arX

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4v1

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Aug

202

1

Force Detection of Electromagnetic Beam Chirality at the

Nanoscale

Abid Anjum Sifat1, Filippo Capolino1, Eric O. Potma1,2

1Department of Electrical Engineering and Computer Science,

University of California, Irvine and

2Department of Chemistry, University of California, Irvine

Abstract

Many nanophotonic applications require precise control and characterization of electromagnetic

field properties at the nanoscale. The chiral properties of the field are among its key characteristics,

yet measurement of optical chirality at dimensions beyond the diffraction limit has proven difficult.

Here we theoretically show that the chiral properties of light can be characterized down to the

nanometer scale by means of force detection. We demonstrate that the photo-induced force exerted

on a sharp chiral tip, subjected to sequential illumination by two circularly polarized beams of

opposite handedness, provides a useful probe of the chirality of the electromagnetic field. The

gradient force difference ∆〈Fgrad,z〉 is found to have exclusive correspondence to the time-averaged

helicity density, whereas the differential scattering force provides information about the spin angular

momentum density of light. We further characterize and quantify the helicity-dependent ∆〈Fgrad,z〉

using a Mie scattering formalism complemented with full wave simulations, underlining that the

magnitude of the difference force is within an experimentally detectable range.

1

I. INTRODUCTION

Light can exist in a chiral state, and this property makes it possible to study chiral ob-

jects, such as molecules and nanostructures, in an optical manner.[1–6] Optical detection

of chirality is important in a myriad of scientific fields, with key applications in molecu-

lar biology and pharmacology.[7–10] In common applications, the specimen is illuminated

sequentially with right-handed and left-handed circularly polarized light (RCP and LCP),

and the chiral information is inferred from the differential response of the material. The

chiral state of circularly polarized light is well understood, which facilitates the analysis of

such measurements. However, performing similar measurements at the nanoscale can be

challenging [11–13], because the chiral state of light in the near zone, close to a nano-object,

is generally unknown and cannot be directly assessed experimentally. Without experimen-

tal tools for extracting information about the chirality of the electromagnetic field, optical

examination of chirality at the nanoscale is severely complicated.

The chiral state of light is related to the curled character of the electric and magnetic

fields. The quantity commonly used for quantifying this ‘degree of curliness’ is the time-

averaged helicity density [14, 15], which is defined as h = 1

2ωcIm(E · H∗), where E and H

are the phasor electric and magnetic field at angular frequency ω, and c is the speed of light

(∗ denotes complex conjugation) [16, 17]. The helicity density is a time-even, pseudo-scalar

conserved quantity that corresponds to the difference between the numbers of RCP and

LCP photons [14, 15, 18]. The flux of the helicity density, i.e. the chiral momentum density,

is related to the spin angular momentum of the beam [14, 18]. The latter quantity is a

time-odd, pseudo-vector quantity [15] and its density is defined as σ = − ε04ωIm(E × E∗) −

µ0

4ωIm(H×H∗), where ε0 and µ0 are the absolute permittivity and permeability of free space

[15, 19], respectively. Although both quantities describe angular momentum associated with

the polarization state of light[14], only the time-averaged helicity density is classified as a

conserved property of the electromagnetic field [1, 20], and, therefore, considered to be the

proper descriptor of optical chirality.

The question arises whether a quantity like the helicity density can be properly measured

at the nanoscale. Unlike quantities such as energy, mechanical force and torque carried

or performed by the electromagnetic field, properties like linear and angular momentum of

light are more difficult to measure and are commonly deduced from other measurable quan-

2

tities [21]. Helicity density of light falls into the latter category, and measuring it requires a

connection to a quantity that can be experimentally assessed. One possibility is to detect he-

licity density through a photo-induced force. For instance, the technique of photo-induced

force microscopy (PiFM) has been used to map electric and magnetic field distributions

through the optical force exerted on a metallic tip [22] or a specially designed magnetic

nanoprobe [23]. The photo-induced force has also been used for in sorting and trapping of

chiral nanoparticles[24–28]. In addition, PiFM has been employed to determine enantios-

elective chirality of nanosamples [29], and for measuring the geometric chirality of broken

symmetry structures with differential force measurements under RCP/LCP illumination[30]

.

