Abstract

We evaluate fundamental bounds for using aperture antennas to harvest sunlight based on a generalized analysis of the partial coherence of solar radiation.

© 2011 Optical Society of America

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References

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  1. R. L. Bailey, J. Eng. Power 94, 73 (1972).
    [CrossRef]
  2. R. Corkish, M. A. Green, T. Puzzer, and T. Humphrey, in Proceedings of the 3rd World Conference on Photovoltaic Energy Conversion (2003), Vol.  3, p. 2682.
  3. M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge University, 1999).
  4. R. Winston, Y. Sun, and R. G. Littlejohn, Opt. Commun. 207, 41 (2002).
    [CrossRef]
  5. G. S. Agarwal, G. Gbur, and E. Wolf, Opt. Lett. 29, 459(2004).
    [CrossRef] [PubMed]
  6. Visible evidence that sunlight possesses a transverse coherence length of the order of 102 μm includes the speckle pattern from surfaces with a structure of the order of tens of micrometers, as many matte metallic coins do, as well as the glory observed around one’s shadow when standing in a mist with the sun at one’s back (due to water droplet diameters being of the order of tens of micrometers).
  7. W. L. Stutzman, Antenna Theory and Design (Wiley, 1981).
  8. The term “intensity” is avoided in referring to the EMCF because “intensity” often connotes a measurable nonnegative quantity, whereas the EMCF can be negative.
  9. The terrestrial solar spectrum can vary significantly with atmospheric conditions. We note, however, that when a typical clear-sky midday midlatitude spectrum is used in Eqs. , there is no noticeable change in the results plotted in Figs. .
  10. R. Winston, J. C. Miñano, and P. Benítez, with contributions from N.Shatz and J.Bortz, Nonimaging Optics(Elsevier, 2005).
  11. If the untruncated blackbody spectrum of Eq.  is used, whereby atmospheric attenuation is ignored and λmax→∞, then the solar intercepted power in the large-radius (asymptotic pure-incoherence) limit is larger. However, this is weighted by quite a low coherence efficiency in that regime (see Fig. ), such that the overall error in estimating antenna power harvesting turns out to be of the order of 1%. [Using λmin=0 instead of 0.3 μm introduces a negligible difference, in part due to the λ2 weighting in Eq. .]
  12. J. W. Goodman, Statistical Optics (Wiley, 1985).

2004

2002

R. Winston, Y. Sun, and R. G. Littlejohn, Opt. Commun. 207, 41 (2002).
[CrossRef]

1972

R. L. Bailey, J. Eng. Power 94, 73 (1972).
[CrossRef]

Agarwal, G. S.

Bailey, R. L.

R. L. Bailey, J. Eng. Power 94, 73 (1972).
[CrossRef]

Benítez, P.

R. Winston, J. C. Miñano, and P. Benítez, with contributions from N.Shatz and J.Bortz, Nonimaging Optics(Elsevier, 2005).

Born, M.

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

Corkish, R.

R. Corkish, M. A. Green, T. Puzzer, and T. Humphrey, in Proceedings of the 3rd World Conference on Photovoltaic Energy Conversion (2003), Vol.  3, p. 2682.

Gbur, G.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Green, M. A.

R. Corkish, M. A. Green, T. Puzzer, and T. Humphrey, in Proceedings of the 3rd World Conference on Photovoltaic Energy Conversion (2003), Vol.  3, p. 2682.

Humphrey, T.

R. Corkish, M. A. Green, T. Puzzer, and T. Humphrey, in Proceedings of the 3rd World Conference on Photovoltaic Energy Conversion (2003), Vol.  3, p. 2682.

Littlejohn, R. G.

R. Winston, Y. Sun, and R. G. Littlejohn, Opt. Commun. 207, 41 (2002).
[CrossRef]

Miñano, J. C.

R. Winston, J. C. Miñano, and P. Benítez, with contributions from N.Shatz and J.Bortz, Nonimaging Optics(Elsevier, 2005).

Puzzer, T.

R. Corkish, M. A. Green, T. Puzzer, and T. Humphrey, in Proceedings of the 3rd World Conference on Photovoltaic Energy Conversion (2003), Vol.  3, p. 2682.

Stutzman, W. L.

W. L. Stutzman, Antenna Theory and Design (Wiley, 1981).

Sun, Y.

R. Winston, Y. Sun, and R. G. Littlejohn, Opt. Commun. 207, 41 (2002).
[CrossRef]

Winston, R.

R. Winston, Y. Sun, and R. G. Littlejohn, Opt. Commun. 207, 41 (2002).
[CrossRef]

R. Winston, J. C. Miñano, and P. Benítez, with contributions from N.Shatz and J.Bortz, Nonimaging Optics(Elsevier, 2005).

Wolf, E.

