Mesoscale cavities in hollow-core waveguides for quantum optics with atomic ensembles

3.1 ARROW Principles and Analysis

ARROWs are an interference-based waveguide in which the core is surrounded by multiple cladding layers collectively acting as a highly reflective mirror. For a given wavelength, the layer thicknesses can be chosen such that there is constructive interference in the core and destructive interference in the cladding, which results in low loss propagation in the core even though the modes are leaky. Looking at the 1D profile shown in Figure 3A, the glancing angle is given by Equation 1, and the optimal thickness for a specific layer i is given by Equation 2 derived using the phase conditions mentioned in Ref. [28]. The optimal layer thickness depends on the desired wavelength of operation, core height, core refractive index, and the refractive index of the layer itself.

The propagation loss of ARROWs can be estimated using the numerical analysis technique developed by Chen et al. [34]. This method takes a 1D refractive index profile and calculates the propagation constants and profiles of the supported electromagnetic modes by using characteristic transfer matrices for each layer. Given that ARROWs actually have a 2D refractive index profile, the overall loss can be estimated by averaging the losses from the two 1D cross-sectional profiles. This analysis can be used to design waveguides that are polarization sensitive (TE or TM) as well. The materials used for the antiresonant layers have included Si₃N₄, SiO₂, TiO₂, and Ta₂O₅. The most common material pair is Si₃N₄ (n = 1.7–2.0, depending on the quality of the deposited film)/SiO₂(n = 1.44), although using TiO₂ (n = 2.4) [29] or Ta₂O₅ (n = 2.1) [35] along with SiO₂ has been explored as well. Si₃N₄ is typically used due to its ease of deposition using plasma-enhanced chemical vapor deposition (PECVD), fairly high refractive index contrast, and nonabsorptivity in the desired optical range of operation [22].

The characteristics of ARROWs, including loss and mode area, may be modeled using photonic simulation software such as the finite element solver FemSIM by RSoft. The main source of losses is due to the fundamental leaky mode of these waveguides. We have found that this can be minimized by using higher index contrast materials, or increasing the number of antiresonant layers surrounding the core. The core size also affects the propagation loss. Decreasing the width of the core will yield a smaller optical mode area but a higher propagation loss. This is due to the fact that for TE modes, the TM like polarization relative to the side walls has lower reflectivity where the angle of propagation is larger (Equation 1), similar to the Brewster angle phenomenon that occurs between two optical interfaces. As a result, cores are typically much wider than they are high. Figure 4A shows how the propagation loss varies as a function of the mode area for an ARROW with 3 periods and Figure 4B shows the propagation loss as a function of period for various antiresonant material pairs.

Fig. 4

(A) Plot of loss versus mode field area for a three-period SAP ARROW using Si₃N₄/SiO₂ layers for varying width and height. (B) Plot of loss versus number of periods of antiresonant layers for a similar ARROW as (A) for material combinations Si₃N₄/SiO₂, where the refractive indices for Si₃N₄, TiO₂ and SiO₂ are 1.4, 2.4, and 1.44 respectively.

3.2 Fabrication

There are two main methods to fabricate hollow core ARROWs. The first method reported by Bernini et al. [36] uses silicon to silicon bonded together to create the hollow core. A square groove is etched into the silicon wafer, followed by deposition of the bottom antiresonant layers. While the top layers are deposited onto another wafer. The two pieces are then wafer bonded to create the hollow core waveguide. These devices have achieved losses as low as 2 dB/cm, but the large core size of 130 µm × 130 µm is not particularly suitable for applications trying to achieve optical mode areas closer to those found in single-mode optical fibers. The second method, explored extensively by the Schmidt group at UC Santa Cruz and the Hawkins group at BYU, uses a bottom-up fabrication process with a sacrificial core material acting as a scaffolding which is later removed. The first step is to deposit the bottom antiresonant layers using PECVD at a temperature of around 250^∘C. This is followed by deposition of a sacrificial layer and using photolithography to pattern the core. Core materials investigated have included aluminum [37], which tended to have a trapezoidal instead of the desired rectangular shape, and photoresist [33]. Multiple positive and negative photoresists have been investigated for the core material, but SU-8 [38] is typically used due to the fact that it can be hard baked which prevents it from reflowing and distorting when placed in the high temperature PECVD chamber [22]. This step is followed by depositing the top antiresonant layers. The last step is to remove the sacrificial core material to create the hollow core. One drawback of using PECVD to deposit the top antiresonant layers is the nonconformal deposition process, whereby layers on top of the core are thicker than those on the sides. Characterizing this ratio t_h/t_v allows one to account for it when modeling the waveguide. There have been some variations of this process to decrease the loss of the waveguide by having an air terminating layer instead the outermost antiresonant layer material on the sides. One variant is known as the prealigned pedestal (PAP) ARROW. Here, a pedestal with the core width is created on the substrate using a photolithography and etch process as the first step before depositing the bottom layers. Another variant is the self-aligned pedestal (SAP) ARROW. After the core is patterned, it is used as a mask whereby the pedestal is created by etching through the bottom antiresonant layers and a portion of the substrate depending on the desired pedestal height. The core is protected using a thin metal layer such as chromium. These variants have been thoroughly investigated by Lunt [39] where it was found that the SAP variant had lower losses than the PAP variant and both were lower than the regular ARROW. The lowest reported losses in that work for fabricated structures with core sizes of a width of 12 µm and a height of 5–6 µm (corresponding to mode-field area of 18 µm²) were around 2.2 cm⁻¹ or 9.6 dB/cm [22].

