Abstract

Following the development of the companion paper [J. Opt. Soc. Am. 36, 1433 (2019) [CrossRef]  ], additional models for the coherence of quasi-monochromatic light are constructed by placing incoherent monopole and dipole sources inside a spherical ball and on a circular plane disk. The paraxial approximation, which is valid for sunlight, reduces the number of different models from eight to three. Based on the solar images corresponding to these three models, the ball model is considered the most appropriate. The ball model can be modified to match the limb-darkening observations.

© 2019 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Coherence of radiation from incoherent sources: I. Sources on a sphere and far-field conditions

Mikhail Charnotskii
J. Opt. Soc. Am. A 36(8) 1433-1439 (2019)

Coherence properties of sunlight

Girish S. Agarwal, Greg Gbur, and Emil Wolf
Opt. Lett. 29(5) 459-461 (2004)

References

  • View by:
  • |
  • |
  • |

  1. M. Charnotskii, “Coherence of radiation from incoherent sources: I. Sphere and ball sources,” J. Opt. Soc. Am. 36, 1433–1439 (2019).
    [Crossref]
  2. S. Divitt and L. N. Novotny, “Spatial coherence of sunlight and its implications for light management in photovoltaics,” Optica 2, 95–103 (2015).
    [Crossref]
  3. H. Neckel and D. Labs, “Solar limb darkening 1986–1990 (λλ303 to 1099  nm),” Sol. Phys. 153, 91–114 (1994).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 1999).
  5. H. Mashaal and A. Goldstein, “Fundamental bounds for antenna harvesting of sunlight,” Opt. Lett 36, 900–902 (2011).
    [Crossref]
  6. 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. Nanophoton. 9, 093044(2015).
    [Crossref]
  7. S. S. Agraval, G. Gbur, and E. Wolf, “Coherence properties of sunlight,” Opt. Lett. 29, 459–461 (2004).
    [Crossref]
  8. R. Borghi, F. Gori, O. Korotkova, and M. Santarsiero, “Propagation of cross-spectral densities from spherical sources,” Opt. Lett. 37, 3183–3185 (2012).
    [Crossref]
  9. S. Sundaram and P. K. Panigrahi, “On the origin of the coherence of sunlight on the earth,” Opt. Lett. 41, 4222–4224 (2016).
    [Crossref]
  10. 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]
  11. 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]
  12. M. Charnotskii, “Eight models for coherence of radiation from incoherent sources and coherence of sunlight,” arXiv:1904.05944 (2019).
  13. F. Gori, “Far-zone approximation for partially coherent sources,” Opt. Lett. 30, 2840 (2005).
    [Crossref]

2019 (1)

M. Charnotskii, “Coherence of radiation from incoherent sources: I. Sphere and ball sources,” J. Opt. Soc. Am. 36, 1433–1439 (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. Nanophoton. 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 (1)

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]

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: I. Sphere and ball sources,” J. Opt. Soc. Am. 36, 1433–1439 (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. Nanophoton. 9, 093044(2015).
[Crossref]

Divitt, S.

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

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.

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. Nanophoton. 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 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. Nanophoton. 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. Nanophoton. (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. Nanophoton. 9, 093044(2015).
[Crossref]

J. Opt. Soc. Am. (1)

M. Charnotskii, “Coherence of radiation from incoherent sources: I. Sphere and ball sources,” J. Opt. Soc. Am. 36, 1433–1439 (2019).
[Crossref]

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

Opt. Commun. (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]

Opt. Lett (1)

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

Opt. Lett. (4)

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

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

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 (8)

