Absolute Measurement of the Refractive Index of Water by a Mode-Locked Laser at 518 nm

In this paper, we demonstrate a method using a frequency comb, which can precisely measure the refractive index of water. We have developed a simple system, in which a Michelson interferometer is placed into a quartz-glass container with a low expansion coefficient, and for which compensation of the thermal expansion of the water container is not required. By scanning a mirror on a moving stage, a pair of cross-correlation patterns can be generated. We can obtain the length information via these cross-correlation patterns, with or without water in the container. The refractive index of water can be measured by the resulting lengths. Long-term experimental results show that our method can measure the refractive index of water with a high degree of accuracy—measurement uncertainty at 10−5 level has been achieved, compared with the values calculated by the empirical formula.


Introduction
During the past decades, marine technology has attracted great attention both as a field of research, and for economical reasons [1]. The resources hidden underwater, such as coal, oil, and natural gas, are receiving increasing interest because of depleting resources on land. Scientists all over the world have developed a wealth of marine instruments, so as to make sailing, as well as resource exploration, convenient and efficient. The acoustic method is the most-widely used technique, owing to a lower attenuation when travelling underwater; however, due to the multipath effect, considerable noise could exist in the signal returns [2][3][4], resulting in a very complicated data process. In addition, the measurement resolution of the acoustic wave is strongly limited by beam size and divergence. The electro-magnetic wave (~GHz) can be exploited in short distance measurements underwater (tens of meters), based on phase measurement and power attenuation [5,6]. In the case of the phase measurement, accuracy and precision cannot be very high on account of the larger electro-magnetic wavelength. In terms of power attenuation, power-based measurements are negatively affected by extreme environments underwater. In short-path underwater measurements, optical waves, with a transmission window of between 480 nm and 540 nm, are an ideal candidate [7][8][9]. With excellent laser alignment and inherent high power, the underwater optical technique has shown itself to be highly effective in marine applications [10]. Different from acoustic and electro-magnetic waves, one basic issue associated with the optical method is the correction of the refractive index of water [11].
Measurement of the refractive index of water is of fundamental importance in the application of optical underwater techniques, and various methods have been proposed to address the issue. Analogous with measurements of the refractive index of air, indirectly, the refractive index of water

Materials and Methods
The experimental setup is shown in Figure 1. The frequency comb (Menlosystem FC1000 (Menlosystem, Martinsried, Germany), 100 MHz repetition frequency, 500 mW maximum output power, 518 nm center wavelength) emits a pulsed laser, which is split at a beam splitter. One part is directed into the Michelson interferometer, which can introduce a measured length L. Please note that the entire Michelson interferometer is placed in the water container, which means that the reference path and the measurement path are thermally expanded at the same time. In addition, the thermal expansion coefficient of quartz glass is relatively lower. Therefore, the system can automatically compensate for the thermal expansion of the water container. The other part is scanned by using a moving stage (PI M521, PI, Karlsruhe, Germany), 200 mm travel range), and is then combined with the output of the Michelson interferometer at a beam splitter. A photo-detector (PD1, Thorlabs APD430A, Thorlabs, NJ, USA) is used to detect the combined beam, and an oscilloscope (LeCroy 640Zi, LeCroy, NYC, USA) is used to measure and store the cross-correlation patterns. To obtain the precise position of the scanning mirror (MS), a fringe counting interferometer based on a cw laser (Thorlabs HRS015B, Thorlabs, NJ, USA), 632.991 nm, 2 MHz frequency stability) is used as the distance meter. The temperature and the density of water are measured by Valeport Mini SVP in real time, which can be used to calculate the refractive index of water with the empirical formula (i.e., Harvey formula) [13]. Please note that the Harvey formula is used in the case of pure water. In our experiments, we used the tap water, which is sufficiently clean for the Harvey formula to be applicable. automatically compensate for the thermal expansion of the water container. The other part is scanned by using a moving stage (PI M521, PI, Karlsruhe, Germany), 200 mm travel range), and is then combined with the output of the Michelson interferometer at a beam splitter. A photo-detector (PD1, Thorlabs APD430A, Thorlabs, NJ, USA) is used to detect the combined beam, and an oscilloscope (LeCroy 640Zi, LeCroy, NYC, USA) is used to measure and store the cross-correlation patterns.
To obtain the precise position of the scanning mirror (MS), a fringe counting interferometer based on a cw laser (Thorlabs HRS015B, Thorlabs, NJ, USA), 632.991 nm, 2 MHz frequency stability) is used as the distance meter. The temperature and the density of water are measured by Valeport Mini SVP in real time, which can be used to calculate the refractive index of water with the empirical formula (i.e., Harvey formula) [13]. Please note that the Harvey formula is used in the case of pure water. In our experiments, we used the tap water, which is sufficiently clean for the Harvey formula to be applicable. We measure the refractive index of water with two steps. First, the water container is not filled with water. We can obtain a length value La in air, and La = naL. na is the group refractive index of air, which can be corrected by using empirical formulae, and L is the geometrical length difference between the reference and measurement beams of the Michelson interferometer. Second, the water container is full with water. Considering the group refractive index of water, we can obtain a length value Lgw, i.e., ngwL, where ngw is the group refractive index of water. In the case of phase refractive index at a certain wavelength, the corresponding length value Lp can be expressed as npL, where np is the corresponding phase refractive index of water. Consequently, the group refractive index of water ngw can be calculated as: For a certain wavelength, the phase refractive index of water can be indicated as: After we obtain the values of Lgw, Lp, La and na, the group and phase refractive indices of water can be measured. We measure the refractive index of water with two steps. First, the water container is not filled with water. We can obtain a length value L a in air, and L a = n a L. n a is the group refractive index of air, which can be corrected by using empirical formulae, and L is the geometrical length difference between the reference and measurement beams of the Michelson interferometer. Second, the water container is full with water. Considering the group refractive index of water, we can obtain a length value L gw , i.e., n gw L, where n gw is the group refractive index of water. In the case of phase refractive index at a certain wavelength, the corresponding length value L p can be expressed as n p L, where n p is the corresponding phase refractive index of water. Consequently, the group refractive index of water n gw can be calculated as: For a certain wavelength, the phase refractive index of water can be indicated as: After we obtain the values of L gw , L p , L a and n a , the group and phase refractive indices of water can be measured.

