Optics Communications 321 (2014) 96–99
Contents lists available at ScienceDirect
Optics Communications
journal homepage: www.elsevier.com/locate/optcom
Photonic Zitterbewegung effect: Asymmetric spatio-temporal filtering
near the Dirac point
Xiaohui Ling a,b,n, Zhixiang Tang a,n, Liezun Chen b
a
Key Laboratory for Micro-/Nano-optoelectronic Devices of Ministry of Education, College of Information Science and Engineering, Hunan University,
Changsha 410082, China
b
Department of Physics and Electronic Information Science, Hengyang Normal University, Hengyang 421002, China
art ic l e i nf o
a b s t r a c t
Article history:
Received 19 November 2013
Received in revised form
13 January 2014
Accepted 31 January 2014
Available online 7 February 2014
We present a classical explanation of the photonic Zitterbewegung (ZB) effect near the Dirac point in this
paper. Due to the asymmetric transmitted spectrums near the Dirac point, a material with the Dirac point
actually can function as a spatio-temporal filter. When an optical pulse with its spectrum centered near
the Dirac point passing through the “filter”, its spatio-temporal spectrum will be modified dramatically.
As a result, the temporal shape of the transmitted pulse is distorted with tailed oscillations, i.e., the
photonic ZB effect. The influence of the temporal and spatial widths of the input pulse on the tailed
oscillations of the transmitted pulse has also been discussed. Our results may pave the way to the
experimental research of the photonic ZB effect and provide a viewpoint for investigating other abnormal
transmission properties near the Dirac point.
& 2014 Elsevier B.V. All rights reserved.
Keywords:
Optical Zitterbewegung
Dirac point
Spatio-temporal filtering effect
1. Introduction
Recently, the intensive investigation of some two-dimensional
materials, such as graphene and topological insulator, refreshes
interests of the Dirac point where the conduction and valence
bands touch each other, forming a pair of cones [1–4]. Near the
Dirac points, the dispersion is linear and the valence electron
dynamics is governed by the massless Dirac equation. Analogous
to the electronic system, in the optical context, the Dirac point has
also been realized in photonic crystals and metamaterials [5–13].
Many abnormal transmission properties have been revealed such
as one-way waveguides [14,15], pseudo-diffusive scaling [16–19],
and conical diffraction [20]. Of particular interest is the photonic
Zitterbewegung (ZB) effect [5,21,22].
The ZB effect referring to the rapid trembling motion of a free
Dirac electron was first proposed by Erwin Schrödinger, and the
interference between the positive and negative energy states was
considered as the origin. Due to a formal similarity between the
double-cone bands in solid and the Dirac equation for relativistic
electrons in vacuum, some controllable physics systems, such as
trapped ions, ultra cold atoms, and graphene, have been proposed
as candidates for directly observing the ZB effect [23–26]. But
experimental observation of ZB in these systems is still a challenge
because of the difficulty to maintain a homogeneous electron
n
Corresponding authors.
E-mail addresses: xhling@hnu.edu.cn (X. Ling), tzx@hnu.edu.cn (Z. Tang).
http://dx.doi.org/10.1016/j.optcom.2014.01.073
0030-4018 & 2014 Elsevier B.V. All rights reserved.
density throughout the system [27]. For the optical system, no
such difficulty exists. The ZB effect for optical pulses in twodimensional photonic crystals [5], zero-index metamaterials [21]
and binary waveguide arrays [22] has been demonstrated, which
offers a direct analogy of the ZB in the electronic system. For
optical pulse, the ZB effect manifests itself as a temporally tailed
oscillation [5,21], and the origin is attributed to the interference of
the two components of the pulses belonging to negative and
positive refractive indices.
In this work, we present a classical explanation of the photonic ZB
effect. We attributed the photonic ZBs, i.e., pulses with tailed
oscillations, to the asymmetric spatio-temporal filtering near the
Dirac point. Due to the asymmetrical transmittance spectrum distribution near the Dirac point, a material with such point functions as
a spatio-temporal filter. So, when a pulse propagates through the
“filter”, its spatio-temporal spectrum will be modified dramatically,
which results in the modifications of the temporal shape of the
output pulses and thus producing the ZB effect.
