Magnetic Localized Surface Plasmons: Supplemental Material.

Paloma A. Huidobro,1, ∗ Xiaopeng Shen,2, ∗ Javier Cuerda,1 Esteban Moreno,1

Luis Martin-Moreno,3 F.J. Garcia-Vidal,1, † Tie Jun Cui,2, ‡ and John B. Pendry4

1Departamento de Fisica Teorica de la Materia Condensada

and Condensed Matter Physics Center (IFIMAC),

Universidad Autonoma de Madrid, 28049, Spain.

2State Key Laboratory of Millimetre Waves,

School of Information Science and Engineering,

Southeast University, Nanjing 210096, China.

3Instituto de Ciencia de Materiales de Aragon and

Departamento de Fisica de la Materia Condensada,

CSIC-Universidad de Zaragoza, E-50009, Zaragoza, Spain

4Imperial College London, Department of Physics,

The Blackett Laboratory, London SW7 2AZ, UK

(Dated: October 22, 2013)

∗ These two authors contribute equally to this work.† Corresponding author: fj.garcia@uam.es‡ Corresponding author: tjcui@seu.edu.cn

1

CONTENTS

I. Modal Expansion 2

II. Contributions to the Scattering Cross Section 3

III. Frequency ordering of the EM modes 4

IV. Metamaterial Approximation 8

V. Disks Corrugated with Spiral-Shaped Grooves 10

VI. Fabricated Sample and Measurement 12

References 13

I. MODAL EXPANSION

Here we detail the analytical model that yields to Eq. 1 in the main text. In order to deal

with the EM response of PEC cylinders corrugated with grooves of parallel walls, we make

use of the Modal Expansion (ME) technique. The ME is based on the matching of the EM

modes outside the cylinder (radial coordinate ρ > R) and inside the grooves (r < ρ < R)

by means of the appropriate boundary conditions at ρ = R. Given that the grooves have a

rectangular shape, we first consider an array of N grooves with parallel walls on a flat surface

and then apply Born von Karman boundary conditions for a super-cell of length Λ = 2πR

to take account of the circular geometry. This procedure leads to the following field inside

the grooves:

HIz = Aeik0ngy′

Λ2π +Be−ik0ngy′

Λ2π (1)

where A and B are constants, x′ = 2πx/Λ and y′ = 2πy/Λ are the renormalized coordinates

and we have taken only the fundamental waveguide mode based on the assumption that the

particle is subwavelength. On the other hand, in the outer region the fields can be expanded

in terms of the Hankel function of the first kind, H(1)n :

HIIz =

∞∑−∞

Cneinx′H(1)

n

(k0y

′ Λ

2π

), (2)

2

the set Cn being constants. By applying the boundary conditions for each mode n we arrive

to the following transcendental equation:

S2n

H(1)n (k0R)

H(1)′n (k0R)

tan (k0ngh) = −ng (3)

which is Eq. 1 of the main text. Here, Sn =√a/d sinc [nπa/(Nd)] and H

(1)′n (x) ≡ dH

(1)n (x)d x

.

The solution of the equation above for a given n yields the complex resonance frequency

of the localized SP mode of order n. Physically, our model represents an EM mode that

runs around the cylinder surface, with each resonance appearing when an integer number of

modal wavelengths fits into the perimeter.

II. CONTRIBUTIONS TO THE SCATTERING CROSS SECTION

The analytical model that yields Eq. (1) of the main text can be used to calculate the

Scattering Cross Section (SCS) of corrugated particles as well as the relative contributions

of each mode n to it. For illustration purposes, we consider here a PEC cylinder corrugated

with radial grooves filled with a dielectric (ng), as sketched in Fig. S 1. This system shows

the same physics as the particles corrugated with rectangular grooves considered in the

main text while allowing for a simpler analytical treatment. Based on the Modal Expansion

procedure, an analytical expression for the SCS of a cylinder textured with radial grooves

can be derived assuming that the field within the grooves is given by only one waveguide

mode (see Ref. [24] of the main text):

