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Plasmonic quasicrystals
Venu Gopal Achanta
PII: S0079-6727(14)00058-5DOI: http://dx.doi.org/10.1016/j.pquantelec.2014.12.002Reference: JPQE184
To appear in: Progress in Quantum Electronics
Cite this article as: Venu Gopal Achanta, Plasmonic quasicrystals, Progress in QuantumElectronics, http://dx.doi.org/10.1016/j.pquantelec.2014.12.002
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Plasmonic Quasicrystals
Venu Gopal Achanta
DCMP&MS, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005
INDIA
Abstract
Plasmonic quasicrystals consisting of quasi-periodic metal-dielectric patterns offer several advantages
compared to the periodic patterns or plasmonic crystals. This paper reviews the present status in
theoretical design, modeling, fabrication and basic and applied results on plasmonic quasicrystals. In
addition to the current status, possible future prospects of plasmonic quasicrystals are also discussed.
Keywords: Plasmonic quasicrystal, surface plasmon polariton, broadband transmission enhancement
1. Introduction:
In modern scientific literature, Penrose tiles and their construction are well discussed in Mathematics.
Several ways to generate non-periodic tilings have been proposed starting from darts and kites getting
replaced by aces and bows to arranging thick and thin rhombi and finite automaton based up-down
generation method [1-5]. However, after the discovery of Al-Mn alloy with icosahedral symmetry
(considered could never exist as they violate the crystallographic principles) by Shechtman et al [6], these
structures caught the imagination of Physicists. Levine and Steinhardt had proposed the term
“quasicrystals” around 1984 to describe crystals that are not periodic [7,8]. From crystallographic
principles, crystals are periodic having rotational order 1, 2, 3, 4, or 6. After the discovery of alloys
showing icosahedral symmetry, quasicrystals are considered with rotational symmetry of 5, or > 6 (which
are not allowed by the crystallographic rules). Higher rotational orders and their equivalent Penrose
structures will be available. Due to the long range ordering and rotational symmetry, quasicrystals show
clear diffraction patterns like crystals. One early review article on quasicrystals is by Goldman and Kelton
[9]. For further reading on crystallography one may refer to [10,11]. Figure 1 shows the schematic of
lattices with different rotational symmetries. Also shown are the thick and thin rhombi with identically
long sides used as building blocks to generate Penrose tiles. Each side is either marked with a single
arrow or two arrows. The rule to follow is, two touching sides of rhombi should have the same number of
arrows and the arrows should also point in the same direction. The rhombi were latter cut along the
diagonals (short diagonal of thin rhombus and long diagonal of thick rhombus) to form 4 triangles (kites)
with 4 different arrow markings [5].
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Figure 1: Schematic of the lattices having different rotational symmetries - (a) 4- ,(b) 6- ,(c) 5- ,(d) 8-
and (e) 12- fold. Insets in (a) and (b) show the square and hexagonal lattices. (f) shows the thick and thin
rhombi which are the building blocks of Penrose tilings and (g) shows the half rhombi used as building
blocks.
Another interesting concept which is known for decades and is also well studied is the existence of
surface plasmons, charge density wave at the metal-dielectric interface [12-14]. The concept of surface
plasmon polaritons (SPPs) is often explained by Fig. 2 which is reproduced from Barnes et al [15].
Figure 2: Surface plasmon polaritons at an interface (a) shown has having both electromagnetic wave
like and surface charge like properties. (b) Their surface nature is shown as the exponentially decaying
field on either side of the interface. (c) Comparison of dispersion of SPPs and light showing the
momentum mismatch problem (reproduced from Ref.15).
The surge in the ever growing area of Plasmonics is often linked to the demonstration of enhanced
transmission through patterned metal which is understood to be due to the interference of specular
transmission and the surface plasmon mediation [14,16-18]. Plasmon mediated effects, using either metal
nanoparticles assembled in predesigned fashion or metal-dielectric nanostructures, are being studied
extensively for applications in biology, chemistry, physics and engineering. Various sub-branches in
plasmonics are emerging like, graphene-, quantum-, active-, magneto- and nonlinear-plasmonics [19-24].
To overcome the momentum mismatch shown in Fig. 2(c), special coupling techniques are required for
optically exciting surface plasmon polaritons [14].
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Different excitation schemes like prism couplers and gratings are widely employed to optically excite
SPPs [14,17]. Periodic lattices in the form of 1-d or 2-d gratings used to excite SPPs offer wide
opportunities in designing and the structures designed for specific applications are the so-called
plasmonic crystals. Owing to the surface nature of propagating plasmons and the properties of plasmonic
crystals like periodicity and short range ordering, optical excitation of SPPs is limited by polarization and
launch angle dependence as well as discrete modes. Aperiodic arrays formed from metal nanoparticles,
metal discs and other shapes are studied to overcome some of these problems. For a recent review on
layered (1-dimensional) aperiodic structures one may see [25]. However, designing and realizing
aperiodic structures for specific applications is not as easy.
Quasicrystals, unlike periodic and aperiodic structures, offer long range ordering and higher order
rotational symmetry. The consequences of the dense k-space and high rotational symmetry include
modification of discrete spectral features to continuum and polarization independence. Inspite of their
strong potential, relatively fewer reports exist on plasmonic quasicrystals, metal-dielectric sub-
wavelength structures conforming to quasiperiodic patterns. Plasmonic quasicrystals were first reported in
2006 [26]. In the visible to near infrared regions, focused ion beam (FIB) was used to drill a few 100 air
holes in thin gold film on a substrate. The difficulty in preparing large area patterns include requirement
of high stitching accuracy. An easier alternative is to use Moire patterning or interference lithography to
write large area quasicrystal patterns in resist which can be transferred to the metal by wet or dry etching
techniques. In general, predicting the spectral response of the plasmonic quasicrystal is not straight
forward.
