Download Port - LIGO

Wavefront Sensing in
Dual-Recycled Interferometers
• LIGO
• What is Wavefront Sensing?
• How does it work?
– Detection of Misalignments
– Control System
• Why do we need it?
• Model of wavefront sensing in a
dual-recycled interferometer
• Consequences for the design of
length and alignment control
systems for the 40m prototype
and Advanced LIGO
Marcus Benna, University of Cambridge
LIGO – Laser Interferometer
Gravitational-Wave Observatory
• Gravitational Waves are
predicted by Einstein’s general
theory of relativity
• Due to their weak interaction
with matter, they have never
been observed experimentally
• Gravitational waves effect tiny
lengths changes between two
test masses (strains of the order
of h~10-21)
• LIGO employs interferometric
techniques to measure those
strains
Marcus Benna, University of Cambridge
Advanced LIGO Layout
- requires numerous control systems
 Five length degrees
of freedom
 Twelve angular
degrees of freedom
(six for each of
pitch and yaw)
 Beam splitter
serves as reference
 Input beam
direction needs to
be controlled
Marcus Benna, University of Cambridge
What is Wavefront Sensing ?
 Mirror endowed test masses are
suspended from a single loop of
wire
 Length sensing detects changes in
the separation of the mirrors,
which can hence be corrected for
to keep the light in resonance in
the cavity
 Alignment sensing (=wavefront
sensing) detects angular
misalignments of the mirrors in
pitch and yaw
Marcus Benna, University of Cambridge
Angular degrees of freedom
Marcus Benna, University of Cambridge
Hermite-Gaussian Modes
 Hermite-Gaussian modes define a complete orthogonal set
of functions describing the transverse beam profile
 The fundamental TEM00 has a Gaussian profile in both
transverse directions
 Higher order modes have more complex profiles with a
number of zeroes given by the mode indices for the two
transverse axes
Marcus Benna, University of Cambridge
Hermite-Gaussian Modes II
As compared to plane waves, Hermite-Gaussian modes
acquire an additional phase shift when passing through their
waist, which is the point of minimal beam diameter. This is
called a Guoy phase shift.
Marcus Benna, University of Cambridge
Guoy Phase Telescope
•
Adjusts beam
diameter to
make spot fit
onto photodiode
• Introduces
Guoy-phase
advance of our
choice
Marcus Benna, University of Cambridge
Length Sensing and
Pound-Drever-Hall Locking
 Lengths need to be
controlled to within 10-10m
 Want a linear, zero-crossing
error signal
 But cannot measure phase
shift of carrier light directly
because frequency is much
too high (300THz)
 Hence we need some
trickery…
Marcus Benna, University of Cambridge
Length Sensing and
Pound-Drever-Hall Locking II
 Modulate carrier light with a Pockels cell to apply radio-frequency (RF)
sidebands before it enters the cavity
 Place RF-photodiode at output port and observe radio-frequency beats
between carrier and sidebands
 Demodulate signal by “mixing” (i.e. multiplying and time-averaging) it with
signal from same local oscillator that drives Pockels cell
 This gives you a linear, zero-crossing error signal -> active null instrument
Marcus Benna, University of Cambridge
Extension to Alignment Sensing
• Misaligned optics tilt reflected beams with respect to nominal optical
axis of cavity
• For small angles (<< far field diffraction angle) this is
mathematically equivalent to adding a TEM10 component to the
fundamental Gaussian TEM00 mode
• Hence we can describe any tilted
mirror as a source of TEM10 mode
• On a normal photodiode the TEM10
mode wouldn’t show up, because odd
functions are averaged to zero
• Hence use split photodetector to
observe beats between carrier TEM00
and sideband TEM10 for example
Marcus Benna, University of Cambridge
Misalignment of a simple
Fabry-Perot cavity
1)
2)
3)
4)
5)
6)
7)
Incident laser beam
TEM00 mode resonant in cavity
Input Test Mass
Tilted End Test Mass
Split Photodiode
Reflected light without misalignment
Reflected light is displaced due to
misalignment. Transverse intensity
profile (solid) with modal
decomposition
(dashed)
Marcus Benna, University
of Cambridge
Why do we need alignment sensing ?
• As always in LIGO, the
answer is:
To reduce noise!
• Misaligned optics
introduce excess noise
• Coupling into higher
order modes represents
effective loss of beam
power
Marcus Benna, University of Cambridge
Current Design of the Advanced LIGO
Control System
•
•
•
•
Two pairs of resonant
sidebands
One resonant in both
power recycling and signal
recycling cavity, the other
one only in power
recycling cavity
One pair of sidebands as
low as possible in
frequency (limited by
dimensions of vacuum
envelope) the other as
high as possible (limited
by current demodulation
technology)
Three output ports: Bright
port, dark port, pick-off
Marcus Benna, University of Cambridge
Building a model of alignment sensing in a
dual-recycled interferometer
• Calculate steady state field of TEM00 mode in perfectly aligned
state
• Treat any tilted mirror as a source of TEM10 mode
• Propagate both modes to the various output ports and perform
demodulation
• Construct the most favorable wavefront sensing matrix, which
relates misalignments in the six different degrees of freedom to
signal levels in the six wavefront sensors, by choosing appropriate
locations, demodulation frequencies and phases for the sensors
• The goal is to make this sensing matrix as diagonal as possible
• Implement all this in a Mathematica notebook
• Involves numerous approximations…
Marcus Benna, University of Cambridge
How do you construct a reasonable wavefront
sensing scheme?
