Digital holographic microscope for measuring three-dimensional particle distributions and motions

doi:10.1364/AO.45.003893

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Digital holographic microscope for measuring three-dimensional particle distributions and motions

Jian Sheng, Edwin Malkiel, and Joseph Katz »View Author Affiliations

Applied Optics, Vol. 45, Issue 16, pp. 3893-3901 (2006)
http://dx.doi.org/10.1364/AO.45.003893

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▸ Article Outline
– Abstract
1. Introduction
2. Methodology
3. Results and Discussions
4. Considerations Related to Bandwidth Limitations
5. Summary and Conclusions
– References and links

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Abstract

Better understanding of particle–particle and particle–fluid interactions requires accurate 3D measurements of particle distributions and motions. We introduce the application of in-line digital holographic microscopy as a viable tool for measuring distributions of dense micrometer $(3.2 μm)$ and submicrometer $(0.75 μm)$ particles in a liquid solution with large depths of $1 - 10 mm$ . By recording a magnified hologram, we obtain a depth of field of $\sim 1000$ times the object diameter and a reduced depth of focus of approximately 10 particle diameters, both representing substantial improvements compared to a conventional microscope and in-line holography. Quantitative information on depth of field, depth of focus, and axial resolution is provided. We demonstrate that digital holographic microscopy can resolve the locations of several thousand particles and can measure their motions and trajectories using cinematographic holography. A sample trajectory and detailed morphological information of a free-swimming copepod nauplius are presented.

1. Introduction

There is growing interest in understanding microscale biophysical processes such as the kinematics and dynamics of swimming microorganisms, e.g., bacteria, dinoflagellates, or nauplii, and their interactions with surrounding fluids.[1

D. DeAngelis and L. Gross, Individual-Based Models and Approaches in Ecology (Chapman & Hall, 1992).

, 2

W. Fennel and T. Neumann, Introduction to the Modelling of Marine Ecosystems , Vol. 72 of the Elsevier Oceanography Series (Elsevier, 2004).

, 3

W. Fennel and T. Osborn, A Unifying Framework for Marine Ecological Model Comparison Deep Sea Research II, Rep. DSR-WTD-06, 2004.

] Direct observations of such processes require suitable tools that are capable of resolving both temporal and spatial scales at the appropriate levels. The readily available candidate is the optical microscope. However, as shown in Fig. 1, as the power of the microscope increases and the lateral resolution (

1.22 λ / NA

, where λ is the wavelength and NA is the numerical aperture of the entire optical system) improves, the field of view and depth of field decrease nonlinearly to a very thin layer. For example, increasing the power from 10× to 40× reduces the theoretical depth of field from 12 to

2 μm

, greatly limiting the size of the resolvable volume.

Holography, on the other hand, is capable of recording a 3D volumetric field on a single plane (hologram plane) and later reconstructing it. With the recent development of in- line digital holography for particulate flows,[4

E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206, 3657–3666 (2003). [CrossRef] [PubMed]

, 5

G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827–833 (2003). [CrossRef] [PubMed]

] it is now possible to record a hologram on a digital medium and then reconstruct the sample volume numerically.[6

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002). [CrossRef]

, 7

J. H. Milgram and W. C. Li, “Computational reconstruction of images from holograms,” Appl. Opt. 41, 853–864 (2002). [CrossRef] [PubMed]

] However, previous implementations of digital holography have been hindered by the spatial resolution of the digital media, i.e., the pixel resolution of a camera. To overcome this limitation, Xu et al.[8

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A. 98, 11301–11305 (2001). [CrossRef] [PubMed]

] used a point laser source for illuminating the sample volume and recorded the hologram on a lensless camera. They successfully resolved

(\sim 60 μm)

D. brightwellii in great detail, and subsequently used this method for 4D (space and time) tracking of microstructures and organisms.[9

L. Xu, X. Peng, J. Miao, and A. K. Asundi, “Studies of digital microscopic holography with applications to microstructure testing,” Appl. Opt. 40, 5046–5051 (2001). [CrossRef]

, 10

W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett. 28, 164–166 (2003). [CrossRef] [PubMed]

] Point source and lensless recording do seem to improve the resolution in all directions, but this approach still has a limited depth of field, and is inherently limited by the pixel resolution of the recording medium. Several recent studies[11

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, “Parameter-optimized digital holographic microscope for high-resolution living-cell analysis,” Appl. Opt. 43, 6536–6544 (2004). [CrossRef] [CrossRef]

, 12

T. Colomb, F. Durr, E. Cuche, P. Marquet, H. G. Limberger, R. P. Salathe, and C. Depeursinge, “Polarization microscopy by use of digital holography: application to optical-fiber birefringence measurements,” Appl. Opt. 44, 4461–4469 (2005). [CrossRef] [PubMed]

, 13

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15, 529–539 (2004). [CrossRef]

, 14

F. Dubois, M. L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. 43, 1131–1139 (2004). [CrossRef] [PubMed]

, 15

P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, “Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time,” Opt. Lett. 28, 1257–1259 (2003). [CrossRef] [PubMed]

