Volker Freudenthaler Meteorological Institute, Ludwig-Maximilians … · 2017-06-28 · 28th...

28th International Laser Radar Conference - Lidar tutorials, Bucharest, 25 June 2017, Volker Freudenthaler, Polarizing lidars and the instrument function 1

Polarizing lidars and the instrument function

Volker Freudenthaler

Meteorological Institute, Ludwig-Maximilians-UniversityMunich, Germany

[email protected]


Comparison of the VLDR and PLDR of two lidar systems


Lidar equation – Linear depolarisation ratio (LDR)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )0

2

exp 2

0,

r

2

2

CP r r r LR r dr background r

r

Cr T r background r

r

b b

b

¢é ù= - +¢ê ú

ë û

= +

òLidar equation

P(r) lidar power incident on receiver from range r

Problem: absolute calibration => C Solution: relative calibration with reference value (Klett, ...)


Lidar equation

Separation of explicit r-dependence

Polarisation measurements

=> Linear depolarisation ratio δ, LDR *

Is,p(r) = signals recorded with data acquisition in channels s and p

Lidar equation – Linear depolarisation ratio (LDR)

( ) ( ) ( )2 0,2

CP r r T r

rb=

( ) ( ){ } ( ){ } ( )

( ) ( ){ } ( ){ } ( )

( ) ( )( )

( ){ }( ){ }

( )( )

2

2

0,

0,

=

2

2

CI r Filter P r Filter r T r

rC

I r Filter P r Filter r T rr

Filter rI r rr

I r rFilter r

b

b

b bd

bb

^ ^ ^

^^ ^

= = ´

= = ´

= =

P P P

P PP

* Gimmestad, G. G.: Reexamination of depolarization in lidar measurements, Appl. Opt., 47(21), 3795–3802, 2008.


Typical lidar setup => The Model

S S S O E Lh= CM M FMI I

Stokes vectors describe the state of polarisation of a parallel light beam Müller matrices describe the transformation by optical media

Freudenthaler, V. About the effects of polarising optics on lidar signals and the Δ90-calibration Atmos. Meas. Tech., 9, 4181-4255, 2016.


p- & s-polarisation with respect to laser or incidence plane?

ideal polarsing beamsplitter (cube)

Tp

,Ts

Rp ,R

s

p and s polarization are defined with respect to the plane of incidencefor each optical element

