Scanning holographic microscopy with resolution
exceeding the Rayleigh limit of the objective by
superposition of off-axis holograms
Guy Indebetouw, Yoshitaka Tada, Joseph Rosen, and Gary Brooker
We present what we believe to be a new application of scanning holographic microscopy to superresolu-
tion. Spatial resolution exceeding the Rayleigh limit of the objective is obtained by digital coherent
addition of the reconstructions of several off-axis Fresnel holograms. Superresolution by holographic
superposition and synthetic aperture has a long history, which is briefly reviewed. The method is
demonstrated experimentally by combining three off-axis holograms of fluorescent beads showing a
transverse resolution gain of nearly a factor of 2. © 2007 Optical Society of America
OCIS codes: 090.0090, 180.0180, 180.2520, 110.0180, 110.6880.
1. Introduction
Extending the resolution beyond the theoretical Ray-
leigh limit of the objective has been, and still is, an
important quest in high-resolution microscopy. In
the recent past, many different approaches toward
achieving this goal have been proposed and demon-
strated. A number of superresolution methods are
based on the rearrangement and smart use of the
degrees of freedom of the image and of the optical
system.
1–6
The general idea is to utilize degrees of
freedom that are deemed unnecessary.
7–11
The de-
grees of freedom that can be sacrificed to improve the
spatial resolution are numerous. For example, they
can be dimensional,
12
temporal,
13,14
or polarization.
15
Different approaches to superresolution are based on
analytic extrapolation, iterative use of a priori knowl-
edge,
16,17
or deconvolution.
18
Closest to the method
described in this paper, structured illumination, spa-
tial encoding, and methods based on aperture syn-
thesis have also been successfully demonstrated for
superresolution. These are based on the early idea
that an object’s spatial frequencies exceeding the
limit of the pupil of the imaging system can be ac-
cessed by illuminating the object with gratings or
interference patterns that shift the otherwise inac-
cessible spatial frequencies back into the pupil’s
transmission disk.
19,20
This idea has been imple-
mented in a large number of ways.
21–27
Many superresolution methods deal with the
improvement of a single planar image. For 3D spec-
imens (as is often the case in biological studies, for
example), it is thus necessary to process each axial
section separately, and sequentially, which can be
time consuming. Holographic methods alleviate this
limitation by offering the possibility of improving the
resolution of the entire 3D information recorded on
the hologram. The idea of improving the resolution by
adding holograms dates from the early developments
of electronic holography.
28,29
Recent advances in de-
tector and computer power have triggered renewed
interest in applying these ideas with modern digital
holography.
30–37
Digital holography has been shown capable of high-
resolution imaging of 3D biological specimens as well
as of quantitative measurement of their phase infor-
mation.
38,39
However, one limitation of digital holog-
raphy is that it requires a minimum degree of spatial
and spectral coherence in order to encode the holo-
gram phase in the form of interference fringes be-
tween the light scattered by the specimen and a
reference wave. This necessity precludes the use of
G. Indebetouw (gindebet@vt.edu) and Y. Tada are with the
Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-
0435, USA. J. Rosen and G. Brooker are with the Department of
Biology, Integrated Imaging Center, Johns Hopkins University,
Montgomery County Campus, 9605 Medical Center Drive, Rock-
ville, Maryland 20850, USA. J. Rosen is also with the Department
of Electrical and Computer Engineering, Ben-Gurion University of
the Negev, P.O. Box 653, Beer-Sheva 84105, Israel, from which he
is presently on leave.
Received 30 August 2006; revised 13 October 2006; accepted 16
October 2006; posted 16 October 2006 (Doc. ID 74610); published 2
February 2007.
