# Demonstration with very-cold neutrons of developments in Fourier time-of-flight

Thomas Neulinger, Karina Bernert, Peter Fierlinger, Hanno Filter, Masahiro

Hino, Tobias Jenke, Franck Lapeyre, Oliver Zimmer

► To cite this version:

Thomas Neulinger, Karina Bernert, Peter Fierlinger, Hanno Filter, Masahiro Hino, et al. Demonstration with very-cold neutrons of developments in Fourier time-of-flight. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2024, 1066, pp.169570. 10.1016/j.nima.2024.169570. hal-05441988

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Full Length Article

# Demonstration with very-cold neutrons of developments in Fourier time-of-flight

Thomas Neulinger<sup>a, ID</sup>,*, Karina Bernert<sup>b, ID</sup>, Peter Fierlinger<sup>b</sup>, Hanno Filter<sup>a</sup>, Masahiro Hino<sup>c, ID</sup>, Tobias Jenke<sup>a</sup>, Franck Lapeyre<sup>a</sup>, Oliver Zimmer<sup>a</sup>

<figure><img src="image_3.png" /></figure>

ARTICLE INFO

Keywords:

Time-of-flight

Fourier time-of-flight

Very-cold neutrons

ABSTRACT

We demonstrate recent developments in neutron Fourier time-of-flight in an experiment using very-cold neutrons. The implementation of a novel sinusoidal stator design eliminates a systematic effect associated with triangular modulation in existing Fourier choppers. Although transmission is inherently higher for triangular modulation, it is demonstrated that only about 3% additional measurement time is needed with sinusoidal modulation to achieve the same statistical uncertainty. In addition, a new neutron-based technique that does not require additional instrumentation is employed to determine the chopper phase offset.

1. Introduction

The neutron time-of-flight (TOF) method enables the determination of the distribution of neutron beam velocities parallel to the axis between a chopper and a detector. In the standard TOF approach, the chopper admits short pulses of neutrons so that the event rate at the detector varies in time roughly as the TOF spectrum plus background. In this case, the TOF resolution is proportional to the opening time of the chopper $t_{open}$, as is the rate at which measurement statistics are accumulated. To avoid frame overlap, measurements are repeated with period no shorter than $t_{max}$, the maximum TOF in the experiment. In standard TOF therefore, the duty cycle is limited to $t_{open}/t_{max}$, which can result in long measurement times when high TOF resolution is required.

In an alternative approach, known as Fourier TOF, the beam is periodically modulated at angular frequency $\omega$, resulting in an oscillating beam intensity with amplitude and phase that depend on $S(\omega)$, the Fourier transform of the TOF spectrum $s(t)$. By performing measurements over a grid of discrete frequencies, $s(t)$ can be reconstructed by inverse discrete Fourier transform. The resolution in Fourier TOF is inversely proportional to the maximum modulation frequency, which can be arbitrarily large within technical limits.

The original papers on experimental studies with Fourier TOF, published in 1969 and 1970 [1,2], implemented a chopper design that periodically modulated the beam with triangular intensity. Such modulation is associated with a systematic error, known as harmonic

distortion, in which measurements of $S(\omega)$ are contaminated by higher-frequency artifacts. One publication in 1974 reported a significant reduction in harmonic distortion using a chopper design that produced approximately sinusoidal modulation [3]. Despite this success, the prevailing opinion appears to have been that the size of the harmonic distortion is negligible in many cases of experimental interest [4,5], and that the associated reduction in beam intensity with sinusoidal modulation is undesirable [5], presumably because it should necessitate longer measurement times in order to reach the same statistical uncertainty. In any case, triangular modulation is employed at present by two of the most prominent Fourier TOF instruments, the High-Resolution Fourier Diffractometer (HRFD) [6] and the Neutron Fourier Stress Diffractometer (FSS) [7], both at the IBR-2 pulsed reactor at the Frank Laboratory of Neutron Physics in Dubna, Russia.

In a recent work by T.N. [8], the theoretical analysis of harmonic distortion is revisited, and a family of chopper stator hole shapes are introduced to produce exact sinusoidal modulation. It is predicted that, despite a 36% reduction in beam intensity compared to triangular modulation, only about 3% additional measurement time should be needed with sinusoidal modulation to achieve equal statistical uncertainty on measurements of $S(\omega)$, and consequently $s(t)$, when there is good signal-to-background. Ref. [8] also introduces a technique, taking advantage of sinusoidal modulation, by which the Fourier chopper phase offset $\phi$ can be determined by modulating the beam at low frequencies. This phase offset, which is a necessary input in the reconstruction of

<footnote><sup>a</sup> Institut Laue-Langevin, Grenoble, 38042, France</footnote>

<footnote><sup>b</sup> Technische Universität München, Garching, 80805, Germany</footnote>

<footnote><sup>c</sup> Institute for Integrated Radiation and Nuclear Science, Kyoto University, Osaka, 590-0494, Japan</footnote>

<footnote>* Corresponding author.
E-mail address: neulinger@ill.fr (T. Neulinger).</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Diagram of the experimental setup, not to scale. After exiting the guide (1), VCN pass a primary aperture (2) and are incident on a multilayer mirror (3) mounted on a stage with two rotation axes. The reflected beam, collimated by the pair of apertures (4) and (5) and modulated by the stator (6) and rotor (7) of the Fourier chopper, is incident on the detector (8). The reflected beam points along the z-axis, parallel to the surface of the optical table; the y-axis points against gravity, perpendicular to the table surface. Not shown is a helium-gas-filled tube placed between the apertures (4) and (5) to reduce VCN loss in air.</figcaption></figure>

$s(t)$ from measurements of $S(\omega)$, results from a timing offset between transmission maxima and acquisition trigger signals.

In this paper, we present the results of measurements with the sinusoidal chopper design described in Ref. [8], and confirm the prediction regarding the statistical efficiency of sinusoidal Fourier choppers. The chopper phase offset is also measured by the new technique, and is found to be in agreement with the results of optically-based measurements. The remainder of the paper is organized as follows. In Section 2, the experimental setup for the Fourier TOF measurements is described. In Section 3, the new technique to determine the phase offset is demonstrated. In Section 4, estimates of the Fourier transform $S(\omega)$ are used to reconstruct the TOF spectrum $s(t)$. In Section 5, the statistical efficiencies of the sinusoidal and triangular modulation choppers are compared. The paper is concluded in Section 6 with a summary of the main results. Several appendices containing miscellaneous details of the analysis and experiments are also included.

## 2. Experimental setup

Fourier TOF experiments [9] were carried out at the recently-upgraded PF2 very-cold neutron (VCN) beamline [10] at the Institut Laue-Langevin (ILL) in Grenoble, France. Measurements were conducted on an approximately monochromatic spectrum prepared by reflection of the primary VCN beam from a multilayer neutron mirror. A simplified diagram of the experimental setup, mounted on the optical table at the PF2 VCN beamline, is shown in Fig. 1.

The primary beam exiting the PF2 VCN guide consists of a broad range of wavelengths roughly between $\lambda = 2$ nm and 40 nm [11]. In the present experimental setup, this beam passes a wide aperture, 34 mm horizontal by 52 mm vertical opening, and is incident on a multilayer mirror. The mirror consists of 100 bilayers of nickle carbide (Ni₃C) and titanium, having nominal thicknesses of 1 nm and 9 nm, respectively, deposited on a 2 inch diameter polished silicon wafer. The mirror used in the TOF experiments was one of several fabricated in a vacuum deposition system capable of simultaneously coating multiple wafers [12]. The neutron reflectivity of a mirror produced in the same batch is shown in Fig. 2. This reflectivity was measured at the MINE2 instrument at the Japan Research Reactor-3 (JRR3). There, a monochromatic beam having peak wavelength $\lambda_{peak} = 0.88$ nm and full-width half maximum 2.7% of $\lambda_{peak}$ was registered by a detector after reflection from the mirror at glancing angles between $\theta = 0.1^{\circ}$ and $6.1^{\circ}$.