, 16

P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. , 30 468–470 (2005). [CrossRef] [PubMed]

, 17

L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Lett. 29, 1132–1134 (2004). [CrossRef] [PubMed]

, 18

F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, J. C. Legros, and P. Kischel, “Pattern recognition with a digital holographic microscope working in partially coherent illumination,” Appl. Opt. 41, 4108–4119 (2002). [CrossRef] [PubMed]

] utilized a Mach–Zehnder interferometer configuration with a coherent or partially coherent light source to record intensity and phase-contrast digital microscopic holograms of a 3D object on a glass slide. They successfully demonstrated the ability of digital holographic microscopy (DHM) to resolve a microstructure of the phase object at high spatial resolution, often down to hundreds of nanometers. These studies focused on a few isolated stationary objects located within micrometers from the imaging plane. To the best of our knowledge, DHM has never been applied to examine the spatial distribution and velocity of a dense cloud of particles with an extended depth.

In this paper we combine in-line digital holography and a conventional microscope objective in order to circumvent the obstacles associated with the limited resolution of a digital recording medium. Using the same setup as an optical microscope, we replace the light source with a collimated, coherent laser beam, and record a stream of magnified holograms on a CCD camera. The 3D fields can be reconstructed from these magnified holograms at almost the same resolution as from optical microscopes. Reconstructed holograms of sample volumes with depths of

1 - 10 mm

, containing particles ranging in size between 0.75 and

3 μm

, demonstrate the efficacy of the DHM as a viable means of extending the depth of field of a microscope by almost 3 orders of magnitude. Detailed data on the spatial resolution in all directions confirms that a DHM maintains the lateral resolution of the microscope and substantially reduces the depth of focus, enabling reconstruction of the 3D coordinates of thousands of particles in a dense cloud. Finally, we also demonstrate the feasibility of using this technique to study particle dynamics by recording a time series of particle traces and the trajectory of a nauplius.

2. Methodology

2A. Optical Setup

As illustrated in Fig. 2, the optical setup is very similar to a conventional transmission light microscope, but instead of using white light we replace it with a coherent laser beam. In the current setup, we spatially filter a

3 mW

He–Ne laser beam by using a

25 μm

pinhole, expand and collimate the beam to

30 mm

diameter, and then illuminate the sample volume. Since the resulting intensity,

0.33 mW / {cm}^{2}

, is still too high, we use a variable neutral density (ND) filter to further attenuate the beam. In most of the tests, a filter of

ND = 1

is used, reducing the illumination intensity on the specimen to

\sim 30 μ W / {cm}^{2}

. A bright-field microscope objective with proper tube length is used to image the optical field (hologram) onto the digital recording medium (CCD sensor). Note that the object plane is located outside the sample volume. A 15 frames∕s digital CCD camera (Kodak ES4.0) with a 2048 × 2048 pixels sensor and 7.4 μm pixel pitch is used as the recording medium. The theoretical background provided in Subsection 2.B proves that the microscope objective enables us to overcome the limited pixel resolution. In fact, it is the magnification of objective that determines the resolution of the recording and reconstruction process.

2B. Analysis of Microscopic Holography

A hologram is a record of interference between light scattered from objects, e.g., micrometer or submicrometer particles, and a reference beam with known phase distribution.[19

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

] One can represent the optical field at the hologram plane (x, y, 0) as

U_{H} (x, y) = A e^{j k_{r} n_{H}} + \sum_{i} \int \int a_{i} (x', y'; z_{i}) h_{z} (x - x', y - y'; z_{i}) d x' d y',

(1)

where

k_{r}

is the propagation vector of the reference beam and

n_{H}

is the norm vector of the hologram plane. The first term represents the optical field of the reference beam, where the phase accounts for its angle with the scattering light, assuming that the hologram is perpendicular to the scattering axis. In the following analysis we assume that this angle is zero. The second term is the superposition of light scattered from discrete particles located at a distance z_i from the hologram plane, and produce (by being illuminated) fields with local distributions of

a_{i}

(x', y')

. Thus each particle is considered as a superposition of point sources, whose individual fields are

h_{z} (x, y; z_{i})

. Using a paraxial approximation for particles much smaller than

z_{i}

h_{z} (x - x', y - y'; z_{i}) = \frac{1}{j λ z_{i}} \exp {j \frac{k}{2 z_{i}} [{(x - x')}^{2} + {(y - y')}^{2}]} .

