Pattern Multiplication in Aluminium Nanoantenna Arrays
N. Dorh, J. L. Stokes, G. S. Hilton and M. J. Cryan*
Department of Electrical and Electronic Engineering,University of Bristol, Bristol BS8 1UB, UK
e-mail: n.dorh@bristol.ac.uk
e-mail: m.cryan @bristol.ac.uk
Abstract
Like radio antennas, nanoantennas have an effective area much larger than their physical cross-section, couple high
near field intensities into the far field and have tunable resonances. These properties have made them highly desirable
across many applications such as optical probes, photonic circuits and single molecule detection. In this paper we present
another RF technique, pattern multiplication, and investigate its validity within the optical regime, using Finite
Difference Time Domain (FDTD) modelling of a 2 x 2 aluminium dipole nanoantenna array resonant around 420 nm.
1. Introduction
Nanoantenna research in recent years has involved proving and promoting the use of Radio Frequency (RF) design
techniques within the optical regime. These devices hold significant promise for sub-wavelength manipulation of light,
which could lead to breakthroughs for the fields of nano-imaging, nano-photonics and sensors, to list a few. Among the
developmental challenges are fabrication, characterisation and testing of such devices; in fact, to date, there is still no
consensus on an appropriate figure of merit for performance comparisons. At the heart of nanoantenna operations are
Localised Surface Plasmon Resonances (LSPRs). At optical frequencies, metal losses become significant and their
classification as perfect conductors breaks down. When light impinges on a metal surrounded by some dielectric
material, the waves can potentially be trapped at the interface between the two materials causing the collective oscillation
of the metal’s electrons at the surface. If the metal structure is much larger than the wavelength of impinging light, this
oscillation tends to propagate along the surface and is generally referred to as a Surface Plasmon Polariton (SPP) or
simply Surface Plasmon (SP). If the SP is confined to a metal of comparable size to the wavelength of the SPP, then the
oscillation forms a standing wave on the surface and is referred to as a Localised Surface Plasmon (LSP). In general, for
an interface mode to exist, the normal component of the wavevector should be purely imaginary, forcing the fields in the
two materials to decay exponentially[1]. For best coupling to plasmon modes, the real part of the dielectric function of
the metal should be opposite in sign and twice the absolute value of the surrounding dielectric, which is known as the
FrÖhlich condition. The material and frequency dependence of this requirement means that not all metals can support
plasmons and that metals could be separated by spectral regions for which they were best suited [2].Initially noble metals
such as gold, were front runners for nanoantennas; however, aluminium and other similarly termed ‘poor metals’ have
recently been targeted due to their relatively low absorption losses in the blue-UV spectral range[3]. LSPRs are extrinsic
and as such can be designed to enhance both the absorption and emission of a fluorescent emitter[4]. The true value and
utility of these structures lies not only in the plasmonic enhancement and confinement of local electric fields but equally
in the control and manipulation of the far fields as with radiowaves. In this paper, we present further support for the RFOptical analogue through a demonstration of pattern multiplication.
2. Principle of Operation
For an interface mode such as an LSP to exist the normal component of the wavevector has to be purely imaginary
in both the metal and dielectric [1]. It can be shown that the parallel component of the wavevector for the LSP mode,
assuming propagation in the x-direction, is given by:
(1)
kx =k1 +jk2
where k1 =
εm1 εd ω
εm1 +εd c
and k2 =
εm1 εd
εm2 εd
ω
2εm1 εm1 +εd c
The variable ε refers to the complex dielectric functions such that subscripts m1 and m2 represent that of real and
imaginary parts for metal and subscript d refers to that of the dielectric. and c refer to the angular frequency of the
incident light and the speed of light in a vacuum respectively.
