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Received 11 Nov 2013 | Accepted 22 Jan 2014 | Published 6 Mar 2014

Light can exert radiation pressure on any object it encounters and that resulting optical force can be used to manipulate particles. It is commonly assumed that light should move a particle forward and indeed an incident plane wave with a photon momentum :k can only push any particle, independent of its properties, in the direction of k. Here we demonstrate, using full-wave simulations, that an anomalous lateral force can be induced in a direction perpendicular to that of the incident photon momentum if a chiral particle is placed above a substrate that does not break any leftright symmetry. Analytical theory shows that the lateral force emerges from the coupling between structural chirality (the handedness of the chiral particle) and the light reected from the substrate surface. Such coupling induces a sideway force that pushes chiral particles with opposite handedness in opposite directions.

DOI: 10.1038/ncomms4307 OPEN

Lateral optical force on chiral particles near a surface

S.B. Wang1 & C.T. Chan1

1 Department of Physics and Institute for Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China. Correspondence and requests for materials should be addressed to C.T.C. (email: mailto:[email protected]

Web End [email protected] ).

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Electromagnetic (EM) waves carry linear momentum as each photon has a linear momentum of :k in the direction of propagation. Circularly polarized light carries angular

momentum due to the intrinsic spin angular momentum (SAM) of photons16. When light is scattered or absorbed by a particle, the transfer of momentum can cause the particle to move. Thus light can be used to manipulate particles725. Light will push a particle in the direction of light propagation (as illustrated in Fig. 1a) irrespective of the polarization of light and irrespective of the particles own properties, even if it has chirality (Fig. 1c,e), as long as we have a plane wave incidence (that is, a well-dened k). Let us now consider the conguration shown in Fig. 1b, which shows a particle placed close to a surface made of an ordinary material (for example, Au or Si or silica). In this case, one might still expect the particle to be pushed in the direction of light propagation, as the surface does not break the leftright symmetry. We will show that this is true only if the particle is non-chiral. If the particle has chiral character, however, it will experience an additional lateral force in a direction that depends on its own chirality as shown in Fig. 1d,f. This counter-intuitive force comes from a lateral radiation pressure and an optical spin density force that couple the chirality of the particle to both the lateral linear momentum and SAM generated by the scattered wave of the chiral particle. The time-averaged SAM densities, dened as /LeS e0/(4oi)(E E*) and /LmS m0/

(4oi)(H H*), respectively, for the electric and magnetic parts,

are associated with the polarization of light26,27. The lateral force could move particles with chirality of different signs in different directions as shown in Fig. 1d,f. A good example of a chiral particle is the helix shown in Fig. 1g. Interactions between such kind of chiral objects and EM waves have been extensively studied28 and are shown to give rise to interesting phenomena such as polarization conversion2934, photonic topological edge states35 and negative refractive metamaterials3639. We will show that the coupling of EM near eld and structural chirality (the handedness of a chiral particle) will induce an anomalous lateral force that pushes the particle sideways rather than forward.

We rst show the full-wave numerical results of the optical forces acting on helical gold particles induced by a linearly polarized plane wave. We then consider a simpler system consisting of a model chiral sphere placed above a substrate. To reveal the underlying physics and trace the origin of this intriguing phenomenon, we further simplify the conguration by considering a dipolar chiral particle above a substrate, in which case the problem can be analytically addressed.

ResultsLateral optical force on a gold helix placed above a substrate. Consider a gold helix with inner radius r 50 nm, outer radius

R 150 nm and pitch P 300 nm, as shown in Fig. 1g. We

are interested in the optical force acting on such a particle induced by a linearly polarized plane wave of the form Einc ^zE0 exp ik0x iot

. The particle is located above a sub

strate with a gap distance of 10 nm and its axis is along the x direction (see Fig. 1d,f). The substrate has the dimensions of l w t (see Fig. 1b) and it can be metallic (for example, gold) or

dielectric.

For an isolated helical particle, the scattering force induced by the plane wave will push it in the direction of k0 independent of the handedness of the helix (see Fig. 1c,e). However, if the helix is placed above a substrate (as shown in Fig. 1d,f), an additional lateral force (Fy) will emerge and push it sideways. The lateral force acting on a left-handed (LH) and a right-handed (RH) particle takes opposite signs. The optical force is calculated numerically by the Maxwell stress tensor method (see Methods).

