Abstract

This paper presents an analysis of the fringes obtained by time-average holographic interferometry of a generalized time-dependent optical phase function. The generalized optical phase function considered is the sum of a series of sinusoidal functions of time having arbitrary amplitudes, frequencies, and relative phases. Characteristic functions are determined for various optical phase functions of interest in time-average holography. In general, the characteristic functions are sums of products of Bessel functions (zero order and higher orders) and exponential phase factors. Rationally and irrationally related frequencies are included in this analysis. An example of vibrating string is considered, to illustrate the application of the results of this paper to objects vibrating at a multitude of frequencies.

© 1970 Optical Society of America

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References

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  1. M. L. Horman, Appl. Opt. 4, 333 (1965).
    [CrossRef]
  2. J. M. Burch, Prod. Eng. 44, 431 (1965).
    [CrossRef]
  3. R. J. Collier, E. T. Doherty, and K. S. Pennington, Appl. Phys. Letters 7, 223 (1965).
    [CrossRef]
  4. R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, Appl. Phys. Letters 7, 248 (1965).
    [CrossRef]
  5. R. L. Powell and K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
    [CrossRef]
  6. N-E. Molin and K. A. Stetson, J. Sci. Instr. (J. Phys. E) 2, 609 (1969).
    [CrossRef]
  7. K. A. Stetson, Optik 29, 386 (1969).
  8. At the time of the drafting of this paper, Refs. 4 and 7 of the Molin and Stetson (1969) paper were not available to the author. Consequently, there may be some unavoidable overlap of the present paper with the following “in press” articles: R. L. Powell, in 1968 Symposium on the Engineering Uses of Holography, Glasgow (Cambridge University Press, Cambridge, 1970); K. A. Stetson, in same collection.
  9. See Refs. 5–7.
  10. A. Kozma, ICO Symposium on Applications of Coherent Light (Florence, 1968).
  11. G. Goertzel and N. Tralli, Some Mathematical Methods of Physics (McGraw–Hill, New York, 1960).
  12. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, Cambridge, 1966).

1969 (2)

N-E. Molin and K. A. Stetson, J. Sci. Instr. (J. Phys. E) 2, 609 (1969).
[CrossRef]

K. A. Stetson, Optik 29, 386 (1969).

1965 (5)

M. L. Horman, Appl. Opt. 4, 333 (1965).
[CrossRef]

J. M. Burch, Prod. Eng. 44, 431 (1965).
[CrossRef]

R. J. Collier, E. T. Doherty, and K. S. Pennington, Appl. Phys. Letters 7, 223 (1965).
[CrossRef]

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, Appl. Phys. Letters 7, 248 (1965).
[CrossRef]

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

Brooks, R. E.

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, Appl. Phys. Letters 7, 248 (1965).
[CrossRef]

Burch, J. M.

J. M. Burch, Prod. Eng. 44, 431 (1965).
[CrossRef]

Collier, R. J.

R. J. Collier, E. T. Doherty, and K. S. Pennington, Appl. Phys. Letters 7, 223 (1965).
[CrossRef]

Doherty, E. T.

R. J. Collier, E. T. Doherty, and K. S. Pennington, Appl. Phys. Letters 7, 223 (1965).
[CrossRef]

Goertzel, G.

G. Goertzel and N. Tralli, Some Mathematical Methods of Physics (McGraw–Hill, New York, 1960).

Heflinger, L. O.

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, Appl. Phys. Letters 7, 248 (1965).
[CrossRef]

Horman, M. L.

Kozma, A.

A. Kozma, ICO Symposium on Applications of Coherent Light (Florence, 1968).

Molin, N-E.

N-E. Molin and K. A. Stetson, J. Sci. Instr. (J. Phys. E) 2, 609 (1969).
[CrossRef]

Pennington, K. S.

R. J. Collier, E. T. Doherty, and K. S. Pennington, Appl. Phys. Letters 7, 223 (1965).
[CrossRef]

Powell, R. L.

