- Email: [email protected]

Physica B 385–386 (2006) 1395–1401 www.elsevier.com/locate/physb

Phase imaging with thermal neutrons$ Brendan E. Allmana, Keith A. Nugentb, a

Iatia Ltd. @/935 Station St., Box Hill Nth 3129, Australia School of Phyics, University of Melbourne 3010, Australia

b

Abstract Across four decades, Sam Werner has built and performed elegant neutron interferometry experiments to measure a variety of quantum mechanical phases. These experiments have stringent requirements on experimental conditions and neutron beam conditioning. However, since refractive variations within a sample redistribute neutron intensity transverse to the propagation direction, a simple experimental geometry permits non-interferometric phase measurement and relaxes beam-conditioning requirements. This phase imaging technique, based on the transport of intensity equation, has advantages of allowing weakly absorbing samples to be radiographed at greatly reduced radiation doses, and enabling the use of polychromatic neutrons to increase ﬂux and speed imaging. Furthermore, using other radiations, phase vortices like those of the Aharonov–Bohm effects have been observed. r 2006 Elsevier B.V. All rights reserved. Keywords: Phase imaging; Neutron interferometry; Phase contrast; Neutron radiography

1. Introduction The measurement of the ‘‘unseen’’ component of the wave, its phase, has made possible understanding of subtleties within key theories of physics. Not surprisingly, this capacity has been exploited with many of the techniques and ideas from classical optics ﬁnding interesting and useful analogies in quantum physics and speciﬁcally matter wave optics [1]. In this vein, neutron phase measurement has a long and distinguished history in the exploration of the fundamental properties of quantum mechanics [2]. The perfect single silicon crystal neutron interferometer was ﬁrst used in 1974 [3]. Over the ensuing four decades, Sam Werner has built and participated in about half of all the elegant neutron interferometry experiments. Starting with the now famous observation of the gravitationally induced quantum interference experiment [4] and most recently a series of high-precision (high-precision measurements being where Sam sees the new science being found) coherent neutron scattering

$ Paper presented as part of the Festschrift honouring Samuel A. Werner. Corresponding author. Tel.: +61 3 83445446; fax: +61 3 83445445. E-mail address: [email protected] (K.A. Nugent).

0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.05.203

length experiments [5] signiﬁcant to the nuclear (strong interaction) two-, three- and four-body models. Uniquely placed as the neutron is to probe all four of the fundamental forces in nature (gravitational, electromagnetic, weak and strong), the use of a perfect crystal neutron interferometer to measure the neutron’s phase has some practical drawbacks. They are difﬁcult to fabricate, delicate to handle, require stringent experimental vibration and thermal isolation and a high degree of neutron coherence [6]. Historically, neutron interferometry experiments entail long and arduous experiment schedules. This work describes an alternative phase measurement technique that relaxes some of these requirements and affords an extension to conventional thermal neutron radiography using phase as the contrast mechanism. 2. Phase measurement The Poynting vector describes the direction and magnitude of the energy ﬂow in the wave. In the case of coherent light with intensity Ið~ rÞ, phase Fð~ rÞ and wavelength l, the Poynting vector is given by ~ rÞ ¼ l Ið~ Sð~ rÞrFð~ rÞ. 2p

(1)

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We immediately see that the phase gradient directly inﬂuences the ﬂow of energy. In the case of quasimonochromatic partially coherent light, the Poynting vector ﬂuctuates with time; however, its average remains well deﬁned. We may therefore introduce a partial coherent phase which is defined via the expression: ~ rÞi ¼ hSð~

l Ið~ rÞrFð~ rÞ, 2p

~ rÞi 2p hSð~ l Ið~ rÞ

(3)

and it can be explicitly seen that this phase is also not deﬁned at points of zero intensity. Furthermore, as an arbitrary vector ﬁeld, the (average) Poynting vector may contain vorticity so that, in analogy with the scalar and vector potentials of electromagnetic theory, we may write [7] ~ rÞi ¼ Ið~ ~V ð~ hSð~ rÞ½rFS ð~ rÞ þ r F rÞ,

