Light Reflectors and Optical Resonators
Outline
Review of Wave Reflection
Reflection and Interference
Fiber for Telecommunications
Optical Resonators
1
Reflection of a
Normally Incident
EM Wave from a
Perfect Conductor
x
H
Reflected
wave ejkz
E
z
E
i = x̂Eo e−jkz
E
i = yˆ Eo e−jkz
H
ηo
H
Incident
wave ejkz
|
|E
2Eo
Standing wave
pattern
of the E-field
kz = −2π
Image by Theogeo http://www.
flickr.com/photos/theogeo/1102
816166/ on flickr
kz = −π
0
z
|H|
Standing wave
pattern
of the H-field
2 (Eo /ηo )
2
kz = −3π/2 kz = −π/2
z
Reflection & Transmission of EM Waves at Boundaries
2 = E
t
E
1 = E
i + E
r
E
Medium 1
1 = H
i + H
r
H
Medium 2
2 = H
r
H
3
Reflection of EM Waves at Boundaries
1 (z = 0) = E
2 (z = 0)
E
Eoi + Eor = Eot
1 (z = 0) = H
2 (z = 0)
H
Eoi
η1
−
Eor
η1
=
Eot
η2
η=
Eor
η 2 − η1
r= i =
η2 + η1
Eo
Eot
2η2
t= i =
η2 + η1
Eo
REFLECTION COEFFICIENT
TRANSMISION COEFFICIENT
(note that sign of r depends on
the relative values of η2 and η1)
4
μ
x
MEDIUM 1
Examples of Light Reflection
MEDIUM 2
z The image below indicates the standing wave patterns ( |Ei + Er |) resulting when an
incident wave in medium 1 with amplitude equal to 1 V/m is incident on an interface.
Label the graphs (a)-(g) to match them with the description detailed below.
2.0
c
2.0
1.33
2.0
a) Plot of |Ey,total | with medium 1 being air,
medium 2, n2 =2. Normal incidence.
e
b) Plot of |Ey,total | with medium 1 having n1 = 2,
medium 2 being air. Normal incidence.
2.0
b
0.95
1.6
c) Plot of |Ey,total |with medium 1 being air,
medium 2 being a perfect conductor
f
d) Plot of |Hy,total | with medium 1 being air,
medium 2 being a perfect conductor
0.67
1.33
a
0.67
2.0
2.85
e) Plot of |Ey,total | with medium 1 having n1 =2,
medium 2 being air. Angle of incidence greater
than critical angle.
g
f) Plot of |Ey,total | with angle of incidence equal
to the Brewster angle.
0.32
d
g) Plot of |Ey,total | with angle of incidence equal
to the Brewster angle. Medium 2 is air.
2.0
5
Remote Sensing of the Environment
1
2
d
T12
E0
E5
E1
E4
E2
E3
… using radar
ε = εo μ = μo
σ=0
T21
ε = 3.5εo μ = μo
σ = 10−6 S/m
EXAMPLE: MEASUREMENT OF THICKNESS OF POLAR ICE CAPS
6
Reflectometry
… measurement of distance to a target by
identifying the nodes in the standing wave pattern
transmitter
receiver
d1
Conducting
surface
d
7
Today’s Culture Moment
Ice is more reflective than water
10% reflected
by ocean water
70-80% of sunlight
reflected by snow
8
20% reflected
by vegetation
and dark soil
The Greenhouse Effect
Sunlight is reradiated as
heat and trapped by
greenhouse gasses such as
carbon dioxide. Too much
carbon dioxide, however,
causes the planet to heat
up more than usual.
19°
Celsius
18°
Future
Temperatures?
17°
16°
15°
57.97°F
56.79°F
14°
13°
1880
1920
1960
2000
Year
2040
2080
9
Deploy Aluminum Rafts over Dead Ocean Areas ?
Net excess energy input into planet Earth 1.6 W/m2.
Illuminance on ground level is ~1000 W/m2
We need to reflect 1.6/1000 of energy back to Balance the Energy IN/OUT
Oceans are 90% absorptive (10% reflective)
Aluminum is 88% reflective on the shiny side and 80% reflective on the dull side.
(Frosted silica might also be able to be used as a reflector)
How much of ocean area do we need to cover
with 80% reflective sheets of aluminum to balance the energy IN/OUT ?
(1.6/1000) / (80% - 10%) = 0.23% (of the Earth’s surface area)
Earth surface area = 5100 million km2
We need to cover = 1.2 million km2
equivalent to ~100 years
of today’s Aluminum production
(assuming 50 μm thick Al foil)
Dead ocean zones = 0.24 million km2
Ice fields:
North Pole = 9 to 12 million km²
Greenland ice sheet = 1.7 million km²
South Pole = 14 million km²
10
Image is in the public domain
METAL REFLECTION
© Kyle Hounsell. All rights reserved.
