Phasor Calculations in LIGO Physics 208, Electro- Peter Beyersdorf

Document info Case study 1, 1 LIGO interferometer

“Power Recycled Fabry-Perot Michelson” interferometer

10m modecleaner filters noise from

4 km Fabry-Perot arm cavities increase the effective length of the arms

Michelson arm lengths are set so that output port is dark

“Power recycling mirror” resonantly enhances the power in the interferometer

Case study 1, 2 Interferometer Control

Requirements Signal – a beam whose phase sensitive to the length to be controlled Local Oscillator – a beam whose phase is insensitive to that length. Detection of the phase between signal and local oscillator

Pound-Drever-Hall method Transmission of cavity ΕΟΜ X P.D.H. input spectrum

Case study 1, 3 Mode Cleaner l Triangular modecleaner has a perimeter Er r,t p=20 m, unit reflectivity end mirror and r,t Et equal, lossless input/output couplers. Ein Ec Illuminated with a steady wave of wavelength λ. The fields transmitting (Et) reflecting from (Er) and circulating in (Ec) the cavity are proportional to the input field (Ein) with the relations r=1

2 ikp ikl ik(p l) Ec = tEin + ( r) e Ec , Et = te Ec , and Er = rEin rte − Ec − − 2 ikl t2eik(p l) giving tEin t e Ein − Ec = , E = , and Er = r 1 Ein 1 r2eikp t 1 r2eikp − 1 r2eikp − − ! − " Ein ikl for kp=2πn Ec = Et = e Ein , and Er = 0 t ,

i.e. it has 100% transmission Case study 1, 4 Mode Cleaner Transmission

Plot of the transmission spectrum for a Fabry-Perot cavity for various values of t 1

0.5 Et/Ein t2=0.9 t2=0.8 … 2 -1.5 -1 -0.5 0 0.5 1 1.5 t =0.1

-4 1!10 0.001 0.01 0.1 x=p/λ=pf/c

0.1 Cavity acts like a low-pass filter for noise with 0.01 cavity pole at δf/2 log-log plot of transmission 0.001

Case study 1, Modecleaner

1 4 5 6 1 10 100 1000 1!10 1!10 1!10

0.1

0.5

0.01

7 7 7 6 6 7 7 7 -2!10 -1.5!10 -1!10 -5!10 0 5!10 1!10 1.5!10 2!10 0.001

Finesse=550 (T=0.006), FSR=15 MHz, δf=27 kHz

Modecleaner control sidebands must not be at integer multiples of 15 MHz so that they reflect from the modecleaner

All other control sidebands must be at integer multiples of 15 MHz to pass through mode-cleaner

Laser noise above f≈13 kHZ is blocked by the modecleaner

Case study 1, 6 Fabry-Perot Arm Cavities

r1,t1 r2,t2 Consider a Fabry-Perot cavity with Ein Ec Et identical, lossless mirrors, illuminated with a steady wave of wavelength λ. The fields transmitting (Et) reflecting Er from (Er) and circulating in (Ec) the cavity are proportional to the input L field (Ein) with the relations

E = t E + ( r )( r )e2ikLE E = r E r t e2ikLE E = t eikLE c 1 in − 1 − 2 c , r 1 in − 1 1 c , and t 2 c ikL 2 2ikL t1Ein t1t2e Ein t1r2e giving Ec = = 2ikL , Et = , and Er r1 2ikL Ein 1 r1r2e 1 r r e2ikL − 1 r1r2e − 1 2  −  −   E 2 E  E  c t = 0  r = 1  for 2kL=2πn and r2=1 E ≈ 1 r , E , and E − in ! − 1 in in Case study 1, 7 The power builds up by a factor of 2/(1-r1) Arm Cavities

Plot of the circulating power in the LIGO arm cavities 1 100

10

0.5 4 1 10 100 1000 1!10

0.1

0 0.01 Finesse=110 (T=0.06), FSR=37.5 kHz,δf=340 Hz

Arm cavity power build-up is 66x

Signal response is constant below about 100 Hz

High frequency signals do not build up in the arms

Case study 1, 8 Signal Generation

r1,t1 r2,t2 The phase accumulated in a round Ein Ec Et trip in the presence of a strain h(t) is t φ(t) = ω0 (1 + h(t)) dt t 2L/c ! − t Er φ(t) = ω0 (1 + h0 cos ωt) dt for h(t)=h0cos(ωt) t 2L/c ! − L 2L ω 2L φ(t) = ω + h 0 sin ωt sin ωt ω 0 c 0 ω − − c ! " #$ so the field in the arm cavity after a round trip is

iω t+φ(t) iω t iω t+ 2L ω0 2L E e 0 = tE e 0 + rE e 0( c ) 1 + ih sin ωt sin ωt ω c in c 0 ω − − c ! ! " #$$ iω t+φ iω t iω t+ 2L ω0 iωt iωt iωt iω 2L iωt+iω 2L E e 0 = tE e 0 + re 0( c ) 1 + h e e− e − c + e− c or c in 0 2ω − − ! " #$ Case study 1, 9 Signal Generation

iω t+φ iω t iω t+ 2L ω0 iωt iωt iωt iω 2L iωt+iω 2L E e 0 = tE e 0 + re 0( c ) 1 + h e e− e − c + e− c c in 0 2ω − − ! " #$ iω t+φ(t) iω t+ 2L ω0 iωt iω L L iωt iω L L E e 0 = tE + rE e 0( c ) 1 + h e e− c sin ω + e− e c sin ω c in c 0 2ω c c ! ! " # " #$$ Which is more instructive in the form

