Abstract

The theory of photon correlation spectroscopy for a flowing system of diffusing particles was extended to include the Gaussian shape properties of an incident light beam. We found that the characteristic time of the coherent translational term caused by the bulk flow of the system was independent of the distance between the scattering volume and the focal point of the incident beam. The flow velocity and beam radius at the focus determined the characteristic time of this term. We also studied the effect of defocusing on the amplitude of the incoherent or number density term of the homodyne-detected intensity autocorrelation function. We verified our results experimentally using a flowing dioctylphthalate aerosol.

© 1986 Optical Society of America

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References

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  1. W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy-I. Theory and Experimental Result for Homogeneous Aerosols,” Aerosol Sci. 3, 501 (1972).
    [CrossRef]
  2. G. B. King, C. M. Sorensen, T. W. Lester, J. F. Merklin, “Photon Correlation Spectroscopy Used as a Particle Size Diagnostic in Sooting Flames,” Appl. Opt. 21, 976 (1982).
    [CrossRef] [PubMed]
  3. W. L. Flower, “Optical Measurements of Soot Formation in Premixed Flames,” Combust. Sci. Technol. 33, 17 (1983).
    [CrossRef]
  4. S. M. Scrivner, T. W. Taylor, C. M. Sorensen, J. F. Merklin, “Soot Particle Size Distribution Measurements in a Premixed Flame Using Photon Correlation Spectroscopy,” Appl. Opt. 25, 291 (1986).
    [CrossRef] [PubMed]
  5. D. P. Chowdhury, C. M. Sorensen, T. W. Taylor, J. F. Merklin, T. W. Lester, “Application of Photon Correlation Spectroscopy to Flowing Brownian Motion Systems,” Appl. Opt. 23, 4149 (1984).
    [CrossRef] [PubMed]
  6. R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Dopper Flowmeter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
    [CrossRef]
  7. H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 485 (1965).
  8. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  9. L. E. Estes, L. M. Narducci, R. A. Tuft, “Scattering of Light from a Rotating Ground Glass,” J. Opt. Soc. Am. 61, 1301 (1971).
    [CrossRef]

1986 (1)

1984 (1)

1983 (1)

W. L. Flower, “Optical Measurements of Soot Formation in Premixed Flames,” Combust. Sci. Technol. 33, 17 (1983).
[CrossRef]

1982 (1)

1972 (2)

W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy-I. Theory and Experimental Result for Homogeneous Aerosols,” Aerosol Sci. 3, 501 (1972).
[CrossRef]

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Dopper Flowmeter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[CrossRef]

1971 (1)

1965 (1)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 485 (1965).

Angus, J. C.

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Dopper Flowmeter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[CrossRef]

Berne, B. J.

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

Chowdhury, D. P.

Dunning, J. W.

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Dopper Flowmeter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[CrossRef]

Edwards, R. V.

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Dopper Flowmeter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[CrossRef]

Estes, L. E.

Flower, W. L.

W. L. Flower, “Optical Measurements of Soot Formation in Premixed Flames,” Combust. Sci. Technol. 33, 17 (1983).
[CrossRef]

French, M. J.

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Dopper Flowmeter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[CrossRef]

Hinds, W.

W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy-I. Theory and Experimental Result for Homogeneous Aerosols,” Aerosol Sci. 3, 501 (1972).
[CrossRef]

King, G. B.

Kogelnik, H.

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 485 (1965).

Lester, T. W.

Merklin, J. F.

Narducci, L. M.

Pecora, R.

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

Reist, P. C.

W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy-I. Theory and Experimental Result for Homogeneous Aerosols,” Aerosol Sci. 3, 501 (1972).
[CrossRef]

Scrivner, S. M.

Sorensen, C. M.

Taylor, T. W.

Tuft, R. A.

Aerosol Sci. (1)

W. Hinds, P. C. Reist, “Aerosol Measurement by Laser Doppler Spectroscopy-I. Theory and Experimental Result for Homogeneous Aerosols,” Aerosol Sci. 3, 501 (1972).
[CrossRef]

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 485 (1965).

Combust. Sci. Technol. (1)

W. L. Flower, “Optical Measurements of Soot Formation in Premixed Flames,” Combust. Sci. Technol. 33, 17 (1983).
[CrossRef]

J. Appl. Phys. (1)

R. V. Edwards, J. C. Angus, M. J. French, J. W. Dunning, “Spectral Analysis of the Signal from the Laser Dopper Flowmeter: Time-Independent Systems,” J. Appl. Phys. 42, 837 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Propagation of a Gaussian beam through a lens. Definition of beam radii and distances.

Fig. 2
Fig. 2

Characteristic time τ2 vs z, the distance between the scattering volume and the focal point of the incident beam, for light focused by two different lenses. The solid lines are theoretical predictions.

Fig. 3
Fig. 3

Characteristic time τ3 vs z, the distance between the scattering volume and the focal point of the incident beam, for light focused by two different lenses. The solid lines are theoretical predictions.

