We use a Mach-Zehnder interferometer to perform neutron holography of a spiral phase plate. The object beam passes through a spiral phase plate, acquiring the phase twist characteristic of orbital angular momentum states. The reference beam passes through a fused silica prism, acquiring a linear phase gradient. The resulting hologram is a fork dislocation image, which could be used to reconstruct neutron beams with various orbital angular momenta. This work paves the way for novel applications of neutron holography, diffraction and imaging.
© 2016 Optical Society of America
Holography was introduced by Dennis Gabor in 1948 , who showed that a far-field electron micrograph of an object could be used to make a transmission mask that allows the object to be reconstructed with visible light. Due to subsequent advances in the brightness of electron sources and, notably, the development of techniques for numerical reconstruction , holography has since become a significant branch of electron microscopy . Atomic-resolution neutron holography has been demonstrated and applied to the determination of crystal structures [4–6]. The advent of coherent laser light sources in the 1960s made all–optical holography practical [7, 8], to a degree that optical security holograms are now routinely printed on many paper currencies, credit cards, and identification documents [9,10]. Holography remains a vibrant field of research in imaging science, as shown by numerous recent research papers [11–17].
Here we report the first demonstration of holography using neutron beams and macroscopic optical elements. Although deployed in a neutron interferometer (NI) with a Mach-Zehnder configuration , our method is a simple adaptation of the two-beam wedge technique introduced by Leith and Upatnieks [7,8], and we discuss it using the conventional terminology of object, reference and reconstruction beams.
Our object is a spiral phase plate (SPP) that we previously used to impart orbital angular momentum (OAM) to neutron waves . In this respect our experiment is a neutron analogue of the holography of braided optical fibers performed by Bazhenov, Vasnetsov and Soskin . Our neutron holograms resemble the fork dislocation gratings synthesized by them and by Heckenberg, et al. , which have since been used to transfer angular momentum to small particles,  atoms, , electrons [24, 25] and light . Digital reconstruction of this hologram provides information about the phase generated by the SPP, which should be useful in the design of the next generation of neutron OAM experiments .
Holography provides a direct connection between the refractive and diffractive mechanism of OAM imprinting of particles and waves. This first demonstration of neutron holography of a macroscopic object suggests that the complex grating methods used to produce OAM, Bessel, and Airy beams for light and electrons [28–30] may also be extended to neutrons.
2. Schematic of neutron holography
We use a NI to effect the coherent superposition of an object beam and a reference beam. As shown in Fig. 1(b), our object beam consists of neutrons that have passed through an aluminum SPP in one arm of the NI. The SPP imprints a spatially-varying phase of qϕ upon the neutron input beam, where q is the topological charge of the SPP  and ϕ is the azimuthal angular coordinate on the neutron wavefront, with coordinate origin being the center of the SPP surface. The other arm of the NI provides the reference beam. The holographic image is constructed from the interferogram of object and reference beams, as recorded by a neutron sensitive digital camera. When no optical device is present in the reference arm, this interferogram displays the topological charge of the object beam, which is the same as that of a beam with orbital angular momentum (OAM) of ℓħ, where ℓ = q . By placing a prism into the reference beam we introduce a linear gradient which effectively tilts the wavefronts [31,32].
The experiment is an expectation valued measurement over many events, each of which involves only a single neutron. In optics one would manipulate fully coherent beams and in the neutron case where an interferometer is used the coherence need only be larger than the deviation due to the prism and the object. There is no advantage in the neutron case in having the coherence beyond that because there is one neutron at a time in the interferometer and the hologram is built up from an incoherent superposition of many events. The neutron coherence length is a function of the beam momentum spread, and in our experimental setup it is on the order of microns, while the incident neutron beam is several millimeters in size, and the deviation due to the SPP and the prisms is on the order of nanometers. Hence the coherence length we use captures the physics needed to perform holography. We have combined neutron interferometery with holography and an appropriate coherence length must be used as the modification of the neutron coherence is extremely difficult and time consuming .
The transverse wavefunction of the incoming neutron wavepacket, which is centered at an arbitrary position (x0, y0) in the beam, can be taken to be:Fig. 1(b). In addition, there is an intrinsic phase θ between the two paths. The last NI blade coherently combines the two paths and the intensity at the camera is then given by: 22]. Figure 2(a) displays the calculated interferograms when only the SPP is present in the NI, and Fig. 2(b) shows the interferograms when both the SPP and the prism are present. The embedded video ( Visualization 1) shows the data of Fig. 2(b) as a continuous function of the topological charge q. We complete the holographic process by applying digital reconstructions to the measured forked dislocation structures, showing that they yield the phase and intensity profiles associated with the SPP.