This content is excluded from our
Creative Commons license. For more
information, see
http://ocw.mit.edu/fairuse
Three Ways to Make a Mirror
MULTILAYER REFLECTION
Image is in the public domain
Image is in the public domain:
http://en.wikipedia.org/wiki/
File:Dielectric_laser_mirror_from_a_dye_las
er.JPG
TOTAL INTERNAL REFLECTION
11
nhi nlo
Dielectric Mirrors
… can be >99% reflective
Simple dielectric mirrors consist of
stacked of layers of high and low
refractive index. The layers are chosen
such the path-length differences of
reflections from low to high index layers
are integer multiples of wavelengths.
Similarly, reflections from low-index layers
have path length difference of half a
wavelength, but add constructively
because of 180 degree phase shift from
the reflection. For normal incidence,
these optimized thicknesses are a quarter
of a wavelength
d1
⎛1− b ⎞
R =⎜
⎟
⎝ 1+ b ⎠
2
b=Π
N
i=0
d2
⎡ nLO ⎤
⎢
⎥
⎣ nHI ⎦
d1 = λ/(4nhi )
d2 = λ/(4nlo )
Thin layers with a high refractive index nHI
are interleaved with thicker layers with a
lower refractive index nLO.
The path lengths lA and lB differ by
exactly one wavelength, which leads to
constructive interference.
Source: wikipedia.com
12
Opals
… are an example of dielectric mirrors
Colors with λ = 2dsin(α) have
constructive interference
Image is in the public domain
Precious opal consists of spheres of
silica of fairly regular size, packed
into close-packed planes that are
stacked together with characteristic
dimensions of several hundred nm.
λ
Silica
spheres
d
α
© Unique Opals. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
13
Fiber to the Home
2.4 Gbps shared by
up to 128 users
Small Businesses
10-100 Mbps
service rates
ONT
Splitter
2.4 Gbps out
1.2 Gbps in
1
OLT
Splitter
2
3
ONT
ONT
2
3
ONT
4
ONT
1
20 km reach
Splitter
ONT
4
Time Division Multiplexing
Splitter
ONT
New Buried
Development
Image by Dan Tentler http://www.flickr.com/
photos/vissago/4634464205/ on flickr
14
Fiber to the Home
Time Division Multiplexing
Small Businesses
2.4 Gbps shared by up to 128 users
Splitter
10-100 Mbps
service rates
ONT
Splitter
2
3
ONT
ONT
ONT
2.4 Gbps out
1.2 Gbps in
1
2
OLT
3
4
20 km reach
Splitter
ONT
ONT
1
4
(located in Verizon's
central switching office)
Splitter
ONT
An ONT (Optical Network Terminal) is a media converter that is installed by Verizon
either outside or inside your premises, during FiOS installation. The ONT converts
fiber-optic light signals to copper/electric signals. Three wavelengths of light are
used between the ONT and the OLT (Optical Line Terminal):
• λ = 1310 nm voice/data transmit
• λ = 1490 nm voice/data receive
• λ = 1550 nm video receive
Each ONT is capable of delivering:
Multiple POTS (plain old telephone service) lines, Internet data, Video
15
ONT
Fiber to the Home
Image by uuzinger
http://www.flickr.com/photos/uuzinger/
411425461/ on flickr
Image by uuzinger
http://www.flickr.com/photos/uuzinger/
411425452/ on flickr
POTS
Bandwidths & Services
Upstream
1310 nm
Voice and
Data
Downstream
1490 nm
Voice and
Data
ONT
Data
1550 nm
Video
ONT
Video
Channels downstream to each home
λ = 1490 and λ = 1550 nm
Channel upstream from each home
λ = 1310 nm
Power &
Battery
Image of ONT by Josh Bancroft
http://www.flickr.com/photos/joshb/87167324/ on flickr
16
Passive Optical Network
(PON) Assembly
Optical
TO Can Laser
TO Can PINS
Single mode
fiber
Ball lens
Thin film filters
Upstream
Transmission
Channels downstream to each home
λ = 1490 and λ = 1550 nm
Downstream
Channelλupstream from each home
λ = 1310 nm
1310
1490 1550
Wavelength (nm)
17
Separating Wavelengths
Dispersion
Image by Ian Mackenzie
http://www.flickr.com/photos/
madmack/136237003/
on flickr
Diffraction
Sunlight diffracted through a 20 μm slit
Image is in the public domain
Image by wonker
http://www.flickr.com/photos/wonker/
2505350820/ on flickr
18
Resonators
RESONATORS
STANDING WAVE
H
|Eo |
E
z
E
H
kz = −2π
kz = −π
Terminate the
standing wave with
a second wall to
form a resonator
19
z
Thin Film Interference
Ei
Image by Yoko Nekonomania http://www.
