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Grating Light Reflection Spectroscopy of Colloids and Suspensions Brian B. Anderson,† Anatol M. Brodsky, and Lloyd W. Burgess* Center for Process Analytical Chemistry, Department of Chemistry, University of Washington, Box 351700, Seattle, Washington 98195-1700 Received June 24, 1996. In Final Form: May 22, 1997X
The interaction of light with scattering matrixes, particularly particles suspended in a liquid, is described as it relates to the use of a technique known as grating light reflection spectroscopy (GLRS). Analysis of the singular behavior of light reflected from a transmission diffraction grating in contact with a scattering matrix has shown that the effective dielectric function of the medium in contact with the grating can be measured using a reflection method. The relative contributions of the real and imaginary components of the effective dielectric function are probed as mean particle radius increases, shifting the forward scattering regime from the Rayleigh limit to the Fraunhofer diffraction approximation. It is shown that GLRS provides substantial information regarding the characteristics of colloids and suspensions. Further investigations could, in theory, yield particle shape and density distribution information as well.
Introduction We have recently described an optical technique for analytical chemical sensing termed grating light reflection spectroscopy (GLRS).1,2 The basis of this technique is the measurement of the intensity and phase of light reflected from a transmission diffraction grating that is in contact with a liquid sample (see Figure 1). GLRS allows for the determination of bulk sample properties without relying on the measurement of transmitted light intensity, and this feature is based on the fact that at specific parameter combinations (thresholds) one of the diffracted transmitted waves is transformed from a traveling wave to an evanescent one.2 The characteristics of all reflected and transmitted diffraction orders, including the specular reflection, abruptly change at these thresholds. It has been demonstrated that the reflection spectrum contains features that, upon appropriate analysis, allow for the separation of refractive index and absorbance effects and, in addition, surface and bulk effects. As was mentioned in refs 1 and 2, GLRS may provide analytically relevant information regarding mesoscopic suspensions and colloids in liquid samples. Such measurements are based on the effect of light scattering by particles in a liquid matrix, where scattering events that change the properties of evanescent waves near the threshold lead to the change of threshold characteristics. These changes occur as a result of the appearance of a specific imaginary component in the effective dielectric constant due to coherence distortion even if the particles do not absorb light in the exploited frequency interval. The measurement of particle characteristics by the GLRS experimental technique bears some analogous relationship to the “frustrated total internal reflection” method used previously for the measurement of static and dynamic properties of colloid particles in electrolyte solutions.3,4 Optical scattering in a heterogeneous matrix is influenced by such particle characteristics as shape, size, † Present address: Westinghouse Savannah River Co., Savannah River Technology Center, Building 773-22A, Aiken, SC 29808. X Abstract published in Advance ACS Abstracts, July 15, 1997.
(1) Anderson, B. B.; Burgess, L. W.; Brodsky, A. M. Anal. Chem. 1996, 68(7), 1081. (2) Anderson, B. B.; Burgess, L. W.; Brodsky, A. M. Phys. Rev. E 1996, 54(1), 912. (3) Prieve, D.; Bike, S.; Frej, N. Faraday Discuss., Chem. Soc. 1990, 90, 209 and literature therein.
S0743-7463(96)00624-5 CCC: $14.00
Figure 1. Physical representation of the GLRS sensor response to scattering systems. Here the m ) 1 transmitted diffraction undergoes a transformation from a traveling wave to an evanescent one. The GLRS response is dependent upon the nature of the coherent and incoherent scatter in the transmitted diffraction order.
