Abstract

The non-paraxial phase-space representation of diffraction of optical fields in any state of spatial coherence has been successfully modeled by assuming a discrete set of radiant point sources at the aperture plane instead of a continuous wave-front. More than a mere calculation strategy, this discreteness seems to be a physical feature of the field, independent from the sampling procedure of the modeling.

©2013 Optical Society of America

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References

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  1. M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 2005); Eq. (17) in Sec. 8.3.2 is the Fresnel-Kirchhoff diffraction formula.
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); Eq. (4.4-25) is the Wolf’s integral equation.
  3. R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express 19(25), 25022–25034 (2011).
    [Crossref] [PubMed]
  4. R. Castañeda, The Optics of Spatial Coherence Wavelets, Vol 164 of Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 2010), pp. 29–255.
  5. R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A 27(6), 1322–1330 (2010).
    [Crossref] [PubMed]
  6. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (Mc Graw-Hill, 2010).
  7. K. Wolf, M. Alonso, and G. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16(10), 2476–2487 (1999).
    [Crossref]
  8. E. Marchand and E. Wolf, “Walther’s definition of generalized radiance,” J. Opt. Soc. Am. 64(9), 1273–1274 (1974).
    [Crossref]
  9. R. Castañeda, “Generalised radiant emittance in the phase-space representation of planar sources in any state of spatial coherence,” Opt. Commun. 284(19), 4259–4262 (2011).
    [Crossref]

2011 (2)

R. Castañeda, “Generalised radiant emittance in the phase-space representation of planar sources in any state of spatial coherence,” Opt. Commun. 284(19), 4259–4262 (2011).
[Crossref]

R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express 19(25), 25022–25034 (2011).
[Crossref] [PubMed]

2010 (1)

1999 (1)

1974 (1)

Alonso, M.

Cañas-Cardona, G.

Castañeda, R.

Forbes, G.

Garcia-Sucerquia, J.

Marchand, E.

Wolf, E.

Wolf, K.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

R. Castañeda, “Generalised radiant emittance in the phase-space representation of planar sources in any state of spatial coherence,” Opt. Commun. 284(19), 4259–4262 (2011).
[Crossref]

Opt. Express (1)

Other (4)

M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 2005); Eq. (17) in Sec. 8.3.2 is the Fresnel-Kirchhoff diffraction formula.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); Eq. (4.4-25) is the Wolf’s integral equation.

R. Castañeda, The Optics of Spatial Coherence Wavelets, Vol 164 of Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 2010), pp. 29–255.

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (Mc Graw-Hill, 2010).

Supplementary Material (2)

» Media 1: MOV (5019 KB)     
» Media 2: MOV (5458 KB)     

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Figures (8)

Fig. 1
Fig. 1 Power spectrum propagation along 0≤z≤10 μm of the field of λ=0.632 μm emitted by two radiant point sources at the AP (z = 0), separated by 1 μm (upper row) and 5 μm (bottom row), for (a) | μ( +,− ) |=0 (spatial incoherence), (b) μ( +,− )=0.3, α( +,− )=π (partial coherence) and (c) μ( +,− )=1, α( +,− )=0 (spatial coherence).

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Fig. 2
Fig. 2 Distribution of radiant, virtual and dual point sources for modeling diffraction of spatially coherent light by a slit.

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Fig. 3
Fig. 3 Conceptual diagram of the non-paraxial propagation modes emitted by pure virtual point sources (a) to (c) and dual point sources (d) to (e). Each emitting source has a twin source that does not appear in the sketch, placed at the symmetric position with respect to the array midpoint (except the point source at the midpoint of the array, in (c)).

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Fig. 4
Fig. 4 Conceptual diagram of the classes of radiator pairs of even orders (including the 0-order class) in the column on the left and of odd orders in the column on the right.

