Photometric immersion refractometry: a method for determining the refractive index of marine...

$: Photometric immersion refractometry: a method for determining the refractive index of marine microbial particles from beam attenuation$
Photometric immersion refractometry:a method for determining the refractive index ofmarine microbial particles from beam attenuation

Miroslaw Jonasz, Georges Fournier, and Dariusz Stramski

Photometric immersion refractometry is a technique for determining the refractive index of particulatematerial. In this technique, the attenuation of light by a suspension of particles is measured as afunction of the refractive index of the immersion medium. A minimum attenuation occurs at therefractive index of the medium equal to that of the particles. This technique can serve as a benchmarkmethod for the refractive index determination because it is independent of assumptions invoked by othertechniques, such as those based on the inversion of the spectral attenuation data. We present a rigorousmodel of the photometric immersion refractometry based on the anomalous diffraction approximation forthe attenuation efficiency of particles. This model permits one to determine the average value of the realpart of the refractive index of the particles, its variance, and the average imaginary part of the refractiveindex of the particles. In addition, the fourth moment of the particle size distribution can be determinedif the concentration and shape of the particles are known. We analyze the sensitivity of this model toexperimental errors and discuss the applicability of photometric immersion refractometry to marinemicrobial particles. We also present experimental results of this technique as applied to heterotrophicmarine bacteria. The results indicate that the refractive index of these bacteria was narrowly distrib-uted about the average value of 1.3886. © 1997 Optical Society of America

1. Introduction

The refractive index is an important characteristic ofsmall particles in suspensions and aerosols. Thisimportance stems essentially from two sources.First, the refractive index of a particle is related to,and thus can be used to infer, the composition of theparticle. This relationship is reflected in a close cor-relation of the refractive index of minerals with theirdensity.1 Similar correlation has been establishedbetween the refractive index of biological cells andtheir content.2,3 It was shown that the refractiveindex of biological cells is a measure of the intracel-lular solids concentration4 and, in particular, of theintracellular carbon concentration.5,6 In the case of

M. Jonasz is with M. Jonasz Consultants, 217 Cadillac Street,Beaconsfield, Quebec H9W 2W7 Canada. G. Fournier is with theDefence Research Establishment Valcartier, P.O. Box 8800,Courcelette, Quebec G0A 1A0 Canada. D. Stramski is with theMarine Physical Laboratory, Scripps Institution of Oceanography,University of California, San Diego, La Jolla, California 92093-0701.

Received 15 July 1996; revised manuscript received 2 December1996.

0003-6935y97y184214-12$10.00y0© 1997 Optical Society of America

4214 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997

biological cells, the refractive index is also affected by,and thus can be used to monitor, the physiologicalstate of the cells.6,7

Second, the refractive index is an essential param-eter in the modeling of light scattering and absorp-tion by particles. In complex suspensions, such asseawater, better knowledge of the refractive indicesof various particle classes is required to assess accu-rately the contributions of these classes to light scat-tering. For example, it has been postulated8,9 thatbacteria may contribute significantly to the opticalproperties of seawater, especially to the fraction oflight backscattering associated with living micro-organisms. The backscattering coefficient of seawa-ter is an important parameter to the remote sensingof the marine environment and to the optimization ofunderwater imaging systems.

Among several methods of determining the refrac-tive index of particles, a technique of immersion re-fractometry has gained an important place in biologyand mineralogy. In the original version of that tech-nique, as applied in biology, cells are suspended in anisotonic protein solution whose refractive index is ad-justed to match that of the cells.4 That techniquerequires tedious microscopic observations that werereplaced, in photometric immersion refractometry,

with measurements of the optical density ~OD! of asuspension.10–14 In photometric immersion refrac-tometry, the OD of a suspension of cells is measuredas a function of the refractive index of the immersionmedium. When the latter index matches that of thecells, the minimum OD is observed. In few pub-lished investigations, the OD measurements weremade with use of general purpose spectrophotom-eters that are not optimized for use with suspensions.Consequently, it is difficult to interpret the OD interms of relevant properties of particles in the sus-pension, even though a simple model of the photo-metric immersion refractometry has been proposed.13

In this paper we derive generalized relationshipsbetween the beam attenuation of light by particles ina suspension and the refractive index of the immer-sion medium. These relationships are based on theanomalous diffraction approximation15 for the atten-uation efficiency of particles. We account for thepolydispersity of the size and refractive index of theparticles and also for the absorption of light by theparticles. We also describe the results of a photo-metric immersion refractometry experiment and as-sociated determinations of the refractive index ofheterotrophic marine bacteria.

2. Significant Notations and Abbreviations

a2, a1, a0 Coefficients of a second-order poly-nomial used to approximate theOD of a suspension as a function ofthe refractive index of the immer-sion medium @Eq. ~22!#.

A ~cm21! A parameter expressing the ratioof the thickness of a suspensionlayer to the square of the wave-length of light in vacuum @Eq.~15!#.

c ~cm21! Attenuation coefficient of a me-dium @Eq. ~6!#.

Cat ~cm2! Attenuation cross section of a par-ticle @Eq. ~7!#.

Cs ~nondimensional! Particle shape coefficient @Eq.~13!#.

D ~cm! Diameter of a particle—a charac-teristic particle size @Eq. ~3!#.

^D3& (cm3! The third moment of the particlesize distribution @Eq. ~29!#.

^D4& ~cm4! The fourth moment of the particlesize distribution @Eq. ~27!#.

Dpeak ~cm! Peak diameter of the log-normaldistribution @Eq. ~B1!#.

FDM ~D, m!~cm24! Generalized size refractive-indexdistribution function

FDmax ~cm24! A scale constant of the log-normalsize distribution @Eq. ~B1!#.

g~D! ~cm24! Frequency size distribution of par-ticles @Eq. ~4!#.

G A factor in the expression for ^k&@Eq. ~31!#.

k ~nondimensional! Imaginary part of the refractiveindex of a particle relative to vac-uum @Eq. ~11!#.

