Track: Advanced Matter and Energy Physics
Master Thesis
Spatially programming gain in plasmonic lattices and
metasurfaces
by
Stefanos Kovaios
Student ID: 12314390
August 14, 2020
60 ECTS
September 2019 - August 2020
Supervisor :
Second Examiner
prof. dr. A.F. Koenderink
prof. dr. P. Schall
Daily Supervisor :
dr. R. Kolkowski
Plasmonic lattices and metasurfaces have been thoroughly studied in recent years, and served as a platform for manipulating the properties of incident light fields on demand. Among those, it has been established that plasmonic lattices embedded in a gain medium can perform as lasers, despite the significant lossy nature of plasmonic particles. More-over, the field of non-Hermitian photonics has attracted the interest of researchers due to the non-trivial behavior that can emerge in systems with alternating gain and loss. Such complex systems have been developed mostly by the introduction of point defects in pho-tonic crystals, microring resonators and coupled waveguides, and whether this behavior could be realized in plasmonic lattices by controlling the optical gain in a programmable fashion remains an open question. In this thesis, we examine whether the optical gain can be programmed in plasmonic lattices, by means of structured illumination. In particular, after presenting the essential theoretical concepts for the description of honeycomb plas-monic lattices with alternating gain and loss, we proceed into quantifying the optical gain of systems for different gain media used in plasmonic lattice lasers. Finally, we perform a series of experiments in order to establish that lasing occurs in honeycomb plasmonic ar-rays, and examine whether structured illumination can affect the lasing behavior of these systems, and potentially induce the emergence of novel effects in their optical properties.
Abstract i
Table of contents iii
1 Introduction 1
2 Theoretical Background 3
2.1 Plasmonic scatterers - Gain . . . 3
2.2 Theory of honeycomb plasmonic lattices: The free-photon model . . . 7
2.3 Optical response of a plasmonic array: Multiple scattering . . . 10
2.4 Simulations . . . 11
2.5 Band folding. . . 15
2.6 Discussion . . . 21
3 Gain measurements 23 3.1 Gain coefficient . . . 23
3.2 The Variable Stripe Method . . . 24
3.3 Experimental Setup - Sample preparation. . . 25
3.4 Measurements - Analysis . . . 27
3.5 Discussion . . . 28
4 Lasing in plasmonic lattices 32 4.1 Sample fabrication . . . 32
4.2 Experimental setup . . . 33
4.3 Lasing at the Band edges of a honeycomb lattice. . . 36
4.4 Measurements for dense nanoparticle arrays . . . 37
4.5 Results - Discussion. . . 37
5 SI in plasmonic lattices 42 5.1 Principles of SI . . . 42
5.2 Experimental setup . . . 43
5.4 Results - Discussion. . . 46
6 Conclusions - Outlook 48
Acknowledgments 51
Introduction
The significant advance in material science and manufacturing processes during the 20th century has lead to the implementation of concepts that were previously technologi-cally unattainable. The capability of developing structures with high precision in a scale of a few tens of nanometers and manipulating physical properties in the nanometer scale led the race towards groundbreaking discoveries, such as the development of nanorobots with applications in medicine [1] and the emergent technology of thin films with appli-cations in sustainable energy and photovoltaics [2]. In addition to all these important developments, the field of nanophotonics, the study of the interaction of light and matter at the nanoscale, has emerged, with the outbreak of a wide range of fundamental research being initiated, in subjects like optomechanics, plasmonics and microscopy [3].
One of the most popular subjects of nanophotonics are optical metasurfaces. An optical metasurface is defined as a two-dimensional periodic arrangement of structures, referred to as meta-atoms, usually at a scale comparable to, and smaller than, the light’s wavelength. The careful design of the meta-atoms and their proper spatial arrangement has led to unprecedented properties that can not be found in nature [4],[5],[6]. Even though the primary goal of metasurfaces is to shape the properties of the transmitted, reflected and diffracted light on demand (such as the amplitude, the phase and the angu-lar momentum), it can be considered under certain aspects as a possible nanoscale light source. The breakthrough in nanoscale light sources came in 2003, when D. Bergman and M. Stockman proposed that the oscillations of electrons in metals, referred to as plasmons, could serve as a subwavelength cavity, making possible the construction of a nanoscale lasers [7]. This led the revelation towards novel light sources, which due to their property of acting as lasers, were identified as nanolasers. Development in nanolasers has been rapid the last twenty years, and several proposals of microlasers have been realized in experiments [8].
Lasing is defined as the process at which coherent monochromatic radiation, with high intensity and directionality is produced from a system through stimulated emission [9]. The main constituents for lasers are: a gain medium and a cavity to ensure single (or
multi) mode operation and provide a mechanism of optical feedback. In this thesis, we will mostly consider the case of distributed feedback lasers (DFB). In this class of lasers, a periodic structure serves as the cavity of the laser, where the optical feedback is sup-plied by multiple scattering, induced by the grating. Even though the DFB lasers have been initially developed with dielectric materials, similar principles of operation have been established for semiconducting materials, and recently similar nanolasers have been devel-oped for the case of plasmonic arrays [10]. Those systems consist of nanoparticles’ arrays made of metals (such as Ag) embedded in a waveguide slab, doped with a fluorescent dye. The nanoparticle array serves a cavity for the lasing to take place, and the optical feedback is supplied though multiple scattering of light between the particles. Depsite the significant losses of metals, these systems have been reported to have similar performance to regular DFB lasers, and lasing has been established in those systems for several types of arrays, gain media and array dimensions [11],[12],[13],[14].
Apart from the development of novel light sources, nanophotonic structures have served as a platform for studying ground-breaking physics. In recent years, the field of parity time (PT) photonics has emerged as a widely studied field. The seminal work of C.M Bender and S. Boettcher [15] on the theory of complex potentials in quantum me-chanics has initiated the research for systems that could exhibit novel phase transitions, and present peculiar properties. Photonics served as a favourable field to study these emerging physical concepts, since the appearance of complex potentials could be assigned to the (familiar) complex nature of the refractive index of materials. Since then several breakthroughs have emerged, both in terms of theoretical predictions and experimental results, ranging from the appearance of topological edge states in photonic crystals [16], the prediction of broken PT symmetry in photonic graphene [17] and even the demon-stration of reversing the pump dependence of a microcavity laser [18].
In more detail, studying the effect of PT-symmetry breaking in optics relies on al-ternating gain and loss in an optical system. These two quantities are encoded in the refractive index of the system, which serves as a complex potential of the system. Manip-ulating the gain and loss in plasmonic lattices turns out to be significantly challenging, since losses due to the metallic nature of the particles can negate the effect of an alternat-ing gain distribution. In addition, it is an open question whether the gain profile of such a system can be dynamically controlled, by modifying the illumination pattern instead of a static implementation of structured gain and loss.
The thesis is outlined as follows: In Chapter 2, a brief theoretical background is pre-sented, including the most important concepts for the description of plasmonic lattices, supported by simulations that predict the emergence of non-trivial features in the optical response of the arrays. Chapter 3 is dedicated to the estimation of the gain coefficient, for two gain media that have been widely used in plasmonic lasers’ experiments. Finally, the experiments concerning lasing in plasmonic lattices, and the study of the effect of structured illumination by means of a spatial light modulator are presented and discussed in Chapters 4,5 respectively.
