# Developments in neutron Fourier time-of-flight

Thomas Neulinger

► To cite this version:

Thomas Neulinger. Developments in neutron Fourier time-of-flight. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2024, 1065, pp.169513. 10.1016/j.nima.2024.169513. hal-05441987

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Full Length Article

# Developments in neutron Fourier time-of-flight

Thomas Neulinger

Institut Laue-Langevin, Grenoble, 38042, France

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

## ARTICLE INFO

**Keywords:**

Time-of-flight

Fourier time-of-flight

## ABSTRACT

Two main developments in neutron Fourier time-of-flight (TOF) are presented. The first is a new sinusoidal stator design that eliminates a systematic effect associated with triangular modulation in existing Fourier choppers. Despite sinusoidal modulation being associated with a reduction in average beam transmission, it is mathematically demonstrated that, on average and for good signal-to-background, only about 3% additional measurement time is required to achieve equal statistical uncertainty. The second development is a new method to measure the chopper phase offset. This method relies only on symmetries inherent to the Fourier transform of any physical TOF spectrum.

## 1. Introduction

Many neutron time-of-flight (TOF) instruments use mechanical choppers to transmit short pulses of neutrons to a detector at a known distance. Measurement of the neutron arrival times with the detector enables calculation of the longitudinal (i.e., along the axis of the flight path) velocity distribution of the incident beam. The event rate at the detector $r(t)$ is the sum of the background rate $b$ and the convolution of the TOF spectrum $s(t)$ of the beam with the chopper's transmission function $A(t)$, i.e.,

$$r(t) = b + \int_{-\infty}^{\infty} dt' A(t - t')s(t'). \qquad (1)$$

The quantities $A(t)$ and $s(t)$ can be defined in various ways so that their product has units of rate per time. In this paper, $A(t)$ is defined to be the area of the chopper opening at time $t$, and $s(t) dt$ is the rate per area (assumed to be spatially uniform) of neutrons incident on the chopper with TOF between $t$ and $t + dt$. If the form of $s(t)$ is known from theoretical considerations, then it is relatively straightforward to determine the spectrum from measurement by refining unknown model parameters via the method of least squares, iteratively convolving the known $A(t)$ with test spectra and comparing the result to measured data. However, when the form of $s(t)$ is not known, other means must be used to deconvolve $A(t)$ and $s(t)$. Although several approaches may be taken, see, e.g., Refs. [1–3], the problem is mathematically ill-posed, meaning that there are many different estimates $\hat{s}(t)$ that, when convolved with $A(t)$, will agree with measured data to within experimental error.

In the standard TOF technique, convolution effects can be mitigated by operating the chopper with a small opening time $t_{open}$ so that $r(t)$ is

approximately equal to $s(t)$ plus the background. However, a small $t_{open}$ results in low counting statistics per measurement. As measurements must be repeated with period $t_{frame}$ at least as long as the longest TOF $t_{max}$ to avoid frame overlap, the duty cycle is limited to $t_{open}/t_{max}$. This can result in long measurement times when high TOF resolution is required.

The Fourier TOF technique [4,5] is an alternative approach in which one uses a so-called Fourier chopper to modulate the neutron beam at frequency $\omega/(2\pi)$ so that $r(t)$ oscillates with amplitude and phase determined by $S(\omega)$, the Fourier transform of $s(t)$. By performing measurements at a set of discrete frequencies, the spectrum $s(t)$ may be reconstructed by inverse discrete Fourier transform. Whereas $t_{open}$ determines the TOF resolution in the standard TOF technique, the resolution in the Fourier TOF technique is determined by the maximum modulation frequency, which can be arbitrarily large within technical limits. In this paper, two main developments to the Fourier TOF technique are presented.

The first development is the introduction of a family of parameterized stator hole shapes that produce sinusoidal beam modulation. Triangular Fourier choppers in use at present suffer from a systematic effect in which the amplitude and phase of the rate $r(t)$ depend not only on $S(\omega)$, but on $S(n\omega)$ for odd integers $n$. This systematic effect, known as harmonic distortion, is not present for sinusoidal beam modulation. Despite this advantage, the literature appears to offer two main reasons for the continued usage of triangular modulation. First, for a narrow and singly-peaked TOF spectrum, the error due to harmonic distortion is at most about 3% of $|S(\omega)|$ [6], which is considered by some to be negligible. In fact, it is claimed in Ref. [6] that “there should be no serious reasons in practice” to employ a sinusoidal Fourier chopper, “except possibly in spectra featuring some very prominent and

<footnote>E-mail address: neulinger@ill.fr.</footnote>
sharp peaks or cut-offs". It is interesting to note that both the High-Resolution Fourier Diffractometer (HRFD) [7] and the Neutron Stress Fourier Diffractometer (FSS) [8] instruments at the IBR-2 pulsed reactor, which use triangular modulation Fourier choppers, are intended to reconstruct TOF spectra having many sharp peaks. The second apparent justification for continued usage of triangular modulation is that the average beam intensity is lower for sinusoidal modulation; a reduction of 30% was reported in Ref. [9] for an approximately sinusoidal Fourier chopper. It is thus presumably to be understood that an undesirably-long measurement time would be required with sinusoidal modulation in order to achieve the same statistical uncertainty as with triangular modulation [10]. In the present paper, it is shown that, in situations with good signal-to-background, only about 3% additional measurement time is required.

The second development to the Fourier TOF technique is a new method to determine the chopper phase offset $\phi$, which arises from a nonzero timing offset between the chopper transmission maxima and trigger signals used to gate acquisition. Various ad hoc techniques have been proposed to correct for the phase offset. For example, in Ref. [11], $\phi$ is estimated by assuming that harmonic distortion is negligible, but no estimate of the error resulting from such an approximation is given. In Ref. [7], the unknown $\phi$ is fitted so that diffraction peak "profiles are symmetrical", but no physical justification of this approach is reported. In Refs. [12,13], a phase correction technique was developed specifically for a particular Fourier TOF diffractometer. In the present paper, a general method is described by which the exact least-squares estimate of $\phi$ may be determined from a small number of measurements.

The rest of the paper is organized as follows. In Section 2, the theory of the Fourier TOF technique is reviewed and applied to the cases of triangular and sinusoidal modulation. This is followed in Section 3 by a comparison of the measurement efficiency for the two types of modulation. Section 4 presents the new technique to determine the phase offset $\phi$. In Section 5, the reconstruction of $s(t)$ from the measured $S(\omega)$ is discussed. The main body of the paper is concluded in Section 6 with a summary and outlook. Several appendices follow: the statistical properties of various estimators are discussed in Appendix A; analytical expressions for the harmonic distortion are derived for the case of triangular modulation in Appendix B; the family of sinusoidal stator hole shapes is presented in Appendix C; the algorithm for calculating the exact least-squares estimate of $\phi$ is given in Appendix D; some mathematical details of the reconstruction of the TOF spectrum by inverse discrete Fourier transform are covered in Appendix E; the measurement efficiencies of Fourier and standard TOF are compared in Appendix F. Together with the main body of the paper, these appendices are intended to support a self-contained discussion.

## 2. Theory

The theoretical formulation presented in this section will mainly follow that developed in Ref. [6], although with the inclusion of the chopper phase offset $\phi$, which was neglected in that treatment. In order to facilitate later measurement efficiency calculations, the notation here also differs from Ref. [6] in that the TOF spectrum is defined so that $s(t)dt$ is the rate per area of neutrons at the chopper entrance, rather than the fraction of the spectrum, with TOF between $t$ and $t+dt$. Similarly, $A(t)$ is defined as the area of the chopper opening, and not as the beam intensity.

Call $\omega$ the angular frequency of the beam modulation. The transmission function $A(t)$ is periodic in time and can be represented by the Fourier series

$$A(t) = \sum_{n=-\infty}^{\infty} A_n e^{in\omega t}, \qquad (2)$$

where $A_{-n} = A_n^*$ so that $A(t)$ is real. In the following, only those $A(t)$ that are symmetric with respect to time $t=0$ will be considered, so that $A_{-n} = A_n$ also. Inserting (2) into (1), one obtains

$$r(t | \omega) = \sum_{n=-\infty}^{\infty} r_n(\omega) e^{in\omega t}, \qquad (3)$$

where (1) has been written as $r(t | \omega)$ to emphasize the dependence of the rate on $\omega$, and where, for constant background rate $b$,

$$r_0(\omega) = b + A_0 S(0), \qquad (4a)$$

$$r_n(\omega) = A_n S(n\omega), \quad n \neq 0, \qquad (4b)$$

with $S(\omega)$ being the Fourier transform of $s(t)$,

$$S(\omega) = \int_{-\infty}^{\infty} dt \, s(t) e^{-i\omega t}, \qquad (5a)$$

$$s(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} d\omega \, S(\omega) e^{i\omega t}. \qquad (5b)$$

Note that $S(-\omega) = S^*(\omega)$ follows from (5a) as $s(t)$ is real, i.e., the real and imaginary parts of $S(\omega)$ are even and odd functions, respectively.

In an experiment, detection events (counts) are the observable rather than the neutron rate. It is therefore necessary to develop an expression for the former in terms of $S(\omega)$. In the Fourier TOF technique, counts are accumulated in $K$ equal time intervals (channels) that divide the beam modulation period $T = 2\pi/\omega$. Thus, the quantity

$$c_k(\omega) = \int_{(k-1)T/K + t_{\text{offset}}(\omega)}^{kT/K + t_{\text{offset}}(\omega)} dt \, r(t | \omega), \qquad (6)$$

for $k = 1, 2, \dots, K$, is both the expectation value and the variance of the Poisson-distributed random number of counts $C_k(\omega)$ in the $k$th channel [14]. The timing offset is

$$t_{\text{offset}}(\omega) = T \frac{\phi}{2\pi} = \frac{\phi}{\omega}, \qquad (7)$$

where the phase offset $\phi$ is defined for one sense of rotation, e.g., $\omega > 0$. Inserting (3) into (6), one obtains

$$c_k(\omega) = \frac{T}{K} \sum_{n=-\infty}^{\infty} e^{in\phi} r_n(\omega) e^{i\pi n(2k-1)/K} \frac{\sin(\pi n/K)}{\pi n/K}, \qquad (8)$$

which is identical to Equation 6 in Ref. [6] except for the appearance of $e^{in\phi}$. One would like to invert this expression to obtain the $r_n(\omega)$ in terms of the $c_k(\omega)$. That way, using (4b) for $n=1$ together with $A_1$, one would be able to construct an estimate of $S(\omega)$ from the measured data $C_k(\omega)$. Unfortunately, this is not possible in general, as such an inversion couples $r_n(\omega)$ spaced by $K$. Following Ref. [6], one can define

$$\tilde{r}_0(\omega) = r_0(\omega), \qquad (9a)$$

$$\tilde{r}_n(\omega) = \sum_{q=-\infty}^{\infty} e^{iqK\phi} \frac{n}{qK+n} r_{qK+n}(\omega), \qquad (9b)$$

where $n=1, \dots, K-1$ in the second line. Then (8) becomes

$$c_k(\omega) = \frac{T}{K} \sum_{n=0}^{K-1} e^{in\phi} \tilde{r}_n(\omega) e^{i\pi n(2k-1)/K} \frac{\sin(\pi n/K)}{\pi n/K}. \qquad (10)$$

This expression can now be inverted to obtain

$$e^{in\phi} \tilde{r}_n(\omega) = \frac{\omega}{2\pi} \frac{\pi n/K}{\sin(\pi n/K)} \sum_{k=1}^{K} c_k(\omega) e^{-i\pi n(2k-1)/K}. \qquad (11)$$

The content of this equation is identical to Equation 9 in Ref. [6] except for the presence of the phase offset $\phi$.

The relationship between $e^{i\phi}\tilde{r}_1(\omega)$ and $S(\omega)$ is found by inserting (4b) into (9b) for $n=1$. After multiplying by $e^{i\phi}$, one obtains

$$\begin{aligned} e^{i\phi}\tilde{r}_1(\omega) &= \sum_{q=-\infty}^{\infty} \frac{e^{i(qK+1)\phi}}{qK+1} A_{qK+1} S[(qK+1)\omega] \\ &= A_1 e^{i\phi} [S(\omega) + S_{\text{harm}}(\omega | K)], \end{aligned} \qquad (12)$$

in which the harmonic distortion is

$$S_{\text{harm}}(\omega | K) = \sum_{q \neq 0} \frac{e^{iqK\phi}}{qK+1} \frac{A_{qK+1}}{A_1} S[(qK+1)\omega], \qquad (13)$$

where the sum is performed over all integers $q \neq 0$. As with several of the previous expressions, the content of (13) here is the same as that
of Equation 13 in Ref. [6], except for the presence of the phase offset. Consequently, $S_{\text{harm}}(-\omega | K)$ and $S_{\text{harm}}^*(\omega | K)$ are not equal, and so, unlike with $S(\omega)$, the real and imaginary parts of $S_{\text{harm}}(\omega | K)$ are not even and odd functions, respectively.

