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Multiple interference

Multiple interference. Optics, Eugene Hecht, Chpt. 9. Multiple reflections. Multiple reflections give multiple beams First reflection has different sign Interior vs. exterior reflection. n 1 n f n 1. Multiple reflection analysis. Path difference between reflections

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Multiple interference

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  1. Multiple interference Optics, Eugene Hecht, Chpt. 9

  2. Multiple reflections • Multiple reflections give multiple beams • First reflection has different sign • Interior vs. exterior reflection

  3. n1 nf n1 Multiple reflection analysis • Path difference between reflections • L = 2 nf d cos qt • Special case 1 -- L = m l • Er = E0r - (E0trt’+ E0tr3t’+ E0tr5t’+...) • = E0r - E0trt’(1 + r2 + r4 +...) = E0r (1 - tt’/(1-r2)) • assumed r’ = -r • From formulas for r & t, tt’=1-r2 • result: Er = 0 • Special case 2 -- L = (m+1/2) l • reflections alternate sign • Er = E0r + E0trt’(1 - r2 + r4 -...) • = E0r (1 + tt’/(1+r2)) = E0r /(1+r2)

  4. n1 nf n1 General case -- resonance width • Round trip phase shift d • Er = E0r + (E0tr’t’e-id+ E0tr’3t’e -2id + E0tr’5t’e -3id +...) • = E0r - E0tr’t’e-id(1 + r’2e -id+ (r’2e -id)2 +...) • = E0 (r + r’tt’ e-id /(1-r’2 e-id)) • Assume r’= - r and tt’=1-r2 • Define Finesse coeff (not Finesse)

  5. Interpret resonance width Airy function • Recall Finesse coeff • For large r ~ 1 • F ~ [2/(1-r2)]2 • Half power full-width ~ d1/2 = 1-r2 • Number of bounces: N ~ 1/(1- r2) • Half power width d1/2 = 1/N Path length sensitivity of etalon Dx/l = d1/2 = 1-r2 = 1/N Transmission = Airy function Reflection = 1 - Airy function

  6. Define Finesse and Q • Important quantity is: Q = Free Spectral Range (FSR) / linewidth • Q = Finesse/2 = (p/4) F = (p/2) [R / (1-R)] • Finesse = p [R / (1-R)] ~ p / (1-R), when R ~1 FSR linewidth

  7. Include loss Loss term • Conservation of energy T + R + A = 1 • R = r2, T = tt’ FSR linewidth

  8. Etalons • Multiple reflections • If incidence angle qsmall enough, reflections overlap – interference • Max. number of overlapping beamlets = w / 2 l cot q , where w = beam diameter • Round trip phase determines whether interference constructive or destructive • round trip path length must be multiple of wavelength • Resonance condition: 2 l sin q = n l • fixed angle gives limited choices for l (resonance spacing) • fixed l gives limited choices for angle (rings) Multiple reflections in etalon Round trip conditions Input Mirror Mirror q Mirror Mirror l Input q 1st reflection: p phase shift Walk off per pass: 2 l cot q Round trip path: 2l sin q l

  9. Resonance width of etalon • Sum of round trip beamlets interfere destructively • Occurs when phase difference between first and last beamlet is 2 p • (1 + exp(i f) + exp(2i f) + … + exp((N-1)i f) ) • = (1 + exp(2pi/N) + exp(4pi/N) + … + exp(2(N-2)pi/N) + exp(2(N-1)pi/N) ) • = (1 + exp(2pi/N) + exp(4pi/N) + … + exp(-4pi/N) + exp(-2pi/N) ) • Resonance width • 2 N(l+Dl) sin (q+Dq) = (nN+1) l • assumed reflectivity high • Angle -- sin Dq = l / 2Nl cos q • Spacing -- Dl = l / 2Nsin q • path length Dx = 2 Dl sin q = l / N • agrees with exact equation • Depends on distance and angle • rings become sharp • Quality factor • Q ~ resonance-freq / linewidth • Q ~ N -- Field amplitude ~ N 2 -- Intensity cancellations Multiple reflections in etalon Input Mirror Mirror After N round trips: total path length = 2 Nl sin q q l 1 exp(i f) exp(2 i f) exp(3 i f) exp(4 i f) exp(5 i f)

  10. Summing waves • Add series of waves having different phases • Special case of equally spaced phases Sum of 5 waves with phases up to p Sum of 9 waves with phases up to 4p / 5