morphology, and distribution in the matrix. In addition, the particle material composition and bulk dielectric function affect the phase and amplitude of scattered light. GLRS measures the changes in bulk dielectric function of the matrix under study, where the values of the bulk dielectric are influenced by the matrix and the particles in that matrix. Therefore, it is possible, in theory, to either directly or indirectly measure the aforementioned particle characteristics using GLRS as a noninvasive, optical reflection method. In this article, we describe the theoretical effects of scattering matrixes on the GLRS signal and correlate particle size and concentration information with GLRS measurements made using monodispersed systems of polystyrene in water as model systems. The article is organized as follows: We first describe the theory of the GLRS effect for systems in which the mean distance between particles is larger than the incident light wavelength. We then describe the experiments and experimental results for a range of suspensions of polystyrene spheres with fixed radii. In the conclusion, we compare the theoretical predictions with the experimental results and discuss possible further application of the described method. (4) Heeten, M.; North, A. Meas. Sci. Technol. 1991, 2, 441.
© 1997 American Chemical Society
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Theory We will first discuss the general expressions that describe the effective dielectric function of a system containing particles in suspension. Following this, we will show the relationship between the effective dielectric function of a system and the optical response, in general, of various reflected diffracted waves as defined by theory2 and demonstrated experimentally for homogenous systems.1 To conclude the theoretical section we will discuss the expressions that describe the forward scattering amplitudes and the relative changes in sample effective dielectric function due to particle size and concentration changes. Light scattering in random media, a very complex phenomenon, is a function of different light scattering events by nonuniformities which include coherence changes and the absorption of incident light. The analytical theory of such scattering in random media is fully developed only for the limiting cases of dilute systems or systems where the wavelength of light is larger than the characteristic distance of density variations, where it is possible to use the quasi-classical approximation.5 In the case of GLRS, the theoretical interpretation is substantially simplified by the fact that we probe only that fraction of light scattered by particles that travels in the forward direction and that remains coherent with the incident light.2 Since the diffraction patterns of GLRS are dominated by the interference of the scattered beams with the incident beam, the observed diffraction peaks correspond to the directions of constructive interference. The fraction of the scattered energy departing from its original propagation direction (but still at the same frequency since we are considering here only elastic scattering) becomes incoherent and effectively lost for constructive interference. The possibility of a certain part of the incoherent wave energy being returned to the coherent state may be disregarded here since to be coherent means that propagation is not only in the original direction but also with the same phase, and these combined requirements make the return probability essentially zero. This allows us to describe the bulk sample medium containing particles by the following effective dielectric constant: (2) eff =
[
c2
x(2) + 2π (2)
=
+ 4π
c2
ω2x(2)
∑R
∑R NRAR(0)
ω2
]
2
NRAR(0)
c2 2π | ω2(2)
∑R NRAR(0)| < 1
which is fulfilled in solutions with moderate particle concentrations. To illustrate the restrictions imposed by (2) we consider the case when the particle radii RR are larger than the wavelength λ. In this case
|AR(0)| = R2Rω/(2)c
c/ωR h R < dp(2)/ds
(5) Newton, R. Scattering Theory of Waves and Particles; McGrawHill: New York, 1966. (6) Champion, J.; Meeten, G.; Senior, M. J. Colloid Interface Sci. 1979, 72, 471.
