the sun in polarized light - leiden observatorykeller/teaching/china_2008/cuk_l0… · lecture 1:...

Lecture 1: Basic Concepts of Polarized Light

Outline

1 The Sun in Polarized Light2 Fundamentals of Polarized Light3 Electromagnetic Waves Across Interfaces4 Fresnel Equations5 Brewster Angle6 Total Internal Reflection

Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 1

The Sun in Polarized Light

Magnetic Field Maps from Longitudinal Zeeman Effect


Second Solar Spectrum from Scattering Polarization


Fundamentals of Polarized Light

Electromagnetic Waves in MatterMaxwell’s equations⇒ electromagnetic wavesoptics: interaction of electromagnetic waves with matter asdescribed by material equationspolarization of electromagnetic waves are integral part of optics

Maxwell’s Equations in Matter

∇ · ~D = 4πρ

∇× ~H − 1c∂~D∂t

=4πc~j

∇× ~E +1c∂~B∂t

= 0

∇ · ~B = 0

Symbols~D electric displacementρ electric charge density~H magnetic fieldc speed of light in vacuum~j electric current density~E electric field~B magnetic inductiont time


Linear Material Equations

~D = ε~E~B = µ~H~j = σ~E

Symbolsε dielectric constantµ magnetic permeabilityσ electrical conductivity

Isotropic and Anisotropic Mediaisotropic media: ε and µ are scalarsanisotropic media: ε and µ are tensors of rank 2isotropy of medium broken by

anisotropy of material itself (e.g. crystals)external fields (e.g. Kerr effect)


Wave Equation in Matterstatic, homogeneous medium with no net charges: ρ = 0for most materials: µ = 1combine Maxwell, material equations⇒ differential equations fordamped (vector) wave

∇2~E − µε

c2∂2~E∂t2 −

4πµσc2

∂~E∂t

= 0

∇2~H − µε

c2∂2~H∂t2 −

4πµσc2

∂~H∂t

= 0

damping controlled by conductivity σ~E and ~H are equivalent⇒ sufficient to consider ~Einteraction with matter almost always through ~Ebut: at interfaces, boundary conditions for ~H are crucial


Plane-Wave Solutions

Plane Vector Wave ansatz ~E = ~E0ei(~k ·~x−ωt)

~k spatially and temporally constant wave vector~k normal to surfaces of constant phase|~k | wave number~x spatial locationω angular frequency (2π× frequency)t time

~E0 (generally complex) vector independent of time and space

could also use ~E = ~E0e−i(~k ·~x−ωt)

damping if ~k is complexreal electric field vector given by real part of ~E


Complex Index of Refractiontemporal derivatives⇒ Helmholtz equation

∇2~E +ω2µ

c2

(ε+ i

4πσω

)~E = 0

dispersion relation between ~k and ω

~k · ~k =ω2µ

c2

(ε+ i

4πσω

)complex index of refraction

n2 = µ

(ε+ i

4πσω

), ~k · ~k =

ω2

c2 n2

split into real (n: index of refraction) and imaginary parts (k :extinction coefficient)

n = n + ik


Transverse Waves

plane-wave solution must also fulfill Maxwell’s equations

~E0 · ~k = 0, ~H0 · ~k = 0, ~H0 =nµ

~k

|~k |× ~E0

isotropic media: electric, magnetic field vectors normal to wavevector⇒ transverse waves~E0, ~H0, and ~k orthogonal to each other, right-handed vector-tripleconductive medium⇒ complex n, ~E0 and ~H0 out of phase~E0 and ~H0 have constant relationship⇒ consider only ~E


Energy Propagation in Isotropic MediaPoynting vector

~S =c

4π

(~E × ~H

)|~S|: energy through unit area perpendicular to ~S per unit timedirection of ~S is direction of energy flowtime-averaged Poynting vector given by⟨

~S⟩

=c

8πRe(~E0 × ~H∗0

)Re real part of complex expression∗ complex conjugate〈.〉 time average

energy flow parallel to wave vector (in isotropic media)

⟨~S⟩

=c

8π|n|µ|E0|2

~k

|~k |


Quasi-Monochromatic Lightmonochromatic light: purely theoretical conceptmonochromatic light wave always fully polarizedreal life: light includes range of wavelengths⇒quasi-monochromatic lightquasi-monochromatic: superposition of mutually incoherentmonochromatic light beams whose wavelengths vary in narrowrange δλ around central wavelength λ0

δλ

λ� 1

measurement of quasi-monochromatic light: integral overmeasurement time tmamplitude, phase (slow) functions of time for given spatial locationslow: variations occur on time scales much longer than the meanperiod of the wave


Polarization of Quasi-Monochromatic Light

electric field vector for quasi-monochromatic plane wave is sum ofelectric field vectors of all monochromatic beams

