Journal of Biomedical 7(3), 291–299 (July 2002)

Optical rotation and linear and circular depolarization rates in diffusively scattered from chiral, racemic, and achiral turbid media

Kevin C. Hadley Abstract. The properties of light scattered in a lateral University of Waterloo direction from turbid media were studied. Polarization modulation Department of Chemistry and synchronous detection were used to measure, and Mueller calcu- Waterloo, Ontario N2L 3G1, Canada lus to model and derive, the degrees of surviving linear and circular I. Alex Vitkin polarization and the induced by turbid samples. Poly- Ontario Cancer Institute/University of Toronto styrene microspheres were used as scatterers in water solutions con- Departments of Medical Biophysics and taining dissolved chiral, racemic, and achiral . The preser- Radiation Oncology vation of circular polarization was found to exceed the linear Toronto, Ontario M5G 2M9, Canada polarization preservation for all samples examined. The optical rota- tion induced increased with the chiral concentration only, whereas both linear and circular polarizations increased with an in- crease in the concentrations of chiral, racemic, and achiral molecules. This latter effect was shown to stem solely from the refractive index matching mechanism induced by the solute molecules, independent of their chiral nature. © 2002 Society of Photo-Optical Instrumentation Engineers. [DOI: 10.1117/1.1483880]

Keywords: polarization; multiple scattering; ; polarization modulation. Paper JBO TP-14 received Mar. 8, 2002; revised manuscript received Mar. 14, 2002; accepted for publication Mar. 31, 2002.

1 Introduction Optical investigations of turbid media that contain chiral 23–31 Many natural and synthetic systems have disordered proper- components have also been reported. The interest in chi- ties that render them optically turbid. Examples abound in ral turbid media is driven by the attractive possibility of non- invasive optical monitoring of the glucose level in diabetic science, technology, medicine, and the natural environment, patients. This is a very difficult unresolved problem in clinical and include such diverse systems as interstellar gases, biologi- medicine and an active area of research that utilizes different cal tissues, turbulent flows, colloidal suspensions, and micro- biomedical optics approaches.32,33 For polarized light re- crystalline solids. Optical examinations of these random me- search, detection of glucose and quantification of its concen- dia are challenging due to extensive multiple light scattering tration rely on the molecule’s chirality that stems from its which scrambles potentially useful sample information en- asymmetric molecular structure, resulting in a number of coded in the light beam. For example, most biological tissues characteristic effects generically called optical activity.34 A ͑ ͒ are highly scattering at visible and near-infrared IR well-known manifestation of optical activity is the ability of a 1 , so quantitative tissue and imaging glucose solution to rotate the plane of linearly polarized light are difficult. In recent investigations researchers have at- about the axis of propagation. The amount of rotation depends tempted to utilize the polarization properties of light to probe on the molecular concentration, the pathlength through the biological tissues and other random media, since diffusely medium, and the optical rotatory power at the interrogation scattered light is often partially polarized to an extent that can .35 This and other chiral asymmetries encoded in be experimentally detected. Several research groups are thus the polarization properties of light transmitted through largely investigating the response of turbid media to polarized light transparent ͑nonscattering͒ media enable very sensitive and by examining the polarization properties of multiply scattered accurate determination of glucose concentrations. With turbid light.2–19 These polarization methodologies have been devel- materials, however, the scattering effects dominate, loss of oped to study various aspects of polarized light interaction polarization information is significant, pathlengths get with turbid media, including the investigation of the depolar- scrambled, and chiral effects stemming from a small amount ization mechanisms, quantification of depolarization length of dissolved glucose are much more difficult to discern. scales, determination of system Mueller matrices, and deriva- We have previously developed polarization modulation tion of the of scattered light. In several and synchronous detection methods to measure surviving po- recent publications polarization imaging studies of multiply larization fractions in diffusive scattering even in the presence scattering media were described.6,11,20–22 of large noisy depolarized backgrounds, such as in the case of synthetic turbid media25,27,28,30,31 and ex vivo28 and in vivo Address all correspondence to I. Alex Vitkin.Tel: 416-946-4501, Ext. 5742; Fax: 416-946-6529; E-mail: [email protected] 1083-3668/2002/$15.00 © 2002 SPIE

Journal of Biomedical Optics ᭹ July 2002 ᭹ Vol. 7 No. 3 291 Hadley and Vitkin

Fig. 1 Schematic of the apparatus for measuring degrees of polarization and optical rotation of scattered light. Arrangement B is shown. In arrangement A, the quarter wave plate is removed.

