Abstract

A method for determining particle size, number density, and velocity utilizing a laser interferometer is analyzed. The results show that when the fringe spacing is comparable to a particle diameter, size can be estimated; and when the fringe spacing is much greater than the average particle diameter, number density can be measured. Since the optical arrangment for the interferometer is identical to that for a number of laser velocimeters, the effects of particle size on the velocimeter signal are discussed.

© 1972 Optical Society of America

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References

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  1. C. M. Penny, IEEE J. Quantum Electron. QE-5, 318 (1968).
  2. D. B. Brayton, W. H. Goethert, Trans. Instrum. Soc. Am. 10, 41 (1971).
  3. L. Lading, Appl. Opt. 10, 1943 (1972).
    [CrossRef]
  4. A. E. Lennert, D. B. Brayton, F. L. Crosswy et al., AEDC-TR-70-101.
  5. F. H. Smith, J. A. Parsons, AEDC-TR-70-119.
  6. J. W. Dunning, “Application of Laser Homodyne Spectrometer to Particle Size Measurements,” Ph.D. Thesis (School of Engineering, Case Western Reserve University, Cleveland, Ohio, 1967).
  7. D. G. Andrews, H. S. Seifert, “Investigation of Particle-Size Determination from the Optical Response of a Laser Doppler Velocimeter,” SUDAAR No. 435, November1971.
  8. M. J. Rudd, J. Phys. E 2, 55 (1969).
    [CrossRef]
  9. M. Kerker, The Scattering of Light (Academic Press, New York, 1969), p. 46.
  10. D. B. Brayton, W. M. Farmer, “An Analysis of the Probe Volume in a Dual Scatter Laser Doppler Velocimeter,” to be submitted to Applied Optics.
  11. H. Kogelnik, Bell Syst. Tech. J. 44, 468 (1965).
  12. R. S. Adrian, R. S. Goldstein, J. Phys. E 4, 505 (1971).
    [CrossRef]
  13. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1965), p. 488.
  14. Ref. 13, p. 683.
  15. W. M. Farmer, D. B. Brayton, Appl. Opt. 10, 2319 (1971).
    [CrossRef] [PubMed]
  16. C. P. Wang, Appl. Phys. Lett. 18, 522 (1971).
    [CrossRef]
  17. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 633.

1972 (1)

1971 (5)

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

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

W. M. Farmer, D. B. Brayton, Appl. Opt. 10, 2319 (1971).
[CrossRef] [PubMed]

D. G. Andrews, H. S. Seifert, “Investigation of Particle-Size Determination from the Optical Response of a Laser Doppler Velocimeter,” SUDAAR No. 435, November1971.

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

1969 (1)

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

1968 (1)

C. M. Penny, IEEE J. Quantum Electron. QE-5, 318 (1968).

1965 (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 468 (1965).

Adrian, R. S.

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

Andrews, D. G.

D. G. Andrews, H. S. Seifert, “Investigation of Particle-Size Determination from the Optical Response of a Laser Doppler Velocimeter,” SUDAAR No. 435, November1971.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 633.

Brayton, D. B.

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

W. M. Farmer, D. B. Brayton, Appl. Opt. 10, 2319 (1971).
[CrossRef] [PubMed]

A. E. Lennert, D. B. Brayton, F. L. Crosswy et al., AEDC-TR-70-101.

D. B. Brayton, W. M. Farmer, “An Analysis of the Probe Volume in a Dual Scatter Laser Doppler Velocimeter,” to be submitted to Applied Optics.

Crosswy, F. L.

A. E. Lennert, D. B. Brayton, F. L. Crosswy et al., AEDC-TR-70-101.

Dunning, J. W.

J. W. Dunning, “Application of Laser Homodyne Spectrometer to Particle Size Measurements,” Ph.D. Thesis (School of Engineering, Case Western Reserve University, Cleveland, Ohio, 1967).

Farmer, W. M.

W. M. Farmer, D. B. Brayton, Appl. Opt. 10, 2319 (1971).
[CrossRef] [PubMed]

D. B. Brayton, W. M. Farmer, “An Analysis of the Probe Volume in a Dual Scatter Laser Doppler Velocimeter,” to be submitted to Applied Optics.

