Abstract

A general light-scattering spectrometer for measuring surface fluctuations is analyzed by using the methods of Fourier optics. Optical configurations can be designed so that the autocorrelation of the photocurrent is given by either the first- or the second-order correlation functions for the surface fluctuations. Fluctuations in capillary-wave amplitude and in surface density can be detected separately for a number of cases of practical interest. A method is proposed for unambiguous detection of capillary waves with amplitudes larger than the optical wavelength.

© 1989 Optical Society of America

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  1. J. A. Mann, “Dynamics, structure and function of interfacial regions,” Langmuir 1, 10–23 (1985).
    [CrossRef]
  2. L. T. Mandel’shtam, “Roughness of free liquid surfaces,” Ann. Phys. Leipzig 41, 609–624 (1918).
  3. D. Langevin, J. Meunier, “Light scattering by liquid interfaces,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977), p. 501.
  4. S. Hard, R. D. Neuman, “Laser light scattering measurements of viscoelastic monomolecular films,”J. Colloid Interface Sci. 83, 315–334 (1981).
    [CrossRef]
  5. R. V. Edwards, R. S. Sirohi, J. A. Mann, L. B. Shih, L. Lading, “Surface fluctuation scattering using grating heterodyne spectroscopy,” Appl. Opt. 21, 3555–3568 (1982).
    [CrossRef] [PubMed]
  6. D. Langevin, “Light-scattering study of monolayer viscoelasticity,”J. Colloid Interface Sci. 80, 412–425 (1980).
    [CrossRef]
  7. T. E. Furtak, J. Reyes, “Critical analysis of theoretical models for the giant Raman effect from adsorbed molecules,” Surf. Sci. 91, 351–382 (1980).
    [CrossRef]
  8. L. B. Shih, “Surface fluctuation spectroscopy: a novel technique for characterizing liquid interfaces,” Rev. Sci. Instrum. 55, 716–726 (1984); J. A. Mann, R. V. Edwards, “Surface fluctuation spectroscopy: comments on experimental technique and capillary ripple theory,” Rev. Sci. Instrum. 55, 727–732 (1984).
    [CrossRef]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  10. M. J. Rudd, “A new theoretical model for the laser Dopplermeter,”J. Phys. E 2, 55–58 (1969).
    [CrossRef]
  11. L. Lading, “A Fourier optical model for the laser Doppler velocimeter,” Opto-electronics 4, 385–398 (1972).
    [CrossRef]
  12. H. T. Davis, L. E. Scriven, Adv. Chem. Phys. 49, 357–454 (1982).
    [CrossRef]
  13. J. Meunier, “Liquid interfaces: role of the fluctuations and analysis of ellipsometry and reflectivity measurements,”J. Phys. 48, 1887–1931 (1987).
  14. A. Papoulis, Probabilistic Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 481.
  15. A. J. F. Siegert, “Passages of stationary processes through linear and non-linear devices,”IRE Trans. Inf. Theory TT-3, 4–25 (1954).
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 424–428.
  17. R. Hoffman, L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169–1176 (1975).
    [CrossRef] [PubMed]
  18. R. A. Sprague, B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469–1479 (1972).
    [CrossRef] [PubMed]
  19. See, e.g., pp. 42–46 in T. E. Bell, “Optical computing: a field in flux,” Proc. IEEE Spectrum 23 (8), 34–38 (1986). (This is a review paper with a subsection on spatial light modulators.)
  20. P. Hariharan, Optical Holography (Cambridge U. Press, Cambridge, 1984), Chap. 4.
  21. S. Hanson, “Broadening of the measured frequency spectrum in a differential laser anemometer due to interference plane gradients,”J. Phys. D 6, 164–171 (1973).
    [CrossRef]
  22. A. W. Adamson, Physical Chemistry of Surfaces, 4th ed. (Wiley, New York, 1982).
  23. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 374–378.

1987 (1)

J. Meunier, “Liquid interfaces: role of the fluctuations and analysis of ellipsometry and reflectivity measurements,”J. Phys. 48, 1887–1931 (1987).

1986 (1)

See, e.g., pp. 42–46 in T. E. Bell, “Optical computing: a field in flux,” Proc. IEEE Spectrum 23 (8), 34–38 (1986). (This is a review paper with a subsection on spatial light modulators.)

