The Fabry Perot Cavity
!What is a Fabry Perot Cavity?
•Optical cavity where light is stored
!How can we measure distance with a Fabry Perot cavity?
•How does this depend upon mirror properties?
!How does Fabry Perot Cavity act as filter to frequency noise?
!How does Fabry Perot Cavity act as filter to intensity noise?
!How does Fabry Perot Cavity act as a spatial filter to beam shape?
!How does Fabry Perot Cavity act as filter to beam jitter noise?
Cavity Description
dielectric coating
input mirror
output mirror
Eincident
Etransmitted
Ereflected
cavity length L
t1
t1
r1
E incident (t ) = E0 e
-r1
r= R
t= T
-r2
R+T+L=1
iωt
E transmitted (t ) = E0t1t 2 e
E reflected (t ) = E0 r1e
t2
iωt
iω ( t − Lc )
+ E0t1t 2 r1r2 e
2 iω ( t − 2cL )
0 2 1
−E rt e
iω ( t − 3cL )
2 2 iω ( t − 5cL )
0 1 2 1 2
+E tt r r e
2 2 iω ( t − 4cL )
0 1 2 1
−E rr t e
+ ...
2 3 2 iω ( t − 6cL )
0 1 2 1
−E r r t e
− ...
Results for Electric field
[
]
E transmitted
− iω Lc
−iω 2cL
2 2 −iω 4cL
1 + r1r2 e
= t1t 2 e
+ r1 r2 e
+ ...
E incident
=
t1t 2 e
−iω Lc
1 − r1r2 e
− i 2ω Lc
[
]
E reflected
− iω 2cL
−iω 2cL
2
2 2 −iω 4cL
1 + r1r2 e
= r1 − t1 r2 e
+ r1 r2 e
+ ...
E incident
= r1 −
=
−i 2ω Lc
2
1 2
−i 2ω Lc
1 2
t re
1− r r e
(
)
2 −i 2ω Lc
1
−i 2ω Lc
r1 − r2 r + t e
2
1
1 − r1r2 e
Alternative Approach
Eincident
E2
Ereflected
E5 cavity length L
E4
t1
t2
t1
r1
-r1
E 2 = t1E incident − r1E 5
E3 = e
− iω
L
c
E2
E 4 = −r2 E 3
E5 = e
− iω
L
c
E4
E transmitted = t 2 E 3
E reflected = r1E incident + t1E 5
E3
Etransmitted
-r2
!Six equations; six unknowns
•Solve for each electric field
!Good technique for multiple
mirror cavities
•Like recycled-recombined LIGO
Important results
! Intensity
! Phase
I transmitted E transmitted
=
Iincident
E incident
Phase transmitted
2
E transmitted E*transmitted
=
E incident E*reflected
 Im(E transmitted ) 
= Arctan

Re(E
)
transmitted 

!Similarly for the reflected light
!Simplify for
2ωL
<< 1
c
!Important to remember e
i
2 ωL
c
=e
i
2 ωL
+ i 2 nπ
c
for n = 0,1,2,3,4,...
L ≈ 10 −6 parts per million (ppm)
!For good optics:
T ≈ few percent to ppm
I transmitted
t12t 22
t12t 22
=
≈
2
2
2 2 2ωL 2
2ωL 2
I incident
(1 − r1r2 ) + r1 r2 ( c ) (1 − r1r2 ) + ( c )
phasetransmitted
− (1 + r1r2 )  2ωL 
− 2  2ωL 
=
≈



1 − r1r2  c  1 − r1r2  c 
2t12t 22 − (1 − r1r2 )
c
Bandwidth(FWHM) ∆υ =
2πL
r12 r22
2
Finesse F =
c
2 L∆υ
F
EquivalentBounces n =
π
Transmitted Light
!Phase about 0 gives linear measure
of distance
0.25
Imax
Itransmitted/Iincident
0.20
0.15
Full Width at Half Maximum
(FWHM)
0.10
0.05
0.00
-11
-1.0x10
-5.0x10
-12
0.0
5.0x10
length in metres
-12
1.0x10
-11
Itransmitted/Iincident
•Depends upon mirror properties
0.1
0.01
1E-3
1E-4
-11
-1.0x10
-5.0x10
-12
0.0
-12
5.0x10
1.0x10
-11
length in metres
Transmitted Phase in radians
!Light is transmitted through cavity
over narrow range of distance
L1=L2=1ppm
L1=L2=10ppm
L1=L2=100ppm
T1=10 ppm
T2=1 ppm
L1=L2=1ppm
L1=L2=10ppm
L1=L2=100ppm
T1=10 ppm
T2=1 ppm
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-1.0x10
-11
-5.0x10
-12
0.0
5.0x10
length in metres
-12
1.0x10
-11
1
Reflected Light
Ireflected/Iincident
!Reflected phase gives signal for
locking cavity
L1=L2=1ppm
L1=L2=10ppm
L1=L2=100ppm
T1=10 ppm
T2=1 ppm
•Reflection locking
•Pound-Drever-Hall Locking
-1.0x10
-11
-5.0x10
-12
0.0
5.0x10
-12
1.0x10
-11
length in metres
1.0
Reflected Phase in radians
0.9
0.8
Ireflected/Iincident
0.7
0.6
0.5
0.4
0.3
0.2
L1=L2=1ppm
L1=L2=10ppm
L1=L2=100ppm
T1=10 ppm
T2=1 ppm
3
visibility=(Imax-Imin)/Imax
0.1
0.0
-11
-1.0x10
-12
-5.0x10
0.0
length in metres
-12
5.0x10
-11
1.0x10
2
1
0
-1
-2
-3
-1.0x10
-11
-5.0x10
-12
0.0
5.0x10
length in metres
-12
1.0x10
-11
Storage Time
!Light makes many trips in cavity
•Light escapes through mirror
•Light is absorbed or scattered
!Processes that occur on a faster time scale than storage time are
filtered
•Intensity fluctuations of light
•Frequency changes of light
3L
5L
L
+ t1t 2 r1r2
+ t1t 2 r12 r22
+ ...
c
c
c
L
= t1t 2 (1 + 3r1r2 + 5r12 r22 + ..)
c
∞
L

= t1t 2 1 + ∑ (2n + 1)3r1n r2n 
c  n =1

τ storage = t1t 2
L  3r1r2 − r12 r22 
L  r1r2 + 1 
= t1t 2 1 +
= t1t 2 
2 
2
c
c
(r1r2 − 1) 
 (r1r2 − 1) 
LLO 4k Optical Layout
ISCT1
Y Trans.
Michelson Refl.(Bright)
[symmetric]
Telescope
HAM1
ETMY
Optical
Lever
SM1
Y 4 km arm
ISCT3
HAM2
Optical
Lever
Y Pickoff
ITMY
Faraday
BSC1
MMT2
MMT1
Optical
Lever
BSC5
Optical
Optical Lever
Lever
Telescope
MMT3
ETMX
4 km arm
RM
Optical
Lever
MC1
Telescope
X
MC3
PSL
BSC4
MC2
HAM3
HAM4
BS
ITMX
BSC2
X Trans.
BSC3
Optical
Lever
Telescope
Michelson (Dark)
[anti-symmetric]
Beam Splitter
Pickoff
X Pickoff
IOT1
MC Refl.
MC Trans.
Faraday
ISCT4