Abstract

Scatter of a two-dimensional Gaussian beam of a rectangular cross section by individual particles suspended in a fluid in a cylindrical channel is modeled by using a full-wave approach. First, the internal and scattered fields associated with the cylindrical channel and the two-dimensional Gaussian beam are computed. The spatial variations of the computed electromagnetic field inside the channel indicate that particles and cells of sizes relevant to flow cytometry are subjected to essentially plane-wave illumination, and hence Lorenz–Mie theory is applicable for spherical particles. Further, it is assumed that the perturbation of the electromagnetic field in the channel that is due to the presence of a particle is negligible, allowing us to ignore the interactive scatter of the particle and the channel (they are electromagnetically uncoupled). This approximation is valid when the particle intercepts a small fraction of the total energy inside the channel and when the particle or cell has a low relative refractive index. Measurements of scatter from the channel agree with the analytical model and are used to determine the location of detectors to measure scatter from particles in the channel. Experimental results of accumulated scatter from single latex spheres flowing in the channel show good agreement with computed results, thereby validating the internal field and uncoupled scatter models.

© 2006 Optical Society of America

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  1. A. V. Chernyshev, V. I. Prots, A. A. Doroshkin, and V. P. Maltsev, "Measurement of scattering properties of individual particles with a scanning flow cytometer," Appl. Opt. 34, 6045-6311 (1995).
    [CrossRef]
  2. A. N. Shvalov, I. V. Surovtsev, A. V. Chernyshev; J. T. Soini, and V. P. Maltsev, "Particle classification from light scattering with the scanning flow cytometer," Cytometry 37, 215-220 (1999).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. K. F. Ren, G. Grehan, and G. Gouesbet, "Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory: formulation and numerical results," J. Opt. Soc. Am. A 14, 3014-3025 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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1999 (2)

A. N. Shvalov, I. V. Surovtsev, A. V. Chernyshev; J. T. Soini, and V. P. Maltsev, "Particle classification from light scattering with the scanning flow cytometer," Cytometry 37, 215-220 (1999).
[CrossRef] [PubMed]

G. Gouesbet, "Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres," J. Opt. Soc. Am. A 16, 1641-1650 (1999).
[CrossRef]

1997 (2)

1995 (4)

G. Gouesbet, "Interaction between Gaussian beams and infinite cylinders by using the theory of distributions," J. Opt. (Paris) 26, 225-239 (1995).
[CrossRef]

G. Gouesbet, "The separability theorem revisited with applications to light scattering theory," J. Opt. (Paris) 26, 123-125 (1995).
[CrossRef]

J. A. Lock, "Interpretation of extinction in Gaussian-beam scattering," J. Opt. Soc. Am. A 12, 929-938 (1995).
[CrossRef]

A. V. Chernyshev, V. I. Prots, A. A. Doroshkin, and V. P. Maltsev, "Measurement of scattering properties of individual particles with a scanning flow cytometer," Appl. Opt. 34, 6045-6311 (1995).
[CrossRef]

1994 (4)

1991 (1)

1987 (1)

1982 (1)

S. Kozaki, "Scattering of a Gaussian beam by a homogeneous dielectric cylinder," IEEE Trans. Antennas Propag. AP-30, 881-887 (1982).
[CrossRef]

1981 (1)

1979 (1)

T. J. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by a cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

1978 (1)

1972 (1)

N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
[CrossRef]

Alexopoulos, N. G.

N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
[CrossRef]

Barber, P. W.

Benincasa, D. S.

Bohren, C. F.

E. D. Hirleman and C. F. Bohren, "Optical particle sizing: an introduction by the feature editors," Appl. Opt. 30, 4685-4687 (1991).
[CrossRef] [PubMed]

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

Chang, R. K.

Chernyshev, A. V.

