Light scattering of small particles is important invarious scientific fields such as astrophysics, atmo-spheric sensing, combustion research, and microbiol-ogy. In most cases, light scattering is observed fromensembles of statistically oriented, nonspherical par-ticles of various sizes, hampering the interpretationof the observed intensity of the scattered light.Angular-resolved light scattering from single ori-ented particles or particle agglomerates yields infor-mation on the shapes of the scattering objects, ontheir internal structures, and on their indices of re-fraction. In addition, knowledge of the angular dis-tribution of light scattered from such particles wouldfacilitate theoretical analyses of light scattered byensembles of statistically oriented, arbitrarily shapedparticles.
Recently two-dimensional angular-resolved pat-
When this research was conducted, the authors were with theDivision of Medical Physics and Metrological Information Technol-ogy, Physikalisch-Technische Bundesanstalt, Abbestrasse 2–12,10587 Berlin, Germany. C. Gohlke is now with LINOS PhotonicsSociete a Responsabilite Limitee, Avenue de Lanessan 90, 69410Champagne au Mont d’Or, France. T. Wessel is now with Schaad,Balass, Menzl & Partner AG, Dufourstrasse 101, 8034 Zurich,Switzerland. J.Neukammer’se-mailaddress is [email protected].
Received 28 April 2003; revised manuscript received 7 August2003.
terns of light scattered from airborne clusters consist-ing of various numbers of polystyrene microsphereswere recorded in the near-forward and near-backward directions. Although the results could bequalitatively explained by a theoretical analysis,quantitative comparison of experimental and theo-retical data was not possible because the size andorientation of each cluster were unknown.1
Previously, differential or angular-resolved lightscattering of living bacterial or mammalian cells wasinvestigated as well. Approximately three decadesago it was shown that cells that belong to selectedspecies exhibit characteristic angular distributions2–4
of scattered-light intensity. Suspensions of cellswere studied; hence the specific angular distributionof light scattered from a single oriented cell could notbe measured because of orientational and size aver-aging and contributions from agglomerates. To ex-plain such experiments a theoretical study of lightscattering by aggregated red blood cells was per-formed recently.5 Scattering of laser light by singlebiological cells contained in the liquid sample flow ofa flow cytometer6 and passing single file through the�common� focus of one or several laser beams is rou-tinely used to count and differentiate blood cells. Todistinguish blood cells, integrated light scatter is ob-served, i.e., scattered light collected by several detec-tors within suitably chosen opening angles orientedperpendicularly to the sample flow and along or per-pendicularly to the laser beams. In addition, depo-larization of the scattered light, which depends on theinternal structure of cells, was shown to be charac-
teristic of eosinophilic granulocytes,7 a subpopulationof white blood cells. Apart from their morphologicalproperties derived from light-scatter experiments,8blood cells are distinguished according to their bio-logical functions by use of specific staining proce-dures, e.g., tagging with fluorescent monoclonalantibodies followed by observation of laser-inducedfluorescence. Because of its high throughput of�blood� cells and its high specificity and sensitivity toidentification of subpopulations of interest, flow cy-tometry has evolved into a powerful tool for researchand routine applications.9
Whereas cell differentiation by integral forwardand orthogonal light-scattering measurements incombination with specific fluorescent staining is rou-tinely used for medical diagnostics, other light-scattering signals such as backscattered light anddifferential light scatter have barely been explored,as was indicated in a review chapter by Hoekstra andSloot.10 First experiments with one-dimensionaldifferential light scatter of single mammalian cellswere performed by Salzman et al.,11 who used a flowcytometer. Recently this technique was improved toyield higher throughput and better angular resolu-tion12,13 and was applied for classifying particles in-cluding Escherichia coli14 and red blood cells.15
However, interpretation of the observed angular dis-tributions was limited to particles with high symme-try because angular resolution was restricted to thepolar angle only. Apart from flow cytometry mea-surements, scattering cross sections of single biolog-ical cells contained in an optical trap were measuredas function of polar angle.16 In that experiment,measuring times of more than 10 min did not allow alarge number of cells to be examined, as would berequired for reliable classification of subpopulationsof blood cells, for example.
