# Optimized signal deduction procedure for the MIEZE neutron spectroscopy
technique

J. K. Jochum  Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universität
München, D-85748 Garching, Germany  L. Spitz  Heinz Maier-Leibnitz Zentrum
(MLZ), Technische Universität München, D-85748 Garching, Germany  C. Franz
Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universität München, D-85748
Garching, Germany  Physik Department, Technische Universität München, D-85748
Garching, Germany  Jülich Centre for Neutron Science JCNS-MLZ,
Forschungszentrum Jülich GmbH Outstation at MLZ FRM-II, 85747 Garching,
Germany  J. C. Leiner  Physik Department, Technische Universität München,
D-85748 Garching, Germany  C. Pfleiderer  Physik Department, Technische
Universität München, D-85748 Garching, Germany  O. Soltwedel  Physik
Department, Technische Universität München, D-85748 Garching, Germany
Institut für Physik Kondensierter Materie, Technische Universität Darmstadt,
D-64289 Darmstadt, Germany<EMAIL_ADDRESS>
###### Abstract

We report a method to determine the phase and amplitude of sinusoidally
modulated event rates, binned into 4 bins per oscillation. The presented
algorithm relies on a reconstruction of the unknown parameters. It omits a
calculation intensive fitting procedure and avoids contrast reduction due to
averaging effects. It allows the current data acquisition bottleneck to be
relaxed by a factor of 4. Here, we explain the approach in detail and compare
it to the established fitting procedures of time series having 4 and 16 time
bins per oscillation. In addition we present the empirical estimates of the
errors of the three methods and compare them to each other. We show that the
reconstruction is unbiased, asymptotic, and efficient for estimating the
phase. Reconstructing the contrast, which corresponds to the amplitude of the
modulation, is roughly $10\%$ less efficient than fitting 16 time binned
oscillations. Finally, we give analytical equations to estimate the error for
phase and contrast as a function of their initial values and counting
statistics.

###### pacs:

keywords: neutron spectroscopy, neutron resonance spin-echo, mieze, error
estimation

##  1 Introduction

MIEZE (Modulation of IntEnsity with Zero Effort) spectroscopy is a hybrid
technique combining neutron resonance spin-echo (NRSE) and neutron Time-of-
Flight (ToF) spectroscopy. It is routinely available at the NRSE-Spectrometer
RESEDA at the Heinz Maier-Leibnitz Zentrum [1]. In Fig. 1 we present a basic
MIEZE set-up. It uses neutron spin precession generated by two resonant
(neutron) spin flippers ($RSF_{1}$ and $RSF_{2}$) separated by a distance
$L_{\mathrm{1}}$ and operated at individual frequencies ($f_{1}\,<\,f_{2}$),
to manipulate the spin eigenstates [2, 3]. The resulting interference pattern
of the superposition of the spin states corresponds to a sinusoidal intensity
as a function of time akin an optical heterodyne interferometer.

Also, in MIEZE, the modulation frequency of the intensity is given by twice
the difference of the $RSF$ frequencies
$f_{\mathrm{MIEZE}}\,=\,2(f_{2}-f_{1})$. The coherence volume of the
interference is indirectly proportional to the wavelength of the incoming
neutron beam, the width of the wavelength band, and $f_{\mathrm{MIEZE}}$ [4,
5]. Its center is located at a point $L_{2}$ downstream $RSF_{2}$. It is
exclusively determined by the frequency ratio and distance of both flippers
$\frac{f_{2}-f_{1}}{f_{1}}=\frac{L_{1}}{L_{2}}$ [6]. In practice these
frequencies are limited at the lower end by the neutron spin flip efficiency
generated by the Bloch-Siegert-shift to $f_{\mathrm{min}}\,=\,35\,$kHz [7].
The limitations at the upper end are due to skin and proximity effects in the
resonant flippers, as well as parasitic capacities in the resonant circuits,
which currently sets the maximum $RSF$ frequency to
$f_{\mathrm{max}}\,=\,3.6\,$MHz [8, 3]. Without loss of generality, the
experimental description given here includes only those components necessary
to produce the intensity as modulated in time. Further details and a
description of the MIEZE setup may be found in Refs. [9, 10].

