Tekever drone português espionagem industrial 1.pdf

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Fig. 1 A chiral material slab with thickness d is illuminated by a normalincident circularly polarized plane wave. A perfect electric conductorscreen is laced at a distance l from the chiral slab.

Optical Tractor Beam with Chiral LightDavid E. Fernandes, Mário G. Silveirinha

University of Coimbra, Department of Electrical Engineering – Instituto de Telecomunicações, 3030-290, Coimbra, Portugal,

[email protected] , [email protected]

Abstract—We propose a novel mechanism to promoteoptomechanical interactions that can induce the motion of achiral material body towards the optical source. Our solution isbased on the interference between a chiral light beam and itsreflection on an opaque mirror. Surprisingly, we theoreticallyshow that it is possible to tailor the response of the chiralmetamaterial such that independent of its specific location withthe respect to the mirror, it is always pushed downstream againstthe photon flow associated with the incoming wave. Moreover,we prove that by controlling the handedness of the incomingwave it may be possible to harness the sign of the optical force,and switch from a pushing force to a pulling force

Index Terms—tractor beam, chirality, metamaterials.

I. I NTRODUCTION

Light transports both energy and momentum and thus theradiation field and matter can exchange momentum [1-3]. This

property is manifested macroscopically in terms of a radiation pressure that may be used to control the motion and position ofmicrometer-sized particles and neutral atoms [4]. In particular,optical forces have had a tremendous impact on thedevelopment of atom cooling and trapping technologies (e.g.[5]). Moreover, focused laser beams (optical tweezers [6]) arecommonly used in molecule manipulations for biological and

physical chemistry applications [7-8]. These advances were

made possible by the advent of lasers, which provide themeans to effectively boost the usually weak optomechanicalinteractions.

Recently, there has been a great interest in new techniquesfor optical manipulation of nanoparticles based on non-structured (gradientless) light. Intuitively, one might expectthat when a uniform light beam impinges on a polarizable

body, the translational force acting on the body should always push it towards the direction of propagation. Notably, severalgroups demonstrated theoretically, and in some cases alsoexperimentally, that tailored travelling light beams may allow

pulling micron-sized particles towards the light source [9-24].

Such solutions are known as optical “tractor beams”, whichmay be defined as engineered light beams that exert a negativescattering force on a polarizable body forcing it to move alonga direction counter-propagating with the photon flow. A

pulling force can be obtained with interfering non-diffractive beams [10], based on optical-conveyor belts [9], with

structured media supporting backward waves [12], with thehelp non-paraxial Bessel beams [13, 14, 16, 24], and usinggain media [22, 23].

In this article, we put forward a different paradigm for atractor beam based on unstructured light and on suitablydesigned chiral metamaterials. It is shown that by controllingthe light polarization state helicity it is possible to switch froma pushing force (upstream motion) to a pulling force(downstream motion). The emergence of pulling forcesresulting from the interaction of chiral particles with lightBessel beams has been recently discussed in [24]. Moreover, ithas been shown that light can exert a lateral optical force onchiral particles above a dielectric substrate [25]. As detailed inwhat follows, our proposal to harness the sign of the opticalforce is based on entirely different physical mechanisms, anddoes not require complex Bessel beams.For the sake of clarity, we consider a simple canonicalgeometry wherein the body of interest is a planar slab withthickness d (Fig. 1). Nevertheless, the ideas that follow can bereadily generalized to other geometries, e.g. to smallnanoparticles. As a starting point, we calculate the force thatlight exerts on the polarizable body when it stands alone in avacuum.

In a stationary time-harmonic regime

( , Re i t t e E r E r ) the time-averaged force V F is

given by ˆ ReV c

V

dV

F n T where cT is the (complex-

valued) Maxwell stress tensor

This work was funded by Fundação para Ciência e a Tecnologia under

project PTDC/EEI-TEL/2764/2012. D. E. Fernandes acknowledges support

by Fundação para a Ciência e a Tecnologia, Programa Operacional Potencial

Humano/POPH, and the cofinancing of Fundo Social Europeu under the

fellowship SFRH/BD/70893/2010.