4.1 Gratings in HCPC fibers

Fiber Bragg gratings (FBGs) have traditionally been created by periodically modifying the refractive index of a fiber by exposure to CW or pulsed laser beams, usually in the UV range. This technique is generally intended to produce photo-induced refractive index change in silica, which can range between ~ 10³ and 10⁴, although material removal and correspondingly higher index change have been reported with optical nanofibers [41]. The exposure has been done by scanning a beam across the fiber [42] or by employing interferometric techniques [43–46]. In the latter approach, the Bragg grating pitch is controlled either by adjusting the angle of intersection between the two writing beams, adjusting the wavelength of the laser acting as a writing source, by selecting a phase-mask with appropriate period of the mask’s grating [46], or a combination of these three methods.

However, since the design of HCPC fibers tries to minimize the overlap between the propagating fundamental mode and the silica material of the photonic crystal, the photo-induced modification of the refractive index of the original fiber structure might not be sufficient to create a grating of desired properties. Rather than modifying the silica in the fiber by photo-induced processes, we consider injecting a photoresist or other UV-curable polymer into selected hollow regions of the fiber and exposing such a modified fiber to an interference pattern. The resulting polymer film will introduce a spatially varying refractive index and as a result will act as a Bragg grating integrated into the HCPC fiber. This approach is in essence similar to the Bragg gratings demonstrated in solid-core photonic-crystal fibers previously [47]. A similar approach has been also used to implement long-period Bragg gratings with pitches of up to several hundred micrometers utilizing higher-order bandgaps when the refractive index of liquids injected into the fibers was modulated with electric fields or mechanical stress, such as in Ref. [48].

In our first proposed approach [49], this is accomplished by injecting a suitable photoresist into the core of the fiber in a way that results in coating the inner walls of the hollow core with a thin layer of additional dielectric. Exposing the photoresist to the grating pattern and flushing the fiber with a developer will remove sections of this coating in a periodic manner along the fiber (Figure 5A). This will act to modulate the effective index of the fiber, while still allowing unobstructed access of the hollow core for loading vapors or liquids. Our second proposed approach to produce a Bragg grating in HCPC fiber is to introduce a polymer into selected holes of the photonic crystal, as shown in Figure 6A. The disadvantage of this method is that because of the small diameter of the photonic crystal holes, the polymer will fill them completely and the holes will not be flushed with a developer after exposure to remove the exposed (or unexposed) sections of the resist. The Bragg grating formed in this approach will then rely on the index contrast between the exposed and unexposed sections of the resist, which is relatively low. Assuming the holes are filled with a UV-curable epoxy (e.g., Norland optical adhesives), the index contrast between the exposed and unexposed sections will be ~ 10⁻². This method is expected to result in lower reflectivity compared to the first approach because of the significantly lower resulting effective index modulation (~ 10⁻⁴). The low effective index modulation will lead to rather large penetration depths into the grating combined with high attenuation due to the partial disruption of the finely tuned photonic-crystal guiding mechanism.

Fig. 5

(A) A cutaway of a fiber Bragg grating is shown, in which a thin film of dielectric material is patterned via photolithography in the hollow core of the fiber to periodically modulate the effective index, allowing for Bragg reflection. (B) The corresponding maximum reflectivities that can be achieved using a perfect Bragg reflector with optical losses associated with the fiber are shown for polymer in the hollow core. Method 1 calculates reflectivity by finding the average penetration depth of light into the Bragg Mirror. The reflectivity will then be reduced by the amount of loss caused by the fiber as the light travels this distance, as shown in Equation 5. Method 2 uses the method of single expression (MSE) [50] whereby reflectivity is calculated from the forward and backward propagating field amplitudes at the beginning of the grating.

