Three-dimensional Photonic Band Gap Research Articles

Symmetries and wave functions of photons confined in three-dimensional photonic band gap superlattices

Symmetries and wave functions of photons confined in three-dimensional photonic band gap superlattices

Diamond-lattice photonic crystals assembled from DNA origami.

Colloidal self-assembly allows rational design of structures on the micrometer and submicrometer scale. One architecture that can generate complete three-dimensional photonic bandgaps is the diamond cubic lattice, which has remained difficult to realize at length scales comparable with the wavelength of visible or ultraviolet light. In this work, we demonstrate three-dimensional photonic crystals self-assembled from DNA origami that act as precisely programmable patchy colloids. Our DNA-based nanoscale tetrapods crystallize into a rod-connected diamond cubic lattice with a periodicity of 170 nanometers. This structure serves as a scaffold for atomic-layer deposition of high-refractive index materials such as titanium dioxide, yielding a tunable photonic bandgap in the near-ultraviolet.

Scaling Theory of Wave Confinement in Classical and Quantum Periodic Systems.

Functional defects in periodic media confine waves-acoustic, electromagnetic, electronic, spin, etc.-in various dimensions, depending on the structure of the defect. While defects are usually modeled by a superlattice with a typical band-structure representation of energy levels, determining the confinement associated with a given band is highly nontrivial and no analytical method is known to date. Therefore, we propose a rigorous method to classify the dimensionality of wave confinement. Starting from the confinement energy and the mode volume, we use finite-size scaling to find that ratios of these quantities raised to certain powers yield the confinement dimensionality of each band. Our classification has negligible additional computational costs compared to a band structure calculation and is valid for any type of wave, both quantum and classical, and in any dimension. In the quantum regime, we illustrate our method on electronic confinement in 2D hexagonal boron nitride (BN) with a nitrogen vacancy, in agreement with previous results. In the classical case, we study a three-dimensional photonic band gap cavity superlattice, where we identify novel acceptorlike behavior. We briefly discuss the generalization to quasiperiodic lattices.

Enhanced absorption in thin and ultrathin silicon films by 3D photonic band gap back reflectors

Since thin and ultrathin silicon films have limited optical absorption, we explore the effect of a nanostructured back reflector to recycle the unabsorbed light. As a back reflector, we investigate a three-dimensional (3D) photonic band gap crystal made from silicon that is readily integrated with the thin silicon films. We numerically obtain the optical properties by solving the 3D time-harmonic Maxwell equations using the finite-element method, and model silicon with experimentally determined optical constants. The absorption enhancement spectra and the photonic band gap generated current density are obtained by weighting the absorption spectra with the AM 1.5 standard solar spectrum. We study thin films in two different regimes, much thicker (L Si = 2400 nm) or much thinner (L Si = 80 nm) than the wavelength of light. For L Si = 2400 nm thin film, the 3D photonic band gap crystal enhances the spectrally averaged (λ = 680 nm to 880 nm) silicon absorption by 2.22 × (s −pol.) to 2.45 × (p −pol.), which exceeds the enhancement of a perfect metal back reflector (1.47 to 1.56 ×). The absorption is considerably enhanced by the (i) broadband angle and polarization-independent reflectivity in the 3D photonic band gap, and (ii) the excitation of many guided modes in the film by the crystal’s surface diffraction leading to greatly enhanced path lengths. For L Si = 80 nm ultrathin film, the photonic crystal back reflector yields a striking average absorption enhancement of 9.15 ×, much more than 0.83 × for a perfect metal. This enhancement is due to a remarkable guided mode that is confined within the combined thickness of the ultrathin film and the photonic crystal’s Bragg attenuation length. An important feature of the 3D photonic band gap is to have a broad bandwidth, which leads to the back reflector’s Bragg attenuation length being much shorter than the silicon absorption length. Consequently, light is confined inside the thin film and the remarkable absorption enhancements are not due to the additional thickness of the photonic crystal back reflector. We briefly discuss a number of high-tech devices that could profit from our results, including ultrathin film solar cells.

Topology optimization of 3D photonic crystals with complete bandgaps.

