using signals at the 2nd, 4 , and combined pem harmonics ......harmonics to an accuracy of about...
TRANSCRIPT
Poster TP8.000042
Using signals at the 2nd, 4th, and combined PEM harmonics in MSE polarimeters
S. Scott (PPPL), R. Mumgaard, R. Granetz (PSFC / MIT)
55th APS DPP Meeting
Denver, Colorado
November 11-15, 2013
Motivation
• The MSE diagnostic on Alcator C-Mod is ‘photon-starved’: the statistical measurement error is a few tenths of a degre
• Even with long measurement periods (~100 ms)
• Desired accuracy is ~0.1 degree • In plasmas, our measurement error scales with signal intensity
more strongly than expected by Poisson statistics ( ~ I-0.7).
• Like all other MSE diagnostics, we base our measurement on signals at the 2nd harmonic of the frequencies of our photo-elastic modulators (PEMs).
• But the theory underlying MSE polarimeters suggests that we can also measure the polarization angle from signals at the 4th PEM harmonic.
• Can we reduce our measurement error by simultaneously measuring signals at the 2nd and 4th PEM harmonics?
Abstract: The use of signal intensities at various harmonic frequencies of paired photo-elastic
modulators (PEMs) in polarimeters to measure the polarization angle of linearly polarized light is
explored.
The Alcator C-Mod Motional Stark Effect diagnostic has been calibrated at the fourth PEM
harmonics to an accuracy of about 0.05o, which is within a factor ~2 of the calibration accuracy
at the traditionally used second harmonic, but requires additional calibration terms.
A new mode of operation is derived analytically and verified experimentally: the PEM
retardance is set to 3.61 radians, which maximizes the combined signal strength at the 2nd and
4th harmonic, and the polarization angle is deduced from the ratio of the summed signal
amplitudes at the second and fourth harmonic.
The new system is less sensitive to small drifts in the PEM retardance.
This new regime of operation provides a 40% improvement in photon statistics without
compromising the polarimeter calibration or sensitivity to PEM retardance drift.
Polarimeters typically operate at a retardance of 3.054 radians thatmaximizes the signal strength at I(2ω)
* This also makes the system less sensitive to drifts in the actual PEM retardance.
* Operating at a retardance of 3.611 radians maximizes the sum of J2 and J4, which maximizes the signal I(4ω)-I(2ω).
* A ~40% increase in signal strength is realized, and the system retains its insensitivity to drifts in retardance.
0 2 4 6x-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
J2(x)
J2(x) + J4(x)
J4(x)
bese
lj_71
5.pd
f
Bessel function amplitude
operate PEMs here (standard) or here (proposed)?
Using 4th harmonic signals to measure polarization angles: analytic analysis
Mueller matrix analysis of ideal polarimeters
Use standard Mueller matrix arithmetic to compute the time-dependent signal generated when
linearly-polarized light passes through an ideal polarimeter consisting of:
• two photoelastic modulators (pems) oriented at 45o relative to one another;
• a linear polarizer whose transmission axis is midway between the fast axes of the pems.
The Stokes vector for the output light, Inet, is obtained by multiplying together the Mueller matrices
for each element in their order along the optical path,
Inet = Mp ·MPEM2 ·MPEM1 · Sv (1)
where MPEM1 and MPEM2 are the Mueller matrices for the first and second photoelastic modulators,
Mp is the Mueller matrix for a static polarizer at 22.5o. The output intensity, taken from the first
component of the output Stokes vector, is:
4Inet = 2(Iup + Ip) +√
2 Ip(cosB) cos(2γ)−√
2 Ip(cosA + sinA sinB) sin(2γ)
(2)
where the time-dependent pem retardances are given by A = R1 cos(ω1t) and B = R2 cos(ω2t)
where ω1 and ω2 are the angular frequencies of the two pems. Now we make use of the Bessel-
function expansions:
cos(R cos(ωt)) = J0(R) + 2∞∑n=1
(−1)nJ2n(R) cos(2nωt)
sin(R cos(ωt)) = 2∞∑n=1
(−1)n−1J2n−1(R) cos((2n− 1)ωt).
