Phase measurement permeates modern science. New propagation-based
alternatives to interferometry are providing increased opportunities for phase
measurements using x rays, electrons, neutrons, and other waves.
Scintillate, scintillate globule vivific Oft do I wonder your nature specific Loftily poised midst the aether capacious Closely resembling a gem carbonaceous . . .
—Source unknown
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.
The concept of phase is usually introduced as a property
of coherent wavefields. Such wavefields are described by a complex function
ψ(r) = [√I(r)]
exp[iφ(r)], 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 2a)
is illuminated by a uniform plane wave, an intensity distribution such
as that shown in Figure 2b 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
2c shows a sample interferogram of the phase distribution in Figure
2a. 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 images2 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 nR
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, nR - 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 (∇φS
+ ∇ ×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" φS
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.
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 methods2 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.
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.
References
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Physics Today References
February 1992, page 17
December 1994, page 24
January 2000, page 31
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