Figure 1: (a) Schematic representation of the essential parts of the MIEZE
setup. Polarized neutrons travel in the y-direction passing the resonant spin
flippers ($RSF_{1}$ and $RSF_{2}$) and the precession region between them, the
spin analyser, the sample, and finally hitting the detector. (b) The time-of-
flight difference $\Delta t$ of the spin eigenstates as function of distance
along the flight path is shown. The time-of-flight difference at the sample
position is $\tau_{\mathrm{MIEZE}}$

In contrast to conventional neutron spin-echo (NSE), the quantity measured in
MIEZE corresponds to a sinusoidally modulated intensity in time. From a
practical point of view the detector registers events per oscillation and
histograms according to a certain number of time bins [11]. Thus, for a fixed
number of time bins, the length of each time bin is a function of the
modulation frequency. The lower limit of the time bin length is given
naturally by the temporal resolution of the detector, which is limited by the
electron drift time and the clock of the electronics readout of the detector
[11]. Hence, to detect signals with fast modulation, it is necessary for the
number of time bins to be as low as possible. In contrast, an insufficient
number of time bins per oscillation results in a smearing of the recorded
oscillation amplitude.

Without loss of generality and neglecting the average background count rate,
the event rate registered by a detector recording signals at discrete
intervals in time is given by the integral over a harmonic oscillation with
amplitude $I_{0}$ and arbitrary phase $\phi_{0}$

$\displaystyle I^{{}^{\prime}}\,$
$\displaystyle=\,\frac{I_{0}}{\Delta\phi}\,\int_{\mathrm{-\frac{\Delta\phi}{2}}}^{\frac{\Delta\phi}{2}}\,\sin{(\phi-\phi_{0})}\,d\phi$
(1a) $\displaystyle I^{{}^{\prime}}\,$
$\displaystyle=\,I_{0}\sin(\phi_{0})\,\frac{\sin{\frac{\Delta\phi}{2}}}{\frac{\Delta\phi}{2}}.$
(1b)

where $\Delta\phi\,=\,\frac{2\pi}{\\#\,timebins}$. In this resolution function
the sinc-function acts as a damping factor, which assumes a minimum value if
$\Delta\phi\,=\,0$, i.e. infinite time bins representing a trivial but
prohibitively impractical solution. Keeping in mind that at least three
parameters $I_{\mathrm{mean}}$ (the time average), $I_{\mathrm{0}}$ (the
amplitude), and $\phi_{0}$ (the arbitrary phase) must be extracted from the
signal, a minimum of three time bins is necessary for an unambiguous
reconstruction. From this the MIEZE contrast
$C\,=\,\frac{I_{0}}{I_{\mathrm{mean}}}$ may be extracted [6].

A good compromise between the number of time bins and the magnitude of the
reduction factor are 16 time bins, the default value for MIEZE experiments at
RESEDA. In this regime, the damping induced by the sinc-function is only
0.64%. Currently a detector with a clock rate of $10\,$MHz is used to detect
the MIEZE-signals at RESEDA [11]. Therefore, the maximum MIEZE frequency is
limited to $f_{\mathrm{MIEZE}}\,=\,\frac{10\,\mathrm{MHz}}{16}\,=\,625\,$kHz
when using 16 time bins. In order to increase the resolution (e.g.
$\tau_{\mathrm{MIEZE}}$, the Fourier time), $f_{\mathrm{MIEZE}}$ has to be
maximized since it is directly proportional to $\tau_{\mathrm{MIEZE}}$ by the
following equation

$\tau_{\mathrm{MIEZE}}\,=\frac{2\pi\,\hbar\,L_{\mathrm{{S}}}}{m_{\mathrm{n}}v_{\mathrm{n}}^{3}}\,f_{\mathrm{MIEZE}}$
(2)

with the neutron mass $m_{\mathrm{n}}$, its velocity $v_{n}$, and the sample
detector distance $L_{\mathrm{S}}$ (cf. Fig. 1).

