Abstract

It is shown that temporal modulation of the object and/or reference wave in holographic interferometry can advantageously control fringe characteristics. The work presented is based on the concept that the hologram recording process acts as a filter applied to the cross correlation between the conjugate of the reference wave spectrum with the object wave spectrum. The filter is determined by the shuttering of the hologram. Specific types of modulation are considered both experimentally and theoretically. With single-sideband suppressed-carrier modulation of either the reference wave or object wave the hologram process can act as a heterodyne detection system for small-amplitude vibrations or as a system with a controllable number of fringes for large-amplitude vibrations. Certain types of modulation, such as sinusoidal phase modulation of the reference wave, are shown to give the relative phases of the object vibrations. Consideration is given to synthesizing general fringe patterns in multiple exposure holograms. Generalized strobe holography is also analyzed.

© 1971 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. L. Powell, K. K. Stetson, J. Opt. Soc. Amer. 55, 1593 (1965).
    [CrossRef]
  2. P. Waddell, W. Kennedy, New Scientist (1968), p. 633.
  3. E. R. Robertson, J. M. Harvey, The Engineering Uses of Holography (Cambridge U. P., New York, 1970).
  4. D. Denby, J. N. Butters, New Scientist (1970), p. 394.
  5. J. W. Goodman, Appl. Opt. 6, 857 (1967).
    [CrossRef] [PubMed]
  6. P. Shajenko, C. D. Johnson, Appl. Phys. Lett. 13, 44 (1968).
    [CrossRef]
  7. J. T. LaMacchia, J. Appl. Phys. 39, 5340 (1968).
    [CrossRef]
  8. G. M. Mayer, J. Appl. Phys. 40, 2863 (1969).
    [CrossRef]
  9. A. N. Zaidel, L. G. Malkhasyan, G. V. Markova, Y. I. Ostroviskii, Sov. Phys.-Tech. Phys. 13, 1470 (1969).
  10. C. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969).
    [CrossRef]
  11. D. B. Neumann, C. F. Jacobson, G. M. Brown, Appl. Opt. 9, 1357 (1970).
    [CrossRef] [PubMed]
  12. F. M. Mottier, Appl. Phys. Lett. 15, 285 (1969).
    [CrossRef]
  13. C. C. Aleksoff, Appl. Opt. 6, 2192 (1967).
    [CrossRef] [PubMed]
  14. D. B. Neumann, Opt. Soc. Amer. 58, 447 (1968).
    [CrossRef]
  15. It should be noted that χ(0) is very closely related to the mutual coherence function of classical coherence theory and the ambiguity function of radar theory.
  16. L. Bergmann, Ultrasonics (G. Bell and Sons, London, 1938).
  17. G. A. Massey, Proc. IEEE 56, 2157 (1968).
    [CrossRef]
  18. R. L. Whitman, A. Korpel, Appl. Opt. 8, 1567 (1969); A. Korpel, R. L. Whitman, Appl. Opt. 8, 1577 (1969).
    [CrossRef] [PubMed]
  19. A. Macovski, Appl. Phys. Lett. 14, 166 (1969).
    [CrossRef]
  20. M. O. Fein, E. L. Green, Appl. Opt. 7, 1864 (1968).
    [CrossRef] [PubMed]
  21. N. E. Molin, K. A. Stetson, J. Sci. Instrum. 2, 609 (1969).
    [CrossRef]
  22. A. D. Wilson, J. Opt. Soc. Amer. 60, 1068 (1970).
    [CrossRef]
  23. C. C. Aleksoff, “Multi-Mode Lasers in Interferometry and Holography,” Thesis, U. of Michigan, 1969 (order No. 69-17,955 from University Microfilms, Ann Arbor, Mich.).
  24. J. F. Stephany, J. Opt. Soc. Amer. 55, 136 (1965).
    [CrossRef]
  25. D. B. Neumann, H. W. Rose, Appl. Opt. 6, 1097 (1967).
    [CrossRef] [PubMed]
  26. J. W. Goodman, D. W. Jackson, M. Lehmann, J. Knotts, Appl. Opt. 8, 1581 (1969).
    [CrossRef] [PubMed]
  27. J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, Appl. Phys. Lett. 8, 311 (1966).
    [CrossRef]
  28. F. M. Mottier, Appl. Phys. Lett. 15, 44 (1969).
    [CrossRef]
  29. H. J. Caulfield, Appl. Phys. Lett. 16, 234 (1970).
    [CrossRef]
  30. F. M. Mottier, private communication.See also Nouv. Rev. d’Optique Appliquee 1, Suppl. No. 2 (1970).
  31. R. C. Cumming, Proc. IRE 45, 175 (1957).
    [CrossRef]
  32. M. Gottlieb, M. Crarbuory, Appl. Opt. 7, 2238 (1968).
    [CrossRef] [PubMed]
  33. K. A. Stetson, J. Opt. Soc. Amer. 60, 1378 (1970).
    [CrossRef]
  34. B. P. Hildebrand, J. Opt. Soc. Amer. 60, 1511 (1970).
    [CrossRef]

