Abstract

Models for the coherence of quasi-monochromatic light from spherical incoherent sources are constructed by placing incoherent monopole and dipole sources on the surface of a sphere. All models allow for relatively simple numerical calculations of coherence functions for arbitrary source sizes and positions of observation points. We show analytically and confirm numerically that the far-field regime for transverse coherence is formed at distances larger than the source size, regardless of the wavelength.

© 2019 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Partially coherent sources that generate the same far-field spectra as completely incoherent sources

John T. Foley and Emil Wolf
J. Opt. Soc. Am. A 5(10) 1683-1687 (1988)

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).
  2. S. S. Agraval, G. Gbur, and E. Wolf, “Coherence properties of sunlight,” Opt. Lett. 29, 459–461 (2004).
    [Crossref]
  3. R. Borghi, F. Gori, O. Korotkova, and M. Santarsiero, “Propagation of cross-spectral densities from spherical sources,” Opt. Lett. 37, 3183–3185 (2012).
    [Crossref]
  4. F. Gori and O. Korotkova, “Modal expansion for spherical homogeneous sources,” Opt. Commun. 282, 3859–3861 (2009).
    [Crossref]
  5. M. Charnotskii, “Coherence of radiation from incoherent sources: II. Ball and disk sources and coherence of sunlight,” J. Opt. Soc. Am. 36, 1440–1446 (2019).
    [Crossref]
  6. H. Mashaal and A. Goldstein, “Fundamental bounds for antenna harvesting of sunlight,” Opt. Lett 36, 900–902 (2011).
    [Crossref]
  7. P. B. Lerner, P. H. Cutler, and N. M. Miskovsky, “Coherence properties of blackbody radiation and application to energy harvesting and imaging with nanoscale rectennas,” J. Nanophotonics 9, 093044 (2015).
    [Crossref]
  8. S. Divitt and L. N. Novotny, “Spatial coherence of sunlight and its implications for light management in photovoltaics,” Optica 2, 95–103 (2015).
    [Crossref]
  9. H. Neckel and D. Labs, “Solar limb darkening 1986–1990 (λλ303 to 1099 nm),” Sol. Phys. 153, 91–114 (1994).
    [Crossref]
  10. H. Mashaal, A. Goldstein, D. Feuermann, and J. M. Gordon, “First direct measurement of the spatial coherence of sunlight,” Opt. Lett. 37, 3516–3518 (2012).
    [Crossref]
  11. J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, “Coherence and polarization properties of a three-dimensional, primary, quasi-homogeneous, and isotropic source and its far field,” Opt. Commun. 283, 4452–4456 (2010).
    [Crossref]
  12. J. R. Zurita-Sánchez, “Coherence properties of the electric field generated by an incoherent source of currents distributed on the surface of a sphere,” J. Opt. Soc. Am. A 33, 118–130 (2016).
    [Crossref]
  13. S. Sundaram and P. K. Panigrahi, “On the origin of the coherence of sunlight on the earth,” Opt. Lett. 41, 4222–4224 (2016).
    [Crossref]
  14. M. Charnotskii, “Eight models for coherence of radiation from incoherent sources and coherence of sunlight,” arXiv:1904.05944 (2019).
  15. F. Gori, “Far-zone approximation for partially coherent Sources,” Opt. Lett. 30, 2840–2842 (2005).
    [Crossref]

2019 (1)

M. Charnotskii, “Coherence of radiation from incoherent sources: II. Ball and disk sources and coherence of sunlight,” J. Opt. Soc. Am. 36, 1440–1446 (2019).
[Crossref]

2016 (2)

2015 (2)

P. B. Lerner, P. H. Cutler, and N. M. Miskovsky, “Coherence properties of blackbody radiation and application to energy harvesting and imaging with nanoscale rectennas,” J. Nanophotonics 9, 093044 (2015).
[Crossref]

S. Divitt and L. N. Novotny, “Spatial coherence of sunlight and its implications for light management in photovoltaics,” Optica 2, 95–103 (2015).
[Crossref]

2012 (2)

2011 (1)

H. Mashaal and A. Goldstein, “Fundamental bounds for antenna harvesting of sunlight,” Opt. Lett 36, 900–902 (2011).
[Crossref]

