Supplement of “ Polarization lidar for detecting dust orientation : System design and calibration

Dust orientation is an ongoing investigation in recent years. Its potential proof will be a paradigm shift for dust remote sensing, invalidating the currently used simplifications of randomly-oriented particles. Vertically-resolved measurements of dust orientation can be acquired with a polarization lidar designed to target the off-diagonal elements of the backscatter matrix which are non-zero only when the particles are oriented. Building on previous studies, we constructed a lidar system emitting linearlyand elliptically-polarized light at 1064 nm and detecting the linear and circular polarization of the backscattered light. 5 Its measurements provide direct flags of dust orientation, as well as more detailed information of the particle microphysics. The system also employs the capability to acquire measurements at varying viewing angles. Moreover, in order to achieve good signal-to-noise-ratio in short measurement times the system is equipped with two laser sources emitting in interleaved fashion, and two telescopes for detecting the backscattered light from both lasers. Herein we provide a description of the optical and mechanical parts of this new lidar system, the scientific and technical objectives of its design, and the calibration methodologies 10 tailored for the measurements of oriented dust particles. We also provide the first measurements of the system.

S1 Mueller matrices of the atmosphere and of the optical elements of the lidar S1.1 Backscatter Stokes phase matrices of the atmosphere S1.1.1 Oriented dust particles The backscatter Stokes phase matrix of oriented dust particles is shown in Eq. S1

S1.1.2 Atmospheric gases
The backscatter Stokes phase matrix of the gases in the atmosphere is shown in Eq. S2 Where, g ii = G ii F 11 .

S1.1.3 Randomly-oriented particles and atmospheric gases
The backscatter stokes phase matrix of an atmosphere with randomly-oriented particles and gases is F atm (a), where "atm" denotes the atmosphere (aerosols and gases) and a is the polarization parameter of the atmosphere, a = F 22 +G 22 F 11 +G 11 = − F 33 +G 33 F 11 +G 11 .

S1.2 Mueller matrices of the optical elements of the lidar
The Mueller matrices of the optical elements of the lidar are shown in Eq. S4 -S11. The values of the optical element specs are provided by the corresponding manufacturers. S1.2.1 Receiver optics (Eq. S.4.12 in Freudenthaler (2016)) θ is the fast-axis-angle relative to the reference plane c 4θ = cos(4θ), s 4θ = sin(4θ) S1.2.3 Quarter Wave Plate (QW P ) φ is the fast-axis-angle relative to the reference plane c 2φ = cos(2φ), s 2φ = sin(2φ) With the use of cleaning polarizing sheet filters after the P BS, considering "ideal cleaning" we get D T = 1 and Z T = 0 (S.10.10 in Freudenthaler (2016)): With the use of cleaning polarizing sheet filters after the P BS, considering "ideal cleaning" we get D R = −1 and Z R = 0 (S.10.10 in Freudenthaler (2016)):

S2 Calculation of the measured signals I Li Tk S
The measured signals I i k S for laser i = LA, LB, at the detection unit after telescope k = T A, T B, at the detector S = T , R ("Transmitted" and "Reflected" channel after the P BS k , respectively), are shown in Eq. S12 and S13. In Eq. S12 and S13 we consider backgroundcorrected values and we omit the electronic noise at the detectors.
In Eq. S12 and S13, where A k is the area of the telescope k, O i k is the overlap function of the laser beam receiver field-of-view with range 0-1 (for laser i and telescope k), T (0, r) is the transmission of the atmosphere between the lidar at range r = 0 and a specific range in the atmosphere, and E oi is the pulse energy of laser i. η S k is the amplification of the signals at S = T or R detector of the detection unit after telescope k.ẽ = [1, 0, 0, 0] T is used to select the first component of the Stokes vector (the signal measured at the APDs). M S k is the Mueller matrix of the P BS k followed by cleaning polarizing sheet filters (Eq. S9 and S11), M O k is the Mueller matrix of the receiver optics (i.e. telescope k, collimating lenses, bandpass filter; Eq. S5), and M QW T B is the Mueller matrix of the QW P T B (Eq. S7). F (Eq. S1) and G (Eq. S2) are the backscatter Stokes phase matrices of the dust particles and of the gase molecules, respectively, at a certain range in the atmosphere.ĩ i is the Stokes vector of the light from the emission unit of laser i. Figure S1: Sketch of the system design: two lasers shooting alternatively (L A and L B ), with the backscattered signals correspondingly alternatively collected by two telescopes (T A and T B ) and then redirected at two detectors for each telescope (D S j , S = T, R as of "Transmitted" and "Reflected" channels, j = A, B). The polarization of the light emitted from each laser is changed appropriately, using the HW P LA for laser A and the QW P LB followed by the HW P LB for laser B. The laser beam of each laser is expanded with a beam expander (BEX). After the first telescope the light goes through P BS T A and after the second telescope the light goes through QW P T B and P BS T B . The HW P T A at telescope A is used to correct the rotation of the P BS T A (Section 4.1 in the manuscript). The HW P T B at telescope B is used to check the position of the QW P T B with respect to the P BS T B (Section S4)). The shutter at each telescope is used for performing dark measurements. The camera at each telescope is used for the alignment of the laser beams with the field-of-view of the telescope.

