## Abstract

We perform a numerical simulation study of hollow-core anti-resonant reflection optical waveguides (ARROWs) fabricated using lithography and material deposition in the context of their suitability as a platform for on-chip photonic quantum information processing. We explore the effects of the core size, the number of pairs of anti-resonant layers surrounding the hollow core, and the refractive index contrast between the anti-resonant layer materials on propagation losses in the waveguide. Additionally, we investigate the feasibility of integrating these waveguides with Bragg gratings and dielectric metasurfaces to form on-chip cavities that could act as nonlinear optical elements controllable with single photons when loaded with atomic ensembles.

© 2016 Optical Society of America

## 1. Introduction

Hollow-core waveguides filled with atomic ensembles offer a promising platform for realizing mediated interactions [1] between single photons needed to implement a variety of quantum information protocols. The waveguides can combine low losses for the propagating light with small optical mode areas, which in turn can give rise to strong light-matter interactions enabling extremely low-power optical nonlinearities in dilute gasses confined in the waveguides, especially when coherent light-matter interaction techniques are used. Several notable demonstrations of such strong light-matter interaction in hollow-core waveguides have been reported in recent years, enabling for example, all-optical switching with a few-hundred photons in a hollow-core photonic crystal fibre filled with laser-cooled atoms [2], cross-phase modulation with few photons [3], and single-photon broadband quantum memory [4] in a photonic-crystal fibre filled with room-temperature alkali atoms, as well as demonstration of quantum state control of warm alkali vapor in a hollow-core anti-resonant reflecting optical waveguides (ARROW) on a chip [5].

Used extensively for microfluidics applications [6], the lithographically defined hollow-core ARROWs have propagating modes of similar areas (~10*λ*^{2}) as hollow-core photonic-crystal fibres (HCPCFs) [7]. However, compared to HCPCFs, hollow-core ARROWs offer some complicated trade-offs between loss performance and ease of fabrication and monolithic integration. Unlike hollow-core fibres, these on-chip waveguides do not need a drawing tower and can be made with a combination of photolithography and relatively basic material deposition tools. Since they are fabricated in a bottom-up fashion through deposition of alternating layers of two dielectric materials, one can additionally envision including steps into the fabrication process to control the photonic environment of the waveguide, such as adding a Bragg grating as shown in Fig. 1(d). At the same time, propagation losses reported for on-chip air-core ARROWs range between ~ 0.1 cm^{−1} (~ 0.4 dB/cm) for numerical simulations and ~ 1 cm^{−1} (~ 4 dB/cm) for fabricated structures, which does not compare favorably with the losses achieved in HCPCFs ranging from < 250 dB/km for off-the-shelf fibres down to ~ 1 dB/km [8] – a value comparable to high-performance conventional solid-core fibres [9].

Here, we explore the limits of hollow-core ARROWs posed by their propagation loss and evaluate the feasibility of integrating mirrors and cavities with these waveguides, focusing on mirror structures that would leave the hollow core of the waveguide unobstructed and allow loading of atomic vapors.

## 2. Design principles and propagation loss

Antiresonant reflection optical waveguides (ARROWs) were first proposed and demonstrated by Duguay *et al.* [10] in 1986 for light propagation in low index media. ARROWs consist of a core surrounded by multiple dielectric layers whose thicknesses are chosen such that there is destructive interference of the multiple incident and reflected waves in each cladding layer, and constructive interference in the core for a given wavelength. The cladding acts as a highly reflective mirror that confines light to the core and results in low loss propagation. The initial demonstration guided light in a silicon dioxide core using a single pair of *Si*/*SiO*_{2} antiresonant layers below and total internal reflection from the air above for confinement. Hollow core ARROW waveguides were first demonstrated by Delonge and Fouckhardt [11] in 1995. Combining the constructive interference condition in the core with the destructive interference conditions in the cladding, one can obtain an analytic expression for the thickness of each cladding layer *i* of the ARROW with a core of refractive index *n _{c}* and thickness

*d*[10] (Fig. 1(a)):

_{c}*t*=

_{i}*λ*/4

*n*) on a quarter-wavelength stack. One of the advantages of ARROWs is that the cladding layers don’t need to be periodic with respect to the refractive index or their thicknesses as long as they satisfy the antiresonance condition specified in equation (1), although we will only consider two different materials to be used for the cladding layers here.

