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Basic electromagnetics and interference

Basic electromagnetics and interference. Optics, Eugene Hecht, Chpt 3. B. a. I. Maxwell’s equations. Induction. Based on observation -- not derived. Loop voltage. Flux change. Charges give electric field. Capacitor Q = CV Q = e A E = V ( e A /d) C = e A /d

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Basic electromagnetics and interference

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  1. Basic electromagneticsand interference Optics, Eugene Hecht, Chpt 3

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

  3. Maxwell’s eqns -- differential form Electromagnetic field in vacuum Propagating waves E = E0 cos(kx - wt) B = B0 cos(kx - wt) • No sources of electric field, no currents Light speed: c = 1/(me) = w/k B = E / c

  4. Energy and momentum • Electric field UE = e0 E2/2 • Magnetic field UB = B2/ (2 m0) • Since c = 1/(me) -- UB = UE • Poynting vector: • Average energy flow = c e0 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

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

  6. 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

  7. Mirror Beamsplitter Beamsplitter Outputs Inputs Mirror Input Mirror Beamsplitter Outputs Mirror Mirror Input Mirror Beamsplitter Outputs Mirror Input Beamsplitter Beamsplitter Output Output Mirror Interferometers Mach-Zender -- Modulators for fiber communications Michaelson -- FTIR spectrometers Sanac -- Laser gyros for aircraft navigation Mirror 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 Mirror Beamsplitter Beamsplitter Outputs Inputs Mirror

  9. Translation stage option Mirror Beamsplitter Input Optical feedback Mirror Outputs Michaelson Mirror Input Beamsplitter Outputs • 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 Mirror =

  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 Sanac Mirror Input Mirror Beamsplitter Outputs Mirror

  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 Etalon Beamsplitter Beamsplitter Input Output Output Ring Mirror Input Beamsplitter Beamsplitter Output Output Mirror

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

  13. Interference of two-wavelength beams Dual wavelength laser beam Beat length Wavelength #1 Wavelength #2 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

  14. Multi wavelength light wave E field position 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

  15. Linear polarization Time evolution • E-field magnitude oscillates • Direction fixed • Arbitrary polarization angle • superposition of x and y polarized waves • real numbers 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 Rotate linear pol. by angle 2q • 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. Create circular polarization Retardation of one 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

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