Harmonic holography: a new holographic principle.
ABSTRACT The process of second harmonic generation (SHG) has a unique property of forming a sharp optical contrast between noncentrosymmetric crystalline materials and other types of material, which is a highly valuable asset for contrast microscopy. The coherent signal obtained through SHG also allows for the recording of holograms at high spatial and temporal resolution, enabling whole-field four-dimensional microscopy for highly dynamic microsystems and nanosystems. Here we describe a new holographic principle, harmonic holography (H(2)), which records holograms between independently generated second harmonic signals and reference. We experimentally demonstrate this technique with digital holographic recording of second harmonic signals upconverted from an ensemble of second harmonic generating nanocrystal clusters under femtosecond laser excitation. Our results show that harmonic holography is uniquely suited for ultrafast four-dimensional contrast microscopy.
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ABSTRACT: Optical microscopy has played a critical role for discovery in biomedical sciences since Hooke's introduction of the compound microscope. Recent years have witnessed explosive growth in optical microscopy tools and techniques. Information in microscopy is garnered through contrast mechanisms, usually absorption, scattering, or phase shifts introduced by spatial structure in the sample. The emergence of nonlinear optical contrast mechanisms reveals new information from biological specimens. However, the intensity dependence of nonlinear interactions leads to weak signals, preventing the observation of high-speed dynamics in the 3D context of biological samples. Here, we show that for second harmonic generation imaging, we can increase the 3D volume imaging speed from sub-Hertz speeds to rates in excess of 1,500 volumes imaged per second. This transformational capability is possible by exploiting coherent scattering of second harmonic light from an entire specimen volume, enabling new observational capabilities in biological systems.Proceedings of the National Academy of Sciences 10/2013; · 9.74 Impact Factor
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ABSTRACT: In this paper, the detection of a small reflector in a randomly heterogenous medium using second-harmonic generation is investigated. The medium is illuminated by a time-harmonic plane wave at frequency omega. It is assumed that the reflector has a non-zero second-order nonlinear susceptibility, and thus emits a wave at frequency two omega in addition to the fundamental frequency linear scattering. It is shown how the fundamental frequency signal and the second-harmonic signal propagate in the medium. A statistical study of the images obtained by migrating the boundary data is performed. It is proved that the second-harmonic image is more stable with respect to medium noise than the one obtained with the fundamental signal. Moreover, the signal-to-noise ratio for the second-harmonic image does not depend neither on the second-order susceptibility tensor nor on the volume of the particle.06/2013;
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ABSTRACT: We tackle the problem of information recovery and imaging through scattering microfluidic chips by means of digital holography (DH). In many cases the chip can become opalescent due to residual deposits settling down the inner channel faces, biofilm formation, scattering particle uptake by the channel cladding or its damaging by corrosive substances, or even by condensing effect on the exterior channels walls. In these cases white-light imaging is severely degraded and no information is obtainable at all about the flowing samples. Here we investigate the problem of counting and estimating velocity of cells flowing inside a scattering chip. Moreover we propose and test a method based on the recording of multiple digital holograms to retrieve improved phase-contrast images despite the strong scattering effect. This method helps, thanks to DH, to recover information which, otherwise, would be completely lost.Optics Express 10/2013; 21(20):23985-23996. · 3.55 Impact Factor
Research Papers in Physics and Astronomy
Martin Centurion Publications
University of Nebraska - LincolnYear
Harmonic holography: a new holographic
∗California Institute of Technology, firstname.lastname@example.org
†niversity of Nebraska - Lincoln, email@example.com
‡´Ecole Polytechnique F´ ed´ erale de Lausanne
This paper is posted at DigitalCommons@University of Nebraska - Lincoln.
