chapter 7. interference of light - hanyangoptics.hanyang.ac.kr/~shsong/7-interference of...
TRANSCRIPT
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 = = +⎜ ⎟
⎝ ⎠