Two dimensional plasmonic quasicrystal patterns involve either metal nanoparticles arranged in quasi-
periodic fashion or air holes made in metal films conforming to quasi-periodic lattice. Realizing these
patterns involve calculating the coordinates of the particles or holes and then precisely placing them.
Designing the quasicrystal patterns is the first step which is followed by fabrication and characterization.
Unlike periodic structures where distinct theoretical models are available to calculate the spectral
response as well as the band structure, for quasicrystals there are no exact theoretical models so far and
some periodic approximant methods are used as demonstrated for photonic quasicrystal structures
[27,28].
In this article the current status of modeling, fabrication, characterization and applications of plasmonic
quasicrystals are reviewed. In section 2, different models to design are briefly covered. In section 3, the
fabrication techniques are reviewed and section 4 covers the experimental results. Applications of
plasmonic quasicrystals reported are reviewed in Section 5 followed by future prospects in Section 6 and
summary in Section 7.
2. Theoretical models:
2.1. Models to design plasmonic quasicrystals:
Quasiperiodic structures can be designed by different methods like deflation, cut and projection,
approximants [9,29,30]. These are used for both 1-dimensional Fibonacci like sequences as well as for
designing 2-dimensional structures. In addition, for 2-dimensional quasiperiodic patterns, tilings with thin
and thick rhombi [31,32] or the so-called grid [3] methods are well used. Deflation methods are used to
transform patterns by increasing the number of tiles while keeping the area same. On the other hand,
inflation methods help construct patterns over larger areas. The pentagrid or in general the multigrid and
the tiling methods are duals of one another and this has been shown by Kramer and Neri [33] .
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A modification of the pentagrid method was recently proposed and demonstrated by Kasture et al [34.] In
this, each line in the parallel set of lines is replaced by a linear array of dots. The separation between the
dots can be fixed to introduce an inherent length scale (though there is no short scale periodicity in
quasicrystals). This method is shown in Fig. 3. For quasicrystal structure with π/5 rotational symmetry, 5
sets of arrays were superposed, each rotated by π/5 radians before superposition. The points which are
common to the five superpositions correspond to the quasicrystal lattice points. As the coordinates of each
of the points are stored at the beginning, it is easier to sort or filter the points that are common to the
superpositions. In addition, the introduction of the inherent length scale is shown to be advantageous in
designing the central wavelength of the response of the quasicrystal. This is desirable in designing the
quasicrystal patterns for a particular wavelength region. From the Fourier transform of the designed
structure one can extract the k-space and thus the spectral response. In one of the earlier works on
photonic quasicrystals, it has been shown that the spectral response can be calculated from the measured
Bragg peaks in the diffraction pattern [35]. It was argued that this is a direct way to estimate the response
compared to approximate methods to simulate the quasicrystals. Similar technique was also used in the
plasmonic quasicrystal reports in the visible and THz wavelength regions [26,36].
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Figure 3: Schematic showing the five overlay steps used to generate the quasicrystal lattice points is
shown. Pink squares are marked to show the lattice coordinates of the quasicrystal pattern with π/5-
rotational symmetry.
Dual-grid method for photonic quasicrystal design was reported earlier. This is a generalized method to
design 2-dimensional photonic quasicrystals in which the phase matching conditions can be taken care of.
Photonic quasicrystal for nonlinear harmonic generation was demonstrated using this technique [37].
2.2. Models to study plasmonic quasicrystal properties:
Theoretical models to study the properties of quasicrystals are presented here below. A model to calculate
the spectral response of a plasmonic quasicrystal when the corrugations are shallow is given by Milton
Pereira and co-workers [38].
Garcia de Abajo and coworkers [39] have presented an analytical model to calculate the spectral response
from metallic hole array. They proposed that the transmission response in these structures depend on the
interplay between the localized resonances in individual holes and resonances originating due to the
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periodicity of the hole lattice. In their model, the collective response of a hole array is represented in
terms of polarizability that includes the polarization of each hole in response to the external field and also
to the field induced by other holes. So, the polarization of hole at a lattice site R is given by,
( ) ( )ext
R R
R R
p E R G R R pα′
′≠
′= + −
∑
where the field contribution of hole at position R’ on hole at R is given by G(R-R’). Thus the
transmission will be governed by the coherent superposition of the far field produced by all induced
dipoles. They used this model to explain the enhanced transmission through quasicrystal hole arrays in
microwave range [40]. They showed that the equivalent of Wood’s anomaly is possible in quasicrystals
when the corresponding diffraction order grazes the surface. At Wood’s anomalies one would expect
transmission dips.
Wood’s anomalies and field localization in sub-wavelength quasiperiodic hole arrays in metal were also
theoretically shown by J. Bravo-Abad et al [41]. They also consider hole-hole interaction and field
localization inside the hole to calculate the total response. In their model, a linear set of equations is
written considering all the modes. The hole-hole interaction is considered through a propagator. The input
and output field at each hole and the external field are also considered in this model. By taking metal to
be a perfect conductor, the model is compared with the experimental results in THz range.
However, to apply this to the visible to NIR range, the metal dispersion needs to be considered. A
simplistic picture based on interference of the SPPs excited by the holes and the transmitted light through
the film was presented by Gao et al [42]. The total electric field is written as the sum of the exponentially
decaying SPPs generated at two holes and the transmitted light through the metal film. Thus the intensity,
square of the total field, has the interference terms from the SPPs generated at different holes (hole-hole
interaction) as well as the SPP-direct transmission through film.
