Spatially incoherent single channel digital
Roy Kelner* and Joseph Rosen
Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
*Corresponding author: email@example.com
Received June 14, 2012; accepted July 16, 2012;
posted July 20, 2012 (Doc. ID 170643); published August 31, 2012
We present a new method for recording digital Fourier holograms under incoherent illumination. A single ex-
posure recorded by a digital camera is sufficient to record a real-valued hologram that encodes the complete
three-dimensional properties of an object.© 2012 Optical Society of America
OCIS codes:090.0090, 090.1995, 110.6880, 100.3010, 070.6120.
In recent years we have witnessed noteworthy achieve-
ments of incoherent holography techniques such as op-
tical scanning holography  and Fresnel incoherent
correlation holography (FINCH) [2,3]. The latter has fun-
damental system robustness since it is based on a single
channel incoherent interferometer and because it does
not require any scanning or mechanical movement. How-
ever, though a single FINCH image contains the complete
three-dimensional (3D) information of an object, at least
three images are required to solve the twin image pro-
blem . In this Letter, we present a new method for
recording digital Fourier holograms under incoherent il-
lumination. Fourier holograms [5,6] possess some advan-
tages over Fresnel holograms including, but not limited
to, increased space–bandwidth product performance
[7,8] and the relatively easy ability to process and manip-
ulate the hologram, since it is captured in the spatial fre-
quency domain. In addition, the hologram is more robust
to information loss, as each object point is distributed
over the entire hologram plane. Moreover, by recording
a Fourier hologram of a half plane (or space), the twin
image problem is avoided and the object can be recon-
structed from a single exposure. Still, the proposed meth-
od maintains many other advantageous characteristics of
FINCH . We coin our method Fourier incoherent single
channel holography (FISCH).
The FISCH system is shown in Fig. 1. A white-light
source illuminates a 3D object. Light scattered from
the object passes through a band pass filter (BPF), be-
comes partially temporally coherent, and continues to
propagate through a single channel incoherent interfe-
rometer. Eventually, interference patterns are captured
by the CCD. Consider a point source object of complex
amplitude Aspositioned at the coordinate ?xs;ys;zs?, a
distance zsfrom the lens L0. A tilted diverging spherical
wave of the formT?x;y; ⃗rs;zs? ? Asc?⃗rs;zs?L?−⃗rs∕
zs?Q?1∕zs? is induced over the L0
⃗rs? ?xs;ys?, c?⃗rs;zs? is a complex valued constant de-
pendent on the position of the point source, and L?⃗s? ?
exp?i2πλ−1?sxx ? syy?? and Q?s? ? exp?iπsλ−1?x2? y2??
are the linear and the quadratic phase functions, respec-
tively, in which λ is defined as the central wavelength. A
diffractive optical element of the form Q?−1∕f1? is dis-
played on the spatial light modulator (SLM), positioned
at a small angle relative to the optical axis. The SLM is
polarization sensitive, so by introducing the linear polar-
izer P1 the single channel optical apparatus is effectively
split into two beams (see  for a detailed explanation).
With one beam the SLM functions as a converging diffrac-
tive lens with a focal length f1, while in the other it acts
as plane mirror. The second polarizer P2 rejoins the
The unmodulated wave at the SLM plane, right after
passing through it for the second time, is
C1?x;y; ⃗rs;zs??T?x;y; ⃗rs;zs?·Q
while the modulated wave, at the same position, is
C2?x;y; ⃗rs;zs? ? T?x;y; ⃗rs;zs? · Q
where ?Q?1∕zd? denotes a Fresnel propagation of a wave
to a distance of zd[mathematically, ? denotes a two-
dimensional (2D) convolution] and Q?−1∕f? denotes
the influence of a converging lens of focal length f upon
a wave propagating through it. Upon arrival to the CCD
plane the waves are
solution chart; BPF, bandpass filter; BS, beam splitter; SLM,
spatial light modulator; M, mirror; CCD, charge-coupled device;
P1 and P2, polarizers.
