Twisting Electromagnetic Fields with Singular Transformation Optics

0.55 MB·English

by  Yong-Liang Zhang

Twisting Electromagnetic Fields with Singular Transformation Optics
Yong-Liang Zhang1, Li-Na Shi2, Xian-Zi Dong1, Zhen-Sheng Zhao1 and Xuan-Ming Duan1∗
1 Key Laboratory of Functional Crystal and Laser Technology,
Technical Institute of Physics and Chemistry, Chinese Academy of Science, Beijing 100190, China and
2 Key Laboratory of Nano-Fabrication and Novel Devices Integrated Technology,
Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China
Transformation optics is a powerful method for manipulating the electromagnetic fields with
metamaterials which resembles a optical Riemannian space for photons. Analogous to the theory
oftopological defects,weproposeageometricextensionoftransformation opticsbyintroductionof
linear topological defect which produces spatial torsion. In order to illustrate the optical effect of
3
singulartransformationoptics,wepresentanexampleofapplicationtocontrolthetwistingdegreeof
1
0 freedomoflight,theorbitalandspinangularmomentumoflightwithanopticalcosmicstring. Our
2 results provide a systematic and versatile framework which may open intriguing venues in angular
momentum engineering of light with artificial metamaterials.
n
a
J
Introduction.—In recent years, transformation optics over,the vastmajorityofreportedTO designsarebased
9
(TO) has attracted a great deal of attention because it holonomic coordinate transformationwhich preserve the
2
promises a variety of intriguing applications from invisi- topology between the virtual and physical spaces. Here

] bilitycloaking,electromagneticwormholestobroadband weexploretheextensionofTOofPendryandLeonhardt
s
lightharvesting[1–5,8–12]. Theunderlyingphysicalidea to include spatialtorsionand investigatethe physicalef-
c
i of TO stems from the formal invariance of the govern- fects of torsional media on the propagation of electro-
t
p ing Maxwell’s equations [1] and Helmholtz equation [2] magnetic waves,focusing on the generationand physical
o under a spatial coordinate transformation. The consti- effects of electromagnetic torsion by anholonomic coor-
.
s tute parameters of materials ε and µ are renormalized dinate transformation. Inspired by the theory of topo-
c [6]by the transformation’sJacobianmatrixwhichrepre- logical defects in crystalline solid or gravitation [24–29],
i
s sents the metric tensor of the transformed space. Alter- we introduced torsion in terms of the spatial gradient
y
nativeapproacheshavealsobeenproposed,basedon,e.g. of the inverse lattice deformation in subwavelengthscale
h
space-timetransformations[13],directfieldstransforma- andnaturallyarisesinMaxwellequationsintransformed
p
[ tions[14],andspectradomain[15]transformationswhich space as magnetoelectric coupling terms. In addition,

