Computerized simulation of 2-dimensional phase contrast images using spiral phase plates in neutron interferometry

1 Introduction

The first evidence that light can have angular orbital momentum was published as early as 1935, followed by extensive studies on this phenomenon many years later [2], [3], [4], [5]. It was found that light with a spatially varying amplitude and phase distribution due to an object with a spiral phase contains an orbital angular momentum (OAM) and that this should be quantized in units of ℏ. Twisted phase fronts caused by spiral phase plates offer exciting applications. Interference experiments with spiral phase plates were used for quantum entanglement, quantum information science, and imaging [6], [7]. A quantized orbital angular momentum (OAM) has already been realized for photons and electrons [8], [9], X-rays with orbital angular momentum could be generated in a free-electron laser oscillator [10]. The twisting of neutral particles with electric fields and phase vortex gratings in neutron interferometry was studied theoretically [11], and very recently, theoretical work on spin-textured neutron beams with orbital angular momentum was published [12]. A comprehensive review of optical vortices, i.e. OAM manipulations from topological charge to multiple singularities, can be found in [13], [14], another review describes suggestions relevant to vortex states in high energy collisions [15]. Creating twisted wave fronts with SPP seems to be an elegant method to obtain vortex beams of photons, electrons and also of atoms and molecules as were realized in [16].

Consequently, the control of the orbital angular momentum (OAM) of neutrons (n-OAM) is expected to provide new possibilities for the study of the foundations of quantum mechanics. The neutron crystal interferometer seems to be a perfect instrument to realize this. By Laue diffraction, it provides two widely separated coherent beams. A spiral phase plate (SPP) can be inserted in one beam path and the interference of the two converging beams is detected behind the third plate of the interferometer (see Figure 1).

Figure 1:

Layout of a neutron interferometer. By the first crystal plate (S), the incident neutron wave is coherently split into a forward beam (O-direction) and a Bragg diffracted beam (G-direction), which is repeated by the second (M) and third crystal plate (A). Only the interfering beams are depicted. Note, it makes no difference whether the spiral phase plate (SPP) is in the other beam path. The gray area is the scattering plane, defined by the incident wave vector of the neutron and the reciprocal lattice vector G ⃗ .

The use of a crystal interferometer to generate a twisted phase front was first done by [1]. In this article, however, we will show and prove that, in our opinion, the interference patterns were wrongly attributed to quantized n-OAM processes, leading to serious and widespread misinterpretations. One reason for the misinterpretation was ignoring the lack of sufficiently large lateral coherence of the neutron wave in the interferometer for the generation of an n-OAM, which was very clearly shown in [17], [18]. However, there are other serious reasons why with a crystal interferometer of the Mach–Zehnder type n-OAM-based interference cannot be observed. Proper use of a crystal interferometer for neutron interference experiments requires detailed knowledge of the dynamic theory of neutron diffraction in perfect crystals, based on established mathematics covered in several textbooks [19], [20]. Based solely on this theory, we calculated all the interferograms reported in [1] using exactly the same parameters, i.e., a perfect crystal neutron interferometer, the same neutron wavelength, the same spiral phase plate(s) (SPP), and the same neutron detector, and found excellent agreement between our interferograms and those in [1], but no evidence of quantization of the interference process, i.e. n-OAM.

2 Neutron crystal interferometry

The whole mathematical and theoretical background of neutron interferometry can be found in [21], but some details have to be explained here because they are not so well known and described there, but are important to understand the interferograms obtained by a crystal interferometer.

In order to use a neutron interferometer (Figure 1) for a specific purpose, it is necessary to know that the underlying interaction of neutron waves for interference is crystal diffraction. For X-rays this is given by the dynamical theory of diffraction ([19], [20]), which later on was adapted to neutron interferometry (for details see [21], [22]). The fundamental idea of perfect crystal interferometry is coherent splitting of a beam by so-called Laue-diffraction (first interferometer crystal plate, splitter), which is repeated by a second (mirror) crystal and by the third crystal plate (analyser) as shown in Figure 1. The coherent superposition of two converging beams in front of or inside the analyser crystal produces an interference pattern whose fringe spacing d _f is equal to the crystal lattice spacing (for λ = 0.271 nm, d _f ∼ 0.314 nm). This interference pattern cannot be resolved by any X-ray or neutron detector. Instead, the reflecting atomic planes of the analyser crystal are used to create with the interference pattern a Moiré pattern magnifying the interference pattern by about 10⁷, which makes the observation of neutron interference possible at all. This interference method was first realized in 1965 by Bonse and Hart for X-rays [23], [24] and nine years later, in 1974, for neutrons [25]. It must be noted, this behaviour is fundamentally different from that of a light interferometer of the same type (Mach Zehnder).

