!! · !D = "

Ch 1,

Gauss’ Law

Electrical charges are the source of the electric field

For all cases considered in this class, ρ=0

ε is a 3x3 tensor not a scalar (unless the material is isotropic)!

ε may be a function of E and H! (giving rise to non-linear optics)

1

!D = " !E = "0 !E + !P

ch 1, 6corrected

Ch 1,

Gauss’ Law for Magnetism

There are no source of magnetic fields

No magnetic monopoles

Magnetic field lines can only circulate

μ is a 3x3 tensor not a scalar (unless the material is isotropic)!

μ may be a function of E and H! (giving rise to non-linear optics 2

!B = µ !H = µ0!H + !M

ch 1, 7corrected

Ch 1,

Waves and Maxwell’s Equations

A charged particle is a source of an electric field

When that particle moves it changes the (spatial distribution of) the electric field

When the electric field changes it produces a circulating magnetic field

If the particle accelerates this circulating magnetic field will change

A changing magnetic field produces a circulating electric field

The circulating electric field becomes the source of a circulating magnetic field 3ch 1, 8

corrected

Ch 1,

Phasors

The complex amplitude of a sinusoidal function can be represented graphically by a point (often an arrow from the origin to a point) in the complex plane

Re

Im

Re

Im

Re

Im

a(t) = cos !t a(t) = sin !t!A = 1 !A = !i

a(t) = cos (!t) + sin (!t)

4!A =

!2e!i!/4

ch 1, 16corrected

Ch 1,

Phasor Example

5ch 1, 19corrected

E1 + E2 = E10ei!1t + E20e

i!2t

=!

Eavg +!E

2

"ei!1t +

!Eavg !

!E

2

"ei!2t

=!

Eavg +!E

2

"ei(!+!!

2 )t +!

Eavg !!E

2

"ei(!!!!

2 )t

=#2Eavg cos

!!!t

2

"+ i!E sin

!!!t

2

"$ei!t

=#4E2

avg cos2!

!!t

2

"+ (!E)2 sin2

!!!t

2

"$ 12

ei!t!i"

!E ! E10 " E20

Eavg !E10 + E20

2! ! !1 + !2

2!! ! !1 " !2

! ! arctan!

!E

2Eavgtan

"!"t

2

#$

Ch 1,

Shortcuts with Complex Notations

With a plane wave described in complex notation by

thus we can say d/dt→iω and ∇→ik

!E = !E0ei(!k·!r+"t)

d !E

dt= i" !E0e

i(!k·!r+"t)

= i" !E

→

6

→

ch 1, 21corrected

!! · !E =d !Ex

dx+

d !Ey

dy+

d !Ez

dz= i

!kxi + ky j + kz k

"· !E0e

i(!k·!r+"t)

= i!k · !E

Ch 1,

Poynting Vector Example

For electric and magnetic fields given by

where

is the impedance of free space, what is the irradiance of the wave?

E = E0ei!t+"

H =E0

!0ei!t+"

!0 !!

µ0

"0" 377!

7

This is analogous to Pavg=V2/2R for AC circuitsPavg = !Savg · !A ! A

E20

2"0

Savg =!

!E ! !H"

=12Re

#$E $H!

%=

12Re

&E0e

i! E0

"0e"i!

'=

E20

2"0

ch 1, 23corrected

Ch 1,

Derivation of the Wave Equation

Starting with Faraday’s law

take the curl of both sides

use vector calculus relationship to get

Use Ampere’s law (in free space where J=0)and Gauss’ law (in free space where ρ=0)in an isotropic medium

8ch 1, 24corrected

Ch 1,

Derivation of the Wave Equation

In an anisotropic medium

does not simplify as much since ! ! ! ! ! ! ! ! does not implybut rather

where ∇ε≠0. In this case it is usually easiest to write the wave equation as

or9

!! · !D = 0

!! · !D = !! · " !E = "!! · !E + !E ·!"

!!" !!" !E + µ"#2 !E

#t2= 0

!k ! !k ! !E + µ"#2 !E = 0 ch 1, 24½added

Case study 1,

Interferometer Control

Signal – a beam whose phase sensitive to the length to be controlled Local Oscillator – a beam whose phase is insensitive to that length. Detection of the phase between signal and local oscillator

10

Transmissionof cavity ΕΟΜ

XP.D.H. inputspectrum

Pound-Drever-Hall method

Requirements

case study 1, 2½added

Case study 1,

Mode CleanerTriangular modecleaner has a perimeter p=20 m, unit reflectivity end mirror and equal, lossless input/output couplers. Illuminated with a steady wave of wavelength λ. The fields transmitting (Et) reflecting from (Er) and circulating in (Ec) the cavity are proportional to the input field (Ein) with the relations

Ein Ec

EtEr

l

, , and

, , andgiving

, , andfor kp=2πn Er = 0Ec =Ein

t

r,tr,t

i.e. it has 100% transmission

r=1Ec = tEin + (!r)2eikpEc

Ec =tEin

1 ! r2eikp Et =t2eiklEin

1 ! r2eikp

11

Er = rEin ! rteik(p!l)Ec

Er = r!1 ! t2eik(p!l)

1 ! r2eikp

"Ein

Et = teiklEc

Et = eiklEin

case study 1, 3corrected