Linear & Circular Polarization; Stokes...
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Linear & Circular Polarization; Stokes Parameters
Brendan ReedPHYS 72111/17/2014

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Outline● Polarization
– Linear– Elliptical– Circular polarization basis
● Stokes Parameters– Linear Basis– Circular Basis
● Summary● Questions

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Polarization

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Linear Polarization● Linearly polarized plane wave
● 2nd plane wave with ε2 linearly independent of ε1
– Combine → general homogeneous plane wave
– propagating in direction of
E1=ϵ1E1ei k⋅x−iwt
E2=ϵ2 E2ei k⋅x−iwt
k=k n̂
E ( x , t )=(ϵ1E1+ϵ2E2)ei k⋅x−iwt

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Linear Polarization Cont.
● Linear Polarization → E1 & E2 have same phase
– angle θ and magnitude E
E ( x , t )=(ϵ1E1+ϵ2E2)ei k⋅x−iwt (1)

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Elliptical Polarization
● Elliptically polarized → E1 & E2 have different phases
● Simplest case → circular polarization– E1 & E2 have same magnitude but differ
in phase by 90⁰
E ( x , t )=E0(ϵ1±i ϵ2)ei k⋅x−iwt

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Elliptical Polarization Cont.
● Let and take real part
k=k ẑ ϵ1=ϵ1 x̂ ϵ2=ϵ2 ŷ
E ( x , t )=E0(ϵ1±i ϵ2)ei k⋅x−iwt
E x(x , t )=E 0 cos(kz−wt )E y( x , t)=∓E0 sin (kz−wt )

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Circular Polarization Basis
● Two circularly polarized waves– complex orthogonal unit vectors
with properties
E ( x , t )=E0(ϵ1±i ϵ2)ei k⋅x−iwt
ϵ+=1
√ 2(ϵ1±i ϵ2)
ϵ+* ⋅ϵ+=0
ϵ+* ⋅ϵ3 =0
ϵ+* ⋅ϵ+=1

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Circular Polarization Basis Cont.
● Using the previous unit vectors and their properties →
– E+ & E are complex amplitudes
● Recall for a linear basis:
E ( x , t )=(E+ ϵ++E ϵ)ei k⋅x−iwt
(2)
E ( x , t )=(ϵ1E1+ϵ2E2)ei k⋅x−iwt (1)

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Stokes Parameters● Polarization is known if it can be written in
the form of either (1) or (2)– With coefficients of (E1,E2) or (E+,E)
● Converse problem? Given the form:
how can we determine, from observing the beam, the state of polarization.
● Use the four Stokes parameters
E ( x , t )=Ε eik n⋅x−iwt
B (x , t )=Β eik n⋅x−iwt

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Stokes Parameters Cont.
● Stokes Parameters– Can be determined through intensity
measurements only in conjunction with a linear polarizer and a quarterwave plate or equivalents
– Their measurement completely determines the state of polarization

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Stokes Parameters Cont.
● For a wave propagating in the zdirection, the scalar products:
– 1st term → linear polarization in xdir– 2nd term → linear polarization in ydir– 3rd term → positive helicity– 4th term → negative helicity
● Squares of amplitudes → intensity of each type of polarization
● Phase information from cross products
ϵ1⋅E ϵ2⋅E ϵ+* ⋅E ϵ
* ⋅E

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Defining Stokes Parameters
● Define Stokes parameters in terms of– a) the projected amplitudes– b) magnitudes & relative phases of the
components● For b), define each of the scalar
coefficients in (1) & (2) as a magnitude times a phase factor
E1=a1ei δ1 E 2=a2e
i δ2
E+=a+ ei δ+ E =ae
i δ

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Stokes Parameters in Linear Basis
s0=∣ϵ1⋅E ∣2+∣ϵ2⋅E ∣
2=a12+a2
2
s1=∣ϵ1⋅E ∣2−∣ϵ2⋅E ∣
2=a12−a2
2
s2=2 ℜ[(ϵ1⋅E)∗(ϵ2⋅E )]=2a1a2 cos(δ2−δ1)s3=2ℑ[(ϵ1⋅E)∗(ϵ2⋅E )]=2a1a2 sin (δ2−δ1)

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Stokes Parameters in Circular Basis
s0=∣ϵ+* ⋅E ∣2+∣ϵ
* ⋅E ∣2=a+2+a
2
s1=2 ℜ[(ϵ+* ⋅E )∗(ϵ
* ⋅E )]=2a+a cos(δ−δ+)s2=2 ℑ[(ϵ+
* ⋅E )∗(ϵ* ⋅E )]=2a+a sin (δ−δ+)
s3=∣ϵ+* ⋅E ∣2+∣ϵ
* ⋅E ∣2=a+2−a
2

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Meaning of Stokes Parameters
● Linear Basis– s0 → relative intensity of the wave
– s1 → xlinear pol. over ylinear pol.
– s2 → phase information
– s3 → phase information
● Circular Basis– s0 → relative intensity of the wave
– s1 → phase information
– s2 → phase information
– s3 → difference in relative intensity of pos. & neg. helicity

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Properties of Stokes Parameters
● Stokes parameters are not independent
● Satisfy the relation:
● Jackson refers us to Section 13.13 of Stone for details of the operational steps to measure the Stokes parameters
a1 a2 δ2−δ1
s02=s1
2+s22+s3
2

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Summary● There are 2 types of polarization
– Linear → Linear Pol. Basis– Elliptical → Circular Pol. Basis
● Stokes parameters completely determine the state of polarization of a wave

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Questions?

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References
[1] Jackson, John David. Classical Electrodynamics. 3rd ed. New York, NY: Wiley, 1999

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Backup Slides

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Helicity
● At a fixed point in space → the electric vector of the fields described by the above eqn. are constant in magnitude but sweep in a circle @ frequency ω
– '+' → counterclockwise → left circularly polarized → positive helicity
– '' → clockwise → right circularly polarized → negative helicity
E ( x , t )=E0(ϵ1±i ϵ2)ei k⋅x−iwt

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Quasimonochromatic Radiation
● Beams of radiation are not monochromatic● Fourier's Theorem → they contain a range
of frequencies & are not completely monochromatic
– the magnitudes & phases (ai δi) vary slowly compared to the frequency ω
– consequence of the averaging process
s2=2 〈a1a2 cos(δ2−δ1)〉
s02⩾s1
2+s22+s3
2

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Astrophysical Example● Study of optical & radiofrequency
radiation from the pulsar in the Crab nebula
– optical light → small linear polarization– radio emission → ω ≈ 2.5 X 109 s1 →
high degree of linear polarization– neither frequency has circular
polarization● This information helps to illuminate the
mechanism of radiation
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