1. Basic electromagnetics and interference Optics, Eugene Hecht, Chpt 3

2. Maxwell’s equations Based on observation -- not derived Induction Charges give electric field No magnetic monopoles Currents give magnetic field I a B Loop voltage Flux change Electric field flux Charge No net magnetic flux through closed surface Current Changing electric field Capacitor Q = CV Q =  A E = V (  A /d) C =  A /d I = dQ/dt =  A dE/dt

3. Electromagnetic field in vacuum No sources of electric field, no currents E = E 0 cos(kx -  t) B = B 0 cos(kx -  t) Maxwell’s eqns -- differential form Propagating waves Light speed: c = 1/  (  ) =  /k B = E / c

4. Energy and momentum Electric field U E =  0 E 2 /2 Magnetic field U B = B 2 / (2  0 ) Since c = 1/  (  ) -- U B = U E Poynting vector: Average energy flow = c  0 E 0 2 /2 Momentum dp/dt = F = dU/dx -- p = (k/  ) U = U/c Photons Energy is quantized: U =   Momentum also quantized: p =  k

5. 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 E field position Light wave Partial mirror 180° phase shift on reflection Constructive interference light Destructive interference no light Interference

6. 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 Mirror

7. Interferometers Beamsplitter Mirror Inputs Outputs Beamsplitter 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 Input Output Beamsplitter Fabry-Perot -- Lasers and wavelength (ring version shown)

8. 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 Mirror Inputs Outputs

9. 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 Input Outputs Beamsplitter Mirror Input Outputs Translation stage option Optical feedback

10. 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 Input Mirror Outputs Sanac

11. 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 Input Output Beamsplitter Ring Beamsplitter Input Output Beamsplitter Etalon

12. Real interferometers General case Alignment not exact -- fringes Curvatures not exact -- rings constructive destructive constructive destructive Straight fringes Rings -- “bulls eye”

13. 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

14. 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 E field position Multi wavelength light wave

15. Linear polarization E-field magnitude oscillates Direction fixed Arbitrary polarization angle –superposition of x and y polarized waves –real numbers Time evolution Example 45 ° linear polarization

16. 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

17. 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

18. 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 wave Polarizer Reflecting element