~E (t) = ~E0 (t) ei(~k ·~x−ωt)

can write this way because δλ� λ0

measured intensity of quasi-monochromatic beam⟨~Ex ~E∗x

⟩+⟨~Ey ~E∗y

⟩= lim

tm−>∞

1tm

∫ tm/2

−tm/2

~Ex (t)~E∗x (t) + ~Ey (t)~E∗y (t)dt

〈· · · 〉: averaging over measurement time tmmeasured intensity independent of timequasi-monochromatic: frequency-dependent material properties(e.g. index of refraction) are constant within ∆λ


Polychromatic Light or White Light

wavelength range comparable wavelength ( δλλ ∼ 1)incoherent sum of quasi-monochromatic beams that have largevariations in wavelengthcannot write electric field vector in a plane-wave formmust take into account frequency-dependent materialcharacteristicsintensity of polychromatic light is given by sum of intensities ofconstituting quasi-monochromatic beams


Electromagnetic Waves Across Interfaces

Introductionclassical optics due to interfaces between 2 different mediafrom Maxwell’s equations in integral form at interface frommedium 1 to medium 2(

~D2 − ~D1

)· ~n = 4πΣ(

~B2 − ~B1

)· ~n = 0(

~E2 − ~E1

)× ~n = 0(

~H2 − ~H1

)× ~n = −4π

c~K

~n normal on interface, points from medium 1 to medium 2Σ surface charge density on interface~K surface current density on interface


Fields at Interfaces

Σ = 0 in general, ~K = 0 for dielectricscomplex index of refraction includes effects of currents⇒ ~K = 0requirements at interface between media 1 and 2(

~D2 − ~D1

)· ~n = 0(

~B2 − ~B1

)· ~n = 0(

~E2 − ~E1

)× ~n = 0(

~H2 − ~H1

)× ~n = 0

normal components of ~D and ~B are continuous across interfacetangential components of ~E and ~H are continuous across interface


Plane of Incidence

plane wave onto interfaceincident (i ), reflected (r ), andtransmitted (t ) waves

~E i,r ,t = ~E i,r ,t0 ei(~k i,r,t ·~x−ωt)

~H i,r ,t =cµω

~k i,r ,t × ~E i,r ,t

interface normal ~n ‖ z-axis

spatial, temporal behavior at interface the same for all 3 waves

(~k i · ~x)z=0 = (~k r · ~x)z=0 = (~k t · ~x)z=0

valid for all ~x in interface⇒ all 3 wave vectors in one plane, planeof incidence


Snell’s Law

spatial, temporal behavior thesame for all three waves

(~k i ·~x)z=0 = (~k r ·~x)z=0 = (~k t ·~x)z=0∣∣∣~k ∣∣∣ = ωc n

ω, c the same for all 3 wavesSnell’s law

n1 sin θi = n1 sin θr = n2 sin θt


Monochromatic Wave at Interface

~H i,r ,t0 =

cωµ

~k i,r ,t × ~E i,r ,t0 , ~Bi,r ,t

0 =cω~k i,r ,t × ~E i,r ,t

0

boundary conditions for monochromatic plane wave:(n2

1~E i

0 + n21~E r

0 − n22~E t

0

)· ~n = 0(

~k i × ~E i0 + ~k r × ~E r

0 − ~k t × ~E t0

)· ~n = 0(

~E i0 + ~E r

0 − ~E t0

)× ~n = 0(

1µ1

~k i × ~E i0 +

1µ1

~k r × ~E r0 −

1µ2

~k t × ~E t0

)× ~n = 0

4 equations are not independentonly need to consider last two equations (tangential componentsof ~E0 and ~H0 are continuous)


Two Special Cases

TM, p TE, s

1 electric field parallel to plane of incidence⇒ magnetic field istransverse to plane of incidence (TM)

2 electric field particular (German: senkrecht) or transverse to planeof incidence (TE)

general solution as (coherent) superposition of two caseschoose direction of magnetic field vector such that Poynting vectorparallel, same direction as corresponding wave vector


Electric Field Perpendicular to Plane of Incidence

Ei

Er

Et

Hi

Hr

Ht

θr = θi

ratios of reflected and transmitted to incident wave amplitudes

rs =E r

0E i

0=

n1 cos θi − µ1µ2

n2 cos θt

n1 cos θi + µ1µ2

n2 cos θt

ts =E t

0E i

0=

2n1 cos θi

n1 cos θi + µ1µ2

n2 cos θt


Electric Field in Plane of Incidence

Ei E

r

Et

Hi H

r

Ht

ratios of reflected and transmitted to incident wave amplitudes

rp =E r

0E i

0=

n2 cos θiµ1µ2− n1 cos θt

n2 cos θiµ1µ2

+ n1 cos θt

tp =E t

0E i

0=

2n1 cos θi

n2 cos θiµ1µ2

+ n1 cos θt


Summary of Fresnel Equations

eliminate θt using Snell’s law n2 cos θt =√

n22 − n2

1 sin2 θi

for most materials µ1/µ2 ≈ 1electric field amplitude transmission ts,p, reflection rs,p

