Abstract

A theoretical analysis of a recently published method for measurement of small vibrations using electronic speckle pattern interferometry is given. From the general description of the signal processing in electronic speckle pattern interferometry, we derive a theory that applies to ideal operation of the system. For two special cases, including the small-amplitude limit, the theory is extended to include nonideal effects. The calculated response functions show good agreement with experimental results also beyond the linear small-amplitude range.

© 1977 Optical Society of America

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References

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  1. K. Høgmoen and O. J. Løkberg, “Detection and measurement of small vibrations using electronic speckle pattern interferometry,” Appl. Opt. 16, 1869–1875 (1977).
    [Crossref] [PubMed]
  2. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [Crossref]
  3. S. M. Khanna, J. Tonndorf, and W. W. Walcott, “Laser Interferometer for the Measurement of Submicroscopic Displacement Amplitudes and Their Phases in Small Biological Structures,” J. Acoust. Soc. Am. 44, 1555–1565 (1968).
    [Crossref]
  4. H. A. Deferrari, R. A. Darby, and F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
    [Crossref]
  5. P. R. Dragsten, W. W. Webb, J. A. Paton, and R. R. Capranica, “Light-scattering heterodyne interferometer for vibration measurements in auditory organs,” J. Acoust. Soc. Am. 60, 665–671 (1976).
    [Crossref] [PubMed]
  6. C. J. Tranter, Bessel functions with some physical applications (English Universities Press, London, 1968), p. 36.

1977 (1)

1976 (1)

P. R. Dragsten, W. W. Webb, J. A. Paton, and R. R. Capranica, “Light-scattering heterodyne interferometer for vibration measurements in auditory organs,” J. Acoust. Soc. Am. 60, 665–671 (1976).
[Crossref] [PubMed]

1968 (1)

S. M. Khanna, J. Tonndorf, and W. W. Walcott, “Laser Interferometer for the Measurement of Submicroscopic Displacement Amplitudes and Their Phases in Small Biological Structures,” J. Acoust. Soc. Am. 44, 1555–1565 (1968).
[Crossref]

1967 (1)

H. A. Deferrari, R. A. Darby, and F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[Crossref]

Andrews, F. A.

H. A. Deferrari, R. A. Darby, and F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[Crossref]

Capranica, R. R.

P. R. Dragsten, W. W. Webb, J. A. Paton, and R. R. Capranica, “Light-scattering heterodyne interferometer for vibration measurements in auditory organs,” J. Acoust. Soc. Am. 60, 665–671 (1976).
[Crossref] [PubMed]

Darby, R. A.

H. A. Deferrari, R. A. Darby, and F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[Crossref]

Deferrari, H. A.

H. A. Deferrari, R. A. Darby, and F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[Crossref]

Dragsten, P. R.

P. R. Dragsten, W. W. Webb, J. A. Paton, and R. R. Capranica, “Light-scattering heterodyne interferometer for vibration measurements in auditory organs,” J. Acoust. Soc. Am. 60, 665–671 (1976).
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

Høgmoen, K.

Khanna, S. M.

S. M. Khanna, J. Tonndorf, and W. W. Walcott, “Laser Interferometer for the Measurement of Submicroscopic Displacement Amplitudes and Their Phases in Small Biological Structures,” J. Acoust. Soc. Am. 44, 1555–1565 (1968).
[Crossref]

Løkberg, O. J.

Paton, J. A.

P. R. Dragsten, W. W. Webb, J. A. Paton, and R. R. Capranica, “Light-scattering heterodyne interferometer for vibration measurements in auditory organs,” J. Acoust. Soc. Am. 60, 665–671 (1976).
[Crossref] [PubMed]

Tonndorf, J.

S. M. Khanna, J. Tonndorf, and W. W. Walcott, “Laser Interferometer for the Measurement of Submicroscopic Displacement Amplitudes and Their Phases in Small Biological Structures,” J. Acoust. Soc. Am. 44, 1555–1565 (1968).
[Crossref]

Tranter, C. J.

C. J. Tranter, Bessel functions with some physical applications (English Universities Press, London, 1968), p. 36.

Walcott, W. W.