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

At the time of the drafting of this paper, Refs. 4 and 7 of the Molin and Stetson (1969) paper were not available to the author. Consequently, there may be some unavoidable overlap of the present paper with the following “in press” articles: R. L. Powell, in 1968 Symposium on the Engineering Uses of Holography, Glasgow (Cambridge University Press, Cambridge, 1970); K. A. Stetson, in same collection.

Stetson, K. A.

N-E. Molin and K. A. Stetson, J. Sci. Instr. (J. Phys. E) 2, 609 (1969).
[CrossRef]

K. A. Stetson, Optik 29, 386 (1969).

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

Tralli, N.

G. Goertzel and N. Tralli, Some Mathematical Methods of Physics (McGraw–Hill, New York, 1960).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, Cambridge, 1966).

Wuerker, R. F.

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, Appl. Phys. Letters 7, 248 (1965).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Letters (2)

R. J. Collier, E. T. Doherty, and K. S. Pennington, Appl. Phys. Letters 7, 223 (1965).
[CrossRef]

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, Appl. Phys. Letters 7, 248 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Sci. Instr. (J. Phys. E) (1)

N-E. Molin and K. A. Stetson, J. Sci. Instr. (J. Phys. E) 2, 609 (1969).
[CrossRef]

Optik (1)

K. A. Stetson, Optik 29, 386 (1969).

Prod. Eng. (1)

J. M. Burch, Prod. Eng. 44, 431 (1965).
[CrossRef]

Other (5)

At the time of the drafting of this paper, Refs. 4 and 7 of the Molin and Stetson (1969) paper were not available to the author. Consequently, there may be some unavoidable overlap of the present paper with the following “in press” articles: R. L. Powell, in 1968 Symposium on the Engineering Uses of Holography, Glasgow (Cambridge University Press, Cambridge, 1970); K. A. Stetson, in same collection.

See Refs. 5–7.

A. Kozma, ICO Symposium on Applications of Coherent Light (Florence, 1968).

G. Goertzel and N. Tralli, Some Mathematical Methods of Physics (McGraw–Hill, New York, 1960).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, Cambridge, 1966).

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Figures (4)

Fig. 1
Fig. 1

Magnitude of characteristic function of one half of a string vibrating simultaneously with amplitudes of one-half and one-quarter wavelength in fundamental modes, respectively. (a) J0 product only, (b) next two Bessel-function products, and (c) five and greater-number Bessel-function products.

Fig. 2
Fig. 2

J0 product of characteristic function for one half of a string, viewed and illuminated from the direction transverse to its axis and in the plane of vibration. One-half-wavelength amplitude of vibration of (a) fundamental and (b) second-harmonic mode, and (c), (a) and (b) simultaneously.

Fig. 3
Fig. 3

J0 product of characteristic function for one-half wavelength of vibration in (a) fundamental mode, (b) fundamental and third harmonic simultaneously, and (c) fundamental, second, and third-harmonic modes simultaneously.

Fig. 4
Fig. 4

J0 product of characteristic function for a vibration amplitude of (a) one-wavelength fundamental mode and (b) one-wavelength fundamental mode and one-half wavelength second-harmonic mode simultaneously.

Equations (27)

