Abstract

The problem of computing the internal electromagnetic field of a homogeneous sphere from the observation of its scattered light field is explored. Using empirical observations it shown that, to good approximation for low contrast objects, there is a simple Fourier relationship between a component of the internal E-field and the scattered light in a preferred plane. Based on this relationship an empirical algorithm is proposed to construct a spherically symmetric particle of approximately the same diameter as the original, homogeneous, one. The size parameter (ka) of this particle is then estimated and shown to be nearly identical to that of the original particle. The size parameter can then be combined with the integrated power of the scatter in the preferred plane to estimate refractive index. The estimated values are shown to be accurate in the presence of moderate noise for a class of size parameters.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]

2007 (2)

P. L. D. Roberts and J. S. Jaffe, "Multiple angle acoustic classification of zooplankton," J. Acoust. Soc. Am. 121, 2060-2070 (2007).
[CrossRef]

H. R. Gordon, "Rayeigh-Gans scattering approximation: surprisingly useful for understanding backscatter from disk-like particles," Opt. Express 15, 5572-5588 (2007).
[CrossRef] [PubMed]

2006 (3)

V. M. Rysakov, "light scattering by "soft" particles of arbitrary shape and size: II-Arbitrary orientation of particles in the space," J. Quant. Spectrosc. Radiat. Transf. 98, 85-100 (2006).
[CrossRef]

V. V. Berdnik and V. A. Loiko, "Particle sizing by multiangle light-scattering data using the high-order neural networks," J. Quant. Spectrosc. Radiat. Transf. 100, 55-63 (2006).
[CrossRef]

B. Shao, J. S. Jaffe, M. Chachisvilis, and S. C. Esener, "Angular resolved light scattering for discriminating among marine picoplankton: modeling and experimental measurements," Opt. Express 14, 12473-12484 (2006).
[CrossRef] [PubMed]

2004 (3)

P. C. Chaumet, K. Belkebir, and A. Sentenac, "Three-dimensional sub wavelength optical imaging using the coupled dipole method," Phys. Rev. B,  69 (245405): 1-7 (2004).
[CrossRef]

L. M. Shulman, "Analysis of polarimetric data by solving the inverse scattering problem," Quant. Spectrosc. Radiat. Transf. 88, 243-256 (2004).
[CrossRef]

K. A. Semyanov, P. A. Tarasov A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, "Single-particle sizing from light scattering by spectral decomposition," Appl. Opt. 43, 5110-5115 (2004).
[CrossRef] [PubMed]

2003 (2)

Y. L. Pan, K. B. Aptowicz, R. K. Chang, M. Hart, and J. D. Eversole, "Characterizing and monitoring aerosols by light scattering," Opt. Lett. 28, 589-591 (2003).
[CrossRef] [PubMed]

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. K. Shah, H. E. Savage, R. B. Rosen. S. A. McCormick and R, R. Alfano, "Bacteria Size Determination by Elastic Light Scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

2001 (2)

Q. Fu and W. Sun, "Mie theory for light scattering by a spherical particle in an absorbing medium," Appl. Opt. 9, 1354-1361 (2001).
[CrossRef]

W. Rysakov and M. Ston, "Light scattering by spheroids," J. Quant. Spectrosc. Radiat. Transf. 69, 651-665 (2001).
[CrossRef]

2000 (1)

I. K. Ludlow and J. Everitt, "Inverse Mie Problem," J. Opt. Soc. Am. A. 17, 2229 - 2235 (2000).
[CrossRef]

1998 (1)

P. H. Faye, "Spatial light-scattering analysis as a means of characterizing and classifying non-spherical particles," Meas. Sci. Technol. 9, 141-149 (1998).
[CrossRef]

1994 (1)

B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. 11, 1491-1499 (1994).
[CrossRef]

1991 (1)

J. Everitt and I. K. Ludlow, "Particle sizing using methods of discrete Legendre analysis," Biochem Soc. Trans. 19, 504-5 (1991).
[PubMed]