In this paper we report on two-dimensionalangular-resolved light scattering of single blood cells,in particular, on sphered red blood cells �erythro-cytes�, on native erythrocytes elongated by hydrody-namic forces, and on white blood cells �lymphocytes�.We used a laser flow cytometer together with an in-tensified CCD camera as a detector to measure dif-ferential light-scattering cross sections. By thesame technique we recorded angular-resolved lightscattering of an oriented agglomerate �dumbbell� con-sisting of two identical polystyrene microspheres.Experimental differential cross sections were foundto be in good agreement with cross sections calculatedwithin the discrete dipole approximation.17,18 Theseflow cytometric measurements were supplementedby light-scattering experiments in which a three-dimensional optical trap was employed to orient lin-ear chains of as many as six identical polystyrenespheres in water. Microspheres suspended in waterwere arranged by optical tweezers under visual con-trol to form these agglomerates. In our optical trap,particles were stabilized laterally by gradient forcesand longitudinally by compensation for the �reduced�gravitational force by the force that is due to photonpressure. A three-dimensional trap is required for
investigation of light scattering from free particles,i.e., particles that are sufficiently far from any sur-faces. Because of the size of the agglomerates understudy, conventional three-dimensional optical traps,which are based on gradient forces only, cannot beused. Differential cross sections for forward lightscatter of trapped agglomerates were successfully an-alyzed by use of Mie theory together with the arraytheorem.
2. Description of the Experimental Setup
A. Flow Cytometer
We used a home-built flow cytometer with hydrody-namic focusing of the sample flow by a sheath flow torecord two-dimensional ��, �� angular-resolved lightscattering of single blood cells and oriented particleagglomerates. As shown in Fig. 1, the instrumentincluded an Ar� laser �Innova 304, Coherent Deut-schland GmbH, Dieburg, Germany�, operated at488.0 nm, a Kr� laser �Innova 302, Coherent Deut-schland GmbH� tuned to 413.1 nm, and a 632.8-nmHe–Ne laser �Model 106-1, Spectra-Physics, Moun-tain View, Calif.�. Each laser beam was shaped byspherical and cylindrical telescopes, and the laserbeams were superimposed by dichroic beam splittersand focused by a 1�, N.A. � 0.05 �focal length, 73.5mm� microscope objective �Melles Griot, Bensheim,Germany� to a common spot of approximately 20-�mdiameter. The laser beams intersected the flow cellperpendicularly. The polarization vectors of the lin-early polarized laser beams were parallel to the di-rection of the sample flow and the sheath flow. Aflow cell �Hellma GmbH, Mullheim, Germany� with asquare cross section �3.6 mm � 3.6 mm�, a length of6.5 mm, and a 250 �m � 250 �m flow channel wasused. The outer and inner faces of the flow cell werepolished to achieve high optical quality. The thick-ness �2 � 1.8 mm� of the cell was chosen to match thecover-glass correction �1.8 mm� of the objectives �Le-itz GmbH, Wetzlar, Germany� that we used to ob-serve forward �H20�, N.A. � 0.4, , 1.8 planar� andorthogonal �H32�, N.A. � 0.6, , 1.8 quartz glass�
Fig. 1. Flow cytometer modified to observe angular distributionsof light scattered by single particles: IF, interference filter; M,mirror; PMT, photomultiplier tube; BS, beam splitter; ICCD cam-era, intensified CCD camera.
light scatter. Both objectives were designed for in-finite object-to-image distance. For forward lightscattering, the angle of observation �BST � � � MO,0° � � � 2�� was determined by an on-axis circularbeam stop �BST � 3.3°� and by the half-opening angleMO � 17.4° of the H20�, N.A. � 0.4 microscopeobjective. The opening angle of H32�, N.A. � 0.6microscope objective, which was employed to detectintegral and angular-resolved orthogonal light scat-ter, amounted to MO � 27°. Because both objec-tives were highly corrected for aberrations over theentire field of view, including cover-glass corrections,a linear relation exists between angle —subtendedby the incident ray and the optical axis—and distance� of the emerging ray from the optical axis. We con-firmed this linear relation between and � requiredfor data analysis by ray tracing with the technicaldata of the H32�, N.A. � 0.6 microscope objective.19
For angles 22° � � 27°, small deviations from thislinear relation were observed but were neglected inthe data analysis. We achieved correct alignment ofthe objectives by observing the fluorescence of stainedmicrobeads passing through the flow cell. The dis-tance between the objectives and the flow cell wasoptimized for zero divergence of the fluorescenceemerging from the objectives. Tilting angles of theoptical axes of the microscope objectives with respectto the faces of the flow cell were adjusted until ahomogeneous distribution of the fluorescence inten-sity was obtained. Dichroic beam splitters wereused to separate scattered light of different wave-lengths. Interference filters, matched to the corre-sponding wavelengths, served to suppress ambientlight. We equipped the flow cytometer with an in-tensified CCD camera �Princeton Instruments ModelICCD-576; 14 bits, 384 � 576 pixels; from Roper Sci-entific GmbH, Ottobrunn, Germany� to record two-dimensional intensity distributions of scatteredlight. By removing or inserting the light stop anda movable mirror �Fig. 1� in the path of the scat-tered light, we could detect angular-resolved for-ward or orthogonal light scatter.
