Fourier Analysis and Interferometerscourses.washington.edu/phys331/xu/2015/Lecture_2-2015.pdfFourier...
Transcript of Fourier Analysis and Interferometerscourses.washington.edu/phys331/xu/2015/Lecture_2-2015.pdfFourier...
Fourier Analysis and Interferometers
Important: Speed of light repot is due this week.
Phys 331, Lecture 2, October 06, 2014
Today Lecture: Michelson Interferometer
Monday1:30
AD
Tuesday 1:30
AA
Wednesday1:30
AB
Friday 1:30
AC
Fraunhofer &
Fresnel
Diffraction
Fabry-Perot
Interferometer
Michelson
Interferometer
Concave Diffraction
Grating
Reflection from a
Dielectric Surface
Faraday Rotation
Holography
Must complete Fabry-
Perot or Michelson
first. 1-d diffraction
experiment
Optics Lab — Physics 331 Sign-up Sheet
Monday1:30
AD
Tuesday 1:30
AA
Wednesday1:30
AB
Friday 1:30
AC
Fraunhofer &
Fresnel
Diffraction
Fabry-Perot
Interferometer
Michelson
Interferometer
Concave Diffraction
Grating
Reflection from a
Dielectric Surface
Faraday Rotation
Holography
Must complete Fabry-
Perot or Michelson
first. 1-d diffraction
experiment
Optics Lab — Physics 331 Sign-up Sheet
Please remember what you have signed up
Fourier Analysis
Fourier series for periodic functions:
Hecht, Pg 304
Fourier’s Theorem: a function f(x), having a spatial period λ, can be synthesized by a
sum of harmonic functions whose wavelengths are integral of submultiples of λ.
0
0 0
( ) cos( ) sin( )2
m m
m m
Af x A mkx B mkx
0
2( )cos( )mA f x mkx dx
0
2( )sin( )mB f x mkx dx
k=2π/ λ=2 πσ
Determine A and B through Fourier analysis.
2
/mmk k
m
Wave number
(spatial frequency)
m=1 fundamental frequency
m>1 high order harmonics
“EVEN” Square Wave0
4 sin 2 /0;
2 /
0
m
m
m aA A
a m a
B
Decomposing a Square Wave
Decomposing Periodic Functions
+1
f(x)
λ/a-λ/a
0
( ) cos( )mf x A mkx
x
sinc(u)=sin(u)/u
0
0 0
( ) cos( ) sin( )2
m m
m m
Af x A mkx B mkx
Even function: f(x)=f(-x)
Hecht, Pg 312, Fig. 7.34
The square pulse and its transform
Fourier Integrals and Fourier Transforms
0
sin( / 2)( )
/ 2
kLA k E L
kL
The sinc function.
Inverse relationship between widths
FT in single-slit Fraunhofer diffraction
0 0
1( ) ( )cos( ) ( )sin( )f x A k kx dk B k kx dk
( ) ( )cos( )A k f x kx dx
( ) ( )sin( )B k f x kx dx
Wave Propagation of Light
( . )i k r tAe T
C*T=λ
λ
r (distance)
Fix r
Coherence Time and Coherence Length
( . )i k r tAe
2 /k
Ground state
Excited state
o oE
𝐸 = 𝐸𝑜 ± ∆𝐸/2
𝜔 = 𝜔𝑜 ± ∆𝜔/2
A
o
2
is the bandwidth
1/ct is the coherence time
c cc t L is the coherence length
∆𝜔
Wave Propagation of Light
1/ct c cc t L is the coherence length
Simple explanation of Optical Interference
Constructive Interference
Destructive Interference
8 123 10 10 300 2c t m d
8 63 10 10 300 2c t m d Laser:
Hg lamp:
8 143 10 10 3 2c t m d White light:
Laser, Hg Lamp, and white light
Which is the most coherent light source of the three?
Michelson Interferometer
Amplitude splitting interfrometer
(1) Measure wavelength; (2) Coherence length
𝐸𝑖𝑘𝑟 k.r represents the optical phase
https://www.youtube.com/watch?v=j-u3IEgcTiQ
Point light source
M2o
1l
2l
1E
2E
M1
Michelson Interferometer: Conceptual Rearrangement
Optical phase =k.r
d
Point light source
M2
detector
o1l
2l
1E
2E
M1
Optical path difference:
2(l2 – l1)=2d
Optical phase =2k.d
detector
Michelson Interferometer – simplified picture
1.(2 )
1 1
ik lE Ae
2.(2 )
2 2
ik lE A e
2 2 2
1 2 1 2 1 22 cosE E A A A A
Point light source
M1
M2
detector
o
1l
2l
1E
2E
2 2I E A
ie Phase difference
reflection
Built-in π phase shift between
two optical paths (OM1 and
OM2) due to the reflection.
