Abstract

The scattering of a focused laser beam by moving particles is studied. A solution is given for a spherical scatterer that is small compared to the wavelength. Several numerical examples are given. It is concluded that for particles that have a short time of flight within the laser beam, the bandwidth and frequency spectrum of the scattered field is critically dependent on the laser beam shape. This fact is particularly important when information on the motion of small particles or macromolecules is to be inferred from the measured bandwidth and spectrum of the scattered field due to an incident wave that is not a plane wave.

© 1980 Optical Society of America

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References

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  1. L. W. Casperson, C. Yeh, W. F. Yeung, Appl. Opt. 16, 1104 (1977).
    [CrossRef] [PubMed]
  2. S. Colak, C. Yeh, L. W. Casperson, Appl. Opt. 18, 294 (1979).
    [CrossRef] [PubMed]
  3. C. Yeh, S. Colak, P. W. Barber, “Scattering of Focused Beam by Arbitrarily Shaped Particles—Exact Solution,” to be submitted to Appl. Opt.
    [PubMed]
  4. K. S. Shifrin, Scattering of Light in a Turbid Medium (Nauka, Moscow, 1951); NASA Tech. Translation TTF-477, 1968, Chap. 4.
  5. B. J. Berne, R. Pecora, Dynamic Light Scattering (Academic, New York, 1976).
  6. C. T. O’Kanski, Molecular Electrooptics: Theory and Methods (Dekker, New York, 1976).

1979 (1)

1977 (1)

Barber, P. W.

C. Yeh, S. Colak, P. W. Barber, “Scattering of Focused Beam by Arbitrarily Shaped Particles—Exact Solution,” to be submitted to Appl. Opt.
[PubMed]

Berne, B. J.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Academic, New York, 1976).

Casperson, L. W.

Colak, S.

S. Colak, C. Yeh, L. W. Casperson, Appl. Opt. 18, 294 (1979).
[CrossRef] [PubMed]

C. Yeh, S. Colak, P. W. Barber, “Scattering of Focused Beam by Arbitrarily Shaped Particles—Exact Solution,” to be submitted to Appl. Opt.
[PubMed]

O’Kanski, C. T.

C. T. O’Kanski, Molecular Electrooptics: Theory and Methods (Dekker, New York, 1976).

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Academic, New York, 1976).

Shifrin, K. S.

K. S. Shifrin, Scattering of Light in a Turbid Medium (Nauka, Moscow, 1951); NASA Tech. Translation TTF-477, 1968, Chap. 4.

Yeh, C.

S. Colak, C. Yeh, L. W. Casperson, Appl. Opt. 18, 294 (1979).
[CrossRef] [PubMed]

L. W. Casperson, C. Yeh, W. F. Yeung, Appl. Opt. 16, 1104 (1977).
[CrossRef] [PubMed]

C. Yeh, S. Colak, P. W. Barber, “Scattering of Focused Beam by Arbitrarily Shaped Particles—Exact Solution,” to be submitted to Appl. Opt.
[PubMed]

Yeung, W. F.

Appl. Opt. (2)

Other (4)

C. Yeh, S. Colak, P. W. Barber, “Scattering of Focused Beam by Arbitrarily Shaped Particles—Exact Solution,” to be submitted to Appl. Opt.
[PubMed]

K. S. Shifrin, Scattering of Light in a Turbid Medium (Nauka, Moscow, 1951); NASA Tech. Translation TTF-477, 1968, Chap. 4.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Academic, New York, 1976).

C. T. O’Kanski, Molecular Electrooptics: Theory and Methods (Dekker, New York, 1976).

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem: w0 is the beam spot size; v and d0 are, respectively, the velocity and the zero-time position of the scatterer. The figure shows the t = 0 situation. r is the scattering direction and is defined by angles θs and ϕs.

Fig. 2
Fig. 2

Coordinates for the moving-scatterer frame. The scatterer-coordinate system (the primed one) is moving with a velocity v with respect to the stationary-beam system. x′ and z′ are chosen such that the incident electric field Ei(p,q) is along x′ and the propagation vector ki(p,q) is along z′. The scattering vector ks makes a polar angle θ s ( p , q ) and an azimuthal angle ϕ s ( p , q ) with respect to the scatterer frame. Note that the intersection point of the dashed lines is in the xy′ plane.

