Abstract

Rayleigh-Debye-Gans and Mie theory were previously shown to disagree for spherical particles under ideal conditions4. 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

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References

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  1. 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]
  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. 9, 277-287 (2003).
  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. Transf. 55, 535-575 (1996).
    [CrossRef]
  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 14, 8849-8865 (2006).
    [CrossRef] [PubMed]
  5. 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]
  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. 14, 620-626 (1997).
  7. H. C. Van der Hulst, Light Scattering by Small Particles, Dover Publications, Inc. (New York, 1957).
  8. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science Paper Series, New York, 1998).
    [CrossRef]
  9. M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic Press, New York, 1969).
  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).

2006 (1)

2003 (2)

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).

2002 (1)

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]

1997 (1)

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).

1996 (1)

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]

Alfano, R. R.

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).

Alimova, A.

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).

Alupoaei, C. E.

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]

Brock, J. R.

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).

Choi, M. K.

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).

Garcia-Lopez, A.

Garcia-Rubio, L. H.

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]

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]

Katz, A.

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).

Mackowski, D. W.

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]

McCormick, S. A.

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).

Mishchenko, M. I.

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]

Olivares, J. A.

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]

Perel’man, A. Y.

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]

Rosen, R. B.

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).

Rudolf, E.

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).

Savage, H. E.

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).

Shah, M. K.

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).

Snider, A. D.

Travis, L. D.

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]

Voshchinnikov, N. V.

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]

Xu, M.

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).

Biosens. Bioelectron. (1)

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]

J. Opt. Soc. Am. (1)

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).

J. Quant. Spectrosc. Radiat. Transf. (1)

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]

Opt. Express (1)

Opt. Spectrosc. (1)

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]

Quantum Electron. (1)

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).

Other (4)

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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Figures (13)

Fig. 1.
Fig. 1.

Local Unit Vectors with Respect to the Detector

Fig. 2.
Fig. 2.

Comparison of Calculated Turbidity for 50 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories.

Fig. 3.
Fig. 3.

Comparison of Calculated Turbidity for 100 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories.

Fig. 4.
Fig. 4.

Comparison of Calculated Turbidity for 250 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories.

Fig. 5.
Fig. 5.

Comparison of Calculated Turbidity for 500 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories.

Fig. 6.
Fig. 6.

Comparison of Calculated Turbidity for 50 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories.

Fig.. 7.
Fig.. 7.

Comparison of Calculated Turbidity for 100 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories.

Fig. 8.
Fig. 8.

Comparison of Calculated Turbidity for 250 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories.

Fig. 9.
Fig. 9.

Comparison of Calculated Turbidity for 500 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories.

Fig. 10.
Fig. 10.

Comparison of Calculated Turbidity for 50 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories.

Fig. 11.
Fig. 11.

Comparison of Calculated Turbidity for 100 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories.

Fig. 12.
Fig. 12.

Comparison of Calculated Turbidity for 250 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories.

Fig. 13.
Fig. 13.

Comparison of Calculated Turbidity for 500 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories.

Tables (4)

Tables Icon

Table 1. Simulation Parameters for Turbidity using Mie and Rayleigh-Debye-Gans Theory.

Tables Icon

Table 2. Residual Sum of Squares for Relative Refractive index n/n 0 ~1 and Absorption κ=0.

Tables Icon

Table 3. Residual Sum of Squares for Relative Refractive index n/n 0≥1 and Absorption κ>0.

Tables Icon

Table 4. Residual Sum of Squares Relative Refractive index n/n 0 ~1 and Absorption κ>0.