Results
The source spectrum is shown in Figure 2, with a center wavelength of approximately 518 nm and a spectral width of around 4 nm. The environmental parameters are: 19.2 • C temperature, 1032.5 hPa pressure, and 35% humidity. The group refractive index of air n a can be calculated as 1.00029157 by Ciddor formula [37].

Results
The source spectrum is shown in Figure 2, with a center wavelength of approximately 518 nm and a spectral width of around 4 nm. The environmental parameters are: 19.2 °C temperature, 1032.5 hPa pressure, and 35% humidity. The group refractive index of air na can be calculated as 1.00029157 by Ciddor formula [37].

Measurement of Group Refractive Index of Water
In this section, we describe the measurement of the group refractive index of water. The speed of the moving stage is set to 50 mm/s, i.e., travel time for a single stroke is 4 s. First, the water container is not filled with water. We can obtain a pair of cross-correlation patterns, shown in Figure 3a, which correspond to the reference and measurement mirrors respectively. Figure 3b shows a detailed observation of the cross-correlation pattern; in it, we see that the pulse width is about 100 μm (333 fs). Performing a Hilbert transform of the cross-correlation patterns in Figure 3a, we can obtain the results shown in Figure 4, where the inset is the curve expansion. Based on the peak positions, the length La can be measured, using a fringe counting interferometer, at 70.783 mm. Therefore, length L can be calculated as 70.783 mm/1.00029157, i.e., 70.762 mm.
Second, the container is full with water. The obtained cross-correlation patterns are indicated in Figure 5a. Figure 5b is the expansion of a single cross-correlation pattern in the time axis. We find that, the cross-correlation pattern is obviously broadened to about 350 μm (1.2 ps). This broadening is due to water dispersion. The length Lgw can be measured by the same process as that of Figure 4,

Measurement of Group Refractive Index of Water
In this section, we describe the measurement of the group refractive index of water. The speed of the moving stage is set to 50 mm/s, i.e., travel time for a single stroke is 4 s. First, the water container is not filled with water. We can obtain a pair of cross-correlation patterns, shown in Figure 3a, which correspond to the reference and measurement mirrors respectively. Figure 3b shows a detailed observation of the cross-correlation pattern; in it, we see that the pulse width is about 100 µm (333 fs).

Results
The source spectrum is shown in Figure 2, with a center wavelength of approximately 518 nm and a spectral width of around 4 nm. The environmental parameters are: 19.2 °C temperature, 1032.5 hPa pressure, and 35% humidity. The group refractive index of air na can be calculated as 1.00029157 by Ciddor formula [37].

