[email protected] SPATYPE An introduction to spatial dispersion http://www.kinsler.org/physics/ An introduction to spatial dispersion: revisiting the basic concepts Paul Kinsler∗ Cockcroft Institute, Keckwick Lane, Daresbury WA4 4AD, United Kingdom Physics Department, Lancaster University, Lancaster LA1 4YB, United Kingdom and Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom (Dated: Monday 15th February, 2021) I describe three ways that the spatial properties of a wave propagation medium can cause dispersion, and pro- pose that they should form the basics for correctly understanding and naming phenomena described as “spatial dispersion”. In particular, I emphasise the specific spatial properties which generate the resulting dispersive – i.e. spatially dispersive – behaviour. The properties are geometry, structure, and dynamics. I. INTRODUCTION considered topic of temporal dispersion. The first two spa- tial dispersion mechanisms depend on inhomogeneous spatial properties, whereas the third derives from propagation-based Spatial dispersion is the non-local dependence of material properties. properties on both direction and wavelength, and is of particu- The categorization here shows that instances of spatial lar interest in electromagnetic systems and other fields where dispersion can be caused by the geometric, structural, or dy- artificial functional materials (AFMs) are finding technolog- namic properties of a system. It is not based on existing (and ical uses, such as in acoustic and elastic metamaterials. It often somewhat ad hoc) naming conventions or justifications has been long observable in crystals [1], but is also of partic- for spatial dispersion, but is instead grounded in the following ular importance in AFMs where the wavelength of radiation assertion: becomes comparable to the lattice parameters (see e.g. [2]). Further, it is also significant in the region of the material res- Spatial dispersion refers to any dispersive behaviour that onances utilized in metamaterial unit cells, where constitutive occurs (solely) as a result of spatial properties of the system. parameters such as the permittivity and permeability – and likewise their acoustic or elastic counterparts – are maximised This viewpoint is distinct from a widely used one (see e.g. or have near-zero local values [3] or where the response is [5]) where spatial dispersion is characterised as being due non-reciprocal [4]. In some materials its presence is revealed to (non-trivial) wavevector-dependence in the dispersion re- by the shape of the equi-frequency surfaces of the dispersion lations, and is then categorized into two cases ‘weak’ spatial relations – e.g. when they are non-symmetric, not elliptical or dispersion, where the spatial effects have been approximated hyperbolic, or having multiple modes with the same direction as a local effect; and ‘strong’ spatial dispersion, where non- and polarisation. local effects remain important [6, 7]. It is to be noted that Spatial dispersion usually appears as a either spatially non- so-called weak spatial dispersion is not necessarily small in local effect that produces a wavevector dependence of the ma- any practical sense, despite the implications of ‘weak’; and terial parameters, or as a non-trival wavevector dependence likewise so-called strong spatial dispersion need not be par- for the dispersion relations. Since representing spatial proper- ticularly dominant1. ties in a fourier space (here, that of wavevector k ) necessar- ily involves an integral over spatially separated points, spatial In particular, in the interest of both brevity and not repeating dispersion is intrinsically non-local. existing discussions in the literature, I do not: Although spatial dispersion can be modelled in an ad hoc 1. discuss how homogenization schemes might reduce the manner to suit intuition or an empirical fit to data, it is prefer- description of a complicated medium – such as those able to be more systematic. However, the term “spatial disper- with spatial varying temporal responses – into a simpler sion” is often used rather loosely, and so sometimes the origin one, (e.g.) perhaps one based on a series expansion in of the specific phenomenon being discussed is unclear. In an terms of the wavevector k; attempt to rectify this, here I describe three physically distinct ways that spatial dispersion can arise, briefly describing exam- 2. consider dispersion relations that have been artificially ples of each. Each of these mechanisms has (a) no time depen- constructed without reference to some specific physi- dence beyond that required to support a wave, and (b) a spec- cal system with spatial properties; e.g. putative disper- ification of time-independent material properties. The lack of sia based on suggested or “what-if” polynomials in fre- arXiv:1904.11957v4 [physics.optics] 12 Feb 2021 any (non-trivial) time dependence ensures that the phenom- quency w and wavevector k, some other specification ena treated here are entirely unrelated to the more commonly 1 In my view, it is preferrable for terms such as ‘weak’ or ‘strong’ to be only ∗https://orcid.org/0000-0001-5744-8146; used to to indicate the strength or significance of a phenomenon, not its [email protected] origin or type, or whether it might be effectively local or non-local. 