Autofocusing in optical scanning holography
Taegeun Kim
1,
* and Ting-Chung Poon
2
1
Department of Optical Engineering, Sejong University, 98 Kunja-dong,
Kwangjin-gu, Seoul 134-747, South Korea
2
Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA
*Corresponding author: takim@sejong.ac.kr
Received 13 July 2009; revised 12 October 2009; accepted 12 October 2009;
posted 12 October 2009 (Doc. ID 114193); published 3 November 2009
We present autofocusing in optical scanning holography (OSH) with experimental results. We first record
the complex hologram of an object using OSH and then create the Fresnel zone plate (FZP) that codes the
object constant within the depth range of the object using Gaussian low-pass filtering. We subsequently
synthesize a real-only spectrum hologram in which its phase term contains information about a distance
parameter. Finally, we extract the distance parameter from the real-only spectrum hologram using
fringe-adjusted filtering and the Wigner distribution. Using the extracted distance parameter, we recon-
struct a three-dimensional image of the object from the complex hologram using digital convolution,
which bypasses the conventional blind convolution to reconstruct a hologram. To the best of our knowl-
edge, this is the first report with experimental results that autofocusing in OSH is possible without any
searching algorithm or tracking process. © 2009 Optical Society of America
OCIS codes: 090.1995, 070.0070.
1. Introduction
Three-dimensional (3-D) imaging using optical scan-
ning holography (OSH) has a long-standing history
[1–4]. Most recently, OSH has been applied to 3-D mi-
croscopy [3–6], and an achieved resolution of better
than 1 μm has been reported [5]. OSH is a form of
digital holography [7] but it records a comple x holo-
gram by two-dimensional heterodyne scanning.
Identical to conventional reconstruction of digital ho-
lograms, sectioning reconstruction of a 3-D image is
done by convolving a complex conjugate of the Fres-
nel zone plate (FZP) matched to the depth of a section
of a 3-D object [7]. Since we do not have prior knowl-
edge of the depth location of the object, we need to
perform digital reconstruction blindly to various dis-
tances until we find the sectional images, which is a
time-consuming process. Several numerical techni-
ques have been proposed to extract the distance
parameter automatically [8], but these involve a
search algorithm [9,10] or tracking process [8]. Here
we show experimentally that reconstruction of the fo-
cused image sections from OSH is possible without
any searching algorithm or tracking process. First,
we record the complex hologr am of an object by
use of OSH. Second, we perform Gaussian low-pass
filtering with a complex hologram, which makes the
FZP that codes the object constant within the depth
range of the object [11]. Third, we synthesize the
real-only spectrum hologram from the Gaussian
low-pass filtered hologram. We then show that, since
the FZP that codes the object is constant within the
depth range of the object and the intensity of the ob-
ject is positive and real, the phase term of the real-
only spectrum hologram contains information about
the distance parameter. Fourth, we separate the
phase term of the real-only spectrum hologram using
fringe-adjusted filtering [12]. Fifth, we extract the
distance parameter by performing the Wigner distri-
bution of the phase term of the real-onl y spectrum
hologram [13]. Finally, we reconstruct the entire 3-D
image of the object by reconstruction using a distance
parameter, which avoids the blind convolution nor-
mally used for digital reconstruction.
0003-6935/09/34H153-07$15.00/0
© 2009 Optical Society of America
1 December 2009 / Vol. 48, No. 34 / APPLIED OPTICS H153
Copyright by the Optical Society of America. Taegeun Kim and Ting-Chung Poon, "Autofocusing in optical scanning
holography," Appl. Opt. 48, H153-H159 (2009); doi: 10.1364/AO.48.00H153
2. Complex Hologram Recording and Distance
Parameter Extraction
A. Recording with Optical Scanning Holography
In the recording stage, we record a complex hologram
of an object using OSH. The OSH setup shown in
Fig. 1 consists of a Mach–Zehnder interferometer
and an electronic processing unit. Two mirrors (M2
and M3) and two beam splitters (BS1 and BS2) form
the Mach–Zehnder interferometer. The laser bea m
generated by the He–Ne laser is split into two paths.