R intensity reflectanceT intensity transmittance




Tp

,Ts

Rp ,R

s

50/50 beamsplitter

Rp

Tss

ideal polarisation filters




Tp

,Ts

Rp ,R

s

50/50 beamsplitter

Rp

Tss

more general

TR

p , TR

s

TT

p , TTs

transmitted

reflected


Magenta reference plane defines rotation angles and parameter y = ±1


Retarding Diattenuator

0 0 1 0 00 0 1 0 01

0 0 c s0 0 2 cos 2 sin2

0 0 s c0 0 2 sin 2 cos

p s p sT T T T

Tp s p s

T T T T TT Tp s p s

T T T TT T T T T T

p s p sT T T T

T T T T T T

T T T T DT T T T D

TZ ZT T T T

Z ZT T T T

D D

D D

æ ö+ - æ öç ÷- + ç ÷ç ÷ ç ÷= =ç ÷ ç ÷ç ÷ ç ÷-è øç ÷-è ø

M

22, , 1 , c cos , s sin ,

2

p sp s p sT T p sT T T T

T T T T T T T T T T Tp s p sT T T T

T TT T T TT D Z D

T T T TD D D j j+ -= = = = - = = = -

+ +



0 0 1 0 00 0 1 0 01

0 0 c s0 0 2 cos 2 sin2

0 0 s c0 0 2 sin 2 cos

1 0 0 1 0 0 0

1 0 0 0 1 0 0

0 0 0 0 0 c s

0 0 0

p s p sT T T T T

p s p sT T T T

TT Tp s p s

T T T TT T T T T T

p s p sT T T T

T T T T T T

T

TT

T T T

T

T T T T DT T T T D

TZ ZT T T T

Z ZT T T T

D

DT

Z

Z

D D

D D

æ ö+ - æ öç ÷- + ç ÷ç ÷ ç ÷= = =ç ÷ ç ÷ç ÷ ç ÷-è øç ÷-è ø

æ öç ÷ç ÷=ç ÷ç ÷è ø

M

0 0 s cT T

æ öç ÷ç ÷ç ÷ç ÷-è ø

22, , 1 , c cos , s sin ,

2



T TT T T TT D Z D

T T T TD D D j j+ -= = = = - = = = -

+ +

linear diattenuator retarder



0 0 1 0 00 0 1 0 01

0 0 c s0 0 2 cos 2 sin2

0 0 s c0 0 2 sin 2 cos

1 0 0 1 0 0 0

1 0 0 0 1 0 0

0 0 0 0 0 c s

0 0 0

p s p sT T T T T

p s p sT T T T

TT Tp s p s

T T T TT T T T T T

p s p sT T T T

T T T T T T

T

TT

T T T

T

T T T T DT T T T D

TZ ZT T T T

Z ZT T T T

D

DT

Z

Z

D D

D D

æ ö+ - æ öç ÷- + ç ÷ç ÷ ç ÷= = =ç ÷ ç ÷ç ÷ ç ÷-è øç ÷-è ø

æ öç ÷ç ÷=ç ÷ç ÷è ø

M

0 0 s cT T

æ öç ÷ç ÷ç ÷ç ÷-è ø

22, , 1 , c cos , s sin ,

2



T TT T T TT D Z D

T T T TD D D j j+ -= = = = - = = = -

+ +

linear diattenuator retarder


Retarding linear diattenuator mirror

1 0 0

1 0 0

0 0 c s

0 0 s c

1 0 0 1 0 01 0 0 0

1 0 0 1 0 00 1 0 0

0 0 c s 0 0 c s0 0 1 0

0 0 s c 0 0 s c0 0 0 1

T

TT T

T T T T

T T T T

R R

R RR R R

R R R R R R R R

R R R R R R R R

D

DT

Z Z

Z Z

D D

D DT T

Z Z Z Z

Z Z Z Z

æ öç ÷ç ÷=ç ÷ç ÷-è ø

æ ö æ öæ öç ÷ ç ÷ç ÷ç ÷ ç ÷ç ÷= =

- - -ç ÷ ç ÷ç ÷ç ÷ç ÷ ç ÷- --è øè ø è ø

M

M

2

2

2, , 1 , c cos , s sin ,

2

2, , 1 , c cos , s sin ,

2

p sp sT TT T

T T T T T T Tp sT T

p sp sR RR R

R R R R R R R

p sp sT

p sR R

TT T T Tp s

T T

p sp sR R

R R R Rp sR R

T TT TT Z D

T T

T TT TT Z D

T T

T TD

T T

T TD

T T

D j j

D

D D

D D j j

+= = = - = =

+

+= = = - =

-= = -

+

-= = -

+=

+

mirror linear diattenuator

Leipzig Graduate School Clouds, Aerosols and Radiation (LGS-CAR), Leipzig 11-12.01.2014

Polarized Radiative Transfer in the Troposphere: Optical systems to measure polarization; Volker Freudenthaler 17

Quarter wave coating

source: JEFF BLAKE and RICHARD PAYNTON, Choosing optical coatings for medical displays

Leipzig Graduate School Clouds, Aerosols and Radiation (LGS-CAR), Leipzig 11-12.01.2014

Polarized Radiative Transfer in the Troposphere: Optical systems to measure polarization; Volker Freudenthaler 18

Coating with more layers

source: Dielectric mirror, http://en.wikipedia.org/w/index.php?title=Dielectric_mirror&oldid=544165129

MIM-Seminar, München 18.07.2014, Volker Freudenthaler, Optical systems to measure polarization 19

Coatings Semrock beamsplitter

https://www.semrock.com/filters.aspx




http://www.electronicproducts.com/Optoelectronics/Displays/Choosing_optical_coatings_for_medical_displays.aspx





Coatings Semrock MaxMirror Reflection

http://www.semrock.com/FilterDetails.aspx?id=MM2-311-25


Coatings Semrock MaxMirror Phase shift


Coatings Semrock MaxMirror Phase shift

half wave plate

quarter wave plate


Rotated retarding linear diattenuator

( ) ( ) ( )

