Abstract

Wave dynamics on curved surfaces has attracted growing attention due to its close resemblance to the warped space time governed by general relativity. It also opens up opportunities for designing functional optical devices such as geodesic lenses. In this work we study the wave dynamics on the surface of a torus, a shape of considerable interest due to its nontrivial topology. Governed by the conservation of angular momentum, light propagates on the torus in two different types of modes: one is able to twist around and sweep through the whole surface of the torus; the other is confined within a certain angular range along the torus latitude direction. The confined mode exhibits an interesting self focusing or imaging behavior, which, similar to a geometric lens, shows no dependence of wavelength and thus suffers no chromatic aberration. By changing the geometric parameters of the torus, both the focusing point and the focusing distance can be controlled. Our work provides a new approach to manipulation of light propagation on a curved surface under the conservation of angular momentum.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Generalization of Wolf effect of light on arbitrary two-dimensional surface of revolution

Chenni Xu, Adeel Abbas, and Li-Gang Wang
Opt. Express 26(25) 33263-33277 (2018)

Light-driven mass density wave dynamics in optical fibers

Mikko Partanen and Jukka Tulkki
Opt. Express 26(17) 22046-22063 (2018)

Control on helical filaments by twisted beams in a nonlinear CS2 medium

Jia-Qi Lü, Ping-Ping Li, Dan Wang, Chenghou Tu, Yongnan Li, and Hui-Tian Wang
Opt. Express 26(22) 29527-29538 (2018)

References

  • View by:
  • |
  • |
  • |

  1. A. J. Kox, M. J. Klein, and R. Schulmann, The Collected Papers of Albert Einstein, Vol. 6 (Princeton University, 1997).
  2. S. W. Hawking, “Black hole explosions?” Nature 248(5443), 30–31 (1974).
    [Crossref]
  3. W. G. Unruh, “Notes on black-hole evaporation,” Phys. Rev. D Part. Fields 14(4), 870–892 (1976).
    [Crossref]
  4. R. Brout, S. Massar, R. Parentani, and P. Spindel, “A primer for black hold quantum mechanics,” Phys. Rep. 260(6), 329–446 (1995).
    [Crossref]
  5. U. Lenohardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60(6), 4301–4312 (1999).
    [Crossref]
  6. W. G. Unruh, “Experimental Black-Hole Evaporation?” Phys. Rev. Lett. 46(21), 1351–1353 (1981).
    [Crossref]
  7. C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68(5), 053613 (2003).
    [Crossref]
  8. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319(5868), 1367–1370 (2008).
    [Crossref] [PubMed]
  9. R. C. T. da Costa, “Quantum mechanics of a constrained particle,” Phys. Rev. A 23(4), 1982–1987 (1981).
    [Crossref]
  10. U. Leonhardt and P. Piwnicki, “Relativistic Effects of Light in Moving Media with Extremely Low Group Velocity,” Phys. Rev. Lett. 84(5), 822–825 (2000).
    [Crossref] [PubMed]
  11. E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidiectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
    [Crossref]
  12. I. I. Smolyaninov, “Surface Plasmon toy model of a rotating black hole,” New J. Phys. 5, 147 (2003).
    [Crossref]
  13. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
    [Crossref]
  14. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
    [Crossref]
  15. V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
    [Crossref] [PubMed]
  16. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, 1973).
  17. R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-Preserving Accelerating Electromagnetic Wave Packets in Curved Space,” Phys. Rev. X 4(1), 011038 (2014).
    [Crossref]
  18. B. O’neill, Elementary Differential Geometry, 2nd ed. (Academic Press, 2006).