Fig. 6

(A) The second method of index modulation is shown, in which the photonic crystal holes are selectively filled with a polymer. (B) The corresponding maximum reflectivities that can be achieved using a perfect Bragg reflector with optical losses associated with the fiber are shown for polymer in the photonic crystal region (in which the holes in the first layer are filled one at a time in a counter-clockwise direction). See Figure 5 for a description of the Method 1 and Method 2 calculations of the maximum reflectivity.

The resulting effective index contrast in the fiber now acts as an interface in which light is partially reflected and transmitted. The reflected light can interfere constructively, producing high reflectivity over a particular bandwidth of light. The grating period length, the degree of index contrast between Bragg layers, and the number of periods determine this reflectivity. For an infinite number of periods, Bragg gratings can ideally act as perfectly reflective mirrors at wavelengths that satisfy the Bragg condition (wavelengths that are four times the optical thickness of the layers) [51]. Realistically, perfectly reflecting mirrors are not achievable due to practical fabrication challenges. Another limiting factor that reduces the attainable reflectivities is due to losses associated with the fiber itself.

The calculated maximum theoretical reflectivity of an infinite period Bragg mirror at the Bragg condition for polymer in the hollow core and photonic crystal region are shown in Figures 5B and 6B, respectively. In the latter, the first layer of photonic crystal holes are filled one at a time in a counterclockwise direction. There are two different methods to calculate this reflectivity in an infinite Bragg mirror that incorporate the loss associated with the fiber. The loss in the fiber is first found using Lumerical MODE Solutions [52] by solving for the eigenmodes of a HCPC fiber model. This model is qualitatively similar to the HC-800B [53] fiber available commercially in the sense that its hollow core has the same diameter and its transmission was optimized for ~ 860 nm, although due to limited computational resources the model is not fully optimized and the transmission loss is significantly higher than that specified for HC-800B. This higher loss likely comes from our model’s lattice being not exactly identical with the HC-800B fiber and from our simulations being limited by computational resources. The model could be further optimized to match the transmission of the HC-800B fiber, but for now it should provide a conservative estimate for the performance of the Bragg grating structures in HCPC fibers in general. The supported Gaussian modes and their subsequent attenuation coefficient and effective indices can then be found, in which the mostly transverse electric (TE) polarization in the x-direction was found to undergo lower losses. A wavelength of 860 nm was used in the simulations, corresponding to the lowest attenuation values given the photonic crystal dimensions of the fiber. The index of the material coating the hollow core was set at n = 1.61, which is consistent with photoresists such as AZ701. The index of the material added to the photonic crystal region is 1.62 for unexposed and 1.64 for exposed to UV, corresponding to UV-curable glues, such as NOA162 from Norland. The first method shown in Figure 5B and 6B estimates the resulting decrease in effective reflectivity due to loss in the fiber using an approximation for the penetration depth into a Bragg grating. The reflectivity is then found by taking into account the field loss associated with the travel of light a distance equal to this average penetration depth, z_p. The average penetration depth can be found from the Fourier expansion of the dielectric constant, ϵ, which is determined by the effective indices of the Bragg layers found by the numerical simulations. Equation 3 gives the calculated average penetration depth of light of angular frequency, ω into a Bragg grating with bandgap, Δω.

where a is the grating period length and ϵ₀₀ and |ϵ| are the two dominant Fourier coefficients of the dielectric constant, as given by Equation 4. This approximate expression is valid for weak perturbations of the dielectric constant. Equation 5 gives the resulting reflectivity, R, calculated using the simulated values of the attenuation coefficients of the Bragg layers, α₁ and α₂, in which the total path length of light traveled in each layer is z₁ and z₂ (with the light travelling twice this total distance due to light travelling both in and out of the Bragg mirror).

The second method to analyze these one-dimensional gratings is by using the method of single expression (MSE) described by Baghdasaryan et al. [50]. Starting from the transmitting side, the field in each layer is calculated as we travel backward to the illuminated side, solving a set of coupled differential equations (Equations 6–8) as we do so. The forward and backward propagating fields can be calculated at the illuminated side to determine the grating’s reflectivity.

where the electric field is E(z) = U(z)e^−iS(z), Y(z) = dU(z)/d(k₀z), and P(z) = U²(z)/[dS(z)/d(k₀z)].

