2015年度
天体素粒子物理学
特論II
（重力波：その導出・発生・検出・応用）
2015年6月23日~7月14日（火曜）3限
三代木伸二
東京大学宇宙線研究所
宇宙物理学研究部門
重力波推進室
Contents
1. General Introduction of Challenge to Direct
Detection of Gravitational Waves (GWs).
2. Expected Sources of Gravitational Waves.
3. Small signal measurement (General Knowledge)
4. Interferometric Technology for GW Detection (1).
• Power Recycled Fabry-Perot Michelson Interferometer
using Resonant Sideband Extraction Technique
• Practical noise sources and their suppression
5. Interferometric Technology for GW Detection (2).
•
Interferometer Control as GWDs
6. Data Analysis
7. New Fields Driven by the GW detection technique.
Interferometer Control
and
Control Noise
PR-FPMI with RSE
RFPMI With Resonant Sideband Extraction
10-19
10-20
Power
Recycling
Mirror
Strain [1/rHz]
Set Finesse ~ 1500
Resonant
Sideband
Extraction
Mirror
Signal Extraction Gain is also
defined (practically ~ 10)
Optical Noise
10-21
10-22
10-23
10-24
10
100
1k
10k
Frequency [Hz]
This is one of
example of sensitivity
using RSE technique.
Control of IFO as GWD
- Length degrees of freedom is 5 In order to use Interferometer as GWD, many “lengths” between many mirrors
should be controlled.
(y arm) Arm Fabry-Perot Cavities (~ 4km)
Power Recycling Cavity
(PRC)
Resonance (lx+ly)
(Ly)
If dark fringe locking is operated, this
orange area can be regarded as a
compound Mirror that has ~99%
reflectivity to laser.
This violet area can be regarded as a
compound Mirror that has variable
reflectivity to DARM.
ly
Laser
(x arm)
lx
Signal Recycling Cavity
(SRC)
Resonance (lsx+lsy)
(Lx)
lsy
lsx
Total : 5 lengths !!
(FP Cavity : Lx, Ly, lx+ly, lsx+lsy )
(MI
: lx-ly
)
PD
Dark Fringe Locking (lx-ly) at PD
Control of IF as GWD
- signal replacement Convert actual length definition (Lx, Ly, lx, ly, lsx, lsy) to “control” definition (L+,
L-, l+, l-, ls) for convenience
𝐿𝑥 + 𝐿𝑦
𝐿+ =
2
𝐿𝑥 − 𝐿𝑦
𝐿− =
2
𝑙𝑥 + 𝑙𝑦
𝑙+ =
2
𝑙𝑥 − 𝑙𝑦
𝑙− =
2
𝑙𝑠𝑥 + 𝑙𝑠𝑦
𝑙𝑠 =
2
ETMy
FP cavity common motion
( Use as F-stabilization reference)
Ly
FP cavity differential motion
(GW signals are included in)
PRC length motion
(Michelson Part Common)
Michelson Part
Differential motion
SRC length motion
Common : Same motion from BS
Differential : Opposite motion from BS
ITMy
PRM
ly
BS ITMx
Laser
lx
lsy
lsx
SRM
Michelson Part
PD
Lx
ETMx
Michelson Part Control
Dark fringe locking(1st GWD) & Differential Locking & DC Locking (2nd GWD)
Schnupp method (one mod-EOM and 𝒍𝒙 , 𝒍𝒚 are different) is applied for RF-mod.
𝒍𝒙 − 𝒍𝒚
𝒍− =
𝟐
Michelson Part
Differential motion
Schnupp method
EOM
𝒍𝒚
𝒍𝒙
Mid-fringe
At PD
Dark fringe
At PD
Almost Dark
At PD
DC source
RF (wm) : 10 ~20MHz
Lower wm
Carrier
Upper wm
Michelson Part Control
Differential Locking & DC Locking
𝒍𝒚
Symmetric
Port
𝒍𝒙
0
Anti
Symmetric
Port
0
Almost Dark
At PD
DC source
Mid-fringe at each PD
 The Interfered fringes are just opposite at the symmetric port and the antisymmetric pot.
 Take the differential signal of these two ports and obtain linear signal that is
proportional to the michelson arm differential motion (𝒍𝒙 − 𝒍𝒚 ).
Michelson Part Control
Differential and DC control of the michelson fringe,
－
0
PD output @
Symmetric port
=
DC subtraction
0
0
0
PD output @
Anti-Symmetric
port
 Differential Signal.
 Especially, the circle area signal shows the
almost linear relation between the output
and the michelson’s differential arm length.
 This corresponds to the mid-fringe of MI.
 DC locking is just to omit the symmetric
port signal and use DC subtraction to set
the arbitrary locking point (The near dark is
selected in the present GWDs).
Mid-fringe control cannot compatible with the Power Recycling technique. So
the dark fringe locking and DC locking are introduced in the present GWDs.
Michelson Part Control
Differential and DC control of the michelson fringe,
Set the input beam electrical field as,
𝑟𝑥
𝐸𝑖𝑛 = 𝐸0 𝑒 𝑖Ω0 𝑡
𝐸𝑖𝑛
𝐴𝑛𝑡
𝐸𝑃𝐷
= 𝑟𝑥 𝑟𝐵𝑆 𝑡𝐵𝑆 𝐸0 𝑒 𝑖
𝐸0 𝑖
→𝑟
𝑒
2
Ω0 𝑡−𝜙𝑥
𝜙 +𝜙
Ω0 𝑡− + 2 −
=
𝐴𝑛𝑡 2
𝐸𝑃𝐷
−
Ω0 𝑡−𝜙𝑦
𝑟𝐵𝑆
𝑡𝐵𝑆
2 , 𝜙− = 𝜙𝑥 + 𝜙𝑦 , 𝜙− = 𝜙𝑥 − 𝜙𝑦
1 cos 𝜙−
= −
2
2
𝑟𝑦
𝜙 −𝜙
𝑖 Ω0 𝑡− + 2 −
𝑒
𝑆𝑦𝑚
𝑟𝑥 = 𝑟𝑦 = 𝑟, 𝑟𝐵𝑆 = 𝑟𝐵𝑆 = 1
𝐴𝑛𝑡
𝑃𝑃𝐷
− 𝑟𝑦 𝑟𝐵𝑆 𝑡𝐵𝑆 𝐸0 𝑒 𝑖
𝑆𝑦𝑚
𝑃𝑃𝐷
𝐴𝑛𝑡
𝑃𝑃𝐷
𝑃𝑃𝐷
1 cos 𝜙−
= +
2
2
𝑃𝑚𝑎𝑥 + 𝑃𝑚𝑖𝑛 𝑃𝑚𝑎𝑥 − 𝑃𝑚𝑖𝑛
→
−
cos 𝜙−
2
2
𝑃𝑚𝑎𝑥 − 𝑃𝑚𝑖𝑛
C ≡
𝑃𝑚𝑎𝑥 + 𝑃𝑚𝑖𝑛
𝑃𝑚𝑎𝑥
𝑃𝑚𝑖𝑛
0
Michelson Part Control
Drak fringe locking using Schnupp method (one mod-EOM and 𝒍𝒙 , 𝒍𝒚 are
different) is applied for RF-mod.
𝑟𝑥
EOM
Set the input beam electrical field as,
𝐸𝑖𝑛 = 𝐸0 𝑒 𝑖Ω0𝑡
𝒍𝒚
𝐸𝑖𝑛
𝒍𝒙
𝑟𝑦
Dark fringe
At PD
After phase modulation (modulation frequency
𝜔𝑚 , modulation depth 𝑚) using an EOM,
𝐸𝑖𝑛 → 𝐸0 𝑒 𝑖
Ω0 𝑡+𝑚 cos 𝜔𝑚 𝑡
∞
𝑒𝑖
𝑚 cos 𝜔𝑚 𝑡
𝐽𝑛 𝑚 𝑖 𝑛 𝑒 𝑖𝑛𝜔𝑚 𝑡
=
𝑛=−∞
RF (wm) : 10 ~20MHz
𝐽−𝑛 𝑚 = −1 𝑛 𝐽𝑛 𝑚
Lower wm
Carrier
Upper wm
1 𝑚
𝐽𝑛 𝑚 =
𝑛! 2
𝑛
𝑚≪1
Michelson Part Control
Drak fringe locking using Schnupp method
(one mod-EOM and 𝒍𝒙 , 𝒍𝒚 are different) is applied for RF-modulation.
∞
𝐸𝑖𝑛 𝑡 → 𝐸0
~𝐸0 𝐽𝑜 𝑚 𝑒 𝑖Ω0𝑡 + 𝑖𝐽1
𝐽𝑛 𝑚 𝑖 𝑛 𝑒 𝑖
𝑛=−∞
𝑚 𝑒 𝑖(Ω0+𝜔𝑚 )𝑡
Ω0 +𝑛𝜔𝑚 𝑡
+ 𝑖𝐽1 𝑚 𝑒 𝑖(Ω0−𝜔𝑚 )𝑡
 This means that there are mainly three frequency components, named carrier
(Ω0 ) , upper sideband (Ω0 + 𝜔𝑚 ) and lower sideband (Ω0 − 𝜔𝑚 ).
 We expect beat signals between the carrier and sideband if they travel the
same optical path length.
 If the “static arm length difference ” between x and y arms, we cannot obtain
“effective” modulation index because they are cancelled out. So we need
slight arm length difference.
Michelson Part Control
Effective modulation index generated by the static arm length difference in MI.