3. Experimental method
The experiment was performed at the Neutron Interferometry and Optics Facility (NIOF) at the National Institute of Standards and Technology (NIST) Center for Neutron Research (NCNR) . Neutrons generated by a 20 MW nuclear research reactor are cooled via a liquid hydrogen moderator, and pass through a system of beamlines to experimental stations. A monochromator at the NIOF beamline is used to select neutrons with energies around 11 meV, corresponding to a de Broglie wavelength of λdB = 0.271 nm. This cold neutron beam is incident on the NI, which is an example of the devices discussed in [18,33]. The NI was made using a single ingot of silicon machined so that it has three blades supported by a common base. The common base insures sub-arcsecond alignment between the crystal blades. Neutrons entering the interferometer are Bragg diffracted by the (111) lattice planes of the crystal blades forming two spatially separate paths. Phase differences between these two paths cause interference in the last blade of the interferometer. This interference is then detected using either a fully integrating 3He proportional counter or a neutron-sensitive imaging camera.
An aluminum SPP with q = 2 was placed in the object beam. The construction of the SPP is described by Clark, et al. . We note here that the SPP was made with a standard milling machine cutting a spiral staircase into the surface of a segment of aluminum dowel. A topological charge q = 2 is obtained with a staircase with a total vertical descent of 224 μm, which is ≈ 106 × λdB. Those experienced in optical design for visible light may find it surprising that the mechanical figure of an optical component need only be controlled within a few thousand multiples of the operational wavelength. This is possible with neutrons, since neutron indices of refraction for most materials differ from unity by a few parts per million.
In the reference beam a vertical linear gradient was introduced by using two identical fused silica optical wedges  arranged back-to-back. These wedges had a 6° angle and each could be rotated independently a full rotation of 2π. Note that any non-gradient phase shift inside the interferometer would only shift the pattern at the camera.
In our experiments, the integrating counter was used to measure the average rate of neutrons exiting the interferometer, which was about r = 20 s−1. This rate determines an average time interval τ = 1/r = 50 ms between detection of successive neutrons. Since the distance of the NI from the reactor is 30 m, and the neutron velocity is 1,460 m/s it takes 20 ms for a neutron to travel from the reactor to the NI. Thus, just after one neutron has been detected, the next neutron to be detected has not yet been produced by the reactor. The interference fringes that are seen in our data are truly those of neutrons interfering with themselves.
For the images depicted in Fig. 3, the wedges were adjusted to give a theoretical linear phase gradient of 3.4 rad/mm. Reactor fluctuations were monitored using the 3He detector. The interferograms were recorded on a neutron-sensitive digital camera  that has an active area of 25 mm diameter and a spatial resolution of 100 μm . The neutron quantum efficiency of the camera is 18 % and individual images were taken in 28-hour runs. The average of three images was used for analysis.
Figure 3 depicts interferograms for four different configurations. Figure 3(a) shows the grayscale intensity profile obtained with an empty interferometer. It can be seen that the input beam is nonuniform. Figure 3(b) shows the experimental and simulated interferograms for the case in which only the prism is placed in the reference beam of the interferometer. The number of fringes in the simulated interferogram are in agreement with the measured image. Figure 3(c) shows the interferogram when only the SPP is present in the object beam. It is necessary to add a horizontal gradient of 0.3 rad/mm to the simulation for Figs. 3(a) and 3(b) to reproduce the measured figures. This horizontal gradient might explain the nonuniformity of Fig. 3(a) as well. Figure 3(d) shows the hologram obtained when the prism is placed in the reference beam and the SPP is placed in the object beam. The expected fork grating pattern is recognizable in this hologram.
Figure 4 shows the numerical reconstruction  of the experimental images shown in Figs. 3(b) and 3(d). The reconstruction was obtained by computing the Fresnel-Kirchhoff diffraction integral in the Fresnel limit for λd = 70 mm2 using the expression:
For the hologram, it can be seen from the uniform intensity distribution that the zero diffraction order corresponds to ℓ = 0, while the first diffraction order displays the doughnut profile indicative of a beam carrying OAM. By plotting the reconstructed phase we confirm that the first diffraction beams correspond to beams with ℓ = 2 as expected. Note that the spherical phase outside the Fourier transform was dropped in the Eqn. (3) reconstruction integral in order to look down the diffracted OAM beam paths and not along the reconstruction plane normal. The two linked videos show the two-dimensional intensity with the variable λd increasing with time ( Visualization 2) and the phase profile as a function of the overall phase in the reconstruction plane ( Visualization 3). As the value of λd increases, we see the formation of first-order diffraction peaks with doughnut-shaped intensity profiles, corresponding to orbital angular momentum ℓ = 2.
Using a neutron interferometer, we have for the first time demonstrated digitally recorded neutron holography of a macroscopic object: a spiral phase plate. Numerical reconstruction of the recorded hologram reveals the variation of neutron phase over the surface of that plate. This method provides a new tool for interferometric testing of neutron optics and the characterization of coherence of neutron beams. It offers an extension of coherent phase control techniques to applications in neutron radiography and imaging, which are uniquely useful in the analysis of buried interfaces [6,35].