1985 (1)

J. A. Mann, “Dynamics, structure and function of interfacial regions,” Langmuir 1, 10–23 (1985).
[CrossRef]

1984 (1)

L. B. Shih, “Surface fluctuation spectroscopy: a novel technique for characterizing liquid interfaces,” Rev. Sci. Instrum. 55, 716–726 (1984); J. A. Mann, R. V. Edwards, “Surface fluctuation spectroscopy: comments on experimental technique and capillary ripple theory,” Rev. Sci. Instrum. 55, 727–732 (1984).
[CrossRef]

1982 (2)

1981 (1)

S. Hard, R. D. Neuman, “Laser light scattering measurements of viscoelastic monomolecular films,”J. Colloid Interface Sci. 83, 315–334 (1981).
[CrossRef]

1980 (2)

D. Langevin, “Light-scattering study of monolayer viscoelasticity,”J. Colloid Interface Sci. 80, 412–425 (1980).
[CrossRef]

T. E. Furtak, J. Reyes, “Critical analysis of theoretical models for the giant Raman effect from adsorbed molecules,” Surf. Sci. 91, 351–382 (1980).
[CrossRef]

1975 (1)

1973 (1)

S. Hanson, “Broadening of the measured frequency spectrum in a differential laser anemometer due to interference plane gradients,”J. Phys. D 6, 164–171 (1973).
[CrossRef]

1972 (2)

R. A. Sprague, B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469–1479 (1972).
[CrossRef] [PubMed]

L. Lading, “A Fourier optical model for the laser Doppler velocimeter,” Opto-electronics 4, 385–398 (1972).
[CrossRef]

1969 (1)

M. J. Rudd, “A new theoretical model for the laser Dopplermeter,”J. Phys. E 2, 55–58 (1969).
[CrossRef]

1954 (1)

A. J. F. Siegert, “Passages of stationary processes through linear and non-linear devices,”IRE Trans. Inf. Theory TT-3, 4–25 (1954).

1918 (1)

L. T. Mandel’shtam, “Roughness of free liquid surfaces,” Ann. Phys. Leipzig 41, 609–624 (1918).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 374–378.

Adamson, A. W.

A. W. Adamson, Physical Chemistry of Surfaces, 4th ed. (Wiley, New York, 1982).

Bell, T. E.

See, e.g., pp. 42–46 in T. E. Bell, “Optical computing: a field in flux,” Proc. IEEE Spectrum 23 (8), 34–38 (1986). (This is a review paper with a subsection on spatial light modulators.)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 424–428.

Davis, H. T.

H. T. Davis, L. E. Scriven, Adv. Chem. Phys. 49, 357–454 (1982).
[CrossRef]

Edwards, R. V.

Furtak, T. E.

T. E. Furtak, J. Reyes, “Critical analysis of theoretical models for the giant Raman effect from adsorbed molecules,” Surf. Sci. 91, 351–382 (1980).
[CrossRef]

Goodman, J. W.

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

Gross, L.

Hanson, S.

S. Hanson, “Broadening of the measured frequency spectrum in a differential laser anemometer due to interference plane gradients,”J. Phys. D 6, 164–171 (1973).
[CrossRef]

Hard, S.

S. Hard, R. D. Neuman, “Laser light scattering measurements of viscoelastic monomolecular films,”J. Colloid Interface Sci. 83, 315–334 (1981).
[CrossRef]

Hariharan, P.

P. Hariharan, Optical Holography (Cambridge U. Press, Cambridge, 1984), Chap. 4.

Hoffman, R.

Lading, L.

Langevin, D.

D. Langevin, “Light-scattering study of monolayer viscoelasticity,”J. Colloid Interface Sci. 80, 412–425 (1980).
[CrossRef]

D. Langevin, J. Meunier, “Light scattering by liquid interfaces,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977), p. 501.

Mandel’shtam, L. T.

L. T. Mandel’shtam, “Roughness of free liquid surfaces,” Ann. Phys. Leipzig 41, 609–624 (1918).

Mann, J. A.

Meunier, J.

J. Meunier, “Liquid interfaces: role of the fluctuations and analysis of ellipsometry and reflectivity measurements,”J. Phys. 48, 1887–1931 (1987).

