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Single-shot depth-resolved displacement field measurement
using phase-contrast polychromatic speckle interferometry
P D Ruiz1, M de la Torre-Ibarra2 and J M Huntley1
1
Wolfson School of Mechanical and Manufacturing Engineering, Loughborough
University, Ashby Road, LE11 3TU, UK
2
Centro de Investigaciones en Óptica Asociación Civil, Loma del Bosque 115, León,
Guanajuato, CP 37150, México.
Abstract. We describe a system for measuring depth-resolved displacement fields within a
scattering medium based on a speckle interferometer with a broadband light source. Depth
information is encoded in the form of spatial frequencies along the wavelength axis of a
spectrometer. From measurements of changes in the optical phase at different spatial
frequencies, displacement maps are obtained within a 2-D slice extending into the sample with
a sensitivity of order 10 nm, ultimately limited by the phase rms noise. The data for a given
deformation state is recorded in a single image, allowing sub-surface displacement and strain
mapping in dynamic applications.
1. Depth-resolved displacement measurement
Speckle interferometry recently started to merge with techniques from the optical coherence
tomography (OCT) family to provide depth-resolved displacement field information within weakly
scattering materials [1-3]. In Wavelength Scanning Interferometry (WSI), a temporal sequence of
speckle patterns from light coming from different depths within the sample is recorded as the
wavelength of a tuneable laser is scanned over a given spectral range. The intensity signal recorded
along the temporal axis is then Fourier transformed to extract: a) the structure of the object from the
spectral magnitude, and b) through-the-thickness displacement fields from measurements of phase
changes at different frequencies in the spectrum [2]. The scanning nature of this approach is clearly
not suitable for dynamic applications. Spectral OCT systems are based on the combination of an
interferometer and a spectrometer as a detector. They have no moving parts, no time delay is
introduced, they have better SNR as compared with time-domain techniques, and are capable of
achieving high depth resolution [4-7]. These systems can be considered as polychromatic speckle
interferometers due to the bandwidth of their illumination sources. However, previous workers have
used primarily the magnitude information in the interference signal, whereas it is the phase – as in
traditional speckle interferometry – that is the quantity of interest for the measurement of displacement
fields.
The system we describe in this work uses a broadband source in a Michelson arrangement to
illuminate a weakly scattering object and a reference mirror. Upon recombination on the beam splitter,
a diffraction grating spatially separates the different wavelength components of both the object and
reference beams, which interfere on a photodetector array. The system uses cylindrical lenses as in [8]
to illuminate the object and to image the superposition of speckle patterns from scattering centres at
different depths within the sample. The phase difference between the reference beam and light
scattered within the object from a slice j of thickness equal to the depth resolution of the system, z,
can be expressed in terms of the wave-number k=2 as j(k) =j0+2kzj where  ranges from c-/2
to c+/2, j0 is the phase change introduced by reflection in the j-th slice and zj is the optical path
difference between the reference wavefront and the j-th slice within the object. The spatial frequency fk
along the wave-number (or k-) axis is therefore given by fk = zj /. Assuming that multiple reflections
or multiple scattering within the material under study can be neglected, and representing the medium
by independent slices, the intensity distribution I(k) along the k-axis of the spectrometer is modulated
with spatial frequencies fkj=zj/that depend on the optical path difference between the j-th slice within
the object and the reference surface. The phase difference i(k)−j(k) between light coming from slices
i-th and j-th within the sample can be expressed in terms of the spatial frequencies in k-space as
i(k)−i(k)=2(fki−fkj)k. It is clear that the spatial frequencies fki−fkj=(zi−zj)/ depend on the optical path
difference between the corresponding slices within the object. Fourier transformation of the intensity
signal along the k axis leads to a spectrum with three main components: dc, autocorrelation, and a
correlation component that maps the internal structure, or scattering potential of the object under
study. The latter has to be separated from the autocorrelation component so that there is no cross-talk
between them, which can be done by changing the distance from the reference surface to the object. If
the j-th slice within the object moves to a new position so that the optical path difference is now zj+wj,
then the phase j will vary accordingly to j+j, modifying the intensity distribution I(k). The
complex nature of the Fourier transform of I(k), I*(fkj), allows not only for the computation of the
magnitude spectrum to get the structure of the object, as is common use in spectral OCT techniques,
but also the phase change j which occurs at the frequency component fkj can be obtained from the
real and imaginary parts of I*(fkj). The change in the out-of-plane optical path difference of the j-th
slice within the medium is wj =cj/4. It can be shown that the maximum optical path difference z
between a slice within the object and the reference surface –known as the depth range– is z =
(Nc2)/(4, limited by the spectrometer resolution (N pixels). The theoretical depth resolution
assuming a light source with a Gaussian spectrum is given by z=(c2 2ln2)/(n), with n the
refractive index of the material.
2. Experimental results
The technique was validated by measuring the tilt angle of a pair of cover slips (one under the other)
with polychromatic interferometry and two beam laser interferometry. The displacement of the cover
slips was obtained independently with both techniques, and is shown if Fig. 1 The measured tilt angle
was within 0.36% and 0.44% for the two surfaces, which can be regarded as a very good agreement.
Figure 1. Displacement field profiles measured using
Polychromatic speckle interferometry (solid lines) and
standard two-beam interferometry (dashed lines). An
arbitrary offset of 0.1m has been added for clarity
We investigated the applicability of the technique to scattering materials. Polychromatic speckle
interferometry was used to measure phase change maps due to viscoelastic deformation through-thethickness of ex-vivo porcine corneas. The intraocular pressure, mimicked in a pressure chamber, was
changed from 210 to 212 mm H2O. 10 seconds after the pressure change a reference polychromatic
spectral interferogram was recorded, followed by a sequence of interferograms at various framing
rates ranging from 5 to 0.1 frames per second. Video sequences of the phase maps show that initially
the phase change at the epithelium varies as fast as at the endothelium, but then it increases faster at
the endothelium than at the epithelium. This indicates that the cornea first translates as a rigid structure
and then starts to compress subject to the boundary condition of the epithelium tension on the outside
and the hydrostatic pressure in the inside. Variations in the corneal curvature introduce refractioninduced phase errors that need to be compensated before extracting quantitative data from the phase
maps, and further work is being done to tackle this problem.
3. Conclusions
Polychromatic Speckle Interferometry as described in this work has the ability to measure depthresolved displacement fields within the microstructure of a scattering material or tissue with a
displacement sensitivity that is decoupled from the depth resolution, being of the order of some tens of
nm and ultimately limited by the phase noise. Due to the single shot nature of the technique, it is
suitable to the study of dynamic materials and living tissues.
4. Acknowledgements
Manuel De la Torre-Ibarra thanks the Consejo Nacional de Ciencia y Tecnología (grant 42971) and
also the Engineering and Physical Sciences Research Council (grant GR/T25040/01) for partially
supporting this research. J M Huntley is also grateful to the Royal Society and Wolfson Foundation for
a Royal Society-Wolfson Research Merit Award.
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