Abstract

The border between the descriptions of the classical optical fields in any state of spatial coherence and the quantum coherence state of light is revisited in the framework of the phase-space representation. Although it is established that such descriptions are not completely equivalent, the exact calculation of the marginal power spectrum leads to new analogies that suggest that some features exclusively attributed to quantum states of light can be also shared by classical optical fields due to their spatial coherence state.

© 2012 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 University, 1995).
  2. R. Castañeda and J. Garca-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express 19, 25022–25034 (2011). It contains the complete deduction of equation 1 and the analysis of the main features of the nonparaxial phase-space representation of the optical field.
    [CrossRef]
  3. 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]
  4. R. Castañeda, H. Muñuz-Ossa, and J. Garcia-Sucerquia, “Efficient numerical calculation of interference and diffraction of optical fields in any state of spatial coherence in the phase-space representation,” Appl. Opt. 49, 6063–6071 (2010).
    [CrossRef]
  5. R. Castañeda, “The optics of spatial coherence wavelets,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, 2010), Vol. 164, pp 29–255.
  6. R. Castañeda, H. Muñoz, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58, 962–972 (2011).
    [CrossRef]
  7. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).
  8. U. Leonhardt, Measuring the Quantum State of Light(Cambridge University, 1997).
  9. M. Born and E. Wolf, Principles of Optics6th ed. (Pergamon Press, 1993).
  10. T. Young, “The Bakerian lecture: Experiments and calculations relative to physical optics,” Phil. Trans. R. Soc. London 94, 1–16 (1804).
    [CrossRef]
  11. F. M. Grimaldo, Physico-Mathesis de Lumine Coloribus et Iride (Ex Typographia Haeredi Victorij Benatij, Bononiae, 1665).

Born, M.

M. Born and E. Wolf, Principles of Optics6th ed. (Pergamon Press, 1993).

Cañas-Cardona, G.

Castañeda, R.

Garca-Sucerquia, J.

Garcia-Sucerquia, J.

Grimaldo, F. M.

F. M. Grimaldo, Physico-Mathesis de Lumine Coloribus et Iride (Ex Typographia Haeredi Victorij Benatij, Bononiae, 1665).

Hennelly, B.

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

Leonhardt, U.

U. Leonhardt, Measuring the Quantum State of Light(Cambridge University, 1997).

Mandel, L.

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

Muñoz, H.

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

Muñuz-Ossa, H.

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).

Wolf, E.

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

M. Born and E. Wolf, Principles of Optics6th ed. (Pergamon Press, 1993).

Young, T.

T. Young, “The Bakerian lecture: Experiments and calculations relative to physical optics,” Phil. Trans. R. Soc. London 94, 1–16 (1804).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Phil. Trans. R. Soc. London (1)

T. Young, “The Bakerian lecture: Experiments and calculations relative to physical optics,” Phil. Trans. R. Soc. London 94, 1–16 (1804).
[CrossRef]

Other (6)

F. M. Grimaldo, Physico-Mathesis de Lumine Coloribus et Iride (Ex Typographia Haeredi Victorij Benatij, Bononiae, 1665).

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

R. Castañeda, “The optics of spatial coherence wavelets,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, 2010), Vol. 164, pp 29–255.

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

U. Leonhardt, Measuring the Quantum State of Light(Cambridge University, 1997).

M. Born and E. Wolf, Principles of Optics6th ed. (Pergamon Press, 1993).

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Figures (8)

Fig. 1.
Fig. 1.

Diffraction geometry related to the structured spatial coherence support centered at the point ξA on the AP, that determines the marginal power spectrum S(ξA,rA).

Fig. 2.
Fig. 2.

(a) Exactly calculated marginal power spectra that describe the evolution of the free-space diffraction envelope along the propagation of the field emitted by a single radiant point source placed at ξA=0 on the AP. (b) Power spectra at the OP at different distances z from the AP. (c) Comparison between the free-space diffraction envelope and the Lorentzian and the Gaussian distributions. (d) Power spectrum profiles at the OP provided by a set of N spatially radiant point sources of pitch b and total length L under different states of spatial coherence. The profiles in all panels were scaled for presentation purposes.

Fig. 3.
Fig. 3.

Exactly calculated marginal power spectra (top row), power spectra at the OP (midrow), and radiant and virtual power contributions (bottom row) for a one-dimensional spatially partially Young’s experiment with |μ(a+b/2,ab/2)|=0.3, α(a+b/2,ab/2)=π, λ=0.632μm and radiant point sources separation |b|=5μm, for (a) z=0μm, (b) z=3μm, and (c) z=10μm.

Fig. 4.
Fig. 4.

Comparison between the modulating energies at the AP (z=0) of fully spatially coherent and a partially spatially coherent Young’s experiments, realized by identical radiant point sources with separation |b|=5μm that emit a field of λ=0.632μm, for (a) α=0, (b) α=π/2, (c) α=3π/2.

