From Maxwell's Equations to the Polarization Ellipse

2130-1

Preparatory School to the Winter College on Optics and Energy

1 - 5 February 2010

ELECTODYNAMICS: FROM MAXWELL’S EQUATIONS TO THE POLARIZATION ELLIPSE

I. Ashraf Zahid Quaid-i-Azam University Pakistan ELECTODYNAMICS: FROM MAXWELL’S EQUATIONS TO THE POLARIZATION ELLIPSE

Imrana Ashraf Zahid Quaid-i-Azam University, Islamabad Pakistan

9-Feb-10 Preparatory School to the Winter 1 College on Optics and Energy Layout

• Electrostatic : Revisited • Magneto- static : Revisited • Introduction to Maxwell’s equations • Electrodynamics before Maxwell • Maxwell’s correction to Ampere’s law • General form of Maxwell’s equations • Maxwell’s equations in vacuum • Maxwell’s equations inside matter • The Electromagnetic wave • Energy and Momentum of Electromagnetic Waves • Polarization of Light • Polarization Ellipse

9-Feb-10 Preparatory School to the Winter 2 College on Optics and Energy Nomenclature

• E = Electric field • D = Electric displacement • B = Magnetic flux density • H = Auxiliary field • ρ= Charge density • j = Current density -7 • μ0 (permeability of free space) = 4π×10 T-m/A -12 2 2 • ε0 (permittivity of free space) = 8.854×10 N-m / C • c (speed of light) = 2.99792458×108 m/s

9-Feb-10 Preparatory School to the Winter 3 College on Optics and Energy Introduction

• Electrostatics

Electrostatic field : Stationary charges produce electric fields that are constant in time. The theory of static charges is called electrostatics. Stationary charges Constant Electric field;

9-Feb-10 Preparatory School to the Winter 4 College on Optics and Energy Electrostatic :Revisited

Coulombs Law

G Q r Test G 1 qQ Charge F = rˆ 4πε r 2 q 0 Source Charge 2 −12 C ε = 8.85×10 Permittivity of free space 0 N − m2

9-Feb-10 Preparatory School to the Winter 5 College on Optics and Energy The Electric Field

y G G F = QE ri q q2 P n i G Field 1 qi qn ˆ ′ Point E(P) = ∑ 2 ri r i r 4πε0 i=1 ri

E- the electric field of the source charges. x

Physically E(P) Is force per unit charge exerted on a test charge z placed at P.

9-Feb-10 Preparatory School to the Winter 6 College on Optics and Energy The Electric Field: cont’d

P r G 1 rˆ E(P) = πε λdl ∫ 2 4 0 Line r λ is the line charge density P r G 1πε rˆ E(P) = σda ∫ 2 4 0 Surface r

σ is the surface charge density

9-Feb-10 Preparatory School to the Winter 7 College on Optics and Energy The Electric Field: cont’d

r P

G 1πε rˆ E(P) = ρdτ ∫ 2 4 0 Volume r

ρ is the volume charge density

9-Feb-10 Preparatory School to the Winter 8 College on Optics and Energy Electric Potential

The work done in moving a test charge Q in an electric field

from point P1 to P2 with a constant speed. W = Force• dis tan ce P2 G G W = −∫QE • dl p1 negative sign - work done is against the field. For any distribution of fixed charges. G G ∫ E • dl = 0 The electrostatic field is conservative

9-Feb-10 Preparatory School to the Winter 9 College on Optics and Energy Electric Potential: cont’d Stokes’s TheoremG G gives G∇× E =G 0 E = −∇V where V is Scalar Potential The work done in moving a charge Q from infinity to a point

P2 where potential is V W = QV V = Work per unit charge = Volts = joules/Coulomb

9-Feb-10 Preparatory School to the Winter 10 College on Optics and Energy Electric Potential : cont’d

Field due to a single point charge q at origin ∞ πεqdr q V = = ∫ 2 r 4 0r 4πε0r 1 F ∝ r 2 1 E ∝ r 2 1 V ∝ r

9-Feb-10 Preparatory School to the Winter 11 College on Optics and Energy G G 1 Gauss’s Law E • da = Q ∫ enc ε 0 G G ρ Differential form of Gauss’s ∇ • E = Law ε 0 ρ Poisson’s Equation ∇2V = − ε 0

Laplace's Equation ∇2V = 0

9-Feb-10 Preparatory School to the Winter 12 College on Optics and Energy Electrostatic Fields in Matter

Matter: Solids, liquids, gases, metal, wood and glasses - behave differently in electric field.

Two Large Classes of Matter (i) Conductors (ii) Dielectric Conductors: Unlimited supply of free charges. Dielectrics: • Charges are attached to specific atoms or molecules- No free charges. • Only possible motion - minute displacement of positive and negative charges in opposite direction. • Large fields- pull the atom apart completely (ionizing it).

