Modulation of Intensity Emerging from Zero Effort (MIEZE) is a neutron resonant spin echo technique that allows one to measure time correlation scattering functions in materials by implementing radio-frequency (RF) intensity modulation at the sample and the detector. The technique avoids neutron spin manipulation between the sample and the detector and, thus, could find applications in cases where the sample depolarizes the neutron beam. However, the finite sample size creates a variance in the path length between the locations where scattering and detection happen, which limits the contrast in intensity modulation that one can detect, in particular, toward long correlation times or large scattering angles. We propose a modification to the MIEZE setup that will enable one to extend those detection limits to longer times and larger angles. We use Monte Carlo simulations of a neutron scattering beamline to show that by tilting the RF flippers in the primary spectrometer with respect to the beam direction, one can shape the wave front of the intensity modulation at the sample to compensate for the path variance from the sample and the detector. The simulation results indicate that this change enables one to operate a MIEZE instrument at much increased RF frequencies, thus improving the effective energy resolution of the technique. For the MIEZE instrument simulated, it shows that for an incident beam with the maximum divergence of 0.33°, the maximum Fourier time can be increased by a factor of 3.

## I. INTRODUCTION

Neutron Spin Echo (NSE) is a method of neutron scattering that encodes the velocity (energy) of the neutron in the Larmor spin precession phase,^{1}

where *γ* = −1.832 × 10^{8} rad s^{−1} T^{−1} is the gyromagnetic ratio of the neutron and the magnetic field integral [*∫*sign(*B*)*dl*] is over the modulus of the magnetic field along the neutron trajectory. A neutron passing through two identical magnetic field regions with *B*_{1} = −*B*_{2} will have a total Larmor phase *ϕ* = 0 so that the final polarization is the same as the initial polarization. If a sample is placed in between these two regions and exchanges energy with the neutron, then the neutron velocity and, thus, the Larmor phase through the second magnetic field region will be different, resulting in a nonzero Larmor phase, *ϕ* ≠ 0. Using this technique, a very high energy resolution can be achieved by measuring the change in the polarization

where Δ*E* is the energy transfer during scattering and *ℏ* is the reduced Planck constant. The Fourier time *τ* is a parameter that depends on the field integral, given in Eq. (1), and the third power of the neutron wavelength, *λ*^{3}, and is effectively used to tune the energy resolution of the setup. A weakness of the NSE method is in the technical difficulties that arise when the sample depolarizes the beam, or when a very large magnetic field is desired at the sample. With the Modulation of Intensity Emerging from Zero Effort (MIEZE) technique, it can perform all the spin manipulations only before the neutron reaches the sample and is, therefore, able to accommodate depolarizing samples and complicated sample environments.^{2} MIEZE has been demonstrated at both constant wavelength neutron source^{3,4} and pulsed neutron source.^{5–8}

MIEZE involves a pair of compact radio-frequency (RF) spin flippers operated at different angular frequencies of *ω*_{1} and *ω*_{2}, as shown in Fig. 1(a). The phase of the intensity modulation seen at the detector position is given by

where *L*_{1,2} are the distances from the centers of the two flippers to the sample, *L*_{s} is the distance from the sample to the detector, and *v* is the neutron velocity. As one can see, this phase depends on the neutron time-of-flight (*t*) from the first flipper to the detector. Any variation in the time-of-flight will eventually cause a phase aberration and a loss of contrast in the detector signal. Such a variation could be contributed from the wavelength dispersion, the path length variance, and other effects. To minimize the phase aberration due to the dispersion in the neutron wavelength (or the velocity), the velocity-dependent term in Eq. (3) can be minimized by choosing

Equation (4) is also called the time focusing condition. With a polarization analyzer after the second RF flipper, which only picks up the polarization vector along the analyzing direction, an intensity modulation in time can be produced, yielding

where *A* is the amplitude of modulations and *C* is the time-averaged intensity. A variance in the time-of-flight from the sample to the detector due to quasi-elastic scattering will reduce the contrast, which can be used to determine the intermediate scattering function

where **Q** is the scattering vector and *τ*_{MIEZE} is the Fourier time. For MIEZE, *τ*_{MIEZE} is given by

where *m* is the neutron mass, *h* is Planck’s constant, *L*_{s} is the distance from the sample to the detector, and *λ* is the neutron wavelength.

