Abstract

We have studied time variances of the shape of an electromagnetic pulse scattered by a spherical particle. General formulas are derived for a pulse with an arbitrary envelope, for momentary values of scattered-light fields and light intensity, and for efficiencies of extinction and scattering. It is possible, by the use of these formulas, to obtain by routine integration the sensitivity reaction of a receiver with any time dependence. The formulas are illustrated with examples of scattering of a Gaussian pulse with a carrier wave λ0 = 0.6328 μm and of multisized water drops. Pulses of different durations are studied. However, only those pulses that have all significant values of the Fourier density in the domain of positive frequencies ω are considered.

© 1995 Optical Society of America

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References

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  1. K. S. Shifrin, I. G. Zolotov, “Quasi-stationary scattering of an electromagnetic pulse by a spherical particle,” Appl. Opt. 33, 7798–7804 (1994).
    [CrossRef] [PubMed]
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 82.
  3. G. M. Hale, M. R. Querry, “Optical constants of water in the 200 nm–200 μm wavelength region,” Appl. Opt. 12, 553–563 (1973).
    [CrossRef]
  4. K. S. Shifrin, V. A. Punina, “Of light phase function in a small angle interval,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 788–791 (1968).
  5. V. I. Burenkov, O. V. Kopelivich, K. S. Shifrin, “Light scattering by large particles with refractive indexes close to 1,” Atmos. Oceanic Phys. 11, 828–8356 (1975).
  6. A. L. Fymat, K. D. Mease, “Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion,” Appl. Opt. 20, 194–198 (1981).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1991) Chap. 1, p. 33.
  8. K. S. Shifrin, Scattering of Light in a Turbid Medium (Gostekh-teoretizdat, Moscow-Leningrad, 1951), Chap. 2, p. 25;NASA tech. transl. F-477 (NASA, Washington, D.C., 1968).
  9. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9, p. 144.

1994 (1)

1981 (1)

1975 (1)

V. I. Burenkov, O. V. Kopelivich, K. S. Shifrin, “Light scattering by large particles with refractive indexes close to 1,” Atmos. Oceanic Phys. 11, 828–8356 (1975).

1973 (1)

G. M. Hale, M. R. Querry, “Optical constants of water in the 200 nm–200 μm wavelength region,” Appl. Opt. 12, 553–563 (1973).
[CrossRef]

1968 (1)

K. S. Shifrin, V. A. Punina, “Of light phase function in a small angle interval,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 788–791 (1968).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 82.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1991) Chap. 1, p. 33.

Burenkov, V. I.

V. I. Burenkov, O. V. Kopelivich, K. S. Shifrin, “Light scattering by large particles with refractive indexes close to 1,” Atmos. Oceanic Phys. 11, 828–8356 (1975).

Fymat, A. L.

Hale, G. M.

G. M. Hale, M. R. Querry, “Optical constants of water in the 200 nm–200 μm wavelength region,” Appl. Opt. 12, 553–563 (1973).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 82.

Kopelivich, O. V.

V. I. Burenkov, O. V. Kopelivich, K. S. Shifrin, “Light scattering by large particles with refractive indexes close to 1,” Atmos. Oceanic Phys. 11, 828–8356 (1975).

Mease, K. D.

Punina, V. A.

K. S. Shifrin, V. A. Punina, “Of light phase function in a small angle interval,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 788–791 (1968).

Querry, M. R.

G. M. Hale, M. R. Querry, “Optical constants of water in the 200 nm–200 μm wavelength region,” Appl. Opt. 12, 553–563 (1973).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, I. G. Zolotov, “Quasi-stationary scattering of an electromagnetic pulse by a spherical particle,” Appl. Opt. 33, 7798–7804 (1994).
[CrossRef] [PubMed]

V. I. Burenkov, O. V. Kopelivich, K. S. Shifrin, “Light scattering by large particles with refractive indexes close to 1,” Atmos. Oceanic Phys. 11, 828–8356 (1975).

