This Lecture• Two-Beam Interference• Young’s Double Slit Experiment• Virtual Sources• Newton’s Rings• Film Thickness Measurement by Interference• Stokes Relations• Multiple-Beam Interference in a Parallel Plate

Last Lecture• Superposition of waves • Laser

Chapter 7. Interference of Light

Two-Beam Interference

( ) ( ) ( )

( ) ( ) ( )

1 2

1 01 1 1 01 1 1

2 02 2 2 02 2 2

:

, cos cos

, cos cos

Consider two waves E and E that have the same frequency

E r t E k r t E ks t

E r t E k r t E ks t

ω

ω φ ω φ

ω φ ω φ

= ⋅ − + = − +

= ⋅ − + = − +

r r

rr r rr r

rr r rr r

1kr

2kr

S2

S1

Two-Beam Interference

Interferenceterm

Two-Beam Interference

The interference term is given by

The irradiances for beams 1 and 2 are given by :

1 2 01 02 1 1 2 2cos( )cos( )E E E E ks t ks tω φ ω φ⋅ = ⋅ − + − +r r r r

{ }1 1 2 2 , ks ksα φ β φ≡ + ≡ +1 2 01 02

01 02

01 02

01 02

2 2 cos( ) cos( )

cos( 2 ) cos( )

cos( )

cos

E E E E t t

E E t

E E

E E

α ω β ω

α β ω β α

β α

δ

⋅ = ⋅ − −

= ⋅ ⎡ + − + − ⎤⎣ ⎦= ⋅ −

= ⋅

r r r r

r r

r r

r r

{ }2cos( ) cos( ) cos( ) cos( )A B A B A B= + + −

( ) ( )1 2 2 1k s sδ φ φ≡ − + − : phase difference

1 2 1 22 cosI I I I I δ= + + 01 02when (same polarization)E Er r

1 2 1 22 cosI I I I I δ= + + for purely monochromatic light

Interference of mutually incoherent beams

1 2 1 22 cosI I I I I δ= + +

( ){ }1 2 2 1cos cos ( ) ( )k s s t tδ φ φ= − + −

“mutual incoherence” means φ1(t) and φ2(t) are random in time

cos 0δ =

1 2I I I= +

Mutually incoherent beams do not interference with each other

Interference of mutually coherent beams

1 2 1 22 cosI I I I I δ= + +

( ){ }1 2 2 1cos cos ( ) ( )k s s t tδ φ φ= − + −

“mutual coherence” means [ φ1(t) - φ2(t) ] is constant in time

1 2 1 22 cosI I I I I δ= + +

Mutually coherent beams do interference with each other

( ){ }1 2cos cos cos k s sδ δ= = −

The total irradiance is given by

There is a maximum in the interference pattern when

This is referred to as constructive interference.

There is a minimum in the interference pattern when

This is referred to as destructive interference

Interference of mutually coherent beams

Visibility

Visibility = fringe contrast

{ } 10 minmax

minmax ≤≤+−

≡ V IIIIV

When

Therefore, V = 1

max 04I I= min 0I =

Conditions for good visibilitySources must be:

same in phase evolution in terms of time (source frequency) temporal coherencespace (source size) spatial coherence

Same in amplitude

Same in polarization

Very goodNormal

Very bad

Young’s Double Slit Experiment

Hecht, Optics, Chapter 9.

Assume that y << s and a << s.

The condition for an interference maximum is

The condition for an interference minimum is

Relation betweengeometric path differenceand phase difference :

a

s

y

Δ

θ

Young’s Double Slit Interference

2k πδλ

⎛ ⎞= Δ = Δ⎜ ⎟⎝ ⎠

On the screen the irradiance pattern is given by

Assuming that y << s :

Bright fringes:

Dark fringes:

Young’s Double Slit Interference

max minsy y

aλ⎛ ⎞Δ = Δ = ⎜ ⎟

⎝ ⎠

Interference Fringes From 2 Point Sources

Figure 7-5

Interference Fringes From 2 Point Sources

Two coherent point sources : P1 and P2

Interference With Virtual Sources:Fresnel’s Double Mirror

Hecht, Optics, Chapter 9.

Interference With Virtual Sources:Lloyd’s Mirror

mirror

Rotationstage

Lightsource

Figure 7-7

Interference With Virtual Sources:Fresnel’s Biprism

Hecht, Optics, Chapter 9.

7-4. Interference in Dielectric Films

Analysis of Interference in Dielectric Films

Figure 7-12

Analysis of Interference in Dielectric FilmsThe phase difference due to optical path length differences for the front and back reflections is given by

Analysis of Interference in Dielectric Films

Also need to account for phase differencesΔr due to differences in the reflection processat the front and back surfaces

Constructive interference

Destructive interference

ΔrΔ = Δp + Δr

Δr

Fringes of Equal Inclination

Fringes arise as ∆ varies due to changes in the incident angle: θi θt

Constructive interference

Destructive interference

Figure 7-13

7-5. Fringes of Equal Thickness

When the direction of the incoming light is fixed, fringes arise as ∆ varies due to changesin the dielectric film thickness :

Constructive interference Bright fringe

Destructive interference Dark fringe

7-6. Fringes of Equal Thickness: Newton’s Rings

Figure 7-17

Newton’s rings

θ

R-tmR

tm

rm Maxima (bright rings) when,n

tm << R

2 22 2 2( )

2m m

m mm

r tR r R t Rt+

= + − → =

2 2m mr Rt≈

2 ( 1, )2 2m f rt m nλ λλΔ = + = = Δ =

122 2mt mλ λ⎛ ⎞Δ = + = +⎜ ⎟

⎝ ⎠

2mr m Rλ→ =

Minima (dark rings) when,

reflection transmission

2 12mr m Rλ⎛ ⎞→ = −⎜ ⎟

⎝ ⎠

7-7. Film-thickness measurement by interference

7-8. Stokes Relations in reflection and transmission

Ei is the amplitude of the incident light.The amplitudes of the reflected and transmitted beams are given by

From the principle of reversibility

Stokes relationsr = eiπ r’

7-9. Multiple-Beam Interference in a Parallel Plate

0I

RI

TI

0

RIRI

≡

0

TITI

≡

Reflectance Transmittance

Find the superposition of the reflectedbeams from the top of the plate.The phase difference between neighboring beams is

Given that the incident wave is

Δ= kδ 2 cosf tn d θΔ =

Multiple-Beam Interference in a Parallel Plate

r

r’

t

20 0I E=

2T TI E=

2R RI E=

d

t’

The reflected amplitude resulting from the superposition of the reflected beams from the top of the plate is given by

Define

Therefore

Multiple-Beam Interference in a Parallel Plate

The Stokes relations can now be used to simplify the expression

Multiple-Beam Interference in a Parallel Plate

The reflection irradiance is given by

Multiple-Beam Interference in a Parallel Plate

Multiple-Beam Interference in a Parallel Plate

The transmitted irradiance is given by

Multiple-Beam Interference in a Parallel Plate

Minima in transmitted irradiance and maxima in reflected irradiance occur when

Minima in reflected irradiance and maxima in transmitted irradiance occur when

2 cosf tn d mθ λΔ = =

12 cos2f tn d mθ λ⎛ ⎞Δ = = +⎜ ⎟

⎝ ⎠