flickr.com/photos/nekonomania/4827035737/
on flickr
3
t12 (r21 ) t21 Ei e−jβ2 2L
t12 r21 r21 t21 Ei e−jβ2 2L
t12 r21 t21 Ei e−jβ2 2L
r12 Ei
t12 t21 Ei e−jβ2 2L
t12 Ei
Ei
20
Optical Resonator
Ei
Et = t12 t21 e−jβ2 L + t12 e−jβ2 L r21 e−jβ2 L r21 e−jβ2 L t21 ... Ei
−2jβL 2
−2jβL
−jβL
= t12 t21 e
1 + r12 r21 e
+ r12 r21 e
... Ei
t12 t21 e−jβ2 L
=
Ei
−2jβL
1 − r12 r21 e
21
Fabry-Perot Resonance
Transmission
|t|
2
t12 t21 e−jkL
t=
1 − r12 r21 e−2jkL
1
0.8
0.6
0.4
0.2
0
1.5
1.52
1.54
1.56
1.58
Wavelength [μm]
Fabry-Perot Resonance:
max{e−2jk2 L } = 1 maximum transmission
min{e−2jk2 L } = −1 minimum transmission
22
Total Internal Reflection
Beyond the critical angle, θc , a ray within the
higher index medium cannot escape at shallower
angles
n2 sinθ2 = n1 sinθ1 θc = sin−1 (n1 /n2 )
TOTAL INTERNAL REFLECTION
For glass, the critical internal angle is 42°
For water, it is 49°
Image is in the public domain
n1 = 1.0
n2 = 1.5
INCOMING RAY HUGS SURFACE
42°
23
Waveguide Transport Light Between Mirrors
Metal waveguides
Dielectric waveguides
So what kind of
waveguide are
the optical fibers ?
Image by Dan Tentler http://www.flickr.com/
photos/vissago/4634464205/ on flickr
24
Fabry-Perot Modes
Constructive Interference
r21
Standing Wave E-field
n2 − n1
=
n2 + n1
Er = rEi
⇒ Δλo = 2n2 L
1
1
−
m m+1
λo2
2n2 L
=
m+1
2n 2 L
=
m(m + 1)
λo = 1μm, n2 = 3.5, L = 300μm
λ1 = 1μm ⇒ m = 2100
Δλ = 5Å
25
λo1
2n1 L
=
m
Plane Waves in Lossy Materials
˜
˜
Ey = Re{A1 ej(ωtkz) } + Re{A2 ej(ωtkz) }
Ey (z, t) = A1 e−α/2z cos(ωt − kz) + A2 e+α/2z cos(ωt + kz)
Field strength [a.u.]
1.0
e(−α/2)z
0.5
0
-0.5
-1.0
α = 2κβ =
1
2
3
4πκ
2κω
=
c
λ
4
5
6
Propagation distance [cm]
26
Resonators with Internal Loss
ñ1 − ñ2
r̃ =
ñ1 + n
˜2
Image is in the public domain
2ñ1
t̃ =
ñ1 + n
˜2
˜
Et
t̃1 t˜2 e−jkL
t˜1 t˜2 e−jkr L e−αL
=
=
−
2j
k̃L
1 − r̃1 r̃2 e−2jkr L e−2αL
Ei
1 − r̃1 r̃2 e
...the EM wave loss is what heats the water inside the food
27
Laser Using Fabre-Perot Cavity
Resonant modes
Gain profile
Image is in the public domain
28
Resonators with Internal Gain
What if it was possible to make a material with “negative absorption”
so the field grew in magnitude as it passed through a material?
Field strength [a.u.]
1.0
i egz gz
E(z)
=E
e
0.5
0
-0.5
-1.0
1
2
3
4
5
6
Propagation distance [cm]
˜
Et
t˜1 t˜2 e−jkL
t̃1 t̃2 e−jkr L e−αL
=
=
˜
−
2jkL
1 − r̃1 r̃2 e−2jkr L e−2αL
Ei
1 − r̃1 r̃2 e
29
Resonance:
e2jkL = 1
Lasers: Something for Nothing (almost)
at resonance
e2jkL = 1
˜
Et
t˜1 t̃2 e−jkL
t̃1 t̃2 e−jkr L e−αL
=
=
˜
−
2jkL
1 − r̃1 r̃2 e−2jkr L e−2αL
Ei
1 − r̃1 r̃2 e
singularity at
1 = r1 r2 eΓgL e−αi L ⇔ 1 = R1 R2 e2ΓgL e−2αi L
Et
→∞
Ei
Image is in the public domain
30
MIT OpenCourseWare
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6.007 Electromagnetic Energy: From Motors to Lasers
Spring 2011
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