Omni-resonant optical micro-cavity

Optical cavities transmit light only at discrete resonant frequencies, which are well-separated in micro-structures. Despite attempts at the construction of planar ‘white-light cavities’, the benefits accrued upon optically interacting with a cavity – such as resonant field buildup – have remained confined to narrow linewidths. Here, we demonstrate achromatic optical transmission through a planar Fabry-Pérot micro-cavity via angularly multiplexed phase-matching that exploits a bio-inspired grating configuration. By correlating each wavelength with an appropriate angle of incidence, a continuous spectrum resonates and the micro-cavity is rendered transparent. The locus of a single-order 0.7-nm-wide resonance is de-slanted in spectral-angular space to become a 60-nm-wide achromatic resonance spanning multiple cavity free-spectral-ranges. The result is an ‘omni-resonant’ planar micro-cavity in which light resonates continuously over a broad spectral span. This approach severs the link between the resonance bandwidth and the cavity-photon lifetime, thereby promising resonant enhancement of linear and nonlinear optical effects over broad bandwidths in ultrathin devices.

Here each bilayer (HL) consists of a high-index (H) and low-index (L) material, which are TiO2 and SiO2, respectively. The measured refractive indices for TiO2 and SiO2 at representative wavelengths of interest are given in Table S1 and Table S2 (provided by Blue Ridge Optics, LLC). The TiO2 films were formed by evaporating Ti2O3 source material under O2 partial pressure. Using these values in Tables S1 and S2, we calculated the spectral transmission of the mirror and cavity at normal incidence, Fig. S1 and Fig. S2, respectively, and the spectral-angular response of the cavity (Fig. S3). The spectral simulations employ the transfer matrix method at a resolution of (π/2)/500. Figure S1 | Spectral transmission through the 5-bilayer Bragg mirror at normal incidence. a, The mirror is surrounded by glass on one side and air from the other. b, Mirror is surrounded by glass on both sides.

S2. Simulation of the achromatic resonances
In this Section, we calculate the transmission characteristics of the FP cavity when it is inserted into a setup that induces achromatic resonances. In particular, we simulate the effect of the grating and lens L1 placed in the path of a collimated broadband beam, as shown in Fig. 3b of the main text.
We assume an ideal grating with TE or TM polarized collimated light directed at an incidence angle = 50° with respect to the normal to the grating. See Fig. S4 for a schematic of the setup that highlights the definition of the relevant angles for our analysis. The angularly dispersed light from the grating is then directed to the sample through the lens L1. We assume that 550 nm is the central wavelength and take it to define the optical axis. The tilt angle of the sample is measured with respect to this optical axis. We define the angle ( ), which is the diffraction angle with respect to the grating normal. The central wavelength c = 550 nm is diffracted at o = ( c = 550 nm) and coincides with the optical axis. The angle any wavelength makes with respect to this optical axis is ( ) − o . This angle is boosted via the lens L1 by a ratio 1 2 , where 1 and 2 are the distances from the grating to L1 and from L1 to the cavity, respectively. The incidence angle made by a wavelength after the lens with respect to the optical axis is thus: with o = ( c = 550 nm) = 0. The distances 1 and 2 are selected such that the illuminated spot on the grating is imaged onto the cavity. If the focal length of L1 is , then 2 = 1 − 1 . When the cavity is oriented such that it is perpendicular to the optical axis, the angle of incidence of each wavelength is ( ). Upon tilting the cavity by , the angle of incidence with respect to the normal to the cavity is ( ) = ( ) + .

Details of the experimental setup for measuring spectral transmission
In the experimental setup shown schematically in Fig. 3b in the main text, white light emitted from a halogen source (Thorlabs, QTH10/M) is spatially filtered by coupling through a 1-m-long multimode fiber (50-µm-diameter) and then collimated via a fiber collimator, followed by a polarizer to control the state of polarization of the beam. A grating (Thorlabs GR25-1850) is placed at ~ 30 cm from the collimator and orientated at 50º with respect to the incident light. Before incidence on the grating, the beam passes through a 1-mm-wide vertical slit. Diffracted light is focused by a lens L1 on the cavity, which is mounted on a rotational stage. The lens is placed 12 cm away from the grating to provide the appropriate angular dispersion for phase-matching of the wave-vector axial component. Light transmitted though the cavity is collimated by a second 25-mm-focal-length lens L2 and then coupled into a multimode fiber using an aspherical lens (15-mm-focal-length) connected to a spectrometer (JAZ, Ocean Optics). The wavelength resolution of the measurement is limited by the multimode fiber to ~ 1 nm.
Light diffracted by the grating is spread horizontally, so transversal displacements of optical components can spectrally shift the resonances. To align the setup for the desired spectral range, care must be taken to ensure that the center wavelength of c = 550 nm passes through the center of the lenses L1 and L2 and thus defines the optical axis. To maximize the achromatic resonance bandwidth, the focusing lens L1 is first placed in the desired distance from the grating obtained from geometrical optics considerations. The collection aspherical lens and fiber are then aligned to collect the maximum spectral bandwidth. The cavity is then mounted on the rotational stage at the focal point of the lens L1. Although axial displacement of the resonator does not affect the resonances, it can alter the angular distribution of the beam after L1, a feature we use to fine-tune the bandwidth of the resonances.

Details of the experimental setup for imaging through the cavity
The cavity appears like a mirror at near normal incidence (Fig. 3b inset in the main text), however it transmits most of the incident light in the achromatic resonance configuration and thus appears transparent. To visually demonstrate the cavity transparent, an object (an opaque letter 'i' on a transparent substrate) is imaged onto a CCD camera through the cavity. We first imaged the object through the cavity alone ( Fig. 4c in the main text). The cavity is oriented normally with respect to the beam path and as a result blocks most of the optical power due to very low optical throughput and instead reflects the incident beam (except on resonance). The setup is sketched in Fig. S7.
We next carry out the measurement in the achromatic resonance configuration; see Fig. S8. An Hpolarized (TE) collimated white light beam is incident on the object (letter 'i' on a transparent substrate), is angularly dispersed by the reflective diffraction grating, followed by passage through the lens L1 with focal length = 25 mm. The beam passes through the cavity mounted on a rotation stage and is collimated by a second 25-mm-focal-length lens L2. A second reflective grating restores the original beam structure and finally a 10-cm-focal lens L3 images the object onto a CCD camera (Imaging Source, DFK 31BU03). The distances from L3 to the grating and from L3 to the CCD are both ~ 10 cm. As a reference, we carry out the experiment using this setup but in absence of the cavity (Fig. 4b in the main text). The results of the imaging experiment in presence of the cavity are presented in Fig. 4d of the main text. At certain values of , the resonance broadens, and when the achromatic resonance condition is met the optical throughput increases.