Phasor Calculations in LIGO

Transcription

1 Phasor Calculations in LIGO Physics 208, Electro-optics Peter Beyersdorf Document info Case study 1, 1

2 LIGO interferometer Power Recycled Fabry-Perot Michelson interferometer 10m modecleaner filters noise from light 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

3 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 Transmission of cavity P.D.H. input spectrum Pound-Drever-Hall method X ΕΟΜ Case study 1, 3

4 Mode Cleaner Triangular modecleaner has a perimeter p=20 m, unit reflectivity end mirror and equal, lossless input/output couplers. Illuminated with a steady wave of wavelength λ. The fields transmitting (E t ) reflecting from (E r ) and circulating in (E c ) the cavity are proportional to the input field (E in ) with the relations giving E c = te in + ( r) 2 e ikp E c E c = te in,, and,, and 1 r 2 e ikp E t = t2 e ikl E in 1 r 2 e ikp E r for kp=2πn E c = E in, E t = e ikl E in, and E r = 0 t E t = te ikl E c i.e. it has 100% transmission l r,t r,t E in E c E t r=1 E r = re in rte ik(p l) E c E r = r (1 t2 e ik(p l) ) E 1 r 2 e ikp in Case study 1, 4

5 Mode Cleaner Transmission Plot of the transmission spectrum for a Fabry-Perot cavity for various values of t 1 E t /E in 0.5 t 2 =0.9 t 2 = x=p/λ=pf/c t 2 =0.1 log-log plot of transmission 0.1 Cavity acts like a low-pass filter for laser noise with 0.01 cavity pole at δf/ Case study 1,

6 Modecleaner !10 4 1!10 5 1! !10 7-1!10 7-5! !10 6 1! !10 7 2! 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

7 Fabry-Perot Arm Cavities Consider a Fabry-Perot cavity with identical, lossless mirrors, illuminated with a steady wave of wavelength λ. The fields transmitting (E t ) reflecting from (E r ) and circulating in (E c ) the cavity are proportional to the input field (E in ) with the relations giving for 2kL=2πn and r 2 =1 E in E c E t E r,, and The power builds up by a factor of 2/(1-r 1 ) r 1,t 1 L r 2,t 2 E c = t 1 E in + ( r 1 )( r 2 )e 2ikL E c E r = r 1 E in r 1 t 1 e 2ikL E c E t = t 2 e ikl E c t 1 E in E c =, 1 r 1 r 2 e, and 2ikL E t = t 1t 2 e ikl E in E r = r 1 t2 1 r 2e 2ikL E 1 r 1 r 2 e 2ikL 1 r 1 r 2 e 2ikL in E c 2 E, t E, and r = 0 = 1 E in 1 r 1 E in E in Case study 1, 7

8 Arm Cavities Plot of the circulating power in the LIGO arm cavities ! 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

9 Signal Generation The phase accumulated in a round trip in the presence of a strain h(t) is φ(t) = ω 0 t φ(t) = ω 0 t t 2L/c t 2L/c 2L φ(t) = ω 0 c + h ω 0 0 ω (1 + h(t)) dt (1 + h 0 cos ωt) dt for h(t)=h 0 cos(ωt) [ ( sin ωt sin ωt ω 2L c )] E in E c E t E r r 1,t 1 r 2,t 2 L so the field in the arm cavity after a round trip is or E c e iω 0t+φ(t) = te in e iω 0t + re c e iω 0(t+ 2L c ) [ ω ih 0 ω E c e iω 0t+φ = te in e iω 0t + re iω 0(t+ 2L c ) [ 1 + h 0 ω 0 2ω [ sin ωt sin ( ωt ω 2L c )]] [ e iωt e iωt e iωt iω 2L c + e iωt+iω 2L c ]] Case study 1, 9

