A processing method for the precise acquisition of 1D interference spectrum from the 2D Fabry-Pérot(F-P) fringe pattern is proposed in the paper. Current methods can only measure the spectrum with full-pixel accuracy. The new method is capable to achieve the sub-pixel accuracy as well as reduce the noise by taking full advantage of the 2D image information and analyzing the statistical values of pixels in a comprehensive way. Experimental results show that the accuracy of the measured Brillouin shift and linewidth is around several MHzs, which is significantly improved compared to methods with full-pixel accuracy.
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Brillouin scattering of light is an important tool for measuring physical parameters and has been used in optical fibers and atmospheric applications [1–4]. In recent years, Brillouin Lidar system has been introduced and verified in the remote sensing of the ocean to measure physical parameters like ocean temperature, sound speed, etc [5–7]. There are mainly three techniques for Brillouin Lidar measurement: the edge technique [8, 9], the scanning F-P interferometer [10, 11] and the F-P etalon combined with Intensified Charge Coupled Device(ICCD) [12, 13]. Among these methods, the F-P etalon with ICCD is proved to have more capability  as it can obtain the scattering spectrum in short time.
However, the backscattered signal obtained by F-P etalon and ICCD is displayed as 2D interference fringe pattern rather than the Rayleigh-Brillouin Scattering(RBS) spectrum, therefore an important process is needed to interpret the 2D image into 1D interference spectrum for the retrieval of RBS spectrum. Several methods have been proposed for extracting the 1D information from the 2D fringe pattern, such as the method based on direct reading [15, 16], the CLIO system [17, 18], or the introduction of cylindrical lens .
Unfortunately, in these methods two major factors will limit the measurement accuracy: the full-pixel precision and the system noise. The full-pixel precision indicates the best possible resolution when a pixel is the minimum unit of measurement. For instance, if three orders of fringes are displayed on a 1024 × 1024 ICCD when the free spectral range (FSR) of F-P etalon is about 20GHz, the minimum spectral resolution could only achieve more than 100MHz, which will correspond to large measurement uncertainty and greater error of the measured ocean physical parameters. The system noise comprises of thermal noise, dark current noise on ICCD and the noise brought forth when the experimental setup is not ideally calibrated. As a result the ICCD fringe pattern will be polluted by background noises, and the intensities of pixels in the same order of fringe ring will fluctuate. Although with the advances in instruments these problems will be less severe in the future, it will be more effective to develop an algorithm that will comprehensively analyze the collected information, reduce the noise and increase the measurement accuracy. In 2011, Hirschberger and Ehret developed a method to evaluate the radii of simulated FPI rings for the Doppler wind lidar . However, the method may not be suitable for real experiments as the FPI patterns are simulated under some ideal conditions. For instance, as a basic step, the center evaluation of fringe pattern requires the center of mass to be close to the ring center, while in real experiments this condition cannot be easily fulfilled.
On the other hand, current studies on the F-P measurement model for ocean physical parameters like temperature show that the model accuracy can achieve 0.06°C when the measurement uncertainty for Brillouin frequency shift is 1MHz  and 0.02 °C when the measurement uncertainty for Brillouin line width is 1MHz . Therefore, when the measured accuracy of ocean temperatures is needed to achieve the magnitude of 1/100 degree Celsius, there is an unsolved problem about how to increase the measurement accuracy from 100MHz to several MHz.
In this paper, we present a novel processing method for extracting the 1D interference spectrum from the real experimental 2D fringe pattern. By taking full use of the statistical information provided by large amount of pixels around Rayleigh and Brillouin rings, the image data is analyzed in a comprehensive way, after a series of signal processing techniques, the noise will be significantly reduced and the sub-pixel accuracy (corresponds to MHz magnitude in spectral accuracy) can be achieved, and finally the accuracy of measured ocean parameters will be improved. This method can also be extended to other spectral measurement applications using F-P and ICCD.
2. Experimental setup
2.1 Experimental configuration
Figure 1 shows the configuration of the Brillouin lidar system using F-P etalon and ICCD. In this experiment, a lidar system based on stimulated Brillouin scattering was used to increase the detection depth. Specifically, a focus lens system was added to the optical path so that the laser beam can concentrate at a certain depth where the stimulated Brillouin scattering is generated. The stimulated Brillouin scattering signal was first collected by optical system and then transmitted through F-P etalon; the 2D interference fringes were recorded by ICCD and displayed on PC screen.
The focus lens system consists of a small diverging lens and a large converging lens, the laser beam is ðrst diverged and propagate in water through longer distance and then converged to excite the stimulated Brillouin scattering, therefore the detection depth is increased. The experimental results show that the detection can reach 7-9 attenuation lengths, which is 58.3-75.0 meters as the attenuation coefficient of clean water used in experiment was 0.12 .