S. M. Khanna, J. Tonndorf, and W. W. Walcott, “Laser Interferometer for the Measurement of Submicroscopic Displacement Amplitudes and Their Phases in Small Biological Structures,” J. Acoust. Soc. Am. 44, 1555–1565 (1968).
[Crossref]

Webb, W. W.

P. R. Dragsten, W. W. Webb, J. A. Paton, and R. R. Capranica, “Light-scattering heterodyne interferometer for vibration measurements in auditory organs,” J. Acoust. Soc. Am. 60, 665–671 (1976).
[Crossref] [PubMed]

Appl. Opt. (1)

J. Acoust. Soc. Am. (3)

S. M. Khanna, J. Tonndorf, and W. W. Walcott, “Laser Interferometer for the Measurement of Submicroscopic Displacement Amplitudes and Their Phases in Small Biological Structures,” J. Acoust. Soc. Am. 44, 1555–1565 (1968).
[Crossref]

H. A. Deferrari, R. A. Darby, and F. A. Andrews, “Vibrational Displacement and Mode-Shape Measurement by a Laser Interferometer,” J. Acoust. Soc. Am. 42, 982–990 (1967).
[Crossref]

P. R. Dragsten, W. W. Webb, J. A. Paton, and R. R. Capranica, “Light-scattering heterodyne interferometer for vibration measurements in auditory organs,” J. Acoust. Soc. Am. 60, 665–671 (1976).
[Crossref] [PubMed]

Other (2)

C. J. Tranter, Bessel functions with some physical applications (English Universities Press, London, 1968), p. 36.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

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Figures (5)

FIG. 1
FIG. 1

Setup for measurement of small vibrations using ESPI (from Ref. 1). The modulating mirror provides phase modulation of the reference wave at the shifted frequency ω + Ω. The detector (e.g., a photodiode) measures intensity variations at a chosen object point on the monitor.

FIG. 2
FIG. 2

Schematic illustration of the ESPI signal processing, showing spatial intensity distribution across a vibrating object: (a) image interferogram; (b) filtered “hologram”; (c) displayed interferogram.

FIG. 3
FIG. 3

Detector output zd as function of object amplitude a0 at different working points ar; low-frequency limit.

FIG. 4
FIG. 4

Modulus (solid line) and phase (dotted) of response Z, as function of a0 with ar as parameter at Ω = 1/Te.

FIG. 5
FIG. 5

(a) Lock-in amplifier output recorded as function of excitation voltage to test object at different ar. (b) Corresponding calculated response functions [Eq. (29)].

Equations (40)

