Abstract

The extended use of ray maps in the phase-space representation of optical fields in any state of spatial coherence has promoted the search of efficient approaches for their computation. In this paper, those maps are calculated via a matrix-based approach that i) allows accounting for complex degrees of spatial coherence of any shape and value, ii) models transmittances with real dimensions and not just point objects, iii) accounts for Fresnel phases in the modeled fields, and iv) consumes reasonably short computation time (few seconds for relatively big matrices). Examples illustrate the mentioned features.

© 2010 Optical Society of America

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References

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  1. R. Castañeda and J. Carrasquilla, “Spatial coherence wavelets and phase-space representation of diffraction,” Appl. Opt. 47, E76–E87 (2008).
    [CrossRef] [PubMed]
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  3. R. Castaneda, R. Betancur, and J. F. Restrepo, “Interference in phase space,” J. Opt. Soc. Am. A 25, 2518–2527 (2008).
    [CrossRef]
  4. A. Luis and L. M. Sanchez-Brea, “Ray picture of diffraction gratings,” Opt. Commun. 282, 2009–2015 (2009).
    [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics, 6th. ed.(Pergamon, 1993).
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  7. R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45–55 (2003).
    [CrossRef]
  8. 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, 1322–1330 (2010).
    [CrossRef]
  9. J. García, R. Castañeda, and F. F. Medina, “Fresnel–Fraunhofer diffraction and spatial coherence,” Opt. Commun. 205, 239–245 (2002).
    [CrossRef]
  10. R. Castañeda, R. Betancur, and D. Hincapie, “Holographic features of spatial coherence wavelets,” J. Opt. Soc. Am. A 25, 1894–1901 (2008).
    [CrossRef]
  11. R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26, 147–155 (2009).
    [CrossRef]
  12. R. Betancur, J. Restrepo, and R. Castaneda, “Beam shaping by spatial coherence modulation based on spatial coherence wavelets,” Opt. Lasers Eng. 47, 1340–1347 (2009).
    [CrossRef]

2010

2009

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

A. Luis and L. M. Sanchez-Brea, “Ray picture of diffraction gratings,” Opt. Commun. 282, 2009–2015 (2009).
[CrossRef]

R. Betancur, J. Restrepo, and R. Castaneda, “Beam shaping by spatial coherence modulation based on spatial coherence wavelets,” Opt. Lasers Eng. 47, 1340–1347 (2009).
[CrossRef]

2008

2003

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

2002

J. García, R. Castañeda, and F. F. Medina, “Fresnel–Fraunhofer diffraction and spatial coherence,” Opt. Commun. 205, 239–245 (2002).
[CrossRef]

Betancur, R.

Born, M.

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

Cañas-Cardona, G.

Carrasquilla, J.

Castaneda, R.

R. Betancur, J. Restrepo, and R. Castaneda, “Beam shaping by spatial coherence modulation based on spatial coherence wavelets,” Opt. Lasers Eng. 47, 1340–1347 (2009).
[CrossRef]

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

Castañeda, R.

García, J.

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

J. García, R. Castañeda, and F. F. Medina, “Fresnel–Fraunhofer diffraction and spatial coherence,” Opt. Commun. 205, 239–245 (2002).
[CrossRef]

Garcia-Sucerquia, J.

Hennelly, B.

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

Hincapie, D.

Luis, A.

A. Luis and L. M. Sanchez-Brea, “Ray picture of diffraction gratings,” Opt. Commun. 282, 2009–2015 (2009).
[CrossRef]

Mandel, L.

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

Medina, F. F.

J. García, R. Castañeda, and F. F. Medina, “Fresnel–Fraunhofer diffraction and spatial coherence,” Opt. Commun. 205, 239–245 (2002).
[CrossRef]

Ojeda-Castaneda, J.

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

Restrepo, J.

R. Betancur, J. Restrepo, and R. Castaneda, “Beam shaping by spatial coherence modulation based on spatial coherence wavelets,” Opt. Lasers Eng. 47, 1340–1347 (2009).
[CrossRef]

Restrepo, J. F.

Sanchez-Brea, L. M.

A. Luis and L. M. Sanchez-Brea, “Ray picture of diffraction gratings,” Opt. Commun. 282, 2009–2015 (2009).
[CrossRef]

Testorf, M.

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

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).

