The average concentration of marine copepods has beenestimated at about one per liter, making them the mostnumerous multicelled animals on earth (Boxshall, 1998).These micro-crustaceans form major links between theplankton world and nektonic organisms. The rate andspecificity of their feeding can control phytoplanktonpopulation growth, while copepods themselves are animportant source of food for fish, euphausiids andcoelenterates (Mann and Lazier, 1996). Along with foodavailability and predator pressure, the copepod populationdynamics are affected by swimming behavior and the types offlow fields they generate to capture food, which is speciesdependent (Strickler, 1982). The quantity of food depends onthe volume of water they can scan and their ability to locateprey (Koehl, 1984). Locomotion and the generation of feedingcurrents increase the encounter rates with food, and canenhance the copepod’s detection ability by stimulatingescapes in motile prey (Yen and Strickler, 1996). Alternately,motionless sinking can make the copepod less conspicuous topredators or prey. Within the species, advertisement isimportant for reproduction (Doall et al., 1998; Yen andStrickler, 1996). Because copepods sense their environment

utilizing antennules studded with arrays of setae that respondto fluid deformation, the flow field that a swimming animalgenerates is a ‘lens’ through which the copepod views itssurroundings. Knowledge about this flow field is important forunderstanding the impact of swimming behavior on thesuccess of a copepod in locating food, avoiding predators andfinding mates.

Current knowledge is largely based on video observations,including a Schlieren system for visualizing wakes (Strickler,1977), high-speed cinematography for understanding how foodparticles are captured and feeding currents are generated(Koehl and Strickler, 1981; Strickler, 1984), and three-dimensional (3-D) video tracking of free-swimming copepodsin particle fields (Paffenhöfer et al., 1995; Strickler, 1985). Thelatter combines particle tracks at different times to observetwo-dimensional (2-D) flow fields in lateral and dorsal views.More recently, Particle Image Velocimetry (PIV) has beenused to map instantaneous 2-D flow fields around tetheredspecimens (van Duren et al., 1998). However, since flow fieldsaround copepods are 3-D, unsteady and vary with swimmingspeeds and orientations (Bundy and Paffenhöfer, 1996;Strickler, 1982; van Duren et al., 1998), 3-D measurements are

3657The Journal of Experimental Biology 206, 3657-3666© 2003 The Company of Biologists Ltddoi:10.1242/jeb.00586

Digital in-line holography is used for measuring thethree-dimensional (3-D) trajectory of a free-swimmingfreshwater copepod Diaptomus minutus, andsimultaneously the instantaneous 3-D velocity field aroundthis copepod. The optical setup consists of a collimatedHe–Ne laser illuminating a sample volume seeded withparticles and containing several feeding copepods. A timeseries of holograms is recorded at 15·Hz using a lensless2Kx2K digital camera. Inclined mirrors on the walls of thesample volume enable simultaneous recording of twoperpendicular views on the same frame. Numericalreconstruction and matching of views determine the 3-Dtrajectories of a copepod and the tracer particles to withinpixel accuracy (7.4·µm). The velocity field and trajectoriesof particles entrained by the copepod have a recirculatingpattern in the copepod’s frame of reference. This pattern

is caused by the copepod sinking at a rate that is lowerthan its terminal sinking speed, due to the propulsive forcegenerated by its feeding current. Consequently, thecopepod sees the same fluid, requiring it to hopperiodically to scan different fluid for food. UsingStokeslets to model the velocity field induced by a pointforce, the measured velocity distributions enable us toestimate the excess weight of the copepod (7.2×10–9·N), itsexcess density (6.7·kg·m–3) and the propulsive forcegenerated by its feeding appendages (1.8×10–8·N).

Movies available on-line

Key words: digital in-line holography, particle image velocimetry,copepod, Diaptomus minutus, flow.

Summary

Introduction

The three-dimensional flow field generated by a feeding calanoid copepodmeasured using digital holography

Edwin Malkiel1, Jian Sheng1, Joseph Katz1,* and J. Rudi Strickler21Johns Hopkins University, Department of Mechanical Engineering, N. Charles Street, Baltimore, MD 21218, USA

and 2Great Lakes WATER Institute, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53204, USA*Author for correspondence (e-mail: Katz@jhu.edu)

Accepted 7 July 2003

3658

essential. Considering the importance of copepods in theaquatic and marine food webs, and 30 years of relatedhypotheses, it seems worthwhile to investigate whether digitalcinematographic holography may break barriers and allow usto test these hypotheses.

Unlike video microscopy, holography maintains the samelateral resolution over a substantial depth (Vikram, 1992). Thisadvantage has led to the development of several submersibleholography systems for studying plankton, starting with asample volume of a few ml (Carder et al., 1982), to samplesof one liter and above (Katz et al., 1999; Malkiel et al., 1999;O’Hern et al., 1988; Watson et al., 2001). The latter utilizepulsed lasers and emulsions as recording media.Cinematographic holography was introduced for laboratoryresearch decades ago (Heflinger et al., 1978; Knox and Brooks,1969), but difficulties in acquiring and processing data resultedin limited applications. Recent development in digital imaging,which simplifies the acquisition, and computing power, whichenables numerical reconstruction, has led to renewed interestin cinematic holography (Kebbel et al., 1999; Owen andZozulya, 2000; Xu et al., 2001). The limited resolution ofdigital imaging, which is at least an order of magnitude lowerthan that of holographic emulsions, restricts us to in-lineholography. This technique (details follow) maximizes thefringe spacing. Because the reference beam typically passesthrough the sample volume, it becomes increasingly degradedwith increasing particle concentration. Consequently, thereconstructed images are noisier and the maximum particleconcentration is lower compared to off-axis holograms (Zhanget al., 1997).

