About the effects of polarising optics on lidar signals and the Δ 90-calibration

This paper provides a model for assessing the effects of polarising optics on the signals of typical lidar systems, which is based on the description of the individual optical elements of the lidar and of the state of polarisation of the light by means of the Müller-Stokes formalism. General analytical equations are derived for the dependence of the lidar signals on polarisation parameters, for the linear depolarisation ratio, and for the signals of different polarisation calibration set-ups. The equations can also be used for the calculation of systematic errors caused by non-ideal optical elements, their rotational misalignment, and by non-ideal laser polarisation. We present the description of the lidar signals including the polarisation calibration in a closed form, which can be applied for a large variety of lidar systems.


S.1 Coordinate system and conventions
Müller matrices describe the effect of optical elements on the Stokes vector with respect to a coordinate system and using a set of definitions about signs and directions.Different sets of definitions can be found in the literature as discussed in detail in Muller (1969); some of which are even inconsistent.The discussions led to the so-called Muller (or Muller-Nebraska-) convention, which we follow in this paper (see also Hauge et al. (1980)).We use a right-handed Cartesian coordinate system (see Fig. 8), in wich angles are defined counterclock wise, i.e. from the x-to the y-axis, when looking against the z-axis.The local z-axis points in the propagation direction of the light.We define the reference coordinate system of the lidar setup by the orientation of the polarising beam-splitter (PBS) in the receiving optics.
Light polarised with its E-vector on the x-axis, i.e. parallel to the incident plane of the PBS in Fig. 7, is mostly transmitted by a usual PBS, while light with polarisation in y-direction, i.e. perpendicularly polarised to the incident plane, is mostly reflected.The incident plane is spanned up by the direction of light propagation (z-axis, propagation vector k) and the normal of the reflecting surface, which means that the incident plane in Fig. 7 is the x-z-plane).The parallel and perpendicular polarisations are also called the p-and s-polarisation, respectively.The orientation of linearly polarised light is defined by the orientation of the plane of vibration, which contains both the electric vector E and the propagation vector k.
Fig. 7 Definition of the reference coordinate system with respect to the incidence plane of the polarising beam-splitter.
Other Müller matrix measurement configurations may have other arrangements for the coordinates.All choices, however, are arbitrary, and lead to different Müller matrices (Chipman 2009b).There is no preferred set of definitions in the literature.According to our choice of orientation, the diattenuation parameter D ist defined as , , , In order to keep the results of the Müller matrix calculations consistent when adding reflecting surfaces as mirrors and beam-splitters in the optical setup, a right-handed xyz-coordinate system is used with the z-axis in the direction of the light propagation.The vertical (perpendicular) polarised light has its E-vector in y-direction, Fig. 8 Reflection of a Stokes vector.

S.2 Stokes vector and Müller matrix
The Stokes vector and the Müller matrix are one representation of the state of polarisation of light, which is a based on measurable quantities.The Stokes vector describes the polarisation state of a light beam, and the Müller matrix describes how the Stokes vector changes when passing through an optical volume, which can be an optical element or an atmospheric path with scattering, absorbing and refracting properties.A Stokes vector can be determined by six measurements of the flux I with ideal linear and circular polarisation analysers at different orientations before a detector (Chipman 2009a;Ch. 15.17 Right-circularly polarised light is defined as a clockwise rotation of the electric vector when the observer is looking against the direction of light propagation (Bennett 2009a).Another representation, the so-called modified Stokes column vector (Mishchenko et al. 2002), uses the horizontally (parallel, p) and vertically (perpendicular, s) polarised fluxes I p and I s as the first two Stokes parameters.They can be transformed from the Stokes vector with a transformation matrix ( ) ( ) 0.5 0.5 0 0 0.5 0.5 0 0 0.5 0 0 1 0 0.5

S.3 Depolarisation
Depolarisation is closely related to scattering and usually has it origins from retardance or diattenuation which is rapidly varying in time, wavelength, or spatially over an optical device (Cornu-, Lyot-, or wedge-depolariser).Depolarisation causes a loss of coherence of the polarisation state (Chipman 2009a).The polarisation vector I F reflected by the atmosphere F(a) with linear polarisation parameter a from a generally polarised laser I L is (van de Hulst 1981; Sect.5.32) (Mishchenko et al. 2002;Sect. 4).