Theoretical study was carried out using Eigen (or spectral) decomposition method to calculate the
spectrum from dipolar response [43]. Symmetry, spatial localization and radiation loss of plasmonic
modes in 2-d quasicrystalline arrays of metal nanoparticles was reported in this work. A 3d parameter
space consisting of plasmon frequency, participation ratio of mode (an indicator of spatial localization
properties of the mode, it is equal to 1 when mode is completely localized at a single site and its value
increases as it gets localized over multiple sites) and the eigen value of the mode (inverse of the lifetime)
is used to generate a map. From this, it was shown that the strong spatially localized and high fidelity
mode is a type of antiphase ring mode and different localized modes have different radiation decay
profiles.
During the early part two models to understand the origin of transmission enhancement were reported. In
one model, the Fabry-Perot resonances of guided modes in nanoholes were shown to result in the
observed transmission properties [44]. In the other model, the interference between the fields through
individual holes is shown to result in the observed transmission enhancement [45].
3. Fabrication methods:
3.1. Large area patterning:
So far optical lithographic techniques are widely employed for large area writing. Soft and nanoimprint
lithographies are other techniques which may be used for large area patterning. Optical lithography
techniques could be based on multiple beams or two beams. In this section, first the multiple beam
techniques using different optical or opto-mechanical components or spatial light modulators (SLM) are
discussed. This is followed by method based on two beams and soft lithography based Moire patterning.
7
3.1.1 Multiple beam interference:
Optical fabrication techniques based on multiple optical beams are suitable for realizing large area as well
as 3-dimensional structures. However, these techniques are suitable for longer wavelengths as the feature
sizes realized are larger and are suitable for near- to far- infrared wavelengths. For example, using
holographic technique consisting of 4 beams, a 3-d face centered cubic (fcc) structure made of voids (air)
in cross linked polymer (SU-8) has been demonstrated [46]. The laser beam was split into four by using
two beam splitters. Then the beams were arranged to realize three beams symmetrically (making an angle
of 38.9°) around a central beam. The lattice constant achieved was 922nm for 355nm laser light. Using
similar technique but the beam splitters replaced by reflection grating, large area Penrose patterns were
shown over 2mm2 area [47]. The smallest feature obtained was about 200nm. Authors mention that the
smallest feature size can further be reduced to 100nm by using thicker resist (polymer) layers.
Figure 4. Different optical elements are employed to realize multiple beams needed to perform
holographic or interference lithography. Shown are some of them (a) Pinhole plate [51] (b)Phase only
SLM [49] (c) Beamsplitters [46] (d) Diffractive optical element/Phase mask [50] (e) Hexagonal prism
[52].
Using a special hexagonal prism that generates the required multiple beams, quasicrystal templates were
prepared in photoresist layers by interfering the multiple beams. When gold nanoparticle colloid was spin
coated on top of the template and annealed, inverse patterns could be obtained. In these structures it was
shown that the coupling between waveguide modes and the particle plasmons enhances with increasing
annealing temperature [48]. There are several other techniques used to generate multiple beams. For
example, phase mask [49] and multi-pinhole plate [50] are also employed to generate the required
multiple beams to prepare photonic quasicrystals. Figure 4 shows schematic of various tools used to
generate the multiple beams.
3.1.2. Multiple beam interference with spatial light modulator (SLM) :
Another method that is widely employed in preparing multiple light beams, that are essential for
generating complex patterns by interference or holographic lithography, is based on spatial light
modulators (SLM). This has been employed in preparing photonic quasicrystals [51]. More recently, n-
fold 3-dimensional axial photonic quasicrystals were reported by using n+1 interfering beams [52]. In
their configuration, instead of realizing n+1 beams using optical or opto-mechanical components, they
were generated by using a spatial light modulator (SLM). The programmable phase-only SLM is fed with
8
the phase of the complex amplitude of the calculated profile to generate the requisite interfering multiple
beams. By making the structures in Cerium doped Sr0.6Ba0.4Nb2O6 (SBN:Ce) photorefractive crystal,
authors have demonstrated reconfigurable optically induced quasicrystalline 3-d nonlinear photonic
structures.
However, one current drawback in the SLM based patterning is that the feature sizes are in the range of
tens of micrometers. Thus, these are not yet suitable for visible wavelength applications. While the
literature on photonic quasicrystal is comparatively much larger and we do not plan to cover them here,
the fabrication techniques mentioned above which are used for photonic quasicrystals can be employed to
prepare plasmonic quasicrystals as well. The references given above could be a good starting point to
choose the appropriate tool. A recent review on photonic quasicrystals is by Vardeny et al [53].
Figure 5. Schematic shows that multiple exposures to 2-beam interference pattern (or soft lithography or
Moire technique) leads to quasicrystal patterns with π/5- rotational symmetry. Higher symmetries are
possible with suitable number of exposures. By careful control of the exposure dose one could get desired
quasicrystalline patterns over large area. Black dots show the intersection points of the five lines which
correspond to the lattice points in (e). Black dots in (c) and (d) are to show the intersection of 3 and 4
lines. The opaqueness of the black dots indicate the dosage, 40% in the diamond regions of (b), 60% in
(c), 80% in (d) and 100% in (e). 100% correspond to optimum dose.
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Figure 6. Schematic showing the interference lithography setup with two beams interfering on a sample
coated with photosensitive polymer (photoresist). The laser beam is spatially filtered and expanded to
required size with a pinhole and two lenses (L1, L2) before splitting into two and made to interfere.
