Abstract

An analysis is made of the vector properties of light scattered from small particles (the diameters are assumed comparable to a wavelength of the illumination) illuminated by two or more coherent beams. The results show that the polarization of the scattered light depends on both the polarization of the incident illumination and the scattered-light observation angle. Scattered light from orthogonally polarized illumination beams can interfere. The limits of applicability of current interference models of the scattering processes observed with laser interferometers used to determine Doppler shifts from moving particles are determined.

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  1. Y. Yeh and H. Z. Cummins, Appl. Phys. Lett. 4, 176 (1964).
    [CrossRef]
  2. R. J. Goldstein and D. K. Kried, J. Appl. Mech. 4, 813 (1967).
    [CrossRef]
  3. J. W. Dunning, Jr., Ph.D. thesis, Dept. of Chem. Engr., School of Engr., Case Western Reserve Univ., Cleveland, Ohio (1967) (University Microfilms, Ann Arbor, Mich., Order No. 68-10 153), p. 58.
  4. R. L. Bond, Ph.D. thesis, Univ. of Arkansas, Little Rock, Ark. (1968) (University Microfilms, Ann Arbor, Mich., Order No. 68-9620).
  5. D. B. Brayton and W. H. Goethert, Trans. Inst. Soc. Am. 10, 41 (1971).
  6. W. T. Mayo, J. Phys. E 3, 235 (1970).
    [CrossRef]
  7. F. Durst and J. H. Whitelaw, Proc. R. Soc. A 324, 157 (1971).
    [CrossRef]
  8. C. M. Penney, IEEE J. Quantum Electron. 5, 318 (1969).
    [CrossRef]
  9. M. J. Rudd, J. Phys. E 2, 55 (1969).
    [CrossRef]
  10. J. C. Owens, Appl. Opt. 11, 2977 (1972).
    [CrossRef] [PubMed]
  11. W. M. Farmer and D. B. Brayton, Appl. Opt. 11, 2978 (1972).
    [CrossRef] [PubMed]
  12. F. Durst, Ph.D. thesis, Heat Transfer Section, Dept. of Mech. Engr. Imperial College, London, S.W.I (1972), pp. 52–57.
  13. L. E. Drain, J. Phys. D 5, 481 (1972).
    [CrossRef]
  14. R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
    [CrossRef]
  15. C. P. Wang, Appl. Phys. Lett. 18, 522 (1971).
    [CrossRef]
  16. W. M. Farmer, Appl. Opt. 11, 2603 (1972).
    [CrossRef] [PubMed]
  17. W. P. Altman, Master's thesis, School of Elec. Engr., Univ. of Pennsylvania, Philadelphia, Pa. (1972).
  18. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969), p. 258.
  19. M. Kerker, The Scattering of Light and Other Electromagnetic Radiations (Academic, New York, 1969), pp. 39–50.
  20. D. Deirmendjian, Electromagnetic Scattering on Polydispersions (Elsevier, New York, 1969), pp. 12–55.
  21. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969), p. 636.
  22. Edward Collett, Am. J. Phys. 39, 1483 (1971).
    [CrossRef]
  23. H. H. Bossel, W. J. Hiller, and G. E. A Meier, J. Phys. E 5, 895 (1972).
  24. R. Dändliker and P. D. Iten, "Direction Sensitive Laser-Doppler-Velocimeter with Polarized Beams," presented at the 1973 IEEE/OSA Conference on Laser Applications and Engineering, 30 May–1 June 1973, Washington, D. C.
  25. D. B. Brayton, H. T. Kalb, and F. L. Crosswy, Appl. Opt. 12, 1145 (1973).
    [CrossRef] [PubMed]

1973 (1)

D. B. Brayton, H. T. Kalb, and F. L. Crosswy, Appl. Opt. 12, 1145 (1973).
[CrossRef] [PubMed]

1972 (7)

H. H. Bossel, W. J. Hiller, and G. E. A Meier, J. Phys. E 5, 895 (1972).

W. M. Farmer, Appl. Opt. 11, 2603 (1972).
[CrossRef] [PubMed]

W. P. Altman, Master's thesis, School of Elec. Engr., Univ. of Pennsylvania, Philadelphia, Pa. (1972).

J. C. Owens, Appl. Opt. 11, 2977 (1972).
[CrossRef] [PubMed]

W. M. Farmer and D. B. Brayton, Appl. Opt. 11, 2978 (1972).
[CrossRef] [PubMed]

F. Durst, Ph.D. thesis, Heat Transfer Section, Dept. of Mech. Engr. Imperial College, London, S.W.I (1972), pp. 52–57.

L. E. Drain, J. Phys. D 5, 481 (1972).
[CrossRef]

1971 (5)