D. Langevin, J. Meunier, “Light scattering by liquid interfaces,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977), p. 501.

Neuman, R. D.

S. Hard, R. D. Neuman, “Laser light scattering measurements of viscoelastic monomolecular films,”J. Colloid Interface Sci. 83, 315–334 (1981).
[CrossRef]

Papoulis, A.

A. Papoulis, Probabilistic Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 481.

Reyes, J.

T. E. Furtak, J. Reyes, “Critical analysis of theoretical models for the giant Raman effect from adsorbed molecules,” Surf. Sci. 91, 351–382 (1980).
[CrossRef]

Rudd, M. J.

M. J. Rudd, “A new theoretical model for the laser Dopplermeter,”J. Phys. E 2, 55–58 (1969).
[CrossRef]

Scriven, L. E.

H. T. Davis, L. E. Scriven, Adv. Chem. Phys. 49, 357–454 (1982).
[CrossRef]

Shih, L. B.

L. B. Shih, “Surface fluctuation spectroscopy: a novel technique for characterizing liquid interfaces,” Rev. Sci. Instrum. 55, 716–726 (1984); J. A. Mann, R. V. Edwards, “Surface fluctuation spectroscopy: comments on experimental technique and capillary ripple theory,” Rev. Sci. Instrum. 55, 727–732 (1984).
[CrossRef]

R. V. Edwards, R. S. Sirohi, J. A. Mann, L. B. Shih, L. Lading, “Surface fluctuation scattering using grating heterodyne spectroscopy,” Appl. Opt. 21, 3555–3568 (1982).
[CrossRef] [PubMed]

Siegert, A. J. F.

A. J. F. Siegert, “Passages of stationary processes through linear and non-linear devices,”IRE Trans. Inf. Theory TT-3, 4–25 (1954).

Sirohi, R. S.

Sprague, R. A.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 374–378.

Thompson, B. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 424–428.

Adv. Chem. Phys. (1)

H. T. Davis, L. E. Scriven, Adv. Chem. Phys. 49, 357–454 (1982).
[CrossRef]

Ann. Phys. Leipzig (1)

L. T. Mandel’shtam, “Roughness of free liquid surfaces,” Ann. Phys. Leipzig 41, 609–624 (1918).

Appl. Opt. (3)

IRE Trans. Inf. Theory (1)

A. J. F. Siegert, “Passages of stationary processes through linear and non-linear devices,”IRE Trans. Inf. Theory TT-3, 4–25 (1954).

J. Colloid Interface Sci. (2)

D. Langevin, “Light-scattering study of monolayer viscoelasticity,”J. Colloid Interface Sci. 80, 412–425 (1980).
[CrossRef]

S. Hard, R. D. Neuman, “Laser light scattering measurements of viscoelastic monomolecular films,”J. Colloid Interface Sci. 83, 315–334 (1981).
[CrossRef]

J. Phys. (1)

J. Meunier, “Liquid interfaces: role of the fluctuations and analysis of ellipsometry and reflectivity measurements,”J. Phys. 48, 1887–1931 (1987).

J. Phys. D (1)

S. Hanson, “Broadening of the measured frequency spectrum in a differential laser anemometer due to interference plane gradients,”J. Phys. D 6, 164–171 (1973).
[CrossRef]

J. Phys. E (1)

M. J. Rudd, “A new theoretical model for the laser Dopplermeter,”J. Phys. E 2, 55–58 (1969).
[CrossRef]

Langmuir (1)

J. A. Mann, “Dynamics, structure and function of interfacial regions,” Langmuir 1, 10–23 (1985).
[CrossRef]

Opto-electronics (1)

L. Lading, “A Fourier optical model for the laser Doppler velocimeter,” Opto-electronics 4, 385–398 (1972).
[CrossRef]

Proc. IEEE Spectrum (1)

See, e.g., pp. 42–46 in T. E. Bell, “Optical computing: a field in flux,” Proc. IEEE Spectrum 23 (8), 34–38 (1986). (This is a review paper with a subsection on spatial light modulators.)

