Abstract

In this paper, circularly curved inhomogeneous waveguides are transformed into straight inhomogeneous waveguides first by a conformal mapping. Then, the differential transfer matrix method is introduced and adopted to deduce the exact dispersion relation for modes. This relation itself is complex and difficult to solve, but it can be approximated by a simpler nonlinear equation in practical applications, which is close to the exact relation and quite easy to analyze. Afterward, optimized asymptotic solutions are obtained and act as initial guesses for the following Newton’s iteration. Finally, very accurate solutions are achieved in the numerical experiment.

© 2013 Optical Society of America

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References

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  1. A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).
  2. A. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
    [CrossRef]
  3. J. Hu and C. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photon. 1, 58–106 (2009).
    [CrossRef]
  4. S. Lee, Y. Chung, L. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
    [CrossRef]
  5. L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus Ltd., 1977).
  6. F. Sporleder and H. G. Unger, Waveguides Tapers Transitions and Couplers (IEE, Peter Peregrinus Ltd., 1979).
  7. S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996).
    [CrossRef]
  8. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
    [CrossRef]
  9. T. Yamamoto and M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by finite-element method,” J. Lightwave Technol. 11, 1579–1583 (1993).
    [CrossRef]
  10. W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, “Scalar BPM analyses of TE and TM polarized fields in bent waveguides,” IEEE Trans. Antennas Propag. 51, 1185–1198 (2003).
    [CrossRef]
  11. R. Pregla, “The method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol. 14, 634–639 (1996).
    [CrossRef]
  12. D. Marcuse, “Bending loss of the asymmetric slab waveguide,” Bell Syst. Tech. J. 50, 2551–2563 (1971).
    [CrossRef]
  13. E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
    [CrossRef]
  14. R. Jedidi and R. Pierre, “Efficient analytical and numerical methods for the computation of bent loss in planar waveguides,” J. Lightwave Technol. 23, 2278–2284 (2005).
    [CrossRef]
  15. W. Kim and C. Kim, “Radiation losses of bent planar waveguides,” Fiber Integr. Opt. 21, 219–232 (2002).
    [CrossRef]
  16. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
    [CrossRef]
  17. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
    [CrossRef]
  18. A. Melloni, F. Carniel, R. Costa, and M. Martinelli, “Determination of bend mode characteristics in dielectric waveguides,” J. Lightwave Technol. 19, 571–577 (2001).
    [CrossRef]
  19. K. Thyagarajan, M. R. Shenoy, and A. K. Ghatak, “Accurate numerical method for the calculation of bending loss in optical waveguides using a matrix approach,” Opt. Lett. 12, 296–298 (1987).
    [CrossRef]
  20. W. Berglund and A. Gopinath, “WKB analysis of bend losses in optical waveguides,” J. Lightwave Technol. 18, 1161–1166 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  25. J. Zhu and Y. Lu, “Leaky modes of slab waveguides: asymptotic solutions,” J. Lightwave Technol. 24, 1619–1623 (2006).
    [CrossRef]
  26. J. Zhu and Z. Shen, “Dispersion relation of leaky modes in nonhomogeneous waveguides and its applications,” J. Lightwave Technol. 29, 3230–3236 (2011).
    [CrossRef]
  27. J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications—asymptotic solutions,” Microelectron. Reliab. 48, 555–562 (2008).
    [CrossRef]
  28. D. Sarafyan, “Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge–Kutta formulae and upgrading of their order,” Comput. Math. Appl. 28, 353–384 (1994).
    [CrossRef]
  29. W. Kahan and R. Li, “Composition constants for raising the orders of unconventional schemes for ordinary differential equations,” Math. Comput. 66, 1089–1099 (1997).
    [CrossRef]
  30. J. C. Butcher, “Numerical methods for ordinary differential equations in the 20th century,” J. Comput. Appl. Math. 125, 1–29 (2000).
    [CrossRef]
  31. R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On the LambertW function,” Adv. Comput. Math. 5, 329–359 (1996).
    [CrossRef]

2011 (1)

2009 (1)

2008 (1)

J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications—asymptotic solutions,” Microelectron. Reliab. 48, 555–562 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (3)

2003 (2)

W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, “Scalar BPM analyses of TE and TM polarized fields in bent waveguides,” IEEE Trans. Antennas Propag. 51, 1185–1198 (2003).
[CrossRef]