1989 (1)

1988 (1)

B. T. Draine, "The Discrete-dipole approximation and its application to the interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

1982 (2)

1968 (1)

Appl. Opt. (4)

Astrophys. J. (1)

B. T. Draine, "The Discrete-dipole approximation and its application to the interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

Biochem Soc. Trans. (1)

J. Everitt and I. K. Ludlow, "Particle sizing using methods of discrete Legendre analysis," Biochem Soc. Trans. 19, 504-5 (1991).
[PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Katz, A. Alimova, M. Xu, E. Rudolph, M. K. Shah, H. E. Savage, R. B. Rosen. S. A. McCormick and R, R. Alfano, "Bacteria Size Determination by Elastic Light Scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

J. Acoust. Soc. Am. (1)

P. L. D. Roberts and J. S. Jaffe, "Multiple angle acoustic classification of zooplankton," J. Acoust. Soc. Am. 121, 2060-2070 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. 11, 1491-1499 (1994).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

I. K. Ludlow and J. Everitt, "Inverse Mie Problem," J. Opt. Soc. Am. A. 17, 2229 - 2235 (2000).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (3)

V. V. Berdnik and V. A. Loiko, "Particle sizing by multiangle light-scattering data using the high-order neural networks," J. Quant. Spectrosc. Radiat. Transf. 100, 55-63 (2006).
[CrossRef]

W. Rysakov and M. Ston, "Light scattering by spheroids," J. Quant. Spectrosc. Radiat. Transf. 69, 651-665 (2001).
[CrossRef]

V. M. Rysakov, "light scattering by "soft" particles of arbitrary shape and size: II-Arbitrary orientation of particles in the space," J. Quant. Spectrosc. Radiat. Transf. 98, 85-100 (2006).
[CrossRef]

Meas. Sci. Technol. (1)

P. H. Faye, "Spatial light-scattering analysis as a means of characterizing and classifying non-spherical particles," Meas. Sci. Technol. 9, 141-149 (1998).
[CrossRef]

Opt. Express (2)

B. Shao, J. S. Jaffe, M. Chachisvilis, and S. C. Esener, "Angular resolved light scattering for discriminating among marine picoplankton: modeling and experimental measurements," Opt. Express 14, 12473-12484 (2006).
[CrossRef] [PubMed]

H. R. Gordon, "Rayeigh-Gans scattering approximation: surprisingly useful for understanding backscatter from disk-like particles," Opt. Express 15, 5572-5588 (2007).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. B (1)

P. C. Chaumet, K. Belkebir, and A. Sentenac, "Three-dimensional sub wavelength optical imaging using the coupled dipole method," Phys. Rev. B,  69 (245405): 1-7 (2004).
[CrossRef]

Quant. Spectrosc. Radiat. Transf. (1)

L. M. Shulman, "Analysis of polarimetric data by solving the inverse scattering problem," Quant. Spectrosc. Radiat. Transf. 88, 243-256 (2004).
[CrossRef]

Other (13)

A. V. Oppenheim, R. W. Schaefer, and J. R. Buck, Discrete-Time Signal Processing A. V. Oppenheim, ed., (Prentice Hall Signal Processing Series, 1999) 2nd Edition.

C. Matzler, "Matlab codes for Mie Scattering and Absorption," http://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.htmla>

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (Society of Industrial and Applied Mathematics, 2001).
[CrossRef]

D. Carlton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, (Springer-Verlag, Berlin, 1998).

D. R. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley-VCH, (1983).

H. C. Van de Hulst, Light Scattering by Small Particles, (Dover, 1981).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, (Cambridge University Press, Cambridge, 2002).