Intensity of each beam and the exposure time are controlled to get different shapes and patterns.
3.1.3. Interference lithography with 2-beams:
Instead of having multiple beams, one can achieve quasicrystal patterns of any required rotational
symmetry by multiple exposers to two interfering beams. For example, a single exposure of two
interfering beams leads to 1-dimensional grating structure. The corresponding dosage (exposure time and
power) required are split between multiple exposures. For example, two exposures with half the dosage
(same power density but half the exposure time) and with sample turned by 90° after first exposure give a
square lattice. By extension, 5 carefully controlled partial exposures with sample turned by 36° after each
exposure lead to a structure of 10-fold rotational symmetry as shown in Figure 5. Similar technique is
also used in soft lithography or Moire lithography. The 2-beam interference lithography is schematically
shown in Fig.6 and the AFM images of the fabricated patterns are shown in Fig.7. In the experimental
setup the laser beam is first passed through a spatial filter to pick the central uniform part of the Gaussian
beam. This is followed by a beam expander to enlarge the beam size. This enlarged beam is split into two
by a beamsplitter and then made to interfere on the sample surface with required angle between the two
interfering beams chosen based on the grating period required. Multiple variations of this interference
lithography technique are employed which use either a prism or different optical element combinations to
generate and make the two beams interfere. A photoresist S1805 was spin coated on quartz substrates and
is exposed to interfering beams that are generated by splitting the 442nm He-Cd laser line by a
beamsplitter. The exposure time for initial laser beam (of size 2 cm each) average power of 2mW each is
15 seconds for writing 1-d grating. To make a 10-fold symmetric pattern, five exposures were made (for 3
secs each) with each exposure followed by a 36° rotation of resist coated substrate. The two images show
resist patterns with different levels of resist development. The smallest feature obtained is about 200nm in
photoresist. This is the so called Moire patterning. By introducing a lateral shift of the sample in addition
to the rotation, one may get different Penrose like patterns.
3.1.4. Moire lithography:
Moire lithography was demonstrated to make large area structures with patterns conforming to rotational
order of 12, 14, 16, 24, 30 or 36 [54]. In this technique, initially molded photomasks having 1d- and 2d-
templates with 400nm lattice spacing were prepared in PDMS. On silicon substrates, a 120nm thick
diluted positive photoresist was spin coated. By placing the PDMS photomasks in contact with the resist
coated substrates, the resist was exposed to broadband mercury vapour lamp and developed to get
photoresist pillars on Silicon. By multiple exposures, different patterns were prepared in photoresist with
sub-micron features of the order of 200nm. It may be noted that, by careful control of exposure conditions
one may get pits or hills in the resist as well [55].
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Figure 7. AFM image of two different samples generated by exposing resist to 442nm He-Cd laser line 5
times with a rotation of the sample each time by 36°.
3.2. Medium area:
For high precision patterning, electron beam lithography is well proven for writing medium area patterns
over few mm2. Two of the techniques used are the lift-off and electron beam exposure followed by wet or
dry etching. In the lift-off process, typically, a bilayer resist is used. The bottom layer has smaller
molecular weight (for example, PMMA 495) and the upper layer larger molecular weight (for example,
PMMA950). When exposed to electron beam and developed (using MIBK), this bilayer structure results
in an undercut. Metal deposition can be done by any of the techniques like sputtering or evaporation
(thermal or electron beam). Final step involves lift-off of resist and unwanted metal layers by rinsing in
acetone. This process is, generally, useful for making metal (or dielectric) dot array patterns. Gold
nanoparticle arrays conforming to aperiodic patterns were earlier fabricated by using the electron beam
writing followed by lift-off process. On Indium Tin Oxide coated quartz layers patterns were first made in
resist PMMA950 followed by 30nm thick gold film deposition by electron beam evaporation and lift-off
using acetone [56]. To study coupled plasmons in quasiperiodic structures, two layers of metal dots were
prepared with a MgF2 spacer layer [57]. These are also used to study the plasmonic-waveguide polariton
modes.
On the other hand, by using a single resist layer, the pattern can be made into the resist by exposure to e-
beam followed by developing. In the next step, the pattern is transferred to the metal layer by etching gold
film by using Argon plasma. In the final step residual resist was stripped by oxygen plasma etching.
In addition to the chromatic aberration and astigmatism, some simple tips that will improve the e-beam
lithography are to use a ramp while spinning the resist to get uniform resist layers. While e-beam writing
on conductive substrates or layers is straight forward, to write on insulating layers, to avoid charging, a
thin layer (< 5 nm thick) of metal like Chromium or Aluminium is used on top of the resist. To avoid
damage to the surface as well as hard baking the resist, it is advisable to use etch stop steps while
transferring pattern from resist to the metal layer. For better gold adhesion, the surface flatness and
cleanliness of the substrate is critical. Very uniform and stable gold films were obtained by sputtering (dc-
and rf-) or thermal evaporation on various substrates including quartz, silicon, ferromagnetic thin films,
gadolinium gallium garnet [34].
3.3. Small area patterning:
11
Most of the earlier reports on plasmonic quasicrystals in the visible region were done on samples prepared
by focused ion beam. At the pre-calculated lattice points, the focused ion beam (FIB) was used to drill
holes in metal film. This is used for writing few hundred to a few thousand lattice points [26,44,45]. One
of the problems in using FIB is the difficulty in getting good stitching accuracies over large areas.