Holographic PIV is the only technique to date that canmeasure a 3-D instantaneous velocity distribution over a finitevolume (Barnhart et al., 1994; Pu and Meng, 2000; Sheng etal., 2003; Tao et al., 2002; Zhang et al., 1997) at a resolutionof millions of vectors. The velocity is obtained by recordingtwo exposures of a flow field seeded with microscopicparticles, and measuring the displacement of these particles.However, the depth coordinate of a reconstructed particle isless accurate than the lateral coordinates, severely reducing theability to estimate the corresponding velocity component. Thisproblem has been solved by recording two inclined holograms,using each for determining a 3-D distribution of two velocitycomponents, and matching the two sets to obtain the 3-Dvelocity. Utilizing emulsion and off-axis holography, Tao et al.(2002) measured 136×130×128 3-D velocity vectors in a cubicsample with sides of about 45·mm. By inserting an inclinedmirror in the path of the illuminating beam inside the testfacility, the incident and reflected beams create two views thatcan be recorded on the same emulsion (Sheng et al., 2003).Using off-axis holography, this method enables measurementof particle locations to within 7·µm, and resolves about200·particles·mm–3. We adopt this approach, but use digital in-line holography to record a time series of a free-swimmingcopepod and the flow field surrounding this animal. To ourknowledge, this is the first time that digital holographic PIVhas been implemented as a tool for simultaneous observation

of the animal’s behavior and measurement of the complex flowaround it.

Materials and methodsDigital in-line holography is a two-step process, consisting

of first recording an in-line hologram on a digital recordingmedium, and then numerical reconstruction. The recordingprocess entails back-illuminating a sample volume filled withobjects by a coherent light source. A diffraction pattern isgenerated as a result of interference between the illuminatingbeam and light scattered from the objects. Coherence of thelight source is important in order to enhance the visibility (orcontrast) of the interference pattern, which conveysinformation on the shape and location of an object in space. Asshown in Fig.·1 we use a standard, single mode, 3·mWHelium–Neon laser, which generates a CW (continuous wave)polarized beam of red light (632.8·nm) with a coherence lengthfar beyond that required for digital in-line holography (whichis on the order of 1·cm). The beam is spatially filtered andexpanded to cover at least the entire test section. Collimatingthe expanded beam enables recording of the diffractionpatterns at a constant magnification. This nonessential choicesimplifies the numerical reconstruction by avoiding acalibration procedure to determine the origin of theillumination source.

The expanded beam illuminates an 80·ml sample volumeconstructed of antireflection-coated windows attached to twosupporting prisms. First-surface mirrors are attached to theinclined side walls of the test chamber to direct the illuminationbeam through the test section. These mirrors provide twoperpendicular views of objects located in regions where theincident and reflected beams overlap (cf. Fig.·2). One view iscreated by light that is first reflected from the mirror and thenincident on the object, while the other (mirrored view) isgenerated by light incident on the object and then reflected offthe mirror. The two views are recorded onto a single recordingplane, but they are laterally separated. During reconstruction,the first view appears in focus at its original location, while thesecond perpendicular view is a mirror image (right side ofFig.·2). Use of two mirrors (Fig.·1) doubles the triangularoverlapping volume without necessitating an increase inrecording area.

E. Malkiel and others

Fig.·1. Optical setup for digital in-line holography and test section.

He–Ne Laser

Kodak ES 4.02Kx2K; 15 fps

First surface mirrorOverlap region

Test section CCD cameraLight source

NDfilter

Spatialfilter

Collimatinglens

32 mm

3659Three-dimensional copepod flow field

The interference patterns are recorded on a lenslessMegaplus ES 4.0 digital camera, which has a 2048×2048 pixelCCD sensor with a 7.4·µm pitch, providing a 15·mm×15·mmfield of view. The in-line system minimizes the angle betweenthe light scattered from the objects and the remaining referencebeam. Consequently, the fringe spacing is maximized (Vikram,1992), overcoming the limited resolution of the CCD. TheCCD’s interline transfer capability eliminates the need for ashutter. Because the experiment involves recording movingobjects (a copepod and seed particles), the exposure time isadjusted to be short enough to prevent the smallest resolvablefringe spacing from smearing. Here, the velocity does notexceed 4·mm·s–1 (540·pixels·s–1), and the electronic shutteringof the camera (0.1·ms exposure time) is sufficient to reduce themovement during exposure to much less than 1·pixel. Fasterflows (~1·m·s–1) would require higher power, even pulsedlasers, to generate the energy required for recording ahologram. Images have been recorded at the maximum framerate of the present camera, 15 frames per second, buffering 10·ssegments (150·frames) to the RAM of the computer hosting theimage acquisition card.