A practical solution is to reduce the number of time bins, fit the points with
a sine function, and stretch the amplitude by normalizing it to the damping
factor given by (1a). Since the measurements are normalized to the
instrumental resolution function (which depends equally on the damping
factor), the damping factor cancels out, and therefore does not need to be
taken into account explicitly. Alternatively, one may find an unbiased
estimate to reconstruct the fitting parameters from the minimum necessary time
bins by taking the time integration of the detector into account. In the
following sections, a reconstruction procedure of the underlying parameters
will be deduced using only four time bins. This relaxes the data acquisition
by a factor of 4 corresponding to
$f_{\mathrm{MIEZE}}^{\mathrm{max}}\,=\,2.5\,$MHz. Although three time bins are
the optimal choice to cover the highest frequencies, we focus here on four
time bins because of their backwards compatibility with older data sets
histogrammed in 16 time bins. In turn, all data recorded at RESEDA previously
may be evenly merged into 4 time bins for direct comparisons.

##  2 Reconstruction of the MIEZE Signal

As starting point for the reconstruction of the MIEZE signal, we give the
mathematical description of the time dependent event rate $I(t)$ as recorded
by the detector. This signal may be split into a time-dependent and a time-
independent contribution ($I_{\mathrm{mean}}$). The latter describes the
intrinsic background and all of the contrast reductions such as incoherent
scattering, spin leakage, and sample dynamics. The sinusoidal time-dependence
is characterized by the amplitude $I_{0}$, the duration
$T=\frac{1}{f_{\mathrm{MIEZE}}}$, and phase shift $\phi_{0}$. These combine to
give $I(t)$ as

$I(t)\,=\,I_{\mathrm{mean}}\,+\,I_{0}\,\sin\left(\frac{2\pi}{T}t-\phi_{0}\right).$
(3)

Since the time binning of events in the detector is equal to an integration
over time of $I(t)$ in the respective interval, one may write the number of
detected events in the $k^{th}$ interval $I_{\mathrm{k}}$ as

$I_{k}\,=\,\frac{1}{T}\int_{\mathrm{\frac{T}{N}(k-1)}}^{\frac{T}{N}k}I(t)dt.$
(4)

with $k=1,2,3,...,N$ for $N$ bins. Normalizing $I_{\mathrm{k}}$ by
$I_{\mathrm{mean}}$ corresponds to the probability of a single event occurring
in the $k^{th}$ interval. This fact will be employed later to mimic the
discrete time events $I_{k}$ in a simulation.

For a subdivision into four intervals ($N\,=\,4$ cf. Fig. 2 gray shaded area
for $I_{1}$) one may rewrite (4) as follows

$\displaystyle I_{1}\,$
$\displaystyle=\,\frac{I_{\mathrm{mean}}}{4}+\frac{I_{0}}{2\pi}\left(\sin(-\phi_{0})+\cos(-\phi_{0})\right),$
(5a) $\displaystyle I_{2}\,$
$\displaystyle=\,\frac{I_{\mathrm{mean}}}{4}+\frac{I_{0}}{2\pi}\left(\cos(-\phi_{0})-\sin(-\phi_{0})\right),$
(5b) $\displaystyle I_{3}\,$
$\displaystyle=\,\frac{I_{\mathrm{mean}}}{4}+\frac{I_{0}}{2\pi}\left(-\sin(-\phi_{0})-\cos(-\phi_{0})\right),$
(5c) $\displaystyle I_{4}\,$
$\displaystyle=\,\frac{I_{\mathrm{mean}}}{4}+\frac{I_{0}}{2\pi}\left(-\cos(-\phi_{0})+\sin(-\phi_{0})\right).$
(5d)

Figure 2: A typical time-dependent sinusoidal intensity variation with phase
$\phi_{0}\,=\,\frac{\pi}{8}$ and a contrast
$C\,=\,\frac{I_{0}}{I_{\mathrm{mean}}}$ that is defined by the amplitude
$I_{0}$ and the mean value $I_{\mathrm{mean}}$. The gray shaded area $I_{1}$
normalized to $I_{\mathrm{mean}}$ is the probability of a single event being
detected in the first interval from the division of each oscillation of $I(t)$
into four equally long time bins.