Measurement of Group Refractive Index of Water
In this section, we describe the measurement of the group refractive index of water. The speed of the moving stage is set to 50 mm/s, i.e., travel time for a single stroke is 4 s. First, the water container is not filled with water. We can obtain a pair of cross-correlation patterns, shown in Figure 3a, which correspond to the reference and measurement mirrors respectively. Figure 3b shows a detailed observation of the cross-correlation pattern; in it, we see that the pulse width is about 100 μm (333 fs). Performing a Hilbert transform of the cross-correlation patterns in Figure 3a, we can obtain the results shown in Figure 4, where the inset is the curve expansion. Based on the peak positions, the length La can be measured, using a fringe counting interferometer, at 70.783 mm. Therefore, length L can be calculated as 70.783 mm/1.00029157, i.e., 70.762 mm.
Second, the container is full with water. The obtained cross-correlation patterns are indicated in Figure 5a. Figure 5b is the expansion of a single cross-correlation pattern in the time axis. We find that, the cross-correlation pattern is obviously broadened to about 350 μm (1.2 ps). This broadening is due to water dispersion. The length Lgw can be measured by the same process as that of Figure 4, Performing a Hilbert transform of the cross-correlation patterns in Figure 3a, we can obtain the results shown in Figure 4, where the inset is the curve expansion. Based on the peak positions, the length L a can be measured, using a fringe counting interferometer, at 70.783 mm. Therefore, length L can be calculated as 70.783 mm/1.00029157, i.e., 70.762 mm.
Second, the container is full with water. The obtained cross-correlation patterns are indicated in Figure 5a. Figure 5b is the expansion of a single cross-correlation pattern in the time axis. We find that, the cross-correlation pattern is obviously broadened to about 350 µm (1.2 ps). This broadening is due to water dispersion. The length L gw can be measured by the same process as that of Figure 4, and is 96.676 mm. Based on Equation (1), the group refractive index of water can be thus calculated as: 96.676/70.762 = 1.36621. The water conditions were: 14.4 • C temperature and 999.2 kg/m 3 density, and based on the empirical formula, the group refractive index of water can be calculated to be 1.36622. Finally, we find a difference of 1 × 10 −5 between our result and that obtained using the empirical formula. Please note that the empirical formula in this work refers to Reference [13]. and based on the empirical formula, the group refractive index of water can be calculated to be 1.36622. Finally, we find a difference of 1 × 10 −5 between our result and that obtained using the empirical formula. Please note that the empirical formula in this work refers to Reference [13].  We performed a measurement over a 5-h period; the results are shown in Figure 6. In Figure 6a, the red solid line indicates the results obtained by our method, and the black dashed line represents results obtained using the empirical formula. Please note that, for convenience of display, both results are shifted by −1.36622. We observed that the group refractive index of water changed by up to around 3 × 10 −4 in a 5-h period. Figure 6b shows the difference between the results obtained using the empirical formula and our method. In long-term experiments, we observed the difference to be be well below 2 × 10 −5 .  and based on the empirical formula, the group refractive index of water can be calculated to be 1.36622. Finally, we find a difference of 1 × 10 −5 between our result and that obtained using the empirical formula. Please note that the empirical formula in this work refers to Reference [13].  We performed a measurement over a 5-h period; the results are shown in Figure 6. In Figure 6a, the red solid line indicates the results obtained by our method, and the black dashed line represents results obtained using the empirical formula. Please note that, for convenience of display, both results are shifted by −1.36622. We observed that the group refractive index of water changed by up to around 3 × 10 −4 in a 5-h period. Figure 6b shows the difference between the results obtained using the empirical formula and our method. In long-term experiments, we observed the difference to be be well below 2 × 10 −5 . We performed a measurement over a 5-h period; the results are shown in Figure 6. In Figure 6a, the red solid line indicates the results obtained by our method, and the black dashed line represents results obtained using the empirical formula. Please note that, for convenience of display, both results are shifted by −1.36622. We observed that the group refractive index of water changed by up to around 3 × 10 −4 in a 5-h period. Figure 6b shows the difference between the results obtained using the empirical formula and our method. In long-term experiments, we observed the difference to be be well below 2 × 10 −5 .

