Abstract

The aim of the present paper is to provide a single analytical expression of the power transmission coefficient for leaky rays in multi-step index (MSI) fibres. This expression is valid for all tunnelling and refracting rays and allows us to evaluate numerically the power attenuation along an MSI fibre of an arbitrary number of layers. We validate our analysis by comparing the results obtained for limit cases of MSI fibres with those corresponding to step-index (SI) and graded-index (GI) fibres. We also make a similar comparison between this theoretical expression and the use of the WKB solutions of the scalar wave equation.

© 2006 Optical Society of America

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  1. V. Levin, T. Baskakova, Z. Lavrova, A. Zubkov, H. Poisel, and K. Klein, "Production of multilayer polymer optical fibers," in Proceedings of the eighth international conference on plastic optical fibers and applications-POF ’99, pp. 98-101 (Chiba (Japan), 1999).
  2. K. Irie, Y. Uozu, and T. Yoshimura, "Structure design and analysis of broadband POF," in Proceedings of the tenth international conference on plastic optical fibers and applications-POF ’01, pp. 73-79 (Amsterdam (The Netherlands), 2001).
  3. J. Zubia, G. Aldabaldetreku, G. Durana, J. Arrue, H. Poisel, and C. A. Bunge, "Geometric optics analysis of multi-step index optical fibers," Fiber Integr. Opt. 23, 121-156 (2004).
    [CrossRef]
  4. A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).
  5. R. Sammut and A. W. Snyder, "Leaky modes on circular optical waveguides," Appl. Opt. 15, 477-482 (1976).
    [CrossRef] [PubMed]
  6. A. W. Snyder and J. D. Love, "Tunnelling leaky modes on optical waveguides," Opt. Commun. 12, 326-328 (1974).
    [CrossRef]
  7. J. D. Love and C. Winkler, "Attenuation and tunneling coefficients for leaky rays in multilayered optical waveguides," J. Opt. Soc. Am. 67, 1627-1632 (1977).
    [CrossRef]
  8. J. D. Love and C. Winkler, "Refracting leaky rays in graded-index fibers," Appl. Opt. 17, 2205-2208 (1978).
    [CrossRef] [PubMed]
  9. J. D. Love and C. Winkler, "A universal tunnelling coefficient for step- and graded-index multimode fibres," Opt. Quantum Electron. 10, 341-351 (1978).
    [CrossRef]
  10. D. Gloge and E. A. J. Marcatili, "Multimode theory of graded-core fibers," Bell Syst. Tech. J. 52, 1563-1578 (1973).
  11. A. Ankiewicz, "Geometric optics theory of graded index optical fibres," Ph.D. thesis, Australian National University (1978).
  12. D. Marcuse, D. Gloge, and E. A. J. Marcatili, "Guiding properties of fibers," in Optical fiber telecommunications, S. E. Miller and A. G. Chynoweth, eds., chap. 3 (Academic Press, Inc., San Diego, California, 1979).
  13. S. Zhu, A.W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: A demonstration and review," Am. J. Phys. 54, 601-606 (1986).
    [CrossRef]
  14. F. P. Zanella, D. V. Magalh˜aes, M. M. Oliveira, R. F. Bianchi, L. Misoguti, and C. R. Mendonc¸a, "Frustrated total internal reflection: A simple application and demonstration," Am. J. Phys. 71, 494-496 (2003).
    [CrossRef]
  15. A. A. Stahlhofen, "Comment on "Frustrated total internal reflection: A simple application and demonstration,"Am. J. Phys. 72, 412 (2004).
    [CrossRef]
  16. O. Bryngdahl, "Evanescent waves in optical imaging," in Progress in optics XI, E. Wolf, ed., chap. 4 (North-Holland, Amsterdam, 1973).
    [CrossRef]
  17. R. J. Black and A. Ankiewicz, "Fiber-optic analogies with mechanics," Am. J. Phys. 53, 554-563 (1985).
    [CrossRef]
  18. A. W. Snyder, D. J. Mitchell, and C. Pask, "Failure of geometric optics for analysis of circular optical fibers," J. Opt. Soc. Am. 64, 608-614 (1974).
    [CrossRef]
  19. A. W. Snyder and D. J. Mitchell, "Leaky rays on circular optical fibers," J. Opt. Soc. Am. 64, 599-607 (1974).
    [CrossRef]
  20. A. W. Snyder and D. J. Mitchell, "Generalized Fresnel’s laws for determining radiation loss from optical waveguides and curved dielectric structures," Optik 40, 438-459 (1974).
  21. A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface - Electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. 23, 134-141 (1975).
    [CrossRef]
  22. J. D. Love and C. Winkler, "Generalized Fresnel power transmission coefficients for curved graded-index media," IEEE Trans. Microwave Theory Tech. 28, 689-695 (1980).
    [CrossRef]
  23. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables (Dover Publications, Inc., New York, 1965).
  24. J. D. Love and C. Winkler, "The step index limit of power law refractive index profiles for optical waveguides," J. Opt. Soc. Am. 68, 1188-1191 (1978).
    [CrossRef]
  25. M. Born and E. Wolf, Principles of optics, 6th ed. (Pergamon Press, New York, 1990).
  26. Mitsubishi Rayon Co., Ltd.: "Eska-Miu," URL http://www.pofeska.com.
  27. D. Marcuse, Principles of optical fiber measurements, chap. 4 (Academic Press, Inc., London, 1981).
  28. Japanese Standards Association, "Test methods for structural parameters of all plastic multimode optical fibers," Tech. Rep. JIS C 6862, JIS, Tokyo, Japan (1990).
  29. A. Ankiewicz and C. Pask, "Tunnelling rays in graded-index fibres," Opt. Quantum Electron. 10, 83-93 (1978).
    [CrossRef]

