Abstract

Far-field intensities of light scattered from a linear centro-symmetric array illuminated by a plane wave of incident light are estimated at a series of detector angles. The intensities are computed from the superposition of E-fields scattered by the individual array elements. An average scattering phase function is used to model the scattered fields of individual array elements. The nature of scattering from the array is investigated using an image (θϕ plot) of the far-field intensities computed at a series of locations obtained by rotating the detector angle from 0° to 360°, corresponding to each angle of incidence in the interval [0° 360°]. The diffraction patterns observed from the θϕ plot are compared with those for isotropic scattering. In the absence of prior information on the array geometry, the intensities corresponding to θϕ pairs satisfying the Bragg condition are used to estimate the phase function. An algorithmic procedure is presented for this purpose and tested using synthetic data. The relative error between estimated and theoretical values of the phase function is shown to be determined by the mean spacing factor, the number of elements, and the far-field distance. An empirical relationship is presented to calculate the optimal far-field distance for a given specification of the percentage error.

© 2010 Optical Society of America

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References

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  1. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
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    [CrossRef]
  4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
    [CrossRef]
  5. T. A. Arias, “Notes on scattering theory,” (Massachusetts Institute of Technology, 1997). http://people.ccmr.cornell.edu/~muchomas/8.04/Lecs/lec_scatt/notes.html.
  6. A. Lupu-Sax, “Quantum scattering theory and applications,” Ph.D. thesis (Harvard University, 1998).
  7. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).
  8. R. D. Mountain, “Spectral distribution of scattered light in a simple fluid,” Rev. Mod. Phys. 38, 205-214 (1966).
    [CrossRef]
  9. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377-445 (1908).
    [CrossRef]
  10. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. W. L. Bragg, “The diffraction of short electromagnetic waves by a crystal,” Proc. Cambridge Philos. Soc. 17, 43-57 (1913).

2008 (1)

2007 (1)

2005 (1)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-Domain Method, 3rd ed. (Artech House, 2005).

1998 (4)

J. G. Maloney and M. P. Kesler, “Analysis of periodic structures,” in Advances in Computational Electrodynamics; A.Taflove, ed. (Artech House, 1998), Chap. 6.

J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, and T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. 37, 3586-3593 (1998).
[CrossRef]

A. Lupu-Sax, “Quantum scattering theory and applications,” Ph.D. thesis (Harvard University, 1998).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

1997 (1)

T. A. Arias, “Notes on scattering theory,” (Massachusetts Institute of Technology, 1997). http://people.ccmr.cornell.edu/~muchomas/8.04/Lecs/lec_scatt/notes.html.

1992 (1)

1983 (1)

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973).
[CrossRef]

1966 (2)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equation in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

R. D. Mountain, “Spectral distribution of scattered light in a simple fluid,” Rev. Mod. Phys. 38, 205-214 (1966).
[CrossRef]

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

1913 (1)

W. L. Bragg, “The diffraction of short electromagnetic waves by a crystal,” Proc. Cambridge Philos. Soc. 17, 43-57 (1913).

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377-445 (1908).
[CrossRef]

Aarnoudse, J. G.

Arias, T. A.

T. A. Arias, “Notes on scattering theory,” (Massachusetts Institute of Technology, 1997). http://people.ccmr.cornell.edu/~muchomas/8.04/Lecs/lec_scatt/notes.html.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Bragg, W. L.

W. L. Bragg, “The diffraction of short electromagnetic waves by a crystal,” Proc. Cambridge Philos. Soc. 17, 43-57 (1913).

Colton, D.

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

de Mul, F. F. M.

Draine, B. T.

Eick, A. A.

Flatau, P. J.

Freyer, J. P.

Graaff, R.

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Greve, J.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Hielscher, A. H.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Johnson, T. M.

Kesler, M. P.

J. G. Maloney and M. P. Kesler, “Analysis of periodic structures,” in Advances in Computational Electrodynamics; A.Taflove, ed. (Artech House, 1998), Chap. 6.

Koelink, M. H.

Kress, R.

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

Kurusingal, J.

Lupu-Sax, A.

A. Lupu-Sax, “Quantum scattering theory and applications,” Ph.D. thesis (Harvard University, 1998).

Maloney, J. G.

J. G. Maloney and M. P. Kesler, “Analysis of periodic structures,” in Advances in Computational Electrodynamics; A.Taflove, ed. (Artech House, 1998), Chap. 6.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377-445 (1908).
[CrossRef]

Mountain, R. D.

R. D. Mountain, “Spectral distribution of scattered light in a simple fluid,” Rev. Mod. Phys. 38, 205-214 (1966).
[CrossRef]

Mourant, J. R.

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Shen, D.

Sloot, P. M. A.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equation in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Zijp, J. R.

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377-445 (1908).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (2)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equation in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

J. Opt. Soc. Am. A (2)

Proc. Cambridge Philos. Soc. (1)

W. L. Bragg, “The diffraction of short electromagnetic waves by a crystal,” Proc. Cambridge Philos. Soc. 17, 43-57 (1913).

Rev. Mod. Phys. (1)

R. D. Mountain, “Spectral distribution of scattered light in a simple fluid,” Rev. Mod. Phys. 38, 205-214 (1966).
[CrossRef]

Other (6)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-Domain Method, 3rd ed. (Artech House, 2005).