(4)

~V are appropriately deﬁned quantities that where FS and F have been termed the scalar- and vector-phase components [7]. In order to remove any ambiguity in the deﬁnitions, we ~V ð~ require r rFS ð~ rÞ ¼ 0 and rdr F rÞ ¼ 0. Thought of in this way, we see that phase can be regarded as having a vector as well as a scalar component. The scalar component is the familiar idea we encounter in our undergraduate curriculum. The vector component, as with electromagnetism and ﬂuid ﬂow, can be associated with vorticity, or angular momentum, in the ﬁeld. Interferometry, as Sam will tell you, is rather more challenging for neutrons than for visible light and so there are considerable advantages in developing non-interferometric phase recovery methods. The University of Melbourne group have developed non-interferometric phase recovery methods based on the so-called transport of intensity equation [8]. This is simply an expression for the conservation of energy on propagation. In general, energy conservation requires ~ rÞ ¼ 0. rdSð~

(5)

In the paraxial regime, which assumes that the energy propagates at a small angle to some direction (the optical axis), Eq. (1) becomes 2p qIð~ r? Þ ¼ rd½Ið~ r? ÞrFð~ rÞ. l qz

qIð~ r? Þ 1 ½Ið~ r? Þjþdz Ið~ r? Þjdz qz 2dz

(7)

and (2)

where h i denotes a time average over a period much longer than the coherence time. That is, the partially coherent phase gradient is given by rFð~ rÞ ¼

If we make an intensity measurement at two planes separated by a small distance dz, then we can make the approximations:

(6)

This is the transport of intensity equation and is an elliptic partial differential equation. Standard results of partial differential equation theory tell us that if we measure qIð~ r? Þ=qz and Ið~ r? Þ, and Ið~ r? Þ40 over a simply connected region, then the phase is uniquely speciﬁed to within an additive constant [9].

r? Þjþdz þ Ið~ r? Þjdz , Ið~ r? Þj0 12 ½Ið~

(8)

and use these to solve Eq. (6) for the phase, which has been shown to be experimentally valid for light, electrons, X-rays, atoms, and neutrons, as will now be discussed. 3. Solving for the phase The details of how the phase may be recovered will not be gone into here, but a simple example will provide the spirit of the methods. Consider a pure phase object, which is to say one that imparts only a phase shift to the wave, illuminated by a uniform plane wave with intensity I0. In this limit, the transport of intensity equation may be written: 2p qIð~ rÞ ¼ I 0 r2 Fð~ rÞ. l qz

(9)

~ ^ kÞ, It is well established that if Fð~ rÞ3Fð then 2^ ~ 2 r Fð~ rÞ3k FðkÞ, where 3 denotes a Fourier transform relationship. Thus, 2p 1 1 qIð~ rÞ Fð~ rÞ ¼ FT1 2 FT . (10) l I0 qz k The matter is made rather more complex when the intensity contains a spatial variation, however it can be seen that the transport of intensity equation lends itself, in some cases at least, to a simple determination of the phase distribution. Note that elementary Fourier theory tells us that the singularity at jkj ¼ 0 indicates an indeterminate absolute phase. This is an inevitable consequence of phase measurements that do not use a reference wave. 4. Neutron imaging Neutrons interact with matter in a way that is quite complementary to X-rays, and so neutron imaging and neutron radiography are important techniques for nondestructive testing, most suited for visualization of light elements in the interior of (heavy) metallic objects. However, in comparison to X-ray and laser sources, neutron sources have very low brightness. A major source of neutrons is the reactor that produces neutrons with essentially a thermal energy distribution and a rather large source size. Thus, the ideas developed here for partially coherent phase recovery have the potential to be very powerful in the neutron context [10]. The partially coherent phase [7] for neutrons has been explicitly deﬁned in a recent paper [11]. This work is based

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on the Wigner function: Z ~ ~ x n x W ð~ r; ~ pÞ ¼ c ~ rþ r c ~ exp½i~ pd~ x d~ x, 2 2

