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We propose and numerically simulate two methods for implementing Bragg gratings in hollow-core photonic-crystal fibers. These two methods leave the hollow-core unobstructed and both are based on controlled selective injection of photosensitive polymers into the photonic-crystal region of the hollow-core fiber, followed by interference photolithography. We report the results of numerical simulations for the hollow core fiber with Bragg gratings formed by the two methods. We find that a reflectivity of > 99.99% should be achievable from such fiber-integrated mirrors.
© 2017 Optical Society of America
1. Introduction
Hollow-core photonic-crystal fibers (HCPCF) offer an excellent platform for enhancement of light-matter interactions, particularly when the matter takes the form of an atomic ensemble [1] such as a dilute atomic vapor. The fiber geometry gives rise to a tight confinement of photons and their overlap with atoms over distances not limited by diffraction, while the hollow core allows introduction of atoms or molecules that would be incompatible with a solid-core fiber.
Over the last decade, hollow-core photonic-crystal fibers loaded with room-temperature gases [2–4] and laser-cooled atoms [5–7] have been used in a number of experiments exploring the fundamental limits of non-linear optics, such as all-optical switching [8], cross-phase modulation [9], and single-photon memory [10]. The hollow-core photonic-crystal fibers have propagation losses from < 250 dB/km for off-the-shelf fibers down to recently reported ∼1 dB/km [11], which approaches the lowest losses reported for conventional solid-core fibers [12]. This gives hollow-core photonic-crystal fibers a significant advantage compared to other hollow-core waveguides with a comparable cross-section and mode-field diameter, such as the on-chip hollow-core anti-resonant reflection optical waveguides (ARROW) [13], where the reported losses for fabricated structures are currently around 9 dB/cm. At the same time, lithographical techniques developed recently for atom-chip applications [14] could be used to integrate the hollow-core fibers into on-chip platforms.
Further enhancement of the light-matter interaction inside hollow-core fibers filled with atomic gases could potentially be achieved by dispersion engineering of the hollow-core fiber or by incorporating a cavity into it. The goal of this would be to increase the probability of interaction between single photons and single atoms inside the fiber to as close as possible to unity [1,15], while at the same time keeping the hollow core unobstructed to allow loading of atoms into the fiber. For solid-core fibers, dispersion engineering and mirror integration can be achieved using fiber Bragg-gratings [16] which are implemented by periodically modulating the refractive index of the fiber material, in particular, of the core. This task, however, is challenging for the hollow-core fiber, as the fiber core is empty and, furthermore, the fiber is designed with an effort to minimize the overlap between the propagating light and the glass material of the cladding.
Here, we propose and numerically simulate two methods to implement a Bragg grating in a HCPCF. These methods are somewhat related to realizations of Bragg gratings in photonic-crystal fibers reported in the past [17,18], although these demonstrations were not done with hollow core fibers and resulted in complete filling of the photonic crystal region.
Our first method (Figs. 1(a) and 1(b)) is based on coating the inner wall of the hollow core with a photoresist which is then exposed to an appropriate UV light interference pattern. The fiber core would then be flushed with resist developer to remove the unexposed resist, leaving behind a ribbed structure acting as a Bragg grating. The initial coating of the hollow core with resist could possibly be obtained by the use of pressurized gas to eject resist loaded into the core. Adhesion of the resist to the core walls would allow for a thin film to remain, in which the film thickness would ideally be controllable by the variable pressure of the gas, the viscosity of the photoresist, and the solvent used. Alternatively, uniform thickness could be achieved through some type of evaporation technique, such as reported in [19], or through the use of chemical growth out of a solution.
Fig. 1 The two proposed methods for integrating Bragg gratings into HCPCFs: (a) the cross section and (b) cutout of a HCPCF with a thin film of resist coating the hollow core. The (c) cross section and (d) cutout of a HCPCF with a UV-curable polymer selectively filled in the first layer of the photonic crystal holes.
In our second proposed method (Figs. 1(c) and 1(d)), one or more of the holes of the photonic crystal (PC) region are filled with UV sensitive epoxy. The fiber would again be exposed to a periodic light pattern to leave behind a periodic modulation of the refractive index of the injected epoxy.
Both of these methods rely on selective filling of the holes of the photonic-crystal fiber. Such filling has, to some extent, been demonstrated previously [20–24], although refinement of these techniques will likely be needed. Alternatively, polymer cold-drawing reported by Shabahang et al. [25] may also provide a possible technique to produce high refractive index modulation in the fiber core.
In the following sections we describe the results of numerical simulations predicting the performance of these two types of Bragg gratings as mirrors integrated into a HCPCF.