2004

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

A. A. Stahlhofen, "Comment on "Frustrated total internal reflection: A simple application and demonstration,"Am. J. Phys. 72, 412 (2004).
[CrossRef]

2003

F. P. Zanella, D. V. Magalh˜aes, M. M. Oliveira, R. F. Bianchi, L. Misoguti, and C. R. Mendonc¸a, "Frustrated total internal reflection: A simple application and demonstration," Am. J. Phys. 71, 494-496 (2003).
[CrossRef]

1986

S. Zhu, A.W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: A demonstration and review," Am. J. Phys. 54, 601-606 (1986).
[CrossRef]

1985

R. J. Black and A. Ankiewicz, "Fiber-optic analogies with mechanics," Am. J. Phys. 53, 554-563 (1985).
[CrossRef]

1980

J. D. Love and C. Winkler, "Generalized Fresnel power transmission coefficients for curved graded-index media," IEEE Trans. Microwave Theory Tech. 28, 689-695 (1980).
[CrossRef]

1978

A. Ankiewicz and C. Pask, "Tunnelling rays in graded-index fibres," Opt. Quantum Electron. 10, 83-93 (1978).
[CrossRef]

J. D. Love and C. Winkler, "Refracting leaky rays in graded-index fibers," Appl. Opt. 17, 2205-2208 (1978).
[CrossRef] [PubMed]

J. D. Love and C. Winkler, "The step index limit of power law refractive index profiles for optical waveguides," J. Opt. Soc. Am. 68, 1188-1191 (1978).
[CrossRef]

J. D. Love and C. Winkler, "A universal tunnelling coefficient for step- and graded-index multimode fibres," Opt. Quantum Electron. 10, 341-351 (1978).
[CrossRef]

1977

1976

1975

A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface - Electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. 23, 134-141 (1975).
[CrossRef]

1974

A. W. Snyder and D. J. Mitchell, "Leaky rays on circular optical fibers," J. Opt. Soc. Am. 64, 599-607 (1974).
[CrossRef]

A. W. Snyder, D. J. Mitchell, and C. Pask, "Failure of geometric optics for analysis of circular optical fibers," J. Opt. Soc. Am. 64, 608-614 (1974).
[CrossRef]

A. W. Snyder and J. D. Love, "Tunnelling leaky modes on optical waveguides," Opt. Commun. 12, 326-328 (1974).
[CrossRef]

A. W. Snyder and D. J. Mitchell, "Generalized Fresnel’s laws for determining radiation loss from optical waveguides and curved dielectric structures," Optik 40, 438-459 (1974).