The design of photonic crystals with complete bandgaps has recently received considerable research focus for numerous reasons. This work leverages well-known nonlinear programming techniques to alleviate the non-smoothness caused by degenerate eigenvalues such that topology optimization problems can be solved with the open-source IPOPT software. A fully-vectorial plane wave expansion technique is used with an iterative eigensolver to efficiently predict dispersion properties of candidate structures. Nonlinear programming is employed to solve the inverse problem of designing three-dimensional periodic structures that exhibit complete two-dimensional (2D) and three-dimensional (3D) photonic bandgaps. Mesh refinement is performed to alleviate the large computational burden of designing and analyzing photonic crystals, and a periodic density filter is implemented to impose a minimum feature size for manufacturability considerations.

Local density of optical states in the three-dimensional band gap of a finite photonic crystal

A three-dimensional (3D) photonic band gap crystal is an ideal tool to completely inhibit the local density of optical states (LDOS) at every position in the crystal throughout the band gap. This notion, however, pertains to ideal infinite crystals, whereas any real crystal device is necessarily finite. This raises the question as to how the LDOS in the gap depends on the position and orientation inside a finite-size crystal. Therefore, we employ rigorous numerical calculations using finite-difference time-domain (FDTD) simulations of 3D silicon inverse woodpile crystals filled with air or with toluene, as previously studied in experiments. We find that the LDOS versus position decreases exponentially into the bulk of the crystal. From the dependence on dipole orientation, we infer that the characteristic LDOS decay length $\ell_{\rho}$ is mostly related to far-field dipolar radiation effects, whereas the prefactor is mostly related to near-field dipolar effects. The LDOS decay length has a remarkably similar magnitude as the Bragg length for directional transport, which suggests that the LDOS in the crystal is dominated by vacuum states that tunnel from the closest interface towards the position of interest. Our work leads to design rules for applications of 3D photonic band gaps in emission control and lighting, quantum information processing, and in photovoltaics.

Reflectivity of three-dimensional GaAs photonic band-gap crystals of finite thickness

We study the optical reflectivity of three-dimensional (3D) photonic band gap crystals with increasing thickness. The crystals consist of GaAs plates with nanorod arrays that are assembled by an advanced stacking method into high-quality 3D woodpile structures. We observe intense and broad reflectivity peak with stop bands that correspond to a broad gap in the photonic band structures. The maximum reflectivity quickly reaches high values even for a few crystal layers. Remarkably, the bandwidth of the stop bands hardly decreases with increasing crystal thickness, in good agreement with FDTD simulations. This behavior differs remarkably from the large changes observed earlier in weakly interacting 3D photonic crystals. The nearly constant bandwidth and high reflectivity are rationalized by multiple Bragg interference that occurs in strongly interacting photonic band gap crystals, whereby the incident light scatters from multiple reciprocal lattice vectors simultaneously, in particular from oblique ones that are parallel to a longer crystal dimension and thus experience hardly any finite size effects. Our new insights have favorable consequences for the application of 3D photonic band gap crystals, notably since even thin structures reveal the full band gap functionality, including devices that shield quantum bits from vacuum fluctuations.

Experimental probe of a complete 3D photonic band gap.

The identification of a complete three-dimensional (3D) photonic band gap in real crystals typically employs theoretical or numerical models that invoke idealized crystal structures. Such an approach is prone to false positives (gap wrongly assigned) or false negatives (gap missed). Therefore, we propose a purely experimental probe of the 3D photonic band gap that pertains to any class of photonic crystals. We collect reflectivity spectra with a large aperture on exemplary 3D inverse woodpile structures that consist of two perpendicular nanopore arrays etched in silicon. We observe intense reflectivity peaks (R>90%) typical of high-quality crystals with broad stopbands. A resulting parametric plot of s-polarized versus p-polarized stopband width is linear ("y=x"), a characteristic of a 3D photonic band gap, as confirmed by simulations. By scanning the focus across the crystal, we track the polarization-resolved stopbands versus the volume fraction of high-index material and obtain many more parametric data to confirm that the high-NA stopband corresponds to the photonic band gap. This practical probe is model-free and provides fast feedback on the advanced nanofabrication needed for 3D photonic crystals and stimulates practical applications of band gaps in 3D silicon nanophotonics and photonic integrated circuits, photovoltaics, cavity QED, and quantum information processing.