2. HCPCF models
We start by implementing a fiber model using the Lumerical MODE Solutions software in which the field eigenmodes of the structure are solved to find its attenuation coefficient and effective index. We simulated two different models of the fiber, which are roughly based on HC-800-02 (Fig. 2(a)), a commercially available HCPCF from NKT Photonics guiding light using a photonic bandgap [26]. We focus on this particular fiber as it can guide wavelengths corresponding to transitions of alkali atoms, such as rubidium and cesium, whose vapors are commonly used to study non-linearities and quantum optics of low light levels. The first model, shown in Fig. 2(b) and referred to as the circular hole model, employs an idealized circular shape for both the hollow core as well as the photonic crystal holes. It resembles one of the first demonstrations of a hollow-core fiber by Cregan et al. [27] in which light was guided by the photonic-bandgap effect in an air core and could be considered the simplest HCPCF shape. The second model, shown in Fig. 2(c) and referred to as the hexagonal fiber model, is based on a theoretical rendering of a HCPCF [28] which has a twelve-sided core with alternately long and short sides arising when the glass nodes of the core are placed at uniform pitch and at the corners of each cladding hexagon. Shape-wise, the second model resembles relatively closely to the scanning electron microscope (SEM) image of the commercial fiber shown in Fig. 2(a). The loss associated with the hexagonal hole model was found using a discretized picture of the fiber shown in Fig. 2(c). The hollow core diameter was scaled to 6 μm, which resulted in a lattice pitch of ∼ 2.1μm and a PC hole diameter of ∼ 2μm. Minimal attenuation in this model occurred at a wavelength of 851 nm and all subsequent simulations for the fiber model are performed at this wavelength unless stated otherwise.
Fig. 2 HCPCF cross sections: The (a) SEM picture of the HC-800-02 fiber. (b) The circular hole model and (c) hexagonal hole model [28] implemented to simulate the fiber structure (optimized lattice pitch, PC hole size and hollow core diameter not drawn to scale). Propagation attenuation in HCPCFs: (d) The manufacturer specification for HC-800-02 and the numerically simulated losses for the (e) circular hole model and (f) hexagonal hole model.
Although not exact, the hexagonal hole model of the fiber qualitatively reproduces the transmission properties of the commercial fiber (Fig. 2(d)) with minimal losses of ∼ 0.150 dB/m. The simulated hexagonal model (Fig. 2(f)), yields a minimum loss of ∼ 0.330 dB/m. Our simulation model has neglected any additional losses that may be caused by surface scattering and this higher loss arises from having fewer than 10 hole layers [26] surrounding the hollow core. While increasing the loss, this reduced number of hole layers allowed us to fit our simulation within our available computational resources. Since the light propagating through this model will experience larger attenuation compared to the actual fiber, we expect to obtain a conservative estimate on the performance of the fiber Bragg mirrors calculated in the later sections of this paper.
The attenuation spectrum for the circular hole model (Fig 2(e)) was minimized by altering the lattice pitch of the triangular photonic crystal pattern, as well as the diameters of the PC holes and hollow core of the fiber, resulting in optimized dimensions roughly similar to HC-800-02 in which the pitch is specified as 2.3 ± 0.1μm and the hollow core diameter as 7.5 ± 1μm. In our circular model, the resulting optimized pitch was 2.3 μm, with a hole diameter of 2.174 μm and hollow core diameter of 6.386 μm. The minimum attenuation found occurred at a wavelength of 860 nm, producing a minimum loss of ∼ 1.78 dB/m.
Although the circular hole model leads to inferior mirrors compared to the hexagonal hole model, we included it to demonstrate that our methods can be applied to form mirrors in more than one design of a hollow-core photonic crystal fiber. For these reasons, this paper will focus on the results from the hexagonal hole simulation model, and give only a summary of the results provided by the circular hole model in the discussion section.
Perfectly matching layer (PML) boundary conditions were used for an eigenmode solver of size 23.6571 μm×22.7641 μm, in order to simulate a symmetric region around the core. The number of mesh cells in this region were set at 2160 × 2160, so as to produce a discretized simulation that highly resembles the actual fiber structures. The subsequent loss and effective index of the supported fundamental Gaussian mode can then be found.
Our simulations assume that the photoresist and UV epoxies that will coat the hollow core walls and fill the PC holes, respectively, are both continuous and homogeneous materials.
3. Hollow core coating method
The first approach we investigated for creating a Bragg grating along the axis of the fiber is based on coating the walls of the hollow core with a thin film of photoresist, as shown in Figs. 1(a) and 1(b). The Bragg grating would be created by exposing the resist to a periodic interference pattern from a UV laser. The exposed sections of the resist film would then be flushed away by injecting resist developer into the fiber, which would produce a longitudinal variation of the fiber’s effective refractive index.
Ideally, for an infinite number of periods, a simple Bragg mirror would produce a reflectivity of 100% for wavelengths at the Bragg condition, where the wavelength is four times the optical path length of each of the layers. In practice though, reflectivity will be reduced due to the loss associated with the fiber itself, as well as due to additional losses arising from the presence of the photoresist, which will partially disrupt the photonic-crystal waveguiding of the fiber.
Propagation attenuation in the fiber is greatly dependent on the film thickness of the resist. Figures 3(a) and 3(b) shows the loss and effective propagation refractive index for various film thicknesses, respectively, using the hexagonal simulation model. The three different resist material indices explored were ∼ 1.61, corresponding to photoresists such as AZ701 specifically, as well as 1.45 and 1.30 which act to span the region of possible resist indices in the hopes to observe a general trend.