0003-6935/07/060993-08$15.00/0
© 2007 Optical Society of America
20 February 2007 兾 Vol. 46, No. 6 兾 APPLIED OPTICS 993
Copyright by the Optical Society of America. Guy Indebetouw, Yoshitaka Tada, Joseph Rosen, and Gary Brooker, "Scanning
holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis
holograms," Appl. Opt. 46, 993-1000 (2007); doi: 10.1364/ao.46.000993
digital holography with fluorescent specimens. Yet
modem biological studies rely heavily on specific flu-
orescence excitation and the emission of specific flu-
orescent probes. In scanning holography a phase
encoding excitation pattern is scanned over the spec-
imen in a 2D raster, and the hologram phase is en-
coded in the temporal domain.
40
The method has
been shown capable of reconstructing high-resolution
holographic images of 3D fluorescent specimens with-
out requiring the spatial coherence of the specimen’s
emission,
41
as well as of measuring the phase of a
specimen quantitatively.
42
The method has also been
shown to offer the possibility of synthesizing different
point spread functions (PSF)
43
and of locating small
fluorescent features in 3D with submicrometer accu-
racy.
44
In this paper we demonstrate experimentally that
synthesizing a pupil exceeding the objective’s pupil
disk is easily implemented in scanning holographic
microscopy and leads to images of 3D fluorescent
specimens reconstructed with a spatial resolution ex-
ceeding the Rayleigh limit of the objective. The re-
sults shown here are illustrative in demonstrating
the principle, but do not attempt to test the limits of
capability of the method. In Section 2, the basic ideas
of scanning holographic microscopy are briefly re-
viewed and how superresolution is achieved by im-
plementing the idea of pupil synthesis is described.
Section 3 specifies the experimental setup, and an
experimental demonstration is presented and dis-
cussed in Section 4. Final remarks and a summary
are given in Section 5.
2. Theoretical Background
In conventional scanning holography, two pupils are
combined by a beam splitter and superposed in the
entrance pupil of the objective. One pupil is a spherical
wave filling the objective’s pupil disk, and having a
curvature chosen to produce, in the back focal plane of
the objective (which is also the object’s space), a spher-
ical wave with a radius of curvature z
0
. The other is a
point at the center of the entrance pupil of the objec-
tive, producing a plane wave in the object’s space.
The interference of these two waves results in a 3D
Fresnel zone pattern (FZP) that is scanned over the
specimen in a 2D raster. The scattered or fluorescent
light is collected by a nonimaging detector (photo-
diode, or photomultiplier). A single-sideband holo-
gram can be obtained from this data in two ways:
either by heterodyne detection using a frequency dif-
ference between the two pupils,
45
or by homodyne
detection
46
using three scans with three fixed phase
differences between the two pupils.
47,48
Other meth-
ods of data acquisition have also been proposed.
49
The hologram can be reconstructed numerically by
Fresnel back propagation or by correlation with an
experimental PSF. The spatial resolution of the re-
constructed image is determined by the numerical
aperture of the FZP scanned over the object. With
an on-line hologram, the spatial frequency spec-
trum of the reconstruction is confined to the disk of
the objective’s pupil, which has a cutoff frequency
MAX
⫽ NA
OBJ
兾, where NA
OBJ
is the numerical aper-
ture of the objective, and is the wavelength of the
radiation.
To extend the spectrum beyond the objective’s cut-
off in a particular direction specified by a unit vector
nˆ , one can record an off-axis hologram by scanning an
off-axis FZP on the specimen. If the spatial frequency
offset of the FZP is
0
nˆ , the spatial frequency spec-
trum of the reconstruction is extended up to a value
MAX
⫹
0
in the direction of nˆ . By combining several
holograms with shifts in different angular directions,
it is thus possible to tile an object’s spatial frequency
spectrum that extends, in principle, far beyond the
objective’s pupil disk.
The two pupils needed to create an off-axis FZP are
P
˜
1
共
ជ
兲
⫽ exp
共
iz
0
2
兲
Disk
共
兾
MAX
兲
,
P
˜
2j
共
ជ
兲
⫽␦
共
ជ ⫺
0
nˆ
j
兲
. (1)
Disk共x兲 ⫽ 1 for x ⬍ 1, and ⫽0 for x ⬎ 1. In these
expressions, ជ is the transverse spatial frequency vec-
tor in the pupil plane, proportional to the transverse
spatial coordinate vector rជ
P
in the pupil: ជ ⫽ rជ
P
兾
f
OBJ
, where f
OBJ
is the focal length of the objective.
