Light Reflectors and Optical Resonators · Light Reflectors and Optical Resonators Outline Review...

Light Reflectors and Optical Resonators

Outline Review of Wave Reflection Reflection and Interference

Fiber for Telecommunications Optical Resonators

1

Standing wave

zkz = −3π/2 kz = −π/2

2 (Eo/ηo)

pattern of the H-field

Standing wave

ld

o �E

�E

�E

�H

�HReflectedwave ejkz

wave ejkzIncident

oe−jkz

Eo

η

)e−jkz

x

z

z

2Eo

| |

kz = −2π kz = −π0

| �H|

pattern of the E-fie

Reflection of a Normally Incident EM Wave from a Perfect Conductor

�Ei = xE

�Hi y= ˆ

(

Image by Theogeo http://www. flickr.com/photos/theogeo/1102 816166/ on flickr

2

Reflection & Transmission of EM Waves at Boundaries

Medium 1 Medium 2

� �2 = Hr

� � �E1 = Ei + Er

� � �H1 = Hi +Hr

� �E2 = Et

H

3

Reflection of EM Waves at Boundaries

� �E1(z = 0) = E2(z = 0)

Ei Ero = Et

o + o

� �H1(z = 0) = H2(z = 0)

Eio Er

η1− o Et

= o η =η1 η

√μ

2 ε

REFLECTION COEFFICIENT TRANSMISION COEFFICIENT (note that sign of r depends on the relative values of η2 and η1)

Er

r = o η2=

Ei

− η1

o η2 + η1

Et

t = o 2η2=

Eio η2 + η1

4

Exz The im

inciden

MEDIUM 1 MEDIUM 2

x

c e

2.0 2.0

f b 1.33

0.67 0.95

g a 1.33

0.67

2.85

0.32

d 2.0 2.0

2.0 2.0

1.6

amples of Light Reflection |Ei + Er|age below indicates the standing wave patterns ( ) resulting when an

t 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.

|Ey,total|a) Plot of with medium 1 being air, medium 2, n2 =2. Normal incidence.

|Ey,total|b) Plot of with medium 1 having n1 = 2, medium 2 being air. Normal incidence.

|Ey,total|c) Plot of with medium 1 being air, medium 2 being a perfect conductor

|Hy,total|d) Plot of with medium 1 being air, medium 2 being a perfect conductor

|Ey,total|e) Plot of with medium 1 having n1 =2, medium 2 being air. Angle of incidence greater than critical angle.

|Ey,total|f) Plot of with angle of incidence equal to the Brewster angle.

|Ey,total|g) Plot of with angle of incidence equal to the Brewster angle. Medium 2 is air.

5

Remote Sensing of the Environment … using radar

EXAMPLE: MEASUREMENT OF THICKNESS OF POLAR ICE CAPS

1

2

T12

T21

E0

E1

E2 E3

E4

E5 ε = εo μ = μo

σ = 0

σ = 10−6 S/mε = 3.5εo μ = μod

6

Reflectometry … measurement of distance to a target by identifying the nodes in the standing wave pattern

d1 d

transmitter receiver Conducting surface

7

70-80% of sunlight reflected by snow

10% reflected by ocean water 20% reflected

by vegetation and dark soil

Today’s Culture Moment Ice is more reflective than water

8

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.

1880 1920 1960 2000 2040 2080 13° 14°

15°

16°

17°

18° 19°

Future Temperatures?

56.79°F 57.97°F

Year

Cels

ius

9

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 k South Pole = 14 million km²

m²

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)

Image is in the public domain

10

Three Ways to Make a Mirror

TOTAL INTERNAL REFLECTION

METAL REFLECTION

MULTILAYER REFLECTION


Image is in the public domain: http://en.wikipedia.org/wiki/File:Dielectric_laser_mirror_from_a_dye_laser.JPG

© Kyle Hounsell. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse

11

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

⎛1 b 2 d1 = λ/(4n− hi)

N ⎡n⎞= LO ⎤R ⎜ ⎟ b = Πi=0 ⎢ ⎥n λ/(4n⎝1+ b d = lo)⎠ ⎣ 2

HI ⎦

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

nhi nlo

d1 d2

12

Opals … are an example of dielectric mirrors

λ = 2dsin(α)Colors with 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.

Silica spheres

α d

λ

© Unique Opals. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.

13

Small Businesses

New Buried Development

Splitter

OLT

ONT

Splitter

ONT

ONT ONT

Splitter

ONT

ONT

Splitter

ONT

.4 Gbps out 1.2 Gbps in

10-100 Mbps service rates

20 km reach 1

2 3

4

1 4

2 3

Time Division Multiplexing Image by Dan Tentler http://www.flickr.com/ photos/vissago/4634464205/ on flickr

2.4 Gbps shared by up to 128 users

2

Fiber to the Home

14

Small Businesses

Splitter

OLT (located in Verizon's

central switching office)

ONT

Splitter

ONT ONT

Splitter

ONT

ONT

Splitter

2.4 Gbps shared by up to 128 users

2.4 Gbps out 1.2 Gbps in

10-100 Mbps service rates

20 km reach 1 2

3 4

1 4

2 3

Time Division Multiplexing

ONT ONT

ONT

Fiber to the Home

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

Power & Battery

ONT

Video

Data

POTS

ONT

1310 nm 1490 nm

Downstream Upstream

Voice and Data

1550 nm

Video

Bandwidths & Services

Voice and Data

  Channels downstream to each home λ = 1490 and λ = 1550 nm

  Channel upstream from each home λ = 1310 nm

Fiber to the Home

Image of ONT by Josh Bancroft http://www.flickr.com/photos/joshb/87167324/ on flickr

