Abstract

The intrinsic and extrinsic coupling losses of step-index polymer optical fibers are statistically examined by Monte Carlo simulations. In contrast to most existing models that linearly scale individual losses, a comprehensive analytic coupling loss model is used that also considers the interdependencies between mismatches in numerical aperture and core diameter, as well as radial and longitudinal offsets. As a typical example, the coupling losses of A4a.2 step-index multimode fibers are analyzed for an equilibrium mode distribution. The results show considerably less conservative coupling loss estimations than with traditional models, improving link power budgeting.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Analysis of intrinsic coupling loss in multi-step index optical fibres

Gotzon Aldabaldetreku, Gaizka Durana, Joseba Zubia, Jon Arrue, Felipe Jiménez, and Javier Mateo
Opt. Express 13(9) 3283-3295 (2005)

Investigation and comparison of analytical, numerical, and experimentally measured coupling losses for multi-step index optical fibers

Gotzon Aldabaldetreku, Gaizka Durana, Joseba Zubia, Jon Arrue, Hans Poisel, and María Angeles Losada
Opt. Express 13(11) 4012-4036 (2005)

Coupling losses in perfluorinated multi-core polymer optical fibers

Gaizka Durana, Gotzon Aldabaldetreku, Joseba Zubia, Jon Arrue, and Chikafumi Tanaka
Opt. Express 16(11) 7929-7942 (2008)

References

  • View by:
  • |
  • |
  • |

  1. O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook – Optical Short Range Transmission Systems (Springer Verlag, 2008), 2nd ed.
  2. A. Grzemba, MOST – The Automotive Multimedia Network (Franzis Verlag, 2011), 2nd ed.
  3. D. H. Richards, M. A. Losada, N. Antoniades, A. López, J. Mateo, X. Jiang, and N. Madamopoulos, “Modeling methodology for engineering SI-POF and connectors in an avionics system,” J. Lightw. Technol. 31(3), 468–475 (2013).
    [Crossref]
  4. S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “An analytic connector loss model for step-index polymer optical fiber links,” J. Lightw. Technol. 31(16), 2769–2776 (2013).
    [Crossref]
  5. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
    [Crossref]
  6. S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quant. Electron. 11(5), 447–457 (1979).
    [Crossref]
  7. D. Gloge, “Offset and tilt loss in optical fiber splices,” Bell Syst. Tech. J. 55(7), 905–916 (1976).
    [Crossref]
  8. D. Opielka and D. Rittich, “Transmission loss caused by an angular misalignment between two multimode fibers with arbitrary profile exponents,” Appl. Opt. 22(7), 991–994 (1983).
    [Crossref] [PubMed]
  9. W. van Etten, W. Lambo, and P. Simons, “Loss in multimode fiber connections with a gap,” Appl. Opt. 24(7), 970–976 (1985).
    [Crossref] [PubMed]
  10. C. Miller, S. Mettler, and I. White, Optical Fiber Splices and Connectors: Theory and Methods (Marcel Dekker, 1986).
  11. C. Gao and G. Farrell, “Power coupling between two step-index multimode fibers of different numerical apertures with an angular misalignment,” Microw. Opt. Tech. Lett. 43(3), 231–234 (2004).
    [Crossref]
  12. S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “Application and evaluation of an analytic connector loss model for SI-POF,” in Proceedings of 22nd ICPOF (Búzios, Rio de Janeiro, Brazil, 2013), pp. 193–198.
  13. G. Aldabaldetreku, G. Durana, J. Zubia, J. Arrue, F. Jiménez, and J. Mateo, “Analysis of intrinsic coupling loss in multi-step index optical fibers,” Opt. Express 13(9), 3283–3295 (2005).
    [Crossref] [PubMed]
  14. G. Aldabaldetreku, G. Durana, J. Zubia, and J. Arrue, “Analytical expression for measurement of intrinsic coupling loss in multistep index optical fibers,” J. Lightw. Technol. 24(3), 1364–1375 (2006).
    [Crossref]
  15. “Optical fibres – part 2–40: Product specifications – Sectional specification for category A4 multimode fibres,” IEC 60793-2-40 (2009).
  16. G. Aldabaldetreku, G. Durana, J. Zubia, J. Arrue, H. Poisel, and M. A. Losada, “Investigation and comparison of analytical, numerical, and experimentally measured coupling losses for multi-step index optical fibers,” Opt. Express 13(11), 4012–4036 (2005).
    [Crossref] [PubMed]
  17. J. Mateo, M. A. Losada, I. Garcés, and J. Zubia, “Global characterization of optical power propagation in step-index plastic optical fibers,” Opt. Express 14(20), 9028–9035 (2006).
    [Crossref] [PubMed]
  18. G. Jiang, R. F. Shi, and A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” Photon. Technol. Lett. 9(8), 1128–1130 (1997).
    [Crossref]
  19. “Fibre optic communication subsystem test procedures – Part 1–4: General communication subsystems – Light source encircled flux measurement method,” IEC 61280-1-4 (2009).
  20. F. S. Tan, O. Sugihara, and T. Kaino, “Encircled flux-based optimized simple launch condition for standardization of multimode polymer optical waveguide evaluations,” Opt. Express 18(23), 23554–23561 (2010).
    [Crossref] [PubMed]
  21. M. Rousseau and L. Jeunhomme, “Numerical solution of the coupled-power equation in step-index optical fibers,” IEEE Trans. Microwave Theory Tech. 25(7), 577–585 (1977).
    [Crossref]
  22. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  23. S. Savović and A. Djordjevich, “Mode coupling in strained and unstrained step-index plastic optical fibers,” Appl. Opt. 45(26), 6775–6780 (2006).
    [Crossref]
  24. J. Mateo, M. A. Losada, and A. López, “POF misalignment model based on the calculation of the radiation pattern using the Hankel transform,” Opt. Express 23(6), 8061–8072 (2015).
    [Crossref] [PubMed]
  25. Y. Ando, “Statistical analysis of insertion-loss improvement for optical connectors using the orientation method for fiber-core offset,” Photon. Technol. Lett. 3(10), 939–941 (1991).
    [Crossref]

