Abstract

In this paper the characteristic fringe function of hologram interferometry has been evaluated for the case of periodic, nonsinusoidal vibrations represented by a Jacobian elliptic function. To consider the reconstructed holographic image of an object, use has been made of an equation derived from considerations of the effect of motion on coherence. Graphical representation of the fringe irradiance distribution in the reconstructed image is given. It is shown that, as the motion departs from pure sinusoidal form, the intensity of the fringes increases considerably as compared with that for a pure sinusoidal case.

© 1975 Optical Society of America

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References

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  1. R. L. Powell, K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
    [CrossRef]
  2. M. Zambuto, M. Lurie, Appl. Opt. 9, 2066 (1970).
    [CrossRef] [PubMed]
  3. D. B. Neumann, J. Opt. Soc. Am. 58, 447 (1968).
    [CrossRef]
  4. D. B. Neumann, Ph.D. Thesis, Ohio State University (1967); also see C. S. Vikram, R. S. Sirohi, Appl. Opt. 10, 672 (1971).
    [CrossRef] [PubMed]
  5. P. A. Fryer, Rep. Prog. Phys. 33, 489 (1970).
    [CrossRef]
  6. C. C. Aleksoff, Appl. Opt. 10, 1329 (1971).
    [CrossRef] [PubMed]
  7. K. Singh, Atti. Fond. Giorgio Ronchi, 27, 323 (1972).
  8. K. Singh, in Proceedings of 2nd Indo-French Seminar, Nov.-Dec. 1971, IIT, Delhi, K. L. Chopra, (Thomson Press, New Delhi, 1973), p. 183.
  9. R. L. Powell, in The Engineering Uses of Holography, E. R. Robertson, J. M. Harvey, Eds. (Cambridge University Press, 1970), p. 333.
  10. N. E. Mölin, K. A. Stetson, J. Phys. E: J. Sci. Instrum. 2, 609 (1969).
    [CrossRef]
  11. A. D. Wilson, J. Opt. Soc. Am. 60, 1068 (1970).
    [CrossRef]
  12. A. D. Wilson, D. H. Strope, J. Opt. Soc. Am. 60, 1162 (1970).
    [CrossRef]
  13. A. D. Wilson, J. Opt. Soc. Am. 61, 924 (1971).
    [CrossRef]
  14. K. A. Stetson, Ref. 9, p. 307.
  15. K. A. Stetson, J. Opt. Soc. Am. 61, 1359 (1971).
    [CrossRef]
  16. K. A. Stetson, J. Opt. Soc. Am. 62, 297 (1972).
    [CrossRef]
  17. K. A. Stetson, P. A. Taylor, J. Phys. E: J. Sci. Instrum. 4, 1009 (1971).
    [CrossRef]
  18. K. A. Stetson, P. A. Taylor, J. Phys. E: J. Sci. Instrum. 5, 923 (1972).
    [CrossRef]
  19. J. Janta, M. Miller, Optik, 36, 185 (1972).
  20. B. S. Thornton, J. C. Kelly, J. Opt. Soc. Am. 46, 191 (1956).
    [CrossRef]
  21. L. M. Milne-Thomson, Jacobian Elliptic Function Tables (Dover Publications, Inc., New York, 1950).
  22. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1965), Art. 8.146(5), p. 911.
  23. M. R. Wall, Opt. Tech. 1, 266 (1969).
    [CrossRef]
  24. N. Jensen, Optical and Photographic Reconnaissance Systems (Wiley, New York, 1968), Chap. 8.
  25. H. Osterberg, J. Opt. Soc. Am. 22, 19 (1932).
    [CrossRef]
  26. F. Bowmann, Introduction to Elliptic Functions with Applications (Dover Publications, New York, 1961).
  27. P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, London, 1954).

1972 (4)

K. Singh, Atti. Fond. Giorgio Ronchi, 27, 323 (1972).

K. A. Stetson, P. A. Taylor, J. Phys. E: J. Sci. Instrum. 5, 923 (1972).
[CrossRef]

J. Janta, M. Miller, Optik, 36, 185 (1972).

K. A. Stetson, J. Opt. Soc. Am. 62, 297 (1972).
[CrossRef]

1971 (4)

1970 (4)

1969 (2)

M. R. Wall, Opt. Tech. 1, 266 (1969).
[CrossRef]

N. E. Mölin, K. A. Stetson, J. Phys. E: J. Sci. Instrum. 2, 609 (1969).
[CrossRef]

1968 (1)

1965 (1)

1956 (1)

1932 (1)

Aleksoff, C. C.