First, the interference pattern in front of or inside the third crystal plate (analyser crystal) contains all information transmitted by the two coherent partial waves along path I and path II, e.g. crystal defects in the interferometer, phase changes and any misalignment.

Second, the Moiré amplification mentioned above works only in the [x] – direction (see Figure 1) [23]. This was clearly demonstrated in test experiments of the neutron interferometer with X-rays (in 1973, [26]) by generating so-called rotational Moiré pattern (lower image in Figure 2) and phase shift by a vertical Beryllium wedge in the [z] – direction (upper image in Figure 2).

Figure 2:

Two different X-ray interferograms (a) and (b). Each interferogram has two pairs of three fringes (because all rays could pass the crystal interferometer), no 2 and no 5 are the interference patterns in the O – and G – direction. Upper image: A vertical Beryllium wedge in one beam path in the interference beam produced horizontal fringe pattern. red (Mo Kα _1,2, exposure time = 3 h, width of each fringe ∼4 mm and height ∼3.8 cm). Lower image: A rotation of analyser crystal (third crystal plate) about 10⁻⁸ rad against the interference pattern produced a so-called rotational Moiré pattern. Note, that X-rays could penetrate all three Si crystal plates, each 4.460 mm thick, only because of the anomalous absorption, the so-called “Borrmann effect” [27]; in the case of neutrons (no absorption), the entire “Borrmann Delta” contributes to neutron interference, but only one “pair of coherent rays” at a time (original images are from [26]).

Third, in the case of X-rays only one wave field is excited, traversing the crystal parallel to the reflecting atomic planes, the other wave field is absorbed. Due to an anomalous low absorption [27], X-rays can pass also “thick” perfect crystals. In an X-ray interferometer, the O-beam and the G-beam have the same intensity distribution as shown in Figure 2 (see also [26], p 70).

For neutrons with a wave length λ, and Δλ/λ ≫ 10⁻⁴ always the whole “Borrmann Delta” is excited (see Figure 3) because for any other neutron wave incident at exactly the same direction but different wave length, the energy flow is split within the “Borrmann Delta” and exits at “a” (O* < a < A) and “b”, (O* < b < B) (see left image in Figure 3). This is also true for neutrons whose angle of incidence in the {x, y} plane deviates from the Bragg angle by less than 10⁻⁷ rad [22], [28] (see right image in Figure 3). Therefore, interference can only occur due to “one-to-one interference” between the wave from path I and the wave from path II (see above Figure 1) and only within their scattering plane (“pair of coherent beams” [24]). Note, that there is only one neutron in the interferometer at a time and only self-interference occurs.

Figure 3:

Laue diffraction: Rays and intensity distribution. Left image: Rays in the crystal in the case of so-called Laue diffraction. Triangle O A B ̂ = “Borrmann Delta” for an incident pencil beam at Bragg angle Θ _B . ‘a’ and ‘b’ are the exit points of waves coming from the α and β branch of the dispersion surface (not shown here, see [21], p 299), see Figure 1. Right image: Laue diffracted intensity distributions behind a crystal plate for three wavelengths, note the rapid intensity oscillations and different change of P _G for different wavelengths, Γ = tan(Ω)/tan(Θ _B ).

All information contained in the interference pattern of such a crystal interferometer is therefore amplified one-dimensionally, only. Moreover, due to the extremely large difference of the lateral coherence lengths σ _x and σ _z and due to the rather complicated intensity structure resulting from the “Borrmann Delta”, observable interferences outside the scattering plane ({x, y}-plane) are extremely unlikely. The neutron wave is laterally limited and can be described by a Gaussian like wave packet but with different lateral coherence lengths σ _x and σ _z . In [1] σ _z and σ _x were estimated as 60 nm and a few μm, respectively. Therefore, any neutron interference process takes place only within an extremely thin layer (≪1 μm) of the scattering plane by a strong one-to-one correlation between rays of path I and path II (see [22], [29], [30]).

Concerning a SPP-induced neutron OAM (n-OAM), experiments with laser light using a SPP have shown that incident wave packets must be able to detect the discontinuity at the center of the vortex phase plate, otherwise the probability of detecting a single particle with uniform OAM around the center of the packet is extremely low [31], which is certainly true for neutrons in a crystal interferometer, and even for X-rays it is difficult to use a division amplitude interferometer (like the used neutron interferometer) to create a vortex wave front with SPP [32].