Our subject animal, a female Diaptomus minutus Lilljeborg1889, was freshly caught in Lake Michigan and transported toBaltimore in a Dewar’s jar. The test section was filled withartesian well water from the Pryor Street well in Milwaukee.It was seeded with monodisperse, 20·µm diameter polystyrenespheres (specific gravity 1.05) at a concentration of 4·mm–3.Although denser seeding would improve the spatial resolutionof the flow, it would also reduce the signal-to-noise ratio ofreconstructed images. Image acquisition to RAM was startedwhen the copepod appeared in the field of view. Successfulimage sequences, which captured the copepod in a region withtwo views, and were sufficiently far from the wall (>4·mm),were stored. Fig.·3 is a sample digital in-line hologramcontaining two interference patterns (‘shadows’) of the samecopepod, along with the very faint diffraction patterns of theseed particles.

Digital reconstruction regenerates the original light intensitydistribution at any desired distance from the recording plane.

The amplitude of light U(x,y,z) at any point in thereconstruction volume is a superposition of light reaching thispoint from an array of source wavelets distributed over theentire recording plane (the hologram). Using theFresnel–Huygens Principle (Hecht, 2002) with a paraxialapproximation,

Here, (ξ,η) are coordinates of the hologram, Σ indicates thearea of the entire hologram, ⊗ indicates convolution andk=(2π/λ), where λ is the reconstruction wavelength. If λ is setequal to the recording wavelength, the volume is reconstructedat its original depth. The intensity of the light, I(x,y,z), iscalculated from I(x,y,z)=UU* , where U* is the complexconjugate of U. Use of the paraxial approximation is justifiedbecause we record predominantly forward scattered light.Furthermore, due to the limited resolution of the recordingmedium, the camera can resolve only near forwardfringes. Equation·1 is actually a 2-D convolution integralparameterized by z, the distance between the hologram and thereconstruction plane. The kernelhz is the 3-D field generatedby an infinitesimal point source located at (x,y,z). Thediscretized version of this convolution is performed over theentire hologram for each desired depth. Because of the largeconvolution kernel and image size, the integration is moreefficiently performed in Fourier space. After calculating the

U(x,y,z) =

hZ(ξ–x,η–y) = exp .[(ξ–x)2 + (η–y)2]j

⌠⌡

⌠⌡

∑

I(ξ,η) hZ(ξ–x,η–y) dξdη = I ⊗ hZ ,

exp(jkz)

jλz

k

2z

(1)

Mirror(reconstructed) Mirrored view

Originalview

Fig.·2. Recording and reconstruction of an object near a mirror.

Fig.·3. A sample digital hologram containing two views of the sameswimming copepod in a seeded test section. Scale bar, 1·mm.

3660

2-D Fourier transforms of both the recorded image and theconvolution kernel, they are multiplied. Then, an inverseFourier transform provides the intensity distribution in thedesired plane. A similar approach is presented in Milgram andLi (2002), where further details on the mathematics can befound. We typically reconstruct 45 planes separated by0.2·mm.

The purpose of the next phase of the analysis is to match thein-focus perpendicular views of particles in order to determinetheir exact locations in space. Recall that a single view canprovide lateral positional accuracy to within a pixel (7.4·µm)in the lateral plane, but is substantially less accurate in the axialdirection (0.5·mm). These views are laterally separated (as inFig.·3). An automated procedure for matching the two viewsin densely seeded flows recorded using off-axis holography (onemulsion) is outlined in Sheng et al. (2003). There, the arrayof reconstructed images is thresholded, the particles aresegmented into 3-D volumes, and then reduced to linesegments that pass through the barycenters of these volumesand have lengths corresponding to their axial extent. Theintersection of two perpendicular line segments determines the3-D coordinates of the particle centroid. The analysis requiresthe calibration of the location and orientation of the mirror,achieved by manually matching corresponding views ofseveral reference particles. The calibrated mirror orientation isused to transform (flip) the mirrored views onto their originallocations in space. For two views of the same particle to bematched, we require that the perpendicular segments are lessthan a particle diameter away from each other. The 3-Dcoordinates of the particle centroid are approximated as themidpoint of the shortest line segment connecting the views.

This approach did not work while analyzing the present in-line digital holograms. The primary reason was excessivebackground noise resulting from the reference (illuminating)beam passing through the sample volume. It should work withless heavily seeded flows and/or an optical setup with aseparate reference beam that does not pass through the samplevolume. The limited resolution of the digital camera preventedthe implementation of off-axis holography, which ischaracterized by much finer fringe spacing. Consequently, wehave used two semi-manual techniques to obtain the 3-Dlocations of particles and velocity fields.

The first task is to identify particles and distinguish themfrom background noise. Three successive exposures aresuperposed, and elongated traces, or three closely spaced spotsresulting from the displacement of particles, are identified. Thisapproach has turned out to be a very effective tool fordistinguishing between real particles and speckle noise. Basedon the location of these traces, we estimate (computationally)the most likely location of the mirrored view. The lineartransformation used for this calculation is calibrated bymatching corresponding views of the copepod’s extremities,which can be easily identified. The mirrored view lies along analmost horizontal line that corresponds to the depth uncertaintyof the first view. If we find three spots (or elongated traces)along this line, the two views of the same particle are matched.

At the present particle concentration, having more than onematch is very unlikely. If we do not find the second view of aparticle along the expected line, for example, when it is locatedin the shadow of the copepod, the unmatched trace isdisregarded.

