The Talbot Effect for two-dimensional massless Dirac fermions

A monochromatic beam of wavelength λ transmitted through a periodic one-dimensional diffraction grating with lattice constant d will be spatially refocused at distances from the grating that are integer multiples of . This self-refocusing phenomena, commonly referred to as the Talbot effect, has been experimentally demonstrated in a variety of systems ranging from optical to matter waves. Theoretical predictions suggest that the Talbot effect should exist in the case of relativistic Dirac fermions with nonzero mass. However, the Talbot effect for massless Dirac fermions (mDfs), such as those found in monolayer graphene or in topological insulator surfaces, has not been previously investigated. In this work, the theory of the Talbot effect for two-dimensional mDfs is presented. It is shown that the Talbot effect for mDfs exists and that the probability density of the transmitted mDfs waves through a periodic one-dimensional array of localized scatterers is also refocused at integer multiples of zT. However, due to the spinor nature of the mDfs, there are additional phase-shifts and amplitude modulations in the probability density that are most pronounced for waves at non-normal incidence to the scattering array.

away from the scattering array, where λ is the wavelength of light and d is the grating's lattice constant 1 . This self-refocussing of the scattered light intensity is now referred to as the Talbot effect. As first explained by Lord Rayleigh 2 , the Talbot effect is the result of constructive interference of a coherent wave scattered from a periodic array. Within the realm of optical physics, the Talbot effect has been used in a variety of applications in nanolithography 3 , optical metrology and imaging 4 , and light field sensors 5 . The Talbot effect has also been observed in experiments on matter waves 6 , electron beams 7,8 , plasmonic devices 9,10 , wave guides, and in photonic crystals 11 , along with a recent proposal 12 to look at a spin Talbot effect in a two-dimensional electron gases (2DEG).
Sir Michael Berry was the first to make a deeper connection between the physics of the Talbot effect and that of quantum revivals observed for confined quantum particles [13][14][15] , where an initial quantum wave packet undergoes spatiotemporal refocussing as a result of quantum interference. With the discovery of new materials that possess electronic structures that can be described by the relativistic Dirac equation, such as monolayer graphene 16 and the two-dimensional surface states of topological insulators [17][18][19] such as Bi 2 Se 3 , theoretical extensions of the Talbot effect to the case of relativistic quantum revivals were also performed [20][21][22] where it was shown that under certain conditions, bound relativistic particles with nonzero mass could also exhibit spatiotemporal revivals. From this theoretical work, however, it was not clear whether quantum revivals or, for that matter, the Talbot effect could exist for massless Dirac fermions (mDfs) since confining such particles is difficult due to Klein tunneling 23,24 . While recent numerical calculations 25 have shown that a Talbot effect can be present in two-dimensional phononic crystals with a dispersion relation that mimics the mDf dispersion relation, a full theory of the Talbot effect for mDfs is still lacking.
In this paper, we consider the relativistic analogue of Talbot's original experiment applied to a monochromatic beam of two-dimensional mDfs transmitted through a periodic one-dimensional potential. In order to place our theoretical results within a physically realizable context, we consider the particular case of intravalley multiple scattering in monolayer graphene 26 from a periodic array of localized scatterers as illustrated in Fig. 1. Our previous theoretical work 27 for the scattering of mDf waves from a one-dimensional periodic array of localized scatterers is generalized and used to demonstrate that a Talbot effect exists for mDfs. Furthermore, the effects of the mDfs' spinor nature on the predicted Talbot effect is shown to generate an additional amplitude modulation and phase shift in the probability density that is most pronounced for mDfs at non-normal incidence to the scattering array.