Particles were analyzed by �integral� forward andorthogonal light scattering �Fig. 2� observed simulta-neously at two wavelengths � � 413.1, 632.8 nm� byuse of superimposed Kr� and He–Ne laser beams.As indicated in Fig. 2�a�, hydrodynamic focusing ofthe sample stream by a second, concentric liquid�sheath� flow forced the particles to cross the laserbeams in single file. Typically, diameters and veloc-ities of the sample flow amounted to 1–5 �m and to5–10 m s�1, respectively. In this way individualparticles were detected at a high throughput rate.
In Fig. 2�b� we depict a scatter diagram of a dilute�1:100� whole blood sample by plotting the effectivescattering cross section for orthogonal light scatter-ing versus forward light scattering at 413.1 nm foreach cell. Four clusters can be identified in Fig. 2�b�,which correspond to red blood cells and three differ-ent kinds of white blood cell, i.e., lymphocytes, mono-cytes, and granulocytes. Kr� laser radiation at413.1 nm was chosen to differentiate red blood cells
from white blood cells by light scattering. The wave-length of 413.1 nm is close to the maximum of theoxyhemoglobin absorption band. The strong ab-sorption of the Kr� laser radiation leads to a reducedscattering cross section of red blood cells comparedwith that of white blood cells, allowing both cell pop-ulations to be distinguished by integral forward andorthogonal light scatter,20 as illustrated in Fig. 2�b�.Blood cells that belong to a subpopulation of interest,e.g., lymphocytes, were selected by their integrallight-scatter intensities before angular-resolved lightscatter was recorded. For this purpose we used two
Fig. 2. �a� Side view of the interaction region. Angular-resolvedlight scatter of single particles, selected upstream by integratedforward and orthogonal light scatter from probing beams, is ob-served by an intensified CCD camera, ICCD; PMT, photomultipliertube. �b� Typical scatter diagram of a dilute whole blood sampleobtained by simultaneous observation of integrated forward andintegrated orthogonal light scatter at 413.1 nm. Red blood cells�RBC� and white blood cells �leukocytes, i.e., lymphocytes �Ly�,monocytes �M�, and granulocytes �G�� are observed. Straightlines, discriminator settings of two single-channel analyzers �SCA1 and SCA 2� for selection of lymphocytes.
single-channel analyzers to select voltage ranges ofthe detector’s output signals corresponding to an areaof interest of the scatter diagram. To record two-dimensional light scatter of lymphocytes, for exam-ple, we set the single-channel analyzers SCA1 andSCA2 to cover the area between the two vertical andthe two horizontal lines, respectively, outlined in Fig.2�b�. A coincidence signal derived from the outputsof the two single-channel analyzers served to triggera generator, providing a properly delayed high-voltage pulse for opening the intensifier of the CCDcamera. This camera was used to detect light thatwas scattered when a selected blood cell crossed anAr� laser beam that was located 25 �m downstream.A typical exposure time of the CCD camera was 500ns; hence the movement of the cell through the laserfocus �20 �m, 1�e2� could be neglected. The sameprocedure was followed when we were investigatingangular-resolved light scattering of single polysty-rene microspheres of various sizes and of their ag-glomerates.
B. Optical Trap
Besides flow cytometry we used an optical trap �Fig.3� to position and orient single particles and particleclusters for subsequent investigations of differentiallight scatter. Single polystyrene microspheres ofidentical size �diameter, 10 �m� were collected fromthe bottom of a cuvette with five faces of optical qual-ity �1 cm � 1 cm � 3 cm; Hellma GmbH� and filledwith an aqueous suspension of polystyrene micro-spheres �100–1000 particles�mL�. As many as sixparticles were arranged by optical tweezers undervisual control to form linear chains. Subsequently,selected particles or agglomerates were trapped by anAr� laser beam � � 488.0 nm� and levitated by low-ering the cuvette by approximately 3 mm. The Ar�
laser beam was focused to a circular spot of �1.5-�mdiameter �1�e2� by means of a long working distancemicroscope objective �40�, N.A. � 0.5, OlympusGmbH Hamburg, Germany�. Depending on particlesize, the power of the Ar� laser beam required forstable trapping ranged from 0.5 to 10 mW. We mon-itored trapping of the particles by observing theirorthogonal light scatter of He–Ne � � 632.8 nm� andAr� laser radiation through an observation micro-
scope �20� objective, 15� ocular�. An ocular wasintegrated into the setup to form an inverted micro-scope that, together with the trapping objective, per-mitted visual control of the arrangement ofmicrospheres to clusters. We achieved bright fieldillumination by guiding the light from a tungstenbulb through an optical fiber �core diameter, 600 �m�,the end of which was positioned just above the watersurface.
