Abstract

A means of calculating optical power distribution in bent multimode optical fibers is proposed. It employs the power-flow equation approximated by the Fokker–Planck equation that is solved by the explicit finite-difference method. Conceptually important steps of this procedure include (i) dividing the full length of the bent optical fiber into a finite number of short, straight segments; (ii) solving the power equation for each segment sequentially to find its output distribution; and (iii) expressing that output distribution in rotated coordinates of the subsequent segment along the curved fiber to determine the input distribution for that subsequent segment and thus enable the calculation of the power flow and output distribution for it. The segment length and bend-induced perturbation of output angles are determined by geometric optics. The resulting power distributions are given at different cross sections along the curved fiber axis. They vary with the radius of fiber curvature and launch conditions. Results are compared to those for straight fiber. Bending loss is calculated as well.

© 2006 Optical Society of America

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References

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  1. J. Zubia and J. Arrue, "Plastic optical fibers: an introduction to their technological processes and applications," Opt. Fiber Technol. 7, 101-140 (2001).
    [CrossRef]
  2. M. Jouguet, "Effects of curvature on the propagation of electromagnetic waves in guides of circular cross section," Cables Transm. 1, 133-153 (1947).
  3. D. Marcuse, Light Transmission Optics, 2nd ed. (von Nostrand Reinhold, 1982).
  4. D. Marcuse, "Radiation loss of a helically deformed optical fiber," J. Opt. Soc. Am. 66, 1025-1031 (1976).
  5. J. Zubia, G. Aldabaldetreku, G. Durana, and J. Arrue, "Geometric optics analysis of multi-step index optical fibers," Fiber Integr. Opt. 23, 121-156 (2004).
    [CrossRef]
  6. I. Salinas, J. I. Garcés, R. Alonso, A. Llobera, and C. Domínguez, "Simple estimation of transition losses in bends of wide optical waveguides by a ray tracing method," IEEE Photon. Technol. Lett. 16, 825-827 (2004).
    [CrossRef]
  7. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).
  8. A. W. Snyder, Optical Waveguide Theory (Chapman and Hall, 1983).
  9. D. Marcuse, "Fluctuation of the power coupling modes," Bell Syst. Tech. J. 51, 1793-1800 (1972).
  10. D. Marcuse, "Coupled mode theory of round optical fibers," Bell Syst. Tech. J. 52, 917-841 (1973).
  11. D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).
  12. G. Herskowitz, H. Kobrinski, and U. Levy, "Optical power distribution in multimode fibers with angular-dependent mode coupling," J. Lightwave Technol. LT-1, 548-554 (1983).
  13. G. Cancellieri and P. Fantini, "Mode coupling effects in optical fibers: perturbative solution of the time-dependent power equation," Opt. Quantum Electron. 15, 119-136 (1983).
    [CrossRef]
  14. S. Savovic and A. Djordjevich, "Optical power flow in plastic-clad silica fibers," Appl. Opt. 41, 7588-7591 (2002).
  15. A. Djordjevich and S. Savovic, "Numerical solution of the power flow equation in step-index plastic optical fibers," J. Opt. Soc. Am. B 21, 1437-1442 (2004).
    [CrossRef]
  16. S. Savovic and A. Djordjevich, "Influence of numerical aperture on mode coupling in step-index plastic optical fibers," Appl. Opt. 43, 5542-5546 (2004).
    [CrossRef]
  17. S. Savovic and A. Djordjevich, "Solution of mode coupling in step index optical fibers by Fokker-Planck equation and Langevin equation," Appl. Opt. 41, 2826-2830 (2002).
  18. A. F. Garito, J. Wang, and R. Gao, "Effect of random perturbations in plastic optical fibers," Science 281, 962-967 (1998).
    [CrossRef]
  19. M. A. Losada, I. Garces, J. Mateo, I. Salinas, and J. Zubia, "Mode coupling in plastic optical fibers of high and low numerical apertures," in Proceedings of the 10th International Conference on POF and Application (IEEE, 2001), pp. 109-112.
  20. F. Jimenez, J. Arrue, M. A. Losada, and J. Zubia, "Simulation of arbitrarily complex 3-D layouts of SI POF. Losses in some non-planar bends of practical interest," in XIII International POF Conference (IEEE, 2004) pp. 210-216.
  21. J. Mateo, M. A. Losada, I. Garces, J. Arrue, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in Proceedings of the 12th International POF Conference (IEEE, 2004), pp. 123-126.
  22. M. A. Losada, J. Mateo, D. Espinosa, I. Garces, and J. Zubia, "Characterisation of the far field pattern for plastic optical fibers," in XIII International POF Conference (IEEE, 2004), pp. 458-465.
  23. G. Aldabaldetreku, G. Durana, J. Zubia, J. Arrue, and F. Jimenez, "Analysis of intrinsic coupling loss in multi-step index optical fibers," Opt. Express 13, 3283-3295 (2005).
    [CrossRef]
  24. G. Aldabaldetreku, G. Durana, J. Zubia, J. Arrue, and F. Jimenez, "Investigation and comparison of analytical, numerical, and experimentally measured coupling losses for multi-step index optical fibers," Opt. Express 13, 4012-4036 (2005).
    [CrossRef]
  25. M. S. Kovacevic, D. Nikezic, and A. Djordjevich, "Modeling of the loss and mode coupling due to an irregular core-cladding interface in step-index plastic optical fibers," Appl. Opt. 44, 3898-3903 (2005).
    [CrossRef]
  26. M. Rousseau and L. Jeunhomme, "Numerical solution of the coupled-power equation in step-index optical fibers," IEEE Trans. Microwave Theory Tech. 25, 577-586 (1977).
    [CrossRef]
  27. A. A. P. Boechat, D. Su, D. R. Hall, and J. D. C. Jones, "Bend loss in large core multimode optical fiber beam delivery system," Appl. Opt. 30, 321-327 (1991).