ts =2n1 cos θi

n1 cos θi +√

n22 − n2

1 sin2 θi

tp =2n1n2 cos θi

n22 cos θi + n1

√n2

2 − n21 sin2 θi

rs =n1 cos θi −

√n2

2 − n21 sin2 θi

n1 cos θi +√

n22 − n2

1 sin2 θi

rp =n2

2 cos θi − n1

√n2

2 − n21 sin2 θi

n22 cos θi + n1

√n2

2 − n21 sin2 θi


Consequences of Fresnel Equationscomplex index of refraction⇒ ts, tp, rs, rp (generally) complexreal indices⇒ argument of square root can still be negative⇒complex ts, tp, rs, rp

real indices, arguments of square roots positive (e.g. dielectricwithout total internal reflection)

therefore ts,p ≥ 0, real⇒ incident and transmitted waves will havesame phasetherefore rs,p real, but become negative when n2 > n1 ⇒ negativeratios indicate phase change by 180◦ on reflection by medium withlarger index of refraction


Other Form of Fresnel Equationsusing trigonometric identities

ts = 2 sin θt cos θisin(θi+θt )

tp = 2 sin θt cos θisin(θi+θt ) cos(θi−θt )

rs = −sin(θi−θt )sin(θi+θt )

rp = tan(θi−θt )tan(θi+θt )

refractive indices “hidden” in angle of transmitted wave, θt

can always rework Fresnel equations such that only ratio ofrefractive indices appears⇒ Fresnel equations do not depend on absolute values of indicescan arbitrarily set index of air to 1; then only use indices of mediameasured relative to air


Relative Amplitudes for Arbitrary Polarization

electric field vector of incident wave ~E i0, length E i

0, at angle α toplane of incidencedecompose into 2 components: parallel and perpendicular tointerface

E i0,p = E i

0 cosα , E i0,s = E i

0 sinα

use Fresnel equations to obtain corresponding (complex)amplitudes of reflected and transmitted waves

E r ,t0,p = (rp, tp) E i

0 cosα , E r ,t0,s = (rs, ts) E i

0 sinα


ReflectivityFresnel equations apply to electric field amplitudeneed to determine equations for intensity of waves

time-averaged Poynting vector⟨~S⟩

= c8π|n|µ |E0|2

~k|~k |

absolute value of complex index of refraction entersenergy along wave vector and not along interface normaleach wave propagates in different direction⇒ consider energy ofeach wave passing through unit surface area on interfacedoes not matter for reflected wave⇒ ratio of reflected andincident intensities is independent of these two effectsrelative intensity of reflected wave (reflectivity)

R =

∣∣E r0

∣∣2∣∣E i0

∣∣2Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 26

Transmissivityintensity of transmitted wave by multiplying ratios of amplitudessquared with

ratios of indices of refractionprojected area on interface

relative intensity of transmitted wave (transmissivity)

T =|n2| cos θt

∣∣E t0

∣∣2|n1| cos θi

∣∣E i0

∣∣2for arbitrarily polarized light with ~E i

0 at angle α to plane ofincidence

R = |rp|2 cos2 α + |rs|2 sin2 α

T =|n2| cos θt

|n1| cos θi

(|tp|2 cos2 α + |ts|2 sin2 α

)R + T = 1 for dielectrics, not for conducting, absorbing materials


Brewster Angle

rp = tan(θi−θt )tan(θi+θt )

= 0 when θi + θt = π2

corresponds to Brewster angle of incidence of tan θB = n2n1

occurs when reflected wave is perpendicular to transmitted wavereflected light is completely s-polarizedtransmitted light is moderately polarized


Total Internal Reflection (TIR)

Snell’s law: sin θt = n1n2

sin θi

wave from high-index medium into lower index medium (e.g. glassto air): n1/n2 > 1right-hand side > 1 for sin θi >

n2n1

all light is reflected in high-index medium⇒ total internal reflectiontransmitted wave has complex phase angle⇒ damped wavealong interface


Phase Change on Total Internal Reflection

TIR induces phase changethat depends on polarizationcomplex ratios:rs,p = |rs,p|eiδs,p

phase change δ = δs − δp

tanδ

2=

cos θi

√sin2 θi −

(n2n1

)2

sin2 θi

relation valid between criticalangle and grazing incidencefor critical angle and grazingincidence, phase difference iszero


the sun in polarized light - leiden observatorykeller/teaching/china_2008/cuk_l0… · lecture 1:...

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