Theoretical Background
Introduction
Modeling the response of plasmonic metasurfaces can turn out to be theoretical chal-lenge. In this chapter we first review the main constituents that can explain the optical behavior of plasmonic lattices in terms of lasing, such as the optical response of a sin-gle plasmonic emitter. After briefly reviewing the most common approximations used to model the optical response of metasurfaces, we describe our theoretical model of gain in plasmonic lattices, and perform a series of simulations of band structures, in which a gain contrast is introduced at the unit cell scale. We observe that by the introduction of large homogeneous gain in our system, the bands that cross at the Dirac points can fulfill the conditions that would lead to lasing, whereas spatially structured gain could lead to the emergence of PT symmetry breaking features. Finally, we examine two pos-sible band folding mechanisms in the context of introduced super-lattices, which could serve as possible solutions in order to observe the Dirac points of the honeycomb lattice in experiments.
2.1
Plasmonic scatterers - Gain
Plasmonics is the field of science targeting study of the optical response of metals under conditions of excitation of surface plasmons. An incident light field to a plasmonic nanoparticle leads to the collective motion of the conduction electrons of the metal, de-fined as a localized plasmon resonance [19]. This interaction is a primary example of coupling between light and matter, and its manipulation could lead to the emergence of novel phenomena and applications [20].
In general, plasmonic nanoparticles can interact with an incident light field through two mechanisms: scattering and absorption [21]. Absorption is closely connected to the metallic nature of the nanoparticles, and is attributed to Ohmic losses of the material,
where the energy from the incident field is eventually transformed into heat. Since plas-monic nanoparticles allow the concentration of light to volumes significantly smaller than the relevant free-space wavelength (by the efficient coupling of the light field with the conduction electrons and the formation of localised plasmon resonances), significant ef-fort has been put into the study of overcoming the lossy nature of the particles in terms of proper design of the scatterers for the sake of novel applications [22].
The accurate description of plasmonic nanoparticles to an external light field can be proven to be a challenging task. In the context of this thesis, we will approach the re-sponse of a plasmonic nanoparticle with a point dipole approximation in the context of classical electrodynamics [21]. In this context, scattering will be treated as the loss of en-ergy of the oscillating electric dipole induced by an external field, whereas absorption will be treated as a loss of optical energy due to the internal Ohmic damping of the plasmonic nanoparticle.
In terms of electrodynamics, an external electric field E will induce a dipole moment
P according to the following relation:
P = 4π0αdynE (2.1)
where 0 is the vacuum permittivity and αdyn is a complex valued quantity defined as the
dynamic electric polarizability, and is closely related to the optical response of an object to an external field. It depends strongly on the shape, composition and surrounding environment of the particle. Even though it can be analytically calculated for a few symmetric shapes, there is not an analytical expression for any arbitrary object, and it is usually treated through numerical calculations. In order to overcome this issue, it is a common strategy to express the dynamic polarizability with an appropriate form of the static polarizability αstat, for which analytical expressions can be derived for certain
shapes. It can be shown that the dynamic polarizability αdyn can be approximated in
terms of the static polariability αstat by the following equation:
α−1dyn = α−1stat− 2i 3k
3 (2.2)
where k is the wavevector into which the induced dipole radiates part of its optical energy. The above expression allows us to convert any expression of the polarizability into an electrodynamically consistent quantity, which satisfies the optical theorem. Apart from a relatively abstract definition of αdyn, it can be shown that it is directly connected to
observables, such as the scattering cross section σscat and the extinction cross section σext
(which is the sum of the scattering and the absorption cross sections):
σext ∝ kIm(αdyn), σscat∝ k4|αdyn|2 (2.3)
All the above relations can be generalized into a more complete form, in order to account the full electromagnetic nature in scattering. In particular, the dipole moment
which was previously introduced solely for an induced electric dipole, can be extended to a vector form, which would include both the electric and the magnetic moments that are induced by any electromagnetic vector in space. Therefore, we can rewrite that.
P = αααdynF (2.4)
where P = (p, m)T is the vector polarization, which includes the induced electric (p) and magnetic (m) moments and F = (E, H) is the incident electromagnetic light field. The dynamic polarizability is now replaced by the dynamic polarizability matrix, which is a 6×6 matrix. It can be written as a block of 4 polarizability tensors which describe the full electromagnetic dipolar response of a scattering object to the external field.
αααstat =
αEE αEH
αHE αHH
!
(2.5)
Similarly to the previous arguments, we can redefine the dynamic polarizability tensor as: α αα−1dyn= ααα−1stat− 2i 3k 3 I6×6 (2.6)
The dynamic polarizability tensor can be similarly connected to observables such as the scattering cross section. We will only mention that if αααdyn is diagonizable, the
eigen-polarizabilities are linked to corresponding cross sections through equivalent relations to eq. 2.3.
The effect of the metallic nature of the particles on their overall behavior of the particles is encoded indirectly through the static polarizability tensor. In particular, for non magnetic materials and by ignoring possible magnetic induction effect, αstat is reduced
to a 3×3 tensor, corresponding to the electric polarizability tensor. This quantity depends strongly on the dielectric function of the nanoparticle’s material, as well as the dielectric function of the surrounding medium. Due to the strong resonances that appear on metallic nanoparticles (caused by the dependence of the dielectric function of metals to the light frequency), such particles differ significantly from their dielectric counterparts, in terms of optical properties.
Several choices for the calculations of the above mentioned quantities can be made, in order to approximate plasmonic particles in simulations. For example, the most simple, and common, choice for the static polarizability would be to approximate the nanoparticles as spherical objects.In the case of spherical objects made of a material with permittivity
2, embedded in a medium with permittivity 1, the static polarizability matrix is expected
to have the familiar form:
αstat = 4π0V
2− 1
2+ 21I
3×3 (2.7)
where V is the volume of the particle. Moreover, the permittivity of a plasmonic particle can be approximated sufficiently by the Drude model, where the permittivity 2can obtain
the analytical form:
2 = ∞+
ωp
ω2− iωγ (2.8)
where ∞is the electric permittivity in the high frequency limit, ωpis the plasma frequency
of the metal and γ is the Drude damping coefficient. As first approximations, the above mentioned quantities could describe the optical response of a plasmonic nanoparticle sufficiently.
In order to introduce the effect of gain in our simulations, we pick an analytical approach in which we approximate the scatterers as core-shell ellipsoids, where the core represents the plasmonic nanoparticle (with permittivity 3), and the shell the active
medium (with permittivity 2), embedded in a medium with permittivity 1. In that case,
the static electric polarizability will be equal to:
α
αα = V Aj (2.9)
where V is the volume of the particle, and Aj = (Ax, Ay, Az) is a diagonal matrix, where
the diagonal elements Aj are quantities defined for each of the Cartesian axes. It can
be shown that the quantities Aj depend on the permittivities 1,2,3, the volume V of the
particles, the volume fraction of the core and the aspect ratios of the ellipsoids [23]. The effect of gain in our simulations, will be inserted through the permittivity of the shell of the nanoparticles. The simplest model for describing the permittivity g of a gain
medium is by approximating it with a Lorentzian function [24]:
g = g0 1 + gmax kg γg ωg − ω + iγg ! (2.10)
where g0 is the permittivity of the medium in the absence of gain, gmax is the gain peak
value, kg is the wavevector at the central gain frequency ωg and γg is the gain bandwidth.
