Proposed Spot Size Monitor Qweak Goes Nonlinear

1
Proposed Spot Size Monitor(Qweak Goes Nonlinear)

• D.J. Mack (TJNAF)
• Qweak Collaboration Meeting at JLab
• March 10, 2006

• Spot size and intensity jitter sensitivity
• The larger nonlinear picture
• Proposed technical solutions
• Summary

2
AbstractWhereupon the JLab PV program is
introduced to a 2nd order formalism which makes
explicit the size of all second order terms,
allows all sources of nonlinearity to be
naturally incorporated, and clarifies the
relationship between beam moments and the
transducers needed to measure them, in the fond
hope of getting the right answer, improving
agreement between modeled and measured
sensitivities, increasing the stability of
measured sensitivities in the face of day-to-day
changes in beam parameter correlations and
jitter, and ultimately helping our experiment be
interpretable in terms of new physics rather
than unmeasured beam parameters.
3
Wake Up Call Regarding (x2) - (x2)-

• Spot Size Sensitivity (J. Birchall, Systematic
  Errors for the Final Collimator, Oct 05
  _at_TRIUMF)
• for Qweak 1 ppb/67 nm
• Not bounded in previous JLab PV measurements.
  The intrinsic beam spot size is 150,000 nm.
  (plug it in for a hoot!)
• Intensity Variance Sensitivity (D. Mack,
  Nonlinearity Specifications of the Qweak
  Detector Chain Sept 2004, Qweak tech note
  172-v1)
• would be a reasonable specification for
  either Qweak or G0 forward angle.
• Again, this is not bounded in previous JLab
  PV measurements.
• Slow reversals with the half-wave plate, and
  frequently updating effective 1st order
  sensitivities, will sweep a lot of 2nd order
  problems under the rug. But dont bet yet that
  all such effects can be ignored. Ill show that
  at nonlinear order, things are complex, subtle,
  and inherently unstable.

4
Formalism the larger picture
5
PV to First Nonlinear Order
Expanding our detector response at arbitrary
bandwidth to 2nd order in beam parameters (x I,
E, x, x, y, and y) and taking the first few
non-time-averaged beam moments
And using the definitions
Then, defining ?x x - x - , to this order
Offsets matter!
Correlations matter!
Size matters too!
Nonlinearity matters!
Even frequencies gt the reversal frequency may
matter!
6
Limitations of Linear Beam Corrections

• So seductively simple. So obviously correct
  unless one truncates!
• No explicit demonstration that all higher order
  terms are negligible.
• Diagonal terms ltx i2gt present even if
  correlations vanish
• Beam ltx i2gt terms arent even measured at
  JLab (out of sight, out of mind?)
• No intellectual framework for collaboration to
  even begin thinking about
• nonlinearities and how changes in the
  accelerator noise spectrum affect them
• the effect of feedback on ltxixjgt terms,
• the effect of half-wave plate reversal on
  ltxixjgt terms,
• etc.
• With luck, the only effect of ignoring
  higher order processes is
• 1) 1st order sensitivities keep changing
  outside their statistical errors,
• 2) 1st order sensitivities dont match model
  without a lot of empirical tuning.
• Would anyone be surprised if we were not so lucky
  at 1 ppb?

7
Diagonal Terms

• Magnitude depends on product of
• Nonlinear response in apparatus,
• and
• Size modulation at frev in beam ltxi 2gt

No significant JLab bounds on ?ltI2gt, ?ltE2gt,
?ltX2gt, ?ltX2gt, ?ltY2gt, or ?ltY2gt.
(Significant means they havent been proven to
be smaller than 1 ppm.) Examples of currently
invisible ltxi2gt
Interaction between scraping and intensity
feedback. Same ltxgt, ltIgt, Different ltx2gt
Simple breathing . Same ltxgt, ltIgt, Different ltx2gt
Differential intensity bounce. Same ltxgt,
ltIgt, Different ltI 2gt
Xi
Xi
time
8
Off-Diagonal Terms

• Mathematically similar to the diagonal terms (can
  be related by a coordinate transformation), so
  expect similar problems.
• Treated separately here because required beam
  diagnositcs will be very different (probably
  simpler)
• Magnitude depends on product of
• Non-linear response of apparatus,
• and
• Size modulation at frev in beam ltxixjgt

