童奕澄，童學(xué)東，張 凱，肖 達(dá)，戎宇航，周雨迪,3，劉 崇，劉 東

（1. 浙江大學(xué) 光電科學(xué)與工程學(xué)院 現(xiàn)代光學(xué)儀器國(guó)家重點(diǎn)實(shí)驗(yàn)室，浙江 杭州 310027；2. 寧波鋼鐵有限公司 浙江 寧波 315807；3. 浙江大學(xué) 寧波研究院 浙江 寧波 315100）

1 Introduction

Polarization lidar is one of the earliest members of lidar family. Since its birth in 1971, polarization lidar has become a research tool widely used in atmospheric cloud and aerosol detection[1]. The depolarization ratio obtained by polarization lidar inversion can be used to distinguish spherical and non-spherical particles, so it is often applied to the identification of aerosol type and the identification of thermodynamic phase state of clouds[2].Moreover, the depolarization ratio can also be used to identify the tropospheric boundary layer and to distinguish polar stratospheric clouds from other types of clouds in terms of morphology[3-5]. At the same time, the depolarization ratio can be used to study the long-distance transmission characteristics of dust[6]. It can be seen that the high-precision detection of depolarization ratio is of great significance to atmospheric science research. However,how to improve the detection accuracy of depolarization ratio has always been the research focus of polarization lidar[7].

The main causes for depolarization ratio error include: the calibration error of polarization lidar gain ratio, the error caused by the impurity of laser ray polarization, the alignment angle error between laser polarization vector and Polarization Beam Splitter (PBS) incidence plane, as well as the polarization crosstalk error caused when the reflectance and transmittance of PBS cannot reach 100%. In order to simplify the description, the above four errors are referred to as gain ratio calibration error,linear polarization error, alignment angle error and polarization crosstalk error respectively. Among them, gain ratio calibration error is particularly important and has a decisive effect on the accuracy of depolarization ratio[8]. The gain ratio calibration error varies with the gain ratio calibration method. For nearly half a century, more and more researchers have proposed new gain ratio calibration methods.However, there is still a lack of effective guidance and suggestions on the selection of gain ratio calibration method in the actual use of polarization lidar.

This paper analyzes the basic principles of various existing gain ratio calibration methods and compares the accuracy and advantages and disadvantages of +45° method, +45° method, ?45° method, rotation fitting method and pseudo-depolarizer method at different misalignment angles through experiments. Through the comparison of theory and experiment, this paper provides a suggestion for the best choice of gain ratio calibration method.

2 Basic principles and structure

The typical polarization lidar is a two-channel lidar[1]. According to the definition of depolarization ratio δ[9], we can obtain

where β represents atmospheric backscattering coefficient,Prepresents the echo signal power, and the subscripts ⊥ and ‖ respectively represent the vertical and parallel components of the above parameters.

Fig. 1 Basic principle and structure diagram of polarization lidar system圖1 偏振激光雷達(dá)系統(tǒng)基本原理與結(jié)構(gòu)圖

As shown in Fig. 1. the vertical componentP⊥and parallel componentP‖of echo signal can be decomposed into the following components after the coordinate rotation transformation relative to the incident plane of PBS:

where the subscriptsS′andP′respectively represent the directions perpendicular to and parallel to the incident plane of PBS, andθrepresents the angle between the laser polarization vector and the incident plane of PBS (here referred to as misalignment angle).

Because there is polarization crosstalk in the actual PBS, the parametersRP,RS,TPandTSrespectively represent the reflectance and transmittance ratios of P and S light in PBS (the above four parameters are generally known, and are labeled by PBS manufacturer). After the light passes through the PBS, the detected powerPRandPTin the reflection channel and transmission channel can be expressed as

whereKRandKTrespectively represent the gain coefficients of reflection channel and transmission channel,G=KR/KT. According to the Eq. (3), the actually measured depolarization ratio δ?(θ) can be obtained:

It can be seen from the equations (1) ～ (4) that the gain ratioGmust be calibrated before the calculation of the depolarization ratio, and that the gain ratio calibration error will affect the calculation result of the depolarization ratio[10]. Next, we will introduce several typical gain-ratio calibration methods.

3 Gain ratio calibration methods

3.1 Method of clean atmospheric molecule

The method of clean atmospheric molecule is a method to calibrate the gain ratio by comparing the actual depolarization ratio of clean atmosphere detected by the system and the theoretical depolarization ratio of clean atmosphere, assuming that only atmospheric molecules (no aerosols and clouds) exist in the high air. The calculation formula of depolarization ratio in the method of clean atmospheric molecule is

where δ?and δ respectively represent the atmo

molspheric molecular depolarization ratio actually measured at the altitudercand the theoretical atmospheric molecular depolarization ratio. δmolcan be calculated according to the theory of atmospheric scattering[11], but it is not fixed. Atmospheric molecular scattering is mainly composed of Rayleigh scattering and vibration Raman scattering (with a negligible intensity). Rayleigh scattering is mainly composed of pure rotational Raman line and central Cabannes line[12]. In the Rayleigh scattering spectrum, Cabannes lines constitute the central peak of Doppler broadening, while pure rotational Raman lines are distributed on both sides of Cabannes lines to constitute the sidebands[13]. The depolarization effect caused by pure rotational Raman lines is much greater than that caused by Cabannes lines. For a li dar system, the value range of δmolis between 0.003 63～0.014 3 when the filters with different bandwidths (BWs) are used[10]. If the filter bandwidth in the lidar system is narrow (BW＜0.3 nm@532 nm), δmolwill be the lower limit, namely δmol=0.003 63. Conversely, if the filter bandwidth is wide (BW=15 nm@532 nm), δmolwill be the upper limit, namely δmol=0.014 3.

Due to convenient operation and no need to add other devices to the light path of the system, the method of clean atmospheric molecule was widely used in the 1980s and 1990s. However, its shortcoming is also very obvious. Because clean atmosphere rarely exists, the calibration result obtained from a calibration area with aerosols or clouds present will have a large error. Moreover, when the filter bandwidth ranges from 1 nm to 15 nm, the proportion of pure rotational Raman lines in the scattered atmospheric molecules cannot be accurately evaluated. This may lead to the inaccurate calculation of the theoretical depolarization ratio of clean atmosphere, resulting in a calibration error. In addition, it should be noted that the method of clean atmospheric molecule is generally applicable to the lidar with a laser wavelength of less than 550 nm.For the lidar with a detection wavelength of more than 800 nm, this calibration method is likely to cause a large calibration error because the Rayleigh scattering intensity is weak[13]. Therefore, this method is rarely used at present.

3.2 +45° method

+45° method is a method of placing a Half-Wave Plate (HWP) in the receiving optical path(generally in front of PBS) and rotating it in one direction (clockwise/counterclockwise) to realize the gain ratio calibration. In particular, this method assumes that the properties of HWP are ideal and that the atmospheric state does not change during calibration. In addition to the above two basic assumptions, two more assumptions should be added to the+45° method, that is, neither a misalignment angle nor polarization crosstalk will exist[14].

As shown in Fig. 2 (Color online), the HWP is placed in the upstream optical path of PBS. In this case, the laser polarization vector is considered to be parallel to the incident plane of PBS, as shown in Fig. 2(a).