When the second view is found, the program proceeds tomeasure the particle displacement in each of the two views,using cross-correlation of the intensity distribution, generatedby the three exposures. Details on the cross-correlationprocedures, including methods of achieving sub-pixelaccuracy, are discussed by Roth and Katz (2001), Roth et al.(1999) and Sridhar and Katz (1995). The lateral displacementin each of the views and the average vertical displacements arecombined into a 3-D velocity vector, positioned at the particlemean location. The uncertainty in velocity of individualparticles is about 0.05·mm·s–1.

Extended particle tracks in the vicinity of the copepod aremore easily identified. Superposing sets of ten exposures nearthe plane of the copepod in each view, and placing them side-by-side, allows identification of corresponding tracks based ontheir elevation at a given time. This identification is verified bycomparing subsequent sets of exposures. Combining the setsgenerates the 3-D trajectory of the particle. Even when one ofthe views is partially blocked by the copepod, it is usuallypossible to match the segments that are not blocked and theninterpolate them to estimate the trajectory in the blockedsection. The endpoints of segments are stored and used formeasuring the velocity along the path of the particle. In regionswith relatively high velocity, the tracks appear as a series ofdots. In this case, individual traces are used for measuring thevelocity as discussed before.

ResultsFig.·4 shows reconstructed perpendicular views of the

copepod and seed particles, computed from the hologramshown in Fig.·3. Note that the two views are reconstructed atdifferent depths. Clearly, digital reconstruction of in-lineholograms can resolve fine details of the copepod structure.Sample instantaneous distributions of 3-D velocity vectors inthe vicinity of this copepod in the ambient and copepodreference frames are presented in Fig.·5A and B, respectively.The graphic representation of the copepod has the correctscales. Vectors that are located within the central two thirdsspan of the antennules are black; vectors outside of this spanare gray. A recirculating flow pattern is evident in the copepodframe of reference, which sinks at an average speed of0.29·mm·s–1. Similar sets of 3-D velocity distributions areobtained from each pair of reconstructed holograms.

The reconstructed images from 15 sequences consistentlyshow that the copepod sinks for several seconds. It thenexecutes a short hop upwards and resumes sinking (seesupporting movie). As it sinks, the copepod generates a feedingcurrent by moving its feeding appendages. The present 15·Hzrecording rate may not be sufficient for following the (highfrequency) motion of the appendages, but different phases of

E. Malkiel and others

3661Three-dimensional copepod flow field

their motion are discernible (Fig.·4). The ‘high’-speed flowgenerated by the appendages (Fig.·5A) is most evident in theventral region of the copepod (z<79.5·mm), extending to itsanterior and posterior regions. The feeding current generates areaction force that propels the copepod. This vertically directedforce acts against the copepod’s excess weight (weight minusbuoyancy) and drag, reducing its sinking rate compared to theterminal speed (speed with no feeding current).

The recirculating flow pattern generated by the copepod isclearly demonstrated in Fig.·6 by combining 130 reconstructedimages, shifted in time, so that the image of the copepod isfixed. Since the copepod velocity is constant during this period,the shift applied to each image is a linear function of time.Blurring of the copepod occurs due to slight variations insinking speed (<5%), and motion of the appendages. Streaksgenerated by the seed particles are clearly evident in both

Fig.·5. Measured instantaneous velocity near the copepod (A) in the ambient frame of reference, and (B) in the copepod frame of reference.

7880

8

6

4

1 mm s–1z (mm)x (m

m)78

808

6

4

1 mm s–1z (mm)x (m

m)

–12

–10

–8

y (m

m)

–12

–10

–8

y (m

m)

A7.35.3

x B7.35.3

x

Fig.·4. In-focus, numerically reconstructed, dorsal (A) and lateral (B) views of the same swimming copepod, from the hologram of Fig.·3. s,setae on antennule; f, feeding appendages; p, tracer particle (there are many). Inserts show the feeding appendages in up-stroke (top) and down-stroke (bottom) positions. Scale bar, 1·mm.

3662

views. Above the copepod, several particles are drawn towardsthe center of the copepod. Below and to the sides of thecopepod, particles are ejected downwards, subsequentlylooping around and migrating upward (relative to thecopepod), some of them touching its antennules. Quantitativedata on the trajectory and velocity along the path of selectedparticles are presented in Figs·7 and 8. The velocity peaks ina narrow domain near the tips of the appendages, reaching3.6·mm·s–1, i.e. 12.5 times the sinking velocity. We use thesedata to estimate the propulsive force generated by the feedingappendages.

Excess weight and propulsive force

The particles slightly beyond the reach of the antennulescircumvent the copepod, creating a closed streamline pattern,without a separated wake. This pattern is characteristic of lowReynolds number flows (ReL=UL/ν, U and L beingcharacteristic velocity and length scales, respectively, and ν,the kinematic viscosity of the liquid), such as Stokes flows withRe<<1, or Oseen flows with Re~1 (Pozrikidis, 1992). Thepresent Re, based on the sinking velocity and prosome length,is 0.29. Based on the diameter of the recirculation zone, Reincreases to 1.2. Thus, the present flow lies in the transitionregion between Stokes and Oseen flows. Here we use thesimpler Stokes flow in order to estimate the magnitude of theforces produced by the copepod.

