Multiple interference Optics, Eugene Hecht, Chpt. 9

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

n 1 n f n 1 Multiple reflection analysis Path difference between reflections –  = 2 n f d cos  t Special case 1 --  = m –E r = E 0 r - (E 0 trt’+ E 0 tr 3 t’+ E 0 tr 5 t’+...) –= E 0 r - E 0 trt’(1 + r 2 + r ) = E 0 r (1 - tt’/(1-r 2 )) assumed r’ = -r –From formulas for r & t, tt’=1-r 2 –result: E r = 0 Special case 2 --  = (m+1/2) –reflections alternate sign –E r = E 0 r + E 0 trt’(1 - r 2 + r ) – = E 0 r (1 + tt’/(1+r 2 )) = E 0 r /(1+r 2 )

General case -- resonance width Round trip phase shift  –E r = E 0 r + (E 0 tr’t’e -i  + E 0 tr’ 3 t’e -2i  + E 0 tr’ 5 t’e -3i  +...) –= E 0 r - E 0 tr’t’e -i  (1 + r’ 2 e -i  + (r’ 2 e -i  ) ) –= E 0 (r + r’tt’ e -i  /(1-r’ 2 e -i  )) Assume r’= - r and tt’=1-r 2 Define Finesse coeff (not Finesse) n 1 n f n 1

Interpret resonance width Recall Finesse coeff For large r ~ 1 –F ~ [2/(1-r 2 )] 2 –Half power full-width ~  1/2 = 1-r 2 –Number of bounces: N ~ 1/(1- r 2 ) Half power width  1/2 = 1/N Transmission = Airy function Reflection = 1 - Airy function Airy function Path length sensitivity of etalon  x/ =  1/2 = 1-r 2 = 1/N

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

Include loss Conservation of energy T + R + A = 1 R = r 2, T = tt’ FSR linewidth Loss term

Etalons Multiple reflections If incidence angle  small enough, reflections overlap – interference Max. number of overlapping beamlets = w / 2 l cot , 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  = n –fixed angle gives limited choices for l (resonance spacing) –fixed l gives limited choices for angle (rings) Multiple reflections in etalon Input  Mirror Input Round trip conditions Mirror l Round trip path: 2 l sin  Walk off per pass: 2 l cot  l 1st reflection:  phase shift 

Resonance width of etalon Sum of round trip beamlets interfere destructively Occurs when phase difference between first and last beamlet is 2  –(1 + exp(i  ) + exp(2i  ) + … + exp((N-1)i  ) ) –= (1 + exp(2  i/N) + exp(4  i/N) + … + exp(2(N-2)  i/N) + exp(2(N-1)  i/N) ) –= (1 + exp(2  i/N) + exp(4  i/N) + … + exp(-4  i/N) + exp(-2  i/N) ) Resonance width –2 N ( l+  l) sin (  +  = (nN+1) –assumed reflectivity high Angle -- sin  = / 2Nl cos  Spacing --  l = / 2Nsin  –path length  x = 2  l sin  = / 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 After N round trips: total path length = 2 N l sin  Multiple reflections in etalon Input  Mirror l 1 exp(i  ) exp(2 i  ) exp(3 i  ) exp(4 i  ) exp(5 i  ) cancellations

Summing waves Add series of waves having different phases –Special case of equally spaced phases Sum of 5 waves with phases up to  Sum of 9 waves with phases up to 4  5

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. Grating planes input output Integer wavelengths Bragg angle:  L = 2d sin    n d = v sound / f microwave vsvs vsvs

Multi-layer analysis L = 2Nd sin   = sin -1 ( / 2d) L  d Nd Bragg angle selectivity find change in angle that changes L by phase angles vary from 0 to 2  sum over all reflected beams adds to zero L+  L = 2Nd sin (   sin -1 (  2Nd cos   L = (2Nd sin  )  cos  Sum of round trip beamlets interfere destructively Occurs when phase difference between first and last beamlet is 2  –(1 + exp(i  ) + exp(2i  ) + … + exp((N-1)i  ) ) –= (1 + exp(2  i/N) + exp(4  i/N) + … + exp(2(N-2)  i/N) + exp(2(N-1)  i/N) ) –= (1 + exp(2  i/N) + exp(4  i/N) + … + exp(-4  i/N) + exp(-2  i/N) ) Bragg selectivity: 2 N d sin   =  /cos  cancellations

Bragg angle selectivity vs Bragg angle For transmission geometry –  ~ 0, cos  ~ 1,  2Nd cos  ) ~  2Nd) –  small –most selectivity For reflection geometry –  ~ 0,  large –not very sensitive to angle  L  Nd difference  Nd  L difference

Etalon vs Bragg hologram Bragg hologram has small r –multiple bounces ignored Etalon has big r –weak beamlet trapped inside –interference gives high intensity Grating planes input output Integer wavelengths r tr t2rt2r t3rt3rt4rt4r t5rt5r Multiple reflections ignored Multiple reflections in etalon Input  Mirror d t 2 r 2 t 2 r 4 t 2 r 6 t 2 r 8 t 2 r 10 t 2 - r ~ -1 r t 2 r 3 t 2 r 5 t 2 r 7 t 2 r 9 t 2  (1 + r 2 + r 4 + …) = 1/ (1 - r 2 ) = 1 / t 2 cancels factor of t 2  (1 + t + t 2 + …) = 1/ (1 - t) = 1 / (1 - sqrt(T)) cancels factor of t 2 2 d sin  = n 2 N d sin   =  /cos  N hologram = t/d, N etalon = 1/(1-R)  d t

Multiple slits or thin gratings Can be array of slits or mirrors –Like multiple interference –Diffraction angles: d sin  = n –Diffraction halfwidth (resolution of grating): N d sin    =  cos  d  Path difference d sin  = n Path difference N d sin    = n N d = D Grating resolution d  Path difference = d sin  Grating diffraction

Angular resolution of aperture First find angular resolution of aperture –Like multiple interference –Diffraction angles: d sin  = n –Diffraction halfwidth (resolution of grating): N d sin    =  cos  Take limit as d --> 0, but N d = a (constant) –Diffraction angle: sin  = n  / d only works for n = 0,  = 0 -- (forward direction) –Angular resolution: sin    = / N d = / D (cos  = 1) d  Path difference d sin  = n Path difference N d sin    = n N d = D Grating resolution Aperture resolution   D

Resonance width summary Factor of 2 transmission vs reflection otherwise identical

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

Laser gyro 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 ~ 10 5 or greater Led to super mirrors –polished to Angstroms –ion beam machining Conventional mirrors –polished to ~ 100 nanometers –limited by grit size Laser gyro developed for aircraft

Wavefront splitting interferometer Young’s double slit experiment –Interference of two spherical waves –Equal path lengths -- linear fringes m = a sin  m = diffraction order