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Fig. 5
Fig. 5 (Media 1) Comparing the diffraction of the spatially coherent optical field of λ=0.632 μm , emitted by 20 identical r-sources uniformly distributed in a slit of width 10 μm, regarding a discrete radiant layer on the upper row and a continuous radiant layer on the bottom row. Propagation distances 0.055 μm≤z≤ 10 3  μm . The blue and the green profiles in the graphs on the right column denote the radiant power and the contribution of the virtual components of the d-sources to the modulating power. The red profile in the upper graph is the contribution of the pure v-sources to the modulating power.

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Fig. 6
Fig. 6 Rms-error of the power spectrum profile at the bottom row on the mid-column in Fig. 4 and the figure of merit (see the text), for (a) z≤ 10 2  μm and (b) z≤ 10 3  μm .

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Fig. 7
Fig. 7 (Media 2) Comparing the interference produced by 10 identical and spatially coherent r-sources uniformly distributed on an array of length 10 μm, that emit at λ=0.632 μm . Propagation distances 0.055 μm≤z≤ 10 3  μm . The upper row model includes pure virtual point sources, while the bottom row model excludes them.

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Fig. 8
Fig. 8 Rms- error of the power spectrum profile at the bottom row on the mid-column in Fig. 6 and the profile at the upper row on the same column. The last one is taken as merit figure, (a) for z≤ 10 2  μm and (b) for z≤ 10 3  μm .

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Equations (8)

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S( b, r A ;ν )= ∫ AP ξ D ≠0 S 0 ( ξ A +b/2 ;ν ) | t( ξ A +b/2 ) | S 0 ( ξ A −b/2 ;ν ) | t( ξ A −b/2 ) | ×| μ( ξ A +b/2 , ξ A −b/2 ) |M( ξ A +b/2 , ξ A −b/2 , r A ) d 2 ξ A
S( 0, r A ;ν )= ∫ AP S 0 ( ξ A ;ν )  |  t( ξ A )  | 2  M( ξ A , ξ A , r A )  d 2 ξ A
M( ξ A , ξ A , r A )= 1 4 λ  2   ( z+ z 2 + |   r A − ξ A | 2 z 2 + |   r A − ξ A | 2 ) 2 ,
M( ξ A +b/2 , ξ A −b/2 , r A )= 1 2 λ  2 ( z+ z 2 + | r A − ξ A | 2 + | b | 2 /4 + ξ A ⋅b− r A ⋅b z 2 + | r A − ξ A | 2 + | b | 2 /4 + ξ A ⋅b− r A ⋅b ) ×( z+ z 2 + | r A − ξ A | 2 + | b | 2 /4 − ξ A ⋅b+ r A ⋅b z 2 + | r A − ξ A | 2 + | b | 2 /4 − ξ A ⋅b+ r A ⋅b ) ×cos[ k z 2 + | r A − ξ A | 2 + | b | 2 /4 + ξ A ⋅b− r A ⋅b −k z 2 + | r A − ξ A | 2 + | b | 2 /4 − ξ A ⋅b+ r A ⋅b +Δφ+α ] ,
S( r A ;ν )= S rad ( r A ;ν )+ S virt ( r A ;ν )=S( 0, r A ;ν )+ ∫ AP ξ D ≠0 S( ξ D , r A ;ν )  d 2 ξ D ,
S( x A ;ν )= ∑ n=0 N−1 S( nb, x A ;ν ) =S( 0, x A ;ν )+ ∑ n=1 P S( 2nb, x A ;ν ) + ∑ n=0 P S( ( 2n+1 )b, x A ;ν ) ,
S( x A ;ν )= S 0 ( ν ){ ∑ n=0 N−1 M( nb,nb, x A ) + ∑ n=1 P ∑ m=n N−n−1 M( ( m+n )b,( m−n )b, x A ) + ∑ n=0 P ∑ m=n N−n−2 M( ( m+n+1 )b,( m−n )b, x A ) },
S ′ ( x A ;ν )=S( 0 ,  x A ;ν )+ ∑ n=1 P S( 2nb ,  x A ;ν ) = S 0 ( ν ) {   ∑ n=0 N−1 M( n b , n b ,  x A ) + ∑ n=1 P ∑ m=n N−n−1 M( ( m+n ) b , ( m−n ) b ,  x A )   } .

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