Enhancing light–matter interactions on a chip is of paramount importance for classical and quantum photonics, sensing, and energy harvesting applications. Several photonic geometries have been developed, allowing high extraction efficiencies, enhanced light–matter interactions, and control over the spontaneous emission dynamics of solid-state quantum light sources. To this end, a device geometry resilient to nanofabrication imperfections, providing high-quality light confinement and control over the emitted light properties, would be desirable. We demonstrate that aperiodic arrangements, whose geometry is inspired by natural systems where scattering elements are arranged following Fibonacci series, represent a platform for enhancing the light–matter interaction in on-chip nanophotonic devices, allowing us to achieve efficient visible light confinement. We use optically active defect centers in silicon nitride as internal light sources to image and characterize, by means of microphotoluminescence spectroscopy, the individual optical modes confined by photonic membranes with Vogel-spiral geometry. By studying the statistics of the measured optical resonances, in combination with rigorous multiple scattering theory, we observe lognormal distributions and report quality factors with values as high as 2201 ± 443. Our findings improve the understanding of the fundamental physical properties of light-emitting Vogel-spiral systems and show their application to active nanophotonic devices. These results set the basis for further development of quantum devices that leverage the unique properties of aperiodic Vogel spiral order on a chip, including angular momentum states, thus producing mode structures for information processing and communications.

Engineered photonic devices play a key role in the development of on-chip sources of classical and quantum light for a variety of applications including nanolasers,1 single-photon sources,2 sensors,3 and energy harvesters.4 Ideally, one would like nanophotonic device geometries that show scalability (i.e., the ability of fabricating more than one device that has exactly the same, as-designed, optical properties), enhanced light–matter interaction (tunable to a broad range of wavelengths, from the visible to the telecommunication band), control over spontaneous emission dynamics (allowing, for instance, enhanced coherence in the emission and increased repetition rates), and control over the properties of the emitted light (such as the emission directionality and its optical angular momentum). Different photonic geometries have been developed to achieve these goals, and, among those, photonic crystals have shown high quality in light confinement at both telecommunications and near-infrared wavelengths5 and strong lifetime control;6 however scalability and reproducibility of the results is an issue since nanometer-scale accuracy in the fabrication is needed.7 This is particularly evident for devices operating at visible wavelengths where, given the dimensions involved, two-dimensional photonic crystal cavities have proved to be much more challenging to fabricate compared to those operating at longer wavelengths, resulting in limited performance that further depart from simulated values.8,9 Micropillar geometries have assured high-brightness and coherent emission,10 and planar gratings have allowed the realization of broadband, bright, and pure single-photon emission.11 On the other hand, disordered photonic crystal waveguides have allowed less stringent fabrication requirements, high-quality light confinement over a wide range of wavelengths,12,13 control over the spontaneous emission dynamics of single emitters,14 and optical sensing.15 Control over the optical angular momentum has also been achieved by integrating emitters within metamaterials.16 

In this work, we propose a photonic platform, based on aperiodic geometries, which allows efficient visible light confinement and the co-existence of several confined optical modes within devices that require less stringent fabrication accuracies, compatible with silicon technology and with the integration of quantum emitters operating at visible wavelengths, like defect centers in diamond and two-dimensional materials and molecules.

Such a design relies on the concept of aperiodic order17–19 that, for instance, characterizes the peculiar geometry of quasicrystals,20,21 i.e., long-range ordered systems that lack translational symmetry, to achieve light confinement in two and three dimensions. Beyond quasicrystals, aperiodic order can be used to engineer photonic devices with well-defined deterministic mathematical rules,18,22,23 providing compatibility with planar nanofabrication technologies,24 as well as distinctive spectral and optical properties,25–28 including optical angular momentum,29 characteristics that make them appealing for quantum photonics applications.30–33 

