Wavepackets
Outline
-  Review: Reflection & Refraction
-  Superposition of Plane Waves
- Wavepackets
- Δ
Δkk −– Δxx Relations
1
Sample Midterm 2
(one of these would be Student X’s Problem)
Q1: Midterm 1 re-mix
(Ex: actuators with dielectrics)
Q2: Lorentz oscillator
(absorption / reflection / dielectric constant /
index of refraction / phase velocity)
Q3: EM Waves
(Wavevectors / Poynting / Polarization /
Malus' Law / Birefringence /LCDs)
Q4: Reflection & Refraction
(Snell's Law, Brewster angle, Fresnel Equations)
Q5: Interference / Diffraction
2
Electromagnetic Plane Waves
The Wave Equation
ω
k=
c
∂ 2 Ey
∂ 2 Ey
= μ 2
2
∂t
∂z
Ey = A1 cos (ωt − kz) + A2 cos (ωt + kz)
A1
A2
cos (ωt − kz) +
cos (ωt + kz)
Hx = −
η
η
E
H
t
3
E
H
Reflection of EM Waves at Boundaries
E1 (z = 0) = E2 (z = 0)
incident wave
•  Write traveling wave terms in
each region
•  Determine boundary condition
transmitted
wave •  Infer relationship of ω1,2 & k1,2
reflected
wave
Medium 1
•  Solve for Ero (r) and Eto (t)
Medium 2
i + Er
E1 = E
−jk1 z
+jk1 z
=a
ˆx Ei0 e
+ Er0 e
−jk1 z
+jk1 z
=a
ˆx Ei0 e
+ rEi0 e
4
2 = E
t
E
=a
ˆx Et0 e−jk2 z
=a
ˆx tEi0 e−jk2 z
E
H
Reflection of EM Waves at Boundaries
1 (z = 0) = E
2 (z = 0)
E
incident
1 (z = 0) = H
2 (z = 0)
wave
H
transmitted
wave
reflected
wave
•  Write traveling wave terms in
each region
•  Determine boundary condition
•  Infer relationship of ω1,2 & k1,2
•  Solve for Ero (r) and Eto (t)
Medium 1
Medium 2
−jk1 z
+jk1 z
ˆx Ei0 e
+ Er0 e
E1 = a
Ei0 −jk1 z Er0 +jk1 z
H1 = a
ˆy
e
−
e
η1
η1
At normal incidence..
5
2 = a
ˆx Et0 e−jk2 z
E
Et0 −jk2 z
H2 = a
ˆy
e
η2
η2 − η1
E0r
r= i =
η2 + η1
E0
Oblique Incidence at Dielectric Interface
E-field
polarization
perpendicular to
the plane of
incidence (TE)
•  Write traveling wave terms in
each region
•  Determine boundary condition
•  Infer relationship of ω1,2 & k1,2
•  Solve for Ero (r) and Eto (t)
i = ây E
i0 e−jkix x−jkiz z
E
E-field
polarization
parallel to
the plane of
incidence (TM)
x
z
y•
i = a
ˆy Hi0 e−jkix x−jkiz z
H
6
Oblique Incidence at Dielectric Interface
•  Write traveling wave terms in
each region
•  Determine boundary condition
•  Infer relationship of ω1,2 & k1,2
r = ây Er0 e−jkrx x+jkrz z
E
•  Solve for Ero (r) and Eto (t)
t = ây Et0 e−jktx x−jktz z
E
x
z
y•
i = ây Ei0 e−jkix x−jkiz z
E
7
Oblique Incidence at Dielectric Interface
•  Write traveling wave terms in
each region
•  Determine boundary condition
•  Infer relationship of ω1,2 & k1,2
r = ây Er0 e−jkrx x+jkrz z
E
•  Solve for Ero (r) and Eto (t)
t = ây Et0 e−jktx x−jktz z
E
x
z
y•
i = ây Ei0 e−jkix x−jkiz z
E
Tangential field is continuous (z=0)..
Ei0 e−jkix x + Er0 e−jkrx x = Et0 e−jktx x
8
Snell’s Law
Tangential E-field is continuous …
Ei0 e−jkix x + Er0 e−jkrx x = Et0 e−jktx x
n1
n2
REMINDER:
ω
ωn
k=
=
vp
c
x
z
y•
kix = krx
n1 sin θi = n1 sin θr
θ i = θr
kix = ktx
n1 sin θi = n2 sin θt
9
Reflection of Light
(Optics Viewpoint … μ1 = μ2)
TE
i
E
E-field
perpendicular
to the plane of
Ht incidence
t
E
r
E
t
H
TM
x
i
E
i
H
TM:
n2 cos θi − n1 cos θt
r =
n2 cos θi + n1 cos θt
t
E
i
H
r
H
n1 cos θi − n2 cos θt
r⊥ =
n1 cos θi + n2 cos θt
y•
E-field
parallel to
the plane of
incidence
θc
Reflection Coefficients
r
E
r
H
TE:
n1 = 1.44
n2 = 1.00
|r⊥ |
θB
|r |
Incidence Angle
z
10
θi
Electromagnetic Plane Waves
The Wave Equation
ω
k=
c
∂ 2 Ey
∂ 2 Ey
= μ 2
2
∂t
∂z
Ey = A1 cos (ωt − kz) + A2 cos (ωt + kz)
A1
A2
cos (ωt − kz) +
cos (ωt + kz)
Hx = −
η
η
E
H
•  When did this plane wave turn on?
t
•  Where is there no plane wave?