Summing up neighbouring intervals and simplifying the results yields

$\displaystyle I_{1}+I_{2}\,$
$\displaystyle=\,\frac{I_{\mathrm{mean}}}{2}+\frac{I_{0}}{\pi}\cos(-\phi_{0})$
(6a) $\displaystyle I_{2}+I_{3}\,$
$\displaystyle=\,\frac{I_{\mathrm{mean}}}{2}-\frac{I_{0}}{\pi}\sin(-\phi_{0})$
(6b) $\displaystyle I_{3}+I_{4}\,$
$\displaystyle=\,\frac{I_{\mathrm{mean}}}{2}-\frac{I_{0}}{\pi}\cos(-\phi_{0})$
(6c) $\displaystyle I_{4}+I_{1}\,$
$\displaystyle=\,\frac{I_{\mathrm{mean}}}{2}+\frac{I_{0}}{\pi}\sin(-\phi_{0}).$
(6d)

Adding the next nearest neighbour intervals (I1 \+ I3 as well as I2 \+ I4)
yields only the direct component (first terms) while the phase information is
lost. This is a direct consequence of the signal’s harmonic periodicity.

Since the equations (6a) - (6d) are sums of neighbouring intervals, one may
use two independent but identical detector read outs $\frac{\pi}{2}$ phase
shifted of two time intervals each to reach the same result. In doing so one
doubles $f_{\mathrm{MIEZE}}^{\mathrm{max}}$.

In analogy to the well established quadrature detection in optical
interferometers [12], the equations (6a) - (6d) may be combined to yield the
reconstructed phase $\phi_{\mathrm{0,rec}}$

$\tan(-\phi_{\mathrm{0,rec}})\,=\,\frac{I_{\mathrm{4}}+I_{\mathrm{1}}-(I_{\mathrm{2}}+I_{\mathrm{3}})}{I_{\mathrm{1}}+I_{\mathrm{2}}-(I_{\mathrm{3}}+I_{\mathrm{4}})}.$
(7)

It is also possible to deduce the phase by subtracting equations (5a) - (5d)
from each other.

$\displaystyle\frac{I_{\mathrm{1}}-I_{\mathrm{2}}}{I_{\mathrm{1}}-I_{\mathrm{4}}}\,=\,\tan(-\phi_{\mathrm{0}})$
(8a)
$\displaystyle\frac{I_{\mathrm{1}}-I_{\mathrm{2}}}{I_{\mathrm{2}}-I_{\mathrm{3}}}\,=\,\tan(-\phi_{\mathrm{0}})$
(8b)
$\displaystyle\frac{I_{\mathrm{4}}-I_{\mathrm{3}}}{I_{\mathrm{1}}-I_{\mathrm{4}}}\,=\,\tan(-\phi_{\mathrm{0}})$
(8c)
$\displaystyle\frac{I_{\mathrm{4}}-I_{\mathrm{3}}}{I_{\mathrm{2}}-I_{\mathrm{3}}}\,=\,\tan(-\phi_{\mathrm{0}})$
(8d)

These equations show that one interval can be neglected. However, for this
approach information in form of counts is ignored within the respective
interval, thus reducing the overall statistic and accuracy. Indeed, upon
closer inspection, the average over equations (8a) - (8d) equals equation (7),
a fact confirmed by simulations.

Based on the calculations presented below the phase, the contrast may be
deduced as well. To do so, the aforementioned relation
$C=\frac{I_{0}}{I_{\mathrm{mean}}}$ is used, where $I_{0}$ is extracted by
combining either equations (6a) and (6c) or (6b) and (6d).

