Self Assembled Photonic-Plasmonic Crystals for Light control at the Nanoscale

Universidade de Santiago de Compostela Facultade de Fisica Departamento de Fisica Aplicada Self Assembled Photonic-Plasmonic Crystals for Light control at the Nanoscale Cristais Fot´onico Plasm´onicos Autoensamblados para o control da luz na Nanoescala Memoria presentada para obter o grao de Doutor en Fisica por Memoria presentada para obtener el grado de doctor en Fisica por : Mart´ın L´opez Garc´ıa Tesis dirixida por / Tesis dirigida por: Prof. Ceferino L´opez Fern´andez Dr. Juan Francisco Galisteo L´opez Titor / Tutor: Prof. Vicente Moreno de las Cuevas Consejo Superior de Investigaciones Cientificas (CSIC) Instituto de Ciencia de Materiales de Madrid (ICMM) Xu˜ no 2011 ´ A Sara ´ Os meus pais Summary This thesis is the result of more than five years of research on the study of several kind of photonic and plasmonic systems. Main body of this research has been performed in the Photonics Crystals Group at the Material Science Institute of Madrid, (center belonging to the national research council of Spain-CSIC) under the supervision of Prof. Ceferino L´opez Fernandez and Dr. Juan F. Galisteo Lopez. As component of a experimentalist team, most of the experimental work presented in this thesis has been performed in the laboratories of the group. All the samples shown in this manuscript were grown and modify by different processes in the Photonic Crystal‘s Group chemical laboratories. The whole optical measurements (as well as the set-up for implementation Fourier imaging spectroscopy described in chapter 3) where performed at the optics group‘s laboratory as a main part of this thesis. The metallic films of gold and silver used as part of the samples were deposited by sputtering by Dr. Jorge Sanchez and Dr. Eva Cespedes from the Group of Electronic and Magnetic Materials and Heterostructures at the ICMM-CSIC. The AFM measurements shown in chapter 2were performed with the help of Dr. Jorge Sanchez. The ellipsometry measurements to determine the refractive index of the materials forming the here studied structures were performed at the Institute of Microelectronics of Madrid by Prof. Gaspar Armelles. The FDTD calculations shown all along this thesis are the result of a collaboration with Dr. Antonio Garcia-Martin from the Institute of Microelectronics of Madrid. Finally, i can‘t help to mention some other important work, not directly related with this thesis but which provided expertise directly related with the results here presented. In the first stage of this five years, research on the properties of opal based photonic crystals doped with metallic nanoparticles (gold) was performed in collaboration with the Colloidal Chemistry Group from the University of Vigo lead by Prof. Luis Marz´an. Much of my background on plasmonics come from the investigation on metallic nanoparticles in opals performed within this collaboration. On the other hand, during the last three years, a permanent collaboration was established with the groups leaded by Prof. Anna Roig and Prof. Josep Fontcuberta in the Material Science Institute of Barcelona. In that research, the magnetoptical properties of colloidal photonic crystals infiltrated with magnetic nanoparticles was achieved. Although this issues are not included in this thesis, very useful knowledge on how to improve the optical spectroscopy set-up described in chapter 3was retrieved from that collaboration and then applied to the subject of this thesis. Finally, i have to mention the collaboration with the group of Dr. Jui`an Rodr´ıguez- L`opez of the Faculty of Chemistry at the Universidad de Castilla la Mancha which reported me experience on fluorescence spectroscopy of organic molecules (in particular fluorescence dendrons) distributed within 3D colloidal photonic crystals. The work described in this v vi thesis has been reported in the following conferences and papers: Publications in international journals: •”Intrinsic losses in self-assembled hybrid metallodielectric systems.” J. F. Galisteo- Lopez, M. Lopez-Garcia, A. Garcia-Mart´ın and C. Lopez. Submitted (2011). •”Enhancement and Directionality of Spontaneous Emission in Hybrid Self-Assembled Photonic-Plasmonic Crystals.” M. Lopez-Garcia, J. F. Galisteo-Lopez, J. S´anchez- Marcos, C. Lopez and A. Garcia-Mart´ın. Small 6(16): 1757-1761 (2010). •”High Degree of Optical Tunability of Self-Assembled Photonic-Plasmonic Crystals by Filling Fraction Modification.” M. L´opez-Garc´ıa, J. F. Galisteo-L´opez, C. Lopez and A. Garcia-Mart´ın. Advanced Functional Materials 20(24): 4338-4343 (2010). •”Magnetophotonic Response of Three-Dimensional Opals.” J.M. Caicedo, O. Pascu, M. Lopez-Garcia, V. Canalejas, A. Blanco, C. Lopez, J. Fontcuberta, A. Roig and G. Herranz. ACS Nano 5(4): 2957-2963 (2010). •”New poly(phenylenevinylene)-methyl methacrylate-based photonic crystals.” S. Achelle, A. Blanco, M. Lopez-Garcia, R. Sapienza, M. Ibisate, C. Lopez and J. Rodriguez- Lopez. Journal of Polymer Science Part A: Polymer Chemistry 48(12): 2659-2665 (2010). •”Facile route to magnetophotonic crystals by infiltration of 3D inverse opals with magnetic nanoparticles.” J.M. Caicedo, E.Taboada, D.Hrabovsky, M. Lopez-Garcia, G. Herranz, A. Roig, A.Blanco, C. L´opez and J. Fontcuberta. Journal of Magnetism and Magnetic Materials 322(9-12): 1494-1496 (2009). Communications at international conferences presented by me: •CLEO 2011. Munich (21-26 May) ”Enhancement and tailoring of emisison by hybrid self-aasembled photonic-plasmonic crystals”. M. L´opez-Garc´ıa, J. F. Galisteo-L´opez, C. Lopez and A. Garcia-Mart´ın. Oral •Imagenano 2011. Bilbao (11-14 April) ”Tailoring the optical response of self-assembled photonic-plasmonic crystals”. M. L´opez-Garc´ıa, J. F. Galisteo-L´opez, A. Blanco, A. Garcia-Mart´ın and C. Lopez. Poster •PECS 2010. Granada (September) ”Tailoring the optical response of self-assembled photonic-plasmonic crystals”. M. L´opez-Garc´ıa, J. F. Galisteo-L´opez, A. Blanco, A. Garcia-Mart´ın and C. Lopez. Poster •CEN 2010. Segovia (15-18 June) ”Easy implementation tuning of self-assembled photonic-plasmonic crystals”. M. L´opez-Garc´ıa, J. F. Galisteo-L´opez, A. Blanco, A. Garcia-Mart´ın and C. Lopez. Poster •SPIE Photonics Europe 2010. Brussels (12-16 May) ”Enhanced emission in selfassembled photonic crystals by hybrid photonic-plasmonic modes”. M. L´opez-Garc´ıa, J.F. Galisteo-L´opez,A. Blanco, C. L´opez and A. Garc´ıa-Mart´ın. Oral •Nanospain 2010. Malaga (23-27 March) ”Enhanced emission in self-assembled photonic crystals by hybrid photonic-plasmonic modes”. M. L´opez-Garc´ıa, J. F. Galisteo- L´opez, A. Blanco, C. Lopez and A. Garcia-Mart´ın. Oral vi vii •CEN 2008. Tarragona (2-4 April) ”Opal based photonic crystals heterostructuration by nanoparticle physical infiltration”. M. L´opez-Garc´ıa, P.D Garcia, A. Blanco and C. Lopez. Poster vii Contents Summary v Table of contest ix 1 Introduction 1 1.1 Photonic crystals in 1D, 2D and 3D ...................... 1 1.1.1 2D PC Slabs ............................... 5 1.2 Surface Plasmon resonances .......................... 12 1.3 Hybrid Plasmonic-Photonic structures ..................... 15 1.4 Spontaneous emission enhancement by photonic, plasmonic and hybrid structures ..................................... 16 1.5 Outline of this thesis ............................... 19 I Experimental 21 2 Fabrication of monolayers of organic spheres 23 2.1 Introduction .................................... 23 2.2 Fabrication method ................................ 24 2.2.1 Vertical deposition method ....................... 24 2.2.2 Wedge-shaped cell method ....................... 28 2.3 Post-fabrication lattice modification ...................... 31 2.3.1 Oxigen plasma etching .......................... 31 2.4 Conclusions .................................... 35 3 Optical characterization by Fourier image spectroscopy 37 3.1 Introduction .................................... 37 3.2 Optical Fourier transform for angular resolved measurements ........ 39 3.2.1 Reflection Fourier imaging ........................ 41 3.3 Set-up description ................................ 42 3.3.1 Microscope output ............................ 44 3.4 Extracting data from the Fourier Image .................... 45 3.5 Application to photonic crystals: spectroscopy of photonic modes ...... 52 3.6 Conclusions .................................... 54 ix Chapter 1. Introduction 4 experimental and theoretically [10][11]. They have also been used in many applications as for example hollow optical fibers [12], sensors [13] or solar cells [14]. However, concerning the confinement of light, 1D PCs are not the most suitable structure given that they act as a homogenous medium for light propagation in the other two directions perpendicular to the periodicity. Unlike the 1D case, 3D PCs can provide light confinement in all directions which can lead to new fundamental phenomena. The first to be proposed were the inhibition of spontaneous emission by means of a PBG [2] and 3D Anderson localization in the optical regime [3] although many other effects originating in the periodicity of the medium have been demonstrated as, for example, superprism effects [15]. Other important difference with respect to the 1D case is that a 3D periodicity allows changes in both symmetry an topology of the lattice. This introduces versatility in order to find a structure matching required photonic properties. Therefore, each different structure presents its own symmetry directions for the reciprocal lattice and hence a 3D FBZ which will depend on the chosen lattice. Also, plenty of structures with different building blocks forming the lattice have been studied and fabricated during the last two decades. Some of the most common fabrication techniques are photolithography [16], focus ion beam (FIB) [17], holographic lithography [18], direct laser writing, [19], direct ink writing [20] or self assembly [21]. Self assembly methods take advantage of the natural tendency of monodisperse colloidal particles to form ordered arrays under the appropriate conditions. Since they were first studied as 3D PC in the mid 90‘s [22], self assembled 3D PCs have become one of the most widespread PC fabrication methods. The resulting structures are known as artificial opals and consist of (mainly) face centered cubic (FCC) arrays of polymeric or inorganic spheres. Their porous nature makes them an appropriate structure to control its optical properties through infiltration with different materials within the lattice and subsequent removing of the original spheres leading to inverse structures. Artificial opals are PCs whose building blocks are dielectric spheres with diameter (Φ) on the order of the operative wavelength and lattice parameter a=p(2)Φ, immersed in a matrix of a different refractive index (typically air) (see Fig 1.1). This kind of structures have been experimentally shown a 3D PBG [9]. During the last years they have shown to be a good playground to understand light propagation in 3D PCs but also their utility for specific applications as, for example, biomedical applications [23], solar cells [24] or light emitting devices (LEDs)[25]. Figure 1.2: Artistic picture of a set of infinite high refractive index material in air forming a square lattice. Band structure for TM modes and FBZ. Dispersion relation taken from [26] An intermediate point in terms of light confinement is that of 2D PC. We will pay special attention to this system as a first approximation to the structures studied in this 4 5 Chapter 1. Introduction thesis. The structure most commonly used to describe 2D PCs is a set of cylinders of infinite height with a refractive index (RI) different from that of the matrix they are immersed in. Fig. 1.2 shows such structure with a square lattice. An example of photonic bands for wavectors propagating in the plane of periodicity is also shown. If the right RI, filling fraction, contrast and lattice parameter are chosen a PBG can open at a given spectral range. This system can be represented as a homogenous medium in the vertical direction and a periodic 2D lattice for light propagating in the plane. It is worth defining here the two types of of modes present in these structures: transversal electric (TE) and transversal magnetic (TM), corresponding to those modes having E or Hfields contained in the plane of periodicity. Therefore, for the case where the wavector is fully contained in the plane of periodicity (k(x, y)), the structure presents well defined TE and TM modes. Although the band structure for TE and TM can be completely different, it is also possible to find a complete PBG, that is, PBG for TE and TM at the same frequencies [6]. However, if kz6= 0, the mirror symmetry is broken and there no longer exists a well defined difference between TE and TM modes. For the same reason, for out of plane propagation no gaps can exist. As a result of the discussion above, these systems can not be used to confine light efficiently since there is no confinement mechanism in the vertical direction and hence light propagating with component κzwill scape the system. Therefore, a mechanism for light confinement in the vertical direction needs to be incorporated to the 2D PC. The most efficient approach to solve this is the use of a design where the non periodic dimension is limited to a thickness of approximately the operating wavelength so that total internal reflection (TIR) can take place in the vertical direction similar to conventional waveguides. Such systems are called PC slabs or planar PCs. 1.1.1 2D PC Slabs The simplest optical waveguide structure is the step-index planar slab waveguide which represents a starting point to model the kind of confinement obtained in 2D PC slab. It consists of a high refractive index dielectric layer (core) (f) surrounded by lower-index materials known as substrate (s) and cladding (c). A diagram of such waveguide is shown in Fig. 1.3. Figure 1.3: Diagram for vertical confinement by total internal reflection for an asymmetric planar waveguide. TE and TM modes are indicated. If f> s, cTIR takes place and light can be trapped inside the slab. Taking into 5 Chapter 1. Introduction 6 account the coordinate system defined in Fig. 1.3 light will zigzag (in a simple ray tracing model) down the zaxis impinging on the interfaces at angles larger than the critical one given by: θTIR =arcsin(rj f )j=s, c (1.5) Taking into account the previous definition for TE and TM modes, Fig. 1.3 shows that, for the coordinate system under consideration here, E=Eybyfor TE while Ey= 0 for TM modes. In a source-free, linear and isotropic medium Eand Hsatisfy the following equations: ∇2E−µ∂2E ∂t2= 0 (1.6) ∇2H−µ∂2H ∂t2= 0 (1.7) We assume the waveguide to be excited by a source with frequency ω0, wavector in vacuum |k0|=ω0/c and polarization along the yaxis (TE). The wave equation in 1.6 can be written in a scalar form for each material in the waveguide: ∇2Ey−k2 0n2 iEy= 0 (1.8) where ni=nf,nsor ncdepending on the region of the structure we consider. As the slab is infinite in the y direction, which means light is not confined in that direction, the trial solution we can write is of the form: Ey(x, z) = Ey(x)e−jβiz(1.9) where βiis the propagation coefficient along the z direction in each region of the slab, related with the total wavector modulus |k0ni|as shown in Fig. 1.4. Figure 1.4: Wavector nomenclature used in the text. βand kzare respectively the longitudinal and transversal components of the wavector k. Using this solution in equation 1.8 we obtain the general wave equation for TE modes: ∂2Ey ∂x2+ (κ2 0n2 i−β2)Ey= 0 (1.10) and introducing the boundary conditions given by the refractive index distribution along the xaxis shown in Fig. 1.3 we obtain: Ey(x) = E0e±√β2−κ2 0n2 ix(1.11) 6 7 Chapter 1. Introduction where E0is the electric field at x= 0. To be physically meaningful one has to consider the negative solution. Therefore, the transverse wave will have an oscillatory behavior if β2< k2 0n2 i(ni=nc,nsand nf). However, if β > κ0nithe solution is exponentially decaying in the corresponding medium. If we extrapolate this general result to the infinite slab waveguide case with nf> ns> ncthree situations can be defined depending on the value of the z direction propagation wavector β. Figure 1.5: Diagram for the three kind of modes available in the planar waveguide depending on their effective propagation wavector β. Figure based on reference [27] The possible situations are represented in Fig. 1.5. We have plotted a simplified ray picture for each case (bottom) while the top diagram shows the wave picture. For β < κ0nc, solutions are oscillatory in all regions (radiative modes) since no TIR happens at any interface and no confinement can then take place. For k0nc< β < k0nsthe exponential decay (equivalent to TIR) take place just at the cladding-core interface. As plotted in the ray picture, light is total internally reflected at that interface while it refracts at the substrate-core interface. We call this type of mode semiradiative and can be thought of as leaking trough the substrate from the guiding structure. The third case is the one we are interested in and takes place when k0ns< β < k0nf. As shown in Fig. 1.5, for this situation the TIR condition is satisfied at both interfaces while it remains oscillatory inside the core. This solution represents the guided modes of the slab which have been depicted as a set of rays trapped between the two interfaces in Fig. 1.5. Furthermore, the general condition to obtain a guided wave in a dielectric slab is a universal condition regardless of its geometry which will be important for us in terms of light confinement in the kind of structures under study in this thesis. Once the different regimes for light propagation have been described two questions remain: which are those values of βsatisfying the TIR condition and how can a propagating wave in a surrounding medium be coupled to the guided modes of the slab and viceversa. The values that βcan take for each frequency define the dispersion relation of the waveguide. If one imposes the boundary conditions and solves the coupled equation system one obtains an eigenvalue problem for both TE and TM modes: tan(hkf) = γc+γs kf[1 −γcγs k2 f ]TE (1.12) 7 Chapter 1. Introduction 8 tan(hkf) = kf[n2 f n2 sγs+n2 f n2 cγc] k2 f−n4 f n2 cn2 sγcγs TM (1.13) with γs=pβ2−k2 0n2 sand γc=pβ2−k2 0n2 cbeing the exponential decay coefficients in cladding and substrate respectively. kfis the transverse component (x direction) of the wave-vector within the guide. These two eigenvalue problems (known as characteristic equation) for a slab waveguide are transcendental and therefore have to be solved numerically or graphically. Their solution provides the eigenvalues βT E and βT M that correspond to the allowed modes in the system. These equations finally give us the relation (β, ω) for each mode. However, in this thesis a normalized frequency ω= (4π)/(p(3)λ0) will be used with λo being the wavelength in vacuum. Choosing this normalization has to do with the main symmetry directions of a hexagonal lattice and will be explained in depth in chapter 4. We can now plot the solutions of equations 1.12 and 1.13 in order to find some fundamental properties of guided modes in the above structure. Fig. A.1 shows an example of calculation of three modes for a symmetric slab waveguide. Guided modes can only exist below the light line defined by the condition for TIR at the boundary between the core and the surrounding material with lower refractive index. Above this line there is a continuum of radiative modes known as the light cone (shown as a shaded area in Fig. 1.6a). (a) (b) Figure 1.6: a). Dispersion relation for T E00,T E01 and TM00 modes of a symmetric waveguide with ns=nc= 1 and nf= 1.38. The thickness of the core is h= 0.52 µm. The shaded area represents the light cone. b) Intensity profile for TE00 and TE01 modes of the same structure in a) at ω= 0.75 (λ0= 0.6µm). For guided modes, another relevant parameter is the modal confinement. That is, how the electric is field distributed inside the structure for each mode. Fig 1.6b shows a calculation 2of the intensity profile for the TE00 and TE01 modes for normalized frequency ω= 0.75 for the same structure used in Fig. 1.6a. It shows the field intensity distribution along the vertical direction of the waveguide with the expected oscillatory profile inside the 2The calculation was performed using the on-line software provided by Dr. Manfred Hammer on the web [28] 8 9 Chapter 1. Introduction core and an exponential decay in the surrounding media. A useful parameter to quantify light confinement for a given mode is the normalized effective thickness, defined as: Π = k0heff qn2 f−n2 s(1.14) with heff =h+ 1/γs+ 1/γcand γs,c =qβ2 s,c −k2 0n2 s,c (exponential decay parameters in substrate and cladding). Π is inversely proportional to the field confinement properties for a given mode. Hence, if one is looking for efficient light confinement one must seek low Π values which take place for low order modes and systems having large differences between modes in nfand ns. From guided to leaky modes in periodic slab waveguides So far we have seen how slab waveguides can sustain truly guided modes. Here we will see how, by using a 2D PC structure, one can strongly modify such guided modes. This patterning introduces modifications in the dispersion relation of the guided modes bringing them above the light cone and allowing this way the coupling to radiative modes. We next present a simple picture of the effects of coupling a grating to the slab as shown in Fig. 1.7a. As already mentioned, exciting guided modes from outside the guiding region can not be done directly. Among the many methods available for that purpose [29], the most widespread is the use of a periodic corrugation (usually a grating) over the dielectric slab. (a) (b) Figure 1.7: a) Diagram of guided mode coupling/decoupling in a grating topped slab waveguide. b) Folding back into the FBZ of the dispersion relation of Fig. 1.6a. Chosen grating period is 400 nm. 9 Chapter 1. Introduction 10 Let‘s consider a grating of period dcoupled to the slab (see Fig. 1.7a). Such corrugation adds the extra momentum needed for coupling between the light cone and guided modes by adding a reciprocal lattice vector to the incident wave-vector. Following the same nomenclature used so far we can now express the coupling condition of the parallel component of the wavector: β=kin || ±m|G|(1.15) where kin || is the component of the incident wavector parallel to the slab‘s mode propagation. Gis the reciprocal lattice vector along the same direction as kin || and takes a value |G|= 2π/Λ where d is the periodicity of the grating. Therefore, the original guided modes are no longer totally confined but loose some energy through radiation to the surrounding media as they propagate. Hence they are called leaky or quasi-guided modes. In terms of light confinement, leaky modes present a finite lifetime for propagation inside the waveguide due to leakage which is determined by the efficiency of light coupling/decouppling by the periodic structure to the slab. It depends on several factors such as the refractive index contrast or the shape of the periodicity building blocks. The parameter that measures how long a mode propagates before leaking out of the structure is the Quality factor (Q). This parameter will be further discussed in chapter 4when applied to the specific structure studied in this thesis. The field intensity inside the structure is also expected to be modified (according to the periodic vertical thickness) with respect to the guided mode of the unstructured guide. Let‘s now study the effects of the periodical patterning on the in-plane propagation modes. From the above discussion, a complete description of the dispersion of the modes of the system can be extracted from the folding of each of those dispersions to the FBZ. The simplest way to obtain the folded dispersion relation is the use of the so-called empty lattice approximation. In this approach, the folding is performed just by translation of the kvector to the FBZ by subtracting the corresponding lattice vector G. Fig. 1.7b shows the result of applying that folding to the modes of a slab waveguide with a corrugation of period Λ in the direction of propagation. However, the empty lattice approximation fails to fully explain the effect of the grating in the propagation of a leaky mode. In a more accurate picture, a leaky mode is multiply scattered during propagation which leads to the formation of Bloch modes for which stop bands can developed. Such photonic gaps will depend on the scattering-strength of the periodic lattice or its symmetry properties. Calculation of the exact shape of the photonic bands requires numerical simulations. If instead of a grating, one considers a 2D periodic patterning along the whole waveguide thickness, strong light propagation control and confinement can be achieved. While coupling from outside the structure is also available due to modes lying above the light cone, the dispersion relation can be further engineered in every direction of propagation in the plane of periodicity as previously explained for infinite thickness 2D PCs. Therefore, these kind of structures provide TIR confinement in the vertical direction combined with 2D PCs properties (as PBGs). They are known as PC slabs. In order to obtain the photonic bands of a 2D PC slab, Fan and co-workers demonstrated that the empty lattice approximation works appropriately [30] [31]. Fig. 1.8a presents an example of such an approximation. In this case plots have been extracted from reference [31]. The modes of a slab were calculated with the above expressions, and folded back into the FBZ along the ΓXdirection in reciprocal space. As can be observed, the empty lattice model does not show interaction between different modes which, spe- 10 11 Chapter 1. Introduction cially for large refractive index contrasts, produces the most important optical features in PCs. Therefore, for an accurate description, it is necessary to use numerical methods as for example the guided-mode expansion method [32] or Finite difference time domain (FDTD) as shown in Fig. 1.8b. In this case, important differences are evident with the empty lattice. The main features of PCs are now evident such as PBGs at the FBZ edge ω∼0.3 or flat bands at the photonic gap edges. FDTD also shows that, in some cases, anticrossings take place for the real case evidencing the effects of multiple scattering at those points where bands cross for the empty lattice. (a) (b) Figure 1.8: a) Band structure for TE modes in a uniform dielectric slab (nf= 2.282 and nc=ns= 1) folded back into the FBZ of a 2D square lattice of lattice constant a. b) Photonic bands (TE modes) for the PC slab made by the slab shown a) with a 2D lattice of cylinders (radius 0.2a) drilled on it. Both plots taken from [31]. As a result, PC slabs have shown to be the most efficient 2D type of PC concerning light confinement [33]. Besides, the experimental implementation of this kind of structures has taken advantage of microelectronic fabrication methods to obtain high quality PCs. These two facts, have allowed PC slabs to achieve an optical performance which has rendered them ideal for applications in many fields of photonics such as slow light waveguides [34], optomechanical systems [35] , high efficiency solar cells [36] or ultra high Q cavities [37] to name a few. PC slabs with complex building blocks As shown above, PC slabs present a very efficient light confinement in the vertical direction by means of TIR. However, TIR conditions are strongly dependent on the refractive index profile in the vertical direction. Therefore, any complex shape introduced on the vertical profile of building blocks will introduce changes in the light confinement properties of the system. The most simple example of this effect are slot waveguides, where the electric field is mainly confined within a low refractive index sub-wavelength region introduced in the core of the waveguide [38]. But PC slabs can also be constructed from individual building blocks ordered in a 2D array rather than from holes drilled in a planar waveguide (WG) For the particular case of a 2D lattice formed by close packed spheres, the most accurate theoretical approach makes use of the Mie theory to calculate the scattering of each single sphere to next couple them in the periodic lattice configuration. This approach was widely studied by Othaka and Miyazaky [1][39] and has proven to correctly describe the optical properties of this kind 11 Chapter 1. Introduction 12 of systems. In this thesis we also deal with hybrid systems where monolayers of spheres are deposited on a metal, which can not be easily described by such an approximation. Although several methods [40] can be used to describe the monolayer metal interaction we have used mainly FDTD (see Appendix A) simulations in this thesis. 