Abstract

The interference signal formed by combining two coherent light beams carries information on the path difference between the beams. When the path difference is a periodic function of time, as, for example, when one beam is reflected from a vibrating surface and the other from a fixed surface, the interference signal is periodic with the same period as the vibrating surface. Bessel functions provide an elegant and efficient means for deconvoluting such periodic interference signals, thus making it possible to obtain the displacement of the moving surface with nanometer resolution. Here we describe the mathematical basis for the signal deconvolution and employ this technique to obtain the amplitude of miniature capillary waves on water as a test case.

© 2006 Optical Society of America

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References

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  1. T. M. Bohanon, J. M. Mikrut, B. M. Abraham, J. B. Ketterson, and P. Datta, "Fiber-optic detection system for capillary waves: an apparatus for studying liquid surfaces and spread monolayers," Rev. Sci. Instrum. 62, 2959-2962 (1991).
    [CrossRef]
  2. See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1958).
  3. J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy (Wiley-VCH, 2002).
  4. J. Markham and J. A. Conchello, "Numerical evaluation of Hankel transforms for oscillating functions," J. Opt. Soc. Am. A 20, 621-630 (2003).
    [CrossRef]
  5. J. Alda, C. Fumeaue, I. Codreanu, J. A. Schaefer, and D. Boreman, "Deconvolution method for two-dimensional spatial-response mapping of lithographic infrared antennas," Appl. Opt. 38, 3993-4000 (1999).
    [CrossRef]
  6. See, for example, A. Runnemalm, N. Molin, and E. Jansson, "On operating deflection shapes of the violin body including in-plane motions," J. Acoust. Soc. Am. 107, 3452-3459 (2000).
    [CrossRef] [PubMed]
  7. F. Behroozi, "Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy," Eur. J. Phys. 25, 115-122 (2004).
    [CrossRef]
  8. F. Behroozi, B. Lambert, and B. Buhrow, "Noninvasive measurement of viscosity from damping of capillary waves," ISA Trans. 42, 3-8 (2003).
    [CrossRef] [PubMed]
  9. F. Behroozi, "A miniature laser interferometer for noninvasive viscometry," presented at the Technical Proceedings of Nanotech 2003 Conference, San Francisco, Calif., February 24-26, 2003, Vol. 1, pp. 166-169.
  10. F. Behroozi, "Apparatus and method for measurement of fluid viscosity," U.S. patent 6,563,588 B2 (May 13, 2003).
  11. F. Behroozi, B. Lambert, and B. Buhrow, "Direct measurement of the attenuation of capillary waves by laser interferometry: noncontact determination of viscosity," Appl. Phys. Lett. 78, 2399-2401 (2001).
    [CrossRef]
  12. M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions, 10th printing (U.S. Government Printing Office, 1972) p. 361.

2004 (1)

F. Behroozi, "Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy," Eur. J. Phys. 25, 115-122 (2004).
[CrossRef]

2003 (2)

F. Behroozi, B. Lambert, and B. Buhrow, "Noninvasive measurement of viscosity from damping of capillary waves," ISA Trans. 42, 3-8 (2003).
[CrossRef] [PubMed]

J. Markham and J. A. Conchello, "Numerical evaluation of Hankel transforms for oscillating functions," J. Opt. Soc. Am. A 20, 621-630 (2003).
[CrossRef]

2001 (1)

F. Behroozi, B. Lambert, and B. Buhrow, "Direct measurement of the attenuation of capillary waves by laser interferometry: noncontact determination of viscosity," Appl. Phys. Lett. 78, 2399-2401 (2001).
[CrossRef]

2000 (1)

See, for example, A. Runnemalm, N. Molin, and E. Jansson, "On operating deflection shapes of the violin body including in-plane motions," J. Acoust. Soc. Am. 107, 3452-3459 (2000).
[CrossRef] [PubMed]

1999 (1)

1991 (1)

T. M. Bohanon, J. M. Mikrut, B. M. Abraham, J. B. Ketterson, and P. Datta, "Fiber-optic detection system for capillary waves: an apparatus for studying liquid surfaces and spread monolayers," Rev. Sci. Instrum. 62, 2959-2962 (1991).
[CrossRef]

Abraham, B. M.

T. M. Bohanon, J. M. Mikrut, B. M. Abraham, J. B. Ketterson, and P. Datta, "Fiber-optic detection system for capillary waves: an apparatus for studying liquid surfaces and spread monolayers," Rev. Sci. Instrum. 62, 2959-2962 (1991).
[CrossRef]

Alda, J.