The intensities at the second and fourth harmonic frequencies are:
I2ω1 =J2(R1)√
2Ip sin(2γ) I2ω2 = −J2(R2)√
2Ip cos(2γ)
I4ω1 = − J4(R1)√2Ip sin(2γ) I4ω2 =
J4(R2)√2Ip cos(2γ). (3)
Evidently, we can obtain the polarization angle from either the ratio of the signal amplitudes at the
second or fourth harmonic:
tan(2γ) =
−I2ω1
I2ω2
J2(R2)
J2(R1)
tan(2γ) =
−I4ω1
I4ω2
J4(R2)
J4(R1)
. (4)
The standard approach is to use the signals only at the second harmonic. Alternately, by subtracting
the signal intensities at the second and fourth harmonics we can also obtain the polarization angle
from the ratio of the summed signal amplitudes:
tan(2γ) =
−(I2ω1 − I4ω1)
(I2ω2 − I4ω2)
J2(R2) + J4(R2)
J2(R1) + J4(R1)
. (5)
This approach obtains maximum signal intensity at a retardance (3.611 radians)
that maximizes the sum of the second and fourth Bessel functions; the sum of
J2(3.611) and J4(3.611) is 37% larger than the original signal strength of J2(3.054).
An important question is whether it is preferable to infer the polarization angles from the signalratios separately (Eq. 4) or summed (Eq. 5).
Effect of slightly unequal PEM retardance
What error is introduced when a polarimeter is calibrated with equal pem retardance, but then the
retardance of one pem drives a small amount?
• It makes a big difference whether the pem retardance is at the maximum of the Bessel function(s).
• At a retardance near 3.054 radians, the spurious changes in angle (in degrees) are
∆γ = 4.097 (∆R)2 sin 4γ ← inferred fromI(2ω1)/I(2ω2) = 0.01o
= 14.0(∆R) sin 4γ ← inferred fromI(4ω1)/I(4ω2) = 0.7o. (6)
where the numerical values assume ∆R = 0.05 radians.
• So a small drift in pem retardance generates a tiny error in the angle measured using signals
at the 2nd pem harmonic, but a large error if the angle is measured using signals at the 4th
harmonic.
• So if the pem retardance is set at its usual value of 3.054 radians, it will be important to stabilize
the pem retardance or else measure it and correct for drift if the signals at the fourth pem
harmonic are to be used to accurately measure the polarization angle.
• Alternately, we can operate the pems at a retardance of 3.611, at which the sum of J2+J4 reaches
a maximum. Near R = 3.611,∆γ = 2.759 sin 4γ (∆R)2 about 33% less than the spurious change
that would obtain if the polarimeter were operated in the usual way.
• By operating the polarimeter at a retardance of 3.611 radians, and by deducing the polarization
angle from the ratio of the sum of the second and fourth pem frequencies, we expect an increase
of 37% in the signal intensity and a 33% reduction in sensitivity to drifts in the pem retardance.
Measuring PEM retardance
Measuring PEM retardance
There is no ‘cross-talk’ in Eq. 2 between the fft amplitudes arising from the terms that involve sinA, cosA, cosB, sinA sinB and
cosA sinB, even when we multiply the expansion of sinA by sinB and cosA by sinB:
sinA: ω1, 3ω1, 5ω1, . . .
cosA: 2ω1, 4ω1, 6ω1, . . .
cosB: 2ω2, 4ω2, 6ω2,,. . .
sinA sinB: [ω1, 3ω1, 5ω1, . . .]× [ω2, 3ω2, 5ω2, . . .]
cosA sinB: [1, 2ω1, 4ω1, 6ω1, . . .]× [ω2, 3ω2, 5ω2, . . .]
i.e. terms that involve sinA sinB will generate fft signals at ω1 + ω2, 3ω1 − ω2 etc. but so long as ω1 6= ω2 these will not overlap the
ω1 harmonics. So there is a fixed relationship among some of the fft amplitudes that is determined only by the retardance and not
by any other parameter.