The reflectivities are highly dependent on the thickness of the dielectric film coating the core, with reflectivities >96% possible at a resist thickness of 700 nm. When the polymer is instead added to the photonic crystal region, the reflectivity is drastically decreased to a maximum value of ~ 31%, as expected from this approach due to the resulting low index contrast and high attenuation in the fiber. However, it should be noted that our simulation model of the fiber structure provided a spectrum of losses that were at least an order of magnitude larger than manufacturer’s specifications for HC-800B, so the actual reflectivities are likely to be higher.

These values can be compared to the reflectivity potentially achieved with a photo-induced index modulation of the fiber material itself by using femtosecond laser pulses, an approach commonly used to create laser-written photonic structures in silica and related materials [54, 55]. This index modulation of ~ 10⁻² in the fiber material results in a maximum reflectivity of 97%. However, due to the extremely low change in effective index, a mirror based solely on this silica material modulation would require a Bragg grating 2.8 cm long (corresponding to ~ 10⁴ Bragg periods). This is in contrast to a Bragg grating of ~ 270 µm in length resulting from ~ 10² different Bragg periods required for the >96% reflectivity possible with 700 nm of resist in the hollow core.

As far as practical implementation of these two approaches in creating a Bragg reflector in the fiber is concerned, several methods for selective injection of liquids into photonic crystal fibers have been reported in the past. Most of them are suitable for selectively injecting liquid into the central hole, such as using a fusion splicer to collapse the honeycomb structure around the central hole [56] or by taking advantage of the different capillary forces and filling speeds in the small diameter photonic crystal holes and the larger diameter hollow core [57, 58]. However, methods based on gentle splicing of an appropriately positioned capillary to the end of the HCPC [59, 60] or on drilling a micron-sized hole using a femtosecond laser into a thin slice of glass attached to the face of a HCPC fiber [61], can be adopted for filling any combination of holes with liquid.

4.2 Gratings in ARROWs

The bottom-up progression in ARROW fabrication offers the possibility to integrate a Bragg grating into the waveguide directly during the fabrication process, by defining gratings in selected layers of the structure using photolithography or electron beam lithography followed by an etching process. The best candidates for this would be the top layer, the first layer above the core, or the first layer beneath the core as shown in Figure 7. For the three-period ARROW formed by Si₃N₄ and SiO₂ layers, the maximum effective index change is 1.2 × 10⁻⁴, 1.3 × 10⁻⁴, and 1.7 × 10⁻⁴ respectively. For a four-period ARROW using the same materials, the maximum effective changes are 8.8 × 10⁻⁵, 1.1 × 10⁻⁴, and 4.2 × 10⁻⁵ respectively. For a three-period arrow formed by TiO₂/SiO₂ layers, the maximum effective index changes are 1.2 × 10⁻⁴, 2 × 10⁻⁴, and 4.3 × 10⁻⁴ respectively. Since the light is mostly confined to the core, etching the layers closer to it creates a larger perturbation in the mode, causing a larger change in the effective index of propagation. Increasing the number of periods lowers the propagation loss but limits the effective index change we can create due to a more confined mode area inside the core.

Fig. 7

Showing the locations in the ARROW structure where periodic modulations may be introduced (A) above the core (B) below the core (C) in the capping layer.

The reflectivity of these integrated Bragg mirrors can be estimated with the average penetration depth approximation used for the gratings in HCPC fibers. Figure 8 shows the reflectivity of this grating for a given number of periods. We can see that there is an upper bound on the reflectivity due to the presence of significant loss in the waveguide. The reflectivity spectrum for a grating with 10,000 periods, and an effective index change of 1.2 × 10⁻⁴ created by etching into the top layer above the core is shown as well. The mirror has a somewhat limited bandwidth (~26 GHz) due to the fairly small change in effective index.

5.1 Fabrication of planar photonic crystal mirrors

The planar PC mirrors are fabricated using Si₃N₄ free-standing films purchased from Norcada. Si₃N₄ was chosen due to its relatively high index and low absorption at wavelengths desired for future experiments based on alkali atoms, such as rubidium or cesium. The photonic crystal holes are first patterned in the Si₃N₄ film.