11
Superposition Example: Reflection
E
H
incident wave
transmitted wave
reflected wave
Medium 1
Medium 2
How do we get a wavepacket (localized EM waves) ?
12
Superposition Example: Interference
Reflected Light Pathways Through Soap Bubbles
Constructive Interference
(Wavefronts in Step)
Incident
Reflected
Light
Light
Path
Paths
Destructive Interference
(Wavefronts out of Step)
Incident
Light
Path
Reflected
Light
Paths
Outer Surface
of Bubble
Image by Ali T http://www.flickr
.com/photos/77682540@N00/27
89338547/ on Flickr.
Thin Part of Bubble
Inner Surface
of Bubble
Thick Part of Bubble
What if the interfering waves do not have the same frequency (ω, k) ?
13
Two waves at different frequencies
… will constructively interfere and destructively interfere at different times
time
In Figure above the waves are chosen to have a 10%
frequency difference. So when the slower wave goes
through 5 full cycles (and is positive again), the faster
wave goes through 5.5 cycles (and is negative).
14
Wavepackets: Superpositions Along Travel Direction
4π
=E
o ej(ω1 t−k1 z) + ej(ω2 t−k2 z) ŷ
E
| k2 + k 1 |
4π
|k2 − k1 |
REMINDER:
n
k= ω
c
SUPERPOSITION OF TWO WAVES OF DIFFERENT FREQUENCIES
(hence different k’s)
15
Wavepackets: Superpositions Along Travel Direction
WHAT WOULD WE GET IF WE SUPERIMPOSED
WAVES OF MANY DIFFERENT FREQUENCIES ?
=E
o
E
+∞
−∞
f (k) e+j(ωt−kz) dk
LET’S SET THE FREQUENCY DISTRIBUTION as GAUSSIAN
f (k)
1
(k − ko )
f (k ) = √
exp −
2σk2
2πσk
2
REMINDER:
σk
ko
k
16
n
k= ω
c
Reminder: Gaussian Distribution
σ
√
50% of data within ±
2
0.4
0.3
0.2
0.1
34.1% 34.1%
2.1%
0.1%
13.6%
13.6%
μ
(x−μ)2
1
f (x) = √
e− 2σ2
2πσ
2.1%
μ specifies the position of the bell curve’s central peak
σ specifies the half-distance between inflection points
FOURIER TRANSFORM OF A GAUSSIAN IS A GAUSSIAN
Fx e
−ax
2
0.1%
(k) =
17
π − π 2 k2
e a
a
Gaussian Wavepacket in Space
t = to
Re{En (z, to )}
f (k)
Re{En (z, to )}
n
SUM OF SINUSOIDS
= WAVEPACKET
1
(k − ko )
f (k) = √
exp −
2σk2
2πσk
σk
ko
k
2
WE SET THE FREQUENCY DISTRIBUTION as GAUSSIAN
18
Gaussian Wavepacket in Time
2
σk
2
E(z, t) = Eo exp − (ct − z) cos (ωo t − ko z)
2
GAUSSIAN ENVELOPE
Re{En (z, to )}
Re{En (z, t1 )}
t = to
z
z
kn
t = t1
k
n
Re{En (z, t1 )}
Re{En (z, to )}
n
n
19
Gaussian Wavepacket in Space
σk2
2
E(z, t) = Eo exp − (ct − z) cos (ωo t − ko z)
2
WAVE PACKET
λo
λo =
2π
ko
GAUSSIAN
ENVELOPE
In free space …
ω
k=
c
… this plot then shows the PROBABILITY OF
WHICH k (or frequency) EM WAVES are
MOST LIKELY TO BE IN THE WAVEPACKET
f (k)
Δz = √
σk
ko
ΔkΔz = 1/2
k
20
1
2σk
Gaussian Wavepacket in Time
E(z, t) = Eo exp −
σk2
2
(ct − z)
2
WAVE PACKET
cos (ωo t − ko z)
λo
λo =
2π
ko
Δk
Δz
UNCERTAINTY
RELATIONS
c
Δz = Δt
n
c
Δk = Δω
n
ΔkΔz = 1/2
ΔωΔt = 1/2
If you want to LOCATE THE WAVEPACKET WITHIN THE SPACE Δz
you need to use a set of EM-WAVES THAT SPAN THE WAVENUMBER SPACE OF Δk = 1/(2Δz)
21
Wavepacket Reflection
E(z, t)
Ec Ec∗
22
Wavepackets in 2-D and 3-D
E(x, z)E ∗ (x, z)
Contours of constant amplitude
E(x, z)E ∗ (x, z)
t = to
2-D
3-D
Spherical probability distribution
for the magnitude of the amplitudes
of the waves in the wave packet.
23
In the next few lectures we will start
considering the limits of
Light Microscopes
incident
photon
and how these might affect our
understanding of the world we live in
electron
Suppose the positions and speeds of all particles in
the universe are measured to sufficient accuracy at a
particular instant in time
It is possible to predict the motions of every particle at
any time in the future (or in the past for that matter)
24
WAVE PACKET
Key Takeaways
λo
λo =
GAUSSIAN WAVEPACKET IN SPACE
2
σ
2
E(z, t) = Eo exp − k (ct − z) cos (ωo t − ko z)
2
GAUSSIAN
ENVELOPE
UNCERTAINTY
RELATIONS
ΔkΔz = 1/2
ΔωΔt = 1/2
25
2π
ko
MIT OpenCourseWare
http://ocw.mit.edu
6.007 Electromagnetic Energy: From Motors to Lasers
Spring 2011
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