The gain coefficient g is a material property, which serves as an indicator of the increase of a light beam while propagating into a medium, and is the contrary value to the loss coefficient. In particular, when a light beam propagates through a gain medium, after covering a distance z it would have increased its intensity by a factor of G = egz.
Through dimensional analysis, the gain coefficient has units of inverse distance, and is usually expressed in cm−1. Reported values of the gain coefficient for typical laser dyes lie in the range of 10 cm−1, whereas values of the order of 1000 cm−1 have been reported for lead halide perovskite thin films. Despite its simplistic formulation, the introduction of gain in our system serves as a starting point for the theoretical description of plasmonic lattices with gain, which we will describe in the following sections.
2.2
Theory of honeycomb plasmonic lattices:
The
free-photon model
The honeycomb lattice is a non-primitive lattice defined as a hexagonal lattice with two atoms per unit cell [23] (fig. 2.1). We define the lattice vactors a1,2 as:
a2 = (a, 0) , a2 = a 2, √ 3 2 a ! (2.11)
where a is the lattice pitch of the array. It is straightforward that the interparticle distance between the scatterers will not be equal to the lattice pitch, but will be equal to |δ| = √a
3.
Based on our definitions, we can define the reciprocal lattice vectors b1,2 as:
b1 = 2π a 1, − √ 3 3 ! , b2 = 2π a 0, 2√3 3 ! (2.12)
The reciprocal vectors form a new lattice, the reciprocal lattice, where each point corresponds to the wavevector matching conditions at which a certain diffraction order would appear (for example, the (1,2) diffraction condition, will be met under the constraint of ∆k = b1+ 2b2). For the honeycomb lattice the reciprocal lattice vector is a hexagonal
lattice. Due to the periodic nature of the system, certain areas of reciprocal space are equivalent. However, one can construct the so called 1st Brillouin zone (BZ), which is a uniquely defined area in reciprocal space, which depends only on the geometry of the lattice. It is formally described as the Wigner-Seitz cell of the reciprocal lattice, and is widely used in photonics and solid state physics. The most important aspect of the BZ, is that every point outside this area can be reached by adding a lattice vector to a point that lies inside it, or alternatively that any point can be "folded" into the BZ without altering its properties. Physical properties of every periodic system can be fully described within their 1st BZ. Moreover, depending on the underlying symmetries of the lattice, high symmetry points can be defined in the BZ. In particular, for the case of the hexagonal lattice, the high symmetry points are the center of the BZ (Γ), the center of an edge (side) of the BZ (M ) and the intersection of two edges (K). For our study, the important symmetry point will be the K point, which can be shown that is positioned in reciprocal space at:
|ΓK| = 4π
3a (2.13)
We assume that our unperturbed system consists of a 2D waveguide slab, with an effective mode index of nW G= 1.55. In this case, we know that the relation between the
wavevector and the wavelength of the waveguide mode will be:
k = nW G
2π
This dispersion curve represents the light cone of our system, and defines the physical limit of momentum space that can be probed in that system by a external light field, incident as propagating plane waves. By the introduction of a periodic perturbation, scattering is allowed between the waveguide modes. In particular, at the limit where the perturbation is small, and the interaction between the scattered modes is negligible, the response of the combined system can be approximated by a "free-photon" model, in resemblance with the known "free-electron" model in solid state physics. The dispersion curve will now be a combination of multiple cones, which are shifted by an integer number of the reciprocal wave vectors bi. My intersecting this complex structure at an arbitrary
direction in k-space, one can obtain a minimal impression of the band structure, which would be expected from this model (figure 2.1).
(a) (b)
(c) (d) (e)
Figure 2.1: The honeycomb lattice fundamentals. (a) The honeycomb lattice. It consists of a hexagonal lattice with two atoms per unit cell. (b) The 1st Brillouin zone of the
honeycomb lattice (shaded area) in reciprocal space. The high symmetry points M, K, K0
are shown, as well as the first few diffraction orders. Their relative positions with respect to the high symmetry points allows us to navigate in reciprocal space, and assign correctly the high symmetry axes in measurements. (c) Expected slice of the free photon dispersion
for a honeycomb lattice with pitch of 450 nm, for a waveguide mode with index of nW G =
1.55, at a frequency of 2.06 eV. The red circle depicts the light cone limit of our waveguide, whereas the green circle depicts the theoretical limit of measurements, defined by the NA of the objective (in this case set at 1.4). (d)-(e): Different dispersion graphs of the free photon model for the same parameters. Depending on the choice of k that we select, different band structures are expected to appear for the case of the honeycomb lattice ((d) across the Γ − K direction; (e) towards the Γ − M direction)
2.3
Optical response of a plasmonic array: Multiple
scattering
Substituting the weak perturbation of our system with a stronger scattering poten-tial (such as a plasmonic nanoparticle) forces us to move to a more complex model for the proper description of our system. Despite the fact that the main features will be maintained in the dispersion curves, several new features could be observed, such as the formation of band gaps. The two most common regimes in which plasmonic arrays are theoretically studied are the diffractive regime and the near field regime. In the first case, the pitch of the lattice is larger than the wavelength of the incident light field, and multiple scattering can be ignored in our system. The overall model is eventually derived by combining the free photon model we described before with the LSPR of the particles [25]. On the other hand, in the near field regime, the coupling between the particles is significantly stronger through the interaction of their near fields. In these cases, alterna-tive models have to be used for the appropriate description of our models, and mostly rely on tight binding approximations.
In the case of optical metasurfaces which do not fulfill the very weak scattering per unit cell, a different model needs to be considered. In particular, especially for lattices with pitches in the range of a few hundred nanometers, the main mechanism of interaction is multiple scattering. In this section, we will note the main features of the theory, on which the simulations of the chapter are relying. The detailed theory of this section can be found in [23].
The strategy towards modeling the collective response of a lattice of scatterers relies on extending the theory of a single scatterer. When an electromagnetic field is incident on an assembly of l = 1, 2, ..N scatterers, each scatterer will respond to the incident field, as well as to the scattered field of the other nanoparticles. In mathematical terms this can be expressed through the definition of the electromagnetic dipole moment Pl:
Pl= αααl F + N X l=1;l6=l0 G GG0(rl− rl0)Pl0 (2.15)
where Pl is the dipole moment induced on the particle position at the position rl and αααl is
the dynamic polarizability. Inside the bracket, the first term is connected to the scattering by a single isolated particle, whereas the second term is such that the response of all the other scatterers at positions rl0 is taken into account. It consists of the dipole moments
of the remaining scatterers, positioned at rl0, and an additional function GGG(rl − rl0),
which is defined as the Green’s function. Even though Green’s functions are primarily a mathematical construction assisting in the solution of differential equations, they can be intuitively though as a mathematical description of the propagation of the scattered fields from position rl0 to position rl.
for explaining the overall response of an assembly of scaterrers to an incident field, due to multiple scattering. Without going through the detailed derivation (which would require the application of Bloch’s theorem to eq. 2.15), one can express the dipole moment P (which consists of the dipole moments Pl of all particles in the unit cell) as:
P = αααeffeffeffF (2.16)
where αααeff is the effective polarizability, and is defined as:
α ααeff = α α α−1− Glatt −1 (2.17)
where ααα is a block diagonal matrix containing the dynamic polarizabilities of all scatterers,
and G is the lattice Green function. The lattice Green’s function is a summation of all the Green;s functions contributions in equation 2.15 over the real space lattice.