A. Taflove, Computational Electrodynamics: The Finite-difference Time-domain Method, (Artech House, Boston, MA, 1995).

T. L. Blundell and L. N. Johnson, Protein Crystallography, (Academic Press, 1976).

A. C. Lavery, T. K. Stanton, D. E. McGehee, and D. Z Chu, "Three-dimensional modeling of acoustic backscattering from fluid-like zooplankton," J. Acous. Soc. Am 111, 1197-1210 (2002).
[CrossRef]

A. Lompado, "Light Scattering by a Spherical Particle," http://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.html>

M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, Seventh Edition, 2003).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles, (Academic Press, 2000).

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Figures (2)

Fig. 1.
Fig. 1.

The observed complex angular light scattered data can be viewed as the coefficients of the Fourier Transform of the structure: ��(k struc ) on the Ewald sphere [20] that is centered - k inc at a radius of |k|=2π/λ.

Fig. 2.
Fig. 2.

Two graphs of the phase of the internal Ex field along the z-axis for the cases where (a,m)=(1 µm, 1.05) (a) or (1 µm, 1.58) (b). The graph depicts a superposition of the z-dependent phase along with the standard deviation of those phases at a given z location. Graphs were produced via the implementation of Mie theory to compute the interior electric field for a spherical particle embedded inside a cubic lattice of dimensions (21)3.

Tables (2)

Tables Icon

Table 1. Averages and standard deviations of retrieved values for the size index when m=1.035+0.00i. The data was obtained by performing 100 repeated tomographic inversions after 10% multiplicative noise was added to the real and imaginary parts of the simulated complex angular scatter.

Tables Icon

Table 2. Averages and standard deviations of retrieved values of the refractive index for the case m=1.035+0.00i. The data was obtained by performing 100 repeated tomographic inversions when 10% multiplicative noise was added to the real and imaginary parts of the simulated complex angular scatter.

Equations (16)

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

E i = ( E 0    e ^ 0 + E 0 e ^ i ) exp ( ik in c    z  i ω t ) .
E s = E | | s e ^ | | s + E s e ^ s where e ^ | | s = e ^ θ ,        e ^ s =    e ^ ϕ ,        e ^ s × e ^ | | s = e ^ r .
( E s E s ) = exp i k ( r z ) ikr ( S 2 S 3 S 4 S 1 ) ( E i E i )
E i = E 0 n = 1 i n 2 n + 1 n ( n + 1 ) ( M o ln ( 1 ) i N e ln ( 1 ) )
E int = E 0 n = 1 i n 2 n + 1 n ( n + 1 ) ( c n M o ln ( 1 ) i d n N e ln ( 1 ) ) ,
E s = E 0 n = 1 i n 2 n + 1 n ( n + 1 ) ( i b n N o ln ( 3 ) a n M e ln ( 3 ) ) ,
( E s E s ) = exp i k ( r z ) ikr ( S 2 0 0 S 1 ) ( E i E i ) .
E s ( d ) = k 2 exp ( i k d ) d j = 1 N exp ( i k d ̂ · r j ) ( d ̂ d ̂ l 3 ) P j .
E s ( k ̂ obs d ) = k 2 exp ( ik d ) d j = 1 N exp ( i k k ̂ obs · r j ) ( k ̂ obs k ̂ obs l 3 ) P j .
E sx ( k ̂ obs ) = k 2 exp ( i k d ) d j = 1 N exp ( i k obs · r j ) ( P jx ) .
E sx ( k obs ) = k 2 exp ( ik d ) d α j = 1 N E x int ( r j ) exp ( i k obs · r j ) .
a n = 1 2 n ( n + 1 ) 0 π [ S ( θ ) τ n ( θ ) + S ( θ ) π n ( θ ) ] sin θ d θ .
b n = 1 2 n ( n + 1 ) 0 π [ S ( θ ) π n ( θ ) + S ( θ ) τ n ( θ ) ] sin θ d θ .
f scat = k 2 4 π V N ( r v ) e i ( k scat k inc ) 2 i · r v d V .
E x int = E x ( r j ) exp ( i k inc · r j ) .
E xs ( k obs ) = k 2 exp ( ik d ) d α j = 1 N E x int ( r j ) exp ( i ( k obs k inc ) int ) · r j

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