Goethert, W. H.

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

Goldstein, R. S.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1965), p. 488.

Kerker, M.

M. Kerker, The Scattering of Light (Academic Press, New York, 1969), p. 46.

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 44, 468 (1965).

Lading, L.

Lennert, A. E.

A. E. Lennert, D. B. Brayton, F. L. Crosswy et al., AEDC-TR-70-101.

Parsons, J. A.

F. H. Smith, J. A. Parsons, AEDC-TR-70-119.

Penny, C. M.

C. M. Penny, IEEE J. Quantum Electron. QE-5, 318 (1968).

Rudd, M. J.

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

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1965), p. 488.

Seifert, H. S.

D. G. Andrews, H. S. Seifert, “Investigation of Particle-Size Determination from the Optical Response of a Laser Doppler Velocimeter,” SUDAAR No. 435, November1971.

Smith, F. H.

F. H. Smith, J. A. Parsons, AEDC-TR-70-119.

Wang, C. P.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 633.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

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

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 468 (1965).

IEEE J. Quantum Electron. (1)

C. M. Penny, IEEE J. Quantum Electron. QE-5, 318 (1968).

J. Phys. E (2)

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

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

SUDAAR No. 435 (1)

D. G. Andrews, H. S. Seifert, “Investigation of Particle-Size Determination from the Optical Response of a Laser Doppler Velocimeter,” SUDAAR No. 435, November1971.

Trans. Instrum. Soc. Am. (1)

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

Other (8)

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), p. 633.

M. Kerker, The Scattering of Light (Academic Press, New York, 1969), p. 46.

D. B. Brayton, W. M. Farmer, “An Analysis of the Probe Volume in a Dual Scatter Laser Doppler Velocimeter,” to be submitted to Applied Optics.

A. E. Lennert, D. B. Brayton, F. L. Crosswy et al., AEDC-TR-70-101.

F. H. Smith, J. A. Parsons, AEDC-TR-70-119.

J. W. Dunning, “Application of Laser Homodyne Spectrometer to Particle Size Measurements,” Ph.D. Thesis (School of Engineering, Case Western Reserve University, Cleveland, Ohio, 1967).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1965), p. 488.

Ref. 13, p. 683.

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Figures (6)

Fig. 1
Fig. 1

Typical optical arrangement for generating a set of well-defined interference fringes.

Fig. 2
Fig. 2

Scattering geometry for determination of Mie cross-section dependence of Es in a two-beam interference fringe system.

Fig. 3
Fig. 3

Intensity distribution in plane of crossed beams (assumes beams of equal intensity and polarization vectors linear and parallel).

Fig. 4
Fig. 4

Phase dependence of the signal on size of a spherical particle.

Fig. 5
Fig. 5

Fringe visibility vs particle size for a sphere and cylinder.

Fig. 6
Fig. 6

Shapes of the probe volume for different signal constraints.

Equations (77)