Another term in Eq. (3) causing a phase aberration is due to the variation in the path length between the sample and the detector ($Ls\u2032$), as shown in Fig. 1(a). For the neutron trajectories emerging from the center or the edge of the sample, the difference in the time-of-flight to the detector is given by

where s is the transverse distance of the scattering event from the centerline, 2*θ* is the scattering angle, and the last expression is to the lowest order in s. In practice, this means that MIEZE is limited to studying small samples at low scattering angles, as the product *s* sin 2*θ* must be small. In the small angle neutron scattering (SANS) regime, one can operate MIEZE at MHz frequency, whereas at larger scattering angles, one finds oneself bound by *ω*_{1,2} ∼ 2*π* · 10 kHz. Previous studies of the resolution function have noted this correlation and concluded that MIEZE is best suited for a SANS configuration.^{9–11}

## II. CORRECTING THE PHASE VARIANCE OF MIEZE

One may attempt to remove the limitation to stay in the SANS regime by tilting the RF flippers relative to the beam. This is a common practice in the neutron resonant spin echo (NRSE) work to achieve the so-called “phonon focusing” to measure the lifetime of dispersive excitations of quasi-particles^{2,12} in inelastic neutron scattering and Larmor diffraction for high resolution measurements of lattice spacing. For MIEZE, Weber *et al.* have tried to incline the RF flippers to minimize the aberrations caused by the axial asymmetry of the setup introduced by the chosen Montel reflecting mirrors.^{10} The simulation was performed at a transmission geometry instead of large scattering angles, and there is no discussion regarding the correction of the aberration introduced by the sample size. Alternatively, one may also rotate a flat, disk-like sample relative to the beam. The latter was put forward previously.^{9–11}

### A. Tilting the RF flippers

Consider the two parallel neutron trajectories shown in Fig. 1(b), where one interacts with the center of the sample and the other with the edge of the sample at a displacement s from the center. With the RF flippers tilted by an angle *β* and the sample perpendicular to the beam direction (*α* = 0), the phase of the neutron spin at the detector is given by

where $Ls\u2212Ls\u2032=s\u22c5sin\u20612\u2061\theta $ from the above was inserted. Thus, one can, to the lowest order in s, reduce the effect of the sample size by tilting the RF flippers such that

In combination with the standard MIEZE time focusing condition, given in Eq. (4), the phase variance due to a finite size sample is minimized (for a parallel beam) at a scattering angle beyond the SANS regime.

### B. Tilting the sample

This was studied analytically in Refs. 9 and 11, and the optimal sample rotation angle was found to be impractical for small angle scattering. Here, we study whether this is a practical method at very large scattering angles. Therefore, we will briefly discuss the effects of rotating the sample. Consider the setup shown in Fig. 1(b), where the sample is tilted by an angle *α*, and the RF flippers are not tilted.

The phase at the detector for a neutron trajectory with a distance s from the center of the sample is now given by

which is similar to Eq. (10). Note, however, that the angle *α* enters with the opposite sign. To the first order, one now has

and from this, it follows that the leading order term proportional to s is canceled by satisfying the following condition:

This condition illustrates that when 2*θ* is beyond the SANS regime, the optimal angle *α* = *θ* ± *π*/2 dictates reflection geometry for a disk-like flat sample, which becomes “practical” when 2*θ* ≳ 60°. Otherwise, *α* is too large, and most of the beam passes on the side of a “typical” sample without intercepting it.