Behroozi, F.

F. Behroozi, "Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy," Eur. J. Phys. 25, 115-122 (2004).
[CrossRef]

F. Behroozi, B. Lambert, and B. Buhrow, "Noninvasive measurement of viscosity from damping of capillary waves," ISA Trans. 42, 3-8 (2003).
[CrossRef] [PubMed]

F. Behroozi, B. Lambert, and B. Buhrow, "Direct measurement of the attenuation of capillary waves by laser interferometry: noncontact determination of viscosity," Appl. Phys. Lett. 78, 2399-2401 (2001).
[CrossRef]

F. Behroozi, "A miniature laser interferometer for noninvasive viscometry," presented at the Technical Proceedings of Nanotech 2003 Conference, San Francisco, Calif., February 24-26, 2003, Vol. 1, pp. 166-169.

F. Behroozi, "Apparatus and method for measurement of fluid viscosity," U.S. patent 6,563,588 B2 (May 13, 2003).

Bohanon, T. M.

T. M. Bohanon, J. M. Mikrut, B. M. Abraham, J. B. Ketterson, and P. Datta, "Fiber-optic detection system for capillary waves: an apparatus for studying liquid surfaces and spread monolayers," Rev. Sci. Instrum. 62, 2959-2962 (1991).
[CrossRef]

Boreman, D.

Buhrow, B.

F. Behroozi, B. Lambert, and B. Buhrow, "Noninvasive measurement of viscosity from damping of capillary waves," ISA Trans. 42, 3-8 (2003).
[CrossRef] [PubMed]

F. Behroozi, B. Lambert, and B. Buhrow, "Direct measurement of the attenuation of capillary waves by laser interferometry: noncontact determination of viscosity," Appl. Phys. Lett. 78, 2399-2401 (2001).
[CrossRef]

Codreanu, I.

Conchello, J. A.

Datta, P.

T. M. Bohanon, J. M. Mikrut, B. M. Abraham, J. B. Ketterson, and P. Datta, "Fiber-optic detection system for capillary waves: an apparatus for studying liquid surfaces and spread monolayers," Rev. Sci. Instrum. 62, 2959-2962 (1991).
[CrossRef]

Fumeaue, C.

Jansson, E.

See, for example, A. Runnemalm, N. Molin, and E. Jansson, "On operating deflection shapes of the violin body including in-plane motions," J. Acoust. Soc. Am. 107, 3452-3459 (2000).
[CrossRef] [PubMed]

Kauppinen, J.

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy (Wiley-VCH, 2002).

Ketterson, J. B.

T. M. Bohanon, J. M. Mikrut, B. M. Abraham, J. B. Ketterson, and P. Datta, "Fiber-optic detection system for capillary waves: an apparatus for studying liquid surfaces and spread monolayers," Rev. Sci. Instrum. 62, 2959-2962 (1991).
[CrossRef]

Lambert, B.

F. Behroozi, B. Lambert, and B. Buhrow, "Noninvasive measurement of viscosity from damping of capillary waves," ISA Trans. 42, 3-8 (2003).
[CrossRef] [PubMed]

F. Behroozi, B. Lambert, and B. Buhrow, "Direct measurement of the attenuation of capillary waves by laser interferometry: noncontact determination of viscosity," Appl. Phys. Lett. 78, 2399-2401 (2001).
[CrossRef]

Markham, J.

Mikrut, J. M.

T. M. Bohanon, J. M. Mikrut, B. M. Abraham, J. B. Ketterson, and P. Datta, "Fiber-optic detection system for capillary waves: an apparatus for studying liquid surfaces and spread monolayers," Rev. Sci. Instrum. 62, 2959-2962 (1991).
[CrossRef]

Molin, N.

See, for example, A. Runnemalm, N. Molin, and E. Jansson, "On operating deflection shapes of the violin body including in-plane motions," J. Acoust. Soc. Am. 107, 3452-3459 (2000).
[CrossRef] [PubMed]

Partanen, J.

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy (Wiley-VCH, 2002).

Runnemalm, A.

See, for example, A. Runnemalm, N. Molin, and E. Jansson, "On operating deflection shapes of the violin body including in-plane motions," J. Acoust. Soc. Am. 107, 3452-3459 (2000).
[CrossRef] [PubMed]

Schaefer, J. A.

