Monte Carlo, Euler and
Quaternions
Jessica C. Ramella-Roman
II Escuela de Optica Biomedica, Puebla, 2011
Stokes vector reference
• Yesterday, Meridian plane
– Three steps
– “Rotate” to scattering plane
• Rotational matrix
– Scatter
• Scattering matrix
– “Rotate” to a new plane
• Rotational matrix
II Escuela de Optica Biomedica, Puebla, 2011
Today Monte Carlo
• Reference frame is a
triplet of unit vectors
• Rotation are about an
axis and follow a
specific order
http://www.grc.nasa.gov/WWW/K-12/airplane
II Escuela de Optica Biomedica, Puebla, 2011
Today Monte Carlo
• Reference frame is a
triplet of unit vectors
• Rotation of the frame is
done in two steps.
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Euler and Quaternion
• The only difference
between today
programs is the way we
handle these rotations
• Euler angles rotation
Quaternions algebra
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Euler Monte Carlo
• The photon polarization reference frame is
tracked at any time via a triplet of unit vectors
that are rotated by an azimuth and scattering
angle according to a predefined order
• Introduced by Bartel et al. (*)
* S. Bartel, A. Hielsher, “Monte Carlo simulations of the diffuse backscattering Mueller
matrix for highly scattering media,” Applied Optics, Vol. 39, No. 10; (2000).
II Escuela de Optica Biomedica, Puebla, 2011
Euler matrices
• Euler's rotation theorem: any rotation may be
described using three angles.
• And three rotation matrices
– About z axis
– About x’ axis
– About z’ axis
Drawings, Wolfram, Mathworld
II Escuela de Optica Biomedica, Puebla, 2011
Euler cnt.
• We only need TWO vectors and TWO rotations
• Two unit vectors v and u
v  [vx , vy , vz ]  [ 0,1,0]
u  [ux , uy , u z ]  [0,0,1]
• The third unit vector is defined by the cross
product of v and u and is calculated only when
a photon reaches a boundary.
II Escuela de Optica Biomedica, Puebla, 2011
Euler cnt.
• Advantage
– Stokes vector is rotated only once for each
scattering event instead of twice as in the
meridian plane method.
– Simple to implement and visualize
• Drawback –
– Gimbal lock - makes the rotation fail for angles
exactly equal to 90˚ (rare event).
II Escuela de Optica Biomedica, Puebla, 2011
Monte Carlo flow chart
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Launch
• The two unit vector v and u
v  [vx , vy , vz ]  [ 0,1,0]
u  [ux , uy , u z ]  [0,0,1]
• v and u define the starting Stokes vector
reference plane.
• The unit vector u represents the direction of
photon propagation.
II Escuela de Optica Biomedica, Puebla, 2011
Move
• The photon is moved a distance Ds, the new
coordinates of the photon are
x x  ux Ds
y y  u y Ds
z z  uz Ds

II Escuela de Optica Biomedica, Puebla, 2011
Drop
• Reduction of photon weight (W)
• According to material albedo
• albedo = ms/(ms+ ma)
• absorbed = W*(1-albedo)
II Escuela de Optica Biomedica, Puebla, 2011
Scatter
• The rejection method establishes the
scattering angle q and azimuth angle f.
• The Stokes vector must be rotated by an angle
f into the scattering plane before scattering
can occur.
II Escuela de Optica Biomedica, Puebla, 2011
Referencing to scattering plane
• Done as in meridian plane Monte Carlo
• Multiply the Stokes vector by matrix R
1
0
0

0 cos(2f ) sin(2f )
R f  
0  sin(2f ) cos(2f )

0
0
0

0

0
0

1

S out  R f S in

II Escuela de Optica Biomedica, Puebla, 2011
Scatter cnt
• The interaction with a spherical particle is
achieved by multiplying the Stokes vector with
a scattering matrix M(q)
s11( q ) s12 ( q )
0
0 


s ( q ) s11( q )
0
0 
M q   12
 0
0
s33 ( q ) s34 ( q )


0
0
s
(
q
)
s
(
q
)


