Abstract

We have introduced a semi-analytical IS technique suitable for multipole, rational function reflection coefficients, and used it for the design of dispersion-engineered planar waveguides. The technique is used to derive extensive dispersion maps, including higher dispersion coefficients, corresponding to three-, five- and seven-pole reflection coefficients. It is shown that common features of dispersion-engineered waveguides such as refractive-index trenches, rings and oscillations come naturally from this approach when the magnitude of leaky poles in increased. Increasing the number of poles is shown to offer a small but measureable change in higher order dispersion with designs dominated by a three pole design with a leaky pole pair of the smallest modulus.

© 2015 Optical Society of America

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References

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  1. S. Ramachandran, “Dispersion-tailored few-mode fibers : a versatile platform for in-fiber photonic devices,” J. Lightwave Technol. 23(11), 3426–3443 (2005).
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  3. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
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    [Crossref]
  5. M. Takahashi, R. Sugizaki, J. Hiroishi, M. Tadakuma, Y. Taniguchi, and T. Yagi, “Low-loss and low-dispersion-slope highly nonlinear fibers,” J. Lightwave Technol. 23(11), 3615–3624 (2005).
    [Crossref]
  6. M. Onishi, T. Okuno, T. Kashiwada, S. Ishikawa, N. Akasaka, and M. Nishimura, “Highly nonlinear dispersion-shifted fibers and their application to broadband wavelength converter,” Opt. Fiber Technol. 4(2), 204–214 (1998).
    [Crossref]
  7. L. Gruner-Nielsen, M. Wandel, P. Kristensen, C. Jorgensen, L. V. Jorgensen, B. Edvold, B. Palsdottir, and D. Jakobsen, “Dispersion-compensating fibers,” J. Lightwave Technol. 23(11), 3566–3579 (2005).
    [Crossref]
  8. R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” J. Quantum Electron. 35(8), 1105–1115 (1999).
    [Crossref]
  9. J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Quantum Electron. 37(2), 165–173 (2001).
    [Crossref]
  10. M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. 12(5), 498–500 (2000).
    [Crossref]
  11. M. N. Zervas and M. K. Durkin, “Physical insights into inverse-scattering profiles and symmetric dispersionless FBG designs,” Opt. Express 21(15), 17472–17483 (2013).
    [PubMed]
  12. S. Lakshmanasamy and A. K. Jordan, “Design of wide-core planar waveguides by an inverse scattering method,” Opt. Lett. 14(8), 411–413 (1989).
    [Crossref] [PubMed]
  13. A. K. Jordan and S. Lakshmanasamy, “Inverse scattering theory applied to the design of single-mode planar optical waveguides,” J. Opt. Soc. Am. A 6(8), 1206–1212 (1989).
    [Crossref]
  14. C. Papachristos and P. Frangos, “Synthesis of single-and multi-mode planar optical waveguides by a direct numerical solution of the Gel’fand-Levitan-Marchenko integral equation,” Opt. Commun. 203(1-2), 27–37 (2002).
    [Crossref]
  15. I. Hirsh, M. Horowitz, and A. Rosenthal, “Design of planar waveguides with prescribed mode-profile using inverse scattering theory,” J. Quantum Electron. 45(9), 1133–1141 (2009).
    [Crossref]
  16. M. Cvijetic, “Dual-mode optical fibres with zero intermodal dispersion,” Opt. Quantum Electron. 16(4), 307–317 (1984).
    [Crossref]
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    [Crossref]
  18. S. Ahn and A. K. Jordan, “Profile inversion of simple plasmas and nonuniform regions: three-pole reflection coefficient,” IEEE Trans. Antenn. Propag. 24(6), 879–882 (1976).
    [Crossref]
  19. M. Reilly and A. K. Jordan, “The applicability of an inverse method for reconstruction of electron-density profiles,” IEEE Trans. Antenn. Propag. 29(2), 245–252 (1981).
    [Crossref]
  20. I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13(3), 371–393 (1960).
    [Crossref]
  21. J. Xia, A. K. Jordan, and J. A. Kong, “Inverse-scattering view of modal structures in inhomogeneous optical waveguides,” J. Opt. Soc. Am. A 9(5), 740–748 (1992).
    [Crossref]
  22. P. Deift and E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. 32(2), 121–251 (1979).
    [Crossref]
  23. K. R. Pechenick and J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand-Levitan equation,” J. Math. Phys. 22(7), 1513–1516 (1981).
    [Crossref]
  24. D. B. Ge, A. K. Jordan, and L. S. Tamil, “Numerical inverse scattering theory for the design of planar optical waveguides,” J. Opt. Soc. Am. A 11(11), 2809 (1994).
    [Crossref]
  25. K. R. Pechenick and J. M. Cohen, “Exact solutions to the valley problem in inverse scattering,” J. Math. Phys. 24(2), 406–409 (1983).
    [Crossref]
  26. A. Akritas and P. Vigklas, “Counting the number of real roots in an interval with Vincent’s theorem,” Bull. Math. Soc. Sci. Math. Roum. 53, 201–211 (2010).
  27. K. Case, “On wave propagation in inhomogeneous media,” J. Math. Phys. 13(3), 360 (1972).
    [Crossref]
  28. Maplesoft, “MAPLE,” (2012).

