The specified “polarization purity” of lasers commonly used in lidars is of the order of 100:1, if it is specified at all. Already the terminology indicates that such specifications are rather vague. Actually, laser manufacturers do not measure the state of polarization of the laser beams, and seem to give values which are on the safe side under all circumstances. Theoretically, non-linear crystals as second and third harmonic generators should provide very clean linear polarization, just depending on the quality and accuracy of alignment of the crystals. Due to a lack of detailed information we neglect this source of errors in this work. However, in order to remove this uncertainty, in some lidar systems high-quality polarizing beam splitters are used to improve the degree of linear polarization of the emitted laser beams. In both cases, the plane of polarization of the laser beam can be rotated by angle α with respect to the incident plane of the polarizing beam splitter in the receiver optics, which results in the Stokes vector IL of the emitted laser beam: IL=IL1cos2αsin2α0, where IL is the laser energy.

The emitting functional block can consist of a set of dichroic beam splitters and steering mirrors, which align the laser beam with the telescope axis, and optionally includes beam expanders to decrease the emitted beam divergence and for eye safety reasons. The depolarizing effect of beam expanders is neglected, since we cannot estimate the uncertainties introduced by possible birefringence as in the case of CaF2 lenses of apochromatic beam expanders. In addition, stress birefringence in windows in the transmitting part (e.g. roof window) is neglected due to its complex analysis and lack of information. The effect of these optical devices has to be investigated in the future.

The general Müller matrix of dichroic beam splitters and steering mirrors is (Lu and Chipman, 1996; Freudenthaler, 2016a) M=T1D00D10000ZcosΔZsinΔ00-ZsinΔZcosΔ, where T and D are the non-polarized transmittance and the diattenuation parameter (Chipman, 2009), respectively, defined by T=Tp+Ts2D=Tp-TsTp+Ts, with Tp and Ts as the parallel and perpendicular intensity transmission coefficients (transmittances) with respect to the incident plane. The phase shift between the parallel (φp) and the perpendicular (φs) components, hereafter called retardance, is noted by Δ=φp-φs, and Z is given by Z=1-D2. Mirrors include an additional 180∘ retardance.

If the emitting optics is formed by different steering/dichroic mirrors, the Müller matrix of the emitting optics block ME is the matrix product of the individual matrices as ME=M1M2…Mn=∏iMi, where the subscript i=1,2,…n indicates number of steering/dichroic mirrors. We assume that all the elements in a functional bock are aligned perfectly with respect to each other, and with that an effective Müller matrix of the emitter optics ME can be obtained from Eq. (2) as follows: ME=TE1DE00DE10000ZEcosΔEZEsinΔE00-ZEsinΔEZEcosΔE, where TE, DE, and ΔE are the effective transmittance, diattenuation parameter, and retardance of ME. For example, considering an emitting functional block made of two dichroic mirrors (M1 and M2), ME is given by ME=T121D1200D1210000Z12cosΔ12ZsinΔ1200-Z12sinΔ12ZcosΔ12, where T12, D12, Z12, and Δ12 are T12=T1T21+D1D2D12=D1+D21+D1D2Z12=Z1Z21+D1D2Δ12=Δ1+Δ2, where T, D, Z, and Δ subscripted by 1 and 2 are the parameters of M1 and M2 according to Eq. (8).

This process can be applied iteratively to obtain the effective Müller matrix, ME composed of more than two elements.

While we assume the absence of rotational misalignments between the elements inside a block, we consider a rotational misalignment of the whole functional block with respect to the incident plane of the polarizing beam splitter (PBS). To this end, the rotation Müller matrix R is defined by Rβ=10000cos2β-sin2β00sin2βcos2β00001, where the angle β describes the rotational misalignment of ME with respect to the PBS incident plane. Thus, MEβ=RβMER-β, resulting in MEβ=TE1DEcos2βDEsin2β0DEcos2β1-WEsin22βWEsin2βcos2β-ZEsinΔEsin2βDEsin2βWEsin2βcos2β1-WEcos22βZEsinΔEcos2β0ZEsinΔEsin2β-ZEsinΔEcos2βZEcosΔE, with WE=1-ZEcosΔE.