9-Feb-10 Preparatory School to the Winter 13 College on Optics and Energy Polarization

A dielectric with charge displacements or induced dipole moment is said to be polarized.

E=0 + E

+ + + + + +

Induced Dipole Moment p = αE

The constant of proportionality α is called the atomic polarizability

P ≡ dipole moment per unit volume

9-Feb-10 Preparatory School to the Winter 14 College on Optics and Energy The Field of a Polarized Object

Potential of single dipole p is G 1prˆ• r dV = 2 4rπε 0 πε G 1 P • rˆ V = dτ ∫ 2 4 0 volume r 11⎡⎤GGG 1 VPdaPd=•−∇•⎢⎥()τ 4 πε ∫∫rr 0 ⎣⎦⎢⎥surface volum e Potential due to dipoles in the dielectric

9-Feb-10 Preparatory School to the Winter 15 College on Optics and Energy The Field of a Polarized Object: cont’d

G Bound charges at surface σb = P•nˆ πε G G Bound charges in volume ρb = −∇ • P 11⎡ 1⎤ Vdad=+⎢ σ ρτ⎥ 4 ∫∫rrbb 0 ⎣⎢ surface volume ⎦⎥

The total field is field due to bound charges plus due to free charges

9-Feb-10 Preparatory School to the Winter 16 College on Optics and Energy Gauss’s law in Dielectric

• Effect of polarization is to produce accumulations of bound charges. • The total charge density ρ ρ G G = + ρ D • da = Q f b ∫ fenc ε From Gauss’s law ρ Q fenc -Free charges enclosed G G ρ 0∇ • E = = b + ρ f Displacement vector G G G G G ∇ •=D ρ f D = ε 0 E + P

9-Feb-10 Preparatory School to the Winter 17 College on Optics and Energy Magnetostatics : Revisited

• Magnetostatics

Steady current produce magnetic fields that are constant in time. The theory of constant current is called magnetostatics.

Steady currents Constant Magnetic field;

9-Feb-10 Preparatory School to the Winter 18 College on Optics and Energy Magnetic Forces

Lorentz Force

G G G G F = q[E + (v × B)]

• The magnetic force on a segment of current carrying wire is G G F = (I × B)dl mag ∫ G G F = I(dl × B) mag ∫

9-Feb-10 Preparatory School to the Winter 19 College on Optics and Energy Equation of Continuity

The current crossingτ a surface s can be written as G K G G I = J • da = (∇ • J )dτ ∫ ∫ρ τ s v ρ G G d ⎛ ∂ ⎞ ∫()∇ • J d = − ∫∫d = − ⎜ ⎟dτ v dt ⎝ ∂t ⎠ Charge is conserved whatever flows out must come at the expense of that remaining inside - outward flow decreases the charge left in v

G G ∂ρ ∇ • J = − This is called equation of continuity ∂t

9-Feb-10 Preparatory School to the Winter 20 College on Optics and Energy Equation of Continuity 1

In Magnetostatic steady currents flow in the wire and its magnitude I must be the same along the line- otherwise charge would be pilling up some where and current can not be maintained indefinitely. ∂ρ = 0 ∂t In Magnetostatic and equationG G of continuity ∇• J = 0

Steady Currents: The flow of charges that has been going on forever - never increasing - never decreasing.

9-Feb-10 Preparatory School to the Winter 21 College on Optics and Energy Magnetostatic and Current Distributions G I Biot and Savart Law dl G G G μ G 0 I × r G dB B()p = G 3 dl r 4π ∫ r

dl is an element of length. G p r vector from source to point p.

μ0 Permeability of free space.

Unit of B = N/Am = Tesla (T)

9-Feb-10 Preparatory School to the Winter 22 College on Optics and Energy Biot and Savart Law for Surface and Volume Currents G G μ G 0 K × r For Surface Currents B = G 3 da 4π ∫ r μ G G G 0π J × r For Volume Currents B = G 3 dτ 4 ∫ r

9-Feb-10 Preparatory School to the Winter 23 College on Optics and Energy Force between two parallel wires

The magnetic field at (2) due to current I is 1 I1 I 2 μ I 0 1 Points inside dF dl B1 = B 2πd 1 Magnetic force law d G G dF = I (dl × B ) ∫ 2 2 1 (1) (2) ⎛⎞G μ I dF=× I dl01 kˆ ∫ 22⎜⎟ ⎝⎠2 πd 9-Feb-10 Preparatory School to the Winter 24 College on Optics and Energy Force between two parallel wires

μ I I dF = 0 1 2 dl 2πd 2

The total force is infinite but force per unit length is dF μ I I = 0 1 2 dl2 2πd

If currents are anti-parallel the force is repulsive.