Observation of Complete Photonic Bandgap in Low Refractive Index Contrast Inverse Rod-Connected Diamond Structured Chalcogenides

Three-dimensional complete photonic bandgap materials or photonic crystals block light propagation in all directions. The rod-connected diamond structure exhibits the largest photonic bandgap known to date and supports a complete bandgap for the lowest refractive index contrast ratio down to $n_{high}/n_{low} \sim 1.9$. We confirm this threshold by measuring a complete photonic bandgap in the infrared region in Sn--S--O $(n\sim1.9)$ and Ge--Sb--S--O $(n\sim2)$ inverse rod-connected diamond structures. The structures were fabricated using a low-temperature chemical vapor deposition process via a single-inversion technique. This provides a reliable fabrication technique of complete photonic bandgap materials and expands the library of backfilling materials, leading to a wide range of future photonic applications.

Cartesian light: Unconventional propagation of light in a three-dimensional superlattice of coupled cavities within a three-dimensional photonic band gap

We explore the unconventional propagation of light in a three-dimensional (3D) superlattice of coupled resonant cavities in a 3D photonic band gap crystal. Such a 3D cavity superlattice is the photonic analogue of the Anderson model for spins and electrons in the limit of zero disorder. Using the plane-wave expansion method, we calculate the dispersion relations of the 3D cavity superlattice with the cubic inverse woodpile structure that reveal five coupled-cavity bands, typical of quadrupole-like resonances. For three out of five bands, we observe that the dispersion bandwidth is significantly larger in the $(k_x, k_z)$-diagonal directions than in other directions. To explain the directionality of the dispersion bandwidth, we employ the tight-binding method from which we derive coupling coefficients in 3D. For all converged coupled-cavity bands, we find that light hops predominantly in a few high-symmetry directions including the Cartesian $(x, y, z)$ directions, therefore we propose the name "Cartesian light". Such 3D Cartesian hopping of light in a band gap yields propagation as superlattice Bloch modes that differ fundamentally from the conventional 3D spatially-extended Bloch wave propagation in crystals, from light tunneling through a band gap, from coupled-resonator optical waveguiding, and also from light diffusing at the edge of a gap.

Three-dimensional photonic band gap cavity with finite support: Enhanced energy density and optical absorption

We study numerically the transport and storage of light in a 3D photonic band gap crystal doped by a single embedded resonant cavity. The crystal has finite support since it is surrounded by vacuum, as in experiments. Therefore, we employ the finite element method to model the diamond-like inverse woodpile crystal that consists of two orthogonal arrays of pores in a high-index dielectric such as silicon and that has experimentally been realized by CMOS-compatible methods. A point defect that functions as the resonant cavity is formed in the proximal region of two selected orthogonal pores with a radius smaller than the ones in the bulk of the crystal. We present a field-field cross-correlation method to identify resonances in the finite-support crystal with defect states that appear in the 3D photonic band gap of the infinite crystal. Out of 5 observed angle-independent cavity resonances, one is s-polarized and 4 are p-polarized for light incident in the X or Z directions. It is remarkable that quality factors up to Q = 1000 appear in such thin structures (3 unit cells). We find that the optical energy density is remarkably enhanced at the cavity resonances by up to 2400x the incident energy density in vacuum or up to 1200x the energy density of the equivalent effective medium. We find that an inverse woodpile photonic band gap cavity with a suitably adapted lattice parameter reveals substantial absorption in the visible range. Below the 3D photonic band gap, Fano resonances arise due to interference between the discrete fundamental cavity mode and the continuum light scattered by the photonic crystal. We argue that the five eigenstates of our 3D photonic band gap cavity have quadrupolar symmetry, in analogy to d-like orbitals of transition metals. We conclude that inverse woodpile cavities offer interesting perspectives for applications in optical sensing and photovoltaics.

Realization of a three-dimensional photonic topological insulator.

