polarized and unpolarised transverse waves · polarized and unpolarised transverse waves t. johnson...

13-02-15 Electromagnetic Processes In Dispersive Media, Lecture 6 1

Polarized and unpolarised transverse waves

T. Johnson


Outline

•  The quarter wave plate •  Set up coordinate system suitable for transverse waves •  Jones calculus; matrix formulation of how wave polarization

changes when passing through polarizing component –  Examples: linear polarizer, quarter wave plate, Faraday rotation

•  Statistical representation of incoherent/unpolarized waves –  Stokes vector and polarization tensor

•  Poincare sphere

•  Muller calculus; matrix formulation for the transmission of partially polarized waves


Modifying wave polarization in a quarter wave plate (1) •  Last lecture we noted that in birefringent crystals:

–  there are two modes: O-mode and X-mode

–  thus if then the O-mode has larger phase velocity

•  Next describe Quarter wave plates –  uniaxial crystal; normal in z-direction –  length L in the x-direction:

–  Let a wave travel in the x-direction, then k is in the x-direction and θ=π/2

€

nO2 = K⊥

nX2 =

K⊥K||

K⊥sin2 θ+K|| cos

2 θ

⎧

⎨ ⎪

⎩ ⎪

€

eO (k) = 0 , 1 , 0( )

eX (k)∝ K|| cosθ , 0 , K⊥ sinθ( )⎧ ⎨ ⎪

⎩ ⎪

€

K⊥ > K||

€

nO ≥ nX

€

nO2 = K⊥

nX2 = K||

⎧ ⎨ ⎪

⎩ ⎪

€

eO (k) = 0 , 1 , 0( )eX (k) = 0 , 0 , 1( )

⎧ ⎨ ⎩ €

L =cω

π /2K|| − K⊥


Modifying wave polarization in a quarter wave plate (2) •  Plane wave ansats has to match dispersion relation

–  when the wave entre the crystal it will move slow, this corresponds to a change in wave length, or k

–  since the O- and X-mode travel at different speeds we write

•  Assume as initial condition a linearly polarized wave

–  the difference in wave number causes the O- and X-mode to drift in and out of phase with each other!

€

E t,x( ) =ℜ eOEO exp(ikOx − iωt) + eX EX exp(ikX x − iωt){ }€

kO =ωnOc

=ωc

K⊥ , kX =ωnXc

=ωc

K||

€

E = 1 1 0[ ] = eO + eX( )⇒ EO = EX =1

⇒ E t,x( ) =ℜ eO exp(ikOx − iωt) + eX exp(ikX x − iωt){ }= eO cos(kOx −ωt) + eX cos(kOx −ωt + Δkx) , Δk ≡ kX − kO


Modifying wave polarization in a quarter wave plate (3) •  The polarization when the wave exits the crystal at x=L

–  This is cicular polarization! –  The crystal converts linear to circular polarization (and vice versa) –  Called a quarter wave plate; a common component in optical systems –  But work only at one wave length – adapted for e.g. a specific laser! –  In general, waves propagating in birefringent crystal change polarization

back and forth between linear to circular polarization –  Switchable wave plates can be made from liquid crystal

•  angle of polarization can be switched by electric control system

•  Similar effect is Faraday effect in magnetoactive media –  but the eigenmode are the circularly polarized components

€

E t,x( ) = eO cos(kOL −ωt) + eX cos(kOL −ωt + ΔkL)[ ] , Δkx = π /2

= eO cos(kOL −ωt) − eX sin(kOL −ωt)[ ]

€

L =cω

π /2K|| − K⊥


Optical systems •  In optics, interferometry, polarometry, etc, there is an interest

in following how the wave polarization changes when passing through e.g. an optical system.

•  For this purpose two types of calculus have been developed; –  Jones calculus; only for coherent (polarized) wave –  Muller calculus; for both coherent, unpolarised and partially polarised

•  In both cases the wave is given by vectors E and S (defined later) and polarizing elements are given by matrixes J and M

€

Eout = J • Ein

Sout = M • Sin


The polarization of transverse waves

•  Lets first introduce a new coordinate system representing vectors in the transverse plane, i.e. perpendicular to the k. –  Construct an orthonormal basis for {e1,e2,κ}, where κ=k/|k| –  The transverse plane is then given by {e1,e2}, where

–  where α=1,2 and ei , i=1,2,3 is any basis for R3 –  denote e1 the horizontal and e2 the vertical directions