9-Feb-10 Preparatory School to the Winter 25 College on Optics and Energy Straight line currents

The integral of B around a circular path The current is out of the page of radius s, centered at the wire is G G μ I Bdl•=0 dl =μ I vv∫∫2π s 0 For bundle of straight wires. Wire that passes throughG G loop contributes only. B •dl = μ I ∫ 0 enc Applying Stokes’ theorem G G G ∇× B = μ0 J

9-Feb-10 Preparatory School to the Winter 26 College on Optics and Energy Divergence and Curl of B

Biot-Savart lawμ for the general case of a volume current reads G G G G π J (r′)× r r B = 0 dτ ′ 4 ∫ r 3

G r

G G G G G and ∇ • B = 0 ∇× B = μ0 J

9-Feb-10 Preparatory School to the Winter 27 College on Optics and Energy Ampere’s Law G G G ∇× B = μ0 J Ampere’s law

Integral form of Ampere’s law

Using Stokes’ theorem G G G G G G G (∇× B)• da = B • dl =μ J • da ∫ ∫ 0 ∫ G G B •dl = μ I ∫ 0 enc

9-Feb-10 Preparatory School to the Winter 28 College on Optics and Energy Vector Potential

The basic differential law of Magnetostatics G G G ∇× B = μ0 J G G ∇ • B = 0 B curl of some vector field called vector potential A(P) G G G Coulomb’s gauge B (PAP)=∇× ( ) G G G G G ∇ • A = 0 ∇× (∇× A)= μ J 2 0 ∇ A = μ0 J

9-Feb-10 Preparatory School to the Winter 29 College on Optics and Energy Magnetostatic Field in Matter

¾ Magnetic fields- due to electrical charges in motion. ¾ Examine a magnet on atomic scale we would find tinny currents. ¾ Two reasons for atomic currents. • Electrons orbiting around nuclei. • Electrons spinning on their axes. ¾ Current loops form magnetic dipoles - they cancel each other due to random orientation of the atoms. ¾ Under an applied Magnetic field- a net alignment of - magnetic dipole occurs- and medium becomes magnetically polarized or magnetized

9-Feb-10 Preparatory School to the Winter 30 College on Optics and Energy Magnetization

If m is the average magnetic dipole moment per unit atom and N is the number of atoms per unit volume, the magnetization is define as JJG JJG JJGJG MNm= mIaAm== 2 or AmA2 m = Mdτ M ==3 mm

9-Feb-10 Preparatory School to the Winter 31 College on Optics and Energy Magnetic Materials

Paramagnetic Materials The materials having magnetization parallel to B are called paramagnets. Diamagnetic Materials The elementary moment are not permanent but are induced according to Faraday’s law of induction. In these materials magnetization is opposite to B. Ferromagnetic Materials Have large magnetization due to electron spin. Elementary moments are aligned in form of groups called domain

9-Feb-10 Preparatory School to the Winter 32 College on Optics and Energy The Field of Magnetized Object Using the vector potential G of current loop r JG G μ0 mr× ˆ A = 2 4 π r JJG JG G JG μμMn×∇×ˆ M A =+00da dτ 44ππ∫∫rr JJJG G K = Mn× ˆ Bound Surface Current JbJG G G Bound Volume Current Jb =∇×M

9-Feb-10 Preparatory School to the Winter 33 College on Optics and Energy Ampere’s Law in Magnetized Material G G G ∇× B = μ0 J G G G J = Jb + J f 1 G G G G G G G ()∇× B = Jb + J f = J f + (∇× M ) μ0 G G G ∇× H = J G f where G B G H = − M μ0 G G Integral form H.dl = I ∫ fenc

9-Feb-10 Preparatory School to the Winter 34 College on Optics and Energy Faraday’s Law of Induction

• Faraday’s Law - a changing -magnetic flux through circuit induces an electromotive force around the circuit.

JGJJGJGJJGddφ ε ==−=−E..dl B da v∫ dt dt ∫ Є – Induced emf E – Induced electric field intensity

Differential form of Faraday’s law JG JGJG ∂ B ∇×E =− ∂t

9-Feb-10 Preparatory School to the Winter 35 College on Optics and Energy Faraday’s Law of Induction Induced Electric field intensity in terms of vector potential JG JGJG∂ A E = −−∇V ∂t For steady currents JGJG EV= −∇ V – Scalar potential Induced emf in a system moving in a changing magnetic field

JG JGJG∂B JGGJG ε = ∇×E =− +∇×()vB × ∂t

9-Feb-10 Preparatory School to the Winter 36 College on Optics and Energy Maxwell’s Equations