Measurement of Phase Refractive Index of Water
In this section, we measure the phase refractive index of water at a particular wavelength. Different from the Section 3.1, here we used a Fourier transform to determine the distance for each wavelength. In fact, as a classical method of data processing, Fourier transforms can also be used in Section 3.1. In that case, the length can be measured through the slope of the unwrapped phase, and the results are nearly the same.
Assuming that the phase of the wavelength λ is φ0 for the reference mirror, and φ1 for the measurement mirror, the length Lp can be calculated as (φ1 − φ0)/(2π) × λ for the wavelength λ [38,39]. Please note that φ1 and φ0 are the unwrapped phases. The data process of the Fourier transform is shown in Figure 7. The first cross-correlation pattern, (A) in Figure 5, is picked up, and zero padding is needed. To enhance process efficiency and save memory, we subsample the raw data, which is a mature technique widely used in the signal processing [40,41]; the resulting curve is shown in Figure  7a. Please note that the subsampling coefficient N0/N1 should be defined carefully; it must meet the condition that the frequency spectrum of the Fourier transform be completely located within a range from 0 to fs/2 (without spectrum leakage), analogous to the demonstration mentioned in Reference [42]. N0 is the sample number of the raw data (3 × 10 6 in our experiments), N1 is the sample number after subsampling (544 in our experiments), and fs is the sampling rate. Figure 7b shows the frequency spectrum, and we find that no spectrum leakage exists. Figure 7c,d shows the wrapped phase and unwrapped phase respectively. The data process of the second cross-correlation pattern (B in Figure 5) is the same as that shown in Figure 7. After the phases are obtained, Lp can be determined, and the phase refractive index can be measured based on Equation (2).

Measurement of Phase Refractive Index of Water
In this section, we measure the phase refractive index of water at a particular wavelength. Different from the Section 3.1, here we used a Fourier transform to determine the distance for each wavelength. In fact, as a classical method of data processing, Fourier transforms can also be used in Section 3.1. In that case, the length can be measured through the slope of the unwrapped phase, and the results are nearly the same.
Assuming that the phase of the wavelength λ is ϕ 0 for the reference mirror, and ϕ 1 for the measurement mirror, the length L p can be calculated as (ϕ 1 − ϕ 0 )/(2π) × λ for the wavelength λ [38,39]. Please note that ϕ 1 and ϕ 0 are the unwrapped phases. The data process of the Fourier transform is shown in Figure 7. The first cross-correlation pattern, (A) in Figure 5, is picked up, and zero padding is needed. To enhance process efficiency and save memory, we subsample the raw data, which is a mature technique widely used in the signal processing [40,41]; the resulting curve is shown in Figure 7a. Please note that the subsampling coefficient N 0 /N 1 should be defined carefully; it must meet the condition that the frequency spectrum of the Fourier transform be completely located within a range from 0 to f s /2 (without spectrum leakage), analogous to the demonstration mentioned in Reference [42]. N 0 is the sample number of the raw data (3 × 10 6 in our experiments), N 1 is the sample number after subsampling (544 in our experiments), and f s is the sampling rate. Figure 7b shows the frequency spectrum, and we find that no spectrum leakage exists. Figure 7c,d shows the wrapped phase and unwrapped phase respectively. The data process of the second cross-correlation pattern (B in Figure 5) is the same as that shown in Figure 7. After the phases are obtained, L p can be determined, and the phase refractive index can be measured based on Equation (2).