1973

D. Gloge and E. A. J. Marcatili, "Multimode theory of graded-core fibers," Bell Syst. Tech. J. 52, 1563-1578 (1973).

Aldabaldetreku, G.

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

Ankiewicz, A.

R. J. Black and A. Ankiewicz, "Fiber-optic analogies with mechanics," Am. J. Phys. 53, 554-563 (1985).
[CrossRef]

A. Ankiewicz and C. Pask, "Tunnelling rays in graded-index fibres," Opt. Quantum Electron. 10, 83-93 (1978).
[CrossRef]

Arrue, J.

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

Black, R. J.

R. J. Black and A. Ankiewicz, "Fiber-optic analogies with mechanics," Am. J. Phys. 53, 554-563 (1985).
[CrossRef]

Bunge, C. A.

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

Durana, G.

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

Gloge, D.

D. Gloge and E. A. J. Marcatili, "Multimode theory of graded-core fibers," Bell Syst. Tech. J. 52, 1563-1578 (1973).

Hawley, D.

S. Zhu, A.W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: A demonstration and review," Am. J. Phys. 54, 601-606 (1986).
[CrossRef]

Love, J. D.

J. D. Love and C. Winkler, "Generalized Fresnel power transmission coefficients for curved graded-index media," IEEE Trans. Microwave Theory Tech. 28, 689-695 (1980).
[CrossRef]

J. D. Love and C. Winkler, "The step index limit of power law refractive index profiles for optical waveguides," J. Opt. Soc. Am. 68, 1188-1191 (1978).
[CrossRef]

J. D. Love and C. Winkler, "A universal tunnelling coefficient for step- and graded-index multimode fibres," Opt. Quantum Electron. 10, 341-351 (1978).
[CrossRef]

J. D. Love and C. Winkler, "Refracting leaky rays in graded-index fibers," Appl. Opt. 17, 2205-2208 (1978).
[CrossRef] [PubMed]

J. D. Love and C. Winkler, "Attenuation and tunneling coefficients for leaky rays in multilayered optical waveguides," J. Opt. Soc. Am. 67, 1627-1632 (1977).
[CrossRef]

A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface - Electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. 23, 134-141 (1975).
[CrossRef]

A. W. Snyder and J. D. Love, "Tunnelling leaky modes on optical waveguides," Opt. Commun. 12, 326-328 (1974).
[CrossRef]

Marcatili, E. A. J.

D. Gloge and E. A. J. Marcatili, "Multimode theory of graded-core fibers," Bell Syst. Tech. J. 52, 1563-1578 (1973).

Mitchell, D. J.

Pask, C.

Poisel, H.

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

Roy, R.

S. Zhu, A.W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: A demonstration and review," Am. J. Phys. 54, 601-606 (1986).
[CrossRef]

Sammut, R.

Snyder, A. W.

R. Sammut and A. W. Snyder, "Leaky modes on circular optical waveguides," Appl. Opt. 15, 477-482 (1976).
[CrossRef] [PubMed]

A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface - Electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. 23, 134-141 (1975).
[CrossRef]

A. W. Snyder, D. J. Mitchell, and C. Pask, "Failure of geometric optics for analysis of circular optical fibers," J. Opt. Soc. Am. 64, 608-614 (1974).
[CrossRef]

A. W. Snyder and D. J. Mitchell, "Generalized Fresnel’s laws for determining radiation loss from optical waveguides and curved dielectric structures," Optik 40, 438-459 (1974).