^k& ~nondimensional! Average k in a population of par-ticles @Eq. ~28!#.

L ~cm! Thickness of a layer of suspension.m ~nondimensional! Complex refractive index of a par-

ticle relative to vacuum @Eq. ~11!#.n ~nondimensional! Real part of the refractive index of

a particle relative to vacuum @Eq.~11!#.

^n& The average refractive index of aparticle population @Eq. ~26!#.

ns ~nondimensional! Refractive index of the mediumsurrounding a particle ~immersionmedium!.

N ~cm23! Number concentration of particleswith diameters D ~cm! betweenDmin and Dmax, and with refrac-tive indices between mmin andmmax @Eq. ~2!#.

OD ~nondimensional! Optical density of a medium percentimeter of the medium thick-ness @Eq. ~1!#.

P ~cm2! Geometric cross section of the par-ticle projected onto a plane per-pendicular to the direction of thepropagation of light.

Pavg ~cm2! Orientation-averaged geometriccross section of the particle pro-jected onto a plane perpendicularto the direction of the propagationof light @for cylinders with hemi-spherical cups, see Eq. ~44!#.

Qat ~nondimensional! Attenuation efficiency of a particle@Eqs. ~8!, ~12!, and ~13!#.

s ~nondimensional! Slope of a power-law approxima-tion to the OD wavelength spec-trum @Eq. ~45!#.

l ~cm! Wavelength of light in vacuum.r ~nondimensional! Dimensionless size of a particle

@Eq. ~9!#.sD ~nondimensional! Width parameter of the log-

normal size distribution @Eq. ~B1!#.sn

2 ~nondimensional! Variance of the refractive index ina particle population @Eq. ~37!#.

ADA Anomalous diffraction approxima-tion.

BSA Bovine serum albumin.LSQ Least-squares approximation.

3. Theoretical Model of Photometric ImmersionRefractometry

A. Single-Wavelength Model

In this section we develop a simple model relating theoptical density ~at a single wavelength! of a suspen-sion of particles to the refractive index of the immer-sion medium. The major observables of our modelare the optical density OD of the suspension and thereal part ns of the refractive index of the immersion

20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4215

medium. The OD can be expressed as follows:

OD 5 cL log e, (1)

where c is the beam attenuation coefficient ~for ex-ample, Ref. 16!, and L is the thickness of the relevantlayer of the suspension. Because the OD is definedin our model in terms of the beam attenuation coef-ficient, the spectrophotometer used to measure theOD must have an appropriate optical path geome-try.14,17 Also, it is assumed that a double-beam spec-trophotometer is used so as to cancel the effect ofresidual light attenuation by the immersion mediumon the OD measurements.

We assume that each particle in a population ishomogeneous but the population may be polydispersein respect to the particle size and refractive index.No assumptions about the polydispersity of the par-ticles are made. However, the accuracy of certainpredictions of this model improves if the particles areweakly polydisperse. The population can be de-scribed with the generalized size refractive-index dis-tribution function FDM~D, m!, defined as follows:

N 5 ** FDM ~D, m!dD dm, (2)

where N is the number concentration of particleswith diameters D between Dmin and Dmax and withrefractive indices between mmin and mmax. The lat-ter four variables serve as the limits of the integrals.The diameter of a particle is a characteristic dimen-sion of the particle. In the present context, this di-mension is defined as follows:

D 5 @~4yp!P#1y2, (3)

where P is the geometric cross section of a particleprojected onto a plane perpendicular to the directionof the propagation of light. We assume that the re-fractive index of particles and their size are not cor-related. With this assumption, discussed further inAppendix A, the function FDM can be expressed asfollows:

FDM~D, m! 5 g~D!h~m!, (4)

where

h~m! ; h~n! ; h~k!. (5)

Equation ~5! applies, for example, when variations inthe refractive index of marine microbial cells arecaused by dilution of cell contents with water.2

Using the generalized distribution function of theparticle population, we can express the beam atten-uation coefficient of the population as follows:

c 5 ** Cat~D, m, l!FDM~D, m!dD dm, (6)

where

Cat 5 Qat~D, m, l!P~D! (7)


is the attenuation cross section and Qat is the atten-uation efficiency of a particle ~for example, Ref. 16!.The attenuation efficiency Qat is a function of therefractive index of the particle m, the refractive indexof the surrounding medium ns, the particle diameterD, and the wavelength of light in vacuum l. Equa-tion 6 is valid under the assumption that single scat-tering dominates the light-scattering process in thesuspension. Such a situation occurs if particles arefar away from each other, that is, if the interparticledistance is much greater than l. We discuss thisfurther later in this section.

By using the anomalous diffraction approxima-tion15 ~ADA!, we have

Qat 5 2 2 ~4yr!exp~2r tan b!cos b sin~r 2 b!

2 ~4yr2!exp~2r tan b!cos2 b cos~r 2 2b!

1 ~4yr2!cos2 b cos 2b, (8)

where

r 5 2p~Dyl!~n 2 ns!, (9)

tan b 5 ky~n 2 ns!, (10)

m 5 n 2 ik, (11)

where n and k are, respectively, real and imaginaryparts of the refractive index of the particle, ns is therefractive index of the medium surrounding the par-ticle, and i 5 ~21!1y2. All indices are given relativeto the vacuum.

The anomalous diffraction theory applies when theum 2 nsu ,, ns and when the particle is “large” com-pared with the wavelength of light in the mediumsurrounding the particle. We use quotation marksbecause the ADA works in a surprisingly wide parti-cle size range.

Equation ~8! can be well approximated15 by thefollowing formula when r ' 0 ~as a consequence ofn 2 ns ' 0!:

Qat < 2~pyl!2D2@~n 2 ns!2 2 k2# 1 ~8y3!~pyl!Dk. (12)

Equation ~12! can be transformed to the followingform:

Qat < Cs4~p2yl2!D2@~n 2 ns!2 2 k2# 1 Cs~16y3!