(2)

If the scattering is diffraction dominated, as in in-line holography, each particle can be considered as a 2D aperture with a shape equal to its cross section normal to the incident light. Thus scattering from an individual particle is a convolution of a 2D aperture with the impulse response function [Eq. (2)]. The resulting interference intensity on the hologram plane,

I_{z} (x, y)

, is

I_{z} (x, y) = U_{H} U_{H} *

= A^{2} - A \sum a * (x, y) \otimes h_{z} * (x, y) - A \sum a (x, y) \otimes h_{z} (x, y) + {\sum \sum | a (x, y) \otimes h_{z} (x, y) |}^{2},

(3)

where ⊗ indicates a convolution integral. To determine the effect of the microscope objective, we model its compound lens system as a perfect thin lens. The optical field at distance

d_{i}

behind the lens, resulting from an optical disturbance,

U_{0} (x_{o}, y_{o}, d_{0})

, where

d_{0}

, the object distance before the lens, is

U_{i} (x_{i}, y_{i}; d_{i}) = \int \int h_{l} (x_{i}, y_{i}; x_{o}, y_{o}) U_{o} (x_{o}, y_{o}) d x_{o} d y_{o},

where

h_{l} (x_{i}, y_{i}; x_{o}, y_{o}) = \frac{1}{M} δ (\frac{x_{i}}{M} + x_{o}, \frac{y_{i}}{M} + y_{o}) \exp [j \frac{k}{2 M^{2} d_{0}} (x_{i}^{2} + y_{i}^{2})] \exp [j \frac{k}{2 d_{i}} (x_{o}^{2} + y_{o}^{2})],

and

M = d_{i} / d_{0}

is the magnification. Replacing

U_{o} (x_{o}, y_{o})

with

U_{H} (x_{o}, y_{o})

and performing the integration, the optical field generated by the hologram at the image plane is

U_{i} (x_{i}, y_{i}, d_{i}) = \frac{1}{M} U_{H} (- \frac{x_{i}}{M}, - \frac{y_{i}}{M}) \exp [j \frac{k}{2 M^{2} d_{0}} (x_{i}^{2} + y_{i}^{2})] \exp [j \frac{k}{2 d_{i}} (x_{o}^{2} + y_{o}^{2})] .

(4)

Thus the image plane contains a magnified hologram plane with a phase correction that becomes unity when the magnification is sufficiently large. The intensity distribution in the image plane simply becomes a magnified hologram,

I_{i} (x_{i}, y_{i}) = \frac{1}{M^{2}} U_{H} (- \frac{x_{i}}{M}, - \frac{y_{i}}{M}) U_{H} * (- \frac{x_{i}}{M}, - \frac{y_{i}}{M}),

(5)

which contains the four contributors presented in Eq. (3). This true magnified hologram enables us to drastically relax the spatial resolution requirement of the recording medium. Furthermore, we can use the magnification as a means of matching the desired resolution with that of the recording medium. As shown in this paper, the magnified holograms have little difficulty in resolving submicrometer particles and following their trajectories in space.

The magnified instantaneous 3D particle field,

φ_{p} (x, y; z)

, is reconstructed numerically using the Fresnel diffraction formula, i.e.,

φ_{p} (x, y; z) = I_{i} (x, y) \otimes h_{z} (x, y; z),

(6)

I_{p} (x, y; z) = φ_{p} φ_{p} * .

(7)

Note that the distance between a reconstructed image and the image plane is equal to the distance between the original plane in the sample volume and the object plane of the microscope as long as the wavelength of recording and reconstruction remains the same. Since Eq. (6) is a convolution of the magnified hologram with an impulse response function, both

φ_{p}

and the irradiance,

I_{p} (x, y; z)

, can be computed efficiently in the Fourier space, as we do in the present study.

Several simple methods can be used for improving the reconstructed image quality. For example, one can reduce the contamination generated by fixed specks on windows or by nonuniform illumination by subtracting a time average of a series of holograms from each instantaneous hologram. A similar effect can be achieved by recording and subtracting a background hologram that does not contain particles. Both methods greatly reduce the background nonuniformities, preserving only the records of interference patterns generated by moving particles.

3. Results and Discussions

3A. Sample Images and Three-Dimensional Resolution

To demonstrate the capability of in-line digital holographic microscopy, this paper focuses on its advantages over a conventional microscope and over lensless in-line holography. Using 10×, 18×, and 40× objectives, we record holograms of sample volumes containing monodispersed, polystyrene spherical particles with nominal sizes of 3.189 and

0.75 μm

. The particles are dispersed in distilled water at a high concentration of approximately

2000 particles / {mm}^{3}

. The dimensions of the rectangular container are

10 mm \times 30
                         
                        mm \times 10
                         
                        mm

(the latter one is depth), but the actual depth of liquid with particles is varied between

1 and 10 mm

in order to examine various aspects of the technique. Table 1 presents all the present test conditions. For clarity, unless specified otherwise, the data and sample images presented hereafter refer to holograms recorded with a sample depth of 1 mm. Figure 3 showcases a typical hologram and several reconstructed images of a sample containing

3.189 μm

particles recorded by using a 10× objective. A 512 × 512 pixel

(400 \times 400 {μm}^{2})

section of the original in-line hologram (2048 × 2048 pixels or

1.5 \times 1.5 {mm}^{2}

), showing interference patterns generated by particles located at various depths, is presented in Fig. 3(a). Sample images reconstructed at three different depths, obtained by using Eqs. (2), (6), and (7), are shown in Figs. 3(b)–3(d). In-focus images of particles, appearing as dark circular spots in the bright background, along with dark concentric rings associated with close-by, out-of-focus particles, can be clearly observed. Close inspections indicate that the in-focus particle traces possess optical properties that are similar to those observed under a conventional microscope, i.e., a sharp dark edge with a bright spot in the middle. A combined and∕or compressed image showing all the particles reconstructed from the sample hologram segment is presented in Fig. 3(e). To bring all the particles in a volume into focus in one plane, we assign each pixel with the lowest intensity obtained over the entire depth, i.e.,

I_{combined} (x, y) = \min_{z} I (x, y, z) .