Harmonic holography: a new holographic principle
Ye Pu,1,* Martin Centurion,1,2and Demetri Psaltis1,3
1Department of Electrical Engineering, California Institute of Technology, Pasadena, California 91125, USA
2Laboratory for Attosecond and High-Field Physics, Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1,
D-85748 Garching, Germany
3École Polytechnique Fédérale de Lausanne (EPFL), Laboratoire d’optique, STI IOA LO, BM 4102 (Bâtiment BM),
Station 17, CH-1015 Lausanne, Switzerland
*Corresponding author: firstname.lastname@example.org
Received 7 May 2007; accepted 5 July 2007;
posted 26 September 2007 (Doc. ID 82771); published 29 November 2007
The process of second harmonic generation (SHG) has a unique property of forming a sharp optical
contrast between noncentrosymmetric crystalline materials and other types of material, which is a highly
valuable asset for contrast microscopy. The coherent signal obtained through SHG also allows for the
recording of holograms at high spatial and temporal resolution, enabling whole-field four-dimensional
microscopy for highly dynamic microsystems and nanosystems. Here we describe a new holographic
principle, harmonic holography ?H2?, which records holograms between independently generated second
harmonic signals and reference. We experimentally demonstrate this technique with digital holographic
recording of second harmonic signals upconverted from an ensemble of second harmonic generating
nanocrystal clusters under femtosecond laser excitation. Our results show that harmonic holography is
uniquely suited for ultrafast four-dimensional contrast microscopy.
090.0090, 090.1000, 180.6900, 190.4180, 190.7220.
© 2008 Optical Society of America
One of the grand open challenges in modern science
is to probe and understand the mechanism and dy-
namics of biological processes inside living cells at the
molecular level. Besides the highly complex three-
dimensional (3D) structures spanning a large range
of length scale , living organisms by their nature
are highly dynamic: Molecular processes such as pro-
tein, DNA, and RNA conformation changes take
place on a time scale ranging from 100 fs to 100 s [2,3]
while the organisms move and metabolize. Essen-
tially, probing these processes requires taking 3D
microscopic pictures in a rapid succession with high
sensitivity and specificity, i.e., a four-dimensional
(4D) microscope is needed.
Since holography was invented 60 years ago , an
aberration-free, 3D microscope has long been hoped
for. In fact, soon after Leith and Upatniek demon-
strated the first 3D image by holography , holo-
graphic microscopy was proposed . With the recent
advent of digital holography [7,8], holographic 3D
microscopy received renewed and rapidly increasing
attention [9–16]. Holography is a “whole-field” tech-
nique, which means recording all 3D pixels simul-
taneously in one laser shot without scanning, an
extremely valuable asset for biomedical imaging.
When combined with modern high-repetition rate
pulsed laser and fast imaging devices, a time se-
quence of consecutive 3D images can be captured,
forming a 4D microscope. Indeed, successful 4D ho-
lographic imaging has been achieved recently in the
context of fluid velocity measurement .
Despite the technical advancements, holographic
microscopes are not widely deployed in biomedical
research because of the lack of specificity. A holo-
graphic microscope would capture all scattering
entities in the viewing field faithfully but indiscrim-
inately. On the other hand, the signals of interest
(often from small nanostructures like protein mol-
ecules) are usually very weak and buried in the
strong ambient scatterings from much larger or-
A key to achieving the specificity required for mo-
lecular biomedical imaging is creating a contrast be-
© 2008 Optical Society of America
1 February 2008 ? Vol. 47, No. 4 ? APPLIED OPTICS A103
This paper was published in Applied Optics 47, A103 (2008) and is made available as an electronic reprint with the permission of
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Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to
penalties under law.
tween the useful signal and the ambient scatterings.
Contrast imaging is routinely achieved through tag-
ging the molecule or nanostructure of interest with
fluorescent agents, such as fluorescent dyes, green
fluorescent proteins, and quantum dots. By convert-
ing the light signal into a different frequency, the
unwanted ambient scattering can be easily removed
with proper optical filters. However, the incoherent
fluorescence signal lacks the capability of 3D repre-
sentation, and the finite scanning speed constrains
these techniques to mostly two dimensions. Attempts
to extend fluorescence microscopy into three spatial
dimensions over time (4D microscopy)  result in a
great sacrifice of framing time and are thus incapable
of capturing dynamic events.
Here we show that the new holographic principle
described below, namely, harmonic holography ?H2?,
provides an ideal solution to meet the ultimate need
for dynamic 4D imaging with high contrast and am-
ple spatial and temporal resolution.
