Abstract

Fresnel lenses are widely used in concentrating photovoltaic (CPV) systems as primary optical elements focusing sunlight onto small solar cells or onto entrance apertures of secondary optical elements attached to the solar cells. Calculations using the Young-Maggi-Rubinowicz theory of diffraction yield analytical expressions for the amount of light spilling outside these target areas due to diffraction at the edges of the concentrator Fresnel lenses. Explicit equations are given for the diffraction loss due to planar Fresnel lenses with small prisms and due to arbitrarily shaped Fresnel lenses. Furthermore, the cases of illumination by monochromatic, polychromatic, totally spatially coherent and partially spatially coherent light (e.g. from the solar disc) are treated, resulting in analytical formulae. Examples using realistic values show losses due to diffraction of up to several percent.

© 2014 Optical Society of America

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References

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  1. M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013).
    [CrossRef]
  2. G. Zubi, J. L. Bernal-Agustin, and G. V. Fracastoro, “High concentration photovoltaic systems applying IIIV cells,” Renew. Sustain. Energy Rev. 13(9), 2645–2652 (2009).
    [CrossRef]
  3. J. R. Egger, “Use Of Fresnel Lenses In Optical Systems: Some Advantages And Limitations,” Proc. SPIE 193, 63–69 (1979).
    [CrossRef]
  4. A. Davis and F. Kühnlenz, “Optical design using Fresnel lenses,” Optik Photonik 2(4), 52–55 (2007).
    [CrossRef]
  5. F. Duerr, Y. Meuret, and H. Thienpont, “Miniaturization of Fresnel lenses for solar concentration: a quantitative investigation,” Appl. Opt. 49(12), 2339–2346 (2010).
    [CrossRef] [PubMed]
  6. F. Duerr, Y. Meuret, and H. Thienpont, “Down scaling of micro-structured Fresnel lenses for solar concentration: a quantitative investigation,” Proc. SPIE 7725, 772509 (2010).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  8. G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Annalen der Physik 254(4), 663–695 (1883).
    [CrossRef]
  9. G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2, 641–669 (1882).
  10. W. Hoßfeld, Tageslichtsteuerung mit prismatischen Mikrostrukturen im Übergangsbereich von diffraktiver und geometrischer Optik (doctoral thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2004).
  11. T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. Lond. 92(0), 12–48 (1802).
    [CrossRef]
  12. G. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21–48 (1888).
  13. A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Annalen der Physik 378(5-6), 339–364 (1924).
    [CrossRef]
  14. A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Annalen der Physik 358(12), 257–278 (1917).
    [CrossRef]
  15. P. B. S. Kumar and G. S. Ranganath, “Geometrical theory of diffraction,” Pramana 37(6), 457–488 (1991).
    [CrossRef]
  16. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave - Part I,” J. Opt. Soc. Am. 52(6), 615–622 (1962).
    [CrossRef]
  17. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave - Part II,” J. Opt. Soc. Am. 52(6), 626–636 (1962).
    [CrossRef]
  18. S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
    [CrossRef]
  19. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).
  20. W. H. Carter, “Coherence theory,” in Handbook of Optics M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995, vol. 1).
  21. K. K. Sharma, Optics: Principles and Applications (Academic, 2006).

2013 (1)

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013).
[CrossRef]

2010 (2)

F. Duerr, Y. Meuret, and H. Thienpont, “Miniaturization of Fresnel lenses for solar concentration: a quantitative investigation,” Appl. Opt. 49(12), 2339–2346 (2010).
[CrossRef] [PubMed]

F. Duerr, Y. Meuret, and H. Thienpont, “Down scaling of micro-structured Fresnel lenses for solar concentration: a quantitative investigation,” Proc. SPIE 7725, 772509 (2010).
[CrossRef]

2009 (1)

G. Zubi, J. L. Bernal-Agustin, and G. V. Fracastoro, “High concentration photovoltaic systems applying IIIV cells,” Renew. Sustain. Energy Rev. 13(9), 2645–2652 (2009).
[CrossRef]

2007 (2)

A. Davis and F. Kühnlenz, “Optical design using Fresnel lenses,” Optik Photonik 2(4), 52–55 (2007).
[CrossRef]

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

1991 (1)