(8)

The 3D distribution of all the particles within the reconstructed volume is presented in Fig. 3(f). To determine the 3D coordinates of each particle, we use a segmentation method discussed in Sheng et al.[20

J. Sheng, E. Malkiel, and J. Katz, “Single beam two-views holographic particle image velocimetry,” Appl. Opt. 42, 235–250 (2003). [CrossRef] [PubMed]

] The reconstructed planes are threshold-based on the local signal-to-noise ratio (SNR) at first, i.e., based on

[I (x, y, z) - {\bar{I}}_{x \in V}] / σ_{x \in V}

, where

{\bar{I}}_{x \in V}

is the mean intensity of a small volume around the particle of interest, and

σ_{x \in V}

is the standard deviation of intensity over this volume. Scanning through the images provides a list of line segments, which are combined into 2D planar blobs using a join operator. Repeating this procedure at different depths and using a join operator for the planar segments, we unite 2D segments into 3D particle traces. The location of a particle center is then estimated by using the centroid of the 3D blob. In the example presented in this paper, the sample volume contains 5679 identified particles, i.e., more than

2500 particles / {mm}^{3}

Clearly, the example demonstrates the efficacy of a DHM in extending the depth of field of a conventional microscope. However, the DHM also inherits the shortcomings of in- line holography, most prominently the elongated depth of focus. In conventional (nonmagnified) in-line holography, reconstruction of a spherical particle creates an elongated ellipsoidal image, whose length in the depth direction is typically 2 orders of magnitude higher than the lateral dimensions.[21

H. Meng and F. Hussain, “In-line recording and off-axis viewing technique for holographic particle velocimetry,” Appl. Opt. 34, 1827–1840 (1995). [CrossRef] [PubMed]

, 22

J. Zhang, B. Tao, and J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997). [CrossRef]

] This inherent problem greatly reduces one's ability to determine the z coordinate of a particle and requires additional means, e.g., multiple views or multiple holograms,[20

J. Sheng, E. Malkiel, and J. Katz, “Single beam two-views holographic particle image velocimetry,” Appl. Opt. 42, 235–250 (2003). [CrossRef] [PubMed]

, 22

J. Zhang, B. Tao, and J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997). [CrossRef]

] to determine the 3D coordinates of a particle at the same accuracy. Thus it is important to quantify the depth of focus of the DHM and determine whether magnifying the hologram helps in reducing the depth of focus. The measurements are performed under all the conditions and parameters presented in Table 1. A sample measured 3D shape of a reconstructed 3 μm particle by using a 10× objective is shown in the inset included in Fig. 4. The total length of the elongated trace is approximately

40 μm

, i.e., approximately 10 particle diameters, much less than the typical

0.5 - 1 mm

(100 particle diameters) depth of focus of a nonmagnified hologram. Figure 4 shows the ensemble-averaged distributions of

Δ I_{center} (z) / Δ I_{center} (z_{\min}) = [{\bar{I}}_{x \in V} (z) - I_{center} (z)] / [{\bar{I}}_{x \in V} (z_{\min}) - I_{center} (z_{\min})]

(9)

along the depth direction over all the particles for each experimental condition. Here

z_{\min}

is the plane with minimum intensity, and the subscript “center” refers to the center of the particle. Each case contains at least 1000 particles, i.e., the database is sufficiently large for obtaining converged statistics. All the profiles peak when the particle is in focus and decrease away from the center. For the

3 μm

particles, the widths of the peaks, defined based on

75 %

of their peak values, decrease from 25 to

4 μm

with increasing magnification, as also specified in Table 1. The width of the peak also decreases with decreasing particle diameter, from

\sim 4 μm

for the

3 μm

particle to

2.5 μm

for the

0
                        .75
                         
                        μm

particle (at 40×). Thus the effect of size is less pronounced compared to that associated with magnification.

The intensity profiles are asymmetric and oscillate at increasing levels (i.e., display improved contrast) with increasing magnification and decreasing particle size. These oscillations arise from interference between light scattered from the particle and the reference light. With increasing magnification, the higher numerical aperture enables the reconstruction process to resolve increasing numbers of consecutive constructive and destructive interference patterns. The profiles in the forward direction

(z - z_{\min} < 0)

are longer than those in the backward direction, consistent with the intensity distribution associated with Mie scattering by a spherical particle. As the particle size approaches the wavelength of the illumination light, the asymmetry becomes less pronounced, as is evident by comparing the profiles of 0.75 and

3 μm

particles at 40× magnification.

