Optics of GW detectors

(1)

Optics of GW detectors

Jo van den Brand

e-mail: jo@nikhef.nl

(2)

Introduction

•

General ideas

•

Cavities

•

Reflection locking (Pound-Drever technique)

•

Transmission locking (Schnupp asymmetry)

•

Paraxial approximation

•

Gaussian beams

•

Higher-order modes

•

Input-mode cleaner

•

Mode matching

•

Anderson technique for alignment

(3)

General ideas



Measure distance between 2 free falling masses using light

–

h=2L/L (~10

^-22

)

–

L= 3 km  L ~10

^-22

x 10

⁶

~10

^-16

(=10

^-3

fm)

–



light

~ 1 m

–

Challenge: use light and measure L/~10

^-12



How long can we make the arms?

–

GW with f~100 Hz  

GW

~c/f=3x10

⁸

km/s / 100 Hz

= 3000 km

–

Optimal would be 

GW

/4 ~ 1000 km

–

Need to bounce light 1000 km / 3 km ~ 300 times



How to increase length of arms?

–

Use Fabri-Perot cavity (now F=50), then L/~10

^-10

–

Measure phase shift 

x



y

LBh

e

~ 10.(3 km).200.10

^-22

/10

^-6

=10

^-9

rad

L + L

L - L

(4)

General ideas



Power needed

– PD measures light intensity

– Amount of power determines precision of phase measurement _et of incoming wave train (phase ft)

– Measure the phase by averaging the PD intensity over a long period of time T_{period GW}/2 = 1/(2f)

– Total energy in light beam E=I₀.1/(2f)=hbar.N__e

– Due to Poisson distributed arrival times of the photons we have N= Sqrt[N]

– Thus, E= N .hbar. _eand t E= (_e).Sqrt[N]. hbar. _e>hbar

– We find Sqrt[N]  N^= 10¹⁸ photons

– Power needed I₀ = Nhbar. _e.2f ~ 100 W



Power is obtained through power-recycling mirror

– Operate PD on dark fringe

– Position PR in phase with incoming light

– GW signal goes into PD!

– Laser 5 W, recycling factor ~40

L + L

L - L

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Cavities



Fabri-Perot cavity (optical resonator)



Reflectivity of input mirror: -0.96908



Finesse = 50



FSR = 50 kHz



Power



Storage time



Cavity pole

-6 -4 -2 2 4 6

10 20 30 40 50 60 70

(6)

Cavity pole

(7)

Overcoupled cavities (r

₁

- r

₂

< 0)



On resonance 2kL=n



Sensitivity to length changes



Note amplification factor



Note that amplitude of reflected light is phase shifted by 90

^o



Reflected light is mostly unchanged |E

_ref

|

²



Imagine that L is varying with frequency f

_GW



Loose sensitivity for f

GW

>f

pole

 

 





 

 

 



  ik L

r r r r E

E E

E

resonance inc

ref inc

ref

2 

1 1

2 1 2 1

Amplification factor (bounce number)

f cL i

e

ⁱ⁴^^c⁽^f ^^^f ⁾^L

 1  4  

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Reflection locking – Pound Drever locking



Dark port intensity goes quadratic with GW phase shift.



How do we get a linear response?



Note, that the carrier light gets p phase shift due to over- coupled cavity.



RFPD sees beats between carrier and sidebands.



Beats contain information about carrier light in the cavity



Phase of carrier is sensitive to L of cavity

Laser EOM

3 x 10¹⁴ Hz

 20 MHz Faraday isolator

carrier L sideband

RFFD

(9)

Reflection locking

Demodulation

Modulation

(10)

Transmission locking



Schnupp locking is used to control Michelson d.o.f.

–

Make dark port dark and bright port bright

–

Not intended to keep cavities in resonance

–

Requires that sideband (reference) light comes out the dark port

(11)

Gaussian beams

P – complex phase

q – complex beam parameter

(12)

Higher-order modes

(13)

Input-mode cleaner

(14)

Applications – Anderson technique

(15)

Summary



Some of the optical aspects

–

Simulate with Finesse



Frequency stabilization

–

Presentation



Control issues

–