𝐴𝑛𝑡
𝐸𝑃𝐷
= 𝑟𝑥 𝑟𝐵𝑆 𝑡𝐵𝑆 𝐸𝑖𝑛 𝑡 − 2
𝐴𝑛𝑡
𝑃𝑃𝐷
=
𝐴𝑛𝑡 2
𝐸𝑃𝐷
= 𝐽0 2
𝑙+ + 𝑙−
𝑙+ − 𝑙−
− 𝑟𝑦 𝑟𝐵𝑆 𝑡𝐵𝑆 𝐸𝑖𝑛 𝑡 − 2
𝑐
𝑐
2Ω0 𝑙−
sin
𝑐
2
− 2𝐽0 𝐽1 sin
𝑙𝑥 + 𝑙𝑦
𝑙𝑥 − 𝑙𝑦
, 𝑙− =
,
2
2
1
𝑟𝑥 = 𝑟𝑦 =1 , 𝑟𝐵𝑆 = 𝑡𝐵𝑆 =
2
𝑙+ =
4Ω0 𝑙−
2𝜔𝑚 𝑙−
2𝜔𝑚 𝑙+
sin
sin 𝜔𝑚 𝑡 −
+ 𝐽1 2 ×
𝑐
𝑐
𝑐
Affects demodulation
phase
𝑙− =
0
𝑙−
+ 𝛿𝑙− 𝛿𝑙− ≪
0
𝑙−
, Ω0 ≫ 𝜔𝑚
0
4Ω0 𝑙−
2𝜔𝑚 𝑙−
2𝐽0 𝐽1 sin
sin
𝑐
𝑐
If we demodulate by signal with 𝜔𝑚 , we can obtain 𝛿𝑙− signal from the under bar term.
0
2𝜔𝑚 𝑙−
𝑐 𝜋
0
The effective signal strength is decided by sin
→ 𝑙− =
+ 𝑛𝜋 , 𝑛 = 0,1, . .
𝑐
2𝜔𝑚 2
Consequently, we can obtain the linear signal around the dark (bright) fringe
point ! (not the mid fringe point).
Control of Fabry-Perot Cavity
 What is Fabry-Perot Cavity??
Laser
𝐸0
Cavity length : 𝐿[𝑚] → 𝜙[deg]
Ref : r1
Tra : t1
Los : A1 = 0
Ref : r2
Tra : t2
Los : A2 = 0
𝐸𝐹𝑃
 Composed of two or more facing mirrors.
 Mirrors normally have one high reflective side and anti-reflection side.
 A standing wave resonates inside the FP cavity, then the FP inside power can be
enhanced by a factor that is calculated by mirrors reflectivity.
 FP cavity property serves as the multi reflection to lengthen the optical path in GWDs.
Photo : Rigid Type Fabry-Perot Cavity
Body : High quality fused silica, which has
a hole along the optical axis.
Mirrors : optically contacted on the both
circular surfaces (polished). 99.99% or
more high reflectivity for a specified laser
frequency, while less than 0.2% reflectivity
for the opposite side.
Control of Fabry-Perot Cavity
 Optical Parameters of FP
Cavity length : 𝐿 𝑚 → 𝜙 =
Laser
[deg]
Ref : 𝜌2
Ref : 𝜌1
𝐸0 𝑒 𝑖Ω𝑡
Ω0 𝐿
𝑐
𝐸𝐹𝑃
Tra : 𝜏2
Los : A2 (= 0)
Tra : 𝜏1
Los : A1 (= 0)
𝜌, 𝜏 ∶ Amplitude Reflectance, Transmittance
𝑅, = 𝜌2 , 𝑇 = 𝜏 2 𝑇: Power Refrectance, Transmittance.
𝜌12 + 𝜏12 + 𝐴1 = 1
If FP cavity resonates, FP cavity can be regarded as “one” mirror (compound mirror).
Laser
𝑡𝐹𝑃
2
2
𝑟𝐹𝑃
+ 𝑡𝐹𝑃
+ 𝐴𝐹𝑃 = 1
𝑟𝐹𝑃
FP cavity Reflectance
𝑟𝐹𝑃
FP cavity Transmittance 𝑡𝐹𝑃
−𝜌1 + (1 − 𝐴1 )𝜌2 𝑒 −2𝑖𝜙
𝜙 or Ω, 𝐿 =
1 − 𝜌1 𝜌2 𝑒 −2𝑖𝜙
𝜏1 𝜏2 𝑒 −𝑖𝜙
𝜙 or Ω, 𝐿 =
1 − 𝜌1 𝜌2 𝑒 −2𝑖𝜙
𝜙 = 0 menas resonance
𝜙=
Ω𝐿
𝑐
Control of Fabry-Perot Cavity
 Optical Parameters of FP
Laser
𝐸0 𝑒 𝑖Ω𝑡
Cavity length : 𝐿[𝑚] → 𝜙[deg]
Ref : 𝜌1
Ref : 𝜌2 ~ 0.9999
𝐸𝐹𝑃
Tra : 𝜏1
Los : A1 (= 0)
Tra : 𝜏2
Los : A2 (= 0)
In GWDs, 𝜌12 < 𝜌22 ~0.9999 (named over coupled cavity) is selected for the arm FP
2
cavities to obtain high FP cavity reflectance ( 𝑟𝐹𝑃
) for the Power Recycling technique.
Power inside the FP vs Cavity Length
𝐸𝐹𝑃 =
𝜏1 𝑒 −𝑖𝜙
𝐸
1 − 𝜌1 𝜌2 𝑒 −2𝑖𝜙 0
𝜌12 = 0.97
𝜌22 = 0.9999
2
2
𝑟𝐹𝑃
Red , 𝑡𝐹𝑃
(Green)
Control of Fabry-Perot Cavity
 What is Fabry-Perot Cavity??
Cavity length : 𝐿[𝑚] → 𝜙[deg]
Laser
𝐸0
 Free Spectral Range : laser frequency can
resonate in FP cavity with every FSR intervals
𝐹𝑆𝑅 =
𝑐
[Hz]
2𝐿
Ref : r1
Tra : t1
Los : A1 = 0
Ref : r2
Tra : t2
Los : A2 = 0
𝐸𝐹𝑃
 Full Width at Half Maximum (FWHM [Hz] )
𝜌12 = 0.7000
𝜌22 = 0.9999
 FP cavity Finesse
ℱ≡
𝐹𝑆𝑅
𝜋 𝜌1 𝜌2
=
𝐹𝑊𝐻𝑀 1 − 𝜌1 𝜌2
Parameter showing inside power enhancement
Control of Fabry-Perot Cavity
 Optical Parameters of FP
Laser
Cavity length : 𝐿[𝑚] → 𝜙[deg]
Ref : 𝜌1 ~ 0.99
Ref : 𝜌2 ~ 0.99
Tra : 𝜏1
Los : A1 (= 0)
Tra : 𝜏2
Los : A2 (= 0)
Transmitted
beam is used
for GWDs
𝜌12 = 𝜌22 (named critical coupling cavity) is selected for Mode Cleaners in GWDs to
obtain a stabilized laser beam in its transverse mode and frequency noise (Intensity
noise).
2
2
𝑟𝐹𝑃
Re𝑑 , 𝑡𝐹𝑃
(Green)
Power inside the FP vs Cavity Length
𝐸𝐹𝑃 =
𝜏1 𝑒 −𝑖𝜙
𝐸
1 − 𝜌1 𝜌2 𝑒 −2𝑖𝜙 0
𝜌12 = 0.99
𝜌22 = 0.99
Control of FP Cavity
FP cavity control by using Pound-Drever-Hall Method (1983) is base
Resonant
EOM
Laser
PZT,
Thermal
Ref : r1
Tra : t1
Los : A1 = 0
PBS
λ/2
Wide
Band
EOM
Ref : r2
Tra : t2
L+, L- Los : A2 = 0
λ/4
Photo
Detector
Oscillator(wm)
In
Phase
Laser Frequency Tuning port
(PZT, Thermal, Outer EOM)
Frequency stabilization servo
using L+ as a reference
Mixer
Demodulated
Signal
Magnet-Coil actuator for Mirror actuation
L- control to derive GW signals
John Hall : Nobel prize award winner.
Ron Drever : Almost interferometric techniques were invented by him.
Pound Drever Hall Method
Pound-Drever-Hall Method (1983) is base
Laser
𝐸0 𝑒 𝑖Ω𝑡
Resonant
EOM
PBS
λ/2
PZT,
Thermal
Ref : r1
Tra : t1
Los : A1 = 0
Ref : r2
Tra : t2
Los : A2 = 0
Cavity length (L)
λ/4
Photo
Detector
Oscillator(wm)
Mixer
In
Phase
Voltage Signal (V)
Pound-Drever-Hall Method obtains a linear voltage signal (V) that is proportional
to the cavity length (L) to keep the Fabry-Perot Cavity on resonance.
