is the water refractive index, C is the corresponding attenuation coefficient. L is the distance to be determined.When the reference pulse and the measurement pulse overlap temporally and are detected by the photodetector, the detected intensity I can be expressed as:
where Td is the integration time of the photodetector. Therefore, in Eq. (4), the term of cross correlation (i.e., the second term) can be expressed as:We rewrite Eq. (5) as:
where Pm is the power spectral density. From Eq. (6), we find that, the intensity amplitude exponentially decreases with increasing the distance. Please note that, in the underwater case, the water dispersion is much stronger than that in air. Therefore, the pulse can be obviously broadened when travelling underwater, and further the width of the cross-correlation pattern will increase when the distance becomes larger.2.2 Dispersive interferometry
In the frequency domain, frequency comb consists of many single modes with fixed frequency interval. Assume that the spectrum of the reference pulse can be expressed as:
The spectrum of the measurement pulse can be expressed as:
where τ is the time delay between the reference pulse and the measurement pulse. The interference spectrum can be calculated as:Considering the strong dispersion of water, the phase of the interference spectrum can be expressed as:
where ω0 is the center angular frequency, and α is the dispersion parameter. Please note that, we consider the refractive index of water is linearly correlated with the optical frequency based on the Harvey formula [37], i.e., n(ω) = n(ω0) + α(ω-ω0). Based on Eq. (10), we find that the spectral phase is a quadratic function of ω, and thus the modulation frequency of the interference fringe is not stable any more. In other words, the second-order dispersion will make great contribution to the shape of the fringe. In this case, a widest fringe will appear, corresponding to a balanced wavelength at which the group optical path difference between the reference and measurement beams is exactly equal to the multiple of the pulse-to-pulse length. This is the so-called dispersive reference interferometry, i.e., chirped pulse interferometry. Therefore, the balanced wavelength will be shifted if the optical path difference between the reference and measurement beams changes. If we shift the reference mirror with a known length l, we assume that the widest fringe is shifted from f1 to f2. For convenience of explanation, we neglect the dispersion of glass elements, including the beam expander and the wall of the water container. The ‘phase’ optical path differences corresponding to f1 and f2 can be respectively expressed as:where n1 and n2 are the phase refractive index of water, corresponding to f1 and f2. n1 = n(ω0) + α(ω1-ω0), n2 = n(ω0) + α(ω2-ω0). Therefore, l can be calculated as:We define the linear coefficient 4παL as χ, and consequently Eq. (12) can be rewritten as:
Therefore, the measured distance L can be calculated as:
where fshift is the shifted frequency of the widest fringe. Based on Eq. (14), we can determine the distance underwater.3. Underwater distance measurement based on the pulse cross correlation
Figure 2 shows the photograph of the practical experiments. Figure 3 shows the spectrum of the laser source. We find that the spectrum is centered at about 518 nm with about 5 nm width. The water conditions are 21.1°C temperature and 998.02 kg/m3 density. We use Harvey formula to evaluate the water refractive index, and the phase refractive index of water corresponding to different wavelengths is shown in Fig. 4. We find the water dispersion is much stronger than air. When the wavelength is tuned from 513 nm to 523 nm, the phase refractive index of water can change from 1.342394 to 1.341942, with more than 4.5 × 10−4 variation. The group refractive index of water can be calculated to be 1.3655989. Please note that, Harvey formula is used in the case of distilled water, while we use the tap water in our experiments. However, the slight difference between distilled water and tap water can only cause the uncertainty of less than 1 × 10−6, which can be neglected.
3.1 Pulse cross correlation at different distances
Figure 5 shows the patterns of pulse cross correlation at different distances. We find that the width of the cross-correlation patterns obviously increases when the distance increases, since the pulse width is strongly broadened due to the water dispersion. Please note that, the intensity in Fig. 5 has been normalized, for convenience of displaying the width of the cross-correlation patterns. In fact, the intensity decreases following the exponential law. This indicates that, the laser power should be sufficiently high, so that the measurable distance can be long up to hundreds of meters.