2015 (1)

2013 (2)

D. H. Richards, M. A. Losada, N. Antoniades, A. López, J. Mateo, X. Jiang, and N. Madamopoulos, “Modeling methodology for engineering SI-POF and connectors in an avionics system,” J. Lightw. Technol. 31(3), 468–475 (2013).
[Crossref]

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “An analytic connector loss model for step-index polymer optical fiber links,” J. Lightw. Technol. 31(16), 2769–2776 (2013).
[Crossref]

2010 (1)

2006 (3)

2005 (2)

2004 (1)

C. Gao and G. Farrell, “Power coupling between two step-index multimode fibers of different numerical apertures with an angular misalignment,” Microw. Opt. Tech. Lett. 43(3), 231–234 (2004).
[Crossref]

1997 (1)

G. Jiang, R. F. Shi, and A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” Photon. Technol. Lett. 9(8), 1128–1130 (1997).
[Crossref]

1991 (1)

Y. Ando, “Statistical analysis of insertion-loss improvement for optical connectors using the orientation method for fiber-core offset,” Photon. Technol. Lett. 3(10), 939–941 (1991).
[Crossref]

1985 (1)

1983 (1)

1979 (1)

S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quant. Electron. 11(5), 447–457 (1979).
[Crossref]

1977 (2)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

M. Rousseau and L. Jeunhomme, “Numerical solution of the coupled-power equation in step-index optical fibers,” IEEE Trans. Microwave Theory Tech. 25(7), 577–585 (1977).
[Crossref]

1976 (1)

D. Gloge, “Offset and tilt loss in optical fiber splices,” Bell Syst. Tech. J. 55(7), 905–916 (1976).
[Crossref]

Aldabaldetreku, G.

Ando, Y.

Y. Ando, “Statistical analysis of insertion-loss improvement for optical connectors using the orientation method for fiber-core offset,” Photon. Technol. Lett. 3(10), 939–941 (1991).
[Crossref]

Antoniades, N.

D. H. Richards, M. A. Losada, N. Antoniades, A. López, J. Mateo, X. Jiang, and N. Madamopoulos, “Modeling methodology for engineering SI-POF and connectors in an avionics system,” J. Lightw. Technol. 31(3), 468–475 (2013).
[Crossref]

Arrue, J.

Bunge, C.-A.

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “An analytic connector loss model for step-index polymer optical fiber links,” J. Lightw. Technol. 31(16), 2769–2776 (2013).
[Crossref]

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “Application and evaluation of an analytic connector loss model for SI-POF,” in Proceedings of 22nd ICPOF (Búzios, Rio de Janeiro, Brazil, 2013), pp. 193–198.