_{i}The loss of the waveguide can be estimated as the average of the losses through the left and right cladding layers, *α _{horizontal}*, and the losses through the top and bottom cladding layers,

*α*, with

_{vertical}*R*,

_{top}*R*,

_{bottom}*R*, and

_{left}*R*of the dielectric stacks surrounding the core given the incident glancing angle [12, 13]. Here,

_{right}*h*is the height of the core,

_{c}*w*is the width of the core, and

_{c}*θ*

_{c,v}_{(}

_{h}_{)}is the glancing angle along the vertical (horizontal) direction in the core defined by sin

*θ*=

_{c,v}*λ*/2

*n*(sin

_{c}h_{c}*θ*=

_{c,h}*λ*/2

*n*), so

_{c}w_{c}*α*= (

_{waveguide}*α*+

_{vertical}*α*)/2.

_{horizontal}Several possible design variants for the hollow-core ARROW are described in the literature, out of which the self-aligned pedestal (SAP) ARROW (Fig. 1(b)) has been reported to have the lowest losses both in numerical predictions (0.36*cm*^{−1}) and for fabricated devices (2.2*cm*^{−1}) [14] thanks to having air as the terminating layer on both sides [15]. The higher experimental losses compared to the results of numerical simulations arise from the deviation of the cladding layer thicknesses from the ideal values and from the surface roughness which results in additional scattering losses.

The modal characteristics of the ARROW can be found using photonic modeling software such as the finite element solver FemSIM by RSoft. Fig. 2(a) shows the electric field distribution of the fundamental mode of the ARROW waveguide and, at at the same time, visualizes the dynamics of the relatively large propagation losses. As the waveguide relies on anti-resonant reflection instead of, e.g. bandgap or inhbited coupling, even the fundamental mode remains inherently leaky, radiating energy into surrounding space as it propagates. The losses of a structure based on the analytical description of equation (1) can be somewhat reduced by numerically optimizing the thickness of each cladding layer, e.g. by using the RSoft MOST optimization tool, for a specific wavelength and the loss spectrum for a waveguide with 5.8*µm* × 12*µm* hollow core surrounded by three pairs of anti-resonant layers designed for 852nm is shown in Fig. 2(b). The waveguide loss is calculated from the imaginary part of the effective propagation index of the fundamental mode as
$\alpha =\frac{4\pi Im\left\{{n}_{eff}\right\}}{\lambda}$

We now proceed to explore the effects of what one could call ’major’ design parameters, namely the core size, the number of anti-resonant layer pairs, and the refractive index contrast between the layers in each pair, on the propagating losses of the fundamental mode in the waveguide. The results are presented in Fig. 2(c) and 2(d). The points in these plots represent waveguides with the thickness of each anti-resonant cladding layer optimized using the RSoft MOST optimization tool.

Fig. 2(c) presents the waveguide loss for a selection of waveguide core sizes, for which the area of the propagating mode is defined by

The dielectrics used for the cladding layers have included *Si*_{3}*N*_{4}(*n* = 1.7 − 2), *SiO*_{2} (*n* = 1.44), *TiO*_{2} (*n* = 2.4), and *Ta*_{2}*O*_{5} (*n* = 2.1). A *Si*_{3}*N*_{4}/*SiO*_{2} pair has been commonly used [14–16], although a combination of *TiO*_{2} [11] or *Ta*_{2}*O*_{5} [17] along with *SiO*_{2} has been explored as well. *Si*_{3}*N*_{4} has been typically used as as the high index antiresonant layer, given its ease of deposition using plasma enhanced chemical vapor deposition (PECVD), relatively high refractive index (*n* = 1.7 − 2, depending on the deposition parameters), and negligible absorption in the desired optical range [18].