(Color online) Schematic of a FISCH recorder: RC, re-
September 1, 2012 / Vol. 37, No. 17 / OPTICS LETTERS3723
0146-9592/12/173723-03$15.00/0 © 2012 Optical Society of America
Dk?x;y; ⃗rs;zs? ? Ck?x;y; ⃗rs;zs? ? Q
where k ? 1, 2 represent the unmodulated and modu-
justification presented in , the recorded intensity
over the CCD plane, I?x;y; ⃗rs;zs? ? jD1?x;y; ⃗rs;zs??
D2?x;y; ⃗rs;zs?j2, is equal to
I?x;y; ⃗rs;zs? ? Is?jb1j2? jb2j2?
where Is? As· A?
fe? d0− f2? d2and c.c. is the complex conjugate of
the left term inside the square brackets. Since each point
source is only spatially coherent to itself, the recorded
hologram due to many point sources is simply a summa-
tion over all point source contributions. That is, the
recorded hologram is
Consider the special case of a point source object po-
sitioned at the front focal plane of the lens L0(i.e.,
zs? f0). In that case, Eq. (4) is easily reduced to
I?x;y; ⃗rs;f0? ? Is?jb1j2? jb2?j2?
Note that according to the third term of Eq. (6), each
source point at ⃗rsis mapped at the ?x;y? plane to a linear
phase with ⃗rsdependent inclination and with amplitude
dependent on Is. Such mapping exactly defines a 2D
Fourier transform (FT) of the object points. Hence, we
have a digital Fourier hologram. Points positioned at
the front focal plane of the lens L0can be reconstructed
by a simple calculation of the inverse FT of H?x;y?.
Points located on other planes may be out of focus,
and can be reconstructed by applying additional Fresnel
propagation by the reconstruction distance zr. The re-
construction procedure (performed either digitally or op-
tically), depicted in Fig. 2 where Lr is a Fourier
transforming lens of focal length fr, gives
where F−1is the inverse FT and v is the scaling operator,
which states that v?a?f?x? ? f?ax?. Based on Eqs. (4) and
(7), the reconstruction distance is
sis the intensity of the point source, b1
fe? f0zs∕?f0− zs?,f3?
According to Eq. (6) and Fig. 2, the reconstruction plane
of the point source ?⃗rs;zs? f0? would contain the point
source image and its twin, located at ??2⃗rsfr∕f0;zr? 0?,
and an additional zeroth order term, located at its origin.
Thus, by recording a Fourier hologram the twin image
problem is avoided if the object is properly positioned
inside a half plane (e.g., ys> 0).
The system shown in Fig. 1 was implemented using a
Holoeye PLUTO SLM (1920 × 1080 pixels, 8 μm pixel
pitch, phase only modulation) and a PixelFly CCD
(1280 × 1024 pixels, 6.7 μm pixel pitch, monochrome).
Other parameters in the system were f0? 30 cm,
f1?17.5cm, f2?20cm, d0?13cm, and d2?12cm.
The resolution test chart (Edmund Optics Negative
NBS 1963A) was set at two different zs positions
(30 cm and 25 cm) and was illuminated using a 150 W
EKE Halogen lamp light filtered through a 650 ?
20 nm BPF. The intersection angle of the optical axes
at the SLM plane was 17°.
We recorded holograms of a single exposure, two ex-
posures, and three exposures. The two latter versions
use the phase-shifting method [2,4] to eliminate most
of the bias term and, in the three exposures version, also
the twin image. Note that the single exposure and two
exposures holograms are real-valued, whereas the three
exposures hologram is complex-valued. The resulting ho-
lograms were digitally reconstructed based on Eq. (7).
The real-valued two exposures holograms were also op-
tically reconstructed using an apparatus based on Fig. 2,
in which the holograms were displayed on a Holoeye
LC2002 SLM (800 × 600 pixels, 32 μm pixel pitch, set
to amplitude modulation through a proper configuration
of an input and output polarizers). No special processing
was applied to the holograms before displayed on the
SLM, besides resizing with accordance to the SLM reso-
lution and linear mapping of the gray levels to discrete
values from 0 to 255 (8 bits). This mapping naturally
requires the introduction of a bias term, since the two
exposure holograms contain both negative and positive
The experimental results are shown in Fig. 3. The first
hologram was recorded with the resolution test chart
positionedatzs? f0? 30 cm.
