Introduction

A methodology suitable for analyzing strong IBG (Integrated Bragg Grating) is presented using the Transfer Matrix Method (TMM) based on the translation of any waveguide's physical structure into a matrix of effective refractive indexes, n_eff, which is wavelength-dependent and describes the behavior of light in the IBG while avoiding the use of approximations like in Coupled Mode Theory.

IBGs have special characteristics that require a methodology that considers any tiny variation in geometry. These have a notable impact on the value of n_eff because of the high refractive index contrast of SOI (Silicon on Insulator) and therefore on the spectrum of the transfer function.

Method

Starting with a given IBG geometry, we mapped the waveguide structure into a matrix, N’, of effective refractive indices, which are functions of the wavelength and position along the IBG, n_eff(λ,z). Subsequently, the TMM was applied using N’ to characterize each layer and interface of the IBG. Finally, the reflectivity and phase spectra were obtained, i.e., the transfer function.

Using this methodology, four apodization methods (corrugation width, lateral delay, duty cycle, and periodic phase modulations) and different chirp methods are presented and compared. Also, the presence of additional Bragg orders is demonstrated as a result of the Fourier transform equivalence between the IBG apodization and its transfer function in reflection. To demonstrate the generality of the methodology, two technological platforms, SOI and Al₂O₃, are analyzed.

Results

Simulations allow us to obtain the reflectivity and phase spectra, i.e., the IBG transfer function. The goodness of each method is observed, as well as the second and third Bragg order spectra. The Al₂O₃ platform shows more robustness against fabrication errors.

Conclusions

TMM is the most suitable methodology for analyzing an IBG. Apodization, chirp methods, and higher Bragg orders are demonstrated to improve transfer functions.