In order to prove that no n-OAM is necessary to arrive at the experimental results of [1], we calculated all the interference patterns of their experiments with exactly the same parameters, i.e., neutron wavelength, SPP material, Si crystal neutron interferometer. We assumed a perfect crystal with the shape as shown in Figure 1 and calculated each interferogram given [1] on the basis of dynamical neutron diffraction, only [29], [30].

The spiral phase plates (SPP) in [1] had different sizes (10 mm and 15 mm diameter), we used a 15 mm diameter SPP with a thickness height of one, two, four and 7.5 λ-thicknesses D _λ (see Section 3) as in [1] for all calculations. Each of the neutron experiments, as described in [1], lasted more than three days due to the low neutron count rate, therefore, despite data processing, the counting rate did not appear to be uniformly distributed over the entire detector area, perhaps due to inhomogeneity of the neutron beam, perhaps due to varying efficiency of individual detector pixels, perhaps due to small changes in the phase stability of the interferometer. This can be seen in all interferograms in [1], best in the figure “Extended data Figure 1 – Raw data”. These deficiencies were not simulated.

Here, the experimental interferograms taken from [1] are presented in Figures 6–8 (left parts) together with the corresponding interferograms obtained using dynamical theory of neutron diffraction (Figures 6–8 right parts). The mathematics is given below.

3 Calculation of interferograms

The coherent splitting of a neutron wave with a crystal interferometer by Laue diffraction is derived from the theory of dynamic diffraction, details for the neutron interferometer are given in [21]. It is important to note that the coherent waves arising from Laue diffraction in all three crystals via path I and path II belong to a scattering plane defined by the incident wave vector of the neutron wave k ⃗ and the reciprocal lattice vector G ⃗ of the first crystal plate. Then one must know two inherent effects of dynamic diffraction and crystal interferometry: The angular amplification of the neutron flux in the Borrmann delta due to small deviations of the incident wave from the exact Bragg angle or due to slightly different incident wavelength (Figure 3) and the magnification of the interference pattern by a factor of 10⁷ due to Moiré imaging.

This Moiré imaging can be explained as follows: The interaction of a neutron wave with a 3-dimensional perfect crystal is described by solving the Schrödinger equation for a periodic potential, details see [21]. The Laue case solution for a plane crystal plate results in a standing wave pattern behind the first crystal plate (i.e., the interferometer beam splitter S) creating two coherent waves in directions O and G. From the middle crystal plate (M, mirror) two waves converge due to Laue-diffraction and overlap in front (or in) the third crystal plate (A, analyser) and form an interference pattern whose shape depends on the optical path difference between path I and path II [23], [24]. The standing wave pattern behind the first crystal plate (S) is the continuation of the internal pattern, which is finally transported via path I and path II to the third crystal plate (A), where they superimpose and form the resulting fringe pattern which has still the same fringe spacing as the Bragg planes, i.e. 0.314 nm. The overlay of this pattern with the simultaneously excited wave field in the third crystal plate (due to Laue diffraction) produces a Moiré pattern with a fringe spacing of ∼4 mm, which is increased by a factor of 10⁷ as compared to the original fringe spacing of the interference pattern. This magnification occurs in the scattering plane only, since the standing interference pattern behind each crystal plate is always parallel to the reflecting lattice planes due to Laue diffraction. Shifting the third crystal plate (A) with respect to the interference pattern parallel to ( G ⃗ ) by one crystal lattice spacing, the fringes of the Moiré pattern also shift by one lattice constant, allowing Angstrom-scale length measurements [33] and confirm the 1-dimensional amplification.

In a Mach–Zehnder light interferometer, the coherence of beam is defined in front of the interferometer, while in a crystal interferometer of Mach–Zehnder type the interference property is directly related to the Laue–Bragg reflections (of neutrons and X-rays) in the crystal interferometer. In both interferometers, the phase of one partial wave can be manipulated with respect to the other partial wave with a phase-shifting object that has a so-called λ-thickness D _λ , for neutrons D _λ = 2π/(Nb _c λ), (b _c is the neutron coherent scattering length, N is the number of atoms/unit volume, λ = 0.271 nm is the neutron wave length). D _λ is the thickness of a material that shifts the phase of one partial wave relative to the phase of the other partial wave by 2π.