50
In the object’s space, P
1
共rជ 兲 is a converging spherical
wave with radius of curvature z
0
, and P
2j
共rជ 兲 is a plane
wave with a transverse spatial frequency
0
nˆ
j
. After
heterodyne detection, or phase-shifted homodyne de-
tection, each object point is encoded as an off-axis
spherical wave S
j
共rជ , z兲, where z is the axial coordinate
in object space measured from the focal plane of the
objective. In Fourier space, we have
S
˜
j
共
ជ , z
兲
⫽ P
˜
1
共
ជ , z
兲
丣 P
˜
2j
共
ជ , z
兲
⫽ exp
兵
i
关
z
0
0
2
⫹
共
z
0
⫺ z
兲
共
2
⫺ 2ជ ·
0
nˆ
j
兲
兴
其
⫻ Disk
共
ⱍ
ជ ⫺
0
nˆ
j
ⱍ
兾
MAX
兲
, (2)
where Q symbolizes a correlation integral, and
P
˜
1,2
共ជ , z兲 are the generalized (defocused) pupils
51
de-
fined as
P
˜
1,2
共
ជ , z
兲
⫽ P
˜
1,2
共
ជ
兲
exp
共
i2z
冑
⫺2
⫺
2
兲
, (3a)
or in paraxial approximation,
P
˜
1
共
ជ , z
兲
⫽ exp
共
i2z兾
兲
exp
关
i
共
z
0
⫺ z
兲
2
兴
⫻ Disk
共
兾
MAX
兲
,
P
˜
2j
共
ជ , z
兲
⫽ exp
共
i2z兾
兲
exp
共
⫺iz
0
2
兲
␦
共
ជ ⫺
0
nˆ
j
兲
.
(3b)
The Fourier transform of the specimen hologram is
given by
H
˜
Oj
⫽
冕
dzI
˜
共
ជ , z
兲
S
˜
j
共
ជ , z
兲
, (4)
994 APPLIED OPTICS 兾 Vol. 46, No. 6 兾 20 February 2007
where I
˜
共ជ , z兲 is the 2D transverse Fourier transform
of the 3D object intensity distribution I共rជ , z兲. The re-
construction of the hologram in a chosen axial plane
of focus z ⫽ z
R
can be obtained by digital Fresnel
back propagation from the hologram for a distance
z
0
⫹ z
R
. In the experiment discussed in Section 4, we
chose to reconstruct the hologram by correlation with
the experimental hologram of a subresolution point
source. As has been shown previously,
41
this method
offers a way of reducing the phase aberrations of the
objective.
41
With high NA objectives, these aberra-
tions are attributable to the fact that the objective is
not (and cannot be) used in the geometry for which it
was designed, and can be severe. In the experiment
described in Section 3, this scheme is implemented by
recording a reference hologram H
Rj
共rជ 兲 of a point
source ␦共rជ , z兲, and propagating it to the desired re-
construction plane by using a propagation factor
P
j
共rជ , z
R
兲. In Fourier space, we have
H
˜
Rj
共
ជ
兲
⫽ exp
关
iz
0
共
ⱍ
ជ ⫺
0
nˆ
j
ⱍ
2
兲
兴
⫻ Disk
共
ⱍ
ជ ⫺
0
nˆ
j
ⱍ
兾
MAX
兲
, (5)
P
˜
j
共
ជ , z
R
兲
⫽ exp
关
⫺iz
R
共
2
⫺ 2ជ ·
0
nˆ
j
兲
兴
. (6)
The Fourier transform of the reconstructed image is
then given by
R
˜
j
共
ជ , z
R
兲
⫽ H
˜
Oj
共
ជ
兲
关
H
˜
Rj
共
ជ
兲
P
˜
j
共
ជ , z
R
兲
兴
*
⫽
冕
dzI
˜
共
ជ , z
兲
exp
关
i
共
z
R
⫺ z
兲
共
2
⫺ 2ជ ·
0
nˆ
j
兲
兴
⫻ Disk
共
ⱍ
ជ ⫺
0
nˆ
j
ⱍ
兾
MAX
兲
, (7)
where the asterisk represents a complex conjugation.