In this work, we theoretically investigate the connection between the photo-induced forces

felt by a chiral tip under differential RCP/LCP illumination and the chiral state of light

at the nanoscale. Using image dipole theory, we have found a direct relation between the

measured differential force and the chiral properties of the incident electromagnetic field in

terms of the time-averaged helicity density and spin-angular momentum density. We further

model the chiral tip as an isotropic chiral sphere to examine several design considerations

and validate our dipole model with full-wave, finite element method (FEM) simulations.

II. THEORETICAL ANALYSIS

We are interested in detecting the mechanical force that acts on the tip due to the presence

of the electromagnetic field. We can model the tip as particle and write the time-averaged

force exerted on this particle as:[31, 32]

〈F〉 = 1

2Re

{

∫

S

[ε(E · n)E+ µ−1(B · n)B− 1

2(ε|E|2 + µ−1|B|2)n]dS

}

(1)

where the integration is over the arbitrary surface S that encloses the particle, n is the

unit vector normal to this surface, and E and B are the total electric field and magnetic

flux density, which include both the incident light and the scattered light contributions. It

is assumed that the particle is embedded in a non-dissipative medium with permittivity

ε and permeability µ. To simplify this expression, we next assume that the tip can be

described as a dipolar particle with an electric and magnetic dipole moment in free space,

written as ptip [Cm] and mtip [Am2], respectively. Adopting the e−iωt time convention for

3

(a)

chiral tip

glass

substrate

cantilever !"

#"

(b)

tip dipole

image dipole !"

#"

,inc inc± ±E H ,

inc inc± ±E H

FIG. 1. (a) Schematic of the PiFM system with a chiral with illumination from the bottom. (b)

Photo-induced chiral tip dipole and image dipole according to image dipole theory.

the time-harmonic field, the time-averaged electromagnetic force becomes [33]:

〈F〉 = 1

2Re

{

ptip ·(

∇Eloc(rtip))∗

+ µ0mtip ·(

∇Hloc(rtip))∗

−cµ0k4

6π(ptip ×m∗

tip)}

(2)

where Eloc(rtip) and Hloc(rtip) are the local electric and magnetic field at the tip dipole

position. The time-averaged force has three distinct contributions: the first term on the

right hand side of equation (2) is recognized as the electric dipolar force 〈Fe〉, the second

term is known as the magnetic dipolar force 〈Fm〉, and the last term is called the interaction

force 〈Fint〉.To calculate the force, we require expressions for the induced electric and magnetic dipole

moments. Assuming that the tip is an isotropic and reciprocal chiral object, the dipole

moments can be related to the local fields through polarizabilities as [34]:

ptip = αeeEloc(rtip) + αemH

loc(rtip) (3)

mtip = −µ−1

0 αemEloc(rtip) + αmmH

loc(rtip) (4)

where αee, αmm and αem are the electric, magnetic and electro-magnetic polarizabilities,

respectively. The local fields at the tip dipole position can be obtained from image dipole

theory [31, 35, 36]. We model the effect of the substrate by substituting the substrate

response by the image of the photo-induced (chiral) dipole in the tip, written as pimg and

mimg and shown in Figure 1(b). The local electric and magnetic fields at the tip position

can then be expressed as:

Eloc(rtip) = Einc(rtip) + Escaimg→tip(rtip) (5)

Hloc(rtip) = Hinc(rtip) +Hscaimg→tip(rtip) (6)

4

where Einc(rtip), Hinc(rtip) are the incident electric and magnetic field at the tip dipole

location, and Escaimg→tip(rtip), H

scaimg→tip(rtip) are the scattered electric and magnetic field by

the image dipoles at the tip dipole location.

