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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 quarter-wave plate or equivalents

– Their measurement completely determines the state of polarization

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Stokes Parameters Cont.

● For a wave propagating in the z-direction, the scalar products:

– 1st term → linear polarization in x-dir– 2nd term → linear polarization in y-dir– 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 -=a-e

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

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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-

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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 → x-linear pol. over y-linear 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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Quasi-monochromatic 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 s-1 →

high degree of linear polarization– neither frequency has circular

polarization● This information helps to illuminate the

mechanism of radiation

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