Abstract

The non-paraxial marginal power spectrum is decomposed in propagation modes, so that the zeroth-order mode is only emitted by the radiant point sources at the aperture plane, while the modes of higher orders than zero are only emitted by the virtual point sources. It allows representing the non-paraxial propagation of optical fields in arbitrary states of spatial coherence and along arbitrary distances from the aperture plane without approximations, by simply using the power distribution and the spatial coherence state at the aperture plane as entries. This modal expansion is potentially useful in micro-diffraction and spatial coherence modulation.

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References

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  1. K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.)123(3), 202–207 (2012).
    [CrossRef]
  2. X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011).
    [CrossRef]
  3. H. Yu, L. Xiong, and B. Lü, “Nonparaxial Lorentz and Lorentz–Gauss beams,” Optik (Stuttg.)121(16), 1455–1461 (2010).
    [CrossRef]
  4. B. Tang and M. Jiang, “Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation,” J. Mod. Opt.56(8), 955–962 (2009).
    [CrossRef]
  5. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett.34(21), 3430–3432 (2009).
    [CrossRef] [PubMed]
  6. Y. Zhang, “Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture,” Opt. Commun.248(4-6), 317–326 (2005).
    [CrossRef]
  7. H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun.147(1-3), 1–4 (1998).
    [CrossRef]
  8. H. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66(7-8), 163–182 (1944).
    [CrossRef]
  9. J. H. Wu, “Modeling of near-field optical diffraction from a subwavelength aperture in a thin conducting film,” Opt. Lett.36(17), 3440–3442 (2011).
    [CrossRef] [PubMed]
  10. K. Duan and B. Lü, “Nonparaxial diffraction of vectorial plane waves at a small aperture,” Opt. Laser Technol.37(3), 193–197 (2005).
    [CrossRef]
  11. G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun.282(2), 167–171 (2009).
    [CrossRef]
  12. H. Wang and W. She, “Modulation instability and interaction of non-paraxial beams in self-focusing Kerr media,” Opt. Commun.254(1-3), 145–151 (2005).
    [CrossRef]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). Equation (4).4–25) is the Wolf’s integral equation.
  14. R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express19(25), 25022–25034 (2011).
    [CrossRef] [PubMed]
  15. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space optics: fundamentals and applications (Mc Graw-Hill, New York, 2010).
  16. A. Torre, Linear ray and wave optics in the phase-space (Elsevier, 2005).
  17. R. Castañeda, “Phase-space representation of electromagnetic radiometry,” Phys. Scr.79, 035302 (10pp) (2009).
  18. R. Castañeda, R. Betancur, J. Herrera, and J. Carrasquilla, “Phase-space representation and polarization domains of random electromagnetic fields,” Appl. Opt.47(22), E27–E38 (2008).
    [CrossRef] [PubMed]
  19. C. J. Sheppard and K. G. Larkin, “Wigner function for nonparaxial wave fields,” J. Opt. Soc. Am. A18(10), 2486–2490 (2001).
    [CrossRef] [PubMed]
  20. 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. A27(6), 1322–1330 (2010).
    [CrossRef] [PubMed]
  21. R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt.58(11), 962–972 (2011).
    [CrossRef]
  22. R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.
  23. Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
    [CrossRef]
  24. Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).
  25. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, 1993).
  26. R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A26(1), 147–155 (2009).
    [CrossRef] [PubMed]

2012 (1)

K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.)123(3), 202–207 (2012).
[CrossRef]

2011 (5)

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011).
[CrossRef]

J. H. Wu, “Modeling of near-field optical diffraction from a subwavelength aperture in a thin conducting film,” Opt. Lett.36(17), 3440–3442 (2011).
[CrossRef] [PubMed]

R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express19(25), 25022–25034 (2011).
[CrossRef] [PubMed]

R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt.58(11), 962–972 (2011).
[CrossRef]

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).

