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Imaging properties of scanning holographic microscopy
Article in Journal of the Optical Society of America A · April 2000
DOI: 10.1364/JOSAA.17.000380 · Source: PubMed
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380
J. Opt. Soc. Am. A / Vol. 17, No. 3 / March 2000
Indebetouw et al.
Imaging properties of scanning
holographic microscopy
Guy Indebetouw and Prapong Klysubun
Department of Physics, Virginia Polytechnic Institute, Blacksburg, Virginia 24061-0435
Taegeun Kim and Ting-Chung Poon
Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute, Blacksburg,
Virginia 24061-0111
Received May 7, 1999; revised manuscript received October 7, 1999; accepted October 15, 1999
Scanning heterodyne holography is an alternative way of capturing three-dimensional information on a scattering or fluorescent object. We analyze the properties of the images obtained by this novel imaging process.
We describe the possibility of varying the coherence of the system from a process linear in amplitude to a process linear in intensity by changing the detection mode. We illustrate numerically the properties of the threedimensional point-spread function of the system and compare it with that of a conventional imaging system
with equal numerical aperture. We describe how it is possible, by an appropriate choice of the reconstruction
algorithm, to obtain an ideal transfer function equal to unity up to the cutoff frequency, even in the presence
of aberrations. Some practical implementation issues are also discussed. © 2000 Optical Society of America
[S0740-3232(00)01703-8]
OCIS codes: 070.2580, 090.0090, 110.0180, 110.6880, 180.5810.
1. INTRODUCTION
Scanning holography was invented as a clever application
of the two-pupil interaction schemes, which are unique in
extending the incoherent optical processing realm to operations requiring bipolar, or even complex, point-spread
functions.1,2 The principles of the method have been described, and its feasibility has been demonstrated.3
More recently, the application of scanning holography to
three-dimensional microscopy has been contemplated.
The holographic recording of three-dimensional fluorescent specimens was shown possible,4 and the technique
proved promising in locating fluorescent anomalies embedded in turbid media.5
The purpose of this paper is to present an analysis of
the imaging properties of scanning holography, define
theoretically expected performances, and discuss some
practical issues associated with the technique. In Section 2 we briefly review the background of threedimensional microscopy and holographic microscopy.
The primary objective here is to identify the main drawbacks of conventional coherent holographic methods applied to microscopy (i.e., speckle noise and a high-spatialbandwidth requirement) and to show how the scanning
heterodyne method may overcome some of the difficulties.
In Section 3 we analyze the properties of the reconstructed image in detail. For this, the imaging process of
scanning holography is described in terms of its pointspread function as well as its transfer function. Numerical examples are provided as an illustration of the expected performance. Many of the formulas given are
restricted to the paraxial approximation, which often
leads to convenient analytical solutions. It must be
stressed, however, that the analysis presented here is
0740-3232/2000/030380-11$15.00
valid beyond these approximations. The main result of
this section is that scanning holography produces reconstructed images with transverse and axial resolution comparable with or better than those of a conventional microscope of equal numerical aperture and, in addition, offers
unique opportunities for postprocessing, which are discussed in Section 4. The coherence of the scanning holographic process depends on the size of the detector and
can be varied from coherent to incoherent, in the same
way that the coherence in a conventional microscope depends on the size of the source. Thus scanning holography can in principle record phase objects, such as unstained biological specimens, can record incoherent
holograms, resulting in speckle-free reconstruction, and
can record holograms of fluorescent specimens. In Section 4 we outline some unique properties of scanning holography, which include a posteriori compensation of aberrations and a posteriori processing during reconstruction. Some practical issues related to the data acquisition time and the required bandwidth are also mentioned in this section.
2. BACKGROUND
A. Three-Dimensional Microscopy
Commonly used methods for three-dimensional imaging
in microscopy make use of optical sectioning, which generally requires a three-dimensional sampling of the specimen’s volume. The two best-known examples are optical
sectioning microscopy and scanning confocal microscopy.
Optical sectioning uses a conventional microscope to sequentially record a series of images focused at different
depths.6 Suitable algorithms are then used to merge the
images into a three-dimensional representation and re© 2000 Optical Society of America
Indebetouw et al.
duce the effects of out-of-focus blur. Scanning confocal
imaging also requires a three-dimensional scan. This
method, however, can achieve true depth slicing. In confocal imaging, the out-of-focus information in a selected
section is rejected before detection by the use of conjugate
pinholes.7
Both methods require precise threedimensional positioning devices. This is particularly
critical for the confocal methods. For certain applications in biology, one of the drawbacks of these instruments is that the data are acquired sequentially in a relatively slow three-dimensional scan of the specimen. This
long data-acquisition time may be a drawback for in vivo
studies. In addition, a long data-acquisition time may
exacerbate the photo-bleaching problem in fluorescence
microscopy.8 Elimination of the need for a threedimensional scan gave us the impetus to revisit holographic methods for three-dimensional microscopy. A
most attractive quality of holography is its well-known
ability to capture high-resolution images either in a
single shot, as in Gabor’s original idea,9 or, as in the case
of scanning holography, in a single two-dimensional scan.
It should be mentioned that a direct comparison of holography and confocal imaging is inappropriate because the
two methods lead to very different types of data. Confocal imaging efficiently extracts the information on a
single section, while a holographic reconstruction focused
at a particular depth remains corrupted by the out-offocus data. The selection of a single slice and the rejection of out-of-focus data have also been successfully demonstrated in white-light interferometric microscopy, also
called correlation microscopy.10 These methods, however, also require the sequential recording of a large number of transverse cross sections through the object.
B. Conventional Holography
In conventional holography an object is illuminated coherently, and the scattered light is made to interfere with
a mutually coherent reference wave. This results in an
encoding of each scatterer into a Fresnel zone pattern
containing the three-dimensional position information on
the scatterer. The interference pattern, or hologram, is
then recorded on a high-spatial-resolution medium.11
Because the recording medium is quadratic (phase insensitive), spatial heterodyning with the use of an off-axis
reference wave at the recording stage and coherent spatial filtering at the reconstruction stage are needed to extract the reconstructed image without twin-image
artifacts.12 The need for spatial heterodyning necessitates a coherent encoding and a recording medium with
high spatial resolution. These are the sources of the two
drawbacks of conventional holographic microscopy.
Namely, the ubiquitous speckle noise is unavoidable in
coherent imaging, and the detection system requires a
high spatial bandwidth.