2013 (1)

2010 (1)

A. Akritas and P. Vigklas, “Counting the number of real roots in an interval with Vincent’s theorem,” Bull. Math. Soc. Sci. Math. Roum. 53, 201–211 (2010).

2009 (1)

I. Hirsh, M. Horowitz, and A. Rosenthal, “Design of planar waveguides with prescribed mode-profile using inverse scattering theory,” J. Quantum Electron. 45(9), 1133–1141 (2009).
[Crossref]

2006 (2)

M. Wandel and P. Kristensen, “Fiber designs for high figure of merit and high slope dispersion compensating fibers,” J. Opt. Fiber Commun. 3(1), 25–60 (2006).
[Crossref]

J. M. Dudley and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

2005 (3)

2002 (1)

C. Papachristos and P. Frangos, “Synthesis of single-and multi-mode planar optical waveguides by a direct numerical solution of the Gel’fand-Levitan-Marchenko integral equation,” Opt. Commun. 203(1-2), 27–37 (2002).
[Crossref]

2001 (1)

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

2000 (1)

M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. 12(5), 498–500 (2000).
[Crossref]

1999 (1)

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” J. Quantum Electron. 35(8), 1105–1115 (1999).
[Crossref]

1998 (1)

M. Onishi, T. Okuno, T. Kashiwada, S. Ishikawa, N. Akasaka, and M. Nishimura, “Highly nonlinear dispersion-shifted fibers and their application to broadband wavelength converter,” Opt. Fiber Technol. 4(2), 204–214 (1998).
[Crossref]

1994 (1)

1992 (1)

1989 (2)

1984 (1)

M. Cvijetic, “Dual-mode optical fibres with zero intermodal dispersion,” Opt. Quantum Electron. 16(4), 307–317 (1984).
[Crossref]

1983 (1)

K. R. Pechenick and J. M. Cohen, “Exact solutions to the valley problem in inverse scattering,” J. Math. Phys. 24(2), 406–409 (1983).
[Crossref]

1981 (2)

M. Reilly and A. K. Jordan, “The applicability of an inverse method for reconstruction of electron-density profiles,” IEEE Trans. Antenn. Propag. 29(2), 245–252 (1981).
[Crossref]

K. R. Pechenick and J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand-Levitan equation,” J. Math. Phys. 22(7), 1513–1516 (1981).
[Crossref]

1979 (2)

P. Deift and E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. 32(2), 121–251 (1979).
[Crossref]

A. K. Jordan and S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126(10), 945–950 (1979).
[Crossref]

1976 (1)

S. Ahn and A. K. Jordan, “Profile inversion of simple plasmas and nonuniform regions: three-pole reflection coefficient,” IEEE Trans. Antenn. Propag. 24(6), 879–882 (1976).
[Crossref]

1972 (1)

K. Case, “On wave propagation in inhomogeneous media,” J. Math. Phys. 13(3), 360 (1972).
[Crossref]

1960 (1)

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13(3), 371–393 (1960).
[Crossref]

Ahn, S.

A. K. Jordan and S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126(10), 945–950 (1979).
[Crossref]

S. Ahn and A. K. Jordan, “Profile inversion of simple plasmas and nonuniform regions: three-pole reflection coefficient,” IEEE Trans. Antenn. Propag. 24(6), 879–882 (1976).
[Crossref]

Akasaka, N.

M. Onishi, T. Okuno, T. Kashiwada, S. Ishikawa, N. Akasaka, and M. Nishimura, “Highly nonlinear dispersion-shifted fibers and their application to broadband wavelength converter,” Opt. Fiber Technol. 4(2), 204–214 (1998).
[Crossref]

Akritas, A.