2010 (1)

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, “Coherence and polarization properties of a three-dimensional, primary, quasi-homogeneous, and isotropic source and its far field,” Opt. Commun. 283, 4452–4456 (2010).
[Crossref]

2009 (1)

F. Gori and O. Korotkova, “Modal expansion for spherical homogeneous sources,” Opt. Commun. 282, 3859–3861 (2009).
[Crossref]

2005 (1)

2004 (1)

1994 (1)

H. Neckel and D. Labs, “Solar limb darkening 1986–1990 (λλ303 to 1099 nm),” Sol. Phys. 153, 91–114 (1994).
[Crossref]

Agraval, S. S.

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

Charnotskii, M.

M. Charnotskii, “Coherence of radiation from incoherent sources: II. Ball and disk sources and coherence of sunlight,” J. Opt. Soc. Am. 36, 1440–1446 (2019).
[Crossref]

M. Charnotskii, “Eight models for coherence of radiation from incoherent sources and coherence of sunlight,” arXiv:1904.05944 (2019).

Cutler, P. H.

P. B. Lerner, P. H. Cutler, and N. M. Miskovsky, “Coherence properties of blackbody radiation and application to energy harvesting and imaging with nanoscale rectennas,” J. Nanophotonics 9, 093044 (2015).
[Crossref]

Divitt, S.

Feuermann, D.

Friberg, A. T.

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, “Coherence and polarization properties of a three-dimensional, primary, quasi-homogeneous, and isotropic source and its far field,” Opt. Commun. 283, 4452–4456 (2010).
[Crossref]

Gbur, G.

Goldstein, A.

H. Mashaal, A. Goldstein, D. Feuermann, and J. M. Gordon, “First direct measurement of the spatial coherence of sunlight,” Opt. Lett. 37, 3516–3518 (2012).
[Crossref]

H. Mashaal and A. Goldstein, “Fundamental bounds for antenna harvesting of sunlight,” Opt. Lett 36, 900–902 (2011).
[Crossref]

Gordon, J. M.

Gori, F.

Kaivola, M.

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, “Coherence and polarization properties of a three-dimensional, primary, quasi-homogeneous, and isotropic source and its far field,” Opt. Commun. 283, 4452–4456 (2010).
[Crossref]

Korotkova, O.

R. Borghi, F. Gori, O. Korotkova, and M. Santarsiero, “Propagation of cross-spectral densities from spherical sources,” Opt. Lett. 37, 3183–3185 (2012).
[Crossref]

F. Gori and O. Korotkova, “Modal expansion for spherical homogeneous sources,” Opt. Commun. 282, 3859–3861 (2009).
[Crossref]

Labs, D.

H. Neckel and D. Labs, “Solar limb darkening 1986–1990 (λλ303 to 1099 nm),” Sol. Phys. 153, 91–114 (1994).
[Crossref]

Lerner, P. B.

P. B. Lerner, P. H. Cutler, and N. M. Miskovsky, “Coherence properties of blackbody radiation and application to energy harvesting and imaging with nanoscale rectennas,” J. Nanophotonics 9, 093044 (2015).
[Crossref]

Lindberg, J.

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, “Coherence and polarization properties of a three-dimensional, primary, quasi-homogeneous, and isotropic source and its far field,” Opt. Commun. 283, 4452–4456 (2010).
[Crossref]

Mashaal, H.

H. Mashaal, A. Goldstein, D. Feuermann, and J. M. Gordon, “First direct measurement of the spatial coherence of sunlight,” Opt. Lett. 37, 3516–3518 (2012).
[Crossref]

H. Mashaal and A. Goldstein, “Fundamental bounds for antenna harvesting of sunlight,” Opt. Lett 36, 900–902 (2011).
[Crossref]

Miskovsky, N. M.

P. B. Lerner, P. H. Cutler, and N. M. Miskovsky, “Coherence properties of blackbody radiation and application to energy harvesting and imaging with nanoscale rectennas,” J. Nanophotonics 9, 093044 (2015).
[Crossref]

Neckel, H.

H. Neckel and D. Labs, “Solar limb darkening 1986–1990 (λλ303 to 1099 nm),” Sol. Phys. 153, 91–114 (1994).
[Crossref]

Novotny, L. N.