iω t+φ(t) iω t iω 2L iω t ω0 L i(ω +ω)t iω L i(ω ω)t iω L E e 0 = tE e 0 + rE e 0 c e 0 + h sin ω e 0 e− c + e 0− e c c in c 0 2ω c ! " # $ %& When the “carrier” resonates in the arm cavity (2ω0L/c=n π)

tEin E ω rt k L L = c( ) 0 Ec this gives ± = h0 sinc ω 1 r Ein(ω0) 1 r 2 c − ! ! " − # " # ! ! 1 10 100 1000 1 104 1 105 1 106 ! ! ! ! ! ! ! 1 10-4 So the effect of a gravitational wa! ve is !to couple !

-5 light into the arm cavity with a frequency 1!10 -6 1!10 dependent input coupling t(ω)≡Ec(ω)/E0 -7 1!10 Case study 1,10 Signal from Arm Cavities

Calculating the signal that is transmitted from the arm cavities requires the use of the cavity transmission

ikL t t e E Ec( ω) rt k L L = 1 2 in t(ω) = ± = h 0 sinc ω Et 2ikL 0 1 r1r2e Ein(ω0) 1 r 2 c − ! ! " − # " # ! ! with t2→t(ω), the effective inpu!t coupl!ing from a , and k→c/(ω0+ω)

13 1!10

12 1!10 ) 0 11 h 1!10 in

10 1!10 )/(E

ω 9 ( 1!10 t E 8 1!10 4 5 6 1 10 100 1000 1!10 1!10 1!10

f=ω/2π (Hz) Case study 1,11 Michelson Interferometer

Imbalanced arm lengths lx≠ly

ly Biased to a dark fringe Δφout=2πn+π

lx E = t ( r )e2ikly [E + E (ω)] + t (r )e2iklx [E E (ω)] out bs − bs r t bs bs r − t

with a 50-50 beamsplitter tbs=rbs=1/√2

1 E = e2iklx [E E (ω)] e2ikly [E + E (ω)] out 2 r − t − r t ! " with l+=(lx+ly)/2 and l-=(lx-ly)/2

1 2ikl+ 2ikl 2ikl E = e e − [E E (ω)] e− − [E + E (ω)] out 2 r − t − r t ! " 2ik0l+ Eout = e [iEr sin(2ik0l ) cos(2ikl )Et(ω)] − − − Case study 1,12 Michelson Interferometer

2ik0l+ Eout = e [iEr sin(2ik0l ) cos(2ikl )Et(ω)] − − −

We want RF sidebands at 15 MHz transmitted ly to the output for heterodyne detection so with

ω0 + ∆ω15 MHz k lx 0 → c we need 2π15 106 π 2l × = nπ + − c 2

for 100% transmission. For n=0 this gives l-=2.5 m. In practice l-≈1m, resulting in about 60% transmission of the sideband fields.

Note for the audio frequency signal sidebands the transmission is virtually 100% since

∆ωaudio 2l π Case study 1,13 − c ! Recycling Cavity

Michelson interferometer reflects virtually all of the laser power (Since interference at output port is destructive for the carrier)

We can treat the Michelson interferometer 2 as a high reflectivity mirror rm +a=1 for the carrier frequency where a is the total round-trip power loss in the interferometer

Power recycling mirror and Michelson “mirror” form a resonant cavity for the carrier

Case study 1,14 Recycling Cavity

Field inside recycling cavity is given by expression for the circulating field in a Fabry- Perot cavity

t1Ein 10 Ec = 1 r r e2ikL rm=1 − 1 m 7.5 on resonance this gives

5 t1Ein Ec = 1 r1rm

− 2.5 rm=0.9 which is maximized for r1=rm rm=0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 giving a maximum power buildup of P 1 1 bs = = P 1 r2 a Case study 1,15 laser − Recycling Cavity

Note that since the signal sidebands exit the interferometer at the output port they do not “see” the power recycling mirror. This mirror has no effect on the interferometer response other than to increase the circulating power

Case study 1,16 Sensitivity

With a laser power of 10 W and power loss in the interferometer of a=0.04, the power at the beamsplitter is 250 W. 4 10 100 1000 1!10 -17 Shot n1!o10ise from 250W of laser power at 1064 nm is 1 ω0 -18 ! 11 1!1∆0 φsn = = 2.7 10− rad/√Hz √N !P ∆t → × -19 ∆φsn SinceE1!1noise0 = Ebs∆φsn andEsigna l = Et(ω)Ebsh0, h(ω) = Et(ω) is the str-20ain sensitivity, the value of h0 that lets Esignal=Enoise 1!10 !

-21 1!10

Hz] -22 1!10 √ / 1 [ -23 ) 1!10 ω ( h ! -24 1!10 f=ω/2π [Hz] Case study 1,17 Sensitivity

Case study 1,18 Summary

Phasor notation can be used to calculate response of cavities and interferometers in a systematic fashion

Complex optical systems can be modeled by determining the behavior of each subsystem independently and linking them together

LIGO’s peak sensitivity is about 10-22 /√Hz at 100 Hz, and matches the results from these simple calculations

Case study 1,19