Figure 5 summarizes the effect of magnification and particle size on the depth of focus,

L_{z}

, and detected size of the reconstructed particle,

D_{d}

, both based on the points with

75 %

of peak intensity. The four rows represent test conditions A–D, whose parameters are provided in Table 1. The columns show (from left to right) the distributions of

D_{d}, L_{z} / D_{d}

, and

L_{z}

. The mean values are indicated by

〈 〉

. The first three rows indicate that for a constant particle diameter,

D_{d}

decreases with increasing magnification, but the change is particularly evident at a magnification of 40×. At the latter magnification, the detected diameter of the

0.75 μm

particles remains equal to the actual particle size. A plausible explanation for this trend is provided in the following paragraph.

The mean depth of focus decreases slightly, by

26 %

, as the magnification increases from 10× to 18×, and then substantially, by

75 %

, as the magnification increases from 18× to 40×. Thus, for conditions A and B,

〈 L_{z} 〉 / 〈 D_{d} 〉

is still in the 6.3–7.6 range, while for condition C,

〈 L_{z} 〉 / 〈 D_{d} 〉

decreases to 3.4. With the 40× objective, the mean depth of focus reduces to slightly above the actual diameter of the

3.2 μm

particle and to 3.3 times the diameter of the submicrometer particles. These values are substantially lower than the typical depth of focus of

\sim 100

particle diameters for a lensless in-line hologram. We believe that the improvement is a direct result of magnifying the hologram, which provides a much better recording of the high-frequency part of the interference pattern generated, as a point source interferes with the reference beam (see Section 4). The depth information is exclusively encoded in the fringe spacing, and should improve by increasing the number of fringes resolved by the recording medium. Clearly a DHM is not only capable of recording and reconstructing micrometer and submicrometer particles, which is normally impossible for lensless in-line holography (especially in liquids over a large depth), it also reduces the depth of focus by at least 1 order of magnitude. With increasing magnification, the depth of focus is reduced to a few particle diameters.

Conclusions based on the depth extent of present tests are summarized in Fig. 6. It shows the largest depth we have successfully recorded and reconstructed for each test case,

Z_{d}

, normalized by the theoretical depth of field

(Φ)

of the corresponding microscope objective (provided in Fig. 1). These numbers are lower than the maximum depth that can be reconstructed, but not substantially. Clearly, a DHM extends the depth of field by at least 2 orders of magnitude, increasing with decreasing magnification, and reaching almost 1000 times at the present lowest magnification. The values of

Z_{d} / Φ

decrease with particle size. At the same time, DHM substantially reduces the depth of focus problem of lensless in-line holography. The mean depth of focus normalized by the actual particle diameters

(D_{p})

ranges between 1 and 6, decreasing as the microscope power increases. On the other hand, the detected particle size based on the

75 %

intensity also decreases with increasing power (Fig. 5). Thus one needs to consider the bandwidth limits of this method while interpreting the shape of particles, as discussed in Section 4.

Before concluding, we would like to demonstrate that DHM can be used for the tracking of particles by recording a sequence of holograms. Figure 7(a) shows combined and∕or compressed five exposure tracks of all the

3 μm

particles located within a

1 mm

deep sample. They are recorded by using a 10× objective, with a delay between exposures of

66 ms

. Variations in the velocity of the particles at different depths can be clearly observed from the differences in displacements. Figure 7(b) shows tracks of

0.75 μm

particles over seven exposures, which are recorded by using a 40× objective and combined over a depth of

100 μm

. In both examples the particle traces are clear, and the latter displays the variations in trajectory due to Brownian motion. Clearly cinematographic DHM is suitable for studying the microscale dynamics of particles. To further demonstrate the capability of DHM in biofluid applications, we used cinematographic DHM with a 10× objective to record the behavior of a free- swimming copepod nauplius (size

\sim 100 μm

) located in a

25 mm

deep container. The 3D trajectory, presented in Fig. 8, shows the nauplius engaged in a 3D downward helical motion extending over 1 mm in all three dimensions. Images of five swimming postures at different phases of the cycle clearly show the complex motion of its swimming appendages. These images demonstrate that the 3D morphology of this marine organism, including the setae at the tip of its swimming appendages, which have typical widths of approximately

1 μm

, is clearly resolved. Note that, due to the size of the nauplius, not all of its parts are in focus in a certain plane. Different parts come into better focus as the reconstruction plane is varied.

4. Considerations Related to Bandwidth Limitations

As noted before and demonstrated in Fig. 5, we can resolve increasingly smaller objects and the depth of focus decreases with increasing magnification. However, the size of the reconstructed image also decreases, requiring us to proceed with care. Especially noticeable is the reduction in the reconstructed size of the

3 μm

particle to approximately one third its actual value at a magnification of 40×. Both the recording and reconstruction processes can contribute to the loss of shape information. In this section we discuss the effect of the field of view on the image. During recording, the particle depth is encoded as a high-frequency “carrier” function, and its shape information is stored as a low- frequency amplitude modulation, i.e., as the “envelope,” as illustrated in Fig. 9. The solid curves show a line cut through a hologram of a

15 mm

wide, 1D slit. The envelope, marked by the dashed curves, is the diffraction pattern of an aperture. The high- frequency real fringes are generated by interference between diffracted and undisturbed planar reference waves, i.e., due to the angle between the inclined diffracted light and the collimated reference beam. The fringe spacing is a function of the origin of the diffracted light.[23

C. S. Vikram, Particle Field Holography (Cambridge U. Press, 1992). [CrossRef]

]

The wavelength of the envelope is inversely proportional to the original object size, but the carrier function is a 2D chirp function, in which the frequency increases linearly with distance from the center of the object [Eqs. (1) and (2)]. Consequently, the choices of recording parameters affect the bandwidth of the results. A reduced field of view due to increasing magnification limits the extent of the envelope being recorded, which would result in a smoothed boundary (gradually diminishing in the background) during reconstruction. If a constant threshold level were applied during the segmentation process that defines the object boundaries, the object would appear smaller.

