## Abstract

Rayleigh-Debye-Gans and Mie theory were previously shown to disagree for spherical particles under ideal conditions^{4}. A Hybrid model for spheres was developed by the authors combining Mie theory and Rayleigh-Debye-Gans. The hybrid model was tested against Mie and Rayleigh-Debye-Gans for different refractive indices and diameter sizes across the UV-Vis spectrum. The results of this study show that the hybrid model represents a considerable improvement over Rayleigh-Debye-Gans for submicron particles and is computationally more effective compared to Mie model. The development of the spherical hybrid model establishes a platform for the analysis of non-spherical particles.

©2008 Optical Society of America

## 1. Introduction

Spectroscopy has been used as a rapid tool for the characterization of micron and submicron biological particles^{1,2}. However, the interpretation of the spectra is difficult due to the complex nature of the particles and to the fact that the spectral data contains information on the size, the shape, the chemical composition, and the structure of the particles. Several methods have been reported the estimation of properties relevant to biological systems such as particle size and particle shape^{3}. Rayleigh-Debye-Gans and Mie theory provide solutions that enable light scattering methods to be effectively used for real time monitoring applications, however each theory has limitations. Mie theory provides the exact light scattering solution for spherical, homogeneous, and layered particles but is limited in terms of providing shape information (i.e. ellipsoids and rods). The Rayleigh-Debye-Gans (RDG) approximation enables the estimation of the particle shape and size, an important feature when characterizing biological systems; however it is limited in its applicability because of the range of refractive indices and particle sizes it is known to be valid for. A comparative evaluation of Rayleigh-Debye-Gans approximation and Mie theory was recently conducted on refractive indices and spherical sizes representative of biological particles^{4}. The theoretical limits and the quality of the approximation were evaluated using multiwavelength spectroscopy (200–900nm) which showed that there was a general disagreement between the two theories under the conditions where the two should coincide (i.e., spherical particles with refractive indices close to unity). Several approaches were evaluated to reconcile the two theories however these provided ineffective4. The results of this comparison stimulated an investigation into the possibility of reformulating the Rayleigh-Debye-Gans approximation to obtain a model that incorporates shape (RDG) and a good estimation of light scattering (Mie) by particles.

Reformulation of Rayleigh-Debye-Gans has not been considered before in the context of multiwavelength spectroscopy. The concept of improving/hybridizing RDG, or combining theories for particle characterization has been reported in the literature^{5,6}. An improvement to the efficiency factors used in the Rayleigh-Debye-Gans approximation is reported by Perel’man et al^{5} for optically soft particles (S-approximation). The application of a hybrid numerical method was used by Choi et al. for light scattering and absorption^{6}. The hybrid numerical method used for inhomogeneous spheres utilizes the finite-element method and boundary element method.

The method proposed herein is a hybridized theory. The customary Rayleigh-Debye-Gans approach assumes that the infinitesimal ellipsoidal volume elements of the scatterer respond to the local incoming (unperturbed) electric field exactly like they would respond to a static uniform field; that is they form dipoles. These dipoles are then assumed to vibrate synchronously with the incoming field, and their (re-)radiation is computed classically. The key to the new, hybrid approach is to use values for these dipole moments that would be induced by the local Mie-solution field (rather than the unperturbed incoming field). This corrects, to some extent, for the attenuation, rotation, and other modifications inflicted by the rest of the scatterer’s body on the incoming field, en route to the volume element in question. The resulting formalism is similar in most ways to Rayleigh-Debye-Gans and therefore new form factors can be generated through the scattering amplitude functions. The hybridized model was compared to Mie theory and Rayleigh-Debye-Gans approximation for spheres of different size and values of refractive indices (for which the Mie theory is exact). The results of this paper show a dramatic improvement over Rayleigh-Debye-Gans approximation and establish the method for calculating the induced dipole moment from the internal field to be a positive approach.

## 2. Materials

The programs for Mie theory, Rayleigh-Debye-Gans theory, and hypochromicity were developed in Matlab v6.5.1. Computations for these programs were conducted using a Dell Inspiron 4100 with 1GHz Pentium III processor and 512 MB RAM. The optical properties (refractive indices) utilized were provided by Dr. Garcia-Rubio and the SAPD laboratory through the College of Marine Science at the University of South Florida^{6}.