In our new holographic scheme, we use the second
harmonic generation (SHG) process to achieve con-
trast imaging. When a nanocrystal of a noncen-
an inversion center) is under the excitation of an
intense optical field E???, the polarization P??? of the
nanocrystal is described by a Taylor expansion series:
???3?·E???·E???·E???? · · ·,(1)
where the coefficient ??n?is the nth-order susceptibil-
ities of the crystal material. The second-order re-
sponse, ??2?· E · E, gives rise to a radiation at exactly
twice the frequency of the pumping field. The capa-
bility of generating second harmonic radiation is
specific to materials with noncentrosymmetric crys-
talline structures only, and ??2?vanishes for all other
types of material. Therefore a sharp contrast is
formed when particles of noncentrosymmetric struc-
tures are dispersed in a medium of other species,
pumped at a fundamental frequency, and imaged at
the second harmonic frequency. In contrast to fluo-
rescence, a second harmonic signal is coherent, en-
abling holography with contrast at doubled spatial
resolution. Additionally, SHG is an ultrafast process,
permitting the probe of dynamics at a femtosecond
level. With only a few exceptions , biological cells
and tissues are incapable of generating endogenous
second harmonic emissions. Therefore the new
scheme described here provides a sound basis for a
new type of contrast microscopy with enormous po-
tential in molecular biomedical imaging.
Figure 1 illustrates the principle of harmonic ho-
frequency ? is delivered to a group of second harmonic
generating nanocrystals, which are tagged to specific
parts of the system under investigation. The nanocrys-
tals emit scattering signals at both the fundamental
(?) and the second harmonic (2?) frequencies. A band-
pass filter centered at 2? rejects the pumping and
scattered light at frequency ? but transmits the 2?
signal to the charge-coupled device (CCD) sensor. A
reference beam at frequency 2?, independently gener-
ated from the same pump laser pulse, is also delivered
tered 2? signals have a deterministic, static phase re-
lationship with the 2? reference. It is worth noting
that our scheme is fundamentally different from ho-
lography obtained in nonlinear crystals , where
contrast is not achieved.
Fast, 3D imaging presents additional challenges.
The second harmonic emissions from nanocrystals
are usually extremely weak, often only a few photons
per pulse. Furthermore, high numerical aperture
(NA) lenses and objectives are often only optimized
for one imaging plane, resulting in substantial aber-
rations when working with a volume of finite depth.
In addition, fast image sensors with small pixels re-
quired in H2are typically much noisier than slower
ones with larger pixels. Therefore an implementation
of H2will seemingly encounter a serious signal to
noise ratio (SNR) problem. However, the use of holog-
raphy is able to address these issues simultaneously.
Experiments [12,21] have shown that objective aber-
rations can be numerically canceled or compensated
in holography to obtain diffraction-limited imaging
performance. Moreover, the reference used in holog-
raphy serves as a coherent bias that is much higher
than the device noise in the imaging sensor, leading
to a shot noise-limited performance in the low photon
count regime .
Figure 2 shows the experimental setup for H2. Fem-
tosecond laser pulses of 2 mJ energy, 150 fs full width
at half-maximum (FWHM) duration, and 810 nm
wavelength (repetition rate 10 Hz) are split into a
pump and a reference beam at beam splitter BS1.
tense pump laser pulse of frequency ? is sent to a group of SHG
tagging nanocrystals that scatter both ? and 2? signals. The band-
pass filter rejects the scatterings and the pump pulse at ? while
passing signals at 2?. An independently frequency-doubled refer-
ence interferes with the 2? signals at the CCD camera and forms
a hologram. Structures that are not capable of generating second
harmonic signals only scatter at ? and will not be recorded on the
(Color online) Principle of harmonic holography. An in-
A104APPLIED OPTICS ? Vol. 47, No. 4 ? 1 February 2008
Approximately 1 mJ of the energy is delivered to the
sample containing randomly dispersed SHG imaging
targets. After the boost in intensity through the re-
versed 3:1 telescope L3 and L4, the pump intensity at
the sample site is approximately 1 ? 1011W?cm2.
While the intense pump beam at the fundamental
frequency is rejected by the filter F2, the weak second
harmonic radiations form the samples (the object
wave) are picked up by the aspheric objective ?N.A.
? 0.5? at a magnification of 25?. The reference, in-
dependently frequency doubled through a ?-barium
borate (BBO) crystal, meets and interferes with the
signal at the beam splitter BS2 to form holograms
that are captured by the CCD camera. The optical
path lengths of the two second harmonic waves are
carefully matched by the variable delay line.