Iω1 : I3ω1 : I5ω1 = J1(A1) : −J3(A1) : J5(A1) (15)
Iω2 : I3ω2 : I5ω2 = J1(A2) : −J3(A2) : J5(A2)
I2ω1 : I4ω1 : I6ω1 = J2(A1) : −J4(A1) : J6(A1)
I2ω2 : I4ω2 : I6ω2 = J2(A2) : −J4(A2) : J6(A2)
Thus by comparing the fft amplitudes at these pem harmonics, we can immediately determine their respective retardances. Similar
relationships can be derived from the cross-product terms in Eq. 2, i.e. the cosA sinB and sinA sinB terms:
I2ω1−ω2
Iω2
=I18.2
I22.2
=−J2(A1)
J0(A1)
I4ω1−3ω2
I3ω2−2ω1
=I14.2
I26.2
=−J4(A1)
J2(A1)
I3ω2−2ω1
I2ω1−ω2
=I26.2
I18.2
=−J3(A2)
J1(A2)
I3ω1−ω2
Iω1+ω2
=I38.3
I42.3
=−J3(A1)
J1(A1)
I5ω1−3ω2
I3ω2−ω1
=I34.3
I46.3
=+J5(A1)
J1(A1)
I3ω2−ω1
Iω1+ω2
=I46.3
I22.2
=−J3(A2)
J1(A2)
I5ω2−3ω1
I3ω1−ω2
=I34.3
I38.3
=+J5(A2)
J1(A2)
I4ω1−ω2
I2ω1+ω2
=I58.5
I62.5
=−J4(A1)
J2(A1)
I5ω2−2ω1
I2ω1+ω2
=I58.5
I62.5
=+J5(A2)
J1(A2)(16)
Figure 1 plots the inferred retardance as a function of the signal ratio at different frequencies, which corresponds to the ratio of different
Bessel functions. Note that using signals that are based on the ratio of the third to the first Bessel function are particularly good for
inferring retardance in the neighborhood of π radians, since the retardance is a weak function of the signal ratio there - so errors in
the measured signal ratio correspond to very small errors in the inferred retardance.
Retardance inferred from fft ratios
0.0 0.5 1.0 1.5 2.0Amplitude ratio
0
1
2
3
4
5
radi
ans
J2/J0 (negative ratio)
J3/J1
J4/J2
J5/J1
J5/J3
reta
rdan
ce_f
ft_3
1_42
_etc
.pd
f
Figure 1. Inferred pem retardance as a function of the signal ratio at different frequencies.
A second approach: signal maxima and zero-crossing
• Measuring the pem retardance from ratios of signal intensities at selected fft harmonics can be done real-time, i.e. while the
polarimeter is measuring polarization angles.
• A second technique – which should be highly accurate - requires a dedicated retardance scan. It may be used to verify the accuracy
of the first technique.
• If we illuminate a polarimeter with linearly polarized light and vary the retardance demand on e.g. pem-1 while holding the
retardance on pem-2, then the signal intensity becomes proportional to a single Bessel function. So we can identify the retardance
at the maxima of J0, J1, J2 or the zero of J1.
• This approach does not require that the electronic amplification be frequency-independent.
Nominal Actual Amplitude
Frequency Frequency
∼ 0 ω2 − ω1 J1(A1)J1(A2)
3ω2 − 3ω1 J3(A1)J3(A2)
2ω1 − ω2 −J2(A1)J1(A2)
ω2 J0(A1)J1(A2)
∼ ω1 4ω1 − 3ω2 −J4(A1)J3(A2)
3ω2 − 2ω1 J2(A1)J3(A2)
3ω1 − ω2 −J3(A1)J1(A2)
ω1 + ω2 J1(A1)J1(A2)
∼ 2ω1 5ω1 − 3ω2 −J5(A1)J3(A2)
3ω2 − ω1 −J1(A1)J3(A2)
4ω1 − ω2 J4(A1)J1(A2)
∼ 3ω1 2ω1 + ω2 −J2(A1)J1(A2)
3ω2 −J0(A1)J3(A2)
5ω1 − ω2 J5(A1)J1(A2)
∼ 4ω1 3ω1 + ω2 −J3(A1)J1(A2)
3ω2 + ω1 −J1(A1)J3(A2)
5ω2 − ω1 J1(A1)J5(A2)
Table 1. Signals to infer the retardance at the maximum of J0, J1, J2 or the zero-crossing of J1.