The computer codes developed for the analysis of the Rayleigh-Debye-Gans and Mie particles were previously tested ^{4}. In testing and exploring the algorithms for Rayleigh-Debye-Gans the refractive indices selected were those of soft bodies and hemoglobin, where soft bodies are defined here as particles whose relative refractive index is close to one with no absorption component. The values of the index of refraction *n*+iκ for biological particles commonly used are soft bodies (1.04≤*n*≤1.45) and hemoglobin (1.48≤*n*≤1.6, 0.01≤*κ*≤0.15)^{2}. Polystyrene (1.5≤*n*≤2.2, 0.01≤*κ*≤0.82), is a material found in industrial applications whose properties are used as standards for optical instruments2. The optical properties of water (1.3≤*n*≤1.4) have been used to characterize the suspending medium.

The range of particle volumes was chosen between 12700 nm^{3} and 0.0654 µm^{3}. The spherical diameter equivalents to the volume range are between 25 nm and 500 nm. Table 1 gives the simulation parameters used to define the suspensions for the analyses conducted in this study.

## 3. Theory

#### 3.1 Geometry and notation

To describe the electric field scattered by a particle in the laboratory system there are two objects of interest, the detector and scatterer. Figure 1 illustrates the detector located at *r⃗* with spherical coordinates (*r*,*θ*,*ϕ*) or Cartesian coordinates (*x*,*y*,*z*). Points within the scatterer are identified by *R⃗* with coordinates (*R*,Θ,Φ) or (*X*,*Y*,*Z*).

The curvilinear unit vectors attached to the detector in Fig. 1 can be expressed in rectangular coordinates through the following equations:

$$y=r\mathrm{sin}\theta \mathrm{sin}\varphi $$

$$z=r\mathrm{cos}\theta $$

$${\stackrel{\rightharpoonup}{e}}_{\theta}={\mathrm{cos}\theta \mathrm{cos}\varphi \stackrel{\rightharpoonup}{e}}_{x}+{\mathrm{cos}\theta \mathrm{sin}\varphi \stackrel{\rightharpoonup}{e}}_{y}-{\mathrm{sin}\theta \stackrel{\rightharpoonup}{e}}_{z}$$

$${\stackrel{\rightharpoonup}{e}}_{\varphi}={\mathrm{sin}\varphi \stackrel{\rightharpoonup}{e}}_{x}+{\mathrm{cos}\varphi \stackrel{\rightharpoonup}{e}}_{y}$$

As indicated, the incident wave moves in the z-direction and is presumed to be plane-polarized in the x-direction. It impinges upon the particle and is scattered. The scattered wave is detected at some angle *θ* and *ϕ* measured from the direction of propagation of the incident wave; see Fig 1. The following section provides a mathematical description of the fields induced by the particle. As will be seen, the scattering dynamics are best described using the vectors *e⃗ _{R}*,

*e⃗*

_{Θ},

*e⃗*

_{Φ}; the scattered radiation is best described by

*e⃗*,

_{r}*e⃗*,

_{θ}*e⃗*; therefore, the transformation Eqs. (1)–(2), play an important role in unifying the description.

_{ϕ}#### 3.2 Internal Mie field

The incoming field for light illuminating a spherical particle, propagating in the z-direction, and polarized in the x-direction, is described in Cartesian coordinates as

where *k* is the wave number in the medium. The time factor *e ^{-iωt}* will be omitted in the following derivations.

The resulting field inside the sphere is given by Mie theory as

where *M⃗* and *N⃗* are the solutions to the vector wave equation in terms of Bessel functions and spherical harmonics and the superscript refers to first order Bessel functions. The series in Eq. (4) is truncated in the following manner

where *a* is the radius of the spherical particle and *λ* is the wavelength. Bohren and Huffman^{8} provide the general expressions for the terms *M⃗* and *N⃗* as series themselves, which are also truncated

$$-\frac{3}{5}{k}_{1}R\mathrm{sin}\Phi \phantom{\rule{.2em}{0ex}}\mathrm{cos}\phantom{\rule{.2em}{0ex}}\Theta {\stackrel{\rightharpoonup}{e}}_{\Phi}+O\left({\left[{k}_{1}R\right]}^{2}\right)$$

where *k*
_{1} is the wave number inside the sphere. The coefficients for *c*
_{n} and *d*
_{n} are calculated through ^{8}

where μ_{1} is the permeability of the sphere and is presumed to equal μ, the permeability of the medium, and *k* is the wave number in the medium. The primes denote differentiation with respect to *ka*.