Figure 2a shows the lateral forces acting on an LH (blue lines) gold helix and an RH (red lines) one consisting of four pitches when they are placed above a gold (described by a Drude model, see Numerical simulation section) substrate (solid circles) and a dielectric substrate with ed 2.5 (hollow triangles). The lateral

forces are evident in a wide range of frequencies, where local resonances associated with the geometric shapes of the particles result in some oscillations. It is important to note that the lateral force takes opposite signs for the LH and RH helices. The existence of a lateral force in the case of a perfect electric conductor substrate is also examined and similar results are obtained (see Supplementary Fig. 1).

Figure 2b shows the dependence of the lateral force on the number of pitches there are in the helix at f 490 THz. The

magnitudes of the lateral forces in the cases of a gold substrate (solid circles) increase with the pitch number, which indicates that the magnitude of the lateral force increases with the chirality.

y x

Figure 1 | Anomalous lateral force in the helix particlesubstrate conguration. (a) A linearly polarized plane wave pushes a normal spherical particle forward. A standalone RH helix (c) and a standalone LH helix (e) will be pushed forward by radiation pressure. When the normal spherical particle is put near a substrate, (b) it still moves forward while the helical particles (d,f) are shifted in opposite directions by an anomalous lateral force. The arrows associated with the helical particles indicate the handedness. The substrate has the dimensions of l w t.

(g) Dimensions of the helix particle. It has inner radius r 50 nm, outer

radius R 150 nm and pitch P 300 nm.

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Metal-LH

F y(pN mW1 m2 )

0.4

0.2

0.0

0.2

0.4

Dielectric-LH

Dielectric-RH

Metal-RH

300 400

Frequency (THz)

500 600

0.4

0.2

0.0

0.0 0.5 1.0 1.5

F y(pN mW1 m2 )

0.3

0.2

0.1

With sub-L

With sub-R

0.2

0.0

0.2

0.4

; 1

where er and mr are the relative permittivity and permeability of the material, respectively; k is the chirality parameter and takes a real value here; e0, m0 and c are the permittivity, permeability and the speed of light in vacuum. The dispersion for a plane wave in such a medium is k

ermr

p k

o=c,

corresponding to circularly polarized states of light. The gap distance between the sphere (er 2, mr 1) and the surface of the gold substrate is

10 nm. The particle is under the excitation of the same linearly polarized plane wave dened before. The wavelength is set at l 600 nm. Figure 4a,b shows the time-averaged Poynting vector

for the cases with and without the gold substrate, computed by the numerical solver COMSOL Multiphysics (http://www.comsol.com/

Web End =http:// http://www.comsol.com/

Web End =www.comsol.com/ ). For the isolated particle (Fig. 4a), the Poynting vector has a rotation pattern due to the chiral property of the sphere and the scattered energy is symmetric with respect to y and y. Such rotating Poynting vector pattern

can also be found in the case of the gold helix (see Supplementary Fig. 2). However, in the presence of the gold substrate, the handedness of the Poynting vector in the near-eld region results in an asymmetric pattern as shown in Fig. 4b and hence a lateral force. Figure 4d,e shows the ow of the time-averaged electric SAM[/LeS e0/(4oi)(E E*)]. Similar to the Poynting

vector distribution, the computed SAM has a symmetric y-component in the isolated sphere case (Fig. 4d). However, in the presence of a gold substrate shown in Fig. 4e, the SAM tends to point in y direction. We will show (through analytical

theory) that the asymmetric distributions of Poynting vector and SAM are the consequences of particlesubstrate coupling and are closely related to the origin of the lateral force. The blue line in Fig. 4g shows the numerically evaluated lateral force as a function of the chirality parameter k for the case with the gold substrate.

The magnitude of the force increases with the magnitude of the chirality. If the chirality is positive, the force is negative, and vice versa. The force vanishes when the medium is non-chiral (k 0).

The chirality determines the direction of EM energy coupling and this property can be used to realize uni-directional excitation of surface waves4547.