(11)

where cð~ rÞ is the neutron wavefunction and the Wigner function is a quasi-probability distribution for position and momentum. The probability current (which is the quantum mechanical analogue of the optical Poynting vector [12]) can be written [13]: Z ~ jð~ rÞ ¼ ~ pW ð~ r; ~ pÞ d~ p, (12)

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NIST interferometer set-up [15] for practical application imaging working fuel cells [16]. Magnetic phase radiographs performed for polarised neutrons passing through precession coils or magnetic materials contain information about both magnetic and nuclear interactions in the sample. Reversing the Larmor precession by reversing either the magnetic ﬁeld or the polarisation state can then separate the two phase effects.

7. Phase contrast radiography which is simply an average of the momentum over the quasi-probability distribution. The appropriate distribution may be used to calculate ~ jð~ rÞ and then deﬁne the phase in Eq. (2). This approach has been tested experimentally (as will be discussed later) and the results found to be in good agreement with expectations. Applied in this way, it possible to increase the phase–space acceptance of phase measurement techniques and so enable relatively low brightness sources to be used with manageable exposure times. 5. Radiography In conventional thermal neutron radiography (and tomography) [14], a single contact image is made of the sample’s neutron absorption and wide-angle scattering structure. Critical to the resolution of these radiographs is the proximity of the imager to the sample and the distance between source and sample, L, and the diameter of the source, D. This ratio L/D is conventionally referred to as the collimation ratio, and is a measure of the limit of obtainable spatial resolution. In conventional radiography, this value is around 100 with the camera placed as close as practicable to the sample. 6. Phase radiography: interferometry The broad application of phase-sensitive radiography by neutron interferometry [14] has been limited by the difﬁculty of fabrication, limited size and delicateness of operation of the interferometers. Successful interferometry requires satisfaction of the angular divergence of the beam to the Darwin width (a few arcseconds) across each of the three (or more) interferometer blades on its journey of some 109 oscillations across the interferometer before overlapping and mixing at the ﬁnal blade (indeed a ‘‘miracle’’, as Sam is prone to say). Even though neutron CCD-scintillator cameras are now available, their image resolution is compromised by the dynamical diffraction (Bormann fan effect) of the neutron within the interferometer crystal blades. Finally, any interference phase is only determined modulo 2p and unwrapping the phase can be a difﬁcult process. That said, a number of works have been performed, the latest of which uses the state-of-the-art

In propagation-based phase contrast radiography, the distance between sample and imager is great (see Fig. 1) so that the weak lateral deﬂections due to non-absorbing, refractive sample components can form, bringing these ﬁne structures into sharp relief. For this method, the collimation ratio is necessarily of order 5000, so that the detail is not washed out in source size blurring. We note that this ‘‘pin-hole’’ camera geometry additionally provides excellent images of the upstream beam-forming optics of the neutron source/guide. This greater collimation necessarily indicates a loss of ﬂux and consequent prolonged exposure times. However, the contrast and information obtained is generally inaccessible by conventional means, as seen by the comparison of contact and phase contrast images of the ‘‘Yellow Jacket’’ (see Fig. 2) [10]. Further, the enhanced visibility afforded by these novel contrast mechanisms offers a potential reduction in exposure, and therefore activation, for weakly absorbing samples over traditional absorption radiography. The development of phase contrast with propagation is shown in Fig. 3. Here a hollow Pb sinker has been mounted on an Al screw. In the contact image, scattering/absorption within the microcrystalline structure of the Pb leads to some little image contrast, but affords no detail. The Al screw is invisible. Images for propagation steps of approximately 220, 450, 900 and 1800 mm are then shown. In the ﬁnal phase contrast image, the hollow structure and ﬂaws within the sinker, the Al screw, and shavings in the axial hole from the threading process are all clearly visible [17].

Beam from Neutron Guide

Point source

Horizontal Scale 1 meter

Neutron Beam

CCD Camera Sample

Plane 1 Contact Absorption contrast image

Plane 2 Propagated Phase contrast image

Fig. 1. Schematic of a phase contrast imaging set up.