2010 (3)

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
[CrossRef]

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. A27(6), 1322–1330 (2010).
[CrossRef] [PubMed]

H. Yu, L. Xiong, and B. Lü, “Nonparaxial Lorentz and Lorentz–Gauss beams,” Optik (Stuttg.)121(16), 1455–1461 (2010).
[CrossRef]

2009 (5)

B. Tang and M. Jiang, “Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation,” J. Mod. Opt.56(8), 955–962 (2009).
[CrossRef]

A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett.34(21), 3430–3432 (2009).
[CrossRef] [PubMed]

R. Castañeda, “Phase-space representation of electromagnetic radiometry,” Phys. Scr.79, 035302 (10pp) (2009).

G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun.282(2), 167–171 (2009).
[CrossRef]

R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A26(1), 147–155 (2009).
[CrossRef] [PubMed]

2008 (1)

2005 (3)

H. Wang and W. She, “Modulation instability and interaction of non-paraxial beams in self-focusing Kerr media,” Opt. Commun.254(1-3), 145–151 (2005).
[CrossRef]

K. Duan and B. Lü, “Nonparaxial diffraction of vectorial plane waves at a small aperture,” Opt. Laser Technol.37(3), 193–197 (2005).
[CrossRef]

Y. Zhang, “Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture,” Opt. Commun.248(4-6), 317–326 (2005).
[CrossRef]

2001 (1)

1998 (1)

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun.147(1-3), 1–4 (1998).
[CrossRef]

1944 (1)

H. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66(7-8), 163–182 (1944).
[CrossRef]

Betancur, R.

Bethe, H.

H. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66(7-8), 163–182 (1944).
[CrossRef]

Cai, Y.

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011).
[CrossRef]

Cañas-Cardona, G.

Carrasquilla, J.

Castañeda, R.

Duan, K.

K. Duan and B. Lü, “Nonparaxial diffraction of vectorial plane waves at a small aperture,” Opt. Laser Technol.37(3), 193–197 (2005).
[CrossRef]

Duelk, M.

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
[CrossRef]

Garcia-Sucerquia, J.

Hamamoto, K.

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
[CrossRef]

Herrera, J.

Hinokuma, Y.

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
[CrossRef]

Jiang, M.

B. Tang and M. Jiang, “Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation,” J. Mod. Opt.56(8), 955–962 (2009).
[CrossRef]

Laabs, H.

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun.147(1-3), 1–4 (1998).
[CrossRef]

Larkin, K. G.

Li, X.

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011).
[CrossRef]

Lu, X.

K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.)123(3), 202–207 (2012).
[CrossRef]

Lü, B.

H. Yu, L. Xiong, and B. Lü, “Nonparaxial Lorentz and Lorentz–Gauss beams,” Optik (Stuttg.)121(16), 1455–1461 (2010).
[CrossRef]

G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun.282(2), 167–171 (2009).
[CrossRef]

K. Duan and B. Lü, “Nonparaxial diffraction of vectorial plane waves at a small aperture,” Opt. Laser Technol.37(3), 193–197 (2005).
[CrossRef]

Minato, T.

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
[CrossRef]

Mukai, K.

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).

Muñoz-Ossa, H.

R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt.58(11), 962–972 (2011).
[CrossRef]

Navaretti, P.

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
[CrossRef]

Novitsky, A. V.

Novitsky, D. V.

She, W.

H. Wang and W. She, “Modulation instability and interaction of non-paraxial beams in self-focusing Kerr media,” Opt. Commun.254(1-3), 145–151 (2005).
[CrossRef]

Sheppard, C. J.

Tang, B.

B. Tang and M. Jiang, “Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation,” J. Mod. Opt.56(8), 955–962 (2009).
[CrossRef]

Velez, C.

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
[CrossRef]

Wang, H.

H. Wang and W. She, “Modulation instability and interaction of non-paraxial beams in self-focusing Kerr media,” Opt. Commun.254(1-3), 145–151 (2005).
[CrossRef]

Wang, K.

K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.)123(3), 202–207 (2012).
[CrossRef]

Wu, J. H.

Xiong, L.