The high-spatial-bandwidth requirement comes from
the fact that for successful extraction of the reconstructed
image by using spatial heterodyning, the spatial carrier
frequency of the hologram must be at least one and a half
times the spatial bandwidth of the object.12 Thus the total hologram bandwidth is at least four times that of the
object. Storage and transmission of such a hologram
may become problematic. This difficulty may be eased
Vol. 17, No. 3 / March 2000 / J. Opt. Soc. Am. A
381
somewhat by using phase-shifting methods.13 The
speckle noise, as is well known, is most severe in coherent
imaging systems.14 The necessity for coherent encoding
means that all spurious scattering from the specimen, its
support, or the optics will interfere with the object wave
that is recorded and gives rise to speckles. In addition,
reconstructing the hologram without artifacts requires coherent spatial filtering, and the coherently reconstructed
image must then be magnified by some optical system for
observation. This coherent imaging process with magnification leads to an image covered with high-contrast
speckles having exactly the same size as that of the resolution limit of the instrument. Consequently, the fine details of the image are irrevocably lost. A great deal of effort has been devoted to reducing speckle noise in
coherent imaging. Proposed methods include spatial averaging, statistical averaging, and other coherencespoiling schemes.14 All these methods either result in a
reduction of spatial resolution or increase the system’s
complexity beyond reasonable limits. There are some exceptions. In particle field analysis, for example, clever
methods have been described to minimize the speckle
noise with the use of multiple beams and to avoid the
twin-image artifacts of on-line hologram reconstruction
with spatial filters.15
C. Scanning Heterodyne Holography
In scanning holography1 a temporally modulated Fresnel
zone pattern is created, for example, by the interference
of a spherical wave and a plane wave shifted in frequency.
This pattern is scanned in a two-dimensional raster over
the object, and the scattered, reflected, or fluorescent light
is collected on a spatially integrating detector. The photocurrent is then heterodyned at the modulation frequency, or demodulated by other means, to produce a holographic record in electronic form. As a consequence of
the spatial scanning and the spatial integration on the detector, each scatterer is again encoded as a Fresnel zone
pattern, as in conventional holography, but the process
may now be either coherent or incoherent. More important, the process occurs in the temporal rather than the
spatial domain. With a spatially integrating detector,
the imaging process is linear in intensity, and thus insensitive to spurious phase fluctuations, even if the scanning
pattern is created by the interference of two coherent laser beams, as is most conveniently done in practice.
With a pinhole detector, the imaging process is linear in
amplitude and thus able to capture phase distributions.
The most important difference between conventional
and scanning holography is that in the latter the resulting hologram is obtained in the form of a temporal rather
than a spatial signal and that the extraction of the reconstructed image makes use of temporal heterodyning
rather than spatial heterodyning. Consequently, the detection system need not be spatially resolving and can be,
as may be needed for weakly scattering or weakly fluorescing specimens, a large spatially integrating detector.
Furthermore, the holographic signal can be directly
downconverted at the recording stage, resulting in a
single-sideband holographic record with considerably reduced bandwidth requirements. The reconstruction and
the subsequent magnification of an image focused at a
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J. Opt. Soc. Am. A / Vol. 17, No. 3 / March 2000
chosen depth within the specimen are performed digitally
by correlation of the hologram with a pattern matched to
the desired depth. Incoherent encoding and digital reconstruction lead to reconstructed images that are
speckle free. In addition, a digital reconstruction scheme
permits the straightforward implementation of various
postprocessing operations to obtain, a posteriori, e.g.,
dark field or gradient images, or to change the magnification, or to scan through the specimen’s depth, without any
optics or mechanical motion.
Indebetouw et al.
object (as shown in Fig. 1, which will be described in Subsection 3.C). The quasi-spherical wave emerges from a
point source created at the focal point of a well-corrected
microscope objective of numerical aperture (NA) uniformly illuminated by a plane wave. Within the domain
of validity of the Debye integral representation of the
field,22,23 the amplitude distribution at a distance z from
the geometrical focus is given by
U 共 r, z, t 兲 ⫽ ⫺ikA exp共 ⫺i ␻ t 兲
冕
NA
exp关 ik 共 1 ⫺ s 2 兲 1/2z 兴
0
D. Holographic Microscope
The principles of scanning holography have been experimentally demonstrated for simple, macroscopic objects,3
and the method has been extended to record holograms of
fluorescent specimens,4,5 thus demonstrating the incoherent nature of the process. A true holographic microscope
remains to be constructed, but the purpose of this paper is
to discuss some of its expected properties. Certain limitations and unique properties can already be mentioned.
A holographic microscope, for example, will not perform
the sharp optical sectioning characteristic of a confocal
scanning microscope. This is simply because the holographic information is acquired in a single twodimensional scan, which prevents the possibility of rejecting the out-of-focus information before detection.
However, reconstruction from the holographic data can
benefit from the application of a number of algorithms
that have been developed to process and improve images
in the conventional optical sectioning methods.16,17 Restoration and eventually ultraresolution methods18,19 can
also be used to advantage. A unique property of the
scanning holographic method is that it offers the possibility of correcting, during the reconstruction, the aberrations that may have affected the scanning pattern used in
recording the hologram. This may be of importance at
wavelengths for which well-corrected, high-numericalaperture optics are difficult or expensive to fabricate.
A conventional microscope has the capability of varying
the degree of coherence by changing the size of the source.
A broad source provides incoherent imaging, which minimizes speckle noise and artifacts but is blind to object
phase variations, whereas a point source provides spatially coherent illumination, making it possible to image
phase distributions such as unstained biological specimens. A holographic microscope presents an equivalent
versatility because the imaging property of a scanning optical system can be varied from incoherent to coherent
mode by changing the size of the detector.20,21 A large,
spatially integrating detector leads to incoherent imaging, results in speckle-free images, and is capable of imaging fluorescent samples, whereas a pinhole detector results in coherent imaging capable of rendering phase
objects visible and enabling the implementation of wellknown microscopic techniques such as the Zernike phase
contrast and Nomarski interference contrast methods.
3. IMAGING PROPERTY OF SCANNING
HOLOGRAPHY
A. Scanning Field
The scanning pattern is formed by the superposition of a
quasi-spherical wave and a plane wave interfering on the
⫻ J 0 共 ksr 兲共 1 ⫺ s 2 兲 ⫺1/2s ds
⫽ ⫺ikAE 共 r, z 兲 exp关 i 共 kz ⫺ ␻ t 兲兴 ,
where
E 共 r, z 兲 ⫽
冕
NA
(1)
exp共 ⫺i 21 ks 2 z 兲 J 0 共 ksr 兲共 1 ⫺ s 2 兲 ⫺1/2s ds.
0
A is the uniform field amplitude in the aperture, k
⫽ 2 ␲ /␭ ⫽ ␻ /c is the wave number of the radiation, ␻ is
its circular frequency, and c is the speed of light in vacuo.
s ⫽ 兩 s兩 , where s is the transverse component of a unit vector pointing from the geometrical focus to the point of observation, so that ␯ ⫽ s /␭ represents the transverse
spatial-frequency coordinate. J 0 is a Bessel function of
the first kind and zero order. r ⫽ 兩 r兩 is a transverse radial coordinate.
In the paraxial approximation, this distribution can be
written as, neglecting an unimportant factor,
U p 共 r, z, t 兲 ⫽ E p 共 r, z 兲 exp关 i 共 kz ⫺ ␻ t 兲兴 ,
with
E p 共 r, z 兲 ⫽
冕
NA
exp共 ⫺i 21 ks 2 z 兲 J 0 共 ksr 兲 s ds.
