PolarizationBasics•The equationsrepresenta pairof planewaves: the twocomponentsof the electricalfieldof anEM wavepropagatingin the z direction, notnecessarilymonochromatic.•The amplitudes

Polarization Basics

• The equations

represent a pair of plane waves: the two components of the electrical field of an EM wave propagating in the z direction, not necessarily monochromatic.

• The amplitudes E_ox,y(t) and phasesδ_x,y(t) fluctuate slowly with respect to the rapid oscillation of the carrier cos(ωt).

⎩⎨

⎧

+

=

+

=

)]

( cos[

) ( ) (

)]

( cos[

) ( ) (

t t t E t E

t t t E t E

y oy

y

x ox

x

δ ω

δ ω

z E_x

Ey E

Polarization Basics

If we eliminate the term cos(ωt) between the two equations, and defineδ(t)= δ_y(t)-δ_x(t), we find the polarization ellipse (valid in general at a given time), which is the locus of points described by the optical field as it propagates:

⎩⎨

⎧

+

=

+

=

)]

( cos[

) ( ) (

)]

( cos[

) ( ) (

t t t E t E

t t t E t E

y oy

y

x ox

x

δ ω

δ ω

) ( sin ) ( )cos ( ) (

) ( ) ( 2 ) (

) ( ) (

)

( 2

2 2

2 2

t t t

E t E

t E t E t E

t E t E

t E

oy oy

y x oy

y ox

x + −

δ

=

δ

fast fast fast

slow slow slow

slow slow

t ) ( ) ( ) (t E t E t

E x y

r r

r = +

z E_x

Ey E

Polarization Basics

• For purely monochromatic waves, amplitudes and phases must be constant with time:

And the polarization ellipse is also constant:

⎩⎨

⎧

+

=

+

=

] cos[

) (

] cos[

) (

y oy

y

x ox

x

t E t E

t E t E

δ ω

δ ω

δ

δ

²

2 2

2 2

sin )cos ( ) ( 2 ) ) (

( + − =

oy ox

y x oy y ox x

E E

t E t E E

t E E

t E

fast fast fast

t ) ( ) ( ) (t E t E t

E x y

r r

r = +

z E_x

E_y E

Polarization Basics

• In general a beam of light is “elliptically polarized”.

• The polarization ellipse degenerates to special forms for special values of the amplitudes and of the phases.

• Linear polarized waves: when the ellipse collapses to a line, i.e. whenδ=0,π. The direction of the E vector remains constant.

• Circularily polarized waves: when the ellipse reduces to a circle, i.e. when δ=π/2, 3π/2 and E_ox=Eoy=Eo.

E x y

E x y

E x y

Polarization Basics

• The polarization ellipse is specified by the amplitude parameters Eox,Eoy,δ.

• But it can be expressed equivalently by the elliptical parameters:

• Orientation angleψ:

• Ellipticity angle χ :

• For linearly polarized light χ=0.

ψ b a

x y’

x’

y

2 2

cos 2 2

tan

oy ox

oy ox

E E

E E

= − δ

ψ

a

±b χ= tan

2 2

sin 2 2

sin

oy ox

oy ox

E E

E E

= + δ

χ

Stokes Parameters

• Our detectors are too slow to follow the time evolution of the EM field. What we can measure are time averages, over periods much longer than 2π/ω.

• Due to the periodicity of the EM waves, it is enough to compute time averages over a single period of oscillation. These are represented by the symbol <…>.

• So we take the time average of the polarization ellipse:

δ

δ

²

2 2

2 2

sin ) cos ( ) ( 2 ) ( )

( + − =

oy ox

y x oy y ox x

E E

t E t E E

t E E

t E

Stokes Parameters

• Multiplying by 4Eox2Eoy2we find

• Since E_x(t) and E_y(t) are sine waves, we can compute their time averages and substitute above:

• Since we want to express this in terms of intensities, we can add and subtract Eox4+Eoy4:

2 2

2 2

2

) sin 2 ( cos ) ( ) ( 2 8

) ( 4 ) ( 4

δ

δ

_{ox oy}

y x oy ox

y ox x

oy

E E t

E t E E E

t E E t E E

=

−

+ +

2 2

2 2 2 2

) sin 2 ( ) cos 2

( 2 2

δ

δ

_{ox oy}

oy ox

oy ox ox oy

E E E

E

E E E E

=

−

+ +

Stokes Parameters

• We find

• We define the Stokes Parameters:

• so that our equation reduces to

2 2

2 2 2 2 2 2

) sin 2 ( ) cos 2 (

) ( ) (

δ

δ

_{ox oy}

oy ox

oy ox ox oy

E E E

E

E E E E

=

−

+

−

− +

δ δ sin 2

cos 2

3 2

2 2 1

2 2

oy ox

oy ox

oy ox

ox oy o

E E S

E E S

E E S

E E S

=

=

−

= +

=

2 3 2 2 2 1

2 S S S

S_o

= + +

Stokes Parameters

• If light is not purely monochromatic, the amplitudes and phases fluctuate with time.