•  The electric field then has different component representations: Ei (for i=1,2,3) and Eα (for α=1,2)

–  similar for the polarization vector, eM

€

Ei = eiαEα

€

eα = eiαe i

€

eM ,i = eiαeM

α

k

e1

e2

The new coordinates provides 2D representations


Some simple Jones Matrixes

•  In the new coordinate system the Jones matrix is 2x2:

•  Example: Linear polarizer transmitting horizontal polarization

•  Example: Attenuator transmitting a fraction ρ of the energy –  Note: energy ~ ε0|E|2

€

Jαβ =J11 J12J21 J22

⎡

⎣ ⎢

⎤

⎦ ⎥

Eoutα = JαβEin

β

€

JαβL,V =1 00 0⎡

⎣ ⎢

⎤

⎦ ⎥ →

1 00 0⎡

⎣ ⎢

⎤

⎦ ⎥ EH

EV

⎡

⎣ ⎢

⎤

⎦ ⎥ =

EH

0⎡

⎣ ⎢

⎤

⎦ ⎥

€

JαβAtt (ρ) = ρ1 00 1⎡

⎣ ⎢

⎤

⎦ ⎥ → ρ

1 00 1⎡

⎣ ⎢

⎤

⎦ ⎥ EH

EV

⎡

⎣ ⎢

⎤

⎦ ⎥ = ρ

EH

EV

⎡

⎣ ⎢

⎤

⎦ ⎥


Jones matrix for a quarter wave plate

•  Quarter wave plates are birefringent (have two different refractive index) –  align the plate such that horizontal / veritical polarization

(corresponding to O/X-mode) has wave numbers k1 / k2

–  let the light entre the plate start at x=0 and exit at x=L

–  where Ph stands for phaser

•  Quarter wave plates chages the relative phase by π/2

–  usually we considers only relative phase and skip factor exp(ik1L)

€

EH (x)EV (x)⎡

⎣ ⎢

⎤

⎦ ⎥ =

EH (0)exp(ik1x)

EV (0)exp(ik2x)

⎡

⎣ ⎢

⎤

⎦ ⎥

€

E(L) =eik

1L 00 eik

2L

⎡

⎣ ⎢

⎤

⎦ ⎥ EH (0)EV (0)⎡

⎣ ⎢

⎤

⎦ ⎥ ≡ JPhE(L)

€

k1L − k 2L = ±π /2→JQ = eik1L 1 00 ±i⎡

⎣ ⎢

⎤

⎦ ⎥

€

E(x) = R(θ)R(−θ) EM 1(x)cos(θ)sin(θ)⎡

⎣ ⎢

⎤

⎦ ⎥ + EM 2(x)

−sin(θ)cos(θ)

⎡

⎣ ⎢

⎤

⎦ ⎥

⎛

⎝ ⎜

⎞

⎠ ⎟ = R(θ)

EM1(x)EM 2(x)

⎡

⎣ ⎢

⎤

⎦ ⎥ =

= R(θ) eik1x 00 eik

2x

⎡

⎣ ⎢

⎤

⎦ ⎥ EM1(0)EM 2(0)

⎡

⎣ ⎢

⎤

⎦ ⎥ → JPh = R(θ) e

ik1x 00 eik

2x

⎡

⎣ ⎢

⎤

⎦ ⎥


Jones matrix for a rotated birefringent media

•  If a birefringent media (e.g. quarter wave plates) is not aligned with the axis of our coordinate system with axis of the media –  then we may use a rotation matrix

–  Eigenmode have directions as in the fig.:

–  apply two rotation: first -θ and then +θ, i.e. no net rotation:

e1

e2

EM1

EM2

θ

€

R(θ) =cos(θ) −sin(θ)sin(θ) cos(θ)⎡

⎣ ⎢

⎤

⎦ ⎥ → R−1(θ) = R(−θ)

no net rotation €

E(x) = EM 1(x)cos(θ)sin(θ)⎡

⎣ ⎢

⎤

⎦ ⎥ + EM 2(x)

−sin(θ)cos(θ)

⎡

⎣ ⎢

⎤

⎦ ⎥


Jones matrix for a Faraday rotation

•  Faraday rotation is similar to birefrigency, except that eigenmodes have circularly polarized eigenvector

–  there is no useful rotation matrix –  instead use a unitary matrix

€

E(x) =U −1U EM1(x)1i⎡

⎣ ⎢ ⎤

⎦ ⎥ + EM 2(x)