H. Yu, L. Xiong, and B. Lü, “Nonparaxial Lorentz and Lorentz–Gauss beams,” Optik (Stuttg.)121(16), 1455–1461 (2010).
[CrossRef]

Yu, H.

H. Yu, L. Xiong, and B. Lü, “Nonparaxial Lorentz and Lorentz–Gauss beams,” Optik (Stuttg.)121(16), 1455–1461 (2010).
[CrossRef]

Zang, Z.

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
[CrossRef]

Zhang, E.

G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun.282(2), 167–171 (2009).
[CrossRef]

Zhang, Y.

Y. Zhang, “Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture,” Opt. Commun.248(4-6), 317–326 (2005).
[CrossRef]

Zhao, C.

K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.)123(3), 202–207 (2012).
[CrossRef]

Zhao, G.

G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun.282(2), 167–171 (2009).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010).
[CrossRef]

IEICE Trans. Elec. (1)

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).

J. Mod. Opt. (2)

R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt.58(11), 962–972 (2011).
[CrossRef]

B. Tang and M. Jiang, “Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation,” J. Mod. Opt.56(8), 955–962 (2009).
[CrossRef]

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

Opt. Commun. (4)

G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun.282(2), 167–171 (2009).
[CrossRef]

H. Wang and W. She, “Modulation instability and interaction of non-paraxial beams in self-focusing Kerr media,” Opt. Commun.254(1-3), 145–151 (2005).
[CrossRef]

Y. Zhang, “Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture,” Opt. Commun.248(4-6), 317–326 (2005).
[CrossRef]

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun.147(1-3), 1–4 (1998).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

K. Duan and B. Lü, “Nonparaxial diffraction of vectorial plane waves at a small aperture,” Opt. Laser Technol.37(3), 193–197 (2005).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttg.) (2)

H. Yu, L. Xiong, and B. Lü, “Nonparaxial Lorentz and Lorentz–Gauss beams,” Optik (Stuttg.)121(16), 1455–1461 (2010).
[CrossRef]

K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.)123(3), 202–207 (2012).
[CrossRef]

Phys. Rev. (1)

H. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66(7-8), 163–182 (1944).
[CrossRef]

Phys. Scr. (1)

R. Castañeda, “Phase-space representation of electromagnetic radiometry,” Phys. Scr.79, 035302 (10pp) (2009).

Other (5)

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space optics: fundamentals and applications (Mc Graw-Hill, New York, 2010).

A. Torre, Linear ray and wave optics in the phase-space (Elsevier, 2005).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). Equation (4).4–25) is the Wolf’s integral equation.

R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.

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

Supplementary Material (5)

» Media 1: MOV (10590 KB)     
» Media 2: MOV (13132 KB)     
» Media 3: MOV (11513 KB)     
» Media 4: MOV (11653 KB)     
» Media 5: MOV (14294 KB)     

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Figures (11)

Fig. 1
Fig. 1

Distribution of radiant, virtual and dual point sources for modelling micro-diffraction of spatially coherent light ( λ=0.632μm ) by a slit.

Fig. 2
Fig. 2

Conceptual sketch 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.

Fig. 3
Fig. 3

Conceptual sketch of the non-paraxial propagation modes emitted by pure virtual point sources (a-c) and dual point sources (d-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).

Fig. 4
Fig. 4

(Media 1) Example of the odd-order classes: the third-order class for three states of spatial coherence denoted by the values of σ . The diagrams of its modes are on the top row and the profiles of its contribution to the power spectrum at the OP are on the bottom row. Each vertical structure of G 3 ( ξ A , x A ) is the third-order mode M 3 ( ξ A , x A ) emitted by the pure virtual point source placed at the ξ A -coordinate of the corresponding mode.

Fig. 5
Fig. 5

(Media 2) Example of the even-order classes: the eighth-order class for the same states of spatial coherence and propagation distance as in Fig. 4. The diagrams of its modes are on the top row and the profiles of its contribution to the power spectrum at the OP are on the bottom row. Each vertical structure of G 8 ( ξ A , x A ) is the eighth-order mode M 8 ( ξ A , x A ) emitted by the virtual component of the dual point source placed at the ξ A -coordinate of the corresponding mode.