To study differential light scatter of trapped parti-cle clusters of known size, shape, and orientation weused linearly polarized He–Ne laser radiation. Thesize of the focus �1�e2, d � 163 �m� of the He–Nelaser beam was chosen to considerably exceed the sizeof the largest agglomerates studied to approach aplane-wave geometry. Light scattered in the for-ward direction was collected on a ground-glassscreen. The direct laser beam was blocked by a cir-cular beam stop in front of the screen. The diffrac-tion pattern of the trapped particle on the back side ofthe screen was recorded by a CCD video camera �notshown in Fig. 3�. The field of view corresponded topolar angles � � 15°, approximately. Each pixel ofsize 11 �m � 11 �m of the CCD camera was theimage of a pixel of size 63.7 �m � 63.7 �m on theground-glass screen. In a separate experiment wecalibrated our detection system consisting of theground-glass screen, the objective, and the CCD chipof the 8-bit video camera that is necessary to deriveabsolute differential scattering cross sections. Forthis purpose the He–Ne laser beam was directed ontothe ground-glass screen and the beam profile wasimaged on the CCD chip. The calibration factor wasobtained from the total number of counts of all pixelsrecorded during the exposure time �40 ms� and themeasured power of the laser beam. At 632.8 nm, anoptical power of 2.7 nW incident upon an area of size63.7 �m � 63.7 �m of the ground-glass screen re-sulted in 255 counts of the corresponding pixel of theCCD detector, exploiting the detector’s full dynamicrange.
3. Experimental Results and Discussion
A. Angular-Resolved Side Scatter
In Fig. 4 we show angular-resolved side scatter of apolystyrene microsphere of d � 1.91 �m diameter�Fig. 4�a��, of an agglomerate of two identical 1.91-�mspheres �Fig. 4�b��, of a sphered �Fig. 4�c�� and of anative �Fig. 4�d�� red blood cell, and of a lymphocyte�Fig. 4�e��. The scattering geometry chosen is illus-trated in Fig. 4�f �; the hatched area indicates thescattering plane defined by the directions of the inci-dent radiation and of the scattered light. Polar an-gle � and azimuth angle � refer to a coordinatesystem whose z axis coincided with the wave vector ofthe incident Ar� laser radiation, which was linearlypolarized parallel to the x axis. The sample flow wasalong the negative x axis of the coordinate system.For each pixel centered at x, z of the CCD chip, lo-
Fig. 3. Experimental setup for observing angular distributions oflight scattering by optically trapped particles; the CCD video cam-era used to image the light scattered onto the ground-glass screennot shown.
cated at y � �y0, the coordinates ��, �� are obtainedfrom
x �l
MOarccos��sin � sin ��
sin � cos �
�1 � sin2 � sin2 ��1�2 ,
(1)
z �l
MOarccos��sin � sin ��
cos �
�1 � sin2 � sin2 ��1�2 ,
(2)
where 2l � 384 �x � 9.6 mm corresponds to the fullwidth quadratic illuminated area of the rectangularCCD chip and �x � �y � 25 �m is the width of eachpixel. To increase the signal-to-noise ratio we used3 � 3 pixel binning, which resulted in images of 128 �128 picture elements. For � � �90°, polar angle �depends linearly on pixel position z according to Eq.�1� and varies from 63° to 117°, corresponding to twicethe opening angle MO � 27° of the microscope objec-tive. Likewise, for � � 90°, azimuth angle � dependslinearly on pixel position x �Eq. �2�� and changes from�63° to �117°. It follows that the width of each�binned� pixel is equivalent to 0.422°. We normalized
the differential scattering cross section of each image�Fig. 4� to the corresponding maximum value to fa-cilitate a comparison among different angular distri-butions. Figure 4�a� exhibits the characteristicpattern of a homogeneous, dielectric sphere, i.e., ver-tical interference stripes, as expected from Mie the-ory21 for orthogonal light scattering. Quantitativeanalysis of the differential cross sections yielded adiameter of 1.91 �m for the particular polystyrenemicrosphere corresponding to a size parameter X ��d� med � 16.4, where med is the wavelength of theAr� laser light in the surrounding medium �water�.As predicted by Mie theory, we observed that thenumber of interference stripes increased with in-creasing size parameters.
Figure 4�b� depicts the angular distribution of lightscattered from an agglomerate �dumbbell� consistingof two identical microspheres �d � 1.91 �m�. Addi-tional interference stripes are observed because ofthe coherent superposition of the light scattered fromthe two spheres, similar to the diffraction of lightfrom two circular apertures. Hydrodynamic forcesexerted by the sheath flow aligned the dumbbellalong the flow �x� axis. It follows that the line con-necting the centers of the two microspheres was ori-ented perpendicularly to the incident laser beam andto the direction of observation. Therefore the addi-tional interference stripes observed are oriented hor-izontally.
To support our interpretation of Fig. 4�b� we usedthe discrete dipole approximation method17,18 to de-rive theoretical differential cross sections for lightscattered by an agglomerate of two identical micro-spheres, each with a diameter of 1.91 �m. The dis-crete dipole approximation method served tocalculate the elements of the scattering matrix orMueller matrix,21 and the linear combination S11 �S12 cos�2�� � S13 sin�2�� was then compared with theintensity distribution of Fig. 4�b� along the verticalcross section �� � 90°�.