0 1 2 3 4x
-0.5
0.0
0.5
1.0
J0(x)
J1(x)
J2(x)
J3(x)
J4(x) J5(x)
bess
el_0
5b.p
df
1.84
3.05
3.83
3.61
2.40
Illuminate a polarimeter with xed-intensity linearly-polarized light and measuresignal intensities in successive shots while varying the retardance of one PEM
* Identify the retardance for which signal intensities cross zero or reach a maxima.
2 3 4 5PEM demand
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
J1_z
ero_
ch4_
1130
7636
42.p
df
Volts
Light source: laserMSE channel 4PEM2 (shots 1130763642 - 661)
0-crossingI(2ω1-ω2) 18.2 kHz 3.847I(ω2) 22.2 kHz 3.784I(4ω1-ω2) 58.5 kHz 3.848I(2ω1+ω2) 62.5 kHz 3.849I(4ω1+ω2) 102.8 kHz 3.850I(ω2-ω1) 2.0 kHz 3.868I(3ω1-ω2) 38.3 kHz 3.875I(ω1+ω2) 42.3 kHz 3.867I(5ω1-ω2) 78.7 kHz 3.873I(3ω1+ω2) 82.7 kHz 3.875
As expected, the measured signal intensities at ten frequencies all go to zero at the same PEM demand retardance
Signals should go to zero at the first zero of J1, R=3.832.Measured zero crossing (excluding I(ω2)) is 3.861 +/- 0.012.Actual retardance at spot where laser crosses PEM is 99.2% of demanded PEM retardance.Using the zero-crossing of J1 is an accurate method to measure PEM retardance.Actual retardance imposed by PEM agrees well with manufacturer specifications.
Peak: 3.082
2 3 4 5PEM demand [ radians ]
0.0
0.1
0.2
0.3Peak: 3.569
Maximizing ( I (4ω2) - I (2ω2) )= 88.6, 44.3 kHz occurs at max(J2+ J4) = 3.611 radians
Maximizing I (2ω2)= 44.3 kHz occurs at max(J2) = 3.054 radians
2 3 4 5PEM demand [ radians ]
-0.22
-0.20
-0.18
-0.16
-0.14
-0.12
-0.10
-0.08
[ vol
ts ]
pem
2_ch
4_m
axim
a_ex
ampl
e_11
3076
3642
PEM-2MSE channel 4
Laser illumination1130763642
We can calibrate the actual retardance imposed by the PEMs from thePEM demand value that maximizes signal strength at selected frequencies
Could also maximize signal strength at I(ω1) or I(ω2) which occurs at maximum J1(x), x = 1.84 radians, but this is outside interesting range for polarimeter applications.
2 3 4 5PEM demand [radians]
2.0
2.5
3.0
3.5
4.0
4.5
5.0
I (3ω2) / I (ω2) (66.6 / 22.2 kHz) I (4ω2) / I (2ω2) (88.7 / 44.3 kHz)
I (5ω2-3ω1) / I (3ω1-ω2) (34.3 / 38.3 kHz) I (3ω2-ω1) / I (ω1+ω2) (46.3 / 22.2 kHz) I (3ω2-2ω1) / I (2ω1-ω2) (26.2 / 18.2 kHz)
PEM-2, MSE channel 4laser illumination
pem
2_ch
4_11
3076
4642
_ve
Five dierent frequency ratios can be used to infer the PEM-2 retardance
Retardance values inferred from I (5ω2-3ω1) / I (3ω1-ω2) and I (3ω2-ω1) / I (ω1+ω2) are always close to one another, and close to the retardance inferred from the maximum of I(2ω2), maximum of I(4ω2)-I(2ω2), and zero of J1.
Retardance values from I (4ω2) / I (2ω2) are typically 0.10 radians lower than from I (5ω2-3ω1) / I (3ω1-ω2) and I (3ω2-ω1) / I (ω1+ω2).
Retardance values from I (3ω2-2ω1) / I (2ω1-ω2) and I (3ω2) / I (ω2) show signicant variability, due to (?) low signal strength.
Retardance values for PEM-1 can be obtained from xx signal intensity ratios.
