Abstract

A novel description of interference and diffraction with fields in arbitrary states of spatial coherence is introduced in the framework of the phase-space representation. The field is modeled as produced by radiant and virtual point sources. The first ones emit the radiant power of the field, independently of its spatial coherence state, and the second ones emit the modulating energy in strong dependence on such state. This energy can take on positive and negative values that produce the interference and diffraction patterns after adding them to the radiant energy. Radiant and virtual point sources at a given plane can be arranged over two distinct layers, which can be brought together to provide a unified structure of point sources for the field at such plane. So, the coincidence of specific radiant and virtual sources at the same point induces a further type: the dual point source. Descriptions of diffraction arrangements, Young’s experiment with diffraction effects, and some implications of this model are discussed.

© 2010 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  2. R. Castañeda, J. Carrasquilla, and J. Garcia-Sucerquia, “Definition and invariance properties of the complex degree of spatial coherence,” J. Opt. Soc. Am. A 26, 2459–2465 (2009).
    [CrossRef]
  3. R. Castañeda and J. Carrasquilla, “Spatial coherence wavelets and phase-space representation of diffraction,” Appl. Opt. 47, 76–87 (2008).
    [CrossRef]
  4. R. Castañeda, R. Betancur, and J. F. Restrepo, “Interference in phase-space,” J. Opt. Soc. Am. A 25, 2518–2527 (2008).
    [CrossRef]
  5. R. Castañeda and J. Garcia-Sucerquia, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45–55 (2003).
    [CrossRef]
  6. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Progress in Optics, Vol. 37, E.Wolf, ed. (Elsevier, 1997).
    [CrossRef]
  7. A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).
  8. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).
  9. R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8–20 (2007).
    [CrossRef]
  10. R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26, 147–155 (2009).
    [CrossRef]
  11. S. B. Mehta1 and C. J. R. Sheppard, “Phase-space representation of partially coherent imaging systems using the Cohen class distribution,” Opt. Lett. 35, 348–350 (2010).
    [CrossRef]
  12. Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of the IEEE International Conference on Computational Photography (IEEE, 2009), pp. 1–10.
    [CrossRef]
  13. S. Baek-Oh, G. Barbastathis, and R. Raskar, “Augmenting light field to model wave optics effects,” MIT internal communication (2009), sboh@mit.edu.

2010 (1)

2009 (4)

R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26, 147–155 (2009).
[CrossRef]

R. Castañeda, J. Carrasquilla, and J. Garcia-Sucerquia, “Definition and invariance properties of the complex degree of spatial coherence,” J. Opt. Soc. Am. A 26, 2459–2465 (2009).
[CrossRef]

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of the IEEE International Conference on Computational Photography (IEEE, 2009), pp. 1–10.
[CrossRef]

S. Baek-Oh, G. Barbastathis, and R. Raskar, “Augmenting light field to model wave optics effects,” MIT internal communication (2009), sboh@mit.edu.

2008 (2)

R. Castañeda, R. Betancur, and J. F. Restrepo, “Interference in phase-space,” J. Opt. Soc. Am. A 25, 2518–2527 (2008).
[CrossRef]

R. Castañeda and J. Carrasquilla, “Spatial coherence wavelets and phase-space representation of diffraction,” Appl. Opt. 47, 76–87 (2008).
[CrossRef]

2007 (1)

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8–20 (2007).
[CrossRef]

2005 (1)

A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).

2003 (1)

R. Castañeda and J. Garcia-Sucerquia, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45–55 (2003).
[CrossRef]

1997 (1)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Progress in Optics, Vol. 37, E.Wolf, ed. (Elsevier, 1997).
[CrossRef]

1995 (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

1993 (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

Baek-Oh, S.

S. Baek-Oh, G. Barbastathis, and R. Raskar, “Augmenting light field to model wave optics effects,” MIT internal communication (2009), sboh@mit.edu.

Barbastathis, G.

S. Baek-Oh, G. Barbastathis, and R. Raskar, “Augmenting light field to model wave optics effects,” MIT internal communication (2009), sboh@mit.edu.