1970 (6)

D. Denby, J. N. Butters, New Scientist (1970), p. 394.

D. B. Neumann, C. F. Jacobson, G. M. Brown, Appl. Opt. 9, 1357 (1970).
[CrossRef] [PubMed]

A. D. Wilson, J. Opt. Soc. Amer. 60, 1068 (1970).
[CrossRef]

H. J. Caulfield, Appl. Phys. Lett. 16, 234 (1970).
[CrossRef]

K. A. Stetson, J. Opt. Soc. Amer. 60, 1378 (1970).
[CrossRef]

B. P. Hildebrand, J. Opt. Soc. Amer. 60, 1511 (1970).
[CrossRef]

1969 (9)

F. M. Mottier, Appl. Phys. Lett. 15, 44 (1969).
[CrossRef]

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

J. W. Goodman, D. W. Jackson, M. Lehmann, J. Knotts, Appl. Opt. 8, 1581 (1969).
[CrossRef] [PubMed]

R. L. Whitman, A. Korpel, Appl. Opt. 8, 1567 (1969); A. Korpel, R. L. Whitman, Appl. Opt. 8, 1577 (1969).
[CrossRef] [PubMed]

A. Macovski, Appl. Phys. Lett. 14, 166 (1969).
[CrossRef]

F. M. Mottier, Appl. Phys. Lett. 15, 285 (1969).
[CrossRef]

G. M. Mayer, J. Appl. Phys. 40, 2863 (1969).
[CrossRef]

A. N. Zaidel, L. G. Malkhasyan, G. V. Markova, Y. I. Ostroviskii, Sov. Phys.-Tech. Phys. 13, 1470 (1969).

C. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969).
[CrossRef]

1968 (7)

P. Shajenko, C. D. Johnson, Appl. Phys. Lett. 13, 44 (1968).
[CrossRef]

J. T. LaMacchia, J. Appl. Phys. 39, 5340 (1968).
[CrossRef]

P. Waddell, W. Kennedy, New Scientist (1968), p. 633.

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

G. A. Massey, Proc. IEEE 56, 2157 (1968).
[CrossRef]

M. O. Fein, E. L. Green, Appl. Opt. 7, 1864 (1968).
[CrossRef] [PubMed]

M. Gottlieb, M. Crarbuory, Appl. Opt. 7, 2238 (1968).
[CrossRef] [PubMed]

1967 (3)

1966 (1)

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, Appl. Phys. Lett. 8, 311 (1966).
[CrossRef]

1965 (2)

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

J. F. Stephany, J. Opt. Soc. Amer. 55, 136 (1965).
[CrossRef]

1957 (1)

R. C. Cumming, Proc. IRE 45, 175 (1957).
[CrossRef]

Aleksoff, C. C.

C. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969).
[CrossRef]

C. C. Aleksoff, Appl. Opt. 6, 2192 (1967).
[CrossRef] [PubMed]

C. C. Aleksoff, “Multi-Mode Lasers in Interferometry and Holography,” Thesis, U. of Michigan, 1969 (order No. 69-17,955 from University Microfilms, Ann Arbor, Mich.).