biological tissues.31 Furthermore, we have experimentally ob- where the equality holds for fully polarized light, and the served chiral asymmetries due to the presence of glucose. inequality applies to partially polarized light. The total degree Both varied the level of polarization preservation and the ro- of polarization ͑DOP͒ is defined as tation of linearly polarized light fraction varied with glucose concentrations in the turbid samples.25,27,28,30 Q2ϩU2ϩV2 DOP ϭͱ . ͑3͒ In comparing turbid microsphere suspensions with and T I2 without chiral molecules, depolarization was diminished in the former case both in the lateral and backscattering direc- The linearly polarized fraction is represented by the degree of tions, although the mechanism responsible for this polariza- , tion enhancement effect was not determined. In this paper, we 2 2 extend our methodology to perform this determination. Spe- Q ϩU DOP ϭͱ . ͑4͒ cifically, we analyze the polarization characteristics of later- L I2 ally scattered light from chiral, racemic ͑equal amounts of The degree of circular polarization is similarly defined as oppositely handed isomers͒, and achiral solutions that contain scattering polystyrene microspheres in an effort to determine V2 whether the chirality or the refractive index matching mecha- ϭͱ ͑ ͒ DOPT 2 . 5 nism is the dominant cause of preservation of the enhanced I polarization. We also develop methods to explicitly derive Of interest in the current study were the linear and circular linear and circular degrees of polarization, and compare these depolarizations that occurred due to multiple scattering within depolarization rates in microsphere suspensions. Finally, the the turbid sample and the optical rotation of the linearly po- dependence of optical rotation of the linearly polarized light larized light that may be induced by the chirality of the me- fraction on the chirality of the turbid medium is quantified. dium. The sample was thus illuminated with a polarized light beam whose incident polarization varied in a time-dependent, controlled fashion. Upon diffusive interactions with the scat- 2 Theory tering medium, the surviving polarization properties of the Light of any arbitrary polarization state may be represented light were detected at 90°, synchronously with the incident by a Stokes vector ͑S͒ of the form polarization modulation. The surviving linear and circular po- ␤ ␤ larization fractions, L and C , respectively, are defined in I terms of the incident ͑subscript i͒ and scattered ͑subscript s͒ Q polarization states as ͫ ͬ , ͑1͒ U DOP ␤ ϭ L,s ͑ ͒ V L , 6 DOPL,i where I represents the total light intensity, Q and U represent the linearly polarized components of the beam, and V repre- DOP ␤ ϭ C,s ͑ ͒ sents the circularly polarized component. The intensity I is C . 7 directly measurable with a photodetector. For an arbitrary DOPC,i light beam, these terms are related by The layout of optical elements is schematically presented in Figure 1. There were two experimental arrangements em- I2уQ2ϩU2ϩV2, ͑2͒ ployed for this study that differed in the analyzing elements