Self-assembly of soft matter has been shown to result in dodecagonal quasicrystals [58-61]. A design
method to self-assemble soft-matter into quasicrystals and their approximants was proposed by Iacovella
et al [62]. They propose that some of the particles used as building blocks could be of non-spherical
shapes. These particles can be assembled into quasicrystals by appropriately functionalizing particles of
different shapes. However, self-assembly techniques have not yet been exploited in realizing plasmonic
quasicrystals. Considering that biosynthesis of metal nanoparticles is an active area [63,64], self-
assembly techniques could pave way for utilizing plasmonic quasicrystals in biology and medicine.
4. Experimental studies on plasmonic quasicrystals:
In order to measure the resonances, typically, transmission and/or reflection spectra with white light are
carried out. In order to measure the plasmon dispersion, one has to lower the beam divergence. In fact, a
collimated beam with precise k-vector would be ideal and the dispersion is measured by changing the
launch angle by moving the light beam or by turning the sample. As collimating the white light (for
example, from a halogen lamp) is near impossible, nearly collimated light is achieved with divergence
angles less than 0.5° in most of the measurements. Measurements were also done by focusing the beam
on the sample and collecting from a narrow region to restrict the k-vectors [65].
4.1. Short-range or long-range:
One of the first questions that was addressed in the earlier part of literature is, what governs the spectral
response of plasmonic quasicrystals. This is interesting especially because quasicrystals have only long
range ordering and no short range ordering. F. Przybilla et al studied the spectral response of Penrose
subwavelength hole arrays in optical regime [26]. They studied different samples with increasing distance
between holes as well as different size arrays. They compared the transmission peaks and spectral widths
in periodic and quasiperiodic structures and concluded that in quasicrystals plasmon propagation length is
reduced thus ruling out long-range interaction.
D. Pacifici et al compared periodic and quasiperiodic hole arrays and concluded that as long as short
range ordering is maintained, periodicity and long range order are not required for transmission
enhancement or suppression [45]. They gave a model based on calculating the constructive and
destructive interference of light incident and transmitted through individual holes to show that the
transmission maxima and minima can be calculated irrespective of the lattice [66].
Further C. Rockstuhl et al have shown that the transmission enhancement is independent of arrangement
of holes when one invokes the cavity modes (originating due to the guided mode getting localized inside
the holes thus forming a standing wave pattern) [44].
All these studies involve small size patterns with the separation between holes varied in the 100-800nm
range. In order to couple light, microscope objectives were used thus, most likely, have launch of multiple
k values. It would be beneficial to have specified k exciting the sample. For example, for air holes made
in gold film over 1mm2 area, it was found that the transmission enhancement is much larger when
identical density of holes (with the fill factor kept constant) arranged in quasiperiodic fashion compared
to holes arranged in square lattice. In these studies near collimated (divergence <0.3°) white light source
was used for the measurements [34].
4.2. Discrete energy levels vs broadband response:
12
In metal films perforated with airhole arrays conforming to aperiodic patterns with n-fold (n=10, 12, 18,
40 and 120) rotational symmetry, it was shown that THz transmission resonances are possible [36]. Sharp
resonances as a consequence of interference of non-resonant broadband transmission through individual
holes and the discrete resonances corresponding to the quasicrystal lattice were shown. Incidentally, in all
the early studies, discrete plasmon modes are visible in the data presented [26,44,45]. Measured normal
incidence transmission enhancement (compared to unpatterned metal) spectrum is shown in Fig. 8 for TM
polarized incident light on pure quasicrystal patterns. These spectra show sharp resonances corresponding
to discrete levels. However, one may also retain the rest of the points (in addition to the pure quasicrystal
lattice points) which shows a smooth broadband transmission enhancement as shown in [34].
Figure 8. Transmission enhancement spectrum measured through a quasicrystal pattern shows 9 times
more transmission at the peak compared to unpatterned metal. Dip at about 500nm could be the Wood’s
anomaly.
4.3. Broadband response:
Unlike the discrete plasmon modes observed in small area patterns studied earlier, in relatively larger
areas of gold nanodots arranged in aperiodic 2-d arrays, broadband response has been observed [55].
They studied arrays conforming to Fibonacci, Thue-Morse and Rudin-Shapiro lattices and used
generalized Mie theory to explain the experimental results. Diffractive coupling to far-field is concluded
to be the reason for inhomogeneously broadened broadband scattering covering the entire visible band.
The broadband response is explained in terms of the multiple length scales present in the arrays of
identical metal particles which result in a collective response. This is consistent with the fact that the high
rotational and point inversion symmetries in quasicrystals result in dense k-space (that is, large number of
reciprocal lattice vectors) as was shown earlier [67]. Also, in case of nanowire antennas, it was shown that
in random structures composed of wires of different sizes and shapes, each nanowire will have its own
mode thus resulting in a broadband response as has been shown by Podolskiy et al. [68]. However, when
identical nanowires are arranged in parallel then the collective response is similar to single nanowire and
the observed huge intensity enhancement corresponds to the SPP mode of individual nanowire.
In 2-d Fibonacci arrays of metal nanoparticles, near-field coupling of nanoparticle dimers as well as
highly localized plasmons leading to strong local field enhancement or hotspots have been reported
[69,70].
13
Figure 9. Measured surface plasmon polariton dispersion in plasmonic quasicrystal pattern that is
normalized with respect to that from unpatterned metal. White band covering about 525 – 725nm shows
the broadband, dispersionless transmission region with maximum enhancement of about 8.5 times. Red
lines are to guide the eye to show the dispersive modes. Modes are calculated by comparing the measured
dispersion with the calculated k-space (Fourier transform of the spatial profile). A, Q stands for gold-air
and gold-quartz interface plasmon modes, respectively.