3 ~pyl!Dk, (13)

where Cs 5 1y2 represents the particle shape factorfor a sphere. We factored out 1y2 and replaced it bya more general coefficient Cs so as to account for theinfluence of the particle shape on Qat. For example,for long, thin, and randomly oriented uniform cylin-ders Cs 5 4y3, as can be inferred from Ref. 18 @sub-stitute for the first Struve function in Eq. ~51! of Ref.18 its small-argument approximation 2r2y~3p sin2 b!,where b is the angle between the direction of theincident radiation and the axis of the cylinder#. Inthis latter case, D represents the diameter of thecylinder.

From Eqs. ~13!, ~7!, ~6!, ~4!, and ~1! we obtain the

following expression for the optical density of a sus-pension:

OD < CsAH* D4g~D!dD * @~n 2 ns!2 2 k2#h~m!dm

1 ~4y3!~lyp! * D3g~D!dD * kh~m!dmJ, (14)

where A is defined as follows:

A 5 p3 log eLl22. (15)

By using the definition of an rth moment, ^xr&, ofdistribution f of variable x:

^xr& 5 * xr f ~x!dxY* f ~x!dx (16)

and making use of Eq. ~5!, we can transform Eq. ~14!as follows:

OD 5 CsAN@^D4&^~n 2 ns!2& 2 ^D4&^k2&

1 ~4y3!~lyp!^D3&^k&#, (17)

where N is the number concentration of particles @Eq.~2!#. Furthermore, because

^x2& 5 sx2 1 ^x&2, (18)

where sx2 is the variance of variable x, we obtain

OD < CsAN^D4&ns2

2 CsAN^D4&2^n&ns

1 CsAN^D4&~sn2 1 ^n&2!

1 CsAN~4y3!~lyp!^D3&^k&, (19)

where sn2 is the variance of the real part of the re-

fractive index of the particles. In deriving Eq. ~19!we neglected a term involving ^k2& that, for realisticvalues of k, is much smaller than the term involving^k&. From Eq. ~19! it follows that sn

2 and ^k& cannotbe separated with single-wavelength measurementsof OD versus ns. We discuss below how these twoparameters can be separated with multiwavelengthimmersion refractometry.

The minimum optical density occurs at ns 5 ^n& andis equal to

ODmin < CsAN@^D4&sn2 1 ~4y3!~lyp!^D3&^k&#. (20)

If the effect of the absorption of light by the particleson the ODmin is dominant, then the ODmin can beexpressed as follows:

ODmin < CsAN~4y3!~lyp!^D3&^k&. (21)

Equation ~19! is an equation of a second-order poly-nomial in the refractive index of the medium ns:

OD 5 a2ns2 1 a1ns 1 a0. (22)

The coefficients of Eq. ~22! can be obtained by onefitting this equation to results of measurements of the

OD of a suspension versus the refractive index of theimmersion medium. A comment about such mea-surements is appropriate. The domain of the pho-tometric immersion refractometry is a small range ofthe refractive indices of the immersion medium aboutns 5 ^n&. This implies relatively low values of theOD and significant relative errors of the OD measure-ments. The importance of these errors can be re-duced when the optical density is increased, i.e., bythe use of a concentrated suspension. However, theparticle concentration cannot be too high, otherwisemultiple scattering of light in the suspension mayprevent one from using the measured OD as an esti-mate of the single-scattering OD assumed in thismodel. From the discussion of multiple scattering~for example, Ref. 19! it follows that the OD of thesuspension should be significantly less than approx-imately 0.4 ~at a path length of 1 cm!.

By comparing Eqs. ~19! and ~22! we can expresscoefficients a2, a1, and a0 as functions of the charac-teristics of the particle population:

a2 5 CsAN^D4&, (23)

a1 5 2CsAN^D4&2^n&, (24)

a0 5 CsAN@^D4&~sn2 1 ^n&2! 1 ~4y3!~lyp!^D3&^k&#.

(25)

Equations ~23! and ~24! can be solved for the averagereal part of the refractive index of the particles:

^n& 5 2a1y~2a2!. (26)

The retrieval of ^n& is not affected by the variability inthe real part of the refractive index of the particlesnor by the absorption of light by these particles.

If we know the number concentration of particles Nand the particle shape coefficient Cs, we can also ob-tain an estimate of the fourth moment of the particlesize distribution from the following equation, result-ing from Eq. ~23!:

^D4& 5 a2y~CsAN!. (27)

Different functional forms of the size distribution canyield the same ^D4&. Thus the form of the size dis-tribution cannot be inferred from photometric immer-sion refractometry measurements without additionalassumptions. The ^D4& is a biased estimate of ^D&with the bias depending on the particle size distribu-tion ~see Appendix B!.

If it is known a priori that the effect of absorptionon the ODmin is dominant, then ^k& can be calculatedfrom Eq. ~21! as follows:

^k& < ODminy@CsAN~4y3!~lyp!^D3&#. (28)

The right side of Eq. ~28! involves the third momentof the size distribution ^D3&, which cannot be ex-tracted from the coefficients a0, a1, and a2 of Eq. ~22!without the knowledge of ^k&. However, for a log-normal size distribution @Eq. ~B1!#, which is an ade-quate approximation of the marine particle size


distribution ~see Appendix B!, it can be shown that

^D3& 5 ^D4&3y4 exp@2~3y2!sD2#, (29)

where sD is the width parameter of the size distribu-tion. By replacing ODmin with a0 2 a1

2y~4a2!, whichdefines the minimum of the polynomial in Eq. ~22!,and by using Eq. ~29!, we obtain

^k& < G@a0ya23y4 2 a1

2y~4a27y4!#, (30)

where factor G is defined as follows:

G 5 $3p exp@~3y2!sD2#%y@4l~CsAN!1y4#. (31)

Considering the variability of sD ~Appendix B!, theexponential term in most cases will assume a valuebetween 1 to approximately 4.5, with the most prob-able value of approximately 1.5. Thus, without priorknowledge of the form of the size distribution, theaverage imaginary part of the refractive index of theparticles can be determined only to a factor of 1 toapproximately 4.5 of the actual value. The accuracyof this determination can be improved if the size dis-tribution is known.