Fig. 4
Fig. 4

Inverse square root of the amplitude of the incoherent term relative to the background (A/B)−1/2 vs z, the distance between the scattering volume and focal point of the incident beam, for light focused by two different lenses. The solid lines are least-squares fits to the data.

Equations (35)

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I 2 ( t ) = N 2 [ 1 + exp ( - 2 D q 2 t ) exp ( - v 2 t 2 / w 2 ) + 2 - 3 / 2 N - 1 exp ( - v 2 t 2 / w 2 ) ] .
P ( r x y ) = E 0 exp ( - r x y 2 / w 2 - i k r x y 2 / 2 R ) ,
P ( r ) = E 0 exp ( r 2 / w 2 - i k r x y 2 / 2 R ) .
1 w 1 2 = ( 1 - z 0 / f ) 2 + ( π w 0 / λ f ) 2 .
( z 1 - f ) = ( z 0 - f ) f 2 ( z 0 - f ) 2 + ( π w 0 2 / λ ) 2 .
w 2 = w 1 2 [ 1 + ( λ z / π w 1 2 ) 2 ] ,
R = z [ 1 + ( π w 1 2 / λ z ) 2 ] .
w R = λ π w 1 [ 1 + ( π w 1 2 / λ z ) 2 ] - 1 / 2 .
E s ( t ) = j = 1 N exp ( i q · r j ) P ( r j ) ,
r j ( t ) = v t + r j ( t ) .
I 1 ( t ) = E s ( t ) E s * ( 0 ) ,
I 2 ( t ) = E s ( t ) 2 E s ( 0 ) 2 ,
I 1 ( t ) = exp ( i q · v t ) j , k exp [ i q · δ r j k ( t ) δ r × P [ r j ( t ) ] P * [ r k ( o ) ] r ,
exp ( i q · δ r ( t ) ) δ r = exp ( - D q 2 t ) .
E 0 2 / V T d 3 r exp [ - ( r + v t 2 ) / w 2 - i k ( r x y + v t ) 2 / 2 R ] × exp [ - r 2 / w 2 + i k r x y 2 / 2 R ] = E 0 2 V s V T exp [ - v 2 t 2 2 w 2 ( 1 + k 2 w 4 / 4 R 2 ) ] ,
I 1 = I 0 N exp ( i q · v t ) exp ( - D q 2 t ) exp [ - v 2 t 2 2 w 2 ( 1 + k 2 w 4 / 4 R 2 ) ] .
exp ( - v 2 t 2 / 2 w 2 ) exp ( - v 2 t 2 k 2 w 2 / 8 R 2 ) .
exp ( - v 2 t 2 / 2 w 2 ) exp { - v 2 t 2 2 w 1 2 [ 1 + ( π w 1 2 / λ z ) 2 ] - 1 } = exp ( - v 2 t 2 / 2 w 2 ) ,
1 w 2 = 1 w 2 + 1 w 1 2 [ 1 + ( π w 1 2 / λ z ) 2 ] .
w = w 1 ,
I 1 = I 0 N exp ( i q · v t ) exp ( - D q 2 t ) exp ( - v 2 t 2 / 2 w 1 2 ) .
j k l m P [ r j ( t ) ] P * [ r k ( t ) ] P [ r l ( o ) ] P * [ r m ( o ) ] .
I 0 2 V T 2 j l N d 3 r 3 exp [ - 2 ( r j + v t ) 2 / w 2 ] d 3 r l exp ( - 2 r l 2 / w 2 ) = I 0 2 j l ( V s / V T ) 2 .
I 0 2 Σ ( V s / V T ) 2 = I 0 2 N s ( N s - 1 ) ,
I 0 2 / V T j N d 3 r j exp [ - 2 ( r j + v t ) 2 / w 2 ] exp ( - 2 r j 2 / w 2 ) = I 0 2 N 2 - 3 / 2 exp ( - v 2 t 2 / w 2 ) .
j l P [ r j ( t ) ] P * [ r j ( o ) ] r j P * [ r l ( t ) ] P [ r l o ) ] r l ,
j l P [ r j ( t ) ] P * [ r j ( o ) ] 2 r j = j l I 0 2 V s 2 V T 2 exp ( - v 2 t 2 / w 1 2 ) ,
I 0 2 N 2 exp ( - v 2 t 2 / w 1 2 ) .
I 2 ( t ) = N 2 [ 1 + exp ( - 2 D q 2 t ) exp ( - v 2 t 2 / w 1 2 ) + 2 - 3 / 2 N - 1 exp ( - v 2 t 2 / w 2 ) ] .
V s w 2 = w 1 2 [ 1 + ( λ z π w 1 2 ) 2 ] .
N z 2 .
C ( t ) = B + C exp ( - n Δ t τ 1 ) exp ( - n 2 Δ t 2 τ 2 2 ) + A exp ( - n 2 Δ t 2 τ 3 2 ) .
Y ( n Δ t ) = - 1 n Δ t ln [ C ( t ) - B C ]
Y ( n Δ t ) = 1 / τ 1 + 1 τ 2 2 n Δ t .
C ( t ) = B + A exp ( - n 2 Δ t 2 / τ 3 ) .

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