The results shown in Fig. 4af also provide an intuitive understanding of the lateral force. From a symmetry point of view, the time-averaged Poynting vectors corresponding to the

No sub-L

No sub-R

E H

40 80 120

d (nm)

t (m)

4 6 8 10

No. of pitches

Figure 2 | Numerically calculated lateral forces acting on the helical gold particles. (a) Lateral forces Fy acting on the LH (blue lines) and RH (red lines) particles in the presence of metal (gold) and dielectric (ed 2.5)

substrates. (b) Lateral forces as a function of the number of pitches for the LH (blue lines) and RH (red lines) particles in the presence (circles) and absence of (triangles) the gold substrate. (c) Lateral force acting onthe LH particle as a function of the gap distance between the particle and the gold substrate. (d) Lateral force as a function of the thickness of the dielectric substrate for the LH particle. In the numerical simulations, the frequency is set at f 490 THz for case of the metallic substrate (b,c)

and at f 380 THz for the case of the dielectric substrate.

In the absence of a substrate, there is a small residual lateral force (solid triangles) due to the end effect and this residual lateral force decreases as the pitch number increases (the residual lateral force is zero if the particle is symmetric on the yz-plane). Figure 2c shows that the magnitude of the lateral force decreases when the gap distance between the particle and the gold substrate is increased, indicating that the force is due to the coupling between the particle and the substrate. Figure 2d shows the lateral force as a function of the thickness t of the dielectric substrate, where the frequency is set at f 380 THz. The lateral force

undergoes oscillations and we will see later that the oscillations are due to the FabryPerot resonances associated with the dielectric substrate. Note that we do not need to consider the effect of thickness in the case of the gold substrate due to the nite penetration depth of elds.

The numerical results are rather counter-intuitive. The standalone helix (that is, without a substrate), whether it be LH or RH and assuming it is long enough so that the end effect can be ignored, scatters the same amount of light to the left ( y) and

to the right ( y) and hence light cannot push it sideways.

However, when the helix is placed above a substrate, it experiences a lateral force in the y direction. To see pictorially how this happens, we examine the magnetic eld (Hx) patterns for both an RH (Fig. 3a) and an LH (Fig. 3b) helix in the case of the gold substrate at f 420 THz. Figure 3a,b shows that the

scattered eld is asymmetrically distributed. The surface waves propagate predominately in the y direction in the case of the

RH particle while they propagate in the y direction in the case of the LH particle. These eld patterns show that the helix-above-substrate conguration produces asymmetric scattering and hence a lateral force, although individually the helix and the substrate produce symmetric scattering.

Lateral optical force on a chiral sphere placed above a substrate. The numerical results on the gold helix demonstrates the existence of the lateral force. We now show that the effect can be observed even for a chiral sphere described by simple constitutive relations. Consider again the conguration shown in Fig. 1b, where the spherical particle with radius r 75 nm is now

made of a model bi-isotropic chiral material described by the constitutive relations4044:

ere0 ik=c

ik=c mrm0

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1103 (Am1)

y 1103 (Am

1103

Figure 3 | Magnetic eld distribution on the gold substrate. (a) Hx eld pattern for the RH particle on the gold substrate. (b) Hx eld pattern forthe LH particle on the gold substrate. Note the asymmetrical pattern and the different directions in which the RH and LH particles scatter EM elds. The plotted elds are on a plane just above the substrate and the frequency is set at f 420 THz.

4 pN mW1 m2 )

630361.0 0.5 0.0

0.5 1.0

F y (10

Figure 4 | Numerical results for a chiral sphere above a gold substrate. Time-averaged Poynting vectors (a) for an isolated chiral sphere, (b) for a chiral sphere above a gold substrate and (c) for a chiral sphere sandwiched symmetrically by two gold substrates. We set k 1 for the sphere. Time-

averaged electric spin density /LeS (d) for the isolated sphere, (e) for the sphere above a gold substrate and (f) for the sphere sandwiched by two gold substrates. The leftright asymmetry is obvious in panels (b) and (e). (g) Lateral force acting on the chiral sphere above a gold substrate (blue line)

and sandwiched by two gold substrates (red line) as a function of the chirality parameter k. The blue line shows that the sign of the lateral force Fy depends on k and Fy 0 if k 0. The red line indicates the lateral force vanishes in the sandwiched case. The frequency is set at f 500 THz.

total elds for a standalone chiral/spiral particle under linearly polarized light excitation must satisfy /S(y, z)S /S( y, z)S due to the rotational symmetry of the system. This can be seen by examining the numerical results in Fig. 4a. This implies that the total momentum ux scattered by the chiral object to the right ( y) and left ( y) should be the same for a standalone

particle, which in turn implies that there cannot be a net force in the lateral (y) direction. However, if we break the z- z

symmetry of the environment by putting a substrate underneath

the particle, there is no symmetry requirement for |/S(y, z)S|

and |/S( y, z)S| to be equal. This is indeed the case for the

numerically calculated Poynting vector pattern as shown in Fig. 4b. A lateral force (Fy) can now exist as the total photon momentum scattered to the left and right are not balanced, and the magnitude of the lateral force depends on the details of near-eld coupling. If we recover the symmetry of the environment by

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adding another substrate above the sphere, the y-components of the Poynting vectors (Fig. 4c) and the SAM (Fig. 4f) become symmetrical and the lateral force vanishes again as numerically veried in Fig. 4g (red line).