2 2 2 2

2 2 2 2

2 2

2

2 2 2 2 2

2

2 2 2 2 2

1 0 01 0 0 0 1 0 0 0

1 0 00 c s 0 0 c s 0

0 0 c s0 s c 0 0 s c 0

0 0 s c0 0 0 1 0 0 0 1

1 c s 0

c 1 s s c s s

s s c 1 c c

O O

O

OO

O O O O

O O O O

O O

O O O O O

O

O O O

D

DT

Z Z

Z Z

D D

D W W ZT

D W W

f f f f

f f f f

f f

f f f f f

f f f f

f f f= -

æ öæ ö æ öç ÷ç ÷ ç ÷-ç ÷ç ÷ ç ÷ =

-ç ÷ç ÷ ç ÷ç ÷ ç ÷ç ÷-è ø è øè ø

- -=

-

=

=

M R M R

2 2

s

0 s s c s cO O

O O O O O O

Z

Z Z Zf

f f

æ öç ÷ç ÷ç ÷ç ÷ç ÷-è ø

22 2c cos2 ,s sin 2 ,c cos ,s sin , 1 , 1 cO O O O O O O O OZ D W Zf ff f D D= = = = º - = -


Two rotated retarding linear diattenuator

( )

( ) ( )

, retarding linear diattenuator

retarding linear diattenuator

But if , are rotated individually:

rotated retarding linear diattenuator

S O

S O

S O

S Oq f

" ÎÎ

Ï

M M

M M

M M

M M

( )

2 2

2

2 2 2 2 2

2

2 2 2 2 2

2 2

1 0 0

1 0 0

0 0 c s

0 0 s c

1 c s 0

c 1 s s c s s

s s c 1 c c s

0 s s c s c

O

OO O

O O O O

O O O O

O O

O O O O O

O O

O O O O O

O O O O O O

D

DT

Z Z

Z Z

D D

D W W ZT

D W W Z

Z Z Z

f f

f f f f f

f f f f f

f f

f

æ öç ÷ç ÷=ç ÷ç ÷-è ø

æ öç ÷- -ç ÷= ç ÷-ç ÷ç ÷-è ø

M

M


Müller matrix for atmospheric backscatter of randomly oriented particles

11

22

22

44

22

11

11

0 0 0 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 1 2

with polarisation parameter 1

polarisation parameter e d

F

F a

F a

F a

Fa d

F

d

Fb

æ ö æ öç ÷ ç ÷ç ÷ ç ÷Þ = =

- -ç ÷ ç ÷ç ÷ç ÷ -è øè ø

= = -

F

e.g.: Mishchenko, M. I., Hovenier, J. W.: Depolarization of light backscattered by randomly oriented nonspherical particles, Opt. Lett., 20(12), 1356–1358, 1995.


SVLE for atmospheric backscatter of randomly oriented particles

11 11

11 22

11 22

1 0 0 0 1

0 0 0

0 0 0 0 0

0 0 0 1 2 0 0

1 1

1 1

, traditional terminology: parallel

1

1

0

0

and per

S LL L

I I

a a I IF F I

a

a

I F Faa

I a F F

I I

I

ddd

^

^

^

^

+æ ö æ ö æ öç ÷ ç ÷ ç ÷-ç ÷ ç ÷ ç ÷= = = =

-ç ÷ ç ÷ ç ÷ç ÷ ç ÷ ç ÷-è ø è ø

æ öç ÷ç ÷ç ÷ç ÷è ø è ø

-- -= = = Þ =

+ + +

FI I

P

P

P

P pendicular to the laser polarisation

Stokes vector of linearly polarised light

SVLE : Stokes Vector Lidar Equation Hayman, M. and Thayer, J. P.: General description of polarization in lidar using Stokes vectors and polar decomposition of Mueller matrices, J. Opt. Soc. Am. A, 29(4), 400–409, doi:10.1364/JOSAA.29.000400, 2012.


● Stokes vector, intensity measurements

IH

horizontal linear polarizer (0°)

IV vertical linear polarizer (90°)

I45

linear polarizer at 45°

I135

linear polarizer at 135°

IRC

right circular polarizer

ILC

left circular polarizer

The Stokes vector S is defined with six intensity (flux) measurements

using ideal polarization analyzers in front of a radiometer

0

1

45 1352

3

H V

H V

RC LC

SI

SQ

SU

SV

I I

I I

I I

I I

æ ö æ öæ öç ÷ ç ÷ç ÷ç ÷ ç ÷ç ÷= = =ç ÷ ç ÷ç ÷ç ÷ ç ÷ç ÷è ø è -ø

-è ø

+-

S


SVLE for atmospheric backscatter of randomly oriented particles

11

11

1

2

1

1

2

1 22

1 0 0 0 1

0 0 0 1

0 0 0 0 0

0 0 0 1 2 0 0

1 1

1 1

, traditional terminology:

1

parallel and pe

0

r

0

S L L L

I I

a I IF I

a

a

I F Fa

aF I

aI a F F

I I

ddd

^

^

^

^

+æ ö æ ö æ öç ÷ ç ÷ ç ÷-ç ÷ ç ÷ ç ÷= = = =

-ç ÷ ç ÷ ç ÷ç ÷ ç ÷ ç

æ öç ÷ç ÷ç ÷ç ÷ ÷-è ø è ø è ø

-- -

è

= = = Þ =+ + +

ø

FI I

P

P

P

P pendicular to the laser polarisation

backscatter from atmosphere


SVLE with instrument filter – polarisation lidar

( )1

1 111

1

1

0

0

1 measured signal with single channel detection

1 0 0

1 0 0

0 0 c s

0 0 s

1

0

c 0

S

S

S S S S

S S S

S

SS S S S S S L

S S S

S

LS L

L S

D

D

Z Z

Z

D a

D aT T F I

I T I a

F

D

aI

F

Z

h h h

h

+ææ öç ÷ç ÷ç ÷ç ÷è

æ öç ÷ç ÷ç ÷ç ÷

öç ÷+ç ÷= = =ç

-è÷

ç ÷è ø

= +

ø ø

M FI I

( ) ( ) ( ) ( )( ) ( ) ( )

2

2

1 1 1 0,

0, lidar equation

1

1

2S

2

R

S S

R R R

T T TT

D D a r

r

I r r T r

P r r T r

I

C

C

I a

D

T

aT

D

h

b

b

h

- £ £ + = +

=

+= =+

¢

( )( ) two channel linear polarisation detection

p sR RRp s

T T T

T T

T T

dhh d

+

+

polarisation filter (splitter)


Calibration factor η and calibration factor correction K

( )1

1R R R

T T

R

T T

TI aa

I a

DK

T D

h hh

+= ´ = ´+

Freudenthaler, V. About the effects of polarising optics on lidar signals and the Δ90-calibration Atmos. Meas. Tech., 9, 4181-4255, 2016.

cross-talk correction K


Particle linear depolarisation ratio δp , PLDR

( ) ( )( ) ( )

11 1 1

11 1 11

m m

m v v m v vp v

mm v

v

m p

m

RR

R R

R

d dd d d d d dd d dd d

db bb

+ -+ - + += = ++ - + -

++

=

δm molecular linear depolarisation ratio MLDR (Rayleigh)δv volume linear depolarisation ratio VLDRδp particle linear depolarisation ratio PLDR

βm molecular backscatter coefficient (Rayleigh)βp particle backscatter coefficientR backscatter ratio

Biele, J.; Beyerle, G. & Baumgarten, G.; Polarization Lidar: Correction of instrumental effects, Opt. Express, OSA, 2000, 7, 427-435


Particle linear depolarisation ratio δp , PLDR

10

1

1 11 1

1

0111

p v

m m

v v

v

v

m

m

m

v

R

R

R

R

d dd d

dd

d

dd

dd

d

ì ® -ïï® Þ Þ =

+ ++ +

+ ®íï -®î +ï

for small backscatter ratios R (little aerosol)the exact value of δm becomes very important


measured / theoretical molecular LDR – POLIS-6 – 355 / 532 nm –

uncertain theoretical LDR due to uncertainties in- laser wavelength- Rot. Raman Lines in interference filter bandwidth

source: Freudenthaler et al., 27th ILRC 2015, Accuracy of linear depolaristion ratios in clear air ranges measured with POLIS-6 at 355 and 532 nm. http://dx.doi.org/10.1051/epjconf/201611925013

Laser polarisation must be cleaner than deviation of measurements from theory


Rayleigh calibration: molecular linear depolarisation ratio (mLDR, RLDR)

http://www.meteo.physik.uni-muenchen.de/~stlidar/earlinet_asos/Rayleigh_scattering/Rayleigh_coefficients.pdf


Laser - polarisation

laser

divergence

pointing / jitter

wavelength

polarisationstate of polarisationorientation

temporal / thermal stability


Laser – wavelengths measured and theory

source: http://www.meteo.physik.uni-muenchen.de/~stlidar/earlinet_asos/Rayleigh_scattering/Rayleigh_coefficients.pdf