P. B. S. Kumar and G. S. Ranganath, “Geometrical theory of diffraction,” Pramana 37(6), 457–488 (1991).
[CrossRef]

1979 (1)

J. R. Egger, “Use Of Fresnel Lenses In Optical Systems: Some Advantages And Limitations,” Proc. SPIE 193, 63–69 (1979).
[CrossRef]

1962 (2)

1924 (1)

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Annalen der Physik 378(5-6), 339–364 (1924).
[CrossRef]

1917 (1)

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Annalen der Physik 358(12), 257–278 (1917).
[CrossRef]

1888 (1)

G. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21–48 (1888).

1883 (1)

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Annalen der Physik 254(4), 663–695 (1883).
[CrossRef]

1882 (1)

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2, 641–669 (1882).

1802 (1)

T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. Lond. 92(0), 12–48 (1802).
[CrossRef]

Bernal-Agustin, J. L.

G. Zubi, J. L. Bernal-Agustin, and G. V. Fracastoro, “High concentration photovoltaic systems applying IIIV cells,” Renew. Sustain. Energy Rev. 13(9), 2645–2652 (2009).
[CrossRef]

Davis, A.

A. Davis and F. Kühnlenz, “Optical design using Fresnel lenses,” Optik Photonik 2(4), 52–55 (2007).
[CrossRef]

Duerr, F.

F. Duerr, Y. Meuret, and H. Thienpont, “Down scaling of micro-structured Fresnel lenses for solar concentration: a quantitative investigation,” Proc. SPIE 7725, 772509 (2010).
[CrossRef]

F. Duerr, Y. Meuret, and H. Thienpont, “Miniaturization of Fresnel lenses for solar concentration: a quantitative investigation,” Appl. Opt. 49(12), 2339–2346 (2010).
[CrossRef] [PubMed]

Dunlop, E. D.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013).
[CrossRef]

Egger, J. R.

J. R. Egger, “Use Of Fresnel Lenses In Optical Systems: Some Advantages And Limitations,” Proc. SPIE 193, 63–69 (1979).
[CrossRef]

Emery, K.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013).
[CrossRef]

Fracastoro, G. V.

G. Zubi, J. L. Bernal-Agustin, and G. V. Fracastoro, “High concentration photovoltaic systems applying IIIV cells,” Renew. Sustain. Energy Rev. 13(9), 2645–2652 (2009).
[CrossRef]

Green, M. A.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013).
[CrossRef]

Hishikawa, Y.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013).
[CrossRef]

Ivanov, C. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Kasarova, S. N.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Kirchhoff, G.

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Annalen der Physik 254(4), 663–695 (1883).
[CrossRef]

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2, 641–669 (1882).

Kühnlenz, F.

A. Davis and F. Kühnlenz, “Optical design using Fresnel lenses,” Optik Photonik 2(4), 52–55 (2007).
[CrossRef]

Kumar, P. B. S.

P. B. S. Kumar and G. S. Ranganath, “Geometrical theory of diffraction,” Pramana 37(6), 457–488 (1991).
[CrossRef]

Maggi, G.

G. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21–48 (1888).

Meuret, Y.

F. Duerr, Y. Meuret, and H. Thienpont, “Miniaturization of Fresnel lenses for solar concentration: a quantitative investigation,” Appl. Opt. 49(12), 2339–2346 (2010).
[CrossRef] [PubMed]

F. Duerr, Y. Meuret, and H. Thienpont, “Down scaling of micro-structured Fresnel lenses for solar concentration: a quantitative investigation,” Proc. SPIE 7725, 772509 (2010).
[CrossRef]

Miyamoto, K.

Nikolov, I. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Ranganath, G. S.

P. B. S. Kumar and G. S. Ranganath, “Geometrical theory of diffraction,” Pramana 37(6), 457–488 (1991).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Annalen der Physik 378(5-6), 339–364 (1924).
[CrossRef]

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Annalen der Physik 358(12), 257–278 (1917).
[CrossRef]

Sultanova, N. G.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Thienpont, H.