The polarization effect of ME is described by the effective diattenuation parameter (DE), the effective retardance (ΔE), and the rotational misalignment angle β of the whole functional block with respect to the PBS polarizing plane. The possible effect of tilted scanning mirrors on depolarization measurements was highlighted by Bissonnette et al. (2001).

This functional block, formed by the telescope and dichroic beam splitters, leads the received signal to the photomultipliers and, in case of multiwavelength lidar, separates the received signal by wavelength. In the same way as the emitting optics, MO can be described by a unique effective Müller matrix as follows: MOγ=TO1DOcos2γDOsin2γ0DOcos2γ1-WOsin22γWOsin2γcos2γ-ZOsinΔOsin2γDOsin2γWOsin2βcos2γ1-WOcos22γZOsinΔOcos2γ0ZOsinΔOsin2γ-ZOsinΔOcos2γZOcosΔO, where TO, DO, and ΔO are the effective transmittance, diattenuation, and retardance of MO, respectively, and γ describes the rotational misalignment angle of MO with respect to the incident plane of the PBS. The polarization effects of telescopes with small incidence angles of the light beam are neglected in this work. This approximation is valid for Cassegrain telescopes, but possibly not for Newtonian telescopes with 90∘ fold mirrors (Clark and Breckinridge, 2011; Di et al., 2015; Lam and Chipman, 2015) as in the case of the PollyXT lidars (Engelmann et al., 2016). This source of error has to be investigated further. Furthermore, windows commonly used to protect the telescope can also affect the polarization due to intrinsic and stress birefringence, but this is also not considered in the simulator since the effect is very difficult to evaluate due to the lack of information about the properties and the time-dependent behaviour.

A polarizing splitter separates the received signal into reflected and transmitted signals. Consequently, two Müller matrices are required to describe the reflection (MR) and transmission (MT) processes. Due to the similarity in the shape of the matrices, both matrices can be described by one matrix with the subscript S=R,T with MS as follows: MS=TS1DS00DS10000ZScosΔSZSsinΔS00-ZSsinΔSZScosΔS, where ΔS is the phase shift of MS, TS, DS are defined by TS=TSp+TSs2DS=TSp-TSsTSp+TSs, with TSp and TSs as the parallel p and perpendicular s transmittance/reflectance, respectively, and ZS=1-DS. Parallel and perpendicular polarization is defined with respect to the incidence plane of the polarizing beam splitter. We attribute any polarization-independent absorption to the channel gain (see Sect. 2.6) and get therewith TSp+TSs=1. Ideal PBS split light in the two orthogonally polarized components without crosstalk, which means Rp=Ts=0. But real PBS always transmit a fraction of the perpendicular polarization and reflect part of parallel polarization. This effect of the crosstalk on the linear depolarization ratio has been studied for example by Álvarez et al. (2006), Freudenthaler et al. (2009), and Snels et al. (2009). As shown in Freudenthaler (2016a), it is relatively easy to reduce the crosstalk by adding polarizer sheet filters after the PBS. The Müller matrix of the cleaned PBS, MS#, is MS#=TS#1DS#00DS#10000000000, where DS#=DT#=1,DR#=-1 and the superscript “#” indicates that the PBS is “cleaned”.

We assume that the extinction ratio of the cleaning polarizing sheet filters is sufficiently small and that they can be oriented with an accuracy much better than ±5∘ with respect to the PBS, and that therefore the resulting error of the DS# can be neglected (See Appendix A).

The parallel polarized component of the emitted laser beam can be detected either in the transmitted or in the reflected path behind the PBS. This depends on the orientation Ψ of the PBS with respect to the laser polarization. We consider this by means of a rotator R(ψ) (Eq. 14) before the PBS (see Fig. 1). For Ψ=90∘, the reflected and transmitted signals correspond to the parallel and perpendicular polarized components, respectively, and vice versa for Ψ=0∘.