( ) ( ) ( )
11 11 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 1 2 The linear depolarisation ratio is defined as With a linearly polarised laser with intensity I L and linear polarisation parameter a L and rotational misalignment α, i.e. without emitter optics, the laser light reflected by the atmosphere with linear polarisation parameter a is The linear depolarisation ratio δ' resulting from a' can be retrieved with Eq. ( 12) For a small linear depolarisation ratio δ L of the laser beam, the resulting linear depolarisation ratio of an atmospheric measurement is about the sum of the lasers and the atmospheres linear depolarisation ratios If δ L is unknown, the uncertainty will cause an absolute error of the finally retrieved atmospheric linear depolarisation ratio.

S.4 Retarding linear diattenuator
The diattenuation magnitude D* of an optical element, usually simply diattenuation, is calculated from the maximum and minimum transmitted intensities I (or transmittances T) (Chipman 2009b), measured by rotating a linearly polarising analyser in front of the element: The diattenuation magnitude D* is always positive, and if D* is deduced from the reflectances T R p and T R s of an optical sample as in Eq. ( 17), Eq. (S.4.1) becomes (Lu and Chipman 1996) * In order to avoid sign changes in the equations between the cases where T R p < T R s and T R p > T R s , we use instead the diattenuation parameter D S (Eq.(S.4.3); see Chipman (2009b)), where it is named d x or d h ), with which all equations can be expressed together for the transmitting (subscript S = T) and the reflecting (subscript S = R) part of a polarising beam-splitter.

T p s p s T T T T T T p s p s T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T
with the short-cuts in Eq. (S.4.8), the intensity transmission coefficients (transmittance) for light polarised parallel (T T p ) and perpendicular (T T s ) to the plane of incidence of the PBS, the diattenuation parameter D T (see Suppl.S.3), and the average transmittance T T for unpolarised light.Δ T is the difference of the phase shifts of the parallel and perpendicular polarised electrical fields.The Müller matrix for the reflecting part of the PBS (see Eqs. 16,17) includes a mirror reflection (Suppl.S.6): with the corresponding intensity reflection coefficients (reflectance) for light polarised parallel (T R p ) and perpendicular (T R s ) to the plane of incidence of the PBS and 2 2 1 , c cos , s sin , In order to simplify the derivation of the equations, we write in the following for both, the reflecting and transmitting matrix of the polarising beam-splitter, M S (subscript S for splitter) where appropriate.
Müller matrices as in Eqs.(S.4.7) and (S.4.9) can be decomposed in matrices of a pure diattenuator M D and a pure retarder M ret : As both are linear, they commute (Eq.(S.4.13).They have a block-diagonal structure.

