Abstract

Squared elementary cells with correlated radiant point sources are presented as basic structures for characterizing the propagation of the field emitted by two-dimensional planar sources of any shape and in arbitrary state of spatial coherence. The field is transported on a finite expansion of nonparaxial modes, whose propagation in the micro-diffraction domain is discussed under both the diffraction and the interference conditions.

© 2014 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  2. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37.
  3. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).
  4. R. Castañeda and H. Muñoz, “Phase–space non-paraxial propagation modes of optical fields in any state of spatial coherence,” Opt. Express 21, 11276–11293 (2013).
    [CrossRef]
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  6. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).
  7. R. Castañeda, “Point sources and rays in the phase-space representation of random electromagnetic fields,” Opt. Commun. 284, 4114–4123 (2011).
    [CrossRef]
  8. R. Castañeda, D. Vargas, and E. Franco, “Discreteness of the set of radiant point sources: a physical feature of the second-order wave-fronts,” Opt. Express 21, 12964–12975 (2013).
    [CrossRef]
  9. R. Castañeda, E. Franco, and D. Vargas, “Spatial coherence of light and a fundamental discontinuity of classical second-order wave-fronts,” Phys. Scr. 88, 035401 (2013).
    [CrossRef]
  10. R. Castaneda, “Phase-space representation of electromagnetic radiometry,” Phys. Scr. 79, 035302 (2009).
    [CrossRef]
  11. R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58, 962–972 (2011).
    [CrossRef]
  12. R. Castañeda, “Generalised radiant emittance in the phase-space representation of planar sources in any state of spatial coherence,” Opt. Commun. 284, 4259–4262 (2011).
    [CrossRef]
  13. R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express 19, 25022–25034 (2011).
    [CrossRef]

2013 (3)

2011 (4)

R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58, 962–972 (2011).
[CrossRef]

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

R. Castañeda, “Point sources and rays in the phase-space representation of random electromagnetic fields,” Opt. Commun. 284, 4114–4123 (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, 25022–25034 (2011).
[CrossRef]

2009 (1)

R. Castaneda, “Phase-space representation of electromagnetic radiometry,” Phys. Scr. 79, 035302 (2009).
[CrossRef]

Born, M.

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

Cañas-Cardona, G.

R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58, 962–972 (2011).
[CrossRef]

Castaneda, R.

R. Castaneda, “Phase-space representation of electromagnetic radiometry,” Phys. Scr. 79, 035302 (2009).
[CrossRef]

Castañeda, R.

R. Castañeda and H. Muñoz, “Phase–space non-paraxial propagation modes of optical fields in any state of spatial coherence,” Opt. Express 21, 11276–11293 (2013).
[CrossRef]

R. Castañeda, D. Vargas, and E. Franco, “Discreteness of the set of radiant point sources: a physical feature of the second-order wave-fronts,” Opt. Express 21, 12964–12975 (2013).
[CrossRef]

R. Castañeda, E. Franco, and D. Vargas, “Spatial coherence of light and a fundamental discontinuity of classical second-order wave-fronts,” Phys. Scr. 88, 035401 (2013).
[CrossRef]

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

R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58, 962–972 (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, 25022–25034 (2011).
[CrossRef]

R. Castañeda, “Point sources and rays in the phase-space representation of random electromagnetic fields,” Opt. Commun. 284, 4114–4123 (2011).
[CrossRef]

Dragoman, D.

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

Franco, E.

R. Castañeda, E. Franco, and D. Vargas, “Spatial coherence of light and a fundamental discontinuity of classical second-order wave-fronts,” Phys. Scr. 88, 035401 (2013).
[CrossRef]

R. Castañeda, D. Vargas, and E. Franco, “Discreteness of the set of radiant point sources: a physical feature of the second-order wave-fronts,” Opt. Express 21, 12964–12975 (2013).
[CrossRef]

Garcia-Sucerquia, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Hennelly, B.

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

Mandel, L.

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

Muñoz, H.

Muñoz-Ossa, H.

R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58, 962–972 (2011).
[CrossRef]

Ojeda-Castaneda, J.

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

Testorf, M.

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

Vargas, D.

R. Castañeda, E. Franco, and D. Vargas, “Spatial coherence of light and a fundamental discontinuity of classical second-order wave-fronts,” Phys. Scr. 88, 035401 (2013).
[CrossRef]

R. Castañeda, D. Vargas, and E. Franco, “Discreteness of the set of radiant point sources: a physical feature of the second-order wave-fronts,” Opt. Express 21, 12964–12975 (2013).
[CrossRef]

Wolf, E.