Calculating the above equation requires method known as Ewald lattice summation, in order to calculate the effective polarizability of our system. It allows the calculation of the lattice Green’s function G as a function of the parallel wavevector k|| and frequency
ω, and despite its elegant underlying mathematics, it will not be discussed further in this
thesis. However, a couple of important features should be noted, that will be crucial to the performed simulations. Firstly, the band structures are diagrams that span the frequency
ω and wavevector k space. From our analysis, we are able to calculate the effective
polarizability for a certain system, which allows the calculation of the eigenpolarizabilities, by diagonalization. Summing the imaginary parts of the eigenpolarizabilities leads to the net extinction, which can be attributed to a band structure calculation. The resulting "bands" that after the calculations are referred to as lattice surface plasmon resonances (LSPR), and appear due the the hybridization of the nanoparticle resonances (encoded in
ααα) and the lattice anomalies (encoded in Glatt). Moreover, the above mentioned formulas
help us to calculate the total field F taking into account the scattered fields, at a distance far from the array (in the far-field). As a result, the transmission and reflection can be calculated for the system, which would serve as an important indicator of observable features.
2.4
Simulations
The simulations that we will perform rely on the theory that we briefly described be-fore, and are performed with a "lattice-sum" (L.S.) code provided by R. Kolkowski. This code allows us to perform calculations of either band structures (calculating the effective polarizability as a function of k||, the in-plane component of the wavevector) or
transmis-sion and reflection as a function of the angle θ (connected to the wavevector through the common relation k = nk0sin θ, where n is the waveguide refractive index and k0 = 2π/λ
Table 2.1: : Simulation parameters for lattice sums code
Quantity Value (if applicable) Lattice pitch [nm] 210
Lattice sums terms Nij 2
Radii of the core (rx, ry, rz) [nm] (32.5, 32.5,15)
Volume fraction of the core (β) 1/8
Core material Ag
Refr. index of shell in the absence of gain ng0 1.56
Gain coefficient (peak value) g -Gain bandwith γg [eV] 0.25
Central gain frequency ωg [eV] 2.1
Surrounding medium refractive index SU8
simulations regarding the study of the honeycomb lattice, is presented in table 2.1 The choice of the parameters can be directly justified. First of all, we chose Ag as the core material, since it is expected to provide the largest scattering cross section among the available metals in experiments. The optical constants of Ag in the frequency range that we are interested in, were set by tabulated data. The dimensions of the core were set in relation to previous studies, in order to have matching dimensions with a cylinder of the same aspect ratio, for which a maximum scattering cross section was predicted through numerical simulations. The choice of a diameter of 70nm and height of 30nm guarantees an overlap between the frequency range of the maximum of the scattering cross section with the gain central frequency ωg, which was preset at 2.1 eV. Finally, the
surround-ing medium of the particles was set to match the effective mode index of an SU8 slab of thickness of approximately 450nm, which can be calculated by effective mode index calculations.
Firstly we calculate the band structures and the transmission and reflection of a hon-eycomb lattice, with the absence of gain in the lattice (both particles in the unit cell will have g = 0 cm−1). We chose as our k−axis the one corresponding to the Γ − K direction. The resulting band structures and the calculated transmission-reflection are presented in Figure2.2.
In accordance to the theory, we can directly observe a couple of features for the passive (g = 0) case. Firstly we observe the formation of various bands, which correspond to the lattice resonances in our system. Several bands appear after the calculations, and their majority does not contribute to the transmission and reflection, an effect that certifies them as guided modes [26]. These modes lie below the light cone, and do not contribute to the optical properties of the lattice in the far-field. Moreover, we notice that at the
K−point (k|| = 4π/3a) there is a separation between the calculated bands, which has
been attributed to the different dispersion properties between the in-plane and out of plane modes [23]. In this thesis, we are interested in the in-plane modes, since they are
expected to contribute significantly to the experimental observables plasmonic lattices. In addition, the in-plane modes are extending into the light cone of our system, where they form a flat band at the Γ−point at 3 eV, which is also observed in the reflection and transmission images.
We proceed in our simulations by studying the effect of uniform gain in the band structures of the honeycomb array at the vicinity of the K−point (Figure 2.3). Since the relevant modes are present below the light cone, we only present the calculated band structures, at which the effect of gain can be depicted. In particular, we set the gain for each individual scatterer in the unit cell equal to g = 2000cm−1 (uniform gain), which was defined as the point where a critical transition was observed in the simulations. At this value of gain, the in-plane modes correspond to negative values of the extinction cross section, and a transition from a "lossy" to a "gain" behavior is apparent. In this case, the crossings of the in-plane modes could serve as lasing points for our lattice, as the relevant modes have overcome the lossy nature of the plasmonic scatterers.
As a final demonstration of the physics in the vicinity of the K−point of the honey-comb lattice, we set an antisymetric distribution of gain and loss into our unit cell (Figure 2.3). In more detail, we set one scatterer of the unit cell to have a gain of g =2000 cm−1, whereas the gain surrounding the second scatterer is set to g = 0 cm−1. By the inclusion of an uneven distribution of gain and loss into the unit cell, a horizontal separation between the bands is induced, in addition to the formation of an area, corresponding to negative scattering cross section, surrounding the K−point. Even though we will mostly focus on the study of lasing in honeycomb lattices, alternating gain and loss in plasmonic arrays could eventually lead to the horizontal separation of the bands at the Dirac points of the honeycomb lattice, which would serve as an experimental observation for the transition of our system to a broken PT-symmetry state [23].
Figure 2.2: Band structure and transmission/reflection results for a plasmonic honeycomb lattice, along the Γ − K direction. Due to hybridization between the particle resonances and the lattice, several bands emerge from the calculations. A separation between in-plane (at 2 eV) and out of in-plane (at 2.2 eV) modes is eminent from the calcuations at the K-point. At 2.7 eV, the extended in-plane modes are forming a flat band at the Γ-point, which leads to observable effects and variations in transmission and reflection of the lattice.
(a) (b)
(c)
(b)
Figure 2.3: Band structures at the vicinity of the K point for different gain values and distributions. (a) Band structure in the absence of gain. The in plane modes at the K point (at k = 4/3 π/a) respond to positive values of extinction cross section (which is proportional to Imα). (b) By adding sufficient isotropic gain to the system (g = 2000
cm−1 for both particles in the unit cell). The modes now correspond to negative values of
the scattering cross section, and therefore their amplitude would grow exponentially upon excitation, and serve as possible lasing points. (c) Adding a similar nonuniform gain in
the unit cell (g = 0 and g = 2000cm−1) leads to the formation of a horizontal gap at the
K point, which can be attributed to PT-symmetry breaking and existence of exceptional
points in our system [23].