The results here show that the lateral force emerges in a very general conguration of a chiral sphere above a substrate. The sign and magnitude of the lateral force are both directly related to the chirality of the particle. In the next section, we show with the help of an analytical theory that the lateral force is related to the chirality-generated lateral photon momentum.

Force on a dipolar chiral particle placed above a substrate. The results shown up to now are full-wave simulations but in order to obtain an intuitive understanding of the origin of the lateral force, we consider the conguration of a dipolar chiral particle above a substrate in which case the lateral optical force can be analytically evaluated and we will show that it comes from the lateral radiation pressure and spin density force arising from reection. The induced dipole moments of such a chiral particle can be expressed as follows28:

p m

aee iaem

iaem amm

Gref

m; 4

Href e0o2

Gref m io r

Gref

p: 5

Here

Gref is the Greens function for reection describing the effect of the substrate and it takes into account the reected elds including the propagating and the evanescent components42. We now refer to equations (35) to reveal the origin of the lateral force in the presence of a substrate. In equation (3), no lateral component can come from the gradient force as the free energy does not change when we apply a virtual in-plane (xy-plane) displacement of the particle. For a lossless particle under long wavelength condition, its polarizabilities are dominated by their real parts and as a consequence, the strengths of the vortex force and the last term are much smaller than that of the spin density force. For example, the strengths of these two terms are, respectively, about 10 6 and 10 3 smaller than that of the spin density force for a dipolar particle (er 2, mr 1, k 1) of

radius a 30 nm placed at d 60 nm above a semi-innite gold

substrate under the excitation of z-polarized plane wave (l 600 nm). We can hence focus on the radiation pressure

term and the terms related to spin densities. In the presence of a substrate, the lateral component of the Poynting vector becomes asymmetric (Fig. 4b) and hence contributes to a lateral radiation pressure. The magnetic spin density is small compared with the contribution of its electric counterpart because the non-magnetic substrate gives a relatively stronger response to the electric eld. Hence the total spin density can be written as L

h i Le

h i Lm

h i Le

h i e0= 4oi

; 2

where p and m are the electric and magnetic moments, respectively; E and H are the elds acting on the particle. Here the polarizability of the particle is specied by parameters aee,

amm and aem. We note that aem is related to the chirality parameter k of the material that the particle is made of and aem

will changes sign if k changes sign and aem 0 corresponds to a

non-chiral particle.

The optical force acting on a dipolar chiral particle in an EM eld can be derived (see Supplementary Note 1) based on the expression of the force acting on a dipolar particle4850, F

h i 1=2 Re rE

p rH

m ck40= 6p

p m

E H

where E and H are the elds acting on the particle. The force expression can be written as follows:

h i rU s

h i

c Im aem

Im Einc E ref

As the

incident electric eld Einc is along the z direction, this expression indicates that the direction of the spin density is mainly parallel to the surface of the substrate (as the numerical results shown in Fig. 4e conrm). Such a spin density can only contribute to a lateral force through the spin density terms but not through the curl-spin terms.

As the spin density is mainly attributed to /LeS, the lateral component of the spin density force can be written as

r S

h i cser Le

h i csmr Lm

h i

oge Le

h i ogm Lm

h i

ck40

12p Im aeea mm

E H

;

where U 1/4(Re[aee]|E|2 Re[amm]|H|2 2 Re[aem]Im[H E*])

is the term due to particleeld interaction; /SS 1/2

Re[E H*] is the time-averaged Poynting vector; /LeS and

/LmS are the time-averaged spin densities dened before; se

k0/e0 Im[aee], sm k0/m0 Im[amm] and s se sm c2k40=6pReaeea mm aema em are the cross-sections; ge 2oIm aem ck40= 3pe0