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S = ( c / 8 π ) Re ( E H * ) ,
sin ( α ) α ,
cos ( α ) 1.
E = E i ( 1 ) + E i ( 2 ) + E s ( 1 ) + E s ( 2 ) ,
H = K i ( 1 ) E i ( 1 ) / K + K i ( 2 ) E i ( 2 ) / K + K s ( 1 ) E s ( 1 ) / K s + K s ( 2 ) E s ( 2 ) / K s ,
I s = | c 8 π Re ( E s H s * ) | , I s = c 8 π { E s ( 1 ) 2 + E s ( 2 ) 2 + 2 Re [ E s * ( 1 ) · E s ( 2 ) ] } .
E s ( 1 ) = [ i exp ( i K s r ) / K s r ] { cos ( ϕ ) S 2 [ cos ( θ ) ] e θ + sin ( ϕ ) S 1 [ cos ( θ ) ] e ϕ } E i ( 1 ) ,
E s ( 2 ) = [ i exp ( i K s r ) / K s r ] { cos ( ϕ ) S 2 [ cos ( θ ) ] e θ + sin ( ϕ ) S 1 [ cos ( θ ) ] e ϕ } E i ( 2 ) ,
I s c { cos 2 ( ϕ ) S 2 [ cos ( θ ) ] 2 + sin 2 ( ϕ ) S 1 [ cos ( θ ) ] 2 } { E i ( 1 ) 2 + E i ( 2 ) 2 + 2 Re [ E i * ( 1 ) E i ( 2 ) ] } / 8 π K s 2 r 2 ,
I 0 = 2 I exp { ( - 2 / b 0 2 ) [ X 2 + Y 2 + ( Z 2 α 2 / 4 ) ] } × [ cosh ( 2 Y Z α / b 0 2 ) + cos ( K α Y - g ) ] ,
b 0 = 2 λ F # / π ,
I ¯ 0 = 1 A p A p I 0 d A p ,
I s = c 8 π K s 2 r 2 { cos 2 ( ϕ ) S 2 [ cos ( θ ) ] 2 + sin 2 ( ϕ ) S 1 [ cos ( θ ) ] 2 } I ¯ 0 .
V = ( I s max - I s max ) / ( I s max + I s max ) ,
V = A p exp [ - 2 b 0 2 ( X 2 + Y 2 ) ] cos ( 2 π Y α λ ) d A p / { cosh ( Z λ 2 b 0 2 ) × A p exp [ - 2 b 0 2 ( X 2 + Y 2 ) ] cosh [ 2 Z α b 0 2 ( Y ± λ 4 α ) ] d A p } .
V = P s 1 / P G 1 ,
X = X 0 - X ,
Y = Y 0 - Y ,
r 2 = X 2 + Y 2 .
V = ( A p exp { - 2 b 0 2 [ r 2 - 2 ( X X 0 + Y Y 0 ) ] } × cos [ 2 π δ ( Y 0 - Y ) ] d A p ) / ( cosh ( m δ b 0 ) A p exp { - 2 b 0 2 × [ r 2 - 2 ( X X 0 + Y Y 0 ) ] } cosh [ 4 m b 0 ( Y 0 - Y ± δ 4 ) ] d A p ) ,
δ = λ / α ,
Z = 2 m b 0 / α .
Y 0 = r 0 cos ( β ) ,
X 0 = r 0 sin ( β ) ,
Y = r cos ( ψ ) ,
X = r sin ( ψ ) .
V ( 0 2 π 0 a exp [ 4 r 0 r b 0 2 cos ( ψ - β ) ] cos { 2 π δ [ r 0 cos ( β ) - r cos ( ψ ) ] } r d r d ψ ) / ( cosh ( δ m b 0 ) 0 2 π 0 a exp [ 4 r 0 r b 0 2 × cos ( ψ - β ) ] cosh { 4 m b 0 [ r 0 cos ( β ) - r cos ( ψ ) ] } r d r d ψ ) .
V [ exp ( i 2 π Y 0 / δ ) I 1 ( ζ a ) / ζ + exp ( - i 2 π Y 0 / δ ) I 1 ( ζ * a ) / ζ * ] / cosh ( δ m b 0 ) { exp [ 4 m b 0 ( Y 0 ± δ 4 ) ] I 1 ( φ + a ) / φ + + exp [ - 4 m b 0 ( Y 0 ± δ 4 ) ] I 1 ( φ - a ) / φ - } ,