A. N. Shvalov, I. V. Surovtsev, A. V. Chernyshev; J. T. Soini, and V. P. Maltsev, "Particle classification from light scattering with the scanning flow cytometer," Cytometry 37, 215-220 (1999).
[CrossRef] [PubMed]

A. V. Chernyshev, V. I. Prots, A. A. Doroshkin, and V. P. Maltsev, "Measurement of scattering properties of individual particles with a scanning flow cytometer," Appl. Opt. 34, 6045-6311 (1995).
[CrossRef]

Doroshkin, A. A.

A. V. Chernyshev, V. I. Prots, A. A. Doroshkin, and V. P. Maltsev, "Measurement of scattering properties of individual particles with a scanning flow cytometer," Appl. Opt. 34, 6045-6311 (1995).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, 1953).

Gouesbet, G.

G. Gouesbet, "Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres," J. Opt. Soc. Am. A 16, 1641-1650 (1999).
[CrossRef]

K. F. Ren, G. Grehan, and G. Gouesbet, "Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory: formulation and numerical results," J. Opt. Soc. Am. A 14, 3014-3025 (1997).
[CrossRef]

G. Gouesbet, "Interaction between Gaussian beams and infinite cylinders by using the theory of distributions," J. Opt. (Paris) 26, 225-239 (1995).
[CrossRef]

G. Gouesbet, "The separability theorem revisited with applications to light scattering theory," J. Opt. (Paris) 26, 123-125 (1995).
[CrossRef]

G. Gouesbet and G. Gréhan, "Interaction between a Gaussian beam and an infinite cylinder with the use of non-Sum-separable potentials," J. Opt. Soc. Am. A 11, 3261-3273 (1994).
[CrossRef]

G. Gouesbet and G. Gréhan, "Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams," Part. Part. Syst. Charact. 11, 299-308 (1994).
[CrossRef]

J. A. Lock and G. Gouesbet, "Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams," J. Opt. Soc. Am. A 11, 2503-2515 (1994).
[CrossRef]

G. Gouesbet and J. A. Lock, "Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams," J. Opt. Soc. Am. A 11, 2516-2525 (1994).
[CrossRef]

Grehan, G.

Gréhan, G.

G. Gouesbet and G. Gréhan, "Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams," Part. Part. Syst. Charact. 11, 299-308 (1994).
[CrossRef]

G. Gouesbet and G. Gréhan, "Interaction between a Gaussian beam and an infinite cylinder with the use of non-Sum-separable potentials," J. Opt. Soc. Am. A 11, 3261-3273 (1994).
[CrossRef]

Hirleman, E. D.

Hsieh, W.-F.

Huffman, D. R.

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

Kojima, T. J.

T. J. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by a cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

Kozaki, S.

S. Kozaki, "Scattering of a Gaussian beam by a homogeneous dielectric cylinder," IEEE Trans. Antennas Propag. AP-30, 881-887 (1982).
[CrossRef]

Lock, J. A.

Maltsev, V. P.

A. N. Shvalov, I. V. Surovtsev, A. V. Chernyshev; J. T. Soini, and V. P. Maltsev, "Particle classification from light scattering with the scanning flow cytometer," Cytometry 37, 215-220 (1999).
[CrossRef] [PubMed]

A. V. Chernyshev, V. I. Prots, A. A. Doroshkin, and V. P. Maltsev, "Measurement of scattering properties of individual particles with a scanning flow cytometer," Appl. Opt. 34, 6045-6311 (1995).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, 1953).

Owen, J. F.

Park, P. K.

N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
[CrossRef]

Prots, V. I.

A. V. Chernyshev, V. I. Prots, A. A. Doroshkin, and V. P. Maltsev, "Measurement of scattering properties of individual particles with a scanning flow cytometer," Appl. Opt. 34, 6045-6311 (1995).
[CrossRef]

Ren, K. F.

Shvalov, A. N.

A. N. Shvalov, I. V. Surovtsev, A. V. Chernyshev; J. T. Soini, and V. P. Maltsev, "Particle classification from light scattering with the scanning flow cytometer," Cytometry 37, 215-220 (1999).
[CrossRef] [PubMed]

Soini, J. T.

