Symmetry-protected mode coupling near normal incidence for narrow-band transmission filtering in a dielectric grating

J. M. Foley, S. M. Young, and J. D. Phillips

Phys. Rev. B 89, 165111 – Published 9 April 2014

Abstract

A narrow-band transmission filter is demonstrated near normal incidence that operates through relaxation of supported-mode selection rules and is explained in the context of group theory. We calculated the transverse magnetic and transverse electric dispersion relations of a dielectric grating in the subwavelength and near-wavelength region using finite element modal analysis and determine the modes' corresponding irreducible representations. Coupling to select transverse magnetic modes at normal incidence is optimized to yield broadband high reflectance that acts as the background for the transmission filter. While some modes couple at normal incidence, others are shown to remain inaccessible due to symmetry mismatch. Away from normal incidence, the reduced symmetry relaxes the selection rules, enabling weak coupling between the incident field and these symmetry-protected modes. This weak coupling produces narrow transmission bands within the opaque background. Furthermore, by choosing the plane of incidence to include or exclude the grating periodicity, we show that orthogonal mode sets can independently be selected to couple to the incident light, yielding separate transmission bands. This spectral filtering is experimentally demonstrated with a suspended silicon grating in the infrared spectrum ( $7 - 14 μ m$ ), which agrees well with simulated transmittance spectra and modal analysis.

Received 1 January 2014
Revised 20 March 2014

DOI:https://doi.org/10.1103/PhysRevB.89.165111

Authors & Affiliations

J. M. Foley^1,2,*, S. M. Young^1,3, and J. D. Phillips^1,2

¹Applied Physics Program, University of Michigan, Ann Arbor, Michigan 48109, USA
²Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, Michigan 48109, USA
³Physics Department, University of Michigan, Ann Arbor, Michigan 48109, USA

^*foleyjm@umich.edu

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Vol. 89, Iss. 16 — 15 April 2014

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Images

Figure 1
Grating schematic and its corresponding Brillouin zone with incident fields, dimensions, and material properties defined. (a) The grating includes its period ( $Λ$ ), height ( $t$ ), width ( $w$ , defined as $Λ \times F F$ where $F F$ is the fill factor), and material permittivities for the grating ( $ε_{g}$ ) and surrounding material ( $ε_{s}$ ). The electric field lies parallel to the $x y$ plane and the magnetic field lies parallel to the $y z$ plane. The angles with respect to normal are $θ$ and $φ$ , which lie in the $x y$ and $y z$ planes, respectively. (b) The grating's Brillouin zone with analyzed incident wave vectors: Point $I$ : normal incidence or $θ = φ = 0^{\circ}$ ( $k_{x} = k_{z} = 0, | k_{y} | > 0$ ), Point $II$ : $θ > 0^{\circ}$ ( $| k_{x} |$ , $| k_{y} | > 0$ , $k_{z} = 0$ ), and Point $III$ : $φ > 0^{\circ}$ ( $k_{x} = 0$ , and $| k_{y} |$ , $| k_{z} | > 0$ ).
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Figure 2
TM dispersion relations and field profiles for a grating with $\frac{t}{Λ} = 0.6$ , $F F = 0.72$ , $ε_{g} = 11.7$ , and $ε_{s} = 1$ . (a) The dispersion relations include even (blue) and odd (green) bands with respect to reflection across $y = 0$ . Solid bands were calculated using a finite element modal analysis and dashed bands are estimated from scattering analysis. The modes at $k_{x} = 0$ are labeled with their $Γ$ point ( $D_{2 h}$ symmetry) irreducible representations and band definitions. The light cone is shown in light gray. (b) TM mode field profiles at $k_{x} = 0$ with a black line indicating the boundary between high and low permittivity regions. Below: grating simulation element, and directions defined.
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Figure 3
TE dispersion relations and select field profiles for a grating with $\frac{t}{Λ} = 0.6$ , $F F = 0.72$ , $ε_{g} = 11.7$ , and $ε_{s} = 1$ . (a) The dispersion relations include even (blue) and odd (green) bands with respect to reflection across $y = 0$ . The modes at $k_{x} = 0$ are labeled with their $Γ$ point ( $D_{2 h}$ symmetry) irreducible representations and band definitions. The light cone is shown in light gray. (b) TE mode field profiles at $k_{x} = 0$ with a black line indicating the boundary between high and low permittivity regions. Below: grating simulation element, and directions defined.
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Figure 4
Simulated grating transmittance profile at normal and off-normal incidence. $k_{x} = k_{z} = 0$ , $| k_{x} | > 0$ , and $| k_{z} | > 0$ correlates to Points $I$ , $II$ , and $III$ , respectively. The top scale shows increasing $θ$ and $φ$ directions corresponding to $| k_{x} | > 0$ , and $| k_{z} | > 0$ , respectively. Transmission bands are labeled with the mode associated with the resonance.
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Figure 5
Experimental and associated simulated transmittance of a grating with various incident wave configurations. The as-built dimensions were $Λ = 4.9 μ m$ , $t = 2.85 μ m$ , $h = 4.05 μ m$ and $F F = 0.72$ . Broadband reflectance, TM selective filtering, TE selective filtering and mixed TE and TM filtering are demonstrated, which are associated with the various points in the Brillouin zone: Point $I$ ( $θ = φ = 0^{\circ}$ , Point $II$ ( $θ = 7^{\circ}$ , $φ = 0^{\circ}$ ), Point $III$ ( $θ = 0^{\circ}$ , $φ = 14^{\circ}$ ), and Point $IV$ ( $θ = 7^{\circ}$ , $φ = 14^{\circ}$ ), respectively. Inset: scanning electron micrograph of a representative suspended grating.
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