Light Reflectors and Optical Resonators. Outline. Review of Wave Reflection.
Reflection and Interference. Fiber for Telecommunications. Optical Resonators. 1
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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 = x ˆEo 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
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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
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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
−jβ2 L
−jβ2 L
−jβ2 L
−jβ2 L
Et = t12 t21 e + t12 e r21 e r21 e t21 ... Ei
2 = t12 t21 e−jβL 1 + r12 r21 e−2jβL + r12 r21 e−2jβL ... 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°
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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
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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 = 5˚ A 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
˜2 n ˜1 − n r˜ = n ˜1 + n ˜2
Image is in the public domain
t˜ =
2˜ n1 n ˜1 + n ˜2
˜ Et t˜1 t˜2 e−jkL t˜1 t˜2 e−jkr L e−αL = = ˜ − 2j kL 1 − r˜1 r˜2 e−2jkr L e−2αL Ei 1 − r˜1 r˜2 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 t˜1 t˜2 e−jkr L e−αL = = ˜ − 2jkL 1 − r˜1 r˜2 e−2jkr L e−2αL Ei 1 − r˜1 r˜2 e 29
Resonance:
e2jkL = 1
Lasers: Something for Nothing (almost) at resonance
e2jkL = 1 ˜ Et t˜1 t˜2 e−jkL t˜1 t˜2 e−jkr L e−αL = = ˜ − 2jkL 1 − r˜1 r˜2 e−2jkr L e−2αL Ei 1 − r˜1 r˜2 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
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