Reflection & Transmission of EM Waves
Reading – Shen and Kong – Ch. 4
Outline
• Everyday Reflection
• Reflection & Transmission (Normal Incidence)
• Reflected & Transmitted Power
• Optical Materials, Perfect Conductors, Metals
1
TRUE or FALSE
1. Destructive interference occurs when two waves are
offset by a phase of ½πm, or half a wavelength.
2. The intensity of a plane wave oscillates in time. This
means it is always constructively and destructively
interfering with itself.
3. In a double-slit
experiment, as
you decrease the
space between
the slits, the
interference
peaks decrease
proportionally.
Coherent
light
Propagation
direction
Barrier with
double slits
destructive
interference
constructive
interference
detector
screen
2
Waves in Materials
k = (n − jκ )
Index of refraction
Absorption coefficient
2κω 4πκ
=
α=
c
λ
3
ω
c
Incident and Transmitted Waves
E
same amplitudes
H
Normal Incidence
incident wave
transmitted wave
reflected wave
Medium 1
Medium 2
Incident Wave
Transmitted Wave
4
EM Wave Reflection
Metal Reflection
Dielectric Reflection
Image in the Public Domain
Thin Film Interference
Metal Reflection
© Kyle Hounsell. All rights reserved. This content
is excluded from our Creative Commons license.
For more information, see
Image by Ali Smiles :)
http://ocw.mit.edu/fairuse.
http:/www.flickr.com/photos/
77682540@N00/2789338547/ on flickr
Cell Phone Reflection
AM Radio Reflection
5
Incidentand Transmitted Waves
E
incident wave
H
Normal Incidence
transmitted wave
reflected wave
Medium 1 Medium 2
Incident Wave
Known
Transmitted Wave
Define reflection
coefficient as
Reflected Wave
Define transmission
coefficient as
6
Key Takeaways
•  Define the reflection coefficient as
•  Define the transmission coefficient as
2
7
E-Field Boundary Conditions
EA⊥
area A
+ + +
EB⊥
+ +δ +
n̂
ρs
surface
Normal
EA
C
n̂
δ
EB
is discontinuous at a surface charge.
surface
Tangential
L
Known
8
is continuous at a surface.
H-Field Boundary Conditions
μo HA⊥
areaA
δ
μo HB⊥
HA
HB
n̂
surface
Normal
C
K
δ
is continuous at a surface.
n̂
Tangential
is discontinuous at
a surface current
.
L
9
Incident EM Waves at Boundaries
E
H
Medium 1
Incident Wave
incident wave
Medium 2
Known
i = x̂E i e−jk1 z
E
o
1
1 i −jk1 z
Hi = ẑ × Ei = yˆ Eo e
η1
η1
10
Normal Incidence
Reflected EM Waves at Boundaries
E
H
Normal Incidence
reflected wave
Medium 1
Reflected Wave
Medium 2
Unknown
DEFINE REFLECTION
COEFFICIENT AS
r = x̂E r e+jk1 z
E
o
r
1
E
r = (−z)
r = −yˆ o e+jk1 z
ˆ ×E
H
η1
η1
11
Transmitted EM Waves at Boundaries
E
H
Normal Incidence
transmitted wave
Medium 1
Transmitted Wave
Medium 2
Unknown
DEFINE TRANSMISSION
COEFFICIENT AS
r = x̂E t e−jk2 z
E
o
t
1
E
t = ẑ × E
t = yˆ o e−jk2 z
H
η2
η2
12
Reflection & Transmission of EM Waves at Boundaries
2 = E
t
E
1 = E
i + E
r
E
Medium 11
1 = H
i + H
r
H
Medium
Medium22
2 = H
r
H
13
Reflection of EM Waves at Boundaries
1(z=0) = E
2(z=0)
E
1(z=0) = H
2(z=0)
H
η=
14
μ
Reflectivity & Transmissivity of Waves
•  Define the reflection coefficient as
•  Define the transmission coefficient as
What are the ranges for r and t? Is energy conserved?
15
Reflection & Transmission of EM Waves at Boundaries
E
H
incident wave
Normal Incidence
transmitted wave
Medium 1
Medium 2
Additional Java simulation at
http://phet.colorado.edu/new/simulations/
16
Reflectivity & Transmissivity of EM Waves
•  Note that
•  The definitions of the reflection and transmission coefficients
do generalize to the case of lossy media.
•  For loss-less media, r and t are real:
•  For lossy media, r and t are complex:
•  Incident Energy = Reflected Energy + Transmitted Energy
R = |r|2 … fraction of incident power that is reflected
T = 1 – R … fraction of incident power that is transmitted
17
Reflectivity of Dielectrics
Consider nearly-lossless optical materials. For typical dielectrics,
μ1 ≈ μ2 ≈ μ0.
r =
μ2
μ1
−
ε2
ε1
μ2
μ1
+
ε2
ε1
Result
≈
ε1 − ε 2
n1 − n2
=
n1 + n2
ε1 + ε 2
μ1 ≈ μ2
μ1 ≈ μ2
Image by Will Montague http://www.flickr.com/
photos/willmontague/3787127610/ on flickr
18
Reflection of EM Waves at Boundaries
REMEMBER:
In terms of
the index of
refraction,
assuming
μ1 = μ2 = μ0
In terms of the
characteristic
impedances
Animations © Dr. Dan Russell, Kettering University. All rights reserved. This
content is excluded from our Creative Commons license. For more information,
see http://ocw.mit.edu/fairuse.
What is different in the two reflected waves ?
Which side is air and which side is glass ?
19
Why does metal reflect light?
© Kyle Hounsell. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
20
Microscopic Lorentz Oscillator Model
21
T-A-R-T
T
A
R
70
T
60
Reflection %
50
40
30
20
T
A
R
T
10
0
T-A-R-T
•
•
•
•
- the material has four distinct regions of optical properties:
Transmissive,
ω < ω0 - γ/2,
Absorptive,
ω0 - γ/2 < ω < ω0 + γ/2
Reflective,
ω0 + γ/2 < ω < ωp
Transmissive,
ω > ωp
22
Reflection of a
Normally Incident
EM Wave from a
Perfect Conductor
i = x̂Eo e−jkz
E
i = ŷ Eo e−jkz
H
ηo
reflected wave
Incident wave
Standing wave pattern
of the E-field
kz = −2π
kz = −π
Standing wave pattern
of the H-field
kz = −3π/2 kz = −π/2
23
Microscopic Lorentz Oscillator Model
… for FREE ELECTRONS IN METALS
0.5
0.4
0.3
γ/2
Drude Model for metals
r
i
6
0.2
ωp
-5
ω
n, κ
r , i
0.1
-10
κ
5
r
R
4
3
2
T
n
n
1
-20
-25
ωp
-30
24
ω
Reflectivity of Silver
γ/2
100
6
κ
R
4
3
2
80
T
n
Reflection(%)
n, κ
5
n
1
ωp
60
40
T
R
20
ω
ωp
25
ω
Ice is more reflective than water
10% reflected
by ocean water
70-80% of sunlight
reflected by snow
26
20% reflected
by vegetation
and dark soil
Thin Film Interference
Constructive
interference
Incident
light
air
oil
water
Image by Yoko Nekonomania http://www.
flickr.com/photos/nekonomania/4827035737/ on flickr
27
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6.007 Electromagnetic Energy: From Motors to Lasers
Spring 2011
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