The expressions of 6, 7 and 8 are translated to rectangular coordinates resulting in

This expression can be written in exponential form to the same order of accuracy; since

resulting in the following approximation for the Mie field inside the sphere.

Note that in the limit as *k*
_{1}→*k*, and *d*
_{1}→1, *c*
_{1}→1, *d*
_{2}→1 then *E⃗*(*R⃗*)→*E _{o}e^{ikZ}e⃗_{x}*, the incoming field value. This is consistent; if the dielectric properties of the scatterer match those of the medium the incoming field is unaltered.

#### 3.3 Dipole scattering approach

Electromagnetic theory states that a dipole located at *R⃗* of intensity *p⃗*(*R⃗*)*e*
^{-iωt} radiates in the far field according to the following Eq.^{8}

where *E⃗ _{s}* is the scattered electric field radiated by the dipole and

*ε*is the permittivity or dielectric constant of the medium. It also states that a small dielectric sphere of radius

*ρ*placed in a

*uniform static*electric field

*E⃗*generates a dipole moment. The induced dipole moment is proportional to the field and is given by

^{8}

where *ε* is the permittivity, *m* is the relative refractive index, and *dV* is the volume of the scatterer.

Rayleigh scattering assumes that an *oscillating*, *nonuniform* field *E⃗*(*R⃗*)*e*
^{-iωt} generates a dipole moment in a spherical volume given by the same expression in Eq. (15) and that the dipole re-radiates according to Eq. (14). Following the RDG approach, it is assumed that each *infinitesimal* volume within the scatterer behaves in this fashion. By substituting Eq. (15) into (14) the following expression is obtained for the incremental electric field radiated by the infinitesimal dipole located at *R⃗*:

For |*R⃗*|≪|*r⃗*|, Rayleigh approximates 1/|*r⃗*-*R⃗*|≈1/*r*, *e*⃗_{r⃗-R⃗}≈*e⃗ _{r}*, and
${e}^{\mathrm{ik}\mid \overrightarrow{r}-\overrightarrow{R}\mid}\approx {e}^{\mathrm{ik}\left\{(r-{\stackrel{\rightharpoonup}{R}\xb7\overrightarrow{e}}_{r})\right\}}={e}^{ikr}{e}^{-\mathrm{ik}{\overrightarrow{R}\xb7\overrightarrow{e}}_{r}}$
.

When substituting these approximations into Eq. (16) the following expression is obtained.

#### 3.4 Hybridized theory

The difference between Rayleigh-Debye-Gans theory and the hybridized theory presented herein is as follows: RDG assumes that the local field *E⃗*(*R⃗*) generating the infinitesimal dipole in Eq. (17) is given by the *incoming field*, whereas the hybrid theory takes the *internal field* Eq. (13) that Mie theory gives for the sphere as the field inducing the dipole moment. By using the internal Mie field, some of the effects of the surrounding dipole field alterations to the incoming field (such as attenuation) are taken into account. The validity of either approach presumes that the incoming electric field is roughly uniform over the sphere, so that the radius *a* of the sphere must be a small fraction of the wavelength (*a*≪*λ*).

By substituting the expression for the internal electric field, Eq. (13), into Eq. (17) an explicit formula for the scattered electric field is obtained.

To evaluate the term *e⃑ _{r}*×[

*e⃑*×

_{r}*e⃑*] and

_{x}*e⃑*×[

_{r}*e⃑*×

_{r}*e⃑*], the identities in Eqs. (1) and (2) are used. The following expression is a result of the conversion and mathematical manipulation, with the identities

_{z}*Z*=

*R⃗*·

*e⃗*and

_{z}*X*=

*R⃗*·

*e⃗*.