Bergmann, L.

L. Bergmann, Ultrasonics (G. Bell and Sons, London, 1938).

Brown, G. M.

Butters, J. N.

D. Denby, J. N. Butters, New Scientist (1970), p. 394.

Caulfield, H. J.

H. J. Caulfield, Appl. Phys. Lett. 16, 234 (1970).
[CrossRef]

Crarbuory, M.

Cumming, R. C.

R. C. Cumming, Proc. IRE 45, 175 (1957).
[CrossRef]

Denby, D.

D. Denby, J. N. Butters, New Scientist (1970), p. 394.

Fein, M. O.

Goodman, J. W.

Gottlieb, M.

Green, E. L.

Harvey, J. M.

E. R. Robertson, J. M. Harvey, The Engineering Uses of Holography (Cambridge U. P., New York, 1970).

Hildebrand, B. P.

B. P. Hildebrand, J. Opt. Soc. Amer. 60, 1511 (1970).
[CrossRef]

Huntley, W. H.

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, Appl. Phys. Lett. 8, 311 (1966).
[CrossRef]

Jackson, D. W.

J. W. Goodman, D. W. Jackson, M. Lehmann, J. Knotts, Appl. Opt. 8, 1581 (1969).
[CrossRef] [PubMed]

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, Appl. Phys. Lett. 8, 311 (1966).
[CrossRef]

Jacobson, C. F.

Johnson, C. D.

P. Shajenko, C. D. Johnson, Appl. Phys. Lett. 13, 44 (1968).
[CrossRef]

Kennedy, W.

P. Waddell, W. Kennedy, New Scientist (1968), p. 633.

Knotts, J.

Korpel, A.

LaMacchia, J. T.

J. T. LaMacchia, J. Appl. Phys. 39, 5340 (1968).
[CrossRef]

Lehmann, M.

J. W. Goodman, D. W. Jackson, M. Lehmann, J. Knotts, Appl. Opt. 8, 1581 (1969).
[CrossRef] [PubMed]

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, Appl. Phys. Lett. 8, 311 (1966).
[CrossRef]

Macovski, A.

A. Macovski, Appl. Phys. Lett. 14, 166 (1969).
[CrossRef]

Malkhasyan, L. G.

A. N. Zaidel, L. G. Malkhasyan, G. V. Markova, Y. I. Ostroviskii, Sov. Phys.-Tech. Phys. 13, 1470 (1969).

Markova, G. V.

A. N. Zaidel, L. G. Malkhasyan, G. V. Markova, Y. I. Ostroviskii, Sov. Phys.-Tech. Phys. 13, 1470 (1969).

Massey, G. A.

G. A. Massey, Proc. IEEE 56, 2157 (1968).
[CrossRef]

Mayer, G. M.

G. M. Mayer, J. Appl. Phys. 40, 2863 (1969).
[CrossRef]

Molin, N. E.

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

Mottier, F. M.

F. M. Mottier, Appl. Phys. Lett. 15, 285 (1969).
[CrossRef]

F. M. Mottier, Appl. Phys. Lett. 15, 44 (1969).
[CrossRef]

F. M. Mottier, private communication.See also Nouv. Rev. d’Optique Appliquee 1, Suppl. No. 2 (1970).

Neumann, D. B.

Ostroviskii, Y. I.

A. N. Zaidel, L. G. Malkhasyan, G. V. Markova, Y. I. Ostroviskii, Sov. Phys.-Tech. Phys. 13, 1470 (1969).

Powell, R. L.

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

Robertson, E. R.

E. R. Robertson, J. M. Harvey, The Engineering Uses of Holography (Cambridge U. P., New York, 1970).

Rose, H. W.

Shajenko, P.

P. Shajenko, C. D. Johnson, Appl. Phys. Lett. 13, 44 (1968).
[CrossRef]

Stephany, J. F.