(2)
(3)
0
The subscript p stands for paraxial approximation. At
sufficiently large distances from the geometrical focus,
where the Fresnel number of the scanning aperture is
large, the amplitude distribution of Eq. (3) is correctly approximated by a spherical wave truncated in space by a
cone of half-angle ␣ ⫽ sin⫺1(NA). 23 In this case the
scanning field takes the simple form
E t 共 r, z 兲 ⫽ exp关 i⌽ 共 r, z 兲兴 circ关 r/a 共 z 兲兴 ,
Fig. 1. Sketch of a scanning holographic microscope. AO1 and
AO2 are acousto-optic modulators. P1 and P2 are point-source
outputs of single-mode fibers. The specimen is on a twodimensional scanning stage.
Indebetouw et al.
Vol. 17, No. 3 / March 2000 / J. Opt. Soc. Am. A
⌽ 共 r, z 兲 ⫽ kr 2 /2 z.
(4)
冕
E 共 ␳, ␰ ; F 0 兲 ⫽
0
The subscript t stands for truncated wave approximation,
circ(x) ⫽ 1 for x ⬍ 1 and 0 otherwise, and
a 共 z 兲 ⫽ z ⫻ NA
F 共 z 兲 ⫽ a 2 共 z 兲 /␭z ⫽ 共 NA兲 2 z/␭.
(6)
(7)
The calculations in Ref. 23 show that the truncatedspherical-wave approximation is excellent for Fresnel
numbers F ⬎ 40 and is already quite good for F ⬎ 10.
Noticeable discrepancies appear only near the boundary
of the pattern, which, in a practical setup such as that
shown in Fig. 1, could be clipped or tapered off by additional apertures. It should be stressed, however, that
this does not mean that a scanning pattern with a smaller
Fresnel number or one that does not satisfy the paraxial
approximation cannot be used. In this case, however, the
amplitude distribution of the scanning field must be calculated exactly to achieve a correct reconstruction. In
the following, but purely for convenience, we assume that
the truncated-spherical-wave approximation is valid.
B. Reduced Coordinates
In the following subsections, we consider a relatively thin,
weakly scattering specimen located at a distance z 0 from
the point source. The depth variable in object space is
measured from that distance, i.e.,
␦z ⫽ z ⫺ z0 .
(8)
If the object is thin compared with its average distance
from the point source, i.e., ␦ z Ⰶ z 0 , the size of the scanning pattern and its Fresnel number are nearly constant
within the object depth and are given by a ⫽ z 0 ⫻ NA
and F 0 ⫽ (NA) 2 z 0 /␭ ⫽ a 2 /␭z 0 , respectively.
If we anticipate that the transverse resolution limit of
the system will be on the order of ␭/NA and that the axial
resolution limit will be on the order of ␭/(NA) 2 , it is natural and useful to use dimensionless transverse and axial
coordinates scaled to these quantities. We thus define
the normalized transverse and axial coordinates
␳ ⫽ r ⫻ NA/␭,
(9)
␰ ⫽ ␦ z ⫻ 共 NA兲 /␭.
2
(10)
Similarly, we define a dimensionless transverse spatialfrequency coordinate ␮ scaled to the expected cutoff frequency ␯ max ⫽ NA/␭. Thus
␮ ⫽ ␯/ ␯ max ⫽ s/NA.
⫻ J 0 共 2 ␲ ␮ ␳ 兲关 1 ⫺ ␮ 2 共 NA兲 2 兴 ⫺1/2␮ d␮ ,
(12)
and becomes, in the paraxial approximation,
E p 共 ␳, ␰ ; F 0 兲 ⫽
冕
1
exp关 ⫺i ␲ ␮ 2 共 F 0 ⫹ ␰ 兲兴 J 0 共 2 ␲ ␮ ␳ 兲 ␮ d␮ .
0
This field is mixed on the object with a plane wave of amplitude E 0 shifted in frequency by ⍀ to produce a scanning pattern with amplitude
P 共 r, z, t 兲 ⫽ E 共 r, z 兲 ⫹ E 0 exp共 i⍀t 兲 .
exp(⫺i2 ␲ 兵 1 ⫺ 关 1 ⫺ ␮ 2 共 NA兲 2 兴 1/2其
⫻ 共 F 0 ⫹ ␰ 兲共 NA兲 ⫺2 )
(5)
is the radius of the scanning pattern at a distance z from
the point source. At this distance the scanning pattern is
characterized by its Fresnel number
1
383
(11)
With these notations the field amplitude of the quasispherical wave becomes, from Eq. (1),
(13)
When F 0 is large enough and the object depth range is
small compared with the average object distance from the
point source (i.e., ␰ Ⰶ F 0 ), the truncated-spherical-wave
approximation leads to a simplified expression for the
field amplitude:
E t 共 ␳, ␰ ; F 0 兲 ⫽ exp关 i ␲␳ 2 共 F 0 ⫹ ␰ 兲 ⫺1 兴 circ共 ␳ /F 0 兲
⯝ exp关 i ␲␳ 2 共 1 ⫺ ␰ /F 0 兲 /F 0 兴 circ共 ␳ /F 0 兲 .
(14)
C. Holographic Record
We now consider the recording of the hologram of a relatively thin, weakly scattering specimen that can be represented by an amplitude transmittance T( ␳, ␰ ). Extension of the following arguments to the case of threedimensional reflecting surfaces is trivial, and their
extension to fluorescent specimens and rough surfaces
will be discussed below.
The scanning pattern with amplitude P( ␳, ␰ , t) given
by Eq. (7) is projected through the specimen, which is
scanned in a two-dimensional raster. If rs ⫽ rs (t) [or
␳s ⫽ ␳s (t) in reduced coordinates] represents the instantaneous position of the object, the field amplitude behind
the object is approximately
E obj共 t 兲 ⫽
冕
d2 ␳ d␰ P 共 ␳, ␰ , t 兲 T 共 ␳ ⫺ ␳s , ␰ 兲 .
(15)
As shown in Fig. 1, this amplitude is then Fourier transformed by a lens of focal length f and falls on a spatially
integrating quadratic detector through a mask with intensity transmittance M( ␳). The resulting detector current, for each instantaneous position ␳s ⫽ ␳s (t) of the object, is proportional to
i 共 ␳s 兲 ⬀
⬀
冕
冕
d2 ␳ 兩 F␳ 兵 E obj其 兩 2 M 共 ␳兲
d2 ␳ d2 ␳ ⬘ d2 ␳ ⬙
冕
d␰ M 共 ␳ 兲
⫻ exp关 ⫺i2 ␲ 共 ␳⬘ ⫺ ␳⬙ 兲 • ␣兴 P 共 ␳⬘ , ␰ 兲
⫻ P * 共 ␳⬙ , ␰ 兲 T 共 ␳⬘ ⫺ ␳s , ␰ 兲 T * 共 ␳⬙ ⫺ ␳s , ␰ 兲 ,
(16)
where * stands for complex conjugate. Here
F␳ s 兵 E obj其 ⫽
冕
E obj共 ␳s 兲 exp共 ⫺i2 ␲␳s • ␣ ␳兲 d2 ␳ s
is the Fourier transform of the field amplitude behind the
object, with ␣ ⫽ 关 f(NA) 2 /␭ 兴 ⫺1 accounting for the scaling
of the Fourier transform in the back focal plane of the
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J. Opt. Soc. Am. A / Vol. 17, No. 3 / March 2000
Indebetouw et al.
lens with focal length f. Expression (16) simplifies if we
write it in terms of the Fourier transform
M共 ␮兲 ⫽
冕
M 共 ␳兲 exp共 ⫺i2 ␲ ␮ • ␳兲 d2 ␳
of the mask intensity transmittance. We then have
i 共 ␳s 兲 ⬀
冕
d2 ␳ ⬘ d2 ␳ ⬙ d␰ M共 ␣ ␳⬘ ⫺ ␣ ␳⬙ 兲 P 共 ␳⬘ , ␰ 兲
⫻ P * 共 ␳⬙ , ␰ 兲 T 共 ␳⬘ ⫺ ␳s , ␰ 兲 T * 共 ␳⬙ ⫺ ␳s , ␰ 兲 .