To demonstrate the effect of field of view, we can compare it to the size of the Airy pattern, i.e., the diameter of the first lobe of the envelope modulation,

D_{a i r y} = 1.22 λ z / d

, where d is the object diameter. For a

3.2 μm

particle located

1 mm

from the hologram plane and

λ = 633 nm

, one obtains

D_{a i r y} = 0.24 mm

, which is only slightly lower than the entire field of view of the camera at a magnification of 40× (

0.375 mm

, Fig. 1). Thus the camera records only the complete Airy patterns of particles located very close to the center of the field of view of the objective. The rest will only register a portion of the pattern, degrading the recorded particle shape. Since the Airy pattern increases with depth, the detected diameter should also decrease with depth. Indeed, as illustrated in Fig. 10, conditionally sampling the present particle sizes based on depth shows a decrease in the characteristic detected diameter with depth. However, in spite of the limitations in determining particle size and∕or shape, its depth is correctly reconstructed since the depth information is encoded in the high-frequency fringe pattern (carrier frequency). The higher the recording resolution is, the better resolved these fringes are, and consequently the axial resolution is improved. Our assertion is partially supported by a recent study by Yang et al.[24

W. D. Yang, A. B. Kostinski, and R. A. Shaw, “Depth-of-focus reduction for digital in-line holography of particle fields,” Opt. Lett. 30, 1303–1305 (2005). [CrossRef] [PubMed]

] They propose to equalize the intensity of the fringe pattern, i.e., demodulate the envelope function from the carrier function, to achieve high accuracy of the particle location. In our case the limited numerical aperture of the microscope objective in fact limits the range of intensity variations associated with the envelope, i.e., it partially demodulates the envelope.

5. Summary and Conclusions

We have introduced the application of in-line digital holographic microscopy to measure the spatial distributions of micrometer

(3.2 μm)

and submicrometer

(0.75 μm)

particles in dense liquid suspensions with depths of

1 - 10 mm

. It consists of illuminating a sample with a collimated laser beam and using a microscope objective to digitally record a magnified image of the optical field created in a plane located outside of the sample volume. Thus the image is a magnified digital in-line hologram, as demonstrated by a brief analysis. A series of tests with different objectives and particle diameters of 0.75 and

3 μm

are used for characterizing the performance of this method. We demonstrate that DHM is capable of recording and reconstructing the traces of several thousands of particles located within a sample whose depth extends to almost 1000 times the depth of field of a conventional microscope, yet it maintains the lateral spatial resolution of the conventional microscope over the entire volume. Clearly, DHM is capable of recording and reconstructing micrometer and submicrometer particles, which is normally impossible for lensless in-line holography, especially in liquid. The typical elongation on the depth direction, the so-called depth of focus problem that characterizes in-line holography, still exists, but it is at least 1 order of magnitude smaller that that of a lensless in-line hologram. The depth of focus decreases with increasing magnification, ranging between 10 and 2 particle diameters.

An automated segmentation method maps the 3D coordinates of the particles with submicrometer resolution in the lateral direction, and 2–10 diameters in the depth direction. With a 10× objective, we successfully reconstruct the coordinates of particles in a dense suspension containing

2000 particles / {mm}^{3}

. With a 40× objective we successfully reconstruct the traces of

0.75 μm

submicrometer particles over a depth of 1 mm. Cinematographic DHM records the particle motion, which can be used for determining their velocity. Clearly, this method is capable of tracking the 3D motion of particles, such as the swimming behavior of microscopic organisms, and studying their interactions with other particles as well as the surrounding flow fields.

Acknowledgments

This research was supported in part by the National Science Foundation (NSF) (P. Taylor, program manager) under grant OCE-0402792 and in part by the Office of Naval Research (R. Joslin, program manager) under grant N000140310361. Funding for the holographic instrumentation used in this study was provided by the NSF, Major Research Instrument grant CTS0079674.