Confining photons in a finite volume is highly desirable in modern photonic devices, such as waveguides, lasers and cavities. Decades ago, this motivated the study and application of photonic crystals, which have a photonic bandgap thatforbids light propagation in all directions1-3. Recently, inspired by the discoveries of topological insulators4,5, the confinement of photons with topological protection has been demonstrated in two-dimensional (2D) photonic structures known as photonic topological insulators6-8, with promising applications in topological lasers9,10 and robust optical delay lines11. However, a fully three-dimensional (3D) topological photonic bandgap has not been achieved. Here we experimentally demonstrate a 3D photonic topological insulator with an extremely wide (more than 25 per cent bandwidth) 3D topological bandgap. The composite material (metallic patterns on printed circuit boards) consists of split-ring resonators (classical electromagnetic artificial atoms) with strong magneto-electric coupling and behaves like a 'weak' topological insulator (that is, with an even number of surface Dirac cones), or a stack of 2D quantum spin Hall insulators. Using direct field measurements, we map out both the gapped bulk band structure and the Dirac-like dispersion of the photonic surface states, and demonstrate robust photonic propagation along a non-planar surface. Our work extends the family of 3D topological insulators from fermions to bosons and paves the way for applications in topological photonic cavities, circuits and lasers in 3D geometries.

Investigation of pass-band characteristics in three-dimensional PBG structure with line defect

The pass-band characteristics of three-dimensional photonic band-gap (PBG) structure with line defect were investigated in this paper, which were fabricated with high-K TiO2 ceramic based on 3D printing technique and sintering process. The transmission characteristics were measured using transmission/reflection method and computed using FDTD method. Experimental results show that pass-band frequency of the waveguide between 13.5 and 14.7 GHz is within the band-gap of PGB structure (12.2–16.4 GHz) when the practical lattice constant of PBGs is 11.34 mm and practical aspect ratio is 0.296, and transmission efficiency is 80%, which increases by 60% than that of low-dielectric Al2O3 ceramic with transmission efficiency of 50%. Moreover, the computing results basically agree with the experimental results. These results indicate that waveguide of PBG structure with high-K materials will be able to obtain higher transmission efficiency, which will provide a method for adjusting transmission characteristics of PBG structure and promote PBG structure into the practical applications.

Rainbow trapping in a chirped three-dimensional photonic crystal

Light localization and intensity enhancement in a woodpile layer-by-layer photonic crystal, whose interlayer distance along the light propagation direction is gradually varied, has been theoretically predicted and experimentally demonstrated. The phenomenon is shown to be related to the progressive slowing down and stopping of the incident wave, as a result of the gradual variation of the local dispersion. The light localization is chromatically resolved, since every frequency component is stopped and reflected back at different positions along the crystal. It has been further discussed that the peculiar relation between the stopping position and the wave vector distribution can substantially increase the enhancement factor to more than two orders of magnitude. Compared to previously reported one- and two-dimensional photonic crystal configurations, the proposed scheme has the advantage of reducing the propagation losses by providing a three-dimensional photonic bandgap confinement in all directions. The slowing down and localization of waves inside photonic media can be exploited in optics and generally in wave dynamics, in many applications that require enhanced interaction of light and matter.

Reflectivity calculated for a three-dimensional silicon photonic band gap crystal with finite support

We study numerically the reflectivity of three-dimensional (3D) photonic crystals with a complete 3D photonic band gap, with the aim to interpret recent experiments. We employ the finite element method to study crystals with the cubic diamond-like inverse woodpile structure. The high-index backbone has a dielectric function similar to silicon. We study crystals with a range of thicknesses up to ten unit cells ($L \leq 10 c$). The crystals are surrounded by vacuum, and have a finite support as in experiments. The polarization-resolved reflectivity spectra reveal Fabry-P{\'e}rot fringes related to standing waves in the finite crystal, as well as broad stop bands with nearly $100~\%$ reflectivity, even for thin crystals. From the strong reflectivity peaks, it is inferred that the maximum reflectivity observed in experiments is not limited by finite size. The frequency ranges of the stop bands are in excellent agreement with stop gaps in the photonic band structure, that pertain to infinite and perfect crystals. The frequency ranges of the observed stop bands hardly change with angle of incidence, which is plausible since the stop bands are part of the 3D band gap. Moreover, this result supports the previous assertion that intense reflection peaks measured with a large numerical aperture provide a faithful signature of the 3D photonic band gap. The Bragg attenuation lengths $L_{B}$ exceed the earlier estimates based on the width of the stop band by a factor $6$ to $9$. Hence crystals with a thickness of $12$ unit cells studied in experiments are in the thick crystal limit ($L >> L_{B}$). In our calculations for p-polarized waves, we also observe an intriguing hybridization of the zero reflection of Fabry-P{\'e}rot fringes and the Brewster angle, which has not yet been observed in experiments.