S.5.1 Rotation about the direction of light propagation
Some confusion can arise because rotation about the optical axis can be done on a Stokes vector, on the coordinate system (coordinate transformation), and on an optical element while keeping the reference coordinate system.The first two rotations don't change the state of the circular polarisation, while a rotated optical element can do that.Additional confusion arises because often in the literature and in textbooks the vector and coordinate rotations are mixed, or the derivation of the presented final equations from first principles are not provided, and sometimes the explanatory text is misleading or inconsistent.We follow the notations in Mishchenko et al. (2002); Chipman (2009b).Rotations are anti-clockwise, from the x-axis towards the y-axis, seen against the direction of light propagation (z-axis).
Fig. 9 Rotation of the xy-coordinate system (left) and of a vector (right).The z-axis, i.e. the direction of light propagation, points out of the paper plane.
A Stokes vector which is physically rotated by an angle ϕ while the coordinate system is fixed becomes (Mishchenko et al. 2002;Ch. 1.5) and the rotation matrix R(ϕ) ( ) With these definitions a formula for one rotation can easily be converted to other angles with Please note, that in Mishchenko et al. (2002;Ch. 1.5) the equations describe a rotation of the Stokes vector, while the text specifies the transformation as "rotation of the two-dimensional coordinate system".The two transformations are called "alibi" and "alias" transformation (Steinborn and Ruedenberg 1973), respectively.The Stokes vector rotates contra-variantly under the change of the basis.If we rotate the coordinate system (alias transformation) by an angle f (see Fig. 9), the original Stokes vector I appears in the rotated coordinate system under the angle −f, and the Stokes vector I' in the rotated coordinate system is Eq.(S.5.1.7).
( ) The rotation of the polarisation of a Stokes vector can be accomplished by means of a λ/2 plate (HWP), which is a 180° linear retarder.An ideal HWP can be derived from Eq. (S.5.( ) A HWP rotates a Stokes vector by twice the own rotation and additionally changes the direction of the circularly polarised component.For a rotation of f = 90° the HWP acts as a mirror but without changing the direction of light propagation.Real HWPs are often made of birefringent crystals.Their retardance depends in general on the wavelength, on the incident angle, and on the temperature.For lidar applications so-called true zero-order HWPs are best suited because of their relative insensitivity to temperature and incidence angle.The matrices for the HWP rotator and for the mechanical rotator can be combined in one matrix M rot as shown in Suppl.S.10.15.

S.5.2 Rotation of a retarding diattenuator
The rotation of an optical element with Müller matrix M O by an angle f about the direction of light propagation is mathematically performed by first rotating the coordinate system before M O by −f, to achieve the description of the Stokes vector in the local coordinate system (eigen-polarisations) of M O , and then rotating the coordinate system behind M O back to the reference coordinate system by f using the rotation matrix R(f) A linear retarding diattenuator M O rotated by f about the z-axis becomes and with ideal diattenuation Rotation of a retarding diattenuator by ±45° + ε and without error angle ε:

S.6 Mirror
For a pure mirror without diattenuation or retardance the Müller matrix is which results from the rotation of the detector (symbolised by the eye) about the y-axis from the rear of the optical element to the front as shown in Fig. 11.To explain the change of the axes, let the plane of vibration of linearly polarised light be rotated in the (xyz) coordinate system by f around the z-axis, indicated by the E -vector in Fig. 11, and the incident angle be χ = 0 for reflection from a mirror.After the mirror the direction of light propagation has changed, but not the orientation of the plane of vibration, indicated by the E'-vector.Hence, the rotation f' in the mirrored coordinate system (xyz)' is f' = 180° − f, which is equivalent to f' = −f .Thus a Stokes vector rotated by R(f) in (xyz) is described in (xyz)' after the mirror M M by ( ) ( ) Furthermore, the circular polarisation has changed its sign from right circular (RC) before to left circular (LC) after the mirror.

S.6.1 Real mirror
Real mirrors are dielectric or metal surfaces which can exhibit considerable phase retardation and diattenuation under oblique incident angles.Hence, a real mirror is a linear retarding diattenuator M O combined with a mirror M M. .
Eq. (S.6.1.1)is also the description of the reflecting part of a polarising beam-splitter or of any dichroic beam-splitter.

S.6.2 Rotation of a reflecting surface
If we rotate M MO , we have to mind the change of the coordinate system after the mirror.Here it is important which element comes first, because, as explained above, applying a mirror means a change of the local coordinate system after the mirror, and rotation of elements are always done with respect to the local coordinate system before the element.Hence, a diattenuator rotated in (xyz) plus a mirror described in (xyz)' is using (S.5.1.5)and (S.6.2) Moving the mirror before the diattenuator we see from Eqs. (S.6.1.2) to (S.6.2.2) that This explains why the rotation of a reflecting diattenuator has to be described as shown by Chipman (2009b) , i.e.: ( ) ( ) ( )