As shown in Fig. 5�a�, the additional interferencesobserved experimentally �solid curves� as functions ofazimuth angle � are in good agreement with theoret-ical differential cross sections �dotted curves�. Thepositions of the calculated and observed maxima andminima do not coincide exactly, because we did not fitthe theoretical curve to the experimental data byvarying diameter d of the spheres of the dumbbell.The value d � 1.91 �m was derived from an analysisof several patterns observed for single spheres of themonodisperse sample and hence represents an aver-age value. The experimental trace was derived fromthe data by integration of the intensities of pixels �i,j� over the corresponding polar angles 88.3° � �i �91.7° for each azimuth angle �j in the range �117° ��j � �63°. Because the integration was carried outfor polar angles close to 90°, deviations from the lin-ear relationship between � and x are negligible com-pared with the width �x of each pixel. When wecompared experimental and calculated cross sections,we took a small misalignment of the CCD camera into
Fig. 4. Angular-resolved orthogonal light scatter �488.0 nm� ofsingle particles. �a� Polystyrene microsphere with diameter d �1.91 �m, �b� agglomerate of two identical polystyrene microspheres�d � 1.91 �m� oriented along the x axis �flow direction�, �c� spheredand �d� native red blood cells, �e� lymphocyte, �f � scattering geom-etry. The hatched area represents the scattering plane.
account by shifting the experimental trace by 2° withrespect to the calculated cross section.
A typical angular distribution of light scattered byan erythrocyte is depicted in Fig. 4�c�; it was spheredby a method22 that was assumed to be volume con-serving. Vertical interference stripes can be recog-nized, proving the nearly spherical shape of the redblood cell. However, discontinuities of these stripesare observed, presumably because of small deviationsof the erythrocyte’s shape from a perfect sphere.Previously, sphered human red blood cells �RBCs�were modeled as homogenous dielectric particles forcalculation of integrated differential light-scattering
cross sections, and good agreement with experimen-tal results was found.20 This approach is justifiedbecause RBCs do not contain cellular nuclei and theinfluence of the membrane can be neglected becausethe membrane is thin. Based on this approxima-tion, we calculated differential cross sections forsphered RBCs by using the generalized Lorenz–Mietheory23,24; i.e., we took the size of the focus of the Ar�
laser beam into account rather than assuming anincident plane wave. We compared the observed an-gular distribution of the scattered intensity with thecalculated scattering cross section �Fig. 5�b�� to de-duce the volume of the sphered erythrocyte. Forthis purpose we integrated scattered-light intensitiesassociated with pixels �i, j� over the correspondingazimuth angles �j of �96° � �j � �84° for each polarangle �i in the range 63° � �i � 117°.
Figure 5�b� compares the resultant trace with the-oretical data obtained from generalized Lorenz Mietheory; the index of refraction of the erythrocyte istaken to be n � 1.41 � i1.097 � 10�3 at 488.0 nm.25
Volume Very of the erythrocyte was varied until thebest correlation between the two traces was achieved.In this way we deduced a volume of Very � 98 �m3.This volume is slightly larger than the average vol-ume �90 �m3� of erythrocytes26 but lies well withinthe biological size distribution. Typically, the half-width of the size distribution of erythrocytes is 30–40�m3. By angular-resolved light scatter, volumes ofindividual sphered RBCs can be determined withhigh precision ��1%� because the observed patternsdepend sensitively on particle size.
The angular distribution of light scattered by anative RBC �Fig. 4�d�� does not show regular struc-tures to the same extent as observed for the spherederythrocyte. Connected regions of high intensity ap-pear in irregular order �Fig. 4�d��. However, theseregions are found predominantly at polar angles � �90°, indicating that light is scattered mostly in theforward direction. In a comparison of the angulardistributions of the native �Fig. 4�d�� and of thesphered �Fig. 4�c�� erythrocytes, the native RBC ex-hibits larger sizes of the connected regions of highintensity and a smaller spatial frequency along thehorizontal direction. Whereas sphered erythrocytescannot be deformed easily, hydrodynamic shearforces elongate native RBCs along the direction of thesheath flow. Hence the optical path of the laserbeam, incident perpendicularly onto the axis of elon-gation, is shorter for native RBCs, and the spatialfrequency decreases accordingly.
