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G LIGO Laser Interferometer Gravitational Wave

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G LIGO Laser Interferometer Gravitational Wave Powered By Docstoc
					                Proposal
Thermal and Thermoelastic Noise Research
        for Advanced LIGO Optics
                                  Norio Nakagawa
                        Center for Nondestructive Evaluation
                                Iowa State University

                                in collaboration with
                           E.K. Gustafson and M.M. Fejer
                       Ginzton Laboratory, Stanford University

November 29-30, 2001               Talk at PAC-11, LHO           1
LIGO-G010415-00-Z
                             Outline
• Objective and Approach
• Output of the Prior NSF Support
   – Laser phase noise formulas for optical resonator and delay line
   – Coating noise estimation
• Proposed Work
   – Coating noise studies for mirrors of edge geometry
   – Thermo-elastic noise of coated mirrors
   – Thermal & thermo-elastic noises of realistic mirror designs
• Tasks & Time Lines
• Summary
• (Broader Impacts)


 November 29-30, 2001       Talk at PAC-11, LHO                    2
 LIGO-G010415-00-Z
                         Objective and Approach
             Objectives                                 Approaches
• To develop laser-phase-noise               • Calculate phase noise via
  formulas                                     Green’s function method
    – Green’s-function-based
                                                   – Elasticity
    – Analytical and computational
                                                   – Thermo-elasticity & thermal
    – two-point laser-phase-fluctuation
      correlations
                                                     diffusion
    – complex test mass objects.             • Analytical mirror models
• To extend the noise estimation                   – Half space
  method to thermo-elastic noises.                 – Quarter space
• To estimate thermal and thermo-                  – Thin coating layer
  elastic noises of coated mirrors for       • Numerical calculations
  advanced LIGO designs.
                                                   – Realistic mirror shapes
• To examine merits of interferometer
  design options for future LIGO.                  – Coating loss
                                                   – Delay-line vs. Fabry Perot


  November 29-30, 2001            Talk at PAC-11, LHO                              3
  LIGO-G010415-00-Z
                            Output to Date
•    Publications
      – N. Nakagawa, Eric Gustafson, P. Beyersdorf and M. M. Fejer
        “Estimating the off resonance thermal noise in mirrors, Fabry-Perot
        interferometers and delay-lines: the half infinite mirror with uniform
        loss,” to appear in Phys. Rev. D
      – N. Nakagawa, A. M. Gretarsson, E.K. Gustafson, and M. M. Fejer,
        “Thermal noise in half infinite mirrors with non-uniform loss: a slab of
        excess loss in a half infinite mirror,” submitted to Phys. Rev. D
      – N. Nakagawa, E.K. Gustafson, and M. M. Fejer, “Thermal phase noise
        estimations for fabry-perot and delay-line interferometers using coated
        mirrors,” in preparation.
      – D. Crooks, et al., “Excess mechanical loss associated with dielectric
        mirror coatings on test masses in interferometric gravitational wave
        detectors,” submitted to Classical and Quantum Gravity.
      – Gregory M. Harry, et al., “Thermal noise in interferometric gravitational
        wave detectors due to dielectric optical coatings,” submitted to Classical
        and Quantum Gravity.


    November 29-30, 2001          Talk at PAC-11, LHO                         4
    LIGO-G010415-00-Z
                      Phase-Noise Correlation
• Requirement
   – Compute laser phase noise correlation


                                                                                        
                                 x1 , t1  x2 , t 2                       S  , x1 , x2 
                                                                    d i t1 t 2 
                                                                       e              
                                                                  2

                          
        x1 ,t1     x2 ,t2 


                          
        x1                 x2                       E, ,



                                                     E,  , 


 Ex. Coating noise model
 November 29-30, 2001                       Talk at PAC-11, LHO                                       5
 LIGO-G010415-00-Z
                                       Intrinsic Thermal Noise
• Phase Noise Formula
                                                                                   w  
            S  , r1 , r2   4k 2 B                             dS  dS  r   r1  00 r   r2 
                                    2k T                                                   w

                                                        
                                                                                           00


                                           d x                  x , r ; ccijkl x  k  nl x , r ; c
                                                                                                  
                                                   3
                                                           i      nj
                                                                                    
                                              V


                                      cijkl  cijkl  icijkl  1  i  cijkl
                                       
                                                                          


            
   S  , r1 , r2                   the two-point laser-beam phase-           w, k                     the laser beam spot size (amplitude
                                      noise power-spectrum correlation                                   radius), and wave number
                
    ui r u j r                 the displacement spectral                 ij                     elastic Green’s function
                         