∞
𝐸𝑖𝑛 𝑡 → 𝐸0
~𝐸0 𝐽𝑜 𝑚
𝑒 𝑖Ω0 𝑡
+ 𝑖𝐽1
𝐽𝑛 𝑚 𝑖 𝑛 𝑒 𝑖
𝑛=−∞
𝑚 𝑒 𝑖(Ω0+𝜔𝑚 )𝑡
Ω0 +𝑛𝜔𝑚 𝑡
+ 𝑖𝐽1 𝑚 𝑒 𝑖(Ω0−𝜔𝑚 )𝑡
Pound Drever Hall Method
Pound-Drever-Hall Method (1983) is base
𝐸𝑖𝑛 = 𝐸0 𝐽𝑜 𝑚 𝑒 𝑖Ω0 𝑡 + 𝑖𝐽1 𝑚 𝑒 𝑖(Ω0 +𝜔𝑚 )𝑡 + 𝑖𝐽1 𝑚 𝑒 𝑖(Ω0 −𝜔𝑚 )𝑡
𝐹𝑃
𝐸𝑟𝑒𝑓
= 𝐸0 𝐽𝑜 𝑚 𝑟𝐹𝑃 Ω0 𝑒 𝑖Ω0 𝑡 + 𝑖𝐽1 𝑚 𝑟𝐹𝑃 Ω0 + 𝜔𝑚 𝑒 𝑖(Ω0 +𝜔𝑚 )𝑡 + 𝑖𝐽1 𝑚 𝑟𝐹𝑃 Ω0 − 𝜔𝑚 𝑒 𝑖(Ω0 −𝜔𝑚 )𝑡
Normally, 𝜔𝑚 is selected well separated from the
resonance area. Ω0 ± 𝜔𝑚 Is not resonating inside FP
cavity
𝑟𝐹𝑃 2 𝛺 − 𝜔𝑚
𝑟𝐹𝑃 2 𝛺 + 𝜔𝑚
𝑟𝐹𝑃 2 Ω
→ 𝐸𝑐𝑎𝑟 𝑒 𝑖Ω0 𝑡 + 𝐸𝑢𝑝 𝑒 𝑖(Ω0 +𝜔𝑚 )𝑡 + 𝐸𝑑𝑤 𝑒 𝑖(Ω0 −𝜔𝑚 )𝑡
−𝜔𝑚
Demodulation Signal in In-phase is proportional to,
∗ + 𝐸∗
∗
𝐸𝑐𝑎𝑟 𝐸𝑢𝑝
𝑑𝑤 + 𝐸𝑐𝑎𝑟 𝐸𝑢𝑝 + 𝐸𝑑𝑤
Demodulation Signal in Quadrature-phase is proportional to,
∗
∗
∗
−𝑖 × 𝐸𝑐𝑎𝑟 𝐸𝑑𝑤
− 𝐸𝑢𝑝
+ 𝐸𝑐𝑎𝑟
𝐸𝑢𝑝 − 𝐸𝑑𝑤
+𝜔𝑚
Pound Drever Hall Method
Pound-Drever-Hall Method (1983) is base
Demodulation Signal in In-phase is
proportional to,
∗
∗
∗
𝐸𝑐𝑎𝑟 𝐸𝑢𝑝
+ 𝐸𝑑𝑤
+ 𝐸𝑐𝑎𝑟
𝐸𝑢𝑝 + 𝐸𝑑𝑤
+𝜔𝑚
−𝜔𝑚
Demodulation Signal in Quadraturephase is proportional to,
∗
∗
∗
−𝑖 × 𝐸𝑐𝑎𝑟 𝐸𝑑𝑤
− 𝐸𝑢𝑝
+ 𝐸𝑐𝑎𝑟
𝐸𝑢𝑝 − 𝐸𝑑𝑤
FPMI Control
Dark fringe locking of MI with arm FP cavity resonances
The control of MI using l- should be minimized (UGF is small) because bad S/N.
ETMy
𝐿𝑥 + 𝐿𝑦
𝐿𝑥 − 𝐿𝑦
𝑙𝑥 − 𝑙𝑦
𝐿+ =
, 𝐿− =
, 𝑙− =
2
2
2
𝐿𝑦
ITMy
Faraday
Isolator
𝜔𝑚
S-polarized Beam
𝑙𝑦
𝐸0 𝑒 𝑖Ω𝑡
EOM
𝑙𝑥
Sym
PD
Quad
phase
𝑙−
In phase
For frequency
stabilization
BS ITMx
Quad phase
𝐿+
Anti-sym
PD
𝐿−
𝐿𝑥
ETMx
FPMI Control
 Input E and what we should know ?
1
𝐽𝑛 𝑚 𝑖 𝑛 𝑒 𝑖
𝐸𝑖𝑛 𝑡 = 𝐸0
Ω0 +𝑛𝜔𝑚 𝑡
= 𝐸0 𝐽𝑜 𝑚 𝑒 𝑖Ω0 𝑡 + 𝑖𝐽1 𝑚 𝑒 𝑖(Ω0 +𝜔𝑚 )𝑡 + 𝑖𝐽1 𝑚 𝑒 𝑖(Ω0 −𝜔𝑚 )𝑡
𝑛=−1
𝑢𝑝
𝑐𝑎𝑟
𝑑𝑤
→ 𝐸𝐴𝑃𝐷,𝑆𝑃𝐷
𝑒 𝑖Ω0 𝑡 + 𝐸𝐴𝑃𝐷,𝑆𝑃𝐷 𝑒 𝑖(Ω0 +𝜔𝑚)𝑡 + 𝐸𝐴𝑃𝐷,𝑆𝑃𝐷
𝑒 𝑖(Ω0 −𝜔𝑚)𝑡 at APD,
SPD should be derived to signal estimation for 𝐿+ , 𝐿− , 𝑙+ , 𝑙− .
For this purposes, it is useful to calculate equivalent reflectance for each
light component.
ETMy
𝐿𝑦
 Equivalent Ref. of FP cavities 𝑟𝐹𝑃𝑥,𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿−
𝑟𝐹𝑃𝑥,𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿− =
−𝜌1 + (1 − 𝐴1 )𝜌2 𝑒 −2𝑖𝜙𝑥,𝑦
ITMy
1 − 𝜌1 𝜌2 𝑒 −2𝑖𝜙𝑥,𝑦
Ω𝐿𝑥,𝑦 Ω 𝐿+ ± 𝐿−
𝜙𝑥,𝑦 =
=
𝑐
𝑐
Ω = Ω0 + 𝑛𝜔𝑚 (𝑛 = −1,0,1)
𝐸𝑖𝑛 𝑡
 Equivalent Ref. of MI Part for APD, SPD
𝐸𝑆𝑃𝐷
𝑟𝑀𝐼𝐴𝑛𝑡,𝑀𝐼𝑆𝑦𝑚 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
Symmetric Port
𝑟𝑀𝐼𝐴𝑛𝑡 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙− =
𝑟𝑀𝐼𝑆𝑦𝑚 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙− =
𝑙𝑦
𝑙 +𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑟𝐵𝑆 𝑡𝐵𝑆 𝑟𝐹𝑃𝑥 Ω, 𝐿+ , 𝐿− 𝑒
𝑙 +𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑡𝐵𝑆 𝑡𝐵𝑆 𝑟𝐹𝑃𝑥 Ω, 𝐿+ , 𝐿− 𝑒
𝑙𝑥
BS ITMx
𝐿𝑥
ETMx
Anti-Symmetric Port
𝐸𝐴𝑃𝐷
−
𝑙 −𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑟𝐵𝑆 𝑡𝐵𝑆 𝑟𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿− 𝑒
+
𝑙 −𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑟𝐵𝑆 𝑟𝐵𝑆 𝑟𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿− 𝑒
FPMI Control
 What should we know ?
𝑟𝑀𝐼𝐴𝑛𝑡 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙− = 𝑟𝐵𝑆 𝑡𝐵𝑆 𝑟𝐹𝑃𝑥 Ω, 𝐿+ , 𝐿− 𝑒
𝑟𝑀𝐼𝑆𝑦𝑚 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙− =
𝑙 +𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑙 +𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑡𝐵𝑆 𝑡𝐵𝑆 𝑟𝐹𝑃𝑥 Ω, 𝐿+ , 𝐿− 𝑒
− 𝑟𝐵𝑆 𝑡𝐵𝑆 𝑟𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿− 𝑒
𝑙 −𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑙 −𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑟𝐵𝑆 𝑟𝐵𝑆 𝑟𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿− 𝑒
+
ETMy
𝑐𝑎𝑟
𝐸𝐴𝑃𝐷,𝑆𝑃𝐷
= 𝐽𝑜 𝑚 𝑟𝑀𝐼𝐴𝑛𝑡,𝑀𝐼𝑆𝑦𝑚 Ω0 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝐸0
𝐿𝑦
𝑢𝑝
𝐸𝐴𝑃𝐷,𝑆𝑃𝐷 = 𝑖𝐽1 𝑚 𝑟𝑀𝐼𝐴𝑛𝑡,𝑀𝐼𝑆𝑦𝑚 Ω0 + 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝐸0
𝑑𝑤
𝐸𝐴𝑃𝐷,𝑆𝑃𝐷
= 𝑖𝐽1 𝑚 𝑟𝑀𝐼𝐴𝑛𝑡,𝑀𝐼𝑆𝑦𝑚 Ω0 − 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝐸0
ITMy
𝑙𝑦
𝐸𝑖𝑛 𝑡
𝑙𝑥
𝐸𝑆𝑃𝐷
Demodulation Signals in In-phase at APD (, SPD) are proportional to,
𝑢𝑝∗
𝑢𝑝
𝑐𝑎𝑟
𝑑𝑤∗
𝑐𝑎𝑟∗
𝑑𝑤
𝐼
𝑆𝐴𝑃𝐷
𝐿+ , 𝐿− , 𝑙+ , 𝑙− = 𝐸𝐴𝑃𝐷
𝐸𝐴𝑃𝐷 + 𝐸𝐴𝑃𝐷
+ 𝐸𝐴𝑃𝐷
𝐸𝐴𝑃𝐷 + 𝐸𝐴𝑃𝐷
Demodulation Signals in Quadrature-phase at APD, SPD are proportional to,
𝑄
𝑢𝑝∗
BS ITMx
𝑢𝑝
𝑐𝑎𝑟
𝑑𝑤∗
𝑐𝑎𝑟∗
𝑑𝑤
𝑆𝐴𝑃𝐷 𝐿+ , 𝐿− , 𝑙+ , 𝑙− = −𝑖 × 𝐸𝐴𝑃𝐷
𝐸𝐴𝑃𝐷
−𝐸𝐴𝑃𝐷 + 𝐸𝐴𝑃𝐷
𝐸𝐴𝑃𝐷 − 𝐸𝐴𝑃𝐷
𝐸𝐴𝑃𝐷
𝐿𝑥
ETMx
FPMI Control
 Signal for 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝐼
𝜕𝑆𝐴𝑃𝐷,𝑆𝑃𝐷
𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝜕𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝑄
𝜕𝑆𝐴𝑃𝐷,𝑆𝑃𝐷 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝜕𝐿+ , 𝐿− , 𝑙+ , 𝑙−
ETMy
𝐼,𝑄
Signal for 𝐿+
𝜕𝑆𝐴𝑃𝐷,𝑆𝑃𝐷 𝐿+ , 0,0,0
𝜕𝐿+
0 means operating point
𝐿𝑦
𝐼,𝑄
Signal for𝑙−
Signal for 𝑙+
𝜕𝑆𝐴𝑃𝐷,𝑆𝑃𝐷 0, 𝐿− , 0,0
𝐼,𝑄
𝜕𝑆𝐴𝑃𝐷,𝑆𝑃𝐷
0,0, 𝑙+ , 0
𝐸𝑖𝑛 𝑡
𝑙𝑦
BS ITMx
𝐿𝑥
𝜕𝑙+
𝐼,𝑄
Signal for 𝑙−
ITMy
𝜕𝐿−
𝜕𝑆𝐴𝑃𝐷,𝑆𝑃𝐷 0,0,0, 𝑙−
𝜕𝑙−
𝐸𝑆𝑃𝐷
Symmetric Port
𝑙𝑥
Anti-Symmetric Port
𝐸𝐴𝑃𝐷
ETMx
We need Concept of Signal Separation
Length signals are desired to be obtained “independently” at detection ports
(REF, DARM, PICK…) with shot noise limited S/N quality, by using In and
Quadrature demodulation phase.