This done by using an aluminum metal hard mask evaporated onto the Si₃N₄ film and ZEP e-beam resist is then spun coated over the hard mask. A hard mask is used in order to drastically increase the reactive ion etching selectivity when transferring the PC hole pattern between the mask and the Si₃N₄ film. The ZEP layer is patterned with PC holes using e-beam lithography and the resist is developed away. The PC pattern is transferred to the hard mask using a Cl₂/O₂ plasma metal etch. This pattern is subsequently etched into the Si₃N₄, with the metal hard mask providing increased selectivity in the C₄F₈/SF₆ plasma etch process [66]. The remaining e-beam resist and hard mask are removed using a wet etch procedure. Figure 11A shows the PC slab pattern, which includes trenches around the membrane allowing for easy removal when attaching to fiber tip.

Fig. 11

(A) An optical image of the sample pattern created in the Si₃N₄ freestanding films. The surrounding trenches are designed to allow removal of the structure from the substrate, while the inset shows a scanning electron microscope (SEM) image of the middle region containing the PC holes. The optical and SEM image scale bars correspond to 20 µm and 2 µm respectively (B) Schematics of a procedure for mounting a PC membrane onto an optical fiber tip using micro-drops of epoxy (Source: [65]).

In order to mount these PC membranes onto the face of a fiber, a method developed by Shambat et al. [65] can be used, in which a sharp tungsten tip is used to apply two small droplets of epoxy to the face of an optical fiber to attach to it a patterned membrane (Figure 11B). A similar approach can be envisioned for combining the PC membrane mirror with an ARROW.

8.1 Single-Photon Transistor

Compared to single-quantum emitters, ensembles of quantum emitters offer significant advantages for implementing controllable single-photon interactions. These advantages include collective enhancement of light– matter interactions, robustness to noise and decoherence, and the inherent ability handle light pulses containing multiple photons.

An excellent illustration of these advantages is provided, for example, in an experiment performed by Chen et al. [77] realizing an all-optical transistor where individual “gate” photons controlled the propagation of a stream of “source” photons. The experimental system, shown in Figure 15B, consisted of a macroscopic Fabry–Perot cavity containing a cloud of laser-cooled¹³³Cs atoms with an N-type atomic level structure with two stable ground states |g〉, |s〉 and two excited states |d〉, |e〉.

A gate photon, resonant with the (|g〉 → |d〉) transition, is stored in the collective spin-wave excitation between |g〉 and |s〉 (Figure 15C). The population in |s〉 from the single stored gate photon is sufficient to shift the resonance of the cavity and blocks the source photons resonant on the bare cavity mode transition |s〉 → |e〉 (Figure 15D). The authors reported the blockage of more than 600 source photons with a single stored gate photon. Although this gain drops significantly, if one also requires the transistor to operate in the quantum memory regime (i.e., being able to retrieve the stored gate photon at the end of the operation as shown in Figure 15E), it remains above unity, making this is an unprecedented demonstration of an all-optical transistor that can operate both in the quantum and classical regime at the fundamental limit of single gate photon.

However, since the gate and control photons are applied from the side of the cavity, the storage efficiency of the gate photon is limited by the small optical depth of the atomic cloud in the transverse direction. Implementing this scheme with a polarization selective cavity based on a hollow-core waveguide and properly designed photonic-crystal membrane acting as dielectric metasurface mirror (Figure 15A) would allow the gate photons to propagate along the same direction as the source photons and to be stored with much higher efficiency.

8.2 Superradiance

Another example of an application that could benefit from development of the mesoscale cavities described here is the ultrastable, superradiant Raman laser demonstrated recently [78]. It has been previously shown that the linewidth of a laser limited by quantum noise can be smaller than the Schawllow–Townes limit when the gain bandwidth is smaller than the cavity loss rate, the so-called “bad-cavity,” or superradiant limit where 2 ≪ [79]. Bohnet et al. [78] created a superradiant Raman laser using a cavity operated in the bad cavity limit with an ensemble of ⁸⁷Rb atoms in an optical lattice. Instead of creating coherent emission by stimulated emission, the collective synchronization of the atomic dipoles in the ensemble creates a radiation enhancement known as super-radiance. In this system, the linewidth and stability of this laser was affected by the mechanical stability of the mirrors forming the cavity by a factor of 10,000 times less than is the case for a conventional laser. This approach leads to an extremely stable laser with applications ranging from technological, such as in telecommunication networks, to precision measurements of fundamental laws governing the universe [80]. Importantly though, the cavity required operation of this laser is of low finesse and typically operated with cooperativities C = g²/κγ > 1. The collective coupling NC ≫ 1, where N is the number of atoms in the cavity, is the determining factor when it comes to creating self-sustained coherent stimulated emission. This regime should be easily obtainable in HCPC fiber-based cavities (Figure 12) and possibly even in cavities integrated into ARROWs (Figure 13B).