A. W. Snyder and D. J. Mitchell, "Leaky rays on circular optical fibers," J. Opt. Soc. Am. 64, 599-607 (1974).
[CrossRef]

A. W. Snyder and J. D. Love, "Tunnelling leaky modes on optical waveguides," Opt. Commun. 12, 326-328 (1974).
[CrossRef]

Stahlhofen, A. A.

A. A. Stahlhofen, "Comment on "Frustrated total internal reflection: A simple application and demonstration,"Am. J. Phys. 72, 412 (2004).
[CrossRef]

Winkler, C.

Yu, A.W.

S. Zhu, A.W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: A demonstration and review," Am. J. Phys. 54, 601-606 (1986).
[CrossRef]

Zanella, F. P.

F. P. Zanella, D. V. Magalh˜aes, M. M. Oliveira, R. F. Bianchi, L. Misoguti, and C. R. Mendonc¸a, "Frustrated total internal reflection: A simple application and demonstration," Am. J. Phys. 71, 494-496 (2003).
[CrossRef]

Zhu, S.

S. Zhu, A.W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: A demonstration and review," Am. J. Phys. 54, 601-606 (1986).
[CrossRef]

Zubia, J.

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

Am. J. Phys.

S. Zhu, A.W. Yu, D. Hawley, and R. Roy, "Frustrated total internal reflection: A demonstration and review," Am. J. Phys. 54, 601-606 (1986).
[CrossRef]

F. P. Zanella, D. V. Magalh˜aes, M. M. Oliveira, R. F. Bianchi, L. Misoguti, and C. R. Mendonc¸a, "Frustrated total internal reflection: A simple application and demonstration," Am. J. Phys. 71, 494-496 (2003).
[CrossRef]

A. A. Stahlhofen, "Comment on "Frustrated total internal reflection: A simple application and demonstration,"Am. J. Phys. 72, 412 (2004).
[CrossRef]

R. J. Black and A. Ankiewicz, "Fiber-optic analogies with mechanics," Am. J. Phys. 53, 554-563 (1985).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

D. Gloge and E. A. J. Marcatili, "Multimode theory of graded-core fibers," Bell Syst. Tech. J. 52, 1563-1578 (1973).

Fiber Integr. Opt.

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

IEEE Trans. Microwave Theory Tech.

A. W. Snyder and J. D. Love, "Reflection at a curved dielectric interface - Electromagnetic tunneling," IEEE Trans. Microwave Theory Tech. 23, 134-141 (1975).
[CrossRef]

J. D. Love and C. Winkler, "Generalized Fresnel power transmission coefficients for curved graded-index media," IEEE Trans. Microwave Theory Tech. 28, 689-695 (1980).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

A. W. Snyder and J. D. Love, "Tunnelling leaky modes on optical waveguides," Opt. Commun. 12, 326-328 (1974).
[CrossRef]

Opt. Quantum Electron.

J. D. Love and C. Winkler, "A universal tunnelling coefficient for step- and graded-index multimode fibres," Opt. Quantum Electron. 10, 341-351 (1978).
[CrossRef]

A. Ankiewicz and C. Pask, "Tunnelling rays in graded-index fibres," Opt. Quantum Electron. 10, 83-93 (1978).
[CrossRef]

Optik

A. W. Snyder and D. J. Mitchell, "Generalized Fresnel’s laws for determining radiation loss from optical waveguides and curved dielectric structures," Optik 40, 438-459 (1974).

Other

O. Bryngdahl, "Evanescent waves in optical imaging," in Progress in optics XI, E. Wolf, ed., chap. 4 (North-Holland, Amsterdam, 1973).
[CrossRef]

A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

V. Levin, T. Baskakova, Z. Lavrova, A. Zubkov, H. Poisel, and K. Klein, "Production of multilayer polymer optical fibers," in Proceedings of the eighth international conference on plastic optical fibers and applications-POF ’99, pp. 98-101 (Chiba (Japan), 1999).