292 Journal of Biomedical Optics ᭹ July 2002 ᭹ Vol. 7 No. 3 Optical Rotation and Linear and Circular Depolarization Rates... placed between the sample and the detector. In arrangement the light incident on the sample has the following Stokes vec- A, the analyzing optics consisted of a linear ͑ana- tor lyzer͒ with pass axis ␪ ° to the horizontal plane. Positive angles indicate rotation of the analyzer counterclockwise as 1 one looks down the beam in the direction of its propagation. 0 ͫ ͬ , This arrangement was useful for determining the optical rota- cos ␦͑t͒ ␣ ␤ tion and the degree of linear polarization L . To extract the sin ␦͑t͒ degree of circular polarization ␤ , arrangement B was em- C ␦(t) is the retardation of the PEM as a function of time, ployed, which included a quarter wave plate with its fast axis where ␦ϭ␦ sin(2␲ft). ␦ is the maximum retardation im- 45° to the horizontal placed between the sample and the linear 0 0 posed by the PEM, and f is the resonant frequency of its analyzer. The time-varying source of incident polarized light oscillation. was furnished by passing linearly polarized laser light through The turbid sample is modeled as an optical rotator ͑impos- a transparent quartz element of a photoelastic modulator ing a rotation of ␣°͒ and a depolarizing element ͓imposing ͑PEM͒. This device provides controlled, time-varying bire- linear and circular depolarizations ␤ and ␤ , defined in Eqs. fringence that converts the incoming linearly polarized light L C ͑6͒ and ͑7͔͒. The transmitted Stokes vector is thus into transmitted elliptically polarized light,36 the polarization states of which vary over time at the PEM’s resonant fre- 1 quency of 50 kHz. The use of this time-varying polarization ␤ sin͑2␣͒cos ␦͑t͒ modulation is useful in turbid media experiments because it ͫ L ͬ ␤ cos͑2␣͒cos ␦͑t͒ enables high sensitivity measurements of weak polarization L ␤ ␦͑ ͒ signals in a large noisy depolarized background using syn- C sin t chronous detection with a lock-in amplifier. After scattering 10 0 0 interactions with the microsphere sample, directed 90° to the incident beam passed through analyzing elements 0 ␤ 00 ϭ L and impinged on a polarization-insensitive detector. ͫ ␤ ͬ 00 0 To model the polarization effects in this experimental sys- L ␤ tem, the various optical components were represented by 00 0 C Mueller matrices. Using Mueller calculus, an optical element 10 00 that acts on a light beam is represented by multiplication of 1 the incident light Stokes vector by the Mueller matrix for that 0 cos͑2␣͒ sin͑2␣͒ 0 0 ϫ ͫ ͬ . ͑8͒ optical element. The result is the Stokes vector of the trans- ͫ Ϫ ͑ ␣͒ ͑ ␣͒ ͬ cos ␦͑t͒ 0 sin 2 cos 2 0 mitted beam. Mueller matrices for common optical elements, sin ␦͑t͒ such as a linear polarizer, a quarter wave plate ͑QWP͒, and 00 01 the PEM, are available in optics textbooks.37,38 Representing Two comments about the sample matrix are in order. First, the turbid chiral sample by a suitable matrix is more challeng- although in general matrix multiplication is not commutative, ing, as described below. in this case the order of multiplication of the two sample The resultant Stokes vector for arrangement A is matrices does not matter. Second, the choice of sample matrix ␤ with different depolarization rates for linear ( L) and circular ͒ ϭ ␪͒ ͓ ͔͑ ͒͑ ͒ ͒ ␤ ͑S A ͓P2͑ ͔ sample PEM P1 ͑Si ; ( C) polarized light is an improvement over that in our pre- vious work which modeled the sample with a generic depo- similarly, for arrangement B, larization parameter, ␤.25,27,28,30,31 Although other matrix formulations can be suggested, the current representa- ͒ ϭ ␪͒ ͑ ͒͑ ͒͑ ͒͑ ͒ ͒ ͑S B ͓P2͑ ͔ QWP sample PEM P1 ͑Si . tion is relatively straightforward and is in line with reported observations of polarization-state dependent degrees of polar- These expressions are now derived. ization in diffusive scattering.6–8,16,19 Following the passage of incident light through the 45° For arrangement A that employs a linear analyzer oriented linear polarizer P1 ␪ ° to the horizontal, information about the circular polariza- tion state of the scattered light is lost, but the optical rotation 1010 ␣ ␤ and the degree of linear polarization L can be determined 0000 with high precision. Multiplying the Stokes vector in Eq. ͑8͒ P1ϭͫ ͬ , by the Mueller matrix for a linear polarizer oriented at ␪ °, 1010 and selecting the resulting Stokes parameter I that represents 0000 the measured light intensity, ϭ ϩ␤ ␣ϩ␪͒ ␦ ͒ ͑ ͒ and the PEM with its modulation axis horizontal is IA kA͕1 L sin͓2͑ ͔cos ͑t ͖, 9

where kA is an unimportant experimental constant. The signal 10 0 0 observed at the detector is thus a function of user-defined 01 0 0 variables ͓the analyzer angle ␪, the time varying PEM retar- PEMϭ , ␦ ϭ␦ ␲ ͔ ͑␣ ͫ ␦͑ ͒ Ϫ ␦͑ ͒ͬ dation (t) 0 sin(2 ft) and the sample unknowns and 0 0 cos t sin t ␤ ͒ ͑ ͒ L that can be obtained by fitting Eq. 9 to the experimental 0 0 sin ␦͑t͒ cos ␦͑t͒ data.