* * * *0 0

0 0

1 1 1 1

2 2 2c

T E E E E 1 B B B B 1 , 1

is the identity tensor, V is the boundary surface of the bodywith volume V , and n̂ is the outward unit normal vector [26,27]. Suppose that a plane electromagnetic wave illuminates the

planar slab along the normal direction, creating in this mannerreflected and transmitted transverse plane waves characterized

by the reflection and transmission coefficients R and T . Astraightforward analysis shows that in such a scenario thetranslational force (along the z-direction) acting on the body is

2 2

,av av/ 1inc

zF A W R T , where A is the area of the slab

cross-section and 2 2

av 0 0

1

4inc inc incW E H is the time-

averaged energy density of the incident plane wave. It isinteresting to note that ,av z

F is equal to the difference between

the time rates of the electromagnetic momentum entering theslab front face and the momentum exiting the back face.Clearly, for passive materials the conservation of energy

implies that the optical force is nonnegative, ,av 0 zF , suchthat the body is dragged upstream by the light wave. Notably,for active materials the transmission coefficient can be greater

than unity and hence it is possible to have2 2

1 0 R T

and a pulling (negative) force [22, 23].Here, however, we are interested in exploring a differentsolution based on the interference of the upstream wave with adownstream wave created by reflection on an opaque mirror(Fig. 1). It is well known that the interference of the two wavescreates a standing wave, and that usually a material body is

pushed to a spot wherein the electric field intensity reaches alocal maximum. Surprisingly, we prove in what follows that it

is possible to engineer the material response such thatindependent of the location of the material body it is always

pulled downstream towards the light source. Thus, in such asystem the light beam behaves as an optical tractor beam.As seen previously, a light wave propagating upstream(downstream) tends to drag the material body along the up(down) direction. Clearly, in the absence of a mirror, theoptical force is minimal ( ,av 0 zF ) when the material body is

transparent to the incoming wave ( 0 R and 1T ). This

suggests that to have a negative optical force the upstreamwave should desirably be perfectly transmitted through thematerial, so that no momentum is imparted on the material

body by this wave. On the other hand, the downstream wave isrequired to be either absorbed or reflected by the material bodyso that it can transfer a negative momentum to the body, andinduce the negative optical force. Thus, the material body mustscatter the waves propagating upstream and downstream in anasymmetric manner. Clearly, this cannot be achieved with astandard dielectric slab because by symmetry thecorresponding scattering response is the same for the twodirections. Moreover, a stack of conventional dielectrics is alsonot a valid option because if 1T for the upstream wave,

then, because of the Lorentz reciprocity theorem [27], T is

also required to be equal to unity for the downstream wave andhence the optical force vanishes.

Notably, chiral metamaterials provide a way out of this bottleneck, and enable us to engineer the scattering propertiessuch that the transmission of the upstream and downstreamwaves is strongly asymmetric. Chiral media are a special class

of isotropic media whose electromagnetic response depends onthe handedness of its structural unities [28], and ischaracterized by a magneto-electric coupling that breaks thedegeneracy between two circularly polarized plane waves sothat the refractive index increases for one circular polarizationand decreases for the other [28]. Chiral materials endow uswith the means to achieve polarization rotation (opticalactivity) [29]-[31], polarization conversion [32], and negativerefraction [33]-[36]. The constitutive relations of a chiral

material are 0 0 0i D E H and

0 0 0i B E H where is the dimensionless

chirality parameter, is the relative permittivity, and is the

relative permeability. The photonic modes in an unboundedchiral medium are right circular polarized (RCP) waves or leftcircular polarized (LCP) waves characterized by distinct

refractive indices n , where n stands for the

refractive index of RCP waves and n for the LCP waves [28].