The signal intensity is highest for I (4ω2) / I (2ω2)
2 3 4 5PEM-2 demand [radians]
0.001
0.010
0.100vo
lts
I (3ω2) / I (ω2)
I (4ω2) / I (2ω2)
I (5ω2-3ω1) / I (3ω1-ω2)
I (3ω2-ω1) / I (ω1+ω2)
I (3ω2-2ω1) / I (2ω1-ω2)
pem
2_ch
4_am
plit
ude_
1131
0763
642.
abs (Signal intensity)
J1(3.83) = 0.
With the exception of I (4ω2) / I (2ω2), the denominator of all ratios is proportional to J1, and so the signal goes to zero and changes sign at 3.83 radians.
The signal strength for ratios I (3ω2-2ω1) / I (2ω1-ω2) and I (3ω2) / I (ω2) is generally lower than the others, which may be resposible for the larger variability in retardance inferred from these ratios.
2 3 4 5PEM demand [radians]
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Retardance from I (5ω2-3ω1) / I (3ω1-ω2)
Retardance from I (4ω2) / I (2ω2)
Retardance from I (4ω2) / I (2ω2),15% frequency-dependent amplication
PEM-2MSE channel 4
Laser illumination1130763642
pem
2_ch
4_ym
ult_
stud
y_b.
Retardance frommaximizing J2
Retardance frommaximizing (J2 + J4)
Retardance from J1 = 0
PEM
Ret
arda
nce
[ rad
ians
]
The oset in retardance measured by I (4ω2) / I (2ω2) compared to I (5ω2-3ω1) / I (3ω1-ω2) and to I (5ω2-ω1) / I (ω1+ω2) could be caused by a small (15%) change in
signal amplication between 44 and 88 kHz
The implied dierence in amplication is the same for all 5 positions of the laser for channel 4.The implied dierence in amplication is the same for channels 4 and 8.Comparing signal intensities that should be identical (next slide) implies a reduction inamplication of 11% between 47 and 87 kHz, but direct bench measurements indicate only a 4% reduction.
2 3 4 5PEM demand [radians]
2.0
2.5
3.0
3.5
4.0
4.5
Laser positions at MSEobjective lens
PEM2, MSE channel 4, laser illumination
= retardance inferred from max I(2ω2), max I(4ω2)-I(2ω2), or J1(x) = 0
Retardance inferred from I (5ω2-3ω1) / I (3ω1-ω2)ra
dian
s
pem
2_35
31_p
ositi
on_s
can_
1130
763.
Optical rays that do not pass thru the center of the PEMs experience signicantly lower retardance than those that pass thru the center
Rays that pass thru center of PEM are imparted a retardance very close to demand value.
Due to mirror reections and other MSE lenses, o-axis rays at lens may pass thru center of PEMs.
Excellent agreement between retardance inferred from signal intensity ratios versus maximumof I(2ω2), I(4ω2)-I(2ω2), or J1(x) = 0.
The system response when the lens is fully illuminated (e.g. by neutral beam emission) will be an average over the ensemble of rays that strike the lens.
Variation in retardance across lens is large, ~0.75 radians.
2 3 4 5PEM demand [radians]
2.0
2.5
3.0
3.5
4.0
4.5
PEM2, MSE channel 4, laser illumination
= retardance inferred from max I(2ω2), max I(4ω2)-I(2ω2), or J1(x) = 0
Retardance inferred fromI (5ω2-3ω1) / I (3ω1-ω2)
radi
ans illuminate lens uniformily
with LED array
Laser positions at MSEobjective lens
As expected, when the lens is illuminated uniformily across its entire surface,the eective retardance averages the values at dierent spots on the lens
LED
_las
er_p
em2_
3531
_vs_
posi
tion.
2 3 4 5PEM demand [radians]
2.0
2.5
3.0
3.5
4.0
4.5
MSE Channel 4(near MSE optical axis)
MSE Channel 8(near edge of MSE view)
Retardance inferred from I (5ω2-3ω1) / I (3ω1-ω2)
pem
2_vs
_cha
nnel
)113
0763
642.
pdfra
dian
s
PEM-2Laser position: r=15 mm, 0o (= 12 o’clock)
For a given spot on the MSE objective lens, the measured retardance is alsoa function of where the light originates - the MSE spatial channel number
Light from dierent spatial channels hits the PEM at dierent locations --> dierent retardance
delta
_ret
arda
tion_
com
bine
d_c.p
df
Reta
rdan
ce [
radi
ans]
Chan
ge in
Ret
arda
nce
[ rad
ians
]
0 5 10 15 20 25-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
20 kHz PEM
22 kHz PEM
CH 1
CH 9
(a)
(b)
Shot number
1206
05
1208
14Run Day
3.18
3.16
3.14
3.10
3.08
3.1220 kHz PEM
22 kHz PEM
Change in PEM retardance over thecourse of a typical run day, relativeto retardance on rst shot.
