1. Introduction

Axially varying waveguides with mode-cutoff sections have been widely used in the microwave domain, to construct spectral filters [1–3], to explore tunneling phenomena [4], and, more recently, as artificial, epsilon-near-zero (ENZ) media [5]. This is made possible by the low-loss, omnidirectional nature of a metallic reflector at microwave frequencies. As now well-known, dielectric Bragg mirrors of sufficiently high refractive index contrast can also provide an omnidirectional reflection band, thereby exhibiting quasi-metallic properties at optical frequencies [6]. This has led to proposals for mode confinement [7] and spectrometry [8] based on omnidirectional-clad optical waveguides. Nevertheless, the translation of microwave hollow waveguide components [1–5] into the optical domain remains a relatively unexplored theme.

Meanwhile, there is great interest in the use of hollow waveguides (both fibers and integrated waveguides) for applications that require close interactions between light and gas- or liquid-phase media. These applications include sensing and atomic spectroscopy [9], and increasingly lie within the realm of quantum information science [10]. Notably, hollow waveguides with interference-based claddings (e.g. Bragg or ARROW waveguides) can also be designed to act as highly selective spectral filters [11,12]. By augmenting these properties with methods for engineering the dispersion and axial confinement of guided modes [10], it is expected that a host of new applications in quantum optics might be plausible [13,14].

We have previously described a thin-film buckling process that can produce axially varying (in both width and height) hollow Bragg waveguides on a chip [15]. In the present work, we used this process to fabricate back-to-back (dual) taper structures. We demonstrate that these dual tapers act as short-pass filters, by reflecting longer wavelength light subject to mode cutoff. We also show preliminary results for a hollow 3-D microcavity formed by cascading two nominally identical dual-taper ‘mirrors’ [16].

2. Device design and fabrication

A schematic diagram of the dual-taper structure, for the case of a slab Bragg waveguide system, is shown in Fig. 1(a). The structure is nominally symmetric, with input and output waveguides designed to support one or more guided modes across a spectral range of interest. A central ‘tunnel’ section is sufficiently narrow such that some longer wavelengths are subject to mode cutoff. Design parameters include the core heights of the input (H) and tunnel (h) sections, the length of the tapers (t), the length of the tunnel section (l), and the number of periods in and composition of the Bragg cladding mirrors.

 

Fig. 1 (a) Conceptual cross-sectional diagram of a dual taper in a slab Bragg waveguide. The red dotted line depicts the trajectory of a guided ray, at a wavelength not subject to cutoff in the tunnel section. (b) Microscope image showing an as-fabricated dual taper. (b) Surface relief plot of a dual taper extracted using an optical profilometer.

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When guided light encounters a tapered waveguide section, the associated ‘bounce’ angle (from a ray optics perspective) becomes increasingly normal relative to the cladding mirrors as the waveguide core narrows [8]. However, provided the wavelength lies within the omnidirectional band of the claddings, and provided the cladding mirrors are sufficiently reflective, radiation loss can be low. For wavelengths that remain guided within the tunnel section, it follows that reasonably efficient transmission through the dual taper is expected, albeit with some slowing and excess radiation of light. On the other hand, for wavelengths subject to cutoff in the tunnel section, back-reflection into reverse propagating modes is anticipated, again with some slowing and excess radiation of light inside the tapered region. Thus, as verified below, one can expect this structure to behave both as a short-pass filter and as a spectrally selective reflector of guided light [16].

While conceptually simple, the proposed structure is not trivial to fabricate for operation at optical frequencies. We employed a buckling self-assembly approach, which is capable of producing axially non-uniform, 3-D hollow waveguides [8,15,17]. Figures 1(b) and 1(c) show microscope and surface relief images, respectively, for a typical dual taper fabricated by appropriate patterning of a low-adhesion layer within a multilayer stack (the pattern is indicated by the perimeter of the waveguide in Fig. 1(b)). Since the height of the resulting delamination buckle is approximately proportional to the width of the low-adhesion feature [15], the core heights h and H are easily varied by adjusting the width of the low adhesion strip inside (w) and outside (W) the tunnel region.

3. Dual-taper ‘mirrors’ – experimental results

The hollow waveguides produced by buckling self-assembly have low height-to-width aspect ratio, such that slab-waveguide models can accurately predict their guiding properties [8,15]. In the following, experimental results on the channel waveguides were corroborated using transfer matrix and finite-element (COMSOL) models of analogous 2-D structures, by equating the slab core thickness to the peak height of the 3-D buckled waveguide. The waveguides studied here have 5-period a-Si/SiO₂ Bragg mirrors satisfying a quarter-wavelength condition at ~1550 nm wavelength. The indices of the films are close to the values reported elsewhere [17] (n ~1.46 and ~3.7), corresponding to layer thicknesses of ~265 and ~104 nm for SiO₂ and a-Si, respectively. These mirrors provide an omnidirectional band spanning the ~1300-1700 nm range.