S.6.3 Beam-splitters and mirrors in the optical path
In order to make the equations developed in this work applicable to a variety of lidar systems, we have to investigate how the equations change when individual elements are changed from transmitting to reflecting.This is also useful when the reflected and the transmitted paths after a beam-splitter are to be described with the same equations.
Above we showed the local coordinate change behind a mirror.But how does this effect the outcome of a lidar measurement and of the calibration measurements?Let's consider a chain of rotated optical elements using the eigen-polarisations of the polarising beam-splitter matrix M S as the reference coordinate system ( ) ( ) ( ) When we exchange M 2 with its reflecting counterpart M M M 2 , we can move the ideal mirror M M step by step to the right in the chain using (S.6.2.3) and see that all rotation angles before the changed element are inversed.In the last step of Eq. (S.6.3.2) the depolarising atmospheric F-matrix is rotational invariant, and the circular polarisation of the input Stokes vector changes its sign, indicated by the star.
In other words: equations, which are derived for the system in Eq. (S.6.3.1), can be used for the system with an additional mirror as in Eq. (S.6.3.2) by inverting in the original equations all rotation angles before the mirror and reversing the circular polarisation of the input Stokes vector.In case two surfaces are changed from transmitting to reflecting as where a mirror is additionally placed behind the emitter optics, only the rotation angles between these two elements are inversed, because M M M M = 1, and the circular polarisation is not changed.
Real mirrors are usually dielectric or metal surfaces which can exhibit considerable phase retardation and diattenuation under oblique incident angles.For incident angles smaller than the Brewster angle the phase changes for p-(parallel) and s-(perpendicular) polarised light are in the same direction.
Analyser vector from Eq. (S.7.1.1)and input Stokes vector from (E.31) yield ) With horizontal linearly polarised emitter input Stokes vector e g e g With rotated, linearly polarised laser and emitter optics ) Comparison with Eq. ( 69):

S.7.3 Lidar signal with rotational error behind the emitter optics
We get the equation for this case directly from the previous one considering that moving the matrices for the rotational error from before the receiving optics to behind the emitter optics just changes the sign of the angle ε using Eq.(S.6.2)

S.8 Attenuated backscatter coefficient
The attenuated backscatter coefficient F 11 can be derived from the transmitted signal I T :

S.9 Rayleigh calibration
Calibration in ranges with presumably known aerosol depolarisation: With some lidar systems the calibration factor is determined in a measurement range with known volume linear depolarisation ratio δ, for example in clean air δ m .Assuming, for the sake of simplicity, an ideal PBS (see Eq. ( 28)), we get with Eq. ( 26) for the calibration factor in clean air η m ( ) Assuming further that the errors in I R and I T are independent, which could be the case if the background subtraction or non-linearities in analogue signal detection are the main error sources for them, we get as a first estimate for the relative error of the calibration factor This error can easily become very large.The linear depolarisation ratio measured in a volume of air molecules δ m depends on the width of the interference filters in the receiver optics, as they transmit or reject some rotational Raman lines, and on the atmospheric temperature (Behrendt and Nakamura 2002).At 355 nm δ m can range from about 0.004 to 0.015.Errors in the order of some 10% in δ m are already possible in case the wavelength dependence of the transmission of the interference filter or its tilt angle in the optical setup are not known accurately.Furthermore, a small contamination of the assumed clean air with strongly depolarising aerosol as Saharan dust or ice particles from sub-visible cirrus can change the volume linear depolarisation ratio dramatically.Assuming, for example, a small backscatter ratio of 1.01 due to particles with δ p = 0.3 and with δ m = 0.004, we get a real δ = 0.01* δ p + δ m = 0.007 (Biele et al. 2000), which would cause a relative error in the calibration factor η m of (0.007-0.004)/0.004= +75%.Better than this "clean" air calibration would even be to use a calibration in a cirrus cloud with δ p between let's say 0.3 and 0.5, with a resulting calibration factor error of "only" ± ½ * (0.5 -0.3) / (0.5 + 0.3) = ±25%.
Summary of this chapter: depolarisation calibration with presumably known atmospheric depolarisation can cause very large calibration errors.