• Given a set of parameters defining the interferometer as such,
there are potentially many ways to construct a sensing scheme
• For each of the six wavefront sensors, we pick:
– A demodulation frequency, which determines whether we detect beats
of sideband 1 against carrier, sideband 2 against carrier or of the two
sidebands against each other
– A location at one of the three output ports
– An RF phase for the demodulation process
– An artificially introduced Guoy phase advance that effectively shifts
TEM00 and TEM10 components with respect to each other
• You have to choose wisely, otherwise it won’t work!
Marcus Benna, University of Cambridge
Wavefront Sensing Matrices
"Port"
"Demodulation"
"Guoy" "RF" "DETM" "DITM" "CETM" "CITM" "PRM" "SRM"
1.97
"Asym. Port" "Carrier- Sideband2"
130
49
1.13
0
0
0
0
-0.06 0.01 0.01
"Sym. Port"
"Carrier- Sideband1"
75
83
0
0.59
-0.04
0.73
-0.51 0.52 0.46
"Sym. Port"
"Carrier- Sideband2"
126
147
0.01
0.05
-0.03 -0.03
"Pickoff"
"Carrier- Sideband2"
102
90
0
0
0
0.07
0.86
"Sym. Port"
"Carrier- Sideband1"
169
173
0.01
0.14
-4.47 4.18 0.14
0.97
-0.61 -0.35
"Sym. Port" "Sideband1- Sideband2"
138
29
0
-0.03
0
For the 40m prototype: Not diagonal, but it could be worse…
"Port"
"Demodulation"
"Guoy" "RF" "DETM" "DITM" "CETM" "CITM" "PRM" "SRM"
"Asym. Port" "Carrier- Sideband2"
128
53
26.12 24.17
0
0
0
0
3.08
"Asym. Port" "Carrier- Sideband1"
164
85
2.85
0
0
0
0
"Sym. Port"
"Carrier- Sideband1"
89
168
0
-0.01
-2.67
-3.13 0.66
0
1.17 -0.51 -0.21
"Sym. Port"
"Carrier- Sideband2"
100
140
0
-0.01
0.81
"Sym. Port"
"Carrier- Sideband1"
163
158
0
-0.04
-0.71
-4.09 2.48 0.01
-0.79 1.51 -0.71
"Sym. Port" "Sideband1- Sideband2"
157
114
0
0.02
0
For Advanced LIGO: This is worse!
Marcus Benna, University of Cambridge
A Possible Wavefront Sensing Scheme
for Advanced LIGO
Marcus Benna, University of Cambridge
Results of Simulation
• Wavefront sensing in a dual-recycled interferometer is more complex
than for the LIGO I configuration (as expected)
• Nevertheless it’s possible to obtain all necessary information by using
just two pairs of resonant sidebands
• No need for non-resonant sidebands
• Most favorable place for pickoff is antireflection coating of beam splitter
• Crucial points:
– Alignment of signal recycling mirror, which has a weak error signal that never appears
in isolation
– Distinguishing between the differential modes of the input and end test masses
– Distinguishing between common input test mass motion and power recycling mirror
motion
• Even though one can construct a manifestly non-singular wavefront
sensing matrix, it will never be perfectly diagonal, i.e. we need a
multiple-input multiple-output control system
Marcus Benna, University of Cambridge
Design of a Control System for
Advanced LIGO
• Design needs to take into account alignment sensing as well as
length sensing
• Current design of control system is optimized only with respect
to sideband power in the recycling cavities
• We find that high power levels do not guarantee optimal
alignment signals
• We propose changes in the parameters for Advanced LIGO, to
reconcile the requirements of length and alignment sensing
• In particular, we suggest altering the Schnupp asymmetry, the
signal recycling cavity length and potentially the sideband
frequencies
• For the 40m prototype we (luckily) find that the current design
offers a reasonable sensing scheme
Marcus Benna, University of Cambridge
Conclusions
• Using our model we can calculate wavefront sensing
matrices for both initial and advanced LIGO for any
given set of parameters
• The model has been tested, compared with other
simulations and was found to be correct
• Given such a tool, we can chose sets of parameters for
Advanced LIGO and the 40m prototype that optimize
both length and alignment sensing
• We determine the wavefront sensing matrix for the
40m interferometer and propose an improved
configuration for the Advanced LIGO control scheme
Marcus Benna, University of Cambridge