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Fig. 2. Neutron phase contrast of a yellow-jacket wasp (genus Vespula): (a) a photograph of wasp; (b) conventional neutron radiograph using an extended source and (c) phase contrast radiograph using a pin-hole source showing greater clarity of ﬁner structures (from Ref. [10]).

Fig. 3. A series of neutron radiographs showing the development of the phase contrast of a Pb sinker mounted on an Al screw. The images (anticlockwise from top left to top right) correspond to propagation distances of approximately 20, 220, 450, 900 and 1.8 m. The visibility of the edge detail within the sinker, screw and shavings trapped in the axial hole are greatly enhanced (from Ref. [17]).

8. Phase radiography: transport of intensity equation

8.2. Nuclear

8.1. Interactions

The nuclear interaction results in a phase shift of the form:

The most readily attainable changes to the neutron probability current (in this case scalar phase shifts) are the result of nuclear, gravitational or magnetic potentials.

Dfnuc ¼ l0 NbL,

(13)

where N is the atom density, b the nuclear scattering

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Fig. 4. (a) Phase image of the longitudinal axis of a Pb sinker and (b) quantitative phase proﬁle CC0 through the sinker and theoretical proﬁle based on the shape and orientation of the sinker. The bright spot at the bottom of the phase image is a zero-intensity artefact from the neutron-absorbing mounting. An interference experiment would not be able to trace such a rapid excursion (from Ref. [10]).

amplitude, l0 the incident neutron wavelength, and L the path length through the material. Neutron scattering length measurements, which have been the major ‘‘industrial’’ application of the interferometers, were ﬁrst performed in the late 1970s and in a series of measurements since then, culminated in a high-precision technique insensitive to systematic alignment uncertainties [18]. In the generalised phase approach, l0 becomes the ensemble average wavelength. The quantiﬁcation of the phase shift was performed using a bullet-type Pb sinker [10] (see Fig. 4). The shape of the wavefront is determined by the ﬂow directions. This shape is then scaled entirely from the acquisition geometry and the dimensions of the physical object using the scattering length and density values. The line proﬁle shows the result of this. An accurate interferometric measurement of the sinker’s fast-changing phase excursion would require submicrometre camera resolution. Generally, the neutron scattering length of an element is positive (refractive index less than unity). However, elements do exist with negative scattering lengths (refractive indices greater than unity). This provides an opportunity to index match [11] samples to accentuate desired features. Such index matching has been observed in the neutron interferometry ‘‘phase echo’’ effect [19] where interference contrast disappears after passage through sufﬁcient thicknesses of either positive or negative scattering length materials, but reappears after passing through matching optical path lengths of both positive and negative scattering length materials. A characteristic property of neutron sources is their very low-phase space density. To improve the practicality of this

phase-sensitive technique, it is possible to take images with polychromatic neutrons in the knowledge that the scattering length does not change much over this wavelength range. Such a quantitative measurement was made of a silicon block [20] shown in Fig. 5. In this case, with negligible absorption and scattering, the phase calculation simpliﬁes to Eq. (9). The phase contrast image constitutes a two-dimensional projection of the real and imaginary parts of the neutron refractive-index distribution through a three-dimensional object. A number of these projections could be used to generate a quantitative tomographic reconstruction. On the assumption that the incident beam distribution is uniform (or can be normalised to be so) and there is minimal absorption and scattering within the sample then by taking just a single projection, a convenient phase imaging and phase tomography system can be envisioned. 8.3. Magnetic The phase shift induced by the magnetic interaction takes the form: Dfmag ¼

sm0 B L , _ v0

(14)

where s ¼ 1 depends on whether the neutron’s magnetic dipole moment m0 is directed up or down with respect to the applied magnetic ﬁeld B, which deﬁnes the quantisation direction. L and v0 deﬁne the path length through the magnetic ﬁeld region and the neutron velocity, respectively. This phase shift has been used regularly to affect spindependent phase shifts, for example to optimise experi-