σ =4

k0

∞∑n=−∞

|an|2 (4)

The coefficient an for each mode n is given by:

an = −inadJn(k0R)f − ngJ ′n(k0R)g

adHn(k0R)f − ngH ′n(k0R)g

(5)

where f = J1(ngk0R)Y1(ngk0r) − J1(ngk0r)Y1(ngk0R) and g = J0(ngk0R)Y1(ngk0r) −

J1(ngk0r)Y0(ngk0R). The SCS given by the above equation for a subwavelength cylin-

der is plotted in Fig. S 1 together with the numerically computed SCS: the two curves

show a very good agreement. Moreover, the analytical expression for the SCS allows us to

separate the contribution of each mode, which can be written as:

σ0 =4

k0|a0|2 , σn = 2

4

k0|an|2 (n 6= 0) (6)

3

In the SCS plot we present the contributions of the n = 0, n = 1 and n = 2 modes (σ0, σ1

and σ2, respectively). It is clear from this figure that the main contributions to the SCS of

2D subwavelength cylinders come from the two lower order modes. As explained in the main

text, the magnetic and electric dipoles (n = 0 and n = 1) are very close in frequencies with

the magnetic resonance being broader. Both of them contribute to the low energy peak.

σ

σ

σ

NumericalAnalytical

x

y

r R

ng

a

d

PEC

Figure S 1. Contribution of each mode to the SCS. SCS for a subwavelength PEC cylinder

corrugated with radial grooves (r = 0.4R, N = 40, a = 0.3d and ng = 8). The solid line shows

the numerically-computed SCS while the red dashed line corresponds to the analytical SCS. The

contributions of the n = 0, n = 1 and n = 2 modes to the analytical SCS are also shown. The

inset panel displays the geometrical parameters of the structure.

III. FREQUENCY ORDERING OF THE EM MODES

In this section we present a physical argument that explains why the electric dipole mode

of corrugated metallic disks has lower frequency than the magnetic dipole mode, whereas

the opposite frequency ordering is displayed for the modes of dielectric disks.

The precise magnitude of the eigenvalue associated to an eigenmode supported by any

of the structures considered in this paper is dictated by the spatial distribution of its cor-

responding electric field E. This fact can be best appreciated by means of the variational

4

formulation of electrodynamics. According to this formulation [1], the lowest frequency

eigenmode minimizes the following functional

U [E] =

∫dr3 |∇ × E|2∫

dr3 ε(r)|E(r)|2. (7)

Similarly, the next eigenmode minimizes the functional within the subspace orthogonal to

the first eigenmode. Moreover, the frequency ω corresponding to a given eigenmode can be

computed by evaluation of the functional at the field E corresponding to the mode:(ωc

)2= U [E] . (8)

Importantly, the functional is directly related to the electric field curl, the latter being linked

to the geometry and curvature of the field lines. Thus, inspection of the field line patterns

of the eigenmodes, e.g. the presence or absence of vortices, provides a physical clue to the

magnitude and ordering of the corresponding eigenvalues.

Since we have already numerically computed the eigenmodes with the Finite Element

Method, it is straightforward to evaluate Eq. (8) to determine the eigenvalues by means of

the variational theorem. Let us first consider the 2D cylinders. For the metallic corrugated

structure presented in Fig. 1 of the main manuscript the eigenvalues retrieved using Eq. (8)

are consistent with those previously found. The eigenvalues corresponding to the magnetic

and electric modes differ by less than 2 parts in 1000, confirming the quasi-degeneracy

already discussed for infinitely long corrugated cylinders. For a 2D dielectric cylinder of

the same radius and refractive index ng = 8 the eigenvalues determined with the FEM

and via Eq. (8) are again consistent, and they present a different behaviour than that for

the corrugated metallic cylinder. Now the magnetic and electric modes are not degenerate.