A. N. Shvalov, I. V. Surovtsev, A. V. Chernyshev; J. T. Soini, and V. P. Maltsev, "Particle classification from light scattering with the scanning flow cytometer," Cytometry 37, 215-220 (1999).
[CrossRef] [PubMed]

Stevenson, W. H.

Surovtsev, I. V.

A. N. Shvalov, I. V. Surovtsev, A. V. Chernyshev; J. T. Soini, and V. P. Maltsev, "Particle classification from light scattering with the scanning flow cytometer," Cytometry 37, 215-220 (1999).
[CrossRef] [PubMed]

Yanagiuchi, Y.

T. J. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by a cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

Zhang, J.-Z.

Appl. Opt. (4)

Cytometry (1)

A. N. Shvalov, I. V. Surovtsev, A. V. Chernyshev; J. T. Soini, and V. P. Maltsev, "Particle classification from light scattering with the scanning flow cytometer," Cytometry 37, 215-220 (1999).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (2)

N. G. Alexopoulos and P. K. Park, "Scattering of waves with normal amplitude distribution from cylinders," IEEE Trans. Antennas Propag. AP-20, 216-217 (1972).
[CrossRef]

S. Kozaki, "Scattering of a Gaussian beam by a homogeneous dielectric cylinder," IEEE Trans. Antennas Propag. AP-30, 881-887 (1982).
[CrossRef]

J. Appl. Phys. (1)

T. J. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by a cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

J. Opt. (2)

G. Gouesbet, "Interaction between Gaussian beams and infinite cylinders by using the theory of distributions," J. Opt. (Paris) 26, 225-239 (1995).
[CrossRef]

G. Gouesbet, "The separability theorem revisited with applications to light scattering theory," J. Opt. (Paris) 26, 123-125 (1995).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Lett. (1)

Part. Part. Syst. Charact. (1)

G. Gouesbet and G. Gréhan, "Interaction between shaped beams and an infinite cylinder, including a discussion of Gaussian beams," Part. Part. Syst. Charact. 11, 299-308 (1994).
[CrossRef]

Other (2)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, 1953).

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

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Figures (9)

Fig. 1
Fig. 1

Channel is represented as a homogeneous dielectric cylinder with the incident laser beam propagating along the x axis. The refractive indices of glass and sheath fluid are n 1 and n 2 , respectively.

Fig. 2
Fig. 2

Variation of the scattered intensity normalized by I 0 [infinite cylinder with peak intensity of the laser beam at ( x 0 , y 0 ) ] as a function of θ at a radial distance of 25   mm and ϕ = 90 ° . z = 0 , a = 125   μm , n 1 = 1.33 , and n 2 = 1.5 .

Fig. 3
Fig. 3

Variation of the scattered intensity from the channel normalized by I 0 [the peak intensity of the laser beam at ( x 0 , y 0 ) ] as a function of z   at   θ = 0 ° . a = 125   μm, r = 250   mm , n 1 = 1.33 , and n 2 = 1.5 .

Fig. 4
Fig. 4

Amplitudes of ( a ) B , ( b ) E 0     ϵ , ( c ) δ E 0     μ as a function of h   and   l .

Fig. 5
Fig. 5

(a) Amplitudes of the internal electric field distribution as a function of z (where E 0     μ + E 0     ϵ = E 0 ), (b) Distribution of the amplitude of E z in the xy plane, (c) Phase of E z in the xy plane. n 1 = 1.33 , n 2 = 1.5 , a = 125   μm,   r = 0 , z = 0 .

Fig. 6
Fig. 6

Top view of the experimental setup used to measure the intensity of scattered light from particles flowing in the channel. (Drawing is not to scale.)

Fig. 7
Fig. 7

(a) Number of spheres versus peak intensity (arbitrary units), (b) number of spheres versus total scattered energy (arbitrary units), (c) peak intensity versus total scattered energy (region R1 represents single spheres in the laser beam) for (top) 1.71   μm spheres and (bottom) 4.369   μm spheres.