(2′)
where dp is the density of the particle material and ds is the mean weight particle concentration in the solution. The bar over RR represents an averaging over all particle types. The amplitudes AR(0) are in general complex with the imaginary part present, even in the absence of light absorption, due to the coherence loss from scattering. The important point to be made here is that, close to the GLRS thresholds, the diffracted transmitted waves that transform from traveling to evanescent penetrate the medium to a distance (2) )c L ∼ ω/Im(eff
(4)
much larger than the incident wave’s wavelength and the interparticle distances, even in dilute systems. This property allows us to use the generalized GLRS theory to describe scattering from the grating in contact with colloids and suspension solutions by introducing the effective dielectric constants of type 1 into the expressions detailed in GLRS theory. In the following description of GLRS theory, (n)(ω), where n ) 0, 1, or 2, corresponds to the frequencydependent dielectric constants of incident medium, substrate medium, and sample medium, without particles, respectively (see Figure 1), and (2) eff corresponds to the effective dielectric constant in the presence of particles. We introduce the quantity
[
where (2) ≡ (2)(ω) is the dielectric function of the medium without particles present, ω is the frequency of incident light, c is the speed of light in vacuum, and NR and AR(0) are, respectively, the number of particles of type R per unit volume and the scattering amplitude in the forward direction for those particles. The sum over R in (1) is taken over all types of particles. Expression 1 is the square of the expression 1.103 in ref 5 (see also ref 6 ) for the effective dielectric function of particles in a vacuum with corrections made in order to take into account the optical response of the solvent, which is described by (2) ≡ (2)(ω). The applicability of (1) is restricted by the following inequality:
(3)
where we have omitted the numerical coefficients that are on the order of unity. After introduction of (3) into (2), condition (2) is reduced to
(2) δmcr ) Re(eff (ω)) - sin2 θ +
(1)
(2)
4πcmcr sin θ cos ψ + ωa 2πcmcr 2 (5) ωa
(
)]
where θ is the incident angle, ψ is the angle between an incidence plane and a normal to grating strips, and a is the grating period. The component of the wave vector in sample medium in the direction normal to the surface is proportional to δmcr. When δmcr is less than zero, this wave vector component is imaginary and the corresponding diffraction beam enumerated by mcr ) 0, (1, ... is an evanescent wave which does not carry energy from the surface. The index cr in (5) indicates that, in the considered parameter interval changes, the quantity δmcr passes through zerosthe point where the transmitted diffraction beam is transformed from a traveling to an evanescent one. According to GLRS theory described in full detail in ref 2, for the case of Im ((2) eff (ω)) < Re((2) eff (ω)), there can be observed specific features in the behavior of reflected light at points δmcr ) 0, which we have termed “singularities” as the behavior of reflected light is discontinuous around δmcr ) 0 in the limit of infinite
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experimental resolution and perfect incident beam collimation. According to (5), δmcr depends on the incident frequency (ω), the angles θ and ψ, and the sample medium composition, which includes the particle concentration. The behavior of mth diffraction peak in reflected light can be described by the following expressions for the reflection coefficients R(s,p)(m), for s- and p-polarized light, for the interval of parameters in the threshold region (where δm lies near zero): (s,p) R(s,p)(m) = C(s,p) 1 (m) + C2 (m) ×
x12(xδ + (Im ) - δ ) + (m) x12(xδ + (Im ) + δ ) as Im) 2 m
C(s,p) 3
C(s,p) 1 (m) +
2 m
(2) 2 eff
(2) 2 eff
m
m
{
(2) eff f0
C(s,p)2(m) x|δm| for δm e 0 , m * mcr C(s,p)3(m) xδm for δm > 0 (6)
(s,p) (m) are approximately constant where coefficients C1,2,3 near the thresholds. An important point to be stressed here is that a separation of bulk and surface properties is possible with appropriate analysis of the reflected light in the GLRS experiments. The positions of the singularities at δmcr ) 0 and the functional dependence of the reflection coefficient on δm in the interval δm ∼ 0 are dependent only on the properties of the sample matrix at distances from the sample-grating interface of the order of the length L introduced in (4). As a result, the position of the singularity, as well as the reflection coefficient’s and phase shift’s behavior in the vicinity of this position, will be unaffected by surface fouling layers and the details of surface-particle interactions at distances much less than L. An important consequence of (6) is the specific dependence of R(s,p)(m) on the imaginary part of the dielectric function Im(2) eff (ω). If
(2) (2) Imeff (ω) R 2πω (2)
(10a)
or to Fraunhofer diffraction, when the following inequality holds:
λ)
c < Rxp 2πω
(10b)
where R is the characteristic dimension of the particles and p is the dielectric function of the particle material. In the limiting case of (10a), the amplitude AR(0) for all polarizations is equal5 to (2) 1 ω 2 (P - ) 1 R x(2) 2 ImAR(0) ) ReA (0) ) R c ( + 2(2)) R R2R R R P 1ω3 2 (2) (P - )RR for ImP ) Im(2) ) 0 (11a) 2 c
()
()
and in the case of (10b) the imaginary part of AR(0) is equal to
1 |ReAR(0)| < SR
1 2R(xp - x′(2))
ω 1 ImAR(0) ) SR 2c
(11b)
where SR is the averaged “radar cross section” of particles with different orientations. Note that (11a) and (11b) are correct for nonspherical particles when the corresponding condition (10a) or (10b) is fulfilled. Expression (11a) represents the optical analog of the “effective radius approximation” of general wave scattering theory with the parameter RR representing, in this case, an averaged (effective) radius. In the transition interval between (10a) and (10b), a reasonable approximation in the form of the Van de Hulst (eikonal) expression5 can be used for the forward scattering amplitudes which, in the case of scattering by particles in a medium, has the following form (see Figures 5-8): (7) Ludlow, I.; Eviert, J. Phys. Rev. E 1996, 53, 2909.