## III. McStas SIMULATION OF THE PHASE CORRECTION

### A. McStas model of the MIEZE setup

To validate this method, simulations were performed using the neutron beamline simulation software McStas.^{13,14} An RF spin flipper component was coded in-house, and the details have been included in the Appendix. As shown in Eq. (7), the Fourier time can be varied by changing the sample to the detector length *L*_{s}, the wavelength *λ*, or the frequency difference between the RF spin flippers Δ*ω* = 2Δ*f* = *ω*_{2} − *ω*_{1}. Changing *L*_{s} would also change the variance in the path length from the sample to the detector so it is preferable for this study to set it to a constant. Changing *λ* will also change the scattering vector ** Q**, which is measured, and so it is favorable to study the maximum Fourier times for a given scattering angle and wavelength of neutron by scanning the difference of Δ

*ω*. A problem with this, in practice, is that the cancellation of the velocity-dependent phase, given in Eq. (4), requires that the ratio

*ω*

_{2}/

*ω*

_{1}be a constant based on the geometry of the beamline. Due to the Bloch–Siegert shift,

^{15}the flipper frequencies cannot be set arbitrarily low, and there is also a practical limit to the maximum operation frequency, which together limit the range of Δ

*ω*achievable with a set geometry. Effectively, these limits set the dynamic range of a MIEZE setup.

It was demonstrated that the velocity-dependent phase can also be subtracted by using a guide field similar to NSE in between the RF flippers, increasing the dynamic range by several orders of magnitude at RESEDA.^{16} To explore and maximize the dynamic range of the setup instead of using a static guide field, we use an additional pair of RF spin flippers operated with the same frequency between the MIEZE RF flippers, which we call phase flippers. In this configuration, the frequency difference of the two MIEZE flippers can be set independently without being bound by Eq. (4). The phase aberration due to the wavelength dispersion can be canceled by operating the phase flippers at the frequency,

where *D* is the distance between the phase flippers, and the RF flipper lengths are assumed to be much smaller than *D*. With the RF flippers we have tested previously,^{17} this method has been demonstrated experimentally with detector frequencies in the 10 mHz–800 kHz range for a single geometry configuration.

A full schematic of the MIEZE setup used for the simulations is shown in Fig. 2. The source was circular in the beam cross section with a radius that scaled with the sample size and maximum divergence settings, focusing neutrons to the sample. The neutron wavelength used is 5.5 Å with a uniform half spread of 0.5 Å. The sample is a disk shape of the generic incoherent and elastic scattering type with a thickness in the beam path of 1 mm and variable radius in the beam cross section. The key dimensions of the beamline, such as *L*_{1}, *L*_{2}, *L*_{s}, and the detector size were chosen to be close to the existing MIEZE beamline RESEDA at the Heinz Maier-Leibnitz Zentrum.^{3} The RF flippers have a dimension similar to Refs. 17 and 18 along the beam direction of 10 cm when not tilted. The detector has a 10 × 10 cm^{2} active area instead of 20 × 20 cm^{2} used at RESEDA, and for simplicity, we assume an infinitesimally thin detector. This is a reasonable simplification for this study because at large scattering angles, the signal at the detector generally has a period greater than 50 *µ*s. The intensity modulations on the detector are integrated spatially across the detector such that the data can be represented as a one-dimensional modulation in the time domain. Note that we did not apply any post-data reduction procedure here to correct for the path variation caused by the size of the detector. The contrast for each simulation is determined by performing a simple sine fit to the intensity as a function of time for 2 periods of the signal in 20 total time bins. For simulations testing the effect of the RF flipper angle, the sample angle is set to *α* = 0, and for simulations testing the effect of the sample angle, the RF flipper angle is set to *β* = 0.