Their frequency difference is large, the electric mode having a frequency 60% higher than the

magnetic one. As mentioned above, this contrasting behaviour can be ascribed to the field

line patterns of the various eigenmodes, which are rendered in Fig. S2. The magnetic mode

of the dielectric cylinder (panel a) displays one vortex with azimuthal symmetry, whereas

the electric mode (panel b) presents two counter-rotating vortices with enhanced electric

field curl which, according to Eq. (8), rises its frequency value. This reasoning explains the

frequency ordering of the magnetic and electric dipole modes for a dielectric cylinder. On

the other hand, the corrugated metallic cylinder presents a very different electric field line

geometry. In this structure the presence of the grooves imposes a circulating pattern around

5

0 21 33.3x1010

x1010

0.2 0.6 10 0.80.4

1.1x1010

x1010

0 21 3 44.6x1010

x1010

Ele

ctric

dip

ole

Mag

netic

dip

ole

b

0 1.60.41.7x1010

x10100.8 1.2

c

d

aCorrugated PEC cylinderDielectric cylinder

Figure S 2. The curl field for 2D structures: a dielectric cylinder and a subwavelength

PEC cylinder corrugated with grooves. The arrows represent the (Ex,Ey) components of the

electric field, while the color map represents the squared rotational of the electric field normalized

to the electromagnetic energy of the mode. Panels a and b render the rotational fields for the n=0

(a) and n=1 (b) modes of a dielectric cylinder, while panels c and d correspond to the n=0 (c)

and n=1 (d) modes supported by a corrugated metallic cylinder.

the inner metallic core for both the magnetic and electric modes (panels c and d). Although

the quasi-degeneracy in the metallic case cannot be understood from the qualitative aspect

of the field pattern s alone, the full evaluation of the functional as well as the analytical

model presented in the main text predict precisely this behaviour.

When going from 2D corrugated metallic cylinders to 3D disks, it remains to be explained

6

0 212.9x1013

x1013

0 1.50.51.7x1013

x10131

Ele

ctric

dip

ole

Mag

netic

dip

ole

60 42

6.3x1011

x1011

0 424.7x1011

x1011

z = 0.5 L

z = 0.5 L z = 1.25 L

z = 1.25 La

b

c

d

Figure S 3. The curl field for a 3D subwavelength PEC disk corrugated with grooves.

The figure shows the electric field lines [(Ex,Ey), arrows] and the normalized squared rotational

of the electric field (color map) for a 3D disk of L = R at two different cut planes parallel to the

disk lid: modes n = 0 (a) and n = 1 (b) at the middle of the disk and modes n = 0 (c) and n = 1

(d) at a plane cut on top of the disk separated a distance 0.25L from it.

why the quasi-degeneracy is lifted and why, as a difference to the dielectric structures, the

magnetic mode is the one which acquires a higher frequency. We have first computed the

eigenvalues for a corrugated metallic disk of radius R = L by means of the variational

theorem. This confirms that both modes rise their frequencies as compared to the 2D

case: the magnetic mode increases its frequency a 22% from its 2D value and the electric

mode frequency increases a 10%. Notice that the evaluation of the functional requires an

integration to all space and it is important to realize that not all regions of space contribute

equally to the integral. In fact, the region comprising the inner part of the grooves is

7

responsible for about 80% of the total amount. Interestingly, if the functional is evaluated by

integration only within this region, the quasi-degeneracy is not lifted. This is a consequence

of the fact that the field within the grooves is almost the same for the 2D and 3D cases

(compare Figs. S2 and S3). As a consequence, we expect that the reason behind the

degeneracy lift when going from 2D to 3D stems from the fields outside the disk. We have

checked this by separating the integrals in Eq. (8) in three contributions corresponding to:

(i) the grooves, (ii) the regions above and below the disk lids, and (iii) the rest. It turns out

that region (ii) is the second in importance when the functional is evaluated. On this basis

we analyze the value of the electric field curl for both the electric and magnetic modes in

this region, i.e., in a plane parallel to the disk lid and above it, as shown in Fig. S3c and

d. The magnetic mode displays a distinct vortex (panel c) which is absent for the electric

mode (panel d). Thus, according to Eq. (8), when the corresponding electric field curls

are integrated on top of the lids, the magnetic mode increases its frequency more than the

electric mode. Finally, the corresponding computation has been performed for dielectric

structures. In a similar way to metallic structures, the transition from 2D to 3D rises the

frequencies of both modes, namely a 35% increase for the magnetic dipole and a 20% for the

electric dipole. However, as we have shown above, the modes of dielectric cylinders are not

degenerate but, instead, display a large frequency difference, the electric one having a 60%

higher frequency. As a consequence, the blueshifts are now unable to revert the frequency

ordering of the modes, and the magnetic dipole remains the ground mode.

IV. METAMATERIAL APPROXIMATION

Since the PEC particles we are concerned with in this work are corrugated at a subwave-

length scale (d � λ0), the region with grooves can be interpreted as an effective medium

layer. The EM tensors of the effective medium under TM incidence can be obtained by

averaging 1/ε in the azimuthal direction and by considering that the mode propagates with

velocity c/ng in the radial and vertical directions. In cylindrical coordinates they read:

εr =∞, εφ = n2gd/a, εz =∞ (9)

µr = a/d, µφ = 1, µz = a/d (10)

8

These permittivity and permeability tensors, ε̂ and µ̂, represent a metamaterial with

anisotropic and inhomogeneous EM properties.

x

y

r R

PEC

ε̂, µ̂ ε̂, µ̂

PEC

Metamaterial 2DCorrugated PEC 2D

Corrugated PEC L=RMetamaterial L=R

Figure S 4. Metamaterial approximation. In the effective medium approximation corrugated

PEC particles behave as an inhomogeneous and anisotropic layer of thickness R − r wrapped

around a PEC inner core. The plot shows the SCS for corrugated PEC cylinders (r = 0.4R,

N = 40, a = 0.8d and ng = 8) and disks (same parameters, thickness L = R) together with the

SCS in the effective medium approximation for both cases. The geometry for the metamaterial

model is depicted in the insets.

We have tested the validity of this approach for infinitely long cylinders and for disks

of finite thickness by means of numerical simulations. The results are illustrated in Fig. S

4. The geometries under consideration are depicted as insets: i) in the 2D case it consists

of a PEC core of radius r surrounded by a metamaterial shell of width R − r; ii) in the

3D case we deal with a shell of width R − r and thickness L surrounding the PEC core.

In the 2D case, the plot shows the SCS for the same subwavelength corrugated cylinder

considered in Fig. S 1 (black solid line) together with the SCS obtained in the metamaterial

approximation (red dashed line). We can observe that the two lines virtually coincide. In

the metamaterial approximation the low energy peak has contributions from the electrical

and magnetic dipoles, as discussed in the main text for corrugated cylinders (see Fig. 1a).

9

Regarding the effective medium approximation for the 3D corrugated disks of thickness L,

we have assumed that the 3D permittivity and permeability tensors can be approximated

by the 2D ones (Eqs. (4) and (5)) and we have implemented them in a shell of width R− r

and thickness L. In the figure, the SCS for the corrugated PEC particle is shown as a blue

solid line, while the effective medium approximation corresponds to the dashed green line.

In both cases the low energy peak corresponds to the electric LSP while the high energy

peak is the magnetic LSP, as studied for corrugated particles in the main text (see Fig.

1b). The agreement between the two lines demonstrates the fact that the EM response of

subwavelength corrugated PEC particles can be understood as that of an anisotropic and

inhomogeneous layer coating a PEC rather than as a isotropic dielectric shell surrounding

a PEC particle.

V. DISKS CORRUGATED WITH SPIRAL-SHAPED GROOVES

0

-0.0002

0.0002

0.18

-0.004

0

-0.004

0.004

0.81

-0.84

r

R

d a

a

c

b

E M

M

E

Figure S 5. Particles corrugated with spiral-shaped grooves. a, The plot shows the SCS for

the 2D case (black line) and disks of thicknesses L = R (red line) and L = R/2 (blue line), with

parameters: d = 0.159R, a = 0.33d, r = 0.063R, hm = 5R and ng = 1. All the parameters are

shown in the geometry sketch. b, c Field pattern at two different phases at k0R = 0.3245 for an

infinitely long spiral structure showing the electric and magnetic dipoles, respectively.