• It can be shown that, in general,

• The = sign is valid for fully polarized light, while the > sign is valid for partially polarized or unpolarized light. P=degree of polarization:

• The intensity is related to So:

• The orientation of the polarization ellipse is related to S1and S2:

• The ellipticity of the polarization ellipse is related to S3:

2 3 2 2 2 1

2 S S S

S_o≥ + +

1 2 2 2

cos 2 2

tan S

S E E

E E

oy ox

oy

ox =

= − δ

ψ

o oy ox

oy ox

S S E E

E

E ₃

2 2

sin 2 2

sin =

= + δ

χ

δ δ sin 2

cos 2

3 2

2 2 1

2 2

oy ox

oy ox

oy ox

ox oy o

E E S

E E S

E E S

E E S

=

=

−

= +

=

2 2

ox oy

o E E

S = +

1 0

2 3 2 2 2 1

≤

≤

+

= +

=

P S

S S S I P I

o total

pol

Stokes Parameters

• Note that, for linear polarized light (δ=0), both parameters S₁ and S2represent the difference in intensity carried by two orthogonal components:

• S1is the difference in intensity between the components along axis x and y

• S₂is the difference in intensity between the components along two axisx’and y’rotated 45^o with respect to x and y.

x y

x y

y’

x’

2 2

1 Ex Ey

S = −

2 ' 2

' ' ' ' '

2 2

1 2 1 2 1 2

2 1

2E_xE_y E_x E_y E_x E_y E_x E_y

S = −

⎥⎦⎤

⎢⎣⎡ +

⎥⎦⎤

⎢⎣⎡ −

=

=

Ex

Ey

E_x’

E_y’ 45^o

Stokes Parameters: examples

• Unpolarized light:

δ=random

<Eox2>=<Eoy2>=Io

• Linearly polarized light:

– Horizontal(Eoy=0) Vertical (Eox=0) +45^o (Eoy= Eoy; δ= 0) θ^o

• Circular polarized light:

– Left Right

δ δ sin 2

cos 2

3 2

2 2 1

2 2

oy ox

oy ox

oy ox

ox oy o

E E S

E E S

E E S

E E S

=

=

−

= +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= 0 0 0 1 2I_o Sr

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= 0 0 1 1

Io

Sr

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= − 0 0 1 1

Io

Sr

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= 0 1 0 1

Io

Sr

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= 0

2 sin

2 cos

1

θ θ Io

Sr

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

= 1 0 0 1

Io

Sr

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= 1 0 0 1

Io

Sr

Stokes Parameters

• The waves can be represented as complex functions:

• This helps in the time-averaging process needed to compute the Stokes Parameters. They can be rewritten as follows (Stokes vector):

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

− +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

− +

− +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

δ δ sin 2

cos 2 ) (

2 2

2 2

*

*

*

*

*

*

*

*

3 2 1

oy ox

oy ox

oy ox

oy ox

x y y x

x y y x

y y x x

y y x x o

E E

E E

E E

E E

E E E E i

E E E E

E E E E

E E E E

S S S S

⎩⎨

⎧

+

= +

=

+

= +

=

)]

( exp[

] cos[

) (

)]

( exp[

] cos[

) (

y oy

y oy

y

x ox

x ox

x

t i E t E t E

t i E t E t E

δ ω δ

ω

δ ω δ

ω

Stokes Parameters

• The Stokes vector can also be expressed in terms of S_o, ψ, χ.

• From

• And from

• Using

we find S1, so we have:

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

χ ψ χ

ψ χ

2 sin

2 sin 2 cos

2 cos 2 cos

1

3 2 1

o o

S S S S S

δ ψ

ψ 2 cos we can write tan2 2

tan _{2 2} S₂ S₁

E E

E E

oy ox

oy

ox =

= −

δ χ

χ 2 sin we can write sin2 2

sin 2 2 3 o

oy ox

oy

ox S S

E E

E

E =

= +

2 3 2 2 2 1

2 S S S

S_o = + +

Poincare’

Classical measurement of the Stokes Parameters

• The measurement of the 4 Stokes Parameters needs two optical components:

– A retarder (wave plate): it is a phase-shifting element, whose effect is to advance the phase of the x component byφ/2and to retard the phase of the y component by -φ/2. So the field emerging from the retarder is E’_x= E_xe^{i φ/2}and E’_y= E_ye^{-i φ/2} – A polarizer. The optical field can pass only along one axis, the

transmission axis. So the total field emerging from the polarizer is E”=E’_xcosθ+E’_ysinθ, where E’ is the incident field and θ is the angle of the transmission axis.