2005

2004

A. Djordjevich and S. Savovic, "Numerical solution of the power flow equation in step-index plastic optical fibers," J. Opt. Soc. Am. B 21, 1437-1442 (2004).
[CrossRef]

S. Savovic and A. Djordjevich, "Influence of numerical aperture on mode coupling in step-index plastic optical fibers," Appl. Opt. 43, 5542-5546 (2004).
[CrossRef]

J. Zubia, G. Aldabaldetreku, G. Durana, and J. Arrue, "Geometric optics analysis of multi-step index optical fibers," Fiber Integr. Opt. 23, 121-156 (2004).
[CrossRef]

I. Salinas, J. I. Garcés, R. Alonso, A. Llobera, and C. Domínguez, "Simple estimation of transition losses in bends of wide optical waveguides by a ray tracing method," IEEE Photon. Technol. Lett. 16, 825-827 (2004).
[CrossRef]

2002

2001

J. Zubia and J. Arrue, "Plastic optical fibers: an introduction to their technological processes and applications," Opt. Fiber Technol. 7, 101-140 (2001).
[CrossRef]

1998

A. F. Garito, J. Wang, and R. Gao, "Effect of random perturbations in plastic optical fibers," Science 281, 962-967 (1998).
[CrossRef]

1991

1983

G. Herskowitz, H. Kobrinski, and U. Levy, "Optical power distribution in multimode fibers with angular-dependent mode coupling," J. Lightwave Technol. LT-1, 548-554 (1983).

G. Cancellieri and P. Fantini, "Mode coupling effects in optical fibers: perturbative solution of the time-dependent power equation," Opt. Quantum Electron. 15, 119-136 (1983).
[CrossRef]

1977

M. Rousseau and L. Jeunhomme, "Numerical solution of the coupled-power equation in step-index optical fibers," IEEE Trans. Microwave Theory Tech. 25, 577-586 (1977).
[CrossRef]

1976

1973

D. Marcuse, "Coupled mode theory of round optical fibers," Bell Syst. Tech. J. 52, 917-841 (1973).

1972

D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).

D. Marcuse, "Fluctuation of the power coupling modes," Bell Syst. Tech. J. 51, 1793-1800 (1972).

1947

M. Jouguet, "Effects of curvature on the propagation of electromagnetic waves in guides of circular cross section," Cables Transm. 1, 133-153 (1947).