We use clusters of 100 nm BaTiO3nanocrystals
as the imaging target. Although tetragonal phase
BaTiO3has an excellent nonlinear coefficient ??2?,
the unprocessed nanocrystals are in a centrosymmet-
ric cubic crystal structure and are unable to produce
second harmonic responses. To obtain tetragonal
phase crystals, we sinter the nanocrystals at 1000 °C
for 1 h and rapidly cool them in deionized water.
While the sintering converts the crystal structures
into tetragonal, large clusters result from such a pro-
cess with a broad variation in size, as revealed by the
scanning electronic microscopy (SEM) images shown
in Fig. 3. For better control of the number density
of particles, we collect the larger clusters through
sedimentation and disperse them randomly on a mi-
croscope coverslip. Because of the “random phase-
matching” condition, we expect to see specklelike,
randomly located spots within the extent of each clus-
ter at the second harmonic frequency.
The position of the sample is adjustable. We cap-
ture a “direct” (on-focus) image when the reference is
shut off and the sample is positioned at the imaging
focal plane. A hologram, on the other hand, is ob-
tained when the sample is moved off focus by a short
distance (30 ?m in this experiment) and the refer-
ence is turned on.
The camera used in the experiment is a thermal-
electrically cooled CCD camera with 6.8 ?m pixels
and 7e?read noise. The quantum efficiency (QE) of
the image sensor is approximately 50% at 400 nm
wavelength. The digitization sensitivity of the cam-
era is factory calibrated to 3e?per digit. The camera
is cooled down to 35 °C below the ambient tempera-
ture throughout the experiment, although the dark
current does not play a significant role in the noise
figure. We set the intensity of the hologram reference
to at least 100 times stronger than the read noise of
the CCD so that the read noise does not contribute
significantly to the overall noise.
The digital image captured by the CCD camera is
reconstructed numerically to obtain the electric field
of the objective wave at the sample plane. Because
the reference is introduced at near-zero angle with
respect to the optical axis, we treat the hologram as
in-line. The reconstruction algorithm is essentially a
convolution between the holographic image intensity
and the system impulse function :
???, ?, ???h?x, y??g?x, y; ??, (2)
where ???, ?, ?? is the reconstructed 3D image inten-
sity at object coordinate ??, ?, ??, h?x, y? is the inten-
sity of the holographic image captured from the CCD
at hologram coordinate ?x, y?, and g?x, y; ?? is the sys-
tem impulse function at object coordinate ? along the
optical axis. This convolution is implemented as a
sequence of fast Fourier transforms (FFTs) and in-
verse FFTs. In our configuration, the aspheric objec-
tive is not designed to accommodate the thick filter
F2 (1.7 mm thickness), which causes significant
spherical aberration in the image. To compensate for
such aberrations, we computationally generate the
impulse function g?x, y; ?? through ray tracing with
precise design parameters published by the manufac-
turer. With this approach, we are able to completely
phy. Femtosecond laser pulses of 2 mJ energy and 150 fs pulse
width FWHM is split into a pump and a reference beam at BS1.
After a variable delay line, the pump carrying the major portion of
the energy is shrunk by 3:1 with a reverse telescope (L3, L4) for
higher intensity and sent to the sample containing SHG nanocrys-
tal clusters. The second harmonic signals scattered from the
nanocrystal clusters are collected by the aspheric objective and
steered to the CCD camera. The reference, frequency doubled
through a BBO crystal, interferes with the signal at the CCD plane
and forms a hologram. BS1, BS2, beam splitters; L1, L2, lenses;
ASP OBJ, aspheric objective; M1–M8, mirrors; F1–F4, filters; S,
(Color online) Experimental setup for harmonic hologra-
clusters are obtained by sintering 100 nm cubic phase BaTiO3
nanocrystalsat1000 °C foronehourandrapidlycoolingthenanoc-
rystals in deionized water. The sintering converts the crystal struc-
ture from cubic phase to tetragonal phase while forming clusters
with large size variations.