Image by uuzinger http://www.flickr.com/photos/uuzinger/411425452/ on flickr

Image by uuzinger http://www.flickr.com/photos/uuzinger/411425461/ on flickr

16

Tran

smis

sion

Wavelength (nm) 1310 1490 1550

Upstream Downstream

Passive Optical Network (PON)

  Channels downstream to each home   λ = 1490 and λ = 1550 nm

  λChannel upstream from each home   λ = 1310 nm

Optical Assembly

Single mode fiber

Ball lens

Thin film filters

TO Can PINS

TO Can Laser

17

Separating Wavelengths

Dispersion Diffraction

Image by Ian Mackenzie http://www.flickr.com/photos/madmack/136237003/ on flickr

Image by wonker http://www.flickr.com/photos/wonker/2505350820/ on flickr


Sunlight diffracted through a 20 μm slit

18

Resonators

STANDING WAVE

�E

�H

�E

�H

z

RESONATORS

z

|Eo|

kz = −πkz = 2π

Terminate the standing wave with

a second wall to form a resonator

−

19

Thin Film Interference

i

Ei

t12Ei

r12Ei

t jβ12r21r21t21Eie

− 22L3

t12 (r21) t21Eie−jβ22L

t12r21t21Eie−jβ22L

t12t21Eie−jβ22L

Image by Yoko Nekonomania http://www. flickr.com/photos/nekonomania/4827035737/ on flickr

E

20

Optical Resonator

Et =[t12t

− + t12e−jβ2Lr jβ2L

21e−jβ2L

21ejβ2L r21e

− t21...

[t12t21

−jβL(

2

]Ei

= e 1 + r12r21e−2jβL +

(r r e−2jβL12 21

)...)]

Ei

t12t21e−jβ2L

= E− i1 r12r21e−2jβL

Ei

21

Fabry-Perot Resonance

t12t21e−jkL

t =1− r12r21e−2jkL

Fabry-Perot R max{e−2jk2Lesonance: maximum transmission

} = 1

min{e−2jk2L} = −1 minimum transmission

1.5 1.52 1.54 1.56 1.580

0.2

0.4

0.6

0.8

1

Wavelength [μm]

Tran

smis

sion |t|

2

22

Total Internal Reflection

Beyond the critical angle, θc , a ray within the higher index medium cannot escape at shallower

angles

n2sinθ2 = n1sinθ 1 θc = sin−1(n1/n2)

For glass, the critical internal angle is 42°

For water, it is 49°

42°

INCOMING RAY HUGS SURFACE n1 = 1.0

n2 = 1.5

TOTAL INTERNAL REFLECTION


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

r21 =n2 − n1

n2 + n1

Er = rEi

λo1 =2n1L

m

λo2 =2n2L

m+ 1

Constructive Interference Standing Wave E-field

Fabry-Perot Modes

22

(1 1⇒ Δλo = 2n Lm

−m+ 1

)n2L

=m(m+ 1)

λo = 1μm, n2 .= 3 5, L = 300μm

λ1 = 1μm ⇒ m = 2100

Δλ = 5A

25

e(−α/2)z

-0.5

0

0.5

1.0

-1.0 1 2 3 4 5 6

α = 2κβ =2κω

c=

4πκ

λ

Propagation distance [cm]

Fiel

d st

reng

th [

a.u.

] Plane Waves in Lossy Materials

Ey = Re{ ˜A1e

j(ωtkz)}+Re{ ˜A2e

j(ωtkz)}

E α/y(z, t) = A1e

− 2z cos(ωt− kz) +A2e+α/2z cos(ωt+ kz)

26

Resonators with Internal Loss

n1r =

− n2

n1 n+ ˜2

2nt

1=

n1 n+ ˜2

t˜

E ˜ jkL ˜ ˜ jkrL αLt 1t2e

− t1t2e− e−

= =E − kLi 1− r1r2e 2j 1− r1r2e−2jkrLe−2αL

...the EM wave loss is what heats the water inside the food


27

Laser Using Fabre-Perot Cavity


Gain profile

Resonant modes

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?

-0.5

0

0.5

1.0

-1.0

1 2 3 4 5 6

egz�E(z) = �Eie

gz

Propagation distance [cm]

˜ ˜ Resonance: E t ˜t 1t2e−jkL t t −jk

1 2e rLe−αL

= =E 2jkL

i 1− r e− ˜1r 2jkL

2 1− r1r2e−2jkrLe−2αL e = 1

Fiel

d st

reng

th [

a.u.

]

29

singularity at

1 = r1r2eΓgLe−αiL ⇔ 1 = R 2ΓgL 2αiL

1R2e e−

Et

Ei→ ∞

Lasers: Something for Nothing (almost)

at resonance e2jkL = 1

E ˜t t1t2e− ˜jkL t t1 2e

−jkrLe−αL

= =Ei 1− r1r e− ˜2jkL

2 1− r1r2e−2jkrLe−2αL


30

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6.007 Electromagnetic Energy: From Motors to LasersSpring 2011

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