In a recent study, it was shown that a broadband response is possible in quasicrystals [34]. Measured SPP
dispersion is shown in Fig. 9 where the angle independent white band covering 525-725nm wavelength
range is the transmission enhancement band that is featureless and independent of launch polarization.
Further, the transmission spectrum can be designed by combining multiple gratings. It has been shown
that the transmission enhancement of up to 15 times is possible in quasicrystal structures made by
superposing two different quasicrystals of different inherent periods (of the initial dot arrays). The
broadband response is shown to be due to dense k-space. To highlight the role of arrangement of holes on
the transmission response, two structures with identical density of similar size and shaped air holes are
considered. In one the holes are arranged in quasiperiodic fashion and in the other the holes are arranged
in periodic square lattice. In the quasiperiodic lattice the measured diffraction pattern shows a large
number of diffraction spots compared to the sparse pattern of square lattice. This shows that for the same
density of identical holes, the arrangement of holes is important and it governs the response of the pattern.
The fill factor could be achieved to be same in both quasicrystal pattern as well as the periodic lattice by
having same inherent period in both. To achieve same fill factor, this implies bigger holes in periodic
lattice. The response will still be narrow band due to sparse diffraction pattern or sparse k-space in such a
structure.
4.4. Launch angle and polarization dependence:
In periodic gratings, the grating expression and the momentum conservation defines the angle at which
the light at a particular wavelength can be coupled to plasmons. Thus, to couple light at different
wavelengths, one would require gratings which exhibit required Fourier components. However, in quasi-
periodic gratings, due to the possibility of having large number of Fourier components, coupling light at
different wavelengths could be easier. Dolev et al [71] have shown that coupling of light at different
wavelengths and angles is possible in 1-d quasi-periodic gratings. In addition, they have shown that the
14
Fourier coefficient amplitudes which dictate the coupling strength between light and SPPs are weaker in
quasiperiodic gratings compared to periodic gratings. It would be interesting to note that compared to
periodic gratings, in dual frequency quasiperiodic gratings they see more number of Fourier components.
One of the advantages of quasicrystal pattern is that it helps couple light at wide range of incident angles
to plasmons. Bauer et al studied launch angle independence of QC patterns realized by arranging metal
nanoparticles in quasiperiodic fashion on a 180 nm dielectric (HfO2) layer [65]. They report particle
plasmon mediated light coupling to waveguide modes in this structure. They have given a model to
theoretically calculate the transmission spectrum for normal and oblique incidence light by calculating the
particle plasmon and the waveguide modes (both TE and TM).
Launch angle and polarization independent dispersionless transmission enhancement band in quasicrystal
patterns was shown by Kasture et al [34]. While in pure quasicrystal lattices, one could see discrete
resonances, in quasicrystal patterns combined with inherent square array coordinates, very broadband
response was seen. Dispersion measurements showed the dispersionless transmission bands. In bigrating
2-d quasicrystal structures, enhancement in transmission of upto 15 times was demonstrated over a broad
wavelength range of 300nm. Overall transmission enhancement compared to unpatterned metal was
found in the range 500-1200nm. Polarization independence has been demonstrated by measuring the SPP
dispersion for different polarizations and the orientation independence has been demonstrated by
measuring at different sample orientations.
4.5. Quasiperiodic structures for coupled plasmon-waveguide modes:
Waveguide-plasmon polaritons are coupled plasmon-waveguide modes in metal-dielectric structures [72].
Giessen’s group has studied 1- and 2-dimensional quasiperiodic gratings on either 180nm thick Hafnium
oxide or Indium Tin Oxide layer over glass substrates. These structures support waveguide modes in the
oxide layers and the top gratings in the form of metal wires or metal disks arranged in 2-d array support
plasmon modes so coupled plasmon-waveguide modes could be studied [56,64,73,74]. The spectra
through quasiperiodic grating structures are compared with the periodic gratings (square in case of 2-d)
for different polarizations and incident angles of light. Theoretical models for coupled waveguide-
plasmon system are presented [56,75] . It would be interesting to note that similar structure but with a
ferromagnetic thin film was employed to show a novel longitudinal magneto-plasmonic intensity effect
(LMPIE) where polarization conversion from TM to TE has been demonstrated [76]. Thus, waveguide-
plasmon polariton structures have interesting applications which need to be exploited for their potential.
Out-of-plane resonances were reported in plasmonic quasicrystals prepared by having patterned dielectric
layer (electron beam resist layer, ZEP) on metal film. Compared to periodic structures they show richer
spectrum due to the denser k-space which allows easy satisfaction of phase matching conditions [77].
These plasmonic-photonic structures are interesting for various applications including solar cells.
In inverse structures prepared by annealing gold colloid deposited dielectric quasiperiodic structures, it
was shown that the coupling between waveguide modes and the particle plasmons enhances with
increasing annealing temperature [74].
4.6. Diffraction pattern and compare with the FFT of spatial profile:
The calculated Fourier transform of the designed quasicrystal pattern can be compared with the measured
light diffraction. Most of the reports on plasmonic quasicrystals present either the calculated (Fourier
transform of the spatial profile) or the measured dispersion or both. A comparison of the calculated and
measured dispersion from quasicrystal patterns of π/5 rotational symmetry was made [34]. From a
comparison of these two, the k-values and thus the full spectral response can be calculated (though
tedious when the k-space is dense).