J. F. Stephany, J. Opt. Soc. Amer. 55, 136 (1965).
[CrossRef]

Stetson, K. A.

K. A. Stetson, J. Opt. Soc. Amer. 60, 1378 (1970).
[CrossRef]

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

Stetson, K. K.

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

Waddell, P.

P. Waddell, W. Kennedy, New Scientist (1968), p. 633.

Whitman, R. L.

Wilson, A. D.

A. D. Wilson, J. Opt. Soc. Amer. 60, 1068 (1970).
[CrossRef]

Zaidel, A. N.

A. N. Zaidel, L. G. Malkhasyan, G. V. Markova, Y. I. Ostroviskii, Sov. Phys.-Tech. Phys. 13, 1470 (1969).

Appl. Opt. (8)

Appl. Phys. Lett. (7)

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, Appl. Phys. Lett. 8, 311 (1966).
[CrossRef]

F. M. Mottier, Appl. Phys. Lett. 15, 44 (1969).
[CrossRef]

H. J. Caulfield, Appl. Phys. Lett. 16, 234 (1970).
[CrossRef]

A. Macovski, Appl. Phys. Lett. 14, 166 (1969).
[CrossRef]

F. M. Mottier, Appl. Phys. Lett. 15, 285 (1969).
[CrossRef]

P. Shajenko, C. D. Johnson, Appl. Phys. Lett. 13, 44 (1968).
[CrossRef]

C. C. Aleksoff, Appl. Phys. Lett. 14, 23 (1969).
[CrossRef]

J. Appl. Phys. (2)

J. T. LaMacchia, J. Appl. Phys. 39, 5340 (1968).
[CrossRef]

G. M. Mayer, J. Appl. Phys. 40, 2863 (1969).
[CrossRef]

J. Opt. Soc. Amer. (5)

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

A. D. Wilson, J. Opt. Soc. Amer. 60, 1068 (1970).
[CrossRef]

K. A. Stetson, J. Opt. Soc. Amer. 60, 1378 (1970).
[CrossRef]

B. P. Hildebrand, J. Opt. Soc. Amer. 60, 1511 (1970).
[CrossRef]

J. F. Stephany, J. Opt. Soc. Amer. 55, 136 (1965).
[CrossRef]

J. Sci. Instrum. (1)

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

New Scientist (2)

P. Waddell, W. Kennedy, New Scientist (1968), p. 633.

D. Denby, J. N. Butters, New Scientist (1970), p. 394.

Opt. Soc. Amer. (1)

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

Proc. IEEE (1)

G. A. Massey, Proc. IEEE 56, 2157 (1968).
[CrossRef]

Proc. IRE (1)

R. C. Cumming, Proc. IRE 45, 175 (1957).
[CrossRef]

Sov. Phys.-Tech. Phys. (1)

A. N. Zaidel, L. G. Malkhasyan, G. V. Markova, Y. I. Ostroviskii, Sov. Phys.-Tech. Phys. 13, 1470 (1969).

Other (5)

E. R. Robertson, J. M. Harvey, The Engineering Uses of Holography (Cambridge U. P., New York, 1970).

It should be noted that χ(0) is very closely related to the mutual coherence function of classical coherence theory and the ambiguity function of radar theory.

L. Bergmann, Ultrasonics (G. Bell and Sons, London, 1938).

C. C. Aleksoff, “Multi-Mode Lasers in Interferometry and Holography,” Thesis, U. of Michigan, 1969 (order No. 69-17,955 from University Microfilms, Ann Arbor, Mich.).

F. M. Mottier, private communication.See also Nouv. Rev. d’Optique Appliquee 1, Suppl. No. 2 (1970).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

The filtering process for a discrete object wave spectrum and for SSSC modulation of the reference wave. (a) The object wave spectrum. (b) The reference wave spectrum for Δf = f2. (c) The c-c spectrum and the sinc Tf filter function. (d) The filtered output (the transform of the exposure function).

Fig. 2
Fig. 2

Plots of some Bessel and related functions. Jn(MN) is the nth-order Bessel function of the first kind.