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[21 + y
A(0) ) ixR
e
-1 -y
Anderson et al.
(
]
+ y-2(e-y - 1)
(i)(xcr 1 ,ω) - sin θ +
)
mcr(2πc/ω) a
2
)0
i ) t or l (16)
where
y ) -2ix(xp - x ) (2)
(12)
x ) (Rω/c)x(2) Approximation 12 gives quantitatively correct results for spherical particles if (2) xp - x > 2πc/ω, it is possible to apply the WKB approximation. In this case, the singularity in reflected light will correspond to the emergence of turning points at some distance xl ) xcr l with corresponding parameter changes for diffraction peaks with m ) mcr where (8) Pollack, J.; Cuzzi, J. Light Scattering by Irregularly Shaped Particles; Scherman, D., Ed.; Plenum Press: New York, 1983; p 113. (9) Ryazanov, M. JETP 1995, 81, 974.
The transmission diffraction grating used in these studies was fabricated via a Lloyd’s mirror configuration11 on a chrome-coated fused-silica disk using standard photoresist development techniques reported earlier.1 The grating period was measured to be 829.4 ( 0.76 nm. The grating substrate was mounted on a dual axis rotation stage with the incident angular resolution of 0.083°, an azimuthal angular resolution of 0.5°, and the incident angle set at 34.84°. The sample was placed in a 2 mL Teflon sample holder, pressure fitted and sealed with a Parafilm gasket against the grating side of the substrate. Sample was introduced via a pipet. The optical system consisted of a dc tungsten-halogen source coupled into a 220 µm core optical fiber (Figure 2). The output of the fiber was collimated via an achromat collimator and the beam size minimized using an iris aperture. Incident polarization state was set via a Glan Taylor cube polarizer to transverse magnetic (TM), or p-polarization. The reference beam was captured via a fiber optic at the output of the collimator, removing a portion of the poorly collimated light at the limit of the clear aperture of the lens but not affecting the light incident on the grating. The grating-substrate interface specular reflection (the zeroth order diffracted beam from the grating) was collected as the sample beam. These two collection fibers were coupled as a stacked pair into a McPherson 218 spectrograph using aperture matching optics and adjustable slit, and the dispersed images were vertically resolved onto a Princeton Instruments LN/CCD camera placed in the image plane of the spectrograph. In order to simplify the comparison of theory and experimental results, polystyrene microsphere dispersions were used in the study. These microspheres were obtained from Polysciences, Inc., and the particle sizes and particle number densities are detailed in Table 1 for the latex samples as received. Dilutions were prepared in the sample cell. Samples were run in random order with rinsing of the Teflon sample cell between samples. Data were collected as five scan averages, 1 s integration time, with sample and source reference spectra obtained simultaneously. Data analysis is similar to that in ref 1, with the exception that a flat field correction was applied prior to the calculation of the GLRS reflection coefficient derivatives using a 51-point Savitsky-Golay filter. Derivative peak positions and magnitudes were used to determine absolute dielectric properties and correlate the experimental data with theory using eqs 1 and 5.