2. Theoretical analysis
There are mainly two kinds of Dirac points in metamaterials
and photonic crystals that have been discussed till now. The first
one is that two bands touch at the center of Brillouin zone
(Γ point) [6–11]. Under the circumstance, the refractive index
increases with frequency from negative to positive. The other is
that the touched point is located at other high symmetry points,
X. Ling et al. / Optics Communications 321 (2014) 96–99
such as K point in two-dimensional triangular-lattice photonic
crystals [5,18,19]. For simplicity, we take a homogenous zero-index
metamaterial (ZIM) as the analysis sample, i.e., the first one as an
example to illustrate our explanation, and the method discussed
here can be extended to the other one. This kind of metamaterial
has already been demonstrated by theoretical and experimental
researches based on liquid crystals and Ω-shaped microstructures
from GHz to visible regions [28–31]. The Drude model is chosen as
the parameter of the permittivity and the permeability of the ZIM
slab:
ϵðωÞ ¼ 1 ω2pe
;
ω2 þ iγ e ω
μðωÞ ¼ 1 ω2pm
;
ω2 þiγ m ω
ð1Þ
where ωpe and ωpm are electronic and magnetic plasma frequencies,
and γe and γm are the damping rates. As shown in [30], with suitable
structure parameters, ωp ¼ ωpe ¼ ωpm can be achieved in a back-toback Ω-shaped metamaterial in the GHz region. When
ω ¼ ωp ¼ ωpe ¼ ωpm and γ e ¼ γ m b ωp , ϵðωÞ and μðωÞ are nearly
equal topzero,
simultaneously. Thus, the effective refractive index
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 ðωÞ ¼ ϵðωÞμðωÞ ¼ 1 ω2p =ðω2 þ iγωÞ of the sample varies from
negative, through zero, then to positive with the increase of frequency.
And the Dirac point is the zero point of effective refractive index.
We consider a paraxial, coherent, TE-polarized, spatially and
temporally Gaussian pulse injecting into the sample along z
direction in vacuum as shown in Fig. 1. It is assumed that the slab
sample is infinite in the xy plane and the thickness of the sample is
h. The spectral function of the pulse at the initial plane z ¼ z0 o 0
[while plane z¼0 is the interface between vacuum (n0 ¼1) and the
sample] is given as
"
#
!
2
Γ 2 ð ω ω0 Þ 2
kx W 20
Ei ðkx ; ω; z0 Þ ¼ A exp exp ;
ð2Þ
4
4
where Γ is the temporal half-width, W0 is the spatial half-width,
kx ¼ ðn0 ω=cÞ sin θ, c is the light velocity in vacuum, ω0 is the
central angular frequency of the pulse, and A is a constant.
The transmission coefficient tðθ; ωÞ ¼ jtðθ; ωÞjexp½iφðθ; ωÞ of
the sample can be deduced from the electromagnetic continuity
conditions of the two interfaces (plane z¼ 0 and z¼ h in Fig. 1) [32].
y
x
z
z=0
h
vacuum
ZIM slab
z=h
vacuum
Fig. 1. Schematic of the ZIM slab in vacuum with the thickness h.
97
For the TE-polarized incident wave, tðθ; ωÞ can be written as
tðθ; ωÞ ¼
2
:
cos θ1
cos θ
þ
2 cos δ i sin δ
cos θ
cos θ1
ð3Þ
Here, θ1 ¼ arcsin½n0 sin θ=n1 ðωÞ is the refractive angle of the
angular spectrum component in the sample which can be obtained
from Snell's law and δ ¼ ωn1 ðωÞh cos θ1 =c stands for the phase
change when electromagnetic waves cross the sample [32].
In this case, the transmission coefficient is equivalent to the
frequency response of a linear filter. Therefore the spectral function at the exit end can be written as Et ðθ; ω; zÞ ¼ Ei ðθ; ω; 0Þtðθ; ωÞ.
Ei ðθ; ω; 0Þ ¼ Ei ðθ; ω; z0 Þexpð ikz z0 Þ is the function of spatio-temporal spectrum of the pulse arriving at the plane z ¼0, where
kz ¼ k0 cos θ (k0 ¼n0ω/c is the wave vector in vacuum). We take
the spatial and temporal half-width of the incident pulse as 10λ0
(λ0 ¼c/f0 is the central wavelength of the pulse in vacuum) and
1 ns, respectively, with the central frequency f0 (¼ 10 GHz) for an
example. The spatio-temporal spectrum of the incident pulse and
the transmittance spectrum distribution jtðθ; ωÞj with slab thickness h ¼40λ0 are plotted in Fig. 2(a) and (b), respectively. Unambiguously, the output spectrum is tailored by the transmittance
spectrum [Fig. 2(c)]. Part of the spatio-temporal spectrum components are filtered by the stop-band of the transmittance spectrum.