The frequencies of the laser in the upper path and
the lower path beams are shifted by Ω and Ω þ ΔΩ,
respectively, through acousto-optic frequency shif-
ters (AOFS1 and AOFS2), and each beam is then col-
limated by a 10× beam expander (BE1 and BE2).
Afterward, the upper path beam becomes a spherical
wave having limited extension through lens L1. The
spherical wave and the lower path collim ated beam
are combined through beam splitter BS2 to form an
interference pattern called the time-dependent (TD)
FZP with limited extension. The TD FZP at z away
from the focal point of lens L1 is given by
I
s
ðx; y; z; tÞ¼
A
s
ðx; y; zÞ
λz
sin
π
λz
ðx
2
þ y
2
Þ − ΔΩt
; ð1Þ
where A
s
ðx; y; zÞ is the size-limiting factor deter-
mined by the numerical aperture of the optics that
was used to create the TD FZP. In practice, we as-
sume a Gaussian envelope with radius aðzÞ, i.e.,
A
s
ðx; y; zÞ¼exp½
−π
aðzÞ
2
ðx
2
þ y
2
Þ:
Here the radius of the TD FZP is determined by
aðzÞ¼NA × z, where NA represents the numerical
aperture defined as the sine of the half-cone angle
subtended by the TD FZP, which scans an object
with intensity transmittance or intensity reflectance,
I
0
ðx; y; zÞ , if the object is diffusely reflecting. Lens L2
collects all the transmitted light through the object
onto the photodetector. Since the lens collects all
the transmitted light, the OSH is in the incoher ent
mode [4], which means that we record the intensity
of the 3-D object holographically. At each scan loca-
tion, the electric current generated by the photode-
tector is provided to the electric processing unit
shown in Fig. 1. In the electronic processing unit,
in-phase and quadrature-phase output currents,
i
I
ðx; y; zÞ and i
Q
ðx; y; zÞ , are generated using a phase-
sensitive detection scheme such as a lock-in ampli-
fier. Afterward, the in-phase and quadrature-phase
output currents are stored in a digital computer ac-
cording to the scanning locations. The complex holo-
gram is constructed subsequently by adding the two
stored outputs in the following manner [14]:
Hðx; yÞ¼i
I
ðx; y; zÞ − ji
Q
ðx; y; zÞ¼
Z
z
0
þð1=2Þδz
z
0
−ð1=2Þδz
I
0
ðx; y; zÞ
⊗
jA
s
ðx; y; zÞ
λz
exp
−j
π
λz
ðx
2
þ y
2
Þ
dz; ð2Þ
where z
0
is the distance from the focused point on
BS2 that is due to lens L1 to the middle of the object,
which we refer to as the distance parameter here-
after, and δz is the depth range of the object. The
⊗ in Eq. (2) denotes a two-dimensional convolution
operation defined as
g
1
ðx; yÞ ⊗ g
2
ðx; yÞ
¼
ZZ
g
1
ðx
0
; y
0
Þg
2
ðx − x
0
; y − y
0
Þdx
0
dy
0
:
Note that, in Eq. (2), the object is coded by a Gaus-
sian FZP with a numerical aperture of NA.
The spectrum of the complex hologram is obtained
by Fourier transformation of Eq. (2). The Fourier
transform operation, Ff:g, is defined as
Ffuðx; yÞg
k
x
;k
y
¼
ZZ
uðx; yÞ exp½jðk
x
x þ k
y
yÞdxdy
¼ uðk
x
; k
y
Þ;
with ðk
x
; k
y
Þ denoting spatial frequencies. Note that
the bold letter u represents the Fourier transform of
u. Since the spatial frequency of the TD FZP is lim-
ited by the NA of the TD FZP [15], the radius of the
spectrum of the TD FZP is given by 2πNA=λ. Thus,
the Fourier transform of Eq. (2) is given by
Fig. 1. OSH: M, mirrors; AOFS1, AOFS12, acousto-optic fre-
quency shifters; BS1, BS2, beam splitters; BE1, BE2, beam expan-
ders; L1, focusing lens; L2, collecting lens; ⊗, electronic multiplier;
LPF, low-pass filter.