Fig. 2 Schematic diagram of +45° method. (a) Before HWP rotation. (b) After HWP rotation by +45°圖2 +45°法原理圖。(a)半波片旋轉(zhuǎn)之前；(b)半波片旋轉(zhuǎn)+45°之后

Since the impact of misalignment angle and polarization crosstalk is not considered, the Eq. (3)can be simplified as

Then, the HWP is rotated clockwise around the optical axis (similar to the counterclockwise case),so the angle between the laser polarization vector and the incident plane of PBS is +90°, as shown in Fi g. 2(b). In the Eq. (7),PT(0°) becomes(90°).The specific formula is not repeated here. In this case, the gain ratioGcan be expressed as

The advantage of this method is easy operation.Its disadvantage is that the alignment angle error and polarization crosstalk error can be easily introduced as the effects of misalignment angle and polarization crosstalk are ignored.

3.3 ±45° method

±45° method is a method of placing a HWP in the receiving optical path and rotating it twice (by+22.5° and ?22.5° respectively relative to the initial position) to realize the gain ratio calibration. This method was proposed by Freudenthaleret al. from the University of Munich, Germany[15]. Based on the+45° method, the ±45° method has considered the effects of both misalignment angle and polarization crosstalk. As shown in Fig. 3 (Color online).

Fig. 3 Schematic diagram of ±45° method. (a) Before HWP rotation. (b) After HWP rotation by +22.5°.(c) After HWP rotation by ?22.5°圖3 ±45°法原理圖。(a)半波片旋轉(zhuǎn)之前；(b)半波片旋轉(zhuǎn)+22.5°之后；(c)半波片旋轉(zhuǎn)?22.5°之后

In Fig. 3, a HWP is placed in the upstream optical path of PBS. It is assumed that there is an initial misalignment angle θinitbetween the laser polarization vector and the incident surface of PBS, and that θhis the quantitative misalignment angle introduced artificially.

After the HWP rotation, the following equation can be derived from Eq. (4):

In order to reduce the influence of initial misalignment angle θinit, the HWP is rotated twice continuously in the ±45° method. At first, the HWP is rotated by +22.5° clockwise around the optical axis relative to its initial position. Then, the HWP is rotated by ?45° counterclockwise around the optical axis based on the first rotation, or by ?22.5° counterclockwise around the optical axis relative to its initial position. In the two rotations, the angles between the laser polarization vector and the incident surface of PBS are +45° and ?45° respectively.At this time, the gain ratioGcan be expressed as

It can be seen from Eq. (10) that in the ±45°method, the polarization crosstalk error of PBS is considered, but the influence of alignment angle error cannot be completely eliminated. It is proved by facts that the error source has little to do with the Signal-to-Noise Ratio (SNR) in the ±45° method[16].If θinit= 1°, the relative error ofGcan be controlled within 5%[15]. With the advantages of simple operation and high accuracy, the ±45° method has been used in MULIS (Multichannel Lidar System), POLIS (Portable Lidar System) and other high-precision polarization lidar systems for gain ratio calibration[16]. Its shortcoming is that the alignment angle error cannot be eliminated.

3.4 ?45° method

?45° method is a method of placing a HWP in the receiving optical path and rotating it by 45° in one direction (clockwise/counterclockwise) to realize the gain ratio calibration. This method has the same operation as +45° method, but different calculation approach. In addition, it doesn’t need initial 0° search. This method was proposed by Luo Jinget al. from Zhejiang University[17], as shown in Fig. 4(Color online).

Fig. 4 Schematic diagram of ?45° method. (a) Before HWP rotation. (b) After HWP rotation by 45°圖4 ?45°法原理圖。(a)半波片旋轉(zhuǎn)之前；(b)半波片旋轉(zhuǎn)45°之后

In Fig. 4, a HWP is placed in the upstream optical path of PBS. It is assumed that there is an initial misalignment angle between the laser polarization vector and the incident surface of PBS.

Before the HWP rotation, the following equation can be derived from Eq. (4):

Then, the HWP is rotated around the optical axis, as shown in Fig. 4(b). After rotation, the termsPR(θinit) andPT(θinit) in Eq. (11) turn intoand. The specific formula is not repeated here. In this case, the gain ratio can be expressed as

3.5 Rotation fitting method

Rotation fitting method is a method of placing a HWP in the receiving optical path and rotating it for several times to realize the gain ratio calibration through the nonlinear least square fitting and the inversion of gain ratio, depolarization ratio and initial misalignment angle θinit[18]. This method was proposed by Alvarezet al. from NASA.

As shown in Fig. 5 (Color online), a HWP is placed in the upstream optical path of PBS. It is assumed that there is an initial misalignment angleθinitbetween the laser polarization vector and the incident surface of PBS.

If the HWP is rotated artificially by θh,j/2 relative to its initial optical axis, a series of misalignment angles θh,jcan be introduced artificially and quantitatively. From Eq. (9), the following equation can be derived:

wherejrepresents thej-th rotation of the HWP. In the Eq. (13), asandTScan be considered as known quantities, only three quantities, namely gain ratioG, initial misalignment angle θinitand theoretical depolarization ratio δ, are unknown. In this case, the three unknowns cannot be solved by a single equation. However, multiple equations can be obtained by rotating the HWP for many times. Then the nonlinear least square method can be used to solve the equation set so as to solve the three unknowns. It should be noted that at least three equations need to be obtained in this method, namelyj≥3.

The advantage of this method is that the three unknowns, namely gain ratio, depolarization ratio and initial misalignment angle, can be inverted simultaneously through one calibration, and no a priori value is required. However, due to time-consuming calibration and complicated operation, this method is only suitable for a relatively stable atmospheric environment.

Fig. 5 Schematic diagram of rotation fitting method. (a)Before HWP rotation. (b) After HWP rotation by θh,j圖5 旋轉(zhuǎn)擬合法原理圖。(a)半波片旋轉(zhuǎn)之前；(b)半波片旋轉(zhuǎn)θ h,j角之后

3.6 Pseudo-depolarizer method

The pseudo-depolarizer method is a method that adds an optical element to the optical path to convert the echo signal received by the system into unpolarized light so as to realize the gain ratio calibration. The gain ratio is calibrated by using the ratio of signal intensities of two detection channels. From Eq. (4), the gain ratioGcan be obtained:

A typical example is found in the CALIOP,where the non-depolarizing signal generated by a depolarizer is used for gain ratio calibration[19]. The specific operation procedure is as follows. A movable depolarizer is placed in the upstream optical path of PBS during the system calibration, and is removed after the calibration completion, as shown in Fig. 6. In the CALIOP, this method is used to calibrate the gain ratio of the system on orbit at night and then the method of clean atmospheric molecule is used to verify the calibration result[20].

Fig. 6 Schematic diagram for CALIOP gain ratio calibration圖6 CALIOP增益比定標(biāo)原理圖

The advantage of this method is that it can be operated easily and calibrated in real time, so as to eliminate the influence of atmospheric state change.Its disadvantage is that other errors will be introduced easily due to the difficulty for a commercial depolarizer to produce completely depolarized light.