In order to obtain the spatio-temporal shape Et ðx; t; zÞ of the
transmitted pulse, the inverse Fourier transform of Et ðθ; ω; zÞ has
been calculated numerically [33]. In our analysis, we only consider
the temporal shape of the transmitted pulses (we consider
Et ðx ¼ 0, t, z)) because the so-called ZB effect here manifests as
the temporally tailed oscillations.
3. Discussions
As is shown in Fig. 2(b), the pass-band of the transmittance
spectrum gradually shrinks to the Dirac point (zero-index point)
from both sides of it, forming a double-cone band. This is due to the
fact that when the frequency approaches the Dirac point from both
sides (positive and negative index regions), the critical angle of the
total reflection at the metamaterial interface z¼0 decreases, thereby
only normal incident angular spectra of the pulse can pass through
the sample at the zero-index point, which can be deduced from
Snell's law. More importantly, the transmittance spectrum is asymmetrical between negative and positive refractive index regions in
respect that n1 ðωÞ ¼ 1 ω2p =ðω2 þ iγωÞ is asymmetrical for the two
regions. The asymmetrical transmittance spectrum gives an asymmetrical filtering effect to the incident spatio-temporal spectrum.
This asymmetry in frequency domain is the origin of the tailed
Fig. 2. (a) The spatio-temporal spectrum of incident Gaussian pulse centered at f0 ¼ 10 GHz with Γ ¼1 ns and W 0 ¼ 10λ0 (λ0 ¼ 0.03 m). (b) Distribution of the spectral
transmittance jðtðθ; ωÞjÞ for a sample with h ¼ 40λ0 and γ ¼ γ e ¼ γ m ¼ 104 Hz, the blue part denotes the stop-band and the rest denotes the pass-band. At the cross point, i.e.,
the Dirac point, the refractive index is zero. (c) The spatio-temporal spectrum of the transmitted pulse.
98
X. Ling et al. / Optics Communications 321 (2014) 96–99
1
f0 = 10 GHz
0.8
f0 = 10.1 GHz
0.6
f0 = 9.9 GHz
0.4
0.2
0
2
4
6
8
Time t (ns)
Relative intensity (a.u.)
Relative intensity (a. u.)
1
10
0.8
Γ=0.5ns
Γ=1.0ns
0.6
Γ=1.5ns
0.4
0.2
0
0
2
4
6
Time t (ns)
Fig. 3. Relative intensity of the transmitted pulse with the central frequency
10 GHz, 10.1 GHz, and 9.9 GHz, respectively, while h ¼40λ0, Γ ¼ 1 ns, and
W 0 ¼ 10λ0 are given.
8
oscillations in time domain. In a sense, a pulse shape with tailed
oscillations could be seen as a kind of asymmetry. At the same time,
the transmittance spectrum is symmetrical for negative and positive
θ, deduced from Eq. (3) and see Fig. 2(b). Thus, the spatial shape of
the pulse is symmetrical.
It is worth noting that the temporally tailed oscillations will
also arise when the central frequency of the pulse diverges from
the Dirac point, on condition that the asymmetrically filtering
effect still holds. In Fig. 3, we plot the relative intensity
jðEt ðx ¼ 0; t; zÞ=Et ðx ¼ 0; t; zÞmax j2 of the transmitted pulses with
the central frequency 9.9 GHz, 10 GHz, and 10.1 GHz, while
h¼40λ0, Γ ¼1 ns, and W 0 ¼ 10λ0 are given. The initial position
of the pulse is assumed at z0 ¼ 20λ0 . As one can see, the tailed
oscillations are different from each other since they experience
different spatio-temporal filtering effects for the three cases.