34
33


S out  M q S in

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M(q) elements
• s11 , s12 , s33 , s34
calculated with Mie
theory
2
2
1
s11( q )  [ S2 ( q )  S1( q ) ]
2
2
2
1
s12 ( q )  [ S2 ( q )  S1( q ) ]
2
1
s33 ( q )  [ S2* ( q )S1( q )  S2 ( q )S1* ( q )]
2
i
s34 ( q )  [ S2* ( q )S1( q )  S2 ( q )S1* ( q )]
2
• Expressed as
• S1, S2* – Mie scattering

*http://omlc.ogi.edu/calc/mie_calc.html
C. Bohren and D. R. Huffman,
Absorption and scattering of light by small
particles, (Wiley Science Paperback Series,1998).
II Escuela de Optica Biomedica, Puebla, 2011
Reference frame
• After scattering the Stokes vector the
reference coordinate system v u must be
updated.
• The two rotations, for the angles f and q can
be obtained using Euler’s rotational matrices.
II Escuela de Optica Biomedica, Puebla, 2011

Rotational matrix ROT
• Rodrigues’s rotational matrix ROT
• Accomplishes the general case of rotating any
vector by an angle y about an axis K
 k x k x v  c k y k x v  k z s k z k x v  k ys 


ROT ( K ,y )  k x k y v  k z s k y k y v  c k y k z v  k x s 

k x k z v  k y s k y k z v  k x s k z k z v  c 

• Where K is the rotational axis,
• c=cos(y), s=sin(y) and v=1-cos(y).
II Escuela de Optica Biomedica, Puebla, 2011
Vector rotation
• First the unit vector v is
rotated about the
vector u by an angle f
• Multiply v by the
rotational matrix
ROT(u,f); u remains
unchanged
II Escuela de Optica Biomedica, Puebla, 2011
Vector rotation
• Second u is rotated
about the newly
generated v by an angle
q.
• This is done multiplying
the unit vector u by the
rotational matrix
ROT(v,q).
II Escuela de Optica Biomedica, Puebla, 2011
Update of direction cosines
• This is done as in
standard Monte Carlo
• q and f
If |uz| ≈ 1
^
u x  sin( q ) cos( f )
^
u y  sin( q ) sin( f )
^
u z  cos( q )
uz
uz
II Escuela de Optica Biomedica, Puebla, 2011
Update of direction cosines
If |uz| ≠ 1
^
ux 
1
1 u2
sin(q )( ux u y cos(f )  u y sin(f )) ux cos(q )
z
^
uy 
1
1 u2
sin(q )( ux u z cos(f )  ux sin(f )) u y cos(q )
z
^
u z  1 u2 sin(q ) cos(f )( u y u z cos(f )  ux sin(f )) u z cos(q )
z
II Escuela de Optica Biomedica, Puebla, 2011
Photon life
• The life of a photon ends when the photon
passes through a boundary or when its weight
W value falls below a threshold.
• Roulette is used to terminate the photon
packet when W  Wth.
– Gives the photon packet one chance of surviving
– If the photon packet does not survive the roulette,
the photon weight is reduced to zero and the
photon is terminated.
II Escuela de Optica Biomedica, Puebla, 2011
Boundaries
• Two final rotations of the Stokes vector are
necessary to put the photon status of
polarization in the detector reference frame.
• This will be achieved with matrix
multiplication of two rotational matrix by
stokes vector
S final  R( y )R(  )S
II Escuela de Optica Biomedica, Puebla, 2011
Boundaries – R()
• w is reconstructed as
the cross product of v
and u.
w  vu
• An angle  is needed to
rotate the Stokes vector
into a scattering plane
II Escuela de Optica Biomedica, Puebla, 2011
Boundaries – R()
  0 when vz  0 and uz  0
vz
  tan (
) in all other cases
wz
1

• This is the ONLY reason we need the vector u,
v, and w
II Escuela de Optica Biomedica, Puebla, 2011
Boundaries – R()
• This rotation is about
the direction of
propagation of the
photon, i.e. the axis u
• Resulting position of v
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Boundaries – R(y)
• A second rotation of an
angle y about the Z axis
• Put the photon
reference frame in the
detector reference
frame
II Escuela de Optica Biomedica, Puebla, 2011
Boundaries – R(y)
• Transmission
uy
1
y  tan (
ux
)
• Reflection
y  tan 1(
uy
ux
)