The reflected and transmitted signals are detected by the photomultipliers which perform the light-to-electrical signal conversion. They affect the depolarization measurements as, in general, different photomultipliers have different gains. Regarding the Stokes–Müller formalism, we define the opto-electronic gains ηR and ηT for the photomultiplier gains of the transmitted and reflected signals including all optical attenuation of the lidar system in the transmitted and reflected path that is independent of polarization. We set them equal to 1 since we only investigate the polarization-dependent errors of the lidar optics.

The calibrator allows the determination of the polarization effect of the optical devices located behind the calibrator and of the gain ratio of the photomultipliers (PMTs). For the lidar set-up in Fig. 1, the calibration factor η includes the effects of the polarizing splitter (MR and MT) and photomultiplier gains, ηR and ηT: η=ηRηTTRTT. Different calibration methods have been proposed either using the theoretical value of molecular depolarization (e.g. Cairo et al., 1999) or using additional optical devices as half-wave plates or polarization filters (e.g. Álvarez et al., 2006; Snels et al., 2009; Freudenthaler et al., 2009). Particularly, the Δ90∘ calibration method has been implemented within EARLINET lidar systems (e.g. Freudenthaler et al. 2009; Nemuc et al., 2013; Mamouri and Ansmann, 2014; Bravo-Aranda et al., 2015). This method uses two measurements, rotating the polarizing plane of the received signal at angles ϕ1 and ϕ2 around the nominal axial rotation (Ψ) with the constraint ϕ2-ϕ1=90∘ (e.g. ϕ1=45∘,ϕ2=-45∘ around the measurement position Ψ=0∘). These rotations allow the equalization of the reflected and transmitted signals and thus, any difference between the reflected and transmitted signals is only due to the polarizing effects of the optical devices between the calibrator and the photomultipliers. The equalization of the reflected and transmitted signals can be made by a physical rotation of the receiving optics including the photomultipliers, by rotating a half-wave plate placed before the PBS, or by using a linear polarizing filter rotated accordingly (see Freudenthaler, 2016a, for more details).

The measured calibration factor, η∗, is η∗Ψ,x45∘+ε=IRΨ,x45∘+εITΨ,x45∘+ε, where the two rotation angles, ϕ1 and ϕ2, are written as x45∘+ε, with x=±1 indicating the rotational direction and ε taking the deviation from the nominal rotation into account. Then, their geometric mean, η±∗Ψ,ε=η∗Ψ,+45∘+εη∗Ψ,-45∘+ε, is calculated since it is less influenced by ε than η∗Ψ,x45∘+ε, as indicated by Freudenthaler et al. (2009). It is possible to show (Freudenthaler, 2016a) that the analytical expressions of η±∗Ψ,ε, corresponding to a different experimental set-up, are always in the form η±∗α,DE,ΔE,β,DO,ΔO,γ,ε,DT,DR=ηfα,DE,ΔE,β,DO,ΔO,γ,ε,DT,DR, where fα,DE,ΔE,β,DO,ΔO,γ,ε,DT,DR is the combined influence of all the parameters which are not corrected by η. Hereafter, the correction factor, fα,DE,ΔE,β,DO,ΔO,γ,ε,DT,DR, will be noted by fα,….

According to the Stokes–Müller formalism, the reflected (IR) and transmitted (IT) signals can be obtained by multiplying the laser beam Stokes vector (IL) by the subsequent Müller matrices, which represent the different functional blocks, and the atmosphere F, which describes the scattering matrix for randomly oriented particles (van de Hulst, 1957; Mishchenko and Hovenier, 1995), by F=F1110000a0000-a00001-2a, where F11 is the backscatter coefficient and a is the polarization parameter. Both parameters are range-dependent, but since this dependency has no influence on our error calculation, we omitted it in the following. Please note that, instead of the polarization parameter a, different but equivalent expressions are used in other publications as described in more detail in Freudenthaler (2016a). Probably most known is the depolarization parameter d= 1-a used in Gimmestadt (2008).

With that, the Stokes vector of the lidar and calibration measurements is described by ISΨ=ηSMSRΨMOFMEIL ISΨ,x45∘+ε=ηSMSRΨCx45∘+εMOFMEIL, where the first element of the Stokes vectors is the energy detected by the photomultipliers. Based on the Stokes–Müller formalism presented, the detected energy value depends on the 18 lidar parameters summarized in Table 1. However, only some of them are considered by PLS for the assessment of the error sources (see “error source” column in Table 1).