Fig. 5 Cross-correlation patterns at different distances. (a): initial position; (b): 1.1 m distance; (c): 2.2 m distance; (d): 7.7 m distance.
We measure the distances based on the method of peak finding. As shown in Fig. 5(a), the black line represents the absolute value of Hilbert transform (the amplitude of the absolute value of Hilbert transform has been normalized), and the peak position can be easily picked up. The distance can be calculated as:
where N is an integer, frep is the repetition frequency, ng is the group refractive index of water, and d is a small displacement. Based on the results shown in Fig. 5, the small displacement d can be measured via the method of peak finding. Therefore, the distances can be measured underwater based on the principle of pulse cross correlation.3.2 Results of distance measurement based on pulse cross correlation
The experimental results of the distance measurement are shown in Fig. 6, compared with the reference values provided by the coordinate measurement machine. Due to the limited dimensional of the water tank, we measure the distances up to about 8 m. At each distance, we fast measure the distance for five times. We find that, the measurement uncertainty can be well within 100 μm.
Fig. 6 Distance measurement results of pulse cross correlation. The black solid points indicate the average value of five measurements, and the red x markers show the scatters of five single measurements. The error bars represent the standard deviation of five measurements. The pink dashed line indicates the uncertainty limit.
Based on Eq. (13), the measurement uncertainty of L can be evaluated as:
The first term of Eq. (16) is related to the repetition frequency of the light source. In our experiments, the repetition frequency is locked to the Rb clock with 3 × 10−11 stability at 1 s averaging time. This part can be estimated as 10−10∙L. The second term is due to the group refractive index of water. In our experiments, the water temperature is well controlled. The temperature uncertainty is 20 mK, and the density uncertainty is 4 g/m3, corresponding to 2 × 10−6 and 1.6 × 10−6 uncertainty of water refractive index. The uncertainty of the empirical formula itself is about 1.5 × 10−5. Therefore, this part can be expressed as 1.5 × 10−5∙L. The third term is related to the fractional part d, and the uncertainty of d is less than 2 μm caused by the moving stage and the program of peak finding. Please note that, based on Harvey formula, the dispersion of water is not strictly linear. This means that the cross-correlation patterns will be not only broadened, but also chirped because of the group velocity dispersion, which could affect the exact peak position. The chirp is potentially related to the water dispersion, i.e., the water refractive index, and thus distance dependent. Based on our calculations, the uncertainty due to the chirp can be estimated to be less than 0.57 × 10−5∙L. Finally, the combined uncertainty can be calculated as [2μm2 + (1.6 × 10−5∙L)2]1/2. We find a good agreement with the results shown in Fig. 6. Based on the uncertainty evaluation, we find that the uncertainty source is mainly from the correction of the water refractive index. Therefore, it is important to precisely measure the water refractive index.
4. Underwater distance measurement based on dispersive interferometry
4.1 Dispersive interferometry at different distances
Figure 7 shows the fringes of dispersive interferometry at the initial position with about 67 cm water path. We find that, there is a widest fringe in the spectral interferogram, where the modulation frequency is the lowest. This actually corresponds to the balanced wavelength. Figures 7(a) and 7(b) are the raw data, and we use a high-pass filter to isolate the DC component to make the widest fringe easy to measure, as shown in Figs. 7(c) and 7(d). When we precisely move the reference mirror by 100 μm, the widest fringe will be shifted as shown in Fig. 7. As shown in Fig. 7(d), the position of the widest fringe, i.e., the exact value of the balanced frequency, can be easily measured as (fleft + fright)/2. Please note that, this step of 100 μm movement is actually the calibration process. In fact, we have tried to use various methods to carry out the time-frequency analysis of the non-stationary signals, i.e., the oscillation frequency is not constant. The results of different methods are almost the same [38]. The widest fringe is shifted from 580.217 THz (516.69 nm) to 578.393 THz (518.32 nm). Consequently, the linear parameter χ can be calculated as:
Fig. 7 Fringe of dispersive interferometry at the initial position. (a): Fringe of dispersive interferometry before shifting 100 μm; (b): fringe of dispersive interferometry after shifting 100 μm; (c): High-pass signal of the fringe in (a); (d): High-pass signal of the fringe in (b);
Figure 8 shows the fringes of dispersive interferometry at 2.2 m distance relative to the initial position. In this case, the reference mirror is moved by 1200 μm, and the position of the widest fringe is shifted from 581.387 THz to 577.434 THz, as shown in Fig. 8. Compared with the fringes in Fig. 7, we find that the modulation frequency of the fringe becomes higher. Please note that, this is because the total dispersion is distance dependent. Further, the width of the widest fringe gets smaller. The same as the calculation in Eq. (17), χ can be calculated to be 3.03 × 10−4 m/THz. Based on Fig. 7(a) and Fig. 8(a), we find that the widest fringe is shifted from 580.217 THz to 581.387 THz, when we move the target mirror from the initial position to the position of about 2.2 m. The distance L can be thus calculated as:
Fig. 8 Fringe of dispersive interferometry at 2.2 m distance. (a): Fringe of dispersive interferometry before shifting 1200 μm; (b): fringe of dispersive interferometry after shifting 1200 μm.