Betancur, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

Carrasquilla, J.

R. Castañeda, J. Carrasquilla, and J. Garcia-Sucerquia, “Definition and invariance properties of the complex degree of spatial coherence,” J. Opt. Soc. Am. A 26, 2459–2465 (2009).
[CrossRef]

R. Castañeda and J. Carrasquilla, “Spatial coherence wavelets and phase-space representation of diffraction,” Appl. Opt. 47, 76–87 (2008).
[CrossRef]

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8–20 (2007).
[CrossRef]

Castañeda, R.

R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26, 147–155 (2009).
[CrossRef]

R. Castañeda, J. Carrasquilla, and J. Garcia-Sucerquia, “Definition and invariance properties of the complex degree of spatial coherence,” J. Opt. Soc. Am. A 26, 2459–2465 (2009).
[CrossRef]

R. Castañeda and J. Carrasquilla, “Spatial coherence wavelets and phase-space representation of diffraction,” Appl. Opt. 47, 76–87 (2008).
[CrossRef]

R. Castañeda, R. Betancur, and J. F. Restrepo, “Interference in phase-space,” J. Opt. Soc. Am. A 25, 2518–2527 (2008).
[CrossRef]

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8–20 (2007).
[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45–55 (2003).
[CrossRef]

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Progress in Optics, Vol. 37, E.Wolf, ed. (Elsevier, 1997).
[CrossRef]

Garcia-Sucerquia, J.

Herrera, J.

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8–20 (2007).
[CrossRef]

Levoy, M.

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of the IEEE International Conference on Computational Photography (IEEE, 2009), pp. 1–10.
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Mehta1, S. B.

Raskar, R.

S. Baek-Oh, G. Barbastathis, and R. Raskar, “Augmenting light field to model wave optics effects,” MIT internal communication (2009), sboh@mit.edu.

Restrepo, J. F.

Sheppard, C. J. R.

Torre, A.

A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

Zhang, Z.

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of the IEEE International Conference on Computational Photography (IEEE, 2009), pp. 1–10.
[CrossRef]

Appl. Opt. (1)

R. Castañeda and J. Carrasquilla, “Spatial coherence wavelets and phase-space representation of diffraction,” Appl. Opt. 47, 76–87 (2008).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

R. Castañeda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8–20 (2007).
[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45–55 (2003).
[CrossRef]

Opt. Lett. (1)

Other (6)

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of the IEEE International Conference on Computational Photography (IEEE, 2009), pp. 1–10.
[CrossRef]

S. Baek-Oh, G. Barbastathis, and R. Raskar, “Augmenting light field to model wave optics effects,” MIT internal communication (2009), sboh@mit.edu.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Progress in Optics, Vol. 37, E.Wolf, ed. (Elsevier, 1997).
[CrossRef]

A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

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Figures (6)

Fig. 1
Fig. 1

Variation of the size of the structured support of spatial coherence within a one-dimensional aperture of width a. R denotes the size of the maximal support.

Fig. 2
Fig. 2

Unified structure of radiant, virtual, and dual point sources applied to (a) a one-dimensional aperture with maximal source density (the gray level of the sources of the second layer represents the strength of the emission and the letters r, v, and d near the sources of the third layer denote radiant, virtual, and dual, respectively) and (b) the Young’s mask.

Fig. 3
Fig. 3

Young’s experiment with diffraction effects. The illumination of each slit is nearly fully spatially coherent, but the illumination of both slits is nearly fully spatially incoherent. (a) Ray map with the mask profile on the top, (b) power spectrum recorded at the OP.

Fig. 4
Fig. 4

Young’s experiment with diffraction effects. The illumination of the whole mask is nearly fully spatially coherent. (a) Ray map with the mask profile on the top, (b) power spectrum recorded at the OP, (c) ray map of the modulating 0-π rays provided by the virtual sources in the region between the slits, (d) modulating (positive and negative) energy provided by such virtual sources onto the OP. Extension to the diffraction of a spatially coherent field through a Ronchi-grating of five slits: (e) ray map with the mask profile on the top, (f) power spectrum pattern at the OP.