1.2 Surface Plasmon resonances SPRs are light waves trapped at the surface of a metal due to the resonant oscillation of its free electrons [41]. The most attractive property of this kind of modes for light propagation is their ability to confine light at a metal-dielectric surface at subwavelenght scales [42]. Such a confinement provides extremely large light-matter interaction which makes systems supporting SPRs the most appropriate to be applied in many different fields of photonics. Some of the best known fields of application for SPR´s are sensors [43] or medicine [44]. If a metal-dielectric interface is considered one can solve Maxwell‘s equations at the interface between two media with 1and 2with the following boundary conditions: •Propagation is fixed along the interface. •Confinement in the normal direction to the interface. Hence, the field intensity must decay exponentially towards both media. Using these two conditions, we can make use of the same equations shown for the slab case but with h= 0 (zero thickness). In that case, for Γ1,2>0 (exponential field decay in medium 1 and 2) it is found that just the TM zero order solution satisfies field continuity at the interface for a light wave propagating along it [41]. With the previous boundary conditions it is obtained that the wavector of the propagating modes must satisfy: κ2 || =12 1+2 κ2 0=12 1+2 ω2 c2(1.16) where k|| is the component of the wavector parallel to the metal surface and k0is the wavector in vacuum at a given wavelength λ0. For propagation along the normal direction: κ2 j,z =j 1+2 κ2 0j= 1,2 (1.17) where κj,z is the wavector component along the normal to the surface within the jmedium. The same nomenclature can be applied to 2. Propagation along the surface requires a real κ|| while the confinement in the normal direction is obtained for imaginary κz. From this condition it is found that one of the dielectric constants must be negative. As it is well known, noble metals, and specially gold and silver have large negative real parts of the dielectric constant along with a small imaginary part and therefore are good candidates to sustain long lived SPRs at their interface. We can next assume a more realistic scenario, with 2being a dielectric medium with negligible absorbtion. For 1we include the imaginary part of a metal in order to take into account losses introduced by the resonances due to electron scattering (ohmic losses): 1=0 1+i00 1(1.18) leading to an imaginary wavenumber where the real part accounts for the phase and the imaginary part for the absorbtion during the plasmon propagation along the interface. If 12 13 Chapter 1. Introduction we assume that |00 1|<< |0 1|, after some further calculations we obtain the wavelength for the plasmon resonance: λSP R =2π k0 || ≈s0 1+2 0 12 λ0(1.19) (a) (b) Figure 1.9: a) SPR dispersion relation and (b) field intensity decaying into each material of the metal-dielectric interface. In this case n=1.38. Fig. 1.9a represents an example of the dispersion relation for a SPR for gold with a dielectric material with n=1.38 deposited on it. As for the guided modes of the PC slab presented in the previous section, the SPR dispersion relation lies below the light cone (grey shaded area) and then it is not possible to couple to it without adding an extra momentum to the incident beam. Prior to introducing the different approaches to couple to SPRs other important points should be considered, such as its propagation distance. Again, similar to slab waveguides, the field confined for an SPR presents an exponential decay into both media forming the interface. Fig. 1.9b shows a calculation of the field distribution into each of the materials considered for the dispersion relation in Fig. 1.9a. For SPRs the exponential decay is always faster in the metallic part than in the dielectric due to the values of for each material. On the other hand, due to the ohmic losses introduced by 00 1, as the SPR propagates along the interface the electric field will decay by a factor 1/e after a given propagating distance. Increasing that distance without affecting the other properties of SPRs is essential for many applications. As an example we can consider the dielectric constant for gold at λ= 633 nm (1=−11.6 + 1.2i) and air (2= 1). In this case we obtain a 1/e propagation distance of ∼10µm and a decaying of 1/e in the normal direction of 28 nm and 328 nm for gold and air respectively. This shows that the decay into the metal is much shorter than into the dielectric as expected from the previous discussion. As mentioned at the beginning of this section, the extremely large field confinement makes SPR suitable for many applications. Besides the extremely short effective wavelengths of SPR allow for novel phenomena such as subwavelength focusing [45] or supertransmission [46] in reduced scales and makes SPR supporting systems ideal platforms to integrate photonic devices [47]. However, as mentioned above, two main issues have to be solved. On the one hand, in order to couple/deouple light to this kind of modes, the momentum mismatch with the radiative modes of the outside world has to be overcome. On the other hand, it is mandatory to reduce losses due to absorbtion of the metal to achieve large propagation lengths. For the latter, many approaches have been proposed 13 Chapter 1. Introduction 20 •In Chapter 3we describe the optical experimental setup based on Fourier Imaging angle resolved spectroscopy used for the optical measurements in this thesis. First we introduce some notions on Fourier imaging. Next, the set-up is described and a complete calibration protocol is detailed. Finally we demonstrate its accuracy by performing angle-resolved measurements on opal based 3D PCs. •In Chapter 4we study from a theoretical point of view the optical properties of monolayers of dielectric spheres for free standing, dielectric and metallic substrate structures. By means of FDTD simulations we obtain the optical response (normal incidence reflectance) and the total field intensity distribution for the different modes of the structure which allow us to divide them between WG-like and SPR-like. •In Chapter 5we study the different channels of losses for a monolayer deposited on a metallic substrate. We divide them in intrinsic losses (unavoidable and related with the leaky character of the modes and with the metal absorbance) and extrinsic losses(related with induced disorder during the fabrication process). •Chapter 6presents angle and polarization resolved spectroscopy measurements performed on monolayers grown on dielectric and metallic substrates. Well resolved dispersion relations are obtained for both gold and silver substrates along the two high symmetry directions of the reciprocal lattice. Experimental results are interpreted with an empty lattice aproximation and the physical origin of the different modes is studied. Finally, we study experimental equifrequency surfaces (EFS) in a reflectance configuration. •In Chapter 7, we study the emission properties of a close-packed monolayer of dye doped spheres. It is shown how spontaneous emission of internal sources is strongly modified, evidencing a spatial and spectral redistribution of the DOS. EFS are also collected in this case and compared with the reflectance configuration corresponding images. •Finally in Chapter 8we show how the post-processing technique presented in chapter 2can be employed to tailor the optical response of the structures. We show that by reducing the diameter of the spheres while keeping the lattice parameter constant it is possible to accurately tune the spectral position of the resonances of the system. The evolution of the optical response both in reflection and emission is studied both numerically and experimentally as the sphere diameter is reduced. 20 Part I Experimental 21 Chapter 2 Fabrication of monolayers of organic spheres In this chapter we study the two different sample fabrication methods used in this thesis: vertical deposition and wedge cell method. They allow us to obtain high quality monolayers of spheres on top of metallic or dielectric substrates. A morphological study of the disorder of samples fabricated over gold substrate is presented. In the last section we show results on the modification of the filling fraction of the 2D lattice by means of oxigen plasma etching. 2.1 Introduction As already mentioned in earlier chapters, one of the main advantages of self assembled photonic crystals (PCs) is their simple and fast fabrication procedures when compared to the usual microelectronics processes used for silicon (Si) based PCs. The well known high refractive index (RI) slab photonic crystals [84] are the best known example of such differences. Regarding this interest on self-assembly photonic crystals working in the ultraviolet (UV) - visible (VIS) and - infrared (IR) spectral regimes, many techniques have been developed during the last two decades to grow both two (2D) and three dimensional (3D) structures using nano and micro spheres as building blocks. Ever since Stober [85] obtained monodisperse inorganic spheres and Philipse et al [86] assembled them into a photonic structure, a large variety of new materials (organic and inorganic) and many assembly methods have been developed, for example: vertical deposition [87], electric field induced assembly [88], robotic manipulation [89], Langmuir-Blodgett deposition [90], spin coating assisted assembly [91] [92] or more recently wedge-cell assembly method [93]. It is important to notice that the natural tendency of a colloidal suspension is to order into hexagonal close-packed (HCP) lattices in 2D or face centered cubic (FCC) type structures in 3D. These two configurations are the most stable ones after dispersion evaporation and have therefore been the most widely studied. However, other periodicities can be forced offering more possibilities [94] [95] [96] if the growth process is directed with patterned substrates. In this chapter we present the results obtained applying two of the aforementioned techniques to grow monolayers of spheres on different substrates, specially on metallic ones. In the first part, we describe the two methods used and the main parameters influencing sample growth. Results on the ordering of the samples are obtained by morfological 23 Chapter 2. Fabrication of monolayers of organic spheres 24 study. Special attention has been pay to on how to hydrofilize the substrates given that very different processes have to be applied depending on whether the substrate is dielectric or metallic. On the last part we present an in depth study on the modification of the filling fraction of the ordered array by means of oxigen plasma etching whose effects on the optics of the sample will be studied in chapter 8. 2.2 Fabrication method As just mentioned two methods have been used to fabricate the samples studied in this thesis: vertical deposition and wedge-cell. We have studied and compared them, and several differences in the quality of the samples have been found. Vertical deposition is a more versatile method (almost not limited by the substrate total area) while wedge-cell shows better results concerning quality of the sample but with a more limited substrate shape. Some of this conclusions have been previously reported by Sun et al. [93]. However, we have extended this study to metallic substrates. Next, samples are presented and morphologically studied. 2.2.1 Vertical deposition method Vertical deposition has become the most widespread method to grow self-assembled photonic crystals, specially when a thick sample is to be grown (as for instance opaline structures). Nagayama et al. [97] first developed this method for monolayers and it has been extended to 3D opal based PCs, specially since 1999 [87]. The standard procedure to grow a monolayer consists in placing a flat substrate in a vial (typically a vial) containing the colloidal suspension of monodisperse spheres (see Fig. 2.1 ). A meniscus is formed at the interface between substrate, air and liquid. At the point where the meniscus is thinner than the diameter (Φ) of the spheres theses are pulled against the substrate and between them by capillary forces and the ordering process begins [98]. As evaporation takes places, the flow of solvent to the meniscus moves the spheres on the extra liquid to the growth region and incorporates them to the lattice. As the liquid of the colloidal suspension evaporates the lattice of spheres remain over the substrate forming the self-assembled structure. If the concentration of spheres is high enough, not a monolayer, but a 3D structure will be grown whose total amount of layers will be mainly dependent on the concentration of the dispersion [98]. . As can be deduced from the mechanism explained above, the main parameters determining the appropriate growth are, on the one hand, the colloidal suspension concentration and on the other hand the temperature and humidity of the environment. These account for the growth velocity of the lattice and the first one determines the thickness of the main structure though all parameters have some effect on growth velocity and sample quality. By choosing the right conditions it is possible to select if one (2D) or several layers (3D) will be obtained for a given sphere diameter [87]. This general picture can be applied with to inorganic or organic colloidal particles. Although the first works on opal growth were performed with silica (SiO2) spheres, polymers have shown better monodisperstity. Furthermore, due to the lower density of PS as compared with silica the former is more suitable to be used with large diameters (¿ 500 nm) given that sedimentation of the spheres before the growth process has finished is an issue with silica spheres opal growth. 24 25 Chapter 2. Fabrication of monolayers of organic spheres Figure 2.1: Diagram of the vertical deposition method. As we will make use of 2D organic systems in this thesis, we will focus on the appropriate conditions for the polymeric structures. The quality of the samples obtained by this method and, in general, for self-assembled PCs refers to how much the real structure differs from the ideal lattice formed by identical building blocks placed at exactly the same distance in the structure. From this definition, there can coexist many types of disorder in the real structures. Most common experimental sources of disorder are vibrations during the evaporation or temperature changes that can lead to different thickness in the sample and produce separations between different domains of the structure (craks). However, if steady environmental conditions are assumed, polidispersity of the spheres and substrate roughness become the two main parameters which affect the crystaline quality. Although controlling the substrate surface can force the colloid to organize following a pre-defined pattern [95] [94], it usually happens that unwanted defects on the substrate surface introduce distortions during the growth process that may end up in lattice disorder. For a more general study we have employed four different materials as substrate: silicon , silica, gold (Au)and silver (Ag). Silicon and silica have been used before as substrates for many different kinds of self-assembled structures, from monolayers [99] to inverse 3D opals [9]. For the case of metallic substrates, we used gold and silver thin films grown by magnetron sputtering over silicon and glass substrates 1. As we used silicon wafers as support for the metal, very high smoothness (around 1 nm) due to the conformal growth of the metallic layer on the silicon surface was obtained. Concerning the thickness of the metallic films, they were characterize by ellipsometry, Scanning Electron Microscopy (SEM) and Atomic Force Microscopy (AFM). Large homogeneity of ∼60nm thickness of the 1Sputtering was performed by Dr. Jorge S´anchez-Marcos in the laboratories of the ICMM-CSIC 25 Chapter 2. Fabrication of monolayers of organic spheres 26 deposited layers were obtained over areas of 2 cm2. Substrate hydrophillization The commercial substrate surfaces we are using (silicon and silica) have very small roughness (less than 1 nm) as compared to the sphere diameter. Therefore, in order to obtain a high quality sample, the main source of disorder concerning the substrate comes from the growth process. From the growth process decribed before if follows that the meniscus where the lattice starts forming has to be positive and always thinner than the sphere diameter during evaporation. To ensure this behaviour the substrate must present a homogeneous hydrofilic surface. For the dielectric substrates the hydrofilization process is well stablished. In our case, silica substrates were clean glass microscope slides. They were hydrofilized by placing them in HCl 33% overnight and then rinsing them with de-ionized water. Finally they were dried in a nitrogen flow. Silicon wafers 2were first introduced in a 1% HF solution for 10 minutes in order to remove the native silicon oxide formed on the surface due to air contact. After that, the wafers were introduced in a H2SO4/H2O2 solution (3:2 vol) for 30 minutes. Again rinsing with de-ionized water and drying with a nitrogen flow were the last steps. For the metallic substrates more complex procedures were needed to achieve a homogenous hydrofilization. We used a temperature controlled oxigen plasma etching which is a cleaner technique than those based on wet chemical reactions. For the gold case we introduced the thin films on the oxygen plasma etching working at room temperature and 0.4 mbar pressure for 30 minutes. The gold surface quality showed no degradation after that process and a good hydrofilization was obtained 3As can be assessed from AFM images in Fig. 2.2a very low roughness is observed after the whole fabrication process. (a) (b) Figure 2.2: a) AFM image of the surface for the silver films after oxygen plasma etching. b) AFM for gold substrate after the same process. As an alternative to gold as metallic substrate we have also used silver. Owing to the 2Silicon wafers provided by ACM 3Bulk gold is oxidation-resistant in air even at elevated temperatures. However, Au2O3can be prepared with radicals provided by an oxygen plasma [100] obtaining the hydrophilic surface character we are looking for. 26 27 Chapter 2. Fabrication of monolayers of organic spheres low absorbance of silver in the visible it suits better our purposses for that spectral range while gold has better optical performance in the IR. However, silver, is not as oxidization resistant as gold. Its surface degrades after some days when exposed to air. In order to avoid oxidation after hydrofilization it is necessary to protect the metallic film from air contact. In our case, a thin (6 nm) Si3N4film was deposited on top of the silver film previously deposited by sputtering on a silicon wafer. Several oxygen plasma etching conditions were tested to obtain the hydrofilization of the protective film while avoiding silver film oxidation. It was found that a 30 second treatment at room temperature in the same conditions of pressure and oxygen purity as for the gold films was enough to obtain a good hydrophilic surface. The resultant substrate did not show any sign of degradation one month after the plasma treatment. Larger oxygen plasma treatments leads to oxidation of silver surface and a fast degradation of the optical properties (as well as the surface quality) was observed. Colloidal suspension Opposite to, for example, silica ones, PS spheres present very high monodispersity with diameter deviation from the mean value below 2%. In our experiments, commercially available [101] [102] PS and polymethylmethacrylate (PMMA) spheres were used. Among the polystyrene spheres, two kind of suspensions were used, on the one hand pure polystyrene ones with very high quality, on the other hand colloidal suspensions of dye-doped polystyrene spheres in order to obtain active samples that will allow emission studies as will be shown on chapter 7. The latter spheres are homogenously dopped with a dye emitting at VIS wavelengths with a broad emission/absortion band 4. Although no differences on the growing conditions were needed for this kind of spheres, a slightly larger polydispersity (∼3%) has to be taken into account for the dopped ones. The liquid where the particles are dispersed is water in order to avoid agregation of the particles. Furthermore, the use of water provides an easy way to manipulate the meniscus properties as it was shown in the previous section. In most of the samples fabricated during this study we have used polystyrene spheres of two diameters: 520 nm and 1 micronmicrometer. However, other diameters were tested too ranging from 250 nm to 2 microns. It is important to notice that due to the sphere fabrication process, the best monodispersity value for PS and PMMA is typically obtained for diameters between 400 and 900 nm. The effect of polydispersity on the crystalline quality of self-assembled systems has been studied in the past [103]. For the case of 2D systems it is well known to be responsible for the introduction of different types of defects which can degrade the optical properties such as dislocations, rotated domains or cracks. By using a temperature/humidity controlled chamber we have applied the same environmental conditions for the growth of monolayers that previously showed good results for artificial opals. That is, humidity values where set to 30% and temperature was 45oC. The hydrofilic substrate was introduced overnight in a vial with 5 ml of colloidal suspension. In order not to get a thick opal but a monolayer we reduced the concentration (in weight) of the dispersion to 0.08 % while the growth of artificial opals usually requires concentrations above 0.1 %. The angle between the substrate and the surface of the dispersion which determines the meniscus size is a critical parameter too so we used the same configuration typically used for opals [104] where the substrate forms an angle of aproximately 20owith the surface of the suspension. 