1−i⎡

⎣ ⎢

⎤

⎦ ⎥

⎛

⎝ ⎜

⎞

⎠ ⎟ =U −1 EM 1(x)

EM 2(x)

⎡

⎣ ⎢

⎤

⎦ ⎥ =

=U −1 eik1x 0

0 eik2x

⎡

⎣ ⎢

⎤

⎦ ⎥ EM1(0)EM 2(0)

⎡

⎣ ⎢

⎤

⎦ ⎥ → JFR =U −1 eik

1x 00 eik

2x

⎡

⎣ ⎢

⎤

⎦ ⎥

€

U =12

1 −i1 i⎡

⎣ ⎢

⎤

⎦ ⎥ → U −1 =

12

1 i1 −i⎡

⎣ ⎢

⎤

⎦ ⎥

€

E(x) = EM 1(x)121i⎡

⎣ ⎢ ⎤

⎦ ⎥ + EM 2(x)

121−i⎡

⎣ ⎢

⎤

⎦ ⎥


Outline

•  The quarter wave plate •  Set up coordinate system suitable for transverse waves •  Jones calculus; matrix formulation of how wave

polarization changes when passing through polarizing component –  Examples: linear polarizer, quarter wave plate, Faraday rotation



•  Muller calculus; matrix formulation for the transmission of partially polarized waves


Incoherent/unpolarised •  Many sources of electromagnetic radiation are not coherent

–  they do not radiate perfect harmonic oscillations (sinusoidal wave) •  over short time scales the oscillations look harmonic •  but over longer periods the wave look incoherent, or even stochastic

–  such waves are often referred to as unpolarised

•  To model such waves we will consider the electric field to be a stochastic process, i.e. it has –  an average: < Eα

(t,x) > –  a variance: < Eα (t,x) Eβ

(t,x) > –  a covariance: < Eα

(t,x) Eβ (t+s,x+y) >

•  In this chapter we will focus on the variance, which we will refer to as the intensity tensor Iαβ = < Eα (t,x) Eβ (t,x) > and the polarization tensor (where eM=E / |E| is the polarization vector) pαβ = < eM

α (t,x) eMβ (t,x) >


The Stokes vector •  It can be shown that the intensity tensor is hermitian

–  thus it can be described by four Stokes parameter {I,Q,U,V} :

•  A basis for hermitian matrixes is a set of unitary four matrixes:

–  where the last three matrixes are the Pauli matrixes •  Define the Stokes vector: SA=[I,Q,U,V]

€

Iαβ =12

I +Q U − iVU + iV I −Q⎡

⎣ ⎢

⎤

⎦ ⎥

€

Iαβ =12I

1 00 1⎛

⎝ ⎜

⎞

⎠ ⎟ +Q

1 00 −1⎛

⎝ ⎜

⎞

⎠ ⎟ +U

0 11 0⎛

⎝ ⎜

⎞

⎠ ⎟ +V

0 −ii 0⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

Iαβ =12τAαβSA with inverse : SA = τA

αβIαβ

€

τ1αβ =

1 00 1⎛

⎝ ⎜

⎞

⎠ ⎟ , τ2

αβ =1 00 −1⎛

⎝ ⎜

⎞

⎠ ⎟ , τ3

αβ =0 11 0⎛

⎝ ⎜

⎞

⎠ ⎟ , τ4

αβ =0 −ii 0⎛

⎝ ⎜

⎞

⎠ ⎟


Representations for the polarization tensor

•  The polarization tensor has similar representation –  Note: trace(pαβ)=1, thus it is described by three parameter {q,u,v} :

•  As we will show in the following slides the four terms above represents different types of polarization –  unpolarised (incoherent) –  linear polarization –  circular polarization

€

pαβ =12

1 00 1⎛

⎝ ⎜

⎞

⎠ ⎟ + q

1 00 −1⎛

⎝ ⎜

⎞

⎠ ⎟ + u

0 11 0⎛

⎝ ⎜

⎞

⎠ ⎟ + v

0 −ii 0⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥


Examples

•  For example consider: –  linearly polarised wave eM

α=[1,0]

