Chapter 7 : Interference of light
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Transcript of Chapter 7 : Interference of light
Chapter 7: Interference of lightChapter 7: Interference of light
2. Sports. Obstruction of an opponent, resulting in penalty.
in·ter·fer·ence
constructive destructive
3. Physics. Superposition of two or more waves, resulting in a new wave pattern.
1. Life. Hindrance or imposition in the concerns of others.
HeNe laser
Radio City Rockettes, New York, NY
J.R. Stroop "Studies of interference in serial verbal reactions" Journal of Experimental Psychology 18:643-662 (1935).
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Peacock
Soap bubbles
2-beam interference
propagation distance from source of disturbance
initial phase (at t=0)
from superposition principle:
)cos(
)cos(
22022
11101
tks
tks
EE
EE
21 EEE
P
EE cI 0
- Electric fields are rapidly varying (n ~ 1014 Hz)
- Quickly averages to 0
- Instead of measuring E directly, measure radiant power density
= irradiance, Ee (W/m2) = time average of the square of the electric field
amplitude
- Note: to avoid confusion, Pedotti3 now uses the symbol I instead of Ee
Measuring interference
Irradiance at point P
20 PcI E
PPc EE 0
21210 EEEE
c
2122110 2 EEEEEE
cI
I1 I2 I12I = + +
- when E1 and E2 are parallel, maximum interference- when orthogonal, dot product = 0; no interference
The interference term I12
21012 2 EE cI
)cos()cos( 2211021021 tkstksEEEE
dot product of electric fields:
simplify by introducing constant phases:
2211 ksks
)cos()cos(22 021021 tt EEEE
use trigonometry: 2cosAcosB = cos(A+B) + cos(B-A) and consider again the time average:
)cos()2cos(2 021021 tEEEE
kills it
The interference term I12
)cos(0210 EE
)cos()2cos(2 021021 tEEEE
))(cos( 12120210 sskEE
simplify by introducing d: 1212 )( d ssk
d cos0210012 EE cI
to yield the interference term of the irradiance:
Irradiance formula1221 IIII
1101 EE cI
)(cos22010 tcE
20101 2
1 cEI
2202 EE cI
)(cos22020 tcE
20202 2
1 cEI
d cos0210012 EE cI
02100210 EEEE
if E1║ E2,
then
dcos2 2112 III
dcos2 2121 IIIII
-where d is the phase difference -for parallel electric fields
Interferencemutually incoherent beams (very short coherence time)
21 III
mutually coherent beams (long coherence time)
dcos2 2121 IIIII
constructive interference
destructive interference
maximum when cos d = 1
2121 2 IIIII
minimum when cos d = -1
2121 2 IIIII
d = (2mp)
d = (2m+1)p
Interference fringesdcos2 2121 IIIII
maximum when I1 = I2 = I0
1 + 1 = 4 !?!
Interference in time and space
Young’s experimentwavefront division
Michelson interferometeramplitude division
The double slit experiment (first performed in 1803)
http://www.youtube.com/watch?v=ZJ-0PBRuthc
Double slit experiment with electrons
Criteria for light and dark bands
conditions for interference:
sinam
- approximate arc S1Q to be a straight line - optical path difference = a sin
sin21 am
constructive
destructive
m = 0, 1, 2, 3, …
Interference from 1 source: reflection
Fresnel’s mirrors
Lloyd’s mirror
part of the wavefront is reflected off each mirror
part of the wavefront is reflected; part goes direct to the screen
Fresnel’s mirrors as solar collectors
part of the incident light is refracted downward and part upward
Interference from 1 source: refraction
Fresnel’s biprism
Interference via amplitude division
- thin films- oil slicks- soap bubbles- dielectric coatings- feathers- insect wings- shells- fish- …
Interference intermezzoInterference intermezzo
The Dancing Couple-1663-Jan Steen
Anatomy of a soap bubble
Soap bubble interference
optical path difference: = nf(AB + BC) = nf (2t)
Thin film interference: normal incidence
= m: constructive interference = (m + ½): destructive interference where m = 0,1,2,…
Thin film interference: non-normal incidence
optical path difference: = nf(AB + BC) – n0(AD) = 2nf t cost
Keep in mind the phase
Simple version: phase of reflected beam shifted by p if n2 > n1
0 if n1 > n2
Correct version: use Fresnel equations!