H inc共 ␳s 兲 ⫽
(17)
From Eq. (7) the scanning amplitude P has a component
oscillating at the frequency ⍀. The photodetector current can then be demodulated to extract the component at
the heterodyne frequency ⍀. For example, this can be
done by mixing the photocurrent with reference signals
cos(⍀t) and sin(⍀t) and low-pass filtering the result, as in
a lock-in amplifier, to obtain two quadrature signals
C( ␳s ) and S( ␳s ), which are then digitized and combined
to form the single-sideband holographic record
H 共 ␳s 兲 ⫽ C 共 ␳s 兲 ⫹ iS 共 ␳s 兲 .
(18)
Equivalently, the signal i( ␳s ) can be digitized directly, by
using a fast analog-to-digital converter (ADC), fast Fourier transformed, and filtered around the modulation frequency ⍀. This of course assumes that the modulation
frequency is large enough, compared with the signal fluctuations resulting from scanning the object, for the demodulation or the filtering to be performed without introducing artifacts. In other words, the Shannon–Nyquist
criterion must be satisfied. This clearly imposes a limit
on the scanning speed, as will be discussed below.
Using the definition of the scanning pattern from Eq.
(7) in expression (16) and extracting the terms oscillating
at the temporal frequency ⍀ lead, to within some constant
factors, to the following holographic record:
H 共 ␳s 兲 ⫽
冕
冕
d2 ␳ ⬘ d␰ E 共 ␳⬘ , ␰ 兲 I 共 ␳⬘ ⫺ ␳s , ␰ 兲 ,
(21)
where I( ␳⬘ , ␰ ) ⫽ 兩 T( ␳⬘ , ␰ ) 兩 2 . The process is linear in intensity and thus, according to conventional wisdom, incoherent. This mode of operation is needed to record holograms of rough objects or rough surfaces without speckle
noise and to record holograms of incoherent objects such
as fluorescent specimens. In both the coherent and the
incoherent mode, a point object is encoded as the same
wave E( ␳, ␰ ), which, for relatively large Fresnel numbers
(F ⬎ 10) and relatively small numerical apertures (NA
⬍ 0.5), is well approximated by a truncated spherical
wave.
D. Hologram Reconstruction
For the reconstruction of an image focused at a distance
z R from the point source used in the recording, that is, a
distance ␰ R into the object, in reduced coordinates, the hologram can be digitally correlated with the pattern
E R ( ␳, ␰ R ) matched to the desired depth. Thus the focused reconstruction is, from Eq. (19),
R 共 ␳, ␰ R 兲 ⫽
⫽
冕
冕
* 共 ␳s ⫺ ␳, ␰ R 兲 d 2 ␳ s
H 共 ␳s 兲 E R
d2 ␳ s d2 ␳ ⬘ d2 ␳ ⬙ d␰ M共 ␣ ␳⬘ ⫺ ␣ ␳⬙ 兲
* 共 ␳s ⫺ ␳, ␰ R 兲 E 共 ␳⬘ , ␰ 兲
⫻ ER
⫻ T 共 ␳⬘ ⫺ ␳s , ␰ 兲 T * 共 ␳⬙ ⫺ ␳s , ␰ 兲 .
d2 ␳ ⬘ d2 ␳ ⬙ d␰ M共 ␣ ␳⬘ ⫺ ␣ ␳⬙ 兲 E 共 ␳⬘ , ␰ 兲
(22)
In the coherent case, we obtain
⫻ T 共 ␳⬘ ⫺ ␳s, ␰ 兲 T * 共 ␳⬙ ⫺ ␳s, ␰ 兲 .
(19)
In the truncated-wave approximation, E( ␳, ␰ ) is given by
Eq. (13), and in more general cases, it can be calculated
from Eq. (1).
Two extreme cases are of interest because they lead to
linear superposition integrals from which one can define
point-spread functions and transfer functions. The first
case is that of a coherent process. It results from using a
pinhole on the axis as a mask. Thus we have M( ␳)
⯝ ␦ ( ␳), where ␦ (␳) is a Dirac delta function, and M( ␮)
⯝ 1, leading to
H coh共 ␳s 兲 ⫽ H 0
smooth three-dimensional surfaces, as met in the microelectronics industry. If the object is rough, however, we
expect the images to be corrupted by speckle noise. The
second extreme case is that of an incoherent process,
which results from using an open mask and a large spatially integrating detector. Here M( ␳) ⯝ 1, and M( ␮)
⯝ ␦ ( ␮). The holographic record is in this case
冕
d2 ␳ ⬘ d␰ E 共 ␳⬘ , ␰ 兲 T 共 ␳⬘ ⫺ ␳ s , ␰ 兲 ,
(20)
where H 0 ⫽ 兰 d2 ␳ ⬙ T * ( ␳⬙ ⫺ ␳s ) is a constant complex factor. In this case the amplitude T( ␳, ␰ ) of each object
point is encoded as a wave E( ␳, ␰ ). The process is linear
in field amplitude and is thus coherent according to conventional wisdom. This hologram is sensitive to object
phase variations and thus is capable of recording phase
objects such as thin unstained specimens, as encountered
in biomedical imaging, as well as the topography of
R coh共 ␳, ␰ R 兲 ⫽
冕
d2 ␳ s d2 ␳ ⬘ d␰ E *
R 共 ␳s ⫺ ␳, ␰ R 兲
⫻ E 共 ␳⬘ , ␰ 兲 T 共 ␳⬘ ⫺ ␳s , ␰ 兲 ,
(23)
and in the incoherent case, we obtain
R inc共 ␳, ␰ R 兲 ⫽
冕
* 共 ␳s ⫺ ␳, ␰ R 兲
d2 ␳ s d2 ␳ ⬘ d␰ E R
⫻ E 共 ␳⬘ , ␰ 兲 I 共 ␳⬘ ⫺ ␳s , ␰ 兲 .
(24)
It is remarkable that the reconstructed data have exactly
the same form whether the system operates in a coherent
or an incoherent mode. The point-spread functions are
identical in both cases and in general are complex. This
of course comes from the fact that the heterodyne detection gives access to the phase of the photocurrent. When
the system operates in a coherent mode (with a pinhole
detector), the current is proportional to the object amplitude and thus also carries information on the object
phase. When the system operates in an incoherent mode
(with a spatially integrating detector), the photocurrent is
proportional to the object intensity and is blind to its
Indebetouw et al.