                                      correlation
                                                                                cijkl [c’ijkl, c”ijkl]   elastic constants [dispersive and
                                    The laser beam reflection points                                   absorptive parts]
  r1 , r2

                                 
                                      (the beam centers)
                2                                                             ()                     loss function
   r   e  2 r
      w
      00
                             w2
                                      the Gaussian laser-beam profile
                                                                                kB , T                   Boltzmann constant, and temperature
                                      function

 November 29-30, 2001                                          Talk at PAC-11, LHO                                                             6
 LIGO-G010415-00-Z
            Fluctuation-Dissipation Relation
• Surface Force density  Strain
          
          w
          00
                                                    
                                                                      w 
                                                                                
     F r   r1   uij x; r1   F  dS   i  nj x , r ; c 00 r   r1 

   S
            
       , r1, r2   4k 2 2kBT
                                
                                          1
                                          F 2 V
                                                 3
                                                      
                                                       
                                                d x uij                  
                                                                           
                                                                                        
                                                        x; r1  cijkl x  ukl x; r2  

• Laudau-Lifshitz

                                     
                          2
    Emech   T0  dV T  2  dV uij ijkl ukl , cijkl  ijkl  
                                               
                                                       
                                2


                                                                      
                                                               structural
               
                     V            V
                                                
                        thermo elastic                   viscous                    viscous


                            
                   S  , r , r   4k 2 4 k BT 1

                                           2 F2
                                                     
                                                    Emech             
 November 29-30, 2001                       Talk at PAC-11, LHO                                7
 LIGO-G010415-00-Z
              Various Optical Configurations
• Phase noise formulas                              
                                 S    S  , r , r 
                                  Single


  Computed explicitly for
                                                                                           1
                                            1  rI 2                   
  – Single-reflection mirror     S    
                                      FP
                                                    2 
                                                           1
                                                              2rI
                                                                                                
                                                                    cos 2  S    rI2 S  
                                                                               E             I
                                                                                                                            
                                            1  rI   1  rI
                                                                  2
                                                                            

                                                                             N n 1
                                 S ( )   S  , rn , rn   2 cos[2( n  q) ]  S  , rn , rq 
                                               N
                                      DL              E                                   E

                                              n 1                        n  2 q 1
                                              N 1                            N 1 n 1

  – Fabry-Perot resonator                     S  ,  n ,  n   2  cos[2( n  q) ]  S  ,  n ,  q 
                                                 I                                              I

                                               n 1                           n  2 q 1



                                rI                    the input mirror reflection coefficient
                                                     the transit time

                                 S  , SI  
                                  E                   the single-reflection phase noises of the input and
                                                      end-point mirrors.
  – Optical delay line
                                                     the positions of the N-time reflections on the end-
                                 rn
                                                      mirror surface
                                 
                                 p                   the positions of the (N-1)-time reflections on the input-
                                                      mirror surface
                                E,                   Young’s modulus, and Poisson ratio
                                If Half space:
                                                                  8k T 
                                                 S  , r1 , r2   B
                                                                        k 2 1   2  r1  r2 2
                                                                      w E
                                                                                   e
                                                                                       
                                                                                                     2 w2
                                                                                                                  2
                                                                                                            I 0 r1  r2  2 w 2   
  November 29-30, 2001         Talk at PAC-11, LHO                                                                                     8
  LIGO-G010415-00-Z
                   Fabry-Perot vs. Delay Line
• Fabry-Perot vs Delay lines                                  15

   – Analytical half-space mirror model                                  1
                                                                                                                 2w
   – Fabry-Perot interferometer vs several
     delay lines
                                                              10                                        R
   – Storage time as proposed for LIGO II.                                            2
                                                                     3
   – Delay line beam centers
      • evenly spaced                                                             4
                                                              5
      • on a circle                                                          5

• When the spots are not overlapping
  appreciably, the delay line is less
                                                                     6
  noisy than the Fabry-Perot.                                 0
                                                                                                          150              200
                                                                                 50        100
   – Noise levels are similar if                                                          Frequency [Hz ]

      • the spot circle radii comparable to the        Figure 1: Comparison of the phase noise from a delay-line and a Fabry-Perot
                                                       interferometer. The solid curve (1) is for a 4 km Fabry-Perot interferometer
        beam spot size                                 with an input mirror power reflectivity of RI=0.97 and an end mirror power
      • the spots are largely overlapping, and         reflectivity of RE=1.00. Curves 2, 3, 4, 5 and 6 correspond to 4 km delay lines
                                                       all with 130 spots on the end mirror and laser beam spots of 1/e field radius w in
        above several hundred Hertz.                   a pattern with their centers on a circles of radius R=w/3 (2), R=2w/3 (3),
                                                       R=5w/2 (4), R=10w (5) and R=20w (6) where w=3.5 cm, the spot size used for
                                                       both mirrors of the Fabry-Perot interferometer. The mirror Q is assumed to be
                                                       3´108 and the material properties are those of Sapphire E=71.8 GPa and s=0.16
                                                       however we are treating sapphire as isotropic for the purpose of this illustration
                                                       and assuming a single loss function.