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S 1 τ 0 τ exp [ i Φ ( r , t ) ] d t ,
Φ ( r , t ) = n ϕ n ( r ) g n ( t ) .
Φ ( r , t ) = ϕ 0 ( r ) + n = 1 N ϕ n ( r ) cos ( ω n t - θ n ) .
S = e i ϕ 0 ( r ) τ 0 τ exp [ n = 1 N ϕ n ( r ) cos ( ω n t - θ n ) ] d t ,
e i α cos B = m = - ( i ) m J m ( α ) e i m B
e i α sin B = m = - J m ( α ) e i m B .
S = e i ϕ 0 ( r ) τ 0 τ n = 1 N { m = - i m J m [ ϕ n ( r ) ] e i m ( ω n t - θ n ) } d t .
S = e i ϕ 0 ( r ) m = 1 M n = 1 N i l n m J l n m [ ϕ n ( r ) ] e - i l n m θ n ,
n = 1 N l n m ω n = 0.
S = e i ϕ 0 J 0 [ ϕ ( r ) ] .
S = e i ϕ 0 ( r ) n = 1 N J 0 [ ϕ n ( r ) ] .
n = 1 N n l n m = 0.
Φ ( r , t ) = ϕ 0 ( r ) + ϕ 1 ( r ) cos ( ω 0 t - θ 1 ) + ϕ 2 ( r ) cos ( 2 ω 0 t - θ 2 ) .
l 2 m = m m = 0 , ± 1 , ± 2 , l 1 m = - 2 l 2 m .
S = e i ϕ 0 ( r ) m = - i - 2 m J - 2 m ( ϕ 1 ) e - 2 i m θ 1 i m J m ( ϕ 2 ) e i m θ 2 .
S = e i ϕ 0 ( r ) m = 0 m ( - 1 ) m { J 4 m ( ϕ 1 ) J 2 m ( ϕ 2 ) × cos [ 2 m ( θ 2 - 2 θ 1 ) ] + i J 2 ( 2 m + 1 ) ( ϕ 1 ) J 2 m + 1 ( ϕ 2 ) × cos ( 2 m + 1 ) ( θ 2 - 2 θ 1 ) } ,
m = { 1 m = 0 2 m > 0.
Φ ( r , t ) = ϕ 0 ( r ) + ϕ p ( r ) sin ( ω p t - θ p ) + ϕ g ( r ) sin ( ω g t - θ g ) ,
S = e i ϕ 0 ( r ) m = 0 m 2 J g m [ ϕ p ( r ) ] J p m [ ϕ g ( r ) ] [ ( - 1 ) g m × e i m ( g θ p - p θ g ) + ( - 1 ) p m e - i m ( g θ p - p θ g ) ] .
Φ ( r , t ) = ϕ 0 ( r ) + ϕ 1 ( r ) sin ( ω t - θ 1 ) + ϕ 3 ( r ) sin ( 3 ω t - θ 3 ) + ϕ 5 ( r ) sin ( 5 ω t - θ 5 ) .
l 1 m + 3 l 3 m + 5 l 5 m = 0.
( 0 , 0 , 0 ) , ( 0 , 5 , ± 3 ) , ( ± 5 , 0 , 1 ) , ( 3 , ± 1 , 0 ) , ( ± 1 , ± 3 , 2 ) , ( ± 2 , ± 1 , 1 ) , ( 8 , ± 1 , ± 1 ) , .
S = J 0 ( ϕ 1 ) J 0 ( ϕ 3 ) J 0 ( ϕ 5 ) + J 0 ( ϕ 1 ) J 5 ( ϕ 3 ) J ± 3 ( ϕ 5 ) e - i ( 5 θ 3 ± 3 θ 5 ) + J ± 5 ( ϕ 1 ) J 0 ( ϕ 3 ) J 1 ( ϕ 5 ) e - i ( ± 5 θ 1 θ 5 ) + J 3 ( ϕ 1 ) J ± 1 ( ϕ 3 ) J 0 ( ϕ 5 ) e - i ( 3 θ 1 ± θ 3 ) + J ± 1 ( ϕ 1 ) J ± 3 ( ϕ 3 ) J 2 ( ϕ 5 ) e - i ( ± θ 1 ± 3 θ 3 2 θ 5 ) + J ± 2 ( ϕ 1 ) J ± 1 ( ϕ 3 ) J 1 ( ϕ 5 ) e - i ( ± 2 θ 1 ± θ 3 θ 5 ) + J 8 ( θ 1 ) J ± 1 ( ϕ 3 ) J ± 1 ( ϕ 5 ) e - i ( 8 θ 1 ± θ 3 ± θ 5 ) .
y ( x , t ) = n Y n ( 0 ) sin ( n π x L ) cos ( n π c t L ) ,
c 2 = tension ÷ mass per unit length ,
Y n ( 0 ) = 2 L 0 L y ( x ) sin n π x L d x ;
S = e i ϕ 0 ( r ) m = 1 M n = 1 N i l n m e - i l n m θ n J l n m × [ 2 π Y n ( 0 ) sin n π x L ] .