_{x}As in Rayleigh-Debye-Gans theory, the total scatterer volume contributions are summed (integrated). Introduced are *f*
_{1} and *f*
_{2} which are “form factors” for the sphere:

where

$$=\frac{1}{V}\int {e}^{i\overrightarrow{R}\xb7\left({k}_{1}\frac{\left({d}_{2}-{c}_{1}\right)}{2}{\overrightarrow{e}}_{x}-k{\overrightarrow{e}}_{r}\right)}dV-\frac{1}{V}\int {e}^{i\overrightarrow{R}\xb7\left(-k{\overrightarrow{e}}_{r}\right)}dV$$

Note that if *k*
_{1}=*k*, since (as noted above) *c*
_{1}=*d*
_{1}=*d*
_{2}=1, *f*
_{1} reduces to
$\frac{1}{V}\int {e}^{ik\overrightarrow{R}\xb7\left({\overrightarrow{e}}_{z}-{\overrightarrow{e}}_{r}\right)}dV$
, the form factor “*f*” in the RDG theory. Furthermore observe that the factor *f*
_{2}, which does not appear in the RDG theory, goes to zero when *k*
_{1}=*k*.

The problem now becomes how to calculate the integrals in Eqs. (21) and (22), which have the form ∫*e*
^{iR⃗·S⃗}
*dV* with constant *S⃗*. Consider a local coordinate system in the sphere with its z’ axis aligned with *S⃗*. The element of volume at height z’ is

However *x*′^{2} +*y*′^{2} +*z*′^{2}=*a*
^{2} and *z’* runs from -*a* to *a*; therefore,

The integrals in Eqs. (21) and (22) were calculated using Maple© and the results are

$$B=\sqrt{{k}^{2}+\frac{{k}_{1}^{2}{({d}_{2}-{c}_{1})}^{2}}{4}-k{k}_{1}({d}_{2}-{c}_{1})\mathrm{cos}\varphi \mathrm{sin}\theta}$$

$$C=\sqrt{{k}^{2}}=k$$

Observe from Fig. 1 that *θ* and *ϕ* are the detector angles and that *e⃗ _{θ}* is in the scattering plane while

*e⃗*is perpendicular. Therefore parallel and perpendicular components of the scattered field, Eq. 20, are expressed in terms of the form factors as

_{ϕ}The scattering intensity is given by

### 3.4.1 Scattering amplitude matrix formulation for the hybrid model

In the new model the scattered field can still be expressed using a scattering matrix in the manner of van de Hulst^{7}, Bohren and Huffman^{8}, and Kerker^{9}. The incoming field must be expressed in terms of its components parallel and perpendicular to the scattering plane. In spherical coordinates the incoming field is given by

Here *e⃗ _{ϕ}* is perpendicular to the scattering plane while the unit vector cos

*θe⃗*+sin

_{θ}*θe⃗*lies in the plane. As a result the incoming field can be written as:

_{r}$${E}_{\perp ,i}={E}_{o}{e}^{\mathrm{ikz}}\mathrm{sin}\varphi $$

After some manipulation the scattered field (Eqs. (28), (29)) can be related to the incident field in a scattering matrix format:

### 3.4.2 Scattering intensity ratio and turbidity

The scattering intensity ratio is expressed using Eq. (20).

Equation (32) can be written in terms of the scattering amplitude matrix Eq. (31); however this is not recommended due to the singularity (cos*ϕ*)^{-1}.

Using the definition of transmission, T=It/I_{0}=exp (-NC_{ext} l)=exp(-τl) the turbidity τ can be expressed as optical density where the scattering cross section *C _{sca}* is calculated using Eq. (34)

^{7,10}

and the absorption cross section *C _{abs}*

^{7}:

the turbidity is then evaluated from the extinction cross section C_{ext}=C_{sca}+C_{abs}
^{10}

