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1Fizeau fringes in resonant photonic structures with spatially varying parameters
D.A. Bvkov1'2*. E.A. Bezus1'2, L.L. Doskolovich1'2
1- Samara National Research University, 34 Moskovskoye shosse, Samara, 443086, Russia 2- Image Processing Systems Institute, NRC "Kurchatov Institute", 151 Molodogvardeyskaya st.,
Samara, 443001, Russia
Fizeau interferometer is a Fabry-Perot interferometer with a wedged central layer (Fig. 1a). Resonant transmission in such structures occurs at different spatial positions depending on the wavelength of the incident light. This allows one to divide the incident radiation into a large number of spectral channels. In this regard, nowadays, such structures are usually referred to as linear variable filters (LVFs) and are used in spectrometers and refractive index sensors [1]. When both the wedge angle and the reflection coefficients of the interfaces are quite large, the spatial profile of the resonant peak has a characteristic non-symmetric shape containing the main peak and several secondary peaks. Resonant peaks with such shape are sometimes referred to as Fizeau fringes. Taking this effect into account is important when designing compact optical filters [1].
Fig. 1. LVF based on Fizeau interferometer (a) and considered GMRGs with varying thickness (b) and period (c).
In this work, we show that similar effects arise in guided-mode resonant gratings (GMRGs) with varying period (Fig. 1c) as well as in GMRGs with varying thickness of the waveguide layer (Fig. 1b). Such structures can also be used as filters in spectrometric optical systems [2]. To describe the appearance of the Fizeau fringes in GMRGs, we developed a spatiotemporal coupled-mode theory (CMT). Such theory is written as two coupled nonuniform unidirectional wave equations [3]. Importantly, all parameters of the proposed CMT - the coupling coefficients - can be obtained by analyzing a strictly periodic GMRG with constant parameters. We show that the CMT predictions are in perfect agreement with the results of rigorous solution of Maxwell's equations using the superperiod approach incorporated into the Fourier modal method. By solving the plane wave diffraction problem, we calculated the reflected and transmitted field distributions exhibiting pronounced Fizeau-like fringes with the main peak accompanied by several secondary peaks. In contrast to the conventional Fizeau interferometer, in which secondary peaks appear on one side of the main peak, in GMRGs the secondary peaks may appear on both sides.
Interestingly, a similar CMT can also be developed for a conventional Fizeau interferometer, e.g., for a wedge with Bragg claddings shown in Fig. 1a. In this case, the coupled-mode theory is written as a second-order partial differential equation. The obtained results can be used to design linear variable filters based on GMRGs as well as for the design of optical resonators, in which the parameters of the structure vary in space in a nonlinear fashion.
This work was funded by the Russian Science Foundation (project 22-12-00120).
[1] R.R. McLeod, T. Honda, Improving the spectral resolution of wedged etalons and linear variable filters with incidence angle, Opt. Lett., vol. 30, pp. 2647-2649 (2005).
[2] H.-Y. Hsu, Y.-H. Lan, C.-S. Huang, A gradient grating period guided-mode resonance spectrometer, IEEE Photon. J., vol. 10, p. 4500109 (2018).
[3] D.A. Bykov, E.A. Bezus, A.A. Morozov, V.V. Podlipnov, L.L. Doskolovich, Optical properties of guided-mode resonant gratings with linearly varying period, Phys. Rev. A, vol. 106, p. 053524 (2022).
