Synthetic Aperture Fourier Holographic Optical Microscopy
Sergey A. Alexandrov,*Timothy R. Hillman, Thomas Gutzler, and David D. Sampson
Optical+Biomedical Engineering Laboratory, School of Electrical, Electronic and Computer Engineering,
The University of Western Australia, Crawley, WA, 6009, Australia
(Received 6 April 2006; published 18 October 2006)
We report a new synthetic aperture optical microscopy in which high-resolution, wide-field amplitude
and phase images are synthesized from a set of Fourier holograms. Each hologram records a region of the
complex two-dimensional spatial frequency spectrum of an object, determined by the illumination field’s
spatial and spectral properties and the collection angle and solid angle. We demonstrate synthetic micro-
scopic imaging in which spatial frequencies that are well outside the modulation transfer function of the
collection optical system are recorded while maintaining the long working distance and wide field of view.
DOI: 10.1103/PhysRevLett.97.168102PACS numbers: 87.64.Rr, 42.40.?i, 42.30.Kq, 42.30.Lr
in which amplitude and phase images are synthesized from
a set of digital holograms of different regions of the two-
dimensional complex spatial Fourier spectrum of a wave
after interaction with a sample. Selection of an area of the
Fourier spectrum is performed via control of the angular
and/or spectral properties of the illumination and collec-
tion light fields. We show high-synthetic-NA images con-
taining spatial frequencies that are well outside the MTF of
the long working distance, low-NA collection optical sys-
tem, while maintaining its wide field of view.
The principle of synthetic aperture Fourier holography
is illustrated in Fig. 1. A sample is located in the object
plane, with coordinates ?, ?, and a recording is performed
in the recording plane, with coordinates x, y, optically
conjugate to the back focal (Fourier) plane of the objective.
Let r??;?? be the complex scalar reflectance or trans-
mission profile of the sample, and the illumination wave
be plane and have polar angle ?iand azimuthal angle ?i,
as shown. The complex scalar field of the reflected or
transmitted wave (sample wave) in the object plane is
given by Ui??;?? ? r??;??Ai??;??, where Ai??;?? ?
A0exp??j2????? ? ????? is the complex scalar field of
the illumination wave in the object plane. The quantity A0
isa complexconstant and??,??represent the illumination
wave spatial frequencies in the object plane, so that, for
wavelength ?, ???? sin?icos?i, and ???? sin?isin?i.
The complex scalar field in the recording plane US?x;y?,
the scaled and phase-shifted Fourier transform of Ui??;??,
is recorded holographically. The sample spatial frequency
magnitude ?sthat yields diffraction or scattering into the
angle ?outin the plane of incidence [as shown in Fig. 1(b)]
can be written as 
Optical microscopy is one of the most extensively used
tools in the life and material sciences. To achieve the
highest resolutions, sophisticated high-numerical-aperture
(NA) objective lenses have been realized. Alternative path-
ways to subwavelength resolution based on near-field [1–
3] and far-field [4,5] techniques have also been developed.
A high-NA lens brings with it constraints on the field of
view and the working distance that are limitations in many
applications. A field of view smaller than the region of
interest is usually dealt with by moving the sample or
objective to record multiple tiled images. Here, we present
an alternative procedure, drawing together three innova-
tions in classical optics to relax the constraints on the
optical components, thereby allowing greatly increased
working distance, field of view, and contrast of the highest
spatial frequencies. The first of these innovations is
Zernike’s phase contrast microscope [6,7], which trans-
formed the microscopic visualization of phase objects.
The second is Gabor’s invention, holography , and its
relatively recent application to phase (and amplitude) mi-
croscopic imaging [9–11]. The final innovation is Ryle’s
concept of the synthetic aperture, first applied to increase
the resolution of the radio telescope  and later to
synthetic aperture radar . We report their combination
in a new synthetic aperture phase and amplitude micros-
copy based on Fourier holography.