H154 APPLIED OPTICS / Vol. 48, No. 34 / 1 December 2009
Hðk
x
;k
y
Þ¼FfHðx;yÞg ¼
Z
z
0
þð1=2Þδz
z
0
−ð1=2Þδz
I
0
ðk
x
;k
y
;zÞ
× exp
−
1
4π
λ
NA
2
þj
λz
4π
ðk
2
x
þk
2
y
Þ
dz:
ð3Þ
The idea of autofocusing is to extract the value of z
0
,
the distance parameter, from Eq. (3) and then to re-
construct the hologram through convolution starting
from z ¼ z
0
, thereby bypassing the blind convolution
starting at z ≈ 0 from the hologram.
B. Distance Parameter Extraction
We first filter the complex hologram using a Gaus-
sian low-pass filter in such a way that the pattern
of the FZP that codes the object remains unchanged
within the depth range of the object. The Gaussian
low-pass filter is given by
A
g
ðk
x
; k
y
Þ¼exp
−π
λ
2πNA
g
2
ðk
2
x
þ k
2
y
Þ
; ð4Þ
where NA
g
determines the cutoff frequency of the fil-
ter. In the frequency domain, the Gaussian low-pass
filter hologram, H
lp
ðk
x
; k
y
Þ, is given by multiplying
the Gaussian low-pass filter with Eq. (3):
H
lp
ðk
x
;k
y
Þ¼Hðk
x
;k
y
Þ × A
g
ðk
x
;k
y
Þ
¼
Z
z
0
þð1=2Þδz
z
0
−ð1=2Þδz
I
0
ðk
x
;k
y
;zÞ
× exp
−
1
4π
λ
NA
lp
2
þ j
λz
4π
ðk
2
x
þ k
2
y
Þ
dz;
ð5Þ
where NA
lp
¼ NA
g
NA=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NA
2
þ NA
2
g
q
. The Gaussian
low-pass filtered hologram in the space domain is gi-
ven by the inverse Fourier transformation of Eq. (5):
H
lp
ðx; yÞ¼F
−1
fH
lp
ðk
x
; k
y
Þg ¼
Z
z
0
þð1=2Þδz
z
0
−ð1=2Þδz
I
0
ðx; y; zÞ
⊗
jA
lp
ðx; y; zÞ
λz
exp
−j
π
λz
ðx
2
þ y
2
Þ
dz; ð6Þ
where F
−1
fgrepresents inverse Fourier transforma-
tion operation and
A
lp
ðx; y; zÞ¼exp½
−π
a
lp
ðzÞ
2
ðx
2
þ y
2
Þ;
with a
lp
ðzÞ¼NA
lp
z. Note that, in the filtered holo-
gram given by Eq. (6), the object is now coded with
a Gaussian FZP with NA
lp
instead of the original
complex hologram in which the object is coded with
a Gaussian FZP with NA. Hence, the Rayleigh range
of the Gaussian FZP is now determined by the NA of
the FZP, which is given by
Δz ¼ 2λ=πNA
2
lp
¼ 2λ=πðNA
2
þ NA
2
g
Þ=ðNA
g
NAÞ
2
:
When we set NA
g
such that the Rayleigh range of the
FZP is larger than the depth range of the object, i.e.,
Δz ≥ δz, the radius of the scanning beam pattern is
approximately constant within the depth range of
the object, i.e., a
lp
ðzÞ ≈ a
lp
ðz
0
Þ¼NA
lp
z
0
. As a result,
the FZP that encodes the complex hologram becomes
constant within the depth range of the object, i.e., the
free-space impulse response now becomes
h
z
ðx; yÞ ≈ h
z
0
ðx; yÞ
¼ jA
lp
ðx; y; z
0
Þ=ðλz
0
Þ exp½−j
π
λz
0
ðx
2
þ y
2
Þ
within the depth range of the object. This is impor-
tant because we can now pull the factor h
z
