Quantum Dots in Photonic Structures Lecture 4: Photonic crystals Jan Suffczyński Wednesdays, 17.00, SDT Projekt Fizyka Plus nr POKL.04.01.02-00-034/11 współfinansowany przez Unię Europejską ze środków Europejskiego Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki

Plan for today 2. Two-dimensional photonic crystals – band structure DBR mirrors 3. Two-dimensional photonic crystal – fabrication methods

Reminder Refractive index: 𝑫= 𝜀 0 𝜀 𝑟 (𝜔)𝑬 Polarization by EM wave 𝑫= 𝜀 0 𝜀 𝑟 (𝜔)𝑬 Polarization by EM wave Complex dielectric function

After: András Szilágyi Reminder Refractive index: 𝑫= 𝜀 0 𝜀 𝑟 (𝜔)𝑬 Polarization by EM wave Complex dielectric function Simultaneous description of refraction and absorption Speed light in a medium: 𝑣= 𝑐 𝑛 𝜀 𝑟 (𝜔) =𝑛 𝜔 +𝑖∙𝜅(𝜔) Dispersion After: András Szilágyi

Reminder Photonic crystal: periodic arrangement of dielectric (or metallic…) objects periodic refractive index contrast the period comparable to the wavelength of light in the material. 1D photonic crystal: Distributed Bragg Reflector (DBR) Example calculation: Transfer Matrix Method

Bragg mirror from the University of Warsaw MgTe High n: Cd0.86Zn0.14Te Cd0.86Zn0.14Te buffer GaAs substrate 20 stack DBR Cd0.86Zn0.14Te Low n: 20 x Superlattice 1:1 1 mm SEM: T. Jakubczyk J.-G. Rousset

Refractive index engineering For a good DBR we need a pair of materials that have: large refractive index contrast Δn = nhigh-nlow lattice paramters as close as possible

Bragg mirror lattice matched to CdTe The structure 15 par ZnTe 53nm superlattice 18 periods ZnTe 53 nm ZnTe 0.7 nm superlattice MgTe 0.9 nm ZnTe buffer 1000 nm ZnTe 0.7 nm MgSe 1.3 nm ZnSe 62 nm GaAs 1 μm W. Pacuski, UW

Bragg mirror lattice matched to CdTe The structure 15 par ZnTe 53nm supersieć 18 powtórzeń ZnTe 53 nm ZnTe 0.7 nm supersieć MgTe 0.9 nm ZnTe buffer 1000 nm ZnTe 0.7 nm MgSe 1.3 nm ZnSe 62 nm GaAs W. Pacuski, UW

Bragg mirror lattice matched to CdTe The structure 15 par ZnTe 53nm supersieć 18 powtórzeń ZnTe 53 nm MgSe 1.3 nm ZnTe 0.7 nm MgTe 0.9 nm 1 nm ZnTe 0.7 nm supersieć MgTe 0.9 nm ZnTe buffer 1000 nm ZnTe 0.7 nm MgSe 1.3 nm ZnSe 62 nm GaAs W. Pacuski, UW

DBR mirror and DBR cavity reflectivity Microcavity W. Pacuski, UW

DBR mirror and DBR cavity reflectivity Q = λ/Δλ = 3600 Microcavity

CdTe based microcavity – 60 pairs But sometimes technology makes jokes…

Planar cavity with DBR mirrors Stop band Δλ=(n1-n2)/π(n1+n2) Δθ ~ 20o typically in the case og GaAs/AlAs DBR Reflectivity Antinode of the field in the center of the cavity lBr λ-cavity Electric field distribution Reflection of a planar DBR microcavity consisting of a top and bottom mirror with 15 and 21 repeats of GaAs/AlAs. In (a) a 240 nm-thick bulk GaAs cavity is incorporated. Field distributions corresponding to (a) at wavelength of (c) 841 nm Cavity mode Exponential decay of the stationary field from the center of the cavity

Towards 2D and 3D photonic crystals Low index of refraction High index of refraction 3D photonic crystal

Photonic crystals – how it works? a>>l incoherent scattering a a~l coherent scattering a a<<l averaging a Photonic crystals

2D, 3D photonic bandgap? J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light

Dispersion relation and origin of the band gap in 1D Consider medium of refractive index n1 and light of wavelenght of l = 2a Light line: free space Light line: medium n1 frequency ω Rob Engelen standing wave in n1 π/a wave vector k

