This Lecture• Two-Beam Interference• Young’s Double Slit Experiment• Virtual Sources• Newton’s Rings• Film Thickness Measurement by Interference

Last Lecture• Wave equations• Maxwell equations and EM waves• Superposition of waves

Chapter 10. Interference of LightChapter 10. Interference of Light

Two-Beam InterferenceTwo-Beam Interference

( ) ( )( ) ( )

1 2

1 01 1 1

2 02 2 2

:

, cos

, cos

Consider two waves E and E that have the same frequency

E r t E k r t

E r t E k r t

ω

ω ε

ω ε

= ⋅ − +

= ⋅ − +

r r

rr rr r

rr rr r

(polarizationdirection)Hecht, Optics, Chapter 9.

Two-Beam InterferenceTwo-Beam Interference

Interferenceterm

Two-Beam InterferenceTwo-Beam Interference

The interference term is given by

The irradiances for beams 1 and 2 are given by :

x

Two-Beam InterferenceTwo-Beam InterferenceThe time averages are given by

The interference term is given by

: phase difference

Two-Beam InterferenceTwo-Beam InterferenceThe 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

When

VisibilityVisibility

Visibility = fringe contrast

{ } 10 minmax

minmax ≤≤+−

≡ V IIIIV

When

Therefore, V = 1

Conditions for good visibilityConditions 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 ExperimentYoung’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 InterferenceYoung’s Double Slit Interference

On the screen the irradiance pattern is given by

Assuming that y << s :

Bright fringes:

Dark fringes:

Young’s Double Slit InterferenceYoung’s Double Slit Interference

Interference Fringes From 2 Point SourcesInterference Fringes From 2 Point Sources

Interference Fringes From 2 Point SourcesInterference Fringes From 2 Point Sources

Two coherent point sources : P1 and P2

Interference With Virtual Sources:Fresnel’s Double Mirror

Interference With Virtual Sources:Fresnel’s Double Mirror

Hecht, Optics, Chapter 9.

Interference With Virtual Sources:Lloyd’s Mirror

Interference With Virtual Sources:Lloyd’s Mirror

mirror

Rotationstage

Lightsource

Interference With Virtual Sources:Fresnel’s Biprism

Interference With Virtual Sources:Fresnel’s Biprism

Hecht, Optics, Chapter 9.

Interference in Dielectric FilmsInterference in Dielectric Films

Analysis of Interference in Dielectric FilmsAnalysis of Interference in Dielectric Films

Analysis of Interference in Dielectric FilmsAnalysis 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 FilmsAnalysis 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

Fringes of Equal InclinationFringes of Equal Inclination

Fringes arise as Δ varies due to changesin the incident angle:

Constructive interference

Destructive interference

Fringes of Equal ThicknessFringes of Equal Thickness

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

Constructive interference

Destructive interference

Fringes of Equal Thickness: Newton’s RingsFringes of Equal Thickness: Newton’s Rings

Fringes of Equal Thickness: Newton’s RingsFringes of Equal Thickness: Newton’s Rings