Measurement of Phase Refractive Index of Water
In this section, we measure the phase refractive index of water at a particular wavelength. Different from the Section 3.1, here we used a Fourier transform to determine the distance for each wavelength. In fact, as a classical method of data processing, Fourier transforms can also be used in Section 3.1. In that case, the length can be measured through the slope of the unwrapped phase, and the results are nearly the same.
Assuming that the phase of the wavelength λ is φ0 for the reference mirror, and φ1 for the measurement mirror, the length Lp can be calculated as (φ1 − φ0)/(2π) × λ for the wavelength λ [38,39]. Please note that φ1 and φ0 are the unwrapped phases. The data process of the Fourier transform is shown in Figure 7. The first cross-correlation pattern, (A) in Figure 5, is picked up, and zero padding is needed. To enhance process efficiency and save memory, we subsample the raw data, which is a mature technique widely used in the signal processing [40,41]; the resulting curve is shown in Figure  7a. Please note that the subsampling coefficient N0/N1 should be defined carefully; it must meet the condition that the frequency spectrum of the Fourier transform be completely located within a range from 0 to fs/2 (without spectrum leakage), analogous to the demonstration mentioned in Reference [42]. N0 is the sample number of the raw data (3 × 10 6 in our experiments), N1 is the sample number after subsampling (544 in our experiments), and fs is the sampling rate. Figure 7b shows the frequency spectrum, and we find that no spectrum leakage exists. Figure 7c,d shows the wrapped phase and unwrapped phase respectively. The data process of the second cross-correlation pattern (B in Figure 5) is the same as that shown in Figure 7. After the phases are obtained, Lp can be determined, and the phase refractive index can be measured based on Equation (2).  With water conditions of 14.4 °C and 999.2 kg/m 3 , the experimental results of phase refractive index measurements in the spectral range from 515 nm to 521 nm are shown in Figure 8. The pink dashed line indicates the results obtained using our method, and the green solid line shows the those from the empirical formula. In the range from 515 nm to 521 nm, the difference between our method and the empirical formula can be less than 1 × 10 −5 . We also find that water dispersion is very significant when green light transmits in water (~2.5 × 10 −4 in the 6-nm spectral range), and that the pulse can be strongly broadened. We carried out 5-h experiments; the experimental results for 518 nm wavelength (vacuum wavelength) are shown in Figure 9. In Figure 9a, the red solid line indicates the results obtained using our method, and the black dashed line represents those results obtained using the empirical formula. Please note that for convenience of display, both the results are shifted by −1.34272. We found that the results obtained using our method and those from the empirical formula changed under the same law, and the phase refractive index of water at 518 nm varied by up to about 2.5 × 10 −4 in a 5-h period. Figure 9b shows the difference between the results obtained using the empirical formula and our method. In long-term experiments, we found measurement uncertainty to be well below 1.2 × 10 −5 . With water conditions of 14.4 • C and 999.2 kg/m 3 , the experimental results of phase refractive index measurements in the spectral range from 515 nm to 521 nm are shown in Figure 8. The pink dashed line indicates the results obtained using our method, and the green solid line shows the those from the empirical formula. In the range from 515 nm to 521 nm, the difference between our method and the empirical formula can be less than 1 × 10 −5 . We also find that water dispersion is very significant when green light transmits in water (~2.5 × 10 −4 in the 6-nm spectral range), and that the pulse can be strongly broadened.  Figure 8. The pink dashed line indicates the results obtained using our method, and the green solid line shows the those from the empirical formula. In the range from 515 nm to 521 nm, the difference between our method and the empirical formula can be less than 1 × 10 −5 . We also find that water dispersion is very significant when green light transmits in water (~2.5 × 10 −4 in the 6-nm spectral range), and that the pulse can be strongly broadened. We carried out 5-h experiments; the experimental results for 518 nm wavelength (vacuum wavelength) are shown in Figure 9. In Figure 9a, the red solid line indicates the results obtained using our method, and the black dashed line represents those results obtained using the empirical formula. Please note that for convenience of display, both the results are shifted by −1.34272. We found that the results obtained using our method and those from the empirical formula changed under the same law, and the phase refractive index of water at 518 nm varied by up to about 2.5 × 10 −4 in a 5-h period. Figure 9b shows the difference between the results obtained using the empirical formula and our method. In long-term experiments, we found measurement uncertainty to be well below 1.2 × 10 −5 . We carried out 5-h experiments; the experimental results for 518 nm wavelength (vacuum wavelength) are shown in Figure 9. In Figure 9a, the red solid line indicates the results obtained using our method, and the black dashed line represents those results obtained using the empirical formula. Please note that for convenience of display, both the results are shifted by −1.34272. We found that the results obtained using our method and those from the empirical formula changed under the same law, and the phase refractive index of water at 518 nm varied by up to about 2.5 × 10 −4 in a 5-h period. Figure 9b shows the difference between the results obtained using the empirical formula and our method. In long-term experiments, we found measurement uncertainty to be well below 1.2 × 10 −5 .