K. Irie, Y. Uozu, and T. Yoshimura, "Structure design and analysis of broadband POF," in Proceedings of the tenth international conference on plastic optical fibers and applications-POF ’01, pp. 73-79 (Amsterdam (The Netherlands), 2001).

M. Born and E. Wolf, Principles of optics, 6th ed. (Pergamon Press, New York, 1990).

Mitsubishi Rayon Co., Ltd.: "Eska-Miu," URL http://www.pofeska.com.

D. Marcuse, Principles of optical fiber measurements, chap. 4 (Academic Press, Inc., London, 1981).

Japanese Standards Association, "Test methods for structural parameters of all plastic multimode optical fibers," Tech. Rep. JIS C 6862, JIS, Tokyo, Japan (1990).

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables (Dover Publications, Inc., New York, 1965).

A. Ankiewicz, "Geometric optics theory of graded index optical fibres," Ph.D. thesis, Australian National University (1978).

D. Marcuse, D. Gloge, and E. A. J. Marcatili, "Guiding properties of fibers," in Optical fiber telecommunications, S. E. Miller and A. G. Chynoweth, eds., chap. 3 (Academic Press, Inc., San Diego, California, 1979).

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Figures (5)

Fig. 1.
Fig. 1.

The linearized profile used in the analysis corresponds to AB ¯ (red line). At rtp the ray is partially reflected and the position rrad in the cladding stands for the point where the transmitted ray reappears.

Fig. 2.
Fig. 2.

Refractive index profiles corresponding to the MSI-POFs used.

Fig. 3.
Fig. 3.

Contour plots of the power transmission coefficient T on the β̃2-2 plane, calculated for both tunnelling and refracting rays and for the wavelength λ = 650 nm. The dashed dotted line shows the limit where a tunnelling ray becomes a refracting ray.

Fig. 4.
Fig. 4.

Evolution of the power transmission coefficient T as a function of the number of layers N of an MSI fibre with constant = 0.255. Results obtained for the wavelength λ = 650 nm. The blue and red solid curves denote the results for the SI and clad-parabolic-profile GI fibres, respectively. The dashed dotted vertical line shows the division between tunnelling and refracting rays, which occurs at β̃ = 1.3786, when β̃2+2 = ncl2 .

Fig. 5.
Fig. 5.

Comparison between the uniform method and the WKB approximation. Plots of the power transmission coefficient T with constant = 0.255 for the wavelength λ = 650 nm. Results obtained for the parabolic-profile MSI fibre of N = 1000 layers and both Eska-Miu and TVER MSI-POFs. Legend: T of Eq. (33): solid curves; T of Eqs. (43) and (48): dashed curves. The dashed dotted vertical line at β̃ = 1.3786 corresponds to the limit β̃2+2 = ncl2 . To the right of this limit value rays are tunnelling, whereas to the left they are refracting.

Tables (1)

Tables Icon

Table 1. Physical dimensions of the different layers (outer radii in μm).

Equations (71)

