Solution of the Phase Problem of
Almost 100 years ago, Max Laue suggested that X-rays may be diffracted by a crystalline medium. 1 This
was confirmed almost immediately by
experiments. Henry Bragg and his son,
William Lawrence Bragg, soon used
this phenomenon to estimate structures
of several crystalline solids from X-ray
diffraction experiments. These investigations were the starting point of a
technique that has found numerous applications in physics, chemistry, biology
Successful as this technique has
been, its usefulness is limited by the fact
that it provides no information about
phases of the diffracted beams. However,
for unambiguous determination of the
structure of crystals, one must know
both the amplitudes and the phases of
Recently, a solution of this “phase
problem” was found, 2, 3 after it was
pointed out that all previous treatments
have made the unrealistic assumption
that the X-ray beams are monochromatic. Actually, amplitudes and phases
of the beams vary randomly in time.
Quantities that are physically meaningful, and that can be measured, are
correlation functions, well known in
coherence theory of light. 4 The correlation functions contain information
about the amplitude and the phase of
an average wave function of a diffracted
beam and from their knowledge the
crystal structure can be unambiguously
determined, provided that the beams
are spatially coherent—a property that
must be distinguished from monochromaticity. 2, 3 Spatially coherent
Schematic of the usual arrangements for determining structure of crystalline solids by
X-ray diffraction experiments (a), and of the new technique (b), which makes it possible to
determine not only the amplitudes, but also the phases of the diffracted beams.
* The knowledge of the amplitudes alone has made it possible to determine the structures of some crystalline media,
at least approximately. The importance of this technique
is evident from the fact that 11 Nobel prizes in physics,
chemistry, medicine and physiology have been awarded
for such contributions.
light beams are routinely generated
in laboratories, and spatially coherent
X-ray beams have also been produced in
recent years. 5
Part (a) of the figure illustrates the
usual layout used in X-ray diffraction
experiments. The intensities of the
diffracted X-ray beams are measured
by square-law detectors D1 and D2.
Part (b) illustrates the essence of the
new method, which makes it possible
to determine the intensities and also
the phases of the diffracted beams. The
detectors D1 and D2 are replaced by
pinholes Q1 and Q2 in an opaque screen
A, and X-rays that pass through them
form an interference pattern on a plane
B, parallel to A, some distance behind
it. From the fringe pattern, both the
amplitudes and the phases of the dif-
fracted beams can be deduced. t
Emil Wolf ( email@example.com) is with the
department of physics and astronomy and The
Institute of Optics at the University of Rochester,
Rochester, N. Y., U.S.A.
1. M. Laue. Proceedings of the Bavarian Academy of Science, 1912, p. 303.
2. E. Wolf. Phys. Rev. Lett. 103, 075501-03, (2009).
3. E. Wolf. Phys. Lett. A 374, 491-5 (2010).
4. E. Wolf. Introduction to the Theory of Coherence and
Polarization of Light, Cambridge University Press, Cambridge, England, 2007.
5. e.g. Y. Liu et al. Phys. Rev. A 63, 033802 (2001).
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