J. G. Maloney and M. P. Kesler, “Analysis of periodic structures,” in Advances in Computational Electrodynamics; A.Taflove, ed. (Artech House, 1998), Chap. 6.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

T. A. Arias, “Notes on scattering theory,” (Massachusetts Institute of Technology, 1997). http://people.ccmr.cornell.edu/~muchomas/8.04/Lecs/lec_scatt/notes.html.

A. Lupu-Sax, “Quantum scattering theory and applications,” Ph.D. thesis (Harvard University, 1998).

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

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Figures (7)

Fig. 1
Fig. 1

Scattering of a plane wave by a collinear array of scattering elements.

Fig. 2
Fig. 2

Typical θ ϕ plot to illustrate ring-shaped contours representing different Bragg orders. The darker shades indicate regions representative of positive Bragg orders ( m > 0 ) and vice-versa. The contours separating the dark and light shaded regions represent points that satisfy Snell’s law of refraction.

Fig. 3
Fig. 3

Relationship between ϕ, θ and α, where (a) ϕ < 180 ° and θ < 180 ° ϕ ; (b) ϕ < 180 ° and θ > 180 ° ϕ ; (c) ϕ > 180 ° and θ < 540 ° ϕ ; (d) ϕ > 180 ° and θ > 540 ° ϕ .

Fig. 4
Fig. 4

Comparison of θ ϕ plots generated using Kurusingal and Pennypacker approaches for the spacing factor ( a < 0.5 ) : (a) θ ϕ plots using the Kurusingal approach, (b) θ ϕ plots using the Purcell and Pennypacker iterative approach, and (c) plot of maximum pixel-wise relative error between the pair of θ ϕ plots against spacing factor a.

Fig. 5
Fig. 5

(a) θ ϕ plots showing diffraction contours (rings) obtained using a = 5 , N = 2 , λ = 1 e 6 m , n = 1.2 , r 0 = 0.1 m ; (a1)–(a3) correspond to g = 0.3 , g = 0 , and g = 0.3 . The intensities are represented by a color scale with red and blue (top and bottom of scale in print) corresponding to the maxima and minima, respectively. (b) Intensity variation along the contour representing θ ϕ pairs that satisfy Snell’s law of refraction for a = 5 , N = 10 , λ = 1 e 6 m , n = 1.2 . Panels (b1)–(b3) correspond to g = 0.3 , g = 0 , and g = 0.3 , respectively. The curves shown in each panel correspond to different far-field distances, expressed as multiples of the array length ( L ) .

Fig. 6
Fig. 6

Intensity versus scattering angle ( α ) simulated using a = 2 , N = 40 , ϕ = 30 ° , λ = 1 e 6 m , r 0 = 0.1 m and (a) g = 0 , (b) g = 0.3 . The locations of each peak correspond to different Bragg orders. Peaks in (a) have the same magnitude. The envelopes of peaks in (b) follow the shape of the scattering phase function used in the simulation.

Fig. 7
Fig. 7

Theoretical and estimated scattering phase function ( p ) versus scattering angle ( α ) . Simulation is performed using a = 10 , N = 40 , g = 0.3 , λ = 1 e 6 m , and (a) r 0 = 0.3 m , (b) r 0 = 0.8 m . The theoretical values computed using Eq. (9) are very close to the ones obtained using direct division; hence the differences are not clearly visible in the figure. Direct division also generates artifacts at some angles. It is clear from the left panel that even if r 0 satisfies the far-field condition, the errors in the estimated phase function can be quite large.

Tables (1)

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Table 1 Dependence of Relative Error in the Estimated Phase Function on a, N, and r 0 a

Equations (13)

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E ̃ = E inc exp ( j ω t ) q = N N exp [ j k ( r q + s q ) ] r q ,
r q r 0 q d sin θ .
E ̃ = E inc exp ( j ω t j k r 0 ) r 0 q = N N exp [ j q Θ ] ,
I ( r 0 , θ ) = E E * 2 = 1 2 ( E inc r 0 ) 2 ( 1 + 2 q = 1 N cos ( k Θ p ) ) 2 .
2 q = 1 N cos ( q Ω ) = sin [ ( N + 1 2 ) Ω ] sin [ Ω 2 ] 1 ,
I ( r 0 , θ ) = 1 2 ( E inc r 0 ) 2 sin 2 [ π a ( 2 N + 1 ) ( sin θ n 0 sin ϕ ) ] sin 2 [ π a ( sin θ n 0 sin ϕ ) ] .
θ = sin 1 ( n sin ϕ m a ) ,
a = m n sin ϕ sin θ .
a < a cr = 1 ( n 0 + 1 ) .
E q = E inc exp ( j k q n d sin ϕ ) + l q P l ( exp ( j k r q l ) r q l 3 ) [ k 2 r q l 2 + j k r q l 1 ] ,
γ = 3 d 3 ( ϵ 1 ) 4 π ( ϵ + 2 ) ,
p ( μ ) = 1 2 1 g 2 ( 1 + g 2 2 g μ ) 3 2 ,
I ( g , ϕ , θ ) = p ( g , 180 ° ( θ + ϕ ) ) I iso ( 0 , ϕ , θ ) .

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