## Finite Bragg grating synthesis by numerical solution of Hermitian Gel’fand–Levitan–Marchenko equations

JOSA B, Vol. 23, Issue 10, pp. 2040-2045 (2006)

http://dx.doi.org/10.1364/JOSAB.23.002040

Enhanced HTML Acrobat PDF (110 KB)

### Abstract

We examine the problem of fiber Bragg grating reconstruction from its reflection coefficient. A direct numerical method of solving the Gel’fand–Levitan–Marchenko integral equations for the problem is developed. The method is based on a bordering procedure, Cholesky decomposition, and piecewise-linear approximation. It is tested using high-reflectance homogeneous and hyperbolic secant profiles. The proposed method is shown to concede the popular discrete layer peeling technique in efficiency but surpasses it in accuracy and stability at high reflectance.

© 2006 Optical Society of America

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(050.2770) Diffraction and gratings : Gratings

(060.2430) Fiber optics and optical communications : Fibers, single-mode

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 22, 2006

Revised Manuscript: June 24, 2006

Manuscript Accepted: June 28, 2006

**Citation**

Oleg V. Belai, Evgeny V. Podivilov, Osip Ya. Schwarz, David A. Shapiro, and Leonid L. Frumin, "Finite Bragg grating synthesis by numerical solution of Hermitian Gel'fand-Levitan-Marchenko equations," J. Opt. Soc. Am. B **23**, 2040-2045 (2006)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-23-10-2040

Sort: Year | Journal | Reset

### References

- G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002). [CrossRef]
- R. Kashyap, Fiber Bragg Gratings (Academic, 1999).
- M. Sumetsky and B. J. Eggleton, "Fiber Bragg gratings for dispersion compensation in optical communication systems," J. Opt. Fiber. Commun. Rep. 2, 256-278 (2005). [CrossRef]
- V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Zh. Eksp. Teor. Fiz. 61, 118-134 (1971).
- L. Poladian, "Simple grating synthesis algorithm," Opt. Lett. 25, 787-789 (2000). [CrossRef]
- L. Poladian, "Simple grating synthesis algorithm," Opt. Lett. 25, 1400 (2000), errata. [CrossRef]
- J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quantum Electron. 37, 165-173 (2001). [CrossRef]
- P. Frangos and D. Jaggard, "A numerical solution to the Zakharov-Shabat inverse scattering problem," IEEE Trans. Antennas Propag. 39, 74-79 (1991). [CrossRef]
- C. Papachristos and P. Frangos, "Design of corrugated optical waveguide filters through a direct numerical solution of the coupled Gel'fand-Levitan-Marchenko integral equations," J. Opt. Soc. Am. A 19, 1005-1012 (2002). [CrossRef]
- 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 equations," Opt. Commun. 203, 27-37 (2002). [CrossRef]
- G. B. Xiao and K. Yashiro, "An efficient algorithm for solving Zakharov-Shabat inverse scattering problem," IEEE Trans. Antennas Propag. 50, 807-811 (2002). [CrossRef]
- P. V. Frangos and D. L. Jaggard, "The reconstruction of stratified dielectric profiles using successive approximations," IEEE Trans. Antennas Propag. 35, 1267-1272 (1987). [CrossRef]
- J. Skaar and R. Feced, "Reconstruction of gratings from noisy reflection data," J. Opt. Soc. Am. A 19, 2229-2237 (2002). [CrossRef]
- J. Capmany and J. Marti, "Design of fibre grating dispersion compensators using a novel iterative solution of the Gel'fand-Levitan-Marchenko coupled equations," Electron. Lett. 32, 918-919 (1996). [CrossRef]
- E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996). [CrossRef]
- L. Poladian, "Iterative and noniterative design algorithms for Bragg gratings," Opt. Laser Technol. 5, 215-222 (1999).
- P. V. Frangos and D. L. Jaggard, "Inverse scattering: solution of coupled Gelfand-Levitan-Marchenko integral equations using successive kernel approximations," IEEE Trans. Antennas Propag. 43, 547-552 (1995). [CrossRef]
- G. H. Song and S. Y. Shin, "Design of corrugated waveguide filters by the Gel'fand-Levitan-Marchenko inverse-scattering method," J. Opt. Soc. Am. A 2, 1905-1915 (1985). [CrossRef]
- F. Ahmad and M. Razzagh, "A numerical solution to the Gel'fand-Levitan-Marchenko equation," Appl. Math. Comput. 89, 31-39 (1998). [CrossRef]
- A. Rosenthal and M. Horowitz, "Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings," IEEE J. Quantum Electron. 39, 1018-1026 (2003). [CrossRef]
- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge U. Press, 1992).
- H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
- H. Bateman and A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.
- D. A. Shapiro, "Family of exact solutions for reflection spectrum of Bragg grating," Opt. Commun. 215, 295-301 (2003). [CrossRef]
- J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber grating," IEEE J. Quantum Electron. 39, 1238-1245 (2003). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.