As shown in the bottom of Fig. 1, the fringe rings is recorded in the form of 2D image, it is then interpreted into a 1D interference spectrum with a processing method, after that the Brillouin spectrum is retrieved and analyzed to measure the physical parameters of the ocean.
The laser used is an stable injection-seeded pulsed Nd:YAG laser (Continuum Powerlite PrecisionII8000) with a wavelength of 532nm, a pulse energy of 650mJ, a repetition rate of 10Hz. The F-P etalon is a CVI solid etalon made of quartz with a free spectral range of FSR = 20GHz and a reflective rate of 99.5%. After calibration, the FSR was calibrated to 19.6GHz. The ICCD system consists of an ICCD camera and an ICCD controller, The ICCD camera is Princeton Instruments PI-MAXIII with a resolution of 1024 × 1024 and a pixel size of 12.8 × 12.8μm. The ICCD controller is Princeton Instruments ST-133. Stanford Research System’s DG535 is also used to synchronize the laser with the ICCD system.
2.2 Fringe patterns on ICCD
The interference pattern obtain by ICCD contains several orders of fringes, and the number of orders shown on the pattern can be adjusted by modifying focus length of the lens used. The positions of rings are directly related to the spectral parameters like Brillouin shift and linewidth, so if more orders of rings are contained, the measurement accuracy can be improved by using data fusion to combine the information provided by several orders of rings. On the other hand, when too many orders are contained, each order of ring will contain fewer pixels, which result in large measurement error. In summary, 2-3 orders of rings should be the most favorable number for the analysis of the spectral data.
One of the patterns taken from the experiment is shown in Fig. 2 . The image has a resolution of 1024 × 1024 and contains three complete orders of fringes. Each order of the fringe has two rings, the inner one represents the Brillouin peak and the outer one represents the Rayleigh peak. It can be observed that the overall image quality is acceptable, but the noise distributed over the whole pattern poses the major challenge for further processing. The outer fringe rings are immersed in noise and can be barely recognized. Additionally, as the perfect alignment of the optical path is difficult, the pattern obtained might not be rotationally symmetric, i.e. the left part of the fringes has larger intensity than the right part. If the center of mass is evaluated as described by Hirschberger, it will deviate a lot from the geometric center, therefore, the method proposed in  is not suitable for these imperfect fringe rings.
The frequency difference between two neighboring orders of Brillouin/Rayleigh peaks is equal to the FSR of F-P etalon. As the image is covered by about three orders of fringes, so the 1024 pixels correspond to about three full FSRs. When FSR = 19.6GHz, it can be estimated that the measurement accuracy of each pixel is about 114.8MHz. In fact, this accuracy is insufficient for the measurement of parameters like ocean temperature in remote sensing applications. For instance, the frequency shift and line width accuracy should reach 1MHz when the temperature accuracy is need to achieve 0.02~0.06 °C [20, 21]. In order to fulfill this requirement a 1/100 pixel accuracy is needed. Consider that the instruments will have limited advancement in the near future, it is more practical to develop specific processing methods to reach the sub-pixel accuracy.
In the following section, we will present a comprehensive processing method to collect and make full use of large amount of pixels on ICCD. The main concept of the method is to combine the information of multiple radial cuts through the rings with a data fusion strategy. A fundamental step of the method is to find out the precise position of the center with an accuracy of 1/100 pixel, as it is indispensable for the measuring of ring radii and spectral parameters. Then a crucial step named data folding is carried out. The coordinates of pixels in a 2D region are converted to the distances of these pixels to the center and combined into a 1D interference spectrum with sub-pixel accuracy. After a de-noising procedure the smoothed spectrum can be obtained for the measurement of Brillouin parameters.
3. The processing method
3.1 Determination of the center
Center evaluation is a crucial step as the accurate radius value of each pixel can only be acquired when the center of the rings is precisely determined. A precise center is also the reliable basis for any further processing steps. The center is determined by the method shown in Fig. 3 . For a complete ring in the fringe pattern, the horizontal position and the vertical position of the center O' can be determined by the horizontal string (e.g. AA') and the vertical string (e.g. AA”) by calculating their midpoints (e.g. B and C).
The position of a point on ICCD can be expressed as x- and y- coordinates shown in Fig. 3. The point at the upper left corner is set as the origin of the coordinates. Figure 3 shows that, ideally, for each pair of points A and A' in the fringe ring, their midpoint should has the same x-coordinate as the center point O'. This relationship can be expressed as
Likewise, the y-coordinate of the center can be determined in a vertical way.