  November 29-30, 2001                  Talk at PAC-11, LHO                                                                9
  LIGO-G010415-00-Z
        Coating Noise; Problem Statement
• Coating noise model
   – Based on half-space mirror model
         • a lossy layer (thickness d) on a lossy host material
   – Requirement: Compute laser phase noise correlation
   – Approach: via analytical Green’s function
                          
        x1 ,t1     x2 ,t2 
                                                                              
                                                             x1 , t1   x2 , t2 
                          
        x1                 x2               E, ,                                              
                                                                d i t1 t2 
                                                                        e              x1   x2  
                                                                2
                                                                                            
                                                                d i t1 t2 
                                                                        e           S  , x1  x2 
                                             E,  ,            2




 November 29-30, 2001                Talk at PAC-11, LHO                                           10
 LIGO-G010415-00-Z
                                                   Coating Noise
• Static Green’s function for layer-on-substrate

          sos
                        
                   ~, z      1 1
                                         21     pz  i 1 G sos  ~,0   1 pz 3 cosh pz
                                                                        p       2
                                                                                                              
                                                                                                               
                                                                                                              
                                                                                                                             
                    p
                                          pz 3  1  2  2 G sos  ~,0   32   1 pzi 1  sinh pz 
                             pE 1                                                 4
                                                                           p              2                   

         sos
           z     ~, z   211  2 pz 3G sos  ~,0  21     pz  i 1 cosh pz
                    p                                  p
                             211   21  pzi 1 G sos  ~,0   pz 3  1  2  2 sinh pz
                                                               p

                                                                                                                     
                                            1    cosh2 pd  411   32    1  2 1  2   2 3  4  sinh 2 pd
                                                                                4                          2
                                                                                                                                       
      where                                
                                            
                                                                         

                                                                                                                                    
                                                                                                                                       
                                             2 1  2  cosh2 pd  1 1  2 sinh 2 pd 
                                                                            1
                                                                                                                                    
                             sos ~      1          1
                                                                                                          2
                                                                                                                                2  
                            G  p,0     81 2 1  2 3  4   2 1  2 3  4    1  2 3  4  sinh pd  2 
                                                                                                                                  
                                             811 2 1   1   3  4  pd                                            
                                                                                         2

                                                                                                                                  
                                            41  pd 1   1   3  4  3
                                                 1
                                                                                                                                       
                                                                                                                                      

                                                                                 
                                cosh pd  211  1  2    3  4 sinh pd  cosh pd  211   1  2    sinh pd 
                                                                                                 

                                                                                          41 2 1      3  4  pd 
                                                                                              1                                   2
                                                                                                            1


                              1 E , 1,2,3  Pauli matrices
                                1
                                    E                                         D. M. Burmister, J. Appl. Phys. 16, 89-94, 1945.
 November 29-30, 2001                                         Talk at PAC-11, LHO                                                          11
 LIGO-G010415-00-Z
                                                     Coating Noise
• Intrinsic Thermal Phase-Noise Estimation
              
 coating
S        , r   4k 2  2kBT              1
                                            w
           1   2  r 2 2 w2                                                                                                 
                 e          I 0 r 2 2 w2                                                                                  
               E
                                                                                                                              
                 1  2 1    1                     1   2 1  2 2 E                                             
                                              2                                                                     
                         1             E                    1 2          E 2  d e r 2               w2
                                                                                                                   O d 2 w2 
                                                         21   1  2  1   w                                         
                                                                                                                           
                                                                1            E                                           

               
  coating
 S        , r    The phase noise two-point correlation for a          2f   Frequency
                     coated half-space mirror; double-sided            
                                                                       r          a relative position vector between the two beam
                                                                                                                  
 E, ,              Young’s modulus, Poisson ratio, and loss                     centers on the coating surface;  0 for a
                                                                                                                 r
                     function of the substrate material                           single reflection.
 E , ,             Those of the coating material                      k, w      The laser beam wave number, and spot size
                                                                                  (amplitude radius)
 d                   The coating thickness
                                                                       kB , T     The Boltzmann constant and the temperature
 I0(z)               The 0-th order modified Bessel function of the
                     first kind; I0(0)=1