It is well known that a chiral slab scatters RCP and LCP wavesdifferently, such that the transmission and reflectioncoefficients for an incident wave are [28], [37]:

02 2 2

4

1 1

ik d

ink d

zeT

z z e

,

0

0

22

2 2 2

1 1

1 1

ink d

ink d

z e R R

z z e

. (1)

In the above, d is the thickness of the chiral slab,

0k k n , 0 /k c , n , and / / z n

is the normalized impedance. Hence, the reflection coefficientis identical for the two polarization states but, crucially, thetransmission coefficient is polarization dependent. Now, thekey observation is that in presence of an opaque mirror, e.g. fora metallic mirror, the wave polarization state is switched uponreflection in the mirror, such that if the upstream wave has

RCP (LCP) polarization then the downstream wave has LCP(RCP) polarization. This implies that the scattering of theupstream and downstream waves can be asymmetric becausethe transmission coefficient for RCP waves ( T ) differs from

the transmission coefficient for LCP waves ( T ). It must be

highlighted that the asymmetric transmission ofelectromagnetic waves through chiral metamaterials wasdiscussed by different groups in other contexts [38, 39, 40].Let us denote 21T as the transmission coefficient for a

circularly polarized wave propagating upstream that impinges



on a chiral slab standing alone in free-space ( 21T T for an

incident RCP/LCP wave; the subscript “21” indicates that theincident wave propagates from the region 1 to the region 2),and 12T as the transmission coefficient for the corresponding

downstream wave ( 12T T

). One important remark is that the

conservation of energy and the fact that R is polarizationindependent imply that

12 21

T T in the absence of material

absorption. Thus, a strong asymmetric transmission requiressome material loss.To put the previous ideas on a firm ground, we calculated theoptical force acting on a chiral material slab when it stands at adistance l from the opaque mirror. For simplicity, the mirror isassumed to be a perfect electric conductor (PEC), but similarresults can be obtained with other mirror types (e.g. with a

photonic crystal wall). For the geometry of Fig. 1, theelectromagnetic fields are of the form:

z C z C z

E e ,

0 z C z C z i

H e (2)

with ˆ ˆ / 2i e x y . The signal determines if the

incoming (upstream) plane wave has RCP (+) or LCP polarization (). The function C z ( C z ) represents the

complex amplitude of the upstream (downstream) wave ineach point of space. Using the Maxwell-stress tensor, it can bereadily found that the translational force acting on the chiralslab is simply

2 2

.av 0/ / 4out

in

z z

z z z

F A C z C z C z C z

, where in z z ( out z z ) are the coordinates of the input(output) interfaces of the slab. The optical force can also bewritten as:

2 2 2 2

.av 0

2 z

out out in in

F C C C C

A

(3)

where in inC C z and out out

C C z . Note that inC is the

complex amplitude of the incident electric field at the chiralslab input interface. The coefficients in

C and out C are linked

by 12in in out C RC T C and 21out in out C T C RC , where 12T ,

21T , and R are the transmission/reflection coefficients

introduced before for a slab standing alone in free-space.Moreover, for a PEC mirror the coefficients out

C are bound to

obey 02i k l

out out C e C . Putting all these results together, it is

possible to express inC and out

C as a function of the incident

wave amplitude inC as follows:

0

0

221 12

21

i k l

in ini k l

e T T C R C

Re

,

0

2121out ini k l

T C C

Re

,

0

0

221

21

i k l

out ini k l

e T C C

Re

. (4)

Substituting this result into Eq. (3), we finally find that the

optical force is:0

0 0

2 22.av 21 12 21

av 2 21 2

1 1

i k l

inc z

i k l i k l

F e T T T W R

A Re Re

, (5)

where2

0av 2inc

inW C is the time-averaged energy density

of the incident plane wave.As previously mentioned, in order to minimize the

upstream light-matter interactions, it is desirable that 0 R

and 21 1T so that the slab is transparent to the upstream

wave. In this ideal case, the optical force becomes 2

.av av 12 1 0inc

zF W T , i.e. consistent with our intuition the

optomechanical interactions force the material slab to movedownstream towards the light source, independent of itslocation with respect to the mirror. Thus, the proposed regimeis fundamentally different from the conventional light-matterinteractions in presence of standing waves, which tend to draga polarizable object to a field intensity maximum.Interestingly, the amplitude of the negative force has amaximum value when 12 0T . A simple picture of this

scenario is that first the upstream beam overtakes the chiral

slab without any momentum transfer, and then, after reflectionin the mirror, it is totally absorbed by the chiral slab,transferring in this way its electromagnetic momentum to theslab. Clearly, this regime requires a strongly asymmetricscattering by the material slab (ideally, 0 R , 21 1T ,

12 0T ). Given that the incoming wave has a positive

momentum, one may wonder how the material slab can gain anegative momentum. The justification is simple: the negativemomentum is indirectly provided by the mirror wall whichupon reflection flips the sign of the incident radiationmomentum.