In our experiment, a light spot occurred in the center of the fringes. The light spot is manually removed before the calculation of the center coordinates to avoid errors. The center in Fig. 2 is calculated as xO' = 449.11 pixel, yO' = 539.58 pixel; the standard deviations of the coordinates are σx = 0.86 pixels and σy = 1.29 pixels, i.e. the position of the center has a standard deviation of ΔO' = 1.55 pixel, which corresponds to a spectral measurement error of 178MHz. This result is compared with the midpoints of several randomly chosen strings, as shown in Table 1 .
xO's and yO's stand for the calculated coordinates of midpoint of the randomly chosen string, Δx(Δy) are the errors between xO's(yO's) and xO'(yO'). It can be seen that the midpoint position of a single string might fluctuate within as many as several pixels. By taking the average of multiple strings this randomness is greatly diminished. The chosen center has a resolution of 1/100 pixel and provides a reliable basis for the following procedures.
3.2 Data folding
As mentioned above, when the calculation for spectral parameters is performed in units of full pixels, the measured uncertainty can only reach 114.8MHz, which is way far from the required accuracy. Current methods only take a single cut through the rings, so a large portion of information on the 2D image is discarded. In order to take full use of the pixels in the pattern, a procedure named “Data folding” is used to extract the 1D interference spectrum, the schematic of the process is shown in Fig. 4 .
In Fig. 4 each square stands for a pixel on ICCD. O' is the center determined in the previous step. Each pixel is noted by its x- and y-coordinates.
The basic idea of data folding is to collect and combine the pixels on multiple radial directions. For example, the distances of the pixels on the diagonal direction (noted as (1,1), (2,2), (3,3), … (i,i), … (i = 0,1,…)) to the center point are so these pixels can provide the intensities information which the pixels on the x-axis((0,0),(1,0),(2,0),(3,0)…)) cannot provide. Imagine that when this diagonal axis is “rotated” by 45° to the x-axis, as shown in Fig. 4, then the coordinate of pixel (i,i) is converted to d’(i,i):
In this way, more pixels with different coordinates are added to the x-axis, therefore the SNR and measurement accuracy of the fringe spectrum is increased. Likewise, pixels on other directions can also be added to the x-axis based on their distances from the center. The distance from a pixel to the center point is:
For the pixels with the same distance from the center point, an average of these pixels is calculated. A SNR-enhanced fringe spectrum is obtained when all pixels in a certain region are converted. The enhanced spectrum will include far more points than pixels the in the original x-axis. Take the 4 × 3 pixels on the left and lower corner of Fig. 4 as an example, originally there are 4 pixels ((0,0), (1,0), (2,0), (3,0)) on the x-axis. After data folding, 5 pixels ((1,1), (2,1), (3,1), (2,2), (3,2)) with different distances are added to the x-axis, therefore eventually there will be 9 pixels on the fringe spectrum.
It should be mentioned that some details need to be taken into consideration in real experiments. First, the coordinates of the center calculated in Fig. 2 are not integers. Therefore, when data folding is performed, the calculations are based on decimals instead of integers. However, the procedure of the data folding algorithm remains the same.
Another problem is that the real fringe pattern is polluted by noise and suffered from shadows which are not easy to diminish, so it is necessary to select a less affected region to process before data folding algorithm. After data folding procedure, the obtained interference spectrum of Fig. 2 is shown in Fig. 5 .
The x-axis of the fringe spectrum in Fig. 5 contains about 350 pixels. After data folding, a total of 110489 points are converted to the x-axis. Compared to number of pixels along a single cut through the ring, there are about 300 times more points available after data folding, so theoretically the measurement accuracy is capable to increase to several MHzs.
3.3 Smooth filtering
As shown in Fig. 5, the noise stemmed from the optical setup and circuit system has covered the whole interference spectrum and need to be addressed. We tested three popular smoothing filters: the wavelet de-noising filter, the mean filter and the FFT low-pass filter.
Wavelet de-noising is a popular method for smoothing. However, excessive number of smoothing parameters makes the definition of the parameters too complex. Mean filter has only one parameter, i.e. the width of smoothing window, but the selection of this parameter is hard as any unsuitable width will probably distort the line shape of the RBS signal in the smoothing procedure. Comparing with the other two smoothing methods, FFT low-pass filter can separate the RBS signal with noise in the frequency domain, and offers a more intuitive and convenient way to select the cut-off frequency. Therefore, FFT low-pass filter is eventually selected as the smoothing. The smoothing result is shown in Fig. 6 .
The de-noised spectrum (see the yellow line in Fig. 6) is significantly smoothed, in the meantime, the shape of the spectrum is less distorted compared to other smoothing techniques. The smoothed spectrum is ready for the retrieval of Brillouin parameters.