15
4.7. Harmonic generation due to local field enhancement:
T. Xu et al compared the SHG from periodic, aperiodic and disordered arrays [78]. From measurements
on FIB patterned samples of size 25x25µm2 and e-beam patterned samples of size 80x80µm
2, they found
strong angular dependence of SHG. For incidence angles roughly corresponding to the transmission
enhancement, they observed strong SHG for 800nm incident light. In addition, while they did not observe
SHG for normal incidence for spherical holes, they observed SHG from square lattice and quasiperiodic
patterns with asymmetric shaped holes. Their results showed that disordered patterns result in SHG along
only the fundamental direction but there is off-axis SHG from periodic and quasi-periodic patterns. Also,
the SHG from quasiperiodic pattern was found to be weaker compared to periodic pattern. It may be
recalled that earlier, Airola et al compared the SHG from a periodic and a random array of (circular and
triangular) holes in metal with patterns over an area of 70x70µm2 [79].
Kasture et al have shown that it is possible to make use of the large local field enhancement due to SPP
excitation in SHG [80]. Though the metal surface nonlinearity is weak, as the SPP mediated local field
enhancement is quite strong, second harmonic generation was observed from thin metal films patterned
with quasicrystal air holes. This is broadband and angle independent as one may expect from the surface
nonlinearity. It may be worth noting that the second harmonic detected is much stronger than that
compared to plain rough metal surface or that from a square lattice with same fill factor (density of holes
and size of air holes same in both square and quasi-periodic patterns). This shows that the SPP excitation
in quasicrystal patterns leads to much stronger local field enhancement over broad wavelength region. In
this structure that has both lattice points corresponding to π/5 rotational symmetry and the points
corresponding to the (5) square lattices that are rotated by 72° from each other, the symmetry seems to be
broken and SHG was observed even for normal incidence [80].
5. Applications demonstrated:
Plasmonic quasicrystals offer broadband plasmon mediated transmission enhancement. The strong local
field related to the plasmon mediated transmission enhancement is useful for various applications. Some
of the applications which have been demonstrated so far are reviewed in this section. The very first
application reported was of focusing of visible light by a nanohole array conforming to quasicrystal by
Huang et al [81]. They also showed hotspots from the near-field studies of plasmonic quasicrystal.
For photovoltaic applications, plasmonics is proposed to be useful [82]. It has been shown theoretically
that plasmonic quasicrystals improve the efficiency of solar cells [73]. For solar cell applications,
aperiodic patterns are shown to enhance the efficiency of solar cells [83] as well as aperiodic spiral
geometries for efficient current extraction [84]. However, there is no experimental demonstration of
plasmonic quasicrystals improving the solar cell efficiency.
In plasmonic quasicrystals by combining different patterns, it was shown that broadband transmission
enhancement is possible with absolute transmission more than 2% through patterned 100nm thick gold
film [34]. This shows designable spectral response in plasmonic quasicrystals and broadband transmission
enhancement. While strong local field is shown by second harmonic generation, it would be interesting to
demonstrate enhancement in absorption as well as actual performance of a solar cell whose top (or
bottom) contact is patterned for light throughput.
Optical impedance matching using arrays of nanoparticles to couple light to absorbing media has been
demonstrated by Spinelli et al [85]. The role of spatial distribution of holes in array has not been studied.
Circularly symmetric light scattering from nanoplasmonic spirals has been demonstrated [86]. In this
context, it would be interesting to see if quasicrystal patterned metasurfaces could be realized for
generating non-Gaussian beams like Bessel beams among others. In addition, phenomena related to phase
change in reflected light at an interface like Goos-Hanchen and Imbert-Federov shifts at quasiperiodic
16
interfaces would be interesting to study. Especially as they offer broadband response, with an active
medium if one could achieve controllable shifts, they would be useful for display applications among
others.
Guided resonances are the coupled diffraction orders and the effective waveguide modes supported by the
grating layer. In plasmonic and photonic quasi-periodic structures, there are several studies to
demonstrate these as mentioned above in the coupled plasmon studies.
Plasmonic-photonic quasicrystals with patterned dielectric on metal have recently been demonstrated for
biosensing applications [87]. In this work, it was shown that the structures have very high surface
sensitivity which is dependent on the thickness of the metal (Al) underneath.
In order to address the issue related to limited tunability of metal properties, it was shown that by having a
graphene layer on top of dielectric pattern conforming to quasicrystal lattice, broadband electrical
tunability of plasmon modes is possible [88]. For the THz wavelength region, graded index lenses with
ultra small focusing spot (λ0/60), collimating, and guiding of graphene plasmons have been demonstrated
in this work.
6. Future:
Plasmonic quasicrystals, in addition to the studies to understand the basic properties like the dispersion of
plasmon modes, coupled plasmon-polaritonic modes, would be useful in wide ranging applications. Some
of the possible applications demonstrated are light harvesting, light control, sensors, and 2nd
harmonic
generation. The field is nascent and there is scope for lot of studies to understand the systems as well as to
find various applications. Some of the possible fundamental and applied problems that can be addressed
are mentioned below.
Polaritonic modes: Coupled plasmon-waveguide modes which are being studied were discussed earlier.
In terms of coupled plasmons, most of the studies focused so far on vertical or out-of-plane coupling of
plasmons at the top and bottom interface of metal layer or coupling between particle plasmons and
surface plasmons [89-100] or coupled localized plasmons resulting in Fano resonances. Comparatively
little work is reported on in-plane coupled plasmon modes and the control of strong coupling [101]. For
planar architectures, for information processing and for manipulating the photon emission from emitters
like quantum dots, it would be interesting to study in-plane coupled plasmon systems. Quasicrystal
patterns could be useful for studying coupled plasmons as well as coupled polaritonic modes like
plasmon- exciton modes.
Quantum plasmonics: Generally high Q cavities coupled with single emitters are used as sources of
single photons [102-104]. But they are limited in terms of speed and bandwidth. Plasmonic quasicrystals,
with their inherent high bandwidth and high density of states could overcome this limitation. Also, these
sources would be much smaller in size.
