## Abstract

We investigate the properties of the resonant modes that occur in the transparency bands of two-dimensional finite-size Penrose-type photonic quasicrystals made of dielectric cylindrical rods. These modes stem from the natural local arrangements of the quasicrystal structure rather than, as originally thought, from fabrication-related imperfections. Examples of local density of states and field maps are shown for different wavelengths. Calculations of local density of states show that these modes mainly originate from the interactions between a limited numbers of rods.

©2006 Optical Society of America

## 1. Introduction

Periodic structures have properties that have been subject of important theoretical
developments in both solid-state physics [1] and
photonics [2]. In solid-state physics, it has
been long believed that periodic crystals were the only ordered structures. The
existence of more complex type of order was found in nature with the discovery of the
icosahedral phase of metallic alloys [3]. D.
Shechtman et al. discovered these structures, which were named quasicrystals. Since
then, *aperiodic order* has been subject of attention from solid-state
physicists, but also mathematicians, and now photonic physicists. Aperiodically-ordered
structures may exhibit a variety of weak (e.g., local and/or statistical) forms of
rotational symmetries, which are not necessarily bounded by the crystallographic
restriction typical of periodic structures [3].
While electronic properties of quasicrystals have been studied thoroughly, their
*photonic* counterparts have been subject of less attention and
deserve further investigations.

The earliest studies of Fibonacci-like one-dimensional quasicrystals have shown the
existence of photonic bandgaps, and localization of the light [4,5]. Bandgaps at large
wavelengths (with respect to the average layer thickness) have been shown to exist
[6]. Field enhancement and group velocity
reduction at the band-edge have been observed experimentally [7]. Numerical studies have shown that two-dimensional photonic
quasicrystals can also exhibit photonic bandgaps [8,9]. Complete bandgaps can be
obtained for relatively low index contrast [10,11] thanks to higher
(*e.g*. 12-fold) statistical symmetries. One advantage of these
aperiodic structures is their capability of exhibiting many inequivalent sites, and
consequently many possible different defects [8,12]. It has also been shown that
long-range interactions play an important role in the formation of bandgaps in such
structures [13]. Recently, M. Notomi *et
al*. have studied the stimulated emission in aperiodically-ordered
structures, considering a Penrose-type quasicrystal laser [14]. Contrary to the typical *extended* modes of the
band-edge photonic crystal lasers, Penrose laser modes were found to be
*localized*. Note that a very interesting theoretical study of
localized modes in defect-free quasiperiodic photonic crystal has shown that
transmission within the bandgap could be attributed to a competition between the
nonperiodicity and self-similarity [15].

Our aim is to provide some physical insight in the nature of these resonant modes via an
analysis based on the calculation of the *local density of states* (LDOS)
for finite-size photonic quasicrystals. We first recall the basic elements of the
numerical method we use, and of the computation of the LDOS. Then, we show that the LDOS
maps present evidence of modes attributable only to the constructive interference of the
field diffracted by a very limited number of rods. This shows the
*localized* nature of the modes, opposite to the
*extended* nature of the resonant modes that can be observed in the
transparency bands of finite-size periodic photonic crystals. Indeed, these latter can
easily be interpreted as Fabry-Perot modes (*i.e*. a standing wave mode)
for a given Bloch mode, whereas the aperiodic-order-induced modes here are found to stem
essentially from the *local* arrangement of rods.

## 2. Modelling and LDOS

We consider a two-dimensional Penrose tiling built as a combination of two types of
rhombus tiles, whose edges have the same length denoted as *a* [16]. This
geometry is characterized by a fivefold symmetry [16]. The quasicrystal of interest here
is generated by placing identical dielectric rods at the vertices of the rhombuses of
the Penrose tiling, as shown in Fig. 1. The rods
are assumed to lay in vacuum and to be made of nondispersive dielectric material, with
relative permittivity *ε _{a}*=12, and with radius

*r*=0.116

_{a}*a*. The electric field is assumed to be parallel to the axis of the rods (E parallel polarization).

A computationally-effective and physically-insightful modeling of such a finite-size structure is addressed here, without any supercell approximation, via the LDOS analysis.