II Escuela de Optica Biomedica, Puebla, 2011
Boundaries End
• Stokes vector in the detector frame of
reference
S final  R( y )R(  )S
• We need vector u, v, and w to obtain these
angles
II Escuela de Optica Biomedica, Puebla, 2011
Quaternion Monte Carlo
• Quaternion Monte Carlo simply uses
Quaternion Algebra to handle vector rotation.
• Advantage
– Stokes vector is rotated only once for each
scattering event instead of twice as in the
meridian plane method.
– No issue with Gimbal lock
– Optimized for computer simulations
II Escuela de Optica Biomedica, Puebla, 2011
Quaternions
• A quaternion is a 4-tuple of real numbers; it is
an element of R4.

q  qo ,q1 ,q2 ,q3

• Quaternion can also be defined as the sum of
a scalar part q0 and a vector part Q in R3 of the
form
q  q0  Q  q0  iq1  jq2  kq3
II Escuela de Optica Biomedica, Puebla, 2011
Quaternions cnt.
• The vector part Q is the rotational axis and the
scalar part is the angle of rotation.
• Multiplication of a vector t by the quaternion
is equivalent to rotating the vector t around
the vector Q of an angle q0.
II Escuela de Optica Biomedica, Puebla, 2011
Vector rotation
• First the unit vector v is
rotated about the
vector u by an angle f
II Escuela de Optica Biomedica, Puebla, 2011
Rotation - f
• The first rotation is about the vector u by an
angle f.
• This is done generating the quaternion qf
qf  f  u  f  iux  juy  kuz
• And then using the quaternion operator
• q∗ vq on the vector v

• q∗ is the complex conjugate of q
II Escuela de Optica Biomedica, Puebla, 2011
Vector rotation
• Second u is rotated
about the newly
generated v by an angle
q.
II Escuela de Optica Biomedica, Puebla, 2011
Rotation - q
• The second rotation is about the vector v by
an angle q.
• This is done generating the quaternion qq
qf  q  v  f  ivx  jv y  kvz
• And then using the quaternion operator
• qq*uqq on the vector u

