Abstract

Extension of space–time optics to objects whose transparency is a function of the temporal frequency ν = c/λ is examined. Considering the effects of such stationary pupils on white light waves, they are called temporal pupils. It is shown that simultaneous encoding both in the space and time frequency domains is required to record pupil parameters. The space–time impulse response and transfer functions are calculated for a dispersive nonabsorbent material. An experimental method providing holographic recording of the dispersion curve of any transparent material is presented.

© 1981 Optical Society of America

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References

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  1. J. C. Viénot, J. P. Goedgebuer, A. Lacourt, Appl. Opt. 16, 454 (1977).
    [CrossRef] [PubMed]
  2. C. Frohely, A. Lacourt, J. C. Viénot, Nouv. Rev. Opt. 4, 183 (1973).
    [CrossRef]
  3. η is considered as a constant in order to neglect the dispersion of the pupil.
  4. J. P. Goedgebuer, A. Lacourt, J. C. Viénot, Opt. Commun. 16, 99 (1976).
    [CrossRef]
  5. Obviously a transparent and nondispersive pupil has a ν dependent transparency function exp[j2π(ν/c)ne(x)]; but the delay does not depend on ν.
  6. Although a geometrical transverse pupil has a ν-independent transparency function, the spectrum of a white light wave is filtered by diffraction on g(x); the diffraction process is ν-dependent and responsible for chromatic encoding of g(x).
  7. J. Calatroni, Opt. Commun. 19, 49 (1976).
    [CrossRef]
  8. In fact a quadratic phase factor should be present in Eq. (13), but it has no effect on the amplitude distribution at the x′ and x″ planes.
  9. H. Bouasse, Z. Carrière, Interferences (Delagrave Ed., Paris, 1923).

1977 (1)

1976 (2)

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

J. Calatroni, Opt. Commun. 19, 49 (1976).
[CrossRef]

1973 (1)

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

Bouasse, H.

H. Bouasse, Z. Carrière, Interferences (Delagrave Ed., Paris, 1923).

Calatroni, J.

J. Calatroni, Opt. Commun. 19, 49 (1976).
[CrossRef]

Carrière, Z.

H. Bouasse, Z. Carrière, Interferences (Delagrave Ed., Paris, 1923).

Frohely, C.

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

Goedgebuer, J. P.

J. C. Viénot, J. P. Goedgebuer, A. Lacourt, Appl. Opt. 16, 454 (1977).
[CrossRef] [PubMed]

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

Lacourt, A.

J. C. Viénot, J. P. Goedgebuer, A. Lacourt, Appl. Opt. 16, 454 (1977).
[CrossRef] [PubMed]

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

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

Viénot, J. C.

J. C. Viénot, J. P. Goedgebuer, A. Lacourt, Appl. Opt. 16, 454 (1977).
[CrossRef] [PubMed]

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

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

Appl. Opt. (1)

Nouv. Rev. Opt. (1)

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

Opt. Commun. (2)

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

J. Calatroni, Opt. Commun. 19, 49 (1976).
[CrossRef]

Other (5)

In fact a quadratic phase factor should be present in Eq. (13), but it has no effect on the amplitude distribution at the x′ and x″ planes.

H. Bouasse, Z. Carrière, Interferences (Delagrave Ed., Paris, 1923).

Obviously a transparent and nondispersive pupil has a ν dependent transparency function exp[j2π(ν/c)ne(x)]; but the delay does not depend on ν.

Although a geometrical transverse pupil has a ν-independent transparency function, the spectrum of a white light wave is filtered by diffraction on g(x); the diffraction process is ν-dependent and responsible for chromatic encoding of g(x).

η is considered as a constant in order to neglect the dispersion of the pupil.

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Figures (9)

Fig. 1
Fig. 1

Effect of a pupil on a white light wave: (a) spatial pupil: it modifies the amplitude of the wave and introduces a delay; the profile of the wave is preserved; (b) temporal pupil: the delay and attenuation vary with the frequency; the profile of the wave is changed.

Fig. 2
Fig. 2

Arrangement to observe the transfer function of a transparent pupil.