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Neutron reflectivity $R$ as a function of the modulus of the transfer wavevector $q = (4\pi/\lambda_{peak})\sin\theta$ ($\lambda_{peak} = 0.88$ nm) of a multilayer mirror fabricated in the same batch as the mirror used in the Fourier TOF experiments. Reflectivity measurements were carried out at the MINE2 instrument at JRR3; data shown in this plot extend to approximately $\theta = 4.8^{\circ}$ glancing angle. The vertical dashed lines mark two maxima, to the right of the region of total reflectivity, at $q \approx 0.54$ nm⁻¹ and its double 1.07 nm⁻¹.</figcaption></figure>

The reflection probability, or reflectivity $R$, was obtained by dividing the rate of reflected neutrons for a given $\theta$ by the rate of the direct beam. This is plotted in Fig. 2 as a function of the modulus of the transfer wavevector $q = (4\pi/\lambda_{peak})\sin\theta$. Two maxima to the right of the region of total reflectivity in Fig. 2 are indicated by vertical dashed lines at $q \approx 0.54$ nm⁻¹ and its double 1.07 nm⁻¹.

The multilayer mirror acts like a prism on the divergent, broadband incident beam, reflecting different wavelengths in different directions depending on its orientation. The collimating pair of apertures, spaced by 125 cm and with 20 mm vertical and 10 mm horizontal openings, act to select a roughly monochromatic spectrum parallel to the surface of the optical table and down its length. Not shown in Fig. 1 between the collimating apertures is a helium-gas-filled tube that serves to reduce VCN loss in air. After collimation, the reflected beam passes the stator and rotor of the chopper. Unlike in the diagram shown in Fig. 1, the rotor of the Fourier chopper has 100 holes, with the angular extent of the stator hole modified accordingly. Finally, the beam is incident on a modified CASCADE-U gas-filled detector [13–15] developed by the
<figure><img src="image_6.png" /><figcaption>Fig. 3. (a) computer model section view of the chopper shown from the perspective of the photo in (b). The rotor axis (1) is clamped by the encoder (2), which is fixed to the chopper housing, and is aligned by the bearings (3) and (4). The rotor wheel (5) is fixed to the shaft. The Fourier version of this wheel, with 100 holes, is shown from the back in (c) and is also visible in (a) through the transparent front housing. The wheel rim is assembled from 10 cards shown from the front in (d), in which the 0.1 mm thick gadolinium foil visible; the foils have been shaded yellow in (a) for increased visibility. The sinusoidal stator insert is shown from the front in (e), and is placed in the position indicated by (6) in (a). The 0.1 mm thick gadolinium foil, held in place by a clamp (not shown in the computer model), is visible in (e). In (f) is shown the exit of the chopper, corresponding to the position indicated by (7) in (a) with the back stator card removed; no back stator card was used in these experiments. At left in (g), the rotor axis is visible on the standard chopper, which has only one hole in its wheel; at right on the Fourier chopper, the axis is covered by the bellows coupling, which is attached to the 25:1 reducing gearbox.</figcaption></figure>

PanEDM collaboration [16]. The detector entrance consists of a 0.1 mm thick AlMg3 alloy foil with a downstream B₄C (96% ¹⁰B) coating of thickness 135 nm. This coating faces the gas-filled sensitive region of the detector, into which nuclear products of the capture reaction ¹⁰B(n, α)⁷Li deposit their energy and are registered. In the experiment, the front (upstream) surface of the stator is 3 mm from the front surface of the rotor, while the distance from the front of the rotor to the detector surface, i.e., the TOF path length, is L = 606(2) mm.

Mechanical details of the chopper are shown in Fig. 3. A computer model section view is shown in Fig. 3a from the perspective of the photo in Fig. 3b. The axis of the chopper is a 12 mm diameter steel shaft aligned by two bearings. The bearings are press-fit into the chopper housing, an aluminum assembly aligned by dowel pins. The rotor wheel, shown from the back in Fig. 3c, is fixed to the chopper axis using a locking-assembly, and consists of a hub and 10 cards aligned by dowel pins. Two versions of the chopper were produced: a Fourier chopper, having 100 rotor holes, and a standard chopper, having only one rotor hole. In the Fourier version, each card is made from a 1 mm thick aluminum sheet onto which is glued a 0.1 mm thick gadolinium foil; an example is shown from the front in Fig. 3d. Into each card, 10 radial holes, annular sections spanning $Δθ = 1.8°$ of inner and outer radii 120 mm and 145 mm, respectively, were cut using wire erosion. In the standard chopper, only one card was made in this manner, but having only a single hole of the same dimensions. In order to block the neutron beam for the rest of the rotor's travel, the remaining 9 cards were coated by hand with a mixture of epoxy resin and Gd₂O₃ powder (65% by weight) no less than 0.2 mm thick.

One of two exchangeable stator inserts was used to achieve either sinusoidal or triangular beam modulation. The sinusoidal stator is shown installed in the chopper housing, aligned with dowel pins, from the front in Fig. 3e and from the back in Fig. 3f. The insert consists of a recessed aluminum plate into which is clamped a 0.1 mm thick gadolinium foil. The plate and foil were cut simultaneously using wire erosion. In polar coordinates, the upper and lower contours of the sinusoidal modulation hole are described, for $|θ| ≤ Δθ/2$, by

$$ \rho_{\text{upper}}(\theta) = \sqrt{\rho_0^2 + (\rho_2^2 - \rho_0^2) \cos(\pi \frac{\theta}{\Delta\theta})}, \quad (1a) $$

$$ \rho_{\text{lower}}(\theta) = \sqrt{\rho_0^2 - (\rho_0^2 - \rho_1^2) \cos(\pi \frac{\theta}{\Delta\theta})}, \quad (1b) $$

where $Δθ = 1.8°$; this value was chosen partly for convenience, so that the rotor would have 100 holes. The minimum and maximum radii at $θ = 0$ are $ρ₁ = 125$ mm and $ρ₂ = 140$ mm, respectively, while the radius at the hole edges, $θ = ±Δθ/2$, is $ρ₀ = 132.5$ mm. The triangular modulation hole is an annular section of inner and outer radii $ρ₁$ and $ρ₂$, respectively, with the same angular extent $Δθ$. Within the 10 mm×20 mm profile of the collimated beam, the maximum width and height of both stator holes are roughly 4 mm and 15 mm, respectively. The areas of the holes are

$$ A \sim \Delta\theta(\rho_2^2 - \rho_1^2)/\pi \approx 39.8 \text{ mm}^2 \quad (2a) $$

$$ A^Δ = \Delta\theta(\rho_2^2 - \rho_1^2)/2 \approx 62.4 \text{ mm}^2. \quad (2b) $$

Here and elsewhere in this paper, tilde and triangle superscripts are used to indicate quantities associated with sinusoidal and triangular modulation, respectively. While the two stator holes have the same angular extent and minimum and maximum radii, the area of the sinusoidal stator opening is smaller by approximately 36%.

The stepper motor is connected to the rotor axis either directly by a bellows coupling, or through a 25:1 reducing gearbox and the coupling. The former configuration, without the coupling or motor, may be seen at left in Fig. 3g on the standard chopper, in which the rotor axis is visible, while the latter may be seen at right in Fig. 3g on the Fourier chopper. The gearbox is installed in order to turn the rotor at low speeds, as is necessary for the measurement of the chopper phase offset, see Section 3.

The gearbox was also used to obtain the data shown in Fig. 4. These data correspond to a series of static measurements in which the rotor was turned in small increments and the neutron rate at the detector was measured for a static position. Fig. 4a shows sinusoidal (blue points) and triangular (black points) modulation of the beam over more than one full period. Fits to the sinusoidal and triangular data were made using the functions (A.1a) and (A.1b), respectively, discussed in Appendix A. The normalized fit residuals, i.e., residual divided by error bar, are shown in Fig. 4b. Fig. 4c shows sinusoidal (brown points) and triangular (magenta points) modulation near the rate minimum at a rotor angle of $1.8°$ from the maximum. Fits to these data were made using the functions (A.4a) and (A.4b), respectively, and normalized fit residuals are shown in Fig. 4d.