Panigrahi, P. K.

Santarsiero, M.

Setälä, T.

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, “Coherence and polarization properties of a three-dimensional, primary, quasi-homogeneous, and isotropic source and its far field,” Opt. Commun. 283, 4452–4456 (2010).
[Crossref]

Sundaram, S.

Wolf, E.

S. S. Agraval, G. Gbur, and E. Wolf, “Coherence properties of sunlight,” Opt. Lett. 29, 459–461 (2004).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

Zurita-Sánchez, J. R.

J. Nanophotonics (1)

P. B. Lerner, P. H. Cutler, and N. M. Miskovsky, “Coherence properties of blackbody radiation and application to energy harvesting and imaging with nanoscale rectennas,” J. Nanophotonics 9, 093044 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

M. Charnotskii, “Coherence of radiation from incoherent sources: II. Ball and disk sources and coherence of sunlight,” J. Opt. Soc. Am. 36, 1440–1446 (2019).
[Crossref]

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

Opt. Commun. (2)

F. Gori and O. Korotkova, “Modal expansion for spherical homogeneous sources,” Opt. Commun. 282, 3859–3861 (2009).
[Crossref]

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, “Coherence and polarization properties of a three-dimensional, primary, quasi-homogeneous, and isotropic source and its far field,” Opt. Commun. 283, 4452–4456 (2010).
[Crossref]

Opt. Lett (1)

H. Mashaal and A. Goldstein, “Fundamental bounds for antenna harvesting of sunlight,” Opt. Lett 36, 900–902 (2011).
[Crossref]

Opt. Lett. (5)

Optica (1)

Sol. Phys. (1)

H. Neckel and D. Labs, “Solar limb darkening 1986–1990 (λλ303 to 1099 nm),” Sol. Phys. 153, 91–114 (1994).
[Crossref]

Other (2)

M. Charnotskii, “Eight models for coherence of radiation from incoherent sources and coherence of sunlight,” arXiv:1904.05944 (2019).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Solid curves depict the DOCs for monopole sources on a sphere numerically calculated using Eq. (15) for three values of ka. Dashed curves represent the far-field result, Eq. (19).
Fig. 2.
Fig. 2. Solid curves depict the DOC for incoherent normal dipoles on a sphere numerically calculated using Eq. (25), for three values of ka. Dashed curves represent the far-field result obtained using Eq. (26).
Fig. 3.
Fig. 3. Solid curves depict the DOCs for incoherent isotropic dipoles on a sphere numerically calculated using Eq. (30), for three values of ka. Dashed curves depict the far-field result obtained using Eq. (31).
Fig. 4.
Fig. 4. Normalized difference of the exact and far-field DOCs Δw(R,α)R2/a2 for monopoles on a sphere for three values of parameter ka calculated by the subtraction of the wide-angle far-field result, Eq. (19), from numerically integrated Eq. (16).
Fig. 5.
Fig. 5. Normalized maximal differences of the exact and far-field DOCs Δw(R,α)R2/a2 for monopoles on a sphere for several values of ka.
Fig. 6.
Fig. 6. Differences of the exact and far-field DOCs Δw(R,α) for monopoles on a sphere, in a ball, and on a disk for ka=3000. The parameter is k(Ra).