4Optical properties of these spheres are presented on chapter 7 27 Chapter 2. Fabrication of monolayers of organic spheres 28 Figure 2.3: SEM image of a monolayer of Φ = 520nm PS spheres fabricated by the vertical deposition method over a silicon substrate (a) and spatial Fourier transform of two different regions the sample (b and c). Fig. 2.3 shows a scanning electron microscopy (SEM) image (a) and the Fourier Transform (FT) of two different areas of the image (b and c) for a monolayer of spheres grown by vertical deposition. By observing these images we conclude that the samples obtained by this method present a good crystaline quality over short distances (∼20µ2). Inspection by SEM shows that random vacancies are present in all the samples but the lattice itself is closepacked. However, when large areas are inspected it is clear that several domains exist separated by abrupt rotation. It is clearly observed if one compares the two fast fourier transform (FFT) images in Fig. 2.3b) and c) where two lines have been plotted showing the orientation of the corresponding region (full line) and the comparison with the other area orientation axis (dashed line). The hexagonal pattern due to the lattice symmetry denotes the good crystaline quality at short distances as already commented. However, the rotation between the hexagonal pattern for each region tell us that the complete lattice is formed by slightly rotated clusters of spheres. This has direct implications on the optical response of the sample. So, if local measurements (just one cluster) are to be carried out, the quality provided by this method is good be enough. However, if measurements or applications where large areas of sample are necessary (for example gain length measurements) other fabrication methods should be used. 2.2.2 Wedge-shaped cell method As shown before one of the main disadvantages of the vertical deposition method for monolayers is the rotation between relatively small domains. In order to improve this point a new method was recently published by Sun et al. [93] providing a simple way for obtaining large area monolayers with single domain lattices. Although they optimized it for 1 µm spheres and dielectric substrates, we have improved it to obtain large monolayers on the different substrates we are interested in and with the appropiate sphere diameter in each case. In this method a wedge cell as the one shown in the diagram of Fig. 2.4 is used. The wedge shaped cell consists of two contact surfaces (a glass slide and the substrate itself) held at a given angle (θ) chosen according to a previous calibration for the substrate ma- 28 29 Chapter 2. Fabrication of monolayers of organic spheres (a) (b) Figure 2.4: Diagram (a) and real structure (b) used to grow large area monolayers by the wedge cell method. terial and the sphere diameter. We obtained that an angle of θ= 2oworks for most of the cases, including metallic substrates. As in the vertical deposition method, both surfaces have to be hydrofilic in order to obtain a good meniscus. Hydrofilization techniques already described before were also applied in this case. The colloidal suspension is then introduced through one of the open sides of the cell using a micropipette. An important advantage of this method is that the area of the substrate can be as large as we want given that we are not limited by the container of the suspension as is in case of vertical deposition. In our case, monolayers as large as ∼6cm2were obtained over a glass substrate. Another advantage of this method is that, due to the reduced space between the cover and the substrate, it is not necessary to use large dispersion volumes. Typically, less than 100 µl are enough to fill the cell. Dispersion concentrations of 0.5 % and 1% in volume were necessary for the case of Φ = 1µm and Φ = 520nm respectively. Again, the evaporation rate and the dispersion concentration will determine the velocity and the quality of the fabrication process. The cell is kept in a horizontal position in the same temperature/humidity controlled chamber used for the vertical deposition method for 6 hours at room temperature and 90% humidity in order to avoid a rapid evaporation. These conditions are slightly different from those described by Sun et al. (20oCand 30 % of humidity) but given that we are dealing with different substrates and sphere sizes the growth parameters had to be re-optimized for our purpose. Once the solvent evaporates the cover is removed and a monolayer deposited over the substrate is obtained. As the cover has to be hydrofilic too, it can happen that a good quality monolayer is obtained in that surface. This may be useful if we are using for 29 Chapter 3 Optical characterization by Fourier image spectroscopy In this chapter an in-depth description of the main experimental technique used to characterize the photonic crystals under study in this thesis is presented. Such technique is based on direct inspection of the spacial optical Fourier Transform of the response of the sample. Both monochromatic and white light spectroscopy configurations are presented. Examples of measurements performed on systems where the angular response is known are used as a means of calibration of the system. 3.1 Introduction Optical spectroscopy is one of the most obvious techniques to characterize one (1D), two (2D) or three (3D) dimensional photonic crystals (PCs). Reflectance and transmittance measurements can reveal how light propagates through/within the system under study. Within the set of techniques available for optical spectroscopy, normal and oblique (angle resolved) reflectance/transmittance is the most commonly used one in PCs characterization given that they allow demonstration of the main features of 2D and 3D PCs. Normal incidence configuration can reveal, for instance, information about confined modes for 2D PCs [111] or pseudogaps [112], full photonic gaps [9] or localized states at defects [113] for 3D self-assembled PCs. On the other hand, if one want to measure the dispersion relation for a given structure, angle and polarization resolved optical spectroscopy is the most straightforward technique. It has been used in the past to map dispersion relations along high symmetry directions of 2D [111] and 3D [114] PCs. Finally, it is worth to mention that for those systems where an emitter is affected by the dispersion relation of a PC, angle and polarization resolved measurements are a must to in-depth characterize the system. This kind of set-ups have been applied for 2D [115] and 3D [116] PCs in the past. For the measurements presented in this thesis a set-up has been designed and realized in order to be able to obtain a full angle/polarization resolved optical characterization of the samples described in the previous chapter in an automated way. The usual and most simplest technique for angle resolved spectroscopy consists in two branches mounted on a goniometer which are able to rotate in a controlled way around a sample. An interesting example of this kind of systems is shown in reference [117] and is plotted in Fig. 3.1. 37 Chapter 3. Optical characterization by Fourier image spectroscopy 38 Figure 3.1: Simplified diagram of a conventional angle-resolved spectroscopy set-up. In this case, signal collected in the rotation stage is taken to an FTIR spectrometer. Image from reference [117] In that diagram, each of the branches holds the illumination and the detection elements respectively. There are many approaches to this technique as for example using fiber coupled ilumination/detection or keeping the mechanical branches fixed while the sample is rotated. However, one main drawback of these systems is the lack of control to choose the point in the sample to be measured. Given that large magnification lenses are required to allow choosing a small region of the sample, it is necessary to work at short working distances. This makes it difficult to rotate the sample for an angle resolved characterization. Another disadvantage is that, as the angle of incidence increases the projection of the incident beam over the sample becomes an ellipse so that the probed area is increased as the angle in increased and therefore the optical response measured at different angles might not be comparable. In order to avoid those problems our approach consisted in the use of a high numerical aperture (HNA) microscope objective coupled to an external optical set-up where the angular response is obtained from the optical Fourier Image formed by the HNA objective. Imaging the back focal plane (BFP) of a HNA objective is a widespread technique in many fields where not just the spectral, but the spatial properties of the probe beam is relevant. As it is well known in optics, an optical system produces in its BFP an image which is the spatial Fourier transform of the object at its focus [118]. Therefore, studying the image of the Fourier plane has been employed to obtained in-sample propagating wavevectors 1. It has been used for example in plasmonic systems [60] where measurements of the effective refractive index of the propagating plasmon can be obtained. Furthermore, the angular response can be easily related with the Fourier Image at the BFP which has made this technique applicable to angle resolved photoluminescence spectroscopy in polaritons [119] or photonic crystals [115]. Recently Cottrell et al.. [120] presented a set-up to measure angle resolved scattering of single polymeric spheres with applications in medicine. The set-up presented in this thesis is similar to that one. In this chapter we will present the improvements we have performed on this kind of set-up and how we have adapted it for the optical study of PCs of different dimensionality. In the first part the set-up is presented and its main characteristics are described. After that we describe how to extract and process data from the direct measurements. Finally, 1Note that reciprocal space is calculated as the Fourier transform of the real one. 38 39 Chapter 3. Optical characterization by Fourier image spectroscopy some examples on its applications to the optical study of photonic structures (mainly self-assembled ones) will be shown. 3.2 Optical Fourier transform for angular resolved measurements The propagation of a light beam in free space can be conveniently described by Fourier analysis. In particular, the complex amplitude of a monochromatic wave can be described as a combination of a set of plane waves of wavector −→ kiand amplitude uiwhose interference will produce the final propagating wavefront (see diagram in Fig. 3.2a). On the other hand, a plane wave impinging on a lens of focal f forming an angle θwith the optical axis of the system will be focused at a point placed a distance f (BFP) of the lens and a distance d from the optical axis [121] as can be observed in the diagram of Fig. 3.2b. (a) (b) Figure 3.2: (a) Any wavefront for a propagating beam in free space can be decomposed in a superposition of plane waves (image taken from [118]). (b) Diagram for focusing by a lens of a plane wave into a point of the BFP. Therefore, if we consider a complex wavefront in a (x,y) plane at a distance d from the lens, it is expected the lens to form an intensity pattern in the BFP. In this pattern each point can be related with a given incident plane wave on the lens with the corresponding amplitude. After some mathematics [118], for a complex amplitude wavefront f(x, y) at a given distance d from the lens the expression for the intensity at the BFP results on a proportional relation to the Fourier Transform of f(x, y) as seen in Fig. 3.3. Figure 3.3: Intensity pattern I(x, y) produced in the BFP by a lens for a wavefront f(x, y) placed at a distance d of the lens. 39 Chapter 3. Optical characterization by Fourier image spectroscopy 40 where F is the Fourier transform of f(x, y) with ( x λf ) and ( y λf ) being the spatial frequencies in the plane at which f(x, y) is evaluated by the Fourier transform. For the particular case of a a wavefront placed at a distance d = f (front focal plane) of the lens, the complex amplitudes at the front and back focal planes of the lens are related by Fourier transform in both magnitude and phase. This case is known as 2-f system and present particular properties concerning the phase of f(x, y). Although we are interested in intensity measurements, we have chosen this configuration to make it compatible with our microscope set-up were the sample is always placed at the focal plane of the objective. If one considers a microscope objective as the lens in the previous study the small size of its BFP has to be considered. Therefore, it is usually necessary to magnify that plane if one wants to inspect the information contained in the Fourier transform of optical signal at the focal of the objective. Figure 3.4: Fourier image plane magnification set-up for a plane situated at the focal plane of the objective. BFP image is magnified by the lens an the new image is formed at a distance Z1+Z2from the BFP. Fig 3.4 shows a diagram of one possible set-up to obtain a magnified image of the BFP of an objective of focal distance f1. Two objects with different sizes have been plotted and the ray tracing has been performed for those rays departing from both objects with the same angle. As previously explained, every wave with the same angle is focused by the objective in the same point of the BFP. An important point has to be stressed from the ray tracing shown in Fig. 3.4. That is, every light ray outgoing the focus plane with the same angle reach the same point at the BFP, no matter the position in the y axis. It is direct from the diagram that X1=f1tan(θ). This discussion can be extrapolated to a 2D case where every ray leaving the (x,y) focus plane with the same θbut also with equal precession angle Φ (angle formed respect to the y axis in the xy polane)) will be focused at the same point in BFP. In order to magnify the angular pattern (Fourier transform) in the BFP an extra lens of focal f2is placed at a given distance Z1. It forms a new image at a position Z2. By using the lens formula one can calculate that the magnification of the system: M=X2/X1=−Z2/Z1=f/(f−Z1) and therefore, the main equation decribing this set-up is: 40 41 Chapter 3. Optical characterization by Fourier image spectroscopy X2=f1tan(θ)M=f1f2 f2−Z1 tan(θ) (3.1) An isotropic point emitter placed at the focal plane of the objective is probably the simplest case but also a very intuitive one. If one consider the emission pattern of an isotropic punctual source at a given wavelength λ, it is straightforward that the spacial Fourier transform is a circular homogeneous pattern and then a homogenously illuminated circle will be collected at position Z2in Fig. 3.4. Experimental implementation of emission angular pattern measurements with this technique [122] consist on pumping the sample with a monochromatic source and collecting the outgoing emission pattern at the required frequency . 3.2.1 Reflection Fourier imaging We have seen that the relation between an outgoing angular pattern and the image collected at the BFP is a straightforward procedure. However, we commented early in this chapter that reflectance measurements are a key technique in PC characterization. Unlike in most of the emission Fourier image experiments, reflectance angular pattern of a PC will be strongly dependent on the incident beam angular distribution. Therefore, it is important to show how we can use BFP imaging to collect a reflectance angular pattern. In order to make a monochromatic collimated beam incident at a given angle over the focus plane one can place a beam splitter at some point between the BFP and the lens as shown in Fig. 3.5a. (a) (b) Figure 3.5: (a) Diagram of angular pattern for a mirror with two incident beams θ1,2collected at BFP. x1,2(θ1,2) are the corresponding position at the BFP. (b) Same example for a grating. θin is the incident angle and θ0,1,2the diffracted orders for m=0,1,2 respectively. As can be observed, the simplest case for reflection is a mirror. The incident beam at angles θ1and θ2are reflected with the same angle and collected at the corresponding distance from the optical axis (x1and x2) in the BFP. Therefore, if one consider an objective with a given numerical aperture defined by the maximum angle collectable (θNA), a collimated beam will be impinging the mirror with all the angles 0 < θ < θNA at once. Then, the resultant Fourier image at BFP is an homogenous pattern. Let‘s assume a more complicated case. Fig. 3.5b shows the same set-up used for the mirror but in this case the object is a grating. Without entering the details one can imagine 41 Chapter 3. Optical characterization by Fourier image spectroscopy 42 an incident beam with a given angle θ. The reflected zero order (θ0) of the grating presents mirror behaviour. However, it is possible that some higher diffractive orders (m= 1,2, ..) are being also excited. They will be reflected on non symmetric angles (θ1,θ2) and then collected at different distances (X1) from the optical axis at BFP. This way, the far field diffractive pattern is obtained. As can be observed in Fig. 3.5b for θ2an special feature take place. If θ2> θNA that diffractive order (and the ones higher than that) is not collected and not shown as a spot in the BFP. Therefore, NA of the objective will determine the maximum angle which we can measure with this kind of set-up. 3.3 Set-up description Our design is based on a modified commercial inverted microscope (Zeiss Axio-Observer). The microscope objective is a Zeiss EC Plan-Neofluar 40X/NA = 0.75 which provides a maximum angular aperture of 48.5oas calculated from NA =nssin(θmax)2. This objective is specially designed to obtain high quality images at visible frequencies which is our working spectral target. For the illumination, the only unmodified part was the transmission arm. Given that transmission spectra are, in most of the cases, not useful for the structures studied in this thesis, that part is not considered in the set-up description. A diagram of the reflectance set-up is shown on Fig. 3.6.As can be seen, an optical fiberbased collimated illumination system (lens L1) replaces the incandescent lamp originally attached to the microscope. The final collimated illumination beam has a diameter of 1 cm. This configuration allows a very simple way of changing the illumination spot size on the sample by just changing the fiber diameter. The collimated beam is resized by the telescope formed by lens L2 and L3. The resultant beam is then focused on the sample by the HNA objective. This configuration is called ”critical illumination” [123] and is probably the most widespread in optical microscopy. In our case the complete optical train has shown to have a conversion factor of 0.1. That is, when illuminating with a 200µm core diameter fiber we shine a circular ∼20µm diameter area of the sample. As mentioned before, by changing the fiber we can choose the total illuminated area over the sample. Given that critical illumination consists in forming the image of the illumination source over the sample, special care has to be taken on the shape of the outgoing beam from the collimation system. Another advantage of this system is that L3 is placed at a distance f3from the BFP of the objective (where f3ifs the focal length of L3). This means that if L2 is removed , the collimated beam will be focused on the objective BFP and K¨ohler illumination will take place (both configurations are shown in Fig. 3.7). This kind of illumination can be useful in some cases, specially if we want to shine as much area as possible as for example in single molecule emission measurements. As this is not our case, we will keep L2 mounted in most cases. In terms of the angle resolved response it is important to note that both illuminations are opposite concepts. The beam coming out of the objective in the Kohler configuration is a collimated one, that is, incidence on the sample is always normal to its surface. In terms of of the optical response of the sample this means that only those states excitable at normal incidence will be probed. However, in critical illumination the incident beam is focused on the sample and then all those angles between θmin = 0oand θmax = 48.5o( determine by NA = 0.75) are being probed at once. This difference will 2nsis the refractive index of the surrounding medium, in this case air. The optical resolution is calculated following the Rayleigh criterion and results in r= 0.61λ/NA = 406 nm for λ= 500 nm 42 43 Chapter 3. Optical characterization by Fourier image spectroscopy Figure 3.6: Diagram of the set-up for Fourier Image direct inspection. Both, white-light and laser illumination are considered. The chosen with or without L6 lens configuration determines if the Fourier or the real image respectively is collected at the CCD camera. be shown as very important in the next chapters in order to understand the reflectance angular patterns of the samples under study. As shown in figure 3.6 a mirror can be placed between L1 and L2 in order to introduce a complementary collimated beam that will be focused over the sample in the same way as that of the the fiber coupled lamp. In our case we have used a laser beam at λ= 485nm usable either in continuous mode (CW) or pulsed with a pulse of 12 ps and repetition rate tunable from 32 kHz to 20 MHz. This allows us to optically pump samples with emission properties. The illuminated spot position on the sample can be controlled in the micrometer scale (spot size is ∼15µm in our case). Emission of samples containing a dye (as those shown in the previous chapter) have been measured in this way. It has to be stressed that the two illuminations (monochromatic and white light) can be interchanged without affecting the rest of the setup. As will be shown in the following sections this allows us to measure reflectance and emission being sure that both measurements are 43 Chapter 3. Optical characterization by Fourier image spectroscopy 44 Figure 3.7: By removing the L2 lens the set-up changes from critical (a) to K¨ohler (b) illumination. The same configuration applies to the collimated laser beam. performed exactly at the same point of the sample, and can therefore be compared. 3.3.1 Microscope output In our set-up, regardless of the chosen illumination, the image collected by the objective is focused out of the microscope cage by lens L4. This forms a magnified image of the sample at the Sample Image plane as shown in Fig. 3.6. Next, lens L5 is placed at a distance f5 from the Sample image plane in such a way that the Optical Fourier transform is obtained at BFP of L5. We have called it Fourier/Real Image in the diagram of the set-up. It has to be noticed that this configuration does not exactly match the diagram of Fig. 3.4 where OFT is formed at the BFP of the objective previously to its magnification with the lens. In the set-up of Fig. 3.6 the magnification is applied by L4 on the sample image previous to the Fourier transform of that plane by L5. This last configuration has shown to be more robust in terms of stability given that imaging the BFP of a 40X objective is a very critical situation due to the reduced size of the Fourier image formed in that plane. Finally, an extra lens L6 can be added after L5 and at a distance f6from the Fourier/Real Image plane (f6being the focal distance of the lens). This lens is placed using a kinematic mount which allow us to remove it easily. When placed, a magnified image of the Sample Image plane is obtained at the Fourier/Real plane (dashed line shows ray tracing in that case). Therefore, both, OFT of the sample image and magnified ”real images are available at the same plane by removing or not L6. We will focus in this chapter in the configuration for the Fourier transform imaging. Therefore, from Fig. 3.3, the Sample Image plane presents a given pattern f(x, y)) while the Fourier Image plane after L5 presents an intensity distribution I(kx, ky) which is proprtional to the Fourier transform of f(x, y). As previously explained, each different point (x,y) at Fourier Plane collects all those waves leaving f(x, y) with the same angle. This way, placing a CCD camera 3at the Fourier plane, the angular response is obtained in one single image. 3In our case the camera is a Hamamatsu C8484-05G01 model 44 45 Chapter 3. Optical characterization by Fourier image spectroscopy If a single working wavelength is considered, the image can be directly related to the angular response. However, if white light is being used (for example in reflectance measurements) the images can be misleading since the colection at the CCD will contain a superposition of the angular response for many different frequencies. To solve this problem a tunable filter (Semrock TBP01-700/13-25x36) was placed in front of the camera which allows us to collect the angular response at a given wavelength in a spectral range between 600-700 nm with a ∆λ=±10nm spectral resolution. Experimental limitations concerning the angular resolution were previously commented to arise in the first place from the NA of the objective used. The larger the NA the more angles can be introduced in the Fourier plane image. However, other important restriction is given by the pixel size of the CCD. As we increase the range of angles included in the Fourier plane, spatial separation between adjacent points decreases and pixel resolution might not be enough to resolve them. As the total size of the CCD is a constant, a compromise must be achieved between the largest NA and the CCD size. In our case the pixel is a square of lateral size 6.45µm. The total size of the CCD chip is 1344x1023 pixels with a total cell of 8.67 ×6.6mm2. The Fourier Image is expanded until it completely covered the CCD. As will be explained in the next section, this values allows an angular resolution of 0.5oin the best of the situations. 