2014 (1)

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-Preserving Accelerating Electromagnetic Wave Packets in Curved Space,” Phys. Rev. X 4(1), 011038 (2014).
[Crossref]

2010 (1)

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

2009 (2)

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidiectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]

2008 (1)

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319(5868), 1367–1370 (2008).
[Crossref] [PubMed]

2006 (1)

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
[Crossref]

2003 (2)

I. I. Smolyaninov, “Surface Plasmon toy model of a rotating black hole,” New J. Phys. 5, 147 (2003).
[Crossref]

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68(5), 053613 (2003).
[Crossref]

2000 (1)

U. Leonhardt and P. Piwnicki, “Relativistic Effects of Light in Moving Media with Extremely Low Group Velocity,” Phys. Rev. Lett. 84(5), 822–825 (2000).
[Crossref] [PubMed]

1999 (1)

U. Lenohardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60(6), 4301–4312 (1999).
[Crossref]

1995 (1)

R. Brout, S. Massar, R. Parentani, and P. Spindel, “A primer for black hold quantum mechanics,” Phys. Rep. 260(6), 329–446 (1995).
[Crossref]

1981 (2)

R. C. T. da Costa, “Quantum mechanics of a constrained particle,” Phys. Rev. A 23(4), 1982–1987 (1981).
[Crossref]

W. G. Unruh, “Experimental Black-Hole Evaporation?” Phys. Rev. Lett. 46(21), 1351–1353 (1981).
[Crossref]

1976 (1)

W. G. Unruh, “Notes on black-hole evaporation,” Phys. Rev. D Part. Fields 14(4), 870–892 (1976).
[Crossref]

1974 (1)

S. W. Hawking, “Black hole explosions?” Nature 248(5443), 30–31 (1974).
[Crossref]

Barcelo, C.

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68(5), 053613 (2003).
[Crossref]

Batz, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

Bekenstein, R.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-Preserving Accelerating Electromagnetic Wave Packets in Curved Space,” Phys. Rev. X 4(1), 011038 (2014).
[Crossref]

Brout, R.

R. Brout, S. Massar, R. Parentani, and P. Spindel, “A primer for black hold quantum mechanics,” Phys. Rep. 260(6), 329–446 (1995).
[Crossref]

da Costa, R. C. T.

R. C. T. da Costa, “Quantum mechanics of a constrained particle,” Phys. Rev. A 23(4), 1982–1987 (1981).
[Crossref]

Dreisow, F.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

Genov, D. A.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]

Hawking, S. W.

S. W. Hawking, “Black hole explosions?” Nature 248(5443), 30–31 (1974).
[Crossref]

Hill, S.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319(5868), 1367–1370 (2008).
[Crossref] [PubMed]

Kaminer, I.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-Preserving Accelerating Electromagnetic Wave Packets in Curved Space,” Phys. Rev. X 4(1), 011038 (2014).
[Crossref]

Kildishev, A. V.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidiectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

König, F.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319(5868), 1367–1370 (2008).
[Crossref] [PubMed]

Kuklewicz, C.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319(5868), 1367–1370 (2008).
[Crossref] [PubMed]

Lenohardt, U.

U. Lenohardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60(6), 4301–4312 (1999).
[Crossref]

Leonhardt, U.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319(5868), 1367–1370 (2008).
[Crossref] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
[Crossref]

U. Leonhardt and P. Piwnicki, “Relativistic Effects of Light in Moving Media with Extremely Low Group Velocity,” Phys. Rev. Lett. 84(5), 822–825 (2000).
[Crossref] [PubMed]

Liberati, S.

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68(5), 053613 (2003).
[Crossref]

Longhi, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

Massar, S.

R. Brout, S. Massar, R. Parentani, and P. Spindel, “A primer for black hold quantum mechanics,” Phys. Rep. 260(6), 329–446 (1995).
[Crossref]

Narimanov, E. E.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidiectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

Nemirovsky, J.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-Preserving Accelerating Electromagnetic Wave Packets in Curved Space,” Phys. Rev. X 4(1), 011038 (2014).
[Crossref]

Nolte, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

Parentani, R.

R. Brout, S. Massar, R. Parentani, and P. Spindel, “A primer for black hold quantum mechanics,” Phys. Rep. 260(6), 329–446 (1995).
[Crossref]

Peschel, U.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

Philbin, T. G.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319(5868), 1367–1370 (2008).
[Crossref] [PubMed]

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
[Crossref]

Piwnicki, P.