S. Khorasani and K. Mehrany, “Differential transfer-matrix method for solution of one-dimensional linear nonhomogeneous optical structures,” J. Opt. Soc. Am. B 20, 91–96 (2003).
[CrossRef]

2002 (2)

C. Kim, Y. Kim, and W. Kim, “Leaky modes of circular slab waveguides: modified Airy functions,” IEEE J. Sel. Top. Quantum Electron. 8, 1239–1245 (2002).
[CrossRef]

W. Kim and C. Kim, “Radiation losses of bent planar waveguides,” Fiber Integr. Opt. 21, 219–232 (2002).
[CrossRef]

2001 (1)

2000 (2)

W. Berglund and A. Gopinath, “WKB analysis of bend losses in optical waveguides,” J. Lightwave Technol. 18, 1161–1166 (2000).
[CrossRef]

J. C. Butcher, “Numerical methods for ordinary differential equations in the 20th century,” J. Comput. Appl. Math. 125, 1–29 (2000).
[CrossRef]

1997 (1)

W. Kahan and R. Li, “Composition constants for raising the orders of unconventional schemes for ordinary differential equations,” Math. Comput. 66, 1089–1099 (1997).
[CrossRef]

1996 (3)

R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On the LambertW function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

R. Pregla, “The method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol. 14, 634–639 (1996).
[CrossRef]

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996).
[CrossRef]

1995 (1)

S. Lee, Y. Chung, L. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

1994 (1)

D. Sarafyan, “Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge–Kutta formulae and upgrading of their order,” Comput. Math. Appl. 28, 353–384 (1994).
[CrossRef]

1993 (1)

T. Yamamoto and M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by finite-element method,” J. Lightwave Technol. 11, 1579–1583 (1993).
[CrossRef]

1987 (1)

1985 (1)

A. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[CrossRef]

1976 (1)

1975 (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

1971 (1)

D. Marcuse, “Bending loss of the asymmetric slab waveguide,” Bell Syst. Tech. J. 50, 2551–2563 (1971).
[CrossRef]

1969 (1)

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[CrossRef]

Ahn, B. H.

W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, “Scalar BPM analyses of TE and TM polarized fields in bent waveguides,” IEEE Trans. Antennas Propag. 51, 1185–1198 (2003).
[CrossRef]

Berglund, W.

Butcher, J. C.

J. C. Butcher, “Numerical methods for ordinary differential equations in the 20th century,” J. Comput. Appl. Math. 125, 1–29 (2000).
[CrossRef]

Carniel, F.

Chang, D. C.

L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus Ltd., 1977).

Chen, Z.

J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications—asymptotic solutions,” Microelectron. Reliab. 48, 555–562 (2008).
[CrossRef]

Chung, Y.

S. Lee, Y. Chung, L. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Coldren, L.

S. Lee, Y. Chung, L. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Corless, R.

R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On the LambertW function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Costa, R.

Ctyroky, J.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

Dagli, N.

S. Lee, Y. Chung, L. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Eghlidi, M.

Ghatak, A.

A. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[CrossRef]

Ghatak, A. K.

Gonnet, G.

R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On the LambertW function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Gopinath, A.

W. Berglund and A. Gopinath, “WKB analysis of bend losses in optical waveguides,” J. Lightwave Technol. 18, 1161–1166 (2000).
[CrossRef]

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996).
[CrossRef]

Hammer, M.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

Hare, D.

R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On the LambertW function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

Hiremath, K. R.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

Hu, J.

Jedidi, R.

Jeffrey, D.

R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On the LambertW function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Kahan, W.

W. Kahan and R. Li, “Composition constants for raising the orders of unconventional schemes for ordinary differential equations,” Math. Comput. 66, 1089–1099 (1997).
[CrossRef]

Kang, M.

W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, “Scalar BPM analyses of TE and TM polarized fields in bent waveguides,” IEEE Trans. Antennas Propag. 51, 1185–1198 (2003).
[CrossRef]

Khorasani, S.

Kim, C.