Angular-resolved light scatter of a lymphocyte isdepicted in Fig. 4�e�. Interference fringes are dis-cernable along vertical curves with large radii of cur-vature. We analyzed the pattern shown in Fig. 4�e�in the same way as described above for the angulardistribution of light scattered by the spherical eryth-rocyte. In particular, we integrated over the samerange of azimuth angles �, obtaining the experimen-tal trace shown in Fig. 5�c�. We calculated theoret-ical differential scattering cross sections bygeneralized Lorenz–Mie theory, assuming an index of
Fig. 5. Comparison of experimental �solid curves, arbitrary units�and theoretical �dotted curves� differential light-scattering crosssections corresponding to the angular distributions shown in Figs.4�b�–4�e�. �a� Vertical cross section through Fig. 4b �dumbbell�compared with �S11 � S12 cos �2�� � S13 sin�2��� calculated by thediscrete dipole approximation. Horizontal cross sections �b�through Fig. 4�c� �sphered red blood cell� and �c� through Fig. 4�e��lymphocyte� compared with the matrix elements �S1����2 calcu-lated by generalized Lorentz–Mie theory.
refraction of the lymphocyte of n � 1.37. This valuewas originally derived for the cytoplasm by phasecontrast microscopy in an investigation of Chinesehamster ovary4 cells and was recently used to simu-late differential light scattering from biological cellsby finite-difference time-domain calculations.27
Varying volume Vly of the lymphocyte from 150 to 500�m3, we obtained the best agreement between exper-imental and theoretical values at Vly � 230 �m3.This value coincides with the mean cellular volume oflymphocytes.26 However, discrepancies between ex-perimental and theoretical data �Fig. 5�c�� were ob-served, as the refractive indices of cell constituentsare not known precisely and our analysis treated thelymphocyte as homogeneous dielectric sphere, an as-sumption that is not strictly justified. In addition,leukocytes from an individual human being exhibit abroad distribution of volumes, indicating that the ob-served agreement is fortuitous.
B. Angular-Resolved Forward Scatter
Angular distributions of light scattered from a He–Nelaser beam in the forward direction by linear chainsof polystyrene microspheres �d � 10 �m� are depictedin Fig. 6 for agglomerates of two �Fig. 6�b��, four �Fig.6�c��, and six �Fig. 6�d�� particles. The linear chainsare oriented approximately along the trapping Ar�
laser beam, propagating along the x direction �Fig.6�a��. The direction of polarization of the He–Ne la-ser coincided with this axis. The shadow of the cir-cular beam stop used to block the direct He–Ne laserbeam appears at the center of the interference ringsin each diffraction pattern. The annular patternthat is due to forward light scatter of each sphere ofthe agglomerate exhibits additional interference
lines, oriented perpendicularly to the particularchain. These interference lines are tilted becausethe linear chains are aligned along the light cone ofthe strongly focused trapping laser beam.
Compared to the angular distribution associatedwith two spheres �Fig. 6�b��, two �four� additionalsecondary maxima are observed in Fig. 6�c� �Fig. 6�d��when the length of the chain is increased to four �six�spheres. Qualitatively, the observed interferencepatterns for p spheres can be interpreted as diffrac-tion images from p identical circular apertures, ar-ranged in a straight line and spaced a distance equalto the diameter of the spheres. For quantitativeanalysis we used Mie theory to account for differen-tial light scatter of a single sphere and applied thearray theorem to calculate light-scattering cross sec-tions of linear chains. This model was adopted be-cause calculations based on the discrete dipoleapproximation did not converge when 1.5 � 106 di-poles were taken to describe a linear chain of 6spheres with a volume of �3000 �m3 and a refractiveindex of 1.6. From the patterns shown in Fig. 6 wederived experimental differential scattering crosssections according to
�dCsca��, ��
d� �exp
��Wsca� x, y�
Iinc��, (3)
where �Wsca�x, y� denotes the power of the scatteredlight incident upon the pixel centered at xy of theground-glass screen, positioned at z � z0. Iinc is theirradiance of the laser beam incident upon the parti-cle chain. We obtained �Wsca�x, y� from the countsof the corresponding pixel of the CCD detector, usingthe calibration factor given in Subsection 2.B. Angleof observation �� was calculated from pixel size�x�y of the ground-glass screen according to the fol-lowing equation:
�� ��F � eR� x, y�
�R� x, y��2 ��x�y cos3�
z02 . (4)
Vector �F is normal to the ground-glass screen, and,pointing along the He–Ne laser beam, its absolutevalue is equal to pixel area �x�y � 63.7 �m � 63.7�m.