The standard deviations of the parameters ^n&, ^k&,and ^D4&1y4 can be estimated from the following equa-tions resulting from a linearized error propagationmodel,20 including the covariances of the coefficientsa0, a1, and a2 of Eq. ~22!:

s~^n&! 5 $@1y~2a2!#2 s2~a1! 1 @a1y~2a2

2!#2 s2~a2!

2 @a1y~2a23!#cov~a1, a2!%

1y2, (32)

s~^k&! 5 G$@1y~a23y4!#2s2~a0! 1 @a1y~2a2

7y4!#2s2~a1!

1 @2~3y4!a0ya27y4

1 ~7y16!a12ya2

11y4#2s2~a2!

2 ~a1ya210y4!cov~a0, a1! 1 @2~3y2!a0ya2

10y4

1 ~7y8!a12ya2

14y4#cov~a0, a2!

1 @~3y4!a0a1ya214y4

1 ~7y16!a13ya2

18y4#cov~a1, a2!%1y2, (33)

s~^D4&1y4! 5 $@~1y4!~AN!21y4a223y4#2s2~a2!%

1y2. (34)

We stress that coefficients a0, a1, and a2 must beknown with sufficient numerical accuracy for Eqs.~30!, ~32!, and ~33! to produce meaningful results,because the right sides of these equations are differ-ences between nearly equal numbers. This is a re-sult of high correlations, typically better than 0.999,between the coefficients a0, a1, and a2. Such corre-lations are mathematical artifacts due to expressingthe OD 5 const ~n 2 ns!

2 as a polynomial in Eq. ~22!.However, the need for a high numerical precision ofa0, a1, and a2 does not imply that the errors of thesecoefficients, as calculated with the least-squares~LSQ! polynomial fitting procedure, must be accord-ingly small for the retrieval of the particle character-istics. Spreadsheet software for planning andanalyzing a single-wavelength photometric immer-


sion refractometry experiment is available for a nom-inal fee from M. Jonasz.

B. Case of Negligible Absorption of Light by Particles

The imaginary part k of the refractive index of het-erotrophic marine bacteria is estimated to be of theorder of 0.0001 or less.9 For a typical size distribu-tion of these bacteria, the contribution of the terminvolving ^k& to the ODmin @Eq. ~20!# is generally be-low the OD measurement error. Such particles areconsidered to exhibit negligible absorption of light inthe context of this model. A minimum k of pig-mented phytoplankton cells in the visible spectralregion, which is located typically in the green, is ofthe order of 0.001 ~Ref. 21!. In such a case, thecontribution of the ^k& term to the ODmin can be com-parable or greater than the OD measurement error.Hence the case of negligible absorption of light wouldnot apply to phytoplankton within the visible spectralregion. However, the absorption of light in phyto-plankton cells ~and consequently also k! rapidly van-ishes toward the long-wavelength end of the visiblerange and in the near infrared.21 Thus the assump-tion of negligible assumption of light would apply tophytoplankton in this spectral region.

By setting ^k& 5 0 in Eq. ~20! we obtain

ODmin < CsAN^D4&sn2. (35)

Similarly, Eq. ~25! reduces to

a0 5 CsAN^D4&~sn2 1 ^n&2!. (36)

From Eqs. ~23!, ~24!, and ~36! we find the variance ofthe refractive index of particles in the population:

sn2 5 a0ya2 2 @a1y~2a2!#

2. (37)

The standard deviation of that estimate can be ex-pressed as follows with the linearized error propaga-tion model20:

s~sn2! 5 $@1y~a2!#

2s2~a0! 1 @a1y~2a22!#2s2~a1!

1 @2a0ya22 1 a1

2y~2a23!#2s2~a2!

2 ~a1ya23!cov~a0, a1! 1 2@2a0ya2

3

1 a12y~2a2

4!#cov~a0, a2! 2 2@2a0a1y~2a24!

1 a13~4a2

5!#cov~a1, a2!%1y2, (38)

s~sn! 5 @1y~4sn2!s2~sn

2!#1y2. (39)

We again stress that coefficients a0, a1, and a2, forreasons discussed above, must be known with suffi-cient numerical accuracy for Eqs. ~37! and ~38! toproduce meaningful results.

C. Multiwavelength Photometric ImmersionRefractometry

Consider first the case of particles that do not absorblight, and assume for simplicity that the distributionof the refractive index of particles is narrow. Then

^k& 5 0, sn2 ' 0, and ^n& ' n. From Eq. ~19! we have

@n~l1! 2 ns~l1!#y@n~l2! 2 ns~l2!#

5 @OD~l1!yOD~l2!#1y2~l1yl2!. (40)

The refractive index of the medium ns can be obtainedseparately as a function of wavelength. Thus, fromthe OD spectrum for a refractive index of the immer-sion medium near that of the OD minimum and fromthe refractive index of particles at a single wave-length ~for example, l1!, we can determine the dis-persion of the refractive index of the particles.

Multiwavelength measurements are necessary forthe retrieval of ^k& and sn

2 in the case of non-negligible absorption of light by the particle. Vari-ables sn

2 and ^k& in Eq. ~20! each have a uniquewavelength spectrum. If these spectra are known,up to a constant factor each, the latter two variablescan be expressed as follows:

sn2 5 KnSn

2~l!, (41)

^k& 5 KkSk~l!, (42)

where factors K are independent of wavelength, andboth Sn~l! ~the spectrum of the real part of the re-fractive index of a particle! and Sk~l!, are assumed tobe the same for all particles in the suspension, as aconsequence of Eq. ~5!. After substituting Eqs. ~41!and ~42! into Eq. ~20! we obtain

ODmin~l! < CsAN@^D4&KnSn2~l! 1 ~4y3!~lyp!