K. S. Shifrin, V. A. Punina, “Of light phase function in a small angle interval,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 788–791 (1968).

K. S. Shifrin, Scattering of Light in a Turbid Medium (Gostekh-teoretizdat, Moscow-Leningrad, 1951), Chap. 2, p. 25;NASA tech. transl. F-477 (NASA, Washington, D.C., 1968).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9, p. 144.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1991) Chap. 1, p. 33.

Zolotov, I. G.

Appl. Opt. (3)

Atmos. Oceanic Phys. (1)

V. I. Burenkov, O. V. Kopelivich, K. S. Shifrin, “Light scattering by large particles with refractive indexes close to 1,” Atmos. Oceanic Phys. 11, 828–8356 (1975).

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

K. S. Shifrin, V. A. Punina, “Of light phase function in a small angle interval,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 788–791 (1968).

Other (4)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, p. 82.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1991) Chap. 1, p. 33.

K. S. Shifrin, Scattering of Light in a Turbid Medium (Gostekh-teoretizdat, Moscow-Leningrad, 1951), Chap. 2, p. 25;NASA tech. transl. F-477 (NASA, Washington, D.C., 1968).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9, p. 144.

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Figures (12)

Fig. 1
Fig. 1

θ component of the envelope of the electric field of the scattered pulse (at T = 1) for a nonabsorbent particle with a radius of 0.01 μm as a function of τ for the scattering angles θ = 0°, 60°, 120°, and 180°: (a) real part Re(Esθ0), (b) imaginary part Im(Esθ0).

Fig. 2
Fig. 2

Imaginary part Im ( E s θ 0 ) of the θ component of the envelope of the scattered pulse (at T = 1) for a nonabsorbent particle with a radius of 5 μm as a function of τ for the scattering angles θ = 0°, 60°, 120°, and 180°: (a) overall picture (done on a small scale), (b) detailed picture (done on a large scale).

Fig. 3
Fig. 3

Real part Re ( E s θ 0 ) of the θ component of the envelope of the scattered pulse (at T = 1) for a nonabsorbent particle with a radius of 5 μm as a function of τ for the scattering angles θ = 0°, 60°, 120°, and 180°: (a) overall picture (done on a small scale), (b) detailed picture (done on a large scale).

Fig. 4
Fig. 4

Real part Re(Esθ0) of the θ component of the envelope of the scattered pulse (at T = 1) for nonabosrbent and absorbent particles with radius of 2 μm as a function of τ for the scattering angles θ = 0°, 60°, 120°, and 180°: (a) nonabsorbent particle (k = 0), (b) absorbent particle (k = 0.1).

Fig. 5
Fig. 5

Stationary part of the scattered-pulse intensity Iso (at T = 1) as a function of τ (at small τ) for the scattering angles θ = 0°, 1°, 2°, and 3° and for nonabsorbent particles with radii a of (a) 2 μm and (b) 5 μm.

Fig. 6
Fig. 6

Stationary part of the scattered-pulse intensity Iso (at T = 1) as a function of τ (for large τ) for the scattering angles θ = 0°, 1°, 2°, and 3° and for nonabsorbent particles with radii a of (a) 2 μm and (b) 5 μm.

Fig. 7
Fig. 7

Stationary part of the scattered-pulse intensity Iso (at T = 1) as a function of τ (for small τ) for the scattering angles θ = 2°, 4°, 6°, and 8° and for nonabsorbent particles with radii a of (a) 2 μm and (b) 5 μm.

Fig. 8
Fig. 8

Stationary part of the scattered-pulse intensity Iso (at T = 1) as a function of τ (for large τ) for the scattered angles θ = 0°, 60°, 120°, and 180° and for nonabsorbent particles with radii a of (a) 2 μm and (b) 5 μm.

Fig. 9
Fig. 9

Efficiencies of extinction Qext and scattering Qsca of a pulse (stationary parts) as functions of τ for a nonabsorbent particle with a radius of 0.01 μm. The pulse durations are T = 0.33 and 1.0.

