Abstract

A study of the Leith–Upatnieks hologram for the time average of the coherent wavefronts scattered from a vibrating object is reported. The image reconstructed by the hologram is found to contain a system of interference fringes which map contours of constant vibration amplitude, thus providing a method of analysis of the vibration of objects with arbitrary surfaces. Experimental results are presented and interpreted for a simple periodic vibration; statistical motion of objects is discussed.

© 1965 Optical Society of America

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References

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  1. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 10 (1962).
    [Crossref]
  2. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 12 (1963).
    [Crossref]
  3. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 11 (1964).
    [Crossref]
  4. M. Born and E. Wolf, Principales of Optics (Pergamon Press, New York and London, 1964), Chap. 9, p. 480.
  5. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (The Focal Press, New York and London, 1964).
  6. H. Osterberg, J. Opt. Soc. Am. 22, 19 (1932).
    [Crossref]

1964 (1)

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 11 (1964).
[Crossref]

1963 (1)

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 12 (1963).
[Crossref]

1962 (1)

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 10 (1962).
[Crossref]

1932 (1)

Born, M.

M. Born and E. Wolf, Principales of Optics (Pergamon Press, New York and London, 1964), Chap. 9, p. 480.

Leith, E.

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 11 (1964).
[Crossref]

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 12 (1963).
[Crossref]

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 10 (1962).
[Crossref]

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (The Focal Press, New York and London, 1964).

Osterberg, H.

Upatnieks, J.

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 11 (1964).
[Crossref]

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 12 (1963).
[Crossref]

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 10 (1962).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principales of Optics (Pergamon Press, New York and London, 1964), Chap. 9, p. 480.

J. Opt. Soc. Am. (4)

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 10 (1962).
[Crossref]

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 12 (1963).
[Crossref]

E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 11 (1964).
[Crossref]

H. Osterberg, J. Opt. Soc. Am. 22, 19 (1932).
[Crossref]

Other (2)

M. Born and E. Wolf, Principales of Optics (Pergamon Press, New York and London, 1964), Chap. 9, p. 480.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (The Focal Press, New York and London, 1964).

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Figures (5)

Fig. 1
Fig. 1

Experimental arrangement for hologram recording. B, Beam from laser; BS, beam splitter; LP, lens and pinhole combination; M, mirror; MT, mount; S, solenoid; FC, film can; H, hologram plate.

Fig. 2
Fig. 2

Reconstructions of of three holograms of a 35 mm film-can bottom with a progressive increase in amplitude of excitation at the lowest resonance frequency of the can bottom.

Fig. 3
Fig. 3

Reconstructions of three holograms of the film-can bottom with three progressively increasing amplitudes at the second resonance of the can bottom.

Fig. 4
Fig. 4

Reconstructions of seven holograms of other resonant frequencies of the same can bottom.

Fig. 5
Fig. 5

Reference frame for analysis of vibrating object.

Equations (22)

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S ( x , y , 0 ) S 0 ( x 0 , y 0 ) .
S 0 ( x 0 , y 0 , t ) = S 0 t S ( x , y , z , t ) = S t .
A ( x , y , z ) e i ϕ ( x , y , z ) .
E ( x , y , z , t ) = A e i ϕ + S t ,
E E * = A 2 + S t 2 + A e - i ϕ S t + A e i ϕ S t * .
1 t r 0 t r E E * d t = A 2 + 1 t r 0 t r S t 2 d t + A e - i ϕ 1 t r 0 t r S t d t + A e i ϕ 1 t r 0 t r S t * d t .
E av ( x , y , z ) = 1 t r 1 N t n S n + A e i ϕ ,
E av ( x , y , z ) = 1 t r 0 t r S t d t + A e i ϕ .
E av E av * = A 2 + | 1 t r 0 t r S t d t | + A e - i ϕ 1 t r 0 t r S t d t . + A e - i ϕ 1 t r 0 t r S t * d t .
A t r { e - i ϕ 0 t r S t d t + e i ϕ 0 t r S t * d t } ,
1 t r 0 t r S 0 t d t .
I ( x 1 , y 1 ) = - Δ Δ { 1 t r S 0 ( x 0 , y 0 , z 0 , t ) d t } × K ( x 1 - x 0 , y 1 - y 0 ) d x 0 d y 0 ,
z 0 = f ( x 0 , y 0 )
S 0 ( x 0 , y 0 , z 0 , t ) = S 0 ( x 0 - x , y 0 - y , z 0 - z ) .
S 0 ( x 0 - x , y 0 - y , z 0 - z ) = S 0 ( x 0 - x , y 0 - y , z 0 ) e i ϕ ( z ) .
K ( x 1 - x 0 , y 1 - y 0 ) K [ x 1 - ( x 0 - x ) , y 1 - ( y 0 - y ) ] .
I ( x 1 y 1 ) = 1 t r 0 t r e i ϕ ( z ) - x k , y k x k , y k S 0 ( x 0 - x , y 0 - z , z 0 ) · K [ x 1 - ( x 0 - x ) , y 1 - ( y 0 - y ) ] d x 0 d y 0 d t ,
I ( x 1 , y 1 ) = { 1 / t r 0 t r e i ϕ ( z ) d t } I st ( x 1 , y 1 ) ,
r = m ( x 0 , y 0 ) cos [ ω t + μ ( x 0 , y 0 ) ] .
ϕ ( z ) = ( 2 π / λ ) ( cos θ 1 + cos θ 2 ) m ( x 0 , y 0 ) × cos [ ω t + μ ( x 0 , y 0 ) ] .
I ( x 1 , y 1 ) = J 0 [ ( 2 π / λ ) ( cos θ 1 + cos θ 2 ) m ( x 0 , y 0 ) ] × I st ( x 1 , y 1 ) .
M x y = 1 t r 0 t r exp [ 2 π i λ ( cos θ 1 + cos θ 2 ) r ] d t = - exp [ 2 π i λ ( cos θ 1 + cos θ 2 ) r ] p ( r ) d r .