$\displaystyle
C_{\mathrm{1,rec}}\,=\,\frac{I_{1}+I_{2}-(I_{3}+I_{4})}{I_{1}+I_{2}+I_{3}+I_{4}}\cdot\frac{\pi}{2\cos{\phi_{\mathrm{0,rec}}}}$
(9a) $\displaystyle
C_{\mathrm{2,rec}}\,=\,\frac{I_{1}+I_{4}-(I_{2}+I_{3})}{I_{1}+I_{2}+I_{3}+I_{4}}\cdot\frac{\pi}{2\sin{\phi_{\mathrm{0,rec}}}}$
(9b)

Of course the accuracy of the evaluated contrast is strongly coupled to the
accuracy of the estimated phase and diverges at the singularities, i.e., when
$\cos{\phi_{\mathrm{0,rec}}}$ or $\sin{\phi_{\mathrm{0,rec}}}$ tend towards
zero. In order to avoid the singularities we apply equations (9a) and (9b) for
the appropriate case.

$C_{\mathrm{rec}}=\begin{cases}C_{\mathrm{1,rec}},&\quad\text{for
}\cos{\phi_{\mathrm{0,rec}}}\geq\sin{\phi_{\mathrm{0,rec}}}\\\
C_{\mathrm{2,rec}},&\quad\text{for
}\cos{\phi_{\mathrm{0,rec}}}<\sin{\phi_{\mathrm{0,rec}}}\end{cases}$ (10)

##  3 Estimation of the confidence interval

Next, we discuss how many events are necessary to determine phase and contrast
with a desired accuracy. As a first attempt, these uncertainties are estimated
using Gaussian error propagation with the relative errors
${\frac{1}{\sqrt{I_{k}}}}$. Deducing the partial derivatives is
straightforward. Less obvious is the estimate of the total errors $\Delta
I_{1},...\,I_{4}$, due to their mutual dependence. Moreover, the total errors
also depend on $C_{0}$ and $\phi_{0}$. At this point, it would be appropriate
to use the generalized Gaussian error propagation, which accounts also for the
covariance between all parameters.

To circumvent this multi-dimensional exercise, we applied simulations and
executed them for various initial phases ($\phi_{\mathrm{0}}=$$,$$,...,$$$)
and contrasts ($C_{\mathrm{0}}=0.05,0.1,...,0.95$). First 10 single events
with the desired sinusoidal distribution are generated using the pseudo-random
generator of Matlab® (the code used was published in Ref. [13]) and
histogrammed subsequently. For a given $C_{0}$ and $\phi_{0}$, the probability
to fall in a certain time bin is determined by equations (5a) - (5d).
Subsequently, the phase and contrast are calculated according to the
reconstruction method and the standard fit over 4 and 16 time bins for the
same data set. Next, new events are added to this time series and the
evaluation is repeated recursively.

The number of added events in such a series increases logarithmically. This
ensures a low computational burden over a large dynamic range of events (here
over five orders of magnitude) and keeps the evaluation equally weighted in a
logarithmic representation. Finally, the results are compared with each other
(cf. Fig. 3 (a) and (d)). It is found that the phases estimated for the 4
point fitting method (red) and the reconstruction (green) are of high accuracy
identical when there are more than 30 events.

Figure 3: For the initial parameters $C_{\mathrm{0}}=0.85$,
$\phi_{\mathrm{0}}=$$$, approximated deviations for phase $\phi$
($(a)$,$(b)$,$(c)$) and contrast $C$ ($(d)$,$(e)$,$(f)$) versus the number of
total count events ($I\,=\,I_{1}+I_{2}+...+I_{k}$) for a single time series
($(a)$ and $(d)$) and averaged over 500 time series ($(b)$ and $(e)$). The
standard deviations ($(c)$ and $(f)$) calculated respectively for the
reconstruction method (red) and generic fitting procedures using 4 (green) and
16 (blue) time bins are displayed as dots together with their fits (solid
lines). As a guide to the eye, the black lines show a power law with an
exponent of $-0.5$. To highlight the significance of the resulting phase as a
function of events, $(a)$ and $(d)$ have been purposefully cropped. Except for
Fig. 3 $(d)$ and $(e)$, the green data points coincide with the red data
points reflecting nearly identical values.