A remarkable property of our system is that if the

handedness of the upstream light is reversed (e.g. from LCP toRCP) then 21T and 12T are interchanged. Hence, if for a certain

light polarization the upstream light wave originates a negativeoptical force (e.g. 0 R , 21 1T , 12 0T ), then an

upstream light wave with opposite handedness originates a positive optical force (e.g. 0 R , 21 0T , 12 1T ). Indeed,

in the latter situation the upstream light is completely absorbed by the chiral slab never reaching the opaque mirror, and thus itimparts a positive momentum to the material body. Therefore,

by controlling the handedness of the upstream light it may be



Fig. 2 (a) Geometry of the conjugated-gammadions planar chiral metamaterial. The length of the central arm of the gammadions is denoted by l and the width by w . The meta-atom is arranged in a periodic square lattice with period a. (b) Side view of the meta-atom: the conjugated-gammadions are made of silver

with thicknessmd and are separated by a distance

gd . The two gammadions are embedded in a polyimide material with thickness 2l and electric permittivity

0 6.25 1 0.03 p i . Two layers of silicon, with permittivity 0 11.9 1 0.004s i and thickness 1l , are placed on each side of the polyimide material.

(c) Amplitude of: i) Transmission coefficient for incident LCP waves, ii) Transmission coefficient for incident RCP waves, iii) Reflection coefficient forincident LCP waves, iv) Reflection coefficient for incident RCP waves (d) Similar to (c) but for the phase of the transmission and reflection coefficients. (e)Circular dichroism.

possible to transport a material particle at will, either upstreamor downstream, similar to an optical conveyer belt.

The asymmetry in the response is determined by2 2

T T ,

which is a measure of the circular dichroism. Therefore, achiral material that enables the light source to behave as atractor beam must have a strong circular dichroism.To demonstrate these ideas, we designed a chiral metamaterial

such that the meta-atoms are made of conjugated-gammadionsas illustrated in Fig. 2a-b. Previous studies have shown thatthis metamaterial is characterized by a strong circulardichroism and large optical activity in the microwave and low-THz regime [30],[31]. Evidently the response of thismetamaterial is not isotropic, but the key requirement to have a

pulling force is the strong circular dichroism, rather than theisotropy.To take advantage of high-power laser sources, the meta-atomwas engineered to operate around the wavelength

0 1.55μm . The length of the central arm of the

gammadions is 151.81nml , the width is 12.19 nmw andits thickness is 25nmmd . The conjugated-gammadions are

made of silver, described by a Drude dispersion model

21m p i , with parameters

2 2175 z p and 2 4.35 z , consistent with

experimental data reported in the literature [41]. Thegammadions are separated by a distance 11.58 nmg

d and

embedded in a polyimide material with thickness 2 88.4nml

and electric permittivity 0 6.25 1 0.03 p i [30]. We also

added two layers of silicon ( 0 11.9 1 0.004s i , one on

each side of the polyimide material, as shown in Figure 2b,with thickness 1 50.6 nml , so that the total thickness of the

meta-atom is 1 2 02 189.6 nm 0.122d l l . The meta-



Fig. 3 Normalized force .av av

inc

zF AW as a function of the frequency for

two fixed positions of the chiral slab: 0 1.5k l (solid curves) and

0 2.0k l (dashed curves) for i) RCP upstream light and ii) LCP upstream

light.

atoms are arranged in a periodic square lattice with a deeplysubwavelength period 0202.41nm 0.131a .

Using the full-wave electromagnetic simulator CST-MWS [42]we calculated the transmission and reflection matrices thatrelate the fields scattered by a metamaterial slab standing alonein free-space with the incident fields as

t inc xx xy x x

t inc yx yy y y

T T E E

T T E E

and

r inc xx xy x x

r inc yx yy y y

R R E E

R R E E

.

