Scintillate, scintillate globule vivific Oft do I wonder your nature specific Loftily poised midst the aether capacious Closely resembling a gem carbonaceous …
We are introduced to the effects of phase from the earliest days of our childhood, from the nursery rhyme above (or its less verbose form, “Twinkle, Twinkle Little Star”) to the shimmer over a hot road and the network of bright lines at the bottom of a swimming pool. These are all manifestations of phase. And there are many more.
Because of its intimate relationship to gauge transformations, as well as the idea that potentials are more fundamental than fields, phase is fundamental to all of physics. Moreover, essentially all we know of the universe is conveyed via waves.
Knowledge of the appropriate (time-independent) wave equation, along with the wave’s intensity and phase over a surface, typically allows the wave to be known everywhere. We know how to measure the intensity directly, corresponding as it does to the intensity of light or the probability distribution of quantum mechanical waves. In this article we review some recent ideas on how to measure the phase.
Phase and phase visualization
The concept of phase is usually introduced as a property of coherent wavefields. Such wavefields are described by a complex function , where r is the position in space, I(r) is the intensity (or probability density) of the wave, and φ(r) is its phase. The surfaces of constant φ(r) can be identified with the wavefronts. In free space, ∇φ(r) describes the local propagation direction, as shown in figure 1.
The phase of an electromagnetic wave is inevitably changed as it passes through an object, although our eyes see only changes in the intensity. Some materials affect the phase of a wave with only minimal effect on the intensity, such as a clear high-quality window that introduces negligible intensity change and a spatially uniform phase shift. A lens, on the other hand, also does not change the intensity but induces a nonuniform phase shift to the wave. On propagation through a suitable distance, these invisible phase shifts are transformed into visible intensity variations (see figure 1). We can conclude that a phase gradient can be visualized by observing the propagation of the intensity. For example, when an object that only affects the phase (see figure 2(a)) is illuminated by a uniform plane wave, an intensity distribution such as that shown in figure 2(b) is created a short distance downstream.
Many physicists automatically connect the measurement of phase with the technique of interferometry. The key to interferometry is to overlay one coherent wave with another and use the resulting interference fringes to deduce the relative phases of the two waves. Figure 2(c) shows a sample interferogram of the phase distribution in figure 2(a). Interferometry can be routinely performed with coherent radiation, which may be sound, electrons, x rays, neutrons, or atoms.
Interferometric techniques are not well suited to imaging, however. For example, for optimum resolution an optical microscope requires partially coherent radiation, which is insufficient for interferometry. So, in order to see phase, optical microscopists have for a long time used a slight defocus of the system, a form of propagation-induced phase contrast. The work of Frits Zernike, published in 1942 and for which he was awarded the 1953 Nobel Prize in Physics, was the first to combine an in-focus image with phase visualization and high resolution.
Defocused images are also used in electron microscopy. Many of the samples of interest to electron microscopists yield only phase information. For fast electrons passing through sufficiently thin crystals, the defocus information yields the Laplacian of the projected potential of the crystal. 1 Consequently, the phase may be estimated quantitatively using a series of defocused images 2 and applying numerical techniques to find a phase distribution consistent with the entire data set.
In what is probably a better-known approach, the image and the far-field (Fraunhofer) diffraction pattern are used as input for iterative phase-retrieval algorithms. This method was first proposed by Owen Saxton and Ralph Gerchberg. 3 The essence of this algorithm is to assume that a wave’s intensity and far-field diffraction pattern are known, but not its phase. An initial guess of the wave’s phase is made and, with the known intensity, Fourier-transformed to obtain the corresponding far-field diffraction pattern. In general, the calculated pattern will be incorrect. But when the measured intensity of the diffraction pattern is substituted for the calculated intensity, keeping the calculated far-field phase the same, a reverse transform provides a refined estimate of the wave’s phase. This phase estimate is then used with the measured intensity for the next iteration. Given certain constraints, the phase distribution converges to the correct value as the iterations proceed. In a related approach, James Fienup subsequently showed that a complex wave may be recovered from its far-field diffraction pattern and knowledge of its “support,” that is, the area outside of which the wave is known to be identically zero. 4
In an interesting extension and demonstration of this idea, Jianwei Miao, in the group led by Janos Kirz and David Sayre, obtained very high-resolution x-ray images of noncrystalline samples by combining a Gerchberg–Saxton-type iterative algorithm with oversampling of the diffraction pattern. 5
There are also other alternatives to interferometric phase determination. Consider, for example, the field of astronomical adaptive optics. The phase induced by atmospheric turbulence, which scintillates stars and inspires poetry, has dire effects on astronomical imaging because it prevents the acquisition of diffraction-limited images. Indeed, the rule of thumb is that the atmosphere limits even the best telescope to the resolution of a perfect 20-cm-aperture instrument. Clearly, if it is possible to measure the phase distribution of the incoming light across the entrance of the telescope, then it might be possible to introduce real-time correction of the distorted wavefront to create a diffraction-limited image. This is the idea underlying astronomical adaptive optics (see Physics Today, February 1992, page 17 and December 1994, page 24).