Laser – wavelength – interference filter - centre wavelength over incidence angle

http://dx.doi.org/10.1051/epjconf/201611925013


Laser – wavelength – interference filter bandwidth – rotational Raman lines (RRL)

laserdivergence

pointing / jitter

wavelength



https://epub.ub.uni-muenchen.de/24942/index.html



theoretically measured molecular LDR

The combination of laser wavelengthIF-filter center wavelengthIF-filter bandwidthIF-filter incidence angle

determines the theoretically measured molecular LDR


Laser – polarisation - orientation

source: http://www.litronlasers.com/pages/nano_series.html



Laser – polarisation – orientation – SHG / THG / beam separation

source: http://www.litronlasers.com/pages/nano_series.html


Laser - polarisation

Linear polariser in the resonator should clean the 1064 polarisation,

- but NdYAG rod birefringence can decrease the DOLP (Degree Of Linear Polarisation) of 1064 nm

- SHG and THG only convert light in certain polarisation planes=> DOLP of 355 should be very clean=> DOLP of 532 could be decreased by THG=> DOLP of the residual 1064 less than original

- Harmonic beam separators can decrease the DOLP

see also: https://en.wikipedia.org/wiki/Second-harmonic_generation https://www.rp-photonics.com/frequency_doubling.html


Laser – polarisation – 532 nm Surelite II, Continuum

Giuseppe d'Amico 2006:

Results from Continuum USA.

The measurements were done using a laser Surelite II - 10Hz with a SHG crystal of Type I and II. (so at 532 nm; Giuseppe)

Using both crystals, the energy of the vertical component of polarization was 3 Watts and the energy of the horizontal component was 2 mWatts corresponding to polarization purity of about 99.93%.

=> LDR = 0.002 / 3 = 0.00067


Laser – Quantel CFR 200 ? Polarisation Purity


Laser – temporal stability

laserdivergence

pointing / jitter

wavelength




Typical lidar setup – emitter and steering optics

emitter and steering optics

wavelength dependencefocal length => divergencetransmissionpolarisationbirefringence

alignment accuracystabilityalignment control

polarisationorientationflatness

https://en.wikipedia.org/wiki/Second-harmonic_generation


Emitter optics – TROPOS beam expander 6.5x (with CaF2 lens ) apochromat

ca. 600 mm

Engelmann, R., et al., The Automated Multiwavelength Raman, Polarization, and Water-Vapor Lidar Polly XT : The NeXT Generation, AMT, 2016. http://www.atmos-meas-tech.net/9/1767/2016/amt-9-1767-2016.html

Big problem: CaF2 and MgF2 lenses are birefringent

Other problems: - residual wavelength dependence- 3 lambda AR-coating- glass solarisation


Stress birefringence






Crossed steering mirrors compensate diattenuation and retardation perfectly


Typical lidar setup – telescope

receiving opticstelescope

focal length, diameter

alignment stability

Newton telescope90° mirror depolarisation

field of view


Newton 90° mirror

receiving opticstelescope

focal length, diameter

alignment stability

Newton telescope90° mirror depolarisation

field of view


Newton 90° mirror, Aluminium + MgF2 + SiO2 coating reflection vs. angle


Newton 90° mirror, Aluminium + MgF2 + SiO2 coating reflection vs. angle


Newton 90° mirror, Aluminium + MgF2 + SiO2 coating retardance vs. angle


Newton 90° mirror diattenuation and retardance

Di, et. al., Polarization analysis and corrections of different telescopes in polarization lidar, Appl. Opt., OSA, 2015, 54, 389-397

considered: raw aluminium coatings

own calculations:

- confirm above values- diattenuation ~ 0.03- retardance ~ 13°- LDR 0.0003

aluminium + MgF2 coating reduces effect

- diattenuation ~ 0.001- retardance ~ 7°

own measurements

- no diattenuation- retardance ~ 30°


Typical lidar setup – receiver optics

receiving opticsbeamsplitters and filters

focal length of collimator=> beam divergence=> beam diameter

accpetance angles of beamsplitters and interference filters

polarisation problemesdiattenuationretardance


Effect of optical elements can be described by Müller matrices


Laser – wavelength – interference filter - centre wavelength over incidence angle