In the plane of focus z ⫽ z
R
, the reconstruction has a
spatial frequency spectrum consisting of the object’s
spatial frequencies located within a disk of radius
MAX
, centered at
0
nˆ
j
. By adding coherently the re-
construction amplitudes of a number of holograms
with different spatial frequency shifts, one can, in
principle, tile a synthetic pupil with an area far ex-
ceeding the pupil disk of the objective.
It must be emphasized that although the pupil rep-
resentation in Eq. (1) is, strictly speaking, valid only
in the paraxial regime,
52
the method of scanning ho-
lographic microscopy is not limited to small NAs. In
fact, since we actually measure the PSF, and recon-
struct the holograms by this experimental PSF, our
method is valid regardless of the system’s NA, and it
is independent of the theoretical representation of
the PSF.
3. Experimental Arrangement
The experimental setup of a scanning holographic
microscope has been described in detail in previous
publications.
40,41
For completeness, a sketch is given
in Fig. 1. The only addition to the previous arrange-
ments is the introduction of a wedge prism in a con-
jugate object plane to shift the spatial frequency of
the plane wave creating the FZP by a chosen amount
in a chosen direction. The wedge is then rotated in
discrete angular steps to cover a desired area of the
object’s spatial frequency spectrum. For experimen-
tal simplicity we chose to introduce both the spherical
wave and the off-axis plane wave forming the FZP
through the pupil of the objective. With this method,
the largest spatial frequency shift achievable is the
objective’s frequency cutoff, namely,
0
ⱕ
MAX
. Opt-
ing for this geometry is not a necessary requirement.
Fig. 1. Schematic of an off-axis scanning holographic microscope. M, mirrors; BS, beam splitters; DBS, dichroic beam splitter; EOM,
electro-optic phase modulator. L
1,2
are achromat doublet lenses, 16 cm focal length, L
3
is a collecting lens, 1 cm focal length. The wedge
on a rotating stage is used to create off-axis Fresnel patterns on the specimen.
20 February 2007 兾 Vol. 46, No. 6 兾 APPLIED OPTICS 995
In principle, it is possible to introduce the off-axis
plane waves externally at any angle, although this
practice may be difficult to implement at high NA.
Nevertheless, the transfer function of such system
could in principle be extended to its theoretical fre-
quency limit of 2兾 and cover the entire spectral re-
gion inside this limit. Additionally, it may also be
possible to introduce several plane waves simulta-
neously by using an array of sources.
35–37
The experiment uses a Mitutoyo Plan Apo objective
20x, 0.42 NA, chosen for its long working distance.
The detector is a commercial Hamamatsu photomul-
tiplier tube (R1166). The specimen is scanned in a 2D
raster using a piezo scanning stage (PI Hera 625).
The FZP modulation is provided by an electro-optic
phase modulator (Linos LM0202) driven by a saw
tooth waveform at 25 kHz. The signal is sampled at
100 kSample兾s by a GageScope 1602. The hologram
is constructed line by line after bandpass filtering
at the modulation frequency, as previously de-
scribed, and reconstructed digitally using MATLAB
codes (MATLAB 7.1.0.144兾R14 SP3) on a standard
personal computer (PC).
It was found that at high NA, the fact that the
off-axis FZP has to cross the cover glass of the spec-
imen before illuminating it introduced a fair amount
of aberrations and distortions. Furthermore it also
changed the spatial frequency offset by an amount
difficult to predict. In principle, if the optical proper-
ties of the specimen and its mounting environment
are known, this effect can be compensated for numer-
ically. We chose to eliminate these difficulties by us-
ing a specimen without cover glass. The specimen
used in the demonstration in Section 4 is a cluster of
1 m diameter fluorescent beads (R0100 Polymer
Microspheres, Duke Scientific, Palo Alto, Calif., USA)
smeared on a microscope slide without a cover slip.
