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Basic electromagneticsand interference
Optics, Eugene Hecht, Chpt 3
Maxwell’s equations• Based on observation -- not derived
A
CSd
t
BldE
V
AdVSdE
0
1
0A SdB
A
CSd
t
EJldB
Induction
Charges give electric field
No magneticmonopoles
Currents givemagnetic field
I a
B
Loopvoltage Flux change
Electricfield flux Charge
No net magnetic fluxthrough closed surface
Current
Changing electric field
Capacitor Q = CVQ = A E = V (A /d)C = A /dI = dQ/dt = A dE/dt
Electromagnetic field in vacuum• No sources of electric field, no currents
A
CSd
t
BldE
0A SdE
0A SdB
A
CSd
t
EldB
00
2
2
002
t
EE
2
2
002
t
BB
E = E0 cos(kx - t)B = B0 cos(kx - t)
0/ E
0 B
t
B
cE
1
t
EJB
Maxwell’s eqns -- differential form
Propagating waves
Light speed: c = 1/() = /kB = E / c
Energy and momentum• Electric field UE = 0 E2/2
• Magnetic field UB = B2/ (2 0)
• Since c = 1/() -- UB = UE
• Poynting vector:
• Average energy flow = c 0 E02 /2
• Momentum dp/dt = F = dU/dx -- p = (k/) U = U/c
Photons• Energy is quantized: U = • Momentum also quantized: p = k
)(cos2000
2 trkBEcS
Light is wave• Electric field oscillates with position
– travelling wave– wavelength = c / ~ 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
Efield
position
Light wave
Partial mirror
180° phase shifton reflection
Constructiveinterference
• light
Destructiveinterference• no light
Interference
Interferometer • Split laser beams -- then recombine• Output light direction depends on path length difference• Path change ~ /2 << 1 micron• Very sensitive
– accurate position measurement– noisy
Interferometer
Beamsplitter Beamsplitter
Mirror
Mirror
Interferometers
Beamsplitter Beamsplitter Mirror
Mirror
Inputs Outputs
Beamsplitter Mirror
Mirror
Input
Outputs
Mirror
Beamsplitter
Mirror Input Mirror
Outputs
Sanac -- Laser gyros for aircraft navigation
Michaelson -- FTIR spectrometers
Mach-Zender -- Modulators for fiber communications
Beamsplitter Mirror
Mirror
Input
Output
Output
Beamsplitter
Fabry-Perot -- Lasers and wavelength (ring version shown)
Mach-Zender• Simplest -- all inputs and outputs separate
– can cascade – ex: quantum computing
• Used for high speed light modulation– fiber communications
Mach-Zender Interferometer
Beamsplitter Beamsplitter
Mirror
Mirror
Inputs Outputs
Michaelson • Like folded Mach-Zender
– beamsplitter serves an input and output– first used to attempt detection of ether– popular in optics courses
• Advantages:– easy to change path length difference– coherence length measurement– FFT spectrometer
• Dis-advantages– some output light goes back to source– optical feedback– problem for laser diodes
=Beamsplitter
Mirror
Mirror
Input
Outputs
Beamsplitter
Mirror
Mirror
Input
Outputs
Translationstage option
Opticalfeedback
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
Beamsplitter
Mirror
Mirror
Input Mirror
Outputs
Sanac
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– interference after round trip– optical standing wave– used in laser cavities, filters– Advantage -- simple– Disadvantage -- optical feedback
Beamsplitter
Mirror
Mirror
Input
Output
Output
Beamsplitter
Ring
Beamsplitter Input
Output Output
Beamsplitter
Etalon
Real interferometersGeneral case• Alignment not exact -- fringes• Curvatures not exact -- rings
constructivedestructiveconstructive
constructive
destructive
Straight fringes
Rings -- “bulls eye”
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
Interference of two-wavelength beams
Wavelength #1
Wavelength #2
Dual wavelength laser beam
Beat length
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
Efield
position
Multi wavelength light wave
Linear polarization• E-field magnitude oscillates
• Direction fixed
• Arbitrary polarization angle– superposition of x and y polarized waves
– real numbers
Timeevolution
Example45 ° linear polarization
Circular polarization• E-field magnitude constant
• Direction rotates
• Complex superposition of x and y polarizations– x and y in quadrature
Timeevolution
Example:right circular polarization
Waveplates• Polarization converters• One linear polarization direction propagates faster• Half wave plate -- phase delay 180°
– 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.
Rotate linear pol. by angle 2
Retardation of one polarization
Create circular 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
Quarter wavePolarizer Reflecting element