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 Light
Chapter 10. Interference of Light
Two-Beam Interference Two-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
r r r r r
r r r r r
(polarization direction)
Hecht, Optics, Chapter 9.
Two-Beam Interference Two-Beam Interference
Interference term
Two-Beam Interference Two-Beam Interference
The interference term is given by
The irradiances for beams 1 and 2 are given by :
x
Two-Beam Interference Two-Beam Interference
The time averages are given by
The interference term is given by
: phase difference
Two-Beam Interference Two-Beam Interference
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 When
Visibility Visibility
Visibility = fringe contrast
{ 0 1 }
min max
min
max
≤ ≤
+
≡ − V
I I
I V I
When
Therefore, V = 1
Conditions for good visibility Conditions for good visibility
Sources must be:
same in phase evolution
in terms oftime (source frequency) Æ temporal coherence space (source size) Æ spatial coherence
Same in amplitude Same in polarization
Very good Normal
Very bad
Young’s Double Slit Experiment 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 between
geometric path difference and phase difference :
a
s
y
Δ
θ
Young’s Double Slit Interference
Young’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 Interference
Young’s Double Slit Interference
Interference Fringes From 2 Point Sources
Interference Fringes From 2 Point Sources
Interference Fringes From 2 Point Sources Interference 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
Rotation stage Light
source
Interference With Virtual Sources:
Fresnel’s Biprism
Interference With Virtual Sources:
Fresnel’s Biprism
Hecht, Optics, Chapter 9.
Interference in Dielectric Films
Interference in Dielectric Films
Analysis of Interference in Dielectric Films
Analysis of Interference in Dielectric Films
Analysis of Interference in Dielectric Films Analysis of Interference in Dielectric Films
The phase difference due to optical path length differences for the front and back reflections is given by
Analysis of Interference in Dielectric Films Analysis of Interference in Dielectric Films
Also need to account for phase differences Δr due to differences in the reflection process at the front and back surfaces
Constructive interference
Destructive interference
Fringes of Equal Inclination Fringes of Equal Inclination
Fringes arise as Δ varies due to changes
in the incident angle:
Constructive interference
Destructive interference
Fringes of Equal Thickness Fringes of Equal Thickness
When the direction of the incoming light is fixed, fringes arise as Δ varies due to changes
in the dielectric film thickness :
Constructive interference
Destructive interference