Fig. 1.
Fig. 1. Solid curves depict the DOC for monopole sources in a ball numerically calculated using Eq. (3) for three values of k a . Dashed curves represent the far-field result obtained using Eq. (6).
Fig. 2.
Fig. 2. Solid curves depict the DOC for incoherent isotropic dipoles on a sphere numerically calculated using Eq. (9) for three values of k a . Dashed curves represent the far-field result obtained using Eq. (11).
Fig. 3.
Fig. 3. Solid curves show the DOC for monopole sources at a circle numerically calculated using Eq. (18) for three values of k a . Dashed curves depict the far-field result obtained using Eq. (18).
Fig. 4.
Fig. 4. Solid curves represent the DOC for normal dipole sources at a disk numerically calculated using Eq. (28) for three values of k a . Dashed curves are the far-field result obtained using Eq. (29).
Fig. 5.
Fig. 5. Solid curves show the DOC for isotropic dipole sources at a circle calculated numerically using Eq. (34) for three values of k a . Dashed curves depict the far-field result acquired using Eq. (35).
Fig. 6.
Fig. 6. Far-field wide-angle DOCs for the eight source models. Heavy solid curves represent sources on a sphere: Eqs. (18), (25), (31) in [1]. Dashed curves represent sources in a ball [Eqs. (6) and (11)], and thin curves represent sources on a disk [Eqs. (22), (29), and (35)]. Blue, green, and red curves demonstrate monopole, normal dipole, and isotropic dipole sources, respectively.
Fig. 7.
Fig. 7. DOC for solar light as a function of the linear point separation d . Green double, brown, and triple blue curves represent spherical, disk, and ball sources, respectively.
Fig. 8.
Fig. 8. Sun image cross sections for three models [Eqs. (37), (38), and (39)] as functions of γ / γ .

Equations (41)