The volume linear depolarization ratio δ which contains the contributions of particles and air molecules, is directly related to the polarization parameter a by (Mishchenko and Hovenier, 1995) δ=1-a1+a. For further details regarding mathematical expressions of depolarization parameters, see Cairo et al. (1999). δ can be retrieved from lidar measurements by the following general equation given by Freudenthalter (2016a): δ=δ∗GT+HT-GR+HRGR-HR-δ∗GT-HT, where the parameters GT, GR, HT, and HR are determined by solving the matrix multiplication of Eq. (24) and separating the measured energy IS into terms with and without the polarization parameter, a, as follows: IS=GS+aHS, (Further details are given by Freudenthaler, 2016.) and δ∗Ψ is the reflected-to-transmitted signal ratio divided by the calibration factor, η (Eq. 23): δ∗Ψ=1ηIRΨITΨ, , where η has to be derived from the measured gain ratio, η±∗ (Eq. 26), estimating fα,… either from information already available (e.g. from technical specifications) or from additional measurements performed for this purpose (see Freudenthaler, 2016a). Without any information, we have to assume fα,…≅1. Therefore, a better characterization of the lidar system through GS, HS, and fα,… decreases the systematic errors of the depolarization measurements.

The laser can introduce errors in the depolarization measurements due to a misalignment angle α of the plane of the laser polarization with respect to the incident plane of the PBS (see Eq. 1). Figure 3 shows Eδ over α with different values of δr. As can be seen in Fig. 3, Eδα increases with α in absolute terms. Δδα ranges between [0, 0.031] and [0, 0.024] for δ values of 0.004 and 0.45, respectively, showing a low δ dependence. Since the smallest atmospheric δm is of the order of 10-3, we assume that Δδα can be neglected (< 1 × 10-4) if α is in the range 0∘±0.6∘.

ME is characterized by the effective diattenuation DE, the retardance ΔE, and the rotational misalignment angle β of ME with respect to the incident plane of PBS. These parameters are interdependent, and thus Fig. 4 shows the Eδ dependence on ΔE and DE for different β. Additionally, the influence of the atmospheric depolarization is assessed for δr values: 0.004 and 0.45 in Fig. 4 (top). As aforementioned, ΔE varies in the range [-180∘, 180∘] because the retardance of steering and dichroic mirrors is generally not provided in the majority of the technical specifications.

As can be seen in Fig. 4 (top), DE introduces systematic errors, except for β=0∘ with ΔδDE,β=0, 0.001. Figure 4 (bottom) shows that the maximum of EδΔE,β is larger than 0.03 for β larger than 5∘ in absolute terms, indicating that the lack of information of ΔE can lead to huge uncertainties even larger than 0.13 for β larger than 10∘. Figure 4 shows also that EδDE,β and EδΔE,β decrease with increasing δ values.

In summary, the total Δδ due to the systematic uncertainty of ME is 0, 0.13 for δr=0.004 and 0, 0.1 for δr=0.45. It is recommended that the laser beam is emitted directly to the atmosphere to avoid this error source. Otherwise, it is crucial to set β=0∘ to keep ΔδME as low as possible, independently of the effective diattenuation and retardance. For the synthetic lidar set-up, β=0∘±2.5∘ would lead to negligible EδDE,β and EδΔE,β with values lower than 1 × 10-4.

The parameters of the receiving optics MO are the effective diattenuation DO, retardance ΔO, and the misalignment angle γ between the receiving optics and the incident plane of the PBS. As in the case of ME (β, DE, ΔE), the influence of any of these parameters on Eδ is not independent of the others. However, the variation of Eδ with γ is very weak, and thus Fig. 5 (top) depicts Eδ vs. DO for different δr, whereas Fig. 5, bottom, depicts Eδ vs. ΔO for different γ.

According to Fig. 5, top, the uncertainty of DO in the range [-0.3, +0.3] has a huge impact on EδDO with errors in the range [+0.4, -0.2] for δr=0.45 and similar relative errors for other δr values.