[Figs. 3(a)–3(d)] are simply the inverse FT of the holo-
gram (discrete transform, in the digital case). Figure 3(a)
demonstrates that a single exposure is sufficient for re-
taining most of the target information. Yet, the zeroth or-
der, clearly visible in the center of the image, dominates
the hologram and reduces the quality of the image. Using
two exposures [Fig. 3(b)] greatly reduces the bias term,
increases the dynamic range of the hologram, and there-
by enhances the quality of the image. A third exposure
[Fig. 3(c)] removes the twin image, but does not enhance
the quality of the image any further, at least not visibly.
(Color online) Schematic of the reconstruction
3724OPTICS LETTERS / Vol. 37, No. 17 / September 1, 2012
Finally, Fig. 3(d) is the optical counterpart of 3(b), and
demonstrates the possibility of optical reconstruction
of the resulting real-valued hologram using a simple
apparatus, as previously described.
To demonstrate the system capability of maintaining
3D information, another hologram was recorded; this
time with zs? 25 cm < f0. Part of the recorded holo-
gram (using two exposures) is shown in Fig. 3(e). A
digital reconstruction at the best plane of focus is shown
in Fig. 3(f), where zrwas calculated according to Eq. (8).
The optical reconstructions in Figs. 3(g) and 3(h) clearly
demonstrate the refocusing capability of the system,
where the image and its twin are in and out of focus, in-
terchangeably. This was simply achieved via back and
forth movement of the CCD within the z-axis.
A simplified schematic of FISCH is presented in Fig. 4.
We note that in the actual system (Fig. 1) the beams pro-
pagate twice through the same SLM, here represented by
SLM1 and SLM2. In addition, the CCD is now positioned
right after SLM2. This configuration may be considered
to be equivalent to Fig. 1 with d2? f2, so that the waves
reaching the CCD are simply a 180° rotated version of the
waves right after exiting the SLM the second time. Here
we consider the case of zs? f0. We further assume that
the recorded hologram is limited by a clear disc aperture
P?RH? of radius RH, which is determined by the overlap
area of the two interfering beams on the CCD plane, and
that the system is only aperture limited by the radius of
SLM1, R0. In this case, RHcan geometrically be shown to
equal R0, regardless of the source distance from the op-
tical axis, RS. The recorded hologram is then equal to
Eq. (6) multiplied by P?R0?. It can be reconstructed as
shown in Fig. 2 so that
h?⃗r? ? v
where ⃗r ? ?x;y?. Equation (9) results with three terms
attributed to the bias, the point source image and its twin,
which are assumed to be properly separated, once RSis
big enough. The point source image is then proportional
hF?⃗r? ∝ Jinc
where Jinc?r? ? J1?r?∕r, J1?r? is the Bessel function of
the first kind andof order one, andMT? ?2⃗rsfr∕f0?∕ ⃗rs?
2fr∕f0is the transverse magnification of the presented
FISCH configuration. A comparison of Eq. (10) to
Eq. (11) of Ref  demonstrates that, under the above
conditions, both FISCH and FINCH possess the same
point spread function (PSF). Therefore, just like FINCH
, FISCH has improved resolution beyond the Rayleigh
limit when compared to conventional imaging.
In this Letter we have presented FISCH, a new method
for recording spatially incoherent digital Fourier holo-
grams. The viability of FISCH and its ability to maintain
the complete 3D information of an object were demon-
strated. We believe that the combination of FISCH inher-
ent robustness with its capability of recording single
exposure holograms under incoherent illumination holds
great potential for many possible applications of FISCH.
We thank Barak Katz for sharing his valuable knowl-
edge and experience. This work was supported by the
Israeli Ministry of Science and Technology (MOST).
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zs? 30 cm from a single exposure, two and three exposures
holograms, respectively; (d) is the optically reconstructed
equivalent of (b). (e) is a two exposures hologram (shown par-
tially to reveal details, recorded with zs? 25 cm) and (f) is its
digitally reconstructed image at the best plane of focus. (g) and
(h) are optical reconstructions of (e) at the best plane of focus
of one of the images and its twin, respectively.
(a), (b), and (c) are digitally reconstructed images for
(Color online) Simplified schematic of a FISCH
September 1, 2012 / Vol. 37, No. 17 / OPTICS LETTERS3725