Virtues of polarization in remote sensing of atmospheres

and oceans

George W. KattawarDepartment of Physics and Institute

for Quantum StudiesTexas A&M University

Research Colleagues:Deric Gray, Ping Yang, Yu You, Peng-Wang Zhai, Zhibo Zhang

Scattering plane

Scattering Geometry

Incident beam

E⊥i

EP

i

Scattering angle

EP

s

E⊥s

Scattering object

Scattering amplitude matrix

EPs

E⊥s

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

eikr

−ikr

S2 S3

S4 S1

⎛

⎝⎜

⎞

⎠⎟

EPi

E⊥i

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

||

||

|| ||

Stokes vector-Mueller matrix formulation

The electric field can be resolved into components. El and Er are complex oscillatory functions.

E = El l + Er r

The four component Stokes vector can now be defined.

They are all real numbers and satisfy the relation

I2 = Q2 + U2 + V2

I =ElEl∗+ ErEr

∗=I l + Ir

Q =ElEl∗−ErEr

∗=I l −Ir

U =ElEr∗+ ErEl

∗

V =i (ElEr∗−ErEl

∗)

Is

Qs

Us

Vs

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

M11 M12 M13 M14M21 M22 M23 M24M31 M32 M33 M34M41 M42 M43 M44

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Ii

Qi

Ui

V i

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

The Mueller matrix relates the incident and scattered Stokes vectors

Stokes vector and polarization parameters

I is the radiance (this is what the human eye sees)

Q is the amount of radiation that is polarized in the 0/90 orientationU is the amount of radiation polarized in the +/-450 orientationV is the amount of radiation that is right or left circularly polarizedDOP= Degree of polarization=

Q2 + U2 + V2 / I

DOLP = Degree of linear polarization =

Q2 + U2 / I

DOCP = Degree of circular polarization = |V|/I

Orientation of plane of polarization = tan-1(U/Q)

Ellipticity= Ratio of semiminor to semimajor axis of polarization ellipse=b/a

=tan[(sin-1(V/I))/2]

Polarizer and analyzer settings for Mueller matrix measurements

S

11

S

12

S

13

S

14

S

21

S

22

S

23

S

24

S

31

S

32

S

33

S

34

S

41

(OO) (HO-VO) (PO-MO) (LO-RO)

(OH-OV)(HH-VV) -

(HV+VH)

(PH-MV) -

(PV+MH)

(LH-RV) -

(LV+RH)

(OP-OM) (HP-VM) -

(HM+VP)

(PP-MM) -

(PM+MP)

(LP-RM) -

(LM+RP)

(OL-OR)

S

42

(HL-VR) -

(HR+VL)

S

43

(PL-MR) -

(PR+ML)

S

44

(LL-RR) -

(RL+LR)

Diagram showing the input polarization (first symbol) and output analyzer orientation (second symbol) to determine each element of the SSMM denoted by Sij. For example, HV denotes horizontal input polarized light and a vertical polarization analyzer. The corresponding symbols denoting polarization are V, vertical; H, horizontal; P, +45, M, -45; R, right handed circular polarization; L, left handed circular polarization; and O, open or no polarization optics. This set of measurements was first deduced by Bickel and Bailey.

Plankton as viewed by a squid

Planktonic animal as seen through "regular" visionAs seen when placed between two crossed linear polarizing filters

As seen by putting the two polarizers at 45° to each other

Contrast enhancement using polarization

Photo taken with a flash lamp and no polarization optics

Photo taken with circular polarized light for illumination and a circular analyzer for viewing

Nissan car viewed in mid-wave infrared

This data was collected using an Amber MWIR InSb imaging array 256x256. The polarization optics consisted of a rotating quarter wave plate and a linear polarizer. Images were taken at eight different positions of the quarter wave plate (22.5 degree increments) over 180 degrees. The data was reduced to the full Stokes vector using a Fourier transform data reduction technique.

I Q U V

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Bacillus anthracis

Experimental Setup

Yong-Le Pan, Kevin B. Aptowicz, Richard K. Chang, Matt Hart, and Jay D. Eversole,Characterizing and monitoring respiratory aerosols by light scattering, Optics Letters,Vol. 28, No. 8, p589 (2003).

Coordinate Setup

z

x

y

Laserbeam

€

ˆ x × ˆ y = ˆ z

source

z

x

ydetector

Z’

The symmetry axis is in the yz plane.

Mueller Image Generating Method

€

r =180 −θ,θ ∈ 90,180( ]

We plot the reduced mueller matrix elements on a circular plane shown as in the figure below. Reduced mueller matrix means all mueller elements except M11 are normalized to M11, which are in the range from -1 to +1. For the M11 element, we use the phase matrix P11 value.

M(,)

r

x

y

90

90

-90

-90

In our mueller image generation, radial position has linear relation to the scattering polar angle. The actual experimental detection may not have same relation. The resultant images will be different than the images shown here.