Journal of Biomedical Optics ᭹ July 2002 ᭹ Vol. 7 No. 3 293 Hadley and Vitkin

␤ For arrangement B, performing the additional multiplica- As mentioned above for arrangement B, the terms L and ␣ ␤ ␣ ͑ ͒ tion with a Mueller matrix for a quarter wave plate with 45° appear together as the product of L•cos(2 ) in Eqs. 15 fast axis orientation yields and ͑16͒. This prevents independent determination of these two parameters. However, if ␣ is small ͑less than a few de- ϭ ϩ␤ ␪͒ ␣͒ ␦ ͒ IB kB͓1 L sin͑2 cos͑2 cos ͑t grees, a reasonable assumption for most biological applica- ͒ ␣ ϳ ␤ tions , then cos(2 ) 1, and L can be determined. Making ϩ␤ cos͑2␪͒sin ␦͑t͔͒. ͑10͒ ␤ ␤ C this assumption, C and L can be found using the 1 f /dc ͑ ͒ This expression contains all three important sample unknowns ratio described in Eq. 15 . Then an additional determination ␤ ͑ ͒ ͑␣ ␤ ␤ ͒ of L can be performed by fitting Eq. 16 to the 2 f /dc signal , L, and C , so the insertion of the quarter wave plate ␤ ratios. permits the determination of C as well. Note, however, that ͑ ͒ ␣ ␤ To summarize, the optical rotation ␣ can only be measured in contrast with Eq. 9 , and L now occur as a product of ␤ cos(2␣), so independent determination of them appears in arrangement A. Also, this arrangement permits the mea- L• ␤ difficult. surement of L without the need to make any assumptions ␣ To facilitate the data analysis with the help of Eqs. ͑9͒ and about the size of . However, no information about circular ͑10͒, we expand the time-dependent retardation terms cos ␦(t) polarization can be obtained from this setup. Insertion of a ͑ and sin ␦(t) in a series of spherical harmonics using the fol- quarter wave plate in front of the linear analyzer arrangement ͒ ␤ ␤ lowing expressions:39 B permits the determination of C and allows L to be evalu- ated with reasonable precision assuming a small induced op- ϱ tical rotation via a 1 f /dc data fit to Eq. ͑15͒. Also, with the ␣ ␤ ␺͒ϭ ͒ϩ ͒ ␺͒ ͑ ͒ same assumption about , L can be independently estimated cos͑z sin J0͑z 2 ͚ J2k͑z cos͑2k , 11a kϭ1 from a 2 f /dc fit to Eq. ͑16͒.

ϱ ␺͒ϭ ͒ ϩ ͒␺ ͑ ͒ 3 Experimental Methods sin͑z sin ͚ J2kϩ1͑z sin͓͑2k 1 ͔, 11b kϭ0 A schematic of the measurement system is shown in Figure 1. Unpolarized light from a HeNe laser (␭ϭ632.8 nm) passed where J are the Bessel functions of the first kind of order n. n through a linear polarizer oriented 45° to the vertical. The Since lock-in amplification isolates signals at specific fre- linearly polarized light traversed the PEM ͑Hinds, PEM-80͒, quencies ͑1 f and 2 f in these experiments͒, we expand the oriented with its modulation axis horizontal and modulation series up to the 2 f terms, with zϭ␦ and ␺ϭ2␲ f , to obtain 0 frequency f of 50 kHz. The resulting