3.4 Extracting data from the Fourier Image Although qualitative information can be obtained from the Fourier image it is from quantitative measurements that we can accurately study the optical response of the sample. To obtain this kind of measurements we need to calibrate the system so that different points in the Fourier plane can be unambiguously associated with exit angles. In order to do so, an object whose Fourier image can be easily calculated is used and compared with the image obtained in the Fourier plane. In our case we consider a 1D grating in reflectance configuration whose Fourier transform is essentially a 1D collection of stripes. We detail below the calibration process which is, with some variations, the one described by Valentine et al [124]. As can be seen in Fig. 3.8, a given outgoing angle θm=0 from the sample impinges in a point at the radial distance (x) respect to the optical axis in the Fourier plane. It is trivial that x is directly proportional to the sine of the scattering angle. However, due to image-deforming optical aberrations caused by the relay lens a nonlinear relation rules this relation in the experimental set-up. We measure this relationship making use of the diffration from a commercial metallic grating (Thorlabs GH25 −24U) with pitch Λ = 417 nm. The illumination correspond to the critical configuration. Figure 3.8 shows a simplified case of the optical system under consideration to calibrate the set-up. A monochromatic focused beam impinges on the sample with an angle θin and the reflected beam produces an angular pattern at the CCD camera placed in the Fourier plane. Light is scattered by the grating according to Bragg condition: −→ κout||(λ) = −→ κin||(λ)±m−→ G(3.2) where kin|| and kout|| are the components of the incident and scattered wave vectors parallel to the surface of the grating, i.e kin|| = (ω/c)sin(θin) and kout|| = (ω/c)sin(θout),θin and θout being the angles measured with respect to the surface normal, ωthe angular frequency and c the speed of light in the dielectric medium on top of the grating. −→ Gis the reciprocal lattice vector of the grating that in this case |−→ G|=Gx= 2π/Λ and which we 45 Chapter 3. Optical characterization by Fourier image spectroscopy 52 with the calculated angular dispersion for the first diffraction order (dotted red line). As a second check, the angle at which re-entering of that order into the angles captured by the objective happen was calculated 7. It is shown as red dashed line and an excellent agrement is obtained. 3.5 Application to photonic crystals: spectroscopy of photonic modes Figure 3.15: Angle resolved reflectivity for an opal made from Φ =320nm PS spheres both for s and p polarization. So far we have shown that it is possible to perform angle resolved measurements with the described set-up with θmax being determined by the NA of the objective. However,in order to fully characterize the dispersion relation of a photonic or plasmonic crystal it is necessary to be able to obtain polarization resolved measurements. To do this a linear polarizer was placed at the output of the microscope before the spectrometer. In the setup shown in Fig. 3.6 every available photonic crystal mode is being excited at the same time. If just one polarization is observed at the Fourier plane it is possible to discriminate between sand ppolarization where those letters stand for perpendicular and parallel 7sin(θout) = λ/∆−sin(48o)) 52 53 Chapter 3. Optical characterization by Fourier image spectroscopy polarization with respect to the diffraction plane (see Fig 3.15 which is defined by the sample normal and the incident and reflected wave-vectors. It is well known that artificial opals provide large normal incidence reflectance peaks at frequencies where Bragg diffraction by sphere planes paralel to the sample surface takes place [21]. It has also been studied by Galisteo et al. that bands away from the reciprocal lattice direction ΓLare not longer degenerate and therefore, the angular reflectance probed in that direction is strongly dependent on the polarization [125]. We have taken advantage of this properties to test the set-up in angle and polarization resolved measurements. We performed such measurements with sand ppolarization for a Φ = 320nm polystyrene opal. S and P positions were found on a first step by using a mirror as background and placing the fiber at a angle matching Brewster condition. After that the polarizer was rotated until a minimum in transmittance was found. Then we take a reflectance background for each polarization in order subtract the system response. Measurements for s and p polarization were performed on the above mentioned sample. From the results in Fig. 3.15 it is clear that large differences on the spectral shape of the reflectance are observed for large angles. While s-polarized reflectance peak width hardly changes with angle, for p-polarization, a pronounced narrowing of the Bragg peak takes place. These measurements agree with previous results found in the literature and provide a powerful test for our system. As a final step, it is possible to apply the conservation of the parallel component of the wavevector to obtain K||out at every pair (θ, ω). If mapped, a direct measurement of the dispersion relation is obtained. As an example, Fig 3.16 shows the dispersion relation for the low energy region of an opal made of polystyrene spheres with a diameter of 490 nm. Figure 3.16: Experimental dispersion relation extracted from reflectance for a PS Φ = 320 nm opal in both p and s polarization resolve measurement. Numerically calculated bands for a polystyrene opals are shown as red lines. Here the relevance of imaging the Fourier plane before carrying out any spectral measurement should be pointed out for two reason. First is the fact that for 2D or 3D PCs, the large anisotropy induced by the periodicity introduces several high symmetry directions. The Fourier image can be used as a way to select the high symmetry direction along which one can scan the fiber and thus carry out the optical characterization. Finally, the Fourier image can provide us information regarding the presence os spurious effects such as dust on the optical path. 53 Chapter 3. Optical characterization by Fourier image spectroscopy 54 3.6 Conclusions In this chapter we have presented a powerful experimental configuration for carrying out angle and polarization resolved measurements in a reflectance/emission configuration. At variance to usual angle resolved measurement set-ups it allows measurement on a reduced area of the sample with fixed dimensions. Given that no mechanic parts act over the sample or the illumination, very large angular and spectral resolution is achieved while the area under study is kept constant. Furthermore, white light or laser sources can be easily interchanged which is relevant in emission measurements where probing reflectance and emission at exactly the same point is essential as will be shown on Chapter 7of this thesis. Also a set-up calibration procedure has been described. From these results, possible future improvements could be introduced. Using larger NA objectives will provide higher collecting angles, as well as higher spatial resolution. In that case, oil immersion objectives should be considered to study those systems that require from extremely large angles of collection. On the other hand, a more advanced improvement would consist on the use of a small micrometric aperture on the BFP of the objective in order to obtain illumination with a single angle. 54 Part II Study of hybrid monolayers as photonic-plasmonic colloidal structures 55 Chapter 4 Optical properties of a monolayer of close-packed spheres In this chapter we study from a numerical point of view the optical properties of a monolayer of organic spheres in a triangular lattice configuration when deposited on both dielectric and metallic substrates. It is shown that, by choosing the appropriate configuration large mode confinement as well as high field enhancements can be achieved with these kind of structures, specially for the metallic substrate case. Furthermore, it is demonstrated that surface plasmon resonances can be excited if the right sphere diameter/metallic material combination is chosen. Total field intensity profiles, spectral response and quality factors of the different modes are discussed. 4.1 Introduction In chapter 1the slab photonic crystal (PC) was introduced as one of the most efficient ways to confine light and modify its propagating properties [6]. In the search for a more low cost and straightforward technology, monolayers of polymeric spheres were presented in previous chapters as building blocks. Lattices of dielectric spheres were first proposed by Inoue et al. [1] as two dimensional (2D) photonic crystals. They also demonstrated that using dielctric spheres as building blocks introduces differences in light confinament compared with the slab PC case. However, despite the high mode confinement provided by these structures, it has been shown in several works [99] [126] that the use of a dielectric substrate under the spheres strongly reduces the potential for applications due to energy leakage towards the substrate. Parallel to the study of 2D self-assembled PCs, colloidal lithography1has spread as a common technique to obtain control on surface plasmon resonance (SPR) propagation [127]. Recent work has pointed out that the structure formed by the monolayer of dielectric spheres and the metallic substrate may provide not just SPR propagation control [59] but modes supported by the monolayer itself [62] [128] [129]. In this Chapter, by means of finite difference time domain (FDTD) simulations, we present a theoretical study on how light couples and propagates in the combined structures monolayer-dielectric and monolayer-metal. We have paid special emphasis to the leaky modes and the differences in light confinement depending on whether they present either plasmonic or waveguided 1Where a monolayer of dielectric spheres is used as a periodic lithographic mask. 57 Chapter 4. Optical properties of a monolayer of close-packed spheres 58 character. By comparison with the ”ideal” case of a free-standing monolayer we have found that the metallic substrate structure provides many advantages. The chapter is structured as follows. We first introduce the dielectric substrate monolayer of spheres and the main propagation channels available for light incident on such a system, with special interest on modes lying above the light line (leaky modes). Then total field intensity profiles are obtained too in order to check how electric field is distributed in the lattice for each resonance. Next, we report on the effect of a dielectric substrate on the losses of each type of mode. Then, we introduce the metal-dielectric system formed by a monolayer deposited over a gold thin film. Radiative and leaky modes are examined in the same way that we did for the dielectric substrate case. Main differences between the two structures are pointed out. As a final result, two different kinds of mode are defined and characterized depending on whether they confine most of the total field intensity in a small volume near the metal (SPP-like) or within the dielectric spheres (WG-like). 4.2 Monolayer on dielectric substrate Let‘s consider one single layer of close-packed dielectric spheres of diameter φand refractive index nsph deposited on a substrate with ns. As the diameter of the spheres is similar to the working wavelength, the scattering of one single sphere can be treated by mean of Mie scattering in the low Mie order. First Othaka [1] and next Miyazaky et al. [39] demonstrated that the optical response of a lattice of dielectric spheres can be calculated by coupling the scattering of the whole set of spheres. However, in order to present a simpler model accounting for the main properties of this system, the slab PC approximation can be an interesting approach. A close packed monolayer of spheres forms a hexagonal lattice with lattice parameter equal to the diameter of the spheres (φ). Therefore, the complete lattice can be defined by the two lattice vector a1=φ 2(√3,1) and a2=φ 2(√3,−1) as shown in the diagram of Fig. 4.1. Those lattice vectors defined by vectors univocally define as G1and G2in reciprocal space shows two major symmetry directions: ΓMand ΓK. It is easily found that points |ΓM|=π/φ and |ΓK|= 2π/(√3φ). Hence, it is useful to define the reduced frequency ω=√3φ/(2λ0) that allow us to make all the results scalable with φ. Figure 4.1: Diagram of the hexagonal lattice formed by close-packed spheres of diameter φand defined by lattice vectors a1and a2.G1and G2define the corresponding reciprocal lattice. In the approximation of the monolayer as a PC slab we found that according to Fig. 4.1 the system may support light confinement as well as dispersion relation modification 58 59 Chapter 4. Optical properties of a monolayer of close-packed spheres for those modes propagating in the plane of the periodicity (x,y). Besides, confinement in the vertical (z) direction due TIR can also happen as described for a slab waveguided in the Introduction of this thesis. Therefore, the modes can be radiative or leaky guided depending on the relation between the effective refractive index of the leaky mode and the substrate media (see Fig. 1.5 in chapter 1). Figure 4.2: Monolayer of spheres on dielectric substrate approximated as a slab waveguide of effective refractive index neff . In this case, the existence of a guided mode is determined by the refractive index of the spheres. Therefore, it is useful to approximate the monolayer as a slab waveguide with an effective refractive index extracted from the volume of the lattice occupied by the spheres, that is, the filling fraction of the dielectric material in the volume occupied by the monolayer. We can calculate neff using the well known approximation: n2 eff =fn2 sph + (1 −f)n2 air (4.1) with nsph and nair being the refractive index of the spheres and air respectively. f is the filling fraction of the lattice 2which for a close packed monolayer results f=π/3p(3) ∼ 0.6. If then, one think the monolayer as a slab waveguide with core index neff and thickness equivalent to the diameter of the spheres, the final refractive index profile shows the form indicated in Fig. 4.2. Within this context, it is easily found that leaky modes presents a given effective refractive index N which correspond to an effective wavector βfullfilling the condition of k0ns< β < k0neff . With k0= 2π/λ0and nsbeing the substrate refractive index. It has to be noticed that under this approximation it may happen that although nsph > nsub, no leaky modes may propagate since it is the effective index which rules such a condition. Moreover, as neff increases does the field confinement as discussed for the case of a slab waveguide in the Introduction of this thesis. 4.2.1 Simplest case: free standing monolayer We define a free-standing monolayer as a hexagonal lattice of spheres surrounded by air with no substrate. Therefore, this case is very similar to the symmetric slab waveguide well described in chapter 1. First consideration is that due to the symmetry of the system, the field intensity profile inside the spheres for each mode must present mirror symmetry with the x-y plane at the center of the spheres. Other important point to be considered is 2Occupied volume for the spheres in each unit cell of the PC over cell‘s total volume 59 Chapter 4. Optical properties of a monolayer of close-packed spheres 60 that due to its symmetric character, there is always at least one confined mode available for the structure regardless of the diameter of the spheres. In order to test this we have considered a free standing monolayer of polystyrene spheres as this is the material to be used in the experiments along this thesis. In that case nsph = 1.58 which implies neff = 1.38. Other parameters are ns=nair = 1 and Φ = 520nm. Concerning modes calculation, in this thesis we have chosen not the exact calculations of the modes but the inspection of FDTD calculated reflectance/transmitance spectra where each leaky mode is shown as a resonance on the background due to non guided propagation [30]. The reason for this choice is mainly that it allows us a direct comparison with the experimental results presented in further chapters. (a) (b) Figure 4.3: a) Theoretical reflectance spectrum in normal incidence for PS spheres (nsph = 1.58) in a close-packed monolayer lattice. b) Electric field intensity profile near the spheres at the two lower resonant frequencies and normal incidence.White lines delimit the spheres. Therefore, let us consider a set o plane waves normally incident (Γ point in the reciprocal lattice) on the sample. Fig. 4.3a shows FDTD simulation3of reflectance spectrum for the hexagonal lattice configuration of PS spheres (nps = 1.58) 4 Three kind of optical feature identified in this spectrum. On the one hand, a broad and low ondulation due to Fabry Perot interference is clearly seen. Secondly, out of plane diffraction due to the periodicity (reflection/transmission grating effect) of the structure happens for values ω≥1 as extracted from the grating equation. That spectral region has not been considered along this discussion given that our interest is mainly focused on resonant states of the system. Finally, the sharp peaks (A1 to A4) correspond to those 3Check appendix Afor in-depth explanation of how simulations were performed 4Reduced frequency has been chosen because by using it we take advantage of the scalability of these kind of PCs. The results that will be shown here can be scaled just by changing the sphere diameter (we consider the spectral dispersion for the polystyrene negligible in the vis-NIR range). From now on ωstands for frequency normalized to the parameter defined in the text. 60 61 Chapter 4. Optical properties of a monolayer of close-packed spheres confined modes with leaky character as they are folded back within the light cone by the periodicity of the system. Such kind of resonances are well known to be shown as peaks in reflectance or dips in transmission for confined modes of a slab PC [30]. It is worthy to comment that there are also a set of guided modes which fall out of the light cone and therefore not shown as reflectance peaks. Although not of much interest in this thesis, it is necessary to comment on the existence of this totally confined resonances given that they have been theoretically demonstrated to be present in this kind of structures [39]. In order to characterize the modes of the system we must look at the field intensity distribution within the PC. As they correspond to confined modes it is expected that large total field intensities in (or close to) the high refractive index part of the PC appears. FDTD calculation of the filed distribution at resonances A1 to A4 are shown in Fig. 4.3b. The shape of the field intensity profile depends on the order of the mode considered and is strongly related to Mie resonances of isolated spheres as well as the coupled array [39]. It can be noticed that the intensity profile is not symmetric with the center of the sphere. It has to do with the fact that we are not solving the modal profile but the direct optical response. As we are incident from upside it is expected for that profile not to be symmetric. This applies for all the intensity profiles simulations of optical coupling performed in this chapter. 4.2.2 Energy leakage with substrate refractive index So far it has been shown that the free standing case of dielectric spheres presents a good optical response in terms of light confinement. However, when the structure is to be fabricated, 2D self-assembled free standing systems become very difficult to be implemented. As a matter of fact, to the best of our knowledge, a large area free standing monolayer of microspheres operating at visible frequencies it has not being reported yet. Therefore, knowing how a dielectric substrate affects the optical properties of the system is essential. During the Introduction of this thesis it was shown that the field confinement for the modes of an asymmetric slab waveguide presents a strong dependence on the refractive index of the substrate. In particular it is demonstrated that the lower the RI contrast between the core and the substrate the larger the leakage thorough the substrate and then the lower the field confinement (equation 1.14). Therefore, assuming the PC slab approximation for the monolayer it is expected that any dielectric substrate wit ns>1 will produce larger leakage of the modes compared to the free standing case. Several works have been publish about this issue specially by Miyazaky et.al [99] [126] . Fig. 4.4 shows the calculation for normal incidence reflectance and its evolution from the free-standing scenario shown in Fig. 4.3 to the large refractive index (nsub = 3.5) substrate case5. The spheres parameters are the same previously shown in the FS case. As the refractive index of the substrate increases the reflectance peaks broaden and become less intense in agreement with the expected behavior from slab WG theory. Getting more into the detail, two different regimes are obtained with respect to the radiative losses introduced by the substrate. On the one hand, for ns< neff , resonances are leaky but nevertheless affected by the presence of the substrate as can be seen in Fig. 4.5 which shows a zoom of Fig. 4.4 over the evolution of the resonant peaks as nsub is increased. As can be observed, as nsincreases, high energy modes are first to disappear from the reflectance spectrum (lack of guidance). The higher the order of one mode the higher 5It is important to notice here that due to the leaky character of each of these modes and to the fact that we are below the onset of of diffraction, reflectance and transmittance are complementary so, if transmission is calculated a dip in transmittance will be obtained if a peak in reflectance is observed. 61 Chapter 4. Optical properties of a monolayer of close-packed spheres 68 Figure 4.10: Calculated normal incidence reflectance for a monolayer of PS spheres over gold substrate (black dotted line). Gray line shows fits for Fano expression on B.1. Every peak was fit separably in order to obtain its main parameters. paramtere q (accounting for the asymmetry of the resonance) and Γ accounting for the spectral width. Finally, quality factor (Q=ω0/Γ) can also be indirectly extracted from the fitting (see appendix B) which provides a quantitative value for the confinament of each mode of the system. Next table shows the values obtained for each mode in the fitting of fig. 4.10. Mode G1G2G3G4G5 ω00.69 0.72 0.80 0.89 0.97 Γ 0.0028 0.00161 0.0031 0.0028 0.0028 q13 22 11 16 17 Q248 448 258 317 346 First thing to be noticed is that qis considerably larger than 1 for each of the peaks which stands for a strong resonance character. If one check at Q values in the table above, large values are obtained taking into account the low refractive index contrast of our structure (∆n= 0.598). Comparison with literature indicates the good performance of the hybrid monolayer as a light confining structure. Moreover, for the same fitting performed over the free-standing spectrum of Fig. 4.3a (not shown) the Q values are approximately twice lower than for the metallic substrate. Therefore, the spectra for both free standing and metallic substrate case can be analyzed in the same terms of resonant and non-resonant states. However, in order to know if we are dealing with similar phenomena one should check the field distribution inside the structure at resonant frequencies. As in the last two sections, FDTD calculations were used to obtain total field intensity profiles after a given propagation time into the structure. Those simulations are shown in Fig. 4.11. These profiles corroborate the conclusions from the fitting of Fig. 4.10. That is, each of the resonances correspond for leaky modes since all of them show very large intensity (for confined field) as large as E/Eincident = 1300. Furthermore, both parameters (Q=ω0/Γ and field enhancement) show large values, well above those shown for the free standing case. This result shows that the the hybrid gold-PC structure performs even better than the ”ideal” dielectric case in terms of light confinement. Similar results have been observed in the visible range using silver substrate by Shi et al. [62]. 68 69 Chapter 4. Optical properties of a monolayer of close-packed spheres Figure 4.11: Field profiles after coupling at normal direction for the resonances shown on spectrum 4.9. Reduced frequency ω1,ω3and ω5correspond to λ= 1.25, 1.1 and 0.94 µm respectively. White line delimits the different interfaces between materials (spheres and gold). If the five resonances profiles are examined in detail it can be observed that two different kinds of field distribution are obtained as in the case of the dielectric substrate. On the one hand G2 and G4 resonances confine the field intensity mainly within the spheres with spatial profiles similar to those shown in Fig. 4.3b for the A2 and A4 modes. Taking into account this similarities we have named this kind of mode WG-like. On the other hand, G1, G3 and G5 present most of the field distributed mainly below the sphere but with a small component within it. In the case of G1, its profile is very similar to the one obtained for the dielectric substrate (A1), specially for the case nsub = 1.4 (Fig. 4.6c). G3 and G5 are not comparable to the dielectric substrate structure due to leakage towards the substrate. Because of this we have named these resonances as SPP-like due to their close confinement to the metallic layer and the large enhancements obtained. 