The phase-sensing methods of adaptive optics typically build on the link between phase and propagation direction described in figure 1. This link is clearly borne out in what is perhaps the best known of the quantitative non-interferometric phase sensors, the Hartmann–Shack sensor (described in Physics Today January 2000, page 31). This device uses a set of small lenslets, each of which senses the phase gradient of the incoming wave averaged over that lenslet’s aperture.
The defocus phase-contrast method of optical and electron microscopy is related to the essential idea underlying the curvature-sensing adaptive optics technique proposed by François Roddier and colleagues at the University of Hawaii. 6 The intensity distribution of a very slightly defocused image of a pure phase object is described by the Laplacian of the phase. Physically, this corresponds to the local phase curvature of the radiation. In an adaptive optics system, one can form a slightly defocused image of the wavefield entering the telescope; the resulting phase contrast yields quantitative information about the phase curvature, which can then be nulled out with adaptive optics.
We are all familiar with the medical x-ray radiograph, which can be regarded as essentially an in-focus image. The film is, for most practical purposes, in contact with the sample, and any phase information the sample imprints on the wave is entirely invisible. Moving the film away is equivalent to defocusing, and will therefore render phase visible. For a sufficiently short distance and with negligible absorption, the intensity distribution at the detector will be described by the Laplacian of the phase. Propagation-based phase visualization has been implemented in x-ray imaging using synchrotron and conventional x-ray sources. 7 That such data could be used to measure the x-ray phase was first reported in 1996, 8 and later work has shown submicron spatial resolution (see figure 3). 9 This work also shows promise for medical applications, in which phase-contrast methods can provide improved image contrast and reduced x-ray dose to the patient (see Physics Today, July 2000, page 23).
Phase as a flow potential
In this article, we have really been using the term “phase” for two distinct physical quantities. The first is the phase of a wave, which is directly measured using interferometry and loses meaning for a partially coherent wave. Indeed, a coherent wave is, by definition, one with a well-defined phase. The second meaning concerns the real part of the refractive index of a medium, an independent and well-defined physical property The concepts are often used interchangeably because the real part n R of the refractive index couples directly to the phase of a wave passing through the medium. When the light is coherent and the refraction is sufficiently weak, the wave will accumulate a phase shift that, at the exit surface of the object, is proportional to the integral of the refractive index increment, n R −1, along each ray. Using the refractive index, we are therefore able to talk sensibly about the phase of the object, even though we may not be able to talk sensibly about the phase of the wave. Recent work at the University of Melbourne has clarified this link between wavefield and object phase, allowing us to generalize what we mean by the phase of a partially coherent wave. 10
This link between wavefield and object phase proceeds via the probability-current or flow vector for the field. This vector describes the flow of energy in the wave and is most familiar in the guise of the Poynting vector for electromagnetic waves. In this article, we use the term “Poynting vector,” but the ideas apply to all waves. In the case of a fully coherent wave, the Poynting vector has the form S = I ∇φ; the flow of energy depends directly on the phase gradient distribution of the wave.
The coherent Poynting vector clearly describes a vector field and so may contain some vorticity, in which case the phase φ is discontinuous (see box 1). It is therefore possible to rewrite the Poynting vector in the general form S = I(∇φ S + ∇ × φV), where we have introduced two phase components in analogy with the scalar and vector potentials of electromagnetism. The vector “potential” φV describes discontinuous phase components, or phase vortices (see box 1), such as photons carrying orbital angular momentum. Indeed, waves of this form have been used to create laser tweezer systems that, by virtue of the angular momentum of the photons, can rotate trapped particles. 11 Such systems have been dubbed “optical spanners.” In the absence of such discontinuous phases, the scalar “potential” φ S is simply the phase with which we are all familiar.
The flow vector is well defined for coherent fields but will fluctuate with time for partially coherent fields. The time average, however, is well defined and, as a vector field, it may also be written as arising from a vector and scalar potential. We may use this formulation as the definition of what we mean by phase. 10 The phase so defined is identical to the conventional phase when the light is coherent, is well defined for partially coherent light, and behaves precisely like the phase of the medium in the sense that we have discussed above.