Watson, G. N.

See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1958).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

F. Behroozi, B. Lambert, and B. Buhrow, "Direct measurement of the attenuation of capillary waves by laser interferometry: noncontact determination of viscosity," Appl. Phys. Lett. 78, 2399-2401 (2001).
[CrossRef]

Eur. J. Phys. (1)

F. Behroozi, "Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy," Eur. J. Phys. 25, 115-122 (2004).
[CrossRef]

ISA Trans. (1)

F. Behroozi, B. Lambert, and B. Buhrow, "Noninvasive measurement of viscosity from damping of capillary waves," ISA Trans. 42, 3-8 (2003).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (1)

See, for example, A. Runnemalm, N. Molin, and E. Jansson, "On operating deflection shapes of the violin body including in-plane motions," J. Acoust. Soc. Am. 107, 3452-3459 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Rev. Sci. Instrum. (1)

T. M. Bohanon, J. M. Mikrut, B. M. Abraham, J. B. Ketterson, and P. Datta, "Fiber-optic detection system for capillary waves: an apparatus for studying liquid surfaces and spread monolayers," Rev. Sci. Instrum. 62, 2959-2962 (1991).
[CrossRef]

Other (5)

See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1958).

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy (Wiley-VCH, 2002).

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions, 10th printing (U.S. Government Printing Office, 1972) p. 361.

F. Behroozi, "A miniature laser interferometer for noninvasive viscometry," presented at the Technical Proceedings of Nanotech 2003 Conference, San Francisco, Calif., February 24-26, 2003, Vol. 1, pp. 166-169.

F. Behroozi, "Apparatus and method for measurement of fluid viscosity," U.S. patent 6,563,588 B2 (May 13, 2003).

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Figures (3)

Fig. 1
Fig. 1

Schematic of the fiber probe and blade above the capillary wave.

Fig. 2
Fig. 2

Interference signal for a half-period.

Fig. 3
Fig. 3

Bessel fit for a sample of generated data ( b = 22 , θ = 2 ) with 15% random error. The fit returns the parameters b and θ to within 1%. Only 1024 data points are used for this fit.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

y ( t ) = a sin ( ω t ) ,
Δ = 2 [ d 0 a sin ( ω t ) ] .
Y ( t ) = A cos [ ( 2 π Δ λ l ) + π ] .
Y ( t ) = A cos [ b sin ( ω t ) + θ ] ,
a = b λ l 4 π .
Y ( t ) = cos [ b sin ( ω t ) + θ ] ,
Y ( t ) = cos θ cos [ b sin ( ω t ) ] sin θ sin [ b sin ( ω t ) ] .
Y ̇ ( t ) = cos θ [ J 0 ( b ) + 2 n = 1 J 2 n ( b ) cos ( 2 n ω t ) ] sin θ [ 2 n = 0 J 2 n + 1 ( b ) sin ( ( 2 n + 1 ) ω t ) ] ,
M 2 l 1 τ t t + τ Y ( t ) cos ( 2 l ω t ) d t = cos θ J 2 l ( b ) ,
M 2 l + 1 1 τ t t + τ Y ( t ) sin [ ( 2 l + 1 ) ω t ] d t = sin θ J 2 l + 1 ( b ) ,
J n 1 ( b ) + J n + 1 ( b ) = ( 2 n b ) J n ( b ) ,
J n 2 ( b ) + J n ( b ) = ( 2 ( n 1 ) b ) J n 1 ( b ) ,
J n 3 ( b ) + J n 1 ( b ) = ( 2 ( n 2 ) b ) J n 2 ( b ) .
J n 1 ( b ) + J n + 1 ( b ) = ( 4 ( n 2 n ) b 2 ) J n 1 ( b ) ( n ( n 2 ) ) [ J n 3 ( b ) + J n 1 ( b ) ] .
b = [ 4 ( n 3 3 n 2 + 2 n ) M n 1 ( n 2 ) ( M n + 1 + M n 1 ) + n ( M n 1 + M n 3 ) ] 1 2 ,
tan ( θ ) = J n 1 ( b ) M n J n ( b ) M n 1 .
x 2 J m ( x ) + x J m ( x ) + ( x 2 m 2 ) J m ( x ) = 0 .
Y ( t ) = ω b cos ( ω t + ϕ ) sin [ b sin ( ω t + ϕ ) + θ ] .

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