Aldabaldetreku, G.

Alonso, R.

I. Salinas, J. I. Garcés, R. Alonso, A. Llobera, and C. Domínguez, "Simple estimation of transition losses in bends of wide optical waveguides by a ray tracing method," IEEE Photon. Technol. Lett. 16, 825-827 (2004).
[CrossRef]

Arrue, J.

G. Aldabaldetreku, G. Durana, J. Zubia, J. Arrue, and F. Jimenez, "Analysis of intrinsic coupling loss in multi-step index optical fibers," Opt. Express 13, 3283-3295 (2005).
[CrossRef]

G. Aldabaldetreku, G. Durana, J. Zubia, J. Arrue, and F. Jimenez, "Investigation and comparison of analytical, numerical, and experimentally measured coupling losses for multi-step index optical fibers," Opt. Express 13, 4012-4036 (2005).
[CrossRef]

J. Zubia, G. Aldabaldetreku, G. Durana, and J. Arrue, "Geometric optics analysis of multi-step index optical fibers," Fiber Integr. Opt. 23, 121-156 (2004).
[CrossRef]

J. Zubia and J. Arrue, "Plastic optical fibers: an introduction to their technological processes and applications," Opt. Fiber Technol. 7, 101-140 (2001).
[CrossRef]

J. Mateo, M. A. Losada, I. Garces, J. Arrue, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in Proceedings of the 12th International POF Conference (IEEE, 2004), pp. 123-126.

F. Jimenez, J. Arrue, M. A. Losada, and J. Zubia, "Simulation of arbitrarily complex 3-D layouts of SI POF. Losses in some non-planar bends of practical interest," in XIII International POF Conference (IEEE, 2004) pp. 210-216.

Boechat, A. A. P.

Cancellieri, G.

G. Cancellieri and P. Fantini, "Mode coupling effects in optical fibers: perturbative solution of the time-dependent power equation," Opt. Quantum Electron. 15, 119-136 (1983).
[CrossRef]

Djordjevich, A.

Domínguez, C.

I. Salinas, J. I. Garcés, R. Alonso, A. Llobera, and C. Domínguez, "Simple estimation of transition losses in bends of wide optical waveguides by a ray tracing method," IEEE Photon. Technol. Lett. 16, 825-827 (2004).
[CrossRef]

Durana, G.

Espinosa, D.

M. A. Losada, J. Mateo, D. Espinosa, I. Garces, and J. Zubia, "Characterisation of the far field pattern for plastic optical fibers," in XIII International POF Conference (IEEE, 2004), pp. 458-465.

Fantini, P.

G. Cancellieri and P. Fantini, "Mode coupling effects in optical fibers: perturbative solution of the time-dependent power equation," Opt. Quantum Electron. 15, 119-136 (1983).
[CrossRef]

Gao, R.

A. F. Garito, J. Wang, and R. Gao, "Effect of random perturbations in plastic optical fibers," Science 281, 962-967 (1998).
[CrossRef]

Garces, I.

M. A. Losada, J. Mateo, D. Espinosa, I. Garces, and J. Zubia, "Characterisation of the far field pattern for plastic optical fibers," in XIII International POF Conference (IEEE, 2004), pp. 458-465.

J. Mateo, M. A. Losada, I. Garces, J. Arrue, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in Proceedings of the 12th International POF Conference (IEEE, 2004), pp. 123-126.

M. A. Losada, I. Garces, J. Mateo, I. Salinas, and J. Zubia, "Mode coupling in plastic optical fibers of high and low numerical apertures," in Proceedings of the 10th International Conference on POF and Application (IEEE, 2001), pp. 109-112.

Garcés, J. I.

I. Salinas, J. I. Garcés, R. Alonso, A. Llobera, and C. Domínguez, "Simple estimation of transition losses in bends of wide optical waveguides by a ray tracing method," IEEE Photon. Technol. Lett. 16, 825-827 (2004).
[CrossRef]

Garito, A. F.