The horizontal (u) and vertical (v) velocity componentsinduced by a vertical point force (Stokeslet) in Stokes flow(Jiang et al., 2002c; Pozrikidis, 1992) are:

where f is the force, µ is the liquid viscosity, x and y arecoordinates, and r is distance from the origin. The Stokesletdescribes the far-field flow realistically, but neglects the factthat the object has a finite size. The far-field net effect of thecopepod on the surrounding flow, its excess weight (wexcess),can be modeled as a Stokeslet.

The flow pattern generated by a Stokeslet in an absoluteframe of reference is illustrated in Fig.·9A, withUref=wexcess/8πµL. A recirculating flow pattern appears in aframe of reference moving downward at a fraction of Uref

(Fig.·9B). For the recirculation to form, there must be aresidual downward velocity component (jet) below the objectin its own reference frame. This flow can only be generatedby a propulsive force. Thus, in reducing its sinking speed andgenerating a propulsive feeding current, the copepod createsa recirculating pattern that extends slightly beyond itsantennae. This combination of sinking and feeding is notdiscussed in detailed conceptual and numerical analyses of theforces acting on a swimming copepod under various

v = + and ,f

8πµL

(y/L)2

(r/L)3

1

r/L(2)

u =f

8πµL

xy/L2

(r/L)3

E. Malkiel and others

Fig.·6. Particle streaks in the copepod reference frame obtained by combining 130 appropriately shifted reconstructed images. In all cases thedorsal and lateral views are in focus.

3663Three-dimensional copepod flow field

conditions (Jiang et al., 2002b,c). They include cases ofstationary bodies producing feeding currents (hovering orconceivably tethered) and freely sinking bodies, both of whichdo not generate recirculating patterns.

The data can be used for estimating wexcess. In a referenceframe sinking at vsink, the vertical velocity component vanishesat y0=±wexcess/4πµvsink. Estimating y0 as half the distancebetween the points with zero velocity in Fig.·6 (y0=2·mm), and

since vsink= 0.29·mm·s–1, one obtains wexcess=7.2×10–9·N. Witha volume of 1.1×10–10·m3 (determined following Chojnacki,1983), the estimated excess density of the copepod is6.7·kg·m–3 (1006.7·kg·m–3). This value falls in the rangemeasured by Svetlichny (1980) and Knutsen et al. (2001). Thesame analysis can be performed in any frame, including one

y

xz

4

Speed(mm s–1)

20

Fig.·7. Selected particle tracks (1–6) in 3 dimensions. A–A, see insetin Fig.·8.

4

3

2

1

0

0

0

0

0–4 –2

Time from capture region (s)

Spee

d (m

m s–1

)0 2 4

0.2

0.15

0.1

0.05

–0.25 0 0.25z (mm)

w (

mm

s–1)

0.50

Fig.·8. Speeds of selected particles (1–6) approaching feedingappendages. Inset: Horizontal velocity component w near tail alongline A–A of Fig.·7. zaxis origin at center of mass.

10A B

5

0

–5

–10–10 –5 0

x/L

u/Uref=1

y/L

5 10 –10 –5 0

x/L

u/Uref=1

5 10

Fig.·9. Flow field generated by a Stokeslet (see Equation·3) at (A) the absolute reference frame and (B) the reference frame sinking at 0.33Uref.

3664

without recirculation, by measuring the velocity and distancebetween points above and below the copepod with the samevelocity.

One can also estimate the propulsive force, P, generated bythe feeding appendages. Averaging v of a Stokeslet at y=0 overa certain radius, R, one obtains v̄=f/(4πµR). Based on Figs·7–8,there are two regions with a radius of 200·µm and characteristicpeak velocity of v̄=3.6·mm·s–1 in the vicinity of the feedingappendages. Combining the two regions, P=2(4πµRv̄), i.e.P=1.8×10–8·N, 2.5 times higher than wexcess. At a constantsinking velocity, P must balance the sum of wexcessand the dragforce (Jiang et al., 2002c). Since the relative velocity aroundthe copepod is downward (Figs·5–8), so is the drag, requiringP to be larger than wexcess. Thus, the estimated drag is about1.5 times the excess weight. The propulsive force alsogenerates a moment (negative x direction), since it is appliedat about 30·µm in front of the centerline of the prosome, basedon the location of maximum v̄. The magnitude of this momentis about 5.4×10–13·Nm. To overcome this moment and avoidtumbling, the copepod turns its tail, creating a velocity bias inthe positive z direction (see insert in Fig.·8). Turning the flowcreates a reaction force and a moment in the positive xdirection. Based on the average velocity bias (~0.1·mm·s–1,Fig.·8), over a radius of 250·µm (half the tail length), the forceis 0.6×10–9·N. Multiplying it by the distance to the center ofmass of the copepod (1·mm), the estimated moment is6×10–13·Nm. Thus, the moment generated by the tail’sredirection of fluid is of sufficient magnitude to compensateagainst the moment of the propulsive force, allowing thecopepod to maintain a relatively vertical orientation.

DiscussionThe discussion is in two parts. The first compares

holography to other 3-D imaging techniques, and the secondfocuses on the behavior of a copepod.

Comparison with other techniques

It is useful to compare the capabilities and difficulties ofdigital in-line holography with other 3-D particle trackingtechniques, such as Stereo PIV and 3-D particle tracking withmultiple cameras. 2-D PIV (Bartol et al., 2003; Drucker andLauder, 2001; van Duren et al., 1998; Wilga and Lauder, 2002)provides high density data, but only in a plane. Typical stereoPIV (Nauen and Lauder, 2002; Prasad, 2000) provides all threevelocity components, but still only in a plane. It also requireselaborate calibration procedures. Existing scanning PIV(Brucker, 1997) and potentially future stereo-scanning PIV,provide data in multiple planes at different times at the cost ofadded complexity. All are unsuitable for following the 3-Dtrajectories of swimming organisms.