The dispersion plot shown in Figure 1 was generated assuming air as the dielectric and using the requisite optical
constants for aluminium. The curve shows the relationship for bulk aluminium but key deductions can still be made from
this diagram. Immediately apparent from the graph is the fact that for a given angular frequency (energy) there is a
mismatch between the momentum of freely propagating photon and the plasmon. In the case of the surface plasmon, it
always has a greater momentum than that of the photon and since there is an inverse relationship to wavelength it is
evident that the plasmon wavelength will invariably be shorter.
kpx
kpy
kp photon k
sp
substrate
Figure 1: Dispersion Relation for Al and air
Figure 2: Momentum Matching
Consider Figure 2, the nano-dipole antenna has an intrinsic plasmonic mode with wavevector ksp. In order to
successfully couple to this plasmon mode, the parallel component of the incident photon wave vector, kpx, must meet the
minimum momentum required (kpx ≥ksp ). As previously discussed, the SP mode tends to have a greater momentum and
thus by the principle of conservation of momentum, the additional momentum needs to be supplied, to excite the
plasmons. The Otto configuration [5] and nanoantennas are two examples of approaches used to bridge this “momentum
gap”. A nanoantenna, therefore, matches the momentum of the incoming photons to that of the LSP mode much like the
impedance matching done by the classical RF antenna.
As the plasmons reflect at the nanoantenna boundary; ‘plasmonic’ standing waves form. Hence the nanoantennas
can be considered resonant cavities, capable of supporting multiple plasmonic modes, each with its own distinct charge
distribution and resulting radiation patterns[6]. The type and number of supported modes is dependent on the physical
dimensions[4]. Figure 3(a) presents the baseline design used during this investigation; dimensions were determined using
scaling rules developed in previous work[7] for resonance at 420 nm. Finite Difference Time Domain (FDTD) modelling
of these aluminium dipole nanoantennas was used to determine the electric field distribution when excited by an electric
dipole in the gap. The resulting E fields at the surface can be seen in Figure 3(b)(d); the field distribution changes from a
single dipolar to a multipolar when the longer antenna arm length was used for the same resonant frequency. The far field
projections in Figure 3 (c) and (e), reflect the differences in the two nanoantenna modes and are in agreement with
findings reported by Taminiau et al[6].
(b)
(a)
(c)
(e)
(d)
Figure 3: (a) Single Al dipole excited by 350-550 nm Electric dipole source, w=40 nm, g=30 nm (b) |E| Field and (c) Far Field
Projection for single dipole nanoantenna l=60 nm, (d) |E| Field and (e) Far field projection for single nanoantenna with l=190
nm
3. Nanoantenna Arrays
Whether for MIMO, diversity, gain or beam control, RF antennas are often used in arrays. An array of antennas is
characterised by a complex vector function called the Array Factor (AF). The AF encapsulates the array size, pitch and
phase dependence of the far field by assuming identical isotropic elements[8]. For an array of N elements, separated by
spacing d, progressive phase weighting β and incident EM wave with wave vector k and phase , the AF can be
expressed as:
AAF= ∑Nn=1 ej
n-1 kd cos θ +β
(2)
The resultant radiation from the array is then the product of the radiation patterns of each element and the AF. In a
uniformed array, that is, made up of a number of identical antenna elements, the resultant field determined using Pattern
Multiplication. This principle states that the total electric field is equal to the product of the field of a single element and
the array factor[8]. The far field radiation pattern can therefore be defined analytically provided the field of the individual
element can also be defined analytically.
Nanoantennas have also begun to be used in arrays. In contrast, their response to external sources is historically
linked to numerical methods such as Mie theory[9]. Richard Gans’ extension to Gustav Mie’s solution to Maxwell’s
Equations has previously been used to account for the scattering of light by the prolate spheriodal particles and
dependence of resonance on the nanoantenna physical dimensions. In a recent publication by Munárriz et al[10], the
angular radiant intensity (U(θ,φ)) on an enclosing sphere of radius R, by a linear array of N silver nanospheres embedded
in a dielectric medium was derived using the following relationship.
UU
,
R2 ∑m Ĝ(R,rm )pm
2
(3)
Operator Ĝ in this expression represented the dyadic Greens function, evaluated at position rm along the enclosing
sphere. pm is the induced dipole moment of the mth nanosphere. As the array size increases, so does the complexity and
computational efforts. Thus an analytic solution to the response of nanoantennas resonant external excitation would assist
in analysis and design of arrays [10].
