Synthesis of Highly Efficient Multilayer Dielectric Diffraction Gratings for Spectral Combining of Laser beams




multilayer diffraction grating, spectral beam combining, incomplete Galerkin method, scattering matrix method, synthesis problem


The paper is devoted to the synthesis of multilayer dielectric reflection diffraction gratings providing high-efficiency spectral combining of the beams with different wavelengths in a given diffraction order. The results are presented for solving the synthesis problems for multilayer dielectric diffraction gratings providing spectral combining in the first or minus first diffraction order. Besides, the synthesis problem for such gratings is solved with account taken of possible technological constraints imposed by the height of the grating profile (etch depth). The solution of the synthesis problem is obtained by means of Nelder-Mead minimization of the merit function depending on the grating parameters. At each minimization step the direct problem is solved using a combination of the incomplete Galerkin method and scattering matrix method.

Author Biography

A.A. Petukhov

Lomonosov Moscow State University,
Faculty of Physics,
Department of mathematics
• Leading Programmer


  1. E. J. Bochove, “Theory of Spectral Beam Combining of Fiber Lasers,” IEEE J. Quantum Electron. 38 (5), 432-445 (2002). doi 10.1109/3.998614
  2. S. J. Augst, A. K. Goyal, R. L. Aggarwal, et al., “Wavelength Beam Combining of Ytterbium Fiber Lasers,” Opt. Lett. 28 (5), 331-333 (2003). doi 10.1364/OL.28.000331
  3. T. H. Loftus, A. M. Thomas, P. R. Hoffman, et al., “Spectrally Beam-Combined Fiber Lasers for High-Average-Power Applications,” IEEE J. Sel. Top. Quantum Electron. 13 (3), 487-497 (2007). doi 10.1109/JSTQE.2007.896568
  4. R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-Doped Fiber Amplifiers,” IEEE J. Quantum Electron. 33 (7), 1049-1056 (1997).doi 10.1109/3.594865
  5. S. Fu, W. Shi, Y. Feng, et al., “Review of Recent Progress on Single-Frequency Fiber Lasers,” J. Opt. Soc. Am. B: Opt. Phys. 34 (3), A49-A62 (2017). doi 10.1364/JOSAB.34.000A49
  6. T. Theeg and T. Pulzer, “High Power Fiber Arrays for Spectral and Coherent Beam Combining of Fiber Laser and Amplifier,” Proc. SPIE 11539 (2020). doi 10.1117/12.2581107
  7. I. Kim, S. So, J. Mun, et al., “Optical Characterizations and Thermal Analyses of HfO_2/SiO_2 Multilayered Diffraction Gratings forHigh-Power Continuous Wave Laser,” J. Phys. Photonics 2 (2020). doi 10.1088/2515-7647/ab7b0f
  8. H. Wang, Y. Song, Y. Yang, et al., “Simulation and Experimental Study of Laser-Induced Thermal Deformation of Spectral Beam Combination Grating,” Opt. Express 28 (22), 33334-33345 (2020). doi 10.1364/OE.408832
  9. J. Neauport, E. Lavastre, G. Razé, et al., “Effect of Electric Field on Laser Induced Damage Threshold of Multilayer Dielectric Gratings,” Opt. Express 15 (19), 12508-12522 (2007). doi 10.1364/OE.15.012508
  10. M. D. Perry, R. D. Boyd, J. A. Britten, et al., “High-Efficiency Multilayer Dielectric Diffraction Gratings,” Opt. Lett. 20 (8), 940-942 (1995). doi 10.1364/OL.20.000940
  11. H. Wei and L. Li, “All-Dielectric Reflection Gratings: A Study of the Physical Mechanism for Achieving High Efficiency,” Appl. Opt. 42 (31), 6255-6260 (2003). doi 10.1364/AO.42.006255
  12. A. G. Sveshnikov, “The Incomplete Galerkin Method,” Dokl. Akad. Nauk SSSR 236 (5), 1076-1079 (1977) [Sov. Math. Dokl. 18, 1331-1334 (1977)].
  13. L. Li, “Formulation and Comparison of Two Recursive Matrix Algorithms for Modeling Layered Diffraction Gratings,” J. Opt. Soc. Am. A: Opt. Image Sci. Vis. 13 (5), 1024-1035 (1996). doi 10.1364/JOSAA.13.001024
  14. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A: Opt. Image Sci. Vis. 20 (4), 655-660 (2003). doi 10.1364/JOSAA.20.000655
  15. P. A. Kuchment, “Floquet Theory for Partial Differential Equations,” Usp. Mat. Nauk 37 (4), 3-52 (1982) [Russ. Math. Surv. 37 (4), 1-60 (1982)].
  16. A. A. Petukhov, “Joint Application of the Incomplete Galerkin Method and Scattering Matrix Method for Modeling Multilayer Diffraction Gratings,” Mat. Model. 25 (6), 41-53 (2013) [Math. Models Comput. Simul. 6 (1), 92-100 (2014)].
  17. Ch. Palmer, Diffraction Grating Handbook (MKS Instruments, Rochester, 2020).
  18. D. Himmelblau, Applied Nonlinear Programming (McGraw-Hill, New York, 1972; Mir, Moscow, 1975).



How to Cite

Петухов А.А. Synthesis of Highly Efficient Multilayer Dielectric Diffraction Gratings for Spectral Combining of Laser Beams // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2021. 22. 200-209. doi 10.26089/NumMet.v22r312



Methods and algorithms of computational mathematics and their applications