Shots are spaced every ~15 minutes.
Change in PEM retardance overthe course of an entire AlcatorC-Mod run campaign.
Each data point represents thedaily-average of the retardanceaveraged over all channels.
We typically observe relatively small (~0.05 radian) drifts in PEM retardance
Using 2nd and 4th harmonic signals separately
θ = Actual Polarization Angle (degrees)-100 0 100 200 300 400
-0.02
-0.01
0.00
0.01
0.02
Deg
rees
Fit error for: θmea = -35.00 + θ+ 0.615 cos(4θ+118.1) + 0.154 cos(2θ -154.9)
average error: 0.008o
Channel: 8 Average error: 0.46o
-1
0
1
Deg
rees
Residual fit error from simple offset-linear fit: θt = Ao + θactual
typical_calibration.pdf
We calibrate MSE by robotically illuminating it with linearly polarized light:θ = 0, 10, ... 360o
MSE
MSE Sightline
Polarization Generation Head
* The system response is very well represented by an offset-linear fit with cos(2θ) and cos(4θ) correction terms.
gg_4
044_
linea
r_t
s_ap
s.pdf
0 2 4 6 8MSE channel
0.0
0.1
0.2
0.3
0.4
0.5
linear-only
linear + cos(2θ)
linear + cos(4θ)
linear + cos(2θ) + cos(4θ)
RMS
t e
rror
[ d
egre
es ]
based on θmea = 0.5 tan-1 ( -I(2ω1) / I(2ω2) )
MSE channel
0.0
0.2
0.4
0.6
0.8
0 2 4 6 8 10
coecient ofcos (2θ)
coecient ofcos (4θ)
[ deg
rees
]
In actual MSE calibrations using signals at the second harmonic, the magnitude of thecos(2θ) and cos(4θ) correction terms varies smoothly with MSE viewing location
For PEM retardance near 3.054, we can still accurately represent the angles inferred from - I(4ω1) / I(4ω2) by including a correction term cos(8θ)
012345
[Deg
rees
] Coefficient of 4 theta
0.0000.002
0.004
0.006
0.008
[Deg
rees
] Coefficient of 2 theta
0.00.1
0.2
0.3
0.4
[Deg
rees
] Coefficient of 8 theta
2.8 2.9 3.0 3.1 3.2PEM-1 retardance [radians]
0.000
0.010
0.020
0.030
[Deg
rees
] RMS fitting error
pp_4
28_t
heta
_aps
x 1
x 1
x 1
x 1
θmea = Ao + θ + A2 cos(2θ + A3) + A4 cos(4θ + A5) + A8 cos(8θ + A9)
numericalsimulation
θmea from - I(4ω1) / I(4ω2)
0 2 4 6 8MSE channel
0.0
0.1
0.2
0.3
0.4
RMS
t e
rror
[deg
rees
]
linear + cos(2θ) + cos(4θ)
linear + cos(4θ) + cos(8θ)
linear + cos(2θ) + cos(4θ) + cos(8θ) gg_8
088_
linea
r_ap
s.pdf
based on θmea = 0.5 tan-1 ( -I4ω1/I4ω2 )
0 2 4 6 8 10MSE channel
0.0
0.2
0.4
0.6
cos (2θ)
cos (8θ)
cos (4θ) / 10
t c
oe
cien
t [ d
egre
es ]
based on θmea = 0.5 tan-1 (-I4ω1 / I4ω2)
Similarly, when MSE is operated at a retardance to maximize signal intensityat the 2nd harmonic, the behavior of the calibration based on signals at
the 4th harmonic varies smoothly with MSE viewing location
0 2 4 6 8MSE channel
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s.pdf
RMS
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[ de
gree
s ]
4th PEM harmonic
2nd PEM harmonic
Optical axis: R(PEM1)=2.93, R(PEM2)=3.36
Summary: when MSE is operated at a retardance to maximize signal strengthat the 2nd harmonic, it can be simultaneously calibrated to an accuracy of:
~0.03o using signals at the 2nd harmonic
~0.05o using signals at the 4th harmonic
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Polarization angle based on I2ω
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Dierence in polarization angle based on I4ω versus I2ω
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RMS = 0.054o
As MSE is illuminated by linearly-polarized light with time-varying polarizationdirection, the polarization angles inferred from signals at the 4th harmonic tracks
the values inferred from the usual signals at the second harmonic to ~0.05o
0
1
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5[D