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

n ( r ) = { n 1 ; r < ρ 1 , n 2 ; ρ 1 r < ρ 2 , n N ; ρ N 1 r < ρ N , n cl ; r ρ N ,
g ( r ) = n 2 ( r ) β ˜ 2 l ˜ 2 ρ N 2 r 2 ,
β ˜ = n i cos θ z i ;
i = 1 N ,
l ˜ = ρ i ρ N n i sin θ z i cos θ ϕ i ;
n cl 2 β ˜ 2 l ˜ 2 < 0 .
Bound rays { n cl β ˜ n 1 , 0 l ˜ l ˜ max ( β ˜ ) .
Tunnelling rays { 0 β ˜ < n cl , ( n cl 2 β ˜ 2 ) 1 2 l ˜ l ˜ max ( β ˜ ) .
l ˜ max 2 ( β ˜ ) = ( n x 2 β ˜ 2 ) ρ x 2 ρ N 2 ,
β ˜ min 2 = { max { 0 , ρ j + 1 2 n j + 1 2 ρ i 2 n i 2 ρ j + 1 2 ρ i 2 } ; { i = 1 N 1 , j = i N 1 , 0 ; i = N .
r rad = l ˜ ρ N ( n cl 2 β ˜ 2 ) 1 2 .
T = 1 power of the reflected ray power of the incident ray .
γ = T z p ,
P ( z ) = P ( 0 ) exp ( γz ) .
n 2 ( r ) = { δ ( r ρ N ) + n N 2 ; r ρ N , n cl 2 ; r ρ N ,
δ = d n 2 ( r ) d r | r = ρ N = n N 2 n N 1 2 ρ N ρ N 1 ,
d 2 Ψ d r 2 + 1 r d r + k r 2 ( r ) Ψ = 0 ,
k r ( r ) = k [ n 2 ( r ) β ˜ 2 l ˜ 2 ρ N 2 r 2 ] 1 2 .
d 2 Φ d r 2 + k 2 [ n 2 ( r ) β ˜ 2 ( l ˜ 2 1 4 k 2 ρ N 2 ) ρ N 2 r 2 ] Φ = 0 ,
d 2 Φ d r 2 + k r 2 ( r ) Φ = 0 .
d 2 Φ d ξ 2 ξ Φ = 0 ,
Φ = C [ ξ ( r ) k r 2 ( r ) ] 1 4 [ Ai ( ξ ( r ) ) + ι Bi ( ξ ( r ) ) ] ,
ξ ( r ) = { [ 3 2 r k r ( r ) d r ] 2 3 ; k r 2 ( r ) > 0 , + [ 3 2 r k r ( r ) d r ] 2 3 ; k r 2 ( r ) < 0 .
ξ 1 ( r ) = { [ 3 2 r r tp k r 1 ( r ) d r ] 2 3 ; k r 1 2 ( r ) > 0 + [ 3 2 r tp r k r 1 ( r ) d r ] 2 3 ; k r 1 2 ( r ) < 0 if r ρ N ,
ξ 2 ( r ) = { [ 3 2 r rad r k r 2 ( r ) d r ] 2 3 ; k r 2 2 ( r ) > 0 + [ 3 2 r r rad k r 2 ( r ) d r ] 2 3 ; k r 2 2 ( r ) < 0 if r ρ N ,
k r 1 ( r ) = { k [ n 2 ( r ) β ˜ 2 l ˜ 2 ρ N 2 r 2 ] 1 2 ; n 2 ( r ) > β ˜ 2 + l ˜ 2 ρ N 2 r 2 ι k [ β ˜ 2 + l ˜ 2 ρ N 2 r 2 n 2 ( r ) ] 1 2 ; n 2 ( r ) < β ˜ 2 + l ˜ 2 ρ N 2 r 2 if r ρ N ,
k r 2 ( r ) = { k ( n cl 2 β ˜ 2 l ˜ 2 ρ N 2 r 2 ) 1 2 ; n cl 2 > β ˜ 2 + l ˜ 2 ρ N 2 r 2 ι k ( β ˜ 2 + l ˜ 2 ρ N 2 r 2 n cl 2 ) 1 2 ; n cl 2 < β ˜ 2 + l ˜ 2 ρ N 2 r 2 if r ρ N .
