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


(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


min 1 2 1 22I I I I I


max 1 2 1 22I I I I I


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


Point light source

M1

M2

detector

o

1l

2l

1E

2E

12

2d m

max 1 2 1 22I I I I I


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


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


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


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

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