3 ^D3&KkSk~l!#. (43)

We can now calculate the factors K by measuring theODmin at two different wavelengths and solving theresulting set of linear equations ~one equation of type43 for each wavelength!. It is assumed here that thereal part n of the refractive index of the particles atthese two wavelengths is nearly the same. Such anassumption can be made because ~1! variations of nare generally estimated at less than approximately0.004 in the visible21 and ~2! the OD varies slowlywith n 2 ns near the OD minimum.

4. Numerical Simulation

A simulation was carried out for particles that do notabsorb light. The size distribution was assumed tobe log normal @Eq. ~B1!# with Dpeak 5 0.4 mm, FDmax5 4 3 108 cm23 mm21, and sD 5 0.57 ~Fig. 1!. Thisdistribution yields the number concentration of theparticles of 2.7 3 108 cm23. Each particle is homo-geneous, but particles in different size classes canhave different real refractive indices. The refractiveindex of particles in a size class was randomly sam-pled from a uniform probability distribution with aspread of 60.0025 around a central value of 1.39.The OD of the suspension was calculated l 5 0.55 mmwith the ADA for spheres in two cases: ~1! the exactoptical density ~Fig. 2! and ~2! the optical densityfrom case 1 with superimposed 10% errors randomlysampled from a uniform probability distribution.The error of numerical integration was approxi-

mately 0.1% of the OD. As seen in Fig. 1, we includeall significant particle size classes in the integrationsize range.

To retrieve the characteristics of the particles, aparabola @Eq. ~22!# was fitted to the OD versus ns datawith the LSQ algorithm. The coefficients of the pa-rabola were then used to calculate ^n&, sn

2, and^D4&1y4 using Eqs. ~26!, ~37!, and ~27!. The resultsare summarized in Table 1.

In our simulation we aimed at reproducing themain features of our experiment: the magnitude ofOD and the dependence of OD on the wavelength oflight ~Fig. 3!. For this reason, we used a size distri-bution with FDmax and sD that are greater than thoseparameters of the experimental size distribution of

Fig. 1. Log-normal particle size distribution ~left axis! used in thesimulation of the photometric immersion refractometry ~Table 1and Fig. 2!: g~D! 5 FDmax exp@2~lnD 2 lnDpeak!2y~2sD

2!#, whereFDmax 5 4 3 108 cm23 mm21, Dpeak 5 0.4 mm, and sD 5 0.57. Thenumber concentration of the particles is 2.7 3 108 cm23. Thecontributions of the various particle size classes to the total opticaldensity ~OD, right axis! were calculated with the anomalous dif-fraction approximation to Qat~ns! for spheres at a wavelength oflight of 0.55 mm in vacuum and at the refractive index of 1.36 of theimmersion medium. The refractive index of particles in each sizeclass was sampled from a uniform probability distribution with amean of 1.39 and a spread of 60.0025. The contributions to theOD obtained with the parabolic approximation to Qat~ns! are alsoshown ~total OD error ' 15%!.

Fig. 2. Optical density OD of a particle suspension ~size distribu-tion in Fig. 1! as a function of the refractive index of the immersionns. The OD was calculated with the anomalous diffraction ap-proximation ~ADA! for spheres. By using the central 7 points ~u^n&2 nsu , 0.02! one obtains much better fit to the parabolic approx-imation to ADA than by using all data points ~u^n& 2 nsu , 0.06!.


Table 1. Summary of the Simulation of the Photometric Immersion Refractometry

Parameter Name

No Errors 10% ErrorsActualValuesAll Data Central 7 Data All Data Central 7 Data

^n& 1.39026 6 0.00016 1.39027 6 1.1E-05 1.39048 6 0.0002 1.39019 6 4.6E-05 1.3902sn

2 5.91E-05 6 1.74E-05 1.9E-06 6 3.9E-07 6.5E-05 6 2.1E-05 3.1E-06 6 1.6E-06 1.3E-06^D4&1y4 ~mm! 1.01977 6 0.00262 1.05554 6 0.00058 1.01854 6 0.0032 1.04891 6 0.00239 1.06

For the columns marked Central 7 Data, the data from within the range of u^n& 2 nsu , 0.02 were used. This reduces the error ofapproximation of the OD versus ns with a second-degree polynomial in ns. A single standard deviation is given following the 6 sign.

single cells so as to account for the contribution to ODof cell aggregates in the experiment. By using theexperimental size distribution of single bacterial cellswe would have obtained OD that is less than thatshown in Fig. 3 by more than an order of magnitude.

As can be seen from Figs. 1 and 2, and from Table1, the accuracy of the parabolic approximation to theOD increases with decreasing u^n& 2 nsu. For theparticle size distribution used, the parabolic approx-imation error decreases from approximately 120% atu^n& 2 nsu 5 0.058 to approximately 15% at u^n& 2 nsu5 0.02. The accuracy of the retrieved particle char-acteristics changes accordingly ~Table 1!. An esti-mate of the average refractive index of the particles isleast dependent on the refractive-index range used toobtain the LSQ estimates of coefficients a0, a1, and a2.However, an estimate of the variance of the refractiveindex of the particles and an estimate of the fourthmoment of the size distribution are affected signifi-cantly by the selection of that range.

5. Photometric Immersion Refractometry Experiment

A. Bacterial Culture

The free-living heterotrophic marine bacterial cul-ture was grown in 70% diluted seawater enrichedwith nutrients. The term free-living is here an ant-onym of the term attached to other particles. Thebacterial cells were isolated from seawater off Ber-muda by Dr. Jed Fuhrman. We prepared the 70%seawater solution by diluting seawater, sampled near

Fig. 3. Wavelength spectrum ~solid squares! of the optical densityOD of a suspension of particles ~size distribution in Fig. 1! calcu-lated with the anomalous diffraction approximation ~ADA!. Therefractive index of the immersion medium equals 1.332. Thepower-law OD spectrum ~const 3 l2s, line! is characteristic of ourexperiment with marine bacteria.