Fig. 3
Fig. 3

Experimental setup for recording dispersion curve of a prism. A hollow Littrow prism was used for liquids. x″ν is the recording plane.

Fig. 4
Fig. 4

White light interferograms for various prisms: (a) distilled water at room temperature, α = 7°23′; (b) optical glass η (5461 Å) = 1.53, α = 7°44″; (c) optical glass η (5461 Å) = 1.53, α = 3°59′; (d) optical glass η (5461 Å) = 1.57, α = 20°55′.

Fig. 5
Fig. 5

Transfer function, impulse response, and recorded intensity for the arrangement Fig. 3.

Fig. 6
Fig. 6

Experimental setting for hologram decoding.

Fig. 7
Fig. 7

Reconstructed image corresponding to dispersion curve of water at room temperature. This image comes from a hologram for which ν0 was chosen near the violet limit of the visible spectrum to obtain high spatial frequencies; the hologram is not shown in Fig. 4.

Fig. 8
Fig. 8

Calculation of fringes of equal intensity.

Fig. 9
Fig. 9

Involved parameters in the recorded hologram.

Equations (25)

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τ ( x ; ν ) = b ( x ; ν ) exp [ j ϕ ( x ; ν ) ] ,
ϕ ( x ; ν ) = 2 π c ν η ( ν ) e ( x ) ,
A ( ξ ; ν ) = j ν c F ( ν ) exp ( j 2 π ν c e 0 ) ( pupil ) × exp { j 2 π ν c [ η ( ν ) 1 ] e ( x ) } exp { j 2 π ν ξ cf x } dx
e ( x ) = α x .
A ( u ; ν ) = exp ( j 2 π ν c e 0 ) F ( ν ) δ [ u α ( η 1 ) ] ,
H ( u ; ν ) = exp ( j 2 π ν c e 0 ) δ [ u α ( η 1 ) ] .
h ( u ; t ) = exp [ j 2 π η 1 ( u α + 1 ) ( t e 0 / c ) ] ,
η 1 ( u α + 1 )
( ν ) = 2 α [ η ( ν ) η ( ν 0 ) ] ,
I ( x ) = ( source spectrum ) 2 | F ( ν ) | 2 [ 1 + cos 2 π ν c Δ ( x ; ν ) ] d ν ,
Δ ( x ; ν ) = x 2 d
x = 2 d + mc / ν ( ν ) .
I ( x ; ν ) = 2 | F ( ν ) | 2 [ 1 + cos 2 π c ν Δ ( ν ; x ) ] .
A ( ξ ; ν ) = F ( ν ) { δ ( ξ ) exp ( j 2 π ν c 2 d ) + δ [ ξ f 2 ( ν ) ] } . Ref . 8
I ( x ; ν ) = | FT ξ ; t { δ ( ξ ) f ( t 2 d c ) + FT 1 [ F ( ν ) δ ( ξ f 2 ) ] } | 2 = | FT ξ ; t [ f ( t ) k ( ξ ; t ) ] | 2 .
k ( ξ ; t ) = δ ( ξ ) δ ( t 2 d c ) + exp { j 2 π [ 1 ( ξ / f 2 ) ] t } ,
R ( ξ ; ν ) = | F ( ν ) | 2 [ δ 2 ( ξ ) + 1 2 δ 2 ( ξ S ) + 1 2 δ 2 ( ξ + S ) ] ,
Δ ( x ; ν ) = 2 α ( η η 0 ) x 2 d ,
Δ ( x ; ν ) = m c ν , m integer ,
x = 2 d + mc / ν 2 α ( η η 0 ) .
η ( ν ) = p + q ν 2 c 2 + + r ν 2 n c 2 n p + q ν 2 c 2 ,
x = 2 d + mc / ν 2 α c 2 q ( ν 2 ν 0 2 )
x ( extreme points ) = d c 2 α q 1 3 ν 2 ν 0 2 ,
x ( center ) = d c 2 2 α q 1 ν 0 2
m ( central ) = 2 d ν 0 c + 1 ν 0 2 ,

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