There have been previous attempts to apply the principle
of the synthetic aperture to optical imaging [14–19], with
the objective of enhancing resolution, as well as the modu-
lation transfer function (MTF) , depth of focus, field of
view, and working distance . Several of these con-
cerned microscopy [14,19] and have employed holography
[15–18] or interferometry . Innovations have included
off-axisillumination [14,16,18,19],Fourier plane synthesis
[15,16], and extended recording planes . Improve-
ments in resolution and other quantities have been demon-
strated, but there is no effective method of synthesizing the
complex amplitude of a sample wave over both a large
range of spatial frequencies and a wide field of view
simultaneously. In this Letter, we introduce and demon-
strate synthetic aperture Fourier holographic microscopy,
Off-axis holography is performed by using a plane refer-
ence beam [Fig. 1(a)], with polar angle ?rand azimuthal
angle ?r, and complex scalar field denoted UR?x;y?. The
recording area, assumed rectangular, is given by ?L=2 ?
x ? L=2, ?H=2 ? y ? H=2, where L, H are, respec-
PRL 97, 168102 (2006)
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© 2006 The American Physical Society
tively,the length and height ofthe detector array (hereafter,
‘‘detector’’). The recorded intensity image is
IH?x;y? ? jUS?x;y? ? UR?x;y?j2rect
When inverse Fourier transformed, the image comprises
zero-order autocorrelation and background images, and
translated twin conjugate complex reconstructions of the
sample field (over the range of spatial frequencies cap-
tured). As demonstrated in Eq. (1), the use of an off-axis
illumination beam ensures that a nonzero sample spatial
frequency gives rise to the on-axis diffracted beam (?out?
0).The range of diffraction angles captured bythe detector,
centered on the axis, corresponds to the following range of
sample spatial frequencies (in the ?, ? directions, respec-
2M? ??;??;min=max? ?H
where M is a constant dependent on the collection system,
and directly proportional to ?. Equation (1) shows that it is
possible to vary the sample spatial frequencies captured by
varying the parameters of both the incident and diffracted
waves, specifically their angles and/or wavelength.
Equation (3) gives the range of recorded spatial frequen-
cies for a given incident angle, assuming the collection
solid angle is limited by the size of the detector. If the
detector is not restricted to the on-axis location, Eq. (3) is
not applicable, and higher spatial frequencies could be
recorded according to Eq. (1). In the limiting case of ?i?
90?and ?out? ?90?, the spatial frequency limit is ?s?
2=?, which is twice the theoretical cutoff spatial frequency
of conventional coherent imaging systems and the equal of
incoherent systems (without immersion). Moreover, our
approach has a dramatically improved spatial frequency
response when compared to the MTF of conventional
microscopy using incoherent illumination, in theory main-
taining the high contrast of coherent illumination up to the
To yield the synthetic image, we firstly reconstructed the
complex sample field from each individual Fourier holo-
gram, by inverse Fourier-transforming IH?x;y?. The result-
ing array was translated by an amount determined by the
reference beam off-axis angle, and postmultiplied by a
carrier function with spatial frequency (??, ??), resulting
in a reconstruction of r??;?? over the frequency range
represented by Eq. (3). The fact that the hologram was
recorded digitally also allowed for the numerical correc-
tion of defocusing in both the object and recording plane.
The synthesized image was then obtained by summing the
resulting complex fields of the sample r??;?? over the
different spatial frequency ranges represented by the mul-
tiple recordings. Anyarea of the image could be selected to
perform this operation.
The technique can be applied to scattering, diffracting,
and pure phase objects, in both transmission and reflection.