F. Duerr, Y. Meuret, and H. Thienpont, “Miniaturization of Fresnel lenses for solar concentration: a quantitative investigation,” Appl. Opt. 49(12), 2339–2346 (2010).
[CrossRef] [PubMed]

F. Duerr, Y. Meuret, and H. Thienpont, “Down scaling of micro-structured Fresnel lenses for solar concentration: a quantitative investigation,” Proc. SPIE 7725, 772509 (2010).
[CrossRef]

Warta, W.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013).
[CrossRef]

Wolf, E.

Young, T.

T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. Lond. 92(0), 12–48 (1802).
[CrossRef]

Zubi, G.

G. Zubi, J. L. Bernal-Agustin, and G. V. Fracastoro, “High concentration photovoltaic systems applying IIIV cells,” Renew. Sustain. Energy Rev. 13(9), 2645–2652 (2009).
[CrossRef]

Ann. Math. (1)

G. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21–48 (1888).

Annalen der Physik (3)

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Annalen der Physik 378(5-6), 339–364 (1924).
[CrossRef]

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Annalen der Physik 358(12), 257–278 (1917).
[CrossRef]

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Annalen der Physik 254(4), 663–695 (1883).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Opt. Mater. (1)

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007).
[CrossRef]

Optik Photonik (1)

A. Davis and F. Kühnlenz, “Optical design using Fresnel lenses,” Optik Photonik 2(4), 52–55 (2007).
[CrossRef]

Philos. Trans. R. Soc. Lond. (1)

T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. Lond. 92(0), 12–48 (1802).
[CrossRef]

Pramana (1)

P. B. S. Kumar and G. S. Ranganath, “Geometrical theory of diffraction,” Pramana 37(6), 457–488 (1991).
[CrossRef]

Proc. SPIE (2)

J. R. Egger, “Use Of Fresnel Lenses In Optical Systems: Some Advantages And Limitations,” Proc. SPIE 193, 63–69 (1979).
[CrossRef]

F. Duerr, Y. Meuret, and H. Thienpont, “Down scaling of micro-structured Fresnel lenses for solar concentration: a quantitative investigation,” Proc. SPIE 7725, 772509 (2010).
[CrossRef]

Prog. Photovolt. Res. Appl. (1)

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013).
[CrossRef]

Renew. Sustain. Energy Rev. (1)

G. Zubi, J. L. Bernal-Agustin, and G. V. Fracastoro, “High concentration photovoltaic systems applying IIIV cells,” Renew. Sustain. Energy Rev. 13(9), 2645–2652 (2009).
[CrossRef]

Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1)

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2, 641–669 (1882).

Other (5)

W. Hoßfeld, Tageslichtsteuerung mit prismatischen Mikrostrukturen im Übergangsbereich von diffraktiver und geometrischer Optik (doctoral thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2004).

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

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

W. H. Carter, “Coherence theory,” in Handbook of Optics M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995, vol. 1).

K. K. Sharma, Optics: Principles and Applications (Academic, 2006).

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Figures (10)

Fig. 1
Fig. 1

Scanning electron micrograph of the resin cast from an industrially produced, silicon-on-glass Fresnel lens. It shows the detail of a cross-section through this negative of the lens form. The structure illustrated here originates from the peak of a Fresnel prism. The radius of the rounded edge is less than 1 µm.

Fig. 2
Fig. 2

Application of Kirchhoff’s integral theorem to the diffraction of an incoming spherical wave by a circular aperture. The field at point P is calculated by integrating over all points Q of the surface S . The boundary conditions are selected such that only the regions within the aperture contribute to the integral.

Fig. 3
Fig. 3

Approach to calculate diffraction by a prism of a Fresnel lens. The inner and outer edges of each Fresnel prism are replaced respectively by a circular disc and an aperture surrounded by completely absorbing material. The form of the wave fronts after refraction is considered, which represent an incoming spherical wave if suitably curved slope facets are assumed. The effect of light that is totally internally reflected by the draft facets is ignored.

Fig. 4
Fig. 4

Nomenclature used in the calculation of the diffraction of an incident spherical wave by a circular aperture according to the Maggi-Rubinowicz diffraction theory. The field at point P is calculated by integrating over all points Q on the edge of the aperture.