Results and Discussion The GLRS response to bulk dielectric fluctuations is demonstrated by orthogonal responses to the real and imaginary parts of the dielectric for samples in contact with the grating. An increase in effective real part of the dielectric, Reeff should yield a threshold shift to higher wavelengths for constant grating period and incident angle. This would appear as a peak in the derivative of the reflection coefficient. Conversely, an increase in the imaginary part of the effective dielectric Imeff will yield a decrease in the derivative peak amplitude. Figure 3 (10) Sarchenko, A.; Zeldovich, B. Phys. Rev. E 1994, 50, 2287. (11) Mai, X.; Moshrefzadeh, R.; Gibson, U. J.; Stegeman, G. I.; Seaton, C. T. Appl. Opt. 1985, 24, 3155. (12) Bowman, E. M. Ph.D. Dissertation, University of Washington, 1992.
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Figure 2. Experimental instrumentation for GLRS: (1) grating substrate in substrate holder, (2) 32 V dc tungsten-halogen source, fiber bundle coupled, (3) achromat collimator, (4) iris aperture, (5) Glan-Taylor polarizer, (6) reference beam fiber optic, (7) sample beam fiber optic, (8) McPherson spectrograph with dual stacked fiber mutliplexed input, (9) Princeton Instruments LN/CCD placed in the image plane of the spectrograph, (10) 486 PC for data acquisition, (11) dual axis rotation stage, and (12) sample cell.
Figure 3. GLRS response to 2.5% (nominal) solids polystyrene microspheres in water (Table 1), p-polarization, 34.84° incident angle. Notice the transition region around a particle radius of 1 micron. The effective real portion of the dielectric dominates for particle sizes smaller than the transition radius, and as the particle size increases to the transition region, the effective imaginary dielectric function begins to presents itself due to noncoherent scattering resulting in loss of coherence.
Table 1. Polystyrene Microsphere Samples in Water: Concentrations and Particle Radiia sample
particle radius, microns
vol % polystyrene
particles/mL
1 2 3 4 5 6 7 8 9 10 11
0.042 ( 0.001 0.101 ( 0.005 0.178 ( 0.007 0.240 ( 0.007 0.349 ( 0.006 0.536 ( 0.009 1.046 ( 0.048 1.418 ( 0.068 5.284 ( 0.517 7.659 ( 0.983 11.006 ( 1.515
2.55 2.38 2.51 2.50 2.56 2.44 2.49 2.38 2.38 2.38 2.56
8.08 × 1015 5.52 × 1014 1.06 × 1014 4.31 × 1013 1.44 × 1013 3.78 × 1012 5.19 × 1011 1.99 × 1011 3.85 × 109 1.27 × 109 4.59 × 108
a
Polystyrene refractive index is 1.577 at 589.3 µm.12
shows the experimental GLRS derivatives for the series of undiluted polystyrene microspheres (2.5 wt % solids). Figure 4 displays the GLRS derivatives for the diluted samples with nominal concentration of polystyrene of 1.875 wt %. The bulk real refractive index, n ) x, for water measured via an Abbe refractometer is 1.3308 at 630 nm and 22 °C. For the experimental conditions given in Figure 4, this bulk refractive index value yields the threshold position at 630 nm predicted using eq 5. The GLRS derivative reflection spectra peak maxima and positions for both experiments are tabulated in Tables 2 and 3. Although the incident angle for each experiment was identical, the singularity position for water for the dilution experiments yielded a slightly lower index of refraction than water in the previous experiment due to the slightly higher ambient temperature of 24 °C. The peak position values were used to predict Reeff values for all samples using eq 5 for comparison with the theoretical real effective dielectric values obtained from the Van de Hulst expression (12). In addition, the peak maxima magnitudes were found, and deviations from the water value were calculated. Theory predicts that the peak height will be inversely proportional to the square root of the imaginary part of the dielectric, and the peak height variations were used to generate normalized Imeff values for comparison with theory.