S2.1 The polarization of the light from the emission units of lasers A and B
The Stokes vector of the light from the emission units of laser A and B is defined with respect to the "frame coordinate system", shown in Fig. S2a, and is provided byĩ LA (Eq. S14) and i LB (Eq. S15), respectively. The light emitted directly from the lasers (ĩ lsr LA andĩ lsr LB ) is considered to be 100% linearly-polarized, with angle of polarization ellipse with respect to the frame coordinate system α LA and α LB for lasers A and B, respectively (Eq. S14 and S15). The polarization of the light from the whole emission units is then defined according to the position of the optical elements in front of the lasers, i.e. the HW P LA in front of laser A, and the QW P LB followed by the HW P LB in front of laser B (Fig. S1, Fig. S2d and e; Eq. S14 and S15).ĩ We use the angles ϑ LA (Eq. S16) and ϕ LB (Eq. S17) to simplify Eq. S14 and S15, as shown in Eq. S18 and S19.
The "DU T A coordinate system" and the "DU T B coordinate system" in Fig. S2b and c are the right-handed coordinate systems of the detection units after telescopes A and B, respectively. The x DU T A and y DU T A axis coincide with the incidence plane of P BS T A , and the x DU T B and y DU T B axis coincide with the incidence plane of P BS T B . The optical elements are considered to be perfectly aligned with eachother in the detection units (because their holders are manufactured and assembled in a mechanical workshop with high accuracy), but the detection units are possibly rotated around the optical axis with respect to the frame coordinate system by angles ω T A and ω T B , respectively ( Fig. S2b and c). The Stokes vectors of the light collected at telescope A and B are consequently described including a multiplication with the rotation matrices R TA (−ω T A ) and R TB (−ω T B ), respectively (see Eq. S.5.1.7 in Freudenthaler˙2016).
This rotation affects the measurements of the polarized components after P BS T A , but not after P BS T B . The rotation of the detection unit after telescope A is corrected using the HW P T A , as shown in Section 4.1 in the manuscript.
Sections S2.2, S2.3, S2.4 and S2.5 provide the analytical calculations for the formulas of I Li T k S , taking into account all the optical elements of the system, including their misalignments. Figure S2: a) The "frame coordinate system" (black) is the reference coordinate system with x Faxis parallel to the horizon. b) The "DU T A coordinate system" (light blue) is the coordinate system of the detection unit after telescope A, which is rotated with respect to the frame coordinate system by an angle ω T A . The effect of this rotation on the signals is corrected using The "DU T B coordinate system" (orange) is the coordinate system of the detection unit after telescope B, which is rotated with respect to the frame coordinate system by an angle ω T B . The rotation does not affect the measured signals. The QW P T B before P BS T B , is placed at φ T B = 45 o with respect to the x DU T B -axis. d) The light emitted directly from laser A is linearly-polarized with unknown angle of polarization α LA . As shown in Eq. S14, using the HW P LA with fast-axisangle θ LA = 22.5 o + α LA 2 , we produce the light emitted from the emission unit of laser A with angle of polarization 2ϑ LA = 45 o . e) The light emitted directly from laser B is linearly-polarized with unknown angle of polarization α LB . As shown in Eq. S15, using the QW P LB with with fast-axis-angle φ LB = α LB − 30 o , and the HW P LB with fast-axis-angle θ LB = α LB 2 − 12.2 o , we produce the elliptically-polarized light emitted from the emission unit of laser B with angle of polarization 5.6 o and degree of linear polarization 0.866.

S2.2 I LA TA S : The signals from laser A, at the detection unit after telescope A
The measurements of the backscatterred light of laser A at the detectors of telescope A are provided in Eq. S20. The rotation of the detection unit after telescope A by an angle ω T A (Fig. S2) is taken into account using the rotation matrix R T A (−ω T A ). The HW P T A (Fig. S1), with Mueller matrix M HW T A is used for the correction of the effect of this rotation on I LA T A S (Section 4.1 in manuscript).
After correcting the effect due to the rotation of the detection unit after telescope A, by setting the fast-axis-angle of HW P T A at θ T A = − ω T A 2 (Section 4.1 in manuscript), Eq. S20 is written as Eq. S21.
S2.3 I LB TA S : The signals from laser B, at the detection unit after telescope A The measurements of the backscatterred light of laser B at the detectors of telescope A are provided in Eq. S22. The HW P T A (Fig. S1), with Mueller matrix M HW T A is used for the correction of the effect of the rotation of the detection unit after telescope A, on I LB T A S (Section 4.1 in manuscript).
After correcting for the rotation of the detection unit after telescope A, by setting the fastaxis-angle of HW P T A at θ T A = − ω T A 2 (Section 4.1 in manuscript), Eq. S22 is written as Eq. S23.
S2.4 I LA TB S : The signals from laser A, at the detection unit after telescope B The measurements of the backscatterred light of laser A at the detectors of telescope B are provided in Eq. S25. The rotation of the detection unit after telescope B by an angle ω T B (Fig.  S2) is taken into account using the rotation matrix R T B (−ω T B ), but as shown in Eq. S25 it does not affect the measurements. The HW P T B (Fig. S1), with Mueller matrix M HW T B is used for checking that the QW P T B is at 45 o with respect of the x DU T B -axis (Fig. S2), as shown in Section S4. The Mueller matrix of the QW P T B with φ T B = 45 o is provided in Eq. S24.
S2.5 I LB TB S : The signals from laser B, at the detection unit after telescope B The measurements of the backscatterred light of laser B at the detection unit after telescope B are provided in Eq. S26.