A. F. Garito, J. Wang, and R. Gao, "Effect of random perturbations in plastic optical fibers," Science 281, 962-967 (1998).
[CrossRef]

Gloge, D.

D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).

Hall, D. R.

Herskowitz, G.

G. Herskowitz, H. Kobrinski, and U. Levy, "Optical power distribution in multimode fibers with angular-dependent mode coupling," J. Lightwave Technol. LT-1, 548-554 (1983).

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, 577-586 (1977).
[CrossRef]

Jimenez, F.

Jones, J. D. C.

Jouguet, M.

M. Jouguet, "Effects of curvature on the propagation of electromagnetic waves in guides of circular cross section," Cables Transm. 1, 133-153 (1947).

Kalymnios, D.

J. Mateo, M. A. Losada, I. Garces, J. Arrue, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in Proceedings of the 12th International POF Conference (IEEE, 2004), pp. 123-126.

Kobrinski, H.

G. Herskowitz, H. Kobrinski, and U. Levy, "Optical power distribution in multimode fibers with angular-dependent mode coupling," J. Lightwave Technol. LT-1, 548-554 (1983).

Kovacevic, M. S.

Levy, U.

G. Herskowitz, H. Kobrinski, and U. Levy, "Optical power distribution in multimode fibers with angular-dependent mode coupling," J. Lightwave Technol. LT-1, 548-554 (1983).

Llobera, A.

I. Salinas, J. I. Garcés, R. Alonso, A. Llobera, and C. Domínguez, "Simple estimation of transition losses in bends of wide optical waveguides by a ray tracing method," IEEE Photon. Technol. Lett. 16, 825-827 (2004).
[CrossRef]

Losada, M. A.

M. A. Losada, J. Mateo, D. Espinosa, I. Garces, and J. Zubia, "Characterisation of the far field pattern for plastic optical fibers," in XIII International POF Conference (IEEE, 2004), pp. 458-465.

M. A. Losada, I. Garces, J. Mateo, I. Salinas, and J. Zubia, "Mode coupling in plastic optical fibers of high and low numerical apertures," in Proceedings of the 10th International Conference on POF and Application (IEEE, 2001), pp. 109-112.

F. Jimenez, J. Arrue, M. A. Losada, and J. Zubia, "Simulation of arbitrarily complex 3-D layouts of SI POF. Losses in some non-planar bends of practical interest," in XIII International POF Conference (IEEE, 2004) pp. 210-216.

J. Mateo, M. A. Losada, I. Garces, J. Arrue, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in Proceedings of the 12th International POF Conference (IEEE, 2004), pp. 123-126.

Marcuse, D.

D. Marcuse, "Radiation loss of a helically deformed optical fiber," J. Opt. Soc. Am. 66, 1025-1031 (1976).

D. Marcuse, "Coupled mode theory of round optical fibers," Bell Syst. Tech. J. 52, 917-841 (1973).

D. Marcuse, "Fluctuation of the power coupling modes," Bell Syst. Tech. J. 51, 1793-1800 (1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).

D. Marcuse, Light Transmission Optics, 2nd ed. (von Nostrand Reinhold, 1982).

Mateo, J.

M. A. Losada, I. Garces, J. Mateo, I. Salinas, and J. Zubia, "Mode coupling in plastic optical fibers of high and low numerical apertures," in Proceedings of the 10th International Conference on POF and Application (IEEE, 2001), pp. 109-112.

J. Mateo, M. A. Losada, I. Garces, J. Arrue, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in Proceedings of the 12th International POF Conference (IEEE, 2004), pp. 123-126.

M. A. Losada, J. Mateo, D. Espinosa, I. Garces, and J. Zubia, "Characterisation of the far field pattern for plastic optical fibers," in XIII International POF Conference (IEEE, 2004), pp. 458-465.

Nikezic, 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, 577-586 (1977).
[CrossRef]

Salinas, I.