Figure 4. GLRS response to 1.875% (nominal) solids polystyrene microspheres in water, p-polarization, 34.84° incident angle. Again, notice the transition region around a particle radius of 1 micron. The shift in the effective real part of the dielectric is smaller than the corresponding shift for the 2.5% concentration as would be expected for scattering close to the Rayleigh limit.
The general trend for observed shifts of the real and imaginary parts of the dielectric function follows what is expected from theory. As seen in Figures 3 and 4, the real refractive index shift (measured as a GLRS threshold wavelength shift) due to Rayleigh scattering is evident for samples having particle radii smaller than the wavelength of incident light. At the same time, the GLRS results for larger radii show the influence of the imaginary part of dielectric function manifested in lowering and widening of corresponding maxima. Figures 5 and 6 show comparison of the experimental and theoretical values for Reeff for the systems studied. The comparison of theory and experiment for the imaginary part of the dielectric function is presented in Figures 5 and 6, showing that the expected GLRS shift due to Reeff, and the Rayleigh limit
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Table 2. GLRS Derivative Peak Positions and Amplitude Differences (from a Water Baseline) and Corresponding Real Dielectric Functions Predicted from the Peak Positionsa sample
position of GLRS singularity, in nm
pred Reeff from GLRS
peak magnitude dev from water
water 1 2 3 4 5 6 7 8 9 10 11
630.1 635.1 635.1 635.1 634.5 633.9 634.3 628.7 630.1 630.1 630.2 630.1
1.7709 1.7869 1.7869 1.7869 1.7850 1.7831 1.7844 1.7662 1.7709 1.7709 1.7712 1.7712
0.000 000 0.000 141 0.001 596 0.003 632 0.003 328 0.004 377 0.004 502 0.001 919 0.000 039 0.000 358 0.001 371 0.000 206
a
Nominal 2.5% by weight concentration of polystyrene.
Table 3. GLRS Derivative Peak Positions and Amplitude Differences (from a Water Baseline) and Corresponding Real Dielectric Functions Predicted from the Peak Positionsa sample
position of GLRS singularity, nm
pred Reeff from GLRS
peak magnitude dev from water
water 1 2 3 4 5 6 7 8 9 10 11
629.9 633.9 633.5 633.5 633.4 631.5 633.3 629.1 629.5 629.8 629.9 629.9
1.7703 1.7831 1.7818 1.7818 1.7815 1.7754 1.7812 1.7677 1.7690 1.7699 1.7702 1.7702
0.000 000 0.000 461 0.001 716 0.002 703 0.003 202 0.004 051 0.004 713 0.001 155 0.000 082 -0.000 020 0.000 174 -0.000 143
a
Figure 6. Comparison of the real part of effective dielectric function for the 1.875% polystyrene samples measured with GLRS (O) and the theoretical values (solid line). Notations are the same as in Figure 5.
Nominal 1.875% by weight concentration polystyrene.
Figure 7. Comparison of the normalized imaginary part of effective dielectric function for the 2.5% polystyrene samples measured with GLRS (O) and the theoretical values (solid line). Notations are the same as in Figure 5.
Figure 5. Comparison of real part of the effective dielectric functions for the 2.5% polystyrene samples measured with GLRS (O) and the theoretical values (solid line) calculated from the Van de Hulst approximation as a function of the ratio of radius R to the critical (threshold) value of wavelength for water λ.