Fig. 10
Fig. 10

Efficiencies of extinction Qext and scattering Qsca of a pulse (stationary parts) as functions of τ for a nonabsorbent particle with a radius of 1 μm. The pulse durations are T = 0.33, 1.0, and 5.0.

Fig. 11
Fig. 11

Efficiencies of extinction Qext and scattering Qsca of a pulse (stationary parts) as functions of τ for a nonabsorbent particle with a radius of 5 μm. The pulse durations are T = 0.33, 1.0, and 5.0.

Fig. 12
Fig. 12

Comparison of the stationary part and the envelope of the variable part of the efficiency of extinction Qext of a pulse for a particle with a radius of 0.01 μm and for pulse duration T = 0.33: (a) momentary values of the stationary part and the envelope of the variable part Qext, (b) values of the stationary part and the envelope of the variable part Qext at different integration times of the receiver, expressed in units of λ0/c.

Equations (13)

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E s ϕ = E 0 ( λ 0 r ) ( 1 2 π ) sin ϕ E s ϕ 0 ( τ ) exp ( i ω 0 T τ ) , E s θ = E 0 ( λ 0 r ) ( 1 2 π ) cos ϕ E s θ 0 ( τ ) exp ( i ω 0 T τ ) ,
E s ϕ 0 ( τ ) = i w ( s ) ( l 0 s + l 0 ) S 1 ( θ , s ) exp ( i 2 π τ s ) d s , E s ϕ 0 ( τ ) = i w ( s ) ( l 0 s + l 0 ) S 2 ( θ , s ) exp ( i 2 π τ s ) d s ,
τ = 1 T ( t r c + ψ ) .
P = c 8 π Re { E s × H s * + E s × H s } .
P r = c 8 π Re { ( E s θ H s ϕ * E s ϕ H s θ * ) + ( E s θ H s ϕ E s ϕ H s θ ) } .
I s ( τ , θ ) ~ | E 0 | 2 [ | S 1 ( θ , τ ) | 2 + | S 2 ( θ , τ ) | 2 ] Re { ( E 0 ) 2 exp ( i 2 ω 0 T τ ) { [ S 1 ( θ , τ ) ] 2 + [ S 2 ( θ , τ ) ] 2 } } ,
S 1 ( θ , τ ) = n 1 [ a n ( τ ) τ n ( θ ) + b n ( τ ) π n ( θ ) ] ,
S 2 ( θ , τ ) = n 1 [ a n ( τ ) π n ( θ ) + b n ( τ ) τ n ( θ ) ] ,
a n ( τ ) = w ( s ) [ l 0 / ( s + l 0 ) ] a n [ ω 0 ( 1 + s / l 0 ) ] × exp ( i 2 π τ s ) d s ,
b n ( τ ) = w ( s ) [ l 0 / ( s + l 0 ) ] b n [ w 0 ( 1 + s / l 0 ) ] × exp ( i 2 π τ s ) d s .
Q sca ( τ ) = 1 2 π 2 ( λ 0 a ) 2 [ n = 1 ( 2 n + 1 ) [ | a n ( τ ) | 2 + | b n ( τ ) | 2 ] Re { ( E 0 ) 2 | E 0 | 2 exp ( i 4 π l 0 τ ) n = 1 ( 2 n + 1 ) × { [ a n ( τ ) ] 2 + [ b n ( τ ) ] 2 } ] ,
Q ext ( τ ) = 1 2 π 2 ( λ 0 a ) 2 Re { { n = 1 ( 2 n + 1 ) [ a n ( τ ) + b n ( τ ) ] } × [ A ( τ ) ( E 0 ) 2 | E 0 | 2 exp ( i 4 π l 0 τ ) A * ( τ ) ] } ,
A ( τ ) = w ( s ) ( l 0 s + l 0 ) exp ( i 2 π τ s ) d s .

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