•  i.e. {q,u,v}={1,0,0} –  rotate linearly polarization by 45o, eM

α=[1,1] / 21/2

•  i.e. {q,u,v}={0,1,0} –  a circularly polarised wave, eM

α=[1,-i] / 21/2

•  i.e. {q,u,v}={0,0,1}

€

pαβ = eMα eM

β* =10⎡

⎣ ⎢ ⎤

⎦ ⎥ 1 0[ ] =

1 00 0⎡

⎣ ⎢

⎤

⎦ ⎥ =

12

1 00 1⎡

⎣ ⎢

⎤

⎦ ⎥ +1

1 00 −1⎡

⎣ ⎢

⎤

⎦ ⎥

⎛

⎝ ⎜

⎞

⎠ ⎟

€

pαβ = eMα eM

β* =1211⎡

⎣ ⎢ ⎤

⎦ ⎥ 1 1[ ] =

1 11 1⎡

⎣ ⎢

⎤

⎦ ⎥ =

12

1 00 1⎡

⎣ ⎢

⎤

⎦ ⎥ +1

0 11 0⎡

⎣ ⎢

⎤

⎦ ⎥

⎛

⎝ ⎜

⎞

⎠ ⎟

€

pαβ = eMα eM

β* =121i⎡

⎣ ⎢ ⎤

⎦ ⎥ 1 −i[ ] =

121 −ii 1⎡

⎣ ⎢

⎤

⎦ ⎥ =

12

1 00 1⎡

⎣ ⎢

⎤

⎦ ⎥ +1

0 −ii 0⎡

⎣ ⎢

⎤

⎦ ⎥

⎛

⎝ ⎜

⎞

⎠ ⎟


The polarization tensor for unpolarized waves (1)

•  What are the Stokes parameters for unpolarised waves? –  Let the eM

1 and eM2 be independent stochastic variable

–  Since eM1 and eM

2 are uncorrelated the offdiagonal term vanish

–  The vector eM is normalised: –  By symmetry (no physical difference between eM

1 and eM2 )

–  the polarization tensor then reads

–  i.e. unpolarised have {q,u,v}={0,0,0}!

€

eM1 2

+ eM2 2

=1

€

pαβ =<eM1

eM2

⎛

⎝ ⎜

⎞

⎠ ⎟

*

eM1 eM

2( ) >=< eM

1 *eM1 > < eM

1 *eM2 >

< eM2 *eM

1 > < eM2 *eM

2 >

⎡

⎣ ⎢

⎤

⎦ ⎥

€

pαβ =<| eM

1 |2> 00 <| eM

2 |2>

⎡

⎣ ⎢

⎤

⎦ ⎥

€

eM1 2

= eM2 2

=1/2

€

pαβ =121 00 1⎡

⎣ ⎢

⎤

⎦ ⎥


The polarization tensor for unpolarized waves (2)

•  Alternative derivation; polarization vector for unpolarized waves –  Note first that the polarization vector is normalised

–  the polarization is complex and stochastic: •  where θ, φ1 and φ2 are

uniformly distibuted in [0,2π]

•  The corresponding polarization tensor

–  here the average is over the three random variables θ, φ1 and φ2

–  i.e. unpolarised have {q,u,v}={0,0,0}!

€

eM2

= eM1 2

+ eM2 2

=1 ~ cos2(θ) + sin2(θ)

€

eM1

eM2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

eiφ 1 cos(θ)eiφ 2 sin(θ)

⎛

⎝ ⎜

⎞

⎠ ⎟

€

pαβ =<eM1

eM2

⎛

⎝ ⎜

⎞

⎠ ⎟

*

eM1 eM

2( ) >=<e− iφ 1 + iφ 1 cos(θ)cos(θ) e− iφ 1 + iφ 2 cos(θ)sin(θ)e− iφ 2 + iφ 1 sin(θ)cos(θ) e− iφ 2 + iφ 2 sin(θ)sin(θ)

⎛

⎝ ⎜

⎞

⎠ ⎟ >

€

pαβ =12π( )3

dθ0

2π

∫ dφ10

2π

∫ dφ20

2π

∫cos2(θ) e− iφ 1 + iφ 2 cos(θ)sin(θ)

e− iφ 2 + iφ 1 sin(θ)cos(θ) sin2(θ)

⎛

⎝ ⎜

⎞

⎠ ⎟ =

121 00 1⎛

⎝ ⎜

⎞

⎠ ⎟


Poincare sphere

•  The polarised part of a wave field describes the normalised vector {q/r,u/r,v/r} where is the degree of polarization –  since this vector is real and normalised it will represent points on a

sphere, the so called Poincare sphere

•  Thus, any transverse wave field can be described by –  a point on the Poincare sphere –  a degree of polarization, r

•  A polarizing element induces a motion on the sphere –  e.g. when passing though a birefringent crystal we trace a circle on the