a r 3 + b r 2 + c = 0 ,
a = δ ,
b = δ ρ N + n N 2 β ˜ 2 ,
c = l ˜ 2 ρ N 2 ,
r tp = 1 3 a [ b + 1 2 1 3 ( L + 3 M ) ] ,
L = ( F 2 + G 2 ) 1 6 cos [ 1 3 arctan ( G F ) ] ,
M = ( F 2 + G 2 ) 1 6 sin [ 1 3 arctan ( G F ) ] ,
F = 2 b 3 27 a 2 c ,
G = 3 3 ( 4 a 2 b 3 c 27 a 4 c 2 ) 1 2 .
Φ i ( r ) = A [ ξ 1 ( r ) k r 1 2 ( r ) ] 1 4 [ Ai ( ξ 1 ( r ) ) + ι Bi ( ξ 1 ( r ) ) ] ,
Φ r ( r ) = R [ ξ 1 ( r ) k r 1 2 ( r ) ] 1 4 [ Ai ( ξ 1 ( r ) ) + ι Bi ( ξ 1 ( r ) ) ] ,
Φ t ( r ) = S [ ξ 2 ( r ) k r 2 2 ( r ) ] 1 4 [ Ai ( ξ 2 ( r ) ) + ι Bi ( ξ 2 ( r ) ) ] ,
Φ i ( r ) | r = ρ N + Φ r ( r ) | r = ρ N = Φ t ( r ) | r = ρ N ,
d d r [ Φ i ( r ) + Φ r ( r ) ] | r = ρ N = d Φ t ( r ) d r | r = ρ N .
T = 1 R A 2 ,
T = 4 π 2 C 2 X 2 + Y 2 ,
X = A 1 A 2 + B 1 B 2 + A 1 ( C 1 A 2 C 2 A 2 ) B 1 ( C 1 B 2 C 2 B 2 ) ,
Y = A 1 B 2 + B 1 A 2 + B 1 ( C 1 A 2 C 2 A 2 ) A 1 ( C 1 B 2 C 2 B 2 ) ,
A 1 Ai ( ξ 1 ) , A 2 Ai ( ξ 2 ) , B 1 Bi ( ξ 1 ) , and B 2 Bi ( ξ 2 ) ,
ξ 1 = { [ 3 2 ρ N r tp k r 1 ( r ) d r ] 2 3 ; k r 1 2 > 0 , + [ 3 2 r tp ρ N k r 1 ( r ) d r ] 2 3 ; k r 1 2 < 0 ,
ξ 2 = { [ 3 2 k ρ N { ( n cl 2 β ˜ 2 l ˜ 2 ) 1 2 l ˜ arccos [ l ˜ ( n cl 2 β ˜ 2 ) 1 2 ] } ] 2 3 ; k r 2 2 > 0 , + [ 3 2 k ρ N { l ˜ ln [ l ˜ + ( β ˜ 2 + l ˜ 2 n cl 2 ) 1 2 ( n cl 2 β ˜ 2 ) 1 2 ] ( β ˜ 2 + l ˜ 2 n cl 2 ) 1 2 } ] 2 3 ; k r 2 2 < 0 ,
C 1 = ( L 2 L 1 ) M 1 , C 2 = M 2 M 1 ,
M 1 = ξ 1 = k r 1 ( ξ 1 ) 1 2 , M 2 = ξ 2 = k r 2 ( ξ 2 ) 1 2 ,
L 1 = 1 4 ( ξ 1 ξ 1 2 k r 1 k r 1 ) = k r 1 4 ( ξ 1 ) 3 2 k r 1 2 k r 1 , L 2 = 1 4 ( ξ 2 ξ 2 2 k r 2 k r 2 ) = k r 2 4 ( ξ 2 ) 3 2 k r 2 2 k r 2 ,
k r 1 d k r 1 ( r ) d r | r = ρ N = { k l ˜ 2 ρ N 2 r 3 + 1 2 d n 2 ( r ) d r [ n 2 ( r ) β ˜ 2 l ˜ 2 ρ N 2 r 2 ] 1 2 | r = ρ N = k l ˜ 2 ρ N + δ 2 ( n N 2 β ˜ 2 l ˜ 2 ) 1 2 ; k r 1 2 > 0 , ι k l ˜ 2 ρ N 2 r 3 1 2 d n 2 ( r ) d r [ β ˜ 2 + l ˜ 2 ρ N 2 r 2 n 2 ( r ) ] 1 2 | r = ρ N = ι k l ˜ 2 ρ N δ 2 ( β ˜ 2 + l ˜ 2 n N 2 ) 1 2 ; k r 1 2 < 0 ,