4 Experimental results and analysis

The polarization lidar used in the experiment is a typical dual-channel lidar, whose structure is shown in Fig. 1 (where the same device is only marked once). The pseudo-depolarizer between the convergent lens and the HWP is only used in the pseudo-depolarizer method. To reduce linear polarization error, a polarizing prism was added to the emergent light path in the experiment system, so that the extinction ratio of the outgoing laser reached 2×105∶1. To reduce the polarization crosstalk error, a PBS was glued with a polarizing film to achieveTP:TS(RS:RP)>30 000∶1. Therefore, the effect of alignment error on gain ratio is discussed instead of the effect of linear polarization error and polarization crosstalk error. The specific parameters of the system are shown in Table 1.

Tab. 1 Main parameters for polarization lidar system表1 偏振激光雷達(dá)系統(tǒng)主要參數(shù)

Due to the large calibration error of clean atmospheric molecules, the experiment in this paper mainly compares the influence of five methods,namely, +45° method, ±45° method, ?45° method,rotation fitting method and pseudo-depolarizer method, on the gain ratio calibration at different misalignment angles (alignment angle errors).

To ensure a high signal-to-noise ratio, the experiment was carried out at night. Meanwhile, in order to reduce the impact caused by atmospheric changes, the detection in horizontal direction (pitch angle: 0°) was adopted. After the optical axis calibration of the system was completed, an electric rotating motor (accuracy: 0.005°) was used to adjust the half-wave plate to the position where the laser polarization vector was parallel to the incident surface of PBS (when 200 signals on average were passed and the maximum power of transmission channel was detected visually). At this point, the HWP angle was the initial 0°. It should be noted that, for the convenience of description, the initial misalignment angle θinitis not included in the actual total misalignment angle when the misalignment angle θhis introduced artificially and quantitatively to the following section. However, each actual misalignment angle contains the initial misalignment angle θinit(an unknown quantity). Then, withθh=0°(the actual misalignment angle is θinit+0°) as the zero point, an electric rotating motor is used to rotate the HWP. In the actual operation, the alignment angle usually does not exceed 15°[21-22]. However, to ensure the experiment integrity, the misalignment angle under discussion is expanded to 45° in this paper.

In the +45° method, ±45° method, ?45° method and rotation fitting method, the HWP was rotated with a step size of 2.5° within the θhrange of?45°～67.5° to obtain a total of 46 sets of original echo signals. By processing the above data, a step size of 5° and a θhrange of ?45° ～ +45° (that is, the difference between the two angles in each group of data before and after the HWP rotation are 45°)were selected in the four methods. After calculation,a total of 19 groups of gain ratio data were obtained.

Before starting the experiment, a pseudo-depolarizer was added to the optical path of the system, as shown in Fig. 1. Since the pseudo-depolarizer was not ideal, the HWP was rotated with a step size of 10° within the θhrange of ?80°～+100° to obtain a total of 19 sets of original echo signals in order to observe the change of pseudo-depolarizer in at least one cycle. By processing the above data, a total of 19 groups of gain ratio data were obtained after calculation.

The calibration results of the above five methods are given in Table 2 (θh=0 °).

Tab. 2 Calibration results of five methods atθh=0?表2 θh=0?時(shí)5種方法的定標(biāo)結(jié)果

As can be seen from Table 2, when there is no misalignment angle (θh=0 °), the calibration results of±45° method, ?45° method and rotation fitting method are close to each other and can be considered closest to the true value. Therefore, the average value of the gain ratios measured by the above three methods at θh=0 ° is defined as the true value.The curves of the relative errors of the five calibration methods changing with the misalignment angle are shown in Fig. 7 (the calculation processes of rotation fitting method and pseudo-deflector method are described in detail in the Sections 4.2 and 4.3).

Fig. 7 Relative errors changing with the misalignment angle for the five calibration methods圖7 5種定標(biāo)方法相對(duì)誤差隨對(duì)準(zhǔn)偏失角的變化曲線(xiàn)

4.1 +45°, ±45° and ?45° methods

As shown in Fig. 7, when |θh|＜15°, the calibration results of ±45° method and ?45° method are similar with a small relative error. However, even if the misalignment angle is 0°, the calibration result of +45° method is still greatly different from those of the above two methods, with a relative error up to 4%. When |θh|>15°, the calibration results of +45°method and ?45° method will remain almost stable.However, with the increase of misalignment angle,the calibration result of ±45° method will become more unstable and its relative error will increase sharply even up to 12.93%. The main reason for such error distribution is that the ±45° method calculates the geometric average of the two measurements before and after rotation, while the ?45°method calculates the arithmetic average of the two measurements before and after rotation based on the+45° method. The reasons for the above phenomena of +45° method will be explained below through specific theoretical analysis.

The relative error between the two methods was analyzed[23-24]. The powers detected in the reflection channel and transmission channel before and after the HWP rotation in the two methods are denoted asand. Then

where δ1and δ2represent the relative errors of the calibration results of ?45° method and ±45° method,and(SisRorTandnisaorb) represents the uncertainty (standard deviation) in each measured value. The photon counting signal in lidar can be considered to be subject to Poisson distribution[25-26],so the statistical error is equal to the root mean square of the mean value of the signal, i.e.Therefore, the equations (15) and (16) can be further deduced as follows

4.2 Rotation fitting method

Considering that the error of misalignment angle was not greater than 15° in the actual process,13 sets of data satisfying |θh|≤ 15° (that is, θh,j= 0°,±2.5°, ±5° ··· ±15°) were selected for fitting. The calculation results of rotation fitting are shown in Fig. 8 (Color online). According to Eq. (13), solving the three unknowns (gain ratioG, initial misalignment angle θinitand theoretical depolarization ratio δ) in the equations is a nonlinear least square problem. Before solvingG, θinitand δ, their initial values need to be predicted. Their optimum initial predicted values are shown in Fig. 8.

Fig. 8 Curve of the actually measured depolarization ratio changing with the misalignment angle, where the blue circle represents the δ?(θ) values measured at different θ angles, and the red and green dotted lines represent the fitting curve and θ init, respectively圖8 實(shí)際測(cè)量退偏比隨對(duì)準(zhǔn)偏失角的變化曲線(xiàn)圖。其中藍(lán)色圓圈代表在不同θ 情況下測(cè)量的 δ?(θ)，紅色虛線(xiàn)代表擬合曲線(xiàn)，綠色虛線(xiàn)代表θinit

It can be found that the relationship betweenand θh,j(blue circle) can be approximately represented by a quadratic polynomial[27].Therefore, the following quadratic polynomial is constructed with the artificially and quantitatively introduced misalignment angle θh,jas an independent variable and the actually measured depolarization ratioas a dependent variable:

whereA0,A1andA2are quadratic polynomial coeffici ents. The fitting result is shown as the red dotted line in Fig. 8. The minimum value of the quadratic polynomial represents the initial misalignment angle θinit, whose result is calculated to beθinit=?A1/(2×A2)= ?0.35°. This value can be used as the optimum initial predicted value of θinit. Then, θinit=?0.35° is substituted into Eq. (13) and the equation set is solved with nonlinear least square method to obtain the gain ratio, namelyG=1.2716.