R�x, y� � R�x, y�eR�x, y� denotes the direction of thescattered He–Ne laser radiation onto the pixel cen-tered at xy, its absolute value R�x, y� is the distanceof the pixel from the origin, i.e., from the center ofgravity of the scattering object; polar angle � is theangle between �F and the direction eR�x, y�; z0 is thedistance between the origin and the screen �Fig. 3�.Combining Eqs. �3� and �4�, we obtained for the dif-ferential cross section along the y axis �x � 0, z � z0�,i.e., perpendicular to the chain axis,
�dCsca��, � � �90°�
d� �exp
��Wsca� x � 0, y�
Iinc��x�y�z02�cos3 �
. (5)
Fig. 6. Angular-resolved forward light scatter �632.8 nm� of par-ticle agglomerates of polystyrene microspheres �d � 10 �m� in anoptical trap. �a� Scattering geometry. The hatched area repre-sents the scattering plane. Angular distributions of scatteredlight recorded at position z � z0 of the ground-glass screen �see Fig.3� for linear chains of �b� two, �c� four, and �d� six identical micro-spheres.
The corresponding theoretical differential scatteringcross section is given by
�dCsca��, � � �90°�
d� �theo
��S1����2
kmed2 p2, (6)
where p is the number of microspheres that form thechain and kmed is the wave vector of the incident laserbeam in the surrounding medium �water�. S1��� isone of the elements of the amplitude scattering ma-trix calculated from Mie theory21 for a single sphere.Along the chain axis, i.e., in the x direction �y � 0, z �z0�, the angular distribution of the light scatteredfrom a single microsphere is modulated because ofthe interference of the light scattered from the otherspheres of the chain. According to the array theo-rem, the interference pattern of p microspheres ofdiameter d can be approximated by introduction ofthe �normalized� grating factor Fgrat:
�dCsca��, � � 0°, 180°�
d� �theo
��S2����2
kmed2 p2Fgrat, (7a)
�dCsca��, � � 0°, 180°�
d� �theo
��S1����2
kmed2 p2Fgrat, (7b)
Fgrat �sin2� p��d� med�sin ��
p2 sin2���d� med�sin ��. (8)
The first factor �shape factor� in Eqs. �7a� and �7b�,which describes the angular distribution of light scat-tered from a single sphere, was calculated from Mietheory. Because forward scatter was observed forpolar angles � � 15°, we used the small-angle approx-imation �S2����2 � �S1����2. Grating factor Fgrat, nor-malized to unity at � � 0°, takes the interference oflight into account, scattered from p parallel slits at aspacing d equal to the diameter of the microspheres.
In Fig. 7 we compare experimentally observed dif-ferential scattering cross sections with differentialcross sections calculated according to relations �6�,�7b�, and �8�. Experimental and theoretical resultswere divided by p2 for normalization. Experimentalresults �dCsca�d��exp were derived from Figs. 6�b�–6�d� for x � 0 �horizontal section� by use of Eq. �5� andfor y � 0 �vertical section� by use of Eq. �5� corre-spondingly modified for y � 0. It should be notedthat the axis connecting the centers of microspheres,rather than the polarization direction, was chosen asthe x axis in each case, because the influence of po-larization can be neglected for small polar angles aswe indicated above by setting �S2����2 � �S1����2. It isevident from the upper three traces of Fig. 7, whichcorrespond to horizontal sections through images6�b�–6�d�, that experimental scattering cross sections�solid curves� exhibit the characteristic angular de-
pendence expected from Mie theory �dotted curves�for a single sphere. In particular, the scattering an-gles at which maxima and minima occur are wellreproduced. Vertical sections through images 6�b�–6�d� are displayed in the lower three traces of Fig. 7.The additional modulation of the differential scatter-ing cross section caused by interference between themicrospheres as described by Eq. �7� is conspicuous.In particular, for p � 6 and � � 0°, four secondarymaxima are observed between the first and secondand between the second and third principal maxima.The intensity of the first principal maximum couldnot be measured because of the insufficient dynamicrange of the CCD array. Positions of minima andmaxima determined experimentally along verticalsections �Figs. 6�b�–�d�� are in good agreement withpositions calculated by use of the array theorem.Small displacements between calculated and ob-served positions occur for p � 2, � � 0°, indicatinglimitations of the simple model used. At larger po-lar angles discrepancies are observed between abso-
Fig. 7. Comparison of experimental differential light-scatteringcross sections �solid curves� derived from Fig. 6 and calculatedcross sections �dotted curves�. Top three traces are along hori-zontal sections �x � 0, z � z0�; lower three traces represent vertical�y � 0, z � z0� sections after rotation of the distributions by thecorresponding small tilting angle.
lute values of experimental and calculateddifferential scattering cross sections. These devia-tions might be caused by the nonisotropic scatteringcharacteristics of the ground-glass screen �Subsec-tion 1.A�. In addition, the dramatic variation of cal-culated scattering cross sections covering more thanfour decades cannot be reproduced experimentallybecause of the limited dynamic range of the videoCCD camera used.