0
ðx; yÞ out-
side the integral of Eq. (6). Under this condition the
Gaussian low-pass filtered hologram becomes
H
lp
ðx; yÞ¼
Z
z
0
þð1=2Þδz
z
0
−ð1=2Þδz
I
0
ðx; y; zÞ d z
⊗
jA
lp
ðx; y; z
0
Þ
λz
0
exp
−j
π
λz
0
ðx
2
þ y
2
Þ
: ð7Þ
Figure 2 is a flow chart. The first two blocks illustrate
the procedures discussed so far. After Gaussian low-
pass filtering, we synthesize the real-only spectrum
hologram to extract distance parameter z
0
. First, we
extract the real and imaginary parts of the hologram,
which are called the sine hologram and the cosine
hologram, respectively. Second, we transform these
holograms into the frequency domain, in which we
synthesize the real-only spectrum hologram by add-
ing the real parts of the sine hologram and the cosine
hologram in the following manner:
H
r-only
ðk
x
; k
y
Þ¼Re½FfRe½H
lp
ðx; yÞg
þ jRe½ FfIm½H
lp
ðx; yÞg; ð8Þ
which can be shown to become [13 ]
H
r-only
ðk
x
; k
y
Þ¼Re
F
Z
z
0
þð1=2Þδz
z
0
−ð1=2Þδz
I
0
ðx; y; zÞ dz
× exp
−
1
4π
λ
NA
lp
2
þ j
λz
0
4π
ðk
2
x
þ k
2
y
Þ
; ð9Þ
where Re½: and Im½: are operators that extract
the real and imaginary parts of a complex number,
respectively. Note that, in Eq. (9), the phase term
of the real-only spectrum hologram contains
1 December 2009 / Vol. 48, No. 34 / APPLIED OPTICS H155
information only about distance parameter z
0
. The
phase term is extracted by the following process.
First, the real-only spectrum hologram is projected
in the k
y
direction. Second, the square of the pro-
jected real-only spectrum hologram is filtered by a
power-fringe-adjusted filter [16], whose output is
given by
H
0
ðk
x
Þ¼
1
R
H
r-only
ðk
x
;k
y
Þdk
y
2
þε
Z
H
r-only
ðk
x
;k
y
Þdk
y
2
≈ exp
j
λz
0
2π
k
2
x
; ð10Þ
where ε is a small value of either a constant or some
function of ðk
x
; k
y
Þ; ε is added to overcome the pos-
sible pole problems of the filter. Note that the output
of the power-fringe-adjusted filter is the chirp signal
along the k
x
. The Wigner distribution of the chirp
signal gives the line impulse with a slope of dk
x
=dx ¼
π=λz
0
on the frequency-space map (x − k
x
plane).
That is,
W
H
o
ðk
x
; xÞ¼
Z
H
o
ðk
x
þ
k
x
0
2
ÞH
o
ðk
x
−
k
x
0
2
Þ
× exp ð −jxk
x
0
Þdk
x
0
∝ δ ð x −
λz
o
π
k
x
Þ:
Since the wavelength is known, we find distance
parameter z
0
by measuring the slope of the line im-
pulse. The last three blocks of the flow chart shown in
Fig. 2 summarize the procedures. We have just now
extracted the distance parameter from the hologram
recorded by OSH. In Section 3 we describe the recon-
struction of the image of the object using the distance
parameter.
