An Overview of Advanced LIGO Interferometry

Transcription

1 An Overview of Advanced LIGO Interferometry Luca Matone Columbia Experimental Gravity group (GECo) Jul 16-20, 2012 LIGO-G

2 Day Topic References Gravitational Waves, Michelson IFO, Fabry-Perot cavity, finesse, freespectral range (FSR), Michelson with Fabry-Perot arms. Group activity: single arm numerical calculations (MATLAB) and plots. Power recycled Michelson IFO, suspended TMs and equation of motion, Laplace transform, modeling of Pound-Drever-Hall locking scheme. Group activity: numerical calculations (MATLAB) of single arm cavity w/o sidebands w/o seismic noise. Initial LIGO: Schnupp asymmetry, degrees of freedom to sense and control, output ports. Advanced LIGO: Resonant Sideband Extraction (RSE), output ports, degrees of freedom to sense and control, sensing matrix, homodyne and DC readout, lock acquisition and green laser locking. Group activity: paper discussion Transverse Electro-Magnetic (TEM) modes, cavity mode, modematching, transverse mode spacing, Gouy phase, Wavefront sensor, Input Mode-Cleaner, frequency stabilization, Output Mode-Cleaner, Thermal Compensation System (TCS). Group activity: paper discussion Amplitude spectral noise, basics on control loops, noise budgeting, paper discussion and summary H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966) K. A. Strain et al., Appl. Opt. 42, (2003) Mullavey A. J. et al, Optics Express, 20, (2011) Morrison E. et al., Appl. Opt. 33, (1994) T P G T T T G M T LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 2

3 Gravitational Waves ΔL L The passage of a Gravitational Wave changes the distance between objects (ΔL). The change in distance L depends on The distance L The gravitational wave amplitude h L = h L LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 3

4 How big is L? Astrophysical motivation Amplitude of GWs produced by binary neutron star systems in the VIRGO cluster is expected to be h ~ For a 4 km baseline (L) L = h L ~ m LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 4

5 Idea: Michelson Interferometer (IFO) to measure ΔL mirror laser Interference fringe mirror LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 5

6 Michelson IFO to measure L Once mirrors move The laser fields in the two arms recombine to interfere Interference can be constructive or destructive The light level changes LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 6

7 How does a Michelson IFO work? LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 7

8 Description of a Laser Beam Any laser beam can be represented in terms of TEM (Propagation) modes Gouy phase U m,n x, y, z = A m,n w z H m 2 x w z Wavefront Spot size Wavefront radius H n 2 y w z e x2 +y 2 e ii x2 +y 2 Hermite polynomial w 2 z 2 R z e i k z φ m,n z LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 8

9 Mode patterns y x TEMmn: Transverse Electro-Magnetic (TEM) modes Figure plot of U m,n x, y, z 2 vs. position x, y LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 9

10 Electromagnetic wave: In general ψ = A e ii Electromagnetic wave function ψ Phase φ Amplitude A Photodiode measures power P P = ψ 2 = ψ ψ = A 2 LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 10

11 Simplifying: Plane-Wave Approximation Flat wavefront (like that of a plane wave) LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 11

12 A laser beam in the plane-wave U m,n x, y, z approximation = A m,n w z H m 2 x w z i k z Ψ z = A e H n 2 y w z e x2 +y 2 e ii x2 +y 2 w 2 z 2 R z e i k z φ m,n z Wave number k = 2π λ Propagation axis z LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 12

13 Modeling a Michelson IFO Assuming no misalignments Assuming plane mirrors Plane-Wave approximation Helpful in conceptualizing the detector >> Powerful model laser mirror photodiode mirror LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 13

14 Objective: To determine what the EM fields look like throughout the IFO Interference port LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 14

15 Beam Propagation l Phase φ = k l Ψ lllll Electromagnetic field right after the laser Electromagnetic field right before the mirror Ψ 1 = e i k l Ψ lllll LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 15

16 Beam Reflection and Transmission Ψ r = ii Ψ ii Reflected electromagnetic field right after the mirror. Mirror s reflection amplitude r. Reflected field gains a 90 0 phase change (i is the imaginary unit) Ψ ii Impinging electromagnetic field t 2 + r 2 = 1 no losses Ψ t = t Ψ ii Transmitted electromagnetic field right after the mirror. Mirror s transmission amplitude t LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 16

17 EM fields part 1 Ψ lllll Ψ 4 = e iil 1 Ψ 3 Ψ 3 = ii Ψ 2 Ψ Ψ 1 = t bb Ψ Ψ 2 = e iil 1 Ψ 1 LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 17

18 EM fields part 2 Ψ 7 = ii Ψ 6 Ψ lllll Ψ 8 = e iil 2 Ψ 7 Ψ 6 = e iil 2 Ψ 5 Ψ Ψ 5 = ir bb Ψ LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 18

19 EM fields: Common (+) and Differential (-) Ports Ψ + = ir bb Ψ 8 + t bb Ψ 4 Ψ 8 Ψ 4 Ψ = t bb Ψ 8 + ir bb Ψ 4 LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 19

20 EM field at the Differential Port Ψ φ 2 Phase in arm 1 φ 1 = 2 k l 1 φ 1 Phase in arm 2 φ 2 = 2 k l 2 Ψ = 1 2 Ψ e i φ 1 + e i φ 2 Assuming no losses, highly reflective end mirrors r 2 = 1 and a symmetric beam-splitter r bb 2 = t bb 2 = 50% LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 20

21 Power at the Differential Port φ 2 Δφ = φ 2 φ 1 φ 1 P = Ψ 2 = 1 2 Ψ cos φ Power level depends on the phase difference Δφ (differential changes) between the two arms. LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 21