For a low number ($<30$) of events, the phase and contrast values have larger
deviations from the true values as the result of insufficient statistics. As
expected from equation (1b), both fitting methods show biased contrast
estimates. For the contrast $C_{0}\,=\,0.85$ presented in Fig. 3, the expected
deviations according to equation. (1b) are $0.64\%\cdot C_{0}\,=\,0.0054$ for
the 16 time bins (blue) and $10\%\cdot C_{0}\,=\,0.085$ for 4 time bins
(green). This allows for a comparison of the contrasts measured under
different binning conditions. It also shows that the estimates inferred from
the reconstruction method are unbiased.

It should be emphasized that in a real measurement, the contrasts detected
experimentally are normalized by a resolution function determined under the
same conditions. Therefore, the errors occurring from unbiased estimators
would cancel out if the same estimator is used for treating resolution and
measurement data.

To estimate the standard deviations of the phase and the contrast, the
simulation was repeated 500 times in order to sample a large number of values
from the random-number generator. From these data, the average phase
($\phi_{\mathrm{avg}}$) and contrast ($C_{\mathrm{avg}}$) as well as their
corresponding standard deviations ($\sigma_{\mathrm{\phi}}$ and
$\sigma_{\mathrm{C}}$) were calculated as a function of events $I$ (cf. Fig. 3
(b), (c), (e) and (f)). Again, while the average phase is estimated correctly,
the unbiased estimate for the contrast bears the expected deviations.
Interestingly though, the estimated standard deviations
$\sigma_{\mathrm{\phi}}$ and $\sigma_{\mathrm{C}}$ (for the reconstruction and
fitting procedures) decrease with the same asymptotic behavior as the total
number of events ($I\,=\,\sum_{\mathrm{k}}^{N}{I_{k}}$) increases (cf. Fig. 3
(c) and (f)). This proves that the procedure used here is asymptotically
consistent.

Moreover, the relationship between standard deviation and events, for both the
phase and contrast, is described by simple power laws

$\displaystyle\sigma_{\mathrm{\phi}}(I)\,$
$\displaystyle=\,10^{\beta_{\mathrm{\phi}}}\cdot I^{\alpha_{\mathrm{\phi}}},$
(11a) $\displaystyle\sigma_{\mathrm{C}}(I)\,$
$\displaystyle=\,10^{\beta_{\mathrm{C}}}\cdot I^{\alpha_{\mathrm{C}}}.$ (11b)

From a linear fit to the log-log plot of the estimated standard deviations
(c.f. Fig. 3 panels (c) and (f)) for more than 30 events, the power law
exponents ($\alpha_{\mathrm{\phi}}$ and $\alpha_{\mathrm{C}}$) may be inferred

$\alpha_{\mathrm{\phi}}\,=\,\alpha_{\mathrm{C}}\,\thickapprox\,-0.5.$ (12)

Figure 4: Fit parameter of the error estimation for phase
($\alpha_{\mathrm{\phi}}$ (a) and $\beta_{\mathrm{\phi}}$ (b)) and contrast
($\alpha_{\mathrm{C}}$ (d) and $\beta_{\mathrm{C}}$ (e)) as a function of the
initial contrast $C_{\mathrm{0}}$ and a fixed phase
$\phi_{\mathrm{0}}\,=\,$$$. Panels (c) and (f) show the scaling parameters
$\tilde{\beta}_{\phi}$ and $\tilde{\beta}_{C}$ using the constraint
$\alpha_{*}\,=\,-0.5$. The color code is the same as for Fig. 3. Apart from
plots $(e)$ and $(f)$, the values deduced from 4-point-fit and reconstruction
method overlap each other.