Noisy signal deteriorate sensitivity as “control noise”.
In case of PR-FPMI, signal separation has good solution. However, In the case of
PR-FPMI-RSE we need some compromise.
Symmetric (REF)
Phase
Anti-sym (DARM)
Pick Off at BS (POX)
I
Q
I
Q
I
L+
~850
0
0
0
1
L-
0
~10-2
0
~274
0
-1.7
0
2.1
l+
lls
2
2
𝑟𝐼𝑇𝑀
= 0.97, 𝑟𝐸𝑇𝑀
= 0.9999, 𝐴𝐼𝑇𝑀 = 𝐴𝐸𝑇𝑀 = 0
Q
FPMI Control
Dark fringe locking of MI with arm FP cavity resonances
The control of MI using l- should be minimized (UGF is small) because bad S/N.
𝐿 𝑥 + 𝐿𝑦
𝐿𝑥 − 𝐿𝑦
𝑙𝑥 − 𝑙𝑦
𝐿+ =
, 𝐿− =
, 𝑙− =
2
2
2
ETMy
𝐿𝑦
ITMy
Faraday
Isolator
𝜔𝑚
S-polarized Beam
𝑙𝑦
𝐸0 𝑒 𝑖Ω𝑡
EOM
𝑙𝑥
SPD
Quad
phase
𝑙−
In phase
For frequency
stabilization
BS ITMx
Quad phase
𝐿+
APD
𝐿−
𝐿𝑥
ETMx
PR-FPMI Control
Dark fringe locking of MI with PRC and Arm FP cavity resonances
RF sideband (wm) is designed to resonate in PRC with Carrier
𝐿𝑥 + 𝐿𝑦
𝐿𝑥 − 𝐿𝑦
𝐿+ =
, 𝐿− =
2
2
𝑙𝑥 + 𝑙𝑦
𝑙𝑥 − 𝑙𝑦
𝑙+ =
, 𝑙− =
2
2
ETMy
𝐿𝑦
Several signal separation methods are proposed.
Faraday
Isolator
EOM
In phase
For frequency
stabilization
SPD
Quad
phase
In phase
Quad phase
𝐿+
PRM
ITMy
𝑙𝑦
BS ITMx
𝐿𝑥
𝑙𝑥
𝑙+
𝑙−
POX
PD
APD
𝐿−
ETMx
PR-FPMI Control
 Condition to realize the PR-FPMI control
• The carrier should resonates inside FP cavities and power-recycling cavity.
• The sideband should resonate inside only the power-recycling cavity.
• We should remind that the phase of the carrier from the resonating FP cavity
changes 180 degrees.
ETMy
 The equivalent PR-cavity length should satisfy the
following condition.
0
𝑙+
𝑐𝜋 1
=
+𝑛
𝜔𝑚 2
𝐿𝑦
(𝑛 = 0,1,2 … )
𝑙𝑥 + 𝑙𝑦
0
𝑙+ =
≡ 𝑙+
+ 𝑑𝑙+
2
static deviation
0
Note : 𝑙−
for the frontal modulation
decreases the reflectivity of the
sideband toward the symmetric port
because of its leak toward Antisymmetric port.
𝑙𝑥 − 𝑙𝑦
0
𝑙− =
≡ 𝑙−
+ 𝑑𝑙−
2
static deviation
ITMy
𝑙𝑦
𝐸𝑖𝑛 𝑡
BS ITMx
𝐿𝑥
𝑙𝑥
𝐸𝑆𝑃𝐷
𝐸𝐴𝑃𝐷
PRM
𝑙+
Comp-M
Power Recycling Cavity
ETMx
PR-FPMI Control
 Input E and what we should know ?
1
𝐽𝑛 𝑚 𝑖 𝑛 𝑒 𝑖
𝐸𝑖𝑛 𝑡 = 𝐸0
𝑛=−1
𝑐𝑎𝑟
𝐸𝐴𝑃𝐷,𝑆𝑃𝐷
𝑒 𝑖Ω0 𝑡
Ω0 +𝑛𝜔𝑚 𝑡
= 𝐸0 𝐽𝑜 𝑚 𝑒 𝑖Ω0 𝑡 + 𝑖𝐽1 𝑚 𝑒 𝑖(Ω0 +𝜔𝑚 )𝑡 + 𝑖𝐽1 𝑚 𝑒 𝑖(Ω0 −𝜔𝑚 )𝑡
𝑢𝑝
𝑑𝑤
→
+ 𝐸𝐴𝑃𝐷,𝑆𝑃𝐷 𝑒 𝑖(Ω0 +𝜔𝑚)𝑡 + 𝐸𝐴𝑃𝐷,𝑆𝑃𝐷
𝑒 𝑖(Ω0 −𝜔𝑚)𝑡 at APD,
SPD should be derived to signal estimation for 𝐿+ , 𝐿− , 𝑙+ , 𝑙− .
For this purposes, it is useful to calculate equivalent reflectance for each
light component.