Bowmann, F.

F. Bowmann, Introduction to Elliptic Functions with Applications (Dover Publications, New York, 1961).

Byrd, P. F.

P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, London, 1954).

Friedman, M. D.

P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, London, 1954).

Fryer, P. A.

P. A. Fryer, Rep. Prog. Phys. 33, 489 (1970).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1965), Art. 8.146(5), p. 911.

Janta, J.

J. Janta, M. Miller, Optik, 36, 185 (1972).

Jensen, N.

N. Jensen, Optical and Photographic Reconnaissance Systems (Wiley, New York, 1968), Chap. 8.

Kelly, J. C.

Lurie, M.

Miller, M.

J. Janta, M. Miller, Optik, 36, 185 (1972).

Milne-Thomson, L. M.

L. M. Milne-Thomson, Jacobian Elliptic Function Tables (Dover Publications, Inc., New York, 1950).

Mölin, N. E.

N. E. Mölin, K. A. Stetson, J. Phys. E: J. Sci. Instrum. 2, 609 (1969).
[CrossRef]

Neumann, D. B.

D. B. Neumann, J. Opt. Soc. Am. 58, 447 (1968).
[CrossRef]

D. B. Neumann, Ph.D. Thesis, Ohio State University (1967); also see C. S. Vikram, R. S. Sirohi, Appl. Opt. 10, 672 (1971).
[CrossRef] [PubMed]

Osterberg, H.

Powell, R. L.

R. L. Powell, K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
[CrossRef]

R. L. Powell, in The Engineering Uses of Holography, E. R. Robertson, J. M. Harvey, Eds. (Cambridge University Press, 1970), p. 333.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1965), Art. 8.146(5), p. 911.

Singh, K.

K. Singh, Atti. Fond. Giorgio Ronchi, 27, 323 (1972).

K. Singh, in Proceedings of 2nd Indo-French Seminar, Nov.-Dec. 1971, IIT, Delhi, K. L. Chopra, (Thomson Press, New Delhi, 1973), p. 183.

Stetson, K. A.

K. A. Stetson, J. Opt. Soc. Am. 62, 297 (1972).
[CrossRef]

K. A. Stetson, P. A. Taylor, J. Phys. E: J. Sci. Instrum. 5, 923 (1972).
[CrossRef]

K. A. Stetson, J. Opt. Soc. Am. 61, 1359 (1971).
[CrossRef]

K. A. Stetson, P. A. Taylor, J. Phys. E: J. Sci. Instrum. 4, 1009 (1971).
[CrossRef]

N. E. Mölin, K. A. Stetson, J. Phys. E: J. Sci. Instrum. 2, 609 (1969).
[CrossRef]

R. L. Powell, K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
[CrossRef]

K. A. Stetson, Ref. 9, p. 307.

Strope, D. H.

Taylor, P. A.

K. A. Stetson, P. A. Taylor, J. Phys. E: J. Sci. Instrum. 5, 923 (1972).
[CrossRef]

K. A. Stetson, P. A. Taylor, J. Phys. E: J. Sci. Instrum. 4, 1009 (1971).
[CrossRef]

Thornton, B. S.

Wall, M. R.

M. R. Wall, Opt. Tech. 1, 266 (1969).
[CrossRef]

Wilson, A. D.

Zambuto, M.

Appl. Opt. (2)

Atti. Fond. Giorgio Ronchi (1)

K. Singh, Atti. Fond. Giorgio Ronchi, 27, 323 (1972).

J. Opt. Soc. Am. (9)

J. Phys. E: J. Sci. Instrum. (3)

N. E. Mölin, K. A. Stetson, J. Phys. E: J. Sci. Instrum. 2, 609 (1969).
[CrossRef]

K. A. Stetson, P. A. Taylor, J. Phys. E: J. Sci. Instrum. 4, 1009 (1971).
[CrossRef]

K. A. Stetson, P. A. Taylor, J. Phys. E: J. Sci. Instrum. 5, 923 (1972).
[CrossRef]

Opt. Tech. (1)

M. R. Wall, Opt. Tech. 1, 266 (1969).
[CrossRef]

Optik (1)

J. Janta, M. Miller, Optik, 36, 185 (1972).

Rep. Prog. Phys. (1)

P. A. Fryer, Rep. Prog. Phys. 33, 489 (1970).
[CrossRef]

Other (9)

D. B. Neumann, Ph.D. Thesis, Ohio State University (1967); also see C. S. Vikram, R. S. Sirohi, Appl. Opt. 10, 672 (1971).
[CrossRef] [PubMed]

K. Singh, in Proceedings of 2nd Indo-French Seminar, Nov.-Dec. 1971, IIT, Delhi, K. L. Chopra, (Thomson Press, New Delhi, 1973), p. 183.