U. Leonhardt and P. Piwnicki, “Relativistic Effects of Light in Moving Media with Extremely Low Group Velocity,” Phys. Rev. Lett. 84(5), 822–825 (2000).
[Crossref] [PubMed]

U. Lenohardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60(6), 4301–4312 (1999).
[Crossref]

Robertson, S.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319(5868), 1367–1370 (2008).
[Crossref] [PubMed]

Schultheiss, V. H.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

Segev, M.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-Preserving Accelerating Electromagnetic Wave Packets in Curved Space,” Phys. Rev. X 4(1), 011038 (2014).
[Crossref]

Smolyaninov, I. I.

I. I. Smolyaninov, “Surface Plasmon toy model of a rotating black hole,” New J. Phys. 5, 147 (2003).
[Crossref]

Spindel, P.

R. Brout, S. Massar, R. Parentani, and P. Spindel, “A primer for black hold quantum mechanics,” Phys. Rep. 260(6), 329–446 (1995).
[Crossref]

Szameit, A.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

Tünnermann, A.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

Unruh, W. G.

W. G. Unruh, “Experimental Black-Hole Evaporation?” Phys. Rev. Lett. 46(21), 1351–1353 (1981).
[Crossref]

W. G. Unruh, “Notes on black-hole evaporation,” Phys. Rev. D Part. Fields 14(4), 870–892 (1976).
[Crossref]

Visser, M.

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68(5), 053613 (2003).
[Crossref]

Zhang, S.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]

Zhang, X.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]

Appl. Phys. Lett. (1)

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidiectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

Nat. Phys. (1)

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009).
[Crossref]

Nature (1)

S. W. Hawking, “Black hole explosions?” Nature 248(5443), 30–31 (1974).
[Crossref]

New J. Phys. (2)

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
[Crossref]

I. I. Smolyaninov, “Surface Plasmon toy model of a rotating black hole,” New J. Phys. 5, 147 (2003).
[Crossref]

Phys. Rep. (1)

R. Brout, S. Massar, R. Parentani, and P. Spindel, “A primer for black hold quantum mechanics,” Phys. Rep. 260(6), 329–446 (1995).
[Crossref]

Phys. Rev. A (3)

U. Lenohardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60(6), 4301–4312 (1999).
[Crossref]

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68(5), 053613 (2003).
[Crossref]

R. C. T. da Costa, “Quantum mechanics of a constrained particle,” Phys. Rev. A 23(4), 1982–1987 (1981).
[Crossref]

Phys. Rev. D Part. Fields (1)

W. G. Unruh, “Notes on black-hole evaporation,” Phys. Rev. D Part. Fields 14(4), 870–892 (1976).
[Crossref]

Phys. Rev. Lett. (3)

W. G. Unruh, “Experimental Black-Hole Evaporation?” Phys. Rev. Lett. 46(21), 1351–1353 (1981).
[Crossref]

U. Leonhardt and P. Piwnicki, “Relativistic Effects of Light in Moving Media with Extremely Low Group Velocity,” Phys. Rev. Lett. 84(5), 822–825 (2000).
[Crossref] [PubMed]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in Curved Space,” Phys. Rev. Lett. 105(14), 143901 (2010).
[Crossref] [PubMed]

Phys. Rev. X (1)

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-Preserving Accelerating Electromagnetic Wave Packets in Curved Space,” Phys. Rev. X 4(1), 011038 (2014).
[Crossref]

Science (1)

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319(5868), 1367–1370 (2008).
[Crossref] [PubMed]

Other (3)

B. O’neill, Elementary Differential Geometry, 2nd ed. (Academic Press, 2006).

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, 1973).