4.3.1 SPP-like modes From Fig. 4.11 one can see that the field profile for the SPP-like modes present a similar profile to the modes of the dielectric substrate case concentrating most of the field under the sphere (see Fig. 4.6). Therefore, we can trace the X profile at two different positions for G1 as shown in Fig. 4.12. In this case, at maximum of field enhancement (a)) the difference between the undersphere concentrated field and the rest of the intensity distributed into the sphere is very large if compared with the dielectric substrate case (check fig 4.7). This is expected since the refractive index contrast at metal-air surface is much larger that in the totally dielectric structure. From this large contrast, field discontinuity provides very large enhancements. It also has to be noticed that the intensity of the mode distributed into the sphere present a similar shape than for the dielectric substrate structure though with a higher total field intensity. In this respect, a more complex model in terms of slot waveguide approximation 69 Chapter 4. Optical properties of a monolayer of close-packed spheres 70 Figure 4.12: Total field intensity for the G1 mode of a Φ = 1µm PS sphere monolayer deposited on gold (a). Vertical sections along two different X positions are shown in (b) and (c). could lead to accurate results. It has been applied in previous works for more simple but similar structures [138]. However, in this case, the spheres lattice plus the metallic susbtrate is much more complex system so it will not be studied here. Despite the lack of a mode coupling model, one would expect that resonances of the PC whose field is mainly concentrated under the sphere, excites SPR of the metallic surface (gold in this case). In previous studies, Cole et al. [59] showed that SPR excitation is possible through coupling to leaky modes of different structures formed originally by dielectric microspheres. In order to expand this study, the empty lattice approximation can be applied to the dispersion of the different kind of modes supported by the system. If a finite dielectric layer with a given thickness is considered on top of the metal, the field concentration on top of the metallic field is known to depart from the infinite dielectric case [58]. To take it into account we have calculated the dispersion of the SPR resonances for a monolayer of PS spheres with Φ = 520nm deposited on gold. However, in that calculation we have not considered the dielectric material refractive index to be the neff = 1.38 obtained from the filling fraction approximation of the monolayer. In order to find the neff that matches the effective refractive index of the hybrid SPP-like mode (N) several values for neff have been tested. The dispersion relation is calculated according to the equation for plasmonic modes (eq. 1.19) using gold dielectric constant at VIS range [136] and a trial neff . After that, the resulting dispersion relation have been folded back into the FBZ along the corresponding direction in the reciprocal space. Fig. 4.13 shows G1 and G3 dispersion relations for the most appropiate values of neff . The dispersion relation is compared in the same figure to reflectance at normal incidence for the corresponding monolayer. neff values have been chosen in order to match the spectral position of the dispersion relation at k= 0µm−1with the reflectance resonances accounting for the modes at normal incidence. As can be observed, the reflectance dips qualitatively agree with the position of the calculated SPR for G1 and G3 considering a dielectric with neff = 1.44 and neff = 1.27 respectively. ΓKfolding has not been plotted as the spectral position of 70 71 Chapter 4. Optical properties of a monolayer of close-packed spheres the modes at k= 0µm fall in the high energy region (ω > 1). Figure 4.13: Calculated reflectance (left) of a PS monolayer of Φ = 520nm on a 55 nm thick gold film. Right panel shows the empty lattice model for SPP-like modes (G1 and G3)and WG-like mode (G2) folded back to the FBZ along the ΓMdirection. As expected, none of the two SPP-like modes present larger values for the neff than that of the polystyrene spheres (nps = 1.58). The empty lattice model has also been plotted for the WG-like mode G2. It is important to noticed that, in this case, the neff to be considered for the empty lattice approximation is that of the monolayer previously calculated with the filling fraction aproximation as we are dealing now with photonic and not plasmonic resonances. A good agrement is also found with the calculated reflectance dips for G2. It has to be notice that the normal incidence spectrum shown Fig. 4.13 corresponds to a monolayer of φ= 510nm which plots the resonances at VIS range while in the in the previous study of this section (see Fig. 4.9) larger spheres (IR range) have been considered. Several differences are found. As the optical absorbance in gold increases for wavelengths below 650 nm (that is ω > 0.7 for Φ = 520nm) a decay is observed on the reflectance of the whole system 9. As this issue is studied in-depth in the next chapter we will not enter in detail here. However, in order to illustrate the effect of absorbance on the modes of the system, it is worth to compare the field intensity profiles in Fig. 4.14 with their counterpart for φ= 1 µm shown in Fig. 4.11. The large reduction can be observed in the total field intensity concentrated below the sphere for G3 which is the one affected by the high absorbance of gold at λ < 600nm. This is an expected effect of the absortion since it mainly affect that filed concentrated closer to the metallic film. Taking into account the large changes for both, spectral response and total field intensity profiles, we can see that the system is not fully scalable with the diameter of the 9We have performed these calculation and measurements with other metals like silver and similar results have been found depending on the absorbance band for each metal. Silver has shown the same results for VIS range than gold for IR spectrum as expectable from its SPR dispersion. 71 Chapter 4. Optical properties of a monolayer of close-packed spheres 72 spheres. Therefore, once the spectral region of interest has been chosen, the other main parameter is the complex refractive index of the metal. As substrate absorbance is expected to strongly affect the optics of the whole system, the metal must work out of the its high absorption spectral region. Therefore, while gold works properly for IR range, the use of silver is more appropriate for the VIS spectrum given that the high absorbtion frequencies for bulk silver take place in the UV. Figure 4.14: Total field intensity profiles for G1 and G3 resonances in a PS sphere monolayer over 55 nm thick gold substrate substrate for Φ = 520nm. 4.3.2 WG-like modes At variance with SPP-like, G2 and G4 resonances present in the profiles of Fig. 4.11 most of the field intensity distributed within the dielectric spheres. That provides this modes with a waveguided character where the metallic substrate plays the role of a metallic boundary in a slab waveguide. Using the empty lattice model considering the lab approximation with neff being that of the filling fraction approximation has shown to properly match the spectral position of the resonances as shown in Fig. 4.13. On the other hand, as mentioned in chapter 1, a metallic boundary in a slab waveguide reduces leakage of the modes through that part of the structure. In particular, if the case of a PC slab is considered, modes can just present leakage through the slab-air interface. Therefore, such a metallic substrate waveguide presents an increasing in the Q value for those modes of the structure when compared with the fully dielectric structure. In this case, a strong enhancement can be observed of the maximum total field intensity for WG-like modes of the structure when compared for those leaky modes of the fully dielectric monolayer. A side effect of using a total reflector under the monolayer is the small blue shift of these resonances when compared to the free standing case. It has to be noticed that for the dielectric substrate increasing of nsproduces low energy shift for every mode. This is a clear evidence of the different nature of both kinds of mode. It is well known in slab PCs theory that as we increase the confinement into the structure a blueshift is produced in the resonances reflectance peaks [130]. Thats the case for G2 and G4 whose resonances in the spectrum moves to higher energies. It has been already commented that the SPP-like G1 and G3 shifts to lower energies (red-shift). This modes are also expected to be affected by spectral variations in the dielectric constant of the metallic film (mainly its absorbance) as previously shown for the SPP-like ones. However, as for WG-like modes most of the field intensity is distributed far from the metallic film the effect of absorbance would be 72 73 Chapter 4. Optical properties of a monolayer of close-packed spheres less intense than for the SPP-like resonances. This point will be studied in depth in the next chapter. 4.4 Conclusions In this chapter, we have presented a theoretical study of the optical properties of a closepacked monolayer of dielectric spheres depending on the nature of the substrate that the structure is resting on. In particular, large attention has been paid to the metallic substrate case. For the case of the dielectric substrate system, it was studied what kind of propagation channels are available on the PC structure. It was found that large field concentration can be obtained for leaky modes. The spectral working region was shown to be scalable with the diameter of the spheres. It was also shown that increasing the substrate‘s refractive index dramatically reduces field confinement. As a main result, it was demonstrated that introducing a metallic substrate under the spheres allows the propagation of hybrid modes mixing SPRs and leaky PC resonances which take us to define two new kind of modes: SPP-like and WG-like. It has been shown that it is possible to efficiently couple to them on an normal incidence configuration. Both kind of modes present large field intensity values and therefor large Q are obtained for both types. As future work, it would be very useful an improved model including expressions for the calculation of the wavectors of the hybrid modes as a relation between the eigenstates of the PC and those of the metallic film. An slot-waveguide approximation in case of SPP-like modes could be interesting too in order to obtain design parameter that could provide some control over the maximum field enhancement confined under the spheres. 73 Chapter 5 Intrinsic and extrinsic losses of self-assembled metal-dielectric systems A study of the mechanisms reducing the light confinement efficiency for the modes of a monolayer of dielectric spheres on metallic substrate is presented in this chapter. These losses can be divided into two types: intrinsic (related with the optical properties of the materials forming the system) and extrinsic (accounting for unwanted defects in the fabricated samples). The former are numerically studied and the best conditions for increasing the confinement efficiency are identified. For the latter, the optical response of fabricated samples is compared with numerical simulations for perfectly ordered samples showing a reduction in the quality factor of the modes due to lattice defects. 5.1 Introduction In previous chapters we showed that, as in a photonic crystal (PC) slab, monolayers of dielectric spheres support leaky modes [6] which can confine light. Therefore, identifying those mechanisms through which light confinement is reduced is key when devising possible applications for these systems. Although energy leakage is intrinsic to the modes supported by a PC slab [139], an appropriate engineering of the structures may lead to an efficient confinement represented by large quality factors (Q) [30] of the modes of the system. Structural disorder introduced during the fabrication process can reduce such confinement. This has been a largely studied issue in PC slabs in the past [33]. In this context, the main sources of losses are positional and size variations of the scattering units as well as surface roughness. For the particular case of 2D self-assembled systems, the main structural defects are polycrystallinity [93] and random vacancies in the lattice inherent to the fabrication process as studied in chapter 2although positional and size disorder also play an important role. In this thesis we are mainly interested in hybrid systems. Therefore, another source of losses must be taken into account which is that related to absorbtion by the metal substrate which is an important issue to be considered in any system supporting surface plasmon resonances (SPR) [60]. If a metal with low absorbtion at a given spectral range is placed under a photonic structure, we have seen in previous chapters that the Q factor can be increased thanks to a reduced leakage through the substrate [83]. However, for those metals with strong absorbance in the spectral region of the SPP-like modes (see chapter 4), one would expect enhanced losses due to absorbtion by the metal. Therefore, this kind 75 Chapter 5. Intrinsic and extrinsic losses of self-assembled metal-dielectric systems 76 of losses can also be considered intrinsic to the system and dependent on the metal chosen as substrate. To illustrate this point, Fig. 5.1 shows a comparison between the complex refractive index of gold and silver showing that for wavelengths λ < 650nm gold presents an anomalous dispersion region while silver works nearly as a lossless mirror. Therefore, placing gold or silver under a given photonic structure may lead to important differences as far as concerns absorbance losses. Figure 5.1: Complex refractive index for bulk gold and silver as extracted from reference [136]. The chapter is divided in two parts: In the first part, the effect of structural disorder on the experimental optical properties is identified by comparison with the calculated ideal case for each metal. Resonances corresponding to SPP-like and WG-like modes are analyzed obtaining lower confinement (measured as variations in their Q) for each mode with respect to the simulated ideal case. In the second part of this chapter we study numerically how the gold optical constants affect the light confinement of each mode for a monolayer of polystyrene (PS) spheres deposited on this metal. This is done by tuning each mode of the system (by varying the sphere diameter) from the strong absorbance spectral region (λ < 600 nm) to the near infrared (NIR) where gold acts as an almost perfect mirror. Large changes in the spectral position of the modes as well as in their Q factors are obtained and correlated to the real and imaginary parts of substrate‘s refractive index (RI). Large differences are found on the dispersion of the modes depending on whether they present SPP-like or WG character. Finally, the Q factor for each mode is analyzed for every spectral position finding a strong dependence between the gold complex refractive index and the confinement factor of the modes. 5.2 Extrinsic losses: effect of lattice defects In chapter 2(section 2.2) it was shown that the main differences between hybrid monolayers fabricated by each of the described methods where related to polycrystallinity. While the vertical deposition method yields samples with randomly oriented domains, the wedgeshaped cell method is almost free from this problem. However, both of them produce 76 77 Chapter 5. Intrinsic and extrinsic losses of self-assembled metal-dielectric systems samples with randomly distributed vacancies, lattice dislocations and positional and size disorder (see Fig. 5.2). As all those defects are expected to introduce losses it is interesting to compare the experimental optical response with numerical simulations to evaluate the effect of structural disorder. Figure 5.2: Three kinds of lattice imperfections are shown on a sample made of PS spheres with Φ = 520 nm and grown on a silicon substrate by the vertical deposition method. In this study we will only consider normal incidence reflectance measurements for monolayers of polystyrene spheres grown on silver and gold substrates. The effect of pollycristallinity is not expected to play an important role in the measured spectra. The reason for this is that, at normal incidence only those modes available at the Γ point of the reciprocal lattice are being excited. As seen in chapter 4those resonances are independent of the orientation of the 2D lattice thus, although rotated among them, domain with good crystalline quality will show the same spectrum at normal incidence. Other important point concerning the existence of rotated domains is what domain size is necessary to obtain a complete Bloch mode. In our case, uniform domains are usually formed by more than 100 spheres even in the vertical deposition case. It was demonstrated by Miyazaky et al. in [140] that for a monolayer formed by ∼100 spheres, the Bloch modes for in-plane propagation are completely formed with no finite size effect in the optical response. Therefore we expect that the only effect of the existence of a polycristalline structure on the optical response will come from scattering by domain boundaries. From the discussion above, we can conclude that the optical response at normal incidence for the samples presented in this thesis should not be strongly affected by domains rotations. However, this kind of lattice deffect can be important for the case of angle resolved measurements if large areas of the sample are probed. This point will be further considered in the next chapter. Fig. 5.3 shows two examples of experimental normal incidence reflection spectra for φ= 520nm spheres monolayers on gold (a) and silver (b) together with numerical simulations. Results are shown in both , wavelength (λ) and normalized frequency ω=√3φ/(2λ) where 77 Chapter 6 Angle and polarization resolved spectroscopy of monolayers of organic spheres In this chapter an in-depth study of the angular response of monolayers of organic spheres is presented. By using the angle-resolved spectroscopy technique described in previous chapters we have obtained the dispersion relation of monolayers deposited over dielectric and metallic substrates. A comparison between the optical properties of each case as well as a study of the dispersion relation regarding the character of each mode (SPP-like or WG-like) with an empty lattice approximation is also presented. Finally, experimentally obtained equifrequency surfaces in reflectance are analyzed. 6.1 Introduction We have shown in chapters 4and 5of this thesis that the best performance as photonic crystal (PC) of a monolayer of organic spheres is obtained when a metallic substrate is used in such a way that energy leakage of the confined modes is minimized. However, we have characterized the system only for normal incidence where symmetry and propagation directions are not being taken into account. In order to fully chracterize the optical properties of the structure the angular dependance of its optical response has to be studied. Given that incident and outgoing angles can be easily related to wavectors with a component contained in the plane of periodicity, angle resolved spectroscopy has shown to provide valuable information about the optical properties of two (2D) [111] and three dimensional (3D)[142] PCs as we pointed out in the introduction of this thesis. If 2D PCs are considered each angle of incidence can be associated with an in-plane propagating wavector. Hence angle-resolved spectroscopy can be used to retrieve the angular dispersion of photonic modes and the presence of forbidden frequency gaps. Furthermore, one can easily retrieve valuable information such as the spectral position of photonic gaps or the spectral regions where slow light effects are expected [143]. Moreover, if measurements are polarization resolved, the symmetry of each mode can also be obtained [6]. This kind of measurement has been applied to monolayers of dielectric spheres in the past mostly with dielectric substrates [93] [99] or freestanding (FS) [140] [144] configurations. In those works authors characterized by means of angle resolved reflectance/trasmittance 85 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres 86 the dispersion relation of the modes available for those systems although their experimental study suffered from the lack of confinement for that kind of structures as shown in previous chapters which resulted in poorly resolved dispersion relations. Other sphere lattices different from the hexagonal close-packed (HCP) [95] [145] as well as monolayers over active dielectric substrates [146] have also been characterized but always finding poorly resolved modes evidencing weak confinement of light. From the surface plasmon resonance (SPR) point of view, angle-resolved spectroscopy is a common technique to obtain information about (SPR) propagation on a metallic surface [41]. If a periodically corrugated surface is considered, a well defined dispersion relation can be obtained from the dips observed in reflectance spectra [50]. For the case of self-assembled hybrid structures, as the one under study here, several works have mapped SPR by means of angle-resolved spectroscopy. Cole et al. [59] used angle-resolved measurements to demonstrate the evolution from propagating plasmonic modes to localized ones in gold nanovoids formed originally by self-assembled organic monolayers. In this chapter we show that in our hybrid system photonic bands can be obtained by this method which completes the study of chapters 4and 5on how light propagates in this kind of structures. We have used the experimental methods described in chapter 3to obtain equifrequency surfaces (EFSs) of light propagating in the plane of periodicity. That is the reciprocal space direct image inspection. This study was performed for the metallic substrate as the dielectric substrate samples presents a not well resolved reflectance pattern. This chapter is structured as follows: First we introduce the dielectric substrate case and show results for monolayers of polystyrene (PS) spheres deposited on silicon and glass in the visible (VIS) spectral range. Next, metallic substrate structures are measured in the same way and complex dispersion relations are obtained by spectrally scanning high symmetry reciprocal lattice directions. We identify each relevant optical feature and relate it with the normal incidence study presented on chapters 4and 5. The physical origin of the modes is further retrieved from a comparison with an empty lattice approximation. After that, experimental EFSs are collected and interpreted taking into account the previously measured dispersion relations. Finally, we present preliminary measurements of the dispersion relation in the high energy spectral range of this system. 6.2 Dielectric substrate As we have already shown in previous chapters monolayers of organic spheres can be considered as 2D PCs slab when grown on metal substrates or in a free-standing configuration and present complex dispersion relations. However, when dielectric substrates are considered angle resolved spectroscopy has not shown, to the best of our knowledge, well defined resonances [95] due to the already explained energy leakage through the substrate. Nevertheless, monolayers grown on dielectric substrates have been optically characterized and their dispersion relations retrieved [99] [140]. To obtain these results, transmission spectra 1are measured at different incidence angles and for different polarizations. After that, the spectral position of each resonance can be plotted as a function of the angle of incidence (θ) to form the dispersion relation as the example taken from [99] and shown in figure 6.1. 1Usually, dispersion relation measurements shown in the literature are extracted from transmission measurements for the monolayer over a dielectric substrate. We have performed reflectance measurements given that our set-up is optimized for the metallic substrate case. 86 87 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres (a) (b) Figure 6.1: a) Calculated (solid) and experimental (dashed) transmission spectra for a monolayer of PS spheres deposited on a glass substrate. Spectra are taken for P polarization and measured along the ΓKdirection. b) Band structure for P polarization. Solid circles correspond to the spectral position of the resonances shown in (a) for different angles of incidence. Open circles show calculated modes. In both cases data have been taken from reference [99] As can be observed in Fig. 6.1a while theoretical results show sharp resonances, experiments do not show well defined dips. This effect is a result of both, leakage through the substrate 2and extrinsic disorder in the fabricated sample in the form of structural defects. As a result, the comparison in Fig. 6.1b of measured and calculated angle resolved spectra do not show a good agrement apart from the lower energy spectral range where the dispersion relation presents a simpler behavior. In order to probe the dispersion relation of the monolayers deposited on dielectric substrates we have performed angle resolved reflectance measurements with the method described in chapter 3and the results are shown in Fig. 6.2. We present below the case 2Simulations of Fig. 6.1 are performed for an optically thin substrate which allow resonances to be well defined as opposed to the case shown in chapter 4for the semi-infinite silicon substrate. 