A. Akritas and P. Vigklas, “Counting the number of real roots in an interval with Vincent’s theorem,” Bull. Math. Soc. Sci. Math. Roum. 53, 201–211 (2010).

Case, K.

K. Case, “On wave propagation in inhomogeneous media,” J. Math. Phys. 13(3), 360 (1972).
[Crossref]

Coen, S.

J. M. Dudley and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

Cohen, J. M.

K. R. Pechenick and J. M. Cohen, “Exact solutions to the valley problem in inverse scattering,” J. Math. Phys. 24(2), 406–409 (1983).
[Crossref]

K. R. Pechenick and J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand-Levitan equation,” J. Math. Phys. 22(7), 1513–1516 (1981).
[Crossref]

Cvijetic, M.

M. Cvijetic, “Dual-mode optical fibres with zero intermodal dispersion,” Opt. Quantum Electron. 16(4), 307–317 (1984).
[Crossref]

Deift, P.

P. Deift and E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. 32(2), 121–251 (1979).
[Crossref]

Dudley, J. M.

J. M. Dudley and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

Durkin, M. K.

M. N. Zervas and M. K. Durkin, “Physical insights into inverse-scattering profiles and symmetric dispersionless FBG designs,” Opt. Express 21(15), 17472–17483 (2013).
[PubMed]

M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. 12(5), 498–500 (2000).
[Crossref]

Edvold, B.

Erdogan, T.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

Feced, R.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” J. Quantum Electron. 35(8), 1105–1115 (1999).
[Crossref]

Frangos, P.

C. Papachristos and P. Frangos, “Synthesis of single-and multi-mode planar optical waveguides by a direct numerical solution of the Gel’fand-Levitan-Marchenko integral equation,” Opt. Commun. 203(1-2), 27–37 (2002).
[Crossref]

Ge, D. B.

Grudinin, A. B.

M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. 12(5), 498–500 (2000).
[Crossref]

Gruner-Nielsen, L.

Hiroishi, J.

Hirsh, I.

I. Hirsh, M. Horowitz, and A. Rosenthal, “Design of planar waveguides with prescribed mode-profile using inverse scattering theory,” J. Quantum Electron. 45(9), 1133–1141 (2009).
[Crossref]

Horowitz, M.

I. Hirsh, M. Horowitz, and A. Rosenthal, “Design of planar waveguides with prescribed mode-profile using inverse scattering theory,” J. Quantum Electron. 45(9), 1133–1141 (2009).
[Crossref]

Ibsen, M.

M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. 12(5), 498–500 (2000).
[Crossref]

Ishikawa, S.

M. Onishi, T. Okuno, T. Kashiwada, S. Ishikawa, N. Akasaka, and M. Nishimura, “Highly nonlinear dispersion-shifted fibers and their application to broadband wavelength converter,” Opt. Fiber Technol. 4(2), 204–214 (1998).
[Crossref]

Jakobsen, D.

Jordan, A. K.

D. B. Ge, A. K. Jordan, and L. S. Tamil, “Numerical inverse scattering theory for the design of planar optical waveguides,” J. Opt. Soc. Am. A 11(11), 2809 (1994).
[Crossref]

J. Xia, A. K. Jordan, and J. A. Kong, “Inverse-scattering view of modal structures in inhomogeneous optical waveguides,” J. Opt. Soc. Am. A 9(5), 740–748 (1992).
[Crossref]

S. Lakshmanasamy and A. K. Jordan, “Design of wide-core planar waveguides by an inverse scattering method,” Opt. Lett. 14(8), 411–413 (1989).
[Crossref] [PubMed]

A. K. Jordan and S. Lakshmanasamy, “Inverse scattering theory applied to the design of single-mode planar optical waveguides,” J. Opt. Soc. Am. A 6(8), 1206–1212 (1989).
[Crossref]

M. Reilly and A. K. Jordan, “The applicability of an inverse method for reconstruction of electron-density profiles,” IEEE Trans. Antenn. Propag. 29(2), 245–252 (1981).
[Crossref]

A. K. Jordan and S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126(10), 945–950 (1979).
[Crossref]

S. Ahn and A. K. Jordan, “Profile inversion of simple plasmas and nonuniform regions: three-pole reflection coefficient,” IEEE Trans. Antenn. Propag. 24(6), 879–882 (1976).
[Crossref]

Jorgensen, C.

