1 Measured Q of sapphire mirrors at KAGRA site
   2 
   3 All results are summarized in klog.
   4 
   5 ETMX : 23.6904 kHz Q=1.1*10^5
   6 80K
   7 GTAT
   8 
   9 ITMX :  23.65325 kHz Q = 6.4*10^4
  10 250K    34.3927 kHz Q = 3.5*10^4
  11 Shikosha
  12 
  13 ITMY : 23.6476 kHz  Q=2.1*10^4
  14 250K   34.3465 kHz Q=1.8*10^5
  15 Shinkosha
  16 
  17 ETMY : 23.7124 kHz Q=2*10^4 
  18 250K   34.4963 kHz Q=2*10^5
  19 GTAT   50.7746kHz Q=1*10^5
  20        51.03675kHz Q= 3*10^4
  21        51.1698 kHz  Q=3*10^4
  22 
  23 Ref.1 
  24 Mirror Q in KAGRA official sensitivity is 10^8.
  25 
  26 Ref.2
  27 T. Uhiyama Ph.D.
  28 Suspended sapphire mirror (100mm in diameter, 60mm in thickness, 
  29 Crystal systems Hemlite grade) 
  30 without ears and nail heads and any bonding techniques.
  31 
  32 
  33 300K
  34 51kHz 3*10^6
  35 68kHz 4*10^6
  36 
  37 77K
  38 51kHz 3*10^7
  39 68kHz 4*10^7
  40 
  41 20K
  42 51kHz 5*10^7
  43 68kHz 1*10^8
  44 
  45 
  46 Thermal noise evaluation by K. Komori (K. Yamamoto checked)
  47 Mirror thermal noise by sapphire substrate is half order of magnitude smaller than O4a sensitivity.
  48 
  49 Note 1 : The mirror temperature is assumed to be room temperature. 
  50 If Q is independent of temperature,
  51 the thermal noise at 20K is 4 times smaller than that at 300K.
  52 -> 20Mpc ? (25 Mpc, Virgo in O2)
  53 
  54 Note 2 : Thermal noise strongly depends on loss distribution in mirror. Homogeneous distribution is assumed 
  55 in above calculation. If loss is inhomogeneous, 
  56 thermal noise amplitude could be 3 or 10 times smaller (K.Yamamoto Ph.D.) except for coating. 
  57 
  58 
  59 Fundamentals
  60 
  61 (1)Measured Q of mirror is limited by sapphire substrate loss 
  62 and other parts (bonding and so on).
  63 
  64 (2)Thermoelastic noise of substrate does not matter for Q 
  65 (Q limited by thermoelastic noise depends on size and shape. 
  66 Q of bulk thermoelastic damping is quite small).
  67 
  68 (3)Here we do not consider coating 
  69 (According to measurement and calculation at Toyama, Q limited by coating is 10^8).
  70 
  71 
  72 What do old theses tell us ? 
  73 
  74 Q limited by substrate loss has no or weak frequency dependence 
  75 (K. Numata, master and Ph.D. theses).
  76 Q limited by inhomogeneous loss has strong frequency dependence 
  77 (N. Ohishi, master thesis) 
  78 because energy density around loss depends on mode.
  79 -> If measured (a few or several or even more) Q-values varies, 
  80 (a)the highest Q indicate the upper limit of substrate loss
  81 (b)the lowest Q indicate the upper limit of inhomogeneous loss (bonding and so on).
  82 
  83 
  84 Example 1
  85 
  86 ITMY : 23.6476 kHz  Q=2.1*10^4
  87 250K   34.3465 kHz Q=1.8*10^5
  88 Shinkosha
  89 
  90 ETMY : 23.7124 kHz Q=2*10^4 
  91 250K   34.4963 kHz Q=2*10^5
  92 GTAT   50.7746kHz Q=1*10^5
  93        51.03675kHz Q= 3*10^4
  94        51.1698 kHz  Q=3*10^4
  95 
  96 -> Only 34kHz mode Q is high. -> Sapphire substrate Q is 2*10^5 AT LEAST. 
  97 No vendor dependence.
  98 
  99 
 100 
 101 Example 2
 102 
 103 ITMX :  23.65325 kHz Q = 6.4*10^4
 104 250K    34.3927 kHz Q=3.5*10^4
 105 Shikosha
 106 
 107 ITMY : 23.6476 kHz  Q=2.1*10^4
 108 250K   34.3465 kHz Q=1.8*10^5
 109 Shinkosha
 110 
 111 ITMX Q (34kHz) is smaller than ITMY Q (34kHz). Vendor is same. 
 112 -> by inhomogenous loss ? or individual differences of substrate from same company ??