R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
[CrossRef]

C. P. Wang, Appl. Phys. Lett. 18, 522 (1971).
[CrossRef]

Edward Collett, Am. J. Phys. 39, 1483 (1971).
[CrossRef]

D. B. Brayton and W. H. Goethert, Trans. Inst. Soc. Am. 10, 41 (1971).

F. Durst and J. H. Whitelaw, Proc. R. Soc. A 324, 157 (1971).
[CrossRef]

1970 (1)

W. T. Mayo, J. Phys. E 3, 235 (1970).
[CrossRef]

1969 (2)

C. M. Penney, IEEE J. Quantum Electron. 5, 318 (1969).
[CrossRef]

M. J. Rudd, J. Phys. E 2, 55 (1969).
[CrossRef]

1968 (1)

R. L. Bond, Ph.D. thesis, Univ. of Arkansas, Little Rock, Ark. (1968) (University Microfilms, Ann Arbor, Mich., Order No. 68-9620).

1967 (2)

R. J. Goldstein and D. K. Kried, J. Appl. Mech. 4, 813 (1967).
[CrossRef]

J. W. Dunning, Jr., Ph.D. thesis, Dept. of Chem. Engr., School of Engr., Case Western Reserve Univ., Cleveland, Ohio (1967) (University Microfilms, Ann Arbor, Mich., Order No. 68-10 153), p. 58.

1964 (1)

Y. Yeh and H. Z. Cummins, Appl. Phys. Lett. 4, 176 (1964).
[CrossRef]

Adrian, R. J.

R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
[CrossRef]

Altman, W. P.

W. P. Altman, Master's thesis, School of Elec. Engr., Univ. of Pennsylvania, Philadelphia, Pa. (1972).

Bond, R. L.

R. L. Bond, Ph.D. thesis, Univ. of Arkansas, Little Rock, Ark. (1968) (University Microfilms, Ann Arbor, Mich., Order No. 68-9620).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969), p. 636.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969), p. 258.

Bossel, H. H.

H. H. Bossel, W. J. Hiller, and G. E. A Meier, J. Phys. E 5, 895 (1972).

Brayton, D. B.

D. B. Brayton, H. T. Kalb, and F. L. Crosswy, Appl. Opt. 12, 1145 (1973).
[CrossRef] [PubMed]

W. M. Farmer and D. B. Brayton, Appl. Opt. 11, 2978 (1972).
[CrossRef] [PubMed]

D. B. Brayton and W. H. Goethert, Trans. Inst. Soc. Am. 10, 41 (1971).

Collett, Edward

Edward Collett, Am. J. Phys. 39, 1483 (1971).
[CrossRef]

Crosswy, F. L.

D. B. Brayton, H. T. Kalb, and F. L. Crosswy, Appl. Opt. 12, 1145 (1973).
[CrossRef] [PubMed]

Cummins, H. Z.

Y. Yeh and H. Z. Cummins, Appl. Phys. Lett. 4, 176 (1964).
[CrossRef]

Dändliker, R.

R. Dändliker and P. D. Iten, "Direction Sensitive Laser-Doppler-Velocimeter with Polarized Beams," presented at the 1973 IEEE/OSA Conference on Laser Applications and Engineering, 30 May–1 June 1973, Washington, D. C.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Polydispersions (Elsevier, New York, 1969), pp. 12–55.

Drain, L. E.

L. E. Drain, J. Phys. D 5, 481 (1972).
[CrossRef]

Dunning, Jr., J. W.

J. W. Dunning, Jr., Ph.D. thesis, Dept. of Chem. Engr., School of Engr., Case Western Reserve Univ., Cleveland, Ohio (1967) (University Microfilms, Ann Arbor, Mich., Order No. 68-10 153), p. 58.

Durst, F.

F. Durst, Ph.D. thesis, Heat Transfer Section, Dept. of Mech. Engr. Imperial College, London, S.W.I (1972), pp. 52–57.

F. Durst and J. H. Whitelaw, Proc. R. Soc. A 324, 157 (1971).
[CrossRef]

Farmer, W. M.