Canadian Excellence Research Chairs (CERC) (215284); Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery (RGPIN-418579); Collaborative Research and Training Experience (CREATE) (414061); National Science Foundation (NSF) (PHY-1205342, PHY-1307426); U.S. Department of Energy (DOE) (DE-FG02-97ER41042); National Institute of Standards and Technology (NIST) Quantum Information Program.
We thank Sean Kelley for his contributions to the figures.
References and links
2. U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002). [CrossRef]
3. H. Lichte and M. Lehmann, “Electron holography–basics and applications,” Rep. Prog. Phys. 71, 016102 (2008). [CrossRef]
6. I. S. Anderson, R. McGreevy, and H. Z. Bilheux, eds., Neutron Imaging and Applications: A Reference for the Imaging Community (SpringerUS, 2009).
7. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52(10), 1123–1130 (1962). [CrossRef]
8. E. N. Leith and J. Upatnieks, “Wavefront reconstruction with continuous-tone objects,” J. Opt. Soc. Am. 53(12), 1377–1381 (1963). [CrossRef]
9. B. Javidi, ed. Optical and Digital Techniques for Information Security (Springer Science & Business Media, 2005). [CrossRef]
10. S. F. Johnston, Holograms: A Cultural History (Oxford University, 2016).
12. O. Backoach, S. Kariv, P. Girshovitz, and N. T. Shaked, “Fast phase processing in off-axis holography by CUDA including parallel phase unwrapping,” Opt. Express 24, 3177–3188 (2016). [CrossRef] [PubMed]
13. R. Isogai, Y. Nakamura, H. Takagi, T. Goto, P. B. Lim, and M. Inoue, “Thermomagnetic writing into magnetophotonic microcavities controlling thermal diffusion for volumetric magnetic holography,” Opt. Express 24, 522–527 (2016). [CrossRef] [PubMed]
14. T. C. Petersen, A. I. Bishop, S. A. Eastwood, D. M. Paganin, K. S. Morgan, and M. J. Morgan, “Singularimetry of local phase gradients using vortex lattices and in-line holography,” Opt. Express 24, 2259–2272 (2016). [CrossRef] [PubMed]
17. D. Bowman, P. Ireland, G. D. Bruce, and D. Cassettari, “Multi-wavelength holography with a single spatial light modulator for ultracold atom experiments,” Opt. Express 23, 8365–8372 (2015). [CrossRef] [PubMed]
18. H. Rauch and S. A. Werner, Neutron Interferometry: Lessons in Experimental Quantum Mechanics, Wave-Particle Duality, and Entanglement (Oxford University, 2015). [CrossRef]
20. V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eks. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1991)].
22. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995). [CrossRef]
23. M. F. Andersen, C. Ryu, P. Cladeé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006). [CrossRef] [PubMed]
24. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331192–195 (2011). [CrossRef] [PubMed]
25. K. Saitoh and M. Uchida, “Electron beam carrying orbital angular momentum,” Nihon Kessho Gakkaishi 58, 79–84 (2016). [CrossRef]
26. A. Yao and M. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). [CrossRef]
27. J. Nsofini, D. Sarenac, C. J. Wood, D. G. Cory, M. Arif, C. W. Clark, M. G. Huber, and D. A. Pushin, “Spin orbit states of neutron wavepackets,” Phys. Rev. A 94, 013605 (2016). [CrossRef]
29. V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4, 011013 (2014).
31. D. A. Pushin, D. G. Cory, M. Arif, D. L. Jacobson, and M. G. Huber, “Reciprocal space approaches to neutron imaging,” Appl. Phys. Lett. 90, 224104 (2007). [CrossRef]
32. D. A. Pushin, M. Arif, D. L. Jacobson, C. K. Doe, and D. G. Cory, “Reciprocal space neutron imaging,” Physica B 385–386, 1402 (2006). [CrossRef]
33. C. B. Shahi, M. Arif, D. G. Cory, T. Mineeva, J. Nsofini, D. Sarenac, C. J. Williams, M. G. Huber, and D. A. Pushin, “A new polarized neutron interferometry facility at the NCNR,” Nuc. Inst. Meth. A 813, 111–122 (2016). [CrossRef]
34. M. Dietze, J. Felber, K. Raum, and C. Rausch, “Intensified CCDs as position sensitive neutron detectors,” Nuc. Inst. Meth. A 377, 320–324 (1996). [CrossRef]
35. J. D. Fairweather, D. Spernjak, A. Z. Weber, D. Harvey, S. Wessel, D. S. Hussey, D. L. Jacobson, K. Artyushkova, R. Mukundan, and R. L. Borup, “Effects of cathode corrosion on through-plane water transport in proton exchange membrane fuel cells,” J. Electrochem. Soc. 160, F980–F993 (2013). [CrossRef]