In the second part of this investigation, we present a comparison of a 2x2 Aluminium
dipole nanoantenna array far field, firstly derived using FDTD modelling and then by
Pattern Multiplication. Figure 4 depicts the array; dimensions were selected to allow for a
spacing of 100 nm between antenna edges in both directions, and such that any effects of
evanescent field can be negated. As with the previous investigation, far field behaviour was
determined by exciting each nanoantenna with identical, coherent electric dipole sources
centred in the gap. For Pattern Multiplication, analytic expressions in elevation angle θ:
sin θ and sin θ cos θ 2 were used to approximate the radiation patterns produced by Figure 4: 2x2 Aluminium
nanoantennas l=60nm and 190 nm respectively. Both approximations assumed omni- dipole nanoantenna array a=
directionality and hence variations in the azimuth plane were omitted. Figure 5 depicts the 510 nm b=140 nm
analytic approximations for the two baseline designs previously explored and the resulting
AF for the given array properties.
(a)
(b)
(c)
Figure 5: Analytic approximations of E plane of radiation pattern from dipole with (a) l=60 nm (b) l=190 nm and (c) AF
calculated using defined array spacings, equal amplitude and phase weightings
The far field obtained from the single nanoantenna with l=60 nm and 190 nm represent the first and second
eigenmodes respectively. Far field projections from Pattern Multiplication and FDTD are shown in Figure 6, taking a
slice through the azimuthal angle =0°. For ease of comparison, a common dB scale was used.
(a)
(b)
Figure 6: (a) Polar plot for 2x2 nanoantenna array for first eigenmode (b) Polar Plot for 2x2 nanoantenna array for the second
eigenmode
Table 1: Key Metrics
Main lobe angle (°)
FDTD
1st Eigenmode
2nd Eigenmode
0
±52.9
Pattern Multiplication
1st Eigenmode
2nd Eigenmode
0
±50
Side Lobe Level (dB)
Side lobe angle (°)
-3 dB Beamwidth (°)
-6.7
±45.7
17.2
-20.9
±18.5
20.8
-1.4
±53
26
-25.7
±20
19
Figure 6 (a) and (b) show remarkable resemblance between the general beamshapes predicted by FDTD and
Pattern Multiplication. From Table 1, it can be seen that there was better overall agreement between FDTD and Pattern
Multiplication for the second eigenmode than the first. However, the main lobe was accurately predicted for the first
eigenmode. A difference of 5.7 dB can be seen in the side lobe levels for the first eigenmode compared to 4.8 dB for the
second. Side lobe angle predictions were within 13.8% and 5.5% for the first and second eigenmodes, respectively which
is significantly more accurate than the previous amplitude predictions. The -3 dB Beamwidth is a commonly used RF
metric which measures the beamwidth of the main lobe, -3 dB from the peak. For the first eigenmode, Pattern
Multiplication overestimated the beamwidth by 8.8° and underestimated the beamwidth by just 1.8° for the second
eigenmode. Pattern Multiplication is clearly a powerful analytic approach to analysing and predicting nanoantenna
arrays, however, differences in the side lobe levels highlight the need for further refining. In the absence of amplitude and
phase weighting of array elements, the beam shape is primarily governed by element spacing. It may be possible,
therefore, to improve these results by improving the analytic approximations and thereafter analysing any residual
differences.
Despite the complex underlying physics governing LSPs themselves, their generation and interactions, the
behaviour of the entire array can be derived, with reasonable accuracy, using straightforward RF techniques. Beyond this,
further control and manipulation of light at subwavelength scales such as beam shaping, steering and gain control via
phased nanoantenna arrays is possible. Subsequent experimental testing is expected to support this inference.
4. Conclusions
Nanoantenna technology shows promise for emerging and established applications alike. Properties such as strong
localised fields, transduction and tunability make them similar to classical RF antennas. FDTD modelling supported
previous postulates of plasmonic standing waves and multipolar resonances which can prove particularly beneficial to the
fabrication of nanoantennas. There was good overall agreement between beam shapes predicted using Pattern
Multiplication and FDTD. Pattern Multiplication predicted side lobe angles within as little as 5.5%. This is a particularly
important result for nanoantenna array design and the development of phased nanoantenna arrays.
5. References
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