Re aeea em

and gm 2oIm aem

oge^y Le

h i gee0=2

^y Im Einc E ref

Since the incident electric eld is polarized along the z direction, the lateral force requires an x-component reected eld. Such a component exists only when there is a substrate and it emerges from the interference between the particle and the substrate that enables the cross-coupling of different Cartesian components of the scattered elds. This can be understood by directly examining the expression of the reected eld. Consider the excitation of the chiral particle under the incident magnetic eld Hinc which

is along y direction. According to equation (2), an electric dipole moment py iaemHinc can be induced, which produces

the reected eld Eyref im0aemo2GyyrefHinc in the presence of a

substrate. Here Gyyref is the corresponding tensor element of the Greens function for reection. The reected eld Eyref further induces a y-direction magnetic dipole moment my iaemEyref

and this magnetic dipole moment produces the required x-component electric eld: Exref im0o3a2emGyyrefr

GrefxyHinc.

ck40= 3pm0

Re amma em

also have the dimension of a cross-

section.

The rst term in equation (3) corresponds to the gradient force. The second term represents the radiation pressure. The third term is a vortex force determined by the energy ow vortex around the particle and the optical activity (aem). The fourth and fth terms are scattering forces associated with the curl of the spin densities and are called curl-spin forces5155. The sixth and seventh terms are referred to as spin density forces as they are directly related to the spin densities.

For an isolated chiral particle acted upon by a linearly polarized plane wave, the gradient force and the vortex force vanish because the elds amplitudes and the time-averaged Poynting vector are constants. The last term contributes nothing as Im[E H*] 0 for a plane wave. The spin density terms vanish

as we are considering linear polarization. As a consequence, the

expression of the force reduces to /FS s/SS/c, which is just a

forward-scattering force.

In the presence of a substrate, the elds in equation (3) consist of both the incident plane wave and the reected elds; that is, E Einc Eref, H Hinc Href, where the reected elds can

be expressed as follows:

Eref m0o2

Gref p io r

h i / ImEinc E ref / a2emand hence the spin

direction remains unchanged when aem changes sign. Consequently, the sign of the lateral spin density force is determined by the sign of the coefcient ge as gepaem.

Note also that Le

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Consider a spherical dipolar chiral particle (a 30 nm)

characterized by er 2.0, mr 1.0 and a chirality parameter k

and the particle is placed at d 60 nm above a semi-innite gold

substrate. One can derive the polarizability tensor elements aee,

amm and aem for the particle from its Mie scattering coefcients (see Supplementary Note 2). The lateral force contributed by the radiation pressure and spin density forces can then be analytically evaluated with the help of equation (3) (see Methods). Figure 5a shows the lateral force as a function of k at wavelength l 600

nm. It is clear that the lateral force versus chirality relationship is similar to those calculated using full-wave simulations for the bigger chiral sphere as shown in Fig. 4g. The analytical theory can also explain the oscillation phenomenon of the lateral force in the gold helix conguration (Fig. 2d). Let us now replace the gold substrate by a dielectric substrate (ed 2.5 0.001i). We show in

Fig. 5b the dependence of the lateral force on the thickness of the dielectric substrate according to the dipole theory. As expected, the force undergoes FabryPerot oscillations induced by the reectance of the nite-thickness substrate. The dielectric substrate reects the scattered eld of the particle and causes constructive or destructive interference. A better understanding can be obtained by examining the reection coefcient of the dielectric substrate described by the expression R 2iz2 1 sinkzd =z 12e ikzd z 12eikzd ; where zTE

kz/(mdk0z), zTM kz/(edk0z) and k2z k2== edmdk20. Here k// is

the wave vector component parallel to the surface of the substrate. For a highly evanescent channel (k//ck0), RTE 0 and

RTM e2d 11 e2k

==t

=ed 12 ed 12e2k ==t

, where

RTM is a monotonic function that increases with t and takes a limit value of (ed 1)/(ed 1). The red line in Fig. 5c shows the

magnitude |RTM| as a function of t for the case of k// 10k0,

where we see that the reection coefcient is a constant for large t. Propagating channels (k//ok0), on the other hand, behave quite

differently. Figure 5c shows the |RTE| (green dotted line) and |RTM| (blue solid line) for the case of k// 0.5k0. Clearly they

undergo oscillations with a period that meets the condition kzDt p (here l 600 nm), indicating the oscillations are due to

the FabryPerot resonances. Combining the properties of both the evanescent and propagating channels, the nal reected eld should oscillate and the resulting lateral force should too.