S.10.10 Cleaned analyser (polarising beam-splitter with additional polarising sheet filters)
The intensity transmission of analysers is proportional to a certain state of polarisation before the analyser, with arbitrary state of polarisation behind, while the output of polarisers is a certain state of polarisation regardless which state of polarisation exists before the polariser (Lu and Chipman 1996).Here we use a polarising sheet filter, which is a depolarising analyser and a depolarising polariser at the same time, to get rid of the cross talk of the polarising beam-splitter.The combined matrix of a polarising sheet filter M A behind the polarising beam-splitter M S is again the matrix of a retarding linear diattenuator, which we call a cleaned polarising beam-splitter.If M A is rotated (misaligned) by  we get from Eq. (S.10.9.3) Transmitted part: ( ) ( ) Typically the manufacturers' terminology for p-and s-polarised transmission is k 1 and k 2 , respectively, as Eq.(S.10.10.4), and their specifications for polarising sheet filters are the transmission of two crossed filters (T cross ) Eq. (S.10.10.6) and that of two parallel filters (T parallel ) Eq. (S.10.10.7) of the same type.
1 2 and For the extinction ratio ρ (Eq.S.10.10.8, see Bennett (2009a), Sect.12.4) and its inverse, i.e. the contrast ratio or transmission rato, different definitions, as in Eq. (S.10.10.9), can be found in manufacturers' descriptions, which is sometimes confusing.However, usually k 2 << k 1 , and the given extinction ratios are then to be understood as "on the order of", irrespective of the used formula.

T T D D D D T D T T D T T T T T D D T T T T D D D T T T T T T T D D T T T T T T T T Z Z Z D D T T T T
(S.10.10.13)

S.10.13 Rotated λ/2 plate (HWP)
Retarder Eq.(S.10.12.1) with T q q q q q q q q q q q q = ae ö ae The rotation of a λ/2 plate by  rotates the Stokes vector by twice the rotation  and additionally inverts the circular polarisation component.This is equivalent to a mirror followed by a rotation of the coordinate system by 2θ.Please note, that the rotator and mirror matrices don't commute (compare Eqs. (S.6.2.1) ff).

T T T T T T T T T T T T D D T T T T T T T T T T T T T T T T T T T T T T T T T T D D T T T T T T T T T T T T T
( ) that the lasers a L and the atmospheres a cannot be discerned in the resulting Stokes vector, and the measured, combined polarisation parameter is L a aa = ¢(S.3.4) 4.6) The optical elements considered here are non-depolarising, linear diattenuators M D , with linear diattenuation parameter D O and average transmission T O for unpolarised light, combined with linear retarders M Ret (linear retardance Δ O , cosΔ O = c O , sinΔ O = s O ).The optical elements with possibly considerable diattenuation and retardation are dichroic beam-splitters, which are used to separate the wavelengths and to analyse the state of polarisation of the collimated beam in the receiver optics.They are used in transmission and reflection.The matrix of the transmitting part is Eq.(S.4.7) (see Eqs. (14 2.3) by setting Δ 0 = 180°, D Ò = 0  Z O = 1 and W O = 2, and T O = 1:

Figure 10
Figure 10 Rotation angles of an optical element.The rotations considered in this work are only f 1 and f 2 .

Figure 11 :
Figure 11: Reflection of light by a mirror.The light propagation is along the z-axis.The plane of vibration of linearly polarised light is indicated by the E-vectors, and right and left circular polarised light by the RC and LC arrows, respectively.
rotational error before the polarising beamsplitter General case with arbitrary laser input and emitter optics I E = M E I L and R(ε,h) from Eq. (S.10.15.1): rotational error before the receiving optics With Eq. (D.7) for the analyser part and (E.26) for the general input vector we get incident light depending on its state of polarisation.For atmospheric depolarisation by randomly oriented, nonspherical particles with rotation and reflection symmetry it can be shown that b = −a and c = (1 − 2a) (van de Hulst 1981;Mishchenko and Hovenier 1995;Mishchenko et al. 2002) (see also Sect.2.1), which results in the backscattering as +-45° calibrator with a QWP, with retardation error ) The substitution Eq. (S.12.1.1)is sometimes called Weierstrass substitution, but it can already be found in Euler's Institutionum calculi integralis (Eneström number E342: Vol. 1 Part 1, Sect. 1, Chap. 5, Problem 29, http://eulerarchive.maa.org/pages/E342.html).