Fig. 2(d) presents the results of our numerical simulations exploring how the waveguide loss changes with the number of layer pairs surrounding the core for three kinds of cladding layer pairs. We chose pairs consisting of *SiO*_{2} (*n* = 1.44) and *Si*_{3}*N*_{4}(*n* = 2), *Si*_{3}*N*_{4}(*n* = 1.7), and *TiO*_{2}(*n* = 2.4) as the high index material. The low index *Si*_{3}*N*_{4} corresponds to the material deposited by the PECVD machine at University of Waterloo Quantum Nanofab, while *n* = 2.0 is the material’s refractive index commonly reported in the literature. We included *TiO*_{2} as its refractive index appears to be one of the highest available for non-absorptive materials at our target wavelength. Note that our numerical optimization included the conformality factor arising in the PECVD process, where vertical and horizontal films are deposited at different rates. In all of our simulations, the ratio between the thicknesses of the horizontal and vertical cladding layers is set to be 1.25 for *Si*_{3}*N*_{4} and *TiO*_{2} and 1.58 for *SiO*_{2} (1.63 for the thick outer layer). Increasing the refractive index contrast between the cladding materials appears to have the strongest effect on reducing the propagation loss and we observe a loss reduction by a factor of approximately three for a given number of layer pairs when the index contrast changes from 0.3 to 0.96. On the other hand, we observe that adding more periods does not necessarily yield a steady decrease in the mode loss – going from 2 to 3 periods decreases the loss more than by going from 3 to 4 periods – and there seems to be a lower bound on the loss for a given material pair. It should be noted that the 4 period *TiO*_{2}/*SiO*_{2} waveguide loss could not be lowered below that of the 3 period version during the multiple optimization runs. However, we believe that with some more tweaking of the optimization parameter space, this should be possible. In absolute values, the propagation losses presented here are somewhat higher that than those reported in Ref. [14] a 5.8*µm* × 12*µm* hollow core waveguide with three *Si*_{3}*N*_{4}(*n* = 2.0)/*SiO*_{2} cladding layer pairs, which is most likely the result of slightly higher (*n* = 2.05) refractive index of *Si*_{3}*N*_{4} and using fewer mesh divisions in the FIMMWAVE simulation used in that work.

Lastly, the waveguides in in the literature [11, 14–19] surround the core with the high index layer as shown in Fig. 1(b) because it is intuitively expected for the higher Fresnel reflection coefficient at the interface to result in a lower overall waveguide loss. However, another possibility that has not yet been explored is to have the layer immediately beneath the core be the low index layer as shown in Fig. 1(c). The propagation losses of the optimized waveguides for this variant are compared with that of the typical configurations in Fig. 2(d) and we find that there is not a significant difference in the propagation loss between the two variants given the same cladding material pair, and number of anti-resonant layers used. In each case, the waveguides can be optimized such that both variants have comparable propagation losses. We were forced to explore this new design variant with a *SiO*_{2} rather than a silicon nitride layer used beneath the core, when we discovered that the SU8 sacrificial core had poor adhesion to silicon nitride films from our PECVD machine and was washed out during the development step. While the SU8 core had excellent adhesion to silicon substrate and silicon oxide films, our silicon nitride films seem to have a lower ratio of silicon to nitrogen that results in problems with resist adhesion and a low refractive index.

## 3. Bragg mirrors in hollow-core ARROWs

The interaction between light and an atomic ensemble confined to the hollow-core ARROW could be further increased by integrating the waveguide with a pair of mirrors to form a cavity. For solid-core waveguides, mirrors may be realized with Bragg gratings which are implemented by periodically modulating the refractive index in the waveguide core [20]. This presents a challenge for hollow-core waveguides as there is no material in the core that could be modified. Here, we explore implementing a grating by modulating the effective index of propagation. We propose to do this by etching a grating in the various anti-resonant layers. The waveguide will then have a slightly different propagation index along the section where one of cladding layers above or below the core has been etched. Given that the ARROW has a bottom up fabrication process, the grating can be implemented in any particular layer. This approach has the advantage of leaving the hollow core unobstructed so that atomic ensembles can be loaded into the waveguide.

As shown in Figure 3(a), we will consider cases where the grating is etched either into the layer immediately above the core, the layer beneath the core, or the very top layer. Figure 3(b) then presents how the effective index of propagation and the propagation loss change for the etched region for the three different locations of the grating, in this case for a *Si*_{3}*N*_{4}(*n* = 1.74)/*SiO*_{2} hollow core ARROW with three cladding layer pairs. We see that changing the thickness of a particular anti-resonant layer increases the propagation loss because the waveguide is no longer optimized. Layers closer to the core have a greater overlap with the fundamental leaky mode, hence etching them creates a larger perturbation relative to the original waveguide. This explains why etching the layer beneath the core causes a large change in the effective index of propagation (~ 5 × 10^{−4}) whereas it is only 2.9 × 10^{−4} when etching the very top layer. Overall, we found that modulating the thickness of the layer beneath the core results in highest Δ*n _{eff}* and the smallest increase in propagation loss. At the same time, increasing the depth of modulation past certain value for all three layers results in complete loss of guiding for the fundamental mode. We found that using higher refractive index contrast cladding materials yields a waveguide with lower loss where the optical mode is more confined to the core. There is then less overlap between the anti-resonant layers and the the fundamental mode, causing a smaller perturbation. For comparison, the same geometry ARROW as the one from Fig. 3(b) but with

*n*= 2 for the

*Si*

_{3}

*N*

_{4}will have maximum effective index change of approximately (~ 4 × 10

^{−4}). Using more periods of anti-resonant layers has the same effect because the lower loss waveguide has a smaller overlap between the cladding layers and the fundamental mode which is again more tightly confined.