  11. output Integer wavelengths input vs vs Grating planes Thick gratings • Many layers • Reflectivity per layer small Examples: • Holograms -- refractive index variations • X-ray diffraction -- crystal planes • Acousto-optic shifters -- sound waves • grating spacing given by sound speed, RF freq. Bragg angle: dL = 2d sin qB = nl d = vsound / fmicrowave

  12. Multi-layer analysis • Sum of round trip beamlets interfere destructively • Occurs when phase difference between first and last beamlet is 2 p • (1 + exp(i f) + exp(2i f) + … + exp((N-1)i f) ) • = (1 + exp(2pi/N) + exp(4pi/N) + … + exp(2(N-2)pi/N) + exp(2(N-1)pi/N) ) • = (1 + exp(2pi/N) + exp(4pi/N) + … + exp(-4pi/N) + exp(-2pi/N) ) • Bragg selectivity: • 2 Nd sin Dq = l /cos q cancellations • Bragg angle selectivity • find change in angle that changes L by l • phase angles vary from 0 to 2p • sum over all reflected beams adds to zero L = 2Nd sin q = Nl L q = sin-1 (l / 2d) L+DL = 2Nd sin (q+Dq)= (N+1)l q DL = (2Nd sinDq)cosq = l d Dq = sin-1 (l /2Nd cos q) Nd

  13. NdL difference difference NdL q q Nd Nd Bragg angle selectivity vs Bragg angle • For transmission geometry • q ~ 0, cosq~ 1,Dq = l /(2Nd cos q) ~ l /(2Nd) • Dq small • most selectivity • For reflection geometry • q ~ 0, Dq large • not very sensitive to angle

  14. Multiple reflections ignored output r tr t2r t3r t4r input t5r Integer wavelengths Grating planes Etalon vs Bragg hologram 2 d sin q = n l 2 Nd sin Dq = l /cos q Nhologram = t/d, Netalon = 1/(1-R) • Bragg hologram has small r • multiple bounces ignored • Etalon has big r • weak beamlet trapped inside • interference gives high intensity Multiple reflections in etalon Input Mirror Mirror q d - r ~ -1 rt2 r3 t2 r5 t2 r7 t2 r9 t2 t2 r2 t2 r4 t2 r6 t2 r8 t2 r10 t2 q d t S (1 + r2 + r4 + …) = 1/ (1 - r2) = 1 / t2 cancels factor of t2 S (1 + t + t2 + …) = 1/ (1 - t) = 1 / (1 - sqrt(T)) cancels factor of t2

  15. Grating diffraction q d Path difference = d sin q Multiple slits or thin gratings • Can be array of slits or mirrors • Like multiple interference • Diffraction angles: d sin q = n l • Diffraction halfwidth (resolution of grating): N d sin q1/2= l / cos q Grating resolution Path difference N d sin q1/2= n l q Path difference d sin q = n l d N d = D

  16. Angular resolution of aperture • First find angular resolution of aperture • Like multiple interference • Diffraction angles: d sin q = n l • Diffraction halfwidth (resolution of grating): N d sin q1/2= l / cos q • Take limit as d --> 0, but N d = a (constant) • Diffraction angle: sin q = n l / d • only works for n = 0, q = 0 -- (forward direction) • Angular resolution: sin q1/2= l/ N d = l/ D (cos q = 1) Aperture resolution Grating resolution Path difference N d sin q1/2= n l q1/2 D q Path difference d sin q = n l d N d = D

  17. Resonance width summary • Factor of 2 transmission vs reflection • otherwise identical

  18. Sagnac interferometer • Light travel time ccw • Travel time cw • Time difference • Number of fringes Fringe shift ~ 4 % for 2 rev/sec

  19. Laser gyro Laser gyro developed for aircraft • Closed loop • Laser can oscillate both directions • High reflectivity mirrors • Improve fringe resolution • Earth rotation = 1 rev/day at poles • 25 ppm of fringe • Need Q ~ 105 or greater • Led to super mirrors • polished to Angstroms • ion beam machining • Conventional mirrors • polished to ~ 100 nanometers • limited by grit size

  20. Wavefront splitting interferometer • Young’s double slit experiment • Interference of two spherical waves • Equal path lengths -- linear fringes m l = a sin q m = diffraction order

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