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.http://www.youtube.com/watch?v=qbQ3o0MkK38
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
Kauai, Hawaii
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 (~ 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 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 : 1212 )( ssk
cos0210012 EE
cI
to yield the interference term of the irradiance:
Irradiance formula
1221 IIII
1101 EE
cI
)(cos22010 tcE
20101 2
1cEI
2202 EE
cI
)(cos22020 tcE
20202 2
1cEI
cos0210012 EE
cI
02100210 EEEE
if E1║ E2,
then
cos2 2112 III
cos2 2121 IIIII
-where is the phase difference -for parallel electric fields
Interferencemutually incoherent beams (very short coherence time)
21 III
mutually coherent beams (long coherence time)
cos2 2121 IIIII
constructive interference
destructive interference
maximum when cos = 1
2121 2 IIIII
minimum when cos = -1
2121 2 IIIII
= (2m)
= (2m+1)
Interference fringes
cos2 2121 IIIII
maximum when I1 = I2 = I0
1 + 1 = 4 !?!What about conservation of energy?
Interference in time and space
Young’s experimentwavefront division
Michelson interferometeramplitude division
The double slit experiment (first performed in early 1800s)
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
sin2
1 am
constructive
destructive
m = 0, 1, 2, 3, …
2-beam 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
2-beam interference from 1 source: refraction
Fresnel’s biprism
Fresnel’s biprism for broadband pulse characterization
Interference intermezzoInterference intermezzo
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 if n2 > n1
0 if n1 > n2
Correct version: use Fresnel equations!
“hard”reflection
“soft”reflection
analogous to wave on a rope
Summary of phase shifts on reflection
TE mode TM mode
airglass
external reflectionn1 < n2
TE mode TM mode
airglass
internal reflectionn1 > n2
n1
n2
n1
n2
Colors indicate bubble thickness
How thick here (yellow 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)
Consider a tapered soap film
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
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
t
trR
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
Exercises
You are encouraged to solve all problems in the textbook (Pedrotti3).
The following may be covered in the werkcollege on 29 September 2010:
Chapter 7:1, 2, 7, 9, 15, 16, 24
Next week’s lecture given by Herman Offerhaus