2.5
Band folding
As we have shown in the previous sections, due to the peculiar properties of the K-point of the honeycomb lattice, interesting features are expected there. However, these
Dirac points lie below the light cone, and therefore are not accessible from the far-field. A few methods have been developed, in order to overcome this issue, ranging from near field probing [27], the use of a coupling prism [28] or the introduction of periodic perturbations which could lead to folding of the bands into the light cone [29]. We examine the latter case, which appears to be the simplest method to be implemented in experiments, since it requires only the addition of controlled perturbations during the fabrication of the plasmonic arrays.
Band folding originates from Bloch’s theorem, which states that the presence of a periodic perturbation leads to the folding of the studied features inside the BZ, defined by the additional periodic structure. If the system that we refer to as unperturbed, is already a periodic lattice, then the additional "superlattice" (SL) that will be implemented should lead to additional band folding in the system. Even though a strict theoretical explanation for the effect of the SL to the lattice concludes that band folding is emerging due to the distortion of the BZ of the primary lattice [30], an intuitive framework of how band folding is achieved will be presented for two cases of periodic perturbations. The periodic modulation is implemented by either modifying the size of a particle in the existing lattice, or by the additional of a small particle in predefined positions in the lattice which retain the underlying symmetries of the honeycomb lattice (fig 2.4).
Let us consider a honeycomb lattice with pitch a. The BZ corresponding to the honeycomb array will be a hexagonal array, and the points that we are aiming to fold into the light cone are the characteristic K-points, which are positioned in reciprocal space across the kx axis at:
K = 4π
3a (2.18)
Let us also assume that the refractive index of the medium is n = 1.55. Then, the wavevector which defines the edge of the light cone is :
kmax= nk0 = n
2π
λ (2.19)
and defines the constraint for the observable features in order to lie inside the light cone. Any point appearing in reciprocal space at kkki, will be lying inside the light cone if it fulfills
the condition:
ki
kmax
< 1 (2.20)
We implement a SL by modifying the size of one particle in the existing lattice. We set the periodic modulation in order to form a rectangular SL, which we will refer to as "asymmetric superlattice" (a-SL). The contrast between the particles should be small, in order to serve as a small perturbation to our system, and for the simulations presented, is set to a ratio between the perturbed/unperturbed radii rSL
x,y/rx,y = 4/5 (leading to a
contrast of 10 nm in diameter). We chose the dimensions of the rectangular a-SL, in such a way that the high symmetry point XSL in the corresponding BZ is set across the kx
axis, and is set to:
XSL=
π
a (2.21)
If band folding is present, the K point of the honeycomb lattice should be folded with respect to the XSL, and appear as a folded point Kf at approximately:
Kf =
2π
3a (2.22)
For our system, the bands that we are interested in appear for a lattice pitch of 210 nm, and at a wavelength of approximately 590-600 nm (approx. 2-2.1 eV). By the choice of this a-SL, we can easily calculate that the folded point Kf should lie inside the light
cone (Kf/kmax ≈ 0.6), and appear as a signature in reflection/transmission when the
incidence angle θ satisfies the condition sin θ = Kf/k0. For the relevant parameters, the
angle of incidence is approximately θ = 68o.
In addition to the previous band folding strategy, a second similar approach is exam-ined, where the perturbation is implemented with the addition of an additional particle in the lattice. The particle has a small volume (rSL
x,y/rx,y = 1/5) and is placed at the
center of every forth hexagon of the honeycomb lattice. The resulting SL is a hexagonal array, with a pitch twice larger than the honeycomb lattice, and will be referred to as the "symmetric superlattice" (s-SL) since it shares the same rotational symmetry as the honeycomb lattice (figure 2.4).
Similarly to the case of the a-SL, the folding point K of the honeycomb lattice will be positioned along the kx axis. Similarly, the BZ zone of the s-SL will be a hexagon, with
the high symmetry point KSL positioned along the kx axis at:
KSL =
4π 3aSL
= 2π
3a (2.23)
where aSL is the pitch of the s-SL, which is equal to 2a. It is straightforward that the
choice of this particular SL will lead directly to the folding of the Dirac point to Kf to the
center of the BZ ( Γ). In addition, due to the shared symmetry between the s-SL and the honeycomb lattice, is expected to not break the symmetry of the modes which contribute to the peculiar properties of the Dirac point, serving as a more favorable candidate for efficient band folding of the symmetry protected features at the K-points.
Figure 2.4: Band folding of the K point in a honeycomb lattice by the introduction of a SL. Top row: Introduction of a periodic perturbation which would form a rectangular superlattice, referred to as a-SL due to its different symmetry with respect to the primary lattice (honeycomb). On the right the corresponding BZ of both lattices. Due to the periodic nature of the a-SL, the K-point of the honeycomb will be folded inside the BZ zone, and depending on the dimensions of the rectangular lattice could in principle
be folded inside the light cone (Kf). Bottom row: Introduction of a different periodic
perturbation, by the addition of a small particle in the unit cell, with a periodicity of two unit cells. The new SL will be a hexagonal lattice with a pitch twice as large as the pitch of the primary lattice and will share the same symmetry as the honeycomb lattice (s-SL).
In this case, the folded Dirac point Kf will be folded exactly to the Γ point.
In order to examine whether band folding can be achieved theoretically, we need to modify the L.S. codes in a proper manner. In particular, since it is necessary to study the effect of a perturbation with a range larger than the original unit cell of the honeycomb, we need to implement a super-cell (SC) in our simulations, instead of the trivial honeycomb unit cell. The SC will contain a larger number of particles, forming a honeycomb lattice with the desired periodic perturbation. The relevant SC used for each of the cases we
mentioned above are presented in figure 2.5. The choice of the unit cells is not unique, and different arbitrary cells could have been chosen, in order to reproduce the desired structures. It should be noted that the choice of the SC will affect the calculated band structures. In more detail, by simply calculating the band structures for a SC which would not include the induced perturbations, additional bands are expected to appear after the calculations. However, those bands are expected to correspond to "Bloch" modes with zero amplitude and should not be considered as the correct result of the process. In order to observe which bands are folded, additional processing should be performed in order to remove the effect of the SC to the calculations, in a similar manner to the widely used computational "unfolding" methods employed in condensed matter physics. However, the effect of band folding can be directly identified by observing the calculated transmission and reflection. In the absence of true perturbation, these quantities are not affected by artificially introducing the SC, since they are calculated by the total scattered fields of the array, which do not depend on the choice of the unit cell but by the relevant position and properties of the scatterers. Therefore, the appearance of features in the transmission/reflection calculations is a sufficient method to establish whether band folding is efficient or not.