The scattered fields are defined through the image dipole moments and their Green’s

functions as: [31, 35]

Escaimg→tip(rtip) = Gee(rtip − rimg) · pimg +Gem(rtip − rimg) ·mimg (7)

Hscaimg→tip(rtip) = Gme(rtip − rimg) · pimg +Gmm(rtip − rimg) ·mimg (8)

where G(r) = Gxxx + Gyyy + Gzzz is the dyadic Green’s function in Cartesian coordi-

nates and the four different electric [Gee(r), Gem(r)] and magnetic [Gme(r), Gmm(r)] dyadic

Green’s functions for an electric and magnetic dipole are defined as: [37]

Gee(r) =1

4πε0|r|3(3rr− I) (9)

Gmm(r) =1

4π|r|3 (3rr− I) (10)

Gem(r) = − 1

iωε0∇×Gmm(r) (11)

Gme(r) =1

iωµ0

∇×Gee(r) (12)

Here I is a unit tensor of rank 2 and r is the unit vector from the source point to the point

of observation.

In the same fashion, we can also define the image dipole moments and the local fields at

the image location. Our goal is to obtain self-consistent expressions for the local fields at tip

location and for the dipole moments at the time in terms of the incident light components.

To achieve this, we first ignore any phase retardation effects in the fields between the tip and

image dipole position. Second, we will only consider terms in the polarizability up to second

order. With these assumptions, we may use equations (3)-(6) to determine the longitudinal

5

(z) component of the time-averaged force in the dipole approximation as:

〈Fz,e〉+ 〈Fz,m〉 =3

2π|z |4

[

1

2

( |αee|2ε0

+ µ0|µ−1

0 αem|2)(1

2|Einc

‖ |2 − |Eincz |2

)

+1

2

( |αem|2ε0

+ µ0|αmm|2)(1

2|Hinc

‖ |2 − |Hincz |2

)

+ Re{αeeα

∗em

ε0− αmmα

∗em

}

Re{1

2Einc

‖ ·Hinc∗‖ − Einc

z ·Hinc∗z

}

− Im{αeeα

∗em

ε0− αmmα

∗em

}

Im{1

2Einc

‖ ·Hinc∗‖ − Einc

z ·Hinc∗z

}

]

(13)

〈Fz,int〉 = − ck4

12π

[

Re{

− α∗emαee

}

Re{

[Einc‖ × Einc∗

‖ ]z

}

+ Re{

µ0α∗mmαem

}

Re{

[Hinc‖ ×Hinc∗

‖ ]z

}

+ 4ωIm{

− α∗emαee

ε0

}

[σincE ]z + 4ωIm

{

− α∗mmαem

}

[σincH ]z

+ Re{

α∗emαem[S

inc∗]z

}

+ Re{

µoα∗mmαee[S

inc]z

}

]

(14)

where Einc‖ = E inc

x x+ E incy y, Hinc

‖ = H incx x+H inc

y y and Eincz = E inc

z z, Hincz = H inc

z z are the

transverse and longitudinal components of the incident fields at the tip dipole location, the

notation (rtip) has been avoided here for brevity; Sinc = Einc ×Hinc∗ is the time-averaged

Poynting vector of the incident field; σincE and σ

incH are the electric and magnetic parts of

the time-averaged total spin angular momentum density of the incident light; and z is the

vertical distance between the tip dipole and its image. See the Supporting Information for

a detailed derivation of equations (13) and (14).

Equation (13) describes the force due to the combined electric and magnetic dipolar

response of the tip. This force shows a distance dependence on the tip-sample distance

which scales as z−4, similar to the distance dependence of the gradient force that is typically

measured in PiFM [31, 38]. The first two lines of equation (13) are recognized as the purely

electric and purely magnetic dipolar force contributions to the gradient force, whereas the

latter two lines describe the forces that arise from a nonzero scalar product of the electric

and magnetic fields, i.e. the helicity density.

