Abstract

Three effects are known with 3-D periodicities: If a coherent wavefield is periodic in the lateral direction, it is also periodic in the longitudinal direction (Talbot). If a coherent wavefield is longitudinally periodic, it may or may not be periodic also in the lateral direction; but also certain aperiodic objects will cause longitudinal periodicity (Montgomery). If a wavefield starts completely incoherent but with lateral periodic intensity distribution, the state of partial coherence will vary periodically in the longitudinal direction (Lau). We study generalizations of these three effects, assuming quasimonochromatic light with arbitrary states of partial coherence.

© 1989 Optical Society of America

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References

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  1. F. Talbot, “Facts Relating to Optical Science No. IV,” Phil. Mag. 9, 401–407 (1836).
  2. W. D. Montgomery, “Self-Imaging Objects of Infinite Apertures,” J. Opt. Soc. Am. 57, 772–778 (1967);“Algebraic Formulation of Diffraction Applied to Self Imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [Crossref]
  3. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. der Physik 6, 417–423 (1948).
    [Crossref]
  4. To maintain continuity in the paper we used νT in Eq. (10). Montgomery obtained the condition of self imaging for some plane z = d along the z-axis. This condition is in the form of concentric circles in the spatial frequency plane.

1967 (1)

1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. der Physik 6, 417–423 (1948).
[Crossref]

1836 (1)

F. Talbot, “Facts Relating to Optical Science No. IV,” Phil. Mag. 9, 401–407 (1836).

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. der Physik 6, 417–423 (1948).
[Crossref]

Montgomery, W. D.

Talbot, F.

F. Talbot, “Facts Relating to Optical Science No. IV,” Phil. Mag. 9, 401–407 (1836).

Ann. der Physik (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. der Physik 6, 417–423 (1948).
[Crossref]

J. Opt. Soc. Am. (1)

Phil. Mag. (1)

F. Talbot, “Facts Relating to Optical Science No. IV,” Phil. Mag. 9, 401–407 (1836).

Other (1)

To maintain continuity in the paper we used νT in Eq. (10). Montgomery obtained the condition of self imaging for some plane z = d along the z-axis. This condition is in the form of concentric circles in the spatial frequency plane.

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Figures (10)

Fig. 1
Fig. 1

Talbot effect. Grating G is illuminated by a monochromatic plane wave. Distribution of dark and bright behind the grating is periodic in x and in z.

Fig. 2
Fig. 2

Lau effect. Grating G1 is illuminated incoherently. Grating G2 serves a detector of partial coherence, which becomes evident behind the second lens.

Fig. 3
Fig. 3

Ewald sphere, where the 3-D Fourier spectrum of the wave-field υ exists.

Fig. 4
Fig. 4

Ewald sphere, crossed by lines of lateral periodicity.

Fig. 5
Fig. 5

Ewald sphere portion, same as in Fig. 4, but elongated in νz-direction.

Fig. 6
Fig. 6

Ewald sphere, crossed by lines of longitudinal periodicity.

Fig. 7
Fig. 7

Ewald sphere, elongated version of Fig. 6.

Fig. 8
Fig. 8

Two-dimensional lateral Fourier spectrum of objects from the Montgomery set.

Fig. 9
Fig. 9

Two Ewald spheres, the one to the left in the ν-domain, the other one in the μ -domain. The horizontal lines represent the condition of lateral periodicity.

Fig. 10
Fig. 10

Two Ewald spheres, as in Fig. 9. The vertical lines represent the condition of longitudinal periodicity.

Tables (1)

Tables Icon

Table I Cases of 3-D Wavefield Periodicities Discussed In this Paper

Equations (25)

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Z T = 2 p 2 / λ .
Δ u ( x , y , z ) + K 2 · u ( x , y , z ) = 0 ; K = 2 π / λ .
u ( x , y , z ) = υ ( x , y , z ) exp ( iKz ) .
Δ υ ( x , y , z ) + 2 i k υ ( x , y , z ) / z = 0 .
υ ( x ) = υ ( ν ) exp ( 2 π i x · ν ) d ν .
υ exp ( .. ) [ ( 2 π ) 2 ν · ν 4 π K ν z ] d ν = 0 .
ν x 2 + ν y 2 + ( ν z + 1 / λ ) 2 = 1 / λ 2 .
ν z = 1 / λ ± 1 / λ 2 m 2 ν 0 2 .
ν z m 2 λ ν 0 2 / 2 = m 2 ν T ; ν T = 1 / Z T .
ν z = n ν T .
ν z = n ν T = ( 1 / λ ) + 1 / λ 2 ρ n 2 λ ρ n 2 / 2 ,
ρ n 2 = n 2 ν T / λ = n ν 0 2 ; ρ n = ν 0 n .
ν x = m ν 0 cos α ; ν y = m ν 0 sin α .
u ( x 1 ) u * ( x 2 ) = Γ ( x 1 , y 1 , z 1 , x 2 , y 2 z 2 ) .
Δ 1 Γ + K 2 Γ = 0 ; Δ 2 Γ + K 2 Γ = 0 .
Γ ( x 1 , x 2 ) = V ( x 1 , x 2 ) exp [ i K ( z 1 z 2 ) ] .
Δ 1 V + 2 i K V / z 1 = 0 ; Δ 2 V + 2 i K V / z 2 = 0 .
V ( ν x , ν y , ν z , x 2 , y 2 , z 2 ) 0 , only where ν x 2 + ν y 2 + ( ν z + 1 / λ ) 2 = ( 1 / λ ) 2 .
V ( x 1 , y 1 , x 1 , μ x , μ y , μ z ) 0 , only where μ x 2 + μ y 2 + ( μ z 1 / λ ) 2 = ( 1 / λ ) 2 .
V ( x 1 , x 2 ) exp [ 2 π ( x 1 · ν + x 2 · μ ) ] d x 1 d x 2 = V ( ν , μ ) 0 , only if | ν + z ̂ ( 1 / λ ) | = 1 / λ and | μ z ̂ ( 1 / λ ) | = 1 / λ .
V ( x 1 , x 2 ) = ( m ) C m ( x 2 ) exp ( 2 π im ν 0 x 1 ) , V ( x 1 , x 2 ) = ( n ) C n * ( x 1 ) exp ( 2 π in ν 0 x 2 ) ,
V ( x 1 , x 2 ) = A m A n * exp [ 2 π i ν 0 ( m x 1 n x 2 ) ] .
A m A n * = | A m | 2 δ m n ; δ m n = 1 if m = n ; = 0 otherwise .
V ( x 1 x 2 ) = | A m | 2 exp [ 2 π im ν 0 ( x 1 x 2 ) ] .
cos [ 2 π ν 0 ( 3 x 1 + x 2 ) ] = cos [ 2 π ν 0 ( x 1 x 2 ) + 4 π ν 0 ( x 1 + x 2 ) ] .

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