Catalina Island off Southern California, with deion-ized water and filtering this solution through a0.22-mm Nuclepore filter. The filtrate was used toprepare the growth medium that contained 0.25 g ofpeptone and 0.25 g of casamino acid per 1 dm3 ofsolution. The growth medium was autoclaved andfiltered through 0.22-mm Nuclepore filters to removeprecipitate formed during the autoclaving. Patchesof bacterial colonies grown on solid agar were trans-ferred to the growth medium and grown for 2 days atT ' 24 °C with continuous stirring.

Bacterial suspension with a concentration of ap-proximately 108 cellsycm3 was filtered through 3-mmNuclepore filters. The filtrate was centrifuged at10 °C and an acceleration of approximately 8000 g for15 min. The supernatant was discarded and thebacteria pellets were dispersed in approximately 20cm3 of filtered 70% seawater. When a standard,1-cm quartz cuvette with the bacterial suspensionwas visually examined, the suspension appearedtranslucent ~no image was transmitted through thecuvette!. This stock bacterial suspension was di-luted 16 times for the OD measurements. At l 50.589 mm, at which we made the measurements, theabsorption of visible light by the bacteria was negli-gible.

The bacterial cells were somewhat larger and moreelongated than typical marine bacteria that grow innatural environments where nutrients are less abun-dant. The cells had cylindrical shapes with adiameter-to-length ratio typically of approximately1:5. The bacterial suspension also contained visu-ally noticeable aggregates. The cells were stainedwith acridine orange and were filtered onto a black0.22-mm Nuclepore filter for enumeration.22 Theenumeration sample was taken in the middle of thesession of the OD measurements. Several photo-graphs ~color slides! of randomly selected areas onthe filter were taken with an epifluorescent micro-scope at a magnification of 1000. The bacteria ap-pear bright in the photographs ~Figs. 4 and 5!.

The size distribution of single cells, expressed as afunction of the spherical equivalent diameter, isshown in Fig. 6. The size distribution was approxi-mated by a log-normal function @Eq. ~B1!# with Dpeak5 0.34 mm, FDmax 5 1.85 3 108 cm23 mm21, and sD5 0.4 for the number concentration of the cells of1.3 3 108 cm23. We determined the size distribu-tion by measuring the lengths and widths of over 400images of single cells ~including small aggregates of 5

to 8 cells! projected onto a screen from the color slides.Thus the size distribution does not account for thelarge aggregates, whose concentration was smallerthan that of single cells by several orders of magni-tude. The spherical equivalent diameter is definedhere as a diameter of a sphere with the projected area~cross section! equal to the orientation average crosssection of the cell. The orientation average crosssection Pavg of a cell was approximated by that of acylinder with hemispherical end cups and was calcu-lated as follows:

Pavg 5 ~1y4!pDcylLcell, (44)

where Dcyl is the diameter of the cylindrical part ofthe cell, and Lcell is the total length of the cell, includ-ing the hemispherical cups. The term ~1y4! in Eq.~44! results from a theorem15,23 stating that the ori-entation average cross section of a convex particleequals 1y4 of the particle’s surface area.

Fig. 4. Marine bacteria used in the immersion refractometry ex-periment. Bacteria were stained with acridine orange and viewedwith an epifluorescent microscope.

Fig. 5. Aggregates of marine bacteria from the immersion refrac-tometry experiment. Bacteria were stained with acridine orangeand viewed with an epifluorescent microscope.

B. Immersion Medium

We used an aqueous solution of bovine serum albu-min ~BSA! as an immersion medium. BSA solutionis a nontoxic high-refractive index liquid suitable foruse with micro-organisms. The molecular weight ofBSA is sufficiently large to prevent the compoundfrom penetrating the cellular wall of bacteria.4,11,12,14

Therefore the refractive index obtained with suchimmersion medium relates to the whole cell. A 30%~wt.yvol.! BSA solution ~Sigma, Inc.! was concen-trated to approximately 46% by slow evaporation at40 °C under ambient pressure. Such concentratedBSA solution had a refractive index nD 5 1.415 ~at l5 0.589 mm; sodium-D line! and served as a stocksolution of the immersion medium in our experiment.

C. Preparation of Samples and Reference Solutions

Using the stock solutions of BSA and deionized ul-trafiltered water, we prepared a series of ten refer-ence solutions with no bacteria. The volume of eachreference consisted of a fixed amount of 70% seawaterand a fixed amount of a BSA solution in deionizedwater. The partial volumes of BSA and deionizedwater in the BSA solution were calculated with theGladstone–Dale mixing rule to yield a desired finalrefractive index of the entire reference. We pre-pared a corresponding series of ten samples by add-ing a volume of concentrated stock of bacteria equalto that of the fixed amount of 70% seawater in thereference solution. The lowest refractive index~1.332! in each series was nearly that of water. Thehighest refractive index ~1.41! was nearly that of the46% BSA stock solution.

For each sample, we added bacterial stock just be-fore the measurements of the OD and refractive indexbecause we were concerned that variations in thesalinity of the solutions may affect the optical prop-erties of the bacteria. Although the bacteria were

Fig. 6. Single-cell size distribution of marine bacteria in our ex-periment. The particle diameter is the diameter of a volume-equivalent sphere. A log-normal function g~D! 5 FDmax

exp@2~lnD 2 ln^Dpeak&!2y~2sD2!# was fitted to the data with the

least squares ~LSQ!. The fit parameters are FDmax 5 1.85 3 108

cm23 mm21, Dpeak 5 0.34 mm, and sD 5 0.4. The number con-centration of cells is 1.3 3 108 cm23. We calculated the LSQ fit byletting the weight of a data point be equal to its y value, as followsfrom the Poisson probability distribution of the number concentra-tion of bacteria.


grown at a salinity of approximately 2.4%, the salin-ity of the samples ranged approximately from 0.85%~the highest refractive index solution! to 0.2% ~thelowest refractive index solution!. Therefore, prior tothe experiment, we tested the effect of variations inthe salinity of a suspension on the OD by transferringcells from the original growth medium to distilledwater that represent the most dramatic change in thesalinity. We observed no effect on the OD of thesuspension during 250 min after the transfer.Later, the OD began to decrease. During the exper-iment, all measurements for a sample were com-pleted within approximately 30–60 min following theaddition of the concentrated bacterial stock. We arethus confident that the effects of changes in the sa-linity on the OD were negligible.