Vortices abound in nature. Examples include eddies in rivers or smoke plumes, spiral galaxies, the red spot of Jupiter, vortices in Bose–Einstein condensates (see Physics Today, November 1999, page 17 and August 2000, page 19), and angular-momentum eigenstates of the hydrogen wavefunction. In a landmark paper written in 1931, P. A. M. Dirac demonstrated the possibility of vortical structures in any complex scalar wavefield, using a general topological argument that is independent of the particular wave equation. He showed that, in free space, the vortex cores are associated with a zero of probability density or intensity, and that the vortex cores themselves must either form closed loops or extend to infinity. He also showed that the wavefunction phase as one walks around a vortex core has the multivalued helical-type structure shown in the figure.
Two important special cases of the complex scalar wavefields considered by Dirac are the complex scalar wavefunction of nonrelativistic quantum mechanics and the complex function describing scalar electromagnetic waves. For both of these wavefields, vortices are ubiquitous in the sense that a propagating wavefunction, even if initially vortex free over a given plane, will almost always develop vortex structures on evolution through a sufficient distance in time or space. Thus, as John Nye and Michael Berry wrote in 1974, “Matter waves contain a tangled ‘cobweb’ of dislocation lines on which the electron density is zero.” 17
The spiral phase structure shown in the figure appears continuous; a plot of the phase on a given plane, however, will show a π phase discontinuity at the axis. This is rather like a spiral staircase: The height of the staircase on its axis is not at all well defined.
These wave structures are important in phase measurement because two waves with the same angular momentum, except for sign, may have identical intensity distributions throughout space. A useful analogy is a rotating deformable sphere. By observing the deformation of the sphere due to the rotation, and knowing its material properties, one can determine the speed and axis of the rotation but not its sign. Note also that if the perfect symmetry of the sphere is broken (such as by putting a mark on the surface), then the ambiguity immediately disappears. Phase measurements may similarly distinguish the underlying vorticity in wavefields.
Since energy may be neither created nor destroyed in free space, the time-averaged Poynting vector S must obey a conservation equation given by ∇ · S = 0. The time-averaged Poynting vector can, in general, be written in the form S = I∇φ, where we define the function φ to be what we mean by the “phase”; if the wave is coherent, then this is equivalent to the conventional phase. The phase obtained in this way need not be continuous, and in general will not be. For example, as discussed in box 1, vortices are described by discontinuous phases. If the phase is not continuous, then components corresponding to the presence of vortices may be present, and the ambiguity described in box 1 may arise. If the phase is known to be continuous, then the continuity equation has a unique solution for the phase.
If we make the approximation of a paraxial wave propagating in the z direction, the continuity equation reduces to the “transport-of-intensity equation”: ∇⊥ · (I∇⊥ φ) = –(2π/λ) ∂I/∂z. Here, ∇⊥ acts in the plane perpendicular to the z-axis of the optical system, and λ is the wavelength. The transport-of-intensity equation can be rapidly and robustly solved for the phase φ, given measurements of the energy density I over a plane and its longitudinal derivative ∂I/∂z. The phase solution is unique inside a given domain if appropriate boundary conditions are provided 18 and there are no phase vortices. Such boundary conditions can usually be derived from a priori knowledge or measurements of the intensity in the vicinity of the domain boundary.
Propagation-based phase measurement
A hydrodynamic formulation of quantum mechanics was proposed in 1926, shortly after the development of the underlying theory. In this formulation, the key equation is that which expresses the conservation of probability on propagation. In his unpublished 1933 PhD thesis, Eugene Feenberg at Harvard University claimed that knowledge of the probability distribution in three dimensions along with its time rate of change was sufficient to fully specify the wavefunction via a solution of the continuity equation. This work did not take into account the effect of vortices, and so the conclusion was not quite correct. Nevertheless, it was the first suggestion that three-dimensional intensity information permits phase determination, and it is indeed true that, for a time-averaged field and in the absence of phase dislocations, knowledge of the intensity distribution is sufficient for the continuity equation ∇ · (I∇φ) = 0 to be uniquely solved for the phase.
The problem of phase determination is simplified if we are able to make the “paraxial” assumption: All of the energy is traveling at a small angle to a given direction. In this case we arrive at what is known as the “transport of intensity” equation. Described in box 2, this differential equation relates the intensity and phase of a wave over a plane to the rate of change of intensity in a direction perpendicular to that plane. It may be interpreted in hydrodynamic terms as stating that the divergence rate of the transverse component of the Poynting vector is proportional to the rate at which the local energy density increases along the direction of propagation. This expression thus encapsulates a good fraction of the physics we have so far discussed.