Dispersion relation and origin of the band gap in 1D Consider the stack of layers of refractive indeces n1 and n2 and light of wavelenght of l = 2a n1 n2 n1 n2 n1 n2 n1 frequency ω a Rob Engelen π/a n1: high index material n2: low index material wave vector k

Dispersion relation and origin of the band gap in 1D Consider the stack of layers of refractive indeces n1 and n2 and light of wavelenght l = 2a standing wave in n2 n1 n2 n1 n2 n1 n2 n1 frequency ω Rob Engelen standing wave in n1 π/a n1: high index material n2: low index material wave vector k

Dispersion relation and origin of the band gap in 1D Consider the stack of layers of refractive indeces n1 and n2 and light of wavelenght l = 2a standing wave in n2 n1 n2 n1 n2 n1 n2 n1 frequency ω bandgap Rob Engelen standing wave in n1 π/a n1: high index material n2: low index material wave vector k

frequency ω wave vector k π/a -π/a Bloch wave with wave vector k is equal to Bloch wave with wave vector k+m2p/a: modified slide from Rob Engelen

k + 2π/a is equivalent to k Band diagram k is periodic: k + 2π/a is equivalent to k

Band diagram frequency ω wave vector k π/a -π/a frequency ω π/a -π/a frequency ω wave vector k π/a -π/a -2π/a 2π/a -2π/a 2π/a This is the first Brillouin zone modified slide from Rob Engelen

Band diagram – one more view Anticrossing of modes leading to formation of the band gap

Another look- Bragg scattering conditions When a wave impinges on a crystal it will be reflected at a particular set of lattice planes characterized by its reciprocal lattice vector g only if the so-called Bragg condition is met If the Bragg condition is not met, the incoming wave just moves through the lattice and emerges on the other side of the crystal (when neglecting absorption)

Photonic crystals - introductory example from the prevous lecture 1. Bragg scattering Regardless of how small the reflectivity r is from an individual scatterer, the total reflection R from a semi infinite structure: Complete reflection when: Propagation of the light in crystal inhibited when Bragg condition satisfied Origin of the photonic bang gap

Reciprocal lattice

Dispersion relation for 2D photonic crystal Dispersion relation for a 1D photonic crystal (solid lines). The boundary of the first Brillouin zone is denoted by two vertical lines. The dispersion lines in the uniform material are denoted by dashed lines. They are folded into the first Brillouin zone taking into account the identity of the wave numbers which differ from each other by a multiple of 2π/a. When two dispersion lines cross, they repel each other and a photonic bandgap appears. 2D square lattice

Dispersion relation for 2D photonic crystal 2D hexagonal lattice Top view of the geometry of the 2D crystal for the numerical calculation of the transmission and the Bragg reflection spectra by means of the plane-wave expansion method Transmittance (right-hand side) and the dispersion relation (left-hand side) of a hexagonal lattice for (a) the E polarization and (b) the H polarization in the Γ-X direction. The ordinate is the normalized frequency. On the left-hand side of these figures, solid lines represent symmetric modes and dashed lines represent antisymmetric modes. The latter cannot be excited by the incident plane wave because of the mismatching of their spatial symmetry, and so they do not contribute to the light transmission. Opaque frequency regions are clearly observed in the transmission spectrum where no symmetric mode exists. (After [49]) Band gap: no propagation possible at that frequency density of optical states (DOS) is 0

Dispersion relation for 2D photonic crystal vs transmission

Photonic crystals in Nature Sea mouse Opal McPhedran et al.

Artificial PC production: Layer-by-Layer Lithography • Fabrication of 2d patterns in Si or GaAs is very advanced (think: Pentium IV, 50 million transistors) …inter-layer alignment techniques are only slightly more exotic So, make 3d structure one layer at a time Need a 3d crystal with constant cross-section layers

A Schematic [ M. Qi, H. Smith, MIT ]

Making Rods & Holes Simultaneously Steven G. Johnson, MIT side view s u b s t r a t e Si top view Steven G. Johnson, MIT

Making Rods & Holes Simultaneously expose/etch holes A A A A s u b s t r a t e A A A A A A A A A A A A A A A A A A A A A Steven G. Johnson, MIT