To test the generality of this power law behavior and to determine the missing
parameters ($\beta_{\mathrm{\phi}}$ and $\beta_{\mathrm{C}}$) for varying
contrasts, the simulations for $C_{\mathrm{0}}\,=\,0.05,0.1,...,0.95$ were
repeated while keeping the initial phase fixed $\phi_{\mathrm{0}}\,=\,$$$. In
Fig. 4, parameters of the reconstruction and fitting methods are deduced for a
comprehensive range of representative contrasts. For each such contrast
($C_{\mathrm{0}}$), ${\alpha_{\mathrm{\phi}}}$ and ${\alpha_{\mathrm{C}}}$
(cf. 4 (a) and (d)) remain nearly unchanged, confirming that use of a normal
distribution to approximate a Poisson distribution is well justified. Further,
the coefficients $\beta_{\mathrm{\phi}}$ and $\beta_{\mathrm{C}}$ show
quantitatively distinct dependencies on the initial parameter $C_{\mathrm{0}}$
(cf. Fig. 4 (b) and (e)). The phase was observed to follow an exponential
decay with increasing $C_{0}$

$\beta_{\phi}\,=\,\beta_{2,\phi}C_{0}^{\beta_{1,\phi}}.$ (13)

with decay constants $\beta_{2,\phi}$ and $\beta_{1,\phi}$ which vary slightly
depending on the method. The functional dependence of the contrast is less
obvious, and the parabolic fits (solid lines) in Fig. 4 (e) are not-optimal.

Figure 5: Color-map plots of parameters $\alpha_{\mathrm{\phi}}$ (a),
$\beta_{\mathrm{\phi}}$ (b), $\alpha_{\mathrm{C}}$ (d), $\beta_{\mathrm{C}}$
(e) for varying initial phases $\phi_{0}$ and contrasts $C_{0}$ using the
reconstruction. All four parameters are nearly independent of the initial
phase $\phi_{\mathrm{0}}$. While the parameters ($\alpha_{\mathrm{\phi}}$ and
$\alpha_{\mathrm{C}}$) remain nearly constant even through varying initial
contrasts, $\beta_{\mathrm{\phi}}$ decays exponentially with increasing
contrast. Analogously to Fig. 4 (e) $\beta_{\mathrm{C}}$ shows a nonphysical
trend with a maximum at $C_{0}\,\approx\,0.4$. Panels (c) and (f) show the
scaling parameters $\tilde{\beta}_{\phi}$ and $\tilde{\beta}_{C}$ using the
constraint $\alpha_{*}\,=\,-0.5$.

Since $\alpha_{*}$ and $\beta_{*}$ ($*\,=\,\phi$ or $C$.) depend on each
other, as the fits are overparameterized, $\beta_{*}$ was recalculated with
the constraint $\alpha_{*}=-0.5$. For the sake of clarity, $\beta_{*}$ is
renamed $\tilde{\beta}_{*}$ in the following if the constraint
$(\alpha_{*}\,=\,-0.5)$ is applied. The resulting fits are plotted in Figs. 4
(c) and (f). Compared to Fig. 4 (b), the exponential dependence of
$\tilde{\beta}_{\phi}$ is maintained. Furthermore, $\tilde{\beta}_{C}$ can now
be described well by a shifted half-normal distribution

$\tilde{\beta}_{C}\,=\,\beta_{2,C}\cdot
e^{-\left(\frac{C_{0}}{\beta_{1,C}}\right)^{2}}-1.$ (14)

To show that these findings hold for the relevant range of phases, this
procedure was repeated for $\phi_{0}\,=\,(0...\,$$)$ in steps of $None$. This
yields a set of curves comparable to the ones in Fig. 4, which are color-
plotted in Fig. 5, highlighting their behavior throughout the entire parameter
space. The plots confirm that the fitting parameters deduced with these
techniques are practically independent of $\phi_{0}$. One may note that due to
the periodicity of the harmonic functions, these findings are valid for all
phases.