ζ = [ ( 16 r 0 2 / b 0 4 ) + ( i 16 π Y 0 / b 0 2 δ ) - ( 4 π / δ 2 ) ] 1 2
φ ± = [ ( 16 r 0 2 / b 0 4 ) ( 32 m Y 0 / b 0 3 ) + ( 16 m 2 / b 0 2 ) ] 1 2 .
P 1 = P s 1 + P G 1 ,
P 1 = P G 1 ( 1 + V ) .
V 2 J 1 ( 2 π a / δ ) / ( 2 π a / δ ) ,
V sin ( 2 π a / δ ) / ( 2 π a / δ ) ,
V ¯ n = P ¯ s n / P ¯ G n ,
P ¯ s n = n P ¯ s 1 ,
P ¯ G n = n P ¯ G 1 .
V ¯ n = V ¯ / n ,
S / N = ( q λ / h c Δ f ) { 1 / [ 1 + P B G / P ¯ G n ) ] } [ ( P ¯ s 1 ) 2 / P ¯ G 1 ] ,
S / N = ( q λ / h c Δ f ) { 1 / [ 1 + P B G / P ¯ G n ) ] } P ¯ G 1 V ¯ 2 ,
E s ( 1 ) = [ i exp ( i K s r ) / K s r ] { cos ( ϕ 1 ) S 2 [ cos ( θ 1 ) ] e θ 1 + sin ( ϕ 1 ) S 1 [ cos ( θ 1 ) ] e ϕ 1 } E i ( 1 ) ,
E s ( 2 ) = [ i exp ( i K s r ) / K s r ] { cos ( ϕ 2 ) S 2 [ cos ( θ 2 ) ] e θ 2 + sin ( ϕ 2 ) S 1 [ cos ( θ 2 ) ] e ϕ 2 } E i ( 2 ) ,
K s ( 1 ) = K s ( 2 ) = K s .
cos ( θ 1 ) = cos ( θ ) cos ( α / 2 ) - sin ( θ ) sin ( ϕ ) sin ( α / 2 ) ,
cos ( θ 2 ) = cos ( θ ) cos ( α / 2 ) + sin ( θ ) sin ( ϕ ) sin ( α / 2 ) .
e ^ ( r , θ , ϕ ) = M ^ e ^ ( X , Y , Z ) ,
M ^ = [ 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 ^ ( X 2 , Y 2 , Z 2 ) = T ^ e ^ ( X , Y , Z ) .
T ^ = [ 1 0 0 0 cos ( α / 2 ) - sin ( α / 2 ) 0 sin ( α / 2 ) cos ( α / 2 ) ] .
e ^ ( X 1 , Y 1 , Z 1 ) = T ^ ˜ ˜ e ^ ( X , Y , Z ) ,
e ^ ( X 1 , Y 1 , Z 1 ) = T ^ ˜ M ^ ˜ e ^ ( r , θ , ϕ ) ,
e ^ ( X 2 , Y 2 , Z 2 ) = T ^ M ^ ˜ e ^ ( r , θ , ϕ ) .
e ^ ( r 1 , θ 1 , ϕ 1 ) = M ^ 1 e ^ ( X 1 , Y 1 , Z 1 ) ,
e ^ ( r 2 , θ 2 , ϕ 2 ) = M ^ e ^ 2 ( X 2 , Y 2 , Z 2 ) ,
e ^ ( r 1 , θ 1 , ϕ 1 ) = B ^ e ^ ( r , θ , ϕ ) ,
e ^ ( r 2 , θ 2 , ϕ 2 ) = B ^ ˜ e ( r , θ , ϕ ) ,
B ^ = M ^ T ^ ˜ M ^ ˜ .
E ^ s = { cos ( ϕ ) S 2 [ cos ( θ ) ] sin ( ϕ ) S 1 [ cos ( θ ) ] } .
E ^ s ( 1 ) = B ^ E ^ s 1 [ i exp ( i K s r ) / K s r ] E i ( 1 ) ,
E ^ s ( 2 ) = B ^ E ^ s 1 [ i exp ( i K s r ) / K s r ] E i ( 2 ) .
S = c 8 π K s Re ( [ E s ( 1 ) 2 K s ( 1 ) + E s ( 2 ) 2 K s ( 2 ) ] + K s ( 1 ) [ E s * ( 1 ) · E s ( 2 ) ] + K s ( 2 ) [ E s * ( 2 ) · E s ( 1 ) ] × { E s * ( 2 ) [ K s ( 2 ) · E s ( 1 ) ] + E s * ( 1 ) [ K s ( 1 ) · E s ( 2 ) ] } ) .