The introduction of additional notation is needed to make clear the physical significance of the fit parameters for the fits in Fig. 4. In
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Triangular and sinusoidal modulation of the neutron rate: (a) modulation of the detector count rate over more than a full period, i.e., one hole of the rotor; (b) normalized residuals of the fits in (a) and values of the reduced chi-squared $\chi_r^2$; (c) separate measurements with finer position resolution showing modulation near a minimum of the rate; (d) normalized residuals of the fits in (c) and $\chi_t^2$.</figcaption></figure>

<caption>**Table 1**</caption>

Parameters of interest for the data shown in Fig. 4. The A column takes the values given in (2), while AS(0) and b are the best fit parameters. S(0) is the quotient of the second and first columns. The last row shows the weighted average estimates of b and S(0) from the two measurements.

<table><thead><tr><th></th><th>A</th><th>AS(0)</th><th>b</th><th>S(0)</th></tr></thead><tbody><tr><td>Sinusoidal</td><td>0.398 cm<sup>2</sup></td><td>24.9(1) s<sup>-1</sup></td><td>0.032(8) s<sup>-1</sup></td><td>62.6(4) s<sup>-1</sup> cm<sup>-2</sup></td></tr><tr><td>Triangular</td><td>0.624 cm<sup>2</sup></td><td>39.2(2) s<sup>-1</sup></td><td>0.03(2) s<sup>-1</sup></td><td>62.9(3) s<sup>-1</sup> cm<sup>-2</sup></td></tr><tr><td>Combined</td><td>-</td><td>-</td><td>0.03(1) s<sup>-1</sup></td><td>62.7(2) s<sup>-1</sup> cm<sup>-2</sup></td></tr></tbody></table>

general, when a chopper is not stationary, the time-dependent rate at the detector is given by

$$r(t) = b + \int_{-\infty}^{\infty} dt' A(t - t')s(t'), \qquad (3)$$

where b is the background rate (assumed to be constant), A(t) is the chopper opening area as a function of time, and s(t) is TOF spectrum, defined so that s(t)dt is the rate per area of neutrons incident on the chopper with TOF between t and t + dt; factors related to neutron loss in air and detector efficiency have been absorbed into s(t). In principle, b = b(t) is possible if, e.g., the chopper does not fully close. However, the data shown in Fig. 4 do not indicate a chopper leak. Time-dependence of the background has not been considered in the present analysis.

When the chopper is stationary at its maximum opening area A, the rate is $r(t) = b + AS(0)$, where S(ω) is the Fourier transform of s(t)

$$S(\omega) = \int_{-\infty}^{\infty} dt s(t) e^{-i\omega t}. \qquad (4)$$

Thus, the fits shown in Fig. 4a enable the count rate maximum AS(0) to be determined, while the fits shown in Fig. 4c, with finer position resolution near the rate minimum enable the background rate b to be determined. In addition, using the values of A from (2) for the sinusoidal and triangular modulation holes, S(0) can be determined. The results are collected in Table 1. In addition, from (2), one expects that $[A \sim S(0)] / [A^\Delta S(0)] = 2/\pi \approx 0.637$. Using the fit parameters given in Table 1, one obtains $[A \sim S(0)] / [A^\Delta S(0)] = 0.634(5)$, which agrees within error bars.

The final aspect of the experimental setup to be discussed is data acquisition. Data needed for the Fourier TOF measurement consist of three signals acquired in list mode and analyzed offline: one analog pulse signal from the neutron detector and two TTL signals from the encoder. The encoder, mounted on the chopper housing and clamped

to the rotor axis, has three output channels: A, B, and R. The A and B channels are 90° out of phase with one another and output 400 pulses per revolution, while the R channel outputs only a single pulse per revolution. An example of the encoder signals is shown in Fig. B.13 of Appendix B, which describes optically-based measurements of the chopper phase offset.

Only channels A and R of the encoder were used for the Fourier TOF measurements, although the choice was arbitrary and B and R could have been used equally well. The R signal serves as an absolute reference for the rotor position, while the 400 TTL pulses per revolution of A serve to divide each beam modulation into K = 4 intervals, or channels. When the beam is modulated at angular frequency ω, the expectation value and variance of the Poisson-distributed counts in the kth channels are [17]

$$c_k(\omega) = \int_{(k-1)T/K + t_{\text{offset}}(\omega)}^{kT/K + t_{\text{offset}}(\omega)} dt r(t | \omega), \qquad (5)$$

where the rate (3) has been written as $r(t | \omega)$ to emphasize the dependence on ω, and where the timing offset $t_{\text{offset}} = \phi/\omega$. Note that $c_k = c_{K+k}$. The phase offset φ arises from noncoincidence of the rate maxima and the arbitrarily chosen k = 1 encoder A pulse, this being in turn a consequence of the unknown angle between the encoder and the rotor axis to which it is clamped. In Ref. [8] it is demonstrated that the $c_k(\omega)$ are related to a quantity Z(ω) by

$$Z(\omega) = \frac{1}{A_1} \frac{\omega}{2\pi} \frac{\pi/K}{\sin(\pi/K)} \sum_{k=1}^{K} c_k(\omega) e^{-i\pi(2k-1)/K}, \qquad (6)$$

where $A_1$ is the n = 1 coefficient in the complex Fourier series representation of A(t). Z(ω) is related to the unknown quantity of interest, namely, the Fourier transform S(ω) of the TOF spectrum, by the relation

$$Z(\omega) = e^{i\phi} [S(\omega) + S_{\text{harm}}(\omega | K)], \qquad (7)$$

where φ and $S_{\text{harm}}(\omega | K)$ are also not known in principle. The latter quantity is the harmonic distortion, being nonzero for triangular modulation and zero for sinusoidal modulation; see Appendix C and Ref. [8].

### 3. Measurement of the phase offset

In the absence of harmonic distortion, i.e., for sinusoidal modulation, (7) becomes $Z(\omega) = e^{i\phi}S(\omega)$. Before S(ω) can be measured however, it remains to determine the phase offset φ. To this end,
the method described in Ref. [8] has been employed, and is briefly reviewed here. Expanding the term $e^{-i\omega t}$ in (4), one obtains

$$S(\omega) = \sum_{n=0}^{\infty} \frac{(-i\omega)^n}{n!} a_n, \qquad (8)$$

where

$$a_n = \int_{-\infty}^{\infty} dt t^n s(t). \qquad (9)$$

For sufficiently low frequencies, it should be the case that only a finite number of terms in (8) are needed to approximate $S(\omega)$. Supposing that this is $N$ terms, one forms the residuals for $J$ measurements of $Z(\omega)$ at low frequencies $\omega_j$

$$\varepsilon(\omega_j | \varphi, \underline{\alpha}) = e^{-i\varphi} \hat{Z}(\omega_j) - \sum_{n=0}^{N-1} \frac{(-i\omega)^n}{n!} a_n, \qquad (10)$$

where, as will be used elsewhere in this paper, the hat denotes an estimator. One then seeks to find the values of the fit parameters $\varphi$ and $\underline{\alpha} = \{\alpha_0, \alpha_1, \dots, \alpha_{N-1}\}$ such that the quantity

$$Q(\varphi, \underline{\alpha}) = \sum_{j=1}^{J} \frac{\varepsilon^*(\omega_j | \varphi, \underline{\alpha}) \varepsilon(\omega_j | \varphi, \underline{\alpha})}{\mathcal{V}(|\hat{Z}(\omega_j)|)} \qquad (11)$$

is minimized, where $\mathcal{V}(|\hat{Z}(\omega_j)|)$ is the estimate of the variance of $|\hat{Z}(\omega_j)|$. These values of $\varphi$ and $\underline{\alpha}$ then give the estimates $\hat{\varphi}$ and $\hat{\underline{\alpha}}$, respectively. The algorithm for the exact calculation of $\hat{\varphi}$ and $\hat{\underline{\alpha}}$ is given in Appendix D of Ref. [8].