Comparing (11) for $n=1$ and (12), one obtains

$$A_1 e^{i\phi} [S(\omega) + S_{\text{harm}}(\omega | K)] = \frac{\omega}{2\pi} \frac{\pi/K}{\sin(\pi/K)} \sum_{k=1}^{K} c_k(\omega) e^{-i\pi(2k-1)/K}, \quad (14)$$

which gives the relationship between the counts $c_k(\omega)$ on the right-hand side and the quantity of interest $S(\omega)$. If neither $\phi$ nor $S_{\text{harm}}(\omega | K)$ is known, then the nearest thing to $S(\omega)$ that can be measured is

$$Z(\omega) = e^{i\phi} [S(\omega) + S_{\text{harm}}(\omega | K)]. \quad (15)$$

The determination of $\phi$ is discussed in Section 4, and the harmonic distortion $S_{\text{harm}}(\omega | K)$ is considered presently. The statistical properties of the estimate $\hat{Z}(\omega)$ of (15), along with that of other related quantities of interest, are discussed in Appendix A. Note that in this paper a hat is used to denote an estimator.

Consider first the harmonic distortion associated with triangular modulation, currently implemented in two of the most prominent Fourier TOF instruments [7,8]. The geometry of one pair of rotor and stator holes is shown in Fig. 1(a). The holes are annular sections with minimum and maximum radii $\rho_1$ and $\rho_2$, respectively, and angular extent $\Delta\theta$ such that $2\Delta\theta$ evenly divides $2\pi$ radians. When the rotor is turned with angular frequency $\omega_0$, the opening area of the chopper is, for one full period $|t| \le \Delta\theta/\omega_0$,

$$A^{\Delta}(t) = A^{\Delta} \left( 1 - \left| \frac{\omega_0 t}{\Delta\theta} \right| \right), \quad (16)$$

where $A^{\Delta} = \Delta\theta(\rho_2^2 - \rho_1^2)/2$ is the maximum area at $t=0$. $A^{\Delta}(t)$ has periodicity $T = 2\Delta\theta/\omega_0$, and can therefore be expressed as the Fourier series (2) in terms of multiples of the angular frequency $\omega = 2\pi/T = \pi\omega_0/\Delta\theta$. The Fourier coefficients of $A^{\Delta}(t)$ are

$$A_0^{\Delta} = \frac{1}{2} A^{\Delta}, \quad (17a)$$

$$A_n^{\Delta} = \frac{1}{2} \left( \frac{2}{\pi n} \right)^2 A^{\Delta}, \quad n \text{ odd}, \quad (17b)$$

where $A_n^{\Delta}$ is zero for all even $n$. Inserting the $A_n^{\Delta}$ into (13) one obtains, for $K > 2$,

$$S_{\text{harm}}^{\Delta}(\omega | K) = \sum_{q \in Q_K} \frac{e^{iqK\phi}}{(qK + 1)^3} S[(qK + 1)\omega] \quad (18)$$

where $A_n^{\Delta}/A_1^{\Delta} = 1/n^2$ has been used, and where the set $Q_K$ contains all integers $q \ne 0$ such that $qK + 1$ is odd. That is, $Q_K = \{\pm 1, \pm 2, \pm 3, ...\}$ for $K$ even, and $Q_K = \{\pm 2, \pm 4, \pm 6, ...\}$ for $K$ odd. As $S_{\text{harm}}^{\Delta}(\omega | K)$ depends on $S(\omega)$, it is not possible to make general statements about the relative magnitude of the two. However, Refs. [6,10] consider the cases of a delta function TOF spectrum and a sharply-peaked TOF spectrum, respectively. Both report the approximate result that, for such spectra, $S_{\text{harm}}^{\Delta}(\omega | K)$ is largest for $K=4$, where it is at most roughly 3% of $|S(\omega)|$. Similar investigations are discussed in Appendix B, where exact expressions for arbitrary $K$ are given.

One possible pair of rotor and stator hole shapes that produces sinusoidal modulation is shown in Fig. 1(b). This and other geometries are derived in Appendix C. The chopper transmission function is

$$A^{-}(t) = A^{-} \left\{ \frac{1}{2} + \frac{1}{2} \cos \left( \pi \frac{\omega_{0} t}{\Delta \theta} \right) \right\}, \quad (19)$$

where $A^{-} = \Delta\theta(\rho_2^2 - \rho_1^2)/\pi$ is the maximum area of overlap at $t=0$. Thus, the only nonzero Fourier series coefficients are

$$A_0^{-} = \frac{1}{2} A^{-}, \quad (20a)$$

$$A_{\pm 1}^{-} = \frac{1}{4} A^{-}, \quad (20b)$$

and the harmonic distortion is zero for $K > 2$, i.e.,

$$S_{\text{harm}}^{-}(\omega | K) = 0. \quad (21)$$

### 3. Measurement efficiency

Despite the absence of harmonic distortion with sinusoidal modulation, the authors of Ref. [10] have pointed to the reduction in intensity as one of its disadvantages compared to triangular modulation. For fully-open choppers, noting that the integrated rate over all TOF is $S(0)$, the relative intensity is

$$\frac{A^{-}S(0)}{A^{\Delta}S(0)} = \frac{\Delta\theta(\rho_{2}^{2} - \rho_{1}^{2})/\pi}{\Delta\theta(\rho_{2}^{2} - \rho_{1}^{2})/2} = \frac{2}{\pi} \approx 0.64. \quad (22)$$

This reduction, by approximately 36%, is greater than the 30% reported in Ref. [9] for an approximately sinusoidal chopper. This is puzzling because in Appendix C it is shown that $A^{-} = \Delta\theta(\rho_2^2 - \rho_1^2)/\pi$ is the opening area for a sinusoidal Fourier chopper irrespective of its shape, so long as $\rho_1$ and $\rho_2$ are the minimum and maximum radii, respectively. Regrettably, insufficient details are provided in Ref. [9], published 50 years ago, to resolve this point.

In any case, it is presumably to be understood by the authors of Ref. [10] that a reduction in intensity implies inferior counting statistics, and therefore would be responsible for undesirably-long measurement times in order to achieve equal statistical uncertainty. A naive calculation might proceed as follows. Using (3) and (4), the expectation value of the total number of counts per modulation period is

$$\int_{t}^{t+T} dt' r(t'|\omega) = T [b + A_0 S(0)]. \quad (23)$$

Thus, in the absence of background, the total counts are proportional to the maximum opening area $A$. If one measures for $M$ modulation periods, then the statistics collected are proportional to $MA$. In order to obtain equal statistics with sinusoidal and triangular modulation, one requires $M \sim A^{-} = M^{\Delta}A^{\Delta}$, or $M \sim M^{\Delta} = A^{\Delta}/A^{-} = \pi/2 \approx 1.57$. Since the required measurement time scales with $M$, one would then conclude that a measurement with a sinusoidal chopper takes 57% longer than with a standard chopper, or, by the same reasoning, 43% according to the value reported in Ref. [9] (1/0.70 ≈ 1.43).

However, for a given modulation frequency $\omega/(2\pi)$, one is mainly interested in the rate at which the statistical uncertainty in a measurement of $|S(\omega)|$ decreases with measurement time. Denote this uncertainty after $M$ modulation periods by

$$\delta(\omega | M) = \sqrt{\Psi_{\text{stat.}}(|\hat{Z}(\omega)|)}, \quad (24)$$

which is the square root of the variance of the estimate $|\hat{Z}(\omega)|$; note that $|\hat{S}(\omega)| = |\hat{Z}(\omega)|$ when the harmonic distortion is unknown. One is interested in the values of the number of sinusoidal and triangular modulations, $M^{-}$ and $M^{\Delta}$ respectively, so that the ratio of the uncertainties is unity, i.e.,

$$\frac{\delta(\omega | M^{-})}{\delta^{\Delta}(\omega | M^{\Delta})} = 1. \quad (25)$$

The variances in the limit of large $M$, due to counting statistics only, are derived in Appendix A.2. For $K > 3$, these expressions greatly simplify, and are

$$\Psi_{\text{stat.}}(|\hat{Z}|) \approx \frac{1}{2M^{-}T} \left( \frac{1}{A_1^{-}} \frac{\pi/K}{\sin(\pi/K)} \right)^2 [b + A_0^{-}S(0)], \quad (26a)$$

$$\Psi_{\text{stat.}}^{\Delta}(|\hat{Z}|) \approx \frac{1}{2M^{\Delta}T} \left( \frac{1}{A_1^{\Delta}} \frac{\pi/K}{\sin(\pi/K)} \right)^2 [b + A_0^{\Delta}S(0)], \quad (26b)$$

where it is additionally required for (26b) that $K$ be even. Inserting (26a) and (26b) into (25), one obtains

$$\sqrt{\frac{M^{\Delta}}{M^{-}} \frac{A_1^{\Delta}}{A_1^{-}}} \sqrt{\frac{b + A_0^{-} S(0)}{b + A_0^{\Delta} S(0)}} = 1, \quad (27)$$

which is independent of the modulation frequency. Further simplification using the $A_n$ given in Section 2 yields

$$\sqrt{\frac{M^{\Delta}}{M^{-}}} \frac{4}{\pi} \sqrt{\frac{1 + SBR/\pi}{1 + SBR/2}} = 1, \quad (28)$$
<figure><img src="image_4.png" /><figcaption>Fig. 1. (a) triangular and (b) sinusoidal Fourier chopper stator hole shapes (thick red lines). The rotor hole shapes (thin black lines) are the same in (a) and (b).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. The relative measurement time needed to achieve equal uncertainties using sinusoidal and triangular modulation (30), plotted as a function of the signal-to-background ratio (29).</figcaption></figure>

where the signal-to-background ratio has been defined as

$$ SBR = \frac{A^{\Delta} S(0)}{b}. \qquad (29) $$

This ratio is the integrated rate of neutrons incident over the area of the standard chopper stator hole divided by the background rate. Rearranging (28), one obtains

$$ \frac{M^{\sim}}{M^{\Delta}} = \left(\frac{4}{\pi}\right)^2 \frac{1 + SBR/\pi}{1 + SBR/2}. \qquad (30) $$

This ratio, plotted in Fig. 2 as a function of the signal-to-background ratio, gives the relative measurement time needed with sinusoidal modulation to achieve equal uncertainty on a measurement of $|Z(\omega)|$ as would be obtained with triangular modulation. The limiting values of (30) are

$$ \lim_{SBR \to 0} \left( \frac{M^{\sim}}{M^{\Delta}} \right) = \frac{16}{\pi^2} \approx 1.62 \qquad (31a) $$

$$ \lim_{SBR \to \infty} \left( \frac{M^{\sim}}{M^{\Delta}} \right) = \frac{32}{\pi^3} \approx 1.03. \qquad (31b) $$

In experimental situations with good signal-to-background, (31b) is relevant, and so a measurement with sinusoidal modulation will only need about 3% additional time to reach equal statistical uncertainty.

The discussion here has focused on measurements of $|S(\omega)|$ rather than of $s(t)$. However, in Appendix A it is shown that the counting statistics contribution to the variance of the TOF spectrum estimate $\hat{s}(t)$

is

$$ \Psi_{\text{stat.}}(\hat{s}) = 4 \left(\frac{\Delta\omega}{2\pi}\right)^2 \sum_{n=1}^{N} \Psi_{\text{stat.}}(|\hat{Z}(\omega_n)|). \qquad (32) $$

Thus, the variance is uniform in $t$, and is proportional to the sum of the variances of the individual $|\hat{Z}(\omega_n)|$. It is therefore also the case that, compared to triangular modulation, only about 3% additional time is needed with sinusoidal modulation to reach equal statistical uncertainty on a measurement of $s(t)$. This would seem to be a small price to pay in order to completely eliminate the harmonic distortion present in triangular modulation data.

## 4. Determination of the phase offset

Before $S(\omega)$ can be determined from $Z(\omega)$, it is necessary to determine the phase offset $\phi$. This can be accomplished in the following way. Expanding $e^{-i\omega t}$ in (5a) and examining the terms, one has

$$ S(\omega) = \int_{-\infty}^{\infty} dt s(t) \sum_{n=0}^{\infty} \frac{(-i\omega t)^n}{n!} = \sum_{n=0}^{\infty} \frac{(-i\omega)^n}{n!} a_n, \qquad (33) $$

where

$$ a_n = \int_{-\infty}^{\infty} dt t^n s(t). \qquad (34) $$

Note that the $a_n$ are both real and positive because $s(t)$ is real and positive and has positive support. Also note that all terms with $n$ even are purely real while those with $n$ odd are purely imaginary. Now consider $\omega$ small enough that $S(\omega)$ is approximated by a finite number of terms in (33). Supposing, for example, that a second-order expansion is sufficient, (15) becomes, for sinusoidal modulation,

$$ Z(\omega) \approx e^{i\phi} (a_0 - i\omega a_1 - \frac{1}{2}\omega^2 a_2). \qquad (35) $$

If $\phi$ is not an integer multiple of $\pi$, the coefficients of the powers of $\omega$ will be complex rather than either purely real for $n$ even or purely imaginary for $n$ odd. Thus, to determine $\phi$, one can search for a phase $\phi$ so that the quantity $e^{-i\phi} Z(\omega)$ is a polynomial in $\omega$ with two real coefficients and one imaginary coefficient.