4.3 Pseudo-depolarizer method

The calculation result of pseudo-depolarizer method is shown in Fig. 9 (Color online), in which the blue circles represent the gain ratios calculated at different angles of the HWP. It can be observed that the distribution of blue circles is in line with the cosine curve distribution law. Therefore, the following cosine function polynomial is constructed with the HWP rotation angle φ (θh=2φ) as an independent variable and the gain ratioGas a dependent variable:

whereB0,B1,B2andB3all represent the coefficients of the cosine function polynomial. The fitting result is shown as the red dotted line in Fig. 9.

Theoretically, if the echo signal is completely unpolarized light, its state will remain unchanged,irrespective of how the HWP angle is changed. In other words, the rotation of HWP will not affect the value of gain ratio. The result of gain ratio calibration should be represented by a line parallel to thexaxis, rather than by a cosine curve as shown in Fig. 9. The reason for this problem is that the existing commercial depolarizer cannot completely transform the polarized light into unpolarized light, so that the echo signal still contains part of the polarized light after passing through the depolarizer.When using laser as the light source to test the depolarizing effect of a depolarizer, Luo Jing et al.from Zhejiang University found that the test result was similar to the cosine distribution in Fig. 9[28].This indicates that the result of gain ratio calibration will still be affected by this portion of polarized light even if the misalignment angle is 0°.

Fig. 9 Gain ratio calibration result of pseudo-depolarizer method. The blue circles represent the gain ratios measured at different misalignment angles, and the red dotted line represents the cosine polynomial fitting curve圖9 退偏器法增益比定標(biāo)結(jié)果。藍(lán)色圓圈代表半波片在不同角度下測(cè)量的增益比，紅色虛線(xiàn)代表余弦函數(shù)多項(xiàng)式擬合曲線(xiàn)

It should be noted that among the measurement data obtained by pseudo-depolarizer method,only 5 sets of data satisfy θh= ?45° ～ +45°. Therefore, only 5 points are marked in Fig. 7 according to this method. At θh= 0°, the relative error of pseudodepolarizer method is 5.6%. This indicates that the result of gain ratio calibration will still be affected by this portion of polarized light even if the misalignment angle is 0°.

4.4 Discussion

Through the analysis of experimental results,we know that the calibration results of ±45° method,?45° method and rotation fitting method are the most accurate. This is consistent with the above theoretical analysis result. However, the error of ±45°method will increase with the misalignment angle.The rotation fitting method is only suitable for a relatively stable atmospheric environment due to timeconsuming calibration and complicated operation. In comparison, ?45° method has obvious advantages:easier operation, robust calibration results not affected by misalignment angle, and no need to search for the initial 0° angle. However, ?45° method cannot eliminate the influence of atmospheric state changes. In contrast, the pseudo-depolarizer method is not only easy to operate, but also capable of eliminating the influence of atmospheric state changes and free from the problem that multiple HWP rotations will increase the accumulated angle error. But so far, the commercial depolarizer still cannot produce completely depolarized light, which will introduce a new error that is difficult to evaluate. In conclusion, we suggest the use of ?45° method for calibration as a general rule and the use of pseudo-depolarizer method for calibration when a high-precision depolarizer is available.

5 Conclusion

The calibration accuracy of gain ratio has a great influence on the detection accuracy of polarization lidar. This paper compares a variety of the existing gain-ratio calibration methods theoretically and experimentally for the first time, and provides some suggestions on the selection of gain-ratio calibration methods.

As far as the operability is concerned, the pseudo-depolarizer method only needs a depolarizer inserted into the system to realize the calibration, and can eliminate the influence of atmospheric state changes. The other four methods require at least two rotations of the HWP in relatively complicated operation and cannot eliminate the influence of atmospheric state changes. In terms of calibration accuracy, this experiment mainly compared the influence of +45° method, ±45° method, ?45°method, rotation fitting method and pseudo-depolarizer method (excluding the method of clean atmospheric molecule due to its large calibration error)on the gain ratio calibration at different misalignment angles. The experimental results show that the calibration accuracy of ±45° method, ?45° method and rotation fitting method is relatively high, but the operation of ±45° method and rotation fitting method is quite complex. The error of ±45° method is big when the misalignment angle is large. In the ?45°method, the result of gain ratio calibration is not affected by the misalignment angle, and there is no need to search for the initial 0° angle. Compared with the first three methods, +45° method has a larger error when the misalignment angle is 0°. The pseudo-depolarizer method is greatly affected by the use of non-ideal depolarizer. If an ideal depolarizer is available, this method will be an ideal gain-ratio calibration method. By comparing different gain-ratio calibration methods theoretically and experimentally, this paper gives the best choice of gain ratio calibration method, suggesting the use of ?45°method for calibration as a general rule and the use of pseudo-depolarizer method for calibration when a high-precision depolarizer is available.

——中文對(duì)照版——

1 引 言

偏振激光雷達(dá)是激光雷達(dá)家族中最早的成員之一，自1971年誕生以來(lái)，其已廣泛應(yīng)用于大氣云及氣溶膠探測(cè)[1]。偏振激光雷達(dá)反演得到的退偏比可用于區(qū)分球形粒子和非球形粒子，故其常被應(yīng)用于氣溶膠的類(lèi)型識(shí)別及云的熱力學(xué)相態(tài)識(shí)別[2]。不僅如此，退偏比也可用于識(shí)別對(duì)流層的邊界層以及從形態(tài)學(xué)上區(qū)分極地平流層云與其它種類(lèi)云[3-5]。同時(shí)，退偏比還可以用于研究沙塵的長(zhǎng)距離傳輸特性[6]?？梢?jiàn)，實(shí)現(xiàn)退偏比的高精度探測(cè)對(duì)大氣科學(xué)研究具有重要的意義。然而，如何提高退偏比的探測(cè)精度，一直以來(lái)都是偏振激光雷達(dá)的研究重點(diǎn)[7]。

退偏比誤差產(chǎn)生的主要原因包括：偏振激光雷達(dá)增益比的定標(biāo)誤差、發(fā)射激光線(xiàn)偏振度不純而引起的誤差、激光偏振矢量與偏振分光棱鏡（Polarization Beam Splitter，PBS）入射面的對(duì)準(zhǔn)角誤差以及PBS反射、透過(guò)率無(wú)法達(dá)到100%而引起的偏振串?dāng)_誤差。為了簡(jiǎn)化描述，本文分別簡(jiǎn)稱(chēng)上述4種誤差為增益比定標(biāo)誤差、線(xiàn)偏振度誤差、對(duì)準(zhǔn)角誤差與偏振串?dāng)_誤差，其中增益比定標(biāo)誤差大小對(duì)退偏比的精度起著決定作用[8]。不同增益比定標(biāo)方法所產(chǎn)生的增益比定標(biāo)誤差也不同。近半個(gè)世紀(jì)，不斷有研究人員提出新的增益比定標(biāo)方法，然而到目前為止，偏振激光雷達(dá)在實(shí)際使用過(guò)程中對(duì)增益比定標(biāo)方法的選擇依舊缺少有效的指導(dǎo)與建議。

本文介紹了現(xiàn)存多種增益比定標(biāo)方法的基本原理，并通過(guò)實(shí)驗(yàn)對(duì)比分析了+45°法、±45°法、?45°法、旋轉(zhuǎn)擬合法與退偏器法在不同對(duì)準(zhǔn)偏失角情況下增益比定標(biāo)的準(zhǔn)確性以及各自的優(yōu)缺點(diǎn)。通過(guò)理論與實(shí)驗(yàn)的對(duì)比，本文給出了增益比定標(biāo)方法的最佳選擇。