W. M. Farmer and D. B. Brayton, Appl. Opt. 11, 2978 (1972).
[CrossRef] [PubMed]

W. M. Farmer, Appl. Opt. 11, 2603 (1972).
[CrossRef] [PubMed]

Goethert, W. H.

D. B. Brayton and W. H. Goethert, Trans. Inst. Soc. Am. 10, 41 (1971).

Goldstein, R. J.

R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
[CrossRef]

R. J. Goldstein and D. K. Kried, J. Appl. Mech. 4, 813 (1967).
[CrossRef]

Hiller, W. J.

H. H. Bossel, W. J. Hiller, and G. E. A Meier, J. Phys. E 5, 895 (1972).

Iten, P. D.

R. Dändliker and P. D. Iten, "Direction Sensitive Laser-Doppler-Velocimeter with Polarized Beams," presented at the 1973 IEEE/OSA Conference on Laser Applications and Engineering, 30 May–1 June 1973, Washington, D. C.

Kalb, H. T.

D. B. Brayton, H. T. Kalb, and F. L. Crosswy, Appl. Opt. 12, 1145 (1973).
[CrossRef] [PubMed]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiations (Academic, New York, 1969), pp. 39–50.

Kried, D. K.

R. J. Goldstein and D. K. Kried, J. Appl. Mech. 4, 813 (1967).
[CrossRef]

Mayo, W. T.

W. T. Mayo, J. Phys. E 3, 235 (1970).
[CrossRef]

Meier, G. E. A

H. H. Bossel, W. J. Hiller, and G. E. A Meier, J. Phys. E 5, 895 (1972).

Owens, J. C.

J. C. Owens, Appl. Opt. 11, 2977 (1972).
[CrossRef] [PubMed]

Penney, C. M.

C. M. Penney, IEEE J. Quantum Electron. 5, 318 (1969).
[CrossRef]

Rudd, M. J.

M. J. Rudd, J. Phys. E 2, 55 (1969).
[CrossRef]

Wang, C. P.

C. P. Wang, Appl. Phys. Lett. 18, 522 (1971).
[CrossRef]

Whitelaw, J. H.

F. Durst and J. H. Whitelaw, Proc. R. Soc. A 324, 157 (1971).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969), p. 636.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969), p. 258.

Yeh, Y.

Y. Yeh and H. Z. Cummins, Appl. Phys. Lett. 4, 176 (1964).
[CrossRef]

Other (25)

Y. Yeh and H. Z. Cummins, Appl. Phys. Lett. 4, 176 (1964).
[CrossRef]

R. J. Goldstein and D. K. Kried, J. Appl. Mech. 4, 813 (1967).
[CrossRef]

J. W. Dunning, Jr., Ph.D. thesis, Dept. of Chem. Engr., School of Engr., Case Western Reserve Univ., Cleveland, Ohio (1967) (University Microfilms, Ann Arbor, Mich., Order No. 68-10 153), p. 58.

R. L. Bond, Ph.D. thesis, Univ. of Arkansas, Little Rock, Ark. (1968) (University Microfilms, Ann Arbor, Mich., Order No. 68-9620).

D. B. Brayton and W. H. Goethert, Trans. Inst. Soc. Am. 10, 41 (1971).

W. T. Mayo, J. Phys. E 3, 235 (1970).
[CrossRef]

F. Durst and J. H. Whitelaw, Proc. R. Soc. A 324, 157 (1971).
[CrossRef]

C. M. Penney, IEEE J. Quantum Electron. 5, 318 (1969).
[CrossRef]

M. J. Rudd, J. Phys. E 2, 55 (1969).
[CrossRef]

J. C. Owens, Appl. Opt. 11, 2977 (1972).
[CrossRef] [PubMed]

W. M. Farmer and D. B. Brayton, Appl. Opt. 11, 2978 (1972).
[CrossRef] [PubMed]

F. Durst, Ph.D. thesis, Heat Transfer Section, Dept. of Mech. Engr. Imperial College, London, S.W.I (1972), pp. 52–57.

L. E. Drain, J. Phys. D 5, 481 (1972).
[CrossRef]

R. J. Adrian and R. J. Goldstein, J. Phys. E 4, 505 (1971).
[CrossRef]

C. P. Wang, Appl. Phys. Lett. 18, 522 (1971).
[CrossRef]

W. M. Farmer, Appl. Opt. 11, 2603 (1972).
[CrossRef] [PubMed]

W. P. Altman, Master's thesis, School of Elec. Engr., Univ. of Pennsylvania, Philadelphia, Pa. (1972).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969), p. 258.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiations (Academic, New York, 1969), pp. 39–50.

