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