Application of the homogenization approximation to rough one-dimensional photonic crystals

Abstract

As previously reported [Opt. Lett. 29, 2791 (2004)], one-dimensional photonic crystals exhibit a decrease in their normal reflectivity if their interfaces are not flat. We show that the homogenization approximation accurately predicts this diminished optical response by comparing results with finite-difference time-domain (FDTD) simulations applied to the same roughened structures. Within the parameter range tested (rms roughness < 20% and rms wavelengths < 100% of the photonic crystal periodicity), the homogenization approximation accurately reproduces the reflectivities obtained by the FDTD simulations, which are much more computationally expensive.

Effect of waveguide side-wall roughness on the performance of quantum cascade lasers

“…During QC laser processing there are two main fabrication steps that introduce waveguide side-wall roughness, (i) photolithographic pattern transfer where roughness is introduced due to imperfections in the photolithography mask and/or sample surface and irregularities created in the photoresist pattern while developing, and (ii) dry etching, where roughness is introduced due to material degradation induced by high-ion energy plasma and slow etching rates. Several studies on the effects of surface roughness on performance of optical devices have been conducted in the fields of Si/SiO 2 waveguides for shorter near-infrared (near-IR) wavelengths of 0.63-1.63 μm [1][2][3][4][5], Si-and III-V-based microdisk resonators [6][7][8][9], and photonic crystal structures [10][11][12][13]. Recently we reported on a study on determining the effects of side-wall roughness on QC laser performance parameters, such as threshold current density and slope efficiency for the longer mid-IR wavelengths of 7-12 μm [14]; here we review this work and extend the study by showing the effects of waveguide side-wall roughness on the far-field beam pattern and group refractive index of the waveguides.…”

Experimental verification of the applicability of the homogenization approximation to rough one-dimensional photonic crystals using a holographically fabricated reflection grating

“…Until recently, the real optical behavior of such structures was not easily predictable using traditional techniques, such as the transfer matrix method. 6 However, in several recent papers, 7,8 we have proposed a method to predict the normal incidence reflectivity of rough 1DPCs using the homogenization approximation 9 in conjunction with the transfer matrix technique. As of yet, this theory has not been verified through comparison with experimental data.…”

“…Therefore, traditional simulation techniques, such as one-dimensional transfer matrix, 6 cannot be used to accurately predict the real optical behavior of this structure directly. Thus, we have used our previously proposed method 7,8 to calculate the theoretical reflectivity spectrum of this structure at normal incidence. This was done by using a modified version ͑to account for the dispersion of the glass slide͒ of the code in Appendix A of Ref.…”

“…This was done by using a modified version ͑to account for the dispersion of the glass slide͒ of the code in Appendix A of Ref. 13. Our theory 7,8 posits that the optical behavior of a rough 1DPC at normal incidence will be equivalent to the response of a 1DPC with a diffuse refractive index profile, where the inhomogeneity of the refractive index at the rough interfaces is averaged out. A three-step process was used to implement this theory on the fabricated structure.…”

“…In conclusion, a 1DPC tuned to reflect the red, green, and blue portions of the spectrum was fabricated through the holographic exposure of a prepolymer syrup. Our previously reported technique 7,8 for calculating the normal incidence reflectance spectrum of rough 1DPCs was employed to calculate the theoretical reflectivity of the fabricated grating. The calculated spectrum matched the experimental one extremely well for wavelengths above about 530 nm.…”