87 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres 88 of monolayers grown on silicon substrates although glass substrates were also used with only small differences as previously mentioned in chapter 4. Figure 6.2: Angle resolved spectra response for a monolayer of PS spheres with diameter Φ = 520nm deposited on a silicon substrate measured along the ΓMdirection using unpolarized light. Full and dashed line shows calculation for Bragg diffraction limit (full line) and diffracted orders reentering the objective (dashed line) in the ΓMdirection. In figure 6.2 we show a contour plot of experimental reflectance for a monolayer of PS spheres with Φ = 520nm deposited on a silicon substrate and measured at angles θ between 0 and 480degrees (notice that unpolarized light is used). By inspection of the Fourier image in the CCD camera we choose the ΓMdirection in reciprocal space as an example (although ΓKwas also inspected). The effect of out-of-plane Bragg diffraction is now also evident as an abrupt decrease in reflectance similar to that shown in figure 3.14 of chapter 3. It appears as two dark bands in the contour plot. A simple calculation of the diffraction limit taking into account the scanning direction (ΓM) is also shown as two sets of grey line delimiting those bands. The full line corresponds to the onset of diffraction of the specular beam extracted from the diffraction condition as explained for the grating case in chapter 3. Above a certain collection angle, light diffracted at smaller angles re-enters the fiber and the collected intensity rises again. This limit is shown as dashed lines. It can be observed that the calculated limits do not agree completely probably due to deviations from the close-packed lattice in the form of small lattice relaxations. Regarding the optical resonances of this system shown in previous chapters for normal incidence, we can see that their angular evolution present a nearly linear trend. As expected from the normal incidence study, just the resonance appearing at reduced frequency ω=p(3)Φ/2λ= 0.71 (A2 in chapter 4) for θ= 0opresents a clear splitting into three different modes. This lack of resonances in the optical response of the system is again related to losses due to leakage through the substrate (nsub > neff ) as well as structural 88 89 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres disorder. This effect increases for oblique incidence for the silicon substrate as compared to the glass one as a result of a larger substrate refractive index. While photonic bands can be qualitatively described with a hexagonal PC slab approximation [147], a more refined theory is needed if one wants to evaluate the coupling efficiency for a given frequency for any available mode of the structure 3. Such a quantitative study would require Mie resonance theory in our case in order to obtain the optical response due to the spherical shape of each building block of the periodic lattice. This study has already been reported in several works by Miyazaky et al [39] [140]. 6.3 Metallic substrate In previous chapters we saw that if a metallic substrate is considered, the modes of the system appear in a reflectance measurement as spectrally narrow dips over a mirror-like background evidencing a more efficient confinement of electromagnetic radiation in this kind of system. Then, for any angle of incidence, the modes available to be excited are expected to appear also as narrow dips at the appropriate frequency according to the dispersion relation of the crystal. Figure 6.3: Reflectance spectra for different angles of incidence and both polarizations for a monolayer of PS spheres (Φ = 520nm) deposited on silver. Measurements where performed along the ΓKdirection in reciprocal space. Inset shows the Brillouin zone of the sample. Figure 6.3 shows several examples of reflectance versus reduced frequency measured for four different angles of incidence on a monolayer of Φ = 520nm PS spheres deposited 3In PCs slab, specially for air-bridged ones, the scattering process is more simple but still exact numerical calculations are necessary if a complete description of the resonances is needed [31] 89 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres 90 on a silver substrate. The spectral range under study is the VIS (400 < λ < 900). The spectra were measured along the ΓKdirection in reciprocal space for both S and P polarizations. As can be observed the evolution of lower order modes (G1, G2 and G3) can be followed as θincreases while higher order ones are hidden in the low reflectance background associated with radiative losses due to out of plane diffraction. From all the radiative channels commented in previous chapters, out of plane Bragg diffraction is the most important for what concerns Q reduction for each resonance. This fact combined with the complex dispersion for each resonance can make following its evolution a non trivial task. It can be observed that, for example, for θ= 20oout of plane diffraction takes place for frequencies as low as ω∼0.8 for both polarizations which hides the response of higher order modes. Besides, as shown in chapter 5, losses increase for higher frequencies and hence decreases the coupling efficiency which reduces the visibility of the resonances in the spectra. Strong differences are also evident when both polarizations are compared. As a step further we have performed measurements with high angular resolution which appear as contour plots in figures 6.4a and 6.4b. Such plots show reflectance for the same sample shown in figure 6.3. In this case reflectance for the two main high symmetry directions in reciprocal space (ΓKand ΓM) and both polarizations are shown. The angular resolution is ∆θ= 0.2o. As can be observed, at variance with the single spectra of Fig. 6.3, this kind of plots allow us to follow the mode dispersion even when we enter the diffraction limit. This effect is shown as a dark broad vertical band as explained in chapter 3. Therefore, it would not appear in a calculated photonic band diagram (where just propagating in the plane of periodicity is considered). As for any periodic 2D or 3D lattice, the scanned direction determines the diffraction limit shown by the black solid line in Fig. 6.4a. In this case such differences can be observed in figure 6.4 for the two high symmetry directions. ΓKrequires larger angles of incidence than ΓMto fulfill the Bragg diffraction condition. Since the latter corresponds to the shorter boundary of the First Brillouin Zone (FBZ), a shorter K|| has to be added to the incoming wave to reach the diffraction limit 4. For the G1 to G3 resonances (see chapter 4) at normal incidence no difference is found with polarization as for this particular symmetry both are equivalent (this can also be observed in Fig. 6.3) . Several anticrossings appear for modes propagating along both directions and polarizations (for example for θ= 18oand ω= 0.78 in ΓK and P polarization). An anticrossing between two modes happens when they have identical symmetry and their dispersion relation crosses at some point in the (k, ω) plot so they present the same energy and wavevector. The importance of anticrossings is stressed by two facts: photonic gaps (if present) may take place in such spectral regions. Also slow light due to almost flat dispersion of the photonic bands at that points can also take place. Moreover, they are very sensitive to variations in the angle of incidence and the scanning direction. If one looks at the plots obtained in Fig. 6.4 anticrossings are well defined in the measurements which is a proof of the high crystalline quality of our structure given that for disordered samples, several lattice directions are integrated in the same measurements. The evolution of each mode is qualitatively similar to the numerical simulations performed for similar systems by P.V. Braun and co-workers in [128]. They employed the Korringa-Kohn-Rostoker (KKR) method which has proved to describe well the optical properties of PCs of different dimensionality fabricated by self-assembly methods [148] . We next compare these contour maps with the one shown for the same sphere size but grown on silicon in Fig. 6.2 (note that no polarization was taken into account in 4|kout|=k0sin(θ) 90 91 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres (a) (b) Figure 6.4: Reflectance spectra as a function of angle and reduced frequency for a monolayer of PS spheres with Φ = 520nm deposited on a silver substrate measured along the ΓM(a) and ΓK(b) directions for both polarizations. Solid and dashed line in a) shows Bragg diffraction and first order diffraction order objective collection respectively. that case). As shown in chapter 4by analysis of the field intensity spatial profiles A1 and A2 modes in the dielectric substrate can be considered analogous to G1 and G2 for the metallic substrate case. We found for silicon that A1 and A2 present a splitting as 91 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres 92 we move away from normal incidence.The same behavior is found for G1 and G2 in Fig. 6.4. The main difference between dielectric and metallic substrate cases has to do with the complexity of the dispersion relation. While for the former case the modes present a nearly linear dispersion, for the latter a more complex behaviour is found evidencing stronger interaction between the different modes as a consequence of stronger light-mater interaction. 6.3.1 Experimental dispersion relation Figure 6.5: Dispersion relation extracted from angle resolved measurements for a monolayer of Φ = 520nm PS spheres deposited on silver. In order to study the above reflectance measurements from a dispersion relation point of view, it is worth to plot them as a function of the component of the incident wave-vector parallel to the plane of periodicity (k||, ω). As shown in chapter 3we can obtain k|| from the measured angle given that k|| =k0sin(θ) where k0is the wavevector of light in vacuum for the frequency considered in each case. This leads to figure 6.5 where the same data shown in fig. 6.4b and 6.4a are plotted as a photonic band diagram. In this case ΓKand ΓMdirections are shown together for each polarization. When the dispersion relations shown in Fig. 6.5 are compared with those reported in literature and previously plotted in Fig. 6.1b we find an overall qualitative agreement with some differences, as already discusses for the normal incidence case. These differences are spectral shifts (specially for the SPP-like modes), as well as mode splitting from A1 and A2 to G1 and G2. The origin of this is again related to the different nature of the substrate as in detailed studied in chapter 4. In the introduction of this thesis it was shown that an empty lattice model has been demonstrated to properly describe the physical origin of the bands in the dispersion relation of a PC slab [31]. Therefore, it is worth to compare the experimental dispersion relation shown in Fig. 6.5 with the calculated photonic bands using empty lattice approximation, taking into account the two different types of mode available for the monolayer on metal. Therefore, the dispersion relations that are folded back into the FBZ of the reciprocal lattice of the monolayer are, on the one hand a wave propagating on an homogeneous medium of effective index neff (WG-like mode) and on the other hand a SPR (SPP-like). For those modes propagating in an homogenous medium the dispersion relation can be expressed in terms of reduced frequency (ω=√3Φ 2λ0). Where λ0is the wavelength 92 93 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres propagation in vacuum. Therefore, in an homogenous medium of refractive index n the propagating is k=nk0=n2π λ0and the dispersion relation (k, ω) is found to be: ω=p(3)φ 4nk(6.1) For a monolayer of PS spheres we consider that n=neff = 1.38. Regarding the SPR, we can write the dispersion relation as a function of the reduced frequency ωresulting in: ω=√3φ 4πsm+n2 d mn2 d (6.2) where mis the real part of the complex dielectric constant of the metal and ndis the refractive index of the dielectric material on top of the metal. In this case that material is not a homogeneous slab but a monolayer of spheres so it is again necessary to consider an effective refractive index. However, as the SPR modes have most of their total field intensity confined close to the metal surface, the calculated neff of the monolayer may not be valid. Figure 6.6: Left: Experimental dispersion relation for a monolayer of Φ = 520 nm PS spheres on silver measured along ΓMdirection and P polarization. Right: Empty lattice model for G1,G2 and G3 modes. Grey band shows the calculated onset of diffraction limited by the NA of the objective. The left panel in Fig. 6.6 shows the experimental dispersion relation for a monolayer of Φ = 520 nm PS spheres deposited on silver along the ΓMdirection in the reciprocal space. The right panel shows the results of applying the empty lattice approximation to the two type of modes. We find that the main features in the spectral region we are considering are reproduced by this simple model. This further confirms the waveguided and SPR origin of the modes we already observed from the numerical calculations for normal incidence in chapter 4. In particular, for the SPP-like modes G1 (ω= 0.65) and G3 (ω= 0.75) it is necessary to assume a value of nd= 1.44 and nd= 1.27 respectively in order fo those resonances to take place at the experimental spectral positions. This values reflect the fact that G1 is more confined into the dielectric region than the G3 as was demonstrated with the field 93 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres 100 Figure 6.12: Experimental EFSs for in plane propagation after excitation of two different (k1and k2) modes of a PS Φ = 520 monolayer at ω= 0.71. The hexagon shows the FBZ for this lattice while the orange circle shows the kmax measurable in our set-up at that frequency. Red lines indicates energy propagation direction at the corresponding points in the EFS. energy region. This is probably a consequence of the fact that we are above the out-of plane diffraction limit which enhance losses (reduction of Q). Another probably reason is the tighter packing of bands in the high energy reason for any kind of PC. It produces too close bands to be measured as one broader resonance. Also the effects of disorder are expected to be larger as the probe wavelenght is smaller than the lattice parameter which increases the measurement sensitivity to structural imperfections. If both polarizations are compared it is observed that the dispersion of the modes is quite different in both cases. For P polarization the interaction between modes at anticrossings is barely noticeable showing the grid-like pattern characteristic of an almost vanishing refractive index contrast structure. For S polarization however a strong interaction is observed, specially at the frequency range between ω= 1.22 and ω= 1.28 and k|| = 6.2. In this case that region can be considered of low dispersion and then, low group velocities are expected. 100 101 Chapter 6. Angle and polarization resolved spectroscopy of monolayers of organic spheres Figure 6.13: Dispersion relation for Φ = 1µm PS spheres deposited on gold substrate along the ΓKdirection. 6.4 Conclusions In this chapter we have presented angle and polarization resolved reflectivity measurements for close packed monolayers of spheres on both dielectric and metallic substrates. We have paid special attention to the metallic case and shown the large dependence of the excitation energy with the angle of incidence. It has also been shown too that different modes can be chosen varying the polarization for the same energy. Potential applications could be those were coupling to the structure needs to be engineered as a function of the angle as for example organic emitting devices (OLED) technology. We have also compared the experimental dispersion relation with an empty lattice model in which we have taken into account modes propagating in an effective homogeneous dielectric material and also SPR modes. The good agreement has allowed us to relate them with the physical origin of the different modes of the system. It has been also demonstrated the feasibility of retrieving EFS in reflection configuration, which provides valuable information about light propagation within the sample. Finally we have obtained dispersion relations in the high energy range. 101 Part III Emission and tuning of the optical properties: adding functionality to hybrid monolayers 103 Chapter 7 Strongly modified spontaneous emission in hybrid self-assembled photonic-plasmonic structures In this chapter we present an in depth study of the emission properties of hybrid photonicplasmonic systems consisting of monolayers of organic spheres deposited on substrates of different nature. By placing an organic molecule within the spheres we show experimental results on the enhancement of the emission as well as the angular and polarization distribution due to the local density of states (LDOS) modification introduced by the periodic structure. Several dielectric and metallic substrates are tested and their performance is studied taking into account the properties of the passive system studied in previous chapters. 7.1 Introduction In the Introduction of this thesis we discussed how 3D and specially 2D photonic crystals (PC) could be key components in many types of light emitting devices since they allow both, enhancement of light-matter interaction [68] as well as extremely efficient light extraction [71]. We have also shown how the combination between plasmonic and photonic modes has recently provided an efficient way of enhancing and controlling light emission [57]. In particular, self-assembly based structures have been extensively used in the last decade to fabricate 2D hybrid metallo-dielectric structures suitable for emission control [82]. However, in most cases, the original bare structure of close-packed spheres is removed as it serves mainly as a mask for further processing [155]. On the contrary to these systems, some approaches have been proposed as structures that do not need further processing after the sphere monolayer is deposited [146]. These structures maintain the fabrication advantages explained in chapter 2and allow the use of organic materials which have shown to work as appropriate emissive devices by means of structuration [156]. However, in the self-assembled bare resulting structures the low refractive index contrast represents an important handicap in emission control through local density of states (LDOS) modification [157]. In this chapter we have employed the hybrid monolayer previously studied to strongly modify the spontaneous emission from the as-grown structure employing dye doped spheres. The system consists of a distribution of organic molecules homogenously 105 Chapter 7. Strongly modified spontaneous emission in hybrid self-assembled photonic-plasmonic structures 106 distributed into the spheres. Therefore, we have obtained an strong modification of the spontaneous emission for those frequencies where the position of the molecules matches the a high LDOS modification due to a mode of the monolayer. An in-depth study of the emission properties of the 2D hybrid self-assembled PC is presented. It is shown that by controlling the LDOS we can effectively control the emission of the active medium placed inside the sample, specially concerning the angular and polarization characteristics. This way we have used the resonances of the metal-dielectric structure to modify the emission of an organic dye homogenously distributed within the spheres. The chapter is structured as follows. First, the emission properties of the monolayer deposited on a dielectric substrate is experimentally studied for the case of glass and silicon substrates by using dye doped spheres as building blocks of the lattice. Next, we study the gold substrate structure for those frequencies where a resonance of the system is available. After that, we present the angle and polarization resolved emission spectra along the two highest symmetry directions. A comparison with the previously measured dispersion relation is presented and emission enhancement is found to take place for some of the modes available in the structure. Next, we have compared emission from samples fabricates with the two metallic substrates considered in this thesis: gold and silver. Finally, emission isofrequency surfaces (IFS) are compared with their reflectance counterpart and the evolution with wavelength is discussed. 7.2 Dielectric substrate In chapter 4we demonstrated how for a monolayer of dielectric spheres no truly leaky modes are confined in the plane of periodicity if neff < nsub with neff and nsub being the effective refractive index of the monolayer and the dielectric substrate, respectively. However, a little field enhancement was shown for the resonances of the system. As nsub is increased the total intensity confined under the spheres grows because of the slot effect between sphere and substrate. We found the most representative case to be that of the silicon substrate (ns= 3.5) as shown in the total field intensity profiles on chapter 4. Figure 7.1: Normalized absorbance and emission spectra for Rodhamine 6G distributed within the polystyrene spheres used for the experiments. Therefore, previous to study the metallic substrate case it is worth to briefly study emission by monolayers made from dye doped polystyrene (PS) spheres with a given 106 107 Chapter 7. Strongly modified spontaneous emission in hybrid self-assembled photonic-plasmonic structures diameter (Φ) and deposited on glass and silicon substrates. The fabrication method was vertical deposition for silicon substrate while the wedge cell method (see chapter 2) was applied to obtain monolayers on glass. In the results presented below we have used Φ = 520nm spheres homogenously doped with Rodhamine 6g (Rh6G) whose tabulated absortion/emission spectra can be seen in Fig.7.1. It presents a broad emission with its maximum at λ= 612nm. We have chosen this molecule due to the compromise between the spectral stokes shift (difference between emission and absorbtion maxima) and emission efficiency which is known to be close to one [158]. High quality samples were obtained and reflection and emission measurements were carried out by means of the set-up described in chapter 3. As it was previously explained, the source used for optical pumping is a pulsed diode laser working at λ= 485nm with a variable repetition rate although in this case we have used the continuous wave (CW) configuration. As can be observed when compared with Fig. 7.1, the pump wavelength does not match the maximum of absorbtion (λ= 542nm) for the Rh6G. The reason to use that wavelength is avoiding gold absorbance at λ= 532nm as will be evident in the next sections of this chapter. Figure 7.2: Reflectance (red line) and emission (black solid line) spectra for a close-packed monolayer of Rh6G doped PS spheres of Φ = 520nm deposited on glass and silicon. Grey dashed line shows emission of a disordered monolayer of Rh6G doped Φ = 160nm PS spheres deposited on each substrate in a disordered way. We have first studied the effect of a structured environment on this emitter. Fig. 7.2 shows reflectance and emission collected for some of the samples described above. In this case we present measurements for a monolayer of Rh6G doped PS spheres of Φ = 520nm deposited on glass (top) and silicon (bottom) with nsub ∼1.45 and nsub ∼3.5 respectively. Reflectance spectra were collected the same way as those shown in previous chapters (see chapter 5). As a reference for emission in this case we have used dye doped spheres with a much smaller diameter (Φ = 160nm) droped on each substrate. Therefore, no effects due to the periodicity are expected. Besides, given that λemiss >> Φ, no Mie resonances due to the spherical shapes are expected to play an important role. The differences in the emission spectra for silicon and glass substrate are probably due to the modifications on the DOS for an emitter close to an interface as shown in this thesis introduction [73]. The grey dotted lines of Fig. 7.2 shown that, when compared with the commercial data shown in Fig. 7.1, the larger change on the shape of the emission spectrum is obtained 107 Chapter 7. Strongly modified spontaneous emission in hybrid self-assembled photonic-plasmonic structures 108 for silicon as it presents a much larger refractive index contrast with the environment of the dye molecules (PS) than the glass substrate. If one takes a look at the emission of the monolayer of spheres with Φ = 520nm deposited on both substrates, the spectral shape differences shown for the reference vanish and both black lines in Fig. 7.2 show an almost identical shape. It is interesting to compare the emission spectra with the reflectance measured at the same point of the sample. As it can be observed, the reflectance spectra do not show any resonance beside the dip corresponding to unconfined resonances (A1 and A2) due to the substrate (nsub > neff ) as already explained in chapter 4(see Fig. 4.4). Because of substrate leakage no emission enhancement is found at that point as expected from the total field intensity distributions for that modes (see chapter 4). Regardless of enhancement at those frequencies of the resonances, the emission shown in Fig. 7.2 present the Rh6G spectrum modulated according with the optical response of the monolayers as can be depicted from comparison with the reflectance spectrums in the same figure. It has to be mentioned that the total intensity for the same pumping conditions was shown to be larger in case of silicon substrate, as expected from the larger refractive index of this material compared to glass. Figure 7.3: Angle resolved reflection (left) and emission (middle panel) measurements for a close-packed monolayer of Rh6G doped PS spheres of Φ = 520nm deposited on silicon and measured in the ΓKdirection. The plot at the right shows normal incidence emission for Rh6G doped Φ = 160nm PS spheres deposited on silicon and in a disordered structure. Let’s consider next the angular emission for the monolayer of spheres here presented. As explained in the introduction of this thesis, emission generated within a waveguiding can be either be emitted directly to the outside medium or propagated as a guided mode. By using an appropriate lattice it is possible to obtain coupling between radiative and leaky modes so the total intensity emitted by the structure can be largely enhanced. It has effects in the total intensity outside the structure but also in the emission angular pattern. The experimental measurements in Fig 7.3 show the angular reflectance map for Φ = 520nm PS-Rh6G spheres over silicon substrate and the angular evolution of its emission. It is known from previous chapters that the two main features measurable in 108 109 Chapter 7. Strongly modified spontaneous emission in hybrid self-assembled photonic-plasmonic structures this structure by reflectance are the onset of diffraction depending on the periodicity of the scanning direction (which is shown as a dark broad band in reflectance map of 7.3) and the semiradiative modes which are the result of system resonances (A1 and A2) shown as weak dips in reflectance/transmitance. As we showed before for normal incidence (Fig. 7.2), the angle resolved emission measurements in Fig 7.3 show a small enhancement for the spectral positions of A1 and A2 at their dispersion relation positions. Although the enhancement is very weak, it can still be observed that the maximums of emission are channeled by the dispersion of the resonances shown as reflectance dips for A1 and A2. As was explained for normal direction emission, the enhancement at specicfic frequencies is due to the small but still present field confinement due to the slot-waveguide effect under the spheres that was shown to take place for a monolayer on a dielectric substrate in chapter 4. However, the angular pattern is determined by the dispersion relation of each mode which brought above the cone of light by the lattice periodicity as demonstrated in chapter 6. Finally, from the study of chapter 4it is known that for A1 and A2 most of the field is concentrated below and not within the PS spheres. It is translated onto the emission properties as a very low enhancement at the PC resonances of the monolayer. Other approaches as for example placing the emitter on top of the substrate, just under the spheres [146] have been tested but the emission enhancements obtained are not larger than the ones shown here for the emitter-inside the spheres system. 