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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
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