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I ( t ) = | u r e i ϕ r ( t ) + u 0 e i ϕ 0 ( t ) | 2 = I r + I 0 + 2 Re ( u r * u 0 exp { i [ ϕ 0 ( t ) ϕ r ( t ) ] } ) ,
E n = t n I ( t ) h ( t n t ) d t ,
E n = I r + I 0 + 2 Re [ u r * u 0 x ( t n ) ] ,
x ( t ) = t h ( t t ) exp { i [ ϕ 0 ( t ) ϕ r ( t ) ] } d t .
Δ E n = E n E n = Δ I r + Δ I 0 + 2 Re [ u r * u 0 x ( t n ) ] .
( Δ E n ) 2 = ( Δ I r ) 2 + ( Δ I 0 ) 2 + { 2 Re [ u r * u 0 x ( t n ) ] } 2 + terms of vanishing mean value .
( 2 Re Z ) 2 = 2 | Z | 2 + 2 Re Z 2 = 2 | Z | 2 ,
I n ( Δ E n ) 2 = σ r 2 + σ 0 2 + 2 I r I 0 | x ( t n ) | 2 ,
y ( t ) = n = 0 | x ( t n ) | 2 δ ( t t n ) .
ϕ 0 ( t ) = a 0 cos ω t ,
ϕ r ( t ) = a r cos ( ω + Ω ) t .
ϕ 0 ( t ) ϕ r ( t ) = a ( t ) cos [ ω t + ϕ ( t ) ] ,
a ( t ) = ( a 0 2 + a r 2 2 a 0 a r cos Ω t ) 1 / 2 ,
ϕ ( t ) = arctan ( a r sin Ω t a 0 a r cos Ω t ) .
x ( t ) = t h ( t t ) exp { i a ( t ) cos [ ω t + ϕ ( t ) ] } d t n = t / τ τ h ( t n τ ) 1 τ × ( n 1 ) τ n τ exp { i a ( n τ ) cos [ ω t + ϕ ( n τ ) ] } d t = n = t / τ τ h ( t n τ ) J 0 [ a ( n τ ) ] t h ( t t ) J 0 [ a ( t ) ] d t ,
J 0 [ ( a 0 2 + a r 2 2 a 0 a r cos Ω t ) 1 / 2 ] = J 0 ( a 0 ) J 0 ( a r ) + 2 k = 1 J k ( a 0 ) J k ( a r ) cos k Ω t .
x ( t ) = J 0 ( a 0 ) J 0 ( a r ) + 2 k = 1 J k ( a 0 ) J k ( a r ) × Re [ H ( k Ω ) e i k Ω t ] ,
H ( Ω ) = 0 h ( t ) e i Ω t d t
x dc 2 = J 0 2 ( a 0 ) J 0 2 ( a r ) + k = 1 J k 2 ( a 0 ) J k 2 ( a r ) | H ( k Ω ) | 2 .
x 1 harm 2 = 4 k = 0 J k ( a 0 ) J k ( a r ) J k + 1 ( a 0 ) J k + 1 ( a r ) × Re { H * ( k Ω ) H [ ( k + 1 ) Ω ] e i Ω t } .
z d = 2 k = 0 J k ( a 0 ) J k ( a r ) J k + 1 ( a 0 ) J k + 1 ( a r ) × Re { G ( Ω ) H * ( k Ω ) H [ ( k + 1 ) Ω ] e i ϕ d } .
z d 2 J 0 ( a 0 ) J 0 ( a r ) J 1 ( a 0 ) J 1 ( a r ) Re [ G ( Ω ) H ( Ω ) e i ϕ d ] .
z d a 0 J 0 ( a r ) J 1 ( a r ) Re [ G ( Ω ) H ( Ω ) e i ϕ d ] = a 0 2 d d a r J 0 2 ( a r ) Re [ G ( Ω ) H ( Ω ) e i ϕ d ] .
z d 2 k = 0 J k ( a 0 ) J k ( a r ) J k + 1 ( a 0 ) J k + 1 ( a r ) cos ϕ d .
z d = 2 Re [ G ( Ω ) Z ( a 0 , a r ; Ω ) e i ϕ d ] ,
Z ( a 0 , a r ; Ω ) = k = 0 J k ( a 0 ) J k ( a r ) J k + 1 ( a 0 ) × J k + 1 ( a r ) H * ( k Ω ) H [ ( k + 1 ) Ω ] .
z d = 2 | G ( Ω ) | | Z ( a 0 , a r ; Ω ) | .
H ( Ω ) = 1 / ( 1 + i Ω T e ) .
| Z ( a 0 , a r ; Ω ) | cos [ ϕ Z ( a 0 ) ϕ Z ( a 0 0 ) ] ,
a a r a 0 cos Ω t ,
y y ( a r ) y ( a r ) a 0 cos Ω t + .
y = f ( x ) ,
x ( t ) = x 0 + Δ x ( t ) ,
x 0 = J 0 ( a 0 ) J 0 ( a r )
Δ x ( t ) = 2 k = 1 J k ( a 0 ) J k ( a r ) Re [ H ( k Ω ) e i k Ω t ]
y = f ( x 0 ) + f ( x 0 ) Δ x + 1 2 f ( x 0 ) Δ x 2 + .
Δ x ( t ) 2 J 1 ( a 0 ) J 1 ( a r ) Re [ H ( Ω ) e i Ω t ] .
y 1 harm f ( x ) Δ x ( t ) 2 f [ J 0 ( a 0 ) J 0 ( a r ) ] J 1 ( a 0 ) J 1 ( a r ) × Re [ H ( Ω ) e i Ω t ] .
z d f [ J 0 ( a 0 ) J 0 ( a r ) ] J 1 ( a 0 ) J 1 ( a r ) × Re [ G ( Ω ) H ( Ω ) e i ϕ d ] .
z d a 0 2 f [ J 0 ( a 0 ) ] J 1 ( a r ) Re [ G ( Ω ) H ( Ω ) e i ϕ d ] = a 0 2 d d a r f [ J 0 ( a r ) ] Re [ G ( Ω ) H ( Ω ) e i ϕ d ] .