1 / 20

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

shay-kim
Download Presentation

Multiple interference

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related