Making Rods & Holes Simultaneously backfill with silica (SiO2) & polish A A A A s u b s t r a t e A A A A A A A A A A A A A A A A A A A A A Steven G. Johnson, MIT

Making Rods & Holes Simultaneously deposit another Si layer l a y e r 1 A A A A s u b s t r a t e A A A A A A A A A A A A A A A A A A A A A Steven G. Johnson, MIT

Making Rods & Holes Simultaneously dig more holes offset & overlapping l a y e r 1 B B B B A A A A s u b s t r a t e B B B B A A A A B B B A A A B B B B A A A A A B A B A B A B A B A B A B B B B A A A Steven G. Johnson, MIT

Making Rods & Holes Simultaneously backfill l a y e r 1 B B B B A A A A s u b s t r a t e B B B B A A A A B B B A A A B B B B A A A A A B A B A B A B A B A B A B B B B A A A Steven G. Johnson, MIT

Making Rods & Holes Simultaneously etcetera (dissolve silica when done) l a y e r 3 A A A A one period l a y e r 2 C C C C l a y e r 1 B B B B A A A A s u b s t r a t e C B C B C B C B A A A A C B C B C B C A A A C B C B C B C B A A A A C A B C A B C A B C C A B C A B C A B C A B C B C B C B C A A A Steven G. Johnson, MIT

Making Rods & Holes Simultaneously etcetera l a y e r 3 A A A A one period l a y e r 2 C C C C l a y e r 1 B B B B hole layers A A A A s u b s t r a t e C B C B C B C B A A A A C B C B C B C A A A C B C B C B C B A A A A C A B C A B C A B C C A B C A B C A B C A B C B C B C B C A A A Steven G. Johnson, MIT

Making Rods & Holes Simultaneously etcetera l a y e r 3 A A A A one period l a y e r 2 C C C C l a y e r 1 B B B B rod layers A A A A s u b s t r a t e C B C B C B C B A A A A C B C B C B C A A A C B C B C B C B A A A A C A B C A B C A B C C A B C A B C A B C A B C B C B C B C A A A Steven G. Johnson, MIT

7-layer E-Beam Fabrication Point out 60nm veins 30nm inter-layer alignment [ M. Qi, et al., Nature 429, 538 (2004) ]

Three-dimensional Si photonic crystal Y. A. Vlasov et al., Nature 414, 289 (2001) S.-Y. Lin et al., Nature 394, 251 (1998)

Two-Photon Lithography 2-photon probability ~ (light intensity)2 lens some chemistry (polymerization) 3d Lithography …dissolve unchanged stuff (or vice versa)

Lithography – the best friend of a man λ = 780nm resolution = 150nm 7µm S. Kawata et al., Nature 412, 697 (2001). 780 nm with a 150-femtosecond pulse width was used as an exposure source. Bull sculpture produced by raster scanning; the process took 180 min. To depict the tiny features of our microbulls, a fabrication accuracy of about 150 nm was needed, which we achieved by using nonlinear processes of the photochemicalreactions involved. Polymerization occurred only in the vicinity of the focal spot and the size of solidified voxels (three-dimensional volume elements) was reduced because of the quadratic dependence of TPA probability on the photon fluence density. Furthermore, photogenerated radicals in this volume were subject to scavenging by dissolved oxygen molecules, so polymerization reactions were not initiated and propagated if the exposure energy was less than a critical value. This property defined a TPA threshold and excluded the low-intensity lobe (the zero-order edge and the subsidiary maxima of the Airy pattern) from polymerization and thus further reduced the voxel size (Fig. 1g). The diffraction limit imposed by Rayleigh criteria was simply a measure of focal-spot size, and did not put any restraint on the actual size of solidified voxels. (3 hours to make) 2µm S. Kawata et al., Nature(2001)

Holographic Lithography Four beams make 3d-periodic interference pattern k-vector differences give reciprocal lattice vectors (i.e. periodicity) absorbing material Photonic crystals for the visible spectrum by holographic lithography AUTHOR: Sharp,-D.-N.; Campbell,-M.; Dedman,-E.-R.; Harrison,-M.-T.; Denning,-R.-G.; Turberfield,-A.-J. SOURCE: Optical-and-Quantum-Electronics. Jan.-March 2002; 34(1-3): 3-12 University of Oxford [ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ] (1.4µm) beam polarizations + amplitudes (8 parameters) give unit cell [ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]