k r 2 d k r 2 ( r ) d r | r = ρ N = { k l ˜ 2 ρ N 2 r 3 ( n cl 2 β ˜ 2 l ˜ 2 ρ N 2 r 2 ) 1 2 | r = ρ N = k l ˜ 2 ρ N ( n cl 2 β ˜ 2 l ˜ 2 ) 1 2 ; k r 2 2 > 0 , ι k l ˜ 2 ρ N 2 r 3 ( β ˜ 2 + l ˜ 2 ρ N 2 r 2 n cl 2 ) 1 2 | r = ρ N = ι k l ˜ 2 ρ N ( β ˜ 2 + l ˜ 2 n N 2 ) 1 2 ; k r 2 2 < 0 .
T = 4 π K 2 Z 2 + Y 2 ,
Z = B 2 + K 1 A 2 K 2 A 2 ,
Y = A 2 K 1 B 2 + K 2 B 2 ,
K 1 = L 2 k r 1 , K 2 = M 2 k r 1 .
Φ i ( r ) = A exp ( ι k r 1 r ) ,
Φ r ( r ) = R exp ( ι k r 1 r ) .
l ͂ max 2 ( β ͂ ) = 4 27 δ 2 ρ N 2 [ δ ρ N ( β ͂ 2 n N 2 ) ] 3 ,
T = i T F i exp [ 2 r tp r rad k r ( r ) d r ] ,
T tunnel = 4 k r x k r x + k 2 ( n x 2 n x + 1 2 ) i = x + 1 N 4 k r i k r i + k r i 2 + k r i + 2 + 2 k r i k r i + exp ( ζ 1 + ζ 2 ) ,
k r i = k ( n i 2 β ͂ 2 l ͂ 2 ρ N 2 ρ i 2 ) 1 / 2 ,
k r i + = k ( n i + 1 2 β ͂ 2 l ͂ 2 ρ N 2 ρ i 2 ) 1 / 2 , ( n N + 1 = n cl )
ζ 1 = 2 k ρ N i = x + 1 N ( { l ͂ ln [ l ͂ ρ N ρ i 1 + ( β ͂ 2 + l ͂ 2 ρ N 2 ρ i 1 2 n i 2 ) 1 / 2 ( n i 2 β ͂ 2 ) 1 / 2 ] [ l ͂ 2 + ρ i 1 2 ρ N 2 ( β ͂ 2 n i 2 ) ] 1 / 2 } { l ͂ ln [ l ͂ ρ N ρ i + ( β ͂ 2 + l ͂ 2 ρ N 2 ρ i 2 n i 2 ) 1 / 2 ( n i 2 β ͂ 2 ) 1 / 2 ] [ l ͂ 2 + ρ i 2 ρ N 2 ( β ͂ 2 n i 2 ) ] 1 / 2 } ) ,
ζ 2 = 2 k ρ N { l ͂ ln [ l ͂ + ( β ͂ 2 + l ͂ 2 n cl 2 ) 1 / 2 ( n cl 2 β ͂ 2 ) 1 / 2 ] ( β ͂ 2 + l ͂ 2 n cl 2 ) 1 / 2 } .
0 4 k r i k r i + k r i 2 + k r i + 2 + 2 k r i k r i + 1 ,
0 4 k r x k r x + k 2 ( n x 2 n x + 1 2 ) 2 ,
T refr = 4 k r N k r N + ( k r N + k r N + ) 2
= 4 [ ( n N 2 β ͂ 2 l ͂ 2 ) ( n cl 2 β ͂ 2 l ͂ 2 ) ] 1 / 2 ( n N 2 β ͂ 2 l ͂ 2 ) + ( n cl 2 β ͂ 2 l ͂ 2 ) + 2 [ ( n N 2 β ͂ 2 l ͂ 2 ) ( n cl 2 β ͂ 2 l ͂ 2 ) ] 1 / 2 .
n MSI , i = n GI ( r ) | r = ρ i 1 i ( ρ 0 = 0 ) .

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