See Hecht 116-117;
1 2 1 22 cosI I I I I
Optical phase =2k.d + π
𝛿 =2𝜋
𝜆. 2 l2 − l1 + 𝜋
Michelson Interferometer – simplified picture
(2 1)
.2 2
m
k d m
2d m
min 1 2 1 22I I I I I
Destructive interference
1 2 1 22 cosI I I I I
2
.2 (2 1)
m
k d m
If
or1
22
d m
max 1 2 1 22I I I I I
Constructive interference
2.2d
Michelson Interferometer – simplified picture
min 1 2 1 22I I I I I
Destructive interference
max 1 2 1 22I I I I I
Constructive interference
Question, in order to get the best contrast
between Imax and Imin, what are the values of
I1 and I2 ? i.e. what type of beam splitter shall we
use?
Interference Contrast
max 1 2 1 22I I I I I min 1 2 1 22I I I I I
1 2max min
max min 1 2
2I II I
contrastI I I I
1 2I I Contrast=1
Michelson Interferometer – simplified picture
Point light source
M1
M2
detector
o
1l
2l
1E
2E
12
2d m
max 1 2 1 22I I I I I
Constructive interference
From the m to the (m+1)
order of constructive
interference, the optical
path dereference is 2d=λ
1 2 1 22 cosI I I I I
Intensity for monochromiatic source:
For a spectrally broad source:
This is a Fourier Transform between position
() and momentum (s).
Inverse Fourier Transform -> optical frequency
Michelson Interferometer (Fourier Transform Spectrometer)
( ) ( ) ( )cos 2I I d I ds s s s s
Constant background Fourier Transform between the frequency
domain and interference pattern
( ) ( ( )) ( ( ))o o oI I s ss s s s s
An example: a doubleto ss o ss s
ab1
0{
a=b
a b
( ) 2 ( )cos 2
2 cos(2 ( )) cos(2 ( ))
2 (1 cos(2 )cos(2 ))
o
o o o o
o o
I I I d
I I s s
I s
s s s
s s
s
Michelson Interferometer
Various I(σ)
Doppler broadened lines
2
2
4ln 2( )
( )( )
o
oI I e
s s
ss
2( )
4ln 2( ) (1 cos 2 )o oI I e
s
vc
sos
( ) ( ( )) ( ( ))o o oI I s ss s s s s
( ) 2 (1 cos(2 )cos(2 ))o oI I s s
Two sharp lines
o ss o ss
Two equal intensity Doppler broadened lines
2( )
4ln 2( ) (1 cos(2 )cos(2 ))o oI I e s
s
so ss o ss
( ) ( ) ( )cos 2I I d I ds s s s s
( ) (1 cos2 )o oI I s Single frequency with infinite
coherence length
Doppler broadened lines
2
2
4ln 2( )
( )( )
o
oI I e
s s
ss
2( )
4ln 2( ) (1 cos 2 )o oI I e
s
vc
1/2
2ln 2
Michelson Interferometer
Two equal intensity Doppler broadened lines
2( )
4ln 2( ) (1 cos(2 )cos(2 ))o oI I e s
s
Video
Is this correct?
Point light source
M2
detector
o1l
2l
1E
2E
M1
Michelson Interferometer: Conceptual Rearrangement
M2O
1l
2l
1E
2E
M1
S
O1
M2’ M1’
O2
2d
(O1O) – (O2O)=2(M1O – M2O) = 2d
Michelson IFM – Finite Light Spot
M2
O
1l
2l
1E
2E
M1 O1M2’ M1’
O2
2d
θ
2 cosd
Destructive interference
when round trip optical
path difference is a
multiple of the wavelength.
When 2dcos = m
screen
No interference when path
difference > coherence length
(Think Fourier)8 123 10 10 300 2c t m d
8 63 10 10 300 2c t m d Laser:
Hg lamp:
8 143 10 10 3 2c t m d White light:
First part of Michelson
Lab. What happens
when d0?
0
1
rpθp
2dcos = m
Destructive Interference