Results
We consider the case of a mDf wave in graphene with energy E = ħv F k 1 ≥ 0 and wave vector , that is incident to a one-dimensional array of localized, cylindrically symmetric, nonmagnetic scatterers as shown in Fig. 1. The subscript, ± K , is the valley index and denotes the corresponding Dirac point that the scattering solutions are expanded about. The transmitted wave function to the right of the scattering array (x ≫ d) is given by: , from a one-dimensional array of localized, cylindrically symmetric, nonmagnetic scatterers. In the Figure where the subscript ∈  m N {1, 2, } s denotes the particular scatterer in the n th unit cell.
, the dimensionless probability density for 2 , can be written as:    ( ) for m ≠ n, respectively. In particular, for a normally incident , and the periodicity along the x-direction in equation (2) and equation (3) can be used to define a set of "Talbot lengths", z m n Talbot ( , ) for |m| > |n| ≥ 0 and  ∈ m n , , which are given by: Scientific RepoRts | 6:26698 | DOI: 10.1038/srep26698 . The traditional Talbot distance defined by Lord . Thus the Talbot length for mDfs and achiral 2DEGs are identical to the traditional Talbot length. Furthermore, similar phase shifts, θ Δ m n T , in equation (2) for a mDf and θ Δ m n ac T , , in equation (3) for an achiral 2DEG, are both the result of the scattering potential, which is reminiscent of the phase shifts associated with the Talbot-Beeby effect 28 . However, due to the spinor nature of the mDfs, the cosine terms in equation (2) possess an additional amplitude factor of φ cos ( ) m n , spinor , and the last term in equation (2) also has an additional phase shift of that is due to the interference between the incident wave, φ ± r ( )  open scattering channels. Similarly, the reflected wave function, in equation (5), are given in Supporting Information. The dimensionless probability density to the left of the scattering . For comparison, a similar calculation of the dimensionless probability density in a 2DEG gives:   (6) and (7), the dimensionless probability densities for both mDfs and 2DEGs consist of a constant plus a sum over cosine terms that are periodic along both the ŷand x-directions with periods either given by spinor that are identical to those found in equation (2) for the transmitted wave. Furthermore, the interference between the incident wave and the n ≠ 0 reflected waves results in an amplitude relative to that found in an achiral 2DEG. As a result, a greater difference in the probability densities between mDfs and achiral 2DEGs will in general be observed for the reflected waves relative to that found for the transmitted waves. In Fig. 2, numerical calculations of the dimensionless probability densities for a mDf and an achiral 2DEG wave normally incident θ = . As seen in Fig. 2, similar periodic patterns in the probability density appear to the right of the scattering array in both the mDf and achiral 2DEG calculations whereas the  amplitude factors and phase shifts of − π θ found in the reflected mDf probability density in equation (2) lead to large differences in the probability to the left of the scattering array relative to that found in an achiral 2DEG [equation (6) vs. equation (7)]. The difference in probability densities between the mDF and an achiral 2DEG was most pronounced for the longest wavelength case λ = = π .
It is known from previous theoretical 26,29,30 and experimental 31-34 work that a particle's spinor nature can significantly affect the observed interference patterns of waves undergoing multiple scattering. However, the observed differences in the probability densities of an mDf and achiral 2DEG in Fig. 2 are due not only to the spinor nature of the mDfs but also due to differences in transmission and reflection coefficients, T n and R n for the mDf versus T n ac and R n ac for the achiral 2DEG. Therefore, to isolate the effects of the spinor nature of the mDfs on the probability density, we can replace T n ac and R n ac by T n and R n in the right hand sides of equation (3) and equa-    . In this case, the relative difference in probability density due solely to the spinor nature of the mDfs, χ, can be calculated using: Plots of the dimensionless probability densities and relative probability density differences, χ [equation (8)], for both regular and "spinless" mDfs waves scattering from the same scattering potentials used in Fig. 2 are given in Fig. 3 (at normal incidence, θ = At normal incidence (Fig. 3), the relative probability density difference to the right of the scattering array, which is mainly due to the cos(φ spinor ) amplitude factors in equation (2), is only significant over a small area. However, at non-normal incidence (θ = π  k 3 1 in Fig. 4), the probability densities are significantly different between the normal and "spinless" mDfs over a larger area, which is consistent with our theoretical predictions. In this case, the difference in probability density is due not only to the cos(φ spinor ) amplitude factors but also the phase shifts generated from the interference between the incident wave and the n ≠ 0 "open" transmitted/reflected waves in equations (2) and (6).
While the results in Figs 2-4 considered a scattering array with a unit cell consisting of a single scatterer, the theory developed in this work can also be applied to arbitrary scatterer configurations within a unit cell. In Figs 5 and 6, ν ψ  r ( )   Fig. 6. For both scatterer arrangements, the sizes of the scatterers were taken either to be equal [r s1 = r s2 = r s3 = r s4 = 4 nm in Figs 5b and 6b] or unequal [r s1 = r s4 = 4 nm, r s2 = 2 nm, and r s3 = 6 nm in Figs 5a and 6a]. The nonlinear arrangement of scatterers led to larger total transmission probabilities relative to the linear arrangement [T tot = 0.5514 in Fig. 6a vs. T tot = 0.3739 in Fig. 5a and T tot = 0.4611 in Fig. 6b vs. T tot = 0.232 in Fig. 5b]. For both types of scatterer arrangements, the total transmission probabilities were also larger when the scatterers were of unequal sizes. Finally, although the Talbot lengths, z m n Talbot ( , ) in equation (4), depend solely upon λ and d, the fine structure in ν ψ  r ( ) F 2 depends sensitively upon the details of the scatterer sizes, potentials, and arrangements within a unit cell, which ultimately determines the various transmission and reflection coefficients, T n and R n in equation (2) and equation (6), respectively.
Finally, we consider an incident wave scattering from a finite scattering array. In this case, previous theoretical work on multiple scattering from a finite number of scatterers 26,27 was applied to calculate ν ψ  r ( ) F 2 . In Fig. 7, ν ψ  r ( )  Fig. 2] cases, some of the finer structures/interference patterns observed in the infinite scattering array were absent for the finite scattering array. The periodic structures in the finite case also became blurrier with increasing distance from the scattering array, particularly at distances x ≫ 10d from the center of the scattering array. This was a consequence of the finite size of the scattering array whereby the interference patterns However, at distances within r Nd 2 from the center of the scattering array, a clear periodic pattern was still observed in the case of a finite scattering array.