We demonstrate the potential of nanophotonic devices characterized by a bio-inspired deterministic aperiodic structure: in particular, we design, fabricate, and characterize bioinspired nanodevices with Vogel-spiral geometry [see Fig. 1(a) and the supplementary material for more details] that were originally introduced in relation to the fascinating geometrical problems of phyllotaxis.34–36 This particular deterministic aperiodic structure was intensively studied in the last decade and was found to have interesting fundamental optical properties for nanophotonic and nanoplasmonic applications, including vector-wave localization,37 polarization-insensitive light diffraction,38 enhanced second-harmonic generation, increased light extraction,31,39,40 control of spontaneous emission dynamics,41 and omni-directional photonic bandgaps.42–44 In this work, we show that Vogel spiral active membranes enable us to achieve efficient light confinement and represent a platform for enhancing the light–matter interaction on a chip.

FIG. 1.

Si3N4 golden-angle Vogel-spiral device designs. (a) Scanning electron micrograph (SEM) of a suspended silicon nitride aperiodic photonic device with golden-angle Vogel-geometry. (b) Local density of optical states (LDOS) map calculated at the center of the spiral as a function of the wavelength λ and of the air-hole diameters d. We have considered 101 different values of the d parameter to design these structures. (c) One dimensional-LDOS line-cuts as a function of wavelength for three different hole diameters: 165 (black line), 190 (blue line), and 215 nm (red line), for devices with an average interparticle separation of 273 nm.

FIG. 1.

Si3N4 golden-angle Vogel-spiral device designs. (a) Scanning electron micrograph (SEM) of a suspended silicon nitride aperiodic photonic device with golden-angle Vogel-geometry. (b) Local density of optical states (LDOS) map calculated at the center of the spiral as a function of the wavelength λ and of the air-hole diameters d. We have considered 101 different values of the d parameter to design these structures. (c) One dimensional-LDOS line-cuts as a function of wavelength for three different hole diameters: 165 (black line), 190 (blue line), and 215 nm (red line), for devices with an average interparticle separation of 273 nm.

Close modal

We experimentally investigate, by means of microphotoluminescence spectroscopy, the optical properties of suspended silicon nitride (Si3N4) membranes, where air holes are arranged in a golden-angle Vogel-spiral geometry [see Fig. 1(a) and the supplementary material]. The devices are designed by evaluating the local density of optical states (LDOS) computed to an accurate spectral method. The LDOS describes the radiation dynamics of a source embedded into an arbitrary structure, from which the spectral location of high-quality factor modes, such as those near the bandgap,28,43,45,46 can be predicted. Specifically, we utilized the rigorous theory of multipolar expansion to evaluate the two-dimensional electromagnetic Green's tensor, from which the LDOS is calculated (for more details, see the supplementary material). The LDOS is evaluated for arbitrary arrays of parallel and nonoverlapping circular cylinders embedded in a nonabsorbing medium. The analysis is limited to the transverse electric polarization only, as this is the polarization for which a bandgap is expected to occur in an air-hole membrane.47 To take into account the out-of-plane losses, we have applied an effective refractive index method where the material dispersion of the dielectric material is replaced by the effective index of the fundamental guided mode in the unperturbed (without air-holes) three-dimensional heterostructure48 (see the supplementary material for more details). Figure 1(b) shows a map of the LDOS as a function of wavelength and air-hole diameters d etched in a silicon nitride membrane, and (c) depicts one-dimensional cuts. Specifically, the LDOS is evaluated at the center of a spiral characterized by 350 air holes and by an averaged interparticle separation of 273 nm. A clear photonic bandgap and pseudogaps (secondary gaps of smaller amplitudes), identified by bright yellow streaks in Fig. 1(b), are visible, and their behavior is determined by the unique multifractal structural properties of the golden-angle Vogel spiral.28 For example, there are several peaks, more clearly visible in Fig. 1(c), which correspond to long-lived modes generated by the first-neighbor distributions of the spiral elements.37 Localized band edge modes are formed when ring-shaped regions with similar hole separation are sandwiched between two regions with different hole separations, creating a photonic heterostructure.28 Moreover, these band edge modes are often spatially extended, long-lived,37 and less sensitive to local perturbations. This makes golden-angle Vogel spirals a very appealing photonic platform, due to more robust fabrication tolerances than traditional photonic crystals.42 The presence of bandgaps despite the relatively low index contrast between silicon nitride and air is related to the long-range order in a nearly isotropic geometry.22,28,43,49 Isotropic gaps also imply reduced group velocity modes and, therefore, increased light–matter interaction, thus making these devices interesting for nonlinear optics applications and for the realization of low-threshold lasers.42,47