Fig. 5
Fig. 5

Light loss due to diffraction as a function of the f-number f/(2rL)of a Fresnel lens. A Fresnel lens with a pitch of 250 µm and a geometrical concentration factor of 400 was taken as an example for the calculation according to Eq. (27) using light with a wavelength of 589 nm. The approximation (28) for large f-numbers is plotted with the dotted line. Depending on the refractive index of the lens material, f-numbers of less than about 0.5 cannot be achieved with a real lens. Typical concentrator Fresnel lenses have f-numbers close to 1.

Fig. 6
Fig. 6

Introduction of the quantity to calculate diffraction for the case that the focal point for the lens region close to the ith edge is not identical to the origin of the coordinate system.

Fig. 7
Fig. 7

Relative power loss due to diffraction as a function of the pitch of the Fresnel lens (separation between adjacent Fresnel prisms). The losses according to Eqs. (27) (solid line), (29) (dashed line) and (41) (dotted line) are plotted. The approximations (29) and (41) are plotted only for the range where they are valid, in which the edge of the target area is located in the geometrical-optical shadow region. This example was calculated for a circular Fresnel lens with conical slope facets, 200 mm diameter, 250 mm focal length and a refractive index of 1.45, which is illuminated with light of wavelength 589 nm and concentrates the light onto a target area with a 10 mm diameter (400x geometrical concentration).

Fig. 8
Fig. 8

Relative power loss due to diffraction as a function of the wavelength. The losses are plotted for calculations according to Eqs. (27) (solid line), (29) (dashed line) and (41) (dotted line). The approximations (29) and (41) are plotted only over the range for which they are valid. The calculations were made using a circular PMMA Fresnel lens with the same geometrical configuration as for Fig. 7, assuming a pitch of 250 µm. The underlying dispersion function for PMMA was published by Kasarova et al. [18].

Fig. 9
Fig. 9

Definition of cases to solve the integral in the second line of Eq. (34). In case a), corresponding to, τ r + r s < r z , one of the multiplied theta functions is always equal to zero; in case b), where, τ r r s < r z < τ r + r s the integral over the theta functions corresponds to that area of the circle with radius rs which is located outside the circle with radius rz and in case c), where, r z < τ r r s the integral is obtained from the area of the circle with radius rs.

Fig. 10
Fig. 10

Calculation of the area for case b) in Fig. 9. The area consists of the circular segment bounded by blue dots with an area of ( π φ S ) r S 2 and the difference τ r r S sin φ S φ Z r Z between the kite-shaped area bounded by green dashed lines and the circular segment which is bounded by red dots and dashes.

Equations (43)