22 Power at the Differential Port Bright Fringe all laser power out Dark Fringe no laser power out Assuming a laser input Ψ 2 = 1 Simplemichelson.m LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 22

23 Power at the Differential Port Periodic function Power level depends on Δφ or Δl General idea Measure power change to determine Δl LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 23

24 Common Port Power level depends on the phase sum φ + (common changes). φ 2 φ 1 P + = 1 2 Ψ cos φ + φ + = φ 2 + φ 1 LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 24

25 So far The passage of a GW induces a change in the arm lengths Δl The arm length change Δl generates a change in the beam phases φ This phase change φ, in turn, causes a change in the light power P. A photodiode then converts this light power to an electrical signal. Measurement of gravitational wave amplitude h! LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 25

26 But Simple Michelson IFO not enough to measure L ~ m Strategy Amplify signal Decrease and/or control noise contribution Noise sources: two categories Displacement noise Ground seismic excitation Thermal excitation of optical elements and suspensions Radiation pressure Phase noise Amplitude and frequency fluctuations of the incoming laser beam Shot-noise, the quantum limit to the counting of photons LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 26

27 Signal amplification Let s look at how to amplify the effect of a GW onto a Michelson IFO Recall: φ 1 is proportional to length of arm 1 Beam coming back to BS has phase φ 1 : Ψ 4 = i r t bb e iφ 1 where φ 1 = 2 k l 1 φ 1 Phase φ 1 carries information about arm length 1 LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 27

28 Michelson arms with Delay-Lines Solution: Michelson arms replaced with optical delaylines Laser beam forced to bounce several times in one arm Effective increase of arm length: L eeeeeeeee = n. oo bbbbbbb L But Difficult to manufacture Number of bounces limited LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 28

29 Alternative: Michelson arms Ψ rrrr with Fabry-Perot cavities Delay-line Beams bounce multiple times (4x in diagram) Beams do not overlap Ψ ii Ψ rrrr Ψ ii Fabry-Perot Partly reflective mirrors Photons bounce multiple times Beams overlap in space LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 29

30 What s the advantage? Ψ 4 = e iii Ψ 3 Ψ rrrr L 1 2 Ψ 3 = ir 2 Ψ 2 Ψ ttttt Ψ ii Ψ 1 Ψ 2 = e iii Ψ 1 LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 30

31 Ψ 1 = ir 1 Ψ 4 + t 1 Ψ ii Ψ 1 = ir 1 Ψ 4 + t 1 Ψ ii = ir 1 e iii Ψ 3 + t 1 Ψ ii = ir 1 e iii ir 2 Ψ 2 + t 1 Ψ ii = ir 1 e iii ir 2 e iii Ψ 1 + t 1 Ψ ii L Ψ rrrr Ψ ii Ψ Ψ 1 Ψ 1 = t r 1 r 2 e iφ Ψ ii φ = 2 k L LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 31

32 Ψ 1 2 Power build-up Ψ 1 (Fast) phase change when light resonates LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 32

33 Power build-up: Just like standing waves LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 33

34 Free Spectral Range (FSR) The spacing in frequency ν fff or wavelength between successive resonances Phase φ [π] 2 π (phase) or λ 2 (length) or ν fff = c 2 L (frequency) LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 34

35 Note Phase φ [π] ν φ = 2 k L = 2 π ν fff where ν is the laser frequency LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 35

36 Finesse F Defined as F = Phase φ [π] FFF FFFF π r 1r 2 1 r 1 r 2 Like quality factor Q Determines power build-up Average number of photon bounces Full-Width-at-Half- Maximum (FWHM) LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 36

37 So what is the advantage of having Fabry-Perot cavities? When light resonates, the field s phase changes rapidly! LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 37

38 Run_fp.m Intra-cavity field as a function of F The rate of phase change depends on F LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 38

39 Ψ rrrr = ir 1 Ψ ii + t 1 Ψ 4 Ψ rrrr Ψ ii Ψ Ψ 1 Ψ rrrr = i r 1 + r 2 e iφ 1 + r 1 r 2 e iφ Ψ ii LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 39

40 Reflected field When in resonance, light leaks into the cavity When out of resonance: 90 0 phase change just like a single mirror Run_fp.m LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 40

41 Reflected field Around resonance: rapid phase change, highly sensitive to ΔL Run_fp.m LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 41

42 Reflected field as a function of F The rate of phase change depends on F Run_fp.m LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 42

43 Phase sensitivity φ arm length φ arm length F Replace Michelson end mirrors with Fabry-Perot cavities Ψ ii LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 43

44 To increase the detector s sensitivity to arm length changes Michelson IFO with Fabry-Perot arms LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 44

45 Strategy Amplify signal Decrease and/or control noise contribution Noise sources: in general two categories Displacement noise Ground seismic excitation Thermal excitation of optical elements and suspensions Radiation pressure Phase noise Amplitude and frequency fluctuations of the incoming laser beam Shot-noise, the quantum limit to the counting of photons LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 45

46 Group Activities 1. Form groups of two or three 2. Using table 2 of T (AdvLIGO LSC final design document), verify AdvLIGO s arm cavity finesse F and FSR 3. Using Matlab and the field equations shown, plot a. The dark port power vs. arm length change for a Michelson IFO b. The reflected and intra-cavity power (with phases) for one of LIGO s arms as a function of cavity length (assume 1W of light in) LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 46

47 From The AdvLIGO Length Sensing and Control Final Design T LIGO-G Matone: An Overview of Advanced LIGO Interferometry (1) 47