Jorgensen, L. V.

Kashiwada, T.

M. Onishi, T. Okuno, T. Kashiwada, S. Ishikawa, N. Akasaka, and M. Nishimura, “Highly nonlinear dispersion-shifted fibers and their application to broadband wavelength converter,” Opt. Fiber Technol. 4(2), 204–214 (1998).
[Crossref]

Kay, I.

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13(3), 371–393 (1960).
[Crossref]

Kong, J. A.

Kristensen, P.

M. Wandel and P. Kristensen, “Fiber designs for high figure of merit and high slope dispersion compensating fibers,” J. Opt. Fiber Commun. 3(1), 25–60 (2006).
[Crossref]

L. Gruner-Nielsen, M. Wandel, P. Kristensen, C. Jorgensen, L. V. Jorgensen, B. Edvold, B. Palsdottir, and D. Jakobsen, “Dispersion-compensating fibers,” J. Lightwave Technol. 23(11), 3566–3579 (2005).
[Crossref]

Lakshmanasamy, S.

Laming, R. I.

M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. 12(5), 498–500 (2000).
[Crossref]

Muriel, M. A.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” J. Quantum Electron. 35(8), 1105–1115 (1999).
[Crossref]

Nishimura, M.

M. Onishi, T. Okuno, T. Kashiwada, S. Ishikawa, N. Akasaka, and M. Nishimura, “Highly nonlinear dispersion-shifted fibers and their application to broadband wavelength converter,” Opt. Fiber Technol. 4(2), 204–214 (1998).
[Crossref]

Okuno, T.

M. Onishi, T. Okuno, T. Kashiwada, S. Ishikawa, N. Akasaka, and M. Nishimura, “Highly nonlinear dispersion-shifted fibers and their application to broadband wavelength converter,” Opt. Fiber Technol. 4(2), 204–214 (1998).
[Crossref]

Onishi, M.

M. Onishi, T. Okuno, T. Kashiwada, S. Ishikawa, N. Akasaka, and M. Nishimura, “Highly nonlinear dispersion-shifted fibers and their application to broadband wavelength converter,” Opt. Fiber Technol. 4(2), 204–214 (1998).
[Crossref]

Palsdottir, B.

Papachristos, C.

C. Papachristos and P. Frangos, “Synthesis of single-and multi-mode planar optical waveguides by a direct numerical solution of the Gel’fand-Levitan-Marchenko integral equation,” Opt. Commun. 203(1-2), 27–37 (2002).
[Crossref]

Pechenick, K. R.

K. R. Pechenick and J. M. Cohen, “Exact solutions to the valley problem in inverse scattering,” J. Math. Phys. 24(2), 406–409 (1983).
[Crossref]

K. R. Pechenick and J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand-Levitan equation,” J. Math. Phys. 22(7), 1513–1516 (1981).
[Crossref]

Ramachandran, S.

Reilly, M.

M. Reilly and A. K. Jordan, “The applicability of an inverse method for reconstruction of electron-density profiles,” IEEE Trans. Antenn. Propag. 29(2), 245–252 (1981).
[Crossref]

Rosenthal, A.

I. Hirsh, M. Horowitz, and A. Rosenthal, “Design of planar waveguides with prescribed mode-profile using inverse scattering theory,” J. Quantum Electron. 45(9), 1133–1141 (2009).
[Crossref]

Skaar, J.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

Sugizaki, R.

Tadakuma, M.

Takahashi, M.

Tamil, L. S.

Taniguchi, Y.

Trubowitz, E.

P. Deift and E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. 32(2), 121–251 (1979).
[Crossref]

Vigklas, P.

A. Akritas and P. Vigklas, “Counting the number of real roots in an interval with Vincent’s theorem,” Bull. Math. Soc. Sci. Math. Roum. 53, 201–211 (2010).

Wandel, M.

M. Wandel and P. Kristensen, “Fiber designs for high figure of merit and high slope dispersion compensating fibers,” J. Opt. Fiber Commun. 3(1), 25–60 (2006).
[Crossref]

L. Gruner-Nielsen, M. Wandel, P. Kristensen, C. Jorgensen, L. V. Jorgensen, B. Edvold, B. Palsdottir, and D. Jakobsen, “Dispersion-compensating fibers,” J. Lightwave Technol. 23(11), 3566–3579 (2005).
[Crossref]

Wang, L.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

Xia, J.