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Y ˜ ( P ) = 1 4 π S ( Y ˜ ( Q ) Ñ e i k s s e i k s s Ñ Y ˜ ( Q ) ) n ^ d S
S
( Y ˜ e i k s s e i k s s Y ˜ ) = Y ˜ 2 e i k s s + e i k s s Y ˜ e i k s s 2 Y ˜ e i k s s Y ˜ = 0.
Y ˜ ( P ) = S ( Y ˜ ( Q ) e i k s s e i k s s Y ˜ ( Q ) ) n ^ d A = S × W ˜ ( Q , P ) n ^ d A = δ S W ˜ ( Q , P ) l ^ d l + i δ S i W ˜ ( Q , P ) l ^ d l .
Y ˜ ( P ) = δ S W ˜ ( Q , P ) l ^ d l + i lim ϵ i 0 δ S i W ˜ ( Q , P ) l ^ d l = Y ˜ b ( P ) + Y ˜ g ( P )
W ˜ (Q,P)= B ˜ 4π e ikq q e iks s s×q sqsq .
Y ˜ g (P)={ B ˜ e ikp p in front of the focal point B ˜ e ikp p behind the focal point 0 in the geometrical-optical shadow
Y ˜ b ( P ) = δ S W ˜ ( Q , P ) l ^ d l = δ S B ˜ 4 π e i k q q e i k s s s × q s q s q l ^ d l .
G ˜ ( x ) e i k S ( x ) d x .
G ˜ (x) e ikS(x) dx i 2π 2 S(x) x 2 | x= x i G ˜ ( x i ) e 1 4 iπ e ikS( x i ) k .
Y ˜ b (P) B ˜ 2 2πk j=0 1 [ (1) j ( 2 s Φ 2 ) 1 2 sin(s,q) 1cos(s,q) e 1 4 iπ e ik(s-q) s sin( θ q ) ] Φ=jπ
s(Φ)= q 2 + p 2 2qp(cosΦsin θ q sin θ p +cos θ q cos θ p ) ,
s(0)= q 2 + p 2 2qpcos( θ p θ q ) and s(π)= q 2 + p 2 2qpcos( θ p + θ q )
2 s(Φ) Φ 2 = qpcosΦsin θ q sin θ p s(Φ) ( qpsinΦsin θ q sin θ p ) 2 s 3 (Φ) ,
2 s(Φ) Φ 2 | Φ=0 = qpsin θ q sin θ p s(0) and 2 s(Φ) Φ 2 | Φ=π = qpsin θ q sin θ p s(π) .
sin(s(0),q)= p s(0) sin( θ p θ q ) , sin(s(π),q)= p s(π) sin( θ p + θ q )
cos(s(0),q)= qpcos( θ p θ q ) s(0) and cos(s(π),q)= qpcos( θ p + θ q ) s(π) .
Y ˜ ( P ) = Y ˜ b ( P ) + Y ˜ g ( P ) B ˜ e i k q 2 2 π k p sin θ q q sin θ p [ sin ( θ p θ q ) s ( 0 ) q + p cos ( θ p θ q ) e 1 4 i π e i k s ( 0 ) s ( 0 ) sin ( θ p + θ q ) s ( π ) q + p cos ( θ p + θ q ) e 1 4 i π e i k s ( π ) s ( π ) ] + Y ˜ g ( P ) .
J(P) prisms c ϵ 0 | Y ˜ b Φ=0 (P) | 2 + prisms c ϵ 0 | Y ˜ b Φ=π (P) | 2 + prisms c ϵ 0 | Y ˜ g (P) | 2 = c ϵ 0 p 8πksin θ p [ i=1 2N1 B ˜ i 2 sin θ q i sin 2 ( θ p θ q i ) q i s i (0) ( s i (0) q i +pcos( θ p θ q i ) ) 2 + i=1 2N1 B ˜ i 2 sin θ q i sin 2 ( θ p + θ q i ) q i s i (π) ( s i (π) q i +pcos( θ p + θ q i ) ) 2 ]+ j=1 N | Y ˜ g,j (P) | 2 .
B ˜ i B ˜ 0 = q i q 0 1 cos θ q i .
J b ( P ) c ϵ 0 2 π k p 3 sin θ p i = 1 2 N 1 B ˜ i 2 sin θ q i ( 1 sin 2 ( θ p θ q i ) + 1 sin 2 ( θ p + θ q i ) ) = c ϵ 0 π k p r i = 1 2 N 1 B ˜ i 2 r l i f i q i f i p r 2 + r l i 2 f i 2 p z 2 ( p r 2 r l i 2 f i 2 p z 2 ) 2 .