Discussion
In this work, the theory of the two-dimensional Talbot effect for massless Dirac fermions (mDfs) was presented. It was shown that the Talbot effect for mDfs exists with Talbot lengths, z m n Talbot ( , ) in equation (4), that were identical to those found for an achiral two-dimensional electron gas (2DEG). The interference patterns seen in the Talbot effect are a result of coherent electron transmission of mDfs through the scattering array, whereby multiple scattering pathways constructively interfere at distances away from the scattering array determined by the periodicity of the scattering array. However, due to the spinor (or pseudospinor in the case of graphene) nature of mDfs, the periodic structures found in the probability density were both amplitude modulated and phase shifted relative to those found in an achiral 2DEG. Such differences were most pronounced for mDf waves at non-normal incidence to the scattering array. Numerical calculations on finite scattering arrays demonstrated that periodic structures in the probability density still exist but that these structures decay with increasing distance from the scattering array. While the probability density is independent of which valley point the scattering states are expanded about [equations (2) and (6)], the use of magnetic scatterers could potentially be used to distinguish the chirality (or in this case, valley index ± K) of the incident waves in monolayer graphene. The mDf Talbot effect predicted in this work should be observable in systems like monolayer graphene and on the surfaces of topological insulators, where phase coherence lengths greater than 5 μm and 1 μm have been experimentally observed in graphene 35 and . The scatter sizes were either chosen to be either (b) equal with r s1 = r s2 = r s3 = r s4 = 4 nm, which resulted in T tot = 0.2320 or (a) unequal with r s1 = r s4 = 4 nm, r s2 = 2 nm, and r s3 = 6 nm, which resulted in T tot = 0.3739. In both calculations, l max = 6 was chosen. . The scatter sizes were chosen either to be (b) equal with r s1 = r s2 = r s3 = r s4 = 4 nm, which resulted in T tot = 0.4611, or (a) unequal with r s1 = r s4 = 4 nm, r s2 = 2 nm, and r s3 = 6 nm, which resulted in T tot = 0.5514. In both calculations, l max = 6 was chosen. topological insulators 36 , respectively. Overall, this work provides yet another example of the fruitful analogy between traditional optics and coherent "electron" optics in graphene and similar systems [37][38][39][40] .
While there exist proposals 9,10 to employ the Talbot effect for nonrelativistic electrons in plasmonic devices, the theory presented in this work could be used as a starting point for designing and understanding the Talbot effect in graphene and topological insulator 41,42 plasmonic devices. It should also be noted that only coherent dynamics was considered in this work. Spatial and spin/pseudospin decoherence, however, will attenuate and destroy the Talbot effect with increasing distance from the scattering array. As a result, comparing the observed spatial decay of the interference patterns in the Talbot carpet with the interference patterns calculated using the theory presented in this work could provide valuable information about both spatial decoherence 43 and spin/ pseudospin decoherence in two-dimensional mDfs.

Methods
The basic results for intravalley scattering of a plane wave incident to a one-dimensional array of localized scatterers in graphene (as illustrated in Fig. 1) and in a 2DEG are derived in Supporting Information 26,27,[44][45][46] . The overall theoretical formalism used in this paper represents a generalization of the case of a single scatterer per unit cell 27 to the case of multiple scatterers within a unit cell. From Fig. 1 denotes the position of the m th scatterer in the n th unit cell, V m and r sm are the potential and radius of the m th scatterer, respectively, and Θ  r ( ) r sm is Heaviside step function given by: r sm sm sm In this work, the potentials of the individual scatterers were taken to be identical in order to avoid the confounding effects of electric fields between the scatterers, i.e., V m = V for all m ∈ {1, 2, … , N s }. The l th partial wave scattering amplitude from the m th cylindrically symmetric scatterer, s m,l with ∈  m N {1, , } s , is given by 26      the scattering solutions for both a mDf and a 2DEG are given in Supporting Information. Finally, all calculations shown in Figs 2-7 were carried out using in-house MATLAB (Mathworks) programs.