Observation of Sub-Bragg Diffraction of Waves in Crystals

Abstract

We investigate the diffraction conditions and associated formation of stopgaps for waves in crystals with different Bravais lattices. We identify a prominent stopgap in high-symmetry directions that occurs at a frequency below the ubiquitous first-order Bragg condition. This sub-Bragg diffraction condition is demonstrated by reflectance spectroscopy on two-dimensional photonic crystals with a centred rectangular lattice, revealing prominent diffraction peaks for both the sub-Bragg and first-order Bragg condition. These results have implications for wave propagation in 2 of the 5 two-dimensional Bravais lattices and 7 out of 14 three-dimensional Bravais lattices, such as centred rectangular, triangular, hexagonal and body-centred cubic.The propagation and scattering of waves such as light, phonons and electrons are strongly affected by the periodicity of the surrounding structure [1,2]. Frequency gaps called stopgaps, emerge for which waves cannot propagate inside crystals due to Bragg diffraction. Bragg diffraction is important for crystallography using X-ray diffraction [3] and neutron scattering [4]. Diffraction determines electronic conduction of semiconductors [1,2] and of graphene [5], and broad gaps are fundamental for acoustic properties of phononic crystals [6,7] and optical properties of photonic metamaterials [9, 10].Bragg diffraction is described in reciprocal space by the Von Laue condition − → Here m is an integer, λ is the wavelength inside the crystal, θ is the angle of incidence with the normal to the lattice planes, and d is the spacing between the lattice planes. A stopgap is also formed when Bragg diffraction occurs on multiple Bragg planes simultaneously, which is called multiple-Bragg diffraction [11], and is fundamental for bandgap formation [2,12,13]. Wave propagation in crystals is described along high-symmetry directions [1]. Multiple-Bragg diffraction has been recognized in highsymmetry directions at frequencies above the first-order simple Bragg diffraction condition: m = 1, λ = 2d orhas not yet been observed at frequencies below simple Bragg diffraction [14].In this Letter we show that for high-symmetry directions multiple-Bragg diffraction can occur at frequencies below the first order simple Bragg condition. As a demonstration we have investigated diffraction conditions for two-dimensional (2D) photonic crystals using reflectance spectroscopy. A broad stopgap is observed below the simple Bragg condition, depending on the symmetry of the lattice. Our findings are not limited to light propagation, but apply for wave propagation in general, and therefore we anticipate similar diffraction for electrons in graphene [5], and sound in phononic crystals [6,7].We have studied light propagation in 2D silicon photonic crystals [17]. Figure 1(a) shows a scanning electron microscope (SEM) image of one of these crystals from the top view. The centred rectangular unit cell has a long side a = 693 ± 10 nm and a short side c = 488 ± 11 nm. The pores have a radius of r = 155 ± 10 nm and are approxima...

Polarization-selective branching of stop gaps in three-dimensional photonic crystals

“…The formation of stop gaps at K (U) points can be well understood through evoking the Laue diffraction condition as shown in Fig. 2 (inset) [14]. At a certain value of θ, the incident wave vector passes through the K point, then the Laue condition is satisfied simultaneously for reciprocal lattice vectors corresponding to (111) and (1 11 There persists a subtle issue on assigning the crystal planes responsible for the origin of stop gaps when the wave vector spans the K or U point.…”

“…This is assisted with branching of stop gaps in the optical reflectivity or transmission measurements [14][15][16]. Multiple Bragg diffraction in 3D photonic crystals has been a topic of intense research [17][18][19][20][21][22] and generally interpreted as the inherent property of photonic crystals using non-polarized light [19,[21][22][23][24][25][26].…”

Method for making a single-step etch mask for 3D monolithic nanostructures

“…The 3D mask for this hexagonal structure shows clearly the flexibility of fabrication method to realise structures with independent patterns on inclined surfaces with an unprecedented out-of-plane alignment better than 5 nm. All described 3D structures except cubic are predicted to reveal sub-Bragg diffraction [43] which makes them interesting subject for optical study, moreover, simple cubic and hexagonal structures have been predicted to reveal 3D band gaps [44,45]. Besides different periodic structures it is possible to fabricate masks that yield nonperiodic three-dimensional structures structures or periodic structures with controlled defects.…”

Reflectivity of three-dimensional GaAs photonic band-gap crystals of finite thickness

“…It appears that the first, simple, Bragg diffraction [similar to that of Fig. 7(a)] is equivalent to multiple Bragg conditions [67][68][69]:…”