2 基本原理與結(jié)構(gòu)

目前常見(jiàn)的偏振激光雷達(dá)為雙通道的激光雷達(dá)[1]，根據(jù)退偏比[9]的定義，可得

式中，β代表大氣后向散射系數(shù)，P代表回波信號(hào)功率，下標(biāo) ⊥與 ‖分別代表上述各參量的垂直與平行分量。如圖1所示，回波信號(hào)垂直分量P⊥與平行分量P‖相對(duì)于PBS入射面在經(jīng)坐標(biāo)旋轉(zhuǎn)變換后可分解為

式中，下標(biāo)S′與P′分別代表與PBS入射面垂直方向及平行方向，θ代表激光偏振矢量與PBS入射面存在的夾角（此處稱(chēng)為對(duì)準(zhǔn)偏失角）。

由于實(shí)際的PBS存在偏振串?dāng)_，因此，定義RP、RS、TP與TS分別代表PBS對(duì)P光與S光的反射率與透過(guò)率（以上4個(gè)參數(shù)一般由PBS生產(chǎn)商標(biāo)稱(chēng)出，屬于已知量）。經(jīng)過(guò)PBS后反射通道與透射通道被探測(cè)到的功率PR、PT分別可表示為

式中KR與KT分別代表反射通道與透射通道的增益系數(shù)。其中G=KR/KT，根據(jù)式（3）可得實(shí)際測(cè)量退偏比δ?(θ)

由式（1）～式（4）可知，在計(jì)算退偏比之前必須完成增益比G的定標(biāo)，而增益比定標(biāo)的誤差會(huì)對(duì)退偏比的計(jì)算結(jié)果造成影響[10]。接下來(lái)本文將介紹幾種常用的增益比定標(biāo)方法。

3 增益比定標(biāo)方法

3.1 潔凈大氣分子法

潔凈大氣分子法是假定高空中只存在大氣分子（無(wú)氣溶膠與云），通過(guò)對(duì)比系統(tǒng)探測(cè)潔凈大氣實(shí)際退偏比與潔凈大氣理論退偏比以完成增益比定標(biāo)的方法。潔凈大氣分子法退偏比的計(jì)算公式為

式中，δ?與 δmol在此處分別代表高度為rc處實(shí)際測(cè)量大氣分子退偏比與理論大氣分子的退偏比。δmol可以根據(jù)大氣散射理論[11]計(jì)算得到，但是該理論值并不固定。由于大氣分子散射主要由瑞利散射與振動(dòng)拉曼散射（該散射強(qiáng)度很小，可忽略不計(jì)）組成，其中瑞利散射主要由純轉(zhuǎn)動(dòng)拉曼線(xiàn)與中心Cabannes線(xiàn)組成[12]。在瑞利散射光譜結(jié)構(gòu)中，Cabannes線(xiàn)屬于多普勒展寬的中央峰，純轉(zhuǎn)動(dòng)拉曼線(xiàn)分布在Cabannes線(xiàn)的兩側(cè)，屬于邊帶[13]，純轉(zhuǎn)動(dòng)拉曼線(xiàn)造成的退偏效果比Cabannes線(xiàn)大得多。對(duì)于激光雷達(dá)系統(tǒng)說(shuō)，使用不同帶寬（Bandwidth，BW）的濾光片， δmol取值范圍在0.003 63～0.014 3之間[10]。如果激光雷達(dá)系統(tǒng)中濾光片的帶寬較窄（BW＜0.3 nm@532 nm）， δmol=0.003 63。反之，如果濾光片的帶寬較寬（BW=15 nm@532 nm），則δmol=0.014 3。

潔凈大氣分子法由于操作比較方便，且不需要在系統(tǒng)光路中添加其它器件，所以該定標(biāo)方法在上世紀(jì)末使用較為廣泛，但其缺點(diǎn)也很明顯，因?yàn)檎嬲凉崈舻拇髿夂苌俅嬖?，若選取的定標(biāo)區(qū)域存在氣溶膠或云，定標(biāo)結(jié)果將會(huì)產(chǎn)生較大誤差，而且當(dāng)濾光片帶寬在1～15 nm時(shí)，由于無(wú)法準(zhǔn)確評(píng)估純轉(zhuǎn)動(dòng)拉曼線(xiàn)在大氣分子散射中所占的比例，可能導(dǎo)致潔凈大氣理論退偏比計(jì)算不準(zhǔn)確，從而造成定標(biāo)誤差。另外，需要注意的是，潔凈大氣分子法一般適用于激光波長(zhǎng)小于550 nm的激光雷達(dá)。對(duì)于探測(cè)波長(zhǎng)大于800 nm的激光雷達(dá)，由于瑞利散射強(qiáng)度較小，使用該定標(biāo)方法易造成較大的定標(biāo)誤差[13]。因此，該方法目前很少被采用。

3.2 + 45°法

+45°法是一種將半波片（Half-Wave Plate，HWP）置于接收光路中（一般置于PBS前），通過(guò)單方向（順/逆時(shí)針均可）旋轉(zhuǎn)半波片以完成增益比定標(biāo)的方法。需要特別說(shuō)明的是，該方法假設(shè)半波片的性質(zhì)理想，且定標(biāo)時(shí)大氣狀態(tài)不發(fā)生改變。對(duì)于+45°法，除了以上兩個(gè)基本假設(shè)外，還需要增加另外兩個(gè)假設(shè)，即不存在對(duì)準(zhǔn)偏失角，也不存在偏振串?dāng)_[14]。

如圖2（彩圖見(jiàn)期刊電子版）所示，首先將半波片放置于PBS上游光路中，此時(shí)認(rèn)為激光偏振矢量與PBS入射面平行，如圖2(a)所示。

此時(shí)由于不考慮對(duì)準(zhǔn)偏失角和偏振串?dāng)_的影響，式（3）可簡(jiǎn)化為

然后將半波片繞光軸順時(shí)針旋轉(zhuǎn)（逆時(shí)針類(lèi)似），使得激光偏振矢量與PBS入射面成+90°，如圖2(b)所示，式（7）中的PT( 0°)變?yōu)?，具體公式不再贅述，此時(shí)增益比G可表示為

該方法的優(yōu)點(diǎn)是操作較簡(jiǎn)便，缺點(diǎn)為忽略了對(duì)準(zhǔn)偏失角和偏振串?dāng)_的影響，易引入對(duì)準(zhǔn)角誤差與偏振串?dāng)_誤差。

3.3 ±45°法

±45°法是一種將半波片置于接收光路中，通過(guò)旋轉(zhuǎn)兩次半波片（相對(duì)于初始位置，分別旋轉(zhuǎn)+22.5°與 ? 22.5°）以完成增益比定標(biāo)的方法。該方法由德國(guó)慕尼黑大學(xué)的Freudenthaler等人提出[15]?！?5°法在+45°法的基礎(chǔ)上同時(shí)考慮了對(duì)準(zhǔn)偏失角和偏振串?dāng)_的影響。