II Escuela de Optica Biomedica, Puebla, 2011
Rotations
• These steps are repeated for every scattering
event.
• At the boundaries the last aligning rotations
are the same as in Euler Monte Carlo.
II Escuela de Optica Biomedica, Puebla, 2011
Testing
• Comparison with Evans Code
• In 1991 Evans designed an adding doubling code
that can calculate both the radiance and flux of a
polarized light beam exiting the atmosphere
• plane parallel slab of thickness L = 4/µs,
• absorption coefficient µa=0,
• unpolarized incident beam
• wavelength l = 0.632 nm.
II Escuela de Optica Biomedica, Puebla, 2011
Evans - Reflectance mode
Diameter (nm)
Evans
This code
Evans
This code
I
I
Q
Q
10
0.6883
0.6886
-0.1041
-0.1042
100
0.6769
0.6769
-0.1015
-0.1009
1000
0.4479
0.4484
0.0499
0.0499
2000
0.2930
0.2931
0.0089
0.0088
Incident I=[1 0 0 0]
Reflectance mode, comparison between Evans adding-doubling code and the
meridian plane Monte Carlo program. The results do not include the final rotation
for a single detector.
II Escuela de Optica Biomedica, Puebla, 2011
Evans – Transmission mode
Diameter (nm)
Evans
This code
Evans
This code
I
I
Q
Q
10
0.33126
0.31133
-0.01228
-0.01233
100
0.32301
0.32305
-0.12844
-0.12770
1000
0.55201
0.55152
0.02340
0.02348
2000
0.70698
0.70689
0.01197
0.01205
Incident I=[1 0 0 0]
Transmission mode, comparison between Evans adding doubling code and the
meridian plane Monte Carlo program. The results are not corrected for a single
detector
II Escuela de Optica Biomedica, Puebla, 2011
How to run the code
• Download the code
• http://faculty.cua.edu/ramella/MonteCarlo/in
dex.html
II Escuela de Optica Biomedica, Puebla, 2011
How to run the code
• Download the code
• http://faculty.cua.edu/ramella/MonteCarlo/in
dex.html
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Download the code
Programs
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Make
Command line
Compilation/linking step
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Run
Command line
This program will run a full Mueller matrix Monte Carlo launching four vectors
[1 1 0 0] H 0 degree polarized
[1 -1 0 0] V 90 degrees polarized
[1 0 1 0] P 45 degrees polarized
[1 0 0 1] R –Right circular
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Memory usage
• Intel (64-bit) – Macbook pro 2.3Ghz I7 8GB
RAM
• Real mem 860 KB
• Virtual mem 17 MB
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Output is a set of Stokes vector
images for each launched Stokes
vector
outHI -> Horizontal incident
polarization and I portion of the
Stokes vector
outHQ -> Horizontal incident
polarization and Q portion of the
Stokes vector
out
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Inside the code
GNU Licence
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Modifiable parameters
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Image related parameters
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Stokes vector launch
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Launch terms
Initial position
Initial orientation
Initial Stokes vector
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We can use the three Monte
Carlo programs to understand
how polarized light travels into
scattering media
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Light transfer –mono disperse particles
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Light transfer –mono disperse particles
• Experimental results
– black circles
• Monte Carlo
– black square
• Diameter 482 nm,
• Wavelength was 543
nm.
• nsphere =1.59
II Escuela de Optica Biomedica, Puebla, 2011
Light transfer –mono disperse particles
• Experimental results
– black circles
• Monte Carlo
– black square
• Diameter 308 nm,
• Wavelength was 543
nm.
• nsphere =1.59
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Poly-disperse solutions
• Gamma distribution of
particles
N (r )  ar exp(br  )

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Poly-disperse solutions
Comparison with Evans
Incident I=[1 0 0 0]
Results from a slab in reflectance
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Poly-disperse solutions -DLP
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Map of polarized light distribution
d
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Map of polarized light distribution
d
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Map of polarized light distribution
d
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d
d
d
y
x
D = 0.01 µm
µs = 0.89 cm-1
l = 0.543µm
d
x
y
II Escuela de Optica Biomedica, Puebla, 2011
d
d
y
x
D = 0.01µm
µs = 0.89 cm-1
l = 0.543µm
y
y
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Puebla, 2011
d
d
d
y
x
D = 0.48 µm
µs = 0.89 cm-1
l = 0.543µm
d
x
y
II Escuela de Optica Biomedica, Puebla, 2011
d
d
y
x
D = 0.48 µm
µs = 0.89 cm-1
l = 0.543µm
y
y
II Escuela de Optica Biomedica,
Puebla, 2011
D = 0.01 µm
µs = 0.89 cm-1
l = 0.543µm
Isosurface Pol= 0.5
d
x
y
II Escuela de Optica Biomedica, Puebla, 2011
D = 0.48 µm
µs = 0.89 cm-1
l = 0.543µm
Isosurface Pol = 0.9
d
x
y
II Escuela de Optica Biomedica, Puebla, 2011
Poincaré sphere analysis, Ellipticity
I  E 2x  E y2
Im()
Q  E 2x  E y2  I cos(2)cos(2f )
U  2E x E y cos()  I cos(2)sin( 2f )
V  2E x E y sin( )  I sin( 2)
+i
-1
+1
Re()

-i
tan( f )  itan( )

1 itan( )tan( f )
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Ellipticity
l = 543nm
2 scattering events
19 scattering events
0.01µm
II Escuela de Optica Biomedica, Puebla, 2011
10 µm
Particle effect on polarization
Small spheres
• Depolarize faster
• Less forward directed
• Turn incident
polarized light into
another linear state
Large spheres
• Depolarize slower
– Large g effect
• More forward directed
• Turn incident
polarized light into
elliptical states
II Escuela de Optica Biomedica, Puebla, 2011