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Fig. 5. (a) Recovered phase image of a projection through a perfect single-crystal silicon block using polychromatic neutrons and (b) computer simulation of the same projection through the block (from Ref. [20]). This phase image is in quantitative agreement with expectations based on Eq. (12) in the text.

mental operating points on the interferogram sinusoid and in polarised neutron interferometry for a long while. When combined with a non-uniform shape, this leads to a difference in lateral deﬂection of the neutron based on the orientation of the neutron spin with respect to the applied magnetic ﬁeld. An experiment can be envisioned where an ensemble of neutrons that on traversing an air-gap prismatic magnetic ﬁeld that a uniform beam would be separated into two components. This lateral redistribution of intensity is a consequence of a non-uniform phase shift, which could then be used to measure the magnetic ﬁeld strength. Such an experimental geometry and perfect crystal analysis of this spin-dependent angular separation has been used as a neutron polariser. A few arcsec separation propagated over 5 m, could be seen as a few pixel lateral shift in intensity at the camera, given 25 mm pixels.

additional geometry in these experiments, entails rotation of the interferometer crystal about its longitudinal axis of symmetry [21] (rather than about the incident beam, which to date has been standard). Such a geometry is achieved by deﬂecting the incident beam by the Si Bragg angle upstream of the interferometer, and would minimise, or at least simplify, bending of the crystal as it tilts. To perform a propagation-based measurement, one would measure the deﬂection of the neutron beam under gravity. In the absence of a known reference level, one or two (or more) harmonics of the neutron beam could be used as the reference. Given a thermal neutron of wavelength 2 A˚ and velocity v2000 ms1 , neutron freefall over a path length of about 10 m would result in a few 25 mm-pixel fall relative to lower and higher harmonics.

8.4. Gravitational

An additional angular momentum term, the Sagnac effect, appears in neutron interferometry due to experiments being performed on a rotating earth, a non-inertial reference frame. The Sagnac phase shift is given by

Termed gravitationally induced quantum interference, the gravitational phase shift is given by g Dfgrav ¼ 2pmi mg 2 l0 A0 ð1 þ ðaÞÞ sinðaÞ, (15) h where mi and mg are the inertial and gravitational masses of the neutron, g is the local acceleration due to gravity, h is Planck’s constant, A0 is the normal area enclosed by the beam paths, ðaÞ is a dynamical diffraction correction term, and a is the tilt angle of the interferometer. This equation, Sam is famously quoted as saying, is the only one where the quantum mechanically fundamental Planck’s constant and the universal gravitational constant, in the form of the local gravitational acceleration g, appear together. Many measurements on the gravitational fall and the gravitationally induced quantum phase of the neutron have been made. These and future experiments are reported at this conference. Another possible experiment, or an

8.5. Motional effects

4pmi OA0 cosðyL Þ cosðaÞ, (16) h where O is the Earth’s rotation frequency, yL the colatitude angle, and A0, mi and a again the enclosed area, inertial mass and tilt angle, respectively. The dependence on an enclosed area would indicate no possible propagationbased phase measurement. However, Coriolis and other motional effects should produce observable neutron deﬂections and phase shifts equivalent to those measured by interferometry. DfSagnac ¼

9. Measurement of quantum mechanical geometric phases As introduced above, the scalar and vector phases correspond to the continuous, single-valued, force-depen-

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dent ‘‘dynamical’’ phase and the ‘‘geometric’’ phases associated with the non-uniqueness of vortices, and intensity zeros. These phases appear in the presence of scalar and vector electromagnetic potentials. To date, neutron interferometric measurements of geometric phases have included the Berry phase, Aharonov–Casher (A–C) and scalar Aharonov–Bohm (A–B), all with Sam’s involvement. In the Berry phase case, and the A–C and scalar A–B effects, the phase shifts are force-free, potentialdependent effects with no deﬂection of the neutron beam, so no change in the probability ﬂow is expected. An observation of a phase vortex has been performed with X-rays [22]. In this experiment, the sample was a stepped spiral around a central axis aligned with the incident X-rays. The discontinuity on the central axis represented the phase vortex. A similar experimental geometry can be envisioned with neutrons.