Continuing with the second-order example, consider for a set of low frequencies $\omega_j/(2\pi)$ the residuals

$$ \epsilon(\omega_j | \phi, \alpha_0, \alpha_1, \alpha_2) = e^{-i\phi} \hat{Z}(\omega_j) - \left( \alpha_0 - i\omega_j \alpha_1 - \frac{1}{2} \omega_j^2 \alpha_2 \right), \qquad (36) $$

where $\hat{Z}(\omega_j)$ is the measured estimate of $Z(\omega_j)$, and where the four real fit parameters are $\phi, \alpha_0, \alpha_1$, and $\alpha_2$. If $\phi$ is close to $\phi + \pi m$ for even (odd) integers $m$, the first term on the right-hand side will be close to $S(\omega) (-S(\omega))$. Thus, the $\alpha_n$ that make $|\epsilon|$ small will be close to the $a_n$ ($-a_n$). As the $a_n$ are positive, one can always put $\phi \to \phi \pm \pi$ to ensure that the $\alpha_n$ are positive. If, on the other hand, $\phi$ is not close to $\phi \pm \pi m$ for even or odd $m$, then $|\epsilon|$ will be large no matter the values of the $\alpha_n$.
In practice, one way to determine the $\varphi$ and $\alpha_n$ that make $|\varepsilon|$ small, and thus to find the estimates $\hat{\varphi}$ and $\hat{a}_n$, is to minimize the weighted sum of the squares of the residuals

$$Q(\varphi, \underline{\alpha}) = \sum_{j=1}^{J} \frac{\varepsilon^*(\omega_j | \varphi, \underline{\alpha}) \varepsilon(\omega_j | \varphi, \underline{\alpha})}{\hat{V}(|\hat{Z}(\omega_j)|)} \quad (37)$$

for the J positive low frequencies $\omega_j/(2\pi)$, where $\underline{\alpha} = \{\alpha_0, \alpha_1, \dots\}$ and the denominator is the estimate of the variance of $|\hat{Z}(\omega_j)|$; see Appendix A. For $Q(\varphi, \underline{\alpha})$ defined in this way, the least-squares estimates $\hat{\varphi}$ and $\hat{a}_n$ may be solved for exactly, and are derived in Appendix D. Note that although the principle goal was to calculate $\hat{\varphi}$, one also obtains, e.g., the estimate $\hat{a}_0 = \hat{S}(0)$, which cannot otherwise be measured with the Fourier TOF technique.

In the application of this method, sinusoidal modulation of the beam is necessary to avoid harmonic distortion. Additionally, it is necessary to qualify the meaning of “low frequency” for the spectrum in question, and to determine how many terms $\alpha_n$ to use in fitting. Regarding the former, low frequency means that $\omega \ll 1/t_{\text{max}}$, where $t_{\text{max}}$ is the maximum TOF present. Regarding the number of terms, the choice is not completely unambiguous. However, one possibility would be to terminate the fit order according to the results of an F-test that compares the chi-squared values from fits using up to $a_n$ and $a_{n+1}$ terms; see, e.g., Section 11.4 in Ref. [15].

## 5. The reconstructed TOF spectrum

This section deals with the practical matter of reconstructing $s(t)$ from the measured $Z(\omega)$. Multiplying $Z(\omega)$ in (15) by $e^{-i\phi}$ and taking the inverse Fourier transform (5b), one obtains

$$\begin{aligned} \frac{1}{2\pi} \int_{-\infty}^{\infty} d\omega e^{-i\phi} Z(\omega) e^{i\omega t} &= \frac{1}{2\pi} \int_{-\infty}^{\infty} d\omega S(\omega) e^{i\omega t} \\ &\quad + \frac{1}{2\pi} \int_{-\infty}^{\infty} d\omega S_{\text{harm.}}(\omega | K) e^{i\omega t} \\ &= s(t) + s_{\text{harm.}}(t | K), \end{aligned} \quad (38)$$

where $s_{\text{harm.}}(t | K)$ is the inverse Fourier transform of the harmonic distortion $S_{\text{harm.}}(\omega | K)$. For sinusoidal modulation, this term vanishes, whereas for triangular modulation, using (18), it is

$$s_{\text{harm.}}^{\Delta}(t | K) = \frac{1}{2\pi} \int_{-\infty}^{\infty} d\omega \sum_{q \in Q_K} \frac{e^{iqK\phi}}{(qK+1)^3} S[(qK+1)\omega] e^{i\omega t}. \quad (39)$$

Defining the integration variable $\omega_q(\omega) = (qK + 1)\omega$, this becomes

$$\begin{aligned} s_{\text{harm.}}^{\Delta}(t | K) &= \sum_{q \in Q_K} \frac{e^{iqK\phi}}{(qK+1)^3} \frac{1}{2\pi} \int_{\omega_q(-\infty)}^{\omega_q(\infty)} \frac{d\omega_q}{qK+1} S(\omega_q) e^{i\omega_q t/(qK+1)} \\ &= \sum_{q \in Q_K} \operatorname{sgn}(qK+1) \frac{e^{iqK\phi}}{(qK+1)^4} s\left(\frac{t}{qK+1}\right), \end{aligned} \quad (40)$$

where $\operatorname{sgn}(x)$ is the sign function, which is 1 for $x > 0$ and $-1$ for $x < 0$. This expression is nearly the same as Equation 18 in Ref. [6] except for the presence of the term $e^{iqK\phi}$, which has the effect of making $s_{\text{harm.}}^{\Delta}(t | K)$ complex rather than purely real. This expression represents the sum of a series of stretched and sometimes inverted (i.e., mirrored about the horizontal axis) copies of $s(t)$. In (40), the inverted terms in the sum, with $qK+1 < 0$, are zero owing to the fact that $s(t)$ has positive support. However, as has been noted in Ref. [6], these terms will in fact contribute to the estimate of $s(t)$, as this estimate is constructed as a discrete approximation using a finite number of frequencies.

To see how this works in practice, suppose that the integration range of (5b) is truncated between $\pm\Omega$ and replaced by a sum over angular frequencies spaced by $\Delta\omega = \Omega/N$. The discrete approximation to (5b) is then

$$s_{\text{disc.}}(t) = \frac{1}{2\pi} \sum_{n=-N}^{N} \Delta\omega S(\omega_n) e^{i\omega_n t}, \quad (41)$$

where $\omega_n = n\Delta\omega$. It follows from this definition that $s_{\text{disc.}}(t)$ is periodic with period $2\pi/\Delta\omega$. Therefore, values of $s_{\text{disc.}}(t)$ for $t$ outside of the range $0 \le t < 2\pi/\Delta\omega$ are to be disregarded on physical grounds. In Appendix E.2, it is shown that $s_{\text{disc.}}(t)$ is related to the TOF spectrum by

$$s_{\text{disc.}}(t) = \frac{\Delta\omega}{2\pi} \int_{-\infty}^{\infty} dt' w(t - t') \sum_{n=-\infty}^{\infty} s\left(t' - n\frac{2\pi}{\Delta\omega}\right), \quad (42)$$

where $w(t)$ is a peaked function given by (E.23) that has width inversely proportional to $\Omega$. Thus, $s_{\text{disc.}}(t)$ is the convolution of $w(t)$ with an infinite number of copies of $s(t)$ shifted by $2\pi/\Delta\omega$. In practice, $\Delta\omega$ should be made small enough so that these copies are non-overlapping.

By the same analysis leading to the result (42), the discrete approximation to (38) for the case of triangular modulation is

$$\frac{1}{2\pi} \sum_{n=-N}^{N} \Delta\omega e^{-i\phi} Z(\omega_n) e^{i\omega_n t} = s_{\text{disc.}}(t) + s_{\text{disc.harm.}}^{\Delta}(t | K), \quad (43)$$

where

$$\begin{aligned} s_{\text{disc.harm.}}^{\Delta}(t | K) &= \sum_{q \in Q_K} \operatorname{sgn}(qK + 1) \frac{e^{iqK\phi}}{(qK + 1)^4} \frac{\Delta\omega}{2\pi} \int_{-\infty}^{\infty} dt' w(t - t') \sum_{n=-\infty}^{\infty} s\left(\frac{t' - n\frac{2\pi}{\Delta\omega}}{qK + 1}\right). \end{aligned} \quad (44)$$

If $w(t)$ is sufficiently narrow, then the integrand will be appreciably nonzero only for values of $t' \approx t$. Consider then a feature of the TOF spectrum at $t_0$. The corresponding term will appear in the sum of (44) and will therefore contribute to $s_{\text{disc.harm.}}^{\Delta}(t | K)$ when

$$t \approx n \frac{2\pi}{\Delta\omega} + t_0(qK + 1) \quad (45)$$

for integer $n$ such that $0 \le t < 2\pi/\Delta\omega$. For a given value of $q$, only one value of $n$ will fulfill this condition. Whereas in (40), terms in the sum with negative values of $q$ are zero and so do not contribute to $s_{\text{harm.}}^{\Delta}(t | K)$, the same is not the case for (44). There, a positive value of $n$ will shift the argument of $s(t)$ so that it is also positive, resulting in a negative contribution to $s_{\text{disc.harm.}}^{\Delta}(t | K)$.

The result (44) is the same as that given in Ref. [6] except for the appearance of the phase offset $\phi$, which has an interesting consequence. Suppose that $K$ is even, so that the sum in $q$ of (44) runs over $\pm 1, \pm 2$, and so on. Then, if one arranges that the phase offset is $\phi = \pi/(2K)$, for example, the lowest order contributions at $q = \pm 1$, which are the largest in magnitude, will be purely imaginary. The same would be true of the lowest order terms for odd $K$, where $q$ runs over $\pm 2, \pm 4$, and so on, if it were arranged that $\phi = \pi/(4K)$. Disregarding the imaginary part of (43) would therefore remove a significant portion of the harmonic distortion. However, such schemes are unnecessary when using sinusoidal beam modulation, which has been shown to require roughly 3% additional measurement time in experiments with good signal-to-background.

## 6. Conclusion

In this paper, the theory of neutron Fourier TOF has been reviewed and extended to include effects related to the phase offset $\phi$. Unlike the results given in Ref. [6], it has been found that, for nonzero $\phi$, the contribution from harmonic distortion to the measured $s(t)$ is complex rather than purely real. This paper has also proposed a family of stator hole shapes that produce exact sinusoidal modulation of the neutron beam; a previous publication reported a geometry that produced only approximate sinusoidal modulation [9]. Despite a reduction in the average beam transmission compared to triangular modulation, it is found that, for good signal-to-background, only 3% additional measurement time is required to achieve equal statistical uncertainty. Finally, the present paper has also proposed a novel technique to determine the exact least-squares estimate of the phase offset using neutron-based
Fourier TOF measurements. A future publication will present the results of measurements with a prototype sinusoidal Fourier chopper. The novel technique to determine the phase offset will be implemented, and the statistical efficiency of sinusoidal modulation will be demonstrated.

### CRediT authorship contribution statement

**Thomas Neulinger:** Conceptualization, Writing – original draft, Writing – review & editing.

### 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

No data was used for the research described in the article.

### Acknowledgments

The author gratefully acknowledges many fruitful discussions of the concepts presented in this paper with Oliver Zimmer and Karina Bernert. 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. Statistical properties of estimators

In this appendix, statistical properties of various estimators are given. Use is made of the formula for the approximate covariance of two functions $f(\underline{X})$ and $g(\underline{X})$ of random variables $\underline{X} = X_1, X_2, \dots$, this being [16]

$$ \mathrm{Cov}(f(\underline{X}), g(\underline{X})) \approx \sum_{ij} \left. \frac{\partial f}{\partial X_i} \right|_{\underline{\mu}} \left. \frac{\partial g}{\partial X_j} \right|_{\underline{\mu}} \mathrm{Cov}(X_i, X_j), \quad (A.1) $$

where the derivatives are evaluated at the expectation value of the random variables $\underline{E}(\underline{X}) = \underline{\mu}$.

The central estimates in Fourier TOF analysis are those of the $c_k(\omega)$, defined in (6). $c_k(\omega)$ is both the expectation value and variance of the Poisson random variable $C_k(\omega)$, which is the observed number of counts in the kth channel. It is important to note that (A.1) is a good approximation only when the deviation of $\underline{X}$ about the mean $\underline{\mu}$ is small, which is not true of the individual $C_k(\omega)$ when $c_k(\omega)$ is small. However, (A.1) holds for the average of a large number $M$ of measurements $C_k(\omega)$. Thus, denoting an estimator by a hat,

$$ \hat{c}_k(\omega) = \frac{1}{M} \sum_{m=1}^{M} C_k^{(m)}(\omega), \quad (A.2) $$

where $C_k^{(m)}(\omega)$ is the mth of $M$ measurements of $C_k(\omega)$. This has expectation value and variance

$$ \underline{E}[\hat{c}_k(\omega)] = c_k, \quad (A.3a) $$

$$ \underline{\Psi}[\hat{c}_k(\omega)] = \frac{1}{M} c_k. \quad (A.3b) $$

In the following, use shall also be made of

$$ \begin{aligned} \mathrm{Cov}(\hat{c}_k, \hat{c}_{k'}) &= \mathrm{Cov} \left( \frac{1}{M} \sum_{m=1}^{M} C_k^{(m)}, \frac{1}{M} \sum_{m'=1}^{M} C_{k'}^{(m')} \right) \\ &= \frac{1}{M} \delta_{kk'} c_k. \end{aligned} \quad (A.4) $$

### A.1. Estimate of $Z(\omega)$

From (14) and (15), the estimate of $Z(\omega)$ is

$$ \hat{Z}(\omega) = \frac{1}{A_1} \frac{\hat{\omega}}{2\pi} \frac{\pi/K}{\sin(\pi/K)} \sum_{k=1}^{K} \hat{c}_k(\omega) e^{-i\pi(2k-1)/K}, \quad (A.5) $$

where $\hat{c}_k(\omega)$ is given by (A.2), and where $\hat{\omega}$ is the estimate of the angular modulation frequency, obtained in some manner by monitoring the chopper rotation speed during the measurement. To condense the notation, define

$$ z(\hat{\omega}) = \frac{1}{A_1} \frac{\hat{\omega}}{2\pi} \frac{\pi/K}{\sin(\pi/K)}. \quad (A.6) $$