Alternatively, multiple pinhole photography (Kieft et al.,2002; Maas et al., 1993; Moroni et al., 2003; Ott and Mann,2000; Stuer et al., 1999; Virant and Dracos, 1997) and itsvariants (Pereira and Gharib, 2002) can be used for 3-Dtracking of particles. The increased depth of focus required by

this method is inherently coupled to reduced resolution and theneed for bright illumination. Conversely, in holography theimage resolution does not have to be compromised. Finedetails, e.g. setae and swimming appendages can be imaged atany depth. Furthermore, in pinhole images, the elongatedtraces of particles extend in depth through the entire volumeof interest. Consequently, matching of perpendicular views canonly be performed at low particle concentrations. Multiplecamera systems partially overcome this effect and can provideas many as 1600 vectors per time step, and may track as manyas several hundred particles over extended periods (Virant andDracos, 1997). Such systems require extensive calibration andrelatively complex processing algorithms. In in-lineholography, the elongated traces extend in depth less than1·mm, enabling matching of views at concentrations as high asseveral particles per mm3 (i.e. 13·500 in the presentoverlapping volumes). The associated calibration process isalso straightforward. On the other hand, one of theshortcomings of in-line holography, as shown in this paper, isthe noise generated in a great part by deterioration of thereference beam. This problem can be partially resolved byusing a separate reference beam that does not pass through thesample volume. A separate reference beam should also allowa much higher seeding concentration, but at the cost of addedcomplexity to the optical setup. When using high-resolutionemulsions, instead of a digital camera, we have successfullyreconstructed 200 particles per mm3. Another shortcoming ofholography is inherent to the use of coherent light, whichmakes the system more sensitive to the quality of windows andvariations of density within the sample volume.

As shown in this paper, digital holographic PIV is arelatively simple, but powerful tool for analysis of 3-Dcopepod (or other organism) dynamics and its interaction withits local environment, be it with other organisms or the local3-D flow field.

Copepod behavior

While generating the feeding current (Strickler, 1982, 1984)the mouthparts, studded with chemoreceptors (Friedman andStrickler, 1975), scan the water flowing by them. When thepresence of a food particle is perceived, additional movementsof the mouthparts capture the particle and bring it to the mouth(Koehl and Strickler, 1981; Strickler, 1984, 1985). Sensorysetae on the stretched out antennules (Fig.·4; Huys andBoxshall, 1991) are mechanoreceptors (Strickler and Bal,1973), as well as chemoreceptors (Bundy and Paffenhöfer,1993). These receptors enlarge the volume of water scannedfor food (Jiang et al., 2002a; Strickler, 1985). Particles that donot smell ‘good enough’ are either actively rejected aftercapture by the mouthparts, or passively rejected (allowed topass without being captured). For a stationary hoveringcopepod, these rejected particles remain in the water below thecopepod, and are not entrained into the feeding current again(Strickler, 1982).

The recirculating pattern in Figs·5–8 is, to our knowledge,the first report of a multi-encounter feeding pattern in calanoid

E. Malkiel and others

3665Three-dimensional copepod flow field

copepods. The combination of feeding and sinking results inparticles being drawn toward the copepod with its feedingcurrent, passively rejected and then recirculated. Therecirculation is interrupted aperiodically, after 8.7·s in thepresent example, when the copepod hops. During a hop, thecopepod jumps about 0.5·mm upward to a position where itsmouthparts are just below the stagnation point of the previousrecirculation zone. If it were not for the hop, the copepod wouldnever encounter new particles, due to the closed recirculationpattern. The recirculation allows the copepod to taste some ofthe particles that have passed near the mouthparts once more,this time with different, perhaps even more sensitivechemoreceptors on its antennules. Considering that the 20·µmpolystyrene, spherical particles are mechanically desirable butchemically undesirable, this additional sensing by the sensorson the antennules ensures that the animal does not forfeitpotentially good food. The available sequences of hologramssuggest that the timing between hops is sufficient for a rejectedparticle to reach the antennule (about half the recirculationcycle). We speculate that once the copepod senses, using itsantennules, a particle that has already been tasted and rejected(and is still not acceptable), it hops to another volume to lookfor different food. These assertions require testing by alteringthe properties of the particles, e.g. replacing them with desirablefood, and observing the behavior of the copepod.

This work was supported by the National ScienceFoundation. J.R.S. dedicates this work to Owen Phillips forhis support at a crucial moment years ago.

ReferencesBarnhart, D. H., Adrian, R. J. and Papen, G. C.(1994). Phase-conjugate

holographic system for high-resolution particle image velocimetry. Appl.Opt. 30, 7159-7170.

Bartol, I. K., Gharib, M., Weihs, D., Webb, P. W., Hove, J. R. and Gordon,M. S. (2003). Hydrodynamic stability of swimming in ostraciid fishes: roleof the carapace in the smooth trunkfish Lactophrys triqueter(Teleostei:Ostraciidae). J. Exp. Biol.206, 725-744.