### B. Tilting the RF flippers

The analytical calculation of Eq. (11) assumed a parallel beam (no divergence), and so it is prudent to study the effect of beam divergence. Figure 3 shows the contrast of the modulation as a function of the RF flipper angle (*β*) for various divergence angles. For a very low divergence, the optimum RF flipper angle *β* well agrees with Eq. (11), whereas for a larger beam divergence, the optimum angle *β* becomes smaller, and the maximum contrast is lower. The reason for this effect is that the frequency difference between the two RF flippers introduces modulations in the time domain, whereas tilting the RF flippers generates a modulation in the space domain. Because both modulations are directly tied to the RF flippers, their focal planes are coupled with each other. With the MIEZE condition satisfied, it means that the focal planes in both the time and space domains superimpose at the detector position. Consequently, as discussed in Ref. 19, the wave front of the time modulation at the sample position for a given scattering angle cannot be precisely shaped, which increases the aberrations at increasing beam divergence.^{19}

While the simulations in Fig. 3 used a detector angle of *θ*_{D} = 2*θ*, the work in Ref. 11 found that it is preferred to maintain *θ*_{D} = 0 such that the detector is always perpendicular to the incoming beam in a conventional MIEZE setup. To investigate the effect of the detector orientation to the contrast of the modulation, a series of scans were performed for two situations: nontilted and tilted RF flippers, with a scattering angle of 2*θ* = 60°. As shown in Fig. 4, it appears that for both cases, to maximize the contrast at a large scattering angle, the detector needs to be orientated to be perpendicular to the scattered beam instead of the incident beam.^{11} In addition, by tilting the RF flippers, the peak contrast is higher and the acceptance angle of the detector can be greatly increased.

With the RF flipper optimally tilted, a series of simulations were performed to determine how much the maximum Fourier time could be improved for a given scattering vector ** Q**. This is carried out by scanning the difference between the MIEZE RF flipper frequencies, observing how the contrast

*P*=

*A*/

*C*at the detector is reduced with an increased Fourier time, and determining the point

*τ*

_{max}, where

*P*(

*τ*

_{max}) = 1/

*e*.

Figure 5 shows the increase in the Fourier time that can be achieved with a 2 cm radius sample with two different incident beam divergences. In Fig. 5, the minimum Fourier times are cutoff for visibility, as only the highest Fourier times are of interest here. For a 0.33° beam divergence, the maximum Fourier time is significantly increased by a factor of 3 by tilting the RF flippers. By comparison, at the 0.67° incident divergence, the gain in the Fourier time is a little less than a factor of 2. Therefore, the balance between the neutron flux and the beam collimation plays a crucial role in the potential improvement in the Fourier time with this method. We have also confirmed that there is less to gain when the sample is smaller but more when the sample is larger, which is expected. For example, when the sample radius is reduced to 1 cm in our chosen configuration, the contrast gained by tilting the RF flippers to optimum angles is a factor of 1.7 for the 0.33° divergence and 1.2 for the 0.67° divergence.

### C. Tilting the sample

The increase in the achievable Fourier time was also studied for a case where the scattering angle is very large, 2*θ* = 120°, and the flat disk-shaped sample is rotated to the reflection geometry. The simulation setup is the same as before, except now the RF spin flippers are not tilted (*β* = 0°), and the sample is tilted in the scattering plane by an angle *α*. Figure 6 shows the results for a scattering angle 2*θ* = 120° and incident beam with the maximum divergence of 0.67°. Here, the theoretical best sample angle is *α* = −30°. Tilting the sample results in an even more significant increase in the maximum Fourier time, a factor of 15 increase for a 2 cm radius sample. We note that this configuration is not strongly affected by the beam divergence by observing a minimal contrast reduction up to 2° divergence.