Daum, W.

O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook – Optical Short Range Transmission Systems (Springer Verlag, 2008), 2nd ed.

Djordjevich, A.

Durana, G.

Farrell, G.

C. Gao and G. Farrell, “Power coupling between two step-index multimode fibers of different numerical apertures with an angular misalignment,” Microw. Opt. Tech. Lett. 43(3), 231–234 (2004).
[Crossref]

Gao, C.

C. Gao and G. Farrell, “Power coupling between two step-index multimode fibers of different numerical apertures with an angular misalignment,” Microw. Opt. Tech. Lett. 43(3), 231–234 (2004).
[Crossref]

Garcés, I.

Garito, A. F.

G. Jiang, R. F. Shi, and A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” Photon. Technol. Lett. 9(8), 1128–1130 (1997).
[Crossref]

Gloge, D.

D. Gloge, “Offset and tilt loss in optical fiber splices,” Bell Syst. Tech. J. 55(7), 905–916 (1976).
[Crossref]

Grzemba, A.

A. Grzemba, MOST – The Automotive Multimedia Network (Franzis Verlag, 2011), 2nd ed.

Jeunhomme, L.

M. Rousseau and L. Jeunhomme, “Numerical solution of the coupled-power equation in step-index optical fibers,” IEEE Trans. Microwave Theory Tech. 25(7), 577–585 (1977).
[Crossref]

Jiang, G.

G. Jiang, R. F. Shi, and A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” Photon. Technol. Lett. 9(8), 1128–1130 (1997).
[Crossref]

Jiang, X.

D. H. Richards, M. A. Losada, N. Antoniades, A. López, J. Mateo, X. Jiang, and N. Madamopoulos, “Modeling methodology for engineering SI-POF and connectors in an avionics system,” J. Lightw. Technol. 31(3), 468–475 (2013).
[Crossref]

Jiménez, F.

Kaino, T.

Krauser, J.

O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook – Optical Short Range Transmission Systems (Springer Verlag, 2008), 2nd ed.

Lambo, W.

López, A.

J. Mateo, M. A. Losada, and A. López, “POF misalignment model based on the calculation of the radiation pattern using the Hankel transform,” Opt. Express 23(6), 8061–8072 (2015).
[Crossref] [PubMed]

D. H. Richards, M. A. Losada, N. Antoniades, A. López, J. Mateo, X. Jiang, and N. Madamopoulos, “Modeling methodology for engineering SI-POF and connectors in an avionics system,” J. Lightw. Technol. 31(3), 468–475 (2013).
[Crossref]

Loquai, S.

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “An analytic connector loss model for step-index polymer optical fiber links,” J. Lightw. Technol. 31(16), 2769–2776 (2013).
[Crossref]

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “Application and evaluation of an analytic connector loss model for SI-POF,” in Proceedings of 22nd ICPOF (Búzios, Rio de Janeiro, Brazil, 2013), pp. 193–198.

Losada, M. A.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Madamopoulos, N.

D. H. Richards, M. A. Losada, N. Antoniades, A. López, J. Mateo, X. Jiang, and N. Madamopoulos, “Modeling methodology for engineering SI-POF and connectors in an avionics system,” J. Lightw. Technol. 31(3), 468–475 (2013).
[Crossref]

Makimoto, T.

S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quant. Electron. 11(5), 447–457 (1979).
[Crossref]

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

Mateo, J.

Mettler, S.

C. Miller, S. Mettler, and I. White, Optical Fiber Splices and Connectors: Theory and Methods (Marcel Dekker, 1986).

Miller, C.

C. Miller, S. Mettler, and I. White, Optical Fiber Splices and Connectors: Theory and Methods (Marcel Dekker, 1986).

Nemoto, S.

S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quant. Electron. 11(5), 447–457 (1979).
[Crossref]

Opielka, D.

Poisel, H.

Richards, D. H.

D. H. Richards, M. A. Losada, N. Antoniades, A. López, J. Mateo, X. Jiang, and N. Madamopoulos, “Modeling methodology for engineering SI-POF and connectors in an avionics system,” J. Lightw. Technol. 31(3), 468–475 (2013).
[Crossref]

Rittich, D.

Rousseau, M.