time-varying elliptically polarized light was incident on a 1-cm-sq quartz spectropho- ϭ ͕ ϩ ͑␦ ͒␤ ͓ ͑␣ϩ␪͔͒ϩ ͑␦ ͒ IA kA 1 J0 0 L sin 2 2J2 0 tometer cuvette that contained the liquid turbid suspension. ϫ␤ ␣ϩ␪͒ ␲ ͒ ͑ ͒ The detector arm was perpendicular to the incident beam, L sin͓2͑ ͔cos͑2 2 f ͖, 12 • and consisted of a linear polarizer ͑analyzer͒ with its pass axis at ␪ ° and a photomultiplier tube ͑PMT͒͑arrangement A͒. For I ϭk ͕1ϩJ ͑␦ ͒␤ sin͑2␪͒cos͑2␣͒ϩ2J ͑␦ ͒ B B 0 0 L 2 0 arrangement B, a quarter wave plate with its fast axis at 45° ϫ␤ ␪͒ ␣͒ ␲ ͒ L sin͑2 cos͑2 cos͑2 2 f was placed between the sample and the linear analyzer. Aper- • tures ͑pinhole diaphragms͒ were located between the sample ϩ ͑␦ ͒␤ ͑ ␪͒ ͑ ␲ ͖͒ ͑ ͒ 2J1 0 C cos 2 sin 2 •1 f . 13 and the analyzing optical elements, and in front of the PMT, In arrangement A, taking the ratio of the signal photocur- to limit the acceptance angle of the detector. The PMT signal ͑ ͒ was fed to a preamplifier ͑Stanford Research Systems, rent component at 2 f to the constant dc signal removes the ͒ ͑ experimental constant k and yields SR570 before entering the lock-in amplifier Stanford Re- A search Systems, SR350͒ which enabled detection of photocur- rents at the PEM modulation frequency and its first harmonic 2J ͑␦ ͒␤ sin͓2͑␣ϩ␪͔͒ ͑ ͒ ϭ 2 0 L ͑ ͒ ͑1 f and 2 f ͒. The dc signal was measured by mechanically 2 f /dc A ϩ ␦ ͒␤ ␣ϩ␪͒ . 14 1 J0͑ 0 L sin͓2͑ ͔ chopping the beam at low frequency of ϳ150 Hz and syn- chronously detecting it with the same preamp/lock-in ampli- Thus, by dividing the signals at 2 f by the signals at dc at fier pair. a range of analyzer angles ␪, parameters ␣ and ␤ can be L Chiral samples consisted of LϪ(ϩ)-arabinose ͑Sigma͒ extracted by fitting the data ratios to the curve described by dissolved in de-ionized water at concentrations up to ϳ2M. Eq. ͑14͒. Racemic samples were prepared from equal quantities of L Similarly, in arrangement B, the ratio of the signal at 1 f to Ϫ(ϩ)-arabinose and DϪ(ϩ)-arabinose by weight. Arabi- dc is nose is a naturally occurring plant similar to glucose. It was selected for these experiments because it has a significant 2J ͑␦ ͒␤ cos͑2␪͒ ͑ ͒ ϭ 1 0 C ͑ ͒ specific rotatory power and because both its chiral isomers ͑L 1 f /dc B ϩ ␦ ͒␤ ␪͒ ␣͒ . 15 1 J0͑ 0 L sin͑2 cos͑2 and D͒ are readily available and are inexpensive. Achiral samples were prepared from glycerol ͑ACP͒ in And the 2 f /dc ratio is distilled water at concentrations of up to ϳ35% glycerol by volume. These glycerol–water solutions were prepared to 2J ͑␦ ͒␤ sin͑2␪͒cos͑2␣͒ ͑ ͒ ϭ 2 0 L ͑ ͒ match the refractive index of the arabinose solutions. By com- 2 f /dc B ϩ ␦ ͒␤ ␪͒ ␣͒ . 16 1 J0͑ 0 L sin͑2 cos͑2 paring chiral and achiral samples of equal refractive index, the