Equation (14) accounts for the interaction force, which lacks a direct dependence on the

tip-sample distance. This contribution to the force resembles the scattering force [31, 38]. In

6

addition to the purely electric and magnetic contributions to the scattering force, described

by the first two lines in equation (14), the latter two lines add force contributions that

scale with the longitudinal component the spin angular momentum density and with the

time-averaged Poynting vector.

We are interested in finding an expression for the differential force, i.e the force difference

between sequential measurements where the tip-sample junction is illuminated with incident

light of opposite handedness. For this purpose, we assume two states for the incident light,

indicated by Einc+, Hinc+ and Einc−, Hinc−, describing the input fields of different handed-

ness. The two illumination states have the same energy densities, |Einc+|2 = |Einc−|2 and

|Hinc+|2 = |Hinc−|2, but exhibit opposite helicity densities, Im(Einc+ ·Hinc+∗) = −Im(Einc− ·Hinc−∗). Note that the longitudinal spin angular momentum density components are related

as [σinc+E ]z = −[σinc−

E ]z and [σinc+H ]z = −[σinc−

H ]z. We also find that Sinc+ = Sinc−, and that

the following relations hold:

Re{

[Einc+‖ × Einc+∗

‖ ]z

}

= Re{

[Einc−‖ × Einc−∗

‖ ]z

}

Re{

[Hinc+‖ ×Hinc+∗

‖ ]z

}

= Re{

[Hinc−‖ ×Hinc−∗

‖ ]z

} (15)

By measuring the force under illumination with incident light of (+) and (−) handedness,

and taking the difference, the differential gradient force ∆〈Fz,grad〉 and the differential scat-

tering force ∆〈Fz,scat〉 can be obtained from equations (13) and (14) as:

∆〈Fz,grad〉 = − 6ωc

π|z |4 Im{αeeα

∗em

εo− αmmα

∗em

}(1

2hinc‖ − hinc

z

)

(16)

∆〈Fz,scat〉 = −2ωck4

3π

[

Im{

− α∗emαee

εo

}

[σincE ]z + Im

{

− α∗mmαem

}

[σincH ]z

]

(17)

where hinc‖ = 1

2ωcIm

{

Einc‖ ·Hinc∗

‖

}

and hincz = 1

2ωcIm {Einc

z ·Hinc∗z } are the tangential and

longitudinal helicity density of the incident chiral light. We see that the differential gra-

dient force depends on the helicity density of the incident beam, whereas the differential

scattering force contains qualitative information about the spin angular momentum. For

small tip-sample distances, the differential gradient force is expected to form the dominant

contribution to the measured force, thus offering a means of experimentally extracting infor-

mation about the helicity density. Note that the differential force in equation (16) depends

on the difference between the helicity density of the transverse and longitudinal components

of the incident field.

7

(a)

(b)

, [pN]z totalF∆

10logee o mm

α ε α−

FIG. 2. (a) The total difference force spectrum in pN. (b) The normalized magnitude spectrum of

αee − ε0αmm. In all calculations, the tip-image dipole distance is |z| = 10 nm. Black lines indicate

resonance condition of c1 and white lines indicate minima due to satisfying the Kerker condition.

III. MIE SCATTERING FORMALISM

We next study the characteristics of the differential gradient force. We see from equa-

tion (16) that ∆〈Fz,grad〉 depends on the polarizability of the tip material, including the

electro-magnetic polarizability. To model the tip polarizability, we assume that the tip can

be described as a spherical chiral nanoparticle (NP). We also assume that the following con-

stitutive relations hold: D = εoεE + i√ε0µ0κH and B = µ0µH − i

√ε0µ0κE, where ε and

µ are the relative permittivity and permeability, respectively.[34] The chirality parameter κ

is an empirical quantity that provides the chiral strength of the material under considera-

tion. We can next relate the electric, magnetic and electro-magnetic polarizabilities through

the material parameters by using Mie scattering theory as:[17, 29, 39] αee = −6πiε0b1/k30,

αmm = −6πia1/k30 and αem = 6πic1/(ck

30) where c is the free space speed of light, k0 is the

wavenumber in free space and b1, a1, c1 are the Mie coefficients. In our calculations, the

material parameters of the isotropic chiral NP such as ε and µ are those of silicon[40] with

µ = 1.