Given that the waveguide has non-negligible loss, the standard transfer matrix method [21] cannot be used to determine the reflectivity of these gratings. We can instead use the Method of Single Expression (MSE) proposed by Baghdasaryan *et al.* [22] to properly account for the absorption loss in each grating layer. Starting from the transmitting side, the field in each Bragg layer is calculated as we travel back to the illuminated side by numerically solving the coupled differential equations below.

*E*(

*z*) =

*U*(

*z*)

*e*

^{−}

^{iS}^{(}

^{z}^{)},

*Y*(

*z*) =

*dU*(

*z*)/

*d*(

*k*

_{0}

*z*), and the power is

*P*(

*z*) =

*U*

^{2}(

*z*)/[

*dS*(

*z*)/

*d*(

*k*

_{0}

*z*)]. Given a grating of length

*L*, the initial conditions are

*U*(

*L*) = 1,

*Y*(

*L*) = 0, and $P(L)=\sqrt{{\mathit{\u03f5}}_{t}}$ where

*ϵ*is the relative permittivity of the transmitting side. The reflectivity is then given by

_{t}*ϵ*is the relative permittivity of the incident side. Now our grating can be thought of as being composed of two alternating dielectric materials

_{i}*n*

_{1}(the effective index of propagation of the waveguide), and

*n*

_{2}(the effective index of propagation when one of the anti-resonant layers has been etched). This approximation is valid because the modes in the two regions are similarly quasi-Gaussian in the core. The corresponding Bragg-condition thicknesses are

*t*

_{1}=

*λ*/4·

*Re*{

*n*

_{1}}, and

*t*

_{2}=

*λ*/4 ·

*Re*{

*n*

_{2}}. For the three period

*Si*

_{3}

*N*

_{4}(

*n*= 1.74)/

*SiO*

_{2}ARROW from Fig. 3(b), the reflectivity of the mirror is shown in Fig. 3(c) for various etched layer cases where the etch depth is chosen to have the highest effective index contrast relative to the original waveguide. While this is an approximate result obtained by assuming plane wave propagation, we have found previously [23] for a different type of hollow-core waveguide that this estimate has an excellent agreement with a full three dimensional simulation. Implementing the grating in the anti-resonant layer beneath the core yields the highest reflectivity (~84%) mirror. The next best mirror with ~ 55% reflectivity can be implemented using the layer above the core, while using the top layer yields a poor mirror (~ 23%) due to its low effective index contrast. Note that for a typical lossless Bragg grating, the reflectivity increases with higher refractive contrast materials and larger number of periods and, regardless of the refractive index contrast, eventually approaches 1 as we add more periods. However, with our ARROW gratings the upper bound on the reflectivity is less than 1 due to the presence of absorption loss.

## 4. Integrated cavities

Depending on the application, several parameters can be used to characterize the performance of an optical cavity, including its mode volume *V _{mode}*, quality factor

*Q*, and cooperativity factor

*C*=

*g*

^{2}/

*κγ*. Cavities with high-cooperativity factor,

*C*≫ 1 or at least

*C*> 1, are of particular interest as the presence of a single atom in such cavity will result in a significant change of the cavity transmission [24]. We now proceed to discuss the feasibility of creating a high-cooperativity cavity a formed by a pair Bragg gratings around a spacer region inside the hollow-core ARROW.

In the expression for the cooperativity factor, *κ* = *ω*/*Q* describes the field decay rate from the cavity, *γ* is the linewidth of the excited state of the atom placed in the cavity, and *g* is the atom-field coupling coefficient or single-photon Rabi frequency:

*µ*is the atomic transition dipole moment,

*ω*is the cavity resonant frequency, and

*ϵ*is the permittivity where the field intensity is highest and the mode volume is given by

_{M}*A*being the waveguide mode area and

_{mode}*L*the effective length of the cavity.

