Gauss’ Law
! ·D
!
∇
= ρ
Electrical charges are the source of the
electric field
! = "E
! = "0 E
! + P!
D
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)
ch 1, 6
Ch 1, 1
corrected
Gauss’ Law for Magnetism
There are no source of magnetic fields
No magnetic monopoles
Magnetic field lines can only circulate
! = µH
! = µ0 H
! +M
!
B
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
ch 1, 7
Ch 1, 2
corrected
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
ch 1, 8
Ch 1, 3
corrected
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
Im
Im
Re
a(t) = cos ωt
!=1
A
Im
Re
a(t) = sin ωt
! = −i
A
Re
a(t) = cos (ωt)
+
sin
(ωt)
√ −iπ/4
! = 2e
A
Ch 1,
ch 1, 16
4
corrected
Phasor Example
E 1 + E2
Eavg
= E10 eiω1 t + E20 eiω2 t
"
!
"
!
∆E
∆E
iω1 t
e
+ Eavg −
eiω2 t
=
Eavg +
2
2
!
"
!
"
∆ω
∆E
∆E
i(ω̄+ ∆ω
)t
i(ω̄−
)t
2
2
=
Eavg +
e
+ Eavg −
e
2
2
"
!
"$
#
!
∆ωt
∆ωt
+ i∆E sin
eiω̄t
=
2Eavg cos
2
2
#
!
"
!
"$ 12
∆ωt
∆ωt
2
2
2
2
=
4Eavg cos
+ (∆E) sin
eiω̄t−iα
2
2
!
"
#$
∆E
∆ωt
ω1 + ω2
E10 + E20
α ≡ arctan
tan
ω̄ ≡
≡
2Eavg
2
2
2
ch 1, 19
Ch 1, 5
∆E ≡ E10 − E20
∆ω ≡ ω1 − ω2
corrected
Shortcuts with Complex Notations
With a plane wave described in complex notation
by
i(!
k·!
r +ωt)
!
!
E = E0 e
!
dE
dt
i(!
!
= iω E0 e k·!r+ωt)
!
= iω E
! ·E
!
∇
!
"
dE!x
dE!y
dE!z
i(!
!
=
+
+
= i kx î + ky ĵ + kz k̂ · E0 e k·!r+ωt)
dx
dy
dz
!
= i!k · E
→
→
thus we can say d/dt→iω and ∇→ik
ch 1, 21
Ch 1, 6
corrected
Poynting Vector Example
For electric and magnetic fields given by
E
= E0 eiωt+φ
E0 iωt+φ
H =
e
η0
!
µ0
η0 ≡
≈ 377Ω
"0
where
is the impedance of free space, what is the
irradiance of the wave?
Savg
!
"
#
1
!
!
$H
$∗
= E × H = Re E
2
%
&
'
2
1
E
E
0
= Re E0 eiφ e−iφ = 0
2
η0
2η0
This is analogous to Pavg=V2/2R for AC circuits
Pavg
2
E
0
!avg · A
!≤A
=S
2η0
ch 1, 23
Ch 1, 7
corrected
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
ch 1, 24
Ch 1, 8
corrected
Derivation of the Wave Equation
In an anisotropic medium
does not simplify as much since
!! ·!D
!! =! 0 ! does not imply
!! ! ∇
but rather
! ·D
! =∇
! · "E
! = "∇
! ·E
! +E
! · ∇"
∇
where ∇ε≠0. In this case it is usually easiest to write the
wave equation as
2!
E
∂
!
!
!
∇ × ∇ × E + µ" 2 = 0
∂t
or
!k × !k × E
! + µ"ω 2 E
!=0
ch 1, 24½
Ch 1, 9
added
Interferometer Control
Requirements
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
Pound-Drever-Hall method
Transmission
of cavity
X
ΕΟΜ
P.D.H. input
spectrum
case study 1, 2½
Case study 1,10
added
Mode Cleaner
l
Triangular 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
Ec = tEin + (−r)2 eikp Ec
giving
tEin
Ec =
1 − r2 eikp
for kp=2πn
,
Et = teikl Ec
Er r,t
Ein
r,t
Ec
Et
r=1
, and
2 ikl
t
in
, and
, Et = e 2Eikp
1−r e
Ein
Ec =
Et = eikl Ein , and
,
t
i.e. it has 100% transmission
Er = rEin − rteik(p−l) Ec
!
"
2 ik(p−l)
t e
Er = r 1 −
Ein
2
ikp
1−r e
Er = 0
case study 1, 3
Case
study 1, 11
corrected