A. J. Kox, M. J. Klein, and R. Schulmann, The Collected Papers of Albert Einstein, Vol. 6 (Princeton University, 1997).

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 (5)

Fig. 1
Fig. 1 (a) Schematic of torus coordinates with usual parametrization. The energy density of (b) reflected wave on a 2D torus. (c) free wave on 2D torus. (d) reflected wave on 3D space torus. (e) free wave on 3D space torus. The geometric parameters used in calculating the effective permittivity and permeability are R1 = 3 m and R2 = 3/2 m, the corresponding frequency is 10GHz.
Fig. 2
Fig. 2 (a) Schematic of light paraxial propagation around the outer radius. (b) The angular focusing length φf dependence on initial propagation angle αs with different geometric parameters of torus. (c) Geometric parameters versus focusing length φf, where general points on curve will not guarantee integer number of focusing points within a single circle, several special points are marked and should correspond to closed geodesics. (d-f) Field distributions on torus with geometric parameters R2 / R1 = 1/9, 1/2 and 7/9, respectively. The wavelength is set as λ = π R1/10 in the calculation.
Fig. 3
Fig. 3 Different number of focusing nodes for geometry parameters of R2/R1 = 1/9, 1/2 and 7/9 for (a-c) respectively, the field patterns are calculated with Comsol Multiphysics after using transformation optics method stated in the Appendix, the geometric parameters used in calculating the effective permittivity and permeability are R2 = 1/3, 3/2, 7/3 m and R1 = 3 m, the corresponding frequency is 10GHz.
Fig. 4
Fig. 4 (a) Schematic of light trajectory with an abaxial point source. (b) The φf-α relation for the point source located at different position. (c-f) Field distributions with different position of point source. The wavelength is set as λ = π R1/10 in the calculation.
Fig. 5
Fig. 5 Wave propagation in the 2D torus for different frequencies: 3 - 6 GHz for (a) - (d) respectively. The geometric parameters used in calculating the effective permittivity and permeability are R1 = 3 m and R2 = 3/2 m.

Equations (20)

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

L = R s sin α s K 0 z ^ ,
φ f = π 1 2 R 2 2 R 1 ,
x = ( R 0 + r 0 cos ( θ ) ) cos ( ϕ ) y = ( R 0 + r 0 cos ( θ ) ) sin ( ϕ ) z = r 0 sin ( θ )
g M = [ ϕ ϕ ϕ θ ϕ R 0 θ ϕ θ θ θ R 0 R 0 ϕ R 0 θ R 0 R 0 ] [ x y z ] = [ g ϕ ϕ g ϕ θ g ϕ R 0 g θ ϕ g θ θ g θ R 0 g R 0 ϕ g R 0 θ g R 0 R 0 ]
g ϕ ϕ = ( R 0 + a R 0 cos ( θ ) ) 2 ( 4 e 2 sin 2 ( 2 ϕ ) + 1 ) g ϕ θ = g θ ϕ = 0 g ϕ R 0 = g R 0 ϕ = 0 g θ θ = a 2 R 0 2 g θ R 0 = g R 0 θ = a R 0 sin ( θ ) ( 1 + a cos ( θ ) ) + a 2 R 0 cos ( θ ) sin ( θ ) g R 0 R 0 = 1 + a 2 + 2 a cos ( θ )
( g g i j E j ) , i = 0 ( g g i j B j ) , i = 0 [ i j k ] E k , j = ( ± g g i j B j ) t [ i j k ] B k , j = 1 c 2 ( ± g g i j E j ) t
ε = μ = ± g g M T
d s 2 = R 0 2 cos 2 θ d φ 2 + r 0 2 d θ 2
d φ = sin α R d s
d θ = cos α r 0 d s
d φ = r 0 R tan α d θ
L s = R s sin α s k 0
L = R sin α k 0
R = R s sin α s sin α
R = R 0 + r 0 cos θ
d R = r 0 sin θ d θ = cos α sin 2 α R s sin α s d α
d θ = R s sin α s cos α r 0 sin 2 α 1 sin θ d α
sin θ = 1 ( R s sin α s r 0 sin α R 0 r 0 ) 2
d φ = 1 1 ( R s sin α s r 0 sin α R 0 r 0 ) 2 d α
φ f = 2 lim α s π 2 α s π 2 1 1 [ ( R 0 + r 0 ) sin α s r 0 sin α R 0 r 0 ] 2 d α = π R 0 + r 0 r 0

Metrics