In the following, we focus mainly on waveguides with W = 60 μm, for which the buckling process produced H ~2 μm. Numerical simulations (COMSOL) were used to assess the air-guided modes for the 3-D structure, and representative modes are shown in Fig. 2(a) (with independently scaled axes, for the sake of clarity). As previously reported [15], the dominant low-loss modes are TE-polarized and have a single lobe in the vertical direction. Thus, a transfer-matrix-based solution [8] for the fundamental TE mode of an analogous slab waveguide provides an accurate prediction of the spectral transmission properties (see Fig. 2(b)). In addition to radiation loss, the models incorporated loss arising from absorption in the a-Si layers, which have extinction coefficient k ~10⁻³ at 1600 nm wavelength [17]. The experimental scan in Fig. 2(b) was obtained using a supercontinuum light source (Koheras, ~600-1700 nm) and an optical spectrum analyzer (OSA, Anritsu). It was normalized to a base scan taken without a sample and scaled using waveguide loss (~3 dB/cm in the 1550 nm wavelength range) estimated by fitting to scattered light streaks (see below). As shown, the straight guides exhibit a low-loss propagation band in the ~1100-1800 nm wavelength range. The notch at ~1300 nm is due to the interaction of TM-polarized light with the ‘sidewalls’ of the buckled guide, as explained elsewhere [15]. It should be possible to eliminate this notch through slight modifications to the Bragg mirrors [18]. In general, there is significant scope for filter customization in waveguides with interference-based claddings [11,12].

 

Fig. 2 (a) Theoretical mode-field profiles for the 3 lowest-loss modes (all TE-polarized) of a waveguide with W = 60 μm and H = 2 μm. The predicted effective indices are ~0.92, 0.91, and 0.90, respectively. (b) Spectrally-dependent transmission for a waveguide with H ~1.8 μm, as measured using a supercontinuum source and OSA (solid), and as predicted by a slab-waveguide model (symbols).

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Next, we explored the properties of waveguides containing dual tapers. Figure 3(a) shows a typical scattered light image obtained using the supercontinuum source. The bright central spot is the location of the dual taper, which causes increased scattering and radiation of guided light. Nevertheless, it is clear that some of the light is transmitted and remains well guided towards the output facet. Conversely, Fig. 3(b) shows a scattered light image obtained using a laser source, where the laser wavelength is subject to cutoff in the tunnel section. A lensed fiber was aligned so as to excite primarily the fundamental mode (see Fig. 2(a)) in this case. Owing to low roughness-induced scatter in these buckled waveguides, we typically observe little coupling between co-propagating modes [15]. As shown, light incident on the taper is efficiently back-reflected into the reverse-propagating mode, forming a standing-wave of high visibility over the entire input section (several mm).

 

Fig. 3 (a) Scattered light image for a waveguide containing a dual taper, excited by the supercontinuum source. The locations of the dual taper (dt) and the output facet (of) are labeled. (b) Scattered light image for excitation by a laser tuned to 1602 nm, which is subject to cutoff within the tunnel section for the case shown. The waveguide boundaries are indicated by the dashed line. Inset: higher magnification image of a portion of the standing wave, with period ~1 μm. (c) Experimental versus theoretically predicted transmission for a dual taper with h ~0.73 μm. (d) Experimental short-pass transition edges for various dual tapers. (e) Cut-off wavelength versus tunnel height, as measured for a variety of dual tapers and as predicted by an analytical slab model.

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The spectrally-dependent transmission for a typical dual-taper waveguide (with h ~0.73 μm, t ~30 μm, and l ~20 μm), measured using the supercontinuum source and the OSA, is shown in Fig. 3(c). Also shown is the predicted transmission obtained using COMSOL for an analogous slab structure (see Fig. 1(a)). While the input waveguide supports low-loss propagation for wavelengths up to ~1800 nm (see Fig. 2(b)), the dual taper introduces an abrupt, short-pass transmission edge, attributable to mode cutoff in the tunnel section for wavelengths above ~1480 nm in the case shown. This behavior is well-replicated by the numerical results; the slight discrepancy in the transition wavelength can be attributed mainly to the neglect of transverse confinement in the slab model as discussed below. Figure 3(d) shows several representative short-pass transmission edges for different values of the tunnel height h. The steepness of the transition is an important filter property [19]; the results shown indicate a 10 dB slope factor (i.e. the width of the 10 dB transition region as a percentage of the transition wavelength) <1%, and edge steepness ~-2 dB/nm. This slope was found to increase with increasing tunnel length l.