Boxshall, G. A. (1998). Mating biology of copepod crustaceans – Preface.Philos. Trans. R. Soc. Lond. B353, 669-670.

Brucker, C. (1997). 3D scanning PIV applied to an air flow in a motoredengine using digital high-speed video. Meas. Sci. Technol.8, 1480-1492.

Bundy, M. H. and Paffenhöfer, G. A. (1993). Innervation of copepodantennules investigated using laser-scanning confocal microscopy. Mar.Ecol. Prog. Ser.102, 1-14.

Bundy, M. H. and Paffenhöfer, G. A. (1996). Analysis of flow fieldsassociated with freely swimming calanoid copepods. Mar. Ecol. Prog. Ser.133, 99-113.

Carder, K. L., Steward, R. G. and Betzer, P. R.(1982). In situ holographicmeasurements of the sizes and settling rates of oceanic particulates. J.Geophys. Res. Oceans Atmos.87, 5681-5685.

Chojnacki, J. (1983). Standard weights of the Pomeranian Bay Copepods. Int.Rev. Gesamten Hydrobiol.68, 435-441.

Doall, M. H., Colin, S. P., Strickler, J. R. and Yen, J.(1998). Locating amate in 3D: the case of Temora longicornis. Philos. Trans. R. Soc. Lond. B353, 681-689.

Drucker, E. G. and Lauder, G. V. (2001). Locomotor function of the dorsalfin in teleost fishes: experimental analysis of wake forces in sunfish. J. Exp.Biol. 204, 2943-2958.

Friedman, M. M. and Strickler, J. R. (1975). Chemoreceptors and feedingin Calanoid Copepods (Arthropoda – Crustacea). Proc. Natl. Acad. Sci. USA72, 4185-4188.

Hecht, E. (2002). Optics. Reading, MA: Addison-Wesley.

Heflinger, L. O., Stewart, G. L. and Booth, C. R.(1978). Holographicmotion-pictures of microscopic plankton. Appl. Optics17, 951-954.

Huys, R. and Boxshall, G. A.(1991). Copepod Evolution. London: RaySociety.

Jiang, H. H., Osborn, T. R. and Meneveau, C.(2002a). Chemoreception andthe deformation of the active space in freely swimming copepods: anumerical study. J. Plankton Res.24, 495-510.

Jiang, H. S., Meneveau, C. and Osborn, T. R.(2002b). The flow field arounda freely swimming copepod in steady motion. Part II: Numerical simulation.J. Plankton Res.24, 191-213.

Jiang, H. S., Osborn, T. R. and Meneveau, C.(2002c). The flow field arounda freely swimming copepod in steady motion. Part I: Theoretical analysis.J. Plankton Res.24, 167-189.

Katz, J., Donaghay, P., Zhang, J., King, S. and Russell, K.(1999).Submersible holocamera for detection of particle characteristics andmotions in the ocean. Deep-Sea Res. I-Oceanogr. Res. Pap.46,1455-1481.

Kebbel, V., Adams, M., Hartmann, H. J. and Juptner, W.(1999). Digitalholography as a versatile optical diagnostic method for microgravityexperiments. Meas. Sci. Technol.10, 893-899.

Kieft, R. N., Schreel, K., van der Plas, G. A. J. and Rindt, C. C. M.(2002).The application of a 3D PTV algorithm to a mixed convection flow. Exp.Fluids 33, 603-611.

Knox, C. and Brooks, R. E.(1969). Holographic motion picture microscopy.Proc. R. Soc. Lond. B174, 115-121.

Knutsen, T., Melle, W. and Calise, L.(2001). Determining the mass densityof marine copepods and their eggs with a critical focus on some of thepreviously used methods. J. Plankton Res.23, 859-873.

Koehl, M. A. R. (1984). Mechanisms of particle capture by copepods at lowReynolds number: Possible modes of selective feeding. In TrophicInteractions within Aquatic Ecosystems(ed. D. G. Meyers and J. R.Strickler), pp. 135-166. Boulder, CO: Westview Press.

Koehl, M. A. R. and Strickler, J. R. (1981). Copepod feeding currents – foodcapture at low Reynolds-number. Limnol. Oceanogr.26, 1062-1073.

Maas, H. G., Gruen, A. and Papantoniou, D.(1993). Particle trackingvelocimetry in 3-dimensional flows.1. Photogrammetric determination ofparticle coordinates. Exp. Fluids15, 133-146.

Malkiel, E., Alquaddoomi, O. and Katz, J. (1999). Measurements ofplankton distribution in the ocean using submersible holography. Meas. Sci.Technol.10, 1142-1152.

Mann, K. H. and Lazier, J. R. N. (1996). Dynamics of Marine Ecosystems:Biological–physical interactions in the oceans. Boston: Blackwell Science.

Milgram, J. H. and Li, W. C. (2002). Computational reconstruction of imagesfrom holograms. Appl. Optics41, 853-864.

Moroni, M., Cushman, J. H. and Cenedese, A.(2003). A 3D-PTV two-projection study of pre-asymptotic dispersion in porous media which areheterogeneous on the bench scale. Int. J. Eng. Sci.41, 337-370.

Nauen, J. C. and Lauder, G. V.(2002). Quantification of the wake of rainbowtrout (Oncorhynchus mykiss) using three-dimensional stereoscopic digitalparticle image velocimetry. J. Exp. Biol.205, 3271-3279.