3. Reconstruction with a Distance Parameter
The reconstruction of the object from the complex ho-
logram is achieved by wave propagation, which cor-
responds physically to optical reconstruction of the
hologram by illuminating the hologram with a plane
reconstruction beam. The image at a depth location
of z
r
is reconstructed by convolution between the
complex hologram and the complex conjugate of the
FZP at the matched depth location, which is given by
I
r
ðx; y; z
r
Þ¼Hðx; yÞ ⊗ h
z
r
ðx; yÞ
¼ I
0
ðx; y; z
r
Þþ
Z
z
0
þð1=2Þδz
z
0
− ð1=2Þδz
z ≠ z
r
I
0
ðx; y; zÞ
⊗
jA
s
ðx; y; zÞ
λðz − z
r
Þ
exp
−j
π
λðz − z
r
Þ
ðx
2
þ y
2
Þ
dz:
ð11Þ
Here we can see that the reconstructed image con-
sists of the focused image at depth location z
r
[the
first term in Eq. (11)] and the defocused noise [the
second term in Eq. (11)]. Since the object is distrib-
uted along the depth axis from z
0
− ð 1=2Þδz to
z
0
þð1=2Þδz, i.e., the depth range of the object, recon-
struction with z
r
within the depth range of the object
yields a full 3-D image.
4. Experimental Results
We record the hologram of a specimen using OSH
(see Fig. 1). As shown in Fig. 3, the specimen or the
3-D object consists of two slides. The front and back
slides are transparencie s of a triangle and a rectan-
gle, respectively. The lateral size of both slides is
approximately 1 cm × 1 cm, and the depth distance
between the two slides is 15 cm. The He–Ne laser
with wavelength λ ¼ 633 nm is used to generate co-
herent light. The diameter of the beam expanded
by beam expanders BE1 and BE2 is D ¼ 15 mm
Fig. 2. Flow chart for extracting distance parameter z
0
from the
hologram recorded by OSH.
H156 APPLIED OPTICS / Vol. 48, No. 34 / 1 December 2009
and the expanded beam from BE1 is focused by lens
L1 with a focal length of f ¼ 400 mm. Thus, the NA of
the scan ning beam is NA ¼ sin½D=ð2f Þ ¼ 0:01875
and the Rayleigh range of the scanning beam is z
R
≈
2λ=πNA
2
¼ 1:15 mm. We positioned the object 87 cm
away from the focal point of the spherical wave and
scanned the 3-D object. The complex hologram of the
object is recorded by adding the two outputs of OSH
as in Eq. (2). Figures 4(a) and 4(b) show the ampli-
tude and phase of the complex hologram. To extract
the distance parameter, we first filter the holo-
gram using a Gaussian low-pass filter with NA
g
¼
0:00116, resulting in
Δz ¼ 2λ=π × ðNA
2
þ NA
2
g
Þ=ðNA
g
NAÞ
2
≈ 30 cm;
which is two times larger than the depth range of the
object and is required to obtain the result in Eq. (7).
Afterward, we synthesized the real-only spectrum
hologram as given by Eq. (8). Finally, we revealed
distance parameter z
0
as a delta line on a space-
frequency map using the Wigner distribution after
power-fringe-adjusted filtering. Figure 5 shows the
delta line on the space-frequency map, and the
slope of the delta line is measured to be π=λz
0
≈
5:7 rad=mm
2
. According to the slope of the delta line,
the distance parameter of the hologram is calculated
to be z
0
≈ 87 cm. Using the distance parameter, we
reconstructed the image at the depth location of
the object according to Eq. (11). The reconstructed
image with z
r
≈ 87 cm is shown in Fig. 6. The subse-
quent focusing within the depth range of the object,
i.e., z
0
− ð 1=2Þδz ≤ z
r
≤ z
0
þð1=2Þδz, is performed by
reconstruction according to Eq. (11), and the 3-D im-
age of the reconstructed image of the object is shown
in Fig. 7. Note that the triangle and rectangle are fo-
cused sharply at the front and the back of the recon-
structed 3-D image, respectively.
5. Concluding Remarks
We have shown that autofocusing in OSH is possible
without any searching algorithm or tracking process.
The idea is to extract the distance parameter from
the complex hologram. The complex hologram of an
Fig. 3. Object with L
x
¼ L
y
¼ 1:5 cm and L
z
¼ 15 cm.
Fig. 4. (a) Amplitude and (b) phase of the complex hologram.
Fig. 5. Line impulse on the frequency-space map.
1 December 2009 / Vol. 48, No. 34 / APPLIED OPTICS H157