Three-dimensional all-dielectric photonic topological insulator

The discovery of two-dimensional topological photonic systems has transformed our views on electromagnetic propagation and scattering of classical waves, and a quest for similar states in three dimensions, known to exist in condensed matter systems, has been put forward. Here we demonstrate that symmetry protected three-dimensional topological states can be engineered in an all-dielectric platform with the electromagnetic duality between electric and magnetic fields ensured by the structure design. Magneto-electric coupling playing the role of a synthetic gauge field leads to a topological transition to an insulating regime with a complete three-dimensional photonic bandgap. An emergence of surface states with conical Dirac dispersion and spin-locking is unimpeded. Robust propagation of surface states along two-dimensional domain walls defined by the reversal of magneto-electric coupling is confirmed numerically by first principle studies. It is shown that the proposed system represents a table-top platform for emulating relativistic physics of massive Dirac fermions and the surface states revealed can be interpreted as Jackiw-Rebbi states confined to the interface between two domains with opposite particle masses.

All-optical dynamical Casimir effect in a three-dimensional terahertz photonic band gap

We identify an architecture for the observation of all-optical dynamical Casimir effect in realistic experimental conditions. We suggest that by integrating quantum wells in a three-dimensional (3D) photonic band-gap material made out of large-scale ($\ensuremath{\sim}200\ensuremath{-}\ensuremath{\mu}\mathrm{m}$) germanium logs, it is possible to achieve ultrastrong light-matter coupling at terahertz frequencies for the cyclotron transition of a two-dimensional electron gas interacting with long-lived optical modes, in which vacuum Rabi splitting is comparable to the Landau level spacing. When a short, intense electromagnetic transient of duration $\ensuremath{\sim}250\phantom{\rule{0.28em}{0ex}}\mathrm{fs}$ and carrying a peak magnetic field $\ensuremath{\sim}5\phantom{\rule{0.28em}{0ex}}\mathrm{T}$ is applied to the structure, the cyclotron transition can be suddenly tuned on resonance with a desired photon mode, switching on the light-matter interaction and leading to a Casimir radiation emitted parallel to the quantum well plane. The radiation spectrum consists of sharp peaks with frequencies coinciding with engineered optical modes within the 3D photonic band gap, and its characteristics are extremely robust to the nonradiative damping which can be large in our system. Furthermore, the absence of continuum with associated low-energy excitations for both electromagnetic and electronic quantum states can prevent the rapid absorption of the photon flux which is likely to occur in other proposals for all-optical dynamical Casimir effect.

Light-trapping for room temperature Bose-Einstein condensation in InGaAs quantum wells.