M. Rousseau and L. Jeunhomme, “Numerical solution of the coupled-power equation in step-index optical fibers,” IEEE Trans. Microwave Theory Tech. 25(7), 577–585 (1977).
[Crossref]

Savovic, S.

Shi, R. F.

G. Jiang, R. F. Shi, and A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” Photon. Technol. Lett. 9(8), 1128–1130 (1997).
[Crossref]

Simons, P.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Sugihara, O.

Tan, F. S.

van Etten, W.

Werzinger, S.

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “An analytic connector loss model for step-index polymer optical fiber links,” J. Lightw. Technol. 31(16), 2769–2776 (2013).
[Crossref]

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “Application and evaluation of an analytic connector loss model for SI-POF,” in Proceedings of 22nd ICPOF (Búzios, Rio de Janeiro, Brazil, 2013), pp. 193–198.

White, I.

C. Miller, S. Mettler, and I. White, Optical Fiber Splices and Connectors: Theory and Methods (Marcel Dekker, 1986).

Zamzow, P. E.

O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook – Optical Short Range Transmission Systems (Springer Verlag, 2008), 2nd ed.

Ziemann, O.

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “An analytic connector loss model for step-index polymer optical fiber links,” J. Lightw. Technol. 31(16), 2769–2776 (2013).
[Crossref]

O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook – Optical Short Range Transmission Systems (Springer Verlag, 2008), 2nd ed.

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “Application and evaluation of an analytic connector loss model for SI-POF,” in Proceedings of 22nd ICPOF (Búzios, Rio de Janeiro, Brazil, 2013), pp. 193–198.

Zubia, J.

Appl. Opt. (3)

Bell Syst. Tech. J. (2)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

D. Gloge, “Offset and tilt loss in optical fiber splices,” Bell Syst. Tech. J. 55(7), 905–916 (1976).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

M. Rousseau and L. Jeunhomme, “Numerical solution of the coupled-power equation in step-index optical fibers,” IEEE Trans. Microwave Theory Tech. 25(7), 577–585 (1977).
[Crossref]

J. Lightw. Technol. (3)

G. Aldabaldetreku, G. Durana, J. Zubia, and J. Arrue, “Analytical expression for measurement of intrinsic coupling loss in multistep index optical fibers,” J. Lightw. Technol. 24(3), 1364–1375 (2006).
[Crossref]

D. H. Richards, M. A. Losada, N. Antoniades, A. López, J. Mateo, X. Jiang, and N. Madamopoulos, “Modeling methodology for engineering SI-POF and connectors in an avionics system,” J. Lightw. Technol. 31(3), 468–475 (2013).
[Crossref]

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “An analytic connector loss model for step-index polymer optical fiber links,” J. Lightw. Technol. 31(16), 2769–2776 (2013).
[Crossref]

Microw. Opt. Tech. Lett. (1)

C. Gao and G. Farrell, “Power coupling between two step-index multimode fibers of different numerical apertures with an angular misalignment,” Microw. Opt. Tech. Lett. 43(3), 231–234 (2004).
[Crossref]

Opt. Express (5)

Opt. Quant. Electron. (1)

S. Nemoto and T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quant. Electron. 11(5), 447–457 (1979).
[Crossref]

Photon. Technol. Lett. (2)

G. Jiang, R. F. Shi, and A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” Photon. Technol. Lett. 9(8), 1128–1130 (1997).
[Crossref]

Y. Ando, “Statistical analysis of insertion-loss improvement for optical connectors using the orientation method for fiber-core offset,” Photon. Technol. Lett. 3(10), 939–941 (1991).
[Crossref]

Other (7)

“Fibre optic communication subsystem test procedures – Part 1–4: General communication subsystems – Light source encircled flux measurement method,” IEC 61280-1-4 (2009).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

S. Werzinger, C.-A. Bunge, S. Loquai, and O. Ziemann, “Application and evaluation of an analytic connector loss model for SI-POF,” in Proceedings of 22nd ICPOF (Búzios, Rio de Janeiro, Brazil, 2013), pp. 193–198.

“Optical fibres – part 2–40: Product specifications – Sectional specification for category A4 multimode fibres,” IEC 60793-2-40 (2009).

C. Miller, S. Mettler, and I. White, Optical Fiber Splices and Connectors: Theory and Methods (Marcel Dekker, 1986).

O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook – Optical Short Range Transmission Systems (Springer Verlag, 2008), 2nd ed.