Vol. 17, No. 3 / March 2000 / J. Opt. Soc. Am. A
phase, but the phase of the photocurrent itself carries the
encoded information on the object location. This is what
makes it possible to record incoherent holograms, insensitive to object phases, but with a point-spread function
that is not necessarily real positive. In fact, a complex
point-spread function of arbitrary shape can in principle
be synthesized by choosing appropriate scanning and reconstruction fields.
When the scanning field is a pure phase function, as it
is, for example, in the truncated-wave approximation, the
optimum choice of reconstructing function is the scanning
field itself. The reconstruction operation, which is then a
correlation with a spherical wave of appropriate curvature, can be interpreted in two different ways. As is
known from the Huygens principle, the correlation of an
optical field with a spherical wave represents a free-space
propagation of that field for a distance equal to the radius
of curvature of the wave. Thus the digital reconstruction
is equivalent to propagating the field that would emerge
from the hologram for a distance z R , or F 0 ⫹ ␰ R in reduced coordinates, where the reconstructed image would
be observed. Correlation is also a pattern recognition
process. Consequently, the reconstruction operation can
be interpreted as a matched filtering of the hologram to
recognize and extract from the hologram all the waves
with a curvature radius F 0 ⫹ ␰ R . The distribution of the
amplitude of these waves is of course identical with the
distribution of scatterers in a plane ␰ R in the object, possibly corrupted by out-of-focus images. The interpretation in terms of pattern recognition may be helpful in designing reconstruction schemes based on nonlinear
reconstruction processes rather than the linear process of
correlation. Such nonlinear operations, which can be
performed digitally, may lead to sharper depth discrimination and sectioning than that provided by a linear imaging process.
E. Point-Spread Function
In the two extreme cases of full coherence or incoherence,
and when the reconstructing field is identical with the
scanning field (a truncated spherical wave in common approximation), the reconstructed data are either a linear
superposition of object amplitudes or a linear superposition of object intensities. With the change of variables
␳s ⫺ ␳ → ␳⬘ ⫺ 12 ␳⬙ , ␳⬘ → ␳⬘ ⫹ 21 ␳⬙ in Eqs. (22) and (23),
the reconstructed image can be written in the usual form:
R coh共 ␳, ␰ 兲 ⫽
冕
d2 ␳ ⬘ d␰ PSF共 ␳⬘ ; ␰ , ␰ R 兲 T 共 ␳⬘ ⫺ ␳, ␰ 兲
(25)
in the coherent case and
R inc共 ␳, ␰ 兲 ⫽
冕
d2 ␳ ⬘ d␰ PSF共 ␳⬘ ; ␰ , ␰ R 兲 I 共 ␳⬘ ⫺ ␳, ␰ 兲
in the incoherent case.
function is
PSF共 ␳; ␰ , ␰ R 兲 ⫽
冕
(26)
In both cases the point-spread
* 共 ␳⬘ ⫺
d2 ␳ ⬘ E R
1
2 ␳,
␰ R 兲 E 共 ␳⬘ ⫹
1
2 ␳,
␰ 兲.
(27)
The point-spread function for the reconstruction at a
depth ␰ R , as a function of the transverse coordinate ␳ and
the axial coordinate ␰, is thus the correlation of the scan-
385
ning field at ␰ with the reconstructing field at ␰ R . In particular, the in-focus point-spread function is the autocorrelation of E( ␳, ␰ R ).
When the truncated-spherical-wave approximation is
valid, both the scanning wave and the reconstructing
function are given by relation (14), and the point-spread
function can be calculated as
PSFt 共 ␳, ␰ 兲
⫽
冦
1
␲ F 02
冕
⫹共 F 0 ⫺␳ /2兲
dx
⫺共 F 0 ⫺␳ /2兲
再
冕
⫹关 F 0 2 ⫺共 ␳ /2⫹x 兲 2 兴 1/2
⫺关 F 0 2 ⫺共 ␳ /2⫹x 兲 2 兴 1/2
dy
冎
i2 ␲
i ␲␰
⫻ exp ⫺
␳x ⫹
关共 ␳ /2 ⫹ x 兲 2 ⫹ y 2 兴 .
F0
F 02
for ␳ ⬍ 2F 0
for ␳ ⬎ 2F 0
0
(28)
For large enough Fresnel numbers, Eq. (28) is well represented by empirical formulas.24 For the transverse distribution in focus, these empirical formulas give, approximately,
PSFe 共 ␳, 0; F 0 兲
⫽
再
P 共 ␳ /2F 0 兲
J 1 关 2 ␲␳ 共 1 ⫺ ␳ /2F 0 兲兴
␲␳ 共 1 ⫺ ␳ /2F 0 兲
for ␳ ⬍ 2F 0
,
for ␳ ⬎ 2F 0
0
(29)
where P(x) ⫽ 1 ⫺ 1.38x ⫹ 0.031x ⫹ 0.344x and J 1 is
a Bessel function of the first kind and first order. For the
axial distribution, Eq. (28) gives
2
PSFe 共 0, ␰ ; F 0 兲 ⫽ sinc共 ␰ /2兲 ,
3
(30)
where the subscript e stands for empirical and sinc(x)
⫽ sin(␲x)/␲x.
It is useful to compare the point-spread function of
scanning holography with that of a conventional imaging
system having the same numerical aperture, which is,12
in the paraxial approximation, PSFcoh( ␳, 0) ⫽ J 1 (2 ␲␳ )/
␲␳ for an aberration-free coherent system and
PSFinc( ␳, 0) ⫽ 关 J 1 (2 ␲␳ )/ ␲␳ 兴 2 for an incoherent system.
As already mentioned in Subsection 3.D, the first striking
difference is that, although the imaging process in scanning holography with a spatially integrating detector is
incoherent and linear in intensity, the point-spread function is bipolar and even complex in general. It can be
shown that for F 0 ⬎ 5 the central lobe of the point-spread
function represented by Eq. (29) is nearly identical with
the central lobe of the conventional coherent point-spread
function (the Airy disk). Thus, in this case, the transverse resolution limit, defined as the radius of the central
lobe of the point-spread function, is, to a good approximation, the same as that of a conventional coherent system.
That is,
⌬ ␳ ⫽ 0.61 or ⌬r ⫽ 0.61␭/NA.
(31)
From Eq. (30) the axial resolution, defined as the distance
between the axial maximum at ␰ ⫽ 0 and the first axial
zero, is found to be
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J. Opt. Soc. Am. A / Vol. 17, No. 3 / March 2000
⌬ ␰ ⫽ 2 or ⌬z ⫽ 2␭/ 共 NA兲 2 .
Indebetouw et al.
(32)
Another interesting feature of the point-spread function
in the truncated-wave approximation is that it vanishes
for ␳ ⬎ 2F 0 (r ⬎ 2a), in contrast to the point-spread
function of a conventional imaging system, which has
sidelobes extending over the entire image. This may
have practical importance because the Fresnel number, in
scanning holography, can be varied easily by changing
the distance between the object and the point source,
without affecting the numerical aperture, and thus keeping the resolution constant. For certain types of objects,
there may be some advantages in using a scanning pattern with a small Fresnel number to reduce the extent of
the sidelobes of the point-spread function. At small
Fresnel numbers, however, the truncated-wave approximation is invalid, and the point-spread function must be
calculated by using Eq. (1). Figures 2–4 illustrate these
points.