DiscussionWhile the lateral optical force is always there if the particle is chiral, it is clear that the lateral force is reasonably strong relative to the forward force only if the chirality is strong. The case with the helical gold particle, with numerical results shown in Fig. 2, is a good example in which the maximum value of the lateral force can reach about 0.4 times that of the forward scattering force if the gold helix is put above a gold substrate. We note that if the incident wavelength is much larger than the helix dimension, the lateral force would become very weak because the total scattering cross-section is small and in addition, the incident EM wave could not resolve the geometric handedness detail of the helix which is crucial for inducing the chiral effect56. Although we have only shown that a linearly polarized plane wave can induce a lateral force on a chiral particle, the force also exists when the plane wave is circularly polarized (see Supplementary Fig. 3 for the case of gold helices on gold substrate). The key point here is the asymmetric coupling of a chiral particle with a substrate, which can also be induced if the incident light is the circularly polarized. In addition, the lateral force also can be induced when the incident plane wave propagates in a direction perpendicular to the axis of the gold helix (see Supplementary Fig. 4).

In summary, we have numerically shown that the EM near-eld coupling can induce a lateral optical force on a chiral particle near a substrate. The anomalous force can laterally push particles with opposite chirality in opposite directions. Using full-wave simulations, we established the existence of this counter-intuitive phenomenon for the helix-above-substrate conguration as well as the case of a chiral sphere with a simple constitutive relationship above a substrate. In the former case, the lateral force takes opposite signs for an LH and an RH helix. In the latter case, the lateral force changes sign when the chirality parameter changes sign from k to k. The asymmetric coupling between

a chiral particle and a substrate breaks the leftright symmetry due to the special handedness distribution of the Poynting vector (Fig. 4b). By analytically deriving and evaluating the lateral force acting on a dipolar chiral particle above a substrate, we show that such lateral force must exist even in the small particle regime, and the force is attributed to the lateral component of the radiation

pN mW1 m2 )

F y (106 pN mW1 m2 )

1.0

0.5

0.0

0.5

1.0

0.0

F y (106

0.5 0.0

0.5 1.0

0.42

0.6

0.3

0.6

0.3

0.0

0.40

0.38

0.2 0.4 0.6 0.8 t (m)

t (m)

1.0

|RTM|-evanescent

|RTM|-propagating

Reflection coefficient

|RTE|-propagating

0.0 0.2 0.4 0.6 0.8 1.0

Figure 5 | Analytical results for a dipolar chiral particle above a substrate. (a) Lateral force acting on a dipolar chiral particle ( a 30 nm,

2.0) as a function of its chirality when the particle is located 60 nm above a semi-innite gold substrate. (b) Lateral force acting on the chiral particle (k 1) as a function of the thickness of a dielectric substrate

(ed 2.5 0.001i), showing oscillating behaviour. (c) Magnitudes of the

reection coefcients for an evanescent channel k// 10k0 (red line) and

a propagating channel k// 0.5k0 (green dotted line and blue line). The

wavelength is set to be l 600 nm.

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pressure and spin density force generated by the reection eld in presence of the substrate. The study reported here may nd applications in the detection of chirality and in the separation of chiral molecules/enantiomers57 using light induced forces. We note that the lateral force here must be accompanied by a torque induced on the chiral particles, the magnitude of which can be enhanced by a nearby surface. This may nd applications in some opto-mechanical systems and as a light-driven rotor or motor58. However, we note that thermodynamic uctuation effects and hydrodynamic properties of the environment should also be considered in such applications.

Methods

The Maxwell stress tensor method. The optical forces acting on the gold helix and the chiral sphere are numerically evaluated using the Maxwell stress tensor59, which is dened as

T e0 EE c2BB

12 E E c2B B I

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; 6

where I is the unit tensor. To calculate the forces, we rst obtain the EM total eld using a commercial nite-element-method package COMSOL Multiphysics (http://www.comsol.com/

Web End =http:// http://www.comsol.com/

Web End =www.comsol.com/ ). Then the tensor in equation (6) is integrated on a closed surface surrounding the particle. The forces shown in the main text are the time-averaged results.

Semi-analytical evaluation of optical forces by dipole theory. When a spherical dipolar chiral particle is placed above a substrate, the elds of the induced dipoles will be reected by the substrate and react on the particle. Hence, the induced dipoles can be expressed as

p m

aee iaem

iaem amm

: 7

For a small particle, the dipole polarizabilities in equation (7) can be related to the material parameters er, mr, k of the particle (see equation (1)) through Mie scattering theory (see Supplementary Note 2 and ref. 40).