Combining equations 11a, 11b, 13, and 14, the analytical equations for an
estimate of the standard deviation may be defined as

$\displaystyle\sigma_{\mathrm{\phi}}\,$
$\displaystyle=\,10^{\left(\beta_{\mathrm{2,\phi}}\cdot
C_{0}^{\beta_{\mathrm{1,\phi}}}\right)}\,\cdot\,\frac{1}{\sqrt{I}},$ (15a)
$\displaystyle\sigma_{\mathrm{C}}\,$
$\displaystyle=\,10^{\left(\,\beta_{\mathrm{2,C}}\,\cdot\,e^{-\left(\frac{C_{0}}{\beta_{1,C}}\right)^{2}}\,-\,1\right)}\cdot\,\frac{1}{\sqrt{I}}.$
(15b)

Typical method dependent parameter values are summarized in Table 1.

Table 1:  Parameters to deduce the standard deviations $\sigma_{\phi}$ and $\sigma_{C}$ using equations 15a and 15b, for the reconstruction and 4- and 16-point fitting methods. $method$ | $\beta_{1,\phi}$ | $\beta_{2,\phi}$ | $\beta_{1,C}$ | $\beta_{2,C}$
---|---|---|---|---
reconstruction | -0.244 | 1.383 | 2.29 | 0.60
4-point-fit | -0.244 | 1.341 | 7.90 | 0.95
16-point-fit | -0.250 | 1.383 | 9.15 | 0.95

##  4 Conclusions

We have presented an algorithm to deduce the contrast and the phase of a
sinusoidally modulated time series sampled at four data points per
oscillation. Both, contrast and phase are recovered in agreement with 16 time
bins. The reconstruction trades in a higher time resolution for less accurate
contrast. Quantitatively, this factor is better than
$\frac{\sigma_{\mathrm{C0,fit,16}}}{\sigma_{\mathrm{C0,rec}}}\,\geq\,0.9$
compared to the fitting method, but may be compensated by increased
statistics, i.e around 20% prolonged counting time. However, using the
reconstruction, there is no fitting procedure involved which significantly
reduces the required computational burden. Thus, this method may be readily
applied to a large number of detector pixels as the measurements proceed in
time.

Most importantly, this new method solves one of the main limitations
afflicting the MIEZE resolution. Using a CASCADE-type detector [11] with a
maximum time resolution of 100 ns (10 MHz), the maximum intensity modulation
frequency for 16 time channels is 625 kHz, which, at 6 Å with the current
dimensions at RESEDA yields a MIEZE (Fourier) time of $\sim$ 3 ns. In stark
juxtaposition, the resolution limit using the four-point method is $\sim$12 ns
at 6 Å or $\sim$ 100 ns at 12 Å . However, for intensity modulation
frequencies at or above 2.5 MHz, extremely flat detector surfaces are needed
to minimize phase differences in a single pixel. A 10B layer on a solid
surface instead of Kapton foil could be a possible solution. Furthermore, a
spherical detector foil shape would suppress the phase rings which occur on
flat surfaces due to variations of path lengths. The only constraint towards
achieving the highest resolution, or equivalently the highest MIEZE (Fourier)
time, remains the undesired contributions of the sample’s size and shape to
the time-of-flight of individual neutrons [14, 15].

Finally, we would like to emphasize that the error bars deduced for the
contrast using the 4 point fitting method must be treated carefully, since the
procedure of inferring the estimate is biased. Re-scaling this contrast and
its error with the damping factor of $0.9$ given by (1a), the same error
observed for the reconstruction method is maintained.

##  5 Acknowledgements

We wish to thank B. Pompe from University of Greifswald and T. Keller from
Max-Planck-Institute for Solid State Research Stuttgart for very useful
discussions and M. Klein from CDT GmbH for discussions and support of the
CASCADE detector system. Financial support through the BMBF projects
‘Longitudinale Resonante Neutronen Spin-Echo Spektroskopie mit Extremer
Energie-Auflösung’ (Förderkennzeichen 05K16W06) and ’Resonante Longitudinale
MIASANS Spin-Echo Spektroskopie an RESEDA’ (Förderkennzeichen 05K19W05) is
gratefully acknowledged.

## References

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