Pulse with different temporal half-width Γ will undergo different
responses when propagating in the ZIM slab sample. The incident
spatio-temporal spectrum varies with Γ, thus results in different
filtering effects. Near the Dirac point, the dispersion is approximately
linear [5–7], however, the linear dispersion approximation is not
valid when Γ becomes very small (few optical cycles in time domain
but extremely large bandwidth in frequency domain), and the highorder dispersion (mainly three-order dispersion) which causes the
tailed oscillation of pulse to dominate [34]. The tailed oscillation
is actually very similar to that resulted by the ZB. In Fig. 4(a), we
plot the relative intensity of the transmitted pulses with Γ ¼0.5 ns,
1.0 ns, and 1.5 ns, while W 0 ¼ 10λ0 and h ¼ 40λ0 are fixed. Fig. 4
(b) illustrates the relationship between the oscillating strength (the
ratios of relative intensities between the tailed peak and the main
peak) and Γ. As Γ o0:8 ns (only several optical cycles), the oscillating strength varies sharply due to the influence of high-order
dispersions. While for Γ 40:8 ns, the high-order dispersions are
negligible and linear dispersion dominates, so the oscillating strength
has a relatively flat dependence on Γ in the framework of spatiotemporal filtering effect alone.
Though the ZB effect occurs in time domain, the spatial width
of the pulse also influences the transmittance spectrum and
thereby the temporally tailed oscillation. When W0 increases, the
spatial part of the incident spatio-temporal spectrum becomes
narrower and narrower [θ-axis direction in Fig. 2(a)]. If we
consider the limiting situation, that is, W0 tends to infinity, the
pulse has only the angular spectra in the normal incident direction
(plane wave pulse), and no spectrum component would be filtered
by the transmittance spectrum of the sample. So the temporally
tailed oscillations would decrease with the increase of W0, and be
too weak (nearly vanished) to observe as long as W0 becomes
Oscillating strength
1
0.8
0.6
0.4
0.2
0
2
4
Γ (ns)
6
Fig. 4. (a) Relative intensity of the transmitted pulse with different temporal halfwidth Γ, while W 0 ¼ 10λ0 and h ¼ 40λ0 are given. (b) Oscillating strength versus Γ.
enough large. In Fig. 5, we plot the relative intensity of the
transmitted pulses with W 0 ¼ 10λ0 , 30λ0, and 30λ0, while Γ ¼1.0 ns
and h ¼ 40λ0 are given. The oscillating strength decreases rapidly,
and finally disappears, which agrees with the above analysis.
It should be pointed out that, although we have only discussed
the photonic ZB effect in metamaterial, similar phenomena in twodimensional photonic crystals could be discussed in the same way
since an extremely similar transmittance spectrum in a twodimensional hexagonal photonic crystal near the Dirac point has
been revealed [19]. Our analysis can be extended to the case as
long as the double-cone transmitted spectrum distribution exists,
and may hold potential for optical pulse shaping.
4. Conclusions
Based on the spatio-temporal filtering analysis, we have presented a classical explanation for the photonic ZB effect. When a
symmetrical pulse passes through a planar material with Dirac
points, its spatio-temporal spectrum will be filtered asymmetrically,
which results in the photonic ZB effect. The influence of the central
frequency f0, temporal half-width Γ, and the spatial half-width W0 of
the pulse on the ZB effect has also been discussed. Besides, we find
that the tailed oscillation may be produced as long as the asymmetric
filtering effect exists. Although we have only considered the Dirac
point in a metamaterial in this paper, our method can be extended to
the Dirac point in photonic crystals. Our research not only provides a
viewpoint for understanding the photonic ZB effect, but also may
X. Ling et al. / Optics Communications 321 (2014) 96–99
References
Relative intensity (a.u.)
1
0.8
W0 =10λ0
0.6
W 0=30λ0
W 0=50λ0
0.4
0.2
0
0
2
4
6
Time t (ns)
8
0.2
Oscillating strength
99
0.1
0
0
20
40
60
W0 (λ0)
Fig. 5. (a) Relative intensity of the transmitted pulse with different spatial halfwidth W0, while Γ ¼ 1.0 ns and h ¼ 40λ0 are given. (b) Oscillating strength versus
W0.
pave the way to the experimental research of the other abnormal
transmission properties near the Dirac point, such as one-way
waveguides, pseudo-diffusive scaling, and conical diffraction.
Acknowledgments
This work was supported by the National Natural Science
Foundation of China (Grant no. 11076011), the Scientific Research
Fund of Hunan Provincial Education Department of China (Grant no.