Combining equation (7) with equations (4) and (5), the reected elds can be written as follows:

Eref

x1 Einc

x2 Hinc;

Href B1 Einc B2 Hinc;

where

Einc Eref

Hinc Href

C1 1

; x2

C1 1

;

D1 1

; B2

D1 1

;

and

C1 ie0o2aem

Gref

I e0o2amm

Gref ioaeer

Gref 1

;

Gref oaemr

C2 e0o2amm

Gref

I e0o2amm

Gref oaemr

Gref 1

;

Gref oaemr

D1 im0o2aem

Gref

I m0o2aee

Gref ioammr

Gref 1

;

Gref oaemr

D2 m0o2aee

Gref

I m0o2aee

Gref oaemr

Gref 1

Gref oaemr

The Greens function can be written as an integral in the momentum space as42

Gref

i 8p2

10 dkxdky

1 k0z

e ik r

h i

RTE^e k0z

eik r^e k0z

e ik r RTM

^h k0ze

ik r^

h k0z

Here, k0 kx^x ky^y k0z^z; k00 kx^x ky^y k0z^z; ^ek0z ^e k0z

^xky ^ykx=k==;

hk0z ^xkx ^ykyk0z=k0k== k==^z=k0,

h k0z

^xkx ^ykyk0z=k0k== k==^z=k0 and k==

k2x k

. The reection coefcients

RTE and RTM are dened in the main text.

In the calculations,

Gref and its curl are rst numerically evaluated and then one can apply equations (810) to evaluate the reected elds Eref and Href. Once the

elds acting on the particle are obtained, one can then apply the analytic results in equation (3) to calculate the radiation pressure and spin density forces which are the source of the lateral force.

Numerical simulation. All the full-wave EM simulations are performed with the package COMSOL Multiphysics. The relative permittivity of gold is described by a

Drude model: eAu 1 o2p=o2 ioot, where op 1.37 1016 rad s 1

and ot 4.084 1013 rad s 1 (ref. 60). For the simulation of the gold helix

systems, we set l w t 4 mm 4 mm 200 nm and for the case of the dielectric

substrate we set l w t 4 mm 2.4 mm 200 nm. We choose a relatively

larger substrate in the former case to reduce the effect caused by the reection (at the edge) of the surface plasmons. For the simulation of the chiral sphere system, we set l w t 2 mm 2 mm 200 nm. The nite substrate does not affect the

physics here; the phenomenon also exists in the case of an innite substrate.

NATURE COMMUNICATIONS | 5:3307 | DOI: 10.1038/ncomms4307 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 7

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4307

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Acknowledgements

This work was supported by AoE/P-02/12 and M-HKUST601/12. We thank Profs. Z.Q.

Zhang and J. Ng and H. Liu and Dr K. Ding for their valuable comments and suggestions.

Author contributions

C.T.C. developed the concept and S.B.W. did the calculations. They wrote the paper

together.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

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Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Wang, S. B. and Chan C. T. Lateral optical force on chiral

particles near a surface. Nat. Commun. 5:3307 doi: 10.1038/ncomms4307 (2014).

This work is licensed under a Creative Commons Attribution-

NonCommercial-NoDerivs 3.0 Unported License. To view a copy of

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Light can exert radiation pressure on any object it encounters and that resulting optical force can be used to manipulate particles. It is commonly assumed that light should move a particle forward and indeed an incident plane wave with a photon momentum hk can only push any particle, independent of its properties, in the direction of k. Here we demonstrate, using full-wave simulations, that an anomalous lateral force can be induced in a direction perpendicular to that of the incident photon momentum if a chiral particle is placed above a substrate that does not break any left-right symmetry. Analytical theory shows that the lateral force emerges from the coupling between structural chirality (the handedness of the chiral particle) and the light reflected from the substrate surface. Such coupling induces a sideway force that pushes chiral particles with opposite handedness in opposite directions.

Details

Title

Lateral optical force on chiral particles near a surface

Author

Wang, S B; Chan, C T

Pages

3307

Publication year

2014

Publication date

Mar 2014

Publisher

Nature Publishing Group

e-ISSN

20411723

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms4307

ProQuest document ID

1504530978

Lateral optical force on chiral particles near a surface

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