1 [email protected] SPATYPE An introduction to spatial dispersion http://www.kinsler.org/physics/ of an w, k interdependence, or a non-local convolution over spatial properties. The main focus here is on spatial dispersion in electromag- w netic systems or artifical functional media such as metamateri- als, but the distinctions apply equally well to waves in acoustic or elastic materials, or indeed potentially any material system which supports wave propagation of some kind. In Section II, I will define what I mean by dispersion (i.e. specifically non- trivial dispersion) by considering the relationships between the frequency w of a wave and the wavevector k associated with that frequency. Following that the next three sections address each of geometric (Section III), structural (IV), and dynamic (V) spatial dispersions in turn. Note that the idea of k spatial dispersion as being grounded in spatial properties as opposed to (spectral) wavevector ones is mirrored in the var- FIG. 1: Dispersionless case, where the dispersion relation is linear, ious treatments of temporal dispersion: although sometimes passes through and includes the origin, and is continuous. The light seen as a consequence of a dynamic time-domain process [8] dotted line indicates that the depicted dispersion behaviour continues particularly when implemented in FDTD algorithms [9], tem- in a similar manner for higher wavevectors and frequencies. poral dispersion is more often discussed in solely terms of a (spectral) frequency response [10, 11]. Of course, in practical terms the categorization of dispersion types is really applied If all these criteria hold, then the wave is dispersionless, to our models or representations of the physical system, and and the phase and group velocities are always the same. If not some perfectly accurate microscopic view of the actual any of the criteria do not hold, then the wave is dispersive, physical system, and this is discussed in Section VI. and an initial pulse shape will change over time as its differ- In what follows, I do not describe the example systems in ent spectral components evolve at different rates. In general, a detailed manner. Only the minimum features necessary to this behaviour might be due to either the time-response of the make the point are presented, as such models are often worked medium (giving rise to temporal dispersion), or to the spatial through both exhaustively and frequently in other sources. properties of the medium (giving rise to spatial dispersion), or Here their role is simply to provide examples that typify their possibly both. role in creating a dispersive response as a result of their par- Here I focus exclusively on dispersion that results solely ticular spatial features or structure. Nonlinear effects are not from spatial properties. However, it is worth keeping in mind considered. that typically the structural elements relied on in a physical device to create those spatial properties may in reality also be temporally dispersive – notably, reflecting walls will not be II. DISPERSION perfectly reflective at all frequencies, and different refractive indices exist primarily because of temporal dispersion; some The definition of dispersion used here is based on three cri- of these issues are discussed in section VI. In what follows teria relating to how the frequency ( f or w) of a wave is related I ignore such complications because I wish to focus solely to its wavelength l or wavevector k. Note that in discrete sys- on spatial properties and their effects; but this is not to say tems, analytic solutions typically depend on an integer index that such time-domain responses may always be ignored in m (and possibly more indices as well). Since these indices practical situations. usually indicate the number or spatial frequency of the os- A dispersionless wave travelling along the z–axis with cillation in a given solution, they play the same role as the propagation speed c follows the wave equation wavevector k, and contain essentially the same physical con- 2 2 tent. In simple cases, the wavevector k can be a straightfor- ¶t E − ¶zc ¶zE = 0; (1) ward multiple of m; in any case here the wavevector associ- where ¶t ≡ d=dt, and ¶z ≡ d=dz. This equation can be Fourier ated with an index m is km, and likewise the related angular transformed in both time and space, and the field strength frequency is wm. The criteria are: terms cancelled, giving the “dispersionless” dispersion rela- tion Linear – the relationship between w and k (or between w and w2 − c2k2 = 0: (2) m or km) is linear, i.e. it lies along a single straight line. A depiction of this relationship between w and k can be seen Origin – if w = 0 then k = 0 (or m = 0), and vice versa.