 113 
 114 
 115 Example 3
 116 
 117 
 118 ETMX : 23.6904 kHz Q=1.1*10^5
 119 80K
 120 GTAT
 121 
 122 ITMX :  23.65325 kHz Q = 6.4*10^4
 123 250K    
 124 Shikosha
 125 
 126 ITMY : 23.6476 kHz  Q=2.1*10^4
 127 250K   
 128 Shinkosha
 129 
 130 ETMY : 23.7124 kHz Q=2*10^4 
 131 250K   
 132 GTAT   
 133 
 134 -> Temperature dependence ? 
 135 
 136 
 137 
 138 Summary
 139 (1)Sapphire substrate Q is 2*10^5 AT LEAST. No vendor dependence. 
 140 Q which is on the order of 10^4 might be limited by inhomogeneous loss. -> First priority ?
 141 (2)Q of 24kHz suggests temperature dependence ? No vendor dependence. 
 142 -> Q measurement after cooling is necessary. 
 143       
 144 
 145 Reason why measured Q was so low can NOT be explain easily.
 146 
 147 (mirror Q limited by inhomogeneous loss)
 148  = (Q of loss)*(total energy of mirror)/(energy in loss)
 149 can be approximated as
 150 (Q of loss)*(total volume of mirror)/(loss region volume)
 151 
 152 Example 
 153 
 154 Let us assume that sapphire ears has large loss and it limit Q.
 155 The ratio of volume of ears to mirror is on the order of 0.01.
 156 If sapphire ears Q is 10^2, the mirror Q is 10^4.
 157 
 158 Loss in smaller parts can not reduce Q as measured.
 159 
 160 Concern : Young's modulus in bonding is smaller than sapphire. 
 161 Although D.Chen (Ph.D.thesis) calculated thremal noise with Levin's method, Q was not calculated.
 162 
 163 
 164 Comparison with T.Uchiyama Ph.D. thesis
 165 
 166 His mirror was smaler than KAGRA mirror -> Size dependence of Q ?
 167 
 168 His sapphire suspension did NOT have 
 169 (1)HCB -> It works in LV well.
 170 (2)Ears -> If ear Q is 10^2, mirror Q is 10^4. 
 171 (3)Gallium -> D. Chen (Ph.D.) investigated Indium.
 172      Q of sapphire sample with indium bonding was about 10^5 at 300K. 
 173      Indium Q is about 10^2.
 174      Weak temperature dependence (Q at 6K is 10 times smaller than that at 300K) 
 175      Glasgow measurement shows Ga Q is comparable with In Q.
 176 (4)Nail head of fiber ->T. Shishido (master thesis). Q of fiber with nail hear is 10^5 ?
 177 (5)Magnet and magnet spacer -> N. Ohishi (master thesis) 
 178 Q of mirror with magnet 10^4 ~ 10^5.
 179 Her mirror is TAMA size (100mm in diameter, 60mm in thickness) 
 180 which is smaller than KAGRA mirror.
 181 Her magnets were larger than KAGRA magnet.
 182 
 183 Coil induced current -> Viscous damping. 
 184 If Q (=10^4) of mirror is limited by this damping, the pendulum mode Q 
 185 must be around unity (Q is proportional to resonant frequency).
 186 
 187 If we assume structure damping of sapphrie substrate, 
 188 sapphire substrate loss at 20K (Q=10^7) is comparable with 
 189 coating thermal noise. The details must be checked.
 190   
 191 
 192 Errors must be checked carefully.
 193 
 194 Experiment
 195 
 196 Loss budget must be clear.
 197 Mesurement of Q of sapphire substrate or sapphire mirror 
 198 with ears in order to where the loss is.
 199 
 200 NAOJ has sapphire bulk with ears.
 201 
 202 Nodal support system -> Coating damage. It is difficult to treat heavy mass. 
 203 
 204 Steel wire suspension -> Stand off is necessary. However it can be removed.
 205 
 206 Electro static actuator -> D. Chen used in his Ph.D. thesis. The gap is 1mm.
 207 KAGRA mirror is much havier than his sample.
 208 
 209 Room temperature ? 
 210  
 211 Team to cosider plan and so on is necessary.
 212 
 213 
 214 
 215 
 216