10. Conclusion In this paper, we have presented a series of neutron phase measurement experiments based on measurement of the neutron probability ﬂow, alongside neutron interferometric experiments to measure equivalent interactions. Neutron phase measurement represents quantum mechanics being played out on a benchtop, and has highlighted the role that Sam Werner has played in substantiating the theory’s validity. The probability ﬂow method allows phase to be viewed in a very geometric manner. In this way, we are able to develop a suite of techniques that allow phase to be recovered in a very direct manner in a variety of different contexts. The approach is now well established for the measurement of the phase of images using light, electrons, X-rays, neutrons and atoms.

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References [1] W.H. Steele, S.A. Werner, Encyclopedia of Applied Physics, VCH, New York, 1994 (Chapter 8). [2] See, for example, H. Rauch, S.A. Werner, Neutron Interferometry, Oxford University Press, London, 2000; A.G. Klein, S.A. Werner, Rep. Prog. Phys. 46 (1983) 259. [3] H. Rauch, W. Treimer, U. Bonse, Phys. Lett. A 47 (1974) 369. [4] R. Colella, A.W. Overhauser, S.A. Werner, Phys. Rev. Lett. 34 (1975) 1472. [5] K. Schoen, D.L. Jacobson, M. Arif, P.R. Huffman, T.C. Black, W.M. Snow, S.K. Lamoreaux, H. Kaiser, S.A. Werner, Phys. Rev. C 67 (2003) 044005. [6] B.E. Allman, D.L. Jacobson, W.-T. Lee, K.C. Littrell, S.A. Werner, Nucl. Instrum. Methods A 412 (1998) 392. [7] D. Paganin, K.A. Nugent, Phys. Rev. Lett. 80 (1998) 2586. [8] M.R. Teague, J. Opt. Soc. Am. 73 (1983) 1434. [9] T.E. Gureyev, A. Roberts, K.A. Nugent, J. Opt. Soc. Am. A 12 (1995) 1942. [10] B.E. Allman, P.J. McMahon, K.A. Nugent, D. Paganin, D.L. Jacobson, M. Arif, S.A. Werner, Nature 408 (2000) 158. [11] P.J. McMahon, B.E. Allman, K.A. Nugent, D.L. Jacobson, M. Arif, S.A. Werner, Appl. Phys. Lett. 78 (2001) 1011. [12] A.G. Klein, S.A. Werner, Rep. Prog. Phys. 46 (1983) 259. [13] K.A. Nugent, D. Paganin, Phys. Rev. A 61 (2000) 063614-1. [14] M. Schlenker, J. Baruchel, Physica 137B (1986) 309. [15] D.L. Jacobson, M. Arif, L. Bergmann, A. Ioffe, SPIE 3767 (1999) 328. [16] R. Satija, D.L. Jacobson, M. Arif, S.A. Werner, J. Power Sources (2004), to be published. [17] These images are previously unpublished, and were captured using the same experimental conﬁguration at NIST as for Ref. [20] with support of P.J. McMahon, D.L. Jacobson, M. Arif, and S.A. Werner. [18] A.I. Ioffe, D.L. Jacobson, M. Arif, M. Vrana, S.A. Werner, P. Fischer, G. Greene, F. Mezei, Physica B 241–243 (1998) 130. [19] R. Clothier, H. Kaiser, S.A. Werner, H. Rauch, H. Woelwitsch, Phys. Rev. A 44 (1991) 5357. [20] P.J. McMahon, B.E. Allman, D.L. Jacobson, M. Arif, K.A. Nugent, S.A. Werner, Phys. Rev. Lett. 91 (2003) 145502. [21] B.E. Allman, Private communication to S.A. Werner, 1996. [22] A.G. Peele, K.A. Nugent, A.P. Mancuso, D. Paterson, I. McNulty, J.P. Hayes, J. Opt. Soc. Am. A 21 (2004) 1575.