Finite element computation of grating scattering matrices and application to photonic crystal band calculations

Dossou, Kokou B.; Byrne, Michael; Botten, L. C.

doi:10.1016/j.jcp.2006.03.029

Cited by 53 publications

(62 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…By setting periodic boundary conditions on two sides of the grating's unit cell, the infinite grating may be simulated using a finite computation domain. In some FEM implementations, including the in-house code [110] used in Chapter 2 and the commercial package COMSOL used in Chapter 3, it is possible to set the other boundaries to be transparent to specified grating orders (so that the grating is effectively surrounded by an infinite uniform dielectric), and calculate the scattering into these from the incident grating order.…”

Section: Simulating Gratingsmentioning

confidence: 99%

Photonic-crystal surface modes found from impedances

et al. 2010

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Section: Simulating Gratingsmentioning

confidence: 99%

Photonic-crystal surface modes found from impedances

et al. 2010

Self Cite

View full text Add to dashboard Cite

show abstract

“…A sudden, sharp rise or fall in transmission is a behavior known to occur near resonant frequencies -frequencies where the linear system of integral equations is singular. Various resonances have been observed in photonic crystals, both theoretically and experimentally [11,14,12,18,29,26]. Our periodic scatterers could model such crystals.…”

Section: Transmissionmentioning

confidence: 81%

“…The analytical method is based on the singular integral methods of J. Thomas Beale [4,6,5], while the scattering problem is motivated by the work of Stephanos Venakides, Mansoor Haider, Stephen Shipman, and Andrew Barnes [31,17,27,26,28,3]. Periodic scattering problems are of interest to electrical engineers and physicists who apply them to photonic crystal lattices [11,24,14,18,29]. The propagation of waves through such lattices is sensitive to the crystals' geometry, their material properties, and on the nature of the incident waves.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A higher order numerical method for 3-d double periodic electromagnetic scattering problem

Nicholas¹

2008

Communications in Mathematical Sciences

View full text Add to dashboard Cite

Abstract. We develop a method for 3D doubly periodic electromagnetic scattering. We adapt the Müller integral equation formulation of Maxwell's equations to the periodic problem, since it is a Fredholm equation of the second kind. We use Ewald splitting to efficiently calculate the periodic Green's functions. The approach is to regularize the singular Green's functions and to compute integrals with a trapezoidal sum. Through asymptotic analysis near the singular point, we are able to identify the largest part of the smoothing error and to subtract it out. The result is a method that is third order in the grid spacing size. We present results for various scatterers, including a test case for which exact solutions are known. The implemented method does indeed converge with third order accuracy. We present results for which the method successfully resolves Wood's anomaly resonances in transmission.

show abstract

“…However, this is a routine calculation and such quantities may be determined to high accuracy using either the multipole method, as described in Ref. [5], or other techniques, including the finite element method [9] which has been used for the computations here. The testing of the exponential dependence of the defect frequency difference on δ ε c , when both are small, is computationally difficult, since the defect mode becomes arbitrarily extended, the more closely we search for modes near to the band edge.…”

Section: Numerical Detailsmentioning

confidence: 99%

Gap-edge Asymptotics of defect modes in 2D Photonic Crystals

Dossou¹,

McPhedran²,

Botten³

et al. 2007

Opt. Express

View full text Add to dashboard Cite

Abstract:We consider defect modes created in complete gaps of 2D photonic crystals by perturbing the dielectric constant in some region. We study their evolution from a band edge with increasing perturbation using an asymptotic method that approximates the Green function by its dominant component which is associated with the bulk mode at the band edge. From this, we derive a simple exponential law which links the frequency difference between the defect mode and the band edge to the relative change in the electric energy. We present numerical results which demonstrate the accuracy of the exponential law, for TE and TM polarizations, hexagonal and square arrays, and in each of the first and second band gaps. 915 -922 (1955).

show abstract

Finite element computation of grating scattering matrices and application to photonic crystal band calculations

Cited by 53 publications

References 37 publications

Photonic-crystal surface modes found from impedances

Photonic-crystal surface modes found from impedances

A higher order numerical method for 3-d double periodic electromagnetic scattering problem

Gap-edge Asymptotics of defect modes in 2D Photonic Crystals

Contact Info

Product

Resources

About