In [1] an “effective angular momentum” is given as L = Nb _c λh _s /2π, which can be rewritten as L = h _s /D _λ , h _s being the step height of the spiral. This means that L in this context is always a linear function of step height h _s scaled by D _λ and therefore has no quantization lℏ. If L is allowed to take only integer values (1, 2, … n), i.e. a multiple of D _λ , as done in [1], then one can only measure interferograms as given in [1], and thus apparently observe a quantization of L, but L can take any real value ∈ R. In our calculations, for example, the value L = 7.5 as in [1], simply leads to the same interferogram as in Figure 6, without assuming any quantization of L.

We first calculated the 2-dimensional phase front of the spiral phase plate (SPP) using the Radon transform RT [34] for the angle of incidence ϑ = 90° (i.e. perpendicular to the SPP, see Figure 4). The SPP was divided into N slices in z-direction (see Figure 4), i.e. the RT{SPP} then provides the spatially dependent transmission lengths of the beam in the y-direction of all (x, y)-planes. For each slice SPP _z , corresponding to a fixed value of z, the RT was calculated as

Figure 4:

Scanning parameter p: For a fixed value of z _N and the angle of incidence ϑ = 90°, the rays pass perpendicularly through the SPP. The Radon transformation is then a one-dimensional function that provides the p-dependent thicknesses of the SPP.

Here p is the sampling parameter for parallel scanning of the (x, y) plane in direction normal to (cos(ϑ), sin(ϑ)).

Finally, the RT{SPP} was scaled by the refractive index of the SPP material (Al), resulting in a 2D phase front behind the SPP (see Figure 5). The left image in Figure 5 shows the spiral phase plate, step height is D _λ = 112 μm, the right image shows the 2D Radon Transform of the SPP, scaled by D _λ of the SPP. This was verified by a different computation, where the 2D thickness function of the SPP (for a scanning angle ϑ = 0) was computed directly from the geometry of the SPP as a phase-shifting object.

Figure 5:

Spiral phase plate (SPP) and its Radon transform. Left image: 3D image of the Aluminum spiral phase plate, step height D _λ = 112 μm, neutrons traverse the SPP in y – direction, y coordinate in units of D _λ . Right image: The Radon transform of the SPP, scaled by D _λ of the SPP with angle of incident ϑ = 90°.

For the calculation of the interference pattern we used the same formula as used in neutron interferometry [21] and thus also by [1]:

Here I(u, v) denotes the two-dimensional intensity distribution that would be measured by a position sensitive neutron detector with detector plane being normal to the O-beam, i.e. in the (x, z)-plane see Figure 1, ϕ _o is the phase of the partial wave, which can be tuned with a phase flag (as was used in [1], Figure 1), D _S (u, v) is the position dependent thickness, at the point, where the neutron wave transmits the SPP, scaled by D _λ . In order to compare with the experimental interference pattern of [1], we used the resolution of the neutron detector as discretization parameter in the (x, z) – plane (100 × 100) and added some random noise. All calculations have been performed with MATLAB [35].

The results of our calculations are shown in Figures 6–8, where we compare the experimental results of [1] with our simulated interference pattern. In Figure 8 we also display the effective ΔD due to rotation ϕ ₀ of the phase flag. It can be seen that for these small rotations, the ΔDs are only a fraction of D _λ = 112 μm and thus a noticeable phase shift is hardly observable (see Figure 8), i.e. one cannot expect a dramatic change of the interference patterns due to such small phase shifts, and also due to the low counting rate of neutrons.

Figure 6:

Comparison of interferograms. Left part: Interference patterns showing “so-called” n-OAM for L = 1, L = 2, etc. presented in [1], Figure 2. Right part: Interference patterns simulated with dynamical neutron diffraction without assuming n-OAM.

Figure 7:

“So-called” addition of n-OAM. Left part: Experimental interference patterns taken from [1], Figure 3. Right part: Interference patterns simulated with dynamical neutron diffraction without assuming n-OAM.

Figure 8:

“So-called” rotational invariance. Left part: Experimental interference patterns taken from [1], Figure 4. Right part: Interference patterns simulated with dynamical neutron diffraction without assuming n-OAM. Note that the resulting change of phase plate thickness ΔD due to a rotation of ϕ _o is only a small fraction of D _λ = 112 μm.

Concerning Figure 4, in [1] one reads: “The average OAM is independent of the position of the phase flag, i.e. L is preserved”. This is a thoroughly misleading interpretation, since similar looking interferograms do not prove L conservation. In fact, the number of neutrons is always preserved if the phase in one partial beam in a neutron interferometer is changed, well known as conservation of the number of particles in neutron interferometry.