_{eff}At the same time, the cooperativity can be expressed independent of the atomic species present in the cavity, such that

and is determined solely by the properties of the cavity. To estimate the*Q*factor of a cavity formed by a pair of Bragg mirrors in a hollow core arrow, we consider the

*Q*factor of a simple Fabry-Perot cavity of length

*L*with perfectly thin mirrors with reflectivities

*R*, which is filled with a medium with effective index

*n*and attenuation

*α*:

The cavity length, *L*, can then be approximated as *L _{eff}* =

*L*+ 2

_{spacer}*z*, with the penetration depth [25] into the Bragg mirrors

_{p}*z*≈

_{p}*a*/Δ

*n*. Here,

_{eff}*a*≈

*λ*/2

*n*is the Bragg grating period and the approximate expression for

_{eff}*z*assumes

_{p}*n*≈ 1 and Δ

_{eff}*n*≪ 1. Further assuming that the propagation loss in the etched sections of the waveguide is roughly the same as in the non-etched sections,

_{eff}*α*, we can estimate the reflectivity of the Bragg mirror as $R\approx {e}^{-2{z}_{p}{\alpha}_{wg}}$ [23]. Combining these, we arrive at an expression estimating the upper limit for the cooperativity of our Bragg grating cavity for

_{wg}*L*≪

_{spacer}*z*:

_{p}*g*

^{2}/

*κγ*> 1 at

*λ*= 852

*nm*in a hollow-core waveguide with

*A*= 18

_{mode}*µm*

^{2}and Δ

*n*= 5 × 10

_{eff}^{−4}, a propagation loss

*α*< 10

_{wg}^{−2}

*cm*

^{−1}would be needed. This value is more than an order of magnitude lower than even the simulated propagation loss that does not account for fabrication imperfections and might be out of reach.

An alternative approach to integrating a cavity into a hollow-core ARROW relies on attaching photonic-crystal membranes acting as dielectric metasurfaces at the two ends of a waveguide section as shown in Fig. 4(a). A dielectric metasurface [26] can be designed to act as a highly reflective mirror for light incident perpendicularly on it [27]. When comprised of a thin dielectric sheet with a pattern of holes, atomic ensembles can be still loaded into the waveguide through this porous membrane. The main parameters used to design such membrane are the refractive index of the dielectric, radius of the holes, spacing between the holes, and the thickness of the membrane.

For a cavity formed by a pair of such membranes, *L _{eff}* is then given just by the length of the waveguide section between the membranes. The cooperativity of such cavity can be then described as

*ln R*| ≪

*αL*the cooperativity will be independent of the cavity length and determined purely by the reflectivity of the metasurface, while for |

_{eff}*ln R*| ≪

*αL*the cooperativity will decrease with increasing

_{eff}*L*. Fig. 4(b) then presents predicted cooperativities for an ARROW with

_{eff}*A*= 18

_{mode}*µm*

^{2}for assorted cavity lengths and metasurface reflectivities. We see that for this type of ARROW-integrated cavity, high cooperativity might achievable for some of the waveguides structures from Fig. 2(d).

## 5. Conclusion

Compared to single quantum emitters, ensembles of quantum emitters can frequently offer significant advantages for implementing controllable single photon interactions. These advantages include collective enhancement of light-matter interaction, robustness to noise and decoherence, and the inherent ability to handle light pulses containing multiple photons. Combining atomic ensembles with on-chip waveguides could open new possibilities for control over the design of the atoms’ photonic environment and for design of scalable devices and architectures utilizing recently demonstrated quantum phenomena, such as vacuum-induced transparency [28] and superradiant lasing [29].

Unfortunately, we found that the propagation loss in hollow-core ARROWs is likely to pose severe limitation for their application in quantum information processing unless reasonably short sections are used. Bragg grating integration in particular results in poor-performance mirrors for wavelengths associated with alkali atoms, such as cesium, although it is possible that decent Bragg mirrors could be implemented, for example, at telecom wavelengths. On the positive side, it appears that combining ARROWs with photonic-crystal membranes acting as high-reflectivity dielectric metasurfaces could yield on-chip high-cooperativity cavities with lengths of ~100*µ*m, which is comparable to the lengths of high-density atomic clouds produced by laser cooling In references [28] and [29].

## Funding

This work was supported by Industry Canada and by Canada’s Natural Sciences and Research Council (NSERC) under the Discovery Grants Program. J. F. was in part supported by Ontario Graduate Fellowship.

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