The real and imaginary parts of (A.5) are

$$ \hat{Z}_1 = \hat{z} \sum_{k=1}^{K} \hat{c}_k \cos(\pi(2k-1)/K), \quad (A.7a) $$

$$ \hat{Z}_2 = -\hat{z} \sum_{k=1}^{K} \hat{c}_k \sin(\pi(2k-1)/K), \quad (A.7b) $$

where $\hat{z} = z(\hat{\omega})$ and $\hat{c}_k = \hat{c}_k(\omega)$ have been written for brevity. As an example of the application of (A.1), the approximate variances are

$$ \underline{\Psi}(\hat{Z}_1) \approx \left( \frac{\partial \hat{Z}_1}{\partial \hat{\omega}} \right)^2 \underline{\Psi}(\hat{\omega}) + \frac{1}{M} \sum_{k=1}^{K} \left( \frac{\partial \hat{Z}_1}{\partial \hat{c}_k} \right)^2 c_k, \quad (A.8a) $$

$$ \underline{\Psi}(\hat{Z}_2) \approx \left( \frac{\partial \hat{Z}_2}{\partial \hat{\omega}} \right)^2 \underline{\Psi}(\hat{\omega}) + \frac{1}{M} \sum_{k=1}^{K} \left( \frac{\partial \hat{Z}_2}{\partial \hat{c}_k} \right)^2 c_k, \quad (A.8b) $$

where (A.4) has been used, and it is to be understood that the derivatives are evaluated as in (A.1). Thus, one obtains

$$ \underline{\Psi}(\hat{Z}_1) \approx \left( \frac{Z_1}{\omega} \right)^2 \underline{\Psi}(\hat{\omega}) + \frac{z^2}{M} \sum_{k=1}^{K} c_k \cos^2(\pi(2k-1)/K), \quad (A.9a) $$

$$ \underline{\Psi}(\hat{Z}_2) \approx \left( \frac{Z_2}{\omega} \right)^2 \underline{\Psi}(\hat{\omega}) + \frac{z^2}{M} \sum_{k=1}^{K} c_k \sin^2(\pi(2k-1)/K). \quad (A.9b) $$

In the following, expressions like (A.8) will not be written out, and only the result will be stated. Thus

$$ \begin{aligned} \mathrm{Cov}(\hat{Z}_1, \hat{Z}_2) \approx \frac{Z_1 Z_2}{\omega \omega} \underline{\Psi}(\hat{\omega}) - \frac{z^2}{M} \sum_{k=1}^{K} c_k \cos(\pi(2k-1)/K) \\ \times \sin(\pi(2k-1)/K). \end{aligned} \quad (A.10) $$

### A.2. Estimate of $|Z(\omega)|$

From (A.5) it follows that

$$ |\hat{Z}| = \hat{z} \left[ \sum_{kk'} \hat{c}_k \hat{c}_{k'} \cos \left( \frac{2\pi(k'-k)}{K} \right) \right]^{\frac{1}{2}}. \quad (A.11) $$

Then

$$ \begin{aligned} \underline{\Psi}(|\hat{Z}|) \approx & \left(\frac{|\underline{Z}|}{\omega}\right)^2 \underline{\Psi}(\hat{\omega}) + \frac{1}{M} \frac{z^4}{|\underline{Z}|^2} \sum_{ijk} c_i c_j c_k \cos\left(\frac{2\pi(j-i)}{K}\right) \\ & \times \cos\left(\frac{2\pi(k-i)}{K}\right). \end{aligned} \quad (A.12) $$

The second right-hand term in (A.12) represents the contribution from counting statistics. Inserting into the denominator of this term an expression for $|Z|$ in terms of the $c_k(\omega)$, one obtains

$$ \underline{\Psi}_{\mathrm{stat.}}(|\hat{Z}|) \approx \frac{z^2 \sum_{ijk} c_i c_j c_k \cos\left(\frac{2\pi(j-i)}{K}\right) \cos\left(\frac{2\pi(k-i)}{K}\right)}{\sum_{ij} c_i c_j \cos\left(\frac{2\pi(i-j)}{K}\right)}. \quad (A.13) $$

Eq. (A.13) can be significantly developed by inserting the $c_k(\omega)$ in (10). To condense the notation, define

$$ \bar{R}_n(\omega) = \frac{T}{K} e^{in\phi} \bar{r}_n(\omega) \frac{\sin(\pi n/K)}{\pi n/K}, \quad (A.14) $$
so that (10) becomes

$$c_i(\omega) = \sum_{n=0}^{K-1} \tilde{R}_n(\omega) e^{i\pi n(2i-1)/K}. \qquad (A.15)$$

Note the difference between $i = \sqrt{-1}$ and the index $i$. Inserting (A.14) into the sum term in the denominator (den.) of (A.13) and simplifying, one obtains

$$ 
\begin{aligned}
\text{den.} &= \sum_{i,j} c_i c_j \cos \left( \frac{2\pi(i-j)}{K} \right) \\
&= \sum_{i,j} c_i c_j \exp \left( \frac{i2\pi(i-j)}{K} \right) \\
&= \sum_{i=1}^{K} \sum_{j=1}^{K} \sum_{m=0}^{K-1} \sum_{K=1}^{K-1} \tilde{R}_n \tilde{R}_m e^{i\pi n(2i-1)/K} e^{i\pi m(2j-1)/K} e^{i2\pi(i-j)/K} \\
&= \sum_{i,j,m,n} \tilde{R}_n \tilde{R}_m \exp \left( \frac{2\pi i}{K} \left[ i(n+1) + j(m-1) - \frac{1}{2}(n+m) \right] \right) \\
&= \sum_{i,m} \tilde{R}_n \tilde{R}_m e^{-i\pi(n+m)/K} \sum_{j} e^{i2\pi i(n+1)/K} \sum_{j} e^{i2\pi j(m-1)/K} \\
&= \sum_{m,n} \tilde{R}_n \tilde{R}_m e^{-i\pi(n+m)/K} K \delta_{n,K-1} K \delta_{m,1} \\
&= -K^2 \tilde{R}_{K-1} \tilde{R}_1,
\end{aligned}
\qquad (A.16) $$

where use has been made of the identity

$$\sum_{i=1}^{K} e^{i2\pi ip/K} = K\delta_{p,0} + K\delta_{p,K} \qquad (A.17)$$

for integer $0 \le p \le K$. The sum term in the numerator (num.) of (A.13) can be evaluated in a similar manner by additionally making use of

$$ 
\begin{aligned}
& \cos \left( \frac{2\pi(j-i)}{K} \right) \cos \left( \frac{2\pi(k-i)}{K} \right) \\
&= \frac{1}{4} \left( e^{i2\pi(j+k-2i)/K} + e^{-i2\pi(j+k-2i)/K} + e^{i2\pi(j-k)/K} + e^{-i2\pi(j-k)/K} \right).
\end{aligned}
\qquad (A.18) $$

Proceeding as before, one obtains

$$ \text{num.} = \frac{1}{4} K^3 (\tilde{R}_2 \tilde{R}_{K-1} \tilde{R}_{K-1} - \tilde{R}_{K-2} \tilde{R}_1 \tilde{R}_1 - \tilde{R}_0 \tilde{R}_{K-1} \tilde{R}_1 - \tilde{R}_0 \tilde{R}_1 \tilde{R}_{K-1}). \qquad (A.19) $$

Further development of the expressions for the denominator and numerator can be obtained via (9b), from which it may be shown that

$$\tilde{r}_{K-n} = -\frac{K-n}{n} e^{-iK\phi} \tilde{r}_{n}^{*}, \qquad (A.20)$$

where

$$r_{-n}(\omega) = A_{-n}S(-n\omega) = A_{n}^{*}S^{*}(n\omega) = r_{n}^{*}(\omega) \qquad (A.21)$$

has been used. From (A.20) then, it may also be shown that

$$\tilde{R}_{K-n} = -\tilde{R}_{n}^{*}. \qquad (A.22)$$

Thus, (A.16) and (A.19) become

$$\text{den.} = K^2 \tilde{R}_1^* \tilde{R}_1, \qquad (A.23a)$$

$$\text{num.} = \frac{1}{4} K^3 (\tilde{R}_2^* \tilde{R}_1^* \tilde{R}_1^* + \tilde{R}_2^* \tilde{R}_1 \tilde{R}_1 + 2 \tilde{R}_0^* \tilde{R}_1^* \tilde{R}_1). \qquad (A.23b)$$

Inserting these into (A.13), one obtains

$$\mathcal{V}_{\text{stat.}}(|\hat{Z}|) = \frac{z^2 \text{ num.}}{M} \frac{\text{den.}}{M} = \frac{z^2}{M} \frac{1}{2} K \left[ \tilde{R}_0 + \frac{\tilde{R}_2^* \tilde{R}_1 \tilde{R}_1 + \tilde{R}_2 \tilde{R}_1^* \tilde{R}_1^*}{2 \tilde{R}_1^* \tilde{R}_1} \right], \qquad (A.24)$$

which is purely real.

From (A.14), (9a), and (4a), the first term in the square brackets of (A.24) is

$$\tilde{R}_0 = \frac{T}{K} \tilde{r}_0 = \frac{T}{K} r_0 = \frac{T}{K} \left[ b + A_0 S(0) \right]. \qquad (A.25)$$

The second term is more involved. Examining $\tilde{R}_2$ and using (4b), one has

$$\tilde{R}_2 = \frac{T}{K} \frac{\sin(2\pi/K)}{2\pi/K} \sum_{q=-\infty}^{\infty} \frac{2e^{i(qK+2)\phi}}{qK+2} A_{qK+2} S[(qK+2)\omega], \qquad (A.26)$$

which is nonzero in general. However, for sinusoidal modulation, so long as $K > 3$, all $A_{qK+2}^-$ will be zero. Similarly, for triangular modulation, all $A_{qK+2}^\Delta$ will also be zero for even $K > 3$. Therefore

$$\tilde{R}_2^\sim = 0, \quad K > 3 \qquad (A.27a)$$

$$\tilde{R}_2^\Delta = 0, \quad K > 3 \text{ and } K \text{ even.} \qquad (A.27b)$$

Under these conditions, the second term in the square brackets of (A.24) vanishes, and so

$$\mathcal{V}_{\text{stat.}}(|\hat{Z}|) \approx \frac{\omega}{4\pi M} \left( \frac{1}{A_1} \frac{\pi/K}{\sin(\pi/K)} \right)^2 [b + A_0 S(0)], \qquad (A.28)$$

provided that the conditions in (A.27a) and (A.27b) are fulfilled for sinusoidal and triangular modulation, respectively.

The variance of $|\hat{Z}(\omega)|$ is directly proportional to $\omega$ and inversely proportional to $M$. However, $M$ is proportional to the product of the measurement time and $\omega$, and so $\mathcal{V}_{\text{stat.}}(|\hat{Z}|)$ decreases with measurement time at the same rate, independent of the modulation frequency.

### A.3. Estimate of $S(\omega)$

From (15),

$$\hat{S}(\omega) = e^{-i\hat{\phi}} \hat{Z}(\omega). \qquad (A.29)$$

The real and imaginary parts of $\hat{S}(\omega)$ are

$$\hat{S}_1(\omega) = \hat{Z}_1(\omega) \cos \hat{\phi} + \hat{Z}_2(\omega) \sin \hat{\phi} \qquad (A.30a)$$

$$\hat{S}_2(\omega) = -\hat{Z}_1(\omega) \sin \hat{\phi} + \hat{Z}_2(\omega) \cos \hat{\phi}. \qquad (A.30b)$$

Applying (A.1) to calculate the quantities of interest, one obtains

$$ \mathcal{V}(\hat{S}_1) \approx (S_2)^2 \mathcal{V}(\hat{\phi}) + \cos^2(\phi) \mathcal{V}(\hat{Z}_1) + \sin^2(\phi) \mathcal{V}(\hat{Z}_2) + \sin(2\phi) \text{Cov}(\hat{Z}_1, \hat{Z}_2), \qquad (A.31a) $$

$$ \mathcal{V}(\hat{S}_2) \approx (S_1)^2 \mathcal{V}(\hat{\phi}) + \sin^2(\phi) \mathcal{V}(\hat{Z}_1) + \cos^2(\phi) \mathcal{V}(\hat{Z}_2) - \sin(2\phi) \text{Cov}(\hat{Z}_1, \hat{Z}_2), \qquad (A.31b) $$

$$ \text{Cov}(\hat{S}_1, \hat{S}_2) \approx -S_1 S_2 \mathcal{V}(\hat{\phi}) + \frac{1}{2} \sin(2\phi) \{-\mathcal{V}(\hat{Z}_1) + \mathcal{V}(\hat{Z}_2)\} + \cos(2\phi) \text{Cov}(\hat{Z}_1, \hat{Z}_2). \qquad (A.31c) $$

The estimate $|\hat{S}(\omega)| = |\hat{Z}(\omega)|$. Thus, the variance of $|\hat{S}(\omega)|$ is given by (A.12).

