Interference

Optical interference: superposition of two or more lightwavesyielding resultant irradiance that deviates from the simple sum of the components

Chapter 9

Phys 322Lecture 24

Example

Two point sources

11011 cos, trkEtrE

22022 cos, trkEtrE

Assume linear polarization:

Irradiance:T

EI 2

v

TTTTT

EEEEEEEI 2122

21

221

2

v2vvvv

~I1 ~I2

interference term, I12

1221 IIII

cos020112 EEI

v

2121 rkrk phase difference:

Two point sources

cos020112 EEI

v

2121 rkrk

1221 IIII

1. Orthogonal polarization: 012 I 21 III

2. Parallel polarization: cos020112 EEI v2

2012

121

EEIT

vv

cos2 2112 III

cos2 2121 IIIII

Two point sources

cos2 2121 IIIII

Total destructive interference: = ±, ±3, …

2121min 2 IIIII

Constructive interference: cos > 0Total constructive interference: = 0, ±2, ±4, …

2121max 2 IIIII

Destructive interference: cos < 0

Special case, : 0201 EE

2

cos4)cos1(2 200

III

0

4I0

Interference and conservation of energy

4I0Is the law of conservation of energy violated?

0The interference term must average out to zero over space!The space average of I is I1+I2.

2cos4)cos1(2 2

00 III

interference fringes

Interference minima and maxima

Note: equation works only when distances r1 and r2 are large compared to the distance between the sources, and also the interference region is small.

2cos4)cos1(2 2

00 III

0201 EE

2121

20 2

1cos4 rrkII

If emitters are in-phase: 21 rrk

maximum: mkmrr /221 , m = 0, ±1, ±2, …

minimum: '21/'21 mkmrr , m’ = ±1, ±3, …

Interference and conservation of energy

2cos4)cos1(2 2

00 III

maximum: mkmrr /221 , m = 0, ±1, ±2, …

minimum: '21/'21 mkmrr , m’ = ±1, ±3, …

hyperboloid of revolution

Young’s double-slit experiment: center

Screen a distance sfrom slits

Single source of monochromatic light

a

2 slits-separated by a

1) Constructive

2) Destructive

3) Depends on s

The rays start in phase, and travel the same distance, so they will arrive in phase.

s

Light waves from a single source travel through 2 slits before meeting on a screen. The interference will be:

Young’s double-slit experiment: screen

Path difference changesacross the screen:Sequence of minima and maxima

At points where the difference in path length is0, , 2, …, the screen is bright. (constructive)

At points where the difference in path

length is

the screen is dark. (destructive)

...2

5,2

3,2

screen

Young’s double-slit experiment: quantitative

Destructive interference

Constructive interference

where m = 0, ±1, ± 2, ... Need < a

Path length difference = a sin

a a

ma sin

21sin ma

ExampleTwo slits 1 mm apart are 2 m away from the screen. What would be the distance between the zero’th and first maximum for light with =500 nm?

a

y

s

ma sin

m=0

m=1

sy tansinGeometry:(for small )

msay

asmy

mm 1m 101

21050013

9

y

Examples

When Young’s double slit experiment is placed under water. The separation y between minima and maxima

1) increases 2) same 3) decreases

Under water decreases so y decreases

ma sin

When distance between the slits in Young’s double slit experiment is decreased,

The separation y between minima and maxima

1) increases 2) same 3) decreases

asmy

Double slit: intensity distributionFar from the sources, s>>a

sa

2cos4 2

0II

2

cos4 2120

rrkII r2 r1 y

syaaarr tansin21

skayII2

cos4 20

sayII 2

0 cos4

a

r1- r2

9.2 Conditions for interference1) For producing stable pattern, the two sources must have

nearly the same frequency.2) For clear pattern, the two sources must have similar

amplitude.3) For producing interference pattern, coherent sources are

required.

Temporal coherence:Time interval in which the light resembles a sinusoidal wave. (~10 ns for nature light)Coherent length: lc= ctc.Spatial coherence:The correlation of the phase of a light wave between different locations.

Double slit interference: conditions

1. Spatial coherence: wave front should be coherent over distance a

2. Spatial coherence: r1-r2<lc

3. Waves should not be orthogonally polarized

a

Reminder: The coherence time is the reciprocal of the bandwidth.

The coherence time is given by:

where is the light bandwidth (the width of the spectrum).

Sunlight is temporally very incoherent because its bandwidth isvery large (the entire visible spectrum).

Lasers can have coherence times as long as about a second,which is amazing; that's >1014 cycles!

1/c v

The Temporal Coherence Time and the Spatial Coherence LengthThe temporal coherence time is the time the wave-fronts remain equally spaced. That is, the field remains sinusoidal with one wavelength:

The spatial coherence length is the distance over which the beam wave-fronts remain flat:

Since there are two transverse dimensions, we can define a coherence area.

Temporal Coherence

Time, c

Spatial Coherence

Length

Spatial and Temporal Coherence

Beams can be coherent or

only partially coherent

(indeed, even incoherent)

in both space and time.

Spatial andTemporal

Coherence:

TemporalCoherence;

Spatial Incoherence

Spatial Coherence;

TemporalIncoherence

Spatial andTemporal

Incoherence

Fresnel-Arago laws (on the interference of polarized light ):

1) Two orthogonal, coherent P-states cannot interfere. 2) Two parallel, coherent P-states will interfere in the same way as will

natural light.3) The two constituent orthogonal P-states of natural light cannot

interfere even if rotated into alignment (because these P-states are incoherent).

4) Two orthogonal P-states obtained from one P-state will interfere if rotated back into alignment (because they were coherent from the start).

Summary: Irradiance of a sum of two waves

2

*2

1

1Rec E E

I I I

Different colors

Different polarizations

Same colors

Same polarizations

1 2I I I

1 2I I I 1 2I I I

Interference only occurs when the waves have the same color and polarization.