Therefore, based on the dispersive interferometry, the distance can be determined underwater.
4.2 Results of distance measurement based on dispersive interferometry
The experimental results of the distance measurement are shown in Fig. 9, compared with the reference values. At each distance, we fast measure the distance for five times. We find that, the measurement uncertainty can be well within 100 μm at 8 m range. Based on Eq. (14), the measurement uncertainty is related to the repetition frequency frep, the group refractive index ng, the linear parameter χ, and the frequency shift fshift, and can be calculated as:
Fig. 9 Distance measurement results of dispersive interferometry. The black solid points indicate the average value of five measurements, and the red x markers show the scatters of five single measurements. The error bars represent the standard deviation of five measurements. The green dashed line indicates the uncertainty limit.
The same as that in Sec. 3.2, the part due to the repetition frequency can be evaluated as 10−10∙L. The part due to the group refractive index can be estimated as 1.5 × 10−5∙L. The third term of Eq. (19) is related to the linear parameter χ. In the case of 1.2 THz frequency shift, this part can be calculated to be 0.9 μm when the uncertainty of χ is 1 × 10−6 m/THz. The fourth term of Eq. (19) is related to the measurement of fshift, where the resolution of the optical spectrum analyzer and the program of the data processing can make contribution. If χ equals to 3.03 × 10−4 m/THz, this part can be calculated to be 8.8 μm with 0.04 THz uncertainty of fshift. We use the standard deviation as the uncertainty of fshift. Finally, the combined uncertainty can be thus expressed as [8.8μm2 + (1.5 × 10−5∙L)2]1/2. The dominating source of the measurement uncertainty is the correction of water refractive index.
5. Conclusion
In this paper, we, for the first time, use frequency comb laser to determine the distance underwater. In both the time and frequency domains, we comprehensively analyze the principle of pulse cross correlation and dispersive interferometry, which can be used to determine the distance. The experimental results show the measurement uncertainty of well below 100 μm in both cases of pulse cross correlation and dispersive interferometry, showing our technique can measure the distance underwater with high accuracy and precision.
Frequency combs have found applications in many fields, and achieved huge success in both the basic science and technology. It is valuable and deployable to use the state-of-the-art technique of frequency comb in the underwater occasions. In our field of measurement, length is a basic physical quantity. This presented work shows that the underwater length can be precisely determined by using frequency comb, and it also implies that the physical quantity related to length can be measured, such as angle, speed, surface topography, thickness, and refractive index [39], etc. The only limitation could be the strong power attenuation of the light when propagating in water. In spite of this, the high peak power of the ultra-short pulses from a low repetition frequency pulsed laser could make the returned signal easy to detect. Our work could open a new opportunity of frequency comb in the marine science and technology, especially in the application of near field.
Funding
National Natural Science Foundation of China (NSFC) (61505140); National Key Research and Development Plan (2016YFC1401203); Natural Science Foundation of Tianjin (18JCYBJC17100); Research projects of Tianjin Municipal Education Commission (JWK1616).
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