Furthermore, due to the small phase shifts, one cannot in any case conclude that L is conserved, because without knowing the counting rate of the O-detector (see Figure 1), one can by no means say unequivocally that anything is conserved. One has to question why only such small flag angles ϕ ₀ were used to demonstrate a so-called “conservation of L”, which should actually hold for all flag angles ϕ ₀. With much larger flag angles, they would have observed with the O and G detectors only a rotation of the whole interference pattern, which again would only prove the conservation of the neutron number and contradicts the assumption of partial prism refraction as the cause of a n-OAM. In neutron interferometry, the O-detector always counts the complementary number of neutrons of the G-detector, so that only the sum of the two counting rates can tell us anything about the conservation of a quantity. The O-detector in [1] was (unfortunately) used as an “integrating counter”, probably to monitor the low neutron flux. Real monitoring is usually done with a transmission detector in front of the interferometer. If Clark et al. had also used the “O detector” for complete neutron counting and not as a monitor (see Figure 1 in [1]), they would have observed complementary neutron interferograms to those of the G detector and could have inferred only conservation of particle number.

One important feature of quantized OAM is the superposition of different, opposite orbital angular momentum, + l + − l resulting in a ring-like structure, containing exactly 2 × l intensity maxima, so the far more interesting neutron experiment, and thus demonstration of the controlling of n-OAM, would have been to image (with no crystal interferometer) the coherent sum of two opposite spiral structures leading to opposite OAMs, as was done in [36].

4 Conclusions

In summary, all neutron interferograms calculated assuming only the dynamical theory of neutron diffraction is in perfect agreement with the interferograms in [1]. These results also show that there is definitely no reason to interpret these neutron interferograms in the context or as the cause of a n-OAM.

Moreover, here are several serious reasons that Clark et al. could not have measured a n-OAM. One important reason is that the lateral coherence is too small, which was already proved in [17] and in [18]. The arguments given there are correct, but incomplete, because the coherence of the partial waves in the interferometer has two laterally very different coherence lengths, that are both too small and moreover extremely different in size, as already stated above. It should be noted that a n-OAM, which requires symmetric lateral coherence lengths, cannot be detected with a Mach–Zehnder crystal neutron interferometer. This is because the available lateral coherence lengths are significantly different and both too small to allow coherent irradiation of an SPP of a few mm, as discussed in Section 2.

An argument that the n-OAM consists of a sum of refracting radial prisms, which in sum yields an observable n-OAM, can be easily falsified by Figures 6 and 7. In both Figures, the phase shift is increased by additional SPPs in one beam path of the interferometer. If refraction were the primary interaction (as assumed above), then the interference pattern would not show additional regions of scattered neutrons, as in Figures 6 and 7. Adding one, two, or more SPPs in one beam path, as is done in [1], causes pure phase shifts, that produce these interferograms, not refraction. A simple calculation of the change in direction of the neutron wave due to an average SPP refraction yields a deviation of about 4 × 10⁻⁹ rad, which is much too small to produce a (twist-based) n-OAM.

The most serious reason for the misinterpretation of the neutron interferograms published by Clark et al. is ignoring the completely different operation of a Mach–Zehnder light interferometer and a crystal neutron interferometer (of Mach–Zehnder type), which have the same topology but operate differently. Both are amplitude splitting interferometers, but in contrast to a Mach–Zehnder light interferometer, in a neutron interferometer each phase change caused by a phase-shifting material can only be detected via Moiré pattern. Only the superposition of the interference pattern (as a standing wave) with the reflecting crystal planes of the third crystal plate of the neutron interferometer makes the observation possible, but this Moiré amplification works only in lateral direction (in this work in x-direction).

This is in contrast to a Mach–Zehnder light interferometer, where the interference image results solely from the superposition of the two partial waves.

The large differences in wavelength, refractive index and coherence properties of neutrons as compared to photons, the different modes of reflection, the extreme different shape of neutron Gaussian wave packets underline the big doubts that the detection of a n-OAM by means of a neutron crystal interferometer as used in [1] is possible at all. We conclude, that all interferograms in [1] are pure phase contrast images of the spiral phase plate and that no control of neutron orbital momentum has been detected in [1]. This is also true for other experiments (e.g. [37], [38]) where one wants to detect or even use n-OAM with a crystal neutron interferometer. This will be discussed elsewhere.

However, there is no doubt about the existence of neutron angular momentum, but its detection must be done in a very different, unambiguous way.