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(a) Coefficient of cos(4θ)
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(b) RMS fitting error
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x 1
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from - I2ω1 / I2ω2
from - I4ω1 / I4ω2
θmea = 0.5 tan-1 [ I (2ω1) / I (2ω2) ] = Ao + θ + A4 cos(4θ + A5) or = 0.5 tan-1 [ I (4ω1) / I (4ω2) ] = Ao + θ + A4 cos(4θ + A5)
If the PEMs are operated with a retardance to optimize the second - harmonic signal intensity(R=3.054), then a small drift in the retardance of one of the PEMs causes:
* a small change in the polarization angles measured from I (2ω1) / I (2ω2) ... good
* big drift in the polarization angles measured from I (4ω1) / I (4ω2) ... bad
R(PEM-2) = 3.14
Using combined 2nd and 4th harmonic signals
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Coecient of cos(4θ)
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RMS tting error
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operate at R=3.054 that maximizes J2(R), using I2ω
operate at R=3.611 that maximizes J2(R)+J4(R), using I4ω - I2ω
numerical simulation
Numerical simulations indicate that it should be possible to calibrate apolarimeter equally well at:
* R=3.611 radians, using combined signals: I(4ω)-I(2ω)
* R=3.054 radians, using I(2ω) (the standard approach)
0 2 4 6 8 10MSE channel
0 2 4 6 8 10MSE channel
0.0
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0.3Amplitude of cos(2θ) correction term
Amplitude of cos(4θ) correction term
Residual t error
θmea = Ao + θ + A2 cos(2θ + A3) + A4 cos(4θ + A5)
angles from I(2ω1)/I(2ω2), R=3.054 radians
angles from (I(4ω1)- I(2ω1)) / (I(4ω2)- I(2ω2))R=3.611 radians
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MSE calibration based on (I(4ω1)- I(2ω1)) / (I(4ω2)- I(2ω2)) at R=3.61 radiansis nearly identical to the customary approach, I(2ω1)/I(2ω2) at R=3.05 radians
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1130774/1130784
A rst-principles numerical model of an ideal PEM-based polarimeter accuratelymodels the dependence of statistical measurement error on signal strength
* Assume a signal intensity and polarization angle.* Compute time-dependent signal (photons/sec) emerging from polarimeter from Eq. (2).* Add in Poission statistics from signal plus dark noise.* Analyze this time-dependent signal using standard MSE analysis code --> obtain polarization angle* Do this over an ensemble of runs, then compute standard deviation of ‘measured’ angles
* Result: statistical measurement error scales as I -0.91, not I -0.5 as expected from pure Poission statistics.
0.0001 0.001 0.01 0.10 1.0I = Polarized signal amplitude [volts]
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measured: y = 0.0022 I -0.910
model: y = 0.0028 I -0.915
dark current = 0, y = 0.00544 I -0.495
stan
dard
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iatio
n of
ens
embl
e of
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les
[ deg
rees
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dark
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rent
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dy.p
df
angles from I(2ω1)/I(2ω2), R=3.054 darkcurrent
typical signalsfrom plasma
0.0001 0.001 0.01 0.1 1.0polarized signal intensity [volts]
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10
using I4ω - I2ω, that maximizes J2+J4, at R=3.611
y = 0.00217 x-0.902
using I2 that maximizes J2 at R=3.054
y = 0.00288 x-0.904
stde
v_30
54_3
611_
082_
083.