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

A. Luis and L. M. Sanchez-Brea, “Ray picture of diffraction gratings,” Opt. Commun. 282, 2009–2015 (2009).
[CrossRef]

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

J. García, R. Castañeda, and F. F. Medina, “Fresnel–Fraunhofer diffraction and spatial coherence,” Opt. Commun. 205, 239–245 (2002).
[CrossRef]

Opt. Lasers Eng.

R. Betancur, J. Restrepo, and R. Castaneda, “Beam shaping by spatial coherence modulation based on spatial coherence wavelets,” Opt. Lasers Eng. 47, 1340–1347 (2009).
[CrossRef]

Other

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

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).

Supplementary Material (1)

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Figures (4)

Fig. 1
Fig. 1

Illustrating the center and difference coordinates at both the AP and the OP, and the geometrical description of the marginal power spectrum as rays along specific paths between such planes and (a) of Eq. (2a) and (b) of Eq. (2b).

Fig. 2
Fig. 2

Diffraction of fully spatially coherent light through a slit of width a. Details in the text.

Fig. 3
Fig. 3

Interference of light with variable Gaussian degree of spatial coherence, produced by a double slit mask (Young’s experiment) (Media 1). The line over the interference pattern represents the diffraction envelope. The following physical parameters were assumed for the calculations: λ = 632.8 nm , slit width a = 100 μm , distance between slit centers b = 200 μm , distance between the slit mask (AP) and the detector plane (OP) z = 1 m .

Fig. 4
Fig. 4

Fraunhofer diffraction of fully spatially coherent light by a Ronchi grating with five identical slits. (a) Ray map S m j with x A j and ξ A m ; matrix size, 1024 × 18000 ; elapsed time, 7.68 s; (b) normalized power distribution S m at the AP; and (c) normalized power distribution S j at the OP. The following physical parameters were assumed for the calculations: λ = 632.8 nm ; slit width, a = 20 μm ; distance between slit centers b = 40 μm ; distance between the slit mask (AP) and the detector plane (OP), z = 1 m .

Equations (14)

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S ( r A , ξ A ) = AP 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 ) exp ( i k z ξ A · ξ D ) exp ( i k z ξ D · r A ) d 2 ξ D ,
S ( ξ A ) = ( 1 λ z ) 2 OP S ( ξ A , r A ) d 2 r A
S ( r A ) = ( 1 λ z ) 2 AP S ( ξ A , r A ) d 2 ξ A ,
S ( ξ A , r A ) = C S ( ξ A ) | t ( ξ A ) | 2 + 2 AP 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 ,
OP S ( r A ) d 2 r A = AP S ( ξ A ) | t ( ξ A ) | 2 d 2 ξ A = ( 1 λ z ) 2 OP AP S ( ξ A , r A ) d 2 ξ A d 2 r A ,
OP 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.
Γ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) = | Γ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) | exp [ i Θ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) ] = | 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 ) | × exp ( i k z ξ A · ξ D + i α ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) + i Δ ϕ ) ,
S ( r A , ξ A ) = C | Γ ( ξ A , ξ A ) | + 2 AP ξ D 0 | Γ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) | cos [ k z ξ D · r A Θ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) ] d 2 ξ D ,
S ( r A , ξ A ) = C | Γ ( ξ A , ξ A ) | + 2 AP ξ D 0 R e [ Γ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) ] cos ( k z ξ D · r A ) d 2 ξ D + 2 AP ξ D 0 I m [ Γ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ) ] sin ( k z ξ D · r A ) d 2 ξ D ,
S m j ( 1 ) = | Γ j j | = Γ j j
S m j ( 2 ) = n = 1 j + 1 2 [ C m , 2 n 1 Γ j + 2 n 1 , j 2 n + 1 ( C ) + S m , 2 n 1 Γ j + 2 n 1 , j 2 n + 1 ( S ) ]
S m j ( 3 ) = Γ j j + n = 1 j 2 [ C m , 2 n Γ j + 2 n , j 2 n ( C ) + S m , 2 n Γ j + 2 n , j 2 n ( S ) ]
S j = ( 1 λ z ) 2 m = 0 M S m j = ( 1 λ z ) 2 m = 0 M S m j ( 1 ) + S m j ( 2 ) + S m j ( 3 )
S m = ( 1 λ z ) 2 j = 0 2 N 1 S m j = ( 1 λ z ) 2 j = 0 2 N 1 S m j ( 1 ) + S m j ( 2 ) + S m j ( 3 ) ,

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