Fig. 5
Fig. 5

Diffraction of nearly spatially coherent light by a slit of width a: (a) ray map with the slit profile on the top, (b) map of the modulating 0-π rays, (c) power spectrum recorded by a one-dimensional detector at the OP.

Fig. 6
Fig. 6

Diffraction of nearly spatially incoherent light by a slit of width a: (a) ray map with the slit profile on the top, (b) power spectrum recorded by a one-dimensional detector at the OP.

Equations (14)

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W ( r A + r D 2 , r A r D 2 ; ξ A ) = S ( ξ A , r A ) exp ( i k z ξ A r D ) ,
S ( ξ A , r A ) = C S ( ξ A ) | t ( ξ A ) | 2 + 2 A P ; ξ D 0 | μ ( ξ A + ξ D 2 , ξ A ξ D 2 ) | S ( ξ A + ξ D 2 ) | t ( ξ A + ξ D 2 ) | × S ( ξ A ξ D 2 ) | t ( ξ A ξ D 2 ) | cos ( k z ξ D r A k z ξ A ξ D α ( ξ A + ξ D 2 , ξ A ξ D 2 ) Δ ϕ ) d 2 ξ D ,
Δ ϕ = ϕ ( ξ A + ξ D 2 ) ϕ ( ξ A ξ D 2 ) ,
μ ( ξ A + ξ D 2 , ξ A ξ D 2 ) = | μ ( ξ A + ξ D 2 , ξ A ξ D 2 ) | exp [ i α ( ξ A + ξ D 2 , ξ A ξ D 2 ) ]
S ( r A ) = ( 1 λ z ) 2 A P S ( ξ A , r A ) d 2 ξ A 0 .
A P S ind ( ξ A ) d 2 ξ A | A P S pairs ( ξ A , r A ) d 2 ξ A |
0 S ( r A ) 2 ( 1 λ z ) 2 A P S ind ( ξ A ) d 2 ξ A
S ind = ( 1 λ z ) 2 A P S ind ( ξ A ) d 2 ξ A = ( 1 λ z ) 2 C A P S ( ξ A ) | t ( ξ A ) | 2 d 2 ξ A 0
S pairs ( r A ) = ( 1 λ z ) 2 A P S pairs ( ξ A , r A ) d 2 ξ A = 2 ( 1 λ z ) 2 A P A P ; ξ D 0 | μ ( ξ A + ξ D 2 , ξ A ξ D 2 ) | S ( ξ A + ξ D 2 ) | t ( ξ A + ξ D 2 ) | × S ( ξ A ξ D 2 ) | t ( ξ A ξ D 2 ) | cos ( k z ξ D r A k z ξ A ξ D α ( ξ A + ξ D 2 , ξ A ξ D 2 ) Δ ϕ ) d 2 ξ D d 2 ξ A
O P S ( r A ) d 2 r A = A P S ( ξ A ) | t ( ξ A ) | 2 d 2 ξ A = ( 1 λ z ) 2 O P A P S ( ξ A , r A ) d 2 ξ A d 2 r A ,
O P cos ( k z ξ D r A k z ξ A ξ D α ( ξ A + ξ D 2 , ξ A ξ D 2 ) Δ ϕ ) d 2 r A = 0 .
F ( ξ A ) = R { 1 , | ξ A | a R 2 1 2 | ξ A | ( a R ) R , a R 2 < | ξ A | a 2 } ,
S ( ξ A , x A ) = S ind ( ξ A ) + S pairs ( ξ A , x A ) = C S 0 rect ( ξ A a 2 ) + 2 S 0 ξ D > 0 ( a 2 ) | ξ A | | μ ( ξ D ) | cos ( k z ξ D x A α ( ξ D ) ) d ξ D ,
S pairs ( ξ A , x A ) 2 S 0 ( a 2 | ξ A | ) [ 1 1 6 ( k z ) 2 ( a 2 | ξ A | ) 2 x A 2 ]

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