SEM image of the BaTiO3nanocrystal cluster sample. The
1 February 2008 ? Vol. 47, No. 4 ? APPLIED OPTICSA105
cancel the aberration effect and obtain a recon-
structed image with near diffraction-limited perfor-
As indicated in previous sections, imaging with sec-
ond harmonic signals often involves a very low pho-
ton count. Thus an accurate estimate of signal levels
for quantifying the SNR becomes difficult because of
the intensity fluctuations due to shot noise. Since the
signal level is directly proportional to the number of
pulses integrated, we estimate the signal intensity at
an integration of n1pulses through an image of n2
pulses by I1? n1I2?n2, where n2? ? n1. An on-focus
(direct) second harmonic image of the nanocrystal
clusters is shown in Fig. 4. The image is acquired
with an integration of 100 pulses and serves as an
accurate measurement of signal intensity. For the
ease of discussions, we assign each major object with
a unique identification number.
To compare the performance of direct and holo-
graphic imaging, we capture direct and holographic
images under identical conditions at integrations of
1, 5, and 10 pulses, which are shown in Fig. 5. Here
we note that the experiments at different integration
times are to investigate the imaging characteristics
as a function of signal intensity rather than the in-
tegration time itself. The femtosecond laser pulses
exclude the use of spatial filters and result in imper-
fections in the reference. Consequently, although the
axial location (z coordinate) of the sample is accu-
rately measured, we find it necessary to numerically
reconstruct a stack of images along the z axis within
a 20 ?m range to find the best focus (location with the
highest image intensity) for the holographic recon-
struction. Because of the large dynamic range, the
as-is image reconstructed from the hologram cannot
be visualized with necessary clarity. Thus for visual-
ization purpose, we histogram stretch every picture
in Fig. 5 so that the darkest 0.01% pixels are black
and the brightest 0.01% pixels are white. The effect of
aberration compensation is clearly visible, as the neb-
ulous images in the direct picture become much
sharper in the reconstructed holographic image, re-
vealing the fine structure of the nanocrystal clusters
through a random phase-matching effect. Figure 6
shows a magnified image of object 7, which is the
smallest object among all identified objects. The re-
markable compactness of the images in Figs. 6(d)–
6(f) suggests that the reconstruction is true focal
images instead of speckles from off-focus objects. Fig-
ure 6 also shows a clear gain in SNR through refer-
ence biasing and aberration compensation.
Walking through the reconstructed image stack
along the z axis reveals the axial intensity profile of
the holographic reconstruction, which is plotted in
Fig. 7. The plot is obtained by averaging the normal-
ized image intensity as a function of the z coordinate
over 50 randomly picked bright pixels at local max-
ima in Fig. 5(f). The position at which Fig. 5 is ob-
tained is set to z ? 0. The inset in Fig. 7 shows a plot
for a typical individual intensity profile. Figure 7 con-
firms that the images shown in Fig. 5 are true focal
images. On average, the depth-of-focus of the recon-
structed images is approximately 4.5 ?m. The depth-
of-focus of an individual, small object is roughly
2.5 ?m. Because of the size of the nanocrystal clus-
slightly different z coordinates, resulting in a broad-
ened mean intensity profile.
Because of the obvious difficulty in defining an im-
age center for each object, we define the signal inten-
sity for every image spot as the mean intensity for
pixels with intensity above 50% of the local intensity
maximum. As stated previously, to minimize the in-
fluence of the shot noise and to obtain accurate mea-
surements, this procedure is performed only on an
image of 100-pulse integration, and the intensity is
then linearly scaled back to the proper integration of
1, 5, and 10 pulses. This approach is applied to both
direct and holographic images to obtain signal inten-
rectly recovered from the hologram recorded with ref-
erence intensity IR.
We quantify the gain of signal intensity from ab-
erration compensation, ? ? ?Ii
approach of ray tracing and wave propagation in the
Fresnel regime. The value of ? is a function of the
objective NA, since a larger NA results in higher
aberrations and thus a higher gain when the aberra-
tion is canceled. In this particular configuration
where a NA of 0.45 is achieved (limited by the size of
the objective), our numerical investigations show
that the gain in signal intensity through aberration
compensation is ? ? 1.63. If we increase the objective
NA to 0.63 while all other parameters remain con-
stant, the gain becomes ? ? 3.0.
Taking the shot noise into account, we now define
the SNR for the direct images as
?H???IRfrom the mean intensity ?Qi
?H?? ? ?Qi
?D??, by a joint
ters obtained directly from the CCD at an integration of 100 laser
pulses. The objects are numbered for ease of discussion.