9-Feb-10 Preparatory School to the Winter 37 College on Optics and Energy Introduction to Maxwell’s Equation • In electrodynamics Maxwell’s equations are a set of four equations, that describes the behavior of both the electric and magnetic fields as well as their interaction with matter • Maxwell’s four equations express

– How electric charges produce electric field (Gauss’s law) – The absence of magnetic monopoles – How currents and changing electric fields produces magnetic fields (Ampere’s law) – How changing magnetic fields produces electric fields (Faraday’s law of induction)

9-Feb-10 Preparatory School to the Winter 38 College on Optics and Energy Historical Background

• 1864 Maxwell in his paper “A Dynamical Theory of the Electromagnetic Field” collected all four equations

• 1884 Oliver Heaviside and Willard Gibbs gave the modern mathematical formulation using vector calculus.

• The change to vector notation produced a symmetric mathematical representation, that reinforced the perception of physical symmetries between the various fields.

9-Feb-10 Preparatory School to the Winter 39 College on Optics and Energy Electrodynamics Before Maxwell

ρ Gauss’s Law (i)∇ε•E= o ∂A No name (ii)∇•B=0 E=−∇V − ∂t Faraday’s Law ∂B (iii)∇×E= B=∇×A ∂t Ampere’s Law (iv)∇×B=μo J

9-Feb-10 Preparatory School to the Winter 40 College on Optics and Energy Electrodynamics Before Maxwell (Cont’d)

Apply divergence to (iii)

⎛ ∂ B ⎞ ∂ ∇ • ()∇ × E = ∇ • ⎜ − ⎟ = − ()∇ • B ⎜ ⎟ ⎝ ∂t ⎠ ∂t The left hand side is zero, because divergence of a curl is zero. The right hand side is zero because ∇ • B = 0.

Apply divergence to (iv)

∇ • (∇ × B )= μ o (∇ • J )

9-Feb-10 Preparatory School to the Winter 41 College on Optics and Energy Electrodynamics Before Maxwell (Cont’d) • The left hand side is zero, because divergence of a curl is zero. • The right hand side is zero for steady currents i.e., ∇ • J = 0 • In electrodynamics from conservation of charge

∂ ρ ∇ • J = − ∂ t ∂ ρ ⇒ = 0 ∂ t ρ is constant at any point in space which is wrong.

9-Feb-10 Preparatory School to the Winter 42 College on Optics and Energy Maxwell’s Correction to Ampere’s Law

Consider Gauss’s Lawε ρ ∇ • o E = ε ∂ ρ∂ ()∇ • o E = ∂t ρ ∂t ∂ ∂ E ⇒ = ∇ • ε ∂t o ∂t ∂ D ∂ E = ε Displacement current ∂ t o ∂ t This result along with Ampere’s law and the conservation of charge equation suggest that there are actually two sources of magnetic field. The current density and displacement current.

9-Feb-10 Preparatory School to the Winter 43 College on Optics and Energy Maxwell’s Correction to Ampere’s Law (Cont’d)

Amperes law with Maxwell’s correction

μ ∂E ∇ × B = J + μ ε o o o ∂t

9-Feb-10 Preparatory School to the Winter 44 College on Optics and Energy General Form of Maxwell’s Equations

Differential Form Integral Form ε ε ρ ρ 1 ∇ • E = ∫ E • d S = ∫ dV o o S V ∇ • B = 0 ∫ B • d S = 0 S ∂ B G d ∇ × E = − E • dl = − B • d S ∂t ∫ dt ∫ μ C μ S ∂ E d μ B • dl = I + μ ε E • d S ∇ × B = o J + o ε o ∫ o enc o o ∫ ∂t C dt S

9-Feb-10 Preparatory School to the Winter 45 College on Optics and Energy Maxwell’s Equations in vacuum

• The vacuum is a linear, homogeneous, isotropic and dispersion less medium ∇ • E = 0 • Since there is no current or electric charge is present in the vacuum, hence ∇ • B = 0 Maxwell’s equations reads ∂B as ∇ × E = − • These equations have a ∂t simple solution interms of traveling sinusoidal waves, μ ∂E with the electric and ∇ × B = oε o magnetic fields direction ∂t orthogonal to each other and the direction of travel

9-Feb-10 Preparatory School to the Winter 46 College on Optics and Energy Maxwell’s Equations Inside Matter Maxwell’s equations are modified for polarized and magnetized materials. For linear materials the polarization P and magnetization M is given by P ε = χo e E

M = χ m H And the D and B fields are related to E and H by

Dε = o E + P = (1 + e ) o E = E G Bμ= o ()H + M = ()1 + m o H = H χ ε Where e is the electric susceptibiε lity of material, χ χ χ m is the magnetic susceptibiμ lity of material and . μ

