On Calculating of Functional Derivative for an Optimal Control Problem

The synthesis problem of a multilayer diffraction grating is formulated as an optimal control problem and consists in minimizing the cost functional depending on the geometric parameters of the grating profile. The gradient method is the most reliable and stable method for solving this problem. The paper deals with a method for calculating the gradient of the cost functional, which is done by solving a cojugate problem with special boundary conditions. Additionally, the paper discusses the numerical implementation of this solution and the calculation of the gradient. The results from a computational experiment are also presented.

Keywords

functional derivative; gradient; conjugate problem; optimal control problem; synthesis problem; diffraction gratings.

References

1. Palmer C., Loewen E. Diffraction Grating Handbook. New York, Newport Corporation, 2005.
2. Loewen E.G., Popov E. Diffraction Gratings and Application. New York, Marcel Dekker, 1997.
3. Fedorenko R.P. Approximate Solution of Optimal Control Problems. Moscow, Nauka, 1978.
4. Artemeva M.V., Bogolyubov A.N., Petukhov A.A. Solving Diffraction Grating Synthesis Problems for Practical Applications. Physical Bases of Instrumentation, 2020, vol. 9, no. 3, pp. 4-13. DOI: 10.25210/jfop-2003
5. Artemeva M.V., Bogolyubov A.N., Petukhov A.A. Transfer Matrix Method for Solving the Problem of Diffraction of a Plane Wave with Te-Polarization on a One-Dimensional Binary Diffraction Grating. Physical Bases of Instrumentation, 2022, vol. 11, no. 2, pp. 40-48. DOI: 10.25210/jfop-2202-040048
6. Petukhov A.A., Bogolyubov A.N., Trubetskov M.K. Mathematical Modeling of Multilayer Diffraction Gratings. Physical Bases of Instrumentation, 2014, vol. 3, no. 4, pp. 20-27. DOI: 10.25210/jfop-1404
7. Lu Nan, Kuang Dengfeng, Mu Guoguang. Design of Transmission Blazed Binary Gratings for Optical Limiting with the Form-Birefringence Theory. Applied Optics, 2008, vol. 47, no. 21, pp. 3743-3750. DOI: 10.1364/ao.47.003743