Download - Multiple interference
Multiple interference
Optics, Eugene Hecht, Chpt. 9
• Multiple reflections give multiple beams
• First reflection has different sign
• Interior vs. exterior reflection
Multiple reflections
n1 nf n1
Multiple reflection analysis• Path difference between reflections
– = 2 nf d cos t
• Special case 1 -- = m – 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 -- = (m+1/2) – reflections alternate sign
– Er = E0r + E0trt’(1 - r2 + r4 -...)
– = E0r (1 + tt’/(1+r2)) = E0r /(1+r2)
General case -- resonance width• Round trip phase shift
– Er = E0r + (E0tr’t’e-i+ E0tr’3t’e -2i + E0tr’5t’e -3i +...)
– = E0r - E0tr’t’e-i(1 + r’2e -i + (r’2e -i )2 +...)
– = E0 (r + r’tt’ e-i /(1-r’2 e-i))
• Assume r’= - r and tt’=1-r2
• Define Finesse coeff (not Finesse)
n1 nf n1
i
i
r er
erEE
20 1
)1(
i
i
t er
ettEE
20 1
'
2
21
2
r
rF
2/sin1
2/sin2
2
F
F
I
I
I
R
2/sin1
12 FI
I
I
T
Interpret resonance width• Recall Finesse coeff
• For large r ~ 1– F ~ [2/(1-r2)]2
– Half power full-width ~ 1/2 = 1-r2
– Number of bounces: N ~ 1/(1- r2)
• Half power width 1/2 = 1/N
2
21
2
r
rF
Transmission = Airy function Reflection = 1 - Airy function
Airy function
2/sin1
2/sin2
2
F
F
I
I
I
R
2/sin1
12 FI
I
I
T
Path length sensitivity of etalonx/ = 1/2 = 1-r2 = 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 = r2, T = tt’
2/sin1
1
)1(1
2 FR
A
I
I
I
T
21
4
R
RF
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 Mirror
Input
Round trip conditions
Mirror Mirror l
Round trippath: 2 l sin
Walk offper 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(2i/N) + exp(4i/N) + … + exp(2(N-2)i/N) + exp(2(N-1)i/N) ) – = (1 + exp(2i/N) + exp(4i/N) + … + exp(-4i/N) + exp(-2i/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 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 45
Thick gratings• Many layers• Reflectivity per layer smallExamples: • Holograms -- refractive index variations• X-ray diffraction -- crystal planes• Acousto-optic shifters -- sound waves
– grating spacing given by sound speed, RF freq.
Gratingplanes
input
output
Integerwavelengths
Bragg angle:L = 2d sin nd = vsound / fmicrowave
vs
vs
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(2i/N) + exp(4i/N) + … + exp(2(N-2)i/N) + exp(2(N-1)i/N) ) – = (1 + exp(2i/N) + exp(4i/N) + … + exp(-4i/N) + exp(-2i/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
Ldifference
Etalon vs Bragg hologram• Bragg hologram has small r
– multiple bounces ignored• Etalon has big r
– weak beamlet trapped inside– interference gives high intensity
Gratingplanes
input
output
Integerw
avelengths
rtr
t2rt3r t4r
t5r
Multiple reflections ignored
Multiple reflections in etalon
Input
Mirror Mirror d
t2
r2 t2
r4 t2
r6 t2
r8 t2
r10 t2
- r ~ -1
r t2
r3 t2
r5 t2
r7 t2
r9 t2
(1 + r2 + r4 + …) = 1/ (1 - r2) = 1 / t2
cancels factor of t2 (1 + t + t2 + …) = 1/ (1 - t) = 1 / (1 - sqrt(T)) cancels factor of t2
2 d sin = n 2 N d sin = /cos Nhologram = t/d, Netalon = 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
differenced sin = n
Path differenceN 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
differenced sin = n
Path differenceN d sin = n
N d = D
Grating resolutionAperture resolution
D
Resonance width summary
Etalon Bragggrating
Diffractiongrating(transmission)
Aperture
Sin /2 N l cos /2 N d cos / N d cos / N d(cos
Factor of 2 transmission vs reflection• otherwise identical
Sagnac interferometer• Light travel time ccw
• Travel time cw
• Time difference
• Number of fringes
Rc
RtCCW
2
8
Rc
RtCW
2
8
2
4
c
At
c
AN
4
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 ~ 105 or greater• Led to super mirrors
– polished to Angstroms– ion beam machining
• Conventional mirrors– polished to ~ 100 nanometers– limited by grit size
Laser gyrodeveloped 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