Reeff agrees very well with the GLRS experimentally determined Reeff. Some deviations from the theory are seen in the transition region, where the sharpness of the decline is evident in the GLRS values. This is because the Van de Hulst expression is an approximation and the particles are not strictly spherically symmetrical with fixed radius. In addition, the theoretical curve is calculated using a
nominal particle concentration of 2.5 wt %, and the actual concentrations of the samples deviate from that nominal value according to Table 1. However, the transition region occurs in the particle radius range that is predicted from theory, and the theoretical limiting values for Reeff and Imeff. are consistent with the GLRS measured values. Thus, the real refractive index shift due to Rayleigh scattering in the limit of small particles is directly measurable using GLRS, even as the contribution from the imaginary part of the dielectric function begins to dominate when the transition region is reached. In addition, notice that the sample containing particles of radius 1.046 µm, which fall in the transition interval, yields a bulk Reeff value that is lower than either the solvent (water) or the particle material. This is due to the interference effects in forward scattering for spherical, nonabsorbing species and further validates the application of GLRS to scattering systems in that the transition region oscillations in Reeff are directly measurable. Figures 7 and 8 show the calculated normalized theoretical Imeff values using the Van de Hulst approximation. The normalized experimental values, calculated from the GLRS peak height modulations by taking the square of the height deviation from a water baseline
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allows for the extraction of information regarding the particle size and dielectric function distribution and morphologies of particles in the analytical matrix. In particular, in the case where the Rayleigh regime is realized, the mean particle radius R h and particle volume fraction F may be calculated from the real and imaginary parts of the dielectric function measured via a GLRS threshold shift as follows
R h )
Imef
λ
F)
Figure 8. Comparison of the normalized imaginary part of effective dielectric function for the 1.875% polystyrene samples measured with GLRS (O) and the theoretical values (solid line). Notations are the same as in Figure 5.
and applying a normalization factor, are also plotted for comparison. It is apparent that the forward scattering approximation agrees very well with the physical phenomena as the monotonic increase in Imeff to the peak of the transition region is seen on both the GLRS measured Imeff and the theoretical Imeff. Similarly, the transition region occurs at roughly the same particle radius as the region for the Reeff values, and this is where the effective imaginary dielectric begins to dominate and the real part reverts back to the bulk solvent value, as in the case of Fraunhofer scattering. It is interesting to note that the finite Imeff expected for larger particles is not evident in the GLRS measurements. It was demonstrated in previous experiments1 that a threshold of sensitivity to Imeff exists for this particular grating-substrate material that is above the Imeff values for the dilute polystyrene concentrations used here. As the weight percent polystyrene is essentially constant for each experiment, the number density of scatterers is reduced by a factor of 107 and this reduces the number of scattering events close to the grating. Conclusions The optical interrogation of scattering systems by GLRS demonstrates the use of a reflection-based optical technique that generates information about the size, nature, and density distribution of scattering materials in a matrix due to dielectric function changes induced by properties of the suspended particles and their concentrations in suspension. The existence of transmitted and reflected diffraction orders are a result of coherent scattering from the grating itself, and the presence of scattering particles in the matrix in contact with the grating disrupts the nature of the coherent scattering at the grating-sample interface. GLRS, due to the coupling of the transmitted and reflected diffraction orders and the presence of singularities in reflected light, allows for the monitoring of coherence loss in the forward scattered light (due to particles present in the analytical matrix) by analyzing reflected light intensity. The deconvolution of the real and imaginary parts of the effective dielectric functions is an important consequence of GLRS theory in that it
; for Im (2) 1), or the angles θ and ψ. In order to demonstrate the possibility of modulating the critical wavelength value experimentally we give an expression for λ in the following form based on (5):
λ)
a [xRe(2)(ω) + sin2 θ sin2 ψ - sin θ cos ψ] |mcr| (18)
All of the described GLRS analytical results are valid for moderately nonuniform particles in concentration ranges where inequality 2 holds. GLRS reflection thresholds, which are dependent on the particle size and concentration as demonstrated by the data in this article, are predicted theoretically for significantly higher concentrations of particles as well, although they will be modified on the basis of additional scattering conditions present. In future work, we plan to further elucidate the methods for extracting quantitative particle size and distribution information from these more concentrated systems using GLRS. LA960624B
