Performance of Arm Locking in LISA Amaldi 8, Columbia University, June 2009 Kirk McKenzie, Robert Spero, and Daniel Shaddock

Jet Propulsion Laboratory, California Institute of Technology

This research was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration

Copyright 2009 California Institute of Technology.

Frequency noise in LISA • Coupling of laser frequency noise to the signal readout must be reduced by 14 orders of magnitude. • Goal: Detailed study • Optimized controller design • Detailed noise model • Include Doppler frequency error effects. • Results: 1) Arm locking + prestabilization has a margin > 100, 2) Arm locking alone can meet requirement 3) Doppler frequency error effects are manageable.

Noise post arm locking

Arm locking Why? • The LISA arms represent the best frequency reference: ∆L/L ~ 10-20/√Hz (100µHz - 1Hz). • Low hardware footprint/cost. How? • The response of a single LISA arm is equivalent to a Mach-Zehnder interferometer with t = 33 s (10 million km) arm length mismatch. • Unusual sensor, first null in the sensor is at 33mHz. • The delay doesn’t necessarily limit control bandwidth to low gain, low bandwidth, but does limits the controller design [1]. • Control scheme verified experimentally [2-4]. [1] B.S. Sheard, M.B. Gray, D.E. McClelland, D.A. Shaddock, Phys. Lett. A 320, 9 (2003). [2] A.F.G. Marin, G. Heinzel, R. Schilling, V. Wand, F.G. Cervantes, F. Steier, O. Jennrich, A. Weidner, and K. Danzman, Class. Quantum Grav. 22, S235. (2005). [3] J.I. Thorpe and G. Mueller, Physics Letters A 342, 199 (2005) [4] B.S. Sheard, M.B. Gray, and D.E. McClelland, Applied Optics 45, 8491 (2006).

Simple model of LISA arm (single arm locking)

!pm (t) = p(t)- p(t-")!

Laser phase impulse.

Phasemeter impulse response.

Dual arm locking • Building on the work of Herz[5], Dual arm locking[6] uses measurements from two arms pushing the first null above 2Hz. • Single/common arm locking cannot suppress frequency noise at 33mHz and integer multiples. • Dual arm locking suppresses laser frequency noise across whole science band, Aggressive controller design below 2Hz. τ2

0

t

p− (t)

− Phasemeter

∆τ = τ1 − τ2 < 0.5 s

G Arm Locking Sensor

[5] M. Herz, Opt. Eng. 44, 090505 (2005) [6] A. Sutton, D.A. Shaddock, PRD 78, 082001 (2008)

+

pD

1 = p+ (t) − ∆τ

!

Laser −

Σ

+

LASER

φL

Phasemeter

+ −

τ1

Frequency responses Common arm sensor

Dual arm sensor

Time domain simulations (common arm locking)

p(t) = !(t)- !(t-")#0

Time domain simulations (common arm locking)

p(t) = !(t)- !(t-")#0

Time domain simulations (common arm locking)

p(t) = !(t)- !(t-")#0

Time domain simulations (dual arm locking)

Limits to performance. 1- frequency pulling • The phase measurements require an estimate of the Doppler frequency to be subtracted.

νDE

νOL

+

νCL

Σ

Sensor PS(ω)

+ +

Σ

−

• An error in the Doppler frequency estimate causes the laser frequency to ramp. • Issue at lock acquisition, and in steady state - Doppler frequencies change throughout the orbit. • To reduce this effect, the control loop is ac coupled.

Controller

Phasemeter

G1(ω)

c = νDE 2L

δνCL δt νOL νCL νDE

− − −

[Hz/s]

open loop laser frequency closed loop laser frequency Doppler error

Limits to performance. 2 - noise sources • Laser frequency noise - free running ∆vlaser = 30 kHz/√Hz x 1/f • Clock noise enters at the phasemeters. Not removed in real time. ∆vclock = 2.4x10-12 1/√Hz x 1/√f

• Spacecraft motion enters as only the interspacecraft phase measurements will be used. ∆x = 2.5nm /√(1+( f/0.3Hz)4) • Shot noise enters at each photoreceiver. Not a significant limit.

Modified dual arm locking • The modified dual arm sensor is a combination of: Common arm sensor at f < 33 mHz, and Dual arm sensor at f > 33 mHz • Provides the stability and gain advantages of dual arm locking and frequency pulling and low frequency noise floor of the common arm locking.

Common High gain Noise performance Frequency pulling

Dual

Modified Dual

Frequency pulling • Errors in Doppler frequency [Hz], and Doppler rate [Hz/s] are significant. • Estimate of Doppler frequency and Dopper rate using laser beatnote • The first time arm locking is engaged, the pulling will be: • No prestabilization: 460 MHz • Mach-Zehnder: 90 MHz • Fabry-Perot: 4 MHz • c.f. 10 GHz mode-hop free range. (200s measurement of Doppler frequency)

• In steady state, pulling is less than 10 MHz p-p

lock acquisition

Steady state

Arm locking noise performance Two regimes: gain limited and noise limited.

300 Hz/√Hz

Maximum arm length mismatch: ∆tau = 0.255s, ∆L = 75,000km.

Minimum arm length mismatch that meets TDI capability: ∆tau =46µs, ∆L = 12km.

Gain limited, noise sources negligible. (Still meets capability)

Noise limited for ∆L<12km. (only see this if an interspacecraft link fails)

Performance as arm lengths evolve

• Noise coupling is inversely proportional to the arm length mismatch. • If an interspacecraft link fails the “center spacecraft” cannot be switched when the arm length mismatch is small. Then, arm locking will not meet TDI capability for ~30 minutes, twice per year. • With all links available maximum noise floor is ~13Hz/√Hz and arm locking alone will meet the TDI capability at all times.

Summary of arm locking performance • Modified dual arm locking sensor • Sophisticated controller design

Noise post arm locking

★ Insight into sensor allows a factor of 10 more gain at 1Hz ★ Limits frequency pulling ★ 30 degree phase margin • Detailed noise budgets • Doppler frequency error effect and solution • Arm locking alone meets TDI capability • Next: A high fidelity time domain simulation (simulink).

Results in: K. McKenzie, R.E. Spero, and D.A. Shaddock, to be submitted to PRD & in FCST whitepaper.

End

Controller design: overview ‘UGF’ = 15kHz

Gain requirement = Laser noise/TDI capability

f -2 f6

Laser noise = 30kHz/√Hz x 1/f TDI capability = 300Hz/√Hz x √(1+(3mHz/f)2) (Assuming 1m ranging) UGF = 5µHz

f -0.66 f f3

Designed using poles and zeroes

Arm locking alone can meet TDI capability

Effective phase of sensor depends on controller gain • When the controller gain, G(f), is high

φsensor → −π/2

for |G| >> 1

• However, when the controller gain is unity the phase from the sensor is zero

φsensor = 0

for |G| = 1

Bode plot of single arm locking sensor

Nyquist plot of single arm locking sensor for various controller gain

New controller: more phase at high frequencies Old

New

UGF

1.2kHz

16kHz

Controller slope

f-0.58

f-0.66

Gain at 1Hz

60

600

30 degree phase margin maintained