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

U ( R ) = 1 4 π P a d 3 P m ( P ) | R P | exp ( i k | R P | ) ,
m ( P ) m * ( P ) = 1 a M ( P ) δ ( P P ) .
W ( R , α ) = M ( 4 π ) 2 a 0 a P 2 d P π π d φ π / 2 π / 2 cos θ d θ exp { i k [ p ( α ) p ( α ) ] } p ( α ) p ( α ) ,
p ( α ) [ P 2 + R 2 2 P R ( cos φ cos θ sin α 2 + sin θ cos α 2 ) ] 1 / 2 .
I ( R ) = W ( R , 0 ) = M 8 π [ 1 ( R 2 a 2 ) 2 a R ln ( R + a R a ) ] .
w FF ( α ) = 3 j 1 ( η ) η = 3 η 2 ( sin η η cos η ) , η = 2 k a sin α 2 .
w PA ( α ) = 3 j 1 ( k a α ) ( k a α ) = 3 [ sin ( k a α ) ( k a α ) 3 cos ( k a α ) ( k a α ) 2 ] ,
U ( R ) = 1 4 π P < a d 3 P R S d I ( P ) d ^ I ( P ) · ( P R ) | R P | 3 [ 1 i k | R P | ] exp ( i k | R P | ) .
W ( R 1 , R 2 ) = D I a 48 π 2 0 a P 2 d P π π d φ π / 2 π / 2 d θ cos θ ( R 1 P ) · ( R 2 P ) | R 1 P | 3 | R 2 P | 3 ( 1 i k | R 1 P | ) ( 1 + i k | R 2 P | ) exp ( i k | R 1 P | i k | R 2 P | ) .
I ( R ) = W ( R , R ) = D I a 24 π r [ ( a R R 2 a 2 + 1 2 ln ( R + a R a ) ) + k 2 r 2 ( a R R 2 a 2 R 2 ln ( R + a R a ) ) ] .
w FF ( α ) = 3 cos α j 1 ( η ) η = 3 cos α ( sin η η 3 cos η η 2 ) , η = 2 k a sin α 2 .
U ( R ) = 1 4 π | R S | a d 2 R S m ( R S ) | R R S | exp ( i k | R R S | ) ,
U ( x , y , 0 ) = 1 4 π x S 2 + y S 2 a 2 d x S d y S m ( x S , y S ) [ ( x x S ) 2 + ( y y S ) 2 ] 1 / 2 exp ( i k [ ( x x S ) 2 + ( y y S ) 2 ] 1 / 2 ) .
U ( x , y , z ) z | z 0 z 4 π x S 2 + y S 2 a 2 d x S d y S m ( x S , y S ) [ ( x x S ) 2 + ( y y S ) 2 + z 2 ] 3 / 2 | z 0 = m ( x , y ) 2 sig ( z ) .
m ( R S ) m * ( R S ) = M δ ( R S R S ) ,
W ( R 1 , R 2 ) = M 16 π 2 | R S | a d 2 R S | R 1 R S | | R 2 R S | exp ( i k | R 1 R S | i k | R 2 R S | ) .
I ( R ) = W ( R , 0 ) = M 16 π ln ( 1 + a 2 R 2 ) R a M 16 π a 2 R 2 .
w ( R , α ) = W ( R , R , α ) I ( R ) = 1 π ln ( 1 + a 2 / R 2 ) 0 2 π d φ 0 a r d r q ( α ) q ( α ) exp [ i k q ( α ) i k q ( α ) ] ,
q ( α ) [ r 2 + R 2 2 r R cos φ sin α 2 ] 1 / 2 .
q ( α ) q ( α ) = R 2 [ 1 + O ( a R ) ] , q ( α ) q ( α ) = 2 ρ cos φ sin α 2 [ 1 + O ( a 2 R 2 ) ] ,
w FF ( α ) = 1 η J 1 ( η ) , η 2 k a sin α 2 ,
w PA ( α ) = 2 J 1 ( k a α ) k a α .
U ( R ) = 1 4 π | R S | a d 2 R S d N ( R S ) ( R S R ) · z ^ | R R S | 3 exp ( i k | R R S | ) [ 1 i k | R R S | ] ,
U ( x , y , z ) | z 0 z 4 π | R S | a d x S d y S d N ( x S , y S ) [ ( x x S ) 2 + ( y y S ) 2 + z 2 ] 3 / 2 | z 0 = d N ( x , y ) 2 sign ( z ) .
d N ( R S ) d N * ( R S ) = a 2 D N δ ( R S R S ) ,
W ( R 1 , R 2 ) = a 2 D N R 1 cos α 1 R 2 cos α 2 16 π 2 × | R S | a d 2 R S [ 1 i k | R 1 R S | ] [ 1 + i k | R 2 R S | ] | R 1 R S | 3 | R 2 R S | 3 exp ( i k | R 1 R S | i k | R 2 R S | ) .
I ( R ) = W ( R , 0 ) = D N a 4 16 π ( a 2 + 2 R 2 2 R 2 ( a 2 + R 2 ) 2 + k 2 ( a 2 + R 2 ) ) R a D N a 4 16 π R 4 ( 1 + k 2 R 2 ) ,
w ( R , α ) = 2 R 4 ( a 2 + R 2 ) 4 cos 2 α 2 π a 2 [ a 2 + 2 R 2 + 2 k 2 R 2 ( a 2 + R 2 ) ] × 0 2 π d φ 0 a r d r [ 1 i k q ( α ) ] [ 1 + i k q ( α ) ] q 3 ( α ) q 3 ( α ) exp [ i k q ( α ) i k q ( α ) ] ,
w ( α ) = cos 2 α 2 J 1 ( η ) η , η 2 k a sin α 2 .
U ( R ) = 1 4 π R S < a d 2 R S d I ( R S ) d ^ I ( R S ) · ( R S R ) | R R S | 3 [ 1 i k | R R S | ] exp ( i k | R R S | ) .
U ( x , y , z ) | z 0 1 2 d I ( x , y ) sin θ d ( x , y ) sign ( z ) ,
W ( R 1 , R 2 ) = a 2 D N 48 π 2 0 a r d r 0 2 π d θ ( R 1 R S ) · ( R 2 R S ) | R 1 R S | 3 | R 2 R S | 3 × ( 1 i k | R 1 R S | ) ( 1 + i k | R 2 R S | ) exp ( i k | R 1 R S | i k | R 2 R S | ) .
I ( R ) = W ( R , 0 ) = D I 48 π ( a 4 R 2 ( a 2 + R 2 ) + k 2 a 2 ln a 2 + R 2 R 2 ) R a D I a 4 48 π R 4 ( 1 + k 2 R 2 ) ,
w ( R , α ) = R 2 ( a 2 + R 2 ) π ( a 2 + k 2 R 2 ( a 2 + R 2 ) ln a 2 + R 2 R 2 ) 0 2 π d φ 0 a r d r ( R 2 cos α + ρ 2 ) q 3 ( α ) q 3 ( α ) × [ 1 i k q ( α ) ] [ 1 + i k q ( α ) ] exp [ i k q ( α ) i k q ( α ) ] ,
w ( α ) = cos α J 1 ( γ ) γ , γ 2 k a sin α 2 .
I IM ( γ ) = C d 2 α W ( α ) exp ( i k R α · γ ) .
I IM ( γ ) { ( 1 γ 2 / γ 2 ) 1 / 2 , γ < γ 0 , γ > γ ,
I IM ( γ ) { ( 1 γ 2 / γ 2 ) 1 / 2 , γ < γ 0 , γ > γ ,
I IM ( γ ) = C { 1 , γ < γ 0 , γ > γ .
M ( P ) = M ( 1 P 2 / a 2 ) n ,
I IM ( γ ) { ( 1 γ 2 / γ 2 ) n + 1 / 2 , γ < γ 0 , γ > γ .

Metrics