R. L. Powell, in The Engineering Uses of Holography, E. R. Robertson, J. M. Harvey, Eds. (Cambridge University Press, 1970), p. 333.

L. M. Milne-Thomson, Jacobian Elliptic Function Tables (Dover Publications, Inc., New York, 1950).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1965), Art. 8.146(5), p. 911.

N. Jensen, Optical and Photographic Reconnaissance Systems (Wiley, New York, 1968), Chap. 8.

K. A. Stetson, Ref. 9, p. 307.

F. Bowmann, Introduction to Elliptic Functions with Applications (Dover Publications, New York, 1961).

P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, London, 1954).

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Figures (2)

Fig. 1
Fig. 1

Function sn (u/m) plotted against u for various values of m.

Fig. 2
Fig. 2

Square of characteristic function (C2) plotted against amplitude of vibration (A/λ).

Equations (28)

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B 1 ( m ) = B 0 ( m ) | γ m R ( 0 ) | 2 ,
g m R ( 0 ) = 1 T 0 T exp [ i ( 2 π / λ ) ( cos α + cos β ) x ( t ) ] d t ,
C = 1 T 0 T exp [ i ( 4 π / λ ) x ( t ) ] d t .
K = K ( m ) = 0 π / 2 d θ ( 1 m sin 2 θ ) 1 / 2 , i K = i K ( m 1 ) = i 0 π / 2 d θ ( 1 m 1 sin 2 θ ) 1 / 2 ,
x ( t ) = A sn ( ω t / m ) ,
C = 1 T 0 T exp [ i ( 4 π A / λ ) sn ( ω t / m ) ] d t .
1 sn u = π 2 K [ 1 sin ( π u / 2 K ) + 4 s = 1 q 2 s 1 1 q 2 s 1 sin ( 2 s 1 ) × ( π u / 2 K ) ] ,
C = 1 T 0 T exp { i [ ( 8 k A ) / λ ] [ 1 sin ( π ω t / 2 K ) + 4 s = 1 q 2 s 1 1 q 2 s 1 } d t , × sin ( 2 s 1 ) ( π ω t / 2 K ) ]
C = 0 1 exp { i ( 8 k A / λ ) [ 1 sin ( π ω T t / 2 K ) + 4 s = 1 q 2 s 1 1 q 2 s 1 } d t , × sin ( 2 s 1 ) ( π ω T t / 2 K ) ]
C = 0 1 exp { i ( 8 k A / λ ) [ 1 sin ( 2 π n t ) + 4 s = 1 q 2 s 1 1 q 2 s 1 } d t . × sin ( 2 s 1 ) ( 2 π n t ) ]
C 2 = | 0 1 cos { ( 8 k A / λ ) [ 1 sin ( 2 π n t ) + 4 s = 1 q 2 s 1 1 q 2 s 1 } d t | 2 × sin ( 2 s 1 ) ( 2 π n t ) ] + | 0 1 sin { ( 8 k A / λ ) [ 1 sin ( 2 π n t ) + 4 s = 1 q 2 s 1 1 q 2 s 1 } d t . | 2 × sin ( 2 s 1 ) ( 2 π n t ) ]
C = 1 4 n K 0 4 n K exp [ i z sn ( u / m ) ] d u , = 1 4 n K 0 4 n K cos [ z sn ( u / m ) ] d u + i 4 n K 0 4 n K sin [ z sn ( u / m ) ] d u ,
cos [ z sn ( u / m ) ] = J 0 ( z ) + 2 n = 1 J 2 n ( z ) [ 1 4 n 2 2 ! sn 2 ( u / m ) + 4 n 2 ( 4 n 2 2 2 ) 4 ! sn 4 ( u / m ) 4 n 2 ( 4 n 2 2 2 ) ( 4 n 2 4 2 ) 6 ! sn 6 ( u / m ) + ] .