A. Grzemba, MOST – The Automotive Multimedia Network (Franzis Verlag, 2011), 2nd ed.

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 (7)

Fig. 1
Fig. 1

Coordinate definitions for the modeling. (a) Coupling interface between two SIPOFs with core diameters d1 and d2 and critical angles Θc1 and Θc2, separated by a longitudinal offset z0. (b) Relative radial offset r0 between the projections of the fiber centers O1 and O2 in the xy-plane with lateral components x0 and y0.

Fig. 2
Fig. 2

Comparison of the theoretical EMD FFP from Eq. (1) with measurements taken from a Toray PFU-CD-1001 of 100 m length and a nominal NA of 0.46. (a) Far field measurements. (b) Corresponding losses by longitudinal offsets z0.

Fig. 3
Fig. 3

Integration of the intensity pattern Iz=z0(r, θ) over the receiving fiber core in the plane z = z0. (a) Case that z 0 tan θ d 1 2 . (b) Case that z 0 tan θ > d 1 2 resulting in a ring shaped intensity pattern.

Fig. 4
Fig. 4

Insertion loss measurements for combinations of discrete radial and longitudinal offsets r0 and z0, respectively. The coupling interface is situated after 50 m of a Toray PFU-CD-1001 fiber with a nominal NA of 0.46 and core diameter d = 0.98 mm. (a) The mesh shows the model predictions using a EMD FFP with NA = 0.46 and blue dots refer to the actually measured coupling efficiencies. (b) Same comparison for only radial or longitudinal offsets.

Fig. 5
Fig. 5

Simulated loss CDFs for mismatches from A4a.2 fiber tolerances. (a) Losses by core diameter mismatch dominate under EMD conditions. Combining both mismatches, results in a mean total loss of Lμ = 0.084 dB and a 97th percentile of L97 = 0.368 dB (b) Under UMD conditions, losses by NA mismatches contribute most to the total loss, with Lμ and L97 increasing to 0.199 dB and 0.702 dB, respectively.

Fig. 6
Fig. 6

Coupling losses by A4a.2 fiber tolerances and lateral and longitudinal offsets under EMD conditions. (a) The mean coupling loss Lμ in dB is nearly independent of the longitudinal offset tolerances (nominal value μ z 0 = 150 μm). (b) For the coupling loss 97th percentile L97 in dB, the influence of the longitudinal offset tolerance decreases with increasing lateral offset tolerances.

Fig. 7
Fig. 7

Comparison of the CDFs obtained from the coupling loss model applied in this paper and the simplified modeling approach that just sums up the coupling losses of the different mechanisms independently (EMD conditions). The intrinsic loss mechanism tolerances correspond to A4a.2 fiber class specifications. The nominal longitudinal offset is μ z 0 = 150 μm and has a tolerance of 4σ z 0 = 50 μm and the lateral offset tolerance is 4σ x 0 y 0 = 50 μm. The improvement compared to the linear model in this example is 0.218 dB (31 %) for the 97th percentile L97 and 0.139 dB (34 %) for the mean loss Lμ.

Tables (1)

Tables Icon

Table 1 Distributions of the random parameters used in the MC simulations. Tolerances of input parameters are specified by their mean μ and 4σ interval.

Equations (5)

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

I FFP ( EMD ) ( θ ) = J 0 ( 2.405 θ Θ c 1 ) ,
η total = Φ 2 Φ 1 = 0 min ( Θ c 1 , Θ c 2 ) η d 1 , d 2 , r 0 , z 0 ( θ ) I FFP ( θ ) sin θ d θ 0 Θ c 1 I FFP ( θ ) sin θ d θ .
η d 1 , d 2 , r 0 , z 0 ( θ ) = 8 π d 1 2 max ( 0 , z 0 tan θ d 1 2 ) min ( r 0 + d 2 2 , z 0 tan θ + d 1 2 ) ϕ ( r ) I z = z 0 ( r , θ ) r d r ,
ϕ ( r ) = Re { arccos ( 4 r 2 + 4 r 0 2 d 2 2 8 r r 0 ) } .
I z = z 0 ( r , θ ) = 1 π Re { arccos ( 4 r 2 + 4 z 0 2 tan 2 θ d 1 2 8 r z 0 tan θ ) } .

Metrics