With the reducing gearbox installed between the motor and rotor, the beam was sinusoidally modulated for 10 min at a time at the nominal frequencies $\omega_n/(2\pi) = n \times (1 \text{ Hz})$ for integers $n$ from 1 to 5, with values drawn in random order without replacement. Once all frequencies were exhausted, the random selection process was repeated. No significant drift over time in the measured data was observed. This procedure was performed once using triangular modulation, and in two instances with sinusoidal modulation. Applying the F-test to experimental data, see the discussion in Section 4 of Ref. [8], it was determined that only $N = 2$ was necessary, giving the $N + 1 = 3$ fit parameters $\varphi, \alpha_0$, and $\alpha_1$.

The results of the fits, which have also been performed on the triangular modulation data, although the method is not strictly applicable there, are listed in Table 2. The penultimate row lists the combined results for the sinusoidal modulation data, and the last row gives the result for $\hat{\varphi}$ obtained by optically-based measurements; see Appendix B for details. Whereas one assumes, e.g., in an optically-based measurement that the stator/rotor pair blocks the light in the same way as it does neutrons, that the digitization electronics behave comparably to the experimental data acquisition system, etc., no such assumptions are necessary in the neutron-based method, and therefore no additional systematic effects are introduced. Thus, the neutron-based measurement of the phase offset is to be favored.

Some disagreement in Table 2 that exceeds error bars may be seen between the values of the $\hat{a}_0$ and $\hat{a}_1$ from the two sinusoidal measurements. This may be due to a lack of repeatability in the positioning of the chopper on the optical table. The second sinusoidal modulation and the triangular modulation measurements were performed after the Fourier chopper had been removed and then reinstalled on the optical table.

As an example of the fitting process, data for the second sinusoidal modulation measurement are shown in Fig. 5. Histograms of the modulation frequencies obtained from the encoder A pulses are shown in Fig. 5a, in which the average frequencies have been subtracted to center the distributions on zero; the legend shows the average values and the standard deviations. The real and imaginary parts of the fit and data are shown in Figs. 5b and 5c. In these two plots, the data have been artificially spread horizontally over 0.2 Hz to enable better visualization of the error bars.

<table><caption>**Table 2**
Estimates of the phase offset, along with the fit parameters $\hat{a}_0$ and $\hat{a}_1$ characterizing the TOF spectrum. The last column gives the reduced chi-square statistic for the fits. The combined estimate of $\varphi$ using the sinusoidal measurement results is given, along with the result of the optically-based measurement described in Appendix B.</caption><thead><tr><th></th><th>φ</th><th>ā₀ = S(0)</th><th>ā₁</th><th>χ²</th></tr></thead><tbody><tr><td>Triangular modulation</td><td>-91.6(3)°</td><td>59.4(1) s⁻¹ cm⁻²</td><td>0.462(6) s⁻¹ cm⁻²</td><td>1.660</td></tr><tr><td>Sinusoidal modulation 1</td><td>-89.9(6)°</td><td>62.6(3) s⁻¹ cm⁻²</td><td>0.47(1) s⁻¹ cm⁻²</td><td>2.075</td></tr><tr><td>Sinusoidal modulation 2</td><td>-90.6(3)°</td><td>61.0(2) s⁻¹ cm⁻²</td><td>0.431(7) s⁻¹ cm⁻²</td><td>1.910</td></tr><tr><td>Combined sinusoidal</td><td>-90.4(3)°</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Optical measurement</td><td>-89(1)°</td><td>—</td><td>—</td><td>—</td></tr></tbody></table>

The negative values of the phase offset given in Table 2 can be seen to agree with the character of the $Z(\omega)$ data in Fig. 5 in the following way. As discussed in Ref. [8], because $s(t)$ is real, its Fourier transform satisfies $S(-\omega) = S^*(\omega)$, i.e., its real and imaginary parts $S_1(\omega)$ and $S_2(\omega)$ are even and odd functions, respectively. Also, because the $a_n$ are all positive, one can see from (8) that, to second order at low frequency, $S_2(\omega)$ is linear with negative slope, while $S_1(\omega)$ is positive and falls off parabolically. In Fig. 5, it is $Z_1(\omega)$ that has apparently negative slope, passing almost through zero, while $Z_2(\omega)$ is roughly constant and negative. However, one recovers the expected behavior of $S(\omega)$ by multiplying $Z(\omega)$ by $e^{i\pi/2} = i$. Thus $\varphi \approx -\pi/2$.

Only the sinusoidal modulation data should be used to determine the phase offset, and potentially to estimate $S(0)$. This is because $Z(\omega) \neq e^{i\varphi}S(\omega)$ in the case of triangular modulation; the influence of harmonic distortion on experimental data is discussed in Appendix C. In the previous section, it was found that $\hat{S}(0) = 62.7(2) s^{-1} \text{ cm}^{-2}$; see Table 1. Combining this with the two measurements from the sinusoidal modulation data in Table 2, the weighted average estimate of $S(0)$ is given by $\hat{S}(0) = 61.7(4) s^{-1} \text{ cm}^{-2}$, where the uncertainty has been expanded to give a reduced chi-squared value of unity.

## 4. Measurement of the TOF spectrum

To determine the TOF spectrum by the Fourier method, $Z(\omega)$ was measured using sinusoidal and triangular modulation at the nominal frequencies $\omega_n/(2\pi) = n \times (0.1 \text{ kHz})$ for integers $n = 1$ to 20; see Appendix D for frequency stability data. Modulations at a given frequency were performed for 1 min, with values of $n$ drawn in random order without replacement in the same manner as for the low-frequency modulation measurements described in the previous section. No significant drift over time in the measured data was observed. The reducing gearbox was not used in these measurements. For both triangular and sinusoidal modulation, the estimate of $S(\omega)$ at a single modulation frequency is

$$\hat{S}(\omega) = e^{-i\hat{\varphi}} \hat{Z}(\omega). \qquad (12)$$

Thus, $|\hat{S}(\omega)| = |\hat{Z}(\omega)|$. Fig. 6 shows the estimates of $|S(\omega)|$, where data corresponding to the same nominal modulation frequency have been combined, together with the value of $\hat{S}(0) = 61.7(4) s^{-1} \text{ cm}^{-2}$ from the previous section.

The set of $\hat{S}(\omega)$ at multiple frequencies is used to estimate the TOF spectrum by discrete approximation of the inverse Fourier transform

$$s(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} d\omega S(\omega) e^{i\omega t}. \qquad (13)$$

Thus

$$\hat{s}(t) = \frac{\Delta\omega}{2\pi} \sum_{n=-N}^{N} [\hat{S}_1(\omega_n) \cos(\hat{\omega}_n t) - \hat{S}_2(\omega_n) \sin(\hat{\omega}_n t)], \qquad (14)$$

where, in the present case, $N = 20$ and $\Delta\omega/(2\pi) = 100 \text{ Hz}$. The $\hat{\omega}_n/(2\pi)$ are listed in Table D.4. Only measurements for one sense of rotation are needed, as the values of $S_1(\omega)$ and $S_2(\omega)$ can be constructed at negative frequencies using $S(-\omega) = S^*(\omega)$. In the ideal case that $\hat{\omega}_n = n\Delta\omega$, (14)
<figure><img src="image_8.png" /><figcaption>Fig. 5. (a) frequency histograms for the low-frequency sinusoidal modulation data; distributions have been centered on their average values, shown in the legend along with the standard deviations; (b) and (c) show $\hat{Z}_1(\omega)$ and $\hat{Z}_2(\omega)$, respectively, along with the real and imaginary parts of the fit.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Estimates of $|S(\omega)|$. The sinusoidal and triangular modulation data points have been shifted horizontally by $\pm 5$ Hz to enable error bars to be seen. The average modulation frequencies are listed in Table D.4 in Appendix D.</figcaption></figure>

is periodic in $t$ with periodicity $2\pi/\Delta\omega$. On physical grounds therefore, one should only consider (14) to be valid for $0 \le t < 2\pi/\Delta\omega = 10$ ms. This same interval should be used in the present situation as the $\hat{\omega}_n$ differ only very slightly from $n\Delta\omega$.

