## Rayleigh-Debye-Gans as a model for continuous monitoring of biological particles: Part II, development of a hybrid model

Optics Express, Vol. 16, Issue 7, pp. 4671-4687 (2008)

http://dx.doi.org/10.1364/OE.16.004671

Acrobat PDF (566 KB)

### 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

^{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.

^{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.

## 2. Materials

^{6}.

^{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.

^{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

*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*).

*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

*k*is the wave number in the medium. The time factor

*e*will be omitted in the following derivations.

^{-iωt}*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

*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

*k*

_{1}is the wave number inside the sphere. The coefficients for

*c*

_{n}and

*d*

_{n}are calculated through

^{8}

_{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*.

*k*

_{1}→

*k*, and

*d*

_{1}→1,

*c*

_{1}→1,

*d*

_{2}→1 then

*E⃗*(

*R⃗*)→

*E*, the incoming field value. This is consistent; if the dielectric properties of the scatterer match those of the medium the incoming field is unaltered.

_{o}e^{ikZ}e⃗_{x}### 3.3 Dipole scattering approach

*R⃗*of intensity

*p⃗*(

*R⃗*)

*e*

^{-iωt}radiates in the far field according to the following Eq.

^{8}

*E⃗*is the scattered electric field radiated by the dipole and

_{s}*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}

*ε*is the permittivity,

*m*is the relative refractive index, and

*dV*is the volume of the scatterer.

*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*:

*R⃗*|≪|

*r⃗*|, Rayleigh approximates 1/|

*r⃗*-

*R⃗*|≈1/

*r*,

*e*⃗

_{r⃗-R⃗}≈

*e⃗*, and

_{r}### 3.4 Hybridized theory

*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*).

*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}*f*

_{1}and

*f*

_{2}which are “form factors” for the sphere:

*k*

_{1}=

*k*, since (as noted above)

*c*

_{1}=

*d*

_{1}=

*d*

_{2}=1,

*f*

_{1}reduces to

*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*.

*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

*x*′

^{2}+

*y*′

^{2}+

*z*′

^{2}=

*a*

^{2}and

*z’*runs from -

*a*to

*a*; therefore,

*θ*and

*ϕ*are the detector angles and that

*e⃗*is in the scattering plane while

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

_{ϕ}#### 3.4.1 Scattering amplitude matrix formulation for the hybrid model

^{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

*e⃗*is perpendicular to the scattering plane while the unit vector cos

_{ϕ}*θe⃗*+sin

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

_{r}#### 3.4.2 Scattering intensity ratio and turbidity

*ϕ*)

^{-1}.

_{0}=exp (-NC

_{ext}l)=exp(-τl) the turbidity τ can be expressed as optical density where the scattering cross section

*C*is calculated using Eq. (34)

_{sca}^{7,10}

*C*

_{abs}^{7}:

_{ext}=C

_{sca}+C

_{abs}

^{10}

## 4. Results

*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

*n*/

*n*=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).

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

*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*≤1.5, 0.01≤

_{0}*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.

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

*n*/

*n*≤1.2, 0.01≤

_{o}*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

^{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

*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;

*S⃗*. The mathematics are time consuming but straightforward.

## Acknowledgement

## References

1. | C. E. Alupoaei, J. A. Olivares, and L. H. Garcia-Rubio, “Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores,” Biosens. Bioelectron. |

2. | A. Katz, A. Alimova, M. Xu, E. Rudolf, M.K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, “Bacteria size determination by elastic light scattering,” Quantum Electron. |

3. | M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-Matrix Computations of Light Scattering by Nonspherical Particles: A Review,” J. Quant. Spectrosc. Radiat. Transfer |

4. | A. Garcia-Lopez, A. D. Snider, and L. H. Garcia-Rubio, “Rayleigh-Debye-Gans as a Model for Continuous Monitoring of Biological Particles: Part I, Assessment of Theoretical Limits and Approximations,” Opt. Express |

5. | A. Y. Perel’man and N. V. Voshchinnikov, “S-Approximation for Spherical Particles with a Complex Refractive Index,” Opt. Spectrosc. |

6. | M. K. Choi and J. R. Brock, “Light scattering and absorption by a radially inhomogenous sphere: application of numerical method,” J. Opt. Soc. Am. |

7. | H. C. Van der Hulst, |

8. | C.F. Bohren and D. R. Huffman, |

9. | M. Kerker, |

10. | A. Garcia-Lopez, “Investigation into the transition between single and multiple scattering for colloidal dispersions,” M.S. thesis, Unviersity of South Florida, Tampa, FL (2001). |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(350.4990) Other areas of optics : Particles

**ToC Category:**

Scattering

**History**

Original Manuscript: November 5, 2007

Revised Manuscript: January 19, 2008

Manuscript Accepted: March 14, 2008

Published: March 20, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Alicia C. Garcia-Lopez and Luis H. Garcia-Rubio, "Rayleigh-Debye-Gans as a model for continuous monitoring of biological particles: Part II, development of a hybrid model," Opt. Express **16**, 4671-4687 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4671

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### References

- C. E. Alupoaei, J. A. Olivares, and L. H. Garcia-Rubio, "Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores," Biosens. Bioelectron. 19, 893-903 (2003). [CrossRef]
- A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," Quantum Electron. 9, 277-287 (2003).
- M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-Matrix Computations of Light Scattering by Nonspherical Particles: A Review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996). [CrossRef]
- A. Garcia-Lopez, A. D. Snider, and L. H. Garcia-Rubio, "Rayleigh-Debye-Gans as a Model for Continuous Monitoring of Biological Particles: Part I, Assessment of Theoretical Limits and Approximations," Opt. Express 14, 8849-8865 (2006). [CrossRef] [PubMed]
- A. Y. Perel’man and N. V. Voshchinnikov, "S-Approximation for Spherical Particles with a Complex Refractive Index," Opt. Spectrosc. 92, 221-226 (2002). [CrossRef]
- M. K. Choi and J. R. Brock, "Light scattering and absorption by a radially inhomogenous sphere: application of numerical method," J. Opt. Soc. Am. 14, 620-626 (1997).
- H. C. Van der Hulst, Light Scattering by Small Particles, Dover Publications, Inc. (New York, 1957).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science Paper Series, New York, 1998). [CrossRef]
- M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic Press, New York, 1969).
- A. Garcia-Lopez, "Investigation into the transition between single and multiple scattering for colloidal dispersions," M.S. thesis, Unviersity of South Florida, Tampa, FL (2001).

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