7.3 Metallic substrate In the previous section it has been shown that an emitter placed within organic spheres forming a monolayer over a dielectric substrate does not experience a noticeable change in its spontaneous emission. However, we have already seen both theoretically and experimentally in previous chapters that the introduction of a metallic film under the spheres strongly modifies the optical response of the system, specially concerning its ability to confine light. To study the effect of a metallic substrate on the spontaneous emission of internal sources, the Rh6G doped PS spheres of Φ = 520nm were deposited over a 60 nm thick gold film. The structure is the same one already studied in previous chapters although in this case, absorbance of the dye molecule has to be taken into account when considering the optical response of the sample. Fig. 7.4a shows the calculated field intensity profiles for this system. Fig. 7.4b shows the unpolarized emission spectra at normal incidence for a sample grown on a gold substrate and on a silicon substrate, together with a reflectance spectrum for the gold substrate sample. At variance with the dielectric substrate case, for the sample grown on the gold substrate, a large enhancement of the emission takes place for the mode located at ω= 0.72. It can be observed that the spectral position matches that of the dips in reflectance which was termed G2 in the study of chapter 4. As G2 is a WG-like resonance the electric field is mainly confined within the spheres, where the emitter is homogeneously distributed. A factor 20 enhancement in emission is observed for this mode when compared with that of the reference sample grown on silicon. This agrees with the large enhancement in the total field intensity within the spheres for the gold case when compared to the silicon substrate structure. Emission is also enhanced over the background for the mode at ω= 0.77, though much less than for the G2 case. This is in agreement with the fact that this resonance corresponds to the SPP-like mode that we named G3 in previous chapters. In general G3 presents most of the field intensity distributed below the spheres (Fig. 4.11). 109 Chapter 7. Strongly modified spontaneous emission in hybrid self-assembled photonic-plasmonic structures 116 concerns the effect of out-of-plane diffraction on the emission measurements we confirm in Fig 7.9 that it is not noticeable for the modes of the system. This agrees with the angle resolved measurements of Fig. 7.5 where reflectance shows the onset of diffraction (dark broad band) while emission is unperturbed by this effect. This is due to the fact that the emission enhancement will be dependent on the Q factor of each mode as previously demonstrated and this emission will be channeled away from the sample mainly through the angular distribution of each mode. If one performs the same measurement at different frequencies it is obtained a set of EFSs that map the emission of the sample in every direction an polarization for the whole dispersion relation of the PC 3. Fig. 7.9 shows a set of images for frequencies ω= 0.64, ω= 0.66, ω= 0.69, ω= 0.71, ω= 0.73 and ω= 0.75 taken for the same sample of Fig. 7.8. The evolution of the emission pattern due to light confinement is obtained 4. As in the reflectance case, it has to be mentioned that due to the low refractive index contrast no discontinuities are observed in the EFSs for any frequency [151] as it would happen in case of an structure showing a photonic gap (large refractive index contrast). This result agrees with the absence of gaps in the photonic bands measured in chapter 6 for this structure. Nevertheless, the refractive index contrast in the samples is high enough to obtain EFSs which depart from the circular shape for those modes in the vanishingly small refractive index contrast approximation studied in section 6.3.2 of chapter 6. This evidences (as we demonstrated for reflectance in chapter 6) that emission will experience anomalous refraction for some propagation directions contained in the plane of periodicity. 7.6 Conclusions We have shown that monolayers of dielectric spheres deposited on metallic substrates can strongly modify the emission of organic dyes contained within the spheres through coupling to the hybrid modes of the structure. Emission enhancement due to strong field confinement inside the spheres has been demonstrated together with its polarization dependence. We have also shown through field intensity profiles calculations that the modes providing larger emission enhancements are those presenting a stronger field confinement at the position of the emitters in the structure. Evidence for the directionality of the emission has been also presented. Finally, the study of the iso-frequency surfaces in emission has provided us with a full characterization on the angular pattern of emission. Therefore, the here studied properties are suitable for those applications where control over emission needs to be achieved using organic materials as, for instance, organic light emission devices (OLED) technology. As future work, other emitter positions could be explored for the same structures in order to use other modes of the system. In particular it could be interesting to place a single emitter in a very controlled position within or at the bottom of the spheres in order to obtain a quantitative measurement of the effect of the LDOS on the emission of the light sources. 3It is the same association between photonic bands and different EFSs that was shown for reflectance in Fig. 6.11 4The broad emission lines are due to the use of a spectrally wider filter in front of the CCD. In this case we have used the tunable ∆λ= 20nm filter while the one used for the images in Fig. 7.8 is a notch filter with a bandwidth below 5 nm 116 Chapter 8 Tuning the optical properties of a monolayer of spheres We present here an in-depth study on the tuning of the optical response of monolayers of dielectric spheres deposited over metallic substrates when their filling fraction is accurately modify. This is achieved by plasma etching to reduce the diameter of organic spheres while the lattice parameter is maintained. We show in this chapter that it is possible to change the spectral position of the optical resonances for this structure as well as their light confinement properties. Finally, by using dye doped spheres, we experimentally demonstrate that changes in the filling fraction can also provide us with an efficient method to control the emission properties of this kind of structures. 8.1 Introduction So far we have demonstrated in this thesis that self-assembled two dimensional (2D) photonic crystals (PCs) deposited over metallic films can provide a good optical performance in terms of its ability for light confinement. However, the functionality of these systems can certainly be enhanced if one can change their the optical response under an external stimulus, turning them into tunable devices. Tuning of the optical properties is a challenging task in both PCs and plasmonics systems. For the latter some possible strategies are optical [160], electrical [161], magnetic [162][163] or acoustic [164] modifications of the plasmonic modes. Alternatively, for the case of PCs one can modify the refractive index or lattice constant of (in our case) the periodic organic lattice to tailor the photonic dispersion and hence the optical response of the system [21] [157]. Certain stimuli could even be employed to simultaneously tune both types of modes and the hybrid modes arising from them. In this chapter we present an in-depth study of the changes in the optical response of self-assembled monolayers of polymeric spheres deposited on metallic films introduced by tuning the filling fraction (ff) following the process previously described in section 2.3 of chapter 2. As already described in that section, the tuning of their optical response is achieved by homogeneously reducing each sphere while keeping the lattice parameter constant, i.e. by changing the filling fraction of the hexagonal lattice as studied previously for similar systems on dielectric substrates [165]. This technique has been used by several authors in the past for example to improve the coupling in 3D opal-based PCs to slow modes [166] or to introduce planar defects in such systems [167]. 117 Chapter 8. Tuning the optical properties of a monolayer of spheres 118 In our study, we use polystyrene (PS) spheres of two different diameters (Φ = 0.52 and Φ = 1µm) forming a close-packed monolayer over a gold substrate for which the two kinds of modes investigated in previous chapters, that is, waveguide-like (WG-like) and plasmon-like (SPP-like), fall in the visible-near infrared (VIS-NIR) spectral region. We demonstrate for both cases that the two kind of modes can be spectrally shifted by reducing the spheres in a controlled manner. A theoretical study of their field intensity profiles is also provided. We show that not just the spectral position of the modes but the emission enhancement associated with the system modes and discussed in chapter 7can be efficiently tuned by this method. The chapter is structured as follows: first we present the evolution of the experimental and theoretical reflectance spectra for PS monolayers with Φ = 1µm spheres as their ff is reduced up to 30 %. Next, the total field intensity is calculated for each resonance at every reduction step and the changes in the nature of each mode is discussed. After that, we characterize the changes introduced by the tuning process in the optical properties of dye doped PS spheres with Φ = 0.52 µm on gold substrates for the VIS range. Finally, measurements of the emission of these samples at each etching step are presented showing an enhancement at the modes spectral position for each reduction step as we previously demonstrated for the close-packed structure in previous chapters (see chapter 7). 8.2 Effect of filling fraction modification in the optical response of a monolayer of spheres on a metal substrate Recalling the study we performed in chapter 2about the homogeneous diameter reduction of PS spheres in a close-packed lattice on gold we can now focus on the changes of the optical response for those systems during the tuning process. In order to track the evolution of the optical response, normal incidence reflectance spectra were collected for a monolayer of Φ = 1µm spheres from the close-packed lattice scenario until a 100 nm diameter reduction (γ= 0.9)1. Figure 8.1 shows both experimental 2and simulated normal incidence reflectance for three different sphere diameters. As explained before, this kind of system supports hybrid modes which can be divided in SPP-like and WG-like depending on whether the total field intensity is distributed mainly below the spheres (SPP-like) or within them (WG-like). The five dips shown in the reflectance spectra correspond to the modes of the sample. The coupling efficiency as well as the field spatial profile are dependent on the nature of each mode. For this study we will consider the first three modes (G1 to G3), representative of the two types of modes sustained by these systems, although the same behavior is obtained for the other two modes available (G4 and G5). A good agreement between simulations and experimental data is found. However, some differences in spectral width and peak position are observed, most likely due to residual disorder introduced during the growth process. Polydispersity of the spheres can also be a source of disorder, though in our case its effect should not be significant due to the low values for the spheres used (less than 3 % according to the manufacturer) 1In chapter 2we defined γ= Φ/Φowhere Φois the original sphere’s diameter and Φ the one at the end of the reduction process. Then, the filling fraction is related with this parameter through ff =γ3π 3√3 2For close-packed Φ = 1µm spheres the optical range of interest falls in the near infrared region (NIR). For such range the microscope set-up shown in chapter 3does not work properly and other techniques must be used. In this case measurements were performed with a microscope coupled to a Bruker FTIR in normal incidence and a maximum angular aperture of ∼50 118 119 Chapter 8. Tuning the optical properties of a monolayer of spheres Figure 8.1: Normal incidence reflectance spectra for three different ffs. Top to bottom: ff= 0.6, ff= 0.52, and ff= 0.44, corresponding to γ= 1.0 , γ= 0.95 and γ= 0.9 respectively. Experimental (black) and theoretical spectra (red) are presented. In chapter 4we investigated the nature of each mode by means of finite difference time domain (FDTD) calculations of the total field intensity profile for each of the resonances of the system. Therefore, in this case we can perform the same study in order to observe how the light confinement changes as the filling fraction of the lattice is reduced. Fig. 8.2 shows the intensity profiles corresponding to modes G1, G2 and G3 (each column shows the same mode) for Φ = 1µm PS spheres deposited over gold film. The plotted filling fractions are ff=0.6 3(upper row), ff=0.52 (middle) and ff=0.44 (bottom). These values correspond to the spectra shown in Fig. 8.1. As explained before the structure is illuminated at normal incidence at those frequencies matching the G1, G2 and G3 modes. For the maximum filling fraction (γ= 1 or ff = 0.6), G1 and G3 present the already studied SPP-like character while G2 shows WG-like nature. If we examine the total field 3The first case corresponds to the close-packed spheres lattice and hence matches the profiles shown in Fig. 4.11 of chapter 4 119 Chapter 8. Tuning the optical properties of a monolayer of spheres 120 Figure 8.2: Total field intensity distribution and its evolution with sphere resizing for the first three modes of the spectra shown in Fig. 8.1: G1 (left column), G2 (middle column) and G3 (right column). Corresponding filling fractions are ff = 0.6 (upper row), ff = 0.52 (middle) and ff = 0.44 (bottom). intensity distribution of the different modes as the filling fraction is reduced we can see that the WG and SP characters of the modes remain unchanged and only a slight decrease in the total field intensity is observed. So far we have considered just three single sphere diameters. However, the tuning procedure can be carried out in a quasi continuous manner allowing a fine degree of control of the sample topology and hence of its optical response. To show this, reflectance 120 121 Chapter 8. Tuning the optical properties of a monolayer of spheres spectra were recorded after each 30 seconds etching step over a 12.5 minute period 4. As a result, Φ = 1µm spheres were reduced by as much as 100 nm in diameter (0.44 < ff < 0.60). Theoretical spectra like those in Fig. 8.1 were calculated for each diameter. For the experimental case, an initial diameter of Φ = 1020nm was required in order to match the theoretical spectra, probably a consequence of the 3 % polydispersity of the spheres. Nevertheless, the filling fraction reduction was comparable in both cases. The two reflectance maps obtained for simulations and experiments are plotted in Fig. 8.3 . Fig. 8.3a shows how as the filling fraction of the sample is reduced, the optical response undergoes two major changes. There is an overall blue-shift of the modes attributed, as already discussed, to a decrease in the effective refractive index of the dielectric part of the system. Also there is a change in the intensity of the dips in the reflectance spectra. While those modes that have a marked SPP-like character (G1, G3) hardly change, those with a WG-like character (G2, G4) present strong variations in intensity. This indicates that for the latter, as we reduce the sphere diameter, the spatial distribution of the field undergoes strong changes in the mode profiles of Fig. 8.2. A clear example is the case of G2, where for γ= 0.94 the associated dip all but vanishes and then recovers upon further reduction. Accompanying this result is the increase of ca. an order of magnitude in the field intensity inside the spheres (see Fig. 8.2). Special attention has to be paid to the G5 resonance. In Fig. 4.11 of chapter 4we demonstrated, by inspecting total field intensity profile of this resonance, that it presents a SPP-like character. However, in Fig. 8.3a it can be observed that it is the only one which completely vanishes for ff < 0.55. This behavior is probably related with two phenomena. On the one hand its high energy spectral position which makes it more sensitive to radiative losses. On the other hand, in spite of the SPP-like character, some field intensity remains distributed within the spheres. Hence it is expected that this mode will be heavily affected by the spheres reduction. When measurements are compared with calculations a good agreement is found. Modes G1, G3 and G4 present an identical linear blue-shift with ffreduction in both cases. Some differences are observed regarding the mode spectral positions for ffbelow 0.5. This is due to the fact that the plasma reduction rate becomes slightly higher as the spheres diameter is reduced as we demonstrated in the calibration for the spheres reduction process shown in chapter 2. For the close-packed structure, G2 can not be appreciated due to the proximity of G1 which presents a spectrally broad and intense dip, hiding the real value for reflectance at γ= 0.71. However, after reduction G1 and G2 become spectrally separated allowing G2 to be clearly visible. This effect can be seen, for instance, in Fig. 8.3b for ff = 0.47 (γ= 0.92). As a final comment on mode evolution, it has to be stressed that the linear blueshit (see Fig. 8.3a) of the modes with reduction can be useful from the point of view of tuning different systems fabricated from the same initial spheres. Once the ratio has been estimated for one sample we can use optical characterization as a means to control the sphere diameter of the rest without employing SEM inspection [108], thereby making the process of tuning the optical features faster. Furthermore, the etching rates obtained could be slowed down or sped up depending on the conditions of the plasma process. 8.3 Tuning the spontaneous emission of dye dopped spheres So far we have considered the tunability of the optical response of optically passive samples as we vary the diameter of the spheres. The same tuning strategy can be adopted to modify 4Check Fig. 2.7 in chapter 2for the calibration on the diameter reduction-time ratio. 121 Chapter 8. Tuning the optical properties of a monolayer of spheres 122 (a) (b) Figure 8.3: Calculated (a) and experimental (b) reflectance spectra represented as a contour plot for a monolayer of Φ = 1µm spheres as the filling fraction is varied from the close-packed lattice (ff = 0.6) to one with a 100 nm reduction in diameter (ff = 0.44). the emission properties of similar samples with luminescent properties (an optically active sample). In this case we have used the same kind of samples used in the emission study of chapter 7: Rhodamine 6G doped spheres with Φ = 520nm deposited over a gold substrate. 