Figure 2 compares the profile of the in-focus pointspread function for different Fresnel numbers and numerical apertures with the Airy pattern of the coherent
paraxial point-spread functions of a conventional
aberration-free imaging system. It is seen that the holographic point-spread function does not vary much with
the Fresnel number, as one would expect theoretically.
For modest numerical apertures, the point-spread function is nearly identical with the Airy pattern, except for a
slightly narrower main lobe and slightly larger sidelobes.
For large numerical apertures, the main lobe is significantly narrower than that of the Airy pattern. As usual,
this gain is obtained at the price of increased sidelobe amplitudes. If the sidelobes are undesirable, it is fairly
simple to apply, a posteriori, an apodizing aperture to the
reconstructing pattern to smooth them out, possibly at
the price of a reduced resolution. Figures 3 and 4 show
other features of the point-spread function for modest
(0.5) and high (0.95) numerical apertures, respectively.
The topographical plots of Figs. 3(c) and 4(c) illustrate the
Fig. 2. Cross sections of the in-focus point-spread function amplitude. For low numerical apertures, the point-spread function
is nearly independent of the Fresnel number and almost identical with the point-spread function of a clear aperture of equal numerical aperture. For high numerical apertures, the pointspread function has a sharper central lobe than the Airy disk but
higher sidelobes.
Fig. 3. Point-spread function (PSF) of a scanning holographic
system with numerical aperture 0.5 and Fresnel number 5. The
PSF is nearly identical with that of a clear aperture of equal numerical aperture. (a) Axial sections through the PSF amplitude,
where ␳ is the radial transverse coordinate and z is the axial coordinate (␰ in the text), (b) three-dimensional representation of
the in-focus PSF, (c) topographical plot of the PSF amplitude.
Indebetouw et al.
Vol. 17, No. 3 / March 2000 / J. Opt. Soc. Am. A
387
as the Fourier–Bessel transform of the point-spread
function.12 The results of Subsection 3.E establish that
the point-spread function with a defocus ␰ is the correlation of the scanning field at ␰ with the reconstructing
function at ␰ ⫽ 0. The transfer function with defocus ␰ is
thus
* 共 ␮; 0 兲 E共 ␮; ␰ 兲 ,
TF共 ␮; ␰ 兲 ⫽ ER
(33)
where ER ( ␮; ␰ ) is the Fourier–Bessel transform of
E R ( ␳, ␰ ):
ER 共 ␮; ␰ 兲 ⫽ 2 ␲
冕
⬁
E R 共 ␳, ␰ 兲 J 0 共 2 ␲ ␮ ␳ 兲 ␳ d␳ .
(34)
0
Using the identity 2 ␲ 兰 0⬁ ␮ ⬘ J 0 (2 ␲ ␮ ⬘ ␳ )J 0 (2 ␲ ␮ ␳ ) ␳ d␳
⫽ ␦ ( ␮ ⫺ ␮ ⬘ ), which simply expresses the fact that the
Fourier transform of a J 0 function is a ␦-ring distribution,
and using the fact that if the domain of ␮ ⬘ is limited to
the range 0 ⬍ ␮ ⬘ ⬍ 1, so will be the domain of ␮, we find
from Eq. (12) that
E共 ␮; ␰ 兲 ⫽ exp(⫺i2 ␲ 兵 1 ⫺ 关 1 ⫺ ␮ 2 共 NA兲 2 兴 1/2其 共 F 0 ⫹ ␰ 兲
⫻ 共 NA兲 ⫺2 )关 1 ⫺ ␮ 2 共 NA兲 2 兴 ⫺1/2circ共 ␮ 兲 .
(35)
The optimum choice for the reconstructing function is
that leading to a perfect in-focus transfer function; i.e., at
␰ ⫽ 0, the transfer function is equal to unity up to the
cutoff frequency. From Eqs. (33) and (35), it is clear that
one must choose a reconstruction function that has a
Fourier–Bessel transform
ER 共 ␮; ␰ 兲 ⬀ E共 ␮; ␰ 兲关 1 ⫺ ␮ 2 共 NA兲 2 兴 .
(36)
The resulting transfer function with defocus ␰ is then,
from Eq. (33),
TF共 ␮; ␰ 兲 ⫽ exp(⫺i2 ␲ 兵 1 ⫺ 关 1 ⫺ ␮ 2 共 NA兲 2 兴 1/2其
⫻ ␰ 共 NA兲 ⫺2 )circ共 ␮ 兲 .
(37)
In particular, the in-focus transfer function is
TF共 ␮; 0 兲 ⫽ circ共 ␮ 兲 ,
(38)
which is the ideal transfer function of a system with cutoff frequency ␮ ⫽ 1.
This result is valid when the Debye integral is a correct
representation of the scanning field and thus is valid beyond the paraxial approximation. In the paraxial approximation, the scanning field is approximated by Eq.
(13), and we have the simplified expressions
Ep 共 ␮; ␰ 兲 ⫽ ERp 共 ␮; ␰ 兲 ⫽ exp关 ⫺i ␲ ␮ 2 共 F 0 ⫹ ␰ 兲兴 circ共 ␮ 兲 ,
(39)
Fig. 4. Same as Fig. 3 but for a system with numerical aperture
0.95 and Fresnel number 5. Compared with the PSF of Fig. 3
and with J 1 (x)/2x, the PSF has a sharper central lobe and larger
sidelobes. The difference is also displayed in (c).
relative advantage in depth resolution that is obtained
with a high-numerical-aperture system.
F. Transfer Function
Additional information on the properties of the holographic images is obtained by defining a transfer function
which leads to
TFp 共 ␮; ␰ 兲 ⫽ exp共 ⫺i ␲ ␮ 2 ␰ 兲 circ共 ␮ 兲 .
(40)
Comparing this result with relation (36) and Eq. (37),
we see that the reconstructing function must be chosen so
as to cancel out the phase of the Fourier–Bessel transform of the scanning field and to level off its amplitude
variations. As is well-known, correcting the phases is
most important because phase variations in the transfer
function are akin to aberrations and always broaden the
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J. Opt. Soc. Am. A / Vol. 17, No. 3 / March 2000
Indebetouw et al.