The numerical method we utilize is based, first, on a multipolar expansion of the fields around (and inside) each rod, giving rise to Fourier-Bessel series that can be separated into two parts: the local incident field and the outgoing scattered fields. Note that the local incident field on a rod includes the field scattered by the other rods. Obviously, the scattered field is related to the local incident field through the diffraction process, i.e. the coefficients of the series representing the incoming and outgoing waves are related to each other by a scattering matrix. Enforcing the suitable matching conditions at the rods interfaces, one is eventually led to a linear system whose solution gives the coefficients of the multipolar expansions. A detailed presentation of the method can be found in Refs. 17 and 18. This method is known as the Korringa-Kohn-Rostocker method in solid states physics [19] and is often called “multipole expansion method” or “scattering matrix method” in the optics community. The method has recently been extended to handle the calculation of the local density of states (LDOS) in finite-size photonic crystals [20,21].

In the following computations, the normalized LDOS
*ρ*(**r**
_{0},*ω*) at any arbitrary location r_{0} is
easily evaluated using the method described above. It is worth noticing that we consider
a two-dimensional geometry invariant along the *z*-axis, and consequently
all the electromagnetic field components are also *z*-invariant. The
symbol r will be used throughout the paper to denote a two-dimensional vector in the
*xOy* plane. It is well known that in the case of interest, i.e. a set
of two dimensional lossless rods, the normalized LDOS,
*ρ*(r_{0},*ω*), is given by the
imaginary part of the Green’s function
*G*(r,*ω*) evaluated at the source
location:

The Green’s function in Eq. (1) is defined via the following equation:

where *ε*(r) is the relative permittivity
(*ε*=*ε _{a}* inside the rods, and

*ε*=1 outside),

*c*the vacuum light celerity,

*ω*the angular frequency, and the standard radiation condition (outgoing field) is implied. The normalization of

*ρ*(r

_{0},

*ω*) has been chosen so as to have

*ρ*(r

_{0},

*ω*)=1 in vacuum. The LDOS has the key feature of being intrinsically related to the response of the structure to any type of excitation. In solid-state physics, the LDOS is informative about the dynamics that an atom would undergo if located at a given point. In our electromagnetic analogy, the LDOS maps practically provide information about the total power emitted versus the excitation point.

## 3. Results and discussion

Figure 1 shows a Penrose quasicrystal structure
made of 530 rods (left), and the LDOS at the center point r_{0}=(0,0) versus the
normalized frequency (right). Several bandgaps can be observed in the frequency range
displayed: A large central one, and two less pronounced (at higher and lower
frequencies). A previous study by the same authors has shown that bandgap formation in
photonic quasicrystals may involve long-range interactions and multiple scattering
[13].

Here, we will concentrate on the behavior of the modes when a frequency in a
transparency band is considered. Figure 2 shows
a typical map of LDOS for the quasicrystal in Fig.
1 at a normalized frequency *a/λ*=0.415 between the two
lower-frequency bandgaps. The map shows that, in a large quasicrystal, several localized
resonances can be observed at the same frequency. A typical nearly-fivefold-symmetric
localized mode is magnified by the zoom in Fig.
2. The deviance from perfect fivefold symmetry is likely attributable to the
finite size of the structure. Figure 3 shows
other typical LDOS maps at a higher normalized frequency
*a/λ*=0.726; this frequency produces the maximum LDOS (with
respect to frequency) at a chosen location *x/a*=1.05 and
*y/a*=4.6. This location has been chosen to be in the region of the
localized mode on which we will focus, and we will show later that its choice is not
critical (see movie). We have chosen these two examples as being representative of the
observed field maps of the modes, but we have found many others analogous situations
corresponding to different wavelengths within the transparency bands.

It should be noticed that, in periodic crystals, a local deviance from periodicity (defect) could induce a transition from the extended modes to localized modes. Indeed, for example, removing a rod in a perfectly periodic structure is a well-known device to create a localized mode. Our numerical computations have confirmed the existence of such localized modes in the transparency bands of quasicrystals (without resizing or displacing any rod). As the authors of Ref. [14] implicitly assumed, we confirm that the behavior observed in their experiments is not a consequence of unavoidable fabrication-related deviance from the quasicrystalline ideal structure, but it rather represents one of its inherent properties.