Fig. 5.
Fig. 5.

Comparison between the phase-space representations of the Young’s experiment with fully spatially coherent radiant point sources separated 2μm, that emit a field with λ=0.632μm and whose interference pattern is observed at the OP placed at z=104μm (top row), and the quantum superposition of two coherence states with 2q0=10 units of generalized coordinate (bottom row). (a) Exactly calculated marginal power spectrum of the Young’s experiment and WDF of the quantum superposition of the coherent states. (b) Exactly calculated power spectrum at the AP for the Young’s experiment and the WDF marginal associated with the probability density of the position of the superposed quantum states on the generalized coordinate axis. (c) Exactly calculated power spectrum at the OP in the Young’s experiment and the WDF marginal associated with the probability density of the generalized momentum of the quantum superposition. (d) Radiant and modulation power contributions to the power spectrum of the field at the OP, provided by the point sources of the radiant and the virtual layers, respectively, in the Young’s experiment; and separate contributions of the single coherent quantum states and their superposition to the marginal of the WDF corresponding to the probability density of the generalized momentum.

Fig. 6.
Fig. 6.

Phase-space representation of the squeezing evolution of a quantum coherent state, with squeezing parameter ζ.

Fig. 7.
Fig. 7.

Exactly calculated phase-space representation for propagation of the fully spatially coherent field of λ=0.632μm, emitted by seven identical and equidistant radiant point sources, in the Fresnel–Fraunhofer domain (z=104μm). The upper row shows the marginal power spectra and the bottom row shows the profiles of the corresponding power spectra at the OP. (a) |b|=2μm>λ and L=12μm>λ, (b) |b|=1μm>λ and L=6μm>λ, and (c) |b|=0.5μm<λ and L=3μm>λ.

Fig. 8.
Fig. 8.

Comparison between the sinc function profile corresponding to the diffraction of a uniform plane and fully spatially coherent wavefront of λ=0.632μm by a slit of width 3μm, and the exact calculated diffraction pattern in the Fresnel–Fraunhofer domain, produced by two arrays of identical and equidistant radiant point sources of total length L=3μm. (a) Seven sources with |b|=0.5μm and (b) 16 sources with |b|=0.2μm.

Equations (6)

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S(ξA,rA)=14λ2APS0(ξA+ξD/2)t(ξA+ξD/2)S0(ξAξD/2)t*(ξAξD/2)μ(ξA+ξD/2,ξAξD/2)×exp[ikz2+|rAξA|2+|ξD|2/4+ξA·ξDrA·ξD]z2+|rAξA|2+|ξD|2/4+ξA·ξDrA·ξDexp[ikz2+|rAξA|2+|ξD|2/4ξA·ξD+rA·ξD]z2+|rAξA|2+|ξD|2/4ξA·ξD+rA·ξD×(z+z2+|rAξA|2+|ξD|2/4+ξA·ξDrA·ξD)(z+z2+|rAξA|2+|ξD|2/4ξA·ξD+rA·ξD)d2ξD,
S(ξA,rA)=S0(ξA)4λ2(z+z2+|rAξA|2z2+|rAξA|2)δ(ξAa).
S(rA)=APS(ξA,rA)d2ξA=S0(a)4λ2(z+z2+|rAa|2z2+|rAa|2)2.
S(ξA,rA)=S0(ξA)4λ2(z+z2+|rAξA|2z2+|rAξA|2)2[δ(ξAa+b/2)+δ(ξAab/2)]+12λ2δ(ξAa)S0(ξA+b/2)S0(ξAb/2)|μ(ξA+b/2,ξAb/2)|(z+z2+|rAξA|2+|b|2/4+ξA·brA·bz2+|rAξA|2+|b|2/4+ξA·brA·b)×(z+z2+|rAξA|2+|b|2/4ξA·b+rA·bz2+|rAξA|2+|b|2/4ξA·b+rA·b)×cos[kz2+|rAξA|2+|b|2/4+ξA·brA·bkz2+|rAξA|2+|b|2/4ξA·b+rA·b+α(ξA+b/2,ξAb/2)]
S(rA)=S0(ab/2)4λ2(z+z2+|rAa+b/2|2z2+|rAa+b/2|2)2+S0(a+b/2)4λ2(z+z2+|rAab/2|2z2+|rAab/2|2)2+12λ2S0(a+b/2)S0(ab/2)|μ(a+b/2,ab/2)|(z+z2+|rAa|2+|b|2/4+a·brA·bz2+|rAa|2+|b|2/4+a·brA·b)×(z+z2+|rAa|2+|b|2/4a·b+rA·bz2+|rAa|2+|b|2/4a·b+rA·b)×cos[kz2+|rAa|2+|b|2/4+a·brA·bkz2+|rAa|2+|b|2/4a·b+rA·b+α(a+b/2,ab/2)]
W(q,p)=exp[(q+q0)2p2]+exp[(qq0)2p2]+2exp[q2p2]cos(2q0p),

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