S3 Calculation of the measured intensities after we place a linear polarizer at 45 o in front of the emission unit of laser A
After placing a linear polarizer in front of the window in front of the emission unit of laser A, at 45 o from x F -axis (Fig. S3), the Stokes vector of the emitted lightĩ LA 45 o is provided by Eq. S27. The measured intensities I LA T A S 45 o at the detectors after telescope A are provided in Eq. S28. In Eq. S27 we consider an ideal linear polarizer. The Mueller matrix of the ideal linear polarizer at 45 o (LP 45 o ) is taken from the Handbook of optics (Table 1 in section 14.11). In Eq. S28 we consider randomly-oriented particles.
S4 Check that the fast-axis-angle of QWP TB is at 45 o with respect to x DU TB -axis

S6 Acronyms and symbols
In the following table a list of acronyms and symbols is provided. In the third column we provide the equation where we first find them. a The polarization parameter of the atmosphere α The angle of the polarization ellipse A k Area of telescope k Eq. S12 b The degree of linear polarization c x Cosine of x Eq. S4 δ Volume linear depolarization ratio ∆ R The retardance of M R Eq. S10 The diattenuation parameter of M R Eq. S10 D T The diattenuation parameter of M T Eq. S8 e [1, 0, 0, 0] T Eq. S12 The pulse energy of the laser i Eq. S12

F
The backscatter Stokes phase matrix of the dust particles in the atmosphere Eq. S1 F atm The backscatter Stokes phase matrix of an atmosphere with randomly-oriented particles and gases Eq. S3

G
The backscatter Stokes phase matrix of the the gases in the atmosphere Eq. S2

HW P
Half Wave Plate Eq. S6 i Laser i = LA, LB Eq. S12 i i The Stokes vector of light from the emission unit of laser i Eq. S12 The Stokes vector of light from the emission unit of laser A, after placing a LP in front of the window in front of the emission unit, at 45 o from x F -axis Eq. S27 I i k S The intensities from laser i at the detector S = T or R after telescope k Eq. S12 The intensities from laser i at the detector S = T or R after telescope A, after correcting for the effect of the rotation of the detection unit, by setting the HW P T A at θ T A = − ω T A 2 Eq. S23 The intensities from laser A at the detector S = T or R after telescope k, after placing a LP in front of the window in front of the emission unit of laser A, at 45 o from x F -axis Eq. S28 The intensities I i T B S , in case the fast-axis-angle of QW P T B is misaligned by ε T B (φ T B = 45 o + ε T B ) Eq. S30 k Telescope k = T A, T B Eq. S12 η k The calibration factor of the ratio of the intensities at the detectors R and T after telescope k θ The fast-axis-angle of the HW P Eq. S6 ϑ LA ϑ LA = θ LA − α LA 2 , where θ LA is the fast-axis-angle of the HW P LA and α LA is the angle of the polarization ellipse of the light emitted directly from laser A Eq. S14

M HW
Mueller matrix of the HW P Eq. S6 M O Mueller matrix of the receiver optics (i.e. telescope k, collimating lenses, bandpass filter) Eq. S4 M T Mueller matrix of the transmitting part of the P BS, followed by cleaning polarizers Eq. S9 M R Mueller matrix of the reflecting part of the P BS, followed by cleaning polarizers Eq. S11 M QW Mueller matrix of the QW P Eq. S7 Mueller matrix of the QW P T B with its fast-axis-angle at φ T B = 45 o Eq. S24 O i k The laser beam receiver field-of-view overlap function, for laser i and telescope k Eq. S12

P BS
Polarizing Beam Splitter QW P Quarter Wave Plate

R k
The rotation matrix used to describe the rotation of the detection unit after telescope k, with respect to the frame coordinate system Eq. S20 S "Transmitted" and "Reflected" channels after the P BS, S = T, R, respectively Eq. ?? s x Sine of x Eq. S4 T (0, r) The transmission of the atmosphere between the lidar at range r = 0 and a specific range r in the atmosphere Eq. S12 The retardation parameter of M R Eq. S10 Z T The retardation parameter of M T Eq. S8 φ The fast-axis-angle of QW P Eq. S7 ϕ LB ϕ LB = φ LB − α LB , where φ LB is the fast-axis-angle of the QW P LB and α LB is the angle of the polarization ellipse of the light emitted directly from laser B Eq. S15 ω k The rotation angle of the detection unit after telescope k, with respect to the frame coordinate system Eq. S20