J b ( P ) c ϵ 0 B ˜ 0 2 π k p r i = 1 2 N 1 r l i f q i 3 f 3 p r 2 + r l i 2 f 2 p z 2 ( p r 2 r l i 2 f 2 p z 2 ) 2 c ϵ 0 B ˜ 0 2 π k p r [ 2 j = 1 N 1 r l, 2 j f q 2 j 3 f 3 p r 2 + r l, 2 j 2 f 2 p z 2 ( p r 2 r l, 2 j 2 f 2 p z 2 ) 2 + r l 2 N 1 q 2 N 1 3 p r 2 + r l, 2 N 1 2 f 2 p z 2 ( p r 2 r l, 2 N 1 2 f 2 p z 2 ) 2 ] = c ϵ 0 B ˜ 0 2 π k p r [ 2 j = 1 N 1 j Δ r l f ( 1 + j 2 Δ r l 2 f 2 ) 3 2 p r 2 + j 2 Δ r l 2 f 2 p z 2 ( p r 2 j 2 Δ r l 2 f 2 p z 2 ) 2 + r L f ( 1 + r L 2 f 2 ) 3 2 p r 2 + r L 2 f 2 p z 2 ( p r 2 r L 2 f 2 p z 2 ) 2 ] .
J b (P) c ϵ 0 B ˜ 0 2 πk p r 3 [ 2 j=1 N1 j Δ r l f ( 1+ j 2 Δ r l 2 f 2 ) 3 2 + r L f ( 1+ r L 2 f 2 ) 3 2 ].
J b ( P ) c ϵ 0 B ˜ 0 2 π k p r 3 [ 2 1 2 N 1 2 j Δ r l f ( 1 + j 2 Δ r l 2 f 2 ) 3 2 d j + r L f ( 1 + r L 2 f 2 ) 3 2 ] c ϵ 0 B ˜ 0 2 π k p r 3 { [ 2 f 5 Δ r l ( 1 + j 2 Δ r l 2 f 2 ) 5 2 ] 1 2 N 1 2 + r L f ( 1 + r L 2 f 2 ) 3 2 } 2 c ϵ 0 B ˜ 0 2 f 5 π k p r 3 r L [ ( 1 + r L 2 f 2 ) 5 2 1 ] N + O ( N 1 ) .
P ( p r r Z ) r Z 0 2 π c ϵ 0 π k p r i = 1 2 N 1 B ˜ i 2 r l i q i f i 2 p r 2 + r l i 2 f i 2 p z 2 ( p r 2 r l i 2 f i 2 p z 2 ) 2 p r d p ϕ d p r = 2 c ϵ 0 k i = 1 2 N 1 B ˜ i 2 r l i q i f i 2 r Z r Z 2 r l i 2 f i 2 p z 2
P( p r r Z ) 4c ϵ 0 B ˜ 0 2 fN 5k r Z r L { ( 1+ r L 2 f 2 ) 5 2 1 }= 4c ϵ 0 B ˜ 0 2 C geo 5kΔ r l ( 1+ tan 2 ϕ ) 5 2 1 tanϕ .
P ( p r r Z ) P in 2 π k r L 2 i = 1 2 N 1 r l i q i 3 f i 2 r Z r Z 2 r l i 2 f i 2 p z 2 = 2 r Z π k r L 2 i = 1 2 N 1 r l i f i r Z 2 r l i 2 f i 2 p z 2 ( 1 + r l i 2 f i 2 ) 3 2 .
P( p r r Z ) P in 4 C geo 5πkΔ r l ( 1+ tan 2 ϕ ) 5 2 1 tan 3 ϕ .
P( p r r Z ) P in tanϕ1 2 C geo πkΔ r l tanϕ .
P ( p r r Z ) P in 2 r Z π k r L 2 i = 1 2 N 1 r l i f i r Z 2 r l i 2 f i 2 ( f i z l i + p z ) 2 ( 1 + r l i 2 f i 2 ) 3 2 .
J b ( P ) c ϵ 0 π k i = 1 2 N 1 [ B ˜ i 2 r l i f i 1 + r l i 2 f i 2 τ x 2 + τ y 2 + r l i 2 f i 2 p z 2 τ x 2 + τ y 2 ( τ x 2 + τ y 2 r l i 2 f i 2 p z 2 ) 2 + ϵ 1 π r S i 2 Θ ( r S i ( p x τ x ) 2 + ( p y τ y ) 2 ) d τ x d τ y ] ,
Θ ( x ) = { 0 if x < 0 1 if x 0 .
r S i cos θ q i 1 q i tan [ arc sin ( n sin ( α i + arc sin ( 1 n sin ϑ in ) ) ) θ q i α i ] sin ϑ in 1 ( f i p z ) 2 + r l i 2 cos α i 1 n 2 sin 2 α i ϑ in .