Fig. 3
Fig. 3

Schematic diagram for making holograms of ultrasonic beams. The reference wave is the first-order diffracted wave from the diffraction cell.

Fig. 4
Fig. 4

Reconstructions showing the path of ultrasonic traveling waves. Both holograms were made with a collimated object wave but the angle of incidence onto the ultrasonic beam was slightly different for the two cases.

Fig. 5
Fig. 5

Picture showing how only a small section of the path of an ultrasonic beam is reconstructed when a diffuser is placed in the object wave. The lower smudge of light is an extraneous reflection in the system.

Fig. 6
Fig. 6

Schematic diagram for making modulated holograms of a vibrating loudspeaker.

Fig. 7
Fig. 7

Detecting small-vibration amplitudes. F ≃ 12 kHz. (a) Normal (SSSC0) hologram of the vibrating speaker. (b) SSSC1 hologram of the same vibrating speaker with the same vibration amplitude as in (a).

Fig. 8
Fig. 8

Comparison of SSSC modulation in interferometry and holography. Left column is the interferometric case and the right column is the holographic case. (a) Nonmodulated reference wave and stationary object; (b) zero order; (c) first order; (d) seventh order; (e) ninth order.

Fig. 9
Fig. 9

SSSC holography of a loudspeaker. (a) zero order; (b) first order; (c) ninth order; (d) twenty-eighth order.

Fig. 10
Fig. 10

SSSC holography of loudspeaker with harder drive than in Fig. 9. (a) Zero order; (b) seventy-fifth order + some zero order.

Fig. 11
Fig. 11

The zeros of the Bessel functions. Jp(jp,q) = 0. Here jp,q, is the qth zero for the pth-order Bessel function and j n , 1 is the primary maximum of the nth-order Bessel function. M = 4π (A/λ).

Fig. 12
Fig. 12

Theoretical fringe synthesis. (a) Ninth order alone; (b) seventh order plus ninth order; (c) seventh order minus ninth order.

Fig. 13
Fig. 13

Experimental fringe synthesis. (a) SSSC0 hologram; (b) J7(M) + J9(M) synthesis; (c) J7(M) − J9(M) synthesis.

Fig. 14
Fig. 14

Schematic of holographic setup for detecting the shear-wave resonances of the ADP crystal.

Fig. 15
Fig. 15

Experimental observations of the ADP crystal resonances. Figures (a) through (c) are of the third-order resonance (109 kHz), and (d) is the thirteenth order (465 kHz). (a) Between crossed polaroids; (b) normal time-average holograms; (c) with PM reference wave; modulation in phase with central lobe and with modulation depth ≈ 1.5; (d) with PM reference wave; modulation in phase with central lobe and with modulation depth ≈ 6; (e) SSSC1 hologram; (f) with PM reference wave; Modulation depth ≈ 1.

Equations (85)