Particle Model-1: SporeOuter coat

Inner coat

Cortex Core

1.0m

0.8m

=0.5m

Philip J. Wyatt, “Differential Light Scattering: a Physical Method for Identifying Living Bacterial Cells”, Applied Optics, Vol.7,No. 10,1879 (1968)

Simulation Models

(a) (b) (c) (d)

1.0μm

0.8μm

1.0 or 2.0 μm

0.5 μm

m=1.34 m=1.34

Homogenous Spore Mueller Image = 0o

m=1.34 1.0m

0.8 m

=0.5m

Spore with Core Mueller Image = 0o

Core 1.0m

0.8 m

=0.5m

Homogenous Cylinder Mueller Image height=1m, =0o

m=1.35

1.0m

0.5m

= 0.532 m

Homogenous Cylinder Mueller Image height = 2m, = 0o

0.5m

= 0.532 m

m=1.35

2.0m

Mueller Image for Three Particles =90o

m=1.34 1.0m

0.8 m

=0.5m

1.0m

0.8 m

0.5m = 0.532 mm=1.35

1.0m

P11 P33 P34 P44

Properties of the TargetThe target reflects light with a Lambertian distribution. The albedo is constant across surface, but each annular region has a different reduced Mueller matrix.R = 0.033

R = 0.067

R = 0.10 m.f.p.s

1 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

2. Depolarizing

1. Polarization Preserving

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

3. Painted Surface1 0 0 0

0 0.61 0 0

0 0 0.58 0

0 0 0 0.51

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥−⎣ ⎦

Geometry of the Imaging System

Target Depth

Target

Incident Beam

Detector field ofview

Detectorresolution

Detector

Medium

Properties of the Model The attenuation coefficient is fixed, but the

absorption and scattering coefficients can be varied. All distances are given in units of photon mean free path (m.f.p.)

The system is assumed to be an active source, so there is full control over all aspects of both source and detector.

The detector details are intentionally left out; only the radiometric quantities at the detector location are calculated.

The problem is solved for a time-independent case.

Inherent Optical Properties of the Medium

Two classes of phase functions are used to describe the scattering by the medium:

Henyey-Greenstein (HG):

Two-Term Henyey-Greenstein (TTH):

2

322

11, ;4

1 2HG

gg

g g

β π

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

−=− +

% cosg

= Θ=

1 2 1 2( , ; , ) ( , ; ) (1 ) ( , ; )TTH HG HGg g g gβ αβ α β = + −% % %

0 1α≤ ≤

The single-scatter reduced Mueller matrix for ocean water is taken to be that of Rayleigh scattering:

2

2 2

1 ( ) 0 0( ) 1 0 0M( )0 0 ( ) 00 0 0 ( )

cos 1 2cos( ) ( )cos 1 cos 1

bb

dd

b d

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

ΘΘΘ =

ΘΘ

Θ− ΘΘ = Θ =Θ+ Θ+

%

Results

The target depth is kept constant at 2 m.f.p.s

The target albedo is kept constant with a value of 0.025

The effective Mueller matrix elements of the system are calculated for different phase functions and single scatter albedos of the medium.

Due to the form of the matrix elements of the target and the backscattering geometry of the imaging system, only the diagonal elements of the effective Mueller matrix are non-zero.

Mueller Matrix ElementsHG g=0.95 phase function Medium Albedo =

0.25

11m

22m%

33m% 44

m%

RadianceHG g=0.5

= 0.5 = 0.25

HG g=0.8 HG g=0.95

= 0.5 = 0.25

= 0.25

= 0.5

= 0.5 = 0.25

= 0.25

= 0.5

TTH SBP TTH LBP

%M22

HG g=0.5

= 0.5 = 0.25

HG g=0.8 HG g=0.95

= 0.5 = 0.25

= 0.25

= 0.5

= 0.5 = 0.25

= 0.25

= 0.5

TTH SBP TTH LBP

%M

33

HG g=0.5

= 0.5 = 0.25

HG g=0.8 HG g=0.95

= 0.5 = 0.25

= 0.25

= 0.5

= 0.5 = 0.25

= 0.25

= 0.5

TTH SBP TTH LBP

%M

44

HG g=0.5

= 0.5 = 0.25

HG g=0.8 HG g=0.95

= 0.5 = 0.25

= 0.25

= 0.5

= 0.5 = 0.25

= 0.25

= 0.5

TTH SBP TTH LBP

Mueller Matrix ElementsMedium Albedo = 0.25

0.0 0.1 0.2 0.3 0.40.000

0.001

0.002

0.003

0.004Radiance

0.0 0.1 0.2 0.3 0.40.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4Radial Distance from Target Center (m.f.p.)

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.0 0.1 0.2 0.3 0.4-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2HG g=0.95HG g=0.8HG g=0.5TTH LBPTTH SBP

%M22

%M33 %M44

%M22

%M33

%M44

0.0 0.1 0.2 0.3 0.40.000

0.002

0.004

0.006

0.008

Radiance

0.0 0.1 0.2 0.3 0.40.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4Radial Distance from Target Center (m.f.p.)

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.0 0.1 0.2 0.3 0.4-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2HG g=0.95HG g=0.8HG g=0.5TTH LBPTTH SBP

%M22

%M33 %M44

Mueller Matrix ElementsMedium Albedo = 0.5

%M22

%M33

%M44

Conclusions

• Only correct way to do radiative transfer calculations• Mueller matrix contains all the elastic scattering information one can obtain from either a single particle or an ensemble of particles• Provides a unique “fingerprint” of particle morphology, orientation, and optical properties

• Can provide information about surface features of objects which can’t be obtained with ordinary radiance measurements

• Can even be used for the detection of precancerous skin lesions