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

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

J. Mod. Opt. (1)

R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58, 962–972 (2011).
[CrossRef]

Opt. Commun. (2)

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

R. Castañeda, “Point sources and rays in the phase-space representation of random electromagnetic fields,” Opt. Commun. 284, 4114–4123 (2011).
[CrossRef]

Opt. Express (3)

Phys. Scr. (2)

R. Castañeda, E. Franco, and D. Vargas, “Spatial coherence of light and a fundamental discontinuity of classical second-order wave-fronts,” Phys. Scr. 88, 035401 (2013).
[CrossRef]

R. Castaneda, “Phase-space representation of electromagnetic radiometry,” Phys. Scr. 79, 035302 (2009).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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

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

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

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

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Figures (16)

Fig. 1.
Fig. 1.

Geometry of the nonparaxial setups for the micro-diffraction domain.

Fig. 2.
Fig. 2.

(a), (c) Cartesian distributions of pure radiant (r), pure virtual (v), and dual (d) point sources in extended planar sources. (b), (d) elementary cells for (a) and (c), respectively (shadowed regions), with the ξA coordinates of the point sources in parentheses.

Fig. 3.
Fig. 3.

Distribution of point sources for characterizing (a) circular planar source (the straight lines delimit cells of different sizes) and (b) squared planar source (the straight lines link correlated pairs of radiant point sources).

Fig. 4.
Fig. 4.

Cell-shaped structures for the 2×2 arrangement of elementary cells in Fig. 1(b). Only the (r, d)-point sources are numbered. (a) There are two adjacent structures with pitches b horizontal and 2b vertical that provide the contributions of the virtual components of the d sources (2,1), (2,2), and (2,3) and the v sources in between. (b) There are two adjacent structures with pitches 2b horizontal and b vertical that provide the contributions of the virtual components of the d sources (1,2), (2,2), and (3,2) and the v sources in between. (c) There is one structure with pitches 2b horizontal and 2b vertical that provide an additional contribution of the d source (2,2).

Fig. 5.
Fig. 5.

Fundamental nonparaxial modes for the elementary cell in Fig. 2(b) and the structures in Fig. 4. The horizontal axis is the propagation coordinate z, and the vertical axis is the position xA on the OP, both in micrometers. (a) Zeroth-order mode. (b)–(e) Under interference condition (pitch b=5λ and separations 52λ, 10λ, 102λ, respectively). (f)–(i) Under diffraction condition (pitch b=0.567λ and separations 0.8λ, 1.13λ, 1.6λ, respectively).

Fig. 6.
Fig. 6.

Light propagation in the micro-diffraction domain for the elementary squared cell in Fig. 2(b) under diffraction condition, i.e., horizontal and vertical spacing b=0.85λ. (a)–(c) Srad(rA;ν), (d)–(f) Svirt(rA;ν), and (g)–(i) S(rA;ν) at subwavelength (left and middle columns) propagation distances and at few wavelengths from the cell (right column). The units of the (xA, yA) axes are micrometers. The dynamic range of some patterns is reduced in order to reveal the structure around the main maxima.

Fig. 7.
Fig. 7.

Light propagation in the micro-diffraction domain for the elementary squared cell in Fig. 2(b) under interference condition, i.e., horizontal and vertical spacing b=3λ. The graphs are corresponding to those in Fig. 2.

Fig. 8.
Fig. 8.

Power spectra S(rA) of the optical field emitted by the elementary squared cell in Fig. 2(d) at specific distances z in the micro-diffraction domain. Graphs (a)–(c) are obtained under the diffraction condition (b=0.85λ) and (d)–(f) under the interference condition (b=3λ). The units of the (xA, yA) axes are micrometers.

Fig. 9.
Fig. 9.

Power spectra S(rA) of the optical field emitted by the 2×2 array of elementary squared cells in Fig. 2(b), as disposed in Fig. 4, at specific distances z in the micro-diffraction domain. Graphs (a)–(c) are obtained under the diffraction condition (b=0.8λ) and (d)–(f) under the interference condition (b=3λ). The units of the (xA, yA) axes are micrometers.

Fig. 10.
Fig. 10.

Conceptual sketch of the 2D Young mask with two elementary squared cells in Fig. 2(d). The distributions of point sources involved (a) in diffraction by the cells and (b) in interference between the cells are shown. The correlated r sources are linked by straight lines.

Fig. 11.
Fig. 11.