4. Experimental Results
To assess the gain in the resolution of the system,
we used the reconstruction of a subresolution pin-
hole (0.5 m diameter) to estimate the size of the PSF.
With a NA of 0.42, the theoretical Rayleigh resolution
limit of the objective is 1.22
EM
兾2NA ⬇ 0.9 m.
EM
⫽ 600 nm is the emission wavelength of the beads.
The wrapped phase of the on-line hologram of the
Fig. 2. (Color online) (a) Wrapped phase of the on-line hologram
of a 0.5 m diameter pinhole. The scale bar is 10 m. The phase
distribution has a radius ⬃18 m, a Fresnel number ⬃12, and a
radius of curvature ⬃50 m. (b) Amplitude of the reconstruction of
the 0.5 m pinhole using the on-line hologram. FWHM ⬃1.0 m.
Fig. 3. (Color online) (a) Wrapped phase of three off-axis holo-
grams of the 0.5 m pinhole illustrating the idea of pupil synthe-
sis. The scale bar is 10 m. (b) Amplitude of the reconstruction of
the 0.5 m pinhole using the composite off-axis holograms. FWHM
⬃0.7 m.
996 APPLIED OPTICS 兾 Vol. 46, No. 6 兾 20 February 2007
0.5 m diameter pinhole is shown in Fig. 2(a), and
its reconstruction is shown in Fig. 2(b). The holo-
gram has a Fresnel number F ⬇ 12, and a radius
a ⬇ 18 m. Using the relation F ⫽ a
2
兾
EX
z
0
with
an excitation wavelength
EX
⫽ 532 nm, the radius
of curvature of the hologram is found to be z
0
⬇
50 m, and its equivalent NA is a兾z
0
⬇ 0.35. This
leads to an expected theoretical Rayleigh limit
⬇0.9 m, which is close to the Rayleigh resolution of
the objective. The observation that the smaller equiv-
alent NA of the hologram leads to the same resolution
limit as that of the objective is attributable to the fact
that the objective forms an image at the emission
wavelength while the hologram is formed at the
shorter excitation wavelength. A dye with a larger
Stokes shift would lead to an even larger gain. The
reconstruction shown in Fig. 2(b) has a FWHM equal
to ⬇1.0 m. Since this is the convolution of the ac-
tual PSF with the 0.5 m pinhole, the size of the
experimental PSF is estimated to be ⬇0.9 m, o r
smaller. Three off-axis holograms of the pinhole
were recorded with a spatial frequency offset
0
⬇
MAX
⬇ 0.66 m
⫺1
in directions 120° apart. The
wrapped phases of the three holograms are combined
in Fig. 3(a) to illustrate the wider spatial frequency
coverage of the composite hologram. The reconstruc-
tion of the pinhole from this hologram is shown in
Fig. 3(b). Its FWHM is ⬇0.7 m. The actual size of
the PSF of the composite hologram is thus estimated
Fig. 4. (Color online) (a) Reconstruction of the on-axis hologram of
a collection of ⬃1.0 m fluorescent beads (excitation兾emission
wavelengths ⫽ 532 nm兾600 nm) at the best focus distance of
47.5 m from the focal plane of the objective. The scale bar is 5 m.
Bead clusters are just barely resolved. (b) Same reconstruction at
a focus distance of 49 m. The two planes are within the Rayleigh
range of the on-axis scanning FZP.
Fig. 5. (Color online) Coherent sum of the complex amplitudes of
the reconstructions of three off-axis holograms recorded with off
sets 120° apart. The scale bar is 5 m. (a) Best focus at 47.5 m
from the focal plane of the objective. (b) Same reconstruction at a
focus distance of 49 m. The distance between the two planes is
close to the Rayleigh range of the synthesized FZP, and different
bead clusters are focused in different planes.
20 February 2007 兾 Vol. 46, No. 6 兾 APPLIED OPTICS 997