13B003), and the Construct Program of the Key Discipline in Hunan
Province.
[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos,
I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666.
[2] A.K. Geim, K.S. Novoselov, Nat. Mater. 6 (2007) 183.
[3] Y.L. Chen, J.G. Analytis, J.-H. Chu, Z.K. Liu, S.-K. Mo, X.L. Qi, H.J. Zhang, D.H. Lu,
X. Dai, Z. Fang, S.C. Zhang, I.R. Fisher, Z. Hussain, Z.-X. Shen, Science 325 (2009)
178.
[4] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y.S. Hor,
R.J. Cava, M.Z. Hasan, Nat. Phys. 5 (2009) 398.
[5] X. Zhang, Phys. Rev. Lett. 100 (2008) 113903.
[6] L.G. Wang, Z.G. Wang, J.X. Zhang, S.Y. Zhu, Opt. Lett. 34 (2009) 1510.
[7] X. Huang, Y. Lai, Z. Hang, H. Zheng, C.T. Chan, Nat. Mater. 10 (2011) 582.
[8] X. Chen, L.G. Wang, C.F. Li, Phys. Rev. A 80 (2009) 043839.
[9] V. Yannopapas, A. Vanakaras, Phys. Rev. B 84 (2011) 045128.
[10] V. Yannopapas, Phys. Rev. B 83 (2011) 113101.
[11] K. Sakoda, Opt. Express 20 (2012) 3898.
[12] V. Yannopapas, New J. Phys. 14 (2012) 113017.
[13] A. Christofi, N. Stefanou, Phys. Rev. B 87 (2013) 115125.
[14] F.D.M. Haldane, S. Raghu, Phys. Rev. Lett. 100 (2008) 013904.
[15] S. Raghu, F.D.M. Haldane, Phys. Rev. A 78 (2008) 033834.
[16] R.A. Sepkhanov, Ya.B. Bazaliy, C.W.J. Beenakker, Phys. Rev. A 75 (2007) 063813.
[17] R.A. Sepkhanov, C.W.J. Beenakker, Opt. Commun. 281 (2008) 5267.
[18] X. Zhang, Phys. Lett. A 372 (2008) 3512.
[19] M. Diem, T. Koschny, C.M. Soukoulis, Physica B 405 (2010) 2990.
[20] O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, D.N. Christodoulides,
Phys. Rev. Lett. 98 (2007) 103901.
[21] L.G. Wang, Z.G. Wang, S.Y. Zhu, Europhys. Lett. 86 (2009) 47008.
[22] S. Longhi, Opt. Lett. 15 (2010) 235.
[23] L. Lamata, J. León, T. Schäz, E. Solano, Phys. Rev. Lett. 98 (2007) 253005.
[24] J.Y. Vaishnav, C.W. Clark, Phys. Rev. Lett. 100 (2008) 153002.
[25] J. Cserti, G. Dávid, Phys. Rev. B 74 (2006) 172305.
[26] T.M. Rusin, W. Zawadzki, Phys. Rev. B 76 (2007) 195439.
[27] E.A. Kim, A.H. Castro Neto, Europhys. Lett. 84 (2008) 54007.
[28] I.C. Khoo, D.H. Werner, X. Liang, A. Diaz, B. Weiner, Opt. Lett. 31 (2006) 2592.
[29] D.H. Werner, D.H. Kwon, I.C. Khoo, A.V. Kildishev, V.M. Shalaev, Opt. Express
15 (2007) 3342.
[30] F. Zhang, G. Houzet, E. Lheurette, D. Lippens, M. Chaubet, X. Zhao, J. Appl. Phys.
103 (2008) 084312.
[31] J.A. Bossard, X. Liang, L. Li, et al., IEEE Trans. Antennas Propag. 56 (2008) 1308.
[32] M. Born, E. Wolf, Principles of Optics, 7th edition, Cambridge University Press,
Cambridge, 1999.
[33] The two-dimensional fast Fourier inverse transform is employed to calculate
the spatio-temporal shape of pulse. According to the sampling theorem, a
suitable frequency mesh is chosen to satisfy the convergence.
[34] G.P. Agrawal, Nonlinear Fiber Optics, 4th edition, Academic Press, San Diego,
2007.