### A.4. Estimate of $s(t)$

From (41), the estimate of $s(t)$ is given by

$$\hat{s}(t) = \frac{1}{2\pi} \sum_{n=-N}^{N} \Delta\omega \hat{S}(\omega_n) e^{i\hat{\omega}_n t}, \qquad (A.32)$$

where $\Delta\omega$ is taken to be a constant. More involved schemes might use the trapezoid rule to approximate (5b) and, e.g., $\Delta\omega_n = \hat{\omega}_{n+1} - \hat{\omega}_n$ inside the sum. In terms of the real and imaginary parts of $\hat{S}(\omega)$, (A.32) is

$$ \hat{s}(t) = \frac{\Delta\omega}{2\pi} \sum_{n=-N}^{N} \left( \begin{aligned} & [\hat{S}_1(\omega_n) \cos(\hat{\omega}_n t) - \hat{S}_2(\omega_n) \sin(\hat{\omega}_n t)] \\ & + i [\hat{S}_1(\omega_n) \sin(\hat{\omega}_n t) + \hat{S}_2(\omega_n) \cos(\hat{\omega}_n t)] \end{aligned} \right). \qquad (A.33) $$

As $s(t)$ is purely real, one should disregard the imaginary part. This is automatically zero if measurements are only made for one sense
of rotation, e.g., $\omega > 0$, and $\hat{S}(\omega)$ is constructed for $\omega < 0$ using $S(-\omega) = S^*(\omega)$. In any case, one takes

$$ \hat{s}(t) = \frac{\Delta\omega}{2\pi} \sum_{n=-N}^{N} \left[ \hat{S}_1(\omega_n) \cos(\hat{\omega}_n t) - \hat{S}_2(\omega_n) \sin(\hat{\omega}_n t) \right]. \quad (A.34) $$

Thus

$$ \begin{aligned} \Psi(\hat{s}) \approx & \left(\frac{\Delta\omega}{2\pi}\right)^2 \sum_{n=-N}^{N} \left( t^2 \left[ S_1(\omega_n) \sin(\omega_n t) + S_2(\omega_n) \cos(\omega_n t) \right]^2 \Psi(\hat{\omega}_n) \right. \\ & \left. + \cos^2(\omega_n t) \Psi[\hat{S}_1(\omega_n)] + \sin^2(\omega_n t) \Psi[\hat{S}_2(\omega_n)] \right. \\ & \left. - \sin(2\omega_n t) \text{Cov}[\hat{S}_1(\omega_n), \hat{S}_2(\omega_n)] \right). \end{aligned} \quad (A.35) $$

As was done for $|\hat{S}(\omega)|$, it is instructive to consider the variance of $\hat{s}(t)$ due to counting statistics only. Inserting (A.9) and (A.10) into (A.31), and then inserting the result into (A.35) while considering only the terms containing $c_k$, one obtains after simplification

$$ \Psi_{\text{stat.}}(\hat{s}) \approx 4 \left(\frac{\Delta\omega}{2\pi}\right)^2 \sum_{n=1}^{N} \sum_{k=1}^{K} c_k(\omega_n) \frac{z_n^2(\omega_n)}{M(\omega_n)} \cos^2 \left[\omega_n t - \phi - \frac{\pi(2k-1)}{K}\right], \quad (A.36) $$

where the number of modulations $M(\omega_n)$ at frequency $\omega_n/(2\pi)$ is chosen by the experimentalist. In (A.36), the $n=0$ term, corresponding to $\hat{S}(0)$, has been removed as it is not directly measured in the same way as the other $\hat{S}(\omega_n)$ for $\omega_n \neq 0$. In addition, the sum over $\omega_n$ ranging from $n=1$ to $N$ is accounted for by the factor of 4 in front of (A.36).

Using (A.15), (A.36) can be simplified in the same manner as was done for the statistical variance of $|\hat{Z}(\omega)|$, discussed earlier in this appendix. Letting $x_n = \omega_n t - \phi$, one has

$$ \cos^2 \left[ x_n - \frac{\pi(2k-1)}{K} \right] = \frac{1}{4} \left( e^{i2x_n - i\pi(2k-1)/K} + e^{-i2x_n + i\pi(2k-1)/K} + 2 \right). \quad (A.37) $$

Inserting this into (A.36) then yields the real quantity

$$ \Psi_{\text{stat.}}(\hat{s}) \approx \left(\frac{\Delta\omega}{2\pi}\right)^2 \sum_{n=1}^{N} \frac{z_n^2 K}{M_n} \left[ \tilde{R}_2(\omega_n) e^{i2x_n} + \tilde{R}_2^*(\omega_n) e^{-i2x_n} + 2\tilde{R}_0(\omega_n) \right], \quad (A.38) $$

where $z_n = z(\omega_n)$ and $M_n = M(\omega_n)$. As before, if $K > 3$ then $\tilde{R}_2^\Delta = 0$ for sinusoidal modulation, and if $K$ is also even, then $\tilde{R}_2^\Delta = 0$. Provided that these conditions hold,

$$ \Psi_{\text{stat}}(\hat{s}) \approx 2 \left(\frac{\Delta\omega}{2\pi}\right)^2 \sum_{n=1}^{N} \frac{z_n^2}{M_n} K \tilde{R}_0, \quad (A.39) $$

or, after inserting $\tilde{R}_0$ from (A.25) and the definition of $z(\omega_n)$,

$$ \Psi_{\text{stat.}}(\hat{s}) \approx 2 \left(\frac{\Delta\omega}{2\pi}\right)^2 \left(\frac{1}{A_1} \frac{\pi/K}{\sin(\pi/K)}\right)^2 \left[b + A_0 S(0)\right] \sum_{n=1}^{N} \frac{\omega_n}{2\pi M_n}. \quad (A.40) $$

which may be compared to (A.28). Thus the statistical variance of $\hat{s}(t)$ is uniform in $t$, and is proportional to the sum of the variances of $|\hat{Z}(\omega_n)|$ over all modulation frequencies.

Note that $2\pi M_n/\omega_n$ is the total measurement time at the modulation frequency $\omega_n/(2\pi)$. Call the total measurement time over all frequencies

$$ t_{\text{meas.}} = \sum_{n=1}^{N} M_n \frac{2\pi}{\omega_n} \quad (A.41) $$

Using the method of Lagrange multipliers, one may show that the choice

$$ M_n = \frac{t_{\text{meas.}}}{N} \frac{\omega_n}{2\pi} \quad (A.42) $$

minimizes the variance (A.40) subject to the constraint that $t_{\text{meas.}}$ is fixed. That is, the measurement time at each modulation frequency is the same. Therefore,

$$ \Psi_{\text{stat.}}(\hat{s}) \approx \frac{2}{t_{\text{meas.}}} \left(\frac{\Omega}{2\pi}\right)^2 \left(\frac{1}{A_1} \frac{\pi/K}{\sin(\pi/K)}\right)^2 \left[b + A_0 S(0)\right], \quad (A.43) $$

where $\Omega = N\Delta\omega$. The statistical variance is thus proportional to the average rate at the detector $b + A_0 S(0)$. If the maximum modulation frequency $\Omega/(2\pi)$ is increased for fixed $t_{\text{meas.}}$, less time is spent measuring at the individual modulation frequencies $\omega_n/(2\pi)$, and the variance thus increases. For increasing $A_1$, the contribution to the count rate from $S(\omega)$ becomes larger, and the variance thus decreases. Finally, for increasing $K$, the sensitivity of the measurement to the phase of the modulated beam is improved, and the variance thus decreases.

## Appendix B. Harmonic distortion for a delta function TOF spectrum

For physical TOF spectra $s(t)$ it is the case that $S(0) \ge |S(\omega)|$ for all $\omega$. This can be seen by examining

$$ |S(\omega)|^2 = \int_{-\infty}^{\infty} dt' \int_{-\infty}^{\infty} dt' s(t) s(t') \cos(\omega(t-t')), \quad (B.1) $$

which follows from (5a), and observing that this expression is maximum when $\omega = 0$. A more restrictive claim is made in Ref. [6], which is that $|S(\omega)|$ “generally tends to decrease more or less rapidly with increasing $\omega$, particularly when the spectrum $[s(t)]$ contains only a few or no sharp features (peaks, edges, etc.)”. This line of reasoning motivated the author of Ref. [6] to consider as a “rough upper limit for the magnitude of the higher-harmonic distortions” the scenario in which $S(0) = |S(\omega)|$ for all $\omega$. Such a relationship is true for the delta function TOF spectrum

$$ s(t) = S(0)\delta(t - \tau), \quad (B.2a) $$

$$ S(\omega) = S(0)e^{-i\omega\tau}, \quad (B.2b) $$

where $\tau$ is the peak location. The approximate result communicated in Ref. [6] was that, for triangular modulation, the harmonic distortion error is largest for $K=4$ channels, being roughly 3%. This and similar calculations in Ref. [6] agree quantitatively with approximate results given in Ref. [10] for a narrow Gaussian spectrum. The treatments in these two previous works reported results for $K=3, 4$, and 8. In this appendix, the phase offset $\phi$ is incorporated into the analysis, and exact results for arbitrary $K$ are given.

The harmonic distortion for a delta function TOF spectrum as would be measured by triangular modulation is found by inserting (B.2b) into (18), whereupon one obtains

$$ S_{\text{harm.}}^{\Delta}(\omega | K) = S(\omega) \sum_{q \in Q_K} \frac{e^{iqK(\phi - \omega\tau)}}{(qK + 1)^3}, \quad (B.3) $$

where the set $Q_K$ contains all integers $q \neq 0$ such that $qK + 1$ is odd, i.e., $Q_K = \{\pm1, \pm2, \pm3, \dots\}$ for $K$ even and $Q_K = \{\pm2, \pm4, \pm6, \dots\}$ for $K$ odd. The relative magnitude of the harmonic distortion is then

$$ \begin{aligned} |f(\omega\tau - \phi | K)| &= \frac{|S(\omega) + S_{\text{harm.}}^{\Delta}(\omega | K)|}{|S(\omega)|} \\ &= \left| 1 + \sum_{q \in Q_K} \frac{e^{-iqK(\omega\tau - \phi)}}{(qK + 1)^3} \right|. \end{aligned} \quad (B.4) $$

The task at hand is thus to find the extrema of this expression. The work may be reduced by noting that the sum in (B.4) is identical for odd $K_{\text{odd}}$ and even $K_{\text{even}} = 2K_{\text{odd}}$, and so only needs to be evaluated for one case. Furthermore, it may be seen that $|f(x | K)|$ is periodic, where $x = \omega\tau - \phi$, with period $2\pi/K$, and so only the interval, e.g., $x \in (-\pi/K, \pi/K]$ needs to be searched. Consider then $K$ even, in which case

$$ |f(x | K)| = \left| \sum_{q=-\infty}^{\infty} \frac{e^{-iqKx}}{(qK + 1)^3} \right| \quad (B.5) $$
and the sum runs over all integers $q$.

The local extrema of $|f(x|K)|$ occur wherever its derivative or, equivalently, that of $|f(x|K)|^2$ vanishes. Calling this latter quantity $g(x|K)$, one may show that

$$g(x|K) = \frac{d}{dx} |f(x|K)|^2 = \sum_{q,p=-\infty}^{\infty} \frac{(p-q)K \sin[(q-p)Kx]}{(qK+1)^3 (pK+1)^3}, \quad (B.6)$$

which may be seen to have zeros at $x = n\pi/K$ for integers $n$. That these are the only zeros of $g(x|K)$ may be demonstrated in the following way. For a particular value, e.g., $K = 4$, it may be verified graphically that $g(x|K)$ is negative on the interval $x \in (-\pi/K, 0)$, positive on the interval $x \in (0, \pi/K)$, and zero at $x = 0$ and $x = \pi/K$, these then being the locations of the minimum and maximum of $|f(x|K)|$, respectively. It may also be shown that

$$g(x|K_1) = \left(\frac{K_2}{K_1}\right)^6 \frac{\sin^2(\pi/K_2)}{\sin^2(\pi/K_1)} g\left(x\frac{K_1}{K_2}\bigg|K_2\right). \quad (B.7)$$

Therefore, the graphs of all $g(x|K)$ are equivalent after appropriate horizontal and vertical scaling, and so what holds for one value of $K$ holds for all values.

It then remains to evaluate $|f(x|K)|$ at $x = 0$ and $x = \pi/K$. At these points, it may be seen from (B.4) that $f(x|K)$ is real. Consider first

$$f(0|K) = 1 + \sum_{q \in Q_K} \frac{1}{(qK + 1)^3} = 1 - \Sigma_1(K), \quad (B.8)$$

where the sign of $\Sigma_1(K)$ has been chosen with the result in mind. For $K$ even, the sum runs over all integers $q \neq 0$, and an exact expression for $\Sigma_1(K)$ may be found. As discussed previously, the sum is identical for $K_{\text{even}} = 2K_{\text{odd}}$. Therefore, the combined result is

$$\Sigma_1(K) = 1 - \left(\frac{\pi}{K'}\right)^3 \cot\left(\frac{\pi}{K'}\right) \csc^2\left(\frac{\pi}{K'}\right), \quad K' = \begin{cases} K, & K \text{ even} \\ 2K, & K \text{ odd} \end{cases}. \quad (B.9)$$

Similarly

$$f(\pi/K|K) = 1 + \sum_{q \in Q_K} \frac{(-1)^q}{(qK+1)^3} = 1 + \Sigma_2(K). \quad (B.10)$$

Again, for $K$ even, the sum runs over all integers $q \neq 0$, and $\Sigma_2(K)$ may be calculated exactly. The combined result is

$$\Sigma_2(K) = -\Sigma_1(2K') - \left(\frac{\pi}{2K'}\right)^3 \cot\left(\frac{\pi K' + 1}{2K'}\right) \csc^2\left(\frac{\pi K' + 1}{2K'}\right), \\ K' = \begin{cases} K, & K \text{ even} \\ 2K, & K \text{ odd} \end{cases}. \quad (B.11)$$

The sums $\Sigma_1(K)$ and $\Sigma_2(K)$, both of which are positive, are plotted in Fig. B.3 for several values of $K$. There is only a small difference in the magnitude of $\Sigma_1(K)$ and $\Sigma_2(K)$, with the former being slightly larger.