Fig. 6
Fig. 6

(Media 3) Modal expansion of the modulating energy emitted by the pure virtual point sources, in the same states of spatial coherence and for the same propagation distances in Figs. 4 and 5. Each vertical structure of their diagrams on the top row results from the superposition of the high odd-order modes ( n=1,3,,9 ) emitted by the pure virtual point source, placed at the ξ A -coordinate. The profiles of the modulating power contributions of all the pure virtual point sources at the AP are shown on the bottom row.

Fig. 7
Fig. 7

(Media 4) Modal expansion of the modulating energy emitted by the virtual components of the dual point sources, in the same states of spatial coherence and for the same propagation distances in Fig. 6. Each vertical structure of their diagrams on the top row results from the superposition of the high even-order modes ( n=2,4,,8 ) emitted by the virtual component of the dual point source, placed at the ξ A -coordinate. The profiles of the modulating power contributions of the virtual components of all the dual point sources at the AP are shown on the bottom row.

Fig. 8
Fig. 8

(Media 5) Modal expansion of the non-paraxial marginal power spectrum under fully spatially coherent illumination: the radiant component is sketched on the left column, the virtual component is on the mid-column and whole expansion is on the right column. The diagrams of the respective expansions are shown on the top row. The profiles of the radiant and the modulating powers as well as the power spectrum at the OP are shown on the bottom row.

Fig. 9
Fig. 9

Illustrating the propagation of a) the radiant power, b) the modulating power and c) the power spectrum of a uniform and spatially coherent field of λ=0.632μm, diffracted by a slit of width L=2.7μm, along the propagation distance 0.1μmz0.5μm<λ<L. Units of axes xA and z are μm, and of the vertical axis are arbitrary.

Fig. 10
Fig. 10

Illustrating the propagation of a) the radiant power, b) the modulating power and c) the power spectrum of a uniform and spatially coherent field of the same attributes as in Fig. 9, along the propagation distance 0.5μmz3.5μm . Units of axes xA and z are μm, and of the vertical axis are arbitrary.

Fig. 11
Fig. 11

Illustrating the propagation of a) the radiant power, b) the modulating power and c) the power spectrum of a uniform and spatially coherent field of the same attributes as in Fig. 9, along the propagation distances λ<L<<8μmz14μm , i.e. in the Fraunhofer domain. Units of axes xA and z are μm, and of the vertical axis are arbitrary.

Tables (1)

Tables Icon

Table 1 Classes of pairs of radiant point sources for the diffraction model in Fig. 1. Even and odd orders are shown in separate groups. The term class population refers to the number of pairs of the each class across the slit.

Equations (10)