No significant JLab bounds on the 15 unique
?ltxixjgt. (Matrix is symmetric.) How big are
they? Does the half-wave plate reverse their
signs without affecting magnitudes? How stable
are they? Is the half-wave plate reversed often
enough to get good common mode rejection of any
instabilities?
Were talking about an accelerator with a million
parts each with 99 reliability theres always
a flakey power supply, a DAC that is
intermittently dropping bits, an RF control
module that is oscillating, a magnet which is off
its hysteresis loop because it didnt get the
memo, etc. Long term instability in beam ltxixjgt
is guaranteed.
9
Frequency Distortion (Another Reason Nonlinearity
Sucks)

• Because Fourier transforms are linear,
  frequencies in the 1st order terms behave
  themselves and stay separate.
• The 1st order correction terms have no
  dependence on non-reversal frequencies.
• In 2nd order no such luck. The signal at fR can
  in general be distorted by the entire frequency
  spectrum. The amplitude of the carrier matters.
  

--
--
--

--
--
--

• There is no suppression of distortion if the
  carrier is at much higher frequency than the
  reversal.

• Distortions are suppressed if the carrier is much
  lower frequency than the reversal (ie., good
  common mode rejection).

10
Convergence

• The fact that helicity-correlated size modulation
  can ride atop a larger background frequency can
  make 2nd terms larger than one might naively
  expect.
• Changes in the background spectrum (or offsets)
  can change the size of these effects.
• These potential instabilities make good
  cancellation with the half-wave plate less likely
  unless we can understand which 2nd order terms
  are important (if any) and monitor them.
• Not all off-diagonal beam moments may reverse
  with the half-wave plate. In principle, any
  correlations for which one beam parameter
  reflects across the axis of the other will
  survive but such pathologies may not be allowed
  by optics.

Order of Correction
11
Application I

• The only place to expect significant nonlinear
  effects is
• Where the experiment sensitivity d2A/dxidxj is
  relatively large
• AND
• The dynamic range of ltXi2gt at high frequencies is
  relatively large (where Xi I, E, x,
  x, y, y, etc.)
• so focus on potential Achilles heels.

Qweak ltI2gt - Expected d2A/dI2 is 10-3 (limited
by the frequency-dependent cryotarget response),
but experiment precision is high. ltx2gt or lty2gt -
Fast raster produces a relatively large spot
size. ltx2gt or lty2gt - Imperfections in FR
chicane may put ghosts in the angle spectrum.
Not large but may still dominate the dynamic
range. G0 forward angle ltI2gt - Since d2A/dI2
was O(10) due to counting dead-time, in relative
terms the experiment was as sensitive to changes
in the intensity spectrum as Qweak will be.
Fine-tuned deadtime corrections would stop
working as soon as the laser intensity spectrum
changed, for example.
12
Application II
After averaging ltxigt over months, under what
circumstances is the 1st order correction a
useful estimate of the net magnitude of false
beam asymmetries?
True only if breathing in all ltxixjgt is zero
or averages to zero. However, why should
anything average to zero? Present feedback
may have no effect on nonzero ?ltxi2gt terms.
Present feedback may preserve most of the
nonzero ?ltxixjgt . If all ltxixjgt are NOT zero,
then the final truncated 1st order error estimate
is only accurate if the machine is perfectly
stable, all nonlinear effects perfectly reverse
sign when the half-wave plate is inserted, and
the integrated luminosities for and - are
equal. I suspect the linear correction in ltxgt
becomes sub-dominant at long integration times.
Someday, somebody is going to hit the systematic
floor. Maybe us.
13
More Predictions

• A 2nd order model is only useful if it
  has predictive power.
• We should observe
• the 1st order sensitivities become more stable,
  and the 1st order sensitivities become closer to
  simulation results which havent been fine-tuned,
  
• the sign of most nonlinear effects reverse with
  the half-wave plate
• collaborators running around in great distress
  when nonlinear effects dont change sign, or
  change magnitude, with the half-wave plate,
• collaborators later relieved to discover that all
  changes in the magnitude of nonlinear effects are
  traceable to measured changes in offsets, the
  background spectrum, or the amplitude at the
  reversal frequency,
• (otherwise, the model needs another beam
  parameter, etc.)
• By the way, significant changes in the
  frequency spectra will tell us when its
  important to initiate an unscheduled slow
  reversal. That alone would be useful!