References and links

1.	D. DeAngelis and L. Gross, Individual-Based Models and Approaches in Ecology (Chapman & Hall, 1992).
2.	W. Fennel and T. Neumann, Introduction to the Modelling of Marine Ecosystems , Vol. 72 of the Elsevier Oceanography Series (Elsevier, 2004).
3.	W. Fennel and T. Osborn, A Unifying Framework for Marine Ecological Model Comparison Deep Sea Research II, Rep. DSR-WTD-06, 2004.
4.	E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206, 3657–3666 (2003). [CrossRef] [PubMed]
5.	G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827–833 (2003). [CrossRef] [PubMed]
6.	U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002). [CrossRef]
7.	J. H. Milgram and W. C. Li, “Computational reconstruction of images from holograms,” Appl. Opt. 41, 853–864 (2002). [CrossRef] [PubMed]
8.	W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A. 98, 11301–11305 (2001). [CrossRef] [PubMed]
9.	L. Xu, X. Peng, J. Miao, and A. K. Asundi, “Studies of digital microscopic holography with applications to microstructure testing,” Appl. Opt. 40, 5046–5051 (2001). [CrossRef]
10.	W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett. 28, 164–166 (2003). [CrossRef] [PubMed]
11.	D. Carl, B. Kemper, G. Wernicke, and G. von Bally, “Parameter-optimized digital holographic microscope for high-resolution living-cell analysis,” Appl. Opt. 43, 6536–6544 (2004). [CrossRef] [CrossRef]
12.	T. Colomb, F. Durr, E. Cuche, P. Marquet, H. G. Limberger, R. P. Salathe, and C. Depeursinge, “Polarization microscopy by use of digital holography: application to optical-fiber birefringence measurements,” Appl. Opt. 44, 4461–4469 (2005). [CrossRef] [PubMed]
13.	G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15, 529–539 (2004). [CrossRef]
14.	F. Dubois, M. L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. 43, 1131–1139 (2004). [CrossRef] [PubMed]
15.	P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, “Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time,” Opt. Lett. 28, 1257–1259 (2003). [CrossRef] [PubMed]
16.	P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. , 30 468–470 (2005). [CrossRef] [PubMed]
17.	L. Repetto, E. Piano, and C. Pontiggia, “Lensless digital holographic microscope with light-emitting diode illumination,” Opt. Lett. 29, 1132–1134 (2004). [CrossRef] [PubMed]
18.	F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, J. C. Legros, and P. Kischel, “Pattern recognition with a digital holographic microscope working in partially coherent illumination,” Appl. Opt. 41, 4108–4119 (2002). [CrossRef] [PubMed]
19.	J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
20.	J. Sheng, E. Malkiel, and J. Katz, “Single beam two-views holographic particle image velocimetry,” Appl. Opt. 42, 235–250 (2003). [CrossRef] [PubMed]
21.	H. Meng and F. Hussain, “In-line recording and off-axis viewing technique for holographic particle velocimetry,” Appl. Opt. 34, 1827–1840 (1995). [CrossRef] [PubMed]
22.	J. Zhang, B. Tao, and J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997). [CrossRef]
23.	C. S. Vikram, Particle Field Holography (Cambridge U. Press, 1992). [CrossRef]
24.	W. D. Yang, A. B. Kostinski, and R. A. Shaw, “Depth-of-focus reduction for digital in-line holography of particle fields,” Opt. Lett. 30, 1303–1305 (2005). [CrossRef] [PubMed]
25.	S. Inoué and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997). [CrossRef]

Table 1 Parameters of Present Tests, and Resulting Mean Diameters and Depth of Focus of Reconstructed Particle Traces

ID	Particle (μm)	Magnification	Resolution of Imaging System (μm∕pixel)	Depth of Sample (mm)	Mean Reconstructed Diameter at 75% of Peak Intensity (μm)	Mean Depth of Focus (um)
A		10×	0.7216	1, 4, 10	2.5238	19.2
B	3.189	18×	0.423	1, 4	2.3454	14.3
C		40×	0.177	1	1.1572	3.89
D	0.75	40×	0.177	0.1, 1	0.7703	2.50

Fig. 1 Theoretical depth of field as a function of resolution or field of view.[25

S. Inoué and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997). [CrossRef]

] M refers to the power of a microscope objective.

Fig. 2 Optical setup of the digital holographic microscope.

Fig. 3 (a) Part of a recorded hologram using a 10× objective, containing 3.189 μm diameter particles in a 1 mm deep solution. (b)–(d) Reconstruction of planes located 120, 580, and 800 μm from the hologram plane. In-focus particles appear as dark spots on the bright background. (e) A combined and∕or compressed image containing all the particles covered by the hologram section shown in (a). (f) Location of all the particles detected within the entire 1.5 × 1.5 × 1 mm³ volume, totaling 5769 particles.

Fig. 4 Ensemble-averaged intensity distribution along the depth direction. For definitions of terms, see Eq. (9). The inset is an isointensity surface plot of a typical reconstructed particle at 75% of its peak intensity. The depths of the sample are 1 mm for the 3 μm particles and 0.1 mm for the 0.75 μm particle.

Fig. 5 Statistics on the properties of reconstructed particle traces. Rows are arranged as experimental conditions A–D (Table 1), with the following magnifications, particle diameters, and sample depth. A, 10×, 3.189 μm, 1 mm; B, 18×, 3.189 μm, 1 mm; C, 40×, 3.189 μm, 1 mm; and D, 40×, 0.75 μm, 0.1 mm. The columns are (from left to right) D_d , L_z∕D_d, L_z , where D_d is the detected particle diameter and L_z is the depth of focus, both based on 75% of the peak intensity. The numbers in 〈〉 above the distributions indicate mean values.

