Abstract

This paper deals with the problem of the scattering of focused laser beams by tenuous particles using an iterative technique. The results are shown to be accurate provided that (a) the polarizability of the particle medium is small and (b) the phase shift of the central ray is less than 2. It was found that when the size of the incident beam waist is close to that of the scatterer, the scattered field deviates significantly from that for the incident plane wave case. Specific examples are given.

© 1979 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. L. W. Casperson, C. Yeh, Appl. Opt. 17, 1637 (1978).
    [CrossRef] [PubMed]
  3. K. S. Shifrin, Scattering of Light in a Turbid Medium (Nauka, Moscow, 1951; NASA, Washington, D.C., 1968), Technical Translation TTF-477.
  4. C. Acquista, Appl. Opt. 15, 2932 (1976).
    [CrossRef] [PubMed]
  5. A. Yariv, Quantum Electronics (Wiley, New York, 1975).
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. P. C. Clemmow, Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, London, 1966).
  8. W. H. Carter, J. Opt. Soc. Am. 62, 1195 (1972).
    [CrossRef]
  9. S. Colak, “Focused Laser Beam Scattering by Stationary and Moving Particles,” Ph.D. Thesis, U. California, Los Angeles (1978).
  10. A. Yariv, Introduction to Optical Electronics (1971).
  11. W. C. Tsai, R. J. Pogorzelski, J. Opt. Soc. Am. 65, 1457 (1975).
    [CrossRef]

1978 (1)

1977 (1)

1976 (1)

1975 (1)

1972 (1)

1971 (1)

A. Yariv, Introduction to Optical Electronics (1971).

Acquista, C.

Carter, W. H.

Casperson, L. W.

Clemmow, P. C.

P. C. Clemmow, Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, London, 1966).

Colak, S.

S. Colak, “Focused Laser Beam Scattering by Stationary and Moving Particles,” Ph.D. Thesis, U. California, Los Angeles (1978).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Pogorzelski, R. J.

Shifrin, K. S.

K. S. Shifrin, Scattering of Light in a Turbid Medium (Nauka, Moscow, 1951; NASA, Washington, D.C., 1968), Technical Translation TTF-477.

Tsai, W. C.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (1971).

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

Yeh, C.

Yeung, W. F.

Appl. Opt. (3)

Introduction to Optical Electronics (1)

A. Yariv, Introduction to Optical Electronics (1971).

J. Opt. Soc. Am. (2)

Other (5)

S. Colak, “Focused Laser Beam Scattering by Stationary and Moving Particles,” Ph.D. Thesis, U. California, Los Angeles (1978).

K. S. Shifrin, Scattering of Light in a Turbid Medium (Nauka, Moscow, 1951; NASA, Washington, D.C., 1968), Technical Translation TTF-477.

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

P. C. Clemmow, Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, London, 1966).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the scattering problem. W0 is the spot size of the beam which is centered on the O system. The beam propagates along the +z direction. The scatterer is centered on the O′ system, which is displaced from the O system by d = (dx,dy,dz), and it may also be rotated with respect to the O system. θs and ϕs define the scattering angles.

Fig. 2
Fig. 2

Placement of the rotated scatterer coordinate system O′ with respect to the stationary beam system O. r and r′ are arbitrary position vectors in the beam and the scatterer system, respectively. d is the vector defining the displacement of the scatterer from the origin of the beam system.

Fig. 3
Fig. 3

Diagram for coordinate transformations. ks, r, and z ^ are vectors all directed to the observation point. The incident field propagation vector ki(p,q) is in the xz′ plane. The scattering plane is chosen to be the xz plane, and thus the vector ks (also r and z ^ ) is in this plane, and it makes a scattering angle θs with the z axis of the beam system O.

Fig. 4
Fig. 4

Beam scattering patterns for centered spheres (solid lines). In each case, the incident laser beam has a large spot size. Dashed lines give the plane wave results for comparison purposes. The dotted lines represent the first iteration results [if E s t ( 2 ) ( r ) is taken to be zero]. For all figures W0/a = 5.0. The scatterer parameters are (a) ka = 1.60, n = 1.10; (b) ka = 5.0, n = 1.10; (c) ka = 1.0, n = 1.55.