We fabricate free-standing 340-nm-thick membranes in silicon nitride grown by plasma-enhanced chemical vapor deposition on a silicon substrate, into which 1000 air holes arranged in a golden-angle spiral geometry are etched (see the supplementary material for more details about the growth and the fabrication of the devices). A scanning electron micrograph image of a fabricated device is shown in Fig. 1(a). As previously reported, Si3N4 can emit radiation over a broad range of wavelengths, typically spanning from ∼600 to ∼850 nm, once excited with external light sources.13 We will use such intrinsic photoluminescence as the internal light source to characterize the confined optical modes13 supported by the golden-angle spiral structure.

The fabricated sample is placed under a confocal optical microscope and illuminated with a 455 nm-light emitting diode that excites the silicon nitride luminescence over an area of about 50 × 50 μm2. The emission is then imaged on an electron-multiplied charge-coupled device [see Fig. 2(a)]. This setup allows us to image the confined optical modes, as shown in Fig. 2(b) where a photoluminescence image of the optical modes with wavelengths above 550 nm confined by the structure is superimposed to the SEM of the fabricated device. We observe a high-intensity region in the center and at the border of the device and confined modes in the photonic crystal area.

FIG. 2.

Microphotoluminescence setup and optical imaging of the confined optical modes. (a) Schematic of the confocal microphotoluminescence setup (not to scale), comprising a light emitting diode (LED) with the emission wavelength centered at 455 nm and a continuous wave (CW) laser emitting at 405 nm as an excitation source, focused by a 50× microscope objective (with numerical aperture NA = 0.65) onto a sample placed on an xy-translation stage. The detection is carried out using an Electron Multiplying Charge Coupled Device (EMCCD) for photoluminescence imaging or using a CCD for spectral characterization (LPF = 550 nm long-pass filter and FM = flip mirror); the squares represent beam splitters. (b) Photoluminescence image, collected under 455-nm light-emitting diode illumination with a power density of 40 W/cm2, of the optical modes with wavelengths above 550 nm confined by the structure, superimposed to the SEM image of the fabricated device.

FIG. 2.

Microphotoluminescence setup and optical imaging of the confined optical modes. (a) Schematic of the confocal microphotoluminescence setup (not to scale), comprising a light emitting diode (LED) with the emission wavelength centered at 455 nm and a continuous wave (CW) laser emitting at 405 nm as an excitation source, focused by a 50× microscope objective (with numerical aperture NA = 0.65) onto a sample placed on an xy-translation stage. The detection is carried out using an Electron Multiplying Charge Coupled Device (EMCCD) for photoluminescence imaging or using a CCD for spectral characterization (LPF = 550 nm long-pass filter and FM = flip mirror); the squares represent beam splitters. (b) Photoluminescence image, collected under 455-nm light-emitting diode illumination with a power density of 40 W/cm2, of the optical modes with wavelengths above 550 nm confined by the structure, superimposed to the SEM image of the fabricated device.

Close modal

The sample is then excited with a 405 nm continuous-wave laser with an excitation spot diameter of ∼2 μm, which allows light excitation and collection from specific areas of the nanophotonic devices. The emitted light is collected and sent to a grating spectrometer equipped with a charge-coupled device for spectral characterization [see Fig. 2(a)].