8

We next place the sphere above a flat and transparent dielectric surface at a tip-surface

distance of 5 nm. The system is subsequently illuminated with a plane wave of field strength

1.5× 106 Vm−1 that is either in the RCP or LCP state, and which propagates through the

transparent dielectric material toward the sphere. The wave induces a polarization in the

sphere, which in turn produces an image dipole in the dielectric material. For simplicity, it

is assumed that the image dipole strength is identical to the induced dipole in the sphere.

In the current configuration, the image dipole is found at a distance z = 10 nm from the

outer diameter of the sphere. We calculate the total difference force spectrum with the

aid of equations (16) and (17), defined as ∆〈Fz,total〉 = ∆〈Fz,grad〉 +∆〈Fz,scat〉, and plotted

in 2(a) as a function of sphere radius and excitation wavelength. In all calculations, the

value of κ is set to 0.1.[41, 42] It is clear that nonzero ∆〈Fz,total〉 is achieved under certain

experimental conditions. In particular, the maxima and minima are seen to co-localize with

the resonant spectral position of the electro-magnetic Mie coefficient c1 of the chiral NP.

The black dotted lines show the location of the peak values of Im(c1). Changing κ does

not alter the spectral resonances, but instead changes the sign and magnitude of the total

differential force proportionally.

A careful inspection of equation 16 suggests that, for any chosen r and κ of the tip,

∆〈Fz,total〉 may approach zero when the first Kerker condition of the chiral NP is met,

irrespective of helicity density of the incident light. The Kerker condition states that αee =

ε0αmm [12, 43, 44], and this condition is plotted in Figure 2(b) as a function of radius and

excitation wavelength, while κ is fixed at 0.1. The white dotted lines depict the conditions

where the relation αee = ε0αmm is satisfied whereas the black dotted lines show the resonant

position of electro-magnetic Mie coefficient c1 as found in panel (a). The finite separation

between the two curves assures that the above relation is less likely to have any impact on

helicity density measurements if the force difference measurement is properly maximized.

IV. FULL WAVE SIMULATIONS

To further validate our analytical findings, we perform 3D full wave simulations to de-

termine the forces between the dipole induced in the chiral sphere and its image in the

dielectric material below. We use the finite element method in COMSOL Multiphysics for

this purpose. In the simulation, the tip and image dipoles are modeled as isotropic chiral

9

spheres [32, 45, 46] with similar parameters as used above in Figure 2(a). The system is

sequentially illuminated by RCP and LCP plane waves of field strength 1.5× 106 Vm−1 in

the bottom illumination scheme. The force exerted on the tip is determined using Maxwell’s

stress tensor formalism.

Figures 3(b), (c) and (d) show the time averaged-force difference spectrum for spheres of

radius of 50 nm, 60 nm and 70 nm. The black curve shows the analytical result whereas the

full wave FEM result is indicated by the green solid line. For r = 50 nm, the FEM simulation

closely corroborates the analytical result obtained in the dipole approximation, where the

dip corresponds to the Mie resonance of the c1 coefficient. The spectra of Figure 4(c) and

(d) display some dissimilarities between the FEM and analytical calculations, which can be

attributed to higher order multipole contributions in the lower wavelength range. Yet, in the

range examined the dipolar contribution remains dominant and provides a maximum force

difference value −0.32 pN for a tip radius of rNP = 70 nm when κ = 0.1. Near resonance,

the differential force is of the order of 0.1 pN. Although such forces are near the noise floor

of a typical PiFM microscope, sensitive experiments are likely able to resolve the targeted

differental force under optimized conditions.