Fig. 5
Fig. 5

Beam scattering patterns showing the effect of decreasing beam waist spot size W0 for centered spherical scatterers. The expansion of the pattern for smaller spot sizes is easily observable. The scatterer parameters are: (a) ka = 5.0, n = 1.10; (b) ka = 3.0, n, = 1.10; (c) ka = 10.0, n = 1.05; (d) ka = 15.0, n = 1.03. The spot sizes are indicated in figures in W0/a ratios.

Fig. 6
Fig. 6

Beam scattering patterns for off-centered spherical scatterers; (a) and (b) are for scatterers off-centered in the z = 0 plane. The magnitude reduction in the pattern is proportional to the incident beam intensity decreases due to off-centering. (c) is for scatterers off-centered out of the z = 0 plane. For this case, the difference between the S11 values for different directions of the scattering plane is shown by the dotted curve to reveal the asymmetry of the pattern. The beam and the scatterer parameters for the figures are as follows: (a) and (c) W0/a = 5.0, ka = 1.0, n = 1.55; (b) W0/a = 5.0, ka = 5.0, n = 1.10, and the displacements of scatterers from the beam system origin are given in the figures in terms of (dx,dy,dz)/a ratios. For (c), S11 values for positive and negative scattering angles are given by dashed curves without and with dots, respectively. The difference between these two cases is plotted by the dotted curve.

Equations (49)