ETMy
𝐿𝑦
 Equivalent Ref. of FP cavities 𝑟𝐹𝑃𝑥,𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿−
𝑟𝐹𝑃𝑥,𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿− =
−𝜌1 + (1 − 𝐴1 )𝜌2 𝑒 −2𝑖𝜙𝑥,𝑦
1 − 𝜌1 𝜌2 𝑒 −2𝑖𝜙𝑥,𝑦
𝐸𝑟 𝑡 ∶ E field
Just after PRM
Ω𝐿𝑥,𝑦 Ω 𝐿+ ± 𝐿−
𝜙𝑥,𝑦 =
=
𝑐
𝑐
Ω = Ω0 + 𝑛𝜔𝑚 (𝑛 = −1,0,1)
𝐸𝑖𝑛 𝑡
 Equivalent Ref. of MI Part for APD, PRM
𝐸𝑆𝑃𝐷
𝑟𝑀𝐼𝐴𝑛𝑡,𝑀𝐼𝑃𝑅 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝑟𝑀𝐼𝐴𝑛𝑡 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙− =
𝑟𝑀𝐼𝑆𝑦𝑚 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙− =
𝑙𝑦
𝑙 +𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑡𝐵𝑆 𝑡𝐵𝑆 𝑟𝐹𝑃𝑥 Ω, 𝐿+ , 𝐿− 𝑒
BS ITMx
𝐿𝑥
ETMx
𝑙𝑥
𝐸𝐴𝑃𝐷
Symmetric Port
𝑙 +𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑟𝐵𝑆 𝑡𝐵𝑆 𝑟𝐹𝑃𝑥 Ω, 𝐿+ , 𝐿− 𝑒
ITMy
−
Anti-Symmetric Port
𝑙 −𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑟𝐵𝑆 𝑡𝐵𝑆 𝑟𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿− 𝑒
+
𝑙 −𝑙
𝑖Ω 𝑡−2 + −
𝑐
𝑟𝐵𝑆 𝑟𝐵𝑆 𝑟𝐹𝑃𝑦 Ω, 𝐿+ , 𝐿− 𝑒
PR-FPMI Control
 Equivalent Transmittance at PRM (𝑔𝑃𝑅𝐶 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙− )
𝑔𝑃𝑅𝐶 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙− ≡
𝐸𝑃𝑅𝐶
𝜏𝑟
=
𝐸𝑖𝑛
1 − 𝜌𝑟 𝑟𝑀𝐼𝑆𝑦𝑚 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
For POX
𝑐𝑎𝑟
𝐸𝑃𝑅
𝑢𝑝
𝐸𝑃𝑅
𝑑𝑤
𝐸𝑃𝑅
= 𝜏𝐵𝑆 𝜌𝐵𝑆𝐴𝑅 𝑟𝐹𝑃𝑥 Ω0 , 𝐿+ , 𝐿−
PRM
𝑙+
𝐸𝑖𝑛 𝜌 𝐸𝑃𝑅𝐶
𝑟
𝜏𝑟
𝐴𝑟
Comp-M
𝑟𝑀𝐼𝑆𝑦𝑚
𝑟𝑒𝑓
𝑙 +𝑙 𝑙𝑝𝑖𝑐𝑘
𝑖Ω 𝑡− + − −
𝑐
𝑐
𝑒
𝐸𝑃𝑅
𝑔𝑃𝑅𝐶 Ω0 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝐽𝑜 𝑚 𝐸0
= 𝜏𝐵𝑆 𝜌𝐵𝑆𝐴𝑅 𝑟𝐹𝑃𝑥 Ω0 , 𝐿+ , 𝐿−
𝑙 +𝑙 𝑙𝑝𝑖𝑐𝑘
𝑖 Ω0 +𝜔𝑚 𝑡− + − −
𝑐
𝑐
𝑒
𝑔𝑃𝑅𝐶 Ω0 + 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝑖𝐽1 𝑚 𝐸0
= 𝜏𝐵𝑆 𝜌𝐵𝑆𝐴𝑅 𝑟𝐹𝑃𝑥 Ω0 , 𝐿+ , 𝐿−
𝑙 +𝑙 𝑙𝑝𝑖𝑐𝑘
𝑖 Ω0 −𝜔𝑚 𝑡− + − −
𝑐
𝑐
𝑒
𝑔𝑃𝑅𝐶 Ω0 − 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝑖𝐽1 𝑚 𝐸0
For APD
𝑐𝑎𝑟
𝐸𝐴𝑃𝐷
= 𝐽𝑜 𝑚 𝐸0 𝑔𝑃𝑅𝐶 Ω0 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝑟𝑀𝐼𝐴𝑛𝑡 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝑢𝑝
𝐸𝐴𝑃𝐷 = 𝑖𝐽1 𝑚 𝐸0 𝑔𝑃𝑅𝐶 Ω0 + 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝑟𝑀𝐼𝐴𝑛𝑡 Ω0 + 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝑑𝑤
𝐸𝐴𝑃𝐷
= 𝑖𝐽1 𝑚 𝐸0 𝑔𝑃𝑅𝐶 Ω0 − 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝑟𝑀𝐼𝐴𝑛𝑡 Ω0 − 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
PR-FPMI Control
 Equivalent Ref. of PR-FPMI (𝑟𝑃𝑅𝐹𝑃𝑀𝐼 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙− )
PRM
𝑟𝑒𝑓
𝑟𝑃𝑅𝐹𝑃𝑀𝐼 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
=
𝐸
≡ 𝑃𝑅𝐶
𝐸𝑖𝑛
−𝜌𝑟 + 1 − 𝐴𝑟 𝑟𝑀𝐼𝑆𝑦𝑚 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
1 − 𝜌𝑟 𝑟𝑀𝐼𝑆𝑦𝑚 Ω, 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝐸𝑖𝑛 𝜌 𝐸𝑃𝑅𝐶
𝑟
𝜏𝑟
𝐴𝑟
𝑟𝑒𝑓
For SPD
𝑐𝑎𝑟
𝐸𝑆𝑃𝐷
= 𝐽𝑜 𝑚 𝐸0 𝑟𝑃𝑅𝐹𝑃𝑀𝐼 Ω0 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝑢𝑝
𝐸𝑆𝑃𝐷 = 𝑖𝐽1 𝑚 𝐸0 𝑟𝑃𝑅𝐹𝑃𝑀𝐼 Ω0 + 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝑑𝑤
𝐸𝑆𝑃𝐷
= 𝑖𝐽1 𝑚 𝐸0 𝑟𝑃𝑅𝐹𝑃𝑀𝐼 Ω0 − 𝜔𝑚 , 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝑙+
𝐸𝑃𝑅𝐶
Comp-M
𝑟𝑀𝐼𝑆𝑦𝑚
PR-FPMI Control
 Signal for 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝑄
𝐼
𝜕𝑆𝐴𝑃𝐷,𝑆𝑃𝐷,𝑃𝑂𝑋
𝐿+ , 𝐿− , 𝑙+ , 𝑙− 𝜕𝑆𝐴𝑃𝐷,𝑆𝑃𝐷,𝑃𝑂𝑋 𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝜕𝐿+ , 𝐿− , 𝑙+ , 𝑙−
𝜕𝐿+ , 𝐿− , 𝑙+ , 𝑙−
ETMy
𝐼,𝑄
Signal for 𝐿+
𝜕𝑆𝐴𝑃𝐷 𝐿+ , 0,0,0
0 means operating point
𝜕𝐿+
Signal for𝑙−
𝜕𝑆𝐴𝑃𝐷 0, 𝐿− , 0,0
𝜕𝐿−
𝐿𝑦
𝐼,𝑄
Signal for 𝑙+
𝐼,𝑄
𝜕𝑆𝐴𝑃𝐷
𝐼,𝑄
Signal for 𝑙−
0,0, 𝑙+ , 0
𝜕𝑙+
𝜕𝑆𝐴𝑃𝐷 0,0,0, 𝑙−
𝜕𝑙−
ITMy
𝐸𝑖𝑛 𝑡
𝐸𝑆𝑃𝐷
Symmetric Port
𝑙𝑦
BS ITMx
𝐿𝑥
𝑙𝑥
𝐸𝐴𝑃𝐷
Anti-Symmetric Port
POX
PD
ETMx
Signal Separation 1 in PR-FPMI
 PRC mirror reflectivity is matched with the equivalent reflectivity of the carrier
in FPMI.
L+ and l+ , L- and l- are not well separated in each port.
Practically, the L+ and L- fluctuation can be suppressed enough by a large control
loop gain to highlight l+ and l- signals.
Symmetric (SPD)
Phase
Anti-sym (APD)
Pick Off at BS (POX)
I
Q
I
Q
I
Q
L+
52644 (850)
0
0
0
~ -2467
0
L-
0
0
0
5006(274)
l+
397
~0
0
0
-14
0
l-
~0
~0
0
38(2.1)
ls
2
2
2
2
𝑟𝐼𝑇𝑀
= 0.97, 𝑟𝐸𝑇𝑀
= 0.9999, 𝐴𝐼𝑇𝑀 = 𝐴𝐸𝑇𝑀 = 0, 𝑟𝑃𝑅
= 0.989653, 𝑟𝐵𝑆𝐴𝑅
= 0.002
Signal Separation 2 in PR-FPMI
 PRC mirror reflectivity is matched with the equivalent reflectivity of the
sideband in FPMI.
L+, l+ , L- and l- are well separated at each port.
However, signal amount is smaller than the previous case 1.
The reflectance prediction of FPMI is a little bit difficult to match it with the
reflectivity of PRM.
Symmetric (SPD)
Phase
Anti-sym (APD)
Pick Off at BS (POX)
I
Q
I
Q
I
Q
L+
-134 (850)
0
0
0
-559
0
L-
0
~0.013
0
5001(274)
l+
52
-0.016
0
0
-1.9
0.011
l-
0.016
-17
0
38(2.1)
ls
2
2
2
2
𝑟𝐼𝑇𝑀
= 0.97, 𝑟𝐸𝑇𝑀
= 0.9999, 𝐴𝐼𝑇𝑀 = 𝐴𝐸𝑇𝑀 = 0, 𝑟𝑃𝑅
= 0.904, 𝑟𝐵𝑆𝐴𝑅
= 0.002
PR-FPMI with RSE Control
Dark fringe locking of MI with PRC, SRC and Arm FP cavity resonances
We need two modulations (w1 PM, w2 PM or AM) to obtain ls signal
w1 resonates in SRC, while w2 in PRC.
ETMy
𝐿𝑥 + 𝐿𝑦
𝐿𝑥 − 𝐿𝑦
𝑙𝑥 + 𝑙𝑦
𝑙𝑥 − 𝑙𝑦
𝐿+ =
, 𝐿− =
, 𝑙+ =
, 𝑙− =
2
2
2
2
𝑙𝑠𝑥 + 𝑙𝑠𝑦
𝑙𝑠 =
2
 Signal extraction strongly depends on RSE style
ITMy
(Broadband-RSE(BRSE) or Detuned-RSE (D-RSE))
Faraday
Isolator
PM or AM Modulation
SPD
PRM
Carrier
w1
ETMx
BS ITMx
S-polarized
Beam
POX
PD
w1
w2
w2
APD
PR-FPMI-RSE Control in KAGRA
PR-FPMI-RSE Control in KAGRA
Selection of Signal Extraction Port
IFO Design of KAGRA
(PR-FPMI with RSE)
ETMy
For DRSE
MC
Pre-MC
ITMy
IFI PRM
EOM
PR2
w3
w1
w2
ETMx
BS ITMx
200W
Laser
PR3
SPD
(REF)
𝐿+
SR2
POX
SR3
SRM
OMC
REF
DDM
𝑙−
𝐿−
OMC
𝑙𝑠
APD
(DARM)
𝑙+
PR-FPMI-RSE Control in KAGRA
Control Noise
Noise of l+, l-, ls signal
should be lower than L+,
L- (Sensitivity) signal.
1 : Take good S/N signal
for l+, l-, ls by selecting
ports and demodulation
method.
2: UGF for l+, l-, ls are
minimized.
