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

w kills it

The interference term I12

)cos(0210 EE

)cos()2cos(2 021021 tEEEE

))(cos( 12120210 sskEE

simplify by introducing d: 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 d 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 d = 1

2121 2 IIIII

minimum when cos d = -1

2121 2 IIIII

d = (2mp)

d = (2m+1)p

Interference fringes

cos2 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 D = a sinq

sin2

1 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: D = nf(AB + BC) = nf (2t)

Thin film interference: normal incidence

D = ml: constructive interferenceD = (m + ½)l: destructive interference where m = 0,1,2,…

Thin film interference: non-normal incidence

optical path difference: D = nf(AB + BC) – n0(AD) = 2nf t cosqt

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 reflection

TE 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)l

At first red band m = 0 t ~ ¼ (700 nm)

Colors indicate bubble thickness

Bright: Colored “monochromatic” stripes occur at (1/4)l 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

''

...

''

''

''

NtiNN

ti

ti

ti

ti

eEttrE

eEttrE

eEttrE

eEttrE

erEE

...]})'(...)'()'(1[''{

...}''...''''{

......

)2(22220

)1(]32[230

321

Niiiiti

NiNiiti

Nrrrrr

erererettrreE

ettrettrettrreE

EEEEE

geometric series 21 ... 1/ 1x x x

kwhere d is the phase difference

tf tn cos2

]'1

''[

20

i

iti

r er

ettrreEE

Multiple beam interference

21 ... 1/ 1x x x

...]})'(...)'()'(1[''{ )2(22220 Niiiiti

r erererettrreEE

Introduce Stokes relations: r’=-r and tt’=1-r2 and simplify to get:

i

iti

r er

ereEE

20 1

)1(

*2

rrrr EEEI

Irradiance:

i

iti

i

iti

r er

ee

er

eerEE

2222

0

2

1

)1(

1

)1(

Working through the math, you’ll arrive at:

Multiple beam interference

cos21

)cos1(224

2

rr

rII ir

where Ii is the irradiance of the incident beam

Likewise for transmission leads to:

cos21

)1(24

22

rr

rII 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:l1 = 0.5890182 mml2 = 0.5896154 mm

mirror reflectivity r = 0.9

mirror separation: 100 - 400 mm

Fabry-Perot interferometer

)2/(sin)2()1(

)1(2222

22

rr

r

I

IT

i

t

)2/(sin1

12 F

T

22

2

1

4

r

rF

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

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 l = 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 = D l/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 5 October 2011:

Chapter 7:1, 2, 7, 9, 15, 16, 24