Here, inc

i E , t

i E and r

i E are the incident, transmitted and

reflected ,i x y field components, respectively. The

calculation of the reflection and transmission matrices requiresilluminating the metamaterial slab with two independentexcitations, in our case with x- and y- linearly polarized waves.The rotation symmetry of the conjugated-gammadions causesthe cross polarized components of the matrices to besymmetric, so that xy yxT T and xy yx R R , whereas the co-

polarized components are equal, i.e. xx yyT T and xx yy R R .

As a consequence, the eigenstates of the scattering problem are

circularly polarized waves, with eigenvectors e . Moreover,the eigenvalues of the transmission and reflection matrices are

xx xyT T iT and xx xy R R iR , respectively [35].

The numerically calculated amplitude and phase of thetransmission and reflection coefficients for incident circularly polarized light are shown in Figure 2c-d. The resultsdemonstrate that, as expected, the reflection coefficients forRCP and LCP waves are the same. Most important, thesimulations show that the chiral metamaterial has a giantoptical activity near 160THz f , 196THz f and

245THz f , where the scattering of the RCP and LCP

waves is strongly asymmetric. This is highlighted in Fig. 2ewhich depicts the circular dichroism of the structure. Notably,near 196THz f there is a spectral region wherein

2 2T T is nearly unity while the reflection coefficient is

near zero. These are exactly the conditions required to have a pulling force with an LCP excitation. Figure 3 depicts thenormalized optical force .av av

inc

zF AW calculated for two

particular positions of the slab, 0 1.5k l (solid curves) and

0 2.0k l (dashed curves) for an upstream wave with either

RCP or LCP polarization. The results confirm that only in afrequency range around 196THz f , the force exerted by an

LCP excitation is negative for the two positions of the PECscreen. Moreover we see that in the same frequency range anupstream RCP wave exerts a positive force on the chiralmetamaterial, in agreement with an earlier theoreticaldiscussion.

Figure 4a and 4c depict the optical forces created by RCPand LCP upstream excitations at two fixedfrequencies 190THz f and 196THz f , respectively. As

seen, while in the former case –due to the interference of theupstream and downstream waves– the optical force signdepends on the specific position of the slab relative to the

mirror, in the latter case an LCP excitation always exerts anegative force on the slab whereas an RCP excitation always pushes the slab towards the mirror, independent of the distancel . Thus, the LCP excitation mimics, indeed, an optical tractor

beam. Here, it is interesting to note that the force varies periodically with the distance between the chiral slab and themetallic screen, with a period 0 / 2l , where 0 is the

wavelength of light in vacuum.

Because the optical force only depends on the position of theslab relative to the mirror, we can define a potential energy as

,av zV F dz , which is depicted in Fig. 4b and 4d for

190THz f and 196THz f , respectively. As seen, in the

former case there are potential wells wherein the material body

will inevitably become “trapped”. Quite differently, at196THz the potential energy has a monotonic dependencewith l and does not have stationary points. Hence, dependingon the light polarization, the material body is steadily pushedeither upstream (RCP polarization) or downstream (LCP polarization). Interestingly, this optical “conveyer belt”operation may occur in a spectral range from

194.419 THz f to 198.57THz f , which corresponds to a

bandwidth of 4.151THz . Finally, we note that a typical laseroperating at 0 1.55μm generates a light beam with a

Poynting vector 289.3MW mincS [43]. In this case, an

optical force with amplitude .av av 1.0inc

zF AW corresponds to

an optical pressure of the order .av / 0.3 Painc

zF A S c ,

which is a significant value in the nanoscale.In summary, it was theoretically demonstrated that by usingcircularly polarized light and a planar chiral metamaterial witha tailored response it may be possible to mimic an opticaltractor beam. By controlling the incident light polarization onemay switch from a pushing force to a pulling force,independent of the specific position of chiral slab. We envisionthat these findings may lead into new inroads in opticalmanipulation of micro and nanoparticles.



Fig. 4 (a) Normalized force .av avinc

zF AW and (b) Potential (in arbitrary units) as a function of the normalized distance 0k l between the dielectric slab and the

PEC screen at a fixed frequency 190 THz f for i) RCP upstream light and ii) LCP upstream light. (c) and (d) Similar to (a) and (b), respectively, but for afixed fre uenc 196 THz f .

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