In keeping with the discussion above, the transition edge is expected to correlate with the onset of cutoff in the tunnel section. In a 2-D approximation, the cutoff wavelength can be assessed analytically [20,21]:

4. Microcavities with cutoff-based axial confinement

A recent theoretical work by Wang et al. [16] described a Bragg-fiber-based optical microcavity, with transverse confinement provided by omnidirectional mirrors and longitudinal confinement provided by mode cutoff sections. Mesoscopic cavities of this type within on-chip hollow waveguides would be of interest for a variety of applications reliant on the quantum optics of gas-phase media [13,14], but are difficult to realize using conventional Bragg grating or photonic crystal mirrors [10].

Figures 4(a) and (b) show images of a nominally symmetric microcavity formed by directly cascading two dual-taper mirrors, with t = 30 μm, l = 0 μm, w = 42 μm, and W = 60 μm. In other cases, a straight section was inserted between the dual tapers. For brevity, we focus on the cavity shown, for which the buckling process resulted in h ~760 nm, corresponding to a cutoff transition at λ ~1480 nm, and a peak height inside the cavity ~1.04 μm, less than the height of the input and output waveguides (H ~2 μm). This implies relatively small mode volume (V_M), but likely sub-optimal coupling efficiency. A more detailed study of these trade-offs is left for future work.

 

Fig. 4 (a) Microscope image of a microcavity bounded by dual-taper mirrors. (b) Surface relief image. (c) Scattered light image for a typical resonant mode. (d) Simulated mode field intensity plot for an analogous slab waveguide cavity. (e) Simulated transmission for the analogous slab structure. The red dashed line shows the predicted transmission for a single dual taper in this case. Insets: experimental long- and short-range transmission scans.

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Experiments were conducted using a tunable laser (Santec, 1520-1620 nm). As shown in Fig. 4(c), it was possible to excite cavity modes by adjusting the laser wavelength and input coupling conditions. For the mode shown, we estimated V_M < 100 λ³. The theoretical mode field profile (COMSOL) for an analogous slab structure is shown in Fig. 4(d), revealing excellent qualitative agreement. Figure 4(e) shows the predicted spectral transmission for the slab structure, with a series of resonant transmission peaks near the cut-off transition edge of the dual tapers. Note that the transition edge has lower slope than above, since l = 0 here. The peaks exhibit decreasing linewidth (increasing Q) and magnitude as the wavelength increases above the cut-off [16], which can be attributed to the dual-taper mirrors providing increased effective reflectance, but also increased radiative loss. The lower inset in Fig. 4(e) shows an experimental scan; discrepancies (in the observed FSR, etc.) can be partly attributed to neglect of transverse confinement and multimodal effects in the slab model. Also, such long-range spectral scans were complicated by experimental instabilities. By isolating individual modes in shorter-range scans and increasing detector gain (for example, see the upper inset of Fig. 4(e)), we have measured cavity linewidths < 0.1 nm, corresponding to cavity Q > 10⁴, in good agreement with the theoretical predictions. Notably, this combination of Q and V_M implies potential for cooperativity C > 1 [10], making these cavities intriguing candidates for quantum information studies.

The lower transmission for the high Q modes is indicative of high absorption and radiation losses in the cutoff sections. This could be improved through the use of higher reflectance cladding mirrors, by increasing the number of periods and reducing material loss [16]. Experimental transmission was even lower than predicted; non-optimized coupling, light scattering, and cavity asymmetry [2] are likely major factors. For example, h of the two tunnel sections was mismatched by ~5 percent for the cavity considered above. Generally speaking, there is significant potential to optimize the fabrication process and the critical cavity parameters – mode volume, finesse, linewidth, etc. – through lithographic control of the cavity dimensions and improvement of the Bragg claddings. These improvements are left for future work.

5. Summary and conclusions

We fabricated back-to-back (dual) tapers within integrated hollow Bragg waveguides, and showed that they can be exploited as short-pass transmission filters or cutoff-based reflectors. Preliminary results for 3-D microcavities having axial confinement provided by these dual-taper reflectors were also reported. With further refinement, applications in atomic spectroscopy and sensing are envisioned.