We present the reflection and transmission images, calculated by implying a s-SL or an a-SL to the honeycomb array (figure2.6). For all cases, we introduce a homogeneous gain to all scatterers, equal to g = 5000 cm−1. We notice that in both cases, distinct regimes of large reflection/transmission appear after the SL implementations, which suggest that the band folding has been achieved. Moreover, we note that in the a-SL case, the folded Dirac point is significantly distorted and presents a significant asymmetric signature, indicating that even though the method is effective in terms of folding, the difference in the underlying symmetries between the honeycomb lattice and the a-SL can lead to severe distortion of the folded K−points.
Figure 2.5: Supercells (SC) implemented in the simulations, for the case of the a-SL and the case of the s-SL. Each SC contains 4 and 9 atoms respectively. The shaded unit cell represents the primary unit cell of the honeycomb lattice. The above choices are not unique, and several different SCs could be chosen. The introduction of SC is affecting the calculations of the band structures, since it introduces additional bands, which are expected to have a zero magnitude and with additional calculations would lead to the correct band structures. However, the choice of the SC does not affect the reflection/transmission simulations, which can serve as an appropriate tool of studying band folding, without the requirement of additional processing in our simulations.
Photon en ergy (eV) Photon en ergy (eV) Photon en ergy (eV) Photon en ergy (eV)
Figure 2.6: Transmission and Reflection calculations for the prediction of band folding for an a-SL (top row) and a s-SL (bottom row). The emergence of distinct features in the results is characteristic of the successful band folding from our arrays.
2.6
Discussion
In this section, we have briefly presented the theoretical concepts that are most relevant to this thesis. Namely, we have introduced the main concepts involved in the theoretical study of plasmonic metasurfaces. In particular, we are interested in approximating the behavior of the honeycomb lattice in the metasurface regime, where the optical response of the system is the result of multiple scattering of light between the scatterers, which leads to the emergence of lattice resonances and their effects on the (observable) reflection and transmittance of the metasurface. Moreover, we examine the peculiar properties of the Dirac points of the honeycomb lattice, which can lead to the formation of bands with large gain, suitable for lasing. These bands result from PT-symmetry breaking at the Dirac points, caused by uneven distribution of gain and loss across the constituent scatterers
of the lattice unit cell. Finally, we introduce the concept of band folding, a method that can be used to fold the Dirac points into the light cone, and distinguish the effect of two different SL approaches, which despite of successfully folding the Dirac points into the light cone, induce different side effects to the calculated reflection and transmission.
Gain measurements
Introduction
It has been demonstrated that alternating the gain and loss in plasmonic lattices can lead to peculiar features in the optical properties of the lattice. However, large values of gain are required, which can be a large challenge for experiments. In this chapter, we introduce the concept of gain and present a methodology to estimate the values of the gain coefficient g for two gain media that have been previously used in lasing experiments: Rh6G, which has an emission frequency of 2.1 eV (590 nm), and a FRET pair of Rh6G and Rh700, with an expected emission frequency of 1.78 eV (700 nm). We measure the gain coefficient of those dyes for different pump powers through the widely used Variable Stripe Length (VSL) method, and conclude on which dye would be the most suitable choice to be used in experiments.
3.1
Gain coefficient
The optical properties of materials depend significantly on their chemical composition. Metals are known for their high reflectivity, whereas glasses can be characterized as the most common example of a transparent medium. It is of particular interest to study how a light beam is affected when it propagates through a particular medium. Depending on the material, different properties of the light beam could be affected, such as the intensity (enhancement, attenuation etc.), the polarization or even the frequency of the photons (generation of harmonics) of the beam. In this thesis, we will focus on the effect of the material gain to the intensity of the propagating light beam. We start by assuming that we have a light beam incident on a transmissive medium, which is homogeneous, but provides loss and/or gain. The light beam is entering the medium at r = 0 as an infinitely extended plane wave of intensity I0. Assuming that the beam has propagated a distance
be equal to:
dI = GIdr (3.1)
The coefficient G is the net gain coefficient which describes the effect of the medium to the light beam. It can be defined as:
G = g − a (3.2)
where g is the gain coefficient and a is the loss coefficient of the medium.
It can be shown that equation would lead to an exponential dependence of the form:
I ∝ I0eGr (3.3)
If G > 0, then the intensity of the light beam is increased while propagating the medium whereas on the opposite case, the light beam is attenuated. Therefore, the rele-vant magnitudes of g, a can define the ability of a medium to amplify/attenuate the light beam propagating in it.
3.2
The Variable Stripe Method
A few techniques have been developed in order to quantify the gain coefficient on different materials. The most common and simple method to implement for this purpose is the widely used Variable Stripe method (VSL) [31],[32],[33], originally developed to study the optical gain properties of semiconductor crystals [34]. Different, more precise methods have been developed in order to quantify the gain properties of materials, but VSL remains one of the most popular choices, due to its simple implementation in experiments.
The procedure relies on pumping the fluorescent sample with a beam focused into a narrow and elongated spot - a stripe of length l, with one end located at the edge of the sample. The emitted light is collected from the edge of the sample. The sample consists of a thin layer of SU8, doped with a fluorescent dye, and has a thickness of approximately 450 nm, in order to ensure the existence of a single mode in the slab. For very short values of l, the collected signal is expected to correspond to the fluorescence signal of the sample, which is dominated by spontaneous emission. For a certain value of the stripe length, the pump power is such that amplified spontaneous emission (ASE) can take place in the slab. ASE is defined as the situation, in which the spontaneous emission is linearly enhanced by excited atoms through stimulated emission [9]. It leads to the emergence of high intensity peaks, with a width ranging from 10-40 nm. It can be thought as case intermediate between of incoherent spontaneous emission and coherent lasing. Under the assumption that the width of the stripe is negligible, and the stripe is homogeneously pumped along its length, the intensity exiting the sample from the side can be directly calculated through 3.3 as a function of stripe length l to be:
I(l) = A G
where G is the net gain coefficient and A is a constant depending on the sample and the pump characteristics. We note that the net gain coefficient cannot be directly attributed to the gain coefficient g, unless the losses inside the medium are negligible. With that in mind, we will simply refer to the net gain coefficient as simply the gain coefficient, without further acknowledging the losses inside the material.
When linearly increasing the stripe length, it is obvious from eq. 3.4 that the ASE intensity will have a non linear increase. Therefore, by varying the stripe length and recording the ASE peak intensity, it is possible to obtain the gain coefficient for our medium, for a specific pump intensity. Finally, by further increasing the stripe length, it is expected that our system will start saturating. The spontaneous emission will be too intense to be further amplified by the material, leading to a divergence from our simplistic relation. The stripe length for which saturation is appearing is referred to as the saturation length. Identifying the possible reach of saturation during the analysis of the data is of huge importance, since neglecting the effect of saturation can lead to underestimation of the gain coefficient.