9-Feb-10 Preparatory School to the Winter 47 College on Optics and Energy Maxwell’s Equations Inside Matter (Cont’d)

• For polarized materials we have bound charges in addition to free charges σ b = P • n

ρb = −∇ • P • For magnetized materials we have bound currents Kb = M × n

J b = ∇ × M

9-Feb-10 Preparatory School to the Winter 48 College on Optics and Energy Maxwell’s Equations Inside Matter (Cont’d) • In electrodynamics any change in the electric polarization involves a flow of bound charges resulting in polarization

current JP

Polarization current density is due ∂P J = to linear motion of charge when the p ∂t Electric polarization changes ρ Total ρcharge density

t = f + ρ b Total current density

J t = J f + J b + J p 9-Feb-10 Preparatory School to the Winter 49 College on Optics and Energy Maxwell’s Equations Inside Matter (Cont’d)

• Maxwell’sρ equations inside matter are written as ε μ B ∂P ∂E ∇× = J f + +∇×Mε+ t o ∇ • E = o ∂t ∂t o μ ⎛ B ⎞ ∂ ∇×⎜ − M ⎟ = J f + ()ε E + P ∇ • B = 0 ⎜ ⎟ o ⎝ o ⎠ ∂t μ ∂B ∇ × E = − ∂ ∂t μ ∇×H = J f + D μ ∂t μ ∂E ∇ × B = J f + J p + J b + ε o o o o o ∂t

9-Feb-10 Preparatory School to the Winter 50 College on Optics and Energy Maxwell’s Equations Inside Matter (Cont’d)

• In non-dispersive, isotropic media ε and µ are time-independent scalars, and Maxwell’sε equations reduces to ρ μ ∇ • E = ∇ • H = 0 μ ∂ H ∇ × E = − ∂ t ∂ E ∇ × H = J + ε ∂ t

9-Feb-10 Preparatory School to the Winter 51 College on Optics and Energy Maxwell’s Equations Inside Matter (Cont’d)

• In uniform (homogeneous) medium ε and µ are independent of position, hence Maxwell’s equations reads as

ρ ∇ • D = D • d S = Q f ∫ f enc S Generally, ε and µ can be rank-2 ∇ • H = 0 H • d S = 0 ∫ tensor (3X3 S matrices) μ ∂H d ∇× E = − E • dl = − H • d S describing ∂t ∫ dt ∫ C S birefringent ∂E μ d anisotropic ∇× H = J f + ε H • dl = I + D • d S ∫ f enc ∫ materials. ∂t C dt S

9-Feb-10 Preparatory School to the Winter 52 College on Optics and Energy Potential Formulation of Electrodynamics 1 • In electrostatic G Putting this in Faraday’s Law ∇× E = 0 G ∂ G G ∇× E = − (∇× A) ⇒ E = −∇V ∂t G ∂ G In electrodynamicsG ⇒ E = −∇V − A ∇ × E ≠ 0 ∂t But G Explain Maxwell’s ii and iii ∇⋅ B = 0 equations G G B = ∇ × A

9-Feb-10 Preparatory School to the Winter 53 College on Optics and Energy Potential Formulation of Electrodynamics 2

As G ρ ∇ ⋅ E = Poisson’s Equation ε o ⎛ ∂A ⎞ ρ ∇ ⋅ ⎜ ∇ V + ⎟ = ε− ⎝ ∂t ⎠ o

2 ∂ ρ This replaces Poisson’s ∇ V + ∇ ⋅ A = − Equation in electrodynamics ∂t ε o

9-Feb-10 Preparatory School to the Winter 54 College on Optics and Energy Potentialμ Formulation of μ Electrodynamicsε 3 μ μ ε ε G G G ⎛ ∂V ⎞ ⎛ ∂ 2 A ⎞ ∇× ()∇× A = J − ∇ − ⎜ ⎟ o o o ⎜ ⎟ o o ⎜ 2 ⎟ ⎝ ∂t ⎠ μ ⎝ ∂t ⎠ G ε ⎛ G ⎛ ∂ 2 A ⎞⎞ ⎛ G ⎛ ∂V ⎞⎞ G ⎜∇2 A − ⎜ ⎟⎟ − ∇⎜∇⋅ A + ⎜ ⎟⎟ = −μ J ⎜ o o ⎜ 2 ⎟⎟ ⎜ o o ⎟ o ⎝ ⎝ ∂t ⎠⎠ ⎝ ⎝ ∂t ⎠⎠

These equation carry all information in Maxwell’s equations

9-Feb-10 Preparatory School to the Winter 55 College on Optics and Energy The Electromagnetic Waves

9-Feb-10 Preparatory School to the Winter 56 College on Optics and Energy Electromagnetic Wave Equation

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. To obtain the electromagnetic wave equation in a vacuum we begin with the modern 'Heaviside' form of Maxwell's equations.