Interference filter - transmission over incidence angle











Diattenuation of HR532 HT607 beamsplitter coating over incident angle

R

T

900

0

1


Retardance of HR532 HT607 beamsplitter coating over incident angle

R

T

900-180

180


Laser – polarisation – POLIS-6 – 355 / 532 nm – measured / theoretical Rayleigh LDR

uncertain theoretical LDR due to uncertainties in- laser wavelength- Rot. Raman Lines in interference filter bandwidth

source: Freudenthaler et al., 27th ILRC 2015, Accuracy of linear depolaristion ratios in clear air ranges measured with POLIS-6 at 355 and 532 nm. http://dx.doi.org/10.1051/epjconf/201611925013


theoretically measured molecular LDR

The combination of laser wavelengthIF-filter center wavelengthIF-filter bandwidthIF-filter incidence angle

determines the theoretically measured molecular LDR


Typical lidar setup – polarisation calibrator

polarisation calibrator

- error of η due to non-ideal linear polarizer calibrator with extinction ratio ρ ρ = 10-5 => Δη/η = 1.3% ρ = 10-4 => Δη/η = 8%

- Δ 90-calibration reduces the calibration error dur to rotational misalignment by orders of magnitude


Typical lidar setup – polarising beam splitter

polarisingbeam splitter

acceptance angles

extinction ratio

cleaning the cross talk with polarizing sheet filters (ρ = 10-3 is sufficient) removes many problems!


Typical lidar setup – detector and optics

detectors and optics

PMT homogeneity of the sensitivity

APDsmall diameter

eyepiece => telescope imaging

neutral density filtersadjust signal level (LICEL)

http://dx.doi.org/10.1051/epjconf/201611925013


Mathematical description – assumptions/limitations

randomly oriented aerosol

modules consist ofn π/2 - rotated retarding linear diattenuators

without depolarization


Mathematical description – SVLE parameters

y

, ,

,

,

11

polarisation parameter / degree of linear polarization

unpolarised transmission

linear diattenuation parameter

extinction ratio of cleaning pol-filters

S S O E LS

R T R T O E L

R T O E

R T

O

T T T IF

a b

D D D

ehh

rD D

= M R FR M M II

,

retardation

module rotation

22 independent parameters

E

R Tf e g b a


Mathematical description – GHK- summary

( )

( )( ) ( ) ( ){ }

y

11

2 2 2 2

2 2 2 2 2 2 2 2

1 y c y s s

c s y c s s s c 2 s

S S S O E L

S S S O E L

S O E S O O E

O E E S E E O E E O O

S

E

S

S

S

I T T F T I a

D D i D Z v

D q u D q

G H

u q v

G

ZH W u

e

g e g e

g g e e g e g g

h

h

+ +

+

=

= +

= + -

é ù= - + + - + -ë û

M R R M FMI I



( )

( )( ) ( ) ( ){ }

y

11

2 2 2 2

2 2 2 2 2 2 2 2

1 y c y s s

c s y c s s s c 2 s

E L

E E

E E E E E

S S S O

S S S O E L S S

S S O S O O

S O S O OE EO

I T T F T I G aH

G D D D Z

H D

i v

q u q vW Zu qD u

e

g e g e

g g e e g e g g

h

h

+ +

+

=

= +

= + -

é ù= - + + - + -ë û

M R R M FM II

( ) ( ) ( )

( )( ) ( )( ) ( )

( )

2 2 2 2

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2

2 2 2 2

,

c s

c c s s s c s

s s c c s c s

s s c c

E Lin

E L E L

L E L L

E L L L E L L E E L

E L L L E L L E E L

E E E E

E E L L E E L

T I T I

i D q u

D i q u W q u Z v

D i q u W q u Z v

Z

i q u v

q u Z v

a b a b

b a a b a b a b

b a a b a b a b

a b a b

b ab a

- -

- -

- -

- -

= =

+ -

é ù+ - + + -ë û= =é ù+ + - + -ë û

- + +

M II



( )

( )( ) ( ) ( ){ }

y

11

2 2 2 2

2 2 2 2 2 2 2 2

1 y c y s s

c s y c s s s c 2 s

S S S O E L

S S S O E L

S O E S O O E

O E E S E E O E E O O

S

E

S

S

S

I T T F T I a

D D i D Z v

D q u D q

G H

u q v

G

ZH W u

e

g e g e

g g e e g e g g

h

h

+ +

+

=

= +

= + -

é ù= - + + - + -ë û

M R R M FMI I

( ) ( )( ) ( )

**

*

*

*

11

1, ,

1

1

R R R R R T R

T

T

T T T T R

T R R

R R T T

R

T

TT R

I G dH T G Ga

I G dH T H H

a

a

F I I

G H G H

G H G H

H H

h dd hh

h

h d

dd

d

+ -= = = =

+ -

+ +- -

--= =+ -

µ -


Mathematical description – Calibration

( )( )