## 4. Results

The performance evaluation of the model in comparison to those of Rayleigh-Debye-Gans and of Mie was achieved through a series of multiwavelength turbidity simulation studies. The connection between light scattering and absorption phenomena and particle’s joint property distribution (size, shape, orientation, chemical composition, and internal structure) is made through the optical properties that are characteristic of the materials contained in the particle. The complex refractive index is given by *N*=*n*+*iκ*, where *n* and *κ* are non negative values, *n* is the refractive index (real), and *κ* is the absorption coefficient (imaginary). The scattering of light is due to differences in refractive indices between the medium and the particle. The refractive index of the particle (*N*
_{1}) relative to the suspending medium (*N*
_{0}) is *m*=*N*
_{1}/*N*
_{0}=*n*
_{1}+*iκ*
_{1}/*n*
_{0}+*iκ*
_{0}. First the validity of the hybrid theory using various particle sizes was tested for relative refractive indices close to one. Second, the effectiveness of the hybrid theory’s is tested by introducing the effects of absorption through the imaginary part of the refractive index. The last study investigates the behavior of the hybrid theory for refractive indices exceeding the conditions required for Rayleigh-Debye-Gans theory (i.e., strong scattering *and* absorption components), for various particle sizes.

#### 4.1 Relative refractive index n/n_{0} ~1 and absorption κ=0

The validity of the hybrid theory was tested against Rayleigh-Debye-Gans and Mie theory using the relative refractive indexes as functions of wavelength of soft bodies (*n*/*n _{0}*=1.04) to calculate the turbidity. The spherical diameter sizes used were 50, 100, 250, and 500 nm. The results of this study are shown in turbidity spectral plots provided in Figs. 2, 3, 4 and 5 including insets. The insets highlight spectral features. Figures 2 (50 nm) and 3 (100 nm) show that the hybrid theory for very small particles at the shorter wavelengths is a much better approximation to Mie theory than is RDG theory. At wavelengths much larger than the particle size, the hybrid spectrum is still superior to Rayleigh-Debye-Gans. In Fig. 4 the hybrid model for 250 nm particles closely approaches Mie theory above 300nm in wavelength and outperforms RDG even down to 200 nm wavelength (which is shorter than the particle diameter).

A significant change in the calculated turbidity spectra is observed in Fig. 5, where the diameter size is 500 nm. Here the hybrid spectrum no longer resembles that of Mie theory or RDG at wavelengths shorter than half the diameter. Nonetheless, for larger wavelengths the hybrid model again provides a better approximation to the exact Mie theory than Rayleigh-Debye-Gans. Clearly, the hybrid theory provides an improved model for estimating the turbidity for nonabsorbing soft particles whose diameter is smaller than the wavelength and where scattering dominates the extinction spectra.

#### 4.2 Relative refractive index n/n_{0}≥1 and absorption κ>0

The previous section showed that the hybrid model provides an improved RDG approximation to Mie theory for non absorbing particles whose relative refractive index is approximately one. In this section the contribution of absorption *κ* is included in the refractive index *n*, while the relative refractive *n*/*n*
_{0} was kept close to one. The optical properties of polystyrene (1.1≤*n*/*n _{0}*≤1.5, 0.01≤

*κ*≤0.6) were selected for this study. The diameter sizes selected for the turbidity calculations were, again, 50, 100, 250, and 500 nm. Figures 6 and 7 have insets plotted on a semi logarithmic scale to highlight features of the spectra. For particles diameters of 50 and 100 nm, Figs. 6 and 7 demonstrate that the calculated turbidity with the hybrid model is a better approximation to Mie theory than the Rayleigh-Debye-Gans.

Figures 8 and 9 show the optical density spectra predicted for particle diameters corresponding to 250nm and 500nm. Notice that the hybrid model deviates from Mie theory in the UV portion of the spectrum (200–400nm), however, it provides a better approximation to Mie theory in the VIS-NIR portion of the electromagnetic spectrum. Close inspection of the spectral features in the wavelength range 200–400nm shows that the hybrid model retains the spectral features predicted by Mie theory but amplifies the predicted intensities. The insets in Figs. 8 and 9 show the portion of the spectra where the particle diameter is of the same order of magnitude or larger than the wavelength; notice that the hybrid model begins to deteriorate at approximately D=λ. It is also interesting to note that the hybrid model amplifies the spectral features predicted by Mie theory when the effects absorption and scattering are important. It is evident that the differences between the incoming field and the Mie field are quite significant when absorption is present.

The simulation results demonstrate that the hybrid model provides both, an improved approximation and an expanded range over Rayleigh-Debye-Gans theory for absorbing scatterers whose relative refractive index is greater than one. The next section studies the behavior of the hybrid model for non-absorbing scatterers whose relative refractive index is approximately 1.