Fig. 6 Summary of the present test conditions, including largest successfully tested sample depth (Z_d ) and depth of focus normalized by the nominal particle diameter and theoretical depth of field (Φ). Letters indicate test conditions (Table 1 and Fig. 5).

Fig. 7 A demonstration of a cinematographic digital holographic microscope. (a) Combined and∕or compressed tracks, consisting of five exposures, of a 3.189 μm particle located within a 1 mm deep sample. (b) Sample tracks, consisting of seven consecutive exposures, of 0.75 μm particles, combined over a depth of 100 μm.

Fig. 8 A 3D, 4 s trajectory, and behavior of a free-swimming copepod nauplius, stage VI, obtained using a cinematographic digital holographic microscope. The images in A–E show reconstructions of planes dissecting the center of the organism at different times, as indicated in each frame.

Fig. 9 Intensity distribution of a hologram generated by a single 15 μm wide, 1D slit, located 1 mm away from the hologram plane. The solid curve represents intensity distribution of fringes in the hologram, while the dashed curve outlines their envelope.

Fig. 10 Detected particle size distribution of Case C (3 μm particles, 40× objective) conditionally sampled based on distance from the hologram plane.

OCIS Codes
(100.6890) Image processing : Three-dimensional image processing
(180.6900) Microscopy : Three-dimensional microscopy

History
Original Manuscript: June 20, 2005
Revised Manuscript: November 23, 2005
Manuscript Accepted: November 25, 2005

Virtual Issues
Vol. 1, Iss. 7 Virtual Journal for Biomedical Optics

Citation
Jian Sheng, Edwin Malkiel, and Joseph Katz, "Digital holographic microscope for measuring three-dimensional particle distributions and motions," Appl. Opt. 45, 3893-3901 (2006)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=ao-45-16-3893

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References

D. DeAngelis and L. Gross, Individual-Based Models and Approaches in Ecology (Chapman & Hall, 1992).
W. Fennel and T. Neumann, Introduction to the Modelling of Marine Ecosystems, Vol. 72 of the Elsevier Oceanography Series (Elsevier, 2004).
W. Fennel and T. Osborn, A Unifying Framework for Marine Ecological Model Comparison Deep Sea Research II, Rep. DSR-WTD-06, 2004.
E. Malkiel, J. Sheng, J. Katz, and J. R. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003). [CrossRef] [PubMed]
G. Pan and H. Meng, "Digital holography of particle fields: reconstruction by use of complex amplitude," Appl. Opt. 42, 827-833 (2003). [CrossRef] [PubMed]
U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002). [CrossRef]
J. H. Milgram and W. C. Li, "Computational reconstruction of images from holograms," Appl. Opt. 41, 853-864 (2002). [CrossRef] [PubMed]
W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001). [CrossRef] [PubMed]
L. Xu, X. Peng, J. Miao, and A. K. Asundi, "Studies of digital microscopic holography with applications to microstructure testing," Appl. Opt. 40, 5046-5051 (2001). [CrossRef]
W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, "Tracking particles in four dimensions with in-line holographic microscopy," Opt. Lett. 28, 164-166 (2003). [CrossRef] [PubMed]
D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004). [CrossRef]
T. Colomb, F. Durr, E. Cuche, P. Marquet, H. G. Limberger, R. P. Salathe, and C. Depeursinge, "Polarization microscopy by use of digital holography: application to optical-fiber birefringence measurements," Appl. Opt. 44, 4461-4469 (2005). [CrossRef] [PubMed]
G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, "A digital holographic microscope for complete characterization of microelectromechanical systems," Meas. Sci. Technol. 15, 529-539 (2004). [CrossRef]
F. Dubois, M. L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004). [CrossRef] [PubMed]
P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, "Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time," Opt. Lett. 28, 1257-1259 (2003). [CrossRef] [PubMed]
P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. , 30468-470 (2005). [CrossRef] [PubMed]
L. Repetto, E. Piano, and C. Pontiggia, "Lensless digital holographic microscope with light-emitting diode illumination," Opt. Lett. 29, 1132-1134 (2004). [CrossRef] [PubMed]
F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, J. C. Legros, and P. Kischel, "Pattern recognition with a digital holographic microscope working in partially coherent illumination," Appl. Opt. 41, 4108-4119 (2002). [CrossRef] [PubMed]
J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
J. Sheng, E. Malkiel, and J. Katz, "Single beam two-views holographic particle image velocimetry," Appl. Opt. 42, 235-250 (2003). [CrossRef] [PubMed]
H. Meng and F. Hussain, "In-line recording and off-axis viewing technique for holographic particle velocimetry," Appl. Opt. 34, 1827-1840 (1995). [CrossRef] [PubMed]
J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997). [CrossRef]
C. S. Vikram, Particle Field Holography (Cambridge U. Press, 1992). [CrossRef]
W. D. Yang, A. B. Kostinski, and R. A. Shaw, "Depth-of-focus reduction for digital in-line holography of particle fields," Opt. Lett. 30, 1303-1305 (2005). [CrossRef] [PubMed]
S. Inoué and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997). [CrossRef]

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