C. Kim, Y. Kim, and W. Kim, “Leaky modes of circular slab waveguides: modified Airy functions,” IEEE J. Sel. Top. Quantum Electron. 8, 1239–1245 (2002).
[CrossRef]

W. Kim and C. Kim, “Radiation losses of bent planar waveguides,” Fiber Integr. Opt. 21, 219–232 (2002).
[CrossRef]

Kim, S.

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. 14, 2085–2092 (1996).
[CrossRef]

Kim, W.

W. Kim and C. Kim, “Radiation losses of bent planar waveguides,” Fiber Integr. Opt. 21, 219–232 (2002).
[CrossRef]

C. Kim, Y. Kim, and W. Kim, “Leaky modes of circular slab waveguides: modified Airy functions,” IEEE J. Sel. Top. Quantum Electron. 8, 1239–1245 (2002).
[CrossRef]

Kim, Y.

C. Kim, Y. Kim, and W. Kim, “Leaky modes of circular slab waveguides: modified Airy functions,” IEEE J. Sel. Top. Quantum Electron. 8, 1239–1245 (2002).
[CrossRef]

Knuth, D.

R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On the LambertW function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Koshiba, M.

T. Yamamoto and M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by finite-element method,” J. Lightwave Technol. 11, 1579–1583 (1993).
[CrossRef]

Kuester, E. F.

L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus Ltd., 1977).

Lee, S.

S. Lee, Y. Chung, L. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

Lewin, L.

L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus Ltd., 1977).

Li, R.

W. Kahan and R. Li, “Composition constants for raising the orders of unconventional schemes for ordinary differential equations,” Math. Comput. 66, 1089–1099 (1997).
[CrossRef]

Love, J.

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

Lu, Y.

Marcatili, E. A. J.

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[CrossRef]

Marcuse, D.

D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
[CrossRef]

D. Marcuse, “Bending loss of the asymmetric slab waveguide,” Bell Syst. Tech. J. 50, 2551–2563 (1971).
[CrossRef]

Martinelli, M.

Mehrany, K.

Melloni, A.

Menyuk, C.

Pierre, R.

Pregla, R.

R. Pregla, “The method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol. 14, 634–639 (1996).
[CrossRef]

Prkna, L.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

Rashidian, B.

Sarafyan, D.

D. Sarafyan, “Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge–Kutta formulae and upgrading of their order,” Comput. Math. Appl. 28, 353–384 (1994).
[CrossRef]

Sarrafi, P.

Shen, Z.

Shenoy, M. R.

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

Song, G. H.

W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, “Scalar BPM analyses of TE and TM polarized fields in bent waveguides,” IEEE Trans. Antennas Propag. 51, 1185–1198 (2003).
[CrossRef]

Song, W. J.

W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, “Scalar BPM analyses of TE and TM polarized fields in bent waveguides,” IEEE Trans. Antennas Propag. 51, 1185–1198 (2003).
[CrossRef]

Sporleder, F.

F. Sporleder and H. G. Unger, Waveguides Tapers Transitions and Couplers (IEE, Peter Peregrinus Ltd., 1979).

Stoffer, R.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

Tang, S.

J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications—asymptotic solutions,” Microelectron. Reliab. 48, 555–562 (2008).
[CrossRef]

Thyagarajan, K.

Unger, H. G.

F. Sporleder and H. G. Unger, Waveguides Tapers Transitions and Couplers (IEE, Peter Peregrinus Ltd., 1979).

Yamamoto, T.

T. Yamamoto and M. Koshiba, “Numerical analysis of curvature loss in optical waveguides by finite-element method,” J. Lightwave Technol. 11, 1579–1583 (1993).
[CrossRef]

Zareian, N.

Zhu, J.

J. Zhu and Z. Shen, “Dispersion relation of leaky modes in nonhomogeneous waveguides and its applications,” J. Lightwave Technol. 29, 3230–3236 (2011).
[CrossRef]

J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications—asymptotic solutions,” Microelectron. Reliab. 48, 555–562 (2008).
[CrossRef]

J. Zhu and Y. Lu, “Leaky modes of slab waveguides: asymptotic solutions,” J. Lightwave Technol. 24, 1619–1623 (2006).
[CrossRef]

Adv. Comput. Math. (1)

R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, “On the LambertW function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

D. Marcuse, “Bending loss of the asymmetric slab waveguide,” Bell Syst. Tech. J. 50, 2551–2563 (1971).
[CrossRef]

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[CrossRef]