9-Feb-10 Preparatory School to the Winter 57 College on Optics and Energy From Maxwell’s Equations to the Electromagnetic Waves 1

The Wave Equation ∇• E = 0 ∇• B = 0 Maxwell’s equation in ∂B free space – no ∇×E = − charge or no current ∂t are given as ∂E ∇×B = μ ε o o ∂t

9-Feb-10 Preparatory School to the Winter 58 College on Optics and Energy From Maxwell’s Equations to the Electromagnetic Waves 2

Take curl of ∂ B ∇ × E = − ∂ t ∂ B ∇ × ∇ × E = ∇ × [ − ] ∂ t Change the order of differentiation on the R.H.S ∂ ∇ × ∇ × E = − [∇ × B] ∂t

9-Feb-10 Preparatory School to the Winter 59 College on Optics and Energy From Maxwell’s Equations to the Electromagnetic Waves 3 As ∂E ∇ × B = μ ε o o ∂t Substituting for ∇ × B we have ∂B ∂ ∂E ∇×[∇× E] = − ⇒ ∇×[∇× E] = − [ ] ∂t ∂t o o ∂t ∂2 E ∇×[∇× E] = −μ ε μ ε o o ∂t 2

•As µo and εo are constant in time

9-Feb-10 Preparatory School to the Winter 60 College on Optics and Energy From Maxwell’s Equations to the Electromagnetic Waves 4 ∂ 2 E Using the vector identity ∇ × ∇ × E = − ∂t 2 ∂ 2 E becomes, ∇(∇ • E) − ∇ 2 E = −μ ε o o ∂t 2 In free space ∇•E=0

And we are left with the wave equation ∂ 2 E ∇2 E − μ ε = 0 o o ∂t 2

9-Feb-10 Preparatory School to the Winter 61 College on Optics and Energy From Maxwell’s Equations to the Electromagnetic Waves 5

Similarly the wave equation for magnetic field G G ∂ 2 B ∇ 2 B − μ ε = 0 o o ∂t 2

where, 1 c = μ oε o

9-Feb-10 Preparatory School to the Winter 62 College on Optics and Energy Electromagnetic Wave Equation in Vacuum

G G G ∂2E G ∂2B ∇2E − μ ε = 0 ∇2B−μ ε =0 o o ∂t 2 o o ∂t2

The solutions to the wave equations, when there is no source charge present can be plane waves - obtained by method of separation of variables

9-Feb-10 Preparatory School to the Winter 63 College on Optics and Energy Solution of Electromagnetic Wave

• Plane electromagnetic waves can be expressed as

G G G itkr(ω −⋅) EEe= o nˆ G G JG G 11itkr()−⋅ ω ˆˆ B = Eeo () k×= nˆ () k × E cc

Where n ˆ is the polarization vector and. kˆ is a propagation vector.

9-Feb-10 Preparatory School to the Winter 64 College on Optics and Energy Electromagnetic Plane waves • Plane wave - a constant-frequency wave whose wave-fronts (surfaces of constant phase) are infinite parallel planes of constant amplitude normal to the phase velocity vector.

9-Feb-10 Preparatory School to the Winter 65 College on Optics and Energy Real Electromagnetic Plane waves

The real electric and magnetic fields in the form of a monochromatic plane wave with propagation vector kˆ and polarization nˆ

JGG GG Ert(),cos()=⋅− Eo krω tnˆ JG() G1 G G G B rt,cos()= Eo k⋅− r t() k × nˆ c ω

9-Feb-10 Preparatory School to the Winter 66 College on Optics and Energy Homogenous Wave Equations Inside Matter The homogeneous form of the equation - written in terms of either the electric field E or the magnetic field B - takes the form: μ ε Vacuum Matter 1 ∂ 2 E 1 ∂ 2 E ∇ 2 E = ∇ 2 E = μ ε 2 με 2 o o ∂t ∂t 2 2 1 2 ∂ B 1 2 ∂ B ∇ B = 2 ∇ B = 2 o o ∂t με ∂t

9-Feb-10 Preparatory School to the Winter 67 College on Optics and Energy Homogenous Wave Equations Inside Matter 1

Permittivity: ε=ε ε (ε is dielectric constant) με r o r Permeability: µ=µ µ (µ is relative permeability ≈1 μr o μr 1 1ε ε 1 1 = = = μ v ε μ r 0 r 0 0 0 rεr c =c = n v = n=Refractive Index n