We demonstrate the possibility of room-temperature, thermal equilibrium Bose-Einstein condensation (BEC) of exciton-polaritons in a multiple quantum well (QW) system composed of InGaAs quantum wells surrounded by InP barriers, allowing for the emission of light near telecommunication wavelengths. The QWs are embedded in a cavity consisting of double slanted pore (SP2) photonic crystals composed of InP. We consider exciton-polaritons that result from the strong coupling between the multiple quantum well excitons and photons in the lowest planar guided mode within the photonic band gap (PBG) of the photonic crystal cavity. The collective coupling of three QWs results in a vacuum Rabi splitting of 3% of the bare exciton recombination energy. Due to the full three-dimensional PBG exhibited by the SP2 photonic crystal (16% gap to mid-gap frequency ratio), the radiative decay of polaritons is eliminated in all directions. Due to the short exciton-phonon scattering time in InGaAs quantum wells of 0.5 ps and the exciton non-radiative decay time of 200 ps at room temperature, polaritons can achieve thermal equilibrium with the host lattice to form an equilibrium BEC. Using a SP2 photonic crystal with a lattice constant of a = 516 nm, a unit cell height of 2a=730nm and a pore radius of 0.305a = 157 nm, light in the lowest planar guided mode is strongly localized in the central slab layer. The central slab layer consists of 3 nm InGaAs quantum wells with 7 nm InP barriers, in which excitons have a recombination energy of 0.944 eV, a binding energy of 7 meV and a Bohr radius of aB = 10 nm. We take the exciton recombination energy to be detuned 35 meV above the lowest guided photonic mode so that an exciton-polariton has a photonic fraction of approximately 97% per QW. This increases the energy range of small-effective-mass photonlike states and increases the critical temperature for the onset of a Bose-Einstein condensate. With three quantum wells in the central slab layer, the strong light confinement results in light-matter coupling strength of ℏΩ = 13.7 meV. Assuming an exciton density per QW of (15aB)-2, well below the saturation density, in a 2-D box-trap with a side length of 10 to 500 µm, we predict thermal equilibrium Bose-Einstein condensation well above room temperature.

Method for making a single-step etch mask for 3D monolithic nanostructures

Current nanostructure fabrication by etching is usually limited to planar structures as they are defined by a planar mask. The realization of three-dimensional (3D) nanostructures by etching requires technologies beyond planar masks. We present a method for fabricating a 3D mask that allows one to etch three-dimensional monolithic nanostructures using only CMOS-compatible processes. The mask is written in a hard-mask layer that is deposited on two adjacent inclined surfaces of a Si wafer. By projecting in a single step two different 2D patterns within one 3D mask on the two inclined surfaces, the mutual alignment between the patterns is ensured. Thereby after the mask pattern is defined, the etching of deep pores in two oblique directions yields a three-dimensional structure in Si. As a proof of concept we demonstrate 3D mask fabrication for three-dimensional diamond-like photonic band gap crystals in silicon. The fabricated crystals reveal a broad stop gap in optical reflectivity measurements. We propose how 3D nanostructures with five different Bravais lattices can be realized, namely cubic, tetragonal, orthorhombic, monoclinic and hexagonal, and demonstrate a mask for a 3D hexagonal crystal. We also demonstrate the mask for a diamond-structure crystal with a 3D array of cavities. In general, the 2D patterns on the different surfaces can be completely independently structured and still be in perfect mutual alignment. Indeed, we observe an alignment accuracy of better than 3.0 nm between the 2D mask patterns on the inclined surfaces, which permits one to etch well-defined monolithic 3D nanostructures.

Design of a three-dimensional photonic band gap cavity in a diamondlike inverse woodpile photonic crystal

We theoretically investigate the design of cavities in a three-dimensional (3D) inverse woodpile photonic crystal. This class of cubic diamondlike crystals has a very broad photonic band gap and consists of two perpendicular arrays of pores with a rectangular structure. The point defect that acts as a cavity is centered on the intersection of two intersecting perpendicular pores with a radius that differs from the ones in the bulk of the crystal. We have performed supercell band structure calculations with up to $5\ifmmode\times\else\texttimes\fi{}5\ifmmode\times\else\texttimes\fi{}5$ unit cells. We find that up to five isolated and dispersionless bands appear within the 3D photonic band gap. For each isolated band, the electric-field energy is localized in a volume centered on the point defect, hence the point defect acts as a 3D photonic band gap cavity. The mode volume of the cavities resonances is as small as 0.8 ${\ensuremath{\lambda}}^{3}$ (resonance wavelength cubed), indicating a strong confinement of the light. By varying the radius of the defect pores we found that only donorlike resonances appear for smaller defect radius, whereas no acceptorlike resonances appear for greater defect radius. From a 3D plot of the distribution of the electric-field energy density we conclude that peaks of energy are found in sharp edges situated at the point defect, similar to how electrons collect at such features. This is different from what is observed for cavities in noninverted woodpile structures. Since inverse woodpile crystals can be fabricated from silicon by CMOS-compatible means, we project that single cavities and even cavity arrays can be realized, for wavelength ranges compatible with telecommunication windows in the near infrared.