Fig. B.4 shows the variation of $|f(x|K)|$ for the values of $K$ reported in Ref. [6,10]. The solid curves are numerical calculations of (B.4), whereas the horizontal dashed lines have been calculated using the exact expressions (B.9) and (B.11). The variations reported in Ref. [6] were approximately $\pm 3\%$ for $K = 4$, $\pm 0.5\%$ for $K = 3$, and $\pm 0.15\%$ for $K = 8$. More precise values for these and other $K$ are given in Table B.1 as percentages.

The formulas given in this appendix are valid for nonzero phase offset $\phi$, which has the effect of horizontally shifting the pattern of maxima and minima in the oscillations of the measured $|Z(\omega)|$ about the true $|S(\omega)|$. It should be noted that, for spectra with multiple sharp peaks, as are present in diffractometry experiments, the harmonic distortion can significantly exceed 3%. This is a consequence of the tendency of the Fourier transforms of the individual peaks to interfere destructively in $|S(\omega)|$, resulting in an increase in the relative magnitude of the harmonic distortion for some frequencies.

<figure><img src="image_6.png" /><figcaption>Fig. B.3. The sums $\Sigma_1(K)$ and $\Sigma_2(K)$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. B.4. Variation in the absolute value of $f(x|K)$ for several $K$.</figcaption></figure>

<table><caption>Table B.1 Values of $\Sigma_1(K)$ and $\Sigma_2(K)$ for several $K$, calculated using (B.9) and (B.11).</caption><thead><tr><th>K</th><th>Σ<sub>1</sub> [%]</th><th>Σ<sub>2</sub> [%]</th></tr></thead><tbody><tr><td>3</td><td>0.547</td><td>0.483</td></tr><tr><td>4</td><td>3.105</td><td>2.772</td></tr><tr><td>5</td><td>0.067</td><td>0.059</td></tr><tr><td>6</td><td>0.547</td><td>0.483</td></tr><tr><td>7</td><td>0.017</td><td>0.015</td></tr><tr><td>8</td><td>0.167</td><td>0.147</td></tr><tr><td>9</td><td>0.006</td><td>0.005</td></tr><tr><td>10</td><td>0.067</td><td>0.059</td></tr></tbody></table>

## Appendix C. Sinusoidal stator shapes

As discussed in Section 2, the stator and rotor holes of a triangular modulation Fourier chopper are annular sections with minimum and maximum radii $\rho_1$ and $\rho_2$, respectively, and angular extent $\Delta\theta$ such that $2\Delta\theta$ evenly divides $2\pi$ radians. The stator hole shape may be modified in various ways so that when combined with the “standard” rotor holes of a triangular modulation chopper, the neutron beam is sinusoidally
<figure><img src="image_8.png" /><figcaption>Fig. C.5. When combined with the standard rotor hole shape (thin black), various stator hole shapes (thick red) can produce pure sinusoidal modulation: (a) upper contour modified, (b) lower contour modified, (c) upper and lower contours modified. All stator holes have the same area. The rotor moves counterclockwise.</figcaption></figure>

modulated. One approach is to deform only the upper contour of the stator hole, as is shown in Fig. C.5(a). Call $\psi$ the rotational coordinate of the rotor. One wishes to determine the upper contour $\rho(\theta)$ so that the area of overlap is

$$A(\psi) = A \sim \left\{ \frac{1}{2} + \frac{1}{2} \cos \left( \pi \frac{\psi}{\Delta\theta} \right) \right\}. \qquad (C.1)$$

Suppose that the rotor rotates so that $\psi$ is increasing. Then for $-\Delta\theta < \psi < 0$, the leading edge of the rotor hole lies along $\theta = \psi + \Delta\theta/2$, and the area of overlap is

$$ 
\begin{aligned}
A(\psi) &= \int_{-\frac{\Delta\theta}{2}}^{\psi + \frac{\Delta\theta}{2}} d\theta \int_{\rho_1}^{\rho(\theta)} \rho \, d\rho \\
&= -\frac{1}{2}(\psi + \Delta\theta)\rho_1^2 + \frac{1}{2}\int_{-\frac{\Delta\theta}{2}}^{\psi + \frac{\Delta\theta}{2}} d\theta \rho^2(\theta).
\end{aligned} 
\qquad (C.2) $$

Equating (C.1) and (C.2) and taking the derivative with respect to $\psi$, one obtains

$$ -A \sim \frac{\pi}{\Delta\theta} \sin \left( \pi \frac{\psi}{\Delta\theta} \right) = -\rho_1^2 + \rho^2 \left( \psi + \frac{\Delta\theta}{2} \right). \qquad (C.3) $$

Rearranging gives

$$ \rho(\theta) = \sqrt{\rho_1^2 + A \sim \frac{\pi}{\Delta\theta} \cos\left(\pi \frac{\theta}{\Delta\theta}\right)}. \qquad (C.4) $$

Calling $\rho_2$ the maximum of $\rho(\theta)$ at $\theta = 0$, one then has that $A \sim = \Delta\theta(\rho_2^2 - \rho_1^2)/\pi$, and so

$$ \rho(\theta) = \sqrt{\rho_1^2 + (\rho_2^2 - \rho_1^2) \cos \left( \pi \frac{\theta}{\Delta\theta} \right)}. \qquad (C.5) $$

Since the problem is symmetric with respect to $\theta$, this solution extends to $0 < \psi < \Delta\theta$ and so is valid for $|\theta| \le \theta/2$.

Another approach to obtain sinusoidal modulation is to deform the lower contour as is shown in Fig. C.5(b). Proceeding as before, one finds that the stator hole area is the same, with the lower contour given by

$$ \rho(\theta) = \sqrt{\rho_2^2 - (\rho_2^2 - \rho_1^2) \cos \left( \pi \frac{\theta}{\Delta\theta} \right)}, \qquad (C.6) $$

and where now $\rho_1 = \rho(0)$ is the minimum radius. It is of course also possible to simultaneously deform the upper and lower contours as in (C.5) and (C.6), respectively, using an intermediate radius $\rho_0$. An example is shown in Fig. C.5(c). Such a stator hole has the same area $A \sim$, independent of the value $\rho_0$.

The most general approach specifies one contour and then deforms the other so that the stator produces sinusoidal modulation. Intersection of the upper and lower contours should only occur at

$$ \rho_{\text{upper}}(\theta = \pm \Delta\theta/2) = \rho_{\text{lower}}(\theta = \pm \Delta\theta/2). \qquad (C.7) $$

Following the steps that led to (C.3), one obtains

$$ -A \sim \frac{\pi}{\Delta\theta} \sin \left( \pi \frac{\psi}{\Delta\theta} \right) = \rho_{\text{upper}}^2 \left( \psi + \frac{\Delta\theta}{2} \right) - \rho_{\text{lower}}^2 \left( \psi + \frac{\Delta\theta}{2} \right). \qquad (C.8) $$

Supposing that $\rho_{\text{lower}}(\theta)$ has been specified, rearrangement yields

$$ \rho_{\text{upper}}(\theta) = \sqrt{\rho_{\text{lower}}^2(\theta) + A \sim \frac{\pi}{\Delta\theta} \cos \left( \pi \frac{\theta}{\Delta\theta} \right)}. \qquad (C.9) $$

Without loss of generality, one may define $\rho_1$ and $\rho_2$ as $\rho_{\text{lower}}(0)$ and $\rho_{\text{upper}}(0)$, respectively. Thus, one recovers $A \sim = \Delta\theta(\rho_2^2 - \rho_1^2)/\pi$, irrespective of the hole shape. The ratio of the areas of a sinusoidal and triangular modulation chopper is therefore

$$ \frac{A \sim}{A^\Delta} = \frac{\Delta\theta(\rho_2^2 - \rho_1^2)/\pi}{\Delta\theta(\rho_2^2 - \rho_1^2)/2} = \frac{2}{\pi} \approx 0.64, \qquad (C.10) $$

which is also the ratio of the beam intensities for the two choppers. It is therefore puzzling that a value of 0.7 was reported in Ref. [9]. Unfortunately, details of the stator geometry in that work, which was published 50 years ago, are not provided.

## Appendix D. Exact least-squares estimate of the phase offset

The quantity $Q(\varphi, \alpha)$ defined in (37) may be written in matrix form as

$$ Q(\varphi, \alpha) = \left( e^{-i\varphi} \underline{Z} - \underline{\Omega} \underline{\alpha} \right)^{\dagger} \underline{V}^{-1} \left( e^{-i\varphi} \underline{Z} - \underline{\Omega} \underline{\alpha} \right), \qquad (D.1) $$

where a single underline denotes a column or row vector, a double underline denotes a matrix, and $\dagger$ denotes the transpose complex conjugate. $\underline{Z}$ is the $J \times 1$ column vector with entries $\hat{Z}(\omega_j)$. $\underline{V}^{-1}$ is the $J \times J$ diagonal matrix

$$ \left( \underline{V}^{-1} \right)_{ij} = \delta_{ij} [\hat{\mathcal{V}}(|\hat{Z}(\omega_j)|)]^{-1}, \qquad (D.2) $$

where $\mathcal{V}(|\hat{Z}(\omega_j)|)$ is given by (A.12); $\hat{\mathcal{V}}(|\hat{Z}(\omega_j)|)$ may be obtained by substituting into that equation the appropriate estimates. In minimizing (D.1), it is necessary to choose $N$, the order of $\underline{\alpha} = \{\alpha_0, \alpha_1, \dots, \alpha_{N-1}\}$. Thus, $\underline{\alpha}$ is a $N \times 1$ column vector. $\underline{\Omega}$ is then the $J \times N$ matrix with entries

$$ \left( \underline{\Omega} \right)_{jn} = \frac{1}{(n-1)!} (-i\omega_j)^{n-1}, \qquad (D.3) $$

where $j = 1, \dots, J$ and $n = 1, \dots, N$; the entries are the coefficients of the powers of $t$ in the expansion of $e^{-i\omega_j t}$. The desired estimates $\hat{\varphi}$ and $\hat{\alpha}$ are obtained via the $N+1$ equations

$$ 0 = \left. \frac{\partial Q}{\partial \varphi} \right|_{\varphi=\hat{\varphi}, \underline{\alpha}=\underline{\hat{\alpha}}}, \qquad (D.4a) $$

$$ 0 = \left. \frac{\partial Q}{\partial \alpha_i} \right|_{\varphi=\hat{\varphi}, \underline{\alpha}=\underline{\hat{\alpha}}}. \qquad (D.4b) $$

Calculating (D.4b) directly, one obtains

$$ \frac{\partial Q}{\partial \alpha_i} = -2 \left( \mathfrak{R} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} e^{-i\varphi} \underline{Z} \right] \right)_i + 2 \left( \mathfrak{R} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{\Omega} \right] \right)_{ij} \left( \underline{\alpha} \right)_j, \qquad (D.5) $$
where repeated indices are summed and where $\Re(x)$ denotes the real part of a complex number $x$. Setting this derivative equal to zero and solving for $\underline{\alpha}$, one finds that

$$ \underline{\alpha}(\varphi) = \left( \mathfrak{R} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{\Omega} \right] \right)^{-1} \mathfrak{R} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} e^{-i\varphi} \underline{Z} \right] \quad (D.6) $$

minimizes $Q$ for a given value of $\varphi$. Substituting this expression back into $Q$, one has

$$ Q(\varphi) = \left( e^{-i\varphi} \underline{Z} - \underline{\Omega} \underline{\alpha}(\varphi) \right)^{\dagger} \underline{V}^{-1} \left( e^{-i\varphi} \underline{Z} - \underline{\Omega} \underline{\alpha}(\varphi) \right), \quad (D.7) $$

which is a function of only $\varphi$. Expanding, this can be written as

$$ Q(\varphi) = Q_0 + Q_1 e^{-i2\varphi} + Q_1^* e^{i2\varphi}, \quad (D.8) $$

where

$$ \begin{aligned} Q_0 = & \underline{Z}^{\dagger} \underline{V}^{-1} \underline{Z} - \left[ \underline{Z}^{\dagger} \underline{V}^{-1} \underline{\Omega} \right] \underline{U} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{Z} \right] \\ & + \frac{1}{4} \left[ \underline{Z}^{\dagger} \underline{V}^{-1} \underline{\Omega} \right] \underline{U} \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{\Omega} \underline{U} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{Z} \right] \\ & + \frac{1}{4} \left[ \underline{Z}^{\dagger} \underline{V}^{-1} \underline{\Omega} \right]^* \underline{U} \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{\Omega} \underline{U} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{Z} \right]^* \end{aligned} \quad (D.9) $$

is real,

$$ \begin{aligned} Q_1 = & -\frac{1}{2} \left[ \underline{Z}^{\dagger} \underline{V}^{-1} \underline{\Omega} \right]^* \underline{U} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{Z} \right] \\ & + \frac{1}{4} \left[ \underline{Z}^{\dagger} \underline{V}^{-1} \underline{\Omega} \right]^* \underline{U} \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{\Omega} \underline{U} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{Z} \right] \end{aligned} \quad (D.10) $$

is complex, and

$$ \underline{U} = \left( \mathfrak{R} \left[ \underline{\Omega}^{\dagger} \underline{V}^{-1} \underline{\Omega} \right] \right)^{-1} \quad (D.11) $$

is a real symmetric matrix. Thus, (D.8) can be further rewritten as

$$ Q(\varphi) = Q_0 + 2|Q_1| \cos(2\varphi - \theta_1), \quad (D.12) $$

where $\theta_1$ is the phase of the complex number $Q_1$.