Fig. 7 shows the resulting estimates of $s(t)$ from the sinusoidal and triangular modulation measurements. In Ref. [8], as elsewhere, e.g., Ref. [17], it is shown that, provided that $\Delta\omega$ is small enough that $2\pi/\Delta\omega$ is greater than the longest present in the experiment $t_{max}$, the expectation value of the TOF spectrum estimate (14) is related to the true TOF spectrum by

$$ \mathbb{E}[\hat{s}(t)] \approx \frac{\Delta\omega}{2\pi} \int dt' w(t' - t) s(t'), \quad (15) $$

where

$$ w(t) = \frac{\sin \left[ \left( N + \frac{1}{2} \right) \Delta\omega t \right]}{\frac{1}{2} \Delta\omega t}. \quad (16) $$

That is, the expectation value of $\hat{s}(t)$ is the convolution of the true TOF spectrum and $w(t)$. This relationship is a consequence of the fact that

(14) is an approximation to the exact inverse Fourier transform (13) in which the integration range is truncated and discretized. The function $w(t)$ is shown in the inset of Fig. 7 for the present case of $N = 20$ and $\Delta\omega/(2\pi) = 100$ Hz; it is this function that is responsible for the “wiggles” in the reconstructed TOF spectrum.

Because of the modest engineering specifications of the Fourier chopper constructed for these experiments, the maximum modulation frequency was limited to $N\Delta\omega/(2\pi) = 2$ kHz. However, in more advanced instruments, see, e.g., Ref. [6], modulation frequencies in excess of 100 kHz are possible. For such a maximum modulation frequency, $w(t)$ would be narrower by a factor of 50. Furthermore, it is also possible to introduce various windowing function into (14), which eliminate the wiggles at the expense of broadening $w(t)$; see, e.g., Ref. [17].

Appendix E of Ref. [8] also discusses the sampling theorem, which implies that

$$ s(t) \approx \sum_{n} s(n\Delta t) \frac{\sin[\Omega(t - n\Delta t)]}{\Omega(t - n\Delta t)}, \quad (17) $$

where $\Delta t = \pi/\Omega$ and $\Omega$ is an angular frequency large enough that $S(\Omega) \approx 0$. This formula states that the value of $s(t)$ for any TOF off the grid $n\Delta t$ can be obtained by interpolating points on the grid via (17). In the present context, $\Omega/(2\pi) = N\Delta\omega/(2\pi) = 2$ kHz, where $S(\omega)$ is small compared to its value at the origin, and $\Delta t = 0.25$ ms. Eq. (17) has been computed over continuous $t$ using (14) to evaluate the $\hat{s}(n\Delta t)$. The relative difference between this and $\hat{s}(t)$ when calculated directly via (14) over continuous values of $t$ is essentially negligible, being at most about 1% within the clearly-nonzero region $6.25$ ms $\le t \le 7.5$ ms. This implies that $N\Delta\omega$, the largest modulation frequency, is sufficiently large to approximately exceed the bandwidth of $S(\omega)$; see Ref. [8] for more details. It is interesting to note that, unlike (14), (17) is not periodic, and gives a valid estimate of the TOF spectrum over the entire domain.

To make connection to the measured reflectivity of the multilayer mirror, it is instructive to transform the TOF spectrum into a wavelength spectrum. This can be accomplished using the mapping between TOF and wavelength

$$ t = \frac{mL}{h} \lambda, \quad (18) $$

where $L$ is the flight path length, $h$ is Planck's constant, and $m$ is the neutron mass. The transformation is thus

$$ s(\lambda) = s\left(\frac{mL}{h}\lambda\right) \frac{mL}{h}, \quad (19) $$

where $s(\lambda)$ and $s(t)$ are understood to be the wavelength and TOF spectra, respectively, depending on their arguments; see, e.g., Section 3.1
<figure><img src="image_10.png" /><figcaption>Fig. 7. Estimates of the TOF spectrum from sinusoidal and triangular modulation measurements using (14). The solid lines show (14) evaluated between $t = 0$ and $t = 2\pi/\Delta\omega = 10$ ms, the latter value being the largest TOF accessible; see the text. Shaded lines indicate 1-sigma uncertainties. Point data correspond to (14) evaluated on the grid $t = n\Delta t$ for integer $n = 0, 1, \dots, 2N$, where $\Delta t = \pi/(N\Delta\omega) = 0.25$ ms. The inset shows $w(t)$, for $\Delta\omega/(2\pi) = 100$ Hz and $N = 20$, on the same horizontal scale as the main plot. This function is responsible for the “wiggles” in the reconstructed TOF spectrum.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. The wavelength spectrum $s(\lambda)$ (black solid line, left vertical axis) with the reflectivity curve $R(\lambda)$ (red dashed line, right vertical axis) overlaid. This $s(\lambda)$ was computed from the sinusoidal modulation TOF spectrum shown in Fig. 7. $R(\lambda)$ has been computed from $R(q)$ in Fig. 2 using $\theta = 11.3^\circ$ in $\lambda = (4\pi/q)\sin\theta$. Unlike in $R(\lambda)$, a second, smaller peak does not appear in $s(\lambda)$ because the incident VCN beam has very little spectral weight below $\lambda = 2.5$ nm.</figcaption></figure>

in Ref. [18] for a general discussion of such transformations. Eq. (19) has been used to plot $s(\lambda)$ in Fig. 8. Additionally, the reflectivity curve versus the modulus of the transfer wavevector $q$ in Fig. 2 has been converted to reflectivity versus wavelength using $\lambda = (4\pi/q)\sin\theta$, and has been overlaid in Fig. 8, where $\theta = 11.3^\circ$ has been adjusted by hand. This angle agrees reasonably well with $10.8(3)^\circ$, the glancing angle of the reflected beam to the multilayer mirror as determined by measurements of the setup geometry. While the similar shape of the measured wavelength spectrum $s(\lambda)$ and reflectivity curve $R(\lambda)$ is compelling, it is not strictly a valid comparison, as $s(\lambda)$ is the product of $R(\lambda)$ and the unknown wavelength-dependent spectrum incident on the mirror. However, the PF2 VCN beam is relatively devoid of sharp features and has very little weight near $\lambda = 2$ nm. The former property implies that, at least for a relatively narrow range of wavelengths near $4.5$ nm, the incident spectrum is approximately constant, meaning that $R(\lambda)$ and $s(\lambda)$ differ only by a multiplicative constant. The latter property implies that the smaller peak in $R(\lambda)$ will not contribute to $s(\lambda)$.

## 5. Measurement efficiency

In Ref. [8] it is shown that, for $K > 3$ channels, the counting-statistics variance of $|\hat{S}(\omega)|$ for a large number of modulations $M$ is

$$ \mathcal{V}_{\text{stat.}}(|\hat{S}(\omega)|) \approx \frac{1}{M} \frac{\omega}{2\pi} \left( \frac{1}{A_1} \frac{\pi/K}{\sin(\pi/K)} \right)^2 \frac{1}{2} [b + A_0 S(0)], \quad (20) $$

where $A_0$ and $A_1$ are coefficients in the Fourier series representation of $A(t)$. For triangular modulation, it is additionally required that $K$ be even. Also in Ref. [8] it is shown that the variance may be estimated

from experimental data by

$$ \hat{\mathcal{V}}_{\text{stat.}}(|\hat{S}(\omega)|) \approx \frac{1}{M} \left( \frac{\hat{\omega}}{2\pi} \frac{1}{A_1} \frac{\pi/K}{\sin(\pi/K)} \right)^2 \frac{\sum_{ijk} \hat{c}_i \hat{c}_j \hat{c}_k \cos\left(\frac{2\pi(j-i)}{K}\right) \cos\left(\frac{2\pi(k-i)}{K}\right)}{\sum_{ij} \hat{c}_i \hat{c}_j \cos\left(\frac{2\pi(i-j)}{K}\right)}, \quad (21) $$

where the summation indices $i, j$, and $k$ run over all channels from 1 to $K$, and where the estimates of the expectation value of the channel counts are

$$ \hat{c}_k(\omega) = \frac{1}{M} \sum_{m=1}^{M} C_k^{(m)}(\omega), \quad (22) $$

i.e., the average over $M$ modulations of the $m$th measured counts $C_k^{(m)}(\omega)$.