Yagi, T.

Zervas, M. N.

M. N. Zervas and M. K. Durkin, “Physical insights into inverse-scattering profiles and symmetric dispersionless FBG designs,” Opt. Express 21(15), 17472–17483 (2013).
[PubMed]

M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. 12(5), 498–500 (2000).
[Crossref]

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” J. Quantum Electron. 35(8), 1105–1115 (1999).
[Crossref]

Bull. Math. Soc. Sci. Math. Roum. (1)

A. Akritas and P. Vigklas, “Counting the number of real roots in an interval with Vincent’s theorem,” Bull. Math. Soc. Sci. Math. Roum. 53, 201–211 (2010).

Commun. Pure Appl. Math. (2)

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13(3), 371–393 (1960).
[Crossref]

P. Deift and E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. 32(2), 121–251 (1979).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Ibsen, M. K. Durkin, M. N. Zervas, A. B. Grudinin, and R. I. Laming, “Custom design of long chirped Bragg gratings: application to gain-flattening filter with incorporated dispersion compensation,” IEEE Photon. Technol. Lett. 12(5), 498–500 (2000).
[Crossref]

IEEE Trans. Antenn. Propag. (2)

S. Ahn and A. K. Jordan, “Profile inversion of simple plasmas and nonuniform regions: three-pole reflection coefficient,” IEEE Trans. Antenn. Propag. 24(6), 879–882 (1976).
[Crossref]

M. Reilly and A. K. Jordan, “The applicability of an inverse method for reconstruction of electron-density profiles,” IEEE Trans. Antenn. Propag. 29(2), 245–252 (1981).
[Crossref]

J. Lightwave Technol. (3)

J. Math. Phys. (3)

K. R. Pechenick and J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand-Levitan equation,” J. Math. Phys. 22(7), 1513–1516 (1981).
[Crossref]

K. Case, “On wave propagation in inhomogeneous media,” J. Math. Phys. 13(3), 360 (1972).
[Crossref]

K. R. Pechenick and J. M. Cohen, “Exact solutions to the valley problem in inverse scattering,” J. Math. Phys. 24(2), 406–409 (1983).
[Crossref]

J. Opt. Fiber Commun. (1)

M. Wandel and P. Kristensen, “Fiber designs for high figure of merit and high slope dispersion compensating fibers,” J. Opt. Fiber Commun. 3(1), 25–60 (2006).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Quantum Electron. (3)

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” J. Quantum Electron. 35(8), 1105–1115 (1999).
[Crossref]

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

I. Hirsh, M. Horowitz, and A. Rosenthal, “Design of planar waveguides with prescribed mode-profile using inverse scattering theory,” J. Quantum Electron. 45(9), 1133–1141 (2009).
[Crossref]

Opt. Commun. (1)

C. Papachristos and P. Frangos, “Synthesis of single-and multi-mode planar optical waveguides by a direct numerical solution of the Gel’fand-Levitan-Marchenko integral equation,” Opt. Commun. 203(1-2), 27–37 (2002).
[Crossref]

Opt. Express (1)

Opt. Fiber Technol. (1)

M. Onishi, T. Okuno, T. Kashiwada, S. Ishikawa, N. Akasaka, and M. Nishimura, “Highly nonlinear dispersion-shifted fibers and their application to broadband wavelength converter,” Opt. Fiber Technol. 4(2), 204–214 (1998).
[Crossref]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

M. Cvijetic, “Dual-mode optical fibres with zero intermodal dispersion,” Opt. Quantum Electron. 16(4), 307–317 (1984).
[Crossref]

Proc. Inst. Electr. Eng. (1)

A. K. Jordan and S. Ahn, “Inverse scattering theory and profile reconstruction,” Proc. Inst. Electr. Eng. 126(10), 945–950 (1979).
[Crossref]

Rev. Mod. Phys. (1)

J. M. Dudley and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

Other (2)

Maplesoft, “MAPLE,” (2012).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

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Figures (12)