D. Measurements of the Refractive Index of theSuspension Medium

We determined the refractive index of the suspensionmedium at l 5 0.589 mm using an Abbe-type refrac-tometer ~Bausch & Lomb! with a 33–45–58 prism.With this instrument, the refractive index can bemeasured to 0.001, with visual interpolation to ap-proximately 0.0002. We calibrated the refractome-ter using three different refractive-index liquids~Cargille Inc.!. Three measurements were made foreach sample and the results were averaged. Foreach refractive-index preparation we verified that therefractive index of the sample with bacteria wasequal to that of the corresponding reference solutionwith no bacteria. This assured that our OD mea-surements with a double-beam spectrophotometerrepresent solely the effect of beam attenuation by thebacteria and not a residual absorption due to possibleslight differences in the concentration of BSA in thesample and reference.

E. Measurements of Optical Density

We measured the OD of cell suspensions with a mod-ified double-beam spectrophotometer ~Uvikon 860,Kontron Instruments, Switzerland!. In this instru-ment, the acceptance angle of the detector was lim-ited to less than 1°. Based on the experimentalscattering function for bacteria of comparable size24

we estimate that the error caused by accepting theforward scattered light should be less than approxi-mately 25%. Comparison of our system with a ma-rine transmissometer ~Sea Tech, Inc.! with use ofsuspensions of milk and magnesium oxide powdershowed that both instruments yield transmission val-ues that agree to within 5%.

We used standard quartz cuvettes with the opticalpath of 1 cm for both the sample with bacteria and thecorresponding reference solution. The OD measure-ments ~Table 2! were made at a single wavelength of0.589 mm to be consistent with the determinations ofthe refractive index of the samples and references.Each OD value was an average of ten measurementsin quick succession. The measurements were madewith a spectral bandwidth of 2 nm. The instrumen-


tal error of the OD measurements is approximately0.002 at the OD of 1.

It took approximately 8 h to complete all measure-ments for ten samples. Therefore, to test whethersignificant changes occurred in the bacterial stocksuspension during this time, we also obtained spectraof the OD at the beginning and at the end of themeasurement session. The spectra were obtainedfor the bacterial concentrate diluted 16 times withdeionized water. This dilution is equivalent to thatof the base sample of the series of ten samples used inthe experiment. The spectra were measured be-tween 0.350 and 0.750 mm at 0.001-mm intervals,with a bandwidth of 0.002 mm. From these mea-surements we found that the OD at l 5 0.589 mmdecreased by approximately 7% during the 8-h mea-surement session. The slopes s of the OD spectraare essentially the same ~1.66 at the beginning and1.61 at the end of the experiment! if these spectra arefitted to the power-law approximation:

OD 5 const l2s. (45)

This approximation gave an excellent fit: r2 5 0.999for log OD versus log l. The small change in theslope s suggests that neither the size distribution ofthe bacteria nor their refractive index changed sig-nificantly during the experiment.

F. Results

The results of OD measurements performed on sus-pensions of bacteria are shown in Fig. 7 and Table 2.We observed a dramatic reduction in the OD at l 5589 nm from 0.204 for ns 5 1.332 to approximately0.001 ~essentially the OD measurement error! for ns5 1.392. This reduction is much higher than thatpreviously reported.10,13 Thus we achieved essen-tially a full clearing of the bacterial suspension, astriking visual effect. Such clearing can be used foraccurate measurements of the absorption of light inbacterial cells14,25 because the effect of light scatter-ing is essentially nullified.

The minimum OD of the bacterial suspension isobserved in our experimental data at a refractiveindex of the medium equal to 1.3886 at l 5 0.589 mm.Thus the mean refractive index of the cells ^n& is

Table 2. Measured OD ~at a wavelength of 0.589 mm in vacuum! of aSuspension of Free-Living Marine Bacteria as a Function of the

Refractive Index ns of the Immersion Medium

ns ODycm

1.3325 0.203501.3431 0.125251.3533 0.063251.3648 0.025751.3704 0.014751.3757 0.007381.3868 0.001501.3920 0.000751.4013 0.007381.4100 0.02075

1.3886 ~Table 3!. Unfortunately, as mentionedabove, without multiwavelength measurements ofthe OD it was not possible to separate the effects of ^k&and sn

2 on the OD. We nevertheless give estimatesof these two latter parameters ~Table 3! assumingthat the variance of the real part of the refractiveindex of the particles vanishes ~for the estimate of ^k&!or that the particles do not absorb light ~for the esti-mate of sn

2!. As seen, both ^k& and sn2 in our exper-

iment are negligible ~at an accuracy of 2 standarddeviations!.

The bimodal size distribution of our bacteria pre-vented us from validation of our model in terms of^D4&1y4. The ^D4&1y4 ' 1.15 mm ~Table 3!, as deter-mined from the LSQ fit to the OD data ~Table 2!, isgreater than the value of 0.55 mm calculated from thelog-normal approximation to the experimental sizedistribution of single cells ~Fig. 6!. This is consistentwith a significant contribution of the cell aggregatesto OD. According to the simulation, approximately50% of the OD is contributed by particles with D .1.7 mm ~Fig. 1!. The significant contribution of theaggregates is also indicated by the small exponentialslope ~approximately 1.6! of OD as a function of lightwavelength @Eq. ~45!#. Typically, bacterial suspen-

Fig. 7. Optical density OD of a suspension of marine bacteria asmeasured in solutions of bovine serum albumin with various re-fractive indices ns. Only seven data points, roughly centeredabout the ODmin, are shown. The horizontal error bars representa 60.001 ns measurement error. The vertical bars represent a60.002 OD measurement error. See Table 2 for the OD values,Table 3 for retrieved characteristics of the bacteria, and Fig. 6 forthe single-cell size distribution of bacteria.