4. Experiments and analysis
In order to evaluate the performance of our method, the processed 1D interference spectrum is used to measure the temperature of water, the result is compared with temperature measured without the processing method. In addition to the experiment whose image is shown in Fig. 2, two more experiments with different temperatures were also carried out. The obtained spectra of the three experiments are shown in Fig. 7 .
By using the retrieving method , the Brillouin shift VB and the Brillouin linewidth ГB are measured, then temperature of water is retrieved either by VB or by ГB, noted as TV and TГ, respectively. The measurement uncertainties are also calculated and expressed as the range of these parameters, as shown in Table 2 .
As shown in Table 2, the measurement accuracy can be evaluated by comparing TV with TГ (their differences are given in Table 2 as ΔT), as the temperature calculated by VB or by ГB should be of the same value in theory. It can be observed that two of the three experiments have very small temperature differences(0.01 °C), while it increased to 0.1 °C in experiment 2 (the reason for the greater error will be discussed later). In general, the results have proved that the accuracy of the measured temperature is sufficient for remote sensing applications.
Another comparison is shown in the last two columns of Table 2. By using the temperature TГ measured by Brillouin linewidth ГB, the theoretical Brillouin shift is obtained and compared with VB, their difference are given as ΔV in Table 2. Likewise the difference between the theoretical linewidth and the measured ГB are given as ΔГ. The values of ΔV and ΔГ in experiment 1 and experiment 3 are less than 1MHz, and those in experiment 2 are around several MHzs, this will ensure the possibility to retrieve the ocean physical parameters with desired accuracy.
The main reason for the lower accuracies of parameters measured in experiment 2 is that the intensities of Brillouin peaks are much weaker than those in the other two experiments as shown in Fig. 7. The lower Brillouin peak will provide fewer effective information, this will result in larger errors in the measurement of VB, ГB, and eventually propagate to TV and TГ. Therefore, in order to avoid unnecessary errors the fringes taken in experiments who have brighter Brillouin rings should be more favorable for further processing’s.
In addition, the experimental results (noted as “folded”) are compared with Brillouin shift and linewidth measured from a single cut through the fringes (noted as “single”), as shown in Table 3 .
As shown in Table 2 and Table 3, the measurement uncertainties of the retrieved Brillouin shift VB and Brillouin linewidth ГB measured by the proposed method are of several MHzs, which is significantly improved than without the processing method. In other words, by using the proposed method, the measurement precision for the spectral parameters has been increased by scores of times. It is also proved that the processing method can improve the measurement accuracy of the interference spectrum from full-pixel to sub-pixel.
The differences in Brillouin shifts between the two methods are 100~200MHz, which match with the measurement accuracy of 114.8MHz reckoned above. The differences in Brillouin linewidths are less than 100MHz and obviously smaller than those in Brillouin shifts, the reason for this is that the frequency shift is mainly influenced by the positions of two pixels (the center and the brightest pixel on Brillouin peak) even using a fitting procedure, while the linewidth is mainly determined by multiple pixels along the Brillouin peak, with more constraints a more accurate linewidth can be obtained. The temperatures measured by using these linewidths are compared with TГs in Table 2, their differences are shown in Table 4 .
The temperatures in Table 4 are calculated by Brillouin linewidth rather than Brillouin shift for two reasons: the Brillouin linewidth is more accurate as discussed above; and the temperature measured by linewidth is also more accurate than that measured by frequency shift in general . As shown in Table 4, the differences of the temperatures are about 1 °C, which also proves that the measuring accuracy are greatly improved by the processing method.
In this paper, processing method of fringe patterns in the Brillouin lidar system using F-P etalon and ICCD is developed. First the center point is determined by taking the average of middle points of multiple strings in the fringe circle, then a data folding operation is performed to convert the 2D fringe pattern to 1D spectrum. The converted spectrum has a measurement accuracy of 1/100 of a pixel. Finally a de-noising filter is chosen to smooth the spectrum. Experimental results showed that with the proposed method, the measuring accuracy of Brillouin shift and linewidth can be as good as several MHz, which corresponds to the measuring accuracy of ocean temperature of less than 0.1 °C. Compared to current measurement methods, our method has significantly increased the measurement capability, this will be very useful for Brillouin lidar systems in the remote sensing of the ocean.
The authors would like to thank National Natural Science Foundation of China (grant No. 61078062 and grant No. 61108074) and Research Fund for the Doctoral Program of Higher Education of China (grant No. 20100142120012). Also, the authors would like to thank Prof. D. Liu for provision of experimental data and his helpful discussions.
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