294 Journal of Biomedical Optics ᭹ July 2002 ᭹ Vol. 7 No. 3 Optical Rotation and Linear and Circular Depolarization Rates... influence of refractive index variations versus the influence of chirality of the molecules could be separated. The turbidity in all samples was provided by suspended polystyrene micro- spheres, 1.4 ␮m in diameter at a concentration of 0.2% ͑2 g/L͒. By measuring signals at 2 f and dc in arrangement A at various angular settings of the analyzer, a plot of 2 f /dc vs ␪ was drawn for each sample. The ␪ range selected was 30°– 150° to encompass a large part of the cyclic function of Eq. ͑14͒. The fitted value of ␣ is particularly sensitive to data ratios in the rapidly changing region around 90°. A commer- cial software package was used for nonlinear curve fitting ͑ ͒ ␣ ␤ SigmaPlot, Jandel Scientific Software to extract and L by ͑ ͒ ␦ fitting to Eq. 14 . Peak PEM retardation 0 was set to 3.469 ␦ ␦ rad to generate relatively large values of J0( 0) and J2( 0). Maximizing these coefficients ensures a large of the 2 f /dc signal variation as a function of ␪. This in turn ␤ makes the curve fit more sensitive to changes in L . In arrangement B, analyzer angles from Ϫ80° to ϩ80° were chosen to capture a large range of 1 f /dc ratios. From these data, and assuming a small value of ␣, a two-parameter ␤ ␤ ͑ ͒ fit of C and L can be performed using Eq. 15 . It should be ␣ ␤ noted that the small assumption for only affects L , and the ␤ accuracy of the C determination remains unchanged. It was also possible to make measurements at 2 f with this arrangement of the components. Using the same assumption ␣ ϳ ␤ of cos(2 ) 1, a single-parameter fit L was performed on a plot of 2 f /dc vs ␪ ͓Eq. ͑16͔͒ as a check of the fit from the ␦ 1 f /dc plot. 0 was set to 1.433 rad when collecting 1 f /dc data and to 1.898 rad when collecting both 1 f /dc and 2 f /dc ␦ ͑ ͒ data to generate relatively large values of Jn( 0) in Eqs. 15 and ͑16͒. Again, large Bessel function values lead to more sensitive fits of the parameters. As a check for the consistency of the three fitting techniques ͓Eqs. ͑14͒–͑16͔͒, the values of ␤ L generated by each one can be compared. Fig. 2 (a) Variation in the 2f/dc values with the angular orientation of analyzer P2 for a turbid sample of 0.2% microspheres in de-ionized 4 Results and Discussion water. The points are the experimental data ratios in arrangement A Figure 2 shows results from a sample of 2 g/L suspension of (linear analyzer in the detector arm), and the line is the best fit using ␦ 1.4 ␮m diam microspheres in pure de-ionized water. From a Eq. (14). The maximum PEM retardation amplitude was 0 ϭ ␣ϭϪ Ϯ ␤ ϭ 40 3.469 rad. The fitted values were 1.43 0.21°, L (23.7 Mie scattering calculation, this corresponds to a scattering 2 ␮ Ϫ1 Ϯ0.2)%, and R ϭ0.9992. (b) 1f/dc ratios as a function of the P2 coefficient s of 73.1 cm , scattering g of 0.936, ␮Ј Ϫ1 angle for the same sample, measured in arrangement B (quarter wave and reduced scattering coefficient s of 4.7 cm . The verti- plate and linear polarizer in the detector arm). The fit is from Eq. (15), ͑ ͒ ␤ ϭ Ϯ ␤ ϭ Ϯ 2ϭ cal axis in Figure 2 a is the ratio of the lock-in amplifier with C (47.4 0.1)%, L (22.5 0.6)%, and R 0.9997, assum- ͑ ͒ ␣ ϳ ␦ ϭ photocurrent at 2 f 100 kHz to the dc signal measured at ing cos(2 ) 1. The PEM retardation setting was 0 1.898 rad. (c) mechanical chopper frequency of 150 Hz. Negative ordinate Corresponding 2f/dc measurements and fit via Eq. (16), using the ␣ ␤ ϭ Ϯ 2 values correspond to negative phase reading of the 2 f signal same assumption of small . This yields L (23.2 0.1)%, and R ␦ ϭ0.9989. on the lock-in amplifier. The PEM retardation 0 was set to 3.469 rad to maximize the magnitude of the 2 f /dc ratio. The plotted points represents the experimental data ratios acquired with arrangement A; the line represents the theoretical best fit from Eq. ͑14͒. The fitting parameters were optical rotation ␣ tical rotation derived is not closer to zero; to investigate this ␤ and the degree of linear polarization L , and the best fit was further, we examined the specular reflection from a 45° me- ␣ϭ Ϯ ␤ ϭ Ϯ obtained with (1.43 0.21)° and L (23.7 0.2)%, tallic mirror placed on the sample platform ͑results not with the nonlinear regression goodness-of-fit parameter R2 shown͒. This arrangement yielded ␣ϭ(Ϫ0.08Ϯ0.21)° and ϭ ␤ ϭ Ϯ 2ϭ 0.9992. The quoted uncertainties represent standard devia- L (100.0 0.2)%, with R 0.9998, very close to the the- tions from the fitting routine. As seen from the close corre- oretical values expected. This result validated the accuracy of spondence between experiment and theory, Eq. ͑14͒ describes the methodology, and ensured sound operation of the optical, ␣ ␤ the data well, and the values of and L derived appear to be electronic, and analytical fitting systems. The apparent offset ͑␣ϳ ␤ Ͻ ␣ ϳϪ reasonable 0°, L 100%, as would be expected for a in of 1.4° from the microspheres-only turbid sample turbid achiral sample͒. It is somewhat surprising that the op- could be caused by some unknown sample or