An example of a collected microphotoluminescence spectrum is shown in Fig. 3(a), where sharp resonances, a signature of light confinement, appear above the broad emission from the silicon nitride material. By fitting the resonant peaks with Lorentzian functions, we extract quality factors reaching 2201 ± 443 [see Figs. 3(b) and 3(c)], thus exceeding values reported for photonic crystal cavities in silicon nitride operating at visible wavelengths.8,9

FIG. 3.

Microphotoluminescence measurements proving efficient light confinement on a Si3N4 chip. (a) Example of a broad-range photoluminescence spectrum showing sharp optical resonances and signature of light confinement in deterministic aperiodic photonic devices. The spectrum was collected at room temperature, under 405 nm CW laser illumination, with a power density of 1.8 kW/cm2. The solid lines represent Lorentzian fits to the data (symbols). (b) and (c) Examples of optical resonances, collected under the same conditions as panel (a), showing Lorentzian fits and the extracted quality factors Q.

FIG. 3.

Microphotoluminescence measurements proving efficient light confinement on a Si3N4 chip. (a) Example of a broad-range photoluminescence spectrum showing sharp optical resonances and signature of light confinement in deterministic aperiodic photonic devices. The spectrum was collected at room temperature, under 405 nm CW laser illumination, with a power density of 1.8 kW/cm2. The solid lines represent Lorentzian fits to the data (symbols). (b) and (c) Examples of optical resonances, collected under the same conditions as panel (a), showing Lorentzian fits and the extracted quality factors Q.

Close modal

The optical resonances supported by open and planar Vogel spirals are embedded in a three-dimensional environment, and they can leak out of the two-dimensional plane, according to their quality factors. As a result, light is not fully confined in the spiral plane and the resulting optical resonances are actually three-dimensional electromagnetic quasimodes.37 The aperiodic photonic devices under study provide a large number of optical resonances, resembling the behavior observed in disordered photonic crystal waveguides, confining light along one dimension in the plane.13 However, the resonances under study are three-dimensional electromagnetic quasimodes with two-dimensional geometrical support of Vogel spiral arrays, with clear advantages for applications since the higher dimensionality of the present devices provides more easily addressable and larger active areas.

In Ref. 41, it was demonstrated that the lifetime of single quantum dots operating at near-infrared wavelength is strongly modified by the modulation of the density of optical states due to the aperiodic photonic structure. In that work, the spectral signature of confined optical modes was not clearly visible in the photoluminescence spectra. In the present work, instead, by using the broad and homogeneous intrinsic photoluminescence of silicon nitride at visible wavelengths,13 we probe the confined optical modes and analyze their wavelength and quality factor distributions as a function of the structural parameters of the photonic devices under investigation.

We have extensively characterized and tested Si3N4 devices with 165, 190, and 215 nm diameters of the air holes. Examples of the statistics of the collected optical resonances, plotted as a function of wavelength and quality factor Q (a factor that estimates the quality in the light confinement, evaluated as the ratio between the optical resonance central wavelength and its linewidth), are shown in Fig. 4. We observe that reducing the air-hole diameter (while keeping the rest of the parameters constant) results in a shift of the optical resonances toward longer wavelengths, in agreement with the bandgap calculations shown in Fig. 1(c), and in an increase in the average quality factors (see the supplementary material for more discussion, including a comparison to simulations). We also show a wide tunability of the wavelength of the confined optical modes over a 100 nm wavelength window [see Fig. 4(a)], proving the suitability of aperiodic systems for the realization of devices with broad-band operation. Concurrently, reducing the air-hole diameters yields an increase in quality factors. This trend is opposite to what is predicted in our simulations, which are only able to predict in-plane quality factors, neglecting the increased out-of-plane losses of more strongly confined modes50 (more details can be found in the supplementary material).

FIG. 4.

Statistical analysis of the optical resonances confined by aperiodic Si3N4 photonic devices. (a) Statistics of the quality factor distributions collected from microphotoluminescence spectra, like the ones shown in Fig. 3, plotted as a function of emission wavelength, for aperiodic photonic devices with hole diameters of 165 (gray), 190 (blue), and 215 (red) nm. (b) Probability distribution of the quality factors shown in panel (a), plotted with the same color coding. Colored continuous lines are lognormal fits to the data.

