Download Optical interferometers

Basic electromagnetics
and interference
Optics, Eugene Hecht, Chpt 3
Maxwell’s equations
• Based on observation -- not derived
Induction

 
B 
CE  dl  A t  dS
Loop
voltage
Charges give
electric field
No magnetic
monopoles
Flux change
  1
 E  dS 
0
A
Electric
field flux
 dV
V
Charge
 
 B  dS  0
A
No net magnetic flux
through closed surface





Currents give
E  


magnetic field CB  dl     J   t   dS

A 
B
a
I
Current
Changing
electric field
Capacitor Q = CV
Q =  A E = V ( A /d)
C =  A /d
I = dQ/dt =  A dE/dt
Electromagnetic field in vacuum
• No sources of electric field, no currents

 
B 
CE  dl  A t  dS
 
 
AE  dS  0 AB  dS  0

 
E
CB  dl  0 0 A t  dS
Light speed: c = 1/() = w/k
B=E/c
Maxwell’s eqns -- differential form
 
  E   / 0

 
1 B
 E  
c t
 
 B  0

 

E
  B  J  
t

2

 E
2 E   0 0 2
t

2

 B
2 B   0 0 2
t
Propagating waves
E = E0 cos(kx - wt)
B = B0 cos(kx - wt)
Energy and momentum
•
•
•
•
Electric field UE = 0 E2/2
Magnetic field UB = B2/ (2 0)
Since c = 1/() -- UB = UE


 
Poynting vector: S  c 2 E  B cos 2 (k  r  wt )
0 0
0
• Average energy flow = c 0 E02 /2
• Momentum dp/dt = F = dU/dx -- p = (k/w) U = U/c
Photons
• Energy is quantized: U =  w
• Momentum also quantized: p =  k
Light is wave
• Electric field oscillates with position
– travelling wave
– wavelength l = c / n ~ 1/2 micron in visible
– electric fields can add or subtract (interference)
• Combine two laser beams
– Incoherent -- equal input intensity -- equal output intensities
– Coherent -- light can go one way, but not other -- intensity = sum of inputs
Light wave
E
field
position
Interference
Partial
mirror
180° phase shift
on reflection
Constructive
interference
• light
Destructive
interference
• no light
Interferometer
•
•
•
•
Split laser beams -- then recombine
Output light direction depends on path length difference
Path change ~ l/2 << 1 micron
Very sensitive
– accurate position measurement
– noisy
Interferometer
Mirror
Beamsplitter
Beamsplitter
Mirror
Interferometers
Mach-Zender -- Modulators for fiber communications
Mirror
Beamsplitter
Beamsplitter
Outputs
Inputs
Michaelson -- FTIR spectrometers
Input
Beamsplitter
Mirror
Mirror
Outputs
Mirror
Sanac -- Laser gyros for aircraft navigation
Input
Mirror
Mirror
Beamsplitter
Outputs
Mirror
Fabry-Perot -- Lasers and wavelength (ring version shown)
Input
Beamsplitter
Mirror
Beamsplitter
Output
Output
Mirror
Mach-Zender
• Simplest -- all inputs and outputs separate
– can cascade
– ex: quantum computing
• Used for high speed light modulation
– fiber communications
Mach-Zender Interferometer
Mirror
Beamsplitter
Beamsplitter
Outputs
Inputs
Mirror
Michaelson
• Like folded Mach-Zender
– beamsplitter serves an input and output
– first used to attempt detection of ether
– popular in optics courses
• Advantages:
Mirror
Input
Beamsplitter
Outputs
Mirror
=
– easy to change path length difference
– coherence length measurement
– FFT spectrometer
• Dis-advantages
Translation
stage option
– some output light goes back to source
– optical feedback
– problem for laser diodes
Mirror
Beamsplitter
Optical
feedback
Input
Outputs
Mirror
Sanac
• Replace 2nd beamsplitter with mirror
– used in rotation sensors -- laser gyro (ex: airplanes)
• Path lengths always equal
– counter-propagating, low noise
• Only non-reciprocal phase shifts important
– magnetic field Zeeman
– general relativity -- rotation
– Fizeau drag
Sanac
Mirror
Input
Mirror
Beamsplitter
Outputs
Mirror
Etalon and ring cavity
• Multi-pass devices
• Ring
–
–
–
–
Mach-Zender with beamsplitters rotated 90°
Interference after round trip
need long coherence length
used in laser cavities
• Etalon
–
–
–
–
–
Etalon
Beamsplitter
interference after round trip
optical standing wave
used in laser cavities, filters
Advantage -- simple
Disadvantage -- optical feedback
Beamsplitter
Input
Output
Output
Ring
Mirror
Input
Beamsplitter
Beamsplitter
Output
Output
Mirror
Real interferometers
General case
• Alignment not exact -- fringes
• Curvatures not exact -- rings
Straight fringes
constructive
destructive
constructive
Rings -- “bulls eye”
constructive
destructive
Coherence length
• Light beam composed of more than one wavelength
• Example: two wavelengths
• Path length difference = 1/2 beat wavelength
– one wavelength deflects downward
– other wavelength deflects upward
– net result -- no interference fringes visible
Dual wavelength laser beam
Interference of two-wavelength beams
Beat length
Wavelength #1
Wavelength #2
General case
•
•
•
•
Many wavelengths
Interference only over limited path difference
Define as “coherence length”
Fringe strength vs. path difference
– related to spectral content of light
– Fourier transform spectrometer
Multi wavelength light wave
E
field
position
Linear polarization
• E-field magnitude oscillates
• Direction fixed
• Arbitrary polarization angle
– superposition of x and y polarized waves
– real numbers
Example
45 ° linear polarization
Time
evolution
Circular polarization
• E-field magnitude constant
• Direction rotates
• Complex superposition of x and y polarizations
– x and y in quadrature
Time
evolution
Example:
right circular polarization
Waveplates
• Polarization converters
• One linear polarization direction propagates faster
• Half wave plate -- phase delay 180°
Rotate linear pol. by angle 2q
– rotate linear polarization up to 90°
– fast axis at 45° to input polarization direction
• Quarter wave plate -- phase delay 90°
– convert linear to circular polarization
– R or L for fast axis +45 or -45 to input pol.
Create circular polarization
Retardation of
one polarization
Isolators -- 1
• Polarizer and quarter waveplate
• Double pass through quarter wave plate
– same as half wave plate
– rotate polarization by up to 90°
• Polarizer blocks reflected light
Polarizer
Quarter wave
Reflecting element