Control of Mirror Alignment
If FP mirror pitch and Yaw motion fluctuate a lot, not only TEM00 but also TEMmn
higher transverse mode can resonate, and this alignment fluctuation generate
noises, spoil the best condition of GWD.
TEM00
TEM10
TEM01
Beam centering is also important because Pitch/Yaw motion can easily couple
with length fluctuation.
Mirror Alignment Control is
Essentially Important
Control of Mirror Alignment
- How to obtain each mirror P/Y motion ? Local optical lever is not useful because local seismic drift generates unreliable
alignment signal.
(This optical level is useful for short time monitoring and shark sensor)
Alignment using local optical lever
Is perfect at the beginning…
QPD (4 segmented PD) fixed on local
position 3km away with each other.
Interference and FP cavity resonance will
be spoiled a lot because of local drift.
Higher mode generation
Imperfect interference
Control of Mirror Alignment
- How to obtain each mirror P/Y motion ? Firstly, find best cavity axis to gives best GWD sensitivity, then take this FP cavity
axis as absolute reference.
In order to obtain alignment signal of mirrors, take beating between TEM00 and
TEM10.01 mode using PDH method using RF/AM modulation (w1 ,w2)
a
a
Flat
Curved
 TEM01 is superimposed on TEM00, then
 Beam center seems to laterally shift.
 4 segmented Photo Detector is used to
derive opposite sign signals in each half
section.
Control of Mirror Alignment
- How to obtain each mirror P/Y motion ? Quadrant photodetector
 4 segmented Photo Detector is used to
derive opposite sign signals in each half
section.
I-V transform
+
−
Yaw
Signal
+
+
0
+
−
Pitch
Signal
0
Control of Mirror Alignment
- How to obtain each mirror P/Y motion ? Firstly, find best cavity axis to gives best GWD sensitivity, then take this FP cavity
axis as absolute reference.
In order to obtain alignment signal of mirrors, take beating between TEM00 and
TEM10.01 mode using PDH method using RF modulation (w)
In order to obtain alignment signal of ITM and ETM separately, use guoy phase
progress difference between TEM00 and TEM10.01 for these mirrors.
UGF for alignment control is desirable to be as small as possible because its
feedback signal will easily couple with length signal as noise.
ITM misalignment case
ETM misalignment case
Flat
Curved
a
a
FP cavity axis moves in lateral
a Flat
Curved
a
FP cavity axis moves angularly.
Helmhortz Equation
Maxwell Equation
1 𝜕2𝐸
∆𝐸 = 2 2
𝑐 𝜕𝑡
𝐸𝑥,𝑦,𝑧 𝒓, 𝑡
 EMW is supposed to have a polarization only in one direction. Set
the part 𝑢.
2𝜋
2
∆𝑢 + 𝑘 𝑢 = 0 𝑘 ≡
𝜆
Search a solution that has the following style
𝑢 = 𝑓 𝑥, 𝑦, 𝑧 𝑒 𝑖𝑘𝑧
Set slowly varying envelope approximation ( f is supposed to change slowly in the
direction of z)
𝜕2𝑓
𝜕𝑓 𝜕 2 𝑓 𝜕 2 𝑓
𝜕𝑓
𝜕𝑧 2
≪𝑘
𝜕𝑧
→
𝜕𝑥 2
+
𝜕𝑦 2
+ 2𝑖𝑘
𝜕𝑧
=0
Solution of Helmhortz Equation
Spherical
𝐴 𝑖𝑘𝑟
𝑢 𝒓 = 𝑒
𝑟
Paraboloid
𝐴
𝑥2 + 𝑦2
𝑢 𝒓 ≈ 𝑒𝑥𝑝 𝑖𝑘 𝑧 +
𝑧
2𝑧
𝐴
𝑥2 + 𝑦2
𝑓 = 𝑒𝑥𝑝 𝑖𝑘
𝑧
2𝑧
Planar
𝑢 𝒓 = 𝐴𝑒 𝑖𝑘𝑧
𝑓=1
𝑟=
𝑥2
+
𝑦2
+
𝑧2
𝑥2 + 𝑦2
= z 1+
𝑧2
Solution of Helmhortz Equation
Paraboloidal
𝐴
𝑥2 + 𝑦2
𝑢 𝒓 ≈ 𝑒𝑥𝑝 𝑖𝑘 𝑧 +
𝑧
2𝑧
𝑟=
𝑥2 + 𝑦2 + 𝑧2
𝑥2 + 𝑦2
=z 1+
𝑧2
𝑥2 + 𝑦2
≅𝑧+
2𝑧
𝐴
𝑥2 + 𝑦2
𝑓 = 𝑒𝑥𝑝 𝑖𝑘
𝑧
2𝑧
𝑧 → 𝑧 − 𝑖𝑧0
𝐴
𝑥2 + 𝑦2
𝑓=
𝑒𝑥𝑝 𝑖𝑘
𝑧 − 𝑖𝑧0
2 𝑧 − 𝑖𝑧0
Set 𝑅 𝑧 and 𝜔 𝑧
1
1
2
1
=
+𝑖
≡
𝑧 − 𝑖𝑧0 𝑅 𝑧
𝑘𝜔 2 𝑧
𝑞 𝑧
𝑧0
𝑅 𝑧 =𝑧 1+
𝑧
2
𝜔 𝑧 = 𝜔0
𝑧0
1+
𝑧
2
𝜔0 ≡
2𝑧0
𝑘
Gaussian Beam
Paraboloidal
𝐴
𝑥2 + 𝑦2
1
2
𝑓=
𝑒𝑥𝑝 𝑖𝑘
+𝑖
𝑧 − 𝑖𝑧0
2
𝑅 𝑧
𝑘𝜔 2 𝑧
2
= 𝐴0
𝜔0
𝑥 +𝑦
𝑒𝑥𝑝 − 2
𝜔 𝑧
𝜔 𝑧
2
2
𝑒𝑥𝑝 𝑖𝑘
2
𝑥 +𝑦
− 𝑖𝜂 𝑧
2𝑅 𝑧
Guoy phase shows the phase
difference between the
transverse modes
𝑧
𝑧0
𝐺𝑢𝑜𝑦 𝑃ℎ𝑎𝑠𝑒 ∶ 𝜂 𝑧 = tan−1
𝑧0
𝑅 𝑧 =𝑧 1+
𝑧
2
𝜔 𝑧 = 𝜔0
𝑧0
1+
𝑧
2𝑧0
𝑘
𝑘≡
2
Finally,
𝑢=
𝑓𝑒 𝑖𝑘𝑧
𝜔0 ≡
𝜔0
𝑥2 + 𝑦2
𝑥2 + 𝑦2
= 𝐴0
𝑒𝑥𝑝 − 2
𝑒𝑥𝑝 𝑖𝑘 𝑧 +
− 𝑖𝜂 𝑧
𝜔 𝑧
𝜔 𝑧
2𝑅 𝑧
𝑃= 𝑢
2
= 𝑃0
𝜔0
𝜔 𝑧
2
𝑥2 + 𝑦2
𝑒𝑥𝑝 −2 2
𝜔 𝑧
2𝜋
𝜆
Parameters of Gaussian Beam
Rayleigh Range Length Beam Waist Beam Divergence Angle
𝜋𝜔02
𝑧𝑅 =
𝜆
Radius of WF
𝑅𝑧
𝜔0 =
𝜆𝑧𝑅
𝜋
𝜔 𝑧
𝜆
=
𝑧→∞ 𝑧
𝜋𝜔0
𝜃0 = lim
BDA is related with Uncertain
Principle
𝜃0
Hermite-Gaussian Mode
Hermite Gaussian Mode
Note : the difference of beam propagation
𝑢𝑙𝑚→ 𝑥, 𝑦, 𝑧 = 𝑢𝑙 𝑥, 𝑧 𝑢𝑚 𝑦, 𝑧 𝑒𝑥𝑝 −𝑖𝑘𝑧 + 𝑖 𝑙 + 𝑚 + 1 𝜂 𝑧
2
𝑢𝑙 𝑥, 𝑧 =
𝜋
1
4
1
2𝑙 𝑙!
1
2
𝐻𝑙
2𝑥
𝑥
𝑒𝑥𝑝 −
𝜔 𝑧
𝜔 𝑧
2
𝑘𝑥 2
−𝑖
2𝑅 𝑧
𝑢𝑙𝑚← 𝑥, 𝑦, 𝑧 = 𝑢𝑙𝑚→ 𝑥, 𝑦, −𝑧
= 𝑢𝑙→ 𝑥, −𝑧 𝑢𝑚→ 𝑦, −𝑧 𝑒𝑥𝑝 −𝑖𝑘(−𝑧) + 𝑖 𝑙 + 𝑚 + 1 𝜂 −𝑧
∗
= 𝑢𝑙𝑚→
𝑥, 𝑦, 𝑧
Two miss-alignments : tilt and lateral shift of the cavity axis.
Find how can the Input TEM00 mode bean can be expressed by
TEM00 and TEM01,10 in the miss-aligned cavity.
Miss alignment of Mirrors
 Tile case
ITM misalignment case
a Flat
𝛼 ∶ 𝑇𝑖𝑙𝑡 𝑎𝑛𝑔𝑙𝑒 ≪ 𝜃0 (𝐷𝐵𝐴)
𝑥
cos −𝛼
=
𝑧
sin −𝛼
− sin −𝛼
cos −𝛼
𝑥′
𝑧′
z’
z
Curved
a
FP cavity axis moves angularly.