122 123 Chapter 8. Tuning the optical properties of a monolayer of spheres As shown in chapter 7this diameter allows us to overlap the dye emission with the main modes we are interested in (G1,G2 and G3 in this case). Figure 8.4: Normal incidence reflectance (red) and emission (black) spectra for Φ = 520nm dye-doped spheres at three different filling fractions corresponding to γ= 1 (a), γ= 0.97 (b), and γ= 0.88 (c). Close-packed samples were subjected to an etching process for up to 7.5 min (corresponding to γ= 0.88 or a final ff= 0.41). Etching times were chosen in order to restrict the process to the linear reduction-rate regime (see Fig. 2.7) as in the Φ = 1µm case. Fig. 8.4 shows normal incidence reflectance and emission spectra for three different steps of the etching process. The reflectance obtained for the close-packed case is the same as the one shown in previous chapters for this diameter. When the diameter of the spheres is reduced under plasma etching the blue-shift of the modes, already studied for the Φ = 1µm case, now takes place in the visible spectral range. However, it has to be mentioned that in this case the absorbtion of gold plays a more relevant role since the blueshift produced by the tuning process will shift each mode towards the spectral region of gold strong absorbance. Unlike the case of the Φ = 1µm spheres (where the field enhancement for the SPP-like modes is barely affected by the tuning) in this case such modes (G1 and G3) will suffer a strong reduction in the total field intensity as we approach the strong absorbance region. This is specially evident for G3. Despite this fact, numerical simulations 123 Chapter 8. Tuning the optical properties of a monolayer of spheres 124 have demonstrated that the conclusions about the linear blueshift evolution for the larger sphere samples are still valid in this case. If we now consider the emission of the monolayer, the peaks of enhanced spontaneous emission follow the trend of dips in reflectance, corresponding to the modes of the structure. From those results it is also expected that in Fig. 8.4a (close-packed sample, ff= 0.6) the stronger emission enhancement takes place for ω= 0.71, corresponding to the G2 mode but with a smaller enhancement taking place for G3. In this way one can effectively tune the sample‘s emission by controlling the plasma process. (a) (b) Figure 8.5: Reflectance (a) and emission (b) as a contour plot for a monolayer of Φ = 520nm dye-doped PS spheres in a continuous filling fraction reduction process. Oxygen-plasma etching was carried out from the closepacked scenario (ff= 0.60) to a final filling fraction of ff= 0.41. Beside the spectral shift, changes in the magnitude of emission enhancement taking 124 125 Chapter 8. Tuning the optical properties of a monolayer of spheres place as the etching process advances are a combination of two factors. The first can be associated with the variations in the field confinement (see Fig. 8.2) and the second to the fact that the modes of the system are swiped across the dye’s broad emission so that enhanced emission should be more noticeable the closer a mode is to the dye‘s emission maximum. A clear example of the above is that of mode G1, of SPP-like character. While for ff = 0.6 (Fig. 8.4a) no enhanced emission takes place at the spectral position of this mode (ω= 0.62) owing to the fact that it is far from the dye’s emission, as we decrease the sphere diameter we shift its spectral position until for γ= 0.95 it eventually overlaps the dye’s emission (ff = 0.51). At that point, emission enhancement is already visible; moreover it is clear that as ffreduces and the mode blue-shifts, the emission increases what can be seen for ff = 0.41 and ω= 0.68 in Fig. 8.4c. The fact that such enhancement is not as large as for the other modes is due to the small spatial overlap between the dye and the field intensity, mainly concentrated near the gold surface. Finally it must be noted that as the etching process takes place the effects of disorder, broadening the peaks in reflectance, become more evident. This is probably to be responsible for the discrepancy between reflectance and emission of the G3 peak in the case of the smallest filling fraction (Figure 4c ). Here the reflectance peak broadens considerably, causing a part of the peak to fall in the high energy tail of the emission (where it is less efficient) and hence causing an asymmetry of the emission peak. A possible reason for this broadening could be a spatially inhomogenious etching of the spheres, which would add up to the intrinsic polydispersity of the spheres. The same way that was done for the Φ = 1µm spheres with reflectance measurements (see Fig. 8.3), as the sphere size is reduced in small steps we have plotted a map of the reflectance and emission with the measured spectra at normal incidence for the Φ = 520nm spheres (Fig. 8.5). Here, several facts are worth mentioning. Firstly, it is observed in the reflectance measurements map (Fig. 8.5a) that modes G2,G3 and G4 are better defined during the spheres reduction than in the experimental map for larger spheres (Fig. 8.3b). This effect is probably due to the experimental set-up. As already mentioned, for monolayers with Φ = 1µm spheres the optical response falls in the NIR spectral region. Hence, we performed the measurements with an FTIR spectrometer coupled to a microscope objective with smaller angular resolution than the Fourier imaging set-up described in chapter 3. As commented in chapter 5the former collects light from a range of ca. 10ofull angle while for the latter this range can be reduced to values lower than 1o, allowing better resolved spectra. Other possible effect is that the larger the spheres, the longer the plasma etching time needed to reduce to the same value the filling fraction of the lattice. As explained in chapter 2longer plasma time can affect the spheres in several ways (for example increasing their porosity) which can introduce differences between these two processes. Nevertheless, the behavior observed in reflectance for the Φ = 520nm qualitatively matches that of Φ=1µm one, indicating that although we are considering different spectral regions, where the dielectric constant of gold changes, the system still presents acceptable scalability. Let us analyze now the evolution of the emission (Fig. 8.5b) which can be observed to follow the reflectance variations for every point in the contour map. In particular we can trace the evolution of mode G2 and see that as we reduce the sphere diameter we can not only continuously shift its spectral position but also modulate its intensity, taking it to a minimum for ω= 0.94 (ff = 0.49). For even larger reductions we see how emission intensity recovers. The SPP-like G1 mode now appears as an enhancement in emission for ω= 0.95, as we shift it towards the dye’s emission. Finally mode G3 is also seen 125 of time. Obviously, the sample in space has to be defined at subwavelenght resolution if the near field is to be resolved. The sampling for the time variable depends on the stability required for the calculation. From Maxwell equations it is obtained that : ∂H ∂t =−1 µ∇×E−ρ µH(A.1) ∂E ∂t =−1 ∇×H−σ E(A.2) then, the vector components of the curl operator can be written in a separate manner yielding a system of six coupled equations: ∂Hi ∂t =−δijk 1 µ(∂Ej ∂k −∂Ek ∂j −ρHi) ∂Ei ∂t =δijk 1 (∂Hk ∂j −∂Hj ∂k −σEi) (A.3) with x, y, z =i, j, k. As both time and space have been discretized with steps (∆i,j,k) one can define any space point to be calculated as part of a three dimensional grid defined by (i, j, k)=(i∆x, j∆y, k∆z) (see diagram of Fig. ??). Any value for time propagation is also discretized to be t=n∆twith N being an integer. Therefore, centered finite differences (central-diifference) expressions can be used for space and time derivatives. The kind of grid and discrete derivative approximation chosen strongly affect the final performance of the simulation in both, accurately and time consuming [172]. In Yee’s notation, the first time partial derivative of a given function u, evaluated at the space point (i,j,k) can be written as: ∂u ∂t (i∆x, j∆y, k∆z, n∆t)∼un+1/2 i,j,k −un−1/2 i,j,k ∆t(A.4) This approximation of the discrete derivative is one of the simplest but still can be applied to the six equation system shown in A.1 A.2, such a way that a system of six discretized equations is obtained. In this case, each of that equations shown that each Hi,j,k or Ei,j,k at a time step n is assume to be simply the arythmetic average of the stored value of Hn−1/2 i,j,k (or E) at a time n−1/2 and the yet to be computed new value of Hn+1/2 i,j,k . Without entering too much detail, and as a result of the previous spatial and temporal discretization, when both fields are known on a single surface (E(x, y, z0) and H(x, y, z0)), the far field can be calculated from the discrete time derivative equations shown before at any discrete position in z (E(x, y, z) and H(x, y, z)). In order to solve that, it has to be notice that the functions for both field have to be known at the initial position. As follows from the previous discussion, the gridding is a fundamental parameter in the resolution which we solve the fields propagation. However, as the considered value must be finite, the boundary conditions at the edge of the grid can introduce significant artifacts. Among the several methods developed to make this boundaries to perform as infinite mediums most known are the absorbing boundary condition (ABC ) and the perfectly matched layer (PML). PML has been demonstrated to show the best results 1. First 1http://math.mit.edu/ stevenj/18.369/pml.pdf (a) (b) Figure A.1: (a) Positions of various field components in the spatial grid taken from reference [170]. The E components are in the middle of the edges and the H components are in the center of the faces. (b) Two dimensional grid representation valid. Shadow area represent different a different material. Figure extracted from reference [173]. defined by Berenger [174], its main difference with total absorbance boundaries is that such waves incident from anon-PML medium are not reflected back into the calculation grid in the analytical case. However, if discretization is being used, as in this case, little reflectance artifacts can be generated although not as lrge as in the ABC case. FDTD method allow for many approximations taking into account the systems to be solved. Due to the large amount of applications in photonics that needs from FDTD simulations, several computational tools have been implemented such a way that just the geometry and optical properties of the structure must be defined before simulation. Some of the most widespread are Meep [26] (free) or Rsoft (commercial). For our simulations we have used the software FDTD solutions provided by Lumerical company [175]. A.1 Geometry and material properties definition Lumerical FDT solutions allows us to define any structure by means of a CAD tool where each component of the structure can be accurately described in both, structure and optical properties. In the case of a monolayer on a given substrate, we first define a single sphere on the required substrate. It has to be notice that, as for any FDTD simulation, the time step is dependent on the size of the smallest finite difference cell in the grid for stability. The grid is defined to be a 3D cube with 3Φ/2 total dimension in the (x,y) plane while the z direction is chosen appropriately depending on the simulation parameters as will be explained below. The dielectric constants of sphere and substrate are set to values obtained either from literature tables (polystyrene) or from ellipsometry measurements (gold, silver and silicon). Surrounding media is always considered to be vacuum in this case. As previously commented, the resolution chosen for the grid is a crucial parameter, specially when working with non rectangular structure. If the grid resolution is not high enough, the shape is not accurately described. Therefore, the chosen gridding strongly affects the final accuracy of the calculated field distribution. Moreover, the grid truncation which define the total volume may introduce artifacts. In this case, we choose the walls in the vertical direction (z) to be PML layers in order Figure A.2: Positions of various field components in the spatial grid taken from reference [170]. The E components are in the middle of the edges and the H components are in the center of the faces. to avoid reflectance in that direction. Concerning the in-plane direction (x,y) there are two possible approaches. The most usual in non-periodic or complex structures is the definition of a total volume including the whole system. In this case, as we want to obtain the effect of an infinite 2D photonic crystal a cell as large as, at least, 100 spheres would be necessary to reproduce the effect of a infinitely large monolayer [140]. However, this approach means solving the field propagation at any point of the grid in a very large volume which is not affordable in terms of computational resources in most of the cases 2. Moreover, effects of finiteness of the lattice may introduce artifacts even if PML layers are used. The other approach, which is the one chosen for the simulations shown in this thesis, take advantage of the in-plane symmetry for 2D PCs and, in particular in the monolayer of spheres. A total volume for simulation of just one cell of the lattice is included in the simulation but the boundary conditions at the periodicity plane are set to reproduce the equivalent condition within the calculate volume. As a result, just one cell volume (approximately) must be calculated in each case. This approach strongly reduces the resources needed in each simulation as well as it increase the accuracy of the results. Fig. ?? shows an example of the main regions involved on the simulation and also an specific set-up for a blaze grating. Although represented in 2D it corresponds to a 3D calculation The general picture (left) define the main regions in the simulation volume. In that figure one find a region A which define the simulation volume of interest. B zone, defined by the yellow area accounting for the boundary condition area. C area is totally ignored. An blaze grating is defined in the right picture. Again, the total volume to be calculated is marked by the yellow box which length in the longitudinal direction match a period of the grating. Four regions are defined as I-IV. For I and IV regions the boundary of the simulation cell is set as PML. It avoids any beam reaching that boundaries to be reflected which ideally would be equivalent to an infinite volume in that directions. Concerning regions II and III, that boundaries are set to be periodic with the period of the grating. The white line correspond to the planar wave source covering the same surface than the grating in the periodicity plane. The arrow shows the angular incidence for the planar waves emitted by the source. 2Remember that, as λ=λ0/n for light propagating in an homogenious media with refractive index n, the higher the refractive index the higher the resolution to be used. A.2 Sources definition As we have shown, FDTD is a time domain method. Therefore, the electromagnetic fields are calculated as a function of time. As a consequence, the system being simulated can be excited by different type of sources (dipole, beam, modes, etc) depending on the problem to be solved. In our case, we are mainly interested on the optical response (in particular reflectance) of the monolayer when light is incident on the sample in normal incident way. As result, the most suitable source for our porpoise is a broadband continuous wave. However, most of the sources used in FDTD are pulsed in their temporal part. In the case of the the software considered here that pulse present a temporal shape given by : s(t) = sin(ω0(t−t0)e−(t−t0)2 2∆2(A.5) where ω0is the central frequency, t0the pulse initio and ∆tthe pulse width. This way, a beam injected into free space at a a position z0is described by E(x, y, z0, x, t) = E(x, y, z0, x)s(t) at a given time tduring the simulation. Taking into account the Fourier transform of eq. A.53, the optical response at any single frequency could be obtained with one single simulation if s(t) is a dirac delta. However, when discretization is considered, that limit can not be achieved and a broad pulse has to be used by simulation stability reasons. Therefore, the width of the pulse determines the frequency range to be calculated while it is the total number of samples (N) that the pulse is divided in that will determine the resolution of the final spectrum obtained from simulation. After the temporal component of the source has been set, it is propagated along the grid which make it necessary to spatially define the source. Fig. ?? shows the spatial configuration chosen for the simulation of this thesis. In that case, the source present a planar shape parallel to the grating surface. The source fulfills is placed at a given distance from the top of the grating. Using an infinite source matches the experimental set-up where a plane wave impinges the sample in its whole structure area at the same time. Moreover, using finite sources can introduce artifacts as diffraction at the source boundaries. The other kind of source very common in FDTD simulations is a punctual source describes as a dipole oscillating at a given frequency. This kind of sources are very useful when a polarized source is required or if the source has to be placed within the structure. This is the most common technique on simulating emissive devices as for example PC cavities with an inside emitter. It is also de usual approach to obtain the dispersion relation of PC slabs photonic crystals CITA. A.3 Retrieving information from simulations Once the source has been set, next step is defining the points of the grid where one is interested on collecting the field profile. This points we call monitors and can be defined as a point but also as a surface or a volume. As we know the vectorial field at each given point, the power at each monitor defined is calculated as the integration of the Pointing vector. Both reflectance and transmittance at a given point are calculated as the power collected at that point divided by the total power injected by the source. As we have performed normal incidence reflectance calculations, we have place monitors above and behind the monolayer of spheres. The chosen size for the monitors in our case 3s(ω) = Rexp(iωt)s(t)dt fulfilling the whole periodicity plane at a given distance from the monolayer in order to match the size of the source an make the reflectance calculation meaningful. Therefore, the pulse is injected at a given position, propagated for time long enough to collect its response in the monitors and plotted as the ratio between the injected and the power collected at the given monitor (see Fig. ??). Finally, one can also define a monitor to collect the value of each filed component at a given point. If storaged, it is easily obtained a map of the field distribution at final step of the simulation. In fact, if field is collected for each temporal step, it is obtained the total evolution of the field while propagating into the system. This kind of analysis has been performed during this thesis for those frequencies matching resonances of the monolayer of spheres obtaining the field distribution after normal incidence in several cases. Appendix B Fano line-shape: extracting Q factor of them modes from reflectance spectra If information about the physical properties of the modes of a slab PC are required (as for example confinement factors or Q-factors) FDTD calculations are routinely employed.An approach is to study the temporal decay of a dipole oscillating at the frequency of the mode at the spatial position where the mode is confined. Alternatively, one can extract the main characteristics of leaky modes directly from the reflectance/transmittance spctrum of the system. This approach is based on the Fano resonances largely studied in solid state and atomic physics [176]. This effect take place whenever a scattering process from any input state can take two different pathways, either directly to a continuum of extended states or resonantly through a discrete energy level. The interference between the two configurations gives rise to an asymmetric scattering lineshape. If light scattered by a periodic system is considered, the continuum of states available are all the radiative modes we can couple to. On the other hand, discrete states are those modes of the system defined as resonances itself (for example a leaky mode in a slab PC). As pointed out by Fan et.al, the optical response of high RI contrast slab PCs [30] account for this kind of resonance. On that study it was shown that the transmisison/reflection of a 2D PC slab can is formed by the sum of a set of sharp resonant features superimposed upon a smoothly varying background. The shape and intensity of those resonances accurately fit the behavior previously reported for Fano resonances which typically present a Lorentzian shape. If one consider a hybrid monolayer of dielectric spheres, the same model can be applied for the reflectance peaks (dips in transmission) shown in Fig. 4.3a. Radiative modes contributing to the mirror-like background can be associated to the Fabry-Perot oscillations described in chapter 4. On the other hand, if the leaky modes of the system are considered, we can fit any dip of a leaky mode to a Fano shape. The Fano resonance can be expressed as : R(ω) = A0+F0 [q+ 2(ω−ω0)/Γ]2 1 + [2(ω−ω0)/Γ]2(B.1) where ω0is the central frequency of the resonance, Γ is the resonance linewidth, A0 correspond to the amplitude of the non-resonant pathways and F0is a constant factor. The dimensionless parameter qis the ratio between resonant and the background amplitudes in the scattering process. This parameter accounts for the asymmetry of the resonance and can take positive or negative values.There are three different regimes. First, for large q, resonant scattering dominates direct scattering pathways and the 137 lineshape tend to a symmetric Lorentzian. Second, for small q, radiative scattering dominates and an inverse Lorentzian is obtained. Finally, when q∼1, the radiative scattering presents a similar amplitude than the guided ones and a strong asymmetric resonance is observed. Probably the most important value available from fitting the reflectance spectrum to a Fano shape is the Q factor of the resonance as shown in literature [177]. As already explained, the Q of leaky modes of a slab PC is related to the lifetime of those modes before they are re-scattered out of the structure. It is defined as : Q= 2πEtotal Edisipated =ω0 Γ(B.2) Although Q can be also calculated directly from spectrums as Q=ω0/FWHM (as shown in chapter ??) where FWHM is the full width half maximum, Fano‘s approximation provides valuable extra information. In particular, qparameter allows an study on the asymmetry of the resonances [178] which provides a measurement of the resonant effect respect to the radiative modes of the system. Bibliography [1] M. Inoue, K. Ohtaka, and S. Yanagawa, “Light scattering from macroscopic spherical bodies. ii. reflectivity of light and electromagnetic localized state in a periodic monolayer of dielectric spheres,” Physical Review B, vol. 25, p. 689, 1982. [2] E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Physical Review Letters, vol. 58, p. 2059, 1987. [3] S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Physical Physics Letters, vol. 58, 1987. [4] C. Kittel, Introduction to Solid State Physics. John Wiley and Sons,Inc, 1971. [5] K. Sakoda, Optical Properties of Photonic Crystals. Springer, 2001. [6] J. D. Joannopoulos, S. Johnson, J. Wing, and R. Meade, Photonic Crystals: Molding the Flow of Light. Princeton University Press, 2008. [7] B. Thompson, Electromagnetic Theory and Applications for Photonic Crystals. Taylor and Francis Group, 2006. [8] E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Physical Review Letters, vol. 67, no. 17, p. 2295, 1991. [9] A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature, vol. 405, p. 437, 2000. [10] C. Mazzoleni and L. Pavesi, “Application to optical components of dielectric porous silicon multilayers,” Applied Physics Letters, vol. 67, p. 2983, 1995. [11] J. B. Pendry, “Symemtry and transport of waves in one-dimensional disrdered systems,” Advances in Physics, vol. 43, p. 461, 1994. [12] B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for co2 laser transmission,” Nature, vol. 420, p. 650, 2002. 139 [13] M. C. Fuertes, F. J. Lopez-Alcaraz, M. C. Marchi, H. E. Troiani, V. Luca, H. Miguez, and G. Soler-Illia, “Photonic crystals from ordered mesoporous thin-film functional building blocks,” Advanced Functional Materials, vol. 17, p. 1247, 2007. [14] S. Colodrero, A. Mihi, L. H¨aggman, M. Oca˜na, G. Boschloo, A. Hagfeldt, and H. M´ıguez, “Porous one-dimensional photonic crystals improve the power-conversion efficiency of dye-sensitized solar cells,” Advanced Materials, vol. 21, p. 764, 2009. [15] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelengthdependent angular beam steering,” Applied Physics Letters, vol. 74, p. 1370, 1999. [16] S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature, vol. 394, no. 6690, p. 251, 1998. [17] A. Chelnokov, K. Wang, S. Rowson, P. Garoche, and J. M. Lourtioz, “Near-infrared yablonovite-like photonic crystals by focused-ion-beam etching of macroporous silicon,” Applied Physics Letters, vol. 77, no. 19, p. 2943, 2000. [18] M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature, vol. 404, p. 53, 2000. [19] M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nature Materials, vol. 3, p. 444, 2004. [20] J. Park, J. Moon, H. Shin, D. Wang, and M. Park, “Direct-write fabrication of colloidal photonic crystal microarrays by ink-jet printing,” Journal of Colloid and Interface Science, vol. 298, p. 713, 2006. [21] J. F. Galisteo-Lopez, M. Ibisate, R. Sapienza, L. S. Froufe-Perez, A. Blanco, and C. Lopez, “Self-assembled photonic structures,” Advanced Materials, vol. 23, p. 30, 2011. [22] V. Astratov, V. Bogomolov, A. Kaplyanskii, A. Prokofiev, L. Samoilovich, S. Samoilovich, and Y. Vlasov, “Optical spectroscopy of opal matrices with cds embedded in its pores: Quantum confinement and photonic band gap effects,” Il Nuovo Cimento D, vol. 17, p. 1349, 1995. [23] V. L. Alexeev, S. Das, D. N. Finegold, and S. A. Asher, “Photonic crystal glucosesensing material for noninvasive monitoring of glucose in tear fluid,” Clinical Chemestry, vol. 50, p. 2353, 2004. [24] S. Guldin, S. Hu` Ittner, M. Kolle, M. Welland, P. Mu` Iller-Buschbaum, R. Friend, U. Steiner, and N. Te` Itreault, “Dye-sensitized solar cell based on a three-dimensional photonic crystal,” Nano Letters, vol. 10, p. 2303, 2010. [25] A. Arsenault, D. Puzzo, I. Manners, and G. Ozin, “Photonic-crystal full-colour displays,” Nature Photonics, vol. 1, p. 468, 2007. [26] D. S. G. Jhonson, “http://ab-initio.mit.edu/photons/tutorial/.” [27] C. Pollok, Fundamentals of Optoelectronics. Tom Casson, 1995. [28] D. M. Hammer, “http://wwwhome.math.utwente.nl/ hammer/index.html.” [29] D. L. Lee, Electromagnetic principles of integrated optics. Wiley, 1986. [30] S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Physical Review B, vol. 65, p. 235112, 2002. [31] K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Physical Review B, vol. 73, p. 115126, 2006. [32] L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Physical Rreview B, vol. 73, p. 235114, 2006. [33] M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Reports on Progress in Physics, vol. 73, p. 096501, 2010. [34] M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Physical Review Letters, vol. 87, p. 253902, 2001. [35] M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature, vol. 462, p. 78, 2009. [36] S. Zanotto, M. Liscidini, and L. C. Andreani, “Light trapping regimes in thin-film silicon solar cells with a photonic pattern,” Optics Express, vol. 18, p. 4260, 2010. [37] Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature, vol. 425, p. 944, 2003. [38] Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-sizelow-refractive-index material,” Optics Letters, vol. 29, p. 1626, 2004. [39] H. Miyazaki and K. Ohtaka, “Near-field images of a monolayer of periodically arrayed dielectric spheres,” Physical Review B, vol. 58, p. 6920, 1998. [40] L. A. Dorado, R. A. Depine, G. Lozano, and H. M´ıguez, “Physical origin of the high energy optical response of three dimensional photonic crystals,” Optics Express, vol. 15, p. 17754, 2007. [41] H. Raether, Surface plasmons on smooth and rough surfaces and on gratings. Springer-Verlag, 1988. [42] W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature, vol. 424, p. 824, 2003. [43] J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nature Materials, vol. 7, p. 442, 2008.