14
What Does This Mean for Qweak?

• Educate ourselves, then set priorities. The sky
  is not falling.
• Its impossible to build an entire 2nd order
  infrastructure by Run I.
• Continue to determine largest 2nd order
  sensitivities from simulation.
• Start monitoring spectrum of each beam parameter
  to 100 KHz(?)
• Think about what feedback is doing to the 2nd
  order terms.
• Study whether there are 2nd order correlations
  which the half-wave plate may not cleanly reverse
  and which we should therefore watch more closely.
• Design and build a fast correlator to measure
  ltxixjgt
• Design and build a device to measure ltx2gt
• Measure the most important diagonal and
  off-diagonal ltxixjgt beam moments.
• Dither to obtain the most important diagonal and
  off-diagonal 2nd order sensitivities d2A/dxidxj
  (Should be quite a challenge.)

15
Building stuff to address 2nd order
16
Off-Diagonal Correlation Monitor ltxixjgt

• More perspiration than invention? Signals
  mostly exist, they would need to be correlated at
  high bandwidth.
• Straw man concept
• take existing analog, high bandwidth versions of
  all xi signals
• (6 of these presently)
• use analog multiplier/divider modules to generate
  normalized, analog versions of ltxixjgt
• (15 unique ones at the moment)
• send resulting signals to TRIUMF ADCs and
  spectrum analyzers.
• (download spectra
  at end of each run?)
• No specifications yet, but will need to
  worry about inter-channel isolation, dynamic
  range, divide by zero, and how to do analog
  calculations for angle and energy.
  Thats the perspiration part!

17
Monitoring Modulation of Beam ParametersltXk2gt
It may be possible to meet most needs for ltx
k2 gt measurements with a generic monitor of
real-space ltx 2gt. plus more spectrum
analyzers.
? Y. Chao thinks 3 locations would be needed to
unambiguously separate modulation in size, angle,
and sizeangle correlation.
18
Spot Size Monitor Specifications

• High bandwidth (to check higher frequencies)
• Modest regressions needed to access ltx2gt
• (so we dont go insane)
• Non-invasive technique preferred
• (concerns are background, adequate coverage
  over the entire run, and potentially long
  integration times)

19
Simple Math of Non-Invasive RF Beam Monitors (One
Nonlinearity Deserves Another)

• Ultimately, the energy for all beam monitor
  signals comes from power coupled from the beam
  into the beam monitors via
• Power/length d(q Ez)/dt I Ez
• If a generic beam monitor had longitudinal
  electric fields of the form (time dependence ei?t
  implicit)

then
yielding
A non-invasive RF monitor of ltx2gt requires
nonlinear Ez. We want the linear term close to
zero so we dont have to make significant
position A normalizing measurement of I0 is
needed.
A linear BPM can never give us ltx2gt!
20
Proposed Cavity Monitor

• Rectangular cavities in the TM310 or TM130 mode
• adequate curvature d2Ez/dx2 gt 0
• dEz/dx approx. zero
• X- and Y- are decoupled
• Normalization with a standard JLab BCM cavity
  (position independent).
• For (l, m 1,2, and n0), we have

TM310
Sensitivity to size and position modulation
estimated in D. Mack and M. Wissman, Qweak tech
note 541-v1. With beam centered within 1 mm of
cavity axis, we should be able to set a limit on
x- or y- modulation at the 100 nm level without
large position regressions.
BCM
Not significantly limited by shot noise or
digitization noise. Probably limited by thermal
noise. Stay tuned.
21
Summary

• To avoid a new surprise every 6 months, I have
  tried to put J. Birchalls discovery of spot
  size sensitivity into a broader context. This
  also synthesizes both discussions with others and
  long-standing questions Ive had about the role
  of nonlinearities in PV experiments.
• The main overhead of a 2nd order experiment is to
  make high bandwidth measurements of real-space
  ltx2gt and lty2gt at several points along the
  beamline.
• (A non-invasive technique preferred.)
• Without bounds on the diagonal ltxi2gt terms, the
  JLab PV program below the ppm-level has
  decreasing discovery potential.
• (We will eventually find most of these
  terms to be negligible, but the interpretability
  gap in a program which is based on grubbing
  around for 1-2 sigma effects is real and
  growing.)
• A rectangular cavity monitor was proposed with
  nonlinear Ez (x,y) near a local maximum. It is
  therefore sensitive to the size of the beam
  without being overly sensitive to beam position.
  Discussions are ongoing with Arne Freyberger
  (JLab beam diagnostics).

22
Acknowledgements

• Nonlinearities R. Carlini, D. Ramsay,
• J. Birchall, M. Gericke, and J. Roche
• Spot Size Monitor M. Wissmann,
• J. Musson, and A. Freyberger