• So the beam arriving on the detector is E”=E_x e^{i φ/2}cosθ+E_ye^{-i φ/2}sinθ

φ θ

source retarder polarizer

detector

Classical measurement of the Stokes Parameters

• E”=Ex e^{i φ/2}cosθ+Eye^{-i φ/2}sinθ

• The detector measures its intensity, i.e. I= E”E”*

• So we get

• Which can be rewritten using the half-angle formulas:

φ θ

source retarder polarizer

detector

θ θ θ

θ

θ θ

φ ϑ

φ

φsin cos sin cos

sin cos

) , (

*

*

2

* 2

*

i y x i

y x

y y x x

e E E e

E E

E E E E I

+ +

+ +

=

−

( ) ( )

( ) ( )

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− + +

+

− +

= +

θ φ θ

φ φ θ

ϑ cos sin2 sin sin2

2 cos )

,

( _{* * * *}

*

*

*

* 2 1

x y y x x

y y x

y y x x y y x x

E E E E i E

E E E

E E E E E E E E I

2 2 sin 2

2 cos 1 2 2

2 cos 1

2 sin sin cos

cosθ=^{+ θ} θ=^{− θ} θ θ= ^θ

[

θ φ θ φ θ

]

φ

ϑ, ) cos2 cos sin2 sin sin2

( ₂¹S S₁ S₂ S₃

I = _o+ + +

Classical measurement of the Stokes Parameters

• This is the formula derived in 1852 by Sir George Gabriel Stokes.

• The first three parameters can be measured by removing the retarder (φ=0) and measuring the intensity with three orientations of the polarizerθ=0^o,45^o,90^o:

• The fourth parameter can be measured by inserting a 90^oretarder (quarter wave plate):

φ θ

source retarder polarizer

detector

[

θ φ θ φ θ

]

φ

ϑ, ) cos2 cos sin2 sin sin2

( 2 1 2 3

1S S S S

I = o+ + +

[ ]

[ ]

[ ]

[ ]

⎪⎪

⎩

⎪⎪

⎨

⎧

+

=

−

= +

= +

=

2 3 1 2 1 1 2 2 1 2 1 1

) 90 , 45 (

) 0 , 90 (

) 0 , 45 (

) 0 , 0 (

S S I

S S I

S S I

S S I

o o o

o o o

o o o

o o o

⎪⎪

⎩

⎪⎪

⎨

⎧

−

−

=

−

−

=

−

= +

=

) 0 , 90 ( ) 0 , 0 ( ) 90 , 45 ( 2

) 0 , 90 ( ) 0 , 0 ( ) 0 , 45 ( 2

) 0 , 90 ( ) 0 , 0 (

) 0 , 90 ( ) 0 , 0 (

3 2

1

o o o o o o

o o o o o o

o o o o

o o o o o

I I I

S

I I I S

I I S

I I S

Classical measurement of the Stokes Parameters

• The great advantage of the Stokes Parameters is that they are observable. The polarization ellipse is not (too fast).

• Moreover, the Stokes parameters can be used to describe unpolarized light: light which is not affected by the rotation of a polarizer or by the presence of a retarder. Stokes was the first one to describe mathematically unpolarized and partially polarized light.

• It is evident from Stokes formula that, for unpolarized light, S₁=S₂=S₃=0, while S_o>0.

• The fully polarized light had

• The intermediate state is partially polarized light, where

φ θ

source retarder polarizer

detector

[

θ φ θ φ θ

]

φ

ϑ, ) cos2 cos sin2 sin sin2

( 2 1 2 3

1S S S S

I = o+ + +

2 3 2 2 2 1

2 S S S

S_o= + +

2 3 2 2 2 1

2 S S S

S_o≥ + +

Partially polarized light

• The Stokes parameters of a combination of independent waves are the sums of the respective Stokes parameters of the separate waves.

• If we combine a fully polarized wave with an

independent, unpolarized one, we find partially polarized light.

• This expression will be useful in the following.

1 0

2 3 2 2 2

1+ + ≤ ≤

=

= P

S S S S I P I

o total

pol

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛ +

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

=

3 2 1

3 2 1

0 0 ) 0 1 (

S S S S P S P S S S S S

o o o

Polarization-active optical components

• When a beam of light interacts with matter its polarization state is almost always changed.

• It can be changed by

– changing the amplitudes – changing the phases – changing the directions

of the orthogonal field components.

• Their effect can be described by means of the Mueller matrices: M is a 4x4 matrix such that the emerging Stokes vector is S’=M S .