Fig. 3
Fig. 3

Three-dimensional plots showing the effect of incident polarization at two different scattering planes. The spot size of the beam is 1 μm, and the velocity of particle is v = (10,0,0) cm/sec and d0 = (0,0,0) μm. Other parameters are given in the text. (a) A0x = 1, A0y = 0, and ϕ = 0°; (b) A0x, = 1, A0y = 0, and ϕs = 90°; (c) A0x = 0, A0y = 1, and ϕs = 0°; (d) A0x = 0, A0y = 1, and ϕs = 90°. The θs axis is 5°/div, and the ω axis is 10 kHz/div, and these values for plots are kept the same in Figs. 4 and 5. The incident frequency indicated in the figures is ωi = 2πci = 37.7 × 1014 Hz.

Fig. 4
Fig. 4

Plots showing the effect of beam spot size and the speed of the scatterer. For these figures A0x = 1, A0y, = 0, and the scattering plane is ϕs = 0 plane and d0 = (0,0,0) μm. (a) is the same as Fig. 3(a) and is included for comparison purposes. (b) W0 = 2 μm, and v = (10,0,0) cm/sec. (c) W0 = 1 μm, and v = (5,0,0) cm/sec. Other common parameters are given in the text and in earlier figures.

Fig. 5
Fig. 5

Plots showing effects of particle velocity and the path it follows through the beam. For these figures A0x = 1 and A0y = 0, W0 = 1 μm, and |v| = 10 cm/sec although the path and direction of the motion change for each plot. (a) is the same as Fig. 3(a) and is included for reference purposes. (b) v = (10,0,0) cm/sec, d0 = (0,1,0) μm; (c) v = (10,0,0) cm/sec, d0 = (0,0,5) μm; (d) v (8,0,6) cm/sec, d0 = (0,0,0) μm; (e) v = (8,0,6) cm/sec, d0 = (0,0,3) μm.

Equations (41)