#### 4.3 Relative Refractive index n/n_{0} ~1 and Absorption κ>

Often, particles and/or scattering elements of interest have large relative indices of refraction and nonzero absorption. Hemoglobin (1≤*n*/*n _{o}*≤1.2, 0.01≤

*κ*≤0.15) is both a strong scatterer and strong absorber and thus a good test case. The particle sizes used to calculate the turbidities were 50, 100, 250, and 500 nm. Figures 10,11,12, and 13 show that at 50, 100 and 250 nm diameters, the spectra calculated from the hybrid theory in general approximate Mie theory better than does Rayleigh-Debye-Gans over a broader range of wavelengths. For the diameter size of 500 nm (fig 13), both, the hybrid and RDG deviate considerably from Mie theory indicating that, as expected, both the hybrid model and RDG break down for the case of strong scatterers and absorbers in the portion of the electromagnetic spectrum under consideration. It is also apparent that the hybrid model considerably extends the range of the RDG approximation.

## 5. Analysis

The overall performance of the hybrid theory has been evaluated through the comparison of the turbidity spectra calculated with the hybrid theory and the spectra evaluated using Mie theory. To provide an index of the performance of the models for the cases analyzed, an error term consisting of the fractional difference between each of the two models (hybrid and RDG) and Mie, theory averaged over the spectral range under consideration, has been calculated and it is reported in Tables 2–4.

As can be appreciated in Table 2, the hybrid theory shows errors that are as much as an order of magnitude smaller than Rayleigh-Debye-Gans for the case in which the relative refractive index is real and close to 1. (i.e., one of the conditions for the applicability of the RDG theory)^{7,9} The error values for the hybrid theory are smaller than those of Rayleigh-Deybe-Gans for sizes above 50 nm and below 500nm. The errors of the hybrid model presented in Table 3 are comparable to those of RDG for relative refractive index values greater than one with an absorption component. The hybrid model appears to perform slightly better for sizes above 50 nm and below 500 nm. In Table 4 the relative refractive index is once again complex, however, the real part is close to one and the values used are beyond the range of applicability of RDG. The error values for the hybrid theory are of the same order of magnitude for all diameter sizes; however the error values are smaller than those of Rayleigh-Debye-Gans except for the smallest particle size.

## 6. Conclusions

The hybrid model for submicron spheres has been shown to approximate Mie theory much better than Rayleigh-Debye-Gans for particle sizes smaller than the wavelength and whose complex refractive index is close to 1. For a wide range of relative refractive indices the improvement is particularly marked for absorbing materials. In the cases were absorption is introduced, the calculations indicate that attenuation becomes more significant for the particles smaller than 500nm diameter, and that the hybrid theory is superior in accommodating attenuation. The results of the simulations conducted demonstrate that the strategy of using the Mie internal field, rather than the incoming field, to energize the RDG dipoles results in significant benefits. One important benefit from the hybrid model is the faster computation time for calculating the extinction spectra.

The hybrid theory can be improved for the spherical model. As the theory stands, the truncation at the first term of the series for the internal field can be extended to include second order terms. These second order terms will influence the series for particles whose size is comparable with the wavelength. In other words, at the shorter wavelengths the hybrid model does not exactly match Mie theory, but by extending the series, terms can be included that will improve the model where *a*/*λ*≈1. Note that the hybrid model can be extended to shapes other than spheres using the RDG Form Factors derived for the various geometries as presented in Kerker. A method for extending the hybrid model to other shapes such as ellipsoids could be developed on the assumption that the internal field of the ellipsoid can be described by mapping the Mie internal field for a volume equivalent sphere evaluated from the induced dipole moment using the postulated field. To account for the shape of the particle, the form factor in Eq. 24;
$\frac{1}{V}\int {e}^{i\overrightarrow{R}\xb7\overrightarrow{S}}dV$
, has to be evaluated for the ellipsoidal volume as has been done for the sphere. The form factors can be directly determined from the tables provided by Kerker, by reinterpreting the constant vector *S⃗*. The mathematics are time consuming but straightforward.

## Acknowledgement

The financial support from Claro Scientific LLC is gratefully acknowledged.

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