We used a reflection configuration in experiments, as
shown in Fig. 1(b), to enable the study of thick and non-
transparent samples . The illumination source was a
red helium-neon laser. Light collected by the objective lens
(focal length 15 mm, working distance 13 mm) was mag-
nified and recorded using a charge-coupled device (CCD)
(1392 ? 1040 pixels) located in a plane conjugate to the
Fourier plane of the objective. The collection solid angle of
the setup is equivalent to an NA of 0.13. As shown in
Fig. 1(a), the optical axis of the collected beam was per-
pendicular to the surface of the sample (?out? 0), thus,
Eq. (3) applies. To change the azimuthal incident angle
and,therefore, the location inthe Fourier plane accessed by
the hologram, samples were rotated around the optical
We recorded four Fourier holograms with azimuthal
angles at 90?intervals, covering the regions of the spatial
frequency spectrum shown in Fig. 1(c). As suggested by
the figure, the four regions were identified with the azimu-
thal angles 0?, 90?, 180?, and 270?. The field of view on
the sample was approximately 1 ? 1 mm. Two samples
containing microstructures significantly smaller than the
resolution limit of the collection optics were chosen. The
first sample was a ruled reflection diffraction grating of
1200 lines=mm (The Optometrics Group). The second
sample was a transmission electron microscopy calibra-
tion grid target (Ladd Research Industries Inc.) consisting
of a germanium-shadowed carbon replica made from a
28800 lines=in diffraction grating mounted on a grid
with 125 ?m pitch, 97 ?m holes, and 28 ?m bars. This
target produces diffraction from the grating structure and
FIG. 1 (color online).
recording (Fourier) planes, and illumination and reference
waves, with respect to the optical axis z; (b) depiction of the inci-
dent and scattered or diffracted waves in the plane of incidence,
and the collection solid angle; (c) range of spatial frequencies
covered in four separate holographic recordings (labeled 0?, 90?,
180?, and 270?). The range covered by a single 0.75 NA lens (for
a coherent imaging system) is also shown.
(a) Orientation of object (sample) and
PRL 97, 168102 (2006)
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wide-band scattering from the grid structure. Both samples
were oriented so that the first-order diffraction spots asso-
ciated with the gratings were imaged at azimuthal angles
Figure 2(a) shows the full-sized amplitude image of the
first sample reconstructed from a single hologram recorded
usinganincident azimuthal angle of0?anda polarangle of
49?, the latter corresponding to a synthetic NA of 0.76.
Figures 2(b) and 2(c) present phase images of selected,
successively magnified areas of the image synthesized
using the four holograms. The dark and light lines on the
synthesized images correspond to the lines on the grating.
For comparison, Fig. 2(d) shows a confocal reflectance
image of the grating obtained using an oil immersion
objective lens (NA ? 1:0).
Synthetic imaging of the second, more complex sample
is demonstrated in Fig. 3. Figure 3(a) shows a confocal
reflectance image of a large portion of the target (10?
objective lens) and Fig. 3(b) shows a differential interfer-
ence contrast (DIC) image (40? objective lens, NA ?
0:75) showing the substructure, including a strip processed
using an edge-detecting filter to enhance the contrast. For
synthetic imaging, an incident polar angle of 46?, cor-
responding to a synthetic NA of 0.72, was used. Re-
constructed full-size images for azimuthal angles of 0?
and 90?are shownin Figs. 3(c) and 3(d). Figure 3(e) shows
an image corresponding to the highlighted region in
Figs. 3(c) and 3(d), synthesized from azimuthal angles
90?and 270?. This image was formed from waves scat-
tered from the grid structure, since diffracted waves were
absent at those angles and, as a result, it is dominated by
speckle. Figure 3(f) shows the synthetic image incorporat-
ing all azimuthal angles. Amplitude and phase images of
the highlighted regions of Fig. 3(f) are shown in
Figs. 3(g)–3(j), respectively. The difference between the
scattering grid structure and the diffracting fine grating
structure is marked. The scattering structure contains spa-
tial frequency components in two distinct orientations,
determined by the four azimuthal angle positions, whereas,
the diffracting structure is dominated by a single spatial
frequency, oriented according to the grating direction.
Increasing the number of holograms used to synthesize
the image would lead to additional spatial frequency com-
ponents in different directions being incorporated into
FIG. 2 (color online).
1200 lines=mm: (a) Reconstructed full-size image for azimuthal
angle 0?; (b), (c) phase images of selected and successively
enlarged areas of synthesized image; (d) confocal microscope
Images of a ruled reflection grating of
FIG. 3 (color online).