O’Hern, T. J., D’Agostino, L. and Acosta, A. J. (1988). Comparison ofholographic and Coulter Counter measurements of cavitation nuclei in theocean. J. Fluids Eng.110, 200-207.

Ott, S. and Mann, J. (2000). An experimental investigation of the relativediffusion of particle pairs in three-dimensional turbulent flow. J. FluidMech.422, 207-223.

Owen, R. B. and Zozulya, A. A.(2000). In-line digital holographic sensorfor monitoring and characterizing marine particulates. Opt. Eng.39, 2187-2197.

Paffenhöfer, G. A., Bundy, M. H., Lewis, K. D. and Metz, C.(1995). Ratesof ingestion and their variability between individual calanoid copepods –direct observations. J. Plankton Res.17, 1573-1585.

Pereira, F. and Gharib, M. (2002). Defocusing digital particle imagevelocimetry and the three-dimensional characterization of two-phase flows.Meas. Sci. Technol.13, 683-694.

Pozrikidis, C. (1992). Boundary Integral and Singularity Methods forLinearized Viscous Flow.Cambridge England; New York: CambridgeUniversity Press.

Prasad, A. K. (2000). Stereoscopic particle image velocimetry. Exp. Fluids29, 103-116.

Pu, Y. and Meng, H.(2000). An advanced off-axis holographic particle imagevelocimetry (HPIV) system. Exp Fluids29, 184-197.

Roth, G. I. and Katz, J.(2001). Five techniques for increasing the speed andaccuracy of PIV interrogation. Meas. Sci. Technol.12, 238-245.

3666

Roth, G. I., Mascenik, D. T. and Katz, J.(1999). Measurements of the flowstructure and turbulence within a ship bow wave. Phys. Fluids11, 3512-3523.

Sheng, J., Malkiel, E. and Katz, J. (2003). Single beam two-viewsholographic particle image velocimetry. Appl. Optics42, 235-250.

Sridhar, G. and Katz, J. (1995). Lift and drag forces on microscopic bubblesentrained by a vortex. Phys. Fluids7, 389-399.

Strickler, J. R. (1977). Observation of swimming performances of planktoniccopepods. Limnol. Oceanogr.22, 165-170.

Strickler, J. R. (1982). Calanoid copepods, feeding currents, and the role ofgravity. Science218, 158-160.

Strickler, J. R. (1984). Sticky water: a selective force in copepod evolution.In Trophic Interactions within Aquatic Ecosystems(ed. D. G. Meyers andJ. R. Strickler), pp. 187-239. Boulder, CO: Westview Press.

Strickler, J. R. (1985). Feeding currents in calanoid copepods: two newhypotheses. In Physiological Adaptations of Marine Animals. Symposia ofthe Society for Experimental Biology, no. 39 (ed. M. S. Laverack), pp. 459-485. Cambridge: Company of Biologists Ltd.

Strickler, J. R. and Bal, A. K. (1973). Setae of the first antennae of thecopepod Cyclops scutifer(Sars): Their structure and importance. Proc. Natl.Acad. Sci. USA70, 2656-2659.

Stuer, H., Maas, H. G., Virant, M. and Becker, J.(1999). A volumetric 3Dmeasurement tool for velocity field diagnostics in microgravity experiments.Meas. Sci. Technol.10, 904-913.

Svetlichny, L. S.(1980). On certain dynamic parameters of tropical copepodpassive submersion. Ekologiya Morya2, 28-33.

Tao, B., Katz, J. and Meneveau, C.(2002). Statistical geometry of subgrid-scale stresses determined from holographic particle image velocimetrymeasurements. J. Fluid Mech.457, 35-78.

van Duren, L. A., Stamhuis, E. J. and Videler, J. J.(1998). Reading thecopepod personal ads: increasing encounter probability withhydromechanical signals. Philos. Trans. R. Soc. Lond. B 353, 691-700.

Vikram, C. S. (1992). Particle Field Holography. Cambridge, NY:Cambridge University Press.

Virant, M. and Dracos, T. (1997). 3D PTV and its application on Lagrangianmotion. Meas. Sci. Technol.8, 1539-1552.

Watson, J., Alexander, S., Craig, G., Hendry, D. C., Hobson, P. R.,Lampitt, R. S., Marteau, J. M., Nareid, H., Player, M. A., Saw, K. et al.(2001). Simultaneous in-line and off-axis subsea holographic recording ofplankton and other marine particles. Meas. Sci. Technol.12, L9-L15.

Wilga, C. D. and Lauder, G. V.(2002). Function of the heterocercal tail insharks: quantitative wake dynamics during steady horizontal swimming andvertical maneuvering. J. Exp. Biol.205, 2365-2374.

Xu, W., Jericho, M. H., Meinertzhagen, I. A. and Kreuzer, H. J.(2001).Digital in-line holography for biological applications. Proc. Natl. Acad. Sci.USA98, 11301-11305.

Yen, J. and Strickler, J. R. (1996). Advertisement and concealment in theplankton: What makes a copepod hydrodynamically conspicuous?Invertebr. Biol.115, 191-205.

Zhang, J., Tao, B. and Katz, J.(1997). Turbulent flow measurement in asquare duct with hybrid holographic PIV. Exp. Fluids23, 373-381.

E. Malkiel and others