size of the point-spread function. In practice, it is less
necessary to equalize the amplitude, but, in fact, it is not
difficult to do so with a digital reconstruction. Problems
would arise only if the Fourier–Bessel transform of the
scanning field had zeros. This may occur if the scanning
beam is corrupted by large aberrations, but with a reasonably well-corrected objective, the scanning field and its
transform have smooth amplitude and phase profiles with
nearly spherical curvature, so that the reconstructing
function also has a smooth amplitude profile with nearly
spherical curvature. Since the reconstructing function is
generated digitally and then digitally correlated with the
hologram to reconstruct an image, it is always possible to
calculate the reconstructing function that will correct the
eventual aberrations of the scanning field and thus realize the ideal transfer function of Eq. (38). Consequently,
the transfer function in scanning holography, whether it
operates in the coherent or the incoherent mode, may be
made flat up to the cutoff frequency ␮ max ⫽ 1, or ␯ max
⫽ NA/␭, even if the scanning beam has some aberrations. This holds, of course, as long as these aberrations
can be duplicated in the reconstruction function. These
attributes are to be contrasted with the transfer function
of a conventional imaging system. For a coherent system, the transfer function is the pupil distribution itself.12
Thus, for an aberration-free system with a numerical aperture NA, all the spatial frequencies lower than the cutoff frequency NA/␭ are transmitted integrally, and the
rest are blocked. Aberrations play a disastrous role in
this case, because they introduce spurious phase distortions in the pupil, which strongly affects the integrity of
the image. In an incoherent system, the transfer function is the autocorrelation of the pupil, which is always
maximum at the origin and tapers off up to the cutoff frequency 2NA/␭. Thus the low frequencies are always emphasized, and the high frequencies are transmitted with
attenuation. Aberrations always result in further attenuation of the high spatial frequencies. For severe aberrations the transfer function may even change sign,
leading to contrast inversion and severe image degradations. In both cases a posteriori correction, or deblurring,
is in general a nontrivial ill-posed inverse problem. In
holography, in contrast, conjugation of phase can readily
be obtained, enabling the application of a variety of aberration compensation schemes. For example, in electron
holography, a posteriori compensation of spherical aberrations has been demonstrated with the use of electronically addressed phase masks.25
4. POSTPROCESSING AND PRACTICAL
CONSIDERATIONS
A. Postprocessing Possibilities
The field function used to reconstruct the image digitally
in scanning holography can in principle be chosen at will.
This degree of freedom can be used to accomplish a number of processing operations while reconstructing the image. In other words, one can synthesize, a posteriori,
various point-spread functions of the form
PSF ⫽
冕
* 共 ␳⬘ ⫺
ER
1
2 ␳,
0 兲 E 共 ␳⬘ ⫹
1
2 ␳,
0 兲 d2 ␳ ⬘ ,
(41)
or, equivalently, one can synthesize in-focus transfer
functions of the form
* 共 ␮; 0 兲 circ共 ␮ 兲 .
TF共 ␮; 0 兲 ⫽ ER
(42)
Most remarkable is that these synthesized point-spread
functions and transfer functions can be made to operate
on the object amplitude, if a pinhole detector was used in
recording the hologram, or on the object intensity, if a
spatially integrating detector was used. For example, it
is easy to synthesize incoherent bipolar point-spread
functions or high-pass incoherent transfer functions.
Such operations cannot be done directly in an incoherent
imaging system. A few examples are discussed in what
follows.
For simplicity, we assume in the following that the
truncated-wave approximation is valid and that the scanning field is given by relation (14) and its Fourier–Bessel
transform is given by Eq. (39). Some examples of possible postprocessing operations are briefly described in
the following paragraphs.
As already discussed in Subsection 3.F, if we choose a
reconstructing function E Rt ( ␳) ⫽ E t ( ␳), the transfer
function is circ(␮), and we obtain a perfect image with a
cutoff frequency ␮ ⫽ 1. It is now easy to add to the reconstructing function an amplitude factor that, for example, enhances the high frequencies for edge enhancement or tapers the high frequencies smoothly for
apodization.
More can be done. If, for example, we choose a reconstructing
function
of
the
form
E Rt ⫽ E t ( ␳ )
⫺ c circ( ␳ /F 0 ), where c is a real constant, the resulting
transfer
function
is
TF( ␮) ⫽ Et ( ␮ ) 关 E t* ( ␮ )
⫺ cJ 1 (2 ␲ ␮ F 0 )/ ␮ 兴 . Thus, if c ⫽ ( ␲ F 0 ) ⫺1 , the transfer
function is TF( ␮ ) ⯝ circ( ␮ ) ⫺ J 1 (2 ␲ ␮ F 0 )/ ␲ ␮ F 0 (use
was made of the fact that the second term has a width
much smaller than unity). The frequency ␮ ⫽ 0 is entirely suppressed since TF(0) ⫽ 0, and the low frequencies up to ⌬ ␮ ⫽ 1.22/F 0 are gradually attenuated from 0
to 1. If F 0 is large enough, this results in a dark field image.
As a final example, one may consider a reconstructing
function of the form E Rt ⫽ E t ( ␳ ⫹ 21 ⑀ x) ⫺ E t ( ␳ ⫺ 21 ⑀ x)
1
⬃ 2 ⑀ (d/dx)E t ( ␳), where ⑀ is smaller than a resolution element and x is a unit vector in the x direction. This results in a reconstruction revealing gradients in the x direction. The transfer function is TF ⬀ Et ( ␮)sin(␲⑀␮x),
where ␮ x is the spatial frequency in the x direction. If
⑀ ⫽ 12 , which corresponds to a reconstructing pattern
made of two patterns E t ( ␳) with opposite polarity and
shifted by half a resolution element in the x direction, the
1
transfer function is TF ⫽ Et ( ␮)sin( 2 ␲␮x). This transfer
function is identical with that obtained with the Nomarski interference contrast method if Et ( ␮) is interpreted
as the spherical curvature in the pupil. When acting on
a phase object recorded in the coherent mode, this
operation reveals the phase gradients along x, as in the
Nomarski method. When acting on an object recorded in
the incoherent mode, it reveals the intensity gradients
along x. Similarly, it is possible to extract axial gradients by choosing a reconstructing function equal to the
Indebetouw et al.
Vol. 17, No. 3 / March 2000 / J. Opt. Soc. Am. A
difference between two scanning patterns corresponding
to a depth difference equal to half the axial resolution,
e.g.,
E Rt ⫽ E t 共 ␳ ⫹
1
2 ⑀␰兲
⫺ E t共 ␳ ⫺
1
2 ⑀␰兲
⬃
1
2 ⑀ 共 d/d␰ 兲 E t 共 ␳,
␰ 兲.
B. Practical Considerations
Since we were unable to secure the funds necessary to actually build a microscope, we will share our thoughts in
trying to design one, in the hope that it may trigger someone’s interest. A possible design for a scanning heterodyne microscope was shown in Fig. 1. The illumination
module produces two quasi point sources of light, P 1 and
P 2 , at the output of two single-mode fibers. An adjustable frequency difference is provided by two acousto-optic
modulators driven in synchronism. The lights from the
two point sources are combined at the beam splitter, and
the beams are shaped to produce, in the approximation
discussed above, a spherical and a plane wave superposed
on the object by means of a microscope objective of numerical aperture NA. An adjustable aperture limiting
the size of the scanning pattern is used to vary its Fresnel
number. The objective images this aperture on the object. Light from P 1 is focused by the objective near its focal point, from which it travels toward the object, where it
forms, in the truncated-spherical-wave approximation, a
spherical wave truncated to a half-cone angle sin⫺1(NA).
Light from P 2 is focused by an intermediate lens at the
center of the objective’s entrance pupil. The objective collimates this light to project a plane wave of limited extent
on the object. The Fresnel number of the scanning pattern can thus be changed without affecting the numerical
aperture or the resolving power.