Figs. 9a and 9b show, as colored lines for sinusoidal and triangular data, estimates of the variance computed via (21) for several modulations frequencies. Also shown are curves of (20), plotted as black dashed lines using $\hat{S}(0) = 61.7(4) \text{ s}^{-1} \text{ cm}^{-2}$ from Section 3, the combined estimate of the background rate $b$ from Table 1, and the Fourier series terms $A_0^\Delta$ and $A_1^\Delta$ known from the chopper geometry; see Ref. [8]. The agreement between (20) and (21) is very reasonable, especially at large $M$.

Another way to compare (20) and (21) is as follows. In Ref. [8] it is shown that (20) leads to the prediction that, on average, the relative number of modulations needed for sinusoidal and triangular measurements to achieve the same statistical uncertainty is

$$ \frac{M}{M^{\Delta}} = \left(\frac{4}{\pi}\right)^2 \frac{1 + \text{SBR}/\pi}{1 + \text{SBR}/2}, \quad (23) $$
<figure><img src="image_12.png" /><figcaption>Fig. 9. The variance of |$\hat{S}(\omega)$| for various modulation frequencies $\omega/(2\pi)$ computed from (20) (black dashed lines) and from (21) (colored lines). Each colored line represents about 10 min worth of measurement time. Note that the agreement between the colored and dashed lines improves at high modulation number $M$, where the approximations (20) and (21) are most accurate.</figcaption></figure>

where the signal-to-background ratio is

$$SBR = \frac{A^{\Delta}S(0)}{b}, \qquad (24)$$

i.e., the total rate of neutrons integrated over the stator hole area divided by the background rate. Using the estimates $\hat{S}(0)$ and $b$ along with $A^{\Delta}$ from (2b), the estimate of the signal-to-background ratio is $1.2(4) \times 10^3$. Thus, $M^{\sim}$ exceeds $M^{\Delta}$ in (23) by approximately 3.3%, where the uncertainty on this percentage is omitted on account of it being very small, which is in turn a consequence of the slope of (23) being very small for large SBR; see Figure 2 in Ref. [8] for a plot of (23). This implies that only about 3% additional measurement time is needed with sinusoidal modulation to achieve the same statistical uncertainty on a measurement of $S(\omega)$ as when using triangular modulation. This can be checked in the following way using the experimental data shown in Fig. 9. From the estimates of the variance computed via (21), one takes the last value of the variance $\hat{V}_{\text{last}}^{\sim}$ for the sinusoidal modulation data, corresponding to the number of modulations $M_{\text{last}}^{\sim}$, and finds $M^{\Delta}$ such that the corresponding value of $\hat{V}^{\Delta}$ for the triangular modulation data is the same, or very nearly the same, as $\hat{V}_{\text{last}}^{\sim}$. One then computes $M_{\text{last}}^{\sim}/M^{\Delta}$. This quotient is shown in Fig. 10 for all modulation frequencies. The average value of $M_{\text{last}}^{\sim}$ exceeds $M^{\Delta}$ by 4.8% with a standard deviation of 2.3%, which agrees within uncertainty with the predicted value 3.3% based on the estimated signal-to-background ratio.

<figure><img src="image_13.png" /><figcaption>Fig. 10. The quotient $M_{\text{last}}^{\sim}/M^{\Delta}$ for all modulation frequencies, shown as black points; see the text for an explanation. The average 1.048 and standard deviation 0.023 are indicated by the blue line and shaded region, respectively. The average agrees within standard deviation with the predicted value of 1.033, indicated by the dashed green line.</figcaption></figure>

## 6. Conclusion

In this paper, we have reported on the experimental realization of developments in Fourier TOF, these being mainly sinusoidal modulation of the beam and a novel method to measure the chopper phase offset. Measurements were carried out at the PF2 VCN beamline on a monochromatic spectrum prepared by reflection from a multilayer mirror. The shape of the wavelength spectrum, converted from the measured TOF spectrum, is found to be consistent with the mirror reflectivity as measured independently at the MINE2 instrument at JRR3. The estimate of the phase offset, determined by the new technique which takes advantage of sinusoidal beam modulation, is found to be in agreement with the result of optically-based measurements. As expected, evidence of harmonic distortion is most apparent in the triangular modulation data at low frequency.

In agreement with the predictions of Ref. [8], we have demonstrated that the statistical efficiency of sinusoidal modulation is very nearly the same as that of triangular modulation provided that signal-to-background is good. The quantitative meaning of “good” can be understood by plotting (23); see Figure 2 in Ref. [8]. In the present

experiment, where SBR ~ $10^3$, the limiting value of (23) for good signal-to-background has more or less been reached. However, even for relatively high background, e.g., SBR = 10, (23) gives only $M^{\sim}/M^{\Delta} \approx 1.13$.

As a final remark, it is perhaps useful to briefly summarize several advantages and disadvantages of Fourier TOF compared to standard TOF, which will be abbreviated here as FTOF and STOF, respectively, when equal TOF resolution is desired. A more detailed discussion is given in Appendix F of Ref. [8]. Let us first consider the case of relatively low background. Roughly speaking, whereas in STOF, the variance on a measurement of $s(t)$ is proportional to $s(t)$, the variance in FTOF is uniform in $t$ and is proportional to $S(0)$, the integrated rate over all TOF. Consequently, for equal-time measurements, FTOF is more efficient when the magnitude of large features in the TOF spectrum, e.g., peaks, are of primary interest. This is especially true in neutron diffraction measurements when the TOF spectrum consists of sharp, isolated peaks. However, as the variance is “shared” equally by all points in FTOF, if one is rather interested in measuring a relatively
small feature of the TOF spectrum, STOF is more efficient. These considerations are modified in the presence of higher background. In general, the variance increases in STOF as the chopper duty cycle decreases. Therefore, in situations with relatively high background and where high TOF resolution is desired, FTOF can be more efficient than STOF regardless of the TOF spectrum under consideration.

The authors hope that their work will promote and renew interest in the Fourier TOF technique, leading to its implementation in experiments that can profit from its unique advantages. The authors also hope that the present work will contribute to the continued development of novel chopper-based spectroscopy techniques, such as those already demonstrated with ultra-cold neutrons [19–23] and cold neutrons [24–26].

### CRediT authorship contribution statement

**Thomas Neulinger:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Karina Bernert:** Writing – review & editing, Investigation, Conceptualization.
**Peter Fierlinger:** Writing – review & editing, Resources, Project administration, Funding acquisition, Conceptualization.
**Hanno Filter:** Writing – review & editing, Software, Resources, Project administration, Investigation, Data curation.
**Masahiro Hino:** Writing – review & editing, Validation, Resources, Methodology, Investigation, Formal analysis, Data curation.
**Tobias Jenke:** Writing – review & editing, Resources.
**Franck Lapeyre:** Resources, Project administration, Methodology.
**Oliver Zimmer:** Writing – review & editing, Writing – original draft, Supervision, Resources, Project administration, Methodology, Investigation, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

Data will be made available on request.

### Acknowledgments

The authors would especially like to thank the physics machine shop at the Technical University of Munich for the rapid and quality fabrication of the chopper housing, and Thomas Brenner at the ILL for crucial last-minute production of mounting parts for the motor gearbox. In addition, thanks are due to the Instrument Control Service at the ILL for technical assistance with the encoder. Finally, the authors wish to thank the staff at the ILL who supported the development of the choppers. This work was supported by the European Union’s Horizon 2020 research and innovation program under the EURIZON project, grant agreement No. 871072.

