Therapeutic applications of polarized light Tissue healing ...
The Sun in Polarized Light - Leiden Observatorykeller/Teaching/China_2008/CUK_L0… · Lecture 1:...
Transcript of 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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 2
Second Solar Spectrum from Scattering Polarization
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 3
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 4
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)
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 5
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 6
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 7
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 8
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 9
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 |
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 10
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 11
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 12
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 13
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 14
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 15
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 16
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 17
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)
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 18
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 19
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 20
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 21
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 22
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 23
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 24
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 25
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 27
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 28
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 29
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
Christoph U. Keller, Utrecht University, [email protected] Lecture 1: Basic Concepts of Polarized Light 30