I. Salinas, J. I. Garcés, R. Alonso, A. Llobera, and C. Domínguez, "Simple estimation of transition losses in bends of wide optical waveguides by a ray tracing method," IEEE Photon. Technol. Lett. 16, 825-827 (2004).
[CrossRef]

M. A. Losada, I. Garces, J. Mateo, I. Salinas, and J. Zubia, "Mode coupling in plastic optical fibers of high and low numerical apertures," in Proceedings of the 10th International Conference on POF and Application (IEEE, 2001), pp. 109-112.

Savovic, S.

Snyder, A. W.

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

Su, D.

Wang, J.

A. F. Garito, J. Wang, and R. Gao, "Effect of random perturbations in plastic optical fibers," Science 281, 962-967 (1998).
[CrossRef]

Zubia, J.

G. Aldabaldetreku, G. Durana, J. Zubia, J. Arrue, and F. Jimenez, "Investigation and comparison of analytical, numerical, and experimentally measured coupling losses for multi-step index optical fibers," Opt. Express 13, 4012-4036 (2005).
[CrossRef]

G. Aldabaldetreku, G. Durana, J. Zubia, J. Arrue, and F. Jimenez, "Analysis of intrinsic coupling loss in multi-step index optical fibers," Opt. Express 13, 3283-3295 (2005).
[CrossRef]

J. Zubia, G. Aldabaldetreku, G. Durana, and J. Arrue, "Geometric optics analysis of multi-step index optical fibers," Fiber Integr. Opt. 23, 121-156 (2004).
[CrossRef]

J. Zubia and J. Arrue, "Plastic optical fibers: an introduction to their technological processes and applications," Opt. Fiber Technol. 7, 101-140 (2001).
[CrossRef]

M. A. Losada, I. Garces, J. Mateo, I. Salinas, and J. Zubia, "Mode coupling in plastic optical fibers of high and low numerical apertures," in Proceedings of the 10th International Conference on POF and Application (IEEE, 2001), pp. 109-112.

M. A. Losada, J. Mateo, D. Espinosa, I. Garces, and J. Zubia, "Characterisation of the far field pattern for plastic optical fibers," in XIII International POF Conference (IEEE, 2004), pp. 458-465.

J. Mateo, M. A. Losada, I. Garces, J. Arrue, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in Proceedings of the 12th International POF Conference (IEEE, 2004), pp. 123-126.

F. Jimenez, J. Arrue, M. A. Losada, and J. Zubia, "Simulation of arbitrarily complex 3-D layouts of SI POF. Losses in some non-planar bends of practical interest," in XIII International POF Conference (IEEE, 2004) pp. 210-216.

Appl. Opt.

Bell Syst. Tech. J.

D. Marcuse, "Fluctuation of the power coupling modes," Bell Syst. Tech. J. 51, 1793-1800 (1972).

D. Marcuse, "Coupled mode theory of round optical fibers," Bell Syst. Tech. J. 52, 917-841 (1973).

D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).

Cables Transm.

M. Jouguet, "Effects of curvature on the propagation of electromagnetic waves in guides of circular cross section," Cables Transm. 1, 133-153 (1947).

Fiber Integr. Opt.

J. Zubia, G. Aldabaldetreku, G. Durana, and J. Arrue, "Geometric optics analysis of multi-step index optical fibers," Fiber Integr. Opt. 23, 121-156 (2004).
[CrossRef]

IEEE Photon. Technol. Lett.

I. Salinas, J. I. Garcés, R. Alonso, A. Llobera, and C. Domínguez, "Simple estimation of transition losses in bends of wide optical waveguides by a ray tracing method," IEEE Photon. Technol. Lett. 16, 825-827 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

M. Rousseau and L. Jeunhomme, "Numerical solution of the coupled-power equation in step-index optical fibers," IEEE Trans. Microwave Theory Tech. 25, 577-586 (1977).
[CrossRef]

J. Lightwave Technol.

G. Herskowitz, H. Kobrinski, and U. Levy, "Optical power distribution in multimode fibers with angular-dependent mode coupling," J. Lightwave Technol. LT-1, 548-554 (1983).

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Express

Opt. Fiber Technol.

J. Zubia and J. Arrue, "Plastic optical fibers: an introduction to their technological processes and applications," Opt. Fiber Technol. 7, 101-140 (2001).
[CrossRef]

Opt. Quantum Electron.