The possibility of using the instrument in transmission
mode, to obtain holograms of unstained specimens or
phase objects, as well as in reflection mode, to form holograms of reflecting objects or of fluorescent specimens, is
illustrated in Fig. 1. A mask in the pupil plane is used to
vary the coherence between an imaging linear in intensity and an imaging linear in amplitude, as discussed
above. For fluorescence imaging the beam splitter is a
dichroic mirror transmitting the laser excitation wavelength and reflecting the fluorescence wavelength. The
reference detector uses a pinhole in a plane conjugate to
the object. The signal from this detector is used as a reference signal for heterodyne detection. In this way,
eventual shifts in signal frequency caused by mechanical
or thermal fluctuations in the illumination stage appear
in both the signal and the reference and can be canceled
out. The sample can be mounted on a computercontrolled, two-dimensional stage, or the scanning can be
accomplished by mirrors for faster scanning rates. The
resulting temporal signal from the detectors is sent to the
data acquisition stage, which can be either digital or analog. For digital acquisition the signal is converted by a
fast analog-to-digital converter (ADC) and filtered digitally to extract the holographic record. In this case we
expect the rate of the ADC to impose a limit on the acceptable scanning rate. For analog acquisition the signal
is mixed with the reference signal and filtered to obtain
two downconverted quadrature signals, which are then
converted from analog to digital and combined digitally to
form the complex holographic record.
389
The design of the data acquisition module involves
some trade-off between its bandwidth, which will eventually limit the scanning rate, and its signal-to-noise ratio.
For weak signals a lock-in amplifier as the phasesensitive detection system may be best because it is specially designed to extract weak signals from large, noisy
backgrounds. But for that same reason, its bandwidth is
small, allowing only very slow scanning rates. Faster
scanning rates can be achieved with a detection system
having a larger bandwidth, but only at the price of a lower
signal-to-noise ratio. For example, let us consider a digital acquisition system equipped with an ADC capable of
acquiring 250 ⫻ 106 samples per second at 8-bit resolution. If the signal is sampled at twice the Nyquist rate,
the maximum signal frequency must not exceed (250
⫻ 106 /2)/2 ⫽ 62.5 MHz. Since, in a well-designed system, the smallest feature of the scanning pattern matches
in size the resolution limit, this frequency cutoff is the
sum of the modulation frequency f m of the scanning pattern and the highest frequency f s resulting from scanning
the sample. From information theory we need f m ⬎ f s ,
and for best bandwidth utilization, we want the largest
possible f s . We may, for example, choose, allowing some
margin for filtering without aliasing, f m ⫽ 2 f s , leading to
f m ⫽ 41.7 MHz and f s ⫽ 20.8 MHz. A frequency difference of 41.7 MHz between the light beams interfering on
the sample can be obtained with a standard acousto-optic
modulator. Of course, the detector bandwidth must be
compatible with this figure. If we specify to collect at
least four samples per resolution element, to exceed the
Nyquist criterion by a factor of 2, the maximum allowable
scanning rate is f s /4 ⫽ 5.2 ⫻ 106 resolution elements per
second. The hologram of a sample with 512 ⫻ 512 resolution elements can thus be captured in (512 ⫻ 4 pixel per
line) ⫻ (512 ⫻ 4 lines)/5.2 ⫻ 10 6 pixels per second
⬃0.8 s, plus the return dead time of the scanning device.
With NA ⬃ 0.6 and ␭ ⬃ 0.5 ␮ m, the resolution limit is
⬃0.5 ␮m, so that the required scanning speed to reach
that rate is ⬃260 cm/s. Such a high speed requires a fast
mirror scanning system. It should be stressed that this
acquisition time is to capture the holographic data, i.e.,
the entire three-dimensional information. For example,
if the sample is 100 focal depths thick (280 ␮m at NA
⬃ 0.6), the three-dimensional data captured in 0.8 s are
512 ⫻ 512 pixels ⫻ 100 slices ⫽ 2.62 ⫻ 107 voxels. But
one must keep in mind that no real sectioning has been
done, as in confocal systems. The method adopted for
scanning the specimen may itself limit the scanning rate.
Mechanical stages, for example, are limited to speeds ⬍10
cm/s. Thus scanning a 500-␮m ⫻ 500-␮m specimen at
four samples per resolution element (0.5 ␮m) takes
⬃(8 ⫻ 500 lines) ⫻ (5 ms/line) ⬃ 20 s. With mirror scanners comparable with those used in confocal scanning microscopes, the acquisition time of the same hologram is on
the order of 1 s. Another important factor that may limit
the signal-to-noise ratio is the dynamic range of the detector. With weakly scattering or fluorescent specimens,
the modulation depth of the signal is expected to be small.
In such cases the signal-to-noise ratio may be limited by
the detector dynamic range or the digitization noise. The
analysis of these limitations requires case-by-case
studies.
390
J. Opt. Soc. Am. A / Vol. 17, No. 3 / March 2000
5. SUMMARY
We have analyzed the imaging properties of a scanning
holographic system and compared it with conventional
imaging. The salient points are the following. Varying
the detection mode from pinhole detection to spatially integrating detection allows one to vary the coherence property of the imaging process from linear in amplitude to
linear in intensity. The method is thus suitable for imaging phase objects and relief surfaces, as well as for obtaining incoherent (and thus speckle-free) holograms and
for imaging fluorescent specimens.
The three-dimensional point-spread function of the system was calculated as a function of two parameters,
namely, the numerical aperture and the Fresnel number
of the scanning pattern. The results are valid beyond the
paraxial approximation and are presented in terms of dimensionless coordinates scaled to the theoretical resolution limits of the system. This allows for direct comparisons of systems with different numerical apertures. As
one might expect from theoretical considerations, it is
found that the amplitude distribution of the point-spread
function is nearly independent of the Fresnel number of
the scanning pattern, within the domain of validity of the
field representation by a Debye integral. The Fresnel
number can thus be used as a free design parameter, the
variation of which leaves the resolution unaffected.
For low and modest numerical apertures, the pointspread function is found to be nearly identical with that of
an aberration-free conventional imaging system of equal
numerical aperture. For higher numerical apertures, the
holographic point-spread function, in reduced coordinates, exhibits improved transverse and axial resolution
limits, compared with the Airy pattern.
An attractive feature of scanning holography is that
the aberrations of the scanning pattern can easily be canceled out by reconstructing the hologram digitally with an
appropriate conjugate pattern. This process leads to an
ideal system transfer function equal to unity up to the
cutoff frequency, independent of the aberrations of the
scanning beam. This is valid for the system operating in
either the coherent or the incoherent mode. Scanning
holography lends itself well to postprocessing operations,
since the images are reconstructed digitally. A few such
possibilities were mentioned, and some practical issues
were considered.
Indebetouw et al.
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ACKNOWLEDGMENTS
Ting-Chung Poon and Taegeun Kim acknowledge the
financial support of the National Science Foundation
(grant ECS-9810158) for parts of this work.
Address correspondence to Guy Indebetouw at the location on the title page or by e-mail, gindebet@vt.edu.
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