[ K s ( 1 ) / K s ] [ E s * ( 2 ) · E s ( 1 ) ] + [ K s ( 2 ) / K s [ E s * ( 1 ) · E s ( 2 ) ] = 2 Re [ E ^ s + B ^ B ^ E ^ s E i * ( 1 ) E i ( 2 ) ] [ B ^ e ^ ( r , θ , ϕ ) / K s 2 r 2 ] ,
E s ( 1 ) 2 [ K s ( 1 ) / K s ] + E s ( 2 ) 2 [ K s ( 2 ) / K s ] = [ E ^ s 2 E i ( 1 ) 2 B ^ + E ^ s 2 E i ( 2 ) 2 B ^ ˜ ] [ e ^ ( r , θ , ϕ ) / K s 2 r 2 ] .
S = ( c / 8 π K s 2 r 2 ) { E ^ s 2 E i ( 1 ) 2 B ^ + E ^ s 2 E i ( 2 ) 2 B ^ ˜ + 2 Re [ E ^ s + B ^ B ^ E ^ s E i * ( 1 ) E i ( 2 ) ] B ^ } e ^ ( r , θ , ϕ ) .
S = ( c / 8 π K s 2 r 2 ) ( E ^ s 2 [ E i ( 1 ) 2 + E i ( 2 ) 2 ] + 2 { cos 2 ( ϕ ) S 2 [ cos ( θ ) ] 2 b 22 + sin 2 ϕ ( S 1 [ cos ( θ ) ] 2 b 33 } Re [ E i * ( 1 ) E i ( 2 ) ] ) B ^ e ^ ( r , θ , ϕ ) ,
b 11 = 2 sin 2 ( θ ) cos 2 ( ϕ ) cos 2 ( α / 2 ) + cos 2 ( θ ) cos ( α ) ,
b 22 = cos 2 ( θ ) [ cos 2 ( ϕ ) + cos ( α ) sin 2 ( ϕ ) ] + cos ( α ) sin 2 ( θ ) ,
b 33 = sin 2 ( ϕ ) + cos ( α ) cos 2 ( ϕ ) .
I s = S , I s = ( c E ^ s 2 / 8 π K s 2 r 2 ) { E i ( 1 ) 2 + E i ( 2 ) 2 + 2 Re [ E i * ( 1 ) E i ( 2 ) ] } ,
V = ( I ¯ 0 max - I ¯ 0 min ) / ( I ¯ 0 max + I ¯ 0 min ) .
I 0 max = 2 I exp ( - 2 m 2 ) exp [ - 2 ( X 2 + Y 2 ) / b 0 2 ] × [ cosh ( 4 Y m / b 0 ) + cos ( 2 π Y / δ ) ] .
I 0 max = 2 I exp ( - 2 m 2 ) exp { - 2 [ X 2 + ( Y ± δ / 2 ) ] 2 / b 0 2 } × { cosh [ 4 m / b 0 ) ( Y ± δ / 2 ) ] - cos ( 2 π Y / δ ) } .
I 0 max - I 0 min 2 I exp ( - 2 m 2 ) exp [ - 2 ( X 2 + Y 2 ) / b 0 2 ] × { cosh ( 4 Y m / b 0 ) - cosh [ ( 4 m / b 0 ) ( Y ± δ / 2 ) ] + 2 cos ( 2 π Y / δ ) } .
I 0 max - I 0 min 2 I exp ( - 2 m 2 ) exp [ - 2 ( X 2 + Y 2 ) / b 0 2 ] × { cosh ( 4 m Y / b 0 ) + cosh [ ( 4 m / b 0 ) ( Y ± δ / 2 ) ] } .
cosh ( 4 m Y / b 0 ) + cosh [ ( 4 m / b 0 ) ( Y ± δ / 2 ) ] = 2 cosh [ ( 4 m / b 0 ) ( Y ± δ / 4 ) ] cosh ( m δ / b 0 ) ,
cosh ( 4 m Y / b 0 ) - cosh [ ( 4 m / b 0 ) ( Y ± δ / 2 ) ] = 2 sinh [ ( 4 m / b 0 ) ( Y ± δ / 4 ) ] sinh ( m δ / b 0 ) 0 ( m δ b 0 ) .
V A p exp [ - 2 ( X 2 + Y 2 ) b 0 2 ] cos ( 2 π Y δ ) d A p / cosh ( m δ b 0 ) A p exp [ - 2 ( X 2 + Y 2 ) b 0 2 ] cosh [ 4 m b 0 ( Y ± δ 4 ) ] d A p ,

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