Andrew M. Weiner, Editor-in-Chief
Vadim Karagodsky, Forrest G. Sedgwick, and Connie J. Chang-Hasnain
Vadim Karagodsky, Forrest G. Sedgwick, and Connie J. Chang-Hasnain*
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA
*Corresponding author: firstname.lastname@example.org
A simple analytic analysis of the ultra-high reflectivity feature of subwavelength dielectric gratings is developed. The phenomenon of ultra high reflectivity is explained to be a destructive interference effect between the two grating modes. Based on this phenomenon, a design algorithm for broadband grating mirrors is suggested.
©2010 Optical Society of America
Connie J. Chang-Hasnain and Weijian Yang
Adv. Opt. Photon. 4(3) 379-440 (2012)
Vadim Karagodsky and Connie J. Chang-Hasnain
Opt. Express 20(10) 10888-10895 (2012)
Fanglu Lu, Forrest G. Sedgwick, Vadim Karagodsky, Christopher Chase, and Connie J. Chang-Hasnain
Opt. Express 18(12) 12606-12614 (2010)
Opt. Express 23(13) 16730-16739 (2015)
Anjin Liu, Werner Hofmann, and Dieter Bimberg
Opt. Express 22(10) 11804-11811 (2014)
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(a) High Contrast Grating (HCG) schematics. The red arrow indicates the direction of wave incidence. The black arrows correspond to the E-field direction in both TE and TM polarizations of incidence. The grating comprises of rectangular dielectric bars having a refractive index in the semiconductor range, surrounded by a low index medium (air or oxide). The high refractive index contrast between the grating bars and the surrounding medium is beneficial for the reflectivity bandwidth and the fabrication tolerance of the grating. The nomenclature for the HCG dimensions is as follows: Λ is the grating period, a is the air-gap width, s is the bar width and tg
is the grating thickness. HCG has subwavelength periodicity (Λ<λ). (b) Example of the broadband high reflectivity spectrum. The HCG parameters for this plot are: n
=3.21, Λ=0.779μm, s/Λ=0.77, tg=0.508μm and TM polarization of incidence.
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Nomenclature for Eqs. (1a)–(1e): The HCG input plane is z=-tg
and the output plane is z=0. ka
are the x-wavenumbers in the air-gaps and in the grating bars respectively. The z-wavenumber β is the same in both the air-gaps and the bars. The x-wavenumber (green arrow) outside the grating (Regions I and III) is determined by the grating periodicity: 2πn/Λ.
Graphic representation of dispersion relations for a HCG with a refractive index n
=3.48, assuming TM polarization of incidence. Lower and upper curves are the first two solutions of Eq. (2), i.e. the first two HCG modes. Dashed lines are the constant wavelength contours, given by Eq. (1g). Black circles indicate the intersections between the dashed lines and the curves, thus describing the modes at a specific wavelength. The mode cutoff (β=0) is given by ks=n
, according to Eq. (1f). Above the cutoff β is real and below the cutoff β is imaginary.
(a) Convergence of the analytical solution in sections 2.1-2.2 towards the RCWA  simulation result as a function of the number of modes taken into consideration. The double-mode solution is in very good agreement with RCWA, especially when the reflectivity is high. The HCG parameters for this plot are: n
=3.214, tg/Λ=0.627, s/Λ=0.62 and TM polarization of incidence. (b) >99% reflectivity contour as a function of wavelength λ and thickness tg. Almost all high-reflectivity configurations are shown to reside within the double-mode region, i.e. between the cutoffs of the second and third modes.
(a-d) Convergence of the intensity profiles as a function of the number of modes taken into consideration, corresponding to the case of ~100% reflectivity. The grating bars are marked by the white dashed squares. When three or more modes are used, the boundary condition matching is almost perfect, but two modes are already a good approximation. The high reflectivity is clearly demonstrated by the standing wave profile created above the grating by the incident and reflected plane waves. (e-h) Contour plots of each mode separately inside the HCG. The grating bars are marked by the white dashed squares. The contours take into account both the forward ( +z) and the backward (-z) propagating components of each mode, including their coefficients. The HCG parameters for this plot are: n
=3.48, λ/Λ=1.509, tg/Λ=0.2, s/Λ=0.4 and TE polarization of incidence.
(a) Double-mode solution exhibiting perfect cancellation at the HCG output plane (z=0-
) leading to 100% reflectivity. At the wavelengths of 100% reflectivity both modes have the same magnitude of the “dc” lateral Fourier component (|(a1+aρ
x,2dx|), but opposite phases: ΔΦ=phase[(a1+aρ
x,2dx]=π. This means that the overall “dc” Fourier component is zero (Eq. (12), which leads to cancellation of the 0th transmissive diffraction order. When two perfect-cancellation points are located in close spectral vicinity of each other, a broad band of high-reflectivity is achieved. HCG parameters for this plot are: n
=3.214, s/Λ=0.62, tg/Λ=0.627 and TM polarization of incidence. (b) Double-mode solution for the overall field profile at the HCG output plane (z=0-
) in the case of perfect cancellation (black curve, right plot). The cancellation is shown to be only in terms of the “dc” Fourier component. The higher Fourier components do not need to be zero, since subwavelength gratings have no diffraction orders other than 0th. The left plot shows the decomposition of the overall field profile into the first two modes, whereby the dc-components of these two modes cancel each other. HCG parameters for this plot are: n
=3.48, s/Λ=0.4, tg/Λ=0.2, λ/Λ=1.563 and TE polarization of incidence.
(a) First stage of broadband reflectivity design: The broadband spectra are found along flat sections in 100% reflectivity contours. The 100% reflectivity contours are the solutions of Eq. (12) for different duty cycles s/Λ, marked by different colors. These contours are shown to form s-type shapes (only the lowest s-shapes are plotted). The optimal dimensions leading to the broadest spectra (indicated by arrows) are chosen. (b) Second stage of broadband reflectivity design: The bandwidth can be further increased through a tradeoff with a spectral dip, by fine-tuning the grating thickness. For example, if the minimal tolerable reflectivity for a particular application is 99%, the bandwidth can be increased by ~35% in comparison to a spectrally-flat configuration. Both figures (a) and (b) use normalized units, since HCG solution is scalable.
Same design as in Fig. 7a repeated for a larger refractive index: n
=4 and four different duty cycles s/Λ. The increased refractive index contrast is shown to have a beneficial effect on the achievable high-reflectivity bandwidths.
Table 1 Differences between TM and TE polarizations of incidence. Check-marks indicate the changes required for each equation. Equations not listed in this table require no changes
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