Polarizer (Diattenuator) Rotator

Wave-plate (Retarder)

1) Polarizer or Diattenuator

• It attenuates the orthogonal components of an optical beam unequally:

• Using the definitions of S and S’

• And inserting the expressions for E’

we get

⎪⎩

⎪⎨

⎧

=

=

y y y

x x x

E p E

E p E

' '

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

− +

− +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

) ( ^{' '* ' '*}

'*

' '*

' '*

' '*

' '*

' '*

'

' 3 ' 2 ' 1 '

x y y x

x y y x

y y x x

y y x x o

E E E E i

E E E E

E E E E

E E E E

S S S S

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

− +

− +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

)

( ^{* *}

*

*

*

*

*

*

3 2 1

x y y x

x y y x

y y x x

y y x x o

E E E E i

E E E E

E E E E

E E E E

S S S S

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

+

−

− +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

3 2 1 2

2 2 2

2 2 2 2

' 3 ' 2 ' 1 '

2 0 0 0

0 2 0 0

0 0

0 0

2 1

S S S S

p p p p p p p p

p p p p

S S S

S _o

y x y x y x y x

y x y x o

Special cases

• If the diattenuator is simply an attenuator, i.e. if p_x=p_y=p we have a neutral density filter:

• If the Polarizer is ideal and horizontal, i.e. if p_y=0 we have

• If the Polarizer is ideal and vertical, i.e.

if p_x=0 we have

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

3 2 1 2

' 3 ' 2 ' 1 '

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

S S S S

p

S S S

So o

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

3 2 1 2

' 3 ' 2 ' 1 '

0 0 0 0

0 0 0 0

0 0 1 1

0 0 1 1

2

S S S S p

S S S

S o

x o

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛ +

−

− +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

3 2 1 2

2 2 2

2 2 2 2

' 3 ' 2 ' 1 '

2 0 0 0

0 2 0 0

0 0

0 0

2 1

S S S S

p p p p p p p p

p p p p

S S S

S o

y x y x y x y x

y x y o x

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

−

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

3 2 1 2

' 3 ' 2 ' 1 '

0 0 0 0

0 0 0 0

0 0 1 1

0 0 1 1

2

S S S S p

S S S

S o

y o

Polarizer:

• The characteristics of the polarizer p_xand p_y can be rewritten in terms of new parameters p and α:

• With these parameters the Mueller matrix of a polarizer is:

• An ideal polarizer converts any incoming beam into a linearly polarized beam:

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛ +

−

− +

=

y x y x y x y x

y x y x

P

p p p p p p p p

p p p p M

2 0 0 0

0 2 0 0

0 0

0 0

2

1 ^{2 2 2 2}

2 2 2 2

sin cos

⎪⎩

⎪⎨

⎧

=

= α α p p

p p

ydef xdef

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

=

α α α

α

2 sin 0 0 0

0 2 sin 0 0

0 0 1 2 cos

0 0 2 cos 1

2 p2

MP

( )

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

± ±

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

±

±

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

0 0 1 1

2 1

0 0 0 0

0 0 0 0

0 0 1 1

0 0 1 1

2 1

1

3 2 1

' 3 ' 2 ' 1 '

S S S S S S

S S S S

o o o

2) Retarder

• It introduces a phase shift between the orthogonal components of an optical beam :

• Using the definitions of S and S’

• And inserting the expressions for E’

we get

⎪⎩

⎪⎨

⎧

=

=

− +

) ( ) (

) ( ) (

2 / '

2 / '

t E e t E

t E e t E

y i y

x i x

ϕ ϕ

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

− +

− +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

) ( ^{' '* ' '*}

'*

' '*

' '*

' '*

' '*

' '*

'

' 3 ' 2 ' 1 '

x y y x

x y y x

y y x x

y y x x o

E E E E i

E E E E

E E E E

E E E E

S S S S

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

− +

− +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

)

( ^{* *}

*

*

*

*

*

*

3 2 1

x y y x

x y y x

y y x x

y y x o x

E E E E i

E E E E

E E E E

E E E E

S S S S

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

3 2 1

' 3 ' 2 ' 1 '

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1

S S S S

S S S

So o

ϕ ϕ

ϕ ϕ

Special cases

• If the retarder is a quarter- wave plate (φ=90^o):

• Such a retarder converts a +45^olinearly polarized beam into a right/left circularly polarized beam:

• If the retarder is a half-wave plate (φ=180^o):

• This reverses the ellipticity and orientation of the incomin polarization state.

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= −

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

3 2 1

' 3 ' 2 ' 1 '

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1

S S S S

S S S

So o

ϕ ϕ

ϕ ϕ

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= −

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

3 2 1

' 3 ' 2 ' 1 '

0 1 0 0

1 0 0 0

0 0 1 0

0 0 0 1

S S S S

S S S

S_{o o}

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟±

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= −

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

± 0

1 0 1

0 1 0 0

1 0 0 0

0 0 1 0

0 0 0 1

1 0 0 1

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

= −

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

3 2 1

' 3 ' 2 ' 1 '

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

S S S S

S S S

So o