## IV. DISCUSSION

We have shown that it is possible to substantially increase the achievable Fourier time with the MIEZE technique by tilting the RF spin flippers in the incident beam. While the Fresnel or Pythagoras coil can correct for the leading quadratic phase aberrations in the regular neutron spin echo instrument caused by the beam divergence and the magnetic field inhomogeneity, the tilting of the RF flippers will correct for the leading first order phase aberration introduced by the transverse size of the sample. The Fresnel or Pythagoras coils will still be required for MIEZE when the quadratic term is a concern at the long spin echo time. Similar to a radar, the tilting of the RF flippers will maximize the contrast of the modulation at a particular scattering angle just like the rotation of the radar antenna does to the area of detection. The proposed modification to an existing experimental setup enables one to go away from the SANS regime with MIEZE. As such, it promises to be an impactful improvement for MIEZE. The effectiveness of the proposed modification increases with larger samples, which provides an additional benefit as it will help to increase the count rate in an experiment. However, a large beam divergence will significantly decrease this benefit again since the wave front cannot be perfectly shaped by the tilting. By employing post-data reduction to further correct the path variation caused by the detector size, the contrast of the modulation in the simulation can be further increased.^{20,21}

We have also demonstrated that, alternatively, at very large scattering angles, the sample may be tilted instead of the RF flippers. While this was originally considered for the SANS regime,^{9,11} we have shown that a substantial increase in the Fourier time may be achieved this way at large angles. The best practice may be to tilt the RF flippers for scattering less than 90° and to tilt the sample for larger scattering angles.

## ACKNOWLEDGMENTS

This work was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy. This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC05-00OR22725.

The manuscript was authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a nonexclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: McSTAS COMPONENT OF THE RF NEUTRON RESONANT SPIN FLIPPER

The RF flipper component in McStas uses a time-dependent magnetic field of the form

where *B*_{0} and *B*_{g} are the static field and the gradient field, respectively. By turning *B*_{g} on and off, one can switch between adiabatic (*B*_{g} ≠ 0) and nonadiabatic (*B*_{g} = 0) modes. The boundaries of the flipper are defined as *x* = ±*l*/2 along the length *l* of the flipper. The angular frequency of the flipper is *ω*, and the amplitude of the RF field is *B*_{RF}. The $x\u0302$ axis is along the beam direction, and so the amplitudes of the RF and gradient fields always vary as a function of *x*. The fields *B*_{0} and *B*_{g} in Eq. (A1) are perpendicular to the beam direction that represents the Transverse Neutron Resonant Spin Echo (TNRSE) mode. In the code, the vector components of the magnetic field in Eq. (A1) can be reconfigured differently so that *B*_{0} is parallel to the beam direction (*x*), which corresponds to the Longitudinal Neutron Resonant Spin Echo (LNRSE) mode.

In the following, the transverse configuration is used as in Refs. 17 and 18. Figure 7 shows the simulation results of the spin flipper operating in nonadiabatic and adiabatic spin flipping modes when scanning *B*_{0}. With the neutron gyromagnetic ratio (*γ*) known previously, for a RF frequency of *f* = 1 MHz, a resonance peak at $B0=2\pi f\gamma =342.8$ G is expected for both modes. The resonance field from the simulation agrees well with the expectation, and one can also see that the resonance peak of the adiabatic mode is wider than that of the nonadiabatic mode.

An additional feature of the new McStas component is the capability to add a random time to the state of the neutron. MIEZE is explicitly a time-dependent method, but most of the sources available in McStas do not produce neutrons in a time window; that is, the initial time for all neutrons is *t*_{0} = 0 rather than *t*_{0} ∈ (0, *t*_{max}). As a result, the time of arrival at the detector is correlated with the neutron wavelength, as opposed to reality at a constant wavelength source with the “white” beam being incident on the detector at all times. A parameter is included in the RF flipper component, which sets *t*_{max}. A time *t*_{0} is added to the McStas time variable for a neutron entering the flipper, where *t*_{0} is sampled from a uniform distribution in the range (0, *t*_{max}). By setting *t*_{max} nonzero for the first flipper and zero for all subsequent flippers, this is equivalent to having a random initialization time at the source, removing the correlation between the time of arrival at the detector and the wavelength.