P ( p r r Z ) c ϵ 0 π k i = 1 2 N 1 [ B ˜ i 2 r l i f i 1 + r l i 2 f i 2 τ x 2 + τ y 2 + r l i 2 f i 2 p z 2 τ x 2 + τ y 2 ( τ x 2 + τ y 2 r l i 2 f i 2 p z 2 ) 2 + ϵ 1 π r S i 2 Θ ( r S i ( p x τ x ) 2 + ( p y τ y ) 2 ) d τ x d τ y ] Θ ( p x 2 + p y 2 r Z ) d p x d p y .
P ( p r r Z ) c ϵ 0 π k i = 1 2 N 1 [ B ˜ i 2 r l i f i 1 + r l i 2 f i 2 τ x 2 + τ y 2 + r l i 2 f i 2 p z 2 τ x 2 + τ y 2 ( τ x 2 + τ y 2 r l i 2 f i 2 p z 2 ) 2 + ϵ 1 π r S i 2 Θ ( r S i ( p x τ x ) 2 + ( p y τ y ) 2 ) Θ ( p x 2 + p y 2 r Z ) d p x d p y d τ x d τ y ] ,
1 π r S i 2 Θ ( r S i ( p x τ x ) 2 + ( p y τ y ) 2 ) Θ ( p x 2 + p y 2 r Z ) d p x d p y = { 0 if τ r + r S i < r Z ( a ) d p x d p y if τ r r S i < r Z < τ r + r S i ( b ) 1 if r Z < τ r r S i ( c ) .
d p x d p y = 1 π r S i 2 ( ( π φ S i ) r S i 2 φ Z i r Z 2 + τ r r S i sin φ S i ) = 1 φ S i π φ Z i π r Z 2 r S i 2 + τ r sin φ S i π r S i .
φ Z i = arc cos r Z 2 + τ r 2 r S i 2 2 r Z τ r and φ S i = arc cos r S i 2 + τ r 2 r Z 2 2 r S i τ r
P ( p r r Z ) 2 c ϵ 0 k i = 1 2 N 1 B ˜ i 2 r l i f i 1 + r l i 2 f i 2 [ r Z r S i r Z + r S i τ r 2 + r l i 2 f i 2 p z 2 ( τ r 2 r l i 2 f i 2 p z 2 ) 2 + ϵ ( φ S i π φ Z i π r Z 2 r S i 2 + τ r sin φ S i π r S i ) d τ r + r Z r S i τ r 2 + r l i 2 f i 2 p z 2 ( τ r 2 r l i 2 f i 2 p z 2 ) 2 + ϵ d τ r ]
P ( p r r Z ) 2 c ϵ 0 k i = 1 2 N 1 B ˜ i 2 r l i f i 1 + r l i 2 f i 2 r Z r S i ( r Z r S i ) 2 r l i 2 f i 2 p z 2 + 2 c ϵ 0 k i = 1 2 N 1 B ˜ i 2 r l i f i 1 + r l i 2 f i 2 r Z r S i r Z + r S i τ r 2 + r l i 2 f i 2 p z 2 ( τ r 2 r l i 2 f i 2 p z 2 ) 2 ( φ S i π φ Z i π r Z 2 r S i 2 + τ r sin φ S i π r S i ) d τ r .
P ( p r r Z ) P in 2 π k r L 2 i = 1 2 N 1 r l i f i ( 1 + r l i 2 f i 2 ) 3 2 r Z r S i ( r Z r S i ) 2 r l i 2 f i 2 p z 2 + 2 π k r L 2 i = 1 2 N 1 r l i f i ( 1 + r l i 2 f i 2 ) 3 2 r Z r S i r Z + r S i τ r 2 + r l i 2 f i 2 p z 2 ( τ r 2 r l i 2 f i 2 p z 2 ) 2 ( φ S i π φ Z i π r Z 2 r S i 2 + τ r sin φ S i π r S i ) d τ r .
P ( p r r Z ) P in 2 π k r L 2 i = 1 2 N 1 r l i f i ( 1 + r l i 2 f i 2 ) 3 2 [ r Z r S i ( r Z r S i ) 2 r l i 2 f i 2 ( f i z l i + p z ) 2 + r Z r S i r Z + r S i τ r 2 + r l i 2 f i 2 ( f i z l i + p z ) 2 ( τ r 2 r l i 2 f i 2 ( f i z l i + p z ) 2 ) 2 ( φ S i π φ Z i π r Z 2 r S i 2 + τ r sin φ S i π r S i ) d τ r ] .
r S i ( z l i p z ) 2 + r l i 2 cos α 1 n 2 sin 2 α ϑ in

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