Power spectra patterns S(rA) at the OP produced by the Young interference with the 2D mask in Fig. 10 for λ=0.632μm, a=0.8λ, and b=5λ. The units of the (xA, yA) axes are micrometers.

Fig. 12.
Fig. 12.

Comparison between the profiles of the exact nonparaxial and the paraxial approached power spectra at the OP, produced by the Young mask in Fig. 10 for λ=0.632μm, a=0.8λ, (a) b=5λ, z=102λ and (b) b=500λ, z=104λ.

Fig. 13.
Fig. 13.

Nonparaxial spatially partially coherent Young interference with μ(ξD)=exp(|ξD|2/2σ2) at z=102λ, for λ=0.632μm, a=0.8λ, and b=5λ and the sizes of the structured supports at the AP given by (a) σ=10λ, (b) σ=5λ, and (c) σ=3λ. The graphs on the bottom row are obtained by σ=3λ at (d) z=λ, (e) z=5λ, and (f) z=10λ. The units of the (xA, yA) axes are micrometers.

Fig. 14.
Fig. 14.

Power spectra S(rA) of the optical field emitted by the distribution of point sources in Fig. 3(b) for λ=0.632μm. (a)–(c) Under the diffraction condition (a=b=0.8λ), and (d)–(f) under the interference condition (a=b=5λ). The units of the (xA, yA) axes are micrometers.

Fig. 15.
Fig. 15.

Power spectra S(rA) of the optical field emitted by the distribution of point sources in Fig. 3(b) for λ=0.632μm. (a) Comparison between the paraxial approached sinc-like profile and the exact nonparaxial profile of the graph in Fig. 14(c) along the axis oriented π/4 with respect to xA. (b) Exact nonparaxial profile of the graph in Fig. 14(f) along the same axis. (c) Paraxial segment of the exact nonparaxial profile for a=b=102λ and z=106λ.

Fig. 16.
Fig. 16.

Graphs of the modulus of Eq. (3a) for λ=0.632μm and 102λz103λ and the structured support centered at (a) xA=0 and (b) xA=±500λ. (c) Comparison of normalized profiles of graphs in (a) and (b) with the paraxial approached sinc-like structured support of spatial coherence at z=104λ. (np) and (p) mean nonparaxial and paraxial, respectively. The planar source projection is composed of a uniform line of 11 identical r-point sources with spacing λ/2.

Equations (10)

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Wlm(r+,r)=APAPWlm(ξ+,ξ)F(ξ+,ξ;r+,r;λ,z)d2ξDd2ξA,
F(ξ+,ξ;r+,r;λ,z)=(k4π)2t(ξ+)t*(ξ)(z+|z+rAξA+(rDξD)/2||z+rAξA+(rDξD)/2|2)(z+|z+rAξA(rDξD)/2||z+rAξA(rDξD)/2|2)×exp(ik|z+rAξA+(rDξD)/2|ik|z+rAξA(rDξD)/2|)
W(r+,r)=APAPμ(ξ+,ξ)S0(ξ+)S0(ξ)F(ξ+,ξ;r+,r;λ,z)d2ξDd2ξA,
Wrad(r+,r)=APS0(ξA)F(ξA;r+,r;λ,z)d2ξA
F(ξA;r+,r;λ,z)=(k4π)2|t(ξA)|2(z+|z+rAξA+rD/2||z+rAξA+rD/2|2)(z+|z+rAξArD/2||z+rAξArD/2|2)×exp(ik|z+rAξA+rD/2|ik|z+rAξArD/2|)
Wvir(r+,r)=2APAPξD0|μ(ξ+,ξ)|S0(ξ+)S0(ξ)Re[F(ξ+,ξ;r+,r;λ,z)exp[iε0(ξ+,ξ)]]d2ξDd2ξA
Srad(rA)=APS0(ξA)F(ξA;rA;λ,z)d2ξA
F(ξA;rA;λ,z)=(k4π)2|t(ξA)|2(z+|z+rAξA||z+rAξA|2)2,
Svir(rA)=2APAPξD0|μ(ξ+,ξ)|S0(ξ+)S0(ξ)Re[F(ξ+,ξ;rA;λ,z)exp[iε0(ξ+,ξ)]]d2ξDd2ξA
F(ξ+,ξ;rA;λ,z)=(k4π)2t(ξ+)t*(ξ)(z+|z+rAξAξD/2||z+rAξAξD/2|2)(z+|z+rAξA+ξD/2||z+rAξA+ξD/2|2)×exp(ik|z+rAξAξD/2|ik|z+rAξA+ξD/2|).

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