Fig. 1
Fig. 1 The physical model for electromagnetic reflection from an inhomogeneous planar waveguide.
Fig. 2
Fig. 2 The allowed regions designed by A and B for the three pole case with a guided mode located at |k3| = 1µm−1 derived by previous authors [12,13].
Fig. 3
Fig. 3 (a) RI profiles with three-pole rational reflection coefficient designs in region A with c1 = 0.1, c2 = 0.1 (design#1) and c1 = 0.85, c2 = 0.499 (design#2). The exact design#2 obtained by Lakshmanasamy and Jordan [12] (dotted green curve) is also shown for comparison. (b) effective index variation with k0 for design#1 and #2 (the design point is demarcated by the dashed lines).
Fig. 4
Fig. 4 Waveguide dispersion D2 (in ps/nm/km), dispersion slope D3 (in ps/nm2/km), and dispersion curvature D4 (in ps/nm3/km) as a function of leaky pole positions. (designs #1 to #5 are designated by yellow dots).
Fig. 5
Fig. 5 (a) Waveguide designs and (b) corresponding TE0 normalized electric field profiles with D2 = −215ps/nm/km and D3 = 0.1ps/nm2/km (design#3), 0.2ps/nm2/km (design#4) and 0.3ps/nm2/km (design#5).
Fig. 6
Fig. 6 Effect of leaky pole modulus R = |k1| = |k2| on IS waveguide RI modulation. (a) R = 3 and (b) R = 4.
Fig. 7
Fig. 7 The dispersion curves (TE0, TE1) for (a) R = 3 and (b) R = 4 designs (the design point is demarcated by the dashed lines).
Fig. 8
Fig. 8 (a) RI modulation profiles for different c1 and fixed c2 = 0.51. (b) neff variation for TE0 and TE1 with k0 (the design point is demarcated by the dashed lines)
Fig. 9
Fig. 9 TE0 effective mode area over the entire (c1,c2) parameter space.
Fig. 10
Fig. 10 Waveguide dispersion map as a function of additional leaky pole positions for a five-pole case. (a) ( c 1 , c 2 )=(0.85,0.4999) and (b) ( c 1 , c 2 )=(1.7,1) | k 5 |=1μ m 1 , n 2 =1.444 and λ=1.55μm .
Fig. 11
Fig. 11 Three-pole ( c 1 , c 2 )=(2.2775,0.52692) and five-pole ( c 1 , c 2 , d 1 , d 2 )=(1.7,1,3.18,0.22) designs with identical D 2 =261 ps/nm/km , D 3 =0.130 ps/nm 2 /km but differing D4 (4.41x10−4 ps/nm3/km & 4.29x10−4 ps/nm3/km). (designs correspond to the ‘white crosses’ in Fig. 4. and Fig. 10 (b)).
Fig. 12
Fig. 12 Seven-pole (e1,e2) allowable region and dispersion map, with fixed ( c 1 , c 2 , d 1 , d 2 )=(1.7,1,3.18,0.22) .

Equations (21)

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E y (x,z,t)= E y0 (x, k 0 )exp(iβz)exp(iωt)
d 2 E y d x 2 +[ k 0 2 n (x) 2 β 2 ] E y =0
d 2 E y d x 2 +[ k 2 q(x) ] E y =0
k 2 = k 0 2 n 2 2 β 2
q(x)= k 0 2 [ n 2 2 n (x) 2 ]
R(x+t)+K(x,t)+ t x K(x,y)R(x+y)dy=0
R(t)= 1 2π r(k)exp(ikt)dki p=1 n r p exp(i k p t)
q(x)=2 dK(x,x) dx
n(x)= n 2 2 q(x) k 0 2
r(k)= k 1 k 2 k 3 (k k 1 )(k k 2 )(k k 3 )
k 1 = c 1 i c 2 ; k 2 = c 1 i c 2 ; k 3 =+ia
D 2 = λ c d 2 n eff d λ 2
D n = d D (n1) /dλ,     n>2
r(k)= k 1 k 2 k 3 k 4 k 5 (k k 1 )(k k 2 )(k k 3 )(k k 4 )(k k 5 )
k 1 = c 1 i c 2 , k 2 = c 1 i c 2 ; k 3 = d 1 i d 2 , k 4 = d 1 i d 2 ; k 5 =+ia
A B 1
A ( c 1 2 + c 2 2 ) 2 ( d 1 2 + d 2 2 ) 2 a 2
B( k 2 +2k c 1 + c 1 2 + c 2 2 )( k 2 2k c 1 2 + c 1 2 + c 2 2 )( k 2 +2k d 1 + d 1 2 + d 2 2 ) ×( k 2 2k d 1 + d 1 2 + d 2 2 )( k 2 + a 2 )
0BA
0p(k)
ρ= ν a ν b

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