Second harmonic on-focus image of the nanocrystal clus-
A106 APPLIED OPTICS ? Vol. 47, No. 4 ? 1 February 2008
(b), (d), (f) Holographic reconstruction of 1, 5, and 10 pulses, respectively. The effect of aberration compensation is clear. For consistent
visualization, all images are histogram stretched so that the darkest 0.01% pixels are black and the brightest 0.01% pixels are white. The
scale bar is 50 ?m.
Comparison between direct imaging and holographic reconstruction. (a), (c), (e) Direct imaging of 1, 5, and 10 pulses, respectively.
1 February 2008 ? Vol. 47, No. 4 ? APPLIED OPTICS A107
where ? is the digitization sensitivity of the CCD
camera in photons per digit, and N?D?is the device
noise after digitization in the direct image. Here we
ignore the effect of QE, which only affects the abso-
lute sensitivity but not the SNR at the same level of
the read noise of the CCD camera dominates N?D?.
The bias provided by the reference beam in H2
renders the device noise negligible, and as such, we
obtain a shot-noise-limited device for all signal levels.
Therefore the SNR for the holographic reconstruction
is simply (again ignoring QE)
The overall gain in the SNR through holography over
direct imaging is then
?Ii?D? ?. (5)
Given that ?Ii
photons digit, and N?D?? 2.0 in the camera used in
this experiment, the measured SNR of the identified
objects is listed in Table 1. The fluctuations in GSNR
for each individual object are a result of the extended
object size, whose scattering wavefront (at second
harmonic frequency) deviates from that of a spherical
wave . In Fig. 8(a), we plot the SNR for both direct
as a function of the number of integrated pulses. The
empty triangles represent the measured data points,
solid triangles with lines show the mean values of the
measurement, and the dashed lines are theoretical
predictions from Eqs. (3) and (4). The gain in SNR,
GSNR, is plotted in Fig. 8(b) as a function of the num-
?D?? ? 2.5 photons?pixel?pulse, ? ? 3
holographic reconstruction. The profile is obtained by averaging
the normalized image intensity as a function of the z coordinate
over 50 randomly picked bright pixels at local maxima in Fig. 5(f).
The inset shows a typical profile of an individual image. The solid
dots are experimental data, and the lines are a visual guide. On
average, the depth-of-focus of the reconstructed images is approx-
imately 4.5 ?m. The depth-of-focus of an individual, small object is
roughly 2.5 ?m. The slightly broadened mean profile is due to the
size of the nanocrystal clusters.
(Color online) Normalized axial intensity profile of the
of 1, 5, and 10 pulses, respectively. The scale bar is 5 ?m.
Magnified images of object 7. (a), (b), (c) Direct imaging of 1, 5, and 10 pulses, respectively. (d), (e), (f) Holographic reconstruction
A108 APPLIED OPTICS ? Vol. 47, No. 4 ? 1 February 2008
ber of integrated pulses, where empty circles are
measurement data points, solid circles with lines are
the mean value, and the dashed line is the theoretical
GSNRcalculated from Eq. (5). Due to the uncontrolled
air flow in the laboratory environment, the value of
GSNRat longer integration falls slightly below what is
predicted by Eq. (5).
Second harmonic generation provides a unique
means to achieve contrast in the coherent domain,
enabling the use of holography in contrast imaging.
We show that a new holographic principle, harmonic
holography ?H2?, is promising for constructing an ul-
trafast four-dimensional contrast microscope with
high spatial and temporal resolution. H2records ho-
lograms using second harmonic signals scattered
from SHG nanocrystals and an independently gener-
ated second harmonic reference. Our experiments
show that the technique of H2has unique advantages
over direct imaging, including numerical aberration
compensation and device noise canceling. The exper-
imental data for the SNR performance is in very
good agreement with the theoretical predictions. H2
has proved to be a powerful technique to achieve
aberration-free, shot-noise-limited performance with
low photon count signals, laying a sound basis for
molecular biomedical imaging applications. The op-
timal performance can be achieved through the use of
high NA objectives (which bear significant aberra-
tions when used to image a volume) and small pixel,
fast frame rate cameras (which typically have high
device noise). This principle can also be extended to
use other coherent optical process, such as sum fre-
quency generation, difference frequency generation,
and coherent anti-Stokes Raman scattering.
This work is supported by the DARPA Center for
Optofluidic Integration at Caltech. The authors
James Adleman for extremely helpful discussions.
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