Figure 4 shows the maps of the modulus and phase
of the field at the normalized frequency *a/λ*=0.726 when the
structure is excited by a single electric line source located at
*x/a*=1.05 and *y/a*=4.6 (see the previous discussion,
pertaining to Fig. 3: For that source location,
the LDOS is maximum at *a/λ*=0.726). As can be expected, the
distribution of the modulus of the field is very similar to that of the LDOS map shown
in Fig. 3, and the phase map also reveals the
fivefold symmetry of the mode. Note that we have checked that the field vanishes outside
the part shown in Fig. 4, thus the source
excites no other localized mode in the structure at this frequency

The quasicrystal laser experiment reported in Ref. [14] has demonstrated that the mode described above could be used for lasing, and that lasers could take advantage of the richness of the possible symmetries of quasicrystals. Performance optimisation of such lasers would need a fine understanding of the quasiycrystals properties and physical origins of the modes. The following observations may hopefully contribute to a deeper understanding.

In order to investigate the process of formation of the localized modes, such that in
Fig. 4, we modified the structure by removing
parts of the quasicrystal outside the resonant region. In a previous paper [13] we have shown that certain bandgaps of a
quasicrystal involve long-range interactions. Thus the question of the role of such
long-range interactions in the appearance of the observed modes arises naturally even if
the two effects are not physically related (the studied modes are outside the bandgap).
Figure 5 shows some examples of LDOS maps
(computed in the same region as in Fig. 4)
pertaining to increasingly smaller quasicrystals obtained by progressively eliminating
certain rods outside the region displayed. The amplitude of the LDOS associated with the
mode slightly changes (about 20 percent) but the distribution does not. This behavior
suggests that the modes have indeed a *highly localized* nature. This is
clearly visible from the last case (rightmost plot), where, in spite of a very drastic
reduction of the number of rods (the quasicrystal has been reduced along both
*x* and *y* dimensions to keep only 47 rods located
inside a square centered at *x/a*=-0.25 and *y/a*=4.25,
with edge length equal to 5.9*a*), a behavior similar to that of the
larger structures is observed. The reduction in the LDOS amplitude can be mainly
attributed to a resonance frequency shift rather than a real decrease in the resonance
strength. The movie in Fig. 6 shows the LDOS for
the smallest quasicrystal made of 47 rods when the frequency
*a/λ* varies from 0.712 to 0.739. It can be observed that the
LDOS maximum arises now at a normalized frequency of 0.729 (instead of the 0.726 value
observed for the larger structure). The frequency shift is accompanied by only a small
weakening of the resonance strength, as shown by the maximum of the LDOS map. Thus we
have shown that long-range order does not play a major role in the lasing effect
observed, contrarily to the hypothesis of the authors of Ref. [14]. This result is fully consistent with the analysis made in Ref.
[15]: Localized modes may appear due to the
non-periodicity of the structure, but in our case they appear outside the bandgap
region, which is unexpected.

## 4. Conclusions

We have presented a numerical study of localized resonant modes in two-dimensional
finite-size Penrose-type photonic quasicrystals. The existence of the localized modes
observed by other authors in the quasicrystal lasers has been numerically confirmed. It
has been shown that these modes probably originate from interactions among a small
number of rods, rather than from undesired fabrication-related defects, and should
accordingly be considered as an inherent property of quasicrystalline geometries.
Indeed, the observed localized modes are only slightly modified by the elimination of
several rods around (and even relatively close to) the localization region. These
conclusions are nontrivial and somehow *counterintuitive*, since it is
well known that band-edge modes in periodic crystals are not localized. The fact that
localization is not much affected by long-range interactions strengthens the difference
with the bandgap effect.

## Acknowledgments

The support of the EC-funded projects PHOREMOST (FP6/2003/IST/2-511616) and METAMORPHOSE (FP6/NMP3-CT-2004-500252) is gratefully acknowledged.

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