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E b ( x , y , z , t ) = ( A x x ^ x + A y y ^ y ) f ( x , y , z ) exp ( i ω t ) ,
f ( x , y , z ) = W 0 W ( z ) exp ( - i k z ) exp [ - i k ( x 2 + y 2 ) / 2 R ( z ) ] · exp [ - ( x 2 + y 2 ) / W 2 ( z ) ] · exp [ - i arcan ( z / z 0 ) ] ,
W ( z ) = W 0 [ 1 + ( z 2 / z 0 2 ) ] 1 / 2 ,
R ( z ) = z [ 1 + ( z 0 2 / z 2 ) ] ,
z 0 = π W 0 2 / λ .
S x ( p , q ) = A x λ 2 - + f ( x , y , 0 ) exp [ i k ( p x + q y ) ] d x d y = A x [ ( b 2 ) / π ] exp [ - b 2 ( p 2 + q 2 ) ] ,
S y ( p , q ) = A y λ 2 - + f ( x , y , 0 ) exp [ - i k ( p x + q y ) ] d x d y = A y [ ( b 2 ) / π ] exp [ - b 2 ( p 2 + q 2 ) ] ,
b = k W 0 / 2.
E b x ( x , y , 0 ) = A x f ( x , y , 0 ) = - + S x ( p , q ) exp [ i k ( p x + q y ) ] d p d q ,
E b y ( x , y , 0 ) = A y f ( x , y , 0 ) = - + S y ( p , q ) exp [ i k ( p x + q y ) ] d p d q ,
2 E b j ( x , y , z ) + k 2 E b j ( x , y , z ) = 0 , j = x , or y ,
E b x ( x , y , z ) = - + S x ( p , q ) exp [ i k ( p x + q y - s z ) ] d p d q ,
E b y ( x , y , z ) = - + S y ( p , q ) exp [ i k ( p x + q y - s z ) ] d p d q ,
s = ( 1 - p 2 - q 2 ) 1 / 2 .
E b ( x , y , z = b 2 π - + ( A x x ^ + A y y ^ + p A x + q A y s z ^ ) · exp [ - b 2 ( p 2 + q 2 ) ] exp [ i k ( p x + q y - s z ) ] d p d q ,
A i ( p , q ) = ( A x x ^ + A y y ^ + p A x + q A y s z ^ ) exp [ - b 2 ( p 2 + q 2 ) ]
k i ( p , q ) = k ( p x ^ + q y ^ + q y ^ - s z ^ ) .
E i ( p , q ) = A i ( p , q ) exp [ i k i ( p , q ) · r ] .
E s ( 1 ) ( r ) = k 2 r exp ( i k r ) E o u ( k r ^ - k ) ,
E s ( 2 ) ( r ) = 2 k 2 ( 2 π ) 2 exp ( i k r ) r V m u ( m + k r ^ ) u ( m + k ) m 2 - k 2 · [ ( 2 3 k 2 + 1 3 m 2 ) E o - ( m · E o ) m ] ] d 3 m ,
E s ( r ) = α E s ( 1 ) ( r ) + α 2 E s ( 2 ) ( r ) ,
u ( v ) = V s exp ( i v · r ) d 3 r ,
α = 3 4 π ( n 2 - 1 n 2 + 2 ) ,
r = [ T ] ( r + d ) ,
r = [ T ] - 1 r - d ,
[ T ] = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] ,
a 31 = sin θ s , a 32 = 0 , a 33 = cos θ s a 21 = - q a 33 / a r , a 22 = ( a 33 p + s a 31 ) / a r , a 23 = q a 31 / a r , a r = [ ( q a 33 ) 2 + ( p a 33 + s a 31 ) 2 + ( q a 31 ) 2 ] 1 / 2 a 11 = a 22 a 33 , a 12 = q / a r , a 13 = - a 22 a 31 , }
E i ( p , q ) = A i ( p , q ) exp [ - i Δ ( p , q ) ] exp [ i k i ( p , q ) · r ] ,
A i ( p , q ) = [ T ] A i ( p , q ) ,
k i ( p , q ) = [ T ] k i ( p , q ) ,
Δ ( p , q ) = k i ( p , q ) · d ,
E 0 A i ( p , q ) exp [ - i Δ ( p , q ) ] , k k i ( p , q ) , r r             and             m m . }
E s t ( 1 ) ( r ) = k 2 r exp ( i k r ) - + E i b ( p , q ) exp [ - i Δ ( p , q ) ] · u [ k r ^ - k i ( p , q ) ] d p d q ,
E s t ( 2 ) ( r ) = 2 k 2 ( 2 π ) 2 exp ( i k r ) r - + · exp [ - i Δ ( p , q ) ] V m u ( m + k r ^ ) u [ m + k i ( p , q ) ] m 2 + k 2 · { ( 2 3 k 2 + 1 3 m 2 ) E i 2 ( p , q ) - [ m · E i ( p , q ) ] m b } d 3 m d p d q ,
E i b ( p , q ) = [ T ] - 1 E i ( p , q ) ,
m b = [ T ] - 1 m ,
E s t ( r ) = α E s t ( 1 ) ( r ) + α 2 E s t ( 2 ) ( r ) .
S 11 = k 2 r 2 { [ I s ( r ) ] / [ I i ( r ) ] } ,
I s ( r ) = α E s t ( 1 ) ( r ) + α 2 E s t ( 2 ) ( r ) 2 ,
I i ( r ) = ( A x 2 + A y 2 ) 1 / 2 ,
S 11 = k 2 α 2 - cos [ Δ ( p , q ) - Δ ( p , q ) ] · { ^ i b ( p , q ) · ^ i b ( p , q ) u [ k r - k i ( p , q ) ] u [ k r - k i ( p , q ) ] + α π 2 V m u [ k r - k i ( p , q ) ] u ( m + k r ) u [ m + k i ( p , q ) ] m 2 - k 2 · { ( 2 3 k 2 + 1 3 m 2 ^ i b ( p , q ) · ^ i b ( p , q ) - [ ^ i b ( p , q ) · m b ] · [ ^ i b ( p , q ) · m ] d 3 m } d p d q d p d q ,
u ( v ) = V s f ( a v ) ,
V s = 4 / 3 π a 3 ,
f ( a v ) = 3 ( a v ) 3 [ sin ( a v ) - ( a v ) cos ( a v ) ] ,
4 k 0 n a + 4 tan - 1 ( a / z 0 ) = ( 2 h + 1 ) π ,
W 0 a = [ 2 n k 0 a tan ( 2 h + 1 4 π - n k 0 a ) ] 1 / 2 ,
S 11 , ( 0 , 0 , 0 ) S 11 , ( 2 , 1 / 2 , 0 ) = 1.71 ,             S 11 , ( 0 , 0 , 0 ) S 11 , ( 4 , 3 , 0 ) = 8.3
I inc , ( 0 , 0 , 0 ) I inc , ( 2 , 1 / 2 , 0 ) = exp [ 2 ( d x 2 + d y 2 ) / W 0 2 ] = 1.65 ,             I inc , ( 0 , 0 , 0 ) I inc , ( 4 , 3 , 0 ) = 7.38 ,
S 11 , ( 0 , 0 , 0 ) S 11 , ( 2 , 2 , 0 ) = 1.88 , I inc , ( 0 , 0 , 0 ) I inc , ( 2 , 2 , 0 ) = 1.90 , S 11 , ( 0 , 0 , 0 ) S 11 , ( 4 , 4 , 0 ) = 12.48 , I inc , ( 0 , 0 , 0 ) I inc , ( 4 , 4 , 0 ) = 12.94.

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