Fig. 8
Fig. 8

Scattered intensity from a distribution of single latex spheres normalized by I i ( 1 / e 2 value of the peak intensity inside the channel).

Fig. 9
Fig. 9

Internal energy distribution at z = 0 for a channel of diameter 20   μm , with a fluid illuminated by a normally incident z-polarized, 16   μm × 20   μm Gaussian laser beam with a free-space wavelength of 488   nm . a = 10   μm,   n 1 = 1.33 , n 2 = 1.5 .

Equations (50)

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E ( r , θ , z ) = e z × ψ ϵ ( r , θ , z ) ,
B ( r , θ , z ) = ( i / ω ) × [ e z × ψ ϵ ( r , θ , z ) ] ,
ψ inc             ϵ ( r , θ , z ) = + d h l = l = + i l + 1 k E 0 A l ( h ) × J l [ k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
A l ( h ) = ( i ) l k ( 2 π ) 2 ( 1 h 2 ) J l [ k r ( 1 h 2 ) 1 / 2 ] × + d z 0 2 π d θ exp ( i k h z ) exp ( i l θ ) × c B z , inc ( r , θ , z ) / E 0 .
B ( r , θ , z ) = ( m / c ) e z × ψ μ ( r , θ , z ) ,
E ( r , θ , z ) = ( i c / m ω ) × [ e z × ψ μ ( r , θ , z ) ] ,
ψ inc             μ ( r , θ , z ) = + d h l = l = + i i + 1 k E 0 B l ( h ) × J l [ k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
B l ( h ) = ( i ) l k ( 2 π ) 2 ( 1 h 2 ) J l [ k r ( 1 h 2 ) 1 / 2 ] × + d z 0 2 π d θ [ exp ( i k h z ) × exp ( i l θ ) ] E z , inc ( r , θ , z ) / E 0 .
E z , inc ( x 0 , y , z ) = E 0 exp [ ( y y 0 ) 2 z 2 ω 0     2 ] .
E z , inc ( x , y , z ) = E 0 ω 0     2 4 π + k d h y × + k d h z exp ( h y     2 4 s 2 ) × exp ( h z     2 4 s 2 ) exp [ i k h x ( x x 0 ) ] × exp [ i k h x ( y y 0 ) ] exp ( i k h z z ) ,
s = 1 / k ω 0 .
h x = ( 1 h y     2 h z     2 ) 1 / 2 ,
E inc = 0.
B l ( h ) = 1 2 π 1 / 2 s ( 1 h 2 ) [ 1 2 i s 2 k x 0 ( 1 h 2 ) 1 / 2 ] 1 / 2 × exp ( h 4 s 2 ) exp [ i k ( 1 h 2 ) 1 / 2 x 0 ] × exp { - s 2 [ l ( 1 - h 2 ) - 1 / 2 + k y 0 ] 2 1 - 2 i s 2 k x 0 ( 1 - h 2 ) - 1 / 2 } ,
A l ( h ) = 0.
A l ( h ) = 1 2 π 1 / 2 s ( 1 h 2 ) [ 1 2 i s 2 k x 0 ( 1 h 2 ) 1 / 2 ] 1 / 2 × exp ( h 4 s 2 ) exp [ i k ( 1 h 2 ) 1 / 2 x 0 ] × exp { - s 2 [ l ( 1 - h 2 ) - 1 / 2 + k y 0 ] 2 1 - 2 i s 2 k x 0 ( 1 - h 2 ) - 1 / 2 } ,
B l ( h ) = 0.
ψ scat               ϵ ( r , θ , z ) = + d h l = l = + i l + 1 k E 0 α l ( h ) × H l     ( 1 ) [ k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
ψ scat               μ ( r , θ , z ) = + d h l = l = + i l + 1 k E 0 β l ( h ) × H l     ( 1 ) [ k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ )
ψ int           ϵ ( r , θ , z ) = + d h l = l = + i l + 1 k E 0 n γ l ( h ) × J l [ n k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
ψ int           μ ( r , θ , z ) = + d h l = l = + i l + 1 k E 0 n δ l ( h ) × J l [ n k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
h = h / n
α l ( h ) = a l ( h ) A l ( h ) + q l ( h ) B l ( h ) ,
β l ( h ) = q l ( h ) A l ( h ) + b l ( h ) B l ( h ) ,
γ l ( h ) = c l ( h ) A l ( h ) + p l ( h ) B l ( h ) ,
δ l ( h ) = n p l ( h ) A l ( h ) + d l ( h ) B l ( h ) ,
a l ( h ) = U 2 W 1 n U 3 W 3 W 2 W 1 n W 3     2 ,
b l ( h ) = U 1 W 2 n U 3 W 3 W 2 W 1 n W 3     2 ,
q l ( h ) = 2 n l h ( y 2 x 2 ) π x 2 y 2 J l     2 ( y ) W 2 W 1 n W 3     2 ,
c l ( h ) = 2 i n x π y 2 W 1 W 2 W 1 n W 3     2 ,
d l ( h ) = 2 i n x π y 2 W 2 W 2 W 1 n W 3     2 ,
p l ( h ) = 2 n x π y 2 W 3 W 2 W 1 n W 3     2 ,
x k a ( 1 h 2 ) 1 / 2 ,
y n k a [ 1 ( h 2 / n 2 ) ] 1 / 2 ,
U 1 = n 2 x y J l ( x ) J l ( y ) J l ( x ) J l ( y ) ,
U 2 = n x y J l ( x ) J l ( y ) n J l ( x ) J l ( y ) ,
U 3 = h l ( y 2 x 2 ) x y 2 J l ( x ) J l ( y ) ,
W 1 = n 2 x y H l     ( 1 ) ( x ) J l ( y ) H l     ( 1 ) × ( x ) J l ( y ) ,
W 2 = n x y  H l     ( 1 ) ( x ) J l ( y ) n H l     ( 1 ) × ( x ) J l ( y ) ,
W 3 = h l ( y 2 x 2 ) x y 2 H l     ( 1 ) ( x ) J l ( y ) .
lim r I scat ( r , θ , z ) = Re { E scat * × B scat 2 μ 0 c } = 2 π k r ( E 0     2 2 μ 0 c ) Re [ ( T 2 * T 4 + T 3 * T 1 ) e r + ( T 5 * T 4 T 3 * T 6 ) e θ + ( T 5 * T 1 + T 2 * T 6 ) e z ] ,
T 1 ( r , θ , z ) = + d h l = l = + ( 1 h 2 ) 1 / 4 β l ( h ) × exp [ i k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
T 2 ( r , θ , z ) = + d h l = l = + ( 1 h 2 ) 1 / 4 α l ( h ) × exp [ i k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
T 3 ( r , θ , z ) = + d h l = l = + ( 1 h 2 ) 3 / 4 β l ( h ) × exp [ i k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
T 4 ( r , θ , z ) = + d h l = l = + ( 1 h 2 ) 3 / 4 α l ( h ) × exp [ i k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
T 5 ( r , θ , z ) = + d h l = l = + h ( 1 h 2 ) 1 / 4 β l ( h ) × exp [ i k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) ,
T 6 ( r , θ , z ) = + d h l = l = + h ( 1 h 2 ) 1 / 4 α l ( h ) × exp [ i k r ( 1 h 2 ) 1 / 2 ] exp ( i k h z ) exp ( i l θ ) .
I s ( θ , ϕ ) = d = 0 A d I s d ( θ , ϕ ) d d ,
A d = 1 σ 2 π exp [ ( d d 0 ) 2 / 2 σ 2 ] ,
I m ( θ , ϕ ) = I s ( θ , ϕ ) + I s ( 180 ° + θ , 180 ° + ϕ ) ,

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