10 Signal Generation E c e iω 0t+φ = te in e iω 0t + re iω 0(t+ 2L c ) [ 1 + h 0 ω 0 2ω E c e iω 0t+φ(t) = te in + re c e iω 0(t+ 2L c ) [ 1 + h 0 ω 0 2ω Which is more instructive in the form E c e iω 0t+φ(t) = te in e iω 0t + re c e iω 0 2L c When the carrier resonates in the arm cavity (2ω 0 L/c=n π) E c = te in 1 r this gives [ e iωt e iωt e iωt iω 2L c + e iωt+iω 2L c ( [e iωt e iω L c sin ω L ) ( + e iωt e iω L c sin ω L )]] c c [ e iω 0t + h 0 ω 0 2ω sin (ω L c E c (±ω) ( E in (ω 0 ) = rt ) 1 r ]] ) [ ]] e i(ω 0 +ω)t e iω L c + e i(ω 0 ω)t e iω L c k 0 L h 0 (ω 2 sinc L ) c So the effect of a gravitational wave is to couple light into the arm cavity with a frequency dependent input coupling t(ω) E c (ω)/e 0 1!10-4 1!10-5 1!10-6 1! !10 4 1!10 5 1!1 Case study 1, 10

11 Signal from Arm Cavities Calculating the signal that is transmitted from the arm cavities requires the use of the cavity transmission E t = t 1t 2 e ikl E in 1 r 1 r 2 e 2ikL t(ω) = E c (±ω) ( E in (ω 0 ) = rt ) 1 r with t 2 t(ω), the effective input coupling from a gravitational wave, and k c/(ω 0 +ω) 1!10 13 k 0 L h 0 (ω 2 sinc L ) c 1!10 12 Et(ω)/(Einh0) 1! ! !10 9 1! !10 4 1!10 5 1!1 f=ω/2π (Hz) Case study 1, 11

12 Michelson Interferometer Imbalanced arm lengths l x l y Biased to a dark fringe Δφ out =2πn+π l y E out = t bs ( r bs )e 2ikl y [E r + E t (ω)] + t bs (r bs )e 2ikl x [E r E t (ω)] l x with a beamsplitter t bs =r bs =1/ 2 E out = 1 ( e 2ikl x [E r E t (ω)] e 2ikl y [E r + E t (ω)] ) 2 with l + =(l x +l y )/2 and l - =(l x -l y )/2 E out = 1 2 e2ikl + ( e 2ikl [E r E t (ω)] e 2ikl [E r + E t (ω)] ) E out = e 2ik 0l + [ie r sin(2ik 0 l ) cos(2ikl )E t (ω)] Case study 1, 12

13 Michelson Interferometer E out = e 2ik 0l + [ie r sin(2ik 0 l ) cos(2ikl )E t (ω)] We want RF sidebands at 15 MHz transmitted to the output for heterodyne detection so with l y k 0 ω 0 + ω 15 MHz c we need 2l 2π c = nπ + π 2 l x 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 2l ω audio c π Case study 1, 13

14 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 as a high reflectivity mirror r m2 +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

15 Recycling Cavity Field inside recycling cavity is given by expression for the circulating field in a Fabry- Perot cavity E c = t 1 E in 1 r 1 r m e 2ikL 10 rm=1 on resonance this gives E c = t 1E in 1 r 1 r m which is maximized for r 1 =r m rm=0.8 rm=0.9 giving a maximum power buildup of P bs = 1 P laser r 2 = 1 a Case study 1, 15

16 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

17 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. Shot noise from 250W of laser power at 1064 nm is φ sn = 1 N = ω0 P t rad/ Hz h(ω) = φ sn Since# E noise # # = E# bs φ # sn # and# E# signal # = # E# t (ω)e # bs # h 0, E t (ω) is the strain sensitivity, the value of h 0 that lets E signal =E noise!10-21 h(ω) [1/ Hz]!10-22!10-23!10-24 f=ω/2π [Hz] Case study 1, 17

18 Sensitivity Case study 1, 18

19 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 / Hz at 100 Hz, and matches the results from these simple calculations Case study 1, 19