( ) ( )( )

( )( )

y

*

gain ratio Theory Measurement

x45 x45x45

x45 x45

we need calibration factor

x45

we

S S

S S

R in R

T i

O ES

S

R

T

R

n

R

i

T

T T

L

nI

I

T

I

T

h

h

e eh e h

e

e e

h

e

h

hh

= =

=

° + ° +° + = =

° + ° +

+

=

°

M R

A C

A C

C M FM

A C

I

I

I

I

I

( )( )

*

*

don't know: but we know:

x45

x45

analytically derived correction factor for the measured gain ratio

R T

T R

R in

T in

T

R

T

T

T

TK

K

h

hehe

h

hÞ

° += =

° +A C

A C

I

I


Δ90-calibration

( ) ( ) ( )( )

( )( )

( )

* * *90

*2

2

*90

+45 45+45 45

+45 45

with 1, 1, 1, 0

x45 1 y 1 xys

1 y 1 xys

1 y

1 y

R R

T T

P T R

O

O

O

O

I I

I I

D D D

D

D

D

D

D

e

e

D

e eh h e h e

e e

gh e

hhh

° + - ° +º ° + - ° + = ×

° + - ° +

= = + = - = Þ

° + - +=+ -

-=+

e.g. Calibration with an ideal linear polariser before the receiving optics

rotation ε error vanishes [AMT, 9, 4181–4255, 2016 Eq.(138)]


POLIS-6 ±45 rotation of the receiver optics => Δ90-calibration


Determination of receiver optis diattenuation with two calibrations

Amodeo et al.,


Determination of receiver optis diattenuation with two calibrations


Determination of GHK-parameters and error calculation

y

, ,

,

,

11

polarisation parameter / degree of linear polarization

unpolarised transmission

linear diattenuation parameter

extinction ratio of cleaning pol-filters

S S O E LS

R T R T O E L

R T O E

R T

O

T T T IF

a b

D D D

ehh

rD D

= M R FR M M II

,

retardation

module rotation

22 independent parameters + calibrator parameters (depending on type)

E

R Tf e g b a

difficult to manage by hand (without errors) =>Freudenthaler, V., 2017: Open source Python code for polarization related error analysis of aerosol lidar signals https://bitbucket.org/iannis_b/atmospheric_lidar_ghk


atmospheric_lidar_ghk input file example# Do you want to calculate the errors? If not, just the GHK-parameters are determined.Error_Calc = True

# Header to identify the lidar system# MUSA configuration http://www.atmos-meas-tech-discuss.net/amt-2015-339/amt-2015-339.pdf Table 5, 532 xcg xpgEID = "po" # Earlinet station IDLID = "MUSA" # Additional lidar ID (short descriptive text)print(" Lidar system :", EID, ", ", LID)

# --- IL Laser IL and +-UncertaintyDOLP, dDOLP, nDOLP = 1.0, 0.00, 0 #degree of linear polarization; default 1RotL, dRotL, nRotL = 3.0, 0.6, 1 #alpha; rotation of laser polarization in degrees; default 0

# --- ME Emitter and +-UncertaintyDiE, dDiE, nDiE = 0., 0.00, 0 # DiattenuationTiE = 1. # Unpolarized transmittanceRetE, dRetE, nRetE = 0., 180.0, 0 # Retardance in degreesRotE, dRotE, nRotE = 0., 0.0, 0 # beta: Rotation of optical element in degrees

# --- MO Receiver Optics including telescope DiO, dDiO, nDiO = -0.055, 0.003, 1TiO = 0.9 RetO, dRetO, nRetO = 0., 180.0, 2 RotO, dRotO, nRotO = 0., 0.1, 0 #gamma