Comput. Math. Appl. (1)

D. Sarafyan, “Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge–Kutta formulae and upgrading of their order,” Comput. Math. Appl. 28, 353–384 (1994).
[CrossRef]

Fiber Integr. Opt. (1)

W. Kim and C. Kim, “Radiation losses of bent planar waveguides,” Fiber Integr. Opt. 21, 219–232 (2002).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

S. Lee, Y. Chung, L. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

C. Kim, Y. Kim, and W. Kim, “Leaky modes of circular slab waveguides: modified Airy functions,” IEEE J. Sel. Top. Quantum Electron. 8, 1239–1245 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, “Scalar BPM analyses of TE and TM polarized fields in bent waveguides,” IEEE Trans. Antennas Propag. 51, 1185–1198 (2003).
[CrossRef]

J. Comput. Appl. Math. (1)

J. C. Butcher, “Numerical methods for ordinary differential equations in the 20th century,” J. Comput. Appl. Math. 125, 1–29 (2000).
[CrossRef]

J. Lightwave Technol. (8)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Math. Comput. (1)

W. Kahan and R. Li, “Composition constants for raising the orders of unconventional schemes for ordinary differential equations,” Math. Comput. 66, 1089–1099 (1997).
[CrossRef]

Microelectron. Reliab. (1)

J. Zhu, Z. Chen, and S. Tang, “Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications—asymptotic solutions,” Microelectron. Reliab. 48, 555–562 (2008).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (2)

A. Ghatak, “Leaky modes in optical waveguides,” Opt. Quantum Electron. 17, 311–321 (1985).
[CrossRef]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

Other (3)

L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures (Peter Peregrinus Ltd., 1977).

F. Sporleder and H. G. Unger, Waveguides Tapers Transitions and Couplers (IEE, Peter Peregrinus Ltd., 1979).

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

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Figures (1)

Fig. 1.
Fig. 1.

Comparing the initial solutions with the modes solved by dispersion relation for r0/d=13000: the initial asymptotic solutions from Eq. (28) are marked by “+”, asymptotic modes from Eq. (29) are marked by “x”, and the accurate modes are marked by “o”.

Tables (6)

Tables Icon

Table 1. Asymptotic Solutions β01 and β03 and Relative Errors Re01 and Re03 for ξ=D/2 when r0/d=13000

Tables Icon

Table 2. Asymptotic Solutions β01 and β03 and Relative Errors Re01 and Re03 for ξ=D/4 when r0/d=13000

Tables Icon

Table 3. Asymptotic Solutions β01 and β03 and Relative Errors Re01 and Re03 for ξ=u0 when r0/d=13000

Tables Icon

Table 4. Asymptotic Solutions β01 and β03 and Relative Errors Re01 and Re03 for ξ=D/5 when r0/d=13000

Tables Icon

Table 5. Asymptotic Solutions β01 and β03 and Relative Errors Re01 and Re03 for ξ=u0 when r0/d=9000

Tables Icon

Table 6. Exact Modes βe

Equations (37)