Poincare sphere

€

r = q2 + u2 + v 2

v

u

q

r

Poincare sphere


Outline

•  Set up coordinate system suitable for transverse waves •  Jones calculus; matrix formulation of how wave

polarization changes when passing through polarizing component –  Examples: linear polarizer, quarter wave plate, Faraday rotation



•  Muller calculus; a matrix formulation for the transmission of arbitrarily polarized waves


Weakly anisotropic media

•  Next we will study Muller calculus for partially polorized waves •  We will do so for weakly anisotropic media:

–  where ΔKαβ is a small perturbation –  although Muller calculus is not restricted to weak anisotropy

•  The wave equation

–  when ΔKij is a small, the 1st order dispersion relation reads: n2≈n02

–  the left hand side can then be expanded to give €

n2 − n02( )Eα = ΔKαβEα

€

Kαβ = n02δαβ + ΔKαβ

x

small, <<1

€

n2 − n02 = (n − n0)(n + n0) = (n − n0)n0 2 +

(n − n0)n0

⎡

⎣ ⎢

⎤

⎦ ⎥ ≈ 2n0(n − n0)

2n0(n − n0)Eα ≈ ΔKαβEα


The wave equation as an ODE

•  Make an eikonal ansatz (assume k is in the x-direction):

•  The wave equation can then be written as

–  describe how the wave changes when propagating through a media! •  Wave equation for the intensity tensor:

€

dIαβ

dx=ddx

< EαE β* >= ... = iω2cn0

ΔKαρδβσ − ΔKβσ*δαρ( )Iρσ€

dEα

dx= −i n0ω

cEα + i ω

2cn0ΔKαβEα€

Eα = E0α (t)exp ikx( ) = E0

α (t)exp iωcn0x

⎛

⎝ ⎜

⎞

⎠ ⎟ exp i

ωcn − n0( )x

⎛

⎝ ⎜

⎞

⎠ ⎟

dEα

dx= iω

cn0E

α + iωcn − n0( )Eα same expression as

on previous page!


The wave equation as an ODE •  The wave equation the intensity tensor is not very convinient •  Instead, rewrite it in terms of the Stokes vector:

–  we may call this the differential formulation of Muller calculus –  symmetric matrix ρAB describes non-dissipative changes in polarization –  and the antisymmetric matrix µAB describes dissipation (absorption)

•  The ODE for SA has the analytic solution (cmp to the ODE y’=ky)

–  where MAB is called the Muller matrix –  MAB represents entire optical components

•  we have a component based Muller calculus

€

dSAdx

= ρAB −µAB( )SB €

SA = τAαβIαβ

€

ρAB =iω4cn0

ΔK H ,αρτAβατB

ρβ − ΔK H ,σβτAρστB

βρ( )

µAB =iω4cn0

ΔK H ,αρτAβατB

ρβ − ΔK H ,σβτAρστB

βρ( )

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

€

SA (x) = δAB + ρAB −µAB( )x +1/2 ρAC −µAC( ) ρCB −µCB( )x 2 + ...[ ]SB (0) =

= exp ρAB −µAB( )x[ ]SB (0) = MABSB (0)


Examples of Muller matrixes

•  For illustration only – don’t memorise!

€

MABL,H =

12

1 1 0 01 1 0 00 0 0 00 0 0 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Linear polarizer (Horizontal Transmission)

€

MABL,45 =

12

1 0 −1 00 0 0 0−1 0 1 00 0 0 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Linear polarizer (45o transmission)

€

MABQ,H =

1 0 0 00 1 0 00 0 0 −10 0 1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Quarter wave plate (fast axis horizontal)

€

MABAtt (0.3) = 0.3

1 0 0 00 1 0 00 0 1 00 0 0 1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Attenuating filter (30% Transmission)


Examples of Muller matrixes

€

SAout = MAB

Q,HMBCL ,45SC

in

€

Sstep1 = MBCL ,45 1 0 0 0[ ]T = 1 0 −1 0[ ]T

€

Sout = MQ,H 1 0 −1 0[ ]T = 1 0 0 −1[ ]T

•  In optics it is common to connect a series of optical elements •  consider a system with:

–  a linear polarizer and –  a quarter wave plate

•  Insert unpolarised light, SAin=[1,0,0,0]

–  Step 1: Linear polariser transmit linear polarised light

–  Step 2: Quarter wave plate transmit circularly polarised light

polarized and unpolarised transverse waves · polarized and unpolarised transverse waves t. johnson...

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