To facilitate comparison of experimental and theo-retical light-scattering cross sections obtained by thediscrete dipole approximation, we measured forwardlight scatter of clusters of microspheres with volumessmaller than 30 �m3, using the modified flow cytom-eter described in Subsection 2.A �Figs. 1 and 2�. Fig-ure 8�a� shows angular-resolved forward light scatterof an agglomerate of two polystyrene microspheres,oriented along the flow direction �x axis� by hydrody-namic forces and with a diameter of 1 �m each. Thecoordinates of a pixel of the intensified CCD camerawere converted into polar and azimuth angles � and� according to x � �l�MO�cos � and y � �l�MO�sin �.As discussed in the preceding paragraph, destructiveinterference of the light scattered from both spheresresulted in two narrow, horizontally oriented stripes.The broad shadow that appears between these inter-ference minima is caused by the rectangular beamstop mounted in front of the microscope objectiveused for observation. The direct laser beam wassuppressed to protect the intensifier of the CCD cam-era. Figure 8�b� shows a vertical cross section
through the image of Fig. 8�a� at the position indi-cated by the arrow in Fig. 8�a�, i.e., along the x axis�see Fig. 6�a��. The solid curve represents the mea-sured differential light-scattering cross section for po-lar angles ranging from � � 6° to � � 24° at azimuthangles � � 0° and � � 180°. For comparison, thedifferential cross section calculated by the discretedipole approximation17,18 is included in the figure asa dashed curve. Good agreement is observed be-tween experimental and calculated results for crosssections exceeding 0.5 �m2 sr�1. In addition, ob-served and calculated interference minima appear atthe same polar angle. However, the dramatic de-crease of the differential scattering cross section pre-dicted by the discrete dipole approximation could notbe observed experimentally because of the limiteddynamic range ��100:1� of the detector. For illus-tration, we have included the corresponding noiseequivalent differential cross section of 0.28 �m2 sr�1
as a vertical dashed straight line �Fig. 8�b��. Thegood agreement between experiment and theorydemonstrates that the discrete dipole approximationis suitable for calculating light-scattering cross sec-tions of irregularly shaped micrometer-sized parti-cles and particle agglomerates.
4. Conclusions and Perspectives
We have measured two-dimensional angular distri-butions of light scattered by single, oriented particlesor particle agglomerates, using a gated intensifiedCCD camera integrated into a flow cytometer. Typ-ical exposure times amounted to 500 ns. The tech-nique presented in our paper is suited to study ofparticles at high throughput to investigate the scat-tering properties of various arbitrarily shaped ob-jects. Two-dimensional angular distributions oflight scattered by single, oriented particles or ag-glomerates permit direct comparison with theoreticalresults. In contrast, information on size, shape, andorientation of individual particles cannot be deducedfrom light-scattering experiments carried out withsuspensions. We calculated the differential light-scattering cross sections for a dumbbell consisting oftwo small microspheres, using the discrete dipole ap-proximation, and found good agreement with exper-imental results for agglomerates with volumes of lessthan 10 �m3.
In addition to investigating polystyrene micro-spheres, we recorded two-dimensional differentialside scatter of single blood cells and derived the vol-umes of sphered red blood cells by employing Mietheory. The angular distribution of the scatteredlight depends sensitively on the particle size, thusallowing us to determine volumes of sphered RBCswith high precision �approximately 1%�. It seemslikely that volumes and shapes of native RBCs can bederived by application of the discrete dipole approx-imation. Furthermore, this theory may allow us tocalculate differential light scatter of more-complexbiological cells, including their intracellular compo-nents. There are effectively no limitations on thesizes of blood cells that can be analyzed by the dis-
Fig. 8. �a� Forward light scatter �488.0 nm� of a dumbbell com-posed of two microspheres, each with a diameter of 1 �m. �b�Comparison of measured �solid curve� and theoretical �dashedcurve� differential cross sections, calculated by the discrete dipoleapproximation.17,18 Vertical dashed line, noise equivalent differ-ential cross section of the detector.
crete dipole approximation, because the refractive in-dices of cells lie in the range 1.35–1.42, close to therefractive index 1.33 of the surrounding water. Forexample, particles with volumes up to 1000 �m3,which are larger than the volumes of RBCs �100 �m3�and white blood cells �150–450 �m3� can be studiedin this way. Different compartments of biologicalcells, such as mitochondria or the nucleus, can beaccounted for by assigning different polarizability orrefractive index to the dipoles within these volumes.Although angular-resolved light scatter has the po-tential to improve flow cytometric discrimination ofvarious subpopulations of blood cells, we made nosystematic study by using various microscope objec-tives or applying pattern recognition for data analy-sis.
Furthermore, we used an optical trap to measurethe distribution of light scattered by particle agglom-erates. Differential forward light scatter of linearchains of 10-�m polystyrene spheres �n � 1.6� wasobserved. Because the discrete dipole approxima-tion cannot be applied to these agglomerates becauseof computational limitations, experimental data werecompared to theoretical cross sections by use of Mietheory together with the array theorem, and satisfac-tory agreement was obtained.
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