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E b ( x , y , z , t ) = exp ( i ω i t ) b 2 π - + ( A 0 x x ^ + A 0 y y ^ + p A 0 x + q A 0 y s z ^ ) × exp [ - b 2 ( p 2 + q 2 ) ] exp [ i k 0 ( p x + q y - s z ) ] d p d q ,
E i ( r , t , p , q ) = A i ( p , q ) exp { i [ ω i t + k i ( p , q ) × r ] } ,
A i ( p , q ) = b 2 ˙ π ( A 0 x x ^ + A 0 y y ^ + p A 0 x + q A 0 y s z ^ ) × exp [ - b 2 ( p 2 + q 2 ) ] ,
k i ( p , q ) = k 0 ( p x ^ + q y ^ - s z ^ ) ,
s = ( 1 - p 2 - q 2 ) 1 / 2 .
E i ( r , t , p , q ) = A i ( p , q ) exp { i [ ω i t + k i ( p , q ) × r + ξ ( t , p , q ) ] } ,
A i ( p , q ) = [ T ] A i ( p , q ) ,
k i ( p , q ) = [ T ] k i ( p , q ) ,
ξ ( t , p , q ) = k i ( p , q ) × d ( t ) .
d ( t ) = d 0 + v t ,
[ T ] = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] ,
[ a 31 a 32 a 33 ] = [ p q - s ] ,
[ a 11 a 12 a 13 ] = 1 A m [ A 0 x A 0 y ( p A 0 x + q A 0 y ) / s ] ,
A m = [ A 0 x 2 + A 0 y 2 + ( p A 0 x + q A 0 y ) 2 s 2 ] 1 / 2 ,
[ a 21 a 22 a 23 ] = [ a 32 a 13 - a 12 a 33 a 33 a 11 - a 31 a 13 a 31 a 12 - a 11 a 32 ] .
E i ( r , t , p , q ) = A x ( p , q ) x ^ × exp { i [ ω i t - k z ( p , q ) z + ξ ( t , p , q ) ] } ,
A x ( p , q ) = [ A 0 x 2 + A 0 y 2 + ( p A 0 x + q A 0 y ) 2 s 2 ] 1 / 2 × b 2 π exp [ - b 2 ( p 2 + q 2 ) ] ,
k z ( p , q ) = k 0 = ω i / c 0 = ω i ( 0 μ 0 ) 1 / 2 ,
E i ( r , t , p , q ) = A x ( p , q ) x ^ × exp { i [ ω i t - k 0 z + ξ ( t , p , q ) ] } ,
E θ s ( r , t , p , q ) = cos [ ϕ s ( p , q ) ] A x ( p , q ) exp { i [ ω i t + ξ ( t , p , q ) ] } k 0 exp ( - i k 0 r ) r ( k 0 a ) 3 { γ ( u ) 2 + cos [ θ s ( p , q ) ] } ,
E ϕ s ( r , t , p , q ) = - sin [ ϕ s ( , q ) ] A x ( p , q ) exp { i [ ω i t + ξ ( t , p , q ) ] } k 0 exp ( - i k 0 r ) r ( k 0 a ) 3 { 1 + γ ( u ) 2 cos [ θ s ( p , q ) ] } ,
u = n k 0 a , γ ( u ) = 1 + 3 u cot ( u ) - 3 u 2 ,
r ^ s = sin θ s cos ϕ s x ^ + sin θ s sin ϕ s y ^ + cos θ s z ^ ,
r ^ s ( p , q ) = [ T ] r ^ s
r ^ s ( p , q ) = [ r s x ( p , q ) r s y p , q ) r s z ( p , q ) ] = [ T ] [ sin θ s cos ϕ s sin θ s sin ϕ s cos θ s ] .
θ s ( p , q ) = cos - 1 [ r s z ( p , q ) ] = sin - 1 [ r x y ( p , q ) ] ,
ϕ s ( p , q ) = cos - 1 [ r s x ( p , q ) r x y ( p , q ) ] = sin - 1 [ r s y ( p , q ) r x y ( p , q ) ] ,
r x y ( p , q ) = { [ r s x ( p , q ) ] 2 + [ r s y ( p , q ) ] 2 } 1 / 2 .
E s ( r , t , p , q ) = [ E x ( r , p , q , ) x ^ + E y ( r , p , q ) y ^ + E z ( r , p , q ) z ^ ] A x ( p , q ) exp { i [ ω i t + ξ ( t , p , q ) ] } k 0 exp [ i η ( t , r ) ] ( k 0 a ) 3 exp ( - i k 0 r ) r ,
η ( t , r ) = k s × d ( t ) ,
[ E x ( r , p , q ) E y ( r , p , q ) E z ( r , p , q ) ] = [ T - 1 ] [ r s z ( r s x r x y ) 2 α θ + ( r s y r x y ) α ϕ r s z r s x r s y ( r x y ) 2 α θ - r s x r s y ( r x y ) 2 α ϕ ] r s x α θ ,
α θ = γ ( u ) 2 + r s z ,
α ϕ = 1 + γ ( u ) 2 r s z .
E S T ( r , t ) = ( k 0 a ) 3 exp ( - i k 0 r ) k 0 r exp { i [ ω i t + η ( t , θ s ) ] } × - + E s c ( r , p , q ) exp [ i ξ ( t , p , q ) ] d p d q ,
E s c ( r , p , q ) = [ E x ( r , p , q ) x ^ + E y ( r , p , q ) y ^ + E z ( r , p , q ) z ^ ] A x ( p , q ) .
E S T ( r , t ) = ( k 0 a ) 3 exp ( i k 0 r ) k 0 r exp [ i ( ω i t + k s V ) t ] × - + E s c ( r , p , q ) exp { - i [ k i ( p , q ) V ] t } d p d q ,
E s c ( r , p , q ) = E s c ( r , p , q ) exp { i d 0 [ k 0 - k i ( p , q ) ] } .
E S T ( r , ω ) = ( 2 π ) 2 ( k 0 a ) 3 exp ( - i k 0 r ) k 0 r δ [ ω - ( ω i + k s v ) ] * { - + E s c ( r , p , q ) δ [ ω - k i ( p , q ) v ] d p d q } ,
ω c = ω i + k s · v .
ω = k i ( p , q ) · v = k 0 ( p v x + q v y - d v z ) .
I = r 2 E S T ( r , ω ) · E S T * ( r , ω ) .

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