# +++++ PBS MT Transmitting path defined with TS, TP, PolFilter extinction ratio ERaT, and +-Uncertainty# --- PBSTP, dTP, nTP = 0.95, 0.01, 1TS, dTS, nTS = 0.001, 0.001, 1RetT, dRetT, nRetT = 0., 180., 0 # Retardance in degrees# --- Pol.Filter behind transmitted path of PBSERaT, dERaT, nERaT = 0.001, 0.001, 1 # Extinction ratioRotaT, dRotaT, nRotaT = 0., 3., 1 # Rotation of the pol.-filter in degreesTiT = 0.5 * (TP + TS)DiT = (TP-TS)/(TP+TS)DaT = (1-ERaT)/(1+ERaT)TaT = 0.5*(1+ERaT)


atmospheric_lidar_ghk input file example continued# --- Parallel signal detected in the transmitted channel => Y = 1, or in the reflected channel => Y = -1Y = -1.# --- Calibrator LocationLocC = 4 #location of calibrator: 1 = behind laser; 2 = behind emitter; 3 = before receiver; 4 = before PBS# --- Calibrator Type used; defined by matrix values below# Type of calibrator: 1 = mechanical rotator; 2 = hwp rotator (fixed retardation); 3 = linear polarizer; # 4 = qwp; 5 = circular polarizer; 6 = real HWP calibration +-22.5°TypeC = 6# --- MC Calibratorif TypeC == 1: #mechanical rotator

DiC, dDiC, nDiC = 0., 0., 0TiC = 1.RetC, dRetC, nRetC = 0., 0., 0RotC, dRotC, nRotC = -2.3, 0.1, 1 #constant calibrator offset epsilon# Rotation error without calibrator: if False, then epsilon = 0 for normal measurementsRotationErrorEpsilonForNormalMeasurements = True # is in general True for TypeC == 1 calibrator

elif TypeC == 2: # HWP rotatorDiC, dDiC, nDiC = 0., 0., 0TiC = 1.RetC, dRetC, nRetC = 180., 0., 0#NOTE: use here twice the HWP-rotation-angleRotC, dRotC, nRotC = -2.3, 0.1, 1 #constant calibrator offset epsilonRotationErrorEpsilonForNormalMeasurements = True # is in general True for TypeC == 2 calibrator

elif TypeC == 3: # linear polarizer calibratorDiC, dDiC, nDiC = 1.0, 0., 0 # ideal 1.0TiC = 0.5 # ideal 0.5RetC, dRetC, nRetC = 0., 0., 0RotC, dRotC, nRotC = 0.0, 0.1, 1 #constant calibrator offset epsilonRotationErrorEpsilonForNormalMeasurements = False # is False for TypeC == 3 calibrator

elif TypeC == 4: # QWP calibratorDiC, dDiC, nDiC = 0.0, 0., 0 # ideal 1.0TiC = 1.0 # ideal 0.5RetC, dRetC, nRetC = 90., 0., 0RotC, dRotC, nRotC = 0.0, 0.1, 1 #constant calibrator offset epsilonRotationErrorEpsilonForNormalMeasurements = False # is False for TypeC == 4 calibrator

elif TypeC == 6: # real half-wave plate calibration at +-22.5° => rotated_diattenuator_X22x5deg.odtDiC, dDiC, nDiC = 0., 0., 0


atmospheric_lidar_ghk output (ANACONDA Spyder => IPhython console)


atmospheric_lidar_ghk output (ANACONDA Spyder => IPhython console)


● Eq. (116) reads

Correct is:

● Supplement S.1, first paragraph, line 10 ff reads: "Light polarised with its E-vector on the x-axis, i.e. parallel to the incident plane of the PBS in Fig. 7,..."

Correct is: "Light polarised with its E-vector on the x-axis, i.e. parallel to the incident plane of the PBS in Fig. 7a,..."

● Supplement S.1, first paragraph, line 14 reads: "…. which means that the incident plane in Fig. 7 is the x-z-plane)."

Correct is: "…. which means that the incident plane in Fig. 7a is the x-z-plane)."

● Supplement S.1, Fig. 8 caption reads: " Reflection of a Stokes vector."Correct is " Reflection of an E-vector."

Errors in Atmos. Meas. Tech., 9, 4181–4255, 2016

( ) ( ) ( )( ) ( ) ( )

2 2 22*2 290

2 2 222 2

1 y y s h c

1 y y s h c

O R E O R E E

O T E O T E E

D D i D D q u

D D i D D q u

e eD

e e

hh

+ - + -=

+ - + -

( ) ( ) ( )( ) ( ) ( )

2 2 22*2 290

2 2 222

22

21 y y s h c

1 y y s h c

O R E O R E E

O T E O T E E

D D i D D q u

D D i D D q u

a

a

e eD

e e

hh

+ - + -=

+ - + -

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Volker Freudenthaler Meteorological Institute, Ludwig-Maximilians … · 2017-06-28 · 28th...

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