D. Deirmendjian, Electromagnetic Scattering on Polydispersions (Elsevier, New York, 1969), pp. 12–55.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1969), p. 636.

Edward Collett, Am. J. Phys. 39, 1483 (1971).
[CrossRef]

H. H. Bossel, W. J. Hiller, and G. E. A Meier, J. Phys. E 5, 895 (1972).

R. Dändliker and P. D. Iten, "Direction Sensitive Laser-Doppler-Velocimeter with Polarized Beams," presented at the 1973 IEEE/OSA Conference on Laser Applications and Engineering, 30 May–1 June 1973, Washington, D. C.

D. B. Brayton, H. T. Kalb, and F. L. Crosswy, Appl. Opt. 12, 1145 (1973).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Example of a multibeam interferometer arrangement and the different observational modes. A laser (A) produces a beam that is split into two equal beams by a dielectric-coated glass prism. A lens causes the beams to focus where they cross (P) to form the probe volume. The dual-scatter observation mode results from observing only scattered light with a lens C and a photon detector E, which generates a current that is sent to a signal processor D. The local-oscillator observation mode (F) results from mixing unscattered light with an unscattered beam at a photon detector, which sends a current to a signal processor.

Fig. 2
Fig. 2

Example of the interference of scattered spherical waves from an array of particles illuminated by two plane waves. A is the spherical surface defined by the radius drawn from the point where the illumination-beam center lines intersect (called the geometric center). B is the center line of a possible illuminating interference fringe. C is the geometric center of the probe volume.

Fig. 3
Fig. 3

Scattering geometry of a spherical particle for two illumination beams.

Fig. 4
Fig. 4

Example of a one-beam scattering interferometer. A laser, A, produces a beam that is focused to some region P. A lens collects and collimates the light scattered from P. An aperture mask at C passes two angularly separated portions of the scattered wave. The apertured wave fronts are folded together and sent to the photon detector at B.

Tables (1)

Tables Icon

Table I Values of the scattered electric-field components for different illumination-beam angles and polarization values.

Equations (47)

