Photonic-crystal surface modes found from impedances

Abstract

We present a rigorous definition of a wave impedance for 2D rectangular and triangular lattice photonic crystals (PCs), in the form of a matrix. Reflection and transmission at an interface between PCs can be represented by matrices that relate the Bloch mode (eigenmode) amplitudes in the two PCs; we show that these matrices, which are multi-mode generalisations of reflection and transmission coefficients, may be calculated from the PCs' impedances that we define.Given the impedances and Bloch factors (propagation constants) of a collection of PCs, the reflection and transmission properties of arbitrary stacks of these PCs may be calculated efficiently using a few matrix operations. Therefore our definition enables PC-based antireflection coatings to be designed efficiently: some computationally expensive simulations are required in an initial step to find a range of PCs' impedances, but then the reflectances of every coating that consists of a stack of these PCs can be calculated without any further simulations.We first define the PC impedance from the transfer matrix of a single PC layer (i.e., a grating). Since transfer matrix methods are not especially widespread, we also present a method and associated source code to extract a PC's propagating and evanescent Bloch modes from a scattering calculation that can be performed by any off-the-shelf field solver, and to calculate impedances from the extracted modal fields.Finally, we put our method to use. We apply it to design antireflection coatings, nearly eliminating reflection at a single frequency for one or both polarisations, or lowering it across a larger bandwidth. We use it to find surface modes at interfaces between PCs and air, and their projected band structures. We use the impedance to define effective parameters for PC homogenisation, and we briefly describe how our definition has been used to dispersion engineer a PC waveguide. Statement of contribution for joint author publicationsLawrence, F. J. et al. "Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition". Appl. Phys. Lett. 93, 1114Lett. 93, (2008 This Letter is partially based on work done in FL's Honours year, and partially on work done during his PhD. Most of the software used was provided by KD and LB, and FL extended it in collaboration with them. In FL's PhD he extended the software implementation of the method (with help from KD and LB) to support non-normal incidence and frequencies above the first Wood anomaly. FL wrote the letter, which was heavily edited and proofread by MdS, KD, and LB.Lawrence, F. J. et al. "Impedance of square and triangular lattice photonic crystals". Phys. Rev. A 80, 23826 (2009) The suggestion of generalising the method to triangular lattices was made by FL's supervisors. LB supplied the orthogonality relations, which as noted in the paper are from previous work, and FL worked out how to use these to define the impedance for these structures. LB suggested the final notation used in the paper. Taking the Ma...

Semianalytical formulations for the surface modes of photonic woodpiles

“…However, if m is odd, then there are specific directions of the Brillouin zone for which the even and odd double-interface modes become degenerate and, hence, for these directions the coupling strength is largely insensitive to changes in m. The degeneracy in the double-interface modes at the edge of the Brillouin zone is also evident in finite 2D PCs [19], although in the case of finite 2D PCs the edge of the Brillouin zone represents just a single Bloch vector.…”

“…Equation (15) was previously derived in [19] for 2D PCs [see, in particular, Eq. (6) of [19]], although the approach used in that study instead relied on impedance matrices.…”

“…(6) of [19]], although the approach used in that study instead relied on impedance matrices. In the case of 2D square and triangular lattices, the internal reflection matrices and Bloch factors satisfy R a = R b and ( ) −1 = s for some phase factor s, and so Eq.…”

“…(19) and (21) as even modes and odd modes, respectively, despite the appearance of the phase factor s n/2 , which depends on both k x and the number of grating pairs. For example, at the M point of the Brillouin zone (k x = k y = π/d), s = 1, hence, c − = ±c + , which are the usual notions of even and odd parity.…”

Strongly localized states at the band-inverting interface with periodic lattice dislocations

“…The difference is attributed to the reverse signs of imaginary acoustic impedances. Therefore, if we put PC1 and PC2 together, we will observe an acoustic resonance state bounded along the interface at some particular frequencies, where the sum of reflection phases exactly equals to zero (Φ 1 + Φ 2 = 0) 21 .…”