如圖3（彩圖見(jiàn)期刊電子版）所示，將一個(gè)半波片放置在PBS上游光路中，假設(shè)此時(shí)激光偏振矢量與PBS入射面存在初始對(duì)準(zhǔn)偏失角 θinit，θh為人為定量引入的對(duì)準(zhǔn)偏失角。

半波片旋轉(zhuǎn)之后，由式（4）可得

為了減小初始對(duì)準(zhǔn)偏失角 θinit的影響，±45°法采用連續(xù)兩次旋轉(zhuǎn)半波片的方式，第一次使半波片相對(duì)于初始位置繞光軸順時(shí)針旋轉(zhuǎn)+22.5°，第二次在第一次旋轉(zhuǎn)的基礎(chǔ)上使半波片繞光軸逆時(shí)針旋轉(zhuǎn)?45°，即相對(duì)于初始位置繞光軸逆時(shí)針旋轉(zhuǎn)?22.5°，分別使得激光偏振矢量與PBS入射面成+45°與?45°，此時(shí)增益比G可表示為

由式（10）可以看出，±45°法雖然考慮了PBS偏振串?dāng)_誤差，但無(wú)法完全消除對(duì)準(zhǔn)角誤差的影響。事實(shí)證明，±45°法的誤差來(lái)源與信噪比關(guān)系不大[16]，當(dāng) θinit= 1°時(shí)，G的相對(duì)誤差可以控制在5%以?xún)?nèi)[15]。由于±45°法操作較為簡(jiǎn)便且精度較高，目前已經(jīng)被MULIS（Multichannel Lidar System）、POLIS（Portable Lidar System）等高精度偏振激光雷達(dá)系統(tǒng)用于增益比的定標(biāo)[16]。其缺點(diǎn)為無(wú)法消除對(duì)準(zhǔn)角誤差。

3.4 ?45°法

?45°法是一種將半波片置于接收光路中，通過(guò)單方向（順/逆時(shí)針均可）旋轉(zhuǎn)45°半波片以完成增益比定標(biāo)的方法（其與+45°法的操作方法一樣，但計(jì)算方式不同，且無(wú)需進(jìn)行初始0°角搜尋）。該方法由浙江大學(xué)羅敬等人提出[17]。

如圖4所示，將一個(gè)半波片置于PBS的上游光路中，假設(shè)此時(shí)激光偏振矢量與PBS入射面存在初始對(duì)準(zhǔn)偏失角。

半波片旋轉(zhuǎn)之前由式（4）可得

然后將半波片繞光軸旋轉(zhuǎn)，如圖4(b)所示，旋 轉(zhuǎn) 之 后 式（11）中 的PR(θinit)與PT(θinit)變 為與，具體公式不再贅述，此時(shí)增益比可表示為

3.5 旋轉(zhuǎn)擬合法

旋轉(zhuǎn)擬合法是一種將半波片置于接收光路中，通過(guò)多次旋轉(zhuǎn)半波片，采用非線(xiàn)性最小二乘法擬合同時(shí)反演增益比、退偏比以及初始對(duì)準(zhǔn)偏失角 θinit以完成增益比定標(biāo)的方法。該方法由美國(guó)宇航局的ALVAREZ等人提出[18]。

如圖5所示，將一個(gè)半波片放置于PBS上游的光路中，假設(shè)此時(shí)激光偏振矢量與PBS入射面存在初始對(duì)準(zhǔn)偏失角θinit。

如果人為控制半波片相對(duì)于其初始光軸位置旋轉(zhuǎn) θh,j/2角，可獲得一系列由人為定量引入的對(duì)準(zhǔn)偏失角θh,j。則由式（9）可得

式中，j代表第j次旋轉(zhuǎn)半波片。觀察式（13），由于與TS可 認(rèn) 為 是 已 知 量，那么式（13）中只有3個(gè)未知量，即增益比G、初始對(duì)準(zhǔn)偏失角 θinit以及理論退偏比δ。此時(shí)一個(gè)方程無(wú)法求解3個(gè)未知數(shù)，但由于多次旋轉(zhuǎn)半波片可得到多個(gè)方程，采用非線(xiàn)性最小二乘法對(duì)該方程組進(jìn)行求解，即可解出3個(gè)未知數(shù)。需要注意的是，該方法要求至少得到3個(gè)方程，即j≥3。

該方法的優(yōu)點(diǎn)為一次定標(biāo)可同時(shí)反演增益比、退偏比以及初始對(duì)準(zhǔn)偏失角3個(gè)未知量，并且不需要知道先驗(yàn)值。其缺點(diǎn)為定標(biāo)耗時(shí)較長(zhǎng)，操作繁瑣，只適用于相對(duì)穩(wěn)定的大氣環(huán)境下。

3.6 退偏器法

退偏器法是一種通過(guò)在光路中添加光學(xué)元件將系統(tǒng)接收的回波信號(hào)轉(zhuǎn)換為非偏振光以完成增益比定標(biāo)的方法。其利用兩路探測(cè)通道信號(hào)強(qiáng)度之比進(jìn)行定標(biāo)，由式（4）可求解增益比G

其中較為典型的是CALIOP利用退偏器產(chǎn)生的非退偏光信號(hào)來(lái)進(jìn)行增益比定標(biāo)[19]。具體操作為在PBS上游光路放置一個(gè)可移動(dòng)的退偏器，系統(tǒng)定標(biāo)時(shí)將其置于光路中，定標(biāo)結(jié)束后移出光路，如圖6所示。CALIOP采用該方法在夜間軌道上對(duì)系統(tǒng)進(jìn)行增益比定標(biāo)，并采用潔凈大氣分子法進(jìn)行驗(yàn)證[20]。

該方法優(yōu)點(diǎn)為操作簡(jiǎn)便并且可進(jìn)行實(shí)時(shí)定標(biāo)，從而排除大氣狀態(tài)改變?cè)斐傻挠绊?。其缺點(diǎn)為商用退偏器還難以產(chǎn)生完全的退偏光，易引入其它誤差。

4 實(shí)驗(yàn)結(jié)果與分析

實(shí)驗(yàn)所采用偏振激光雷達(dá)為典型的雙通道激光雷達(dá)，系統(tǒng)結(jié)構(gòu)如圖1所示（圖中相同的器件只標(biāo)注一次）。其中介于會(huì)聚透鏡與半波片之間的退偏器只有在采用退偏器法定標(biāo)時(shí)才會(huì)使用，其它時(shí)候不需要使用該光學(xué)器件。為了減少線(xiàn)偏振度誤差，實(shí)驗(yàn)系統(tǒng)在出射光路中添加了起偏棱鏡，使得出射激光的消光比達(dá)到了 2×105∶1。為了減少偏振串?dāng)_誤差，實(shí)驗(yàn)系統(tǒng)采用PBS與偏振片膠合的結(jié)構(gòu)，使得TP:TS（RS:RP）>30 000∶1。因此，本文暫不討論線(xiàn)偏振度誤差與偏振串?dāng)_誤差，只討論對(duì)準(zhǔn)誤差對(duì)增益比造成的影響。系統(tǒng)的具體參數(shù)如表1所示

