Three-dimensional Photonic Band Research Articles

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

We perform a computational study of confined photonic states that appear in a three-dimensional (3D) superlattice of coupled cavities, resulting from a superstructure of intentional defects. The states are isolated from the vacuum by a 3D photonic band gap, using a diamondlike inverse woodpile crystal structure, and they exhibit “Cartesian” hopping of photons in high-symmetry directions. We investigate the confinement dimensionality to verify which states are fully 3D-confined, using a recently developed scaling theory to analyze the influence of the structural parameters of the 3D crystal. We create confinement maps that trace the frequencies of 3D-confined bands for select combinations of key structural parameters, namely the pore radii of the underlying regular crystal and of the defect pores. We find that a certain minimum difference between the regular and defect pore radii is necessary for 3D-confined bands to appear, and that an increasing difference between the defect pore radii from the regular radii supports more 3D-confined bands. In our analysis, we find that their symmetries and spatial distributions are more varied than electronic orbitals known from solid-state physics. We surmise that this difference occurs since the confined photonic orbitals derive from global Bloch states governed by the underlying superlattice structure, whereas single-atom orbitals are localized. Based on this realization, we suggest that the extent symmetries of “photonic orbitals” could possibly translate to novel macroscopic behaviors of “photonic solid-state matter,” never before seen in the standard electronic solid-state systems. We also discover pairs of degenerate 3D-confined bands with p-like orbital shapes and mirror symmetries matching the symmetry of the superlattice. Finally, we investigate the enhancement of the local density of optical states for cavity quantum electrodynamics applications. We find that donorlike superlattices, i.e., where the defect pores are smaller than the regular pores, provide greater enhancement in the air region than acceptorlike structures with larger defect pores, and thus offer better prospects for doping with quantum dots and ultimately for 3D networks of single photons steered across strongly coupled cavities. Published by the American Physical Society 2024

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.

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.

Highly efficient GPU eigensolver for three-dimensional photonic crystal band structures with any Bravais lattice

A null-space free method with the FFT-based matrix–vector multiplications was proposed to solve the Maxwell equations that model the three-dimensional photonic crystals. The most time-consuming parts of this method were the FFT-based matrix–vector multiplications. In this article, we propose new mathematical formulas to compute the FFT-based matrix–vector multiplications and derive highly efficient algorithms on top of the NVIDIA GPU architecture. The resulting algorithms are approximately two- to threefold faster than the previous algorithms. We have successfully used a single NVIDIA Tesla P100 GPU to solve a set of generalized eigenvalue problems of 5,184,000 dimensions in 17 to 22 seconds for each problem. Furthermore, we ported the codes to a GPU cluster and achieved near linear scalability. To our knowledge, these GPU implementations of the proposed algorithms are the fastest implementations. The schemes can be applied to simulate a three-dimensional photonic crystal with all 14 Bravais lattices. These highly efficient schemes and codes raise possibilities for large-scale and near real-time numerical simulations for novel physical discoveries and engineering applications of photonic crystals.

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.

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.

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.

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.

Signature of a three-dimensional photonic band gap observed on silicon inverse woodpile photonic crystals

We have studied the reflectivity of CMOS-compatible three-dimensional silicon inverse woodpile photonic crystals at near-infrared frequencies. Polarization-resolved reflectivity spectra were obtained from two orthogonal crystal surfaces using an objective with a high numerical aperture. The spectra reveal broad peaks with maximum reflectivity of $67%$ that are independent of the spatial position on the crystals. The spectrally overlapping reflectivity peaks for all directions and polarizations form the signature of a broad photonic band gap with a relative bandwidth up to $16%$. This signature is supported with stop gaps in plane-wave band-structure calculations and with the frequency region of the expected band gap.

Controllable atom–photon entanglement near a 3D anisotropic photonic band edge

The coherent control of the entanglement between a three-level atom located within a three-dimensional (3D) anisotropic photonic band gap (PBG) material, with one transition frequency near the edge of the PBG, and its radiation field is demonstrated. As a result of quantum interference and photon localization, the entanglement can be decreased or increased depending on the relative phase between the control laser coupling, the two upper levels and the pump laser pulse used to create an excited state of the atom in the form of a coherent superposition of the two upper levels. Unlike the free-space case, the steady-state entanglement is strongly dependent on the externally prescribed initial conditions. This non-zero steady-state entanglement is achieved by virtue of the localization of light in the vicinity of the emitting atom.