Journal of Biomedical Optics ᭹ July 2002 ᭹ Vol. 7 No. 3 295 Hadley and Vitkin

stray reflections from the cuvette walls. Although the true cause of this apparent optical rotation was not determined, its presence did not obscure interesting trends in ␣ that resulted from increasing concentrations of chiral, achiral, and racemic molecules in the turbid suspensions. ␤ To obtain the degree of circular polarization C , and to ␤ independently validate the derived value of L , the same tur- bid microsphere sample was examined in arrangement B. The ␦ PEM retardation 0 was set to 1.898 rad for this arrangement in order to maximize the magnitude of both the 1 f /dc and the 2 f /dc ratios. Figure 2͑b͒ shows the 1 f /dc data ratios and the corresponding theoretical fit from Eq. ͑15͒. There are three ␤ ␤ ␣ unknowns in the fit, C , L , and . As described in Sec. 2, ␤ ␣ the latter two always occur in combination L•cos(2 ), so they cannot be determined independently from this analysis. Fortunately, setting cos(2␣) equal to unity resolves this ambi- Fig. 3 Degree of linear polarization as a function of the refractive ␤ ϳ ␤ index of the aqueous phase of the turbid samples. L was determined guity while introducing less than a 3% error in L for even ␣ϳ from 2f/dc measurements in arrangement A fitted with Eq. (14), as per a reasonably large optical rotation of 4°. When this is Figure 2(a). The results are for chiral media (L-arabinose, squares), done, a two-parameter nonlinear regression fit to the data in racemic media (L/D arabinose mixture, triangles), and achiral media ͑ ͒ ␤ ϭ Ϯ ␤ ϭ Ϯ Figure 2 b yields C (47.4 0.1)%, L (22.5 0.2)%, (glycerol, circles). The microsphere suspension without any added 2ϭ ␤ ϭ and R 0.9997. The error in L is actually somewhat larger solutes (at n 1.331) is represented by hexagons. The error bars are because of the assumption of cos(2␣)ϳ1; nevertheless, the smaller than the size of the symbols. degree of linear polarization obtained is in good agreement with the value obtained above for arrangement A in Figure derived parameters on the properties of the turbid samples. 2͑a͒. Of note is the much larger degree of circular over linear Specifically, the preservation of enhanced polarization in tur- polarization preservation for this turbid sample. This prefer- bid samples caused by the presence of dissolved chiral mol- ential retention of circular polarization has been reported pre- ecules has previously been reported.25,27,28,30 Two putative ex- viously in turbid samples with predominantly forward-peaked 6–8,16,19,25 planations for this observed effect have been proposed. The single scattering phase functions. This is certainly electromagnetic ͑EM͒ wave propagation eigenmodes in a chi- the case for the current investigations, where Mie scattering ral medium are circular polarization states,41 and circular po- ␮ calculations suggest that the 1.4- m-diam polystyrene micro- larization has been shown to be preferentially preserved in a spheres in water exhibit a scattering anisotropy parameter g medium with strong forward anisotropic scattering.4,6 Alterna- ϳ0.94 at 632.8 nm, indicating predominantly forward 40 tively, the solute-induced increase in the refractive index of scattering. The preferential retention of circular over linear the aqueous solution reduces the index mismatch of the sus- polarization states is further explored below within the con- pended scatterers, making the samples less turbid23,24 and thus ͑ text of chiral, achiral, and racemic turbid suspensions Figure lowering the depolarization rates. To determine which expla- ͒ 5 below . nation is correct, we examine the degrees of polarization of For completeness and a check of self-consistency, there is light scattered from turbid suspensions containing ͑i͒ chiral additional analysis that can be performed with arrangement B molecules, ͑ii͒ racemic mixtures of chiral molecules, and ͑iii͒ used in Figure 2͑b͒. Figure 2͑c͒ displays the 2 f /dc ratio ob- achiral molecules, while matching the refractive indices of the tained from the same experiment. The fit is performed using aqueous phase of the three sample types. Figure 3 shows the ͑ ͒ ␤ ␣ the expression in Eq. 16 . Like above, the L cos(2 ) com- degree of linear polarization ␤ derived from measurements • ␣ ␤ L bination precludes independent determination of and L . in arrangement A via Eq. ͑14͒. The three sets of symbols ␣ ϳ ␤ Again, assuming cos(2 ) 1, the one-parameter fit gives L represent ͑i͒ 0–2 M L-arabinose chiral solution, ͑ii͒ a 50–50 ϭ(23.2Ϯ0.1)%, with R2ϭ0.9989. This result further cor- L/D mixture of 0–2 M racemic arabinose solution, and ͑iii͒ roborates the degrees of linear polarization derived in Figures 0%–35% ͑v/v͒ glycerol achiral solution. The horizontal axis 2͑a͒ and 2͑b͒. represents the resultant common refractive index of the solu- ␤ We thus conclude that experimental arrangement B is bet- tions. The trend of increasing L with an increase in the so- ter suited for an accurate determination of the degree of cir- lution refractive index is clear, and it indicates that index ␤ cular polarization C and a reasonably accurate determination matching lowers the turbidity of the medium and provides ␤ ͓ of the degree of linear polarization L by fitting the 1 f /dc improved retention of linear polarization. For the range of ͑ ͒ ␤ ␤ data to Eq. 15 ; L can also be determined by fitting the refractive index values in Figure 3 over which the L is seen 2 f /dc data to Eq. ͑16͔͒; if optical rotation is desired, the best to double from ϳ23% to ϳ45%, the corresponding reduction ␮ approach is to remove the quarter wave plate and fit the re- in the scattering coefficients calculated from Mie theory is s sulting 2 f /dc data ratios to Eq. ͑14͒ like in arrangement A. of 73.1 and 58.7 cmϪ1 ͑reduced scattering coefficient range ␤ ␮Ј Ϫ1͒ Such a setup will also yield an accurate L value, which can s of 4.7 and 2.8 cm . Note that, for a given refractive serve as an additional check of the slightly less accurate val- index, the chiral versus achiral nature of the solute does not ␤ ␭ ues L derived with the /4 plate in place. affect the preservation of linear polarization. It thus appears Having thus determined optimal ways in which to deter- that the chiral nature of the dissolved molecules is unimpor- ␤ ␤ ␣ mine C , L , and , we examine the dependence of these tant in enhancing the degree of linear polarization, and the