Rev. Sci. Instrum. (1)

L. B. Shih, “Surface fluctuation spectroscopy: a novel technique for characterizing liquid interfaces,” Rev. Sci. Instrum. 55, 716–726 (1984); J. A. Mann, R. V. Edwards, “Surface fluctuation spectroscopy: comments on experimental technique and capillary ripple theory,” Rev. Sci. Instrum. 55, 727–732 (1984).
[CrossRef]

Surf. Sci. (1)

T. E. Furtak, J. Reyes, “Critical analysis of theoretical models for the giant Raman effect from adsorbed molecules,” Surf. Sci. 91, 351–382 (1980).
[CrossRef]

Other (7)

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

D. Langevin, J. Meunier, “Light scattering by liquid interfaces,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977), p. 501.

P. Hariharan, Optical Holography (Cambridge U. Press, Cambridge, 1984), Chap. 4.

A. Papoulis, Probabilistic Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 481.

A. W. Adamson, Physical Chemistry of Surfaces, 4th ed. (Wiley, New York, 1982).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 374–378.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 424–428.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of the surface-scattering configuration. The beam from the laser is spatially filtered and expanded to the desired illuminated area on the surface. The grating is imaged on the surface. The filters H1 and H2 determine the operational mode of the system. Transmitter light and receiver light are separated by the polarizing beam splitter (POL B.S.) in conjunction with the quarter-wave plate (λ/4) (efficient only if the surface does not change the polarization state of the light). The illuminated spot on the surface is imaged onto the pinhole, which may help to reduce parasitic light. If this is unnecessary, the detector can be placed directly after the filter H2. PMT, Photomultiplier tube; BEAM EXP., beam expander; AMP., amplifier; PREAMP, preamplifier.

Fig. 2
Fig. 2

Schematic representation of the optical path abstracted from Fig. 1 for analysis. The surface is shown here as a transmission element (rather than as a reflection element) in order to separate the transmitter and the receiver. This has no effect on the analysis. The appropriate field functions are shown just above the corresponding elements. Recall that u = u0g.

Fig. 3
Fig. 3

Attenuation of the filter to be placed in the plane H2 in order to obtain the derivative in the x direction of the phase. kx is proportional to the x coordinate in the plane H2.

Fig. 4
Fig. 4

Configuration for measuring capillary waves with large amplitudes. The filter is specified in Fig. 3. The weights, bi, correspond to a spatial sinusoid of the desired wave number.

Fig. 5
Fig. 5

A correllogram of heptane pressurized to 3010 psi (~20.7 Pa) with helium. The 400-point correllogram is shown fitted with a damped cosine function. The insert displays the tail of the correllogram so that the mismatch between the fit and the data can be seen. The instrument function cannot be ignored in this case. An accurate fit of Ri was obtained by using formula (8) of Ref. 5.

Equations (51)