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

V o ( t ) = U o ( t ) exp ( i ω c t )
V r ( t ) = U r ( t ) exp ( i ω c t ) .
E ( t , T ) = t - ( T / 2 ) t + ( T / 2 ) V o ( t ) + V r ( t ) 2 d t ,
A * ( t ) B ( t ) = - A * ( t ) B ( t + t ) d t ,
E ( t , T ) = rect ( t / T ) [ U o * ( t ) U o ( t ) + U r * ( t ) U r ( t ) + U o * ( t ) U r ( t ) + U r * ( t ) U o ( t ) ] ,
rect ( t / T ) = { 1     if - ( T / 2 ) < t < T / 2 , 0     otherwise .
E r o ( t , T ) = t - ( T / 2 ) t + ( T / 2 ) U r * ( t ) U o ( t ) d t = rect ( t / T ) [ U r * ( t ) U o ( t ) ] .
e r o ( f , T ) = χ ( f ) · T sinc T f ,
sinc T f = ( sin π T f ) / π T f
χ ( f ) = u r * ( f ) u o ( f )
e r o lim T e r o ( f , T ) = χ ( 0 ) δ ( f ) ,
E r o lim T E r o ( t , T ) = χ ( 0 ) .
lim T T sinc T f = δ ( f ) ,
χ ( f ) = u o ( f + Δ f ) exp ( - i ϕ r ) ;
χ ( 0 ) = u o ( Δ f ) exp ( - i ϕ r ) .
u o ( f ) = n a n δ ( f - f n )
U o ( t ) = exp [ i M o sin ( Ω t + ϕ o ) ] .
u o ( f ) = n = - J n ( M o ) exp ( i n ϕ o ) δ ( f - n F ) ,
χ ( 0 ) = J p ( M o ) exp [ i ( p ϕ o - ϕ r ) ]
χ ( 0 ) 2 = J 0 2 ( M o ) .
U = 2 [ ( I o I r ) 1 2 / ( I o + I r ) ] .
U p = v J p ( M o ) .
g ( M ) = n = - A n J n ( M ) ,
J n ( M ) ( 2 / π M ) 1 2 cos [ M - ( π n / 2 ) - ( π / 4 ) ] ,
[ J n ( M ) + J n ± 2 ( M ) ] 0
[ J n ( M ) - J n ± 2 ( M ) ] 2 J n ( M ) .
U o ( t ) = exp [ i ψ ( t ) ] ,
ψ ( t ) = n M n sin ( Ω n t + ϕ n ) .
U o ( t ) = n exp [ i M n sin ( Ω n t + ϕ n ) ] .
A · B · C · D = a * b * c * d * ,
u o ( f ) = n * { p = - J p ( M n ) exp ( i p ϕ n ) δ ( f - p F n ) } ,
ψ ( t ) = M 1 sin ( Ω 1 t + ϕ 1 ) + M 2 sin ( Ω 2 t ) ;
u o ( f ) = [ p = - J p ( M 1 ) exp ( i p ϕ 1 ) δ ( f - p F 1 ) ] * [ p = - J q ( M 2 ) δ ( f - q F 2 ) ] = p = - q = - J p ( M 1 ) J - q ( M 2 ) × exp ( i p ϕ 1 ) δ ( f - p F 1 + q F 2 ) .
| p = - J p ( M 1 ) J - p k + n ( M 2 ) exp ( i p ϕ 1 ) | 2 .
U o ( t ) = n = - a n exp [ i ( n Ω t + α n ) ]
U r ( t ) = n = - b n exp [ i ( n Ω t + β n ) ] ,
χ ( 0 ) = n = - a n b n exp [ i ( α n - β n ) ] .
U o ( t ) = exp [ i n = - a n sin ( n Ω t + α n ) ]
U r ( t ) = exp [ i n = - b n sin ( n Ω t + β n ) ] .
U r * ( t ) U o ( t ) = n = - exp i [ c n sin ( n Ω t + γ n ) ] ,
c n exp ( i γ n ) = a n exp ( i α n ) - b n exp ( i β n ) .
χ ( 0 ) = n = - J 0 ( c n ) .
χ ( 0 ) = J 0 ( c 1 ' ) ,
c 1 2 = a 1 2 + b 1 2 - a 1 b 1 cos ( α 1 - β 1 ) .
U r * ( t ) U o ( t ) d t 2 U r ( t ) 2 d t U o ( t ) 2 d t ,
U r * ( t ) U o ( t ) 2 U r * ( t ) 2 d t U o ( t ) 2 d t .
U o ( t ) = U ( t - t ) U o ( t )
U r ( t ) = U ( t - t ) U r ( t ) ,