     November 29-30, 2001                                   Talk at PAC-11, LHO                                             12
     LIGO-G010415-00-Z
                                          Coating Noise
• Resonator

                                           1  1   2  1  2 1    1
                                                                                d 
 Scoating
              ,0  4k   2
                               
                                   2kBT
                                                                            
                                           w
                                                 E           1         E      w
                                                                                     


• Delay lines
               
  coating
 S        , r   4k 2  2k BT          1
                                          w
                                                                                        
         1   2  r 2 2 w2                    1  2 1    1                     
                             I 0 r 2 2 w2   
                                                                         d
                                                                      e r
                                                                                  2
                                                                                      w2
                 e                                                                       
            E                               1         E     w             
                             r     1
                             
                                    r                                                   

 November 29-30, 2001                          Talk at PAC-11, LHO                    13
 LIGO-G010415-00-Z
                              60

                                                     Fabry-Perot resonator
                                                        w/ coated mirrors
                                                      (Various beam radii)         wo/ coatings
                                                             w=3.5cm                      w=3.5cm
Phase Noise [prad/root-Hz]




                                                             2*w                          2*w             Substrate=Fused silica
                              40                             5*w                          5*w
                                                                                                                E  72.6 GPa
                                                             10*w                         10*w                 
                                                                                                               σ  0.16
                                                                                                               Q  3 10 7
                                                                                                               

                                                                                                          Coating= average of
                                                                                                           Al 2O3 and Ta2O5
                              20
                                                                                                          (Crooks et al.)
                                                                                                                  E  260 GPa
                                                                                                                 
                                                                                                                 σ  0.26
                                                                                                                 Q  1.6 10 4
                                                                                                                 

                                0
                                                50            100                150                200
                                                         Frequency [Hz]

                             November 29-30, 2001                   Talk at PAC-11, LHO                                     14
                             LIGO-G010415-00-Z
                                      60
                                                    Optical Delay Lines
                                                     w/ coated mirrors
                                                        (w=3.5cm)          wo/ coating
                                                            R=w/3                R=w/3
                                                            R=2w/3               R=2w/3
         Phase Noise [prad/root-Hz]


                                      40                    R=5w/2               R=5w/2
                                                            R=10w                R=10w




                                      20




                                      0
                                           50            100              150             200
                                                     Frequency [Hz]
November 29-30, 2001                            Talk at PAC-11, LHO                             15
LIGO-G010415-00-Z
                         Proposed Work
• Edge effects on coated mirror noise
   – Analytical quarter-space model


• Extension to thermo-elastic noise
   – Coated mirrors
   – Delay-line interferometers
   – Mirrors at low temperatures


• Thermal & thermo-elastic noises
   – Mirrors of realistic shapes
         • Numerical Green’s function calculation



 November 29-30, 2001          Talk at PAC-11, LHO   16
 LIGO-G010415-00-Z
                Quarter-Space Mirror Model
• Edge effects on noise calculation
   – Analytical model




                                




 November 29-30, 2001    Talk at PAC-11, LHO   17
 LIGO-G010415-00-Z
        Quarter-Space Mirror Model (con’t)
• Finite-mirror size effect on thermal noise (scalar model)
                                               15
                                                                 Beam (power-)radius=6cm
                                                               Number of Delay-line spots=130
                                                                Diameter of D.-L. circle=27cm
              Phase Noise [prad/sq. root Hz]



                                                                        F.-P. (Mirror dia.=30cm)

                                               10                       D-L #1 (Mirror dia.=30cm)
                                                                        D-L #2 (Mirror dia.=45cm)
                                                                        D-L #3 (Mirror dia.=60cm)
                                                                        D-L #4 (Mirror dia.=300cm)


                                               5




                                               0
                                                    50          100            150            200
                                                           Frequency [Hz]
 November 29-30, 2001                                    Talk at PAC-11, LHO                         18
 LIGO-G010415-00-Z
      Thermo-elasticity & Thermal diffusion
• From previous identification
                                                                                          
                                                                                 2
               
      S  , r , r   4k 2 B2
                            4k T 1    
                                     Emech                Emech   T0  dV T   dV uijijkl ukl
                                                                                                 
                              F  2                                                  V                 V




                                        
                            2
      Emech   T0  dV T  2  dV uij cijkl ukl
                                      
                                                
                             V                                 V


• If the adiabatic condition holds
      T  T0   TC vl2  vt2 ull
                  0
                    p