G. Cancellieri and P. Fantini, "Mode coupling effects in optical fibers: perturbative solution of the time-dependent power equation," Opt. Quantum Electron. 15, 119-136 (1983).
[CrossRef]

Science

A. F. Garito, J. Wang, and R. Gao, "Effect of random perturbations in plastic optical fibers," Science 281, 962-967 (1998).
[CrossRef]

Other

M. A. Losada, I. Garces, J. Mateo, I. Salinas, and J. Zubia, "Mode coupling in plastic optical fibers of high and low numerical apertures," in Proceedings of the 10th International Conference on POF and Application (IEEE, 2001), pp. 109-112.

F. Jimenez, J. Arrue, M. A. Losada, and J. Zubia, "Simulation of arbitrarily complex 3-D layouts of SI POF. Losses in some non-planar bends of practical interest," in XIII International POF Conference (IEEE, 2004) pp. 210-216.

J. Mateo, M. A. Losada, I. Garces, J. Arrue, J. Zubia, and D. Kalymnios, "High NA POF dependence of bandwidth on fibre length," in Proceedings of the 12th International POF Conference (IEEE, 2004), pp. 123-126.

M. A. Losada, J. Mateo, D. Espinosa, I. Garces, and J. Zubia, "Characterisation of the far field pattern for plastic optical fibers," in XIII International POF Conference (IEEE, 2004), pp. 458-465.

D. Marcuse, Light Transmission Optics, 2nd ed. (von Nostrand Reinhold, 1982).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1991).

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

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Figures (9)

Fig. 1
Fig. 1

Ray path on a bent multimode step-index fiber. The bend radius is R.

Fig. 2
Fig. 2

Geometry of the modeled bent fiber. The fiber is divided in the finite number of the segments and all of them are straight as assumed in the model.

Fig. 3
Fig. 3

Unit vector p in the original (left panel) and rotated (right panel) coordinate systems.

Fig. 4
Fig. 4

Normalized power distributions at different locations along the bent fiber, for the input distribution centered at θ 0 = 0 ° . Bend radius is R = 10   m .

Fig. 5
Fig. 5

Normalized power distributions at different locations along the bent fiber, for the input distribution centered at θ 0 = 5 ° . Bend radius is R = 8   m .

Fig. 6
Fig. 6

Normalized power distributions at different locations along the bend fiber for the input distribution centered at θ 0 = 5 ° . Bend radius is R = 5   m .

Fig. 7
Fig. 7

Normalized power distributions at different locations along the bend fiber for the input distribution centered at θ 0 = 5 ° . Bend radius is R = 0.5   m .

Fig. 8
Fig. 8

Normalized power distributions at different locations along the bend fiber for the input distribution centered at θ 0 = 5 ° . Bend radius is R = 0.1   m .

Fig. 9
Fig. 9

Bending loss as a function of bend radius of the fiber. Length of the fiber is L = 3   m .

Equations (12)

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

P ( θ , z ) z = α ( θ ) P ( θ , z ) + ( Δ θ ) 2 1 θ θ × [ θ d ( θ ) P ( θ , z ) θ ] ,
P ( θ , z ) z = α ( θ ) P ( θ , z ) + D θ θ [ θ P ( θ , z ) θ ] .
P ( θ , z ) z = D θ P ( θ , z ) θ + D 2 P ( θ , z ) θ 2 .
P ( θ , z ) z = V P ( θ , z ) θ + D 2 P ( θ , z ) θ 2 ,
p = p x e x + p y e y + p z e z = sin θ cos φ e x + sin θ sin φ e y + cos θ e z .
           p x = p e x = sin θ cos φ ,
p y = p e y = sin θ sin φ cos δϕ
+ cos θ cos ( π / 2 + δ ϕ ) ,
           p z = p e z = sin θ sin φ cos ( π / 2 δ ϕ )
+ cos θ cos δ ϕ .
p i ( z ) = 2 π 0 θ c θ P i ( θ , z ) d θ .
P R = δ P 1 δ P 2 δ P 1 .

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