E r o ( t ) = - U r * ( t ) U o ( t ) d t = - U ( t - t ) 2 U r * ( t ) U o ( t ) d t = S ( t ) [ U r * ( t ) U o ( t ) ] ,
S ( t ) = U ( t ) 2 .
e r o ( f ) = s * ( f ) χ ( f ) ,
χ ( f ) = u r * ( f ) u o ( f ) .
S ( t ) = rect ( t / T ) 2 = rect ( t / T ) .
S ( t ) = ( 1 / T 1 T 2 ) n = - H [ ( 1 / T 1 ) ( t - n T 2 ) ] rect ( t / T ) .
s * ( f ) = T n = - h * ( T 1 n / T 2 ) sinc [ T ( f - n / T 2 ) ] .
E r o = n = - h * ( n T 1 / T 2 ) χ ( n / T 2 ) .
E r o = n = - χ ( n / T 2 ) .
E r o = n = - J n q ( M o ) exp ( i n q ϕ o ) .
q = 1 :             Sampling once each period of vibration E r o 2 = exp ( i M o sin ϕ o ) 2 = 1 ,
q = 2 :             Sampling twice each period E r o 2 = cos ( M o sin ϕ o ) 2 = 1 2 [ 1 - cos ( 2 M o sin ϕ o ) ] ,
q :             High sample rate E r o 2 J 0 2 ( M o ) ,
S ( t ) = | n = 0 N a n exp ( i n M sin Ω t ) | 2
s * ( f ) = n = - b n δ ( f - n F ) ,
b n = m = - N n c m J n ( m M )
c m = { q = 0 N - m a q + m a * q if m 0 , q = 0 N + m a * q - m a q if m 0 .
E r o = n = - b n χ ( n F ) = m = - N N n = - c m J n ( m M ) J n ( M o ) exp ( i n ϕ o ) = m = - N n c m J 0 ( m M - M o exp ( i ϕ o ) ) ,
E r o = m = - N N ( N + 1 - m ) J 0 ( m M - M o ) .
S * ( t ) [ U r * ( t ) · U o ( t ) ] .
F [ A * ( t ) B ( t ) ] = a * ( f ) · b ( f )
F [ C * ( t ) · D ( t ) ] = c * ( f ) d ( f ) ,
F { S * ( t ) [ U r * ( t ) · U o ( t ) ] } = s * ( f ) · F [ U r * ( t ) · U o ( t ) ] = s * ( f ) · [ u r * ( f ) u o ( f ) ] = s * ( f ) · χ ( f ) ,
S ( t ) = [ ( 1 / T 2 ) ( 1 / T 1 ) H ( t / T 1 ) * n = - δ ( t - n T 2 ) ] · rect ( t / T ) .
F [ ( 1 / T 1 ) H ( t / T 1 ) ] = h ( T 1 f ) ,
F [ ( 1 / T 2 ) · n = - δ ( t - n T 2 ) ] = n = - δ ( f - n / T 2 )
s ( f ) = [ h ( T 1 f ) · n = - δ ( f - n / T 2 ) ] * T sinc T f = n = - [ h ( n T 1 / T 2 ) δ ( f - n / T 2 ) * T sinc T f ] .
s ( f ) = T n = - h ( n T 1 / T 2 ) sinc [ T ( f - n / T 2 ) ] .
{ exp [ i a sin ( Ω t + a ) ] } * · { exp [ i b sin ( Ω t + β ) ] } = exp [ i c sin ( Ω t + γ ) ] ,
c exp ( i γ ) = b exp ( i β ) - a exp ( i α ) .
[ n J n ( a ) exp ( i n α ) δ ( f - n F ) ] * [ p J p ( b ) exp ( i p β ) δ ( f - p F ) ] = [ q J q ( c ) exp ( i q γ ) δ ( f - q F ) ] ,
q exp ( i q β ) δ ( f - q F ) n J n ( a ) J n + q ( b ) exp [ i n ( β - α ) ] = q J q ( c ) exp ( i q γ ) δ ( f - q F ) .
J q ( c ) exp [ i q ( γ - β ) ] = n J n ( a ) J n + q ( b ) exp [ i n ( β - α ) ] ,
J 0 ( c ) = n J n ( a ) J n ( b ) exp [ i n ( β - α ) ] .
Z ( t ) = ( M / 2 T ) n = - ( t - n T ) · rect ( F t - n ) .
u ( f ) = n = - sinc ( n - M / Z 1 2 ) δ ( f - n F ) .
sinc ( n - p ) = { 1 if n = p , 0 if n p .

Metrics