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