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

ΔU(R)+k2U(R)=S(R).
U(R)=d3RsS(Rs)G(RRs),
G(R)=14πRexp(ikR),
U(R)=p^·G(R)=p^·R(1ikR)4πR3exp(ikR).
U(r,z)=2d2r0U(r0,0)G(rr0,z)z,z>0.
U(R)=14πRS=ad2RSm(RS)|RRS|exp(ik|RRS|),
W(R1,R2)=U(R1)U*(R2)=116π2RS=ad2RSRS=ad2RSm(RS)m*(RS)|R1RS||R2RS|exp(ik|R1RS|ik|R2RS|).
m(RS)m*(RS)=M(RS)δSPH(RSRS),
R=R(cosθcosφx^+cosθsinφy^+sinθz^),δSPH(RSRS)=δ(φφ)δ(θθ)a2cosθ,
W(R1,R2)=Ma216π2ππdφπ/2π/2dθcosθ|R1RS||R2RS|exp(ik|R1RS|ik|R2RS|).
I(R)=W(R,0)=Ma28ππ/2π/2dθcosθa2+R22aRsinθ=Ma8πRln(R+aRa).
I(Ra)Ma24πR2,
J(R)ImU(R)U*(R)=Im1W(R1,R2)R1=R2=R=kMa24πR2R^,
R1=R(x^sinα2+z^cosα2),R2=R(x^sinα2+z^cosα2),
w(R,α)=W(R,R,α)I(R)=aR2πln(R+aRa)ππdφπ/2π/2cosθdθcos{k[r(α)r(α)]}r(α)r(α),
r(α)[a2+R22aR(cosφcosθsinα2+sinθcosα2)]1/2.
r(α)r(α)=R2[1+O(aR)],r(α)r(α)=2acosφcosθsinα2[1+aRsinθcosα2+O(a2R2)],
wFF(α)=12π/2π/2cosθdθJ0(ηcosθ)=j0(η)=sinηη,η=2kasinα2.
wPA(α)sin(kaα)kaα.
U(R)=14πaRS=ad2RSdN(RS)(a2RS·R)|RRS|3[1ik|RRS|]exp(ik|RRS|),
dN(RS)dN*(RS)=DNδSPH(RSRS),
W(R1,R2)=DNa216π2ππdφπ/2π/2dθcosθ(R1·RSa2)(R2·RSa2)|R1RS|3|R2RS|3(1ik|R1RS|)(1+ik|R2RS|)exp(ik|R1RS|ik|R2RS|),
I(R)=W(R,R)=DN32π[aRln(R+aRa)2a2R23a2(R2a2)2]+DN16πk2a2[2(R2a2)aRln(R+aRa)].
w(R,α)=a2DN16π2I(R)ππdφπ/2π/2cosθdθ[r(α)r(α)]3[1ikr(α)][1+ikr(α)]exp{ik[r(α)r(α)]}[a2ar(cosφcosθsinα2+sinθcosα2)][a2ar(cosφcosθsinα2+sinθcosα2)].
wFF(α)=3cos2α2[sinηη3cosηη2]3sin2α2[sinηη2sinηη3+2cosηη2],η=2kasinα2.
wPA(α)=3j1(kaα)(kaα)=3[sin(kaα)(kaα)3cos(kaα)(kaα)2].
U(R)=14πRS=ad2RSdI(RS)d^I(RS)·(RSR)|RRS|3[1ik|RRS|]exp(ik|RRS|).
W(R1,R2)=DIa448π2ππdφπ/2π/2dθcosθ(R1RS)·(R2RS)|R1RS|3|R2RS|3(1ik|R1RS|)(1+ik|R2RS|)exp(ik|R1RS|ik|R2RS|).
I(R)=W(R,R)=DI12π[a4(R2a2)2+k2a2a2Rln(R+aRa)].
w(R,α)=a2DI48π2I(R)ππdφπ/2π/2cosθdθ[r(α)r(α)]3[1ikr(α)][1+ikr(α)]exp{ik[r(α)r(α)]}×[a22aRsinθcosα2+R2cosα].
wFF(α)=12cosαπ/2π/2cosθJ0(ηcosθ)dθ=cosαsinηη,η=2kasinα2.
Δw(R,α)w(R,α)wFF(α)=a2R2[c1(α)+kasinα2c2(α)+k2a2sin2α2c3(α)]+O(a4r4),
c1(α)=1π0πdφ0π/2cosθdθexp(2ikasinα2cosφcosθ)×(432sin2α2cos2φcos2θ4cos2α2sin2θ),
c2(α)=iπ0πdφ0π/2cosθdθexp(2ikasinα2cosφcosθ)×cosφcosθ(17cos2α2sin2θsin2α2cos2φcos2θ),
c3(α)=2πcos2α20πdφ0π/2cosθdθexp(2ikasinα2cosφcosθ)cos2θsin2θ.
Δw(R,α1ka)a2R2[16sin(kaα)(kaα)12sin(kaα)(kaα)3+12cos(kaα)(kaα)2],

Metrics