Structurally periodic materials appear in nature and can also be synthesized as photonic crystals, self-organized cholesteric liquid crystals, and sculptured thin films.1 Parametric control of the photonic properties of periodic materials is highly desirable, for instance, in realizing tunable photonic bandgaps (PBGs) for optical applications. Work done decades ago on unit cells consisting of two constituent layers of dissimilar materials—so-called Šolc folded filters (ŠFFs)—showed PBGs to be tunable by changing the structural properties of the photonic crystal material.2 Tunability is also achieved through electrically controlled crystalline misalignment using ferroelectric (smectic) liquid crystalline layers. Recently, we demonstrated tunability using 1D magnetophotonic crystals (MPCs),3,4 wherein at least one of the two constituent materials is made of a ferrimagnetic garnet (a mineral material) that displays optical gyrotropy, or helical movement, under the influence of an externally applied magnetic field.
Incorporating helicoidal periodicity in a photonic crystal is yet another mechanism for producing tunable PBGs. Significantly, we are able to devise ‘biperiodic’ 1D photonic crystals that combine both the helicoidal periodicity and the overall periodicity that comes from repetition of unit cells. This morphology is practically feasible and straightforward, as we showed in a previous report of a 1D helicoidal photonic crystal (HPC)5 that contains a structurally chiral (left- or right-handed) material (SCM) in the unit cell.
Here, we consider the case where all three tunability mechanisms—i.e., crystalline misalignment, optical gyrotropy, and helicoidal periodicity—are hosted in a single 1D photonic crystal, and delineate their individual and collaborative contributions to the tunability of PBGs.
Consider a 1D photonic crystal made of two alternating dielectric materials with the overall period along the thickness direction, where t(a, b) are the thicknesses of the two layers of dissimilar materials in the unit cell. Labeling the two materials as a and b, we fix their relative permittivity tensors (essential optical properties of dielectric materials) as
The optical gyrotropy (g) is characterized through the magnetophotonic angle , and the dyadic is used to delineate two tunability mechanisms:5 α′ is the misalignment angle, and the helicoidal morphology is expressed through Ω and h=±1, which denote the helicoidal period and the structural handedness of an SCM, respectively. z and zz refer to components or amplitudes along the z-axis.
Figure 1 shows the dependence of the width of an intra-Brillouin-zone PBG of an ŠFF on α′ and the similar dependences on α of 1D MPC, helicoidal MPC (HMPC), and bicrystalline MPC (BMPC). The bandgap widths increase with α′ and α for the ŠFF and the MPC. Therefore, crystalline misalignment and optical gyrotropy, in essence, function similarly to tune PBGs. But whereas crystal misalignment must be chosen before making the photonic crystal, optical gyrotropy can be dynamically altered after fabrication by applying a low-frequency magnetic field. Magnetic tunability of PBGs is also evinced by a photonic crystal that can be tuned by a combination of mechanisms. Among them is the 1D BMPC, in which the gap widths of PBGs vary with α in general but may also be unaffected by the amplitude of optical gyrotropy when the misalignment angle, α′, equals π2 (see Figure 1). A 1D HMPC demonstrates a bimodal magnetic tunability of PBGs: first the bandgap width decreases to almost the vanishing point and then increases, as the magnetophotonic angle α increases.
Figure 1. The dependences of the widths Δωof intra-Brillouin-zone PBGs on the angle ζ for 1D photonic crystals. ζ=α′ for the Šolc folded filter (ŠFF), but ζ=αfor 1D magnetophotonic crystal (MPC), HMPC, and BMPC. The PBGs calculated for the 1D photonic crystals other than HMPC occur at the fifth branch in the Brillouin diagram, while the PBG for 1D HMPC is in the third lower-frequency bandgap away from the one centered in the circular Bragg regime of the SCM layer. Λ: Overall period. c: Free-space light velocity. α: Magnetophotonic angle. HMPC: Helicoidal MPC. BMPC: Bicrystalline MPC.
Thus, by considering photonic crystals with either one or two tunability mechanisms, we have shown both that different types of trends are possible and that the tunability of photonic crystals with more than one physical mechanism is multifaceted. In the future, we plan to design a class of HMPC materials and structures to elicit new tunability features of photonic bandgaps. One possibility is to use ferroelectric crystals for the magnetophotonic layers in the HMPC, which would introduce additional control by an externally applied DC electric field. Yet another possibility is that the magnetophotonic layers also have a helicoidal morphology. In that case, the additional helicoidal periodicity would introduce new features. All these potential new aspects regarding the control of photonic bandgaps deserve exploring.
Micron Technology Inc.
Pennsylvania State University
University Park, PA
Lake Forest, CA