Negative values of DO cause larger EδDO, in absolute terms, than positive ones (e.g. EδDO=-0.3=0.21 in contrast to EδDO=+0.3=0.12 considering δr=0.25 in both cases) because the parallel signal is stronger than the perpendicular one. In order to make the systematic error caused by MO negligible, the uncertainty of Do should be lower than ±0.0010 for EδDO<10-4, which is a very small value compared to what is available on the market. Thus, we advise the use of accurate calibration methods which can correct for DO, or to measure this value accurately.

Figure 5 (bottom) shows that the effect of the uncertainty in the retardance EδΔOis comparable to that of the emitter block EδΔE with EδΔO larger than 0.03 for γ=±5∘. Therefore, it is recommended that γ is kept as small as possible with little uncertainty.

Summarising, the uncertainties of γ and DO have a huge impact on Δδ, and thus, it is very important to carefully determine these parameters of the receiving optics.

For the synthetic lidar, we consider a non-cleaned polarizing beam splitter which has TTp, TRp, TTs, and TRs values and uncertainties as shown in Table 2. Since TTp+TRp=1 and TTs+TRs=1, TTp, TRp, TTs, and TRs, and their uncertainties are not independent. We choose TTp and TTs as the variable parameters, and show Eδ over TTp in Fig. 6 for different TTs and for δ values of 0.004 and 0.45. Although ΔTTs=±0.01 is much smaller than ΔTTp= ± 0.05, it has a stronger influence on EδTTp,TTs. The ratio EδΔTTs/ΔTTs is about 8 times the ratio EδΔTTp/ΔTTp (i) because the parallel intensity, detected in the reflected path, is much larger than the perpendicular intensity, and (ii) because TTs is small; a small change in TTs makes a big difference in the crosstalk from the parallel to the perpendicular signal. The influence of ΔTTp on EδTTp,TTs increases and that of ΔTTs decreases with increasing δ. The systematic errors due to crosstalk can be avoided by means of a cleaned PBS (see Sect. 2.4).

Here the Δ90∘ calibration with a rotator in front of the PBS is considered. This calibration method is described in Sect. 2.7 (see Freudenthaler, 2016a, for further details). It is worth noting that the uncertainties due to a rotation calibrator affect both calibration and normal measurements, whereas the use of a polarizing filter as calibrator only affects the calibration measurements because it has to be removed for the normal measurement, and hence there is no rotation error ε in that.

As can be seen in Fig. 7, Eδε is 0,0.0008 for δr=0.004 and 0,0.0006 for δr=0.45. Considering the large uncertainty range (ε=±5∘), Δδε is really low. Since ε is usually smaller in real lidars than in the synthetic lidar, we can conclude that Δδε can be neglected. This happens because the Δ90∘ calibration method is used, but for other calibration methods, the uncertainty of this parameter needs to be taken into account.

Despite the fact that ε does not introduce a considerable systematic error, the position of the calibrator within the lidar system has to be considered. For example, the Δ90∘ calibration in front of the receiving optics corrects the calibration factor for diattenuation of the receiving optics DO but it does not if the calibrator is located behind the receiving optics, as is the case for the LB21, IPRAL, and MUSA lidars (see Table 5).

The total systematic error Δδ, including all possible mutual dependencies, is determined with a complete grid search of the error space. In order to keep the number of samples around 106, α, DE, β, γ, and ε are simulated with three values, as e.g. α=-10,0,10∘, and ΔE, D0, ΔO, DT, and DR with five values, as e.g. ΔO=-180∘,-90,0,90,180∘. As a result, there are 3555≅7.6×105 simulated δ values. Figure 8 shows the δ histograms for δr values of 0.004 (left) and 0.45 (right). The minima and maxima of both histograms are provided in Table 3. For both simulations, the values obtained for δ span over quite a large range reaching even impossible values below the molecular δ (0.004) or δ values larger than 1. Since the distribution of simulated δ is displaced to the right of δr, an overestimation of δ is more probable than an underestimation for this synthetic lidar.