generalize the transformation media to include moving, anholonomic coordinate transformation in macroscopic
1
bianisotropic and nonreciprocalmedia. scale can transform light beams with planar wave front
v
4 So far the rapid growing development has proved TO tobeamswithhelicalwavefront. Numericalfullwavecal-
5 asapowerfulandwell-establishedtoolforthesystematic culations basedon a specific example oflinear screw dis-
9 design of artificial metamaterials with specified electro- locations confirm the theoretical predictions. Our model
6
magnetic energy flow distributions, and the same idea notonly extendsthe geometricalandtopologicalaspects
.
1 has made important contributions to advancing the ma- of TO but also provides an experimental approach for
0 nipulation of sound wave [16], matter wave [17], dc elec- simulating of classical and quantum physics in curved
3
tric current [18] and even heat flux [19]. It is usually torsional space in optical laboratory.
1
: accepted that the inhomogeneous optical material acts Topology and geometry of TO.—Similar to the elastic
v as a pseudo-Riemannian manifold, where the propaga-
deformation of perfect crystal, method of TO relies on a
i
X tion ofphotons follows nullgeodesic determined by local spatial coordinate transformation from a prescribed do-
spatial curvature [7, 20–22]. However, the vector nature
r main in Cartesian coordinate system to a deformed one.
a of electromagnetic waves require some extensions of the
The topology of TO will change if the spatial continuity
standardTOtoincludethetwistingdegreeoffreedomof
isbrokenunderthedeformation. Ourproposedapproach
light,suchasopticalspinandorbitalangularmomentum
is based on the anholonomic coordinate transformation
(AM). Horsley [30] has earlier shown that an inhomoge-
inthe spatialdomain,whichis often indicatedthe emer-
neous isotropic Tellegen medium can be interpreted as a
gence of topological defects.
Weitzenb¨ockspacewherethechiralparameterisassigned
Asastartingpoint,weconsideringacoordinatetrans-
with the contorsion tensor.
formation from an initial region D in the flat space,
Although the approaches in [14, 30] seem to capable
identified by r ≡ {xa|(x,y,z)} to a transformed region
to include chiral metamaterial, no further clues with ge-
′
D in physical space, identified by prime coordinates
ometric intuitive were given to extend the standard TO
r′ ≡ {xµ|(x′,y′,z′)}. Here the Latin and Roman letter
toembodythetwistingdegreeoffreedomoflight. More-
areusedforthesuperscripttocoincidewiththesymbolof
tetrad theory of gravity. The action of coordinate trans-
formation on point r is generally described by a locally
∗ Correspondingauthor: [email protected] special Euclidean group SE(3) [31], i.e.
2
partialderivativesofthecoordinatetransformationdon’t
commutable[29],thatimpliesthecoordinatetransforma-
r′ =O(r)r+T(r) (1) tionissubjecttoananholonimicconstraintwhichbreaks
the translational symmetry,
whereO∈SO(3)andT∈T(3). Therefore,therearesix
degreesoffreedomforcoordinatetransformationdescrib- ∂2xa ∂2xa
ing rotationaland translationalsymmetries in R3. From ∂xµ∂xν − ∂xν∂xµ 6=0 (4)
the view point of topology, the virtual space and the
Inequality (4) means the violation of Schwartz inte-
transformed space are generally assumed diffeomorphic
grablecondition[29]. Fromageometricpointofview,the
which means the two space are topologically equivalent.
constrained coordinate transformation leads to a topol-
The material interpretation of TO regard the domains
′
′ ogytransitionformtheauxiliaryspaceD toD. Inother
D andD assomeelasticobjects,whichcarrysystemsof
′
words, the transformed space D must contained some
coordinatestogetherwithelectricdisplacementfieldlines
topological defects, such as dislocations in continuum
in an absolute Euclidean space, the transformed region
′ limit.
D isjustadeformedbodywithcurvedcoordinatedlines
Unlike the well known meaning of curvature which
which is labeled by a metric tensor g (r)= δ ∂xa ∂xb,
µν ab∂xµ∂xν charactershowatangentvectorisrotatedalongaclosed
and the constitutive parameters of transformed media is
loop, torsion tensor measures the degree of how the tan-
given by
gent spaces is twisted about a curve when they are par-
χµν = det(g)gµν; G′ =eaG , (2) allel transported,
µ µ a
p dE′λ =(Rλ E′ρ+Tλ )dSµν (5)
with det(·) denoting the matrix determinant and χ, G ρµν µν
representing (ε, µ) and (E, H), respectively. Here and Ifwedefinethedisplacementvectorfromaperfectlat-
below, we use the Einstein summation convention in tice to a deformed lattice by u = r−r′. Encircling the
which repeated indices are summed. defect with an arbitrary closed contour C, the disconti-
nuity of u is given by
z
(dx·∇)u=b+Ω×(r−r0) (6)
y I
defect core C
x
where the Frank vectorΩ represents a rotationalaction,
theBurgersvectorbdenotesthetranslationalactionand
D C D1 C1 r0 the reference point. Figure 1 schematic shows the
A failure of parallelogramaround a screw dislocation. The
A1