The model predicts that using combined signals at the second andfourth harmonic should decrease statistical errors by a factor 1.32
degr
ees
6 8 10 12Voltage applied to LED light source
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volts
I(2ω1)/I(2ω2),R=3.054
I(4ω1)-I(2ω1) I(4ω2)-I(2ω2) R=3.611
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Does increased signal intensity at I(4ω1)-I(2ω1) / I(4ω2)-I(2ω2) actually reduce error bars?
* Illuminate MSE with linearly polarized light, xed polarization direction* Measure polarization angle at I(2ω1)/I(2ω2) versus (I(4ω1)-I(2ω1)) / (I(4ω2)-I(2ω2))* Vary light intensity over 2+ orders of magnitude.* Compute standard deviation of 70 measurements, 40ms each
* Result #1: operating at (I(4ω1)-I(2ω1)) / (I(4ω2)-I(2ω2)) increases the polarized signal 32%. Good!
0.001 0.01 0.1 1.0 0.001 0.01 0.1 1.0Total light intensity (volts)
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ees
Standard deviation of measurements
Measured signal intensity (volts)
Standard deviation of measurements
Does increased signal intensity at I(4ω1)-I(2ω1) / I(4ω2)-I(2ω2) actually reduce error bars?
* Result #2: surprisingly, at xed incident light intensity, operating at (I(4ω1)-I(2ω1)) / (I(4ω2)-I(2ω2)) does not reduce the measurement variability.
* For xed measured signal intensity (e.g. sqrt( I(2ω1)2+ I(2ω1)2 ), there is more variability based on (I(4ω1)-I(2ω1)) / (I(4ω2)-I(2ω2)). This is not understood.
scan
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Conclusions
• In PEM-based polarimeters, signals at the 4th PEM harmonic can be used to measure linear polarization angles, in conjunction with the usual signals at the 2nd harmonic.
• Operating at the customary retardance of 3.054 radians, the polarization angles can be independently obtained from signals at the 2nd and 4th harmonic. This yields a ~30% increase in signal strength, but the measurement at the 4th harmonic will be very sensitive to small drifts in the PEM retardance.
• Operating at a retardance of 3.611 radians, the polarization angles can be obtained from the sum of the signal strengths at the 2nd and 4th PEM harmonics. This yields a ~40% increase in signal strength, and the measurement is insensitive to small drifts in the PEM retardance.
• The effective PEM retardance can be inferred from various ratios of signal intensities at various mixed PEM harmonic frequencies.
• The statistical measurement error is observed to vary as the I-0.91, not the I-0.5 expected from Poisson statistics. This behavior is confirmed by a first-principles numerical model that includes the effect of dark noise.
• We observe the expected increase in signal strength derived from including signals at the 4th PEM harmonic.
• To date, we have not observed the reduction in statistical measurement error that would be expected from this increased signal intensity.
Extra slides
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0.020.040.060.080.10
2.5 3.0 3.5 4.0PEM1- retardance [radians]
0.00.20.40.60.81.0
(a) I(4Ω1) - I(2Ω1)
(b) I(2Ω1)
(c) I(4Ω1)
(d) - I(4Ω1) / I(2Ω1)
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(a) Coefficient of cos(4θ)
4.8 4.9 5.0 5.1 5.2PEM-1 retardation
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(b) RMS fitting error
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If we operate the PEMs with a retardance near the value that maximizes signal strength at the 4th PEM harmonic (R=5.137), then a simple oset-linear
with a cos(4θ) correction term accurately describes the system response
numericalsimulation
R(PEM2) = 5.137
θmea = Ao + θ + A4 cos(4θ + A5)
2 3 4 5PEM-1 demand [radians]
2.0
2.5
3.0
3.5
4.0
I (4ω1) / I (2ω1)
I (5ω1-ω2) / I (3ω1+ω2)
I (5ω1-3ω2) / I (3ω21ω1)
I (3ω1) / I (ω1)
I (3ω1-ω2) / I (ω1+ω2)
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= retardance inferred from max I(2ω1), max I(4ω1)-I(2ω1), or J1(x) = 0
It pays to measure the retardance: retardance of PEM-1 saturates ata demand value of ~4 radians.
As observed on PEM-2, the retardance inferred from I (4ω1) / I (2ω1) is ~0.1 radians lessthan inferred from other signal ratios