由于潔凈大氣分子法定標(biāo)誤差較大，因此，本文實(shí)驗(yàn)主要對(duì)比分析了采用+45°法、±45°法、?45°法、旋轉(zhuǎn)擬合法與退偏器法等5種方法在不同對(duì)準(zhǔn)偏失角（對(duì)準(zhǔn)角誤差）情況下對(duì)增益比定標(biāo)的影響。

為保證實(shí)驗(yàn)有較高的信噪比，實(shí)驗(yàn)時(shí)間選在夜間。同時(shí)，為了減少大氣變化而帶來(lái)影響，選取水平方向（俯仰角為0°）探測(cè)。在完成系統(tǒng)光軸校準(zhǔn)后，使用電動(dòng)旋轉(zhuǎn)電機(jī)（精度為0.005°）調(diào)整半波片到激光偏振矢量與PBS入射面平行的位置（目測(cè)透射通道平均信號(hào)最大時(shí)即為對(duì)應(yīng)位置），此時(shí)半波片的角度為0°初始值。需要注意的是，為了表述方便，下文在人為定量引入對(duì)準(zhǔn)偏失角 θh時(shí) ，未將初始對(duì)準(zhǔn)偏失角 θinit表示在實(shí)際總的對(duì)準(zhǔn)偏失角中，但實(shí)際上每個(gè)對(duì)準(zhǔn)偏失角都包含了初始對(duì)準(zhǔn)偏失角 θinit（θinit屬于未知量）。然后，以 θh=0°（實(shí)際對(duì)準(zhǔn)偏失角為 θinit+0°）作為零點(diǎn)，使用電動(dòng)旋轉(zhuǎn)電機(jī)旋轉(zhuǎn)半波片。一般來(lái)說(shuō)，在實(shí)際操作過(guò)程中對(duì)準(zhǔn)偏失角不會(huì)超過(guò)15°[21-22]，但為了實(shí)驗(yàn)的完整性，本文將對(duì)準(zhǔn)偏失角的討論擴(kuò)大至45°。

+45°法、±45°法、?45°法與旋轉(zhuǎn)擬合法均以2.5°為 間隔旋轉(zhuǎn)半波片得到 θh=?45°～67.5°情況下共46組原始回波信號(hào)。通過(guò)對(duì)上述數(shù)據(jù)進(jìn)行處理，+45°法、±45°法、?45°法與旋轉(zhuǎn)擬合法4種方法均選擇5°為間隔，選取 θh的范圍為?45° ～ +45°，（即每組數(shù)據(jù)兩個(gè)角度在半波片旋轉(zhuǎn)前后相差45°），計(jì)算后一共可獲得19組增益比數(shù)據(jù)。

采用退偏器法時(shí)，在開(kāi)始實(shí)驗(yàn)之前需要在系統(tǒng)光路中添加退偏器，如圖1所示。由于退偏器是非理想的，為了觀察其至少一個(gè)周期的變化，退偏 器 法 以10°為間隔轉(zhuǎn)半波片得到 θh從?80°到+100°情況下共19組原始回波信號(hào)，通過(guò)對(duì)上述數(shù)據(jù)進(jìn)行處理，一共可獲得19組增益比數(shù)據(jù)。

表2是當(dāng)θh=0 °時(shí)，5種方法的定標(biāo)結(jié)果。觀察表2可以發(fā)現(xiàn)，當(dāng)沒(méi)有對(duì)準(zhǔn)偏失角時(shí)，±45°法、?45°法與旋轉(zhuǎn)擬合法定標(biāo)結(jié)果較為接近，可以認(rèn)為上述3種方法在θh=0 °時(shí)最接近真實(shí)值。令上述3種方法在 θh=0°時(shí)測(cè)得增益比的平均值為真實(shí)值，并繪制出如圖7所示的5種定標(biāo)方法的相對(duì)誤差隨對(duì)準(zhǔn)偏失角的變化曲線(xiàn)（旋轉(zhuǎn)擬合法與退偏器法計(jì)算過(guò)程詳見(jiàn)4.2與4.3節(jié)）。

4.1 +45°、±45°、?45°法

如圖7所示，當(dāng)| θh|＜15°時(shí)，±45°法與?45°法的定標(biāo)結(jié)果相近，相對(duì)誤差也較小，而+45°即使不存在對(duì)準(zhǔn)偏失角時(shí)，定標(biāo)結(jié)果也與上述兩種方法存在較大差異，相對(duì)誤差可達(dá)4%。當(dāng) |θh|>15°時(shí)，+45°法與?45°法的定標(biāo)結(jié)果總體保持穩(wěn)定，而±45°法的定標(biāo)結(jié)果隨對(duì)準(zhǔn)偏失角的增大而變得不穩(wěn)定，相對(duì)誤差急劇增大，甚至高達(dá)12.93%?！?5°法出現(xiàn)這樣誤差分布的原因主要在于其計(jì)算前后兩次測(cè)量數(shù)據(jù)的幾何平均，而?45°法則是在+45°法的基礎(chǔ)上計(jì)算前后兩次測(cè)量數(shù)據(jù)的算術(shù)平均。下文將通過(guò)具體理論分析來(lái)解釋+45°法出現(xiàn)上述現(xiàn)象的原因。

對(duì)兩種方法進(jìn)行相對(duì)誤差分析[23-24]，令兩種方法旋轉(zhuǎn)半波片前后反射通道與透射通道探測(cè)到的功率分別為與，可得

式中 δ1與 δ2分別代表?45°法與±45°法定標(biāo)結(jié)果的相對(duì)誤差，（S為R,T且n為a,b）代表各個(gè)測(cè)量值的不確定度（標(biāo)準(zhǔn)差）。激光雷達(dá)中光子計(jì)數(shù)的信號(hào)可以認(rèn)為服從泊松分布[25-26]，因此統(tǒng)計(jì)誤差等于信號(hào)平均值的均方根，即，因此，式（15）與式（16）可進(jìn)一步推導(dǎo)為與作差可發(fā)現(xiàn)，，即?45°法的不確定度小于等于±45°法。事實(shí)上，根據(jù)實(shí)驗(yàn)結(jié)果，±45°法在對(duì)準(zhǔn)偏失角較大的情況下，定標(biāo)的增益比廓線(xiàn)的振蕩幅度的確要大于?45°法，這也解釋了圖7中±45°法在對(duì)準(zhǔn)偏失角較大的情況下，誤差遠(yuǎn)大于?45°法的原因。

4.2 旋轉(zhuǎn)擬合法

考慮到實(shí)際過(guò)程中對(duì)準(zhǔn)偏失角的誤差不會(huì)超過(guò)15°，選擇 |θh|≤ 15°的13組數(shù)據(jù)（即θh,j=0°, ±2.5°,±5°···±15°）進(jìn)行數(shù)據(jù)擬合，旋轉(zhuǎn)擬合法計(jì)算結(jié)果如圖8（彩圖見(jiàn)期刊電子版）所示，其中藍(lán)色圓圈代表在不同 θh情 況下的 δ?(θ)，由式可知，求解增益比G、初始對(duì)準(zhǔn)偏失角 θinit以及理論退偏比δ 3個(gè)未知數(shù)的方程組屬于非線(xiàn)性最小二乘法問(wèn)題。求解之前需要對(duì)G、θinit與δ 3個(gè)未知數(shù)的初始值進(jìn)行預(yù)測(cè)。為了得到最佳初始預(yù)測(cè)值，由圖8可知，與 θ之間的關(guān)系（藍(lán)色圓圈）可以近似h,j用一個(gè)二次多項(xiàng)式來(lái)表示[27]，故以人為定量引入對(duì)準(zhǔn)偏失角θh,j為自變量，實(shí)際測(cè)量退偏比為應(yīng)變量構(gòu)建如下二次多項(xiàng)式