Fano resonances: Fano resonances are sharp spectral features that arise from the interaction of a discrete
system with a continuum. There is considerable amount of work in the area of Fano resonances in
plasmonic systems [105]. It would be interesting to probe for Fano or Fano-like features in metal-
dielectric structures conforming to quasiperiodic patterns. This would be useful for field localization as
well as high quality factor structures suitable for strong coupling and related studies.
Novel materials: After the advent of graphene, mono- and few layer graphene and graphene-like
materials are being researched for their unique properties. One such material predicted from molecular
17
simulations is, Silicon bilayers compressed between smooth surfaces to 1 atmospheric pressure form
dodecagonal quasicrystals [106]. Such materials when realized could open new possibilities in plasmonics
as active materials.
Near-field studies: There is very little work on near-field features in plasmonic quasicrystals which were
discussed above. One would expect, rich information from near-field studies in systems having coupled
plasmon-plasmon, plasmon-photon, plasmon-excitonic and other systems. Control of plasmon modes and
thus the coupling would be useful for coherent transfer and coherent control applications.
Temporal studies: To our knowledge there are no temporal studies on plasmonic quasicrystals to study
the plasmon dynamics. These would be interesting in coupled plasmon-exciton, plasmon-plasmon,
plasmon-photonic modes. Using time resolved measurements, light pulse transmission in photonic
quasicrystals made of Fibonacci series was studied earlier [107]. It was mentioned that the plasmon
propagation time is short in plasmonic quasicrystals owing to scattering from the dense k-space but there
seems no direct evidence.
Soft-matter techniques for patterning: DNA fabrication is one of the most popular techniques but
unfortunately not much used in plasmonic quasicrystal area. A design method to self-assemble soft-matter
into quasicrystals and their approximants was proposed by Iacovella et al [60]. They proposed that some
of the particles used as building blocks could be of non-spherical shapes. These particles can be
assembled into quasicrystals by appropriately functionalizing particles of different shapes. There is no
experimental demonstration of such proposals.
Light control and manipulation: Waveguide arrays are known for optical soliton propagation both in
weakly coupled waveguide arrays as well as soliton potentials [108,109]. Metal waveguide arrays are also
shown for deep sub-wavelength focusing and steering of light [110]. These ideas could make use of
quasiperiodic arrays where the structures can be designed for specific focusing distances and possibly
spatial profiles. Further, silicon based taper waveguides with metal layers showed coupling efficiency of
about 20% even in the visible region for tapered structures with input width of few micron to 200nm at
the tip [111]. Unlike the Bloch oscillations in periodic arrangement of waveguides, in quasiperiodic arrays
conforming to Fibonacci sequence, light localization to central waveguide was seen in finite difference
time domain simulations [112]. Thus, the potential of metal-dielectric structures for light guiding and
coupling to near-field is yet to be fully exploited.
Nonlinear Plasmonics: Second and third harmonic generations have been reported in plasmonic or
photonic quasicrystals that are designed to satisfy quasi-phase matching [37]. It would be interesting to
extend this to difference frequency generation which would be useful for enhanced THz generation from
nonlinear crystals like ZnTe. Higher harmonics are generally demonstrated in atomic systems and with
high intense lasers. Would it be possible to generate very high harmonics with moderate laser powers
while making use of plasmon mediated local field enhancement? For a review of nonlinear plasmonics
one may see the article by Kauranen and Zayats [24].
By designing structures with specific spatial or k-space profiles one would be able to use them for
applications like near-field excitation, light trapping among others. For lighting applications, dynamically
variable quasicrystal pattern generation based on soft matter assembly techniques would be useful.
Basic studies: There are several questions with no clear answer like, what is the plasmon mode
dispersion in 3 layered quasicrystal systems like the ones studied in plasmon-waveguide polariton modes?
Photonic quasicrystals have been shown on how quasicrystals can be used to simulate topological phase
transitions [113]. However, no such study is available in plasmonic quasicrystals.
18
Theoretical Studies: On the theoretical front, it is still not straight forward to calculate the band structure
and optical properties of quasiperiodic structures. More accurate band structure calculation models need
to be developed for visible wavelength range. In addition to this, models are required for understanding
the properties like density of states for fluorescence enhancement and scattering properties. Extending
Green’s function approach [114] to plasmonic quasicrystals could be one option. Considerable theoretical
support is required for visible range plasmonic quasicrystals which would also help find novel properties
and applications of plasmonic quasicrystals with or without active medium.
Metamaterials: With the knowledge that plasmonic structures offer sub-wavelength photonic
components, it would be interesting to study the metamaterials based on plasmonic quasicrystals.
Negative refraction in photonic quasicrystals has been reported earlier [115]. One of the first works to
report a metasurface based on plasmonic quasicrystal was reported by S.S. Kruk et al [116]. In this work,
isotropic metasurfaces have been reported and a comparison between periodic, quasi-periodic and
amorphous metasurfaces was made.
7. Conclusions:
Plasmonics quasicrystals offer opportunities to extend the advantages of plasmon mediated novel effects
to broadband and omnidirectional applications. Thus, they have the potential to overcome the limitations
of plasmonic crystals. There is considerable scope for both theoretical and experimental studies on the
basic properties as well as the applications of plasmonic quasicrystals. This article reviews the current
status of design, fabrication, characterization and applications of plasmonic quasicrystals.
Acknowledgments
VGA would like to thank S. Kasture, V. J. Yallapragada, A.P. Ravishankar, G. Mulay, R. Patil, Nikesh V.
V.
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