Input TEM00 mode can be expanded with the mode of the cavity at 𝑧 ′ = 0
𝑢𝑙𝑚→ 𝑥, 𝑦, 𝑧 = 𝑢𝑙 𝑥, 𝑧 𝑢𝑚 𝑦, 𝑧 𝑒𝑥𝑝 −𝑖𝑘𝑧 + 𝑖 𝑙 + 𝑚 + 1 𝜂 𝑧
𝑢00 |𝑧 ′ =0 = 𝑢0 𝑥 ′ cos 𝛼 , −𝑥 ′ sin 𝛼
≅ 𝑢00
≅ 𝑢00
𝑢0 𝑦′, −𝑥 ′ sin 𝛼
𝑒𝑥𝑝 𝑖𝑘𝑥 ′ sin 𝛼 + 𝑖𝜂 −𝑥′ sin 𝛼
1
6
2
′
′
′
′
+ 𝑖𝛼
𝑢10 𝑥 , 𝑦 , 0 +
𝛼0 𝑢30 𝑥 , 𝑦 , 0 +
𝛼0 𝑢12 𝑥 ′ , 𝑦 ′ , 0
𝛼0
8
8
𝛼
′
′
𝑥 , 𝑦 , 0 + 𝑖 𝑢10 𝑥 ′ , 𝑦 ′ , 0 , 𝛼0 ≪ 1
𝛼0
𝑥 ′, 𝑦 ′, 0
Miss alignment of Mirrors
 Lateral case
ETM misalignment case
Flat
𝑥 →𝑥−𝑎
a
z’
z
𝑢00 |𝑧 ′=0 = 𝑢0 𝑥 ′ − 𝑎, 0 𝑢0 𝑦′, 0
≅ 𝑢00 𝑥 ′ , 𝑦 ′ , 0 +
Curved
a
FP cavity axis moves in lateral
𝑎
𝑢10 𝑥 ′ , 𝑦 ′ , 0
𝜔0
 We should independently detect tilts and lateral shift of two
mirrors of FP cavity.
PDH method is also used to take beating between TEM00 and
TEM01,01 using quadrant PDs.
Control of Mirror Alignment
- How to obtain each mirror P/Y motion ? We need consider the conversion : TEM00 <-> TEM01,10 for the
carrier and sidebands ??
 Carrier
𝑢00→
- Input Beam : Perfect TEM00 :
𝑎
𝛼
- Input Beam expressed by the Miss
′
𝑢00→ ≅ 𝑢 00→ + −
+𝑖
𝑢′10→
aligned FP modes:
𝜔0
𝛼0
𝑎
𝛼
- Reflected light of the Input carrier
𝑥,𝑐 ′
𝑥,𝑐
𝐽0 𝑚 𝑟00 𝑢 00← + 𝑟10 −
+𝑖
𝑢′10← 𝐸0 𝑒 𝑖𝜔0𝑡
component :
𝜔0
𝛼0
∗
𝑢𝑙𝑚← 𝑥, 𝑦, 𝑧 = 𝑢𝑙𝑚→
𝑥, 𝑦, 𝑧
𝑢𝑙𝑚→ 𝑥, 𝑦, 𝑧 = 𝑢𝑙 𝑥, 𝑧 𝑢𝑚 𝑦, 𝑧 𝑒𝑥𝑝 −𝑖𝑘𝑧 + 𝑖 𝑙 + 𝑚 + 1 𝜂 𝑧
𝑥,𝑐
≅ 𝐽0 𝑚 𝑟00
𝑢00← +
𝑥,𝑐
𝑥,𝑐
𝑟00
− 𝑟10
𝑎
𝑥,𝑐
𝑥,𝑐 𝛼
+ 𝑖 𝑟00
+ 𝑟10
𝑢10← 𝑒 𝑖𝜂𝑑 𝐸0 𝑒 𝑖
𝜔0
𝛼0
𝜔0 𝑡−𝑘𝑐 𝐿+𝜂𝑑
𝑑 ∶ distance between beam waist and detection PD
Control of Mirror Alignment
- How to obtain each mirror P/Y motion ?  Sideband
- Sideband is anti resonant for FP, so we can set
𝑥,𝑐
≅ 𝐽1 𝑚 𝑟00
𝑢00← +
0
𝑥,𝑠
𝑥,𝑐
𝑟00
= 𝑟10
≡ 𝑟𝑠
𝑎
𝛼
𝑖
+ 𝑖 2𝑟𝑠
𝑢10← 𝑒 𝑖𝜂𝑑 𝐸0 𝑒
𝜔0
𝛼0
𝜔0 ±𝜔𝑚 𝑡−𝑘𝑢𝑝,𝑑𝑤 𝐿+𝜂𝑑
Alignment signal using PDH
Same with the length sensing, we can obtain miss-alignment signal from the beating
between (TEM00 carrier and TEM01 sideband), (TEM00 sideband and TEM01 carrier).
𝑉 ∝ 𝐽0 𝑚 𝐽1 𝑚
𝐸02 𝑟𝑠
𝑥,𝑐
𝑟00
−
𝑥,𝑐
𝑟10
𝑎
𝛼
sin 𝜂𝑑 − cos 𝜂𝑑
𝜔0
𝛼0
For convenience, set flat ITM and curved ETM (Curvature : 𝑅 m),
𝑉1 ∝
𝑅 − 𝐿𝐹𝑃
1
sin 𝜂𝑑 − cos 𝜂𝑑 𝛼𝐼𝑇𝑀
𝜔0
𝛼0
𝑉2 ∝
𝑅
sin 𝜂𝑑 𝛼𝐸𝑇𝑀
𝜔0
Obviously, sin 𝜂𝐿 = 0 condition realize independent signal extraction of 𝛼𝐼𝑇𝑀 .
𝛼𝐸𝑇𝑀 is also derived with calculation or selection of 𝑉1 = 0 optical condition.
Control of Mirror Alignment
- How to obtain each mirror P/Y motion ? 𝑉1 ∝
𝑅 − 𝐿𝐹𝑃
1
sin 𝜂𝑑 − cos 𝜂𝑑 𝛼𝐼𝑇𝑀
𝜔0
𝛼0
𝑉
𝑉2 ∝
𝑅
sin 𝜂𝑑 𝛼𝐸𝑇𝑀
𝜔0
𝑉1
Guoy Phase
𝑉2
Control of Mirror Alignment
- How to obtain each mirror P/Y motion ? Alignment Control optical layout example for one Ring FP (CLIO)
MC End and MC In,Out mirror differential motion can be extracted
MC0ut
MCend
MCin
Lenses for Gouy Phase Adjust
QPD1 QPD2
Control of Mirror Alignment
- Complicated for alignment signal extraction of PR-FPMI-RSE-
As length sensing, same control definition is adopted.
D : Differential
C : Common
Control of Mirror Alignment
- Complicated for alignment signal extraction of RFPMI-RSEWe should calculate the how large and what sign (+ / -) signals can be obtained
from each ports and by each demodulation frequencies.
This matrix is an example for Adv-LIGO alignment signal extraction
Control of Mirror Alignment
- Folding mirrors introduction in PRC and SRC We encountered a problem that the alignment signals of PRM and SRM are
highly degenerate with ITMs signals and relatively small by a factor of finesse !
To avoid this difficulties, Adv-GWDs are designed to have PRC and SRC that
involve folding mirrors to adjust “guoy” phase in side them.
ETMy
Guoy phase responses for ITMs and (SRM,
PRM) are almost same. So we cannot
independently separate them.
ITMy
PRM
ETMx
BS ITMx
3km
~ 10 m
SRM
~ 10 m
3km
Control of Mirror Alignment
- Folding mirrors introduction in PRC and SRC We encountered a problem that the alignment signals of PRM and SRM are
highly degenerate with ITMs signals and relatively small by a factor of finesse !
To avoid this difficulties, Adv-GWDs are designed to have PRC and SRC that
involve folding mirrors to adjust “guoy” phase in side them.
ETMy
PR2 PR3, SR2, SR3
have small curvatures
PRM
PR2
ITMy
BS ITMx
ETMx
• Guoy phase responses for ITMs
and (SRM, PRM) are moderately
separated.
• We should be carful of the
astigmatism validation of the
beam.
~ 10 m
convex
3km
PR3
concave
SR2
SR3
concave
SRM
~ 10 m
concave
Practical Problems
of
GWD control
Lock Acquisition Problem of FP
- long optical storage time spoils PDH signals GWD’s FP cavity optical storage time is so long (Finesse ~ 1000, L~4km, so t~ 10
msec) and main mirrors are moving at the speed of ~ 100nm/sec.
Even the amount of doppler shift due to reflection on the moving mirror, the
accumulated phase shift during the long storage time is enough to spoils the PDH
signal as shown the below figure.
Consequently, we are afraid not to lock FP cavities and not to operate GWD !!