式中，A0、A1與A2均為二次多項(xiàng)式系數(shù)。擬合結(jié)果如圖8中紅色虛線(xiàn)所示，該二次多項(xiàng)式的最小值即代表初始對(duì)準(zhǔn)偏失角 θinit，計(jì)算結(jié)果θinit=?A1/(2×A2)= ?0.35°，該值可作為 θinit的最佳初始預(yù)測(cè)值。接著，將θinit= ?0.35°代入式（13）并使用非線(xiàn)性最小二乘法求解方程組，求得增益比G=1.2716。

4.3 退偏器法

退偏器法計(jì)算結(jié)果如圖9（彩圖見(jiàn)期刊電子版）所示，其中藍(lán)色圓圈代表在半波片不同角度下計(jì)算的增益比，觀察各藍(lán)色圓圈的分布情況，可知其較為符合余弦曲線(xiàn)分布規(guī)律。故以半波片旋轉(zhuǎn)角度φ (θh=2φ) 為自變量，增益比G為應(yīng)變量構(gòu)建如下余弦函數(shù)多項(xiàng)式

式中，B0、B1、B2與B3均代表余弦函數(shù)多項(xiàng)式的系數(shù)。擬合的結(jié)果如圖9中紅色虛線(xiàn)所示

從理論上來(lái)說(shuō)，假如回波信號(hào)為完全的非偏振光，那么無(wú)論如何改變半波片的角度都無(wú)法改變非偏振光的狀態(tài)。換句話(huà)說(shuō)，旋轉(zhuǎn)半波片并不會(huì)影響增益比的數(shù)值，增益比定標(biāo)的結(jié)果應(yīng)該表征為一條平行于x軸的直線(xiàn)，而非像圖9中那樣表現(xiàn)為余弦的分布情況。導(dǎo)致該問(wèn)題的原因是目前商用退偏器無(wú)法將偏振光完全轉(zhuǎn)化為非偏振光，回波信號(hào)在經(jīng)過(guò)退偏器之后仍包含一部分的偏振光。浙江大學(xué)羅敬等人以激光為光源，在測(cè)試退偏器的退偏效果時(shí)，發(fā)現(xiàn)測(cè)試結(jié)果也類(lèi)似圖9中余弦分布的情況[28]。這說(shuō)明即使在對(duì)準(zhǔn)偏失角為 0°的情況下，增益比定標(biāo)結(jié)果仍會(huì)受到該部分偏振光的影響。

需要注意的是，退偏器法測(cè)量的數(shù)據(jù)中只有5組數(shù)據(jù)的對(duì)準(zhǔn)偏失角在?45°～ +45°內(nèi)，所以該種方法在圖7中只標(biāo)有5個(gè)點(diǎn)。退偏器法在θh=0°時(shí)，相對(duì)誤差為5.6%。這說(shuō)明即使在對(duì)準(zhǔn)偏失角為0°的情況下，增益比定標(biāo)結(jié)果仍會(huì)受到該部分偏振光的影響。

4.4 討論

通過(guò)對(duì)實(shí)驗(yàn)結(jié)果進(jìn)行分析，可得知±45°法、?45°法與旋轉(zhuǎn)擬合法定標(biāo)結(jié)果最為準(zhǔn)確，這和前文的理論分析一致。但是±45°法在對(duì)準(zhǔn)偏失角較大的情況下誤差較大。旋轉(zhuǎn)擬合法定標(biāo)耗時(shí)較長(zhǎng)，操作繁瑣，只適用于相對(duì)穩(wěn)定的大氣環(huán)境下。相比之下，?45°法操作更為簡(jiǎn)便，定標(biāo)結(jié)果不受對(duì)準(zhǔn)偏失角的影響，而且也無(wú)需進(jìn)行初始0°角的搜尋，優(yōu)勢(shì)明顯。但是，?45°法也無(wú)法排除大氣狀態(tài)變化的影響，相比之下，只有退偏器法同時(shí)具有操作簡(jiǎn)便，可排除大氣狀態(tài)變化影響的能力，而且也不存在多次旋轉(zhuǎn)半波片會(huì)增加角度積累誤差的問(wèn)題，但目前商用退偏器仍無(wú)法產(chǎn)生完全的退偏光，這會(huì)引入新的誤差，且難以評(píng)估。綜上所述，建議偏振激光雷達(dá)研究人員在一般情況下采用?45°法定標(biāo)，在有高精度退偏器的情況下采用退偏器法定標(biāo)。

5 結(jié) 論

增益比的定標(biāo)準(zhǔn)確性對(duì)偏振激光雷達(dá)探測(cè)精度影響很大。本文系首次在原理與實(shí)驗(yàn)上對(duì)比了現(xiàn)存多種增益比定標(biāo)方法，并給相關(guān)研究人員提出了增益比定標(biāo)方法選擇的指導(dǎo)意見(jiàn)。

就可操作性而言，退偏器法只需在系統(tǒng)中插入退偏器即可完成定標(biāo)，可排除大氣狀態(tài)改變?cè)斐傻挠绊?，而其余需要旋轉(zhuǎn)半波片的4種方法至少需要完成兩次旋轉(zhuǎn)，操作相對(duì)復(fù)雜，而且無(wú)法排除大氣狀態(tài)改變?cè)斐傻挠绊?。就定?biāo)準(zhǔn)確性而言，實(shí)驗(yàn)主要對(duì)比了+45°法、±45°法、?45°法、旋轉(zhuǎn)擬合法與退偏器法等5種方法（由于潔凈大氣分子法定標(biāo)誤差較大，本文未作實(shí)驗(yàn)對(duì)比）在不同對(duì)準(zhǔn)偏失角下對(duì)增益比定標(biāo)的影響。實(shí)驗(yàn)結(jié)果表明，±45°法、?45°法與旋轉(zhuǎn)擬合法的定標(biāo)準(zhǔn)確度相對(duì)較高，但±45°法與旋轉(zhuǎn)擬合法操作較為復(fù)雜?！?5°法在對(duì)準(zhǔn)偏失角較大的情況下誤差較大。?45°法增益比定標(biāo)結(jié)果不受對(duì)準(zhǔn)偏失角的影響，而且也無(wú)需進(jìn)行初始0°角的搜尋。+45°法相對(duì)于前3種方法在沒(méi)有對(duì)準(zhǔn)偏失角的情況下誤差更大。退偏器法受到非理想退偏器的影響較大，如果存在理想退偏器，那么這種方法將是一種理想的增益比定標(biāo)方法。通過(guò)對(duì)不同增益比定標(biāo)方法理論與實(shí)驗(yàn)的對(duì)比，本文給出了增益比定標(biāo)方法的最佳選擇，即建議在一般情況下采用?45°法定標(biāo)，有高精度退偏器的情況下采用退偏器法定標(biāo)。