“hard”reflection
“soft”reflection
Summary of phase shifts on reflectionTE mode TM mode
airglass
external reflectionn1 < n2
TE mode TM mode
airglass
internal reflectionn1 > n2
n1
n2
n1
n2
Back to the bubbles
How thick here (red band)?
tn>1
180o phase change
0o phase change
Constructive interference for 2t ~ (m + ½)
At first red band m = 0 t ~ ¼ (700 nm)
Colors indicate bubble thickness
Bright: Colored “monochromatic” stripes occur at (1/4) for visible colors
White: Multiple, overlapping interferences (higher order)
Dark: Super thin; destructive interference for all wavelengths (no reflected light)
pop!
Dark, white, and bright bands
Multiple beam interference
r, t : external reflection
r’, t’ : internal reflection
Note: thickness t !
])1([0
)32(
)3(0
54
)2(0
33
)(02
01
''
...''
''
''
d
d
d
d
NtiNN
ti
ti
ti
ti
eEttrE
eEttrE
eEttrE
eEttrE
erEE
...]})'(...)'()'(1[''{
...}''...''''{
......
)2(22220
)1(]32[230
321
Niiiiti
NiNiiti
Nrrrrr
erererettrreE
ettrettrettrreE
EEEEE
dddd
ddd
geometric series 21 ... 1/ 1x x x
kdwhere d is the phase difference
tf tn cos2
]'1''[ 20 d
d
i
iti
r erettrreEE
Multiple beam interference
21 ... 1/ 1x x x
...]})'(...)'()'(1[''{ )2(22220 Niiiiti
r erererettrreEE dddd
Introduce Stokes relations: r’=-r and tt’=1-r2 and simplify to get:
d
d
i
iti
r erereEE 20 1
)1(
*2rrrr EEEI
Irradiance:
d
d
d
d
i
iti
i
iti
r eree
ereerEE 22
220
2
1)1(
1)1(
Working through the math, you’ll arrive at:
Multiple beam interference
dd
cos21)cos1(2
24
2
rrrII ir
where Ii is the irradiance of the incident beam
Likewise for transmission leads to:
dcos21)1(24
22
rrrII it
Fabry-Perot interferometer (1897)
d
This simulation was performed for the two sodium lines described above, with reflectivity and the separation of the mirrors increasing from 100 microns to 400 microns.
simulation of two sodium lines:1 = 0.5890182 mm2 = 0.5896154 mm
mirror reflectivity r = 0.9
mirror separation: 100 - 400 mm
Fabry-Perot interferometer
)2/(sin)2()1()1(
2222
22
drrr
IIT
i
t
)2/(sin11
2 dFT
22
2
14
rrF
where F is the coefficient of finesse:
see chapter 8
Fabry-Perot interferometer: fringe profiles
Michelson
- transmission maxima occur when d = 2pm- as r approaches 1 (i.e. as F increases), the fringes become very narrow- see Chapter 8 for more on Fabry-Perot:
fringe contrast, FWHM, finesse, free spectral range
d
2/cos2 d
Constructive reflection2d = (m+1/2)λ m=0, 1, 2, 3...
Destructive reflection 2d = mλ m=0, 1, 2, 3...
Fringes of equal thickness
Newton’s rings
pattern depends on contact point: goal is concentric rings
m
mm
ttrR
2
22
white-light illumination
Constructive reflection2d = mλ m=0, 1, 2, 3...
Destructive reflection 2d = (m+1/2)λ m=0, 1, 2, 3...
Oil slick on pavement
Glass: n = 1.5MgF2 coating: n = 1.38
To make an AR coating for = 550 nm, how thick should the MgF2 layer be?
Thin film coatings: anti-reflective
Broadband anti-reflective films
• thin layers with a high refractive index n1,interleaved with thicker layers with a lower refractive index n2
• path lengths lA and lB differ by exactly one wavelength
• each film has optical path length /4: all reflected beams in phase• ultra-high reflectivity: 99.999% or better over a narrow wavelength range
Multilayer mirrors
Anodized titanium
Natural multi-layer reflectors
ExercisesYou are encouraged to solve all problems in the textbook (Pedrotti3).
The following may be covered in the werkcollege on 5 October 2011:
Chapter 7:1, 2, 7, 9, 15, 16, 24