In this form, it is clear that the minima of $Q(\varphi)$ repeat with periodicity $\pi$. The derivative

$$ \frac{\partial Q}{\partial \varphi} = -4|Q_1| \sin(2\varphi - \theta_1) \quad (D.13) $$

has zeros when $2\varphi - \theta_1 = n\pi$ for integer $n$, while the second derivative

$$ \frac{\partial^2 Q}{\partial \varphi^2} = -8|Q_1| \cos(2\varphi - \theta_1) \quad (D.14) $$

is positive, as it must be at the minimum of $Q$, only when $2\varphi - \theta_1 = n\pi$ for $n$ odd. Taking this into account, one obtains

$$ \hat{\phi} = \frac{1}{2}(\pi + \theta_1) + m\pi \quad (D.15) $$

for integer $m$. This determines the phase only up to the factor $\pi$. However, as the $a_n$ are all positive, so too must be the $\hat{a}_n$. Therefore, one may choose $m = 0$ in (D.15) and then compute $\alpha(\hat{\phi})$ via (D.6). If the resulting $\alpha(\hat{\phi}) > 0$, then $\hat{\phi}$ may be accepted as the phase estimate, otherwise $\hat{\phi} \pm \pi$ is correct.

The uncertainties in $\hat{\phi}$ and $\hat{a}$ are obtained from the curvature of $Q(\varphi)$ at its minimum. Thus, from (D.14), the estimate of the variance of $\hat{\phi}$ is

$$ \hat{\Psi}(\hat{\phi}) = \left( \frac{1}{2} \frac{\partial^2 Q}{\partial \varphi^2} \bigg|_{\varphi=\hat{\phi}} \right)^{-1} = \frac{1}{4|Q_1|}, \quad (D.16) $$

which is independent of $\hat{a}$. Similarly, from (D.5), the estimate of the inverse of the covariance matrix of the $\alpha_i$ is

$$ \frac{1}{2} \frac{\partial^2 Q}{\partial \alpha_i \partial \alpha_j} = \left( \underline{U}^{-1} \right)_{ij}, \quad (D.17) $$

which is independent of $\hat{\phi}$ and $\hat{a}$.

## Appendix E. Mathematical considerations of Fourier reconstruction

In an actual Fourier TOF experiment, it is not possible to calculate $s(t)$ via

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

as this would require an infinite number of measurements of $S(\omega)$. Instead one calculates

$$ s_{\text{disc.}}(t) = \frac{1}{2\pi} \sum_{n=-N}^{N} \Delta\omega S(\omega_n) e^{i\omega_n t}, \quad (E.2) $$

in which the frequency range is truncated and discretized compared to (E.1). The consequences of these two modifications are discussed in this appendix.

### E.1. Truncation

Consider an arbitrary function $f(t)$ having Fourier transform $F(\omega)$. Truncating the range of integration of the inverse Fourier transform, one obtains the function

$$ f_{\text{trunc.}}(t | \Omega) = \frac{1}{2\pi} \int_{-\Omega}^{\Omega} d\omega F(\omega) e^{i\omega t}. \quad (E.3) $$

Substituting the Fourier transform into the right-hand side of this expression and integrating over $\omega$, one obtains

$$ f_{\text{trunc.}}(t | \Omega) = \int_{-\infty}^{\infty} dt' f(t') \left\{ \frac{1}{\pi} \frac{\sin[\Omega(t-t')]}{t-t'} \right\}. \quad (E.4) $$

The function in curly braces with which $f(t)$ is convolved has maximum value $\Omega/\pi$, width of the central peak that decreases with increasing $\Omega$, and unit integral over the whole real line; it is a delta function in the limit $\Omega \to \infty$. However, from (E.3), it may be seen that even for finite values of $\Omega$, the functions $f(t)$ and $f_{\text{trunc.}}(t)$ can be equal. This will occur when $\Omega > \omega_{\text{band}}$, where the bandwidth $\omega_{\text{band}}$ of $F(\omega)$ is defined by $F(\omega) = 0$ for $|\omega| > \omega_{\text{band}}$. Furthermore, the sampling theorem [17] states that, provided that $\Omega > \omega_{\text{band}}$, the value of $f(t)$ at any $t$ can be computed by interpolating discrete samples spaced by $\Delta t = \pi/\Omega$. That is,

$$ f(t) = \sum_{n=-\infty}^{\infty} f(n\Delta t) \frac{\sin[\Omega(t - n\Delta t)]}{\Omega(t - n\Delta t)}. \quad (E.5) $$

The second of the two original Fourier chopper papers, Ref. [5], discusses the significance of the grid of sample points $t = n\Delta t$. However, this discussion appears to be somewhat imprecise. The claim is made that “The sampling theorem states that a Fourier inversion of $F(\omega)$ over a finite range of frequencies will yield [the right-hand side of (E.5)]”. On the contrary, it is the Fourier inversion

$$ \tilde{f}_{\text{trunc.}}(t | \Omega) = \frac{1}{2\pi} \int_{-\Omega}^{\Omega} d\omega \tilde{F}(\omega | \Omega) e^{i\omega t} \quad (E.6) $$

that accomplishes this, where

$$ \tilde{F}(\omega | \Omega) = \sum_{n=-\infty}^{\infty} F(\omega - n2\Omega) \quad (E.7) $$

is a sum of copies of $F(\omega)$ spaced by $2\Omega$. The key point is that these copies will overlap if $\Omega \le \omega_{\text{band}}$. To obtain the right-hand side of (E.5), one proceeds as follows. Inserting the definition of the Fourier transform into (E.7), one obtains

$$ \begin{aligned} \tilde{F}(\omega | \Omega) &= \int_{-\infty}^{\infty} dt f(t) e^{-i\omega t} \sum_{n=-\infty}^{\infty} e^{in2\Omega t} \\ &= \int_{-\infty}^{\infty} dt f(t) e^{-i\omega t} \left[ \Delta t \sum_{n=-\infty}^{\infty} \delta(t - n\Delta t) \right] \\ &= \Delta t \sum_{n=-\infty}^{\infty} f(n\Delta t) e^{-i\omega n\Delta t}, \end{aligned} \quad (E.8) $$
where the summation on the first line has been recognized as the Fourier series of the Dirac comb. Thus

$$
\begin{align}
\tilde{f}_{\text{trunc.}}(t | \Omega) &= \frac{\Delta t}{2\pi} \sum_{n=-\infty}^{\infty} f(n\Delta t) \int_{-\Omega}^{\Omega} d\omega e^{i\omega(t-n\Delta t)} \tag{E.9} \\
&= \sum_{n=-\infty}^{\infty} f(n\Delta t) \frac{\sin[\Omega(t-n\Delta t)]}{\Omega(t-n\Delta t)}, \nonumber
\end{align}
$$

which is now the right-hand side of (E.5). Note that when $\tilde{f}_{\text{trunc.}}(t | \Omega)$ is evaluated on the grid $t = m\Delta t$ for integers $m$, that

$$
\frac{\sin[\Omega(m-n)\Delta t]}{\Omega(m-n)\Delta t} = \frac{\sin[\pi(m-n)]}{\pi(m-n)} = \delta_{mn}, \qquad (\text{E.10})
$$

and so one obtains the exact result, for all values of $\Omega$, that

$$
\tilde{f}_{\text{trunc.}}(n\Delta t | \Omega) = f(n\Delta t). \qquad (\text{E.11})
$$

It is only when $\Omega > \omega_{\text{band}}$, that $\tilde{f}_{\text{trunc.}}(t | \Omega) = f(t)$, and (E.5) holds.

The discussion in Ref. [5] also neglects to mention an important detail regarding the application of the sampling theorem to TOF spectra. This is that, because all physical TOF spectra $s(t)$ have finite support, i.e., are nonzero only for a finite range of $t$, all $S(\omega)$ have non-finite bandwidth. Therefore, (E.5) is only approximate when applied to TOF spectra, and

$$
s(t) \approx \sum_{n} s(n\Delta t) \frac{\sin[\Omega(t - n\Delta t)]}{\Omega(t - n\Delta t)} \qquad (\text{E.12})
$$

for $\Omega$ large enough that $S(\Omega) \approx 0$, so that the sum contains a finite number of terms.

E.2. Discretization

In this section, several relationships are derived between the TOF spectrum $s(t)$ and its discrete approximation $s_{\text{disc.}}(t)$. The first is obtained by inserting the definition of the Fourier transform into (E.2). Interchanging the order of summation and integration, one obtains

$$
s_{\text{disc.}}(t) = \frac{\Delta\omega}{2\pi} \int_{-\infty}^{\infty} dt' s(t') \sum_{n=-N}^{N} e^{i\omega_n(t-t')}, \qquad (\text{E.13})
$$

where $\omega_n = n\Delta\omega$. As was noted in Section 5, $s_{\text{disc.}}(t)$ is periodic with period $2\pi/\Delta\omega$. The summation term responsible for the periodic behavior may be evaluated directly. Defining this as $\tilde{w}(t-t')$, one finds

$$
\tilde{w}(t) = \sum_{n=-N}^{N} e^{i\omega_n t} = \frac{\sin\left[\left(N + \frac{1}{2}\right)\Delta\omega t\right]}{\sin\left[\frac{1}{2}\Delta\omega t\right]}. \qquad (\text{E.14})
$$

Thus, $s_{\text{disc.}}(t)$ is the convolution of $s(t)$ with the symmetric periodic function $\tilde{w}(t)$. The latter peaks at zero, having maximum value $2N + 1$, and passes through zero on either side of the origin at

$$
t = \pm \frac{1}{2N+1} \frac{2\pi}{\Delta\omega} = \pm \frac{1}{2 + N^{-1}} \frac{2\pi}{\Omega}. \qquad (\text{E.15})
$$

For large $N$, the width of $\tilde{w}(t)$ decreases mainly with increasing $\Omega$, the maximum angular modulation frequency. An example of $\tilde{w}(t)$ for $N = 20$ is plotted in green in Fig. E.6.

Additional insight can be gained by further developing (E.13) in the
following way. Note that, because $\tilde{w}(t)$ is periodic, one may find $w(t)$,
not unique, so that

$$
\tilde{w}(t) = \sum_{n=-\infty}^{\infty} w\left(t - n \frac{2\pi}{\Delta\omega}\right). \qquad (\text{E.16})
$$

Calling $\tilde{W}(\omega)$ and $W(\omega)$ the Fourier transforms of $\tilde{w}(t)$ and $w(t)$, respectively, one has

$$
\begin{align*}
\tilde{W}(\omega) &= \sum_{n=-\infty}^{\infty} \int_{-\infty}^{\infty} dt \, w\left(t - n \frac{2\pi}{\Delta\omega}\right) e^{-i\omega t} \\
&= \int_{-\infty}^{\infty} dt \, w(t) e^{-i\omega t} \sum_{n=-\infty}^{\infty} e^{-i2\pi n\omega / \Delta\omega} \tag{E.17} \\
&= \Delta\omega \sum_{n=-\infty}^{\infty} W(\omega)\delta(\omega - \omega_n),
\end{align*}
$$

where the summation on the second line has been recognized as
the Fourier series of the Dirac comb. At the same time, the Fourier
transform of the term in the middle of (E.14) yields

$$
\tilde{W}(\omega) = \sum_{n=-N}^{N} 2\pi\delta(\omega - \omega_n). \qquad (\text{E.18})
$$

By comparison of the last line of (E.17) and (E.18), it may be seen that
one requires

$$
W(n\Delta\omega) = \begin{cases} 2\pi/\Delta\omega, & n = -N, -N+1, \dots, N \\ 0, & n \text{ other integers} \end{cases}, \qquad (\text{E.19})
$$

which only constrains $W(\omega)$ for the discrete values $\omega = \omega_n$. Therefore,
one is free to choose

$$
W(n\Delta\omega) = \begin{cases} 2\pi/\Delta\omega, & |n| \le N' \\ 0, & |n| > N' \end{cases} \qquad (\text{E.20})
$$

for continuous values of $n$, where $N'$ is an as yet unknown value
$N \le N' < N + 1$. Thus, taking the inverse Fourier transform of this
expression,

$$
\begin{align*}
w(t) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} d(n\Delta\omega) W(n\Delta\omega) e^{in\Delta\omega t} \\
&= \frac{\Delta\omega}{2\pi} \int_{-N'}^{N'} dn \frac{2\pi}{\Delta\omega} e^{in\Delta\omega t} \tag{E.21} \\
&= \frac{\sin\left[N'\Delta\omega t\right]}{\frac{1}{2}\Delta\omega t},
\end{align*}
$$

so that

$$
\tilde{w}(t) = \sum_{n=-\infty}^{\infty} \frac{\sin\left[N'\Delta\omega\left(t - n - \frac{2\pi}{\omega}\right)\right]}{\frac{1}{2}\Delta\omega\left(t - n - \frac{2\pi}{\omega}\right)}. \qquad (\text{E.22})
$$

The value of $N'$ can be determined by evaluating this expression at $t = 0$. Only the $n = 0$ term contributes on the right-hand side, and one finds that $\tilde{w}(0) = 2N'$. Also, (E.14) can be evaluated at $t = 0$, and one finds that $\tilde{w}(0) = 2N + 1$. Therefore $N' = N + 1/2$ and so

$$
w(t) = \frac{\sin\left[\left(N + \frac{1}{2}\right)\Delta\omega t\right]}{\frac{1}{2}\Delta\omega t}. \qquad (\text{E.23})
$$

As with $\tilde{w}(t)$, the maximum value of $w(t)$ is $2N + 1$, and its first zeros are given by (E.15). An example of $w(t)$ for $N = 20$ is plotted in black in Fig. E.6.