Photonic bands in two-dimensional metallodielectric photonic crystals composed of metal coated cylinders

Photonic bands in two-dimensional metallodielectric (MD) periodic systems composed of metal coated cylinders are investigated theoretically based on frequency dependent plane-wave expansion method. For the case of E-polarization, although the thickness of metal coating is less than half of the cylinder’s radius, most of MD photonic bands are the same as photonic bands composed of pure metal cylinders. This property provides us with a way to substitute metal photonic crystals with MD photonic crystals in many applications. In addition, flatbands are discovered in MD photonic band structures, which can be tuned by changing the thickness of metal coating while other photonic bands do not change their positions. For the case of H-polarization, the lowest frequency band gap (between the first and the second bands) can open up when the thickness of metal coating is thick enough. According to approximate calculation based on Maxwell–Garnett type effective medium theory and comparison with recent studies on three-dimensional MD photonic band structures, we predict that the lowest frequency band gap is not because of Bragg scattering but result from the individual metal coated dielectric cylinders, so that the gap is independent on geometry of photonic crystal lattices. Then, numerical calculation validates that our prediction is right.

Constructing 3D crystal templates for photonic band gap materials using holographic optical tweezers

A simple and robust method is presented for the construction of 3-dimensional crystals from silica and polystyrene microspheres. The crystals are suitable for use as templates in the production of three-dimensional photonic band gap (PBG) materials. Manipulation of the microspheres was achieved using a dynamic holographic assembler (DHA) consisting of computer controlled holographic optical tweezers. Attachment of the microspheres was achieved by adjusting their colloidal interactions during assembly. The method is demonstrated by constructing a variety of 3-dimensional crystals using spheres ranging in size from 3 microm down to 800 nm. A major advantage of the technique is that it may be used to build structures that cannot be made using self-assembly. This is illustrated through the construction of crystals in which line defects have been deliberately included, and by building simple cubic structures.

Photonic Amorphous Diamond Structure with a 3D Photonic Band Gap

We report that a full three-dimensional (3D) photonic band gap (PBG) is formed in a photonic amorphous structure in spite of complete lack of lattice periodicity. It is numerically shown that the structure "photonic amorphous diamond" possesses a sizable 3D PBG (18% of the center frequency for Si-air dielectric contrast) and that it can confine light at a defect as strongly as conventional photonic crystals can. These findings present important new insight into the origin of 3D PBG formation and open new possibilities in developing 3D PBG materials.

Ultrafast optical switching of three-dimensional Si inverse opal photonic band gap crystals

We present ultrafast optical switching experiments on three-dimensional photonic band gap crystals. Switching the Si inverse opal is achieved by optically exciting free carriers by a two-photon process. We probe reflectivity in the frequency range of second order Bragg diffraction where the photonic band gap is predicted. We find good experimental switching conditions for free-carrier plasma frequencies between 0.3 and 0.7 times the optical frequency ω: we thus observe a large frequency shift of up to Δω∕ω=1.5% of all spectral features including the peak that corresponds to the photonic band gap. We deduce a corresponding large refractive index change of ΔnSi′∕nSi′=2.0%, where nSi′ is the refractive index of the silicon backbone of the crystal. The induced absorption length is longer than the sample thickness. We observe a fast decay time of 21 ps, which implies that switching could potentially be repeated at GHz rates. Such a high switching rate is relevant to future switching and modulation applications.

Development of 3D Ceramic Photonic Bandgap Structures

The three-dimensional (3D) photonic band gap material is a material that there exists a full photonic band gap in which waves are forbidden to propagate whatever the polarization or the direction of propagation. In order to obtain photonic bandgap in lower range, we focus on the fabrication of PBG materials of diamond structure with TiO2 powder mixed with SiO2. The inverse epoxy structure with periodic diamond lattices in millimeter order has been fabricated by stereolithographic rapid prototyping. TiO2 slurry was filled into the epoxy structure and then cold isostatic pressing was applied. After sintering at 700K for 5hrs, the epoxy was burnt out and the designed structure was maintained perfectly. The calculated band diagram shows that there exists an absolute photonic band gap for all wave vectors. The measurement of transmission from 10 to 20 GHz in <100> direction shows that a complete band gap is formed at about 14.7-18.5 GHz. The magnitude of the maximum attenuation is as large as 30 dB at 17 GHz.