PDH Signal
v=1um/sec
PDH Signal
v=0.01um/sec
L = 3000m,
Finesse = 312,
Critical Coupling
L = 3000m,
Finesse = 312,
Critical Coupling
Lock Acquisition Problem of FP
- long optical storage time spoils PDH signals Input
Mirror
Ω0 [Hz] 𝐸0
Ref : r1
Tra : t1
Los : A1 = 0
Ref : r2
Tra : t2
Los : A2 = 0
𝜏1 𝐸0
Ω0
𝜌10 𝜌2 𝜏1 𝐸0 𝑒 −𝑖𝜙1
𝑘2Ω
𝜌1 𝜌22 𝜏1 𝐸0 𝑒 −𝑖𝜙2
Cavity
Reflected
Light
Output
Mirror
𝐿1
0
𝑘 4 Ω0
𝜌1𝑛−1 𝜌2𝑛 𝜏1 𝐸0 𝑒 −𝑖𝜙𝑛
𝑘2
0 [m/s]
𝑛−1
Ω0
𝐿2
𝐿𝑛
𝑛=1
𝑛=2
𝑛
𝑣[m/s]
𝑘≡
1−𝛽
1
~ 1− 𝛽
1+𝛽
2
2
~1 − 𝛽
𝛽=
𝑣
≪1
𝑐
Lock Acquisition Problem of FP
- long optical storage time spoils PDH signals 𝑡0 = 0
𝐿0
Ω0 𝑘 2 Ω0
𝜙1 =
+
𝐿1
𝑐
𝑐
𝑡1 =
𝐿0
𝑐+𝑣
𝐿1 = 𝑐𝑡1
𝑘 4 Ω0 𝑘 2 Ω0
𝑘 2 Ω0 Ω0
𝜙2 =
+
𝐿1 +
+
𝐿2
𝑐
𝑐
𝑐
𝑐
𝑡2 =
𝐿1
𝑐+𝑣
𝐿2 = 𝑐𝑡2
𝑘 6 Ω0 𝑘 4 Ω0
𝑘 4 Ω0 𝑘 2 Ω0
𝑘 2 Ω0 Ω0
𝜙3 =
+
𝐿1 +
+
𝐿2 +
+
𝐿3
𝑐
𝑐
𝑐
𝑐
𝑐
𝑐
𝑡3 =
𝐿2
𝑐+𝑣
𝐿3 = 𝑐𝑡3
𝑡𝑛 =
𝐿𝑛−1
𝑐+𝑣
𝐿𝑛 = 𝑐𝑡𝑛
𝑛
𝜙𝑛 =
𝑚=1
𝑛
~
𝑚=1
𝑛
~
𝑚=1
𝑘2
𝑛+1−𝑚
𝑐
Ω0
+
𝑘2
𝑛−𝑚
Ω0
𝑐
2Ω0
1 − 2𝑛 − 2𝑚 + 1 𝛽
𝑐
𝐿𝑚
𝐿0
1
=
𝑐 1+𝛽
𝐿𝑚
~
2Ω0 𝐿0
1 − 2𝑛 − 𝑚 + 1 𝛽
𝑐
𝑘≡
1−𝛽
1
~ 1− 𝛽
1+𝛽
2
𝐿0
1 − 𝑛𝛽
𝑐
2
~1 − 𝛽,
𝑘 𝑝 ~1 − 𝑝𝛽
𝑛
~𝐿0 1 − 𝑛𝛽
Lock Acquisition Problem of FP
- remedies for lock acquisition problem From Experience, this PDH signal “Beating” is proportional to cavity
length and square of Finesse.
PDH Signal
v=1um/sec
L = 3000m,
Finesse = 312,
Critical Coupling
 Problems
• Many nonlinear signals.
• Resonance point dose not show
zero PDH signal.
• Linear range is limited with in
FWHM.
There are many proposed solutions
•
•
•
•
Green Locking (Adv. LIGO and bKAGRA default plan)
Offset Locking
Velocity damping Locking
NQD (near Q-phase demodulation) and OHD (odd-harmonics demodulation signal
combination)
Lock Acquisition Problem of FP
- Green Locking  Green laser(532 nm) that has a half wavelength of 1064nm will be used for
“initial” FP cavity locking by setting the FP cavity finesse for 532nm small (~ 50?).
 532nm is generated by the
original 1064nm source(s1) or is
generated from other 1064nm
source(s2) that is correlated with
(s1) , and the relative frequency
between 532nm and original
1064nm is set to be adjustable to
realize the double resonance of
532nm and 1064nm at the same
time.
 After the length control by using
1064nm, the FP cavity individual
control using 532 nm is turned
off.
1064nm, 532nm
(Innolight Prometheus ?)
Wave length
comparison and
Freq. offset adjust
are necessary.
PR2
PRM
200W
1064nm
Laser
ETMy
ITMy
BS ITMx
PR3
SR2
SR3
SRM
ETMx
Lock Acquisition Problem of FP
- NQD and OHD  Essential problem about PDH signal :
• linear range is limited around FWHM  less feedback impulse cannot
deaccelerate mirrors.
• PDH signal will be deformed a lot when the mirrors moving fast.
 Is it impossible to extend the “linear range” ?
Yes! With NQD : Near Q-phase Demodulation in PDH method
OHD : Sum of Odd harmonic demodulation signal in PDH
 To Obtain wider linear range and beat-less signal by setting RF(AF) modulation
frequency near the resonance width area (not far from it as normal selection)
and use not In-Phase but Near-Q phase.
Lock Acquisition Problem of FP
- NQD and OHD  Normal PDH I-Phase signal
assuming a FP cavity with RF
sideband near resonance.
(assuming 30nm/sec mirror speed for
1550 finesse of 3km arm FP)
Mirror Position in unit of FWHM
 Just set demodulation phase not Inphase but “near Q”. Then, signal linear
range can be extended according to
modulation frequency setting.
Mirror Position in unit of FWHM
Lock Acquisition Problem of FP
- NQD and OHD  Take 1st, 3rd , 5th In-phase demodulation signal and add them at proper ratio.
 Some merits and demerits of NQD and HOD
•
•
•
•
Beating becomes small
Zero of error signal at resonance position.
Linear range can be extended over modulation frequency.
Noisy signal
Lock Acquisition Problem of FP
- NQD and OHD -
We verified NQD signal can be realized by using small rigid FP
Dotted : 1/20 times of I-Phase Signal
Gray : NQD signal (measured)
Black : Theoretical Fitting
Detuned FP Instability
- Optical Spring  A detuned FP Cavity is required in the case of Resonant Sideband
Extraction (RSE) technique.
• Detuned means that the Signal Recycling (SR) cavity
imperfectly resonates to optimize the sensitivity curve for
targeted GW signals.
• Optical spring mechanics arises.
• Optical spring has two situations :
1. Spring with acceleration force
2. Anti-spring with de-acceleration force
Detuned FP Instability
- Optical Spring MechanicsIntra
Cavity
Power
Inside FP cavity
Outside FP cavity
Anti Spring area
Spring Area
Deacceleration
Acceleration
𝐿0
Mirror Position
Ω0
Gravity Force = Radiation Pressure
Detuned FP Instability
- Why (De)Acceleration ??Intra
Cavity
Power
Inside FP cavity
Outside FP cavity
Anti Spring area
Spring Area
Deacceleration
Acceleration
Mirror Position
𝐿0 𝐿+1
• Assuming the mirror is moving toward outside.
Ω0
• The reflected light frequency is doppler shifted (Ω0 → Ω+1 ).
• The cavity resonance curve for Ω+1 will be yellow curve
because Ω+1 < Ω0 .
• The intra-cavity force for Ω+1 becomes larger than for Ω0 .
Mirror Motion • This force accelerates the mirror motion.
Detuned FP Instability
- Why (De)Acceleration ??Intra
Cavity
Power
Inside FP cavity
Outside FP cavity
Anti Spring area
Spring Area
Deacceleration
Acceleration
Mirror Position
𝐿−1𝐿0
• Assuming the mirror is moving toward inside.
Ω0
• The reflected light frequency is doppler shifted (Ω0 → Ω−1 ).
• The cavity resonance curve for Ω−1 will be orange curve
because Ω−1 > Ω0 .
• The intra-cavity force for Ω−1 becomes smaller than for Ω0 .
Mirror Motion • This force also accelerates the mirror motion.
ROC Deformation due to
Thermal Lensing.
Burning of mirror using IR laser is introduced in LIGO
 Troubles in LIGO, VIRGO
•
•
Because of less absorption loss of ITM SiO2 substrate, the planed curvature
deformation due to “thermal lensing” was not realized.
RF sideband resonance inside PRC was spoiled a lot because of “mode mismatch” ,
Miss-match order is ~ several 10m Sensitivity degradation.
 Solutions
•
•
Burn ITMs by a ring heater and additional lasers to change ROC of ITMs and BS at
the order of ~several 10m for km scale ROC.
(Sapphire Cryogenic mirrors in KAGRA have so high thermal conductivity that
substrate thermal lensing is negligible.)
ROC compensation Tech.
Make ROC by coating or Sputter polish
Parametric Instability
Evans et al., Phys. Rev. Lett. 114, 161102 (2015)
Coupling between higher transverse modes of FP cavity and
mirror mechanical resonances
FP Cavity FSR
(L = 3 ~ 4 km)
𝑐
TEM00 ∶
~50𝑘𝐻𝑧
2𝐿
TEMmn ∶ ~10𝑘𝐻𝑧
𝑓𝑚𝑒𝑐ℎ𝑎 ~10𝑘𝐻𝑧
Mechanical Modes
(~30cm x ~20cm SiO2)
Parametric Instability
Evans et al., Phys. Rev. Lett. 114, 161102 (2015)
Resonance growing and Arm cavity Power in time
Resonance growing
Mechanical Mode
Optical Mode
Summary
PR-FPMI with RSE requires the control of 5 degrees of
freedoms for length.
Alignment control for each mirrors is also required.
Both length and alignment signals are designed to obtain
to realize the targeted sensitivity.
Some solutions for practical problems about GWD
control are also investigated.
Presentation Files




http://www.icrr.u-tokyo.ac.jp/~miyoki/2015APPII-miyoki-1.pptx
http://www.icrr.u-tokyo.ac.jp/~miyoki/2015APPII-miyoki-2.pptx
http://www.icrr.u-tokyo.ac.jp/~miyoki/2015APPII-miyoki-3.pptx
http://www.icrr.u-tokyo.ac.jp/~miyoki/2015APPII-miyoki-4.pptx