Holographic Lithography [ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ] Photonic crystals for the visible spectrum by holographic lithography AUTHOR: Sharp,-D.-N.; Campbell,-M.; Dedman,-E.-R.; Harrison,-M.-T.; Denning,-R.-G.; Turberfield,-A.-J. SOURCE: Optical-and-Quantum-Electronics. Jan.-March 2002; 34(1-3): 3-12 University of Oxford [ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ] In this article we describe a new holographic process (Campbell et al. 2000) that is intrinsically well adapted to the production of microstructured photonic materials. A 3D laser interference pattern exposes a photosensitive polymer precursor (photoresist) rendering exposed areas insoluble; unexposed areas are dissolved away leaving a 3D photonic crystal formed of cross-linked polymer with air-ҐҐled voids. This process allows ҐҐxible design of the structure of a unit cell and thus of the optical properties of the microstructured material. We create an interference pattern at the intersection of four non-coplanarbeams from a frequency-tripled, injection-seeded Nd:YAG laser (wavelengthk 1/4 355 nm) (Campbell et al. 2000). Such a pattern has 3D translationalsymmetry; its primitive reciprocal lattice vectors are equal to the diҐҐrencesbetween the wavevectors of the beams. Patterns with any Bravais lattice maybe generated in this way, and interference patterns have been used to createperiodic optical traps for laser-cooled atoms (Grynberg et al. 1993) (we notethat Berger et al. (1997) have independently suggested lithography using 3Dinterference, and that Shoji and Kawata (2000) have used a combination oftwo-dimensional (2D) and one-dimensional patterns to create 3D microstructure).We require a pattern with high intensity contrast that can betranslated into high solubility contrast in the photoresist; we also require theinsoluble polymer to form a connected but open structure. For a particularset of wavevectors, eight parameters describing the intensities and polarizationvectors of the four beams are required to deҐҐe the intensity distributionwithin a unit cell (i.e. the basis of the interference pattern). The eight freebeam parameters give considerable freedom to engineer the content of theunit cell, which in turn determines the photonic band structure (Ho et al.1990; Yablonovitch et al. 1991b). Fig. 3. (a) SEM image of a fcc polymeric photonic crystal generated by exposure of a 10 µm film ofphotoresist to the interference pattern shown in Fig. 2(a). The top surface is a (111) plane; the film hasbeen fractured along (111) cleavage planes of the photonic crystal structure. Scale bar, 10 µm. (b) Close-upof (111) surface and {111} cleavage planes. Scale bar, 5 µm. (c) Simulation of (b) formed by cutting aconstant-intensity surface along narrow bonds. (d) Close-up of a (111) surface. Scale bar, 1 µm. (e) Titaniaphotonic crystal with higher refractive index contrast produced by using the polymeric structure as atemplate. The surface is slightly tilted from the (111) plane. Scale bar, 1 µm. 10µm huge volumes, long-range periodic, fcc lattice…backfill for high contrast

Colloidal photonic crystals Colvin, MRS Bulletin 26, (2001)

3D Bandgap Structure frequency (c/a) for gap at λ = 1.55µm, K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990). frequency (c/a) 11% gap MPB tutorial, http://ab-initio.mit.edu/mpb K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152 (1990). for gap at λ = 1.55µm, sphere diameter ~ 330nm overlapping Si spheres MPB tutorial, http://ab-initio.mit.edu/mpb

Experimental Air-guiding Photonic Crystal Fiber [ R. F. Cregan et al., Science (1999) ] 5µm

How to create a cavity within a photonic crystal? Reminder: Bragg mirror cavity Two equivalent views: Cavity = mirror + resonator+ mirror Cavity = mirror with a „good” defect

Single-Mode Cavity Bulk Crystal Band Diagram A point defect frequency (c/a) A point defect can push up a single mode from the band edge G X M G Γ X M

…here, doubly-degenerate “Single”-Mode Cavity Bulk Crystal Band Diagram frequency (c/a) A point defect can pull down a “single” mode X …here, doubly-degenerate G X M G M Γ X

Tunable Cavity Modes frequency (c/a) Ez: monopole dipole

“1d” Waveguides + Cavities = Devices resonant filters waveguide splitters high-transmission sharp bends channel-drop filters

Outlook: Optical Microcavities Vahala, Nature 424, 839 (2003) Microcavity characteristics: Quality factor Q, mode volume V