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S( ξ A , r A ;ν )= 1 4 λ 2 AP S 0 ( ξ A + ξ D /2 ) t( ξ A + ξ D /2 ) S 0 ( ξ A ξ D /2 ) t * ( ξ A ξ D /2 )μ( ξ A + ξ D /2 , ξ A ξ D /2 ) × exp[ ik z 2 + | r A ξ A | 2 + | ξ D | 2 /4 + ξ A ξ D r A ξ D ] z 2 + | r A ξ A | 2 + | ξ D | 2 /4 + ξ A ξ D r A ξ D exp[ ik z 2 + | r A ξ A | 2 + | ξ D | 2 /4 ξ A ξ D + r A ξ D ] z 2 + | r A ξ A | 2 + | ξ D | 2 /4 ξ A ξ D + r A ξ D , ×( z+ z 2 + | r A ξ A | 2 + | ξ D | 2 /4 + ξ A ξ D r A ξ D )( z+ z 2 + | r A ξ A | 2 + | ξ D | 2 /4 ξ A ξ D + r A ξ D ) d 2 ξ D
F( a+b/2 ,ab/2 , r A ;ν )= 1 4 λ 2 ( z+ z 2 + | r A a | 2 + | b | 2 /4 +ab r A b )( z+ z 2 + | r A a | 2 + | b | 2 /4 ab+ r A b ) × exp[ ik z 2 + | r A a | 2 + | b | 2 /4 +ab r A b ] z 2 + | r A a | 2 + | b | 2 /4 +ab r A b exp[ ik z 2 + | r A a | 2 + | b | 2 /4 ab+ r A b ] z 2 + | r A a | 2 + | b | 2 /4 ab+ r A b ,
S 0 ( a+b/2 ) t( a+b/2 ) S 0 ( ab/2 ) t * ( ab/2 )μ( a+b/2 ,ab/2 )F( a+b/2 ,ab/2 , r A ;ν ) + S 0 ( ab/2 ) t( ab/2 ) S 0 ( a+b/2 ) t * ( a+b/2 )μ( ab/2 ,a+b/2 )F( ab/2 ,a+b/2 , r A ;ν )
2 S 0 ( a+b/2 ) | t( a+b/2 ) | S 0 ( ab/2 ) | t( ab/2 ) || μ( a+b/2 ,ab/2 ) | ×Re{ F( a+b/2 ,ab/2 , r A ;ν )exp[ iΔφ( a+b/2 ,ab/2 )+iα( a+b/2 ,ab/2 ) ] },
Re{ F( a+b/2 ,ab/2 , r A ;ν )exp[ iΔϕ( a+b/2 ,ab/2 )+iα( a+b/2 ,ab/2 ) ] }= 1 4 λ 2 ( z+ z 2 + | r A a | 2 + | b | 2 /4 +ab r A b z 2 + | r A a | 2 + | b | 2 /4 +ab r A b )( z+ z 2 + | r A a | 2 + | b | 2 /4 ab+ r A b z 2 + | r A a | 2 + | b | 2 /4 ab+ r A b ). ×cos[ k z 2 + | r A a | 2 + | b | 2 /4 +ab r A b k z 2 + | r A a | 2 + | b | 2 /4 ab+ r A b +Δϕ+α ]
2Re{ F( a, r A ;ν ) }= 1 2 λ 2 ( z+ z 2 + | r A a | 2 z 2 + | r A a | 2 ) 2
S( a, r A ;ν )= S 0 ( a ) | t( a ) | 2 F( a, r A ;ν )+2 AP ξ D 0 S 0 ( a+ ξ D /2 ) | t( a+ ξ D /2 ) | S 0 ( a ξ D /2 ) | t( a ξ D /2 ) || μ( a+ ξ D /2 ,a ξ D /2 ) | ×Re{ F( a+ ξ D /2 ,a ξ D /2 , r A ;ν )exp[ iΔφ( a+ ξ D /2 ,a ξ D /2 )+iα( a+ ξ D /2 ,a ξ D /2 ) ] } d 2 ξ D .
S virt ( r A ;ν )=2 AP AP ξ D 0 S 0 ( ξ A + ξ D /2 ) | t( ξ A + ξ D /2 ) | S 0 ( ξ A ξ D /2 ) | t( ξ A ξ D /2 ) || μ( ξ A + ξ D /2 , ξ A ξ D /2 ) | ×Re{ F( ξ A + ξ D /2 , ξ A ξ D /2 , r A ;ν )exp[ iΔϕ( ξ A + ξ D /2 , ξ A ξ D /2 )+iα( ξ A + ξ D /2 , ξ A ξ D /2 ) ] } d 2 ξ D d 2 ξ A
S( 0, r A ;ν )= AP S 0 ( ξ A ) | t( ξ A ) | 2 F( ξ A , r A ;ν ) d 2 ξ A ,
S( b, r A ;ν )=2 AP ξ D 0 S 0 ( ξ A +b/2 ) | t( ξ A +b/2 ) | S 0 ( ξ A b/2 ) | t( ξ A b/2 ) || μ( ξ A +b/2 , ξ A b/2 ) | ×Re{ F( ξ A +b/2 , ξ A b/2 , r A ;ν )exp[ iΔϕ( ξ A +b/2 , ξ A b/2 )+iα( ξ A +b/2 , ξ A b/2 ) ] } d 2 ξ A ,

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