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

1rr(rΨr)β2r02r2Ψ+κ02n2(r)Ψ=0,
n(r)={n1,r<r1,n0(r),r1<r<r2,n2,r>r2.
{u=r1ln(rr1),v=r1Φ,
d2Ψdu2+κ02(n2(r(u))exp(2u/r1)N2)Ψ=0,
N=βr0k0r1.
d2Ψdu2+k2(u)Ψ=0.
Ψ(u)=A(u)exp(ik(u)u)+B(u)exp(ik(u)u).
A(u)exp(ik(u)u)+B(u)exp(ik(u)u)=A(u+Δu)exp(ik(u+Δu)u)+B(u+Δu)exp(ik(u+Δu)u),
=ik(u)A(u)exp(ik(u)u)+ik(u)B(u)exp(ik(u)u)ik(u+Δu)A(u+Δu)exp(ik(u+Δu)u)+ik(u+Δu)B(u+Δu)exp(ik(u+Δu)u).
dA(u)duexp(ik(u)u)+dB(u)duexp(ik(u)u)=ik(u)u(A(u)exp(ik(u)u)B(u)exp(ik(u)u)).
=dA(u)duexp(ik(u)u)dB(u)duexp(ik(u)u)(k(u)k(u)exp(ik(u)u)+ik(u)uexp(ik(u)u))A(u)+(k(u)k(u)exp(ik(u)u)+ik(u)uexp(ik(u)u))B(u).
ddu[A(u)B(u)]=U(u)[A(u)B(u)],
U(u)=k(u)2k(u)[1+2ik(u)uexp(2ik(u)u)exp(2ik(u)u)12ik(u)u].
[A(D+)B(D+)]=T0D+[A(0)B(0)].
{dΨdu=ik1Ψ,u=0,dΨdu=ik2Ψu=D,
{T00+=[k0(0)+k1(0)2k0(0)k0(0)k1(0)2k0(0)k0(0)k1(0)2k0(0)k0(0)+k1(0)2k0(0)],TDD+=[k2(D)+k0(D)2k2(D)exp(i(k2(D)k0(D))D)k2(D)k0(D)2k2(D)exp(i(k2(D)+k0(D))D)k2(D)k0(D)2k2(D)exp(i(k2(D)+k0(D))D)k2(D)+k0(D)2k2(D)exp(i(k2(D)k0(D))D)].
[A(D)B(D)]=exp[0+DU0(u)du][A(0+)B(0+)]=exp(M)[A(0+)B(0+)],
U0(u)=ik0(u)u[1001]k0(u)2k0(u)×[1exp(2ik0(u)u)exp(2ik0(u)u)1].
[A(D+)B(D+)]=TDD+·T0+D·T00+[A(0)B(0)],
U0(u)U1(u)=ik0(u)u[1001],
k0(0)+k1(0)k0(0)k1(0)·k0(D)+k2(D)k0(D)k2(D)=exp(2ik0(D)D2i0Dk0(u)udu)=exp(2i0Dk0(u)du).
s0exp(i20Dk0(u)du)=(δ1δ3)14((k0(0)+k1(0))(k0(D)+k2(D))1/2,
s0exp(iDk0(ξ)2)=2k0(ξ)(δ1δ3)14(1+b2k02(ξ)+b4k04(ξ)+),
{ξ1=κ02n02(r(ξ))exp(2ξr1)κ02n12(r(0)),ξ0=κ02n02(r(ξ))exp(2ξr1)κ02n02(r(0)),ξ2=κ02n02(r(ξ))exp(2ξr1)κ02n02(r(D))exp(2Dr1),ξ3=κ02n02(r(ξ))exp(2ξr1)κ02n22(r(D))exp(2Dr1),
is0D4(δ1δ3)14=b0k0(ξ)exp(b0k0(ξ))(1+b2k02(ξ)+b4k04(ξ)+).
{b0k0(ξ)exp(b0k0(ξ))(1+b2k02(ξ)+b4k04(ξ)+)=Wexp(W),W=a0k0(ξ)+a2k02(ξ)+a3k03(ξ)+a4k04(ξ)+a5k05(ξ)+,
W=LambertW(p,iqD4(δ1δ3)14),
k0(ξ)W/a0,
k0(ξ)W/a0a2a0/W2a3a02/W3.
βr1r0(κ02n02(r(ξ))exp(2ξ/r1)+4LambertW2(p,iqD4(δ1δ3)14)D2)12.
Dk0(ξ)=0Dk0(u)du,
k0(u)=iκ0(N2n02(r(u))exp(2u/r1))1/2=iκ0N(1n02(r(u))exp(2u/r1)2N2n04(r(u))exp(4u/r1)8N4),
0Dk0(u)du=iκ0N(D0Dn02(r(u))exp(2u/r1)2N2du0Dn04(r(u))exp(4u/r1)8N4du).
Dn02(r(u0))exp(2u0/r1)=0Dn02(r(u))exp(2u/r1)du.
F(N)=k0(0,N)+k1(0,N)k0(0,N)k1(0,N)·k0(D,N)+k2(D,N)k0(D,N)k2(D,N)exp(2i0Dk0(u,N)du).
R(N)=|v1(N)exp(0DU0(u)du)v2(N)|,
{v1(N)=[(k2(D)+k0(D))exp(i(k2(D)k0(D))D),(k2(D)k0(D))exp(i(k2(D)+k0(D))D)],v2(N)=[k0(0)+k1(0),k0(0)k1(0)]T.

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