Finally, we present the differential force map of the helicity density for a focused chiral

beam, a scenario of direct relevance to microscopy applications. We assume a circularly

polarized beam of 2 mW average power at a wavelength of λ = 520nm as the incident light

source, which is focused by a 1.4 NA oil (n = 1.518) objective lens. The focal plane electric

and magnetic field field components used in the simulation are obtained from [31]. We

first calculate the normalized helicity density distribution at the focal plane for a focused

LCP beam, which is shown in 4(a). As shown in 4(b) and (c), spatial distribution of the

the transverse and longitudinal components of helicity density resemble a circularly and

doughnut-shaped profile, respectively. We next calculate the differential force. In Figure

4(d), ∆〈Fz,total〉 is shown for the case of a tip of radius rNP = 60 nm. The central region

of the beam is characterized by a negative differential force, surrounded by region where

the differential force is positive. We can understand this by realizing that the transverse

and longitudinal helicity density contributions have different spatial profiles with opposite

signs. We can further separate the transverse and longitudinal contributions by proper

design of the tip’s chiral properties. For example, if the chiral tip is designed such that

it exhibits only longitudinal chirality, i.e. αem,xx = αem,yy = 0 and αem,zz = αem, then

10

(b)

(c) (d)

NPr 50nm=

NPr 60nm=

NPr 70nm=

(a)

FIG. 3. (a) Schematic of chiral tip sphere and image sphere with radius rNP and tip-image dipole

distance of |z| = 10 nm used in FEM simulation. Time-averaged differential force from analytical

calculation (black solid curve) and from full wave simulation (green solid curve) for a chiral NP

with radius (b) rNP = 50 nm, (c) rNP = 60 nm, (d) rNP = 70 nm and κ = 0.1 under RCP/LCP

illumination.

the differential force map will track the doughnut-shaped longitudinal helicity density as

shown in 4(e). Conversely, a chiral tip having solely transverse chirality with αem,xx =

αem,yy = αem and αem,zz = 0 produces a map that reports on the central transverse helicity

density, as shown in 4(f). In the supporting information, we also present the helicity density

maps determined with the differential force of a focused azimuthally radially polarized beam

(ARPB)[17, 31, 47].

V. CONCLUSION

In this work, we have studied the information contained in the photo-induced force when

illuminated by chiral light. Our theoretical analysis reveals that the differential force is

directly sensitive to the chiral properties of light. In particular, the differential gradient

force is proportional to the helicity density of the incident light field, whereas the differential

scattering force is sensitive to the spin angular momentum of the applied light. Using realistic

11

(d) (a)

(e) (b)

(f) (c)

[pN]zF∆inc

h

inc

zh

inch!

, , 0em xx em yyα α= =

, 0em zz

α =

[pN]zF∆

[pN]zF∆

FIG. 4. (a) The focal plane distribution of normalized helicity density of a circularly polarized(LCP)

light focused by a 1.4 NA oil objective. Distributions of (b) longitudinal and (c) transverse compo-

nent of the normalized helicity density in (a).(d)The total force difference map at the focal plane for

a chiral isotropic tip of radius rNP = 60 nm under the same focused illumination of LCP and RCP.

Force difference maps for a (b)longitudinally chiral(αem,xx = αem,yy = 0 and αem,zz = αem) and

(c)transversely chiral(αem,xx = αem,yy = αem and αem,zz = 0) tip tracking the respective helicity

densities with opposite sign.

values for the illumination intensity, tip dimension, and the chirality parameter of the tip,

we find that the differential force can reach detectable values of several hundreds of fN,

just above the noise floor of common scan probe microscopy systems. These findings are

significant because a direct characterization of optical chirality at the nanoscale has hitherto

been extremely challenging. The observation that the differential gradient force can map out

12

the local helicity density of the light is relevant for numerous applications where knowledge

of the chiral state of light at sub-diffraction-limited dimensions is important.

ACKNOWLEDGMENTS

The authors thank the Keck Foundation and the National Science Foundation, grant

CMMI-1905582.

SUPPLEMENTARY INFORMATION

Detailed derivations are available in the Supporting Information. Please contact the

corresponding author for a copy.

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