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S = c 8 π Re ( E H * ) ,
E = η N i P E s i η + E LO .
H = ρ N j P ( K s i ρ E s j ρ ) Z K s + K LO E LO Z K LO ,
A ( B C ) = B ( A · C ) C ( A · B ) ,
S = ( c / 8 π K s Z ) Re { K LO | E LO | 2 + ρ N j P [ K s j ρ ( E LO · E s j ρ * ) E s j ρ * ( K s j ρ · E LO ) ] + η N i P [ ( K LO ( E s i η · E LO * ) E LO * ) E LO * ( K LO · E s i η ) ] + η ρ i j [ K s j ρ ( E s i η · E s j ρ * ) E s j ρ * ( K s j ρ · E η ) ] } .
S = S LO + S s ,
S s = ( c / 8 π K s Z ) Re { η N ρ N i P j P [ K s j ρ ( E s i η · E s j ρ * ) E s j ρ * ( K s j ρ · E s i η ) ] } ,
S LO = ( c / 8 π K s Z ) × Re { η N i P [ K LO ( E s i η · E LO * ) E LO * ( K LO · E s i η ) ] + ρ N j P [ K s j ρ ( E LO · E s j ρ * ) E s j ρ * ( K s j ρ · E LO ) ] } .
S s I = ( c / 8 π Z K s ) Re { η N i P [ K s i η | E s i η | 2 E s i η * ( K s i η · E s i η ) ] } ;
S s II = ( c / 8 π Z K s ) × Re { η N i > j P [ ( K s i η + K s j η ) ( E s i η · E s j η * ) E s i η * ( K s i η · E s j η ) } ;
S s III = ( c / 8 π Z K s ) Re { η > ρ N i P [ ( K s i η + K s i ρ ) × ( E s i η · E s i ρ * ) E s i η * ( K s i η · E s i ρ ) ] } ;
S s IV = ( c / 8 π Z K s ) Re { η > ρ N i > j P [ ( K s i η + K s j ρ ) × ( E s i η · E s j ρ ) E s i η * ( K s i η · E s j ρ ) ] } ;
( R , K s max ) λ 4 b 0 2 .
I = e r · S ,
I = e r · S s + e r · S LO ,
I = I s + I LO ,
I s = e r · S s ,
I LO = e r · S LO .
I s = ( c / 8 π Z ) { η N i P | E s i η | 2 + 2 η N i > j P Re ( E s i η · E s j η * ) + 2 η > ρ N i P Re ( E s i η · E s i ρ * ) + 2 η > ρ N i > j P Re ( E s i η · E s j ρ * ) } ,
I LO = ( c / 8 π Z ) { | E LO | 2 + ρ N j P Re ( E LO · E s j ρ * ) + η N i P Re [ cos ( R , K LO ) ( E si η · E LO * ) sin ( R , K LO ) | E LO * | ( E s i η · K LO / K s ) ] } .
I LO = c / 8 π Z { | E LO | 2 + 2 η N i P Re ( E LO · E s i η * ) } .
I LO = ( c / 8 π Z ) { | E LO | 2 + Re η N i P { E LO · ( E s i η * + cos ( R , K LO ) E s i η * ) sin ( R , K LO ) | E LO * | ( K LO K s · E s i η ) ] } .
I s = ( c / 8 π Z ) [ | E s 1 | 2 + | E s 2 | 2 + 2 Re ( E s 1 · E s 2 * ) ] .
e ˆ ( r ) = ( e r e θ e ϕ ) .
e ˆ ( x ) = ( e x e y e z ) .
e ˆ ( r ) = M ˆ e ˆ ( x ) .
M ˆ = ( sin θ cos ϕ sin θ sin ϕ cos θ cos θ cos ϕ cos θ sin ϕ sin θ sin ϕ cos ϕ 0 ) .
e ˆ ( x i ) = T ˆ i e ˆ ( x ) .
e ˆ ( x 1 ) = T ˆ e ˆ ( x ) ,
e ˆ ( x 2 ) = T ˆ e ˆ ( x ) ,
T ˆ = ( 1 0 0 0 cos α / 2 sin α / 2 0 sin α / 2 cos α / 2 ) .
e ˆ ( r 1 ) = M ˆ 1 e ˆ ( x i ) ,
e ˆ ( r 2 ) = M ˆ 2 e ˆ ( x 2 ) ,
e ˆ ( x ) = M ˆ ˜ e ˆ ( r ) ,
e ˆ ( x 1 ) = T ˆ M ˆ ˜ e ˆ ( r ) ,
e ˆ ( x 2 ) = T ˆ ˜ M ˆ ˜ e ˆ ( r ) ,
e ˆ ( r 1 ) = M ˆ 1 T ˆ M ˆ ˜ e ˆ ( r ) ,
e ˆ ( r 2 ) = M ˆ 2 T ˆ M ˆ ˜ e ˆ ( r ) ,
e ˆ ( r 1 ) = M ˆ 1 T ˆ 2 M ˆ ˜ 2 e ˆ ( r 2 ) .
B ˆ = M ˆ 1 T ˆ 2 M ˆ ˜ 2 .
B ˆ = [ B 11 B 12 B 21 B 22 ] ,
B 11 = cos θ 1 cos θ 2 cos ϕ 1 cos ϕ 2 + cos θ 1 sin ϕ 1 [ cos θ 2 sin ϕ 2 cos α + sin θ 2 sin α ] sin θ 1 [ cos θ 2 sin ϕ 2 sin α sin θ 2 cos α ] ,
B 12 = sin ϕ 2 cos ϕ 1 cos θ 1 + cos θ 1 sin ϕ 1 cos ϕ 2 cos α sin θ 1 cos ϕ 2 sin α ,
B 21 = cos θ 2 sin ϕ 1 cos ϕ 2 + cos ϕ 1 ( cos θ 2 sin ϕ 2 cos α + sin θ 2 sin α ) ,
B 22 = sin ϕ 2 sin ϕ 1 + cos ϕ 1 cos ϕ 2 cos α .
E s 1 · E s 2 B 11 E s θ 1 E s θ 2 + B 22 E s ϕ 1 E s ϕ 2 + B 12 E s ϕ 1 E s θ 2 + B 21 E s θ 1 E s ϕ 2 .
E s 1 · E s 2 * = E s ϕ 1 E s ϕ 2 * .