9-Feb-10 Preparatory School to the Winter 68 College on Optics and Energy Energy and Momentum of Electromagnetic Waves

The energy per unit volume stored in electromagnetic field is ε 1 ⎛ 1 ⎞ U = ⎜ E 2 + B 2 ⎟ 2 ⎜ o μ ⎟ ⎝μ ε o ⎠ In the case of monochromatic plane wave 1ε B 2 = E 2 = ε E 2 c 2 o o 2 2 2 ⇒ U = o E = o Eo cos (kx − ωt)

9-Feb-10 Preparatory School to the Winter 69 College on Optics and Energy Energy and Momentum of Electromagnetic Waves 1 As the wave propagates, it carries this energy along with it. The energy flux density (energy per unit area per unit time) transported by the field is given by the poynting vector 1 S = (E × B ) μ ε o For monochromatic plane waves 2 2 ˆ ˆ S = c o Eo cos (kx −ωt)i = cUi

9-Feb-10 Preparatory School to the Winter 70 College on Optics and Energy Polarization of Electromagnetic Waves

•The polarization is specified by the orientation of the electromagnetic field.

•Polarization of electromagnetic waves is very complex- consider the optical (light) part of EM waves.

•Historically, the orientation of a polarized electromagnetic wave has been defined in the optical regime by the orientation of the electric field vector.

•Natural light is generally un-polarized- all planes of propagation being equally probable.

•Light is a transverse electromagnetic wave.

9-Feb-10 Preparatory School to the Winter 71 College on Optics and Energy Linear Polarization

• In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector to a given plane along the direction of propagation.

• The plane containing the electric field is called the plane of polarization.

9-Feb-10 Preparatory School to the Winter 72 College on Optics and Energy Linear Polarization

• Linear polarization can be horizontal or vertical

Electromagnetic Electric Field Wave Magnetic Field

Vertical Polarization Horizontal Polarization y y

x x z E z

9-Feb-10 Preparatory School to the Winter 73 College on Optics and Energy Circular Polarization

• A polarization in which the tip of the electric field vector, at a fixed point in space, describes a circle as time progresses. • The electric vector, at one point in time, describes a helix along the direction of wave propagation. • The magnitude of the electric field vector is constant as it rotates. • Circular polarization is a limiting case of the more general condition of elliptical polarization.

9-Feb-10 Preparatory School to the Winter 74 College on Optics and Energy Circular Polarization

9-Feb-10 Preparatory School to the Winter 75 College on Optics and Energy Elliptical Polarization

• In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. • An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature- with their polarization planes at right angles to each other.

9-Feb-10 Preparatory School to the Winter 76 College on Optics and Energy Elliptical Polarization

9-Feb-10 Preparatory School to the Winter 77 College on Optics and Energy Light as an electromagnetic wave

Light is a transverse wave- an electromagnetic wave

9-Feb-10 Preparatory School to the Winter 78 College on Optics and Energy Polarization Ellipse

• The polarization ellipse appeared in early 19th century- remained the only satisfactory way to visualize polarized light for nearly entire century. • Numerous relations can be derived from the polarization ellipse- like degenerate cases for linearly and circularly polarized light. • The description of light using polarization ellipse- allows us by means of a single equation – to describe any state of completely polarized light.

9-Feb-10 Preparatory School to the Winter 79 College on Optics and Energy The Polarization, Ellipse 1

• States of Polarization The optical (light) field in free space is described by the wave equation. G G 1 ∂ 2 Ert(,) ∇=2 Ert(,) i ixy= , i ct22∂

∇ 2 is the Laplacian operator and c is the velocity of light.

Preparatory School to the Winter 80 College on Optics and Energy The Polarization Ellipse 2

• This equation represents two independent wave equations for field components

Ertxy(,) and Ert (,) • Orthogonal to each other- in a plane perpendicular to the direction of propagation -z-axis. • The simplest solution of wave equation is in terms of sinusoidal functions • For propagation in positive z-direction- the solution of wave equation can be represented

9-Feb-10 Preparatory School to the Winter 81 College on Optics and Energy The Polarization Ellipse 3

are the maximum amplitudes. is the propagator that describes the propagation of the component in the positive z- direction. and are phases of each of the components. The oscillations of abov eeeq equ ations are perpendicular to each otheer- are polarized.

19-Jan-10 Preparatory School to the Winter 82 College on Optics and Energy The Instantaneous Optical Field

• The equations for are said to be “instantaneous ” because the time duration for a propagating wave is of the order of . • The time duration of this magnitude cannot be measured or observed. • The instantaneous view of the propagating optical field is

9-Feb-10 Preparatory School to the Winter 83 College on Optics and Energy The Instantaneouus Optical Field 1

19-Jan-10 Preparatory School to the Winter 84 College on Optics and Energy The Instantaneouus Optical Field 2

19-Jan-10 Preparatory School to the Winter 85 College on Optics and Energy Degenerate States of the Polarization Ellipse

• Consider the equations

1. If , then from

• The light is said to be linearly polarized along the x‐direction called as linear horizontal polarized light.

