Abstract

Assuming some parallelism between space and time variables, namely, in the frequency domain, that occurs in the description of optical signals as functions of space or/and space and time, the Abbe theory applies to temporal distributions. The concept of temporal response of any optical system, space invariant, working at the time frequency ν = c/λ, is then brought out. A method of temporal encoding of optical information is also reported, leading to a consideration of the output of spectroscopic devices as a Temporal Fourier Hologram (TFH). This applies to metrology (interferometry in white light, surface testing, roughness measurements) and image processing by temporal holography. Holograms of extended self-luminous objects light, and reconstructed images are presented.

© 1977 Optical Society of America

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References

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  1. C. Froehly, A. Lacourt, J. C. Viénot, Nouv. Rev. Opt. 4 (4, 183 (1973).
    [CrossRef]
  2. J. C. Viénot, J. Duvernoy, C. Froehly, A. La Court, “Models of Information Transfer,” invited paper, at International Conference on Optical Processing, Sydney, 19–21 August 1974.
  3. P. M. Duffieux, L’Intégrale de Fourier et ses applications à l’optique (Masson Editions, Paris, 1970). The present paper was prepared as a survey of the concepts of temporal holography and represents part of an on-going study. Many references have been given with preliminary results published in recent years.
  4. A. Lacourt, J. C. Viénot, J. P. Goedgebuer, in Proceedings International Conference on Optical Measurements, Tokyo, August1974; Jpn. J. Appl. Phys. (1975), pp. 357–362.
  5. J. C. Viénot, A. Lacourt, J. P. Goedgebuer, in Proceedings International Optical Computing Conference, Washington D.C., April1975, pp. 133–136.
  6. A. Lacourt, Thèse Doctorat, Besançon (1975).
  7. J. P. Goedgebuer, Thèse Doctorat, Besançon (1975).
  8. J. P. Goedgebuer, A. Lacourt, J. C. Viénot, Opt. Commun. 16, 99 (1976).
    [CrossRef]

1976 (1)

J. P. Goedgebuer, A. Lacourt, J. C. Viénot, Opt. Commun. 16, 99 (1976).
[CrossRef]

1973 (1)

C. Froehly, A. Lacourt, J. C. Viénot, Nouv. Rev. Opt. 4 (4, 183 (1973).
[CrossRef]

Duffieux, P. M.

P. M. Duffieux, L’Intégrale de Fourier et ses applications à l’optique (Masson Editions, Paris, 1970). The present paper was prepared as a survey of the concepts of temporal holography and represents part of an on-going study. Many references have been given with preliminary results published in recent years.

Duvernoy, J.

J. C. Viénot, J. Duvernoy, C. Froehly, A. La Court, “Models of Information Transfer,” invited paper, at International Conference on Optical Processing, Sydney, 19–21 August 1974.

Froehly, C.

C. Froehly, A. Lacourt, J. C. Viénot, Nouv. Rev. Opt. 4 (4, 183 (1973).
[CrossRef]

J. C. Viénot, J. Duvernoy, C. Froehly, A. La Court, “Models of Information Transfer,” invited paper, at International Conference on Optical Processing, Sydney, 19–21 August 1974.

Goedgebuer, J. P.

J. P. Goedgebuer, A. Lacourt, J. C. Viénot, Opt. Commun. 16, 99 (1976).
[CrossRef]

J. P. Goedgebuer, Thèse Doctorat, Besançon (1975).

A. Lacourt, J. C. Viénot, J. P. Goedgebuer, in Proceedings International Conference on Optical Measurements, Tokyo, August1974; Jpn. J. Appl. Phys. (1975), pp. 357–362.

J. C. Viénot, A. Lacourt, J. P. Goedgebuer, in Proceedings International Optical Computing Conference, Washington D.C., April1975, pp. 133–136.

La Court, A.

J. C. Viénot, J. Duvernoy, C. Froehly, A. La Court, “Models of Information Transfer,” invited paper, at International Conference on Optical Processing, Sydney, 19–21 August 1974.

Lacourt, A.

J. P. Goedgebuer, A. Lacourt, J. C. Viénot, Opt. Commun. 16, 99 (1976).
[CrossRef]

C. Froehly, A. Lacourt, J. C. Viénot, Nouv. Rev. Opt. 4 (4, 183 (1973).
[CrossRef]

A. Lacourt, J. C. Viénot, J. P. Goedgebuer, in Proceedings International Conference on Optical Measurements, Tokyo, August1974; Jpn. J. Appl. Phys. (1975), pp. 357–362.

J. C. Viénot, A. Lacourt, J. P. Goedgebuer, in Proceedings International Optical Computing Conference, Washington D.C., April1975, pp. 133–136.

A. Lacourt, Thèse Doctorat, Besançon (1975).

Viénot, J. C.

J. P. Goedgebuer, A. Lacourt, J. C. Viénot, Opt. Commun. 16, 99 (1976).
[CrossRef]

C. Froehly, A. Lacourt, J. C. Viénot, Nouv. Rev. Opt. 4 (4, 183 (1973).
[CrossRef]

J. C. Viénot, J. Duvernoy, C. Froehly, A. La Court, “Models of Information Transfer,” invited paper, at International Conference on Optical Processing, Sydney, 19–21 August 1974.

A. Lacourt, J. C. Viénot, J. P. Goedgebuer, in Proceedings International Conference on Optical Measurements, Tokyo, August1974; Jpn. J. Appl. Phys. (1975), pp. 357–362.