## Appendix A. Fits to triangular and sinusoidal beam modulations

In Fig. 4a, the fits to the sinusoidal and triangular rate modulation data over a full period use the functions

$$f_{\text{sinusoidal}}(x|\underline{p}) = p_1 \left[ \frac{1}{2} + \frac{1}{2} \cos \left( \pi \frac{x - p_2}{p_3} \right) \right] + p_4, \quad (A.1a)$$

$$f_{\text{triangular}}(x|\underline{p}) = p_1 \left[ \frac{1}{2} + \frac{1}{2} \text{tri} \left( \pi \frac{x - p_2}{p_3} \right) \right] + p_4, \quad (A.1b)$$

in which $x$ is the independent variable, $p_1, p_2, p_3$, and $p_4$ are four fit parameters, and

$$\text{tri}(x) = \left| 2 - 4 \left( \frac{x}{2\pi} - \left\lfloor \frac{x}{2\pi} \right\rfloor \right) \right| - 1 \quad (A.2)$$

<figure><img src="image_14.png" /><figcaption>Fig. A.11. The functions $f_{\text{sinusoidal}}(x|p_1 = 1, p_2 = 0, p_3 = 1, p_4 = 0)$ and $f_{\text{triangular}}(x|p_1 = 1, p_2 = 0, p_3 = 1, p_4 = 0)$, plotted along with $\cos(\pi x)$ and $\text{tri}(\pi x)$.</figcaption></figure>

is a “triangle wave” of period $2\pi$ spanning $-1 \le \text{tri}(x) \le 1$; the floor function $\lfloor x \rfloor$ returns the greatest integer that is less than or equal to $x$. For the purposes of illustration, the functions $f_{\text{sinusoidal}}(x|p_1 = 1, p_2 = 0, p_3 = 1, p_4 = 0)$ and $f_{\text{triangular}}(x|p_1 = 1, p_2 = 0, p_3 = 1, p_4 = 0)$ are plotted in Fig. A.11 along with $\cos(\pi x)$ and $\text{tri}(\pi x)$.

In the measurements shown in Fig. 4, the stepper motor was driven using an angular resolution of 20,000 microsteps (μstep) per revolution, and was connected to the chopper rotor through a 25:1 reducing gearbox. Thus, the angular positioning resolution of the combined system is

$$\frac{360 \text{ degrees}}{25 \times 20,000 \text{ μstep}} = 7.2 \times 10^{-4} \text{ degrees per μstep}, \quad (A.3)$$

or 5000 microsteps for every $3.6^\circ$, which is the angular span of one modulation. The microstep position values were converted to angles via (A.3) and used as the independent variable in fitting the measured rate data with either (A.1a) or (A.1b). Thus, the fit parameter $p_1$ corresponds to total neutron rate $AS(0)$, $p_2$ corresponds to an arbitrary angular offset, $p_3$ corresponds to $\Delta\theta$, and $p_4$ corresponds to the background rate $b$.

In Fig. 4c, the data near the rate minimum were fit using the models

$$g_{\text{sinusoidal}}(x|\underline{p}) = p_1(x - p_2)^2 + p_3, \quad (A.4a)$$

$$g_{\text{triangular}}(x|\underline{p}) = p_1 |x - p_2| + p_3. \quad (A.4b)$$

Eqs. (A.4a) and (A.4b) are the leading-order expansions to (A.1a) and (A.1b), respectively, and each reduce the number of fit parameters by one. The fit parameter $p_1$ is related to the slopes of these models, $p_2$ corresponds to an arbitrary angular offset, and $p_3$, the fit parameter of interest, corresponds to the background rate $b$.

## Appendix B. Optical measurement of the phase offset

The setup for the measurement of the phase offset by optical methods is shown in Fig. B.12. A laser was sent through an expanding lens and then made parallel to the beam axis by another lens. This beam was sent through the stator as the motor turned the rotor through the 25:1 reducing gearbox. The chopped beam was then focused by a final lens onto a photodiode, the output of which was digitized, along with all signals available from the encoder: A, B, and R; see Section 2. An example of the digitized signals is shown in Fig. B.13. In one measurement, the signals A, B, R, and one modulated light signal
<figure><img src="image_15.png" /><figcaption>**Fig. B.12.** Setup for the optical measurement of the phase offset $\phi$: (1) laser, (2) expanding lens, (3) parallelizing lens, (4) encoder, (5) stator, (6) motor attached to reducing gearbox, (7) focusing lens, and (8) photodiode. The path of the laser is indicated by the dashed red line.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. B.13.** Photodiode voltage signals from separate sinusoidal (red) and triangular (green) modulation measurements. The digitized encoder signals A, B, and R are offset vertically and rescaled for increased visibility; their values do not correspond to the left axis.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. C.14.** Linear fits to $|\hat{Z}(\omega)|$ from the two sinusoidal and one triangular modulation measurements. The fit 1-sigma uncertainty is indicated by the shaded region. For the sinusoidal modulation data, the slopes of the fits are consistent with zero, whereas for the triangular modulation data, the slope is positive.</figcaption></figure>

from the photodiode are recorded, this last being either sinusoidal (red) or triangular (green); two photodiode signals are overlaid in Fig. B.13 for ease of comparison. The A, B, and R TTL logic signals are offset vertically and rescaled for increased visibility, with relative high being logic 1 and relative low being logic zero; their values do not correspond to the left axis. Ambient light is responsible for the nonzero minima of the photodiode signals.

In the Fourier TOF measurements described in the main body of the paper, the zero of time for data analysis was chosen to be at the first rising edge of the encoder A channel after the falling edge of the encoder R channel. It may be seen in Fig. B.13 however that the chopper is fully open at about 25 ms, and thus the zero of time is “too early”, corresponding to a negative phase offset. To determine this phase offset from the digitized data, we use the estimate

$$ \hat{\phi} = 2\pi \frac{\hat{t}_1 - \hat{t}_0}{\hat{t}_2 - \hat{t}_1}, \qquad (B.1) $$

where $t_1$ is time of the first rising edge of the encoder A channel after the falling R edge, $t_2$ is the rising edge of A four periods after this, and $t_0$ is the time of the first maximum of the optical signal after $t_1$; as elsewhere in this paper, a hat denotes an estimate.

The data shown in Fig. B.13 correspond to the motor spinning at a nominal speed of 2.5 rotations per second. However, as the 25:1 gearbox was used, this corresponds to a nominal rotation speed of the rotor of only 0.1 rotations per second. As there are 100 slits on the rotor, the light signal is therefore modulated nominally at 10 Hz (period 0.1 s), and since there are four encoder pulses per modulation period, a

<caption>**Table C.3**</caption>

Best fit values for the linear fits to the data in Fig. C.14; both slopes for the sinusoidal modulation data are consistent with zero, whereas the slope for the triangular modulation data is nonzero and positive.

<table><thead><tr><th></th><th>Intercept</th><th>Slope</th></tr></thead><tbody><tr><td>Sin. mod. 1</td><td>62.8(7) s<sup>-1</sup> cm<sup>-2</sup></td><td>0.17(19) cm<sup>-2</sup></td></tr><tr><td>Sin. mod. 2</td><td>62.0(4) s<sup>-1</sup> cm<sup>-2</sup></td><td>-0.07(11) cm<sup>-2</sup></td></tr><tr><td>Tri. mod.</td><td>59.5(3) s<sup>-1</sup> cm<sup>-2</sup></td><td>0.24(9) cm<sup>-2</sup></td></tr></tbody></table>

frequency of nominally 40 Hz (period 25 ms) for the encoder channels. The locations of the optical signal peaks $t_0$ were found by fitting using the functions (A.1a) and (A.1b) for the sinusoidal and triangular modulation data, respectively. This procedure was repeated multiple times and $\hat{\phi}$ was calculated taking into account the fit uncertainty on $\hat{t}_0$ in addition to the digitization uncertainty and the frequency stability uncertainty, which account for sources of uncertainty on $\hat{t}_1$ and $\hat{t}_2$. The resulting phase offset estimates were averaged, and a combined estimate of the phase uncertainty was found to be $\hat{\phi} = -89(1)^{\circ}$.