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␤ Fig. 4 Optical rotation ␣ a function of the sample refractive index Fig. 5 Degree of circular polarization C (closed symbols) and degree ␤ from the same data and fit as in Figure 3. The error bars represent of linear polarization L (open symbols) from chiral, racemic, and regression-fit uncertainties in the optical rotations derived. The sym- achiral turbid media. The results are from the 1f/dc data ratios in bols are the same as those defined in Figure 3. arrangement B, fitted with Eq. (15) and assuming small ␣ (see the text). The error bars are smaller than the size of the symbols. The symbol shapes have the same meaning as those in Figures 3 and 4. effect is entirely due to refractive index matching. ference decreases the turbidity of the medium, resulting in Figure 4 summarizes the corresponding derived values of less depolarization. the optical rotation ␣ as a function of the solution’s refractive The difference in levels of polarization preservation for index. These results are from the same data set and fit as those linearly versus circularly polarized light is significant; the lat- in Figure 3. There is a small offset of the optical rotations ter appears greater by approximately a factor of 2 across the ͑approximately Ϫ1.5°͒ that we have not been able to explain entire refractive index range ͑corresponding to a scattering ␣ ␮ Ϫ1 nor eliminate. As well, the uncertainty in is considerably coefficient range of s of 73.1– 61.3 cm , and a reduced greater than that in ␤ as is evident by the size of the error bars ␮Ј Ϫ1͒ scattering coefficient range s of 4.7– 3.1 cm . This pref- in Figure 3. This indicates a much more sensitive dependence erential retention of circular over linear polarization states has of functions in Eqs. ͑14͒–͑16͒ on ␤ than on ␣. Nonetheless, of been noted before in media composed of predominantly interest are the trends in optical rotation noted for each solute forward-scattering particles.4,6,7,16,19,25 In the current turbid type. For both the racemic arabinose mixtures and the achiral samples, the anisotropy parameter gϳ0.94, so scattering is glycerol solutions of equal refractive index, the optical rota- ␤ certainly forward peaked, and the larger magnitude of C tion induced by the sample remained essentially constant, in- ␤ compared to L is not surprising. However, one must exercise dependent of the solute concentration. This was expected, caution in extrapolating these microsphere scattering results since racemic mixtures and achiral solutes should not induce to biological tissues. The differences in the nature of refrac- optical rotation. The apparent slight upward trend is probably tive index variation in the two systems, including the potential related to the unexplained ␣ offset noted above. For chiral contribution of dependent scattering, can lead to a variety of solutions of LϪ(ϩ)-arabinose, the optical rotation observed ␤ unexpected effects, including larger magnitudes of L com- increased in a linear fashion with the concentration of solute. ␤ 16,19 pared to C . Even within the microsphere system, the These results imply that the methodology developed is ca- relative magnitude of the two depolarization parameters is pable of measuring small optical rotations of linearly polar- predicted to depend sensitively on the size ͑and thus the scat- ized light even in diffusive scattering. While the preservation tering anisotropy͒ of the inclusions.7,8 Further studies are of linear polarization is insensitive to the chiral nature of the ␤ ␤ planned to measure L and C parameters in a variety of molecules, measurements of optical rotation may reveal if the turbid systems, including those of biological origin. dissolved solute is chiral or achiral/racemic. ␤ Figure 5 shows the dependence of circular polarization C ␤ and linear polarization L as a function of the solution refrac- 5 Summary and Conclusions tive index. These results are from experimental measurement A methodology for determining the optical rotation, the de- with the quarter wave plate in place ͑arrangement B͒, ana- gree of linear polarization, and the degree of circular polar- lyzed with Eq. ͑15͒ at the 1 f /dc signal, as per Figure 2͑b͒. ization of light diffusely scattered from turbid media was de- Both degrees of polarization are seen to increase by ϳ23% in veloped, and was applied to determine the cause of the a strikingly linear fashion. Like in Figure 3, the surviving previously reported chirality-induced increase in the preserva- polarization fractions seems to be solely a function of the tion of polarization. The chiral nature of the dissolved solutes solution refractive index. Chirality of the solute had no visible was only evident in the derived optical rotations, which were ␤ ␤ effect on L or C . Again, this suggests that the increase in seen to increase linearly with chiral molecule concentrations ␤ ␤ L or C is the result of a smaller difference between the but stayed essentially constant with increasing concentrations refractive index of the solution and that of the suspended of achiral or racemic additives. The observed increase in the polystyrene microspheres. This smaller refractive index dif- linear and circular degrees of polarization with dissolved sol-

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