A different way of achieving grooves of large depth is presented here. It consists of a

10

PEC cylinder of radius R drilled with four spiral-shaped grooves that are wrapped around

a small PEC cylinder of radius r (see the geometry sketch in Fig. S 5). Each spiral groove

has width a and depth h = 5R, and two neighbouring arms are separated by a distance d.

The plot in Fig. S 5a presents the computed SCS for 2D cylinders (black line), and 3D disks

of thicknesses L = R (red line) and L = R/2 (blue line). As in the cases shown in the main

text, the dominant peak in the 2D SCS shows an electrical LSP and a magnetic LSP at

two different phases (Hz is depicted in panels b and c). The 3D cases present a shift of the

resonances that takes the magnetic LSP mode to higher frequencies than the electric LSP.

a bHZ

HZ

Ele

ctric

dip

ole

Mag

netic

dip

ole

z=0.5L

z=0.5L

c d

z=1.25L

z=1.25L

EZ

EZ

Figure S 6. Near-field distributions of a 3D PEC disk corrugated with spiral grooves at

the electric and magnetic resonant frequencies on two plane cuts. a, b, Field patterns

at the electric LSP (k0R = 0.32). The magnetic field, Hz is plotted in the x− y planes at z = L/2

(a) and the electric field, Ez, is plotted at z = 1.25L (b). c, d Field patterns at the magnetic LSP

(k0R = 0.364): Hz at z = L/2 (c) and Ez at z = 1.25L (d). The thickness of the disk is L = R/2.

Figure 6 of this Supplementary Information presents the near-field distribution at the

electric (a, b) and magnetic (c, d) LSP for a 3D disk corrugated with spiral-shaped grooves.

Panels a and c show the z component of the magnetic field, Hz, at a plane cut in the middle

of the disk (z = L/2) for the electric (a) and magnetic (b) LSPs. For both resonances, the

near field resembles that of the 2D case (see Fig. S 6b and c), and the electric-dipole and

magnetic-dipole characters are clear. These characters are still clear for both resonances

11

when the electric field, a quantity that is attainable in the experiments, is plotted on top

of the disk, in a plane at a distance 0.25R from its the upper face. The pattern of Ez at

z = 1.25L is shown for the electric LSP and the magnetic LSP in panels b and d, respectively.

We want to stress here that these patterns closely resemble the near-field measured for the

ultra-thin metallic disk with spiral arms shown in Fig. 4 of the main text.

VI. FABRICATED SAMPLE AND MEASUREMENT

A photograph of the fabricated structure can be seen in Fig. S 7a. The sample consists

of a metallic disk wrapped with four spiral arms (radius R = 9.5 mm). The disk has a

thickness, L = 0.035 mm, much smaller than the excitation wavelength. The ultra-thin

structure is printed on top of a FR4 substrate with thickness ts = 0.8 mm and dielectric

permittivity εs = 3.5. The EM response of the sample is measured by using two monopole

antennas, as illustrated in Fig. S 7b. The transmitting antenna is placed 3 mm away from

the sample to excite the EM modes, and the receiving antenna can move freely in a plane

1.5 mm above the sample to measure the near-field response.

Figure S 7. The fabricated sample and measurement of EM responses. a, Fabricated

ultra-thin textured metallic disk: photograph of the structure, with parameters R = 9.5 mm,

r = 0.6 mm, d = 1.508 mm, a = 1.008 mm and L = 0.0037R. b, Photograph of the experimental

system to measure the EM responses of the ultra-thin corrugated disk.

12

[1] Joannopoulos, J. D., Johnson, S. G., Winn, J. N. and Meade R. D., Photonic Crystals: Molding

the Flow of Light, Princeton University press (2008).

13