9-Feb-10 Preparatory School to the Winter 86 College on Optics and Energy Degenerate States of the Polarization Ellipse 1

2. If , then it implies and

• The linear oscillation is along the y‐axis called as linear vertically polarized light. 3. For . Eq. of polarization ellipse reduces to

9-Feb-10 Preparatory School to the Winter 87 College on Optics and Energy Degenerate States of the Polarization Ellipse 2

• It is the equation of straight line with slope and zero intercept. • For special case

The equation of straight line reduces to

9-Feb-10 Preparatory School to the Winter 88 College on Optics and Energy Degenerate States of the Polarization Ellipse 3 • The positive value represents linear polarized light ; called L P. • The negative value represents linear polarized light ; written as L P. 4. For or , the polarization ellipse eq. reduces to

• This is the equation of an ellipse in standard form.

9-Feb-10 Preparatory School to the Winter 89 College on Optics and Energy Degenerate States of the Polarization Ellipse 4

5. For and or , the polarization ellipse equation becomes

• This describes the equation of a circle in which the light is either right or left circularly polarized.

9-Feb-10 Preparatory School to the Winter 90 College on Optics and Energy Degenerate States of the Polarization Ellipse 5

LHP VHP

L+45 L-45

RCP LCP

9-Feb-10 Preparatory School to the Winter 91 College on Optics and Energy The Elliptical Parameters of the polarization ellipse

• Elliptical polarization- in which the two orthogonal components of the electric field have unequal amplitudes and non-zero phase difference. • This is true for fully polarized light- tip of the electric field vector traces an ellipse on a plane perpendicular to the direction of propagation.

9-Feb-10 Preparatory School to the Winter 92 College on Optics and Energy The Elliptical Parameters of the polarization ellipse 1 • Elliptical polarization are characterized by their ellipticity and orientation angles • Linear and circular polarizations are the limiting cases of elliptical polarizations with ellipticity equals zero and 45 degree respectively • The tip of the electric vector traces an ellipse with a semi-major axis “a” and a semi-minor axis “b”.

9-Feb-10 Preparatory School to the Winter 93 College on Optics and Energy ORIENTATION ANGLE • It is the angle of the semi-major axis of the ellipse, measured in counter clockwise from the positive horizontal axis. • The orientation angle ψ of the polarization ellipse is found to be

2EE00xy tan2ψ = 22 cosδ EE00xy−

9-Feb-10 Preparatory School to the Winter 94 College on Optics and Energy Orientation Angle

9-Feb-10 Preparatory School to the Winter 95 College on Optics and Energy Ellipticity Angle

• Ellipticity is used to describe the “flatness” of the locus of the electric field vector of a fully polarized electromagnetic wave. • If the ellipticity is zero the wave is linearly polarized-if it is +45- the wave is circularly polarized. For ellipticities between zero and 45 degree the wave is elliptically polarized. • For polarization ellipse- ellipticity angle is given by

2EE00xy sin 2χ = 22 sinδ EE00xy+

9-Feb-10 Preparatory School to the Winter 96 College on Optics and Energy Ellipticity Angle

9-Feb-10 Preparatory School to the Winter 97 College on Optics and Energy Concluding Polarization Ellipse

• The polarization ellipse can describe any state of completely polarized light. However this representation is inadequate for several reasons. 1.As the beam of light propagates-it traces out an ellipse or some special form of ellipse- circle or st. line- in time interval of the order of 10 − 15sec. This period is too short to measure. 2.The polarization ellipse cannot describe un- polarized and partial polarized light.

9-Feb-10 Preparatory School to the Winter 98 College on Optics and Energy Concluding Polarization Ellipse

3. The polarization ellipse is an amplitude description of polarized light.

The fact that we can only measure the intensity of light and not the amplitudes requires that the polarization ellipse must be transformed so that only intensities are present- observable quantities

9-Feb-10 Preparatory School to the Winter 99 College on Optics and Energy References

1. CLASSICAL ELECTRODYNAMICS By J. D. Jackson (WILEY)

2. INTRODUCTION TO ELECTRODYNAMICS By David. J. Griffiths ( PRENTICE HALL)

3. A GUIDE TO POLARIZED LIGHT By Edward Collett ( SPIE)

9-Feb-10 Preparatory School to the Winter 100 College on Optics and Energy THANK YOU

9-Feb-10 Preparatory School to the Winter 101 College on Optics and Energy