Table 3. Summary of a Photometric Immersion RefractometryExperiment at a Wavelength of 0.589 mm ~in vacuum! withHeterotrophic Marine Bacteria in Bovine Serum Solutions

Parameter and DimensionValue 6 1

Standard Deviation

^n& 1.38859 6 0.00016^k& 6.63E-05 6 4.54E-05sn

2 1.13E-05 6 7.83E-06^D4&1y4 ~mm! 1.14936 6 0.00713

Notes: The number concentration of single cells equals 1.3 3108 cm23. The ^k& was calculated with sD 5 0.4 ~of the experi-mental single-cell size distribution!. The ^D4&1y4 was calculatedassuming the shape factor Cs 5 1y2.

sions with the average cell size of approximately 0.5mm and with negligible cell aggregation yield a slopeof approximately 2 ~Ref. 9!.

6. Conclusions

We have developed and analyzed a rigorous model ofthe photometric immersion refractometry, a tech-nique to characterize particles in suspensions. Thistechnique is based on a reduction in the beam atten-uation of light by a suspension of particles when thereal part of the refractive index of the immersionmedium approaches the real part of the refractiveindex of the particles. Such a technique can serve asa benchmark for the refractive-index determinationbecause it does not rely on assumptions inherent inother techniques, such as those based on the inver-sion of the spectral attenuation data.

We have shown in the numerical simulation thatmeasures of the inhomogeneity of a particle popula-tion, both in regard to the refractive index and size aswell as to the absorption of light by the particles, canbe estimated conveniently with this technique. Todemonstrate the applicability of the model, we havecarried out a photometric immersion refractometryexperiment with heterotrophic marine bacteria.

We successfully clarified the bacterial suspensionusing bovine serum albumin solution as an immer-sion medium, and we determined the average realpart of the refractive index of cells. The variance ofthe real part of the refractive index and the imagi-nary part of the index of cells were found to be neg-ligible in this experiment.

Appendix A: Refractive Index Versus Particle Size

In the model of photometric immersion refractometrywe assumed that the refractive index of the particle isindependent of the particle size. This assumption isbased on microscopic observations with bacteria,which also typically exhibit a low variability of therefractive index.3 However, it has also been postu-lated3,26 that the refractive index may depend on theparticle size. Here we examine how the predictionsof the model would change in the case of such depen-dency. We limit our discussion to the case of parti-cles that do not absorb light, thus n is real.However, this discussion can be extended easily toinclude light-absorbing particles.

From Eqs. ~1!, ~6!, ~7!, and ~13! ~with k set to 0! we have

OD < Cs~p3yl2L log e! * D4~n 2 ns!

2FDM~D, n!dD dn,

< Cs~p3yl2L log e!* ~D4n2 2 2D4n ns

1 D4ns2!FDM~D, n!dD dn. (A1)


Equation ~A1! can be further transformed as follows,with Eqs. ~2!, ~15!, and ~16!:

OD < CsAN^D4n2& 2 2CsAN^D4n&ns 1 CsAN^D4&ns2.

(A2)

Thus the OD can be approximated with a second-degree polynomial @Eq. ~22!#. The coefficients of thispolynomial, which can be obtained by one fitting ex-perimental data of the OD versus ns, can be ex-pressed as functions of the characteristics of theparticles as follows:

a0 5 CsAN^D4n2&, (A3)

a1 5 22CsAN^D4n&, (A4)

a2 5 CsAN^D4&. (A5)

A minimum of the OD occurs at ns min expressed asfollows:

ns min 5 2a1y~2a2!,

5 2~22CsAN^D2n&y~2CsAN^D4&!,

5 ^D4n&y^D4&. (A6)

We have ns min 5 ^n& only if the distribution functionsof D and n are independent, i.e., if F~D, n! 5 g~D!h~n!,for in that case ^D4n& 5 ^D4&^n&. Of course, the ODversus ns could still be fitted with a parabola if D andn are correlated. However, if the coefficients of theparabola a0, a1, and a2 are used in that case to cal-culate ns min @Eq. ~A6!#, the obtained value may poorlyrepresent the average index of the particles ^n&.Similar arguments can be given for other particlecharacteristics discussed in the context of the immer-sion refractometry model.

Appendix B: ^D4&1y4 as an Estimate of ^D& for aLog-Normal Size Distribution

The ^D4&1y4 is a biased estimate of the average par-ticle diameter in a population. The magnitude ofthe bias can be obtained easily, for example, for theparticle size distribution expressed with a 0th orderlog-normal function that was found to adequately de-scribe size distributions of marine particles27:

g~D! 5 FDmax exp@2~lnD 2 lnDpeak!2y~2sD

2!#, (B1)

where FDmax is a scale constant, Dpeak is the geomet-ric mean diameter, and sD is the width parameter ofthe distribution. If there are N particles per unitvolume of the suspension, then Dpeak 5 ~D1 D2 . . .DN!1yN. For the log-normal distribution @Eq. ~B1!#,^D4&1y4 can be expressed as follows:

^D4&1y4 5 ^D& exp@~3y2!sD2#. (B2)

Thus, for a narrow log-normal size distribution ~sD ,0.1!, we have ^D4&1y4 ' ^D&. Most values of sD forlog-normal components of marine size distributions27

are within a range of 0.05 to approximately 1, withthe most probable value of sD at approximately 0.5.


Thus, from Eq. ~B2!, one can expect ^D4&1y4 to rangefrom ^D& to ' 4.5 ^D&, with the most probable value atapproximately 1.5 ^D&.

This research was supported by the Defense Re-search Establishment Valcartier contract W7701-3-1476y01-XSK awarded to M. Jonasz Consultants.Partial support for D. Stramski was also provided bythe Office of Naval Research through grant N00014-95-1-0491. We thank Annick Bricaud and an anon-ymous reviewer for their comments.

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