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

x = r   sin θ   cos ϕ
y = r   sin θ   sin ϕ
z = r   cos   θ
e r = sin θ cos ϕ e x + sin θ sin ϕ e y + cos θ e z
e θ = cos θ cos ϕ e x + cos θ sin ϕ e y sin θ e z
e ϕ = sin ϕ e x + cos ϕ e y
E i ( ξ , η , ς , t ) = E o e ik ς e i ω t e x
E ( R ) = E o n = 1 i n 2 n + 1 n ( n + 1 ) ( c n M O 1 n ( 1 ) id n N E 1 n ( 1 ) )
E ( R ) E o 3 2 ic 1 M O 11 + 3 2 d 1 N E 11 + 5 6 id 2 N E 12 + O ( [ a λ ] 2 )
M O 11 ( R ) k 1 3 R cos Φ e Θ k 1 3 R sin Φ cos Θ e Φ + O ( [ k 1 R ] 2 )
N E 11 ( R ) 2 3 cos   Φ   sin Θ e R + 2 3 cos Φ cos Θ e Θ 2 3 sin Φ e Φ + O ( [ k 1 R ] 2 )
N E 12 ( R ) 6 5 k 1 R cos   Φ sin   Θ cos Θ e R + 3 5 k 1 R cos Φ ( 2 cos 2 Θ 1 ) e Θ
3 5 k 1 R sin   Φ cos Θ e Φ + O ( [ k 1 R ] 2 )
c n = μ 1 j n ( ka ) [ kah n ( 1 ) ( ka ) ] μ 1 h n ( 1 ) ( ka ) [ kaj n ( ka ) ] μ 1 j n ( mka ) [ kah n ( 1 ) ( ka ) ] μ h n ( 1 ) ( k a ) [ mkaj n ( mka ) ]
d n = μ 1 m j n ( k a ) [ kah n ( 1 ) ( ka ) ] μ 1 m h n ( 1 ) ( ka ) [ kaj n ( ka ) ] μ m 2 j n ( mka ) [ kah n ( 1 ) ( ka ) ] μ 1 h n ( 1 ) ( ka ) [ mkaj n ( mka ) ]
E ( R ) E o = [ d 1 ( d 2 + c 1 ) 2 ik 1 Z ] e x + ( d 2 c 1 ) 2 ik 1 X e z + O ( [ k 1 R ] 2 )
e x = 1 + x + O ( x 2 )
E ( R ) E o = d 1 e ik 1 ( d 2 + c 1 ) 2 d 1 Z e x + ( e ik 1 ( d 2 c 1 ) 2 X 1 ) e z
E s = ( e i k r - R i k r - R ik 3 4 π ε e i ω t ) e r R × [ e r R × p ( R ) ]
p = 3 ε m 2 1 m 2 + 2 ( 4 3 π ρ 3 ) E = 3 ε m 2 1 m 2 + 2 E d V
d E s = ( e ik r - R i k r - R i k 3 4 π ε ) e r R × [ e r R × 3 ε m 2 1 m 2 + 2 E ( R ) d V ]
d E s = ( 3 k 2 4 r π m 2 1 m 2 + 2 e i k r e i k R · e r d V ) e r × [ e r × E ( R ) ]
d E s = ( 3 k 2 4 r π m 2 1 m 2 + 2 e i k r e i k R · e r d V ) e r × [ e r × ( d 1 e ik 1 d 2 + c 1 2 d 1 Z e x + ( e ik 1 d 2 c 1 2 X 1 ) e z ) E o ]
d E s = E o ( 3 k 2 4 r π m 2 1 m 2 + 2 e i k r e i k R · e r d V ) d 1 e ik 1 d 2 + c 1 2 d 1 R · e z ( sin ϕ e ϕ cos ϕ cos θ e θ ) + ( e ik 1 d 2 c 1 2 R · e x 1 ) sin θ e θ  
E s = d E s = E o e i k r ( 3 k 2 4 π r m 2 1 m 2 + 2 V ) [ f 1 cos θ cos ϕ e θ + f 2 sin θ e θ + f 1 sin ϕ e ϕ ]
f 1 ( θ , ϕ ) = 1 V d 1 e i k 1 ( d 2 + c 1 ) 2 d 1 R · e z e i k R · e r d V = 1 V d 1 e i R · ( k 1 ( d 2 + c 1 ) 2 d 1 e z k e r ) d V
f 2 ( θ , ϕ ) = 1 V ( e i k 1 ( d 2 - c 1 ) 2 R · e x 1 ) e i k R · e r d V
= 1 V e i R · ( k 1 ( d 2 c 1 ) 2 e x k e r ) d V 1 V e i R · ( k e r ) d V
d V = base × height = π ( x 2 + y 2 ) d z
e i R · S d V = e i S R · e z d V = z = a a e i S z π ( a 2 z 2 ) d z
f 1 = 1 V d 1 2 π ( iAae iAa e iAa + iAae iAa + e iAa i 3 A 3 )
f 2 = 1 V 2 π [ ( iBae iBa e iBa + iBae iBa + e iBa i 3 B 3 ) - ( iCae iCa e iCa + iCae iCa + e iCa i 3 C 3 ) ]
A = k 2 + k 1 2 ( d 2 + c 1 ) 2 4 d 1 2 k k 1 ( d 2 + c 1 ) d 1 cos θ
B = k 2 + k 1 2 ( d 2 - c 1 ) 2 4 k k 1 ( d 2 - c 1 ) cos ϕ sin θ
C = k 2 = k
E , s = E o e ikr 3 k 2 4 r π m 2 1 m 2 + 2 V [ f 1 cos θ cos ϕ + f 2 sin θ ]
E , s = E o e ikr 3 k 2 4 r π m 2 1 m 2 + 2 V [ f 1 sin ϕ ]
I s = 1 2 Re ( ε µ ) [ E , s 2 + E s 2 ]
E i = E o e i kz ( sin θ cos ϕ e ̂ r + cos θ cos ϕ e ̂ θ sin ϕ e ̂ ϕ )
E , i = E o e ikz cos ϕ
E , i = E o e ikz sin ϕ
( E , s E , s ) = 3 k 2 4 π r ( m 2 1 m 2 + 2 ) V [ f 1 cos θ + f 2 sin θ cos ϕ 0 0 f 1 ] ( E , i E , i )
I s I o = E s 2 E o 2 = 9 k 4 32 π 2 r 2 m 2 1 m 2 + 2 2 V 2 [ f 1 cos θ cos ϕ + f 2 sin θ 2 + f 1 sin ϕ 2 ]
C s c a = 0 π 0 2 π I s ( λ , r , θ , ϕ ) I o r 2 sin θ d ϕ d θ
C a b s = 3 k V Im ( m 2 1 m 2 + 2 )
τ = N p l ( C s c a + C a b s )

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