J. C. Viénot, A. Lacourt, J. P. Goedgebuer, in Proceedings International Optical Computing Conference, Washington D.C., April1975, pp. 133–136.

Nouv. Rev. Opt. (1)

C. Froehly, A. Lacourt, J. C. Viénot, Nouv. Rev. Opt. 4 (4, 183 (1973).
[CrossRef]

Opt. Commun. (1)

J. P. Goedgebuer, A. Lacourt, J. C. Viénot, Opt. Commun. 16, 99 (1976).
[CrossRef]

Other (6)

J. C. Viénot, J. Duvernoy, C. Froehly, A. La Court, “Models of Information Transfer,” invited paper, at International Conference on Optical Processing, Sydney, 19–21 August 1974.

P. M. Duffieux, L’Intégrale de Fourier et ses applications à l’optique (Masson Editions, Paris, 1970). The present paper was prepared as a survey of the concepts of temporal holography and represents part of an on-going study. Many references have been given with preliminary results published in recent years.

A. Lacourt, J. C. Viénot, J. P. Goedgebuer, in Proceedings International Conference on Optical Measurements, Tokyo, August1974; Jpn. J. Appl. Phys. (1975), pp. 357–362.

J. C. Viénot, A. Lacourt, J. P. Goedgebuer, in Proceedings International Optical Computing Conference, Washington D.C., April1975, pp. 133–136.

A. Lacourt, Thèse Doctorat, Besançon (1975).

J. P. Goedgebuer, Thèse Doctorat, Besançon (1975).

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Figures (16)

Fig. 1
Fig. 1

Representation of white light: model of wave group.

Fig. 2
Fig. 2

Response of a two-beam interferometer to (a) a Dirac impulse δ(t); (b) a function f(t) representing white light. The frequency responses H(ν), then H(νF(ν) are shown on the right.

Fig. 3
Fig. 3

Frequency response for various optical devices (Michelson, Fabry-Perot, Young’s slits, regular grating): square moduli.

Fig. 4
Fig. 4

Diffraction of the time signal at a geometrical pupil.

Fig. 5
Fig. 5

Time responses given by various apertures.

Fig. 6
Fig. 6

Quadratic response of an ideal spectroscopic device to a pair of impulses of white light: channeled spectrum.

Fig. 7
Fig. 7

Photograph of a channeled spectrum.

Fig. 8
Fig. 8

Formation of a channeled spectrum by means of a spectroscope in cascade with a two-beam interferometer.

Fig. 9
Fig. 9

Images of spectrograms corresponding to (a) a polish spherical surface; (b) a ground surface.

Fig. 10
Fig. 10

Diffraction of a channeled spectrum considered as a grating.

Fig. 11
Fig. 11

Diffracted images in the x‴-plane (see Fig. 8): (a) spherical surface; (b) ground surface.

Fig. 12
Fig. 12

Attaining the autocorrelation function of a pair of wave trains.

Fig. 13
Fig. 13

Access to a spherical surface profile by means of FT spectroscopy. One should note that the quantity 2ΔZ is observed.

Fig. 14
Fig. 14

Temporal coding of information (geometrical rectangular distribution).

Fig. 15
Fig. 15

(a) Temporal hologram of the geometrical rectangular distribution (channeled spectrum modulated by |sinc|-function). (b) Reconstruction of the image of the rectangular aperture from a time Fourier hologram (TFH).

Fig. 16
Fig. 16

(a) THF of a glow lamp filament. (b) +1 and −1 orders observed in the reconstruction of the image from THF.

Equations (22)

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g ( x ) G ( u ) in the transverse domain , and
g ( c t sin θ ) G ( ν sin θ c ) in the longitudinal domain ,
δ ( x t ) F ( u ν ) ( S ) H ( u ν ) h ( x t ) ,
f ( x t ) F ( u ν ) ( S ) F = F . H f = ( f h ) ( x t ) .
F ( u ) = F ( u ) · rect ( u - u 0 Δ u ) = F ( u ) · H ( u ) ,
h ( x ) = FT [ F ( u ) ] s i n π Δ u x π Δ u x · exp ( j 2 π u 0 x ) .
H ( ν ) = rect ( ν - ν 0 Δ ν )
F ( ν ) = F ( ν ) · H ( ν ) ,
h ( t ) = FT [ F ( ν ) ] sin π Δ ν t π Δ ν t · exp ( j 2 π ν 0 t ) .
f ( t ) = - + F ( ν ) exp ( j 2 π ν t ) d ν
F ( ν , u ) = - j λ - + F ( ν ) g ( x ) exp ( - j 2 π u x ) d x ,
F ( ν , u ) = - j ν c F ( ν ) - + g ( x ) exp ( - j 2 π u x ) d x .
F ( ν , u ) = - j ν c F ( ν ) G ( ν c sin θ ) ,
F ( ν , θ ) 2 = ν 2 F ( ν ) 2 | G ( ν c sinc θ ) | 2 .
FT - 1 [ F ] FT - 1 [ - j ν c F · G ] = FT - 1 [ F ] FT - 1 [ - j ν c G ] ,
- j 2 π ν c G ( ν ) = FT [ d d t g ( t ) ] ,
f = f d d t g ( apart from a factor 1 2 π ) .
t = x sin θ c or x = c t sin θ ;
f ( t ) = f ( t ) d d t [ proj θ g ( x ) ] .
rect x w g ( c t sin θ ) = rect ( c t w sin θ ) ,
Δ ( z ) = 2 z ( x )
Δ σ ( x ) = 1 / Δ ( z ) = 1 / 2 z ( x ) .

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