1 4 n K 0 4 n K cos [ z sn ( u / m ) ] d u = J 0 ( z ) + 2 J 2 ( z ) [ 1 2 4 n K × 0 4 n K sn 2 ( u / m ) d u ] + 2 J 4 ( z ) [ 1 8 4 n K 0 4 n K × sn 2 ( u / m ) d u + 8 4 n K 0 4 n K sn 4 ( u / m ) d u ] + 2 J 6 ( z ) [ 1 18 4 n K 0 4 n K sn 2 ( u / m ) d u + 48 4 n K 0 4 n K × sn 4 ( u / m ) d u 32 4 n K 0 4 n K sn 6 ( u / m ) d u ] +
1 4 n K 0 4 n K sin [ z sn ( u / m ) ] d u = 2 J 1 ( z ) [ 1 4 n K 0 4 n K × sn ( u / m ) d u ] + 2 J 3 ( z ) [ 3 4 n K 0 4 n K sn ( u / m ) d u 1 3 n K 0 4 n K sn 3 ( u / m ) d u ] + 2 J 5 ( z ) [ 5 4 n K 0 4 n K × sn ( u / m ) d u 1 n K 0 4 n K sn 3 ( u / m ) d u + 4 5 n K 0 4 n K sn 5 ( u / m ) d u ] + .
0 4 n K sn p ( u / m ) d u A 1 = 0 4 n K sn ( u / m ) d u = 1 m 1 / 2 ln [ d n ( u / m ) m 1 / 2 × cn ( u / m ) ] 0 , = 1 m 1 / 2 [ ln 1 ] , = 0.
1 4 n k 0 4 n K sin [ z sn ( u / m ) ] d u = 0.
A 2 = 0 4 n K sn 2 ( u / m ) d u = 1 m [ u E ( u ) ] 0 4 nK ,
u = F ( φ / m ) = 0 φ d θ ( 1 m sin 2 θ ) 1 / 2 ,
E ( u ) = E ( φ / m ) = 0 φ ( 1 m sin 2 θ ) 1 / 2 d θ ,
φ = a m u .
A 2 = 1 m [ F ( 2 n π / m ) E ( 2 n π / m ) ] , = 4 n m [ F ( π / 2 / m ) E ( π / 2 / m ) ] .
A 4 = 0 4 n K sn 4 ( u / m ) d u = 4 n 3 m 2 { ( 2 + m ) F [ ( π / 2 ) / m ] 2 ( 1 + m ) E [ ( π / 2 ) / m ] } .
A 2 p + 2 = 0 4 n K sn 2 p + 2 ( u / m ) d u , = [ s n 2 p 1 ( u / m ) cn ( u / m ) dn ( u / m ) + 2 p ( 1 + m ) × A 2 p + ( 1 2 p ) A 2 p 2 ( 2 p + 1 ) m ] 0 4 n K .
A 6 = 0 4 n K sn 6 ( u / m ) d u = 1 5 m [ sn 3 ( u / m ) cn ( u / m ) × dn ( u / m ) + 4 ( 1 + m ) A 4 3 A 2 ] 0 4 n K .
0 4 n K sn 6 ( u / m ) d u = 4 n 15 m 3 { ( 8 + 3 m + 4 m 2 ) F [ ( π / 2 ) / m ] ( 2 7 m ) E [ ( π / 2 ) / m ] } .
C = J 0 ( z ) + 2 J 2 ( z ) ( 1 2 K × { F [ ( π / 2 ) / m ] [ E ( π / 2 ) / m ] m } ) + 2 J 4 ( z ) ( 1 8 K { F [ ( π / 2 ) / m ] E [ ( π / 2 ) / m ] m } + 8 K { ( 2 + m ) F [ ( π / 2 ) / m ] 2 ( 1 + m ) E [ ( π / 2 ) / m ] 3 m 2 } + 2 J 6 ( z ) ( 1 18 K { F [ ( π / 2 ) / m ] E [ ( π / 2 ) / m ] m } + 16 K { ( 2 + m ) F [ ( π / 2 ) / m ] 2 ( 1 + m ) E [ ( π / 2 ) / m ] m 2 } ( 8 + 3 m + 4 m 2 ) F [ ( π / 2 ) / m ] 32 K { ( 2 7 m ) E [ ( π / 2 ) / m ] 15 m 3 } ) + .
z = 4 π A / λ .

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