B B1 emergence of topological defects allows us to distinguish
different loops connecting arbitrary two points in trans-
formedspaceviahomotopyclasses. Forthecaseoftrans-
formed media with a single linear topologicaldefect, the
virtual space D transformed space D(cid:146) fundamental group is π (S1)=Z.
1
Physical effects of torsion.—Physically, the permittiv-
FIG. 1. Schematic illustration of anholomonic coordinate
ity ε and permeability µ, i. e. the metric tensor g, fixes
transformation in a discrete lattice model. A closed loop
the causal structure of the space-time as well as the lo-
ABCDA in is mapped toa open curveA1B1C1D1. Thefail-
ureof theclosed loop indicates a line defect along z-axis. cal optical length; the connections describes the parallel
transport of free photons. The idea to relate vortices
The key point for going beyond Riemannian geome- with optical dislocations seems intuitively obvious since
vorticesarealsoknownasopticalphasedislocationswith
try comes from the observation that the Jacobian ma-
an exp(ilθ) azimuthal phase dependence. From TO, the
trix constructed the tetrad fields in the affine space, i.e.
′
e =∂r/∂xµ, which related the basis vector e in D′ electric field E in transformed medium is related the
(µ) (µ)
and the basic vector e in D, original E via
(a)
e =eae (3) E′µ =eaµEa (7)
(µ) µ (a)
Consider a plane wave in virtual space is transformed
whereµ,a=1,2,3arethespatialindices. Torsiontensor under the actionofea, the contourintegralofE′ around
is defined in terms of the asymmetry part of the affine µ
z axis over a closed loop, based on Stokes theorem, can
connection Tλ = (Γλ −Γλ )/2, where the affine con-
µν µν νµ be expressed as
nection is givenby Γλ =eλ·∂ e . Torsionvanishes for
µν µ ν
trivial coordinate transformation which create effective dxµE′ =( dsµνTa )E (8)
Riemannian medium for the photons. By definition, a IC µ ZS µν a
transformedspacewithtorsiondemands thatthe second =baE
a
3
(8)indicatethatamacroscaletopologicaldefectwithtor- (a)
sion can convert a plane wavefront to a dislocated one.
Specifically, a screw dislocation will leading to a optical
vortex with topological invariant b.
To see how torsion is linked with magnetoelectric
coupling in constitutive relations, we follows the idea (b)
of standard TO based on formal invariance. Starting
from Maxwell equations in vacuum, anholonomic coor-
dinate transformation leading to non-vanishing terms
ǫ (∂2x /∂x ∂x )E . A simple calculations give out
µνλ a ν λ a
the following constitutive relations,
(c)
D=εE+iκH (9)
B=µH−iκE
with ε=µ= det(g)gµν and κµρ = 1 ǫµνλTρ .
2ω νλ
Consider thpe local density of optical chirality C which
was firstly introduced by Lipkin [32, 33]
FIG.2. PropagationofaGaussianbeamalonganopticalline
C = ε0E′·∇×E′+ 1 B′·∇×B′ (10) defect with (a) β =0.15,α=1, (b) β =−0.1,α=1 and (c)
2 2µ β=−0.1, α=1.3.
0
According to (2), we have
a result, a gaussian beam is converted to be an optical
C =εµνλgµρTνρλU (11) vortex which carrying orbital angular momentum.
Foraspecificparameterβ =0,coordinatetransforma-
whereU = 1(ε E2+ 1 H2)isthe energydensityoflight
2 0 µ0 tion (1) represents a static cosmic string. In Fig. 3, we
field.
plot the scattering pattern of a vertically incident gaus-
As anillustrativeexample, weconsidera curvedspace
sian beam on a cylinder consist of 2D conical media, i.
with a topological defect which is described by the fol-
e. µ 6= 1, and β = 0 with µ is the deficit angle. For
lowing transformation in cylindrical coordinate[27]
µ = 0, the conical medium account for air. When µ de-
creased,theincidentbeamissplittotwobeamswiththe
β
′ ′ ′
r =r; θ =αθ; z =z+ θ, (12) splitangleincreasedwiththedeficitangle. Wenotedthe
α
observedphenomena is an optical analogy of gravitation
where α=1−4µ, β =b/2π is the Burgersvector. Obvi- lensing effect by ‘cosmic strings’.
ously,thislineelementturnstobevacuumforα=1and
β =0, a conicalspace for β =0, and a screw dislocation (a) (b)
or a static chiral cosmic string for α=1.
Through this transformation, we can obtain the fol-
lowing constitutive parameters for the material
(c) (d)
M
ε=µ= (13)
α3
with
(e) (f)
sin2θ+α2cos2θ sinθcosθ[α2−1] γsinθ
M = cos2θ+α2sin2 −γcosθ 
α2+γ2
 
The material parameters (13) describe an optical (g) (h)
medium with the material distribution breaking trans-
lational and/or rotational symmetries. To explore the
optical properties, we perform fullwave numerical simu-
lations with finite element method (FEM). Figure 2 de-
picts the electromagneticfielddistributionofaGaussian FIG. 3. Scattering of Gaussian beams by cylinders with 2D
beam incident on a cylinder of defect materials with the conical geometry with different µ: (a,b) µ=−1.6, (c,d) µ=
defect line is aligned with z−axis. It is shown the plane −0.8, (e,f) µ=−0.2, (g,h) µ=0, (i,j) µ=0.04.
wavefrontofagaussianwaveistitledinthedefectmedia
region,andthe EMwaveintensity inthe defectcore. As Conclusion.— We show that the geometry framework
4
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