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u 0 g ,
s = - s 0 ( 1 + η ) e j ϕ ,
ϕ ( x , y ) = 2 k λ ( x , y ) ,
u d = [ u s 0 ( 1 + η + j ϕ ) ] * h 2 ,
i = C det . area u d 2 d A ,
i 1 = C 1 det . plane u * h 2 2 d A ,
i 2 = C 1 det . plane u [ ( η + j ϕ ) ] * h 2 2 d A ,
i 3 = 2 C 2 det . plane Re ( ( u * h 2 ) { [ ( η + j ϕ ) u ] * h 2 } * ) ,
ϕ ( r , t ) = i = 0 a i ( t ) cos { K i · [ r - r i ( t ) ] } ,
u = ( 1 / 2 ) ( e j K x + e j K x ) = cos K x .
H 2 = δ ( k x , k y ) ,
R K R ( 2 ) K ( τ ) ( 1 + cos 2 ω K τ ) ,
R ( 2 ) K ( τ ) a K 2 ( ω K ) a K 2 ( t + τ ) ¯ .
R ( 2 ) K ( τ ) = R ( 1 ) K 2 ( 0 ) + 2 × R ( 1 ) K 2 ( τ ) .
R 2 ( τ ) = | S ϕ ( k , 0 ) U ( k ) 2 d k | 2 + 2 | S ϕ ( k , τ ) U ( k ) 2 d k | 2 ,
g = 1 + g 1 ,
U 0 H 2 = 0 ,
u 0 * h 2 = 0
u * h 2 = [ u 0 ( 1 + g 1 ) ] * h 2 = u 0 g 1 .
i 3 = C 1 Re { u 0 2 [ g 1 ( η - j ϕ ) ] } .
R 3 ( τ ) = 4 C 1 π 2 S ϕ ( k , τ ) F i ( k ) 2 d k ,
f i ( r ) u 0 2 ( r ) g 1 ( r ) .
R 3 ( τ ) = 2 C 1 2 π 2 R ( 1 ) K ( τ ) cos ω K τ .
H 2 = 1 - α k x ,
u ip = u 0 e j ϕ - j α x ( u 0 e j ϕ ) ,
u ip = u 0 e j ϕ ( 1 - α ϕ x ) .
u ip 2 = u 0 2 ( 1 - 2 α ϕ x ) .
R ( τ ) = 4 C 1 π 2 k x 2 S ϕ ( k , τ ) F i ( k ) 2 d k
K s = K 0 1 1 + ( z r 0 2 k λ ) 2 ,
i 3 = 2 j C 1 ϕ u 0 2 g 1 d r .
R ( t , t ) = i 3 ( t ) i 3 ( t ) = 4 C 1 × ϕ ( r , t ) u 0 2 ( r ) g 1 ( r ) d r ϕ ( r , t ) u 0 2 ( r ) g 1 ( r ) d r .
R 3 ( τ ) = 4 C 1 R ϕ ( r , τ ) f i ( r - r ) f i ( r ) d r d r ,
f i ( r ) = u 0 2 ( r ) g 1 ( r ) .
f ( r - r ) f ( r ) d r = 1 π 2 F i ( k ) 2 exp ( - j k · r ) d k ,
R 3 ( τ ) = 4 C 1 π 2 R ϕ ( r , τ ) F i ( k ) 2 exp ( - j k · r ) d r d k = 4 C 1 π 2 S ϕ ( k , τ ) F i ( k ) 2 d k ,
i 2 = C 1 u ( ϕ ) * h 2 2 d r .
i 2 ( t 1 ) i 2 ( t 2 ) u ( r ) ϕ ( r , t 1 ) 2 d r u ( r ) ϕ ( r , t 2 ) 2 d r ,
R 2 ( t 1 , t 2 ) = u ( r 1 ) ϕ ( r 1 , t 1 ) d r 1 × u ( r 2 ) ϕ ( r 2 , t 1 ) d r 2 × u ( r 3 ) ϕ ( r 3 , t 2 ) d r 3 × u ( r 4 ) ϕ ( r 4 , t 2 ) d r 4 .
ϕ 1 ϕ 2 ϕ 3 ϕ 4 = ϕ 1 ϕ 2 ϕ 3 ϕ 4 + ϕ 1 ϕ 3 ϕ 2 ϕ 4 + ϕ 1 ϕ 4 ϕ 2 ϕ 3 ,
R 2 ( τ ) = | S ϕ ( k , 0 ) U ( k ) 2 d k | 2 + | S ϕ ( k , τ ) U ( k ) 2 d k | 2 ,
u ip 2 = u 0 2 + α 2 u 0 2 ( ϕ x ) 2 - 2 α u 0 ϕ x + α 2 | u 0 x | 2 .
u 0 = exp [ - 1 2 ( r r 0 ) 2 ] ,             ϕ = ϕ 0 cos K x .
α 2 ϕ 0 2 K 2 ( second term ) 2 α ϕ 0 K ( third term ) , ( α / r 0 ) 2 ( fourth term ) .
K r 0 > 1             and             K α < 1 ;             thus r 0 α .
exp [ j ϕ i cos ( K i x ) ] ,
s = s r * i δ ( r - r i ) ,
s r = exp [ - 1 2 r 2 ( 1 r s 2 + 1 j r ϕ 2 ) ]
Δ k = [ 1 + ( r s / r ϕ ) 4 ] 1 / 2 r s .
Δ ϕ r s < k λ             or that             | ϕ r | < k λ .
u = exp [ - 1 2 r 2 r 0 2 + j ( z 1 / k λ ) ] ( e j K x + e - j K x ) ,
K s = K 0 r 0 4 + ( z 1 / k λ ) ( z 1 / k λ + z 2 / k λ ) r 0 4 + ( z 1 / k λ + z 2 / k λ ) 2 .

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