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