With certain caveats concerning phase vortices (see box 1), the transport-of-intensity equation may be solved uniquely for the phase in a plane given measurements of the intensity in that plane together with the intensity derivative normal to that plane. This possibility was first realized by Michael Teague in 1983, 12 and brought to fruition at the University of Melbourne with the development of efficient, rapid, and robust algorithms for phase retrieval using intensity and intensity derivative data. 8,10 On a related note, one of us (Gureyev) and Stephen Wilkins of CSIRO have pointed out that a similar equation may be solved using defocused intensity data taken over a single plane using multiple wavelengths. 13 Significantly, due to the broader idea of what we mean by phase, these propagation-based methods are able to work with partially coherent radiation of insufficient coherence for interferometric phase determination, 10 and do not have the 2π phase ambiguities often associated with interferometry.
Phase imaging is of widespread importance in microscopy. Since a small defocus of an imaging system is equivalent to propagation over a small distance, it is straightforward to acquire the necessary data set for propagation-based phase measurement using the transport-of-intensity equation. For example, phase measurement in electron microscopy has usually required electron holography (see Physics Today, April 1990, page 22) or iterative methods. The transport-of-intensity equation naturally describes the effect of a small defocus on a sample and so permits the recovery of phase without the need for holography or interferometry. The result of such an experiment is given in figure 4, which shows the phase map of electrons that have passed through a magnetic sample. 14 The map was acquired by solving the transport-of-intensity equation using data that were collected on a conventional transmission electron microscope yet were still sufficient for quantitative recovery of the phase information. When the same sample was independently imaged with electron holography, the two phase measurements were found to be in excellent agreement.
Another area that commonly uses interferometry is neutron optics (see Physics Today, December 1980, page 24). Many fundamental explorations of the bases of physics have been made using neutron interferometry. These experiments are typically very time consuming, as they require a coherent beam of neutrons, which, in turn, requires extensive filtering. Drawing on the ability of propagation-based methods to determine phase using partially coherent waves, phase measurement with a conventional neutron source has also been demonstrated. 15 This work opens up avenues for phase-sensitive neutron radiography, as demonstrated in figure 5.
As a final application of the various methods of quantitative propagation-based phase imaging, we mention the goal, recently achieved by several groups, of extending the phase imaging from two to three dimensions. We single out the work of Peter Cloetens and collaborators at the European Synchrotron Radiation Facility in Grenoble, France. 16 They used iterative methods 2 to achieve quantitative 3D phase tomography of a polystyrene foam sample using images taken in the intermediate field (see box 3) with high-energy x rays from a third-generation synchrotron source. The result of their work is shown in figure 6.
The ideas underlying the theory and practice of non-interferometric propagation-based phase measurement provide a broader perspective on what we mean by phase and so extend phase measurement to encompass areas not previously thought possible. Of course, there is much beautiful work falling outside the scope of this article, examples of which include the measurement of strain fields in crystals and experimental approaches to phase-sensitive x-ray imaging. We anticipate that this broadening of the ambit of phase measurement will open up applications in a wide range of contexts.
A fundamental physical quantity determining the nature of phase contrast is the Fresnel number, N F = a 2/λz, where a is the size of the object being imaged, λ is the radiation wavelength, and z is the defocus distance. Different (near, intermediate, and far) regions (or fields) of imaging are usually defined depending on this number being respectively much larger than, of the same order as, or much smaller than 1. It turns out that the nature of phase contrast (and, correspondingly, methods for quantitative phase measurement) are very different in the different regions. In particular, near-field image contrast depends linearly on the phase shifts introduced by the object into the transmitted wave. The dependence can become quasi-linear in the intermediate field, and strongly nonlinear in the far (Fraunhofer) field (see figure 3). The near-field phase contrast also depends linearly on the defocus distance, and often has a relatively simple (power) dependence on the wavelength.
The transport-of-intensity equation method of phase measurement (box 2) exploits the simplicity of the near-field phase contrast by providing a tool for rapid and accurate phase retrieval. The advantage of working in this region is that the vortices discussed in box 1, unless they arise from vortical structures in the object, will not arise in the near field and so do not normally present a problem. On the downside, near-field imaging may impose stricter requirements on the signal-to-noise ratio and spatial resolution of the imaging system compared to intermediate- and far-field imaging.
The authors would like to thank Stephen Wilkins and John Spence for many helpful comments.
Keith Nugent is a professor and head of the school of physics and David Paganin is a postdoctoral research fellow at the University of Melbourne in Melbourne, Australia. Tim Gureyev is a principal research scientist at Australia’s Commonwealth Scientific and Industrial Research Organisation (CSIRO) in Melbourne.