FIG. 4.

Statistical analysis of the optical resonances confined by aperiodic Si3N4 photonic devices. (a) Statistics of the quality factor distributions collected from microphotoluminescence spectra, like the ones shown in Fig. 3, plotted as a function of emission wavelength, for aperiodic photonic devices with hole diameters of 165 (gray), 190 (blue), and 215 (red) nm. (b) Probability distribution of the quality factors shown in panel (a), plotted with the same color coding. Colored continuous lines are lognormal fits to the data.

Close modal

Figure 4(b) shows that the probability distributions of the measured quality factors follow lognormal statistics. Interestingly, the lognormal distribution of Q-factors has been predicted and observed in disordered systems in the Anderson-localized regime.51–53 However, different from traditional random media where the lognormal distribution of Q-factors is associated with exponentially localized modes, in Vogel spirals, this behavior is related to the multilength scale decay of optical resonances, which reflects the multifractal complexity of the Vogel-spiral geometry and its LDOS28 (see the supplementary material for more details).

The quality factor values are limited by the relatively small index contrast between silicon nitride and air, and we expect that higher values could be obtained using other materials and/or by operating at longer wavelengths. In Ref. 41, it was shown that the light–matter enhancement allowed one to reach the weak coupling regime where the recombination mechanism of the excitations is an irreversible process whose temporal dynamics can be modified. The achievement of the light–matter strong coupling regime, where the system cycles between the excited and the ground state of the photonic and matter excitations, would require small mode volumes that might not be provided by all the confined optical modes in the structure under study since some of them are spatially extended [see Fig. 2(b) for examples of far-field distributions of some of the modes sustained by the aperiodic devices].

In conclusion, our results show that nanophotonic devices based on the bioinspired Vogel spiral geometry provide an effective platform for efficient light confinement on a chip, particularly important in nano- and quantum photonics given the availability of high-quality visible quantum emitters, including atoms, defect centers (in diamond, silicon carbide, and two-dimensional materials), molecules, droplet, and colloidal quantum dots,2 and for cavity quantum electrodynamics experiments with quantum emitters41 in the visible range of wavelengths. We foresee the application of engineered aperiodic geometries for integrated quantum photonics, nanolasers, optical sensors, and nonlinear optics17 and the development of active devices operating from the visible to the telecommunication range of wavelengths. In particular, the efficient light confinement that we have demonstrated in Vogel spiral structures could be utilized for the development of single-photon devices where optical angular momentum is imparted to the emitted radiation and used as an extra degree of freedom to encode information for quantum information technology applications on a chip.29,38,39,54

See the supplementary material for more in-depth discussion of the device geometry, fabrication details, and simulations.

L.S. conceived the optical setup and built it together with O.J.T. O.J.T. grew the silicon nitride material, fabricated the devices, carried out the experiments, and analyzed the data, together with L.S. C.M. contributed to the optical characterization and data analysis. L.S., F.A.P., and L.D.N. conceived the research activities and discussed the results, together with the other authors. L.S. supervised the experimental part of the project and wrote this manuscript with contributions from the other authors. L.D.N. supervised the design and modeling contributions. S.G. developed the numerical tools utilized in this paper. S.G. and F.S. performed, analyzed, and organized the simulation results. S.G. wrote the supplementary material with the help of F.S. and L.D.N. O.J.T. and S.G. contributed equally to this work.

F.A.P. acknowledges financial support from CNPq, CAPES, and FAPERJ. L.D.N. acknowledges partial support from the Army Research Laboratory under Cooperative Agreement No. W911NF-12-2-0023 for the development of theoretical methods utilized in this paper. L.S. acknowledges partial support from the Royal Society, Grant No. RG170217, the Leverhulme Trust, Grant No. IAF-2019-013, and EPSRC, Grant No. EP/P001343/1.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material