3.3
Experimental Setup - Sample preparation
. The setup we used is presented in figure 3.1. As a pump source, we are using a 532 nm pulsed laser (Teem Photonics, type STG-03E-1S0) with a maximum energy per pulse of 4.5 µJ and a pulse duration of 500 ps. The power of the laser is controlled with an acousto-optic modulator. The sample is positioned vertically with respect to the beam axis, and an air objective is used to collect the fluorescence signal from the side of the sample (Nikon Plan 10x / 0.25 NA, WD 10.5). The pump beam is converted into a stripe with the use of a cylindrical lens, and the stripe length is controlled with a slit, positioned between the sample and the lens. The resulting thickness of the stripes is approximately 110 µm. In order to center the stripe on the edge of the sample, we set the width of the slit to the minimum possible length (0.5 mm), such as to appear as a dot on the sample. We then properly align the sample and the cylindrical lens that focuses the beam in such a way that the edge of the sample is in focus with the objective, and the dot is on the edge of the sample. With this arrangement, we are able to generate stripes on the sample which correspond to half of the width of the slit. In the detection path, we are using a spectrometer (Shamrock 303i equipped with an iVAC CCD detector) to capture the fluorescence spectrum from the edge of the sample, and a camera (Andor Clara CCD) in order to image the edge of the sample, ensuring that it is indeed in the focus of the objective.
Laser 532nm Camera f2 f3 f4 f5 f1 Slit
Figure 3.1: Variable stripe method (VSL). The sample is illuminated with a stripe of
length l, formed by a cylindrical lens (f1). The stripe length is controlled by a slit. The
signal is collected from the edge of the sample, and is directed to a spectrometer. (Inset
figure from [33]).
The samples we are interested in are thin films of SU8 (450 nm thickness) on glass (0.17 mm thickness) doped with two different gain media, in different concentrations. The first medium we will be studying is the Rh6G dye (emission peak/ maximal gain at 2.1 eV / 590 nm). We dissolve Rh6G in cyclopentanone, with different ratios ranging from 5 mg per 1 mL (5:1 concentration) down to 1 mg per 1 mL (1:1 concentration). We proceed with ultrasonicating the mixtures for 10 minutes, before mixing them with SU8-cyclopentanone in a 1:1 volume ratio. We clean our glass slides (0.17 µm thickness) with ethanol, and spincoat our solution at 3000 rpm, in order to obtain a layer of approximately 400 nm. Finally we break the samples in the middle, and restrict our measurement area to the center of the broken slide, where the thickness of the spincoated layer is expected to be constant (not affected by the vicinity of the edges).
The second gain medium we will be using is a mixture of Rh6G and Rh700. This mix-ture has been established previously to act as a FRET pair, and has been shown to present maximum FRET efficiency for a balanced content of the two dyes in a mixture. Therefore, we prepare new solutions, where the initial mixture of Rh6G with a concentration of 5 mg/L is mixed with Rh700 of the same concentration in a one to one volume ratio. By mixing the two solutions at equal quantities, we end up with a solution with a content 10 mg. By diluting this solution properly, we end up with different dye concentrations,
which vary from 2mg/mL up to 10mg/mL, we proceed with the sample preparation in a similar manner as in the previous case, in order to obtain a 400- 450 nm thick layer. We will refer to these as the Rh700 samples, in order to annotate their emission frequency.
3.4
Measurements - Analysis
We perform single shot measurements for each of the above samples, for different pump powers. The smallest stripe length that we can obtain on the sample is 0.05 cm. We start our measurement with the smallest stripe at which ASE has started to emerge. We vary the stripe length at steps of 0.0025 cm, and record the spectra for 30 different values. For our analysis, we record the peak intensity of the ASE, and the corresponding full width at half maximum (FWHM). After completing the measurements at a certain pump power, we shift the position of the stripes on the sample, to avoid the effect of dye degradation in our results. By inspecting the measurements, we estimate that the average variation of the peak intensity (due to single shot measurements) is of the order of shot noise, and is neglected from the post analysis of the data that we will introduce.
Representative measurements for the two selected dyes are presented in figure 3.2. For both samples, the emergence of high intensity peaks with a FWHM ranging from 10-20 nm (0.1-0.2 eV) indicates that ASE is taking place in our samples. We report that the ASE peak for the Rh700 samples is broader than the Rh6G samples. Moreover, the apparent exponential increase of the ASE peak with respect to increasing stripe length justifies that our simple theoretical model could serve as a sufficient model in order to estimate the gain coefficient in our measurements, where a nonlinear fit of our data with equation 3.4 would yield the value of the gain coefficient g for a certain pump power density.
One of the primary factors that can affect the estimation of gain with the VSL method is gain saturation. In terms of our measurements, gain saturation would lead to a discrep-ancy between our theoretical model and our data, and the behavior of the peak intensity would decline from the exponential growth expected. In order to validate whether the saturation length of each sample has been reached in our samples, we employ a similar analysis as in [35],[36],[37]. In particular, we fit our data with equation 3.4 for different final stripe lengths, and record the resulting value of the gain coefficient g, as well as the relevant fitting error ∆g resulting from the fitting procedure. If the saturation length is reached in our measurements, then a decrease of g is expected after that value. Moreover, this increase could be accompanied by a relevant "bump" in ∆g, which would further indi-cate the deviation from our theoretical model. The stripe value at which this discrepancy is observed is defined as the saturation length, and sets the correct fitting range which should be used in the analysis for determining the gain coefficient. We perform the selec-tion of the saturaselec-tion length manually, and report that only for some of the samples the saturation length has been exceeded, depending on the dye concentration and the pump
energy density. Examples of the mentioned methodology are presented in figure 3.3. Following the above analysis, we are able to determine the gain coefficient for dif-ferent dye concentrations in our samples, under difdif-ferent pump powers. The results are presented in Figure 3.4. The corresponding error bars have been set equal to the errors resulting from the fitting of our model (∆g), whereas a linear dependence is apparent between the gain coefficient and the pump power.
3.5
Discussion
From the results presented in Figure 3.4, it is apparent that several (expected) trends are present. First of all, a clear linear dependence is observed between the gain coef-ficient and the pump power. Moreover, the larger estimated gain values for larger dye concentrations is expected intuitively (more dye molecules per unit volume correspond to more events of stimulated emission), and indicates that concentration quenching is not significantly affecting the optical gain in our samples. Moreover, the larger errors in lower pump powers and concentrations can be justified by the low ASE intensity observed under those conditions. Finally, it is evident that the largest gain values for each dye is observed for the largest concentrations, under all pump powers. In between the different dyes, the Rh6G samples present larger gain values than their Rh700 counterparts (both in the range of several cm−1 as expected for laser dyes), and therefore can be attributed as the most reasonable choice for our experiments, when higher gain values would be re-quired. However, the possible effect and estimation of losses due to the plasmonic nature of the particles should be considered in future studies, which might make the Rh700 dye a more suitable choice for implementation in experiments.
Figure 3.2: Typical results of VSL measurements for a Rh6G sample (top) and Rh700 (bottom). In both cases, the emergence of a high intensity peak, with a FWHM ranging from 10-40 nm (0.1-0.2 eV) in the fluorescence signal is a characteristic signature of ASE. The apparent exponential increase of the peak intensity verifies that we are in the correct regime of stripe lengths for gain measurements.