(b) DIC; (c), (d) reconstructed full-size images for azimuthal
angles 0?and 90?[scale bar in (d)]; (e) Synthetic image using
90?and 270?azimuthal angles; (f) Synthetic image using all
azimuthal angles [scale bar in (e)]—grid lines are a guide to the
eye; (g), (h) Magnified amplitude images of upper/lower high-
lighted regions of (f), respectively, [scale bar in (g)]; (i),
(j) Magnified phase images of highlighted regions of (f) [scale
bar in (g)].
Images of a grid target: (a) Confocal;
PRL 97, 168102 (2006)
20 OCTOBER 2006
Figs. 3(h) and 3(j), whereas Figs. 3(g) and 3(i) would Download full-text
remain essentially unchanged. Asa result,chaotic structure
would be observed in the grid image, but regular structure
maintained in the grating image.
The reconstructed images in Figs. 2 and 3 display the
submicron-period structures (0.83 and 0:88 ?m, re-
spectively) with high contrast over a 1 ? 1 mm field of
view, and validate the concept. The ruled lines of the
1200 lines=mm grating are clearly resolved in Fig. 2(c),
although the maximum spatial frequency of the theoretical
MTF of the objective lens is 400 lines=mm. Since probing
such high spatial frequencies depends on the illumination
or collection angles and not on the collection solid angle,
higher resolution than demonstrated here over much larger
samples should be feasible. The choice of collection solid
angle represents a trade off between the number of holo-
grams to be recorded and the working distance or detector
In contrast to other methods , holographic recording
in the Fourier plane enables both amplitude and phase
imaging to be performed with greatly relaxed constraints
on the field of view for a given number of available
detector pixels. As mentioned above, recording close to
on-axis diffraction leads to a coherent transfer function
over the synthetic aperture that can be controlled and
greatly enhanced at high spatial frequencies relative to
the MTF of an objective of the same NA . The en-
hanced spatial frequency response is in evidence in com-
paring the contrast in Fig. 3(b) (NA ? 0:75) and Fig. 3(g)
(synthetic NA ? 0:72); in Fig. 3(b) the contrast of the
microstructure is low because its frequency is close to
To correctly synthesize the complex Fourier field from
the set of recorded holograms, it is necessary in general to
correctly set the relative amplitude and phase factors be-
tween the separately recorded complex amplitudes.
Uncontrolled phase factors in our experiments did not
affect the quality of the reconstructed images because the
high-spatial frequency components of our samples con-
sisted of the special case of uniform grating structures.
Phases may be controlled interferometrically during ad-
justment of the incident wave, or inferred from partially
overlapping Fourier holograms [15,16,18] or a priori sam-
ple knowledge. The system is intrinsically tolerant to lat-
eral object translations (which may occur, for example, if
the object rotation and collection optical axes do not
coincide). Such translations contribute only a linear phase
ramp to the field distribution in the Fourier plane, but all
Fourier components will remain in the same positions.
Misalignments of the setup during reconstruction can be
corrected by using a reference object with known structure
Numerical synthesis of the whole Fourier spectrum of a
sample would make it possible to form an image with
unique properties which endow our technique with several
advantages over existing microscopic techniques. The syn-
thetic image has a spatial resolution set byitssynthetic NA,
which would otherwise require use of an immersion ob-
jective or similar high-NA lens, but preserves the large
field of view (and long working distance) of the low-NA
collection optics. Selective recording of spatial frequencies
could be used to efficiently image certain features of a
sample, for example, microstructures of a given size. Since
we record the full complex amplitude of the object wave,
our technique is well suited to performing phase imaging.
Another interesting application of this approach could be
high-resolution imaging of three-dimensional objects, es-
pecially those with significant height variations over the
region of interest. Synthetic aperture Fourier holographic
microscopy promises wide application in medicine, bi-
ology, and the material sciences.
S.A.A. is supported by the Raine Medical Research
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