                                                            w  
      S el  , r1 , r2   4k 2 B        dS  dS  00 r   r1  00 r   r2 
                                  2k T
                                                    w             
       th

                                  
                        
                            
                            
                              T    v
                                  
                                  Cv
                                         2
                                         l           d x  
                                              vt2 
                                                       2

                                                           V
                                                                   3
                                                                       i   l
                                                                               
                                                                               nl x , r ; c i  k  nk x , r ; c
                                                                                                           




 November 29-30, 2001                    Talk at PAC-11, LHO                                                            19
 LIGO-G010415-00-Z
                           Numerical elasticity
•    Noise study:
                                                                            2ui   jTij  0
         Cylindrical mirrors
                                                                                                 
      – Numerical Green’s function                              2 ik   j  ijk   ik   x  x P 
        computation
            • Betti-Rayleigh-Somigliana
                                                                     
                                                                                         S
                                                                                           ijk    
                                                       V  xP uk  xP    dS j  ui    Tij ik
                                                                                                     
                                                                                                                    
              formula
                                                       
            • Nodal discretization by the         ui  x       Displacement
              boundary element method.
                                                                stress tensor Tij  cijkl  k ul 
                                                        
                                                  Tij  x 

                                                              the fundamental solution (Green’s function)
                                                  ij  x 
      – Cylindrical mirror model
            • Single reflection                        
                                                   x 
                                                   ijk            cijlm  l mk
            • Fabry-Perot                         cijkl         elastic constants
            • Delay-line
                                                               Density
      – Demonstrate the delay-line vs.           V, S           volume of a region and its boundary surface
        resonator assertion                        V x 
                                                               the characteristic function of V (=1 inside V, =0
                                                                outside)
      – Effects of mirror aspect ratios.

    November 29-30, 2001             Talk at PAC-11, LHO                                                      20
    LIGO-G010415-00-Z
                                  Static Elasticity
                                                                                                   
• Needs to avoid rigid-body motions                                                                f
   – Mass and moment of inertia                                                  
                                                                                 n
          M   dV  , I ij   dV  x 2 ij  xi x j 
                   V             V


   – Given surface forces f can be counter-                               S              
     balanced by volume forces                                                           g
                                        
                a   V F
                                       1
                                
           g x   a  b  x ,                                         V
                                       M 1
                                b   V I  K
                                                                                            
                                                                                             K
   where
                                
        F   dS f x , K   dS x  f x                       
              S                   S                                                      O
                                                       F  Fg  0, K  K g  0               
                                                                                     x
        Fg   dV g x , K g   dV x  g x                                       b
               V                      V                                                           
                                                                                                 bx
 November 29-30, 2001                      Talk at PAC-11, LHO                                   21
 LIGO-G010415-00-Z
                         Tasks & Time Lines
T1. Coating noise estimation                 T4. Thermo-elastic noise estimation
    – spatial de-correlation                       – delay-line vs. resonator
T2. Numerical noise estimations                    – coating
   – Realistic mirror shapes                 T5. Quarter space II
   – Validation                                    – coating
T3. Quarter space                            T6. Anisotropy effects
   – Analytical model                        T7. Dielectric loss study
   – Finite size effect                      T8. Off-beam-axis scatterings
    Task                  3/31/2003                     3/31/2004         3/31/2005
     T1
     T2
     T3
     T4
     T5
  T6,T7,T8

  November 29-30, 2001            Talk at PAC-11, LHO                           22
  LIGO-G010415-00-Z
                             Summary
Accomplishments                           Proposed Activities
• Obtained two-point phase noise          • Studies of intrinsic noises of
  correlation formulas                      coated mirrors
    – Resonator                                 – Thermal noise
    – Delay line                                – Thermo-elastic noise
    – Delay lines can be quieter                – Relative significance of
                                                  coating noise
                                                – Realistic mirror shapes
• Given coating noise formulas
    – Relative magnitudes between
      coating and substrate noises.
    – Their behaviors against the
      beam size and other optical
      parameters




  November 29-30, 2001         Talk at PAC-11, LHO                           23
  LIGO-G010415-00-Z
                        Broader Impacts
• Education
   – Student involvement through the collaboration with Stanford
     Group
• Contribution to LSC
   – The thermal noise estimation methodology
   – Impact on sorting out advanced optics designs
   – Impact on mirror material selection
• Impacts on other federally funded programs
   – Through advancement of computational physics methodology
         • NSF Industry/University Cooperative Research Center program
         • DOE/NERI project; “On-line NDE for advanced reactors”




 November 29-30, 2001         Talk at PAC-11, LHO                        24
 LIGO-G010415-00-Z

				
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