Figure 3.3: Typical analysis of VSL data. Top row: Rh6G sample: In the Rh6G sample presented, the saturation length is clearly reached in the stripe range that we selected. After varying the final fitting point of our data, a clear decrease of the estimated gain coefficient is observed, accompanied by a slight increase of the error, before stabilization. The saturation length is identified at 0.0825 cm, and the final fitted curve further indi-cates the saturation in the latter data. Bottom row: Rh700 sample: In this sample, the saturation length has not been reached in our measurements. In particular, the estimated gain coefficient remains constant with respect to the stripe length. Despite the stabiliza-tion and slight increase of the error ∆g in larger stripe lengths, the expected decrease of the gain coefficient is not resulting from the analysis, and therefore the saturation length is not reached in the measurements.
Figure 3.4: Gain coefficient results for Rh6G and Rh700 samples as a function of pump fluence.
Lasing in honeycomb plasmonic
lattices
Introduction
Lasing in plasmonic lattices has been established for several types of plasmonic nanopar-ticles’ arrays, notably for simple square, rectangular and hexagonal arrays . Motivated by the theoretical background, we fabricate honeycomb arrays of Ag nanoparticles, aiming for lasing at the band edges as well as for band folding in dense nanoparticle arrays, in an attempt to exploit the physics of the Dirac points. In particular, by measuring the energy distribution (spectra) as well as the angular distribution (Fourier imaging) of the emit-ted fluorescence signal, the lasing transition in our system can be identified. While the expected lasing behavior by arrays designed for lasing at the band edges is observed, the densely packed arrays exhibit a peculiar lasing behavior, which cannot validate whether the band folding mechanisms are reproducible in real life experiments.
4.1
Sample fabrication
Our samples typically consist of an array of metal nanoparticles embedded in a slab waveguide with gain. In principle these systems should be optimally designed in such a way that the plasmonic scatterers have a significant scattering strength, to provide distributed feedback for lasing, and the waveguide layer thickness ensures single mode operation [10]. Firstly, for the rest of the following experiments, we choose as the gain medium the Rh6G dye (maximum emission at 590nm - 2.1 eV), for which the largest gain values were observed in Chapter 3. The thickness of our waveguide layer (in our case SU8) is chosen to be approximately 450 nm, in order to support only the fundamental guided modes. Finally, we aim to fabricate Ag nanoparticles, with height of 30nm and a diameter of approximately 60-100 nm, which have been reported to provide a maximum
scattering cross section in the emission range of Rh6G [38].
The fabrication procedure of the plasmonic metasurfaces on glass relies on e-beam lithography, a recipe provided by A. Berkhout [39] and the full procedure is sketched in Figure4.1. We start by spincoating 100 nm of PMMA 495-A8 on glass slides (thickness of 0.17 mm, cleaned with base Piranha), which will play the role of a spacer layer dur-ing fabrication. A thin Ge layer (20 nm) is deposited on top of the PMMA by e-beam evaporation (Polyteknik Flextura M508 E), before the spincoating of the photoresist layer (positive photoresist CSAR 6200:09, 1:1 in anisole), with a thickness of approximately 50 nm, in order to allow high resolution during fabrication. For the e-beam exposure of the samples we are using a Voyager e-beam system (Raith 50kV) to write 150 x 150 µm size arrays with pitches varying from 170 - 250nm and 400 - 450nm. We are using dot exposure during the e-beam process, with doses ranging from 0.002 - 0.020 pC, depending on the desired pitch of honeycomb lattices, aiming for nanoparticles with a diameter of 60-100 nm. Introducing particles of different size (for the case of band folding arrays) requires the modification of the dot dose by a factor of at least 0.5 for the modified particles with respect to the one used for the rest of the array.
Following the e-beam exposure we develop the photoresist in pentyl acetate (60 s), and subsequently rinse the samples in O-xylene (6 s), MIBK:IPA 9:1 (15 s) and IPA (15 s). Then, the Ge and the PMMA are etched and removed from the developed samples through plasma etching (Ge: etching for 60 s in a 1:5 O2:SF6 mixture; PMMA: 60 s with
O2 plasma in a 30 mTorr chamber pressure). In this step, the Ge layer works as a hard
etch mask for the PMMA, and reduces the damage of the spacer layer during etching, al-lowing small separation distances between the fabricated features. Next, a 30 nm layer of Ag is deposited on the sample through e-beam evaporation (evaporation rate: 0.1 nm/s) in order to obtain the desired plasmonic particles. Finally, we perform lift off in acetone at 45o, and obtain the desired plasmonic arrays on glass.
In order to obtain a waveguide with gain, we are using the same spincoating procedure as mentioned in Chapter 3. In particular, we are focusing on the gain medium with the largest measured gain, which is the Rh6G:cyclopentanone mixture, with a concentration of 5mg of Rh6G to 1 ml of cyclopentanone (5mg/m1 mixture). We dilute the gain mixture with SU8-2005 in a 1 to 1 ratio, and eventually spincoat a layer of approximately 450 nm on top of our arrays.
4.2
Experimental setup
The setup we use is an inverted fluorescence microscope [10], [38], [40] , and is presented in figure 4.2. It consists of a home-built microscope tower, on which our oil objective is mounted (Nikon, Plan Apo VC, 100x / 1.4 NA). The samples are placed in the setup
Figure 4.1: Sample fabrication procedure stages, for plasmonic arrays embedded in gain medium. The fabrication is based on e-beam lithography, where the an appropriate etch mask is used (Ge) in order to allow the printing of features with small separation gaps.
with the glass side facing towards the objective, and their position can be controlled by an appropriate XYZ piezo controller. As a pump source, we are using a 532 nm pulsed laser (Teem Photonics, type STG-03E-1S0) with a maximum energy per pulse of 4.5 µJ and a pulse duration of 500 ps. The power of the laser is controlled with an acousto-optic modulator. Right before the objective, a dichroic mirror is used (Chroma HHQ545lp) in order to filter the unwanted reflections of the glass, in combination with an additional long pass filter (532 nm) in the imaging path.
In the pump path, with the use of an appropriate epi-lens, we can produce a beam spot on the sample with a diameter of approximately 40 µm. The fluorescence signal, after passing the beamsplitter and the long pass filter, is propagating in the imaging path, and can be directed either to a camera (Andor Clara CCD) or to an imaging spectrometer (Shamrock 303i equipped with an iVAC CCD detector). In each path, three mirrors are encountered: a Fourier lens (f2), in focus with the back focal plane of the objective, a
tube lens f3 and an imaging lens f4,5 in order to focus properly the fluorescence signals on
the relevant detectors. By flipping the lens f3 we can have access to the Fourier space or
the real space images for the fluorescence of the samples.
Fourier imaging gives us direct access to the angular dependence of the emission [41]–[44] . When the signal is directed to the Andor Camera, we can record a Fourier image which due to the high NA objective, spans a large range of k|| space. An additional
filter can be used to obtain sharper features from the panchromatic Fourier images (band pass filter: 575 - 620 nm). The exposure time of the Andor camera is set to 30 ms. On the other hand, by directing the Fourier image to the spectrometer, a direct map of the dispersion diagram can be observed, by imaging a "slice" of the Fourier image on the slit of the spectrometer. For the rest of the measurements, we will be measuring band diagrams along the ky direction in the imaging coordinate system, always having centered