## Appendix C. Harmonic distortion at low frequency

In Ref. [8] it is shown that the harmonic distortion associated with triangular modulation is

$$ S_{\text{harm.}}^{\Delta}(\omega | K) = \sum_{q \in Q_K} \frac{e^{iqK\phi}}{(qK + 1)^3} S[(qK + 1)\omega], \qquad (C.1) $$
<figure><img src="image_18.png" /><figcaption>Fig. D.15. Frequency histograms for the (a) sinusoidal and (b) triangular modulation data in Table D.4.</figcaption></figure>

<caption>Table D.4</caption>

Frequency stability data for sinusoidal and triangular measurements: *f* is the nominal modulation frequency; *Δ* is the difference between the average frequency and *f*; *σ* is the standard deviation of the frequency.

<table><thead><tr><th rowspan="2">f [Hz]</th><th colspan="4">Sinusoidal</th><th colspan="4">Triangular</th></tr><tr><th>Δ [Hz]</th><th>σ [Hz]</th><th>Δ/f [%]</th><th>σ/f [%]</th><th>Δ [Hz]</th><th>σ [Hz]</th><th>Δ/f [%]</th><th>σ/f [%]</th></tr></thead><tbody><tr><td>100</td><td>0.395</td><td>0.598</td><td>0.395</td><td>0.598</td><td>0.394</td><td>0.554</td><td>0.394</td><td>0.554</td></tr><tr><td>200</td><td>0.014</td><td>1.615</td><td>0.007</td><td>0.808</td><td>0.012</td><td>1.514</td><td>0.006</td><td>0.757</td></tr><tr><td>300</td><td>-0.379</td><td>1.695</td><td>-0.126</td><td>0.565</td><td>-0.380</td><td>1.592</td><td>-0.127</td><td>0.531</td></tr><tr><td>400</td><td>0.397</td><td>1.218</td><td>0.099</td><td>0.305</td><td>0.397</td><td>1.183</td><td>0.099</td><td>0.296</td></tr><tr><td>500</td><td>0.007</td><td>1.409</td><td>0.001</td><td>0.282</td><td>0.007</td><td>1.404</td><td>0.001</td><td>0.281</td></tr><tr><td>600</td><td>-0.381</td><td>1.783</td><td>-0.064</td><td>0.297</td><td>-0.381</td><td>1.798</td><td>-0.064</td><td>0.300</td></tr><tr><td>700</td><td>0.401</td><td>1.969</td><td>0.057</td><td>0.281</td><td>0.401</td><td>1.953</td><td>0.057</td><td>0.279</td></tr><tr><td>800</td><td>0.015</td><td>2.779</td><td>0.002</td><td>0.347</td><td>0.015</td><td>2.810</td><td>0.002</td><td>0.351</td></tr><tr><td>900</td><td>-0.376</td><td>2.727</td><td>-0.042</td><td>0.303</td><td>-0.376</td><td>2.734</td><td>-0.042</td><td>0.304</td></tr><tr><td>1000</td><td>0.405</td><td>2.780</td><td>0.041</td><td>0.278</td><td>0.405</td><td>2.774</td><td>0.041</td><td>0.277</td></tr><tr><td>1100</td><td>0.016</td><td>3.149</td><td>0.001</td><td>0.286</td><td>0.016</td><td>3.130</td><td>0.001</td><td>0.285</td></tr><tr><td>1200</td><td>-0.371</td><td>3.780</td><td>-0.031</td><td>0.315</td><td>-0.371</td><td>3.733</td><td>-0.031</td><td>0.311</td></tr><tr><td>1300</td><td>-0.365</td><td>4.684</td><td>-0.028</td><td>0.360</td><td>-0.366</td><td>4.597</td><td>-0.028</td><td>0.354</td></tr><tr><td>1400</td><td>0.039</td><td>6.567</td><td>0.003</td><td>0.469</td><td>0.038</td><td>6.426</td><td>0.003</td><td>0.459</td></tr><tr><td>1500</td><td>0.120</td><td>12.982</td><td>0.008</td><td>0.865</td><td>0.113</td><td>12.596</td><td>0.008</td><td>0.840</td></tr><tr><td>1600</td><td>-0.185</td><td>17.589</td><td>-0.012</td><td>1.099</td><td>-0.188</td><td>17.406</td><td>-0.012</td><td>1.088</td></tr><tr><td>1700</td><td>0.084</td><td>10.916</td><td>0.005</td><td>0.642</td><td>0.078</td><td>10.420</td><td>0.005</td><td>0.613</td></tr><tr><td>1800</td><td>0.043</td><td>7.258</td><td>0.002</td><td>0.403</td><td>0.041</td><td>7.015</td><td>0.002</td><td>0.390</td></tr><tr><td>1900</td><td>-0.325</td><td>9.843</td><td>-0.017</td><td>0.518</td><td>-0.331</td><td>9.321</td><td>-0.017</td><td>0.491</td></tr><tr><td>2000</td><td>0.031</td><td>5.673</td><td>0.002</td><td>0.284</td><td>0.030</td><td>5.543</td><td>0.002</td><td>0.277</td></tr></tbody></table>

where $Q_K$ is the set of all integers $q \neq 0$ such that $qK + 1$ is odd. In the present experiment, $K = 4$, and so $Q_K = \{\pm 1, \pm 2, \dots\}$. In Section 3, the phase offset is found to be very nearly $\phi \approx -\pi/2$, in which case $e^{iqK\phi} \approx e^{-i2\pi q} = 1$. Under these considerations, the value of $S_{\text{harm}}^{\Delta}(\omega|K)$ in the limit as $\omega$ approaches zero is

$$ \lim_{\omega \to 0} S_{\text{harm}}^{\Delta}(\omega|4) = \sum_{q=\pm 1, \pm 2, \dots} \frac{S(0)}{(4q+1)^3} \approx -0.031 S(0); \quad (C.2) $$

see Ref. [8]. Thus, the measured value of $S(0)$ from a determination using triangular modulation should be too small by about 3%. Applying this to the value of $\hat{S}(0) = 61.7(4)\text{s}^{-1}\text{cm}^{-2}$, which combines the sinusoidal modulation results in Section 3 and the transmission measurements in Section 2, one expects to find $\hat{S}(0) = 59.8(4)\text{s}^{-1}\text{cm}^{-2}$ by triangular modulation, which is within error bars of the measured result $59.4(1)\text{s}^{-1}\text{cm}^{-2}$ given in Table 2.

Another sign of harmonic distortion in the low-frequency triangular modulation data can be found by fitting lines to the $|\hat{Z}(\omega)|$ data. In the present case of a singly-peaked TOF spectrum, the triangular modulation harmonic distortion is such that $|Z(\omega)|$ is an increasing function of $\omega$ near the origin. Thus, for low-frequency modulation,

fitting a line to the triangular data, one should find a positive slope, and fitting a line to the sinusoidal modulation data, one should find a slope of zero. This is indeed the case, and may be seen in Fig. C.14, in which the data have been spread over 0.3 Hz to enable better visualization of the error bars; as in Fig. 5, the actual frequency spread of the data is on the level of tens of millihertz.

The fit parameters are given in Table C.3. There, the values of the slopes are consistent with zero for the sinusoidal modulation data, whereas a nonzero, positive slope is obtained from a fit to the triangular modulation data. This nonzero slope is evidence of harmonic distortion.

## Appendix D. Frequency stability data

Frequency stability data are listed in Table D.4. Histograms of the frequency during the sinusoidal and triangular measurements are shown in Figs. D.15a and D.15b, respectively. Marked oscillation of the motor speed was seen at high frequencies, perhaps as a result of insufficient rotor balancing. For normally-distributed frequencies, the associated uncertainty in $\hat{Z}(\omega)$ goes like $\sigma/f$ in Table D.4, see Ref. [8]. In the present case, however, at some high frequencies, the distributions are not approximately normal. Attempts to account for this are not pursued in the present paper.

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