Uncertainty evaluation
Based on Equation (1), the measurement uncertainty of ngw is related to Lgw, La and na, and can be calculated as: The first term of Equation (3) is related to the refractive index of air na based on the Ciddor formula. This part is related to the uncertainty of the Ciddor formula itself, the uncertainty of the sensor network, and the stability of the environment. Considering that the total optical path is relatively short, the inhomogeneity of the environment can be neglected. The average value of Lgw is 96.664 mm in 5-h experiments, and La equals to 70.783 mm. This part, dependent upon na, can thus be evaluated to 1.4 × 10 −8 , which is negligible. The second term of Equation (3) is related to the uncertainty of Lgw, which can be affected by the stability of the water and the algorithm of the data process. We fast performed five measurements; the stability of the measurement of Lgw is 1.3 μm (standard deviation). The second term can be calculated to 1.8 × 10 −5 . The third term of Equation (3) is related to the uncertainty of La, and the short-term stability of La is 0.6 μm (standard deviation). This part can be estimated to be 1.2 × 10 −5 . The expansion coefficient is 5 × 10 −7 /°C. The temperature change is about 3.5 °C (from 14.4 °C to 17.9 °C ) in long-term experiments. Thermal expansion can be calculated at 0.1 μm, corresponding to 1.8 × 10 −6 uncertainty. Finally, the combined uncertainty can be calculated to be 2.2 × 10 −5 , with a coverage factor of k=1, which shows a good agreement with the results in Figure 6b.
Based on Equation (2), the measurement uncertainty of np is related to Lp, La and na, and can be calculated as: As mentioned above, the first term of Equation (4) can be neglected. Considering the second term, the short-term stability of Lp is 0.45 μm, corresponding to 0.64 × 10 −5 uncertainty of the refractive index. The third term of Equation (4) can be calculated to 1.1 × 10 −5 uncertainty. The uncertainty due to thermal expansion is also 1.8 × 10 −6 . Therefore, combined uncertainty can be evaluated to be 1.3 × 10 −5 , with a coverage factor of k = 1, which shows a good agreement with the results in Figure 9b.

Uncertainty evaluation
Based on Equation (1), the measurement uncertainty of n gw is related to L gw , L a and n a , and can be calculated as: The first term of Equation (3) is related to the refractive index of air n a based on the Ciddor formula. This part is related to the uncertainty of the Ciddor formula itself, the uncertainty of the sensor network, and the stability of the environment. Considering that the total optical path is relatively short, the inhomogeneity of the environment can be neglected. The average value of L gw is 96.664 mm in 5-h experiments, and L a equals to 70.783 mm. This part, dependent upon n a , can thus be evaluated to 1.4 × 10 −8 , which is negligible. The second term of Equation (3) is related to the uncertainty of L gw , which can be affected by the stability of the water and the algorithm of the data process. We fast performed five measurements; the stability of the measurement of L gw is 1.3 µm (standard deviation). The second term can be calculated to 1.8 × 10 −5 . The third term of Equation (3) is related to the uncertainty of L a , and the short-term stability of L a is 0.6 µm (standard deviation). This part can be estimated to be 1.2 × 10 −5 . The expansion coefficient is 5 × 10 −7 / • C. The temperature change is about 3.5 • C (from 14.4 • C to 17.9 • C) in long-term experiments. Thermal expansion can be calculated at 0.1 µm, corresponding to 1.8 × 10 −6 uncertainty. Finally, the combined uncertainty can be calculated to be 2.2 × 10 −5 , with a coverage factor of k=1, which shows a good agreement with the results in Figure 6b.
Based on Equation (2), the measurement uncertainty of n p is related to L p , L a and n a , and can be calculated as: u n p 2 = L p L a u n a 2 + n a L a u L p 2 + n p L a u L a 2 (4) As mentioned above, the first term of Equation (4) can be neglected. Considering the second term, the short-term stability of L p is 0.45 µm, corresponding to 0.64 × 10 −5 uncertainty of the refractive index. The third term of Equation (4) can be calculated to 1.1 × 10 −5 uncertainty. The uncertainty due to thermal expansion is also 1.8 × 10 −6 . Therefore, combined uncertainty can be evaluated to be 1.3 × 10 −5 , with a coverage factor of k = 1, which shows a good agreement with the results in Figure 9b.

Conclusions
In this work, we present a method which enables the absolute measurement of the refractive index of water using a frequency comb. Based on changes of the optical path in air and water, the refractive index of water can be determined with high precision, which is of fundamental importance in the optical method when used underwater. Our results show a measurement uncertainty of 10 −5 , that is to say, the corresponding measurement uncertainty is only several mm when we measure a distance of up to 100 m underwater. This kind of performance can satisfy most applications. Thanks to the low power attenuation of blue-green light underwater, frequency-comb instruments show great promise for marine technology, significantly improving upon the current technologies. However, the entire system is relatively expensive on account of the cost of the frequency comb. The development of a portable and low-cost frequency comb would solve this problem [43,44].