G. S. Agarwal, G. Gbur, and E. Wolf, Opt. Lett. 29, 459(2004).
[CrossRef] [PubMed]

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

J. Eng. Power

R. L. Bailey, J. Eng. Power 94, 73 (1972).
[CrossRef]

Opt. Commun.

R. Winston, Y. Sun, and R. G. Littlejohn, Opt. Commun. 207, 41 (2002).
[CrossRef]

Opt. Lett.

Other

R. Corkish, M. A. Green, T. Puzzer, and T. Humphrey, in Proceedings of the 3rd World Conference on Photovoltaic Energy Conversion (2003), Vol.  3, p. 2682.

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

Visible evidence that sunlight possesses a transverse coherence length of the order of 102 μm includes the speckle pattern from surfaces with a structure of the order of tens of micrometers, as many matte metallic coins do, as well as the glory observed around one’s shadow when standing in a mist with the sun at one’s back (due to water droplet diameters being of the order of tens of micrometers).

W. L. Stutzman, Antenna Theory and Design (Wiley, 1981).

The term “intensity” is avoided in referring to the EMCF because “intensity” often connotes a measurable nonnegative quantity, whereas the EMCF can be negative.

The terrestrial solar spectrum can vary significantly with atmospheric conditions. We note, however, that when a typical clear-sky midday midlatitude spectrum is used in Eqs. , there is no noticeable change in the results plotted in Figs. .

R. Winston, J. C. Miñano, and P. Benítez, with contributions from N.Shatz and J.Bortz, Nonimaging Optics(Elsevier, 2005).

If the untruncated blackbody spectrum of Eq.  is used, whereby atmospheric attenuation is ignored and λmax→∞, then the solar intercepted power in the large-radius (asymptotic pure-incoherence) limit is larger. However, this is weighted by quite a low coherence efficiency in that regime (see Fig. ), such that the overall error in estimating antenna power harvesting turns out to be of the order of 1%. [Using λmin=0 instead of 0.3 μm introduces a negligible difference, in part due to the λ2 weighting in Eq. .]

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (4)

Fig. 1
Fig. 1

EMCF (normalized to its maximum val ue at zero radius) for individual wavelengths (broken colored curves) and sunlight (solid black curve). For spectrum- averaged solar radiation, the oscillations are essentially averaged out, and the first null is at a radius of 200 μm .

Fig. 2
Fig. 2

Intercepted power (normalized to the asymptotic QM value for λ = 0.5 μm ) as a function of detector radius b. For each QM curve, the asymptotic power is λ 2 I [same I for each λ, with I taken as the spectrum-averaged solar radiance of Eq. (5)]. At small radii, all curves converge to the universal result of proportionality to b 2 , independent of λ. At antenna radii up to 60 μm , the solar result is well approximated by the λ = 0.65 μm QM curve. (The intercepted solar power transitions from three-dimensional to one-dimensional blackbody radiation as the radius increases from zero to asymptotically large.)

Fig. 3
Fig. 3

Intercepted power normalized to its own maximum value plotted against averaged intensity normalized to its own maximum (these normalizations allow a comparison of the QM and broadband results). The three distinct conventional definitions of coherence length are noted for the QM case and elaborated on in the text.

Fig. 4
Fig. 4

Coherence efficiency (intercepted power relative to its value in the pure-coherence limit) as a function of detector radius for solar radiation. For perspective, the null in the EMCF is at 200 μm .

Equations (9)

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P t = I A eff t ,
A eff = | A E ( r , t ) d A | 2 A | E ( r , t ) | 2 d A = | A E ( r , t ) d A | | A E * ( r , t ) d A | A | E ( r , t ) | 2 d A = A A E ( r , t ) E * ( r , t ) d A d A A | E ( r , t ) | 2 d A ,
P t = 1 A ap A A E ( r , t ) E * ( r , t ) t d A d A .
E ( r , t ) E * ( r t ) t = 2 π θ 2 I J 1 ( k θ | r r | ) k θ | r r | ,
E ( r , t ) E * ( r t ) t = 2 π θ 2 λ min λ max I B B ( λ ) J 1 ( k θ | r r | ) k θ | r r | d λ ,
I B B ( λ ) = 2 h c 2 λ 5 ( e h c k T λ 1 ) 1 ,
P t = 2 π θ 2 I A ap A A J 1 ( k θ | r r | ) k θ | r r | d A d A = I λ 2 [ 1 J 0 2 ( k θ b ) J 1 2 ( k θ b ) ] ,
P t = λ min λ max I B B ( λ ) λ 2 [ 1 J 0 2 ( k θ b ) J 1 2 ( k θ b ) ] d λ ,
A eff Ω = π 2 R c 2 θ 2 = λ 2 or R c = λ π θ 67.7 λ .

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