Substituting (E.23) into (E.13), one has

$$
\begin{align}
s_{\text{disc.}}(t) &= \frac{\Delta\omega}{2\pi} \int_{-\infty}^{\infty} dt' s(t') \tilde{w}(t-t') \nonumber \\
&= \frac{\Delta\omega}{2\pi} \int dt' s(t') \sum_{n=-\infty}^{\infty} w\left(t-t' - n\frac{2\pi}{\Delta\omega}\right), \tag{E.24}
\end{align}
$$

which can be further rearranged into the form

$$
s_{\text{disc.}}(t) = \frac{\Delta\omega}{2\pi} \int dt' w(t-t') \sum_{n=-\infty}^{\infty} s\left(t'-n\frac{2\pi}{\Delta\omega}\right). \qquad (\text{E.25})
$$

Thus, $s_{\text{disc.}}(t)$ is also the convolution of the aperiodic function $w(t)$ with copies of $s(t)$ spaced by $2\pi/\Delta\omega$. If the longest TOF present in
<figure><img src="image_9.png" /><figcaption>Fig. E.6. Plots of $\tilde{w}(t)$ and $w(t)$ for $2\pi/\Delta\omega = 10$ ms, i.e., $\Delta\omega/(2\pi) = 100$ Hz, and $N = 20$, corresponding to the maximum measurement frequency $\Omega/(2\pi) = 2$ kHz.</figcaption></figure>

the experiment is less than $2\pi/\Delta\omega$, then the copies of $s(t)$ will be non-overlapping, giving

$$s_{\text{disc.}}(t) \approx \frac{\Delta\omega}{2\pi} \int dt' w(t-t')s(t') \quad (\text{E.26})$$

for large $N$. The result (E.25) is especially important in the analysis of harmonic distortion, and leads to the result (44), which explains the presence of so-called negative “ghost peaks” [6] in the reconstructed TOF spectrum.

## Appendix F. Comparison to standard TOF

In this appendix, Fourier and standard TOF, which will be abbreviated as FTOF and STOF, respectively, are compared in the use case that the same TOF resolution is desired. Consider then a Fourier chopper and a standard chopper consisting of a stator and rotor disk of the same geometries, with the only difference being that the standard chopper rotor has a single hole. In this case, the standard chopper is open for an isolated interval of time $t_{\text{open}}$ once every rotation, or “frame”, having period $t_{\text{frame}}$, this being greater than the maximum TOF $t_{\text{max}}$ to avoid frame overlap. Then, for example, one might have over one full period $-t_{\text{open}}/2 \le t \le t_{\text{frame}} - t_{\text{open}}/2$

$$A(t) = \begin{cases} A \sim \left\{ \frac{1}{2} + \frac{1}{2} \cos \left( \frac{2\pi t}{t_{\text{open}}} \right) \right\}, & |t| \le t_{\text{open}}/2 \\ 0, & t_{\text{open}}/2 \le t \le t_{\text{frame}} - t_{\text{open}}/2 \end{cases} \quad (\text{F.1})$$

for comparison to a sinusoidal Fourier chopper, or, e.g.,

$$A(t) = \begin{cases} A^{\Delta} \left( 1 - \left| \frac{2t}{t_{\text{open}}} \right| \right), & |t| \le t_{\text{open}}/2 \\ 0, & t_{\text{open}}/2 \le t \le t_{\text{frame}} - t_{\text{open}}/2 \end{cases} \quad (\text{F.2})$$

for comparison to a triangular Fourier chopper.

From (1), the expectation value and variance of the number of counts in a time bin of width $\Delta t$ centered on time $t$ is

$$c(t) = b\Delta t + \int_{t-\Delta t/2}^{t+\Delta t/2} dt' \int_{-\infty}^{\infty} dt'' A(t' - t'')s(t''), \quad (\text{F.3})$$

where the chopper timing offset in this case has been taken to be zero. If $t_{\text{frame}}$ is long enough to avoid frame overlap, then in the limits

$$\Delta t \ll s(t) \left| \frac{ds}{dt} \right|^{-1}, \quad (\text{F.4a})$$

$$t_{\text{open}} \ll s(t) \left| \frac{ds}{dt} \right|^{-1}, \quad (\text{F.4b})$$

(F.3) becomes

$$c(t) \approx b\Delta t + \Delta t A_0 t_{\text{open}} s(t), \quad 0 \le t \le t_{\text{frame}}, \quad (\text{F.5})$$

where $A_0$ is the $n=0$ coefficient of the Fourier series of the periodic chopper opening function; for example, if $A(t)$ is given by (F.1), $A_0 = A_0^\Delta$ in (20a), and if $A(t)$ is (F.2), $A_0 = A_0^\Delta$ in (17a).

As the measurement is repeated with each rotation of the chopper, the expectation value of the counts $c(t)$ can be estimated by

$$\hat{c}(t) = \frac{1}{F} \sum_{n=1}^{F} C^{(n)}(t), \quad (\text{F.6})$$

where $F$ is the number of frames, and the $C^{(n)}(t)$ are the observed counts in the bin at time $t$ after the beginning of the $n$th frame. Similarly, the estimate of the background rate $b$ is

$$\hat{b} = \frac{1}{F |\mathcal{T}|} \sum_{n=1}^{F} \sum_{t \in \mathcal{T}} \frac{C^{(n)}(t)}{\Delta t}, \quad (\text{F.7})$$

where $\mathcal{T}$ is the set of size $|\mathcal{T}|$ consisting of all time bins within the frame in which $s(t) \approx 0$, i.e., in which there is only background. Thus, in STOF, the estimate of $s(t)$ from (F.5) is

$$\hat{s}(t) = \frac{-\hat{b}\Delta t + \hat{c}(t)}{A_0 t_{\text{open}} \Delta t}. \quad (\text{F.8})$$

When a measurement of the TOF spectrum is desired, an important consideration is the measurement efficiency, i.e., the duration of time required to achieve a given level of statistical uncertainty. One is therefore interested in the contribution from counting statistics to the variance of the estimate (F.8). Using (A.1), this is

$$\Psi_{\text{stat.}}^{\text{STOF}} [\hat{s}(t)] \approx \frac{1}{(A_0 t_{\text{open}})^2} \Psi(\hat{b}) + \frac{1}{(A_0 t_{\text{open}} \Delta t)^2} \Psi[\hat{c}(t)]. \quad (\text{F.9})$$

From (F.6) and (F.7), this becomes

$$\Psi_{\text{stat.}}^{\text{STOF}} [\hat{s}(t)] \approx \frac{1}{F(A_0 t_{\text{open}})^2} \left( \frac{b}{|\mathcal{T}| \Delta t} + \frac{c(t)}{\Delta t^2} \right), \quad (\text{F.10})$$

which, after inserting (F.5), becomes

$$\Psi_{\text{stat.}}^{\text{STOF}} [\hat{s}(t)] \approx \frac{1}{F(A_0 t_{\text{open}})^2 \Delta t} \left[ b \left( 1 + \frac{1}{|\mathcal{T}|} \right) + A_0 t_{\text{open}} s(t) \right]. \quad (\text{F.11})$$

One would like to compare this to the statistical variance of $\hat{s}(t)$ (A.43) obtained using FTOF, i.e.,

$$\Psi_{\text{stat.}}^{\text{FTOF}}(\hat{s}) \approx \frac{2}{t_{\text{meas.}}} \left(\frac{\Omega}{2\pi}\right)^2 \left(\frac{1}{A_1} \frac{\pi/K}{\sin(\pi/K)}\right)^2 \left[b + A_0 S(0)\right], \quad (\text{F.12})$$

where $t_{\text{meas.}}$ is the total measurement time over all modulation frequencies, $\Omega = N\Delta\omega$ is the maximum angular modulation frequency in the $n=1, \dots, N$ measurements of the Fourier transform of the TOF spectrum $S(n\Delta\omega)$, and where this expression holds for sinusoidal modulation if $K > 3$, and for triangular modulation if $K$ is also even.

To compare (F.11) and (F.12) for equal measurement time, one should set $F = t_{\text{meas.}}/t_{\text{frame}}$ in the former. In addition, recall that in FTOF, $\Delta\omega$ is made small enough that $2\pi/\Delta\omega$ exceeds the longest TOF present in the measurement. Thus, one should set $\Delta\omega = 2\pi/t_{\text{frame}}$. The TOF resolution is determined by the widths of $A(t)$ and $w(t)$ (E.23) for STOF and FTOF, respectively. Take then the first zeros of $A(t)$ and $w(t)$, respectively, as measures of the widths. The zeros of $A(t)$ closest to $t=0$ occur at $t = \pm t_{\text{open}}/2$, while the same is true of $w(t)$ at $t = \pm \pi/\Omega$. Therefore, for $A(t)$ and $w(t)$ to have approximately
<figure><img src="image_10.png" /><figcaption>Fig. F.7. Hypothetical TOF spectrum consisting of multiple isolated peaks. In the absence of background and for K = 4 with FTOF, all measurements of s(t) above the dashed red line will have higher variance with STOF for equal measurement times if similar TOF resolution is required. See the text for details.</figcaption></figure>

the same widths, one can take $\Omega/(2\pi) = 1/t_{open}$, which results also in $N = t_{frame}/t_{open}$. Finally, since the width of the time bins in the STOF measurement should be on the order of the width of $A(t)$, this suggests that it is reasonable also to take $\Delta t = t_{open}$. Therefore, the number of time bins in the STOF measurement will be the same as the number of modulation frequencies in the FTOF measurement. With these replacements, dividing (F.11) by (F.12) for the case that $|\vec{\tau}| \gg 1$, one obtains

$$ \frac{\psi_{\text{stat.}}^{\text{STOF}}[\hat{s}(t)]}{\psi_{\text{stat.}}^{\text{FTOF}}(\hat{s})} \approx \frac{1}{2} \left( \frac{A_1}{A_0} \frac{\sin(\pi/K)}{\pi/K} \right)^2 t_{\text{frame}} \frac{b/t_{\text{open}} + A_0 s(t)}{b + A_0 S(0)}. \quad (\text{F.13}) $$

Call the first factor on the right-hand side of (F.13)

$$ \alpha = \frac{1}{2} \left( \frac{A_1}{A_0} \frac{\sin(\pi/K)}{\pi/K} \right)^2 . \quad (\text{F.14}) $$

Then for the smallest value $K = 4$, $\alpha \sim \approx 0.10$ and $\alpha^\Delta \approx 0.067$ for sinusoidal and triangular modulation, respectively, while in the limit of large $K$, $\alpha \to 1/8 = 0.125$ and $\alpha^\Delta \to 8/\pi^4 \approx 0.082$. The magnitude of (F.13) is therefore mainly controlled by the remaining terms.

The behavior of (F.13) is particularly simple in the case of zero background. One has

$$ \frac{\psi_{\text{stat.}}^{\text{STOF}}[\hat{s}(t)]}{\psi_{\text{stat.}}^{\text{FTOF}}(\hat{s})} \approx \frac{\alpha t_{\text{frame}} s(t)}{S(0)}, \quad b = 0. \quad (\text{F.15}) $$

This expression has the following simple geometric interpretation. Considering the graph of $s(t)$, the variance of an STOF measurement of $s(t)$ will exceed that of an FTOF measurement when the rectangular area $\alpha t_{frame}s(t)$ exceeds $S(0)$, which is the area under the curve $s(t)$. An example is shown in Fig. F.7 with a hypothetical spectrum plotted using arbitrary units. Suppose that one wished to measure the height of the peak $s(t^*)$, indicated by the blue point, using either FTOF with $K = 4$ or STOF. In this example, the black shaded area under $s(t)$ is unity, while the blue shaded area $t_{frame}s(t^*)$ is roughly 40 times this. Thus, $\alpha \sim t_{frame}s(t^*)/S(0) \approx 4$, and so for equal measurement times, the uncertainty on a measurement of $s(t^*)$ with STOF would be double that for FTOF. The dashed horizontal red line indicates the value of $s(t)$ below which the variance is smaller for STOF measurements. Thus, if

one were instead interested in the height of the peak below this level, STOF would be preferred.

For nonzero background, the situation is slightly different, and depends also on the desired TOF resolution, which is of order $t_{open}$. In this case

$$ \frac{\psi_{\text{stat.}}^{\text{STOF}}[\hat{s}(t)]}{\psi_{\text{stat.}}^{\text{FTOF}}(\hat{s})} \approx \frac{\alpha \frac{b}{A_0} \left( \frac{t_{\text{open}}}{t_{\text{frame}}} \right)^{-1} + \alpha t_{\text{frame}} s(t)}{\frac{b}{A_0} + S(0)}, \quad (\text{F.16}) $$

where this expression has been arranged to emphasize the connection to (F.15). One sees in (F.16) that the contribution from the background increases as the duty cycle $t_{open}/t_{frame}$ decreases. It may therefore be the case that the STOF variance is significantly influenced in situations where there is relatively large background and in which high TOF resolution is also desired. In situations dominated by background, the ratio of the variances is controlled by the duty cycle, so that if even moderate TOF resolution is required, FTOF is to be preferred.

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