Fig. 3 Simulation results for attenuation and effective refractive index in the fiber with a layer of photoresist coating the inside walls of its hollow core: (a) and (b) show results obtained for the hexagonal hole model at a wavelength of 851 nm. Three different indices for the resist material were used: 1.61, 1.45 and 1.30, providing a range of possible resist indices, such as that of AZ701. Simulations were done using a custom mesh size of ∼ 1.6 nm×1.6 nm×1.6 nm for the resist film in order to accurately model the relatively thin layer. All plot lines are not extrapolations of the data, but instead provide an easier visual distinction between data sets.
The corresponding effective fiber loss reduces the reflectivity of the Bragg mirror as the light penetrates into the grating region. Two separate methods are used to estimate the attainable maximum reflectivity based on the fiber attenuation. The first method, which we refer to as the penetration depth method (PDM) in Fig. 4, approximates the reduction in reflectivity as due to the loss that occurs for travelling twice the penetration depth, zp. This is a result of reflected light having travelled, on average, a total round trip of 2zp due to propagation in and out of the Bragg mirror. The assumption of an infinite number of Bragg periods is still used, however the resulting approximate reflectivity, R, now becomes
in which the two Bragg layer attenuation coefficients are α1 and α2, and the respective total travel distance in each layer is z1 and z2, such that zp = z1 + z2. This approximation can be further refined by considering a sinusoidally varying function of the resist layer thickness that is likely to result from the interference lithography process, rather than a step function. However, it was found to give negligible corrections to the reflectivity values.Fig. 4 The calculated maximum reflectivities for a Bragg grating caused by periodic films of resist coating the hollow core. The three different material indices used are (a) 1.61, (b) 1.45, and (c) 1.3 for the hexagonal hole model (in which the 1-Reflectivity value is plotted on a logarithmic scale). The penetration depth method (PDM) calculates the reflectivity by determining the approximate average penetration depth of light into the gratings using Eq. (2). The reflectivity, found by using Eq. (1), is then reduced by the amount of fiber attenuation corresponding with this average light travel distance. The method of single expression (MSE), as described by Baghdasaryan et al. [30], is also used to analyze the grating reflectivity. Some of the reflectivity data points are absent for MSE because the numerical algorithm did not always converge to a finite value. All plot lines are not extrapolations of the data, but instead provide an easier visual distinction between data sets.
The average penetration depth can be estimated using the Fourier coefficients of the spatially varying dielectric constant ϵ. We extract those from the numerical results obtained for the effective indices of the Bragg layers with a given thickness of the resist film. For light at a certain angular frequency, ω, the penetration depth into a Bragg grating with a bandgap, Δω, is given by [29]
in which a is the length of an individual Bragg period. The dominant Fourier coefficients of the dielectric constant, ϵ00 and |ϵ1|, are given byThe above penetration depth expression (Eq. (2)) is an approximation valid for small perturbations in the dielectric constant, which is the case here.
The second method for finding the reflectivities, referred to as the method of single expression (MSE) in Fig. 4, starts from the transmitting side of a mirror and calculates the electric field while iterating in discrete steps towards the illuminated side, by solving the set of coupled differential Eq. (5)–(7) given below [30]:
Here, the electric field is E(z) = U(z)e−iS(z), and P(z) = U2(z)/[dS(z)/d(k0z)]. The free space wave vector is given by k0, and ϵl is the dielectric constant of the medium at position, z. The reflectivity of the mirror is then given by Eq. (8) with z = 0 denoting the illuminated side of the mirror.
Both methods for calculating the maximum reflectivities of the fiber Bragg grating result in similar trends. Figures 4(a)–4(c) show these reflectivity trends for the hexagonal simulated model, although there are noticeable discrepancies between the calculated reflectivities obtained by the two methods for small thicknesses of the photoresist (1–5 nm). This is due to the extremely low effective index contrast between the Bragg layers that occurs when using these particular film thicknesses, which presents a challenge to the numerical implementation of MSE. Maximum reflectivities, using the hexagonal hole model, were found to occur using a 700 nm thick resist film (at all three resist indices) with a value of > 99.99%.
To put this result in perspective, we can compare it to reflectivities potentially achievable by index modulation of just the silica material of the photonic crystal region. Such refractive index modulation of silica can be achieved, for example, by exposing the fiber to femtosecond laser pulses [31,32]. The result can be up to ∼ 10−3 index change of the material and forming Bragg gratings in solid-core PCFs have been reported previously [33,34]. Such index modulation of the glass material would result in only minor additional losses and would produce a relatively high maximum reflectivity value of ∼ 99.2%. However, to obtain such a high reflectivity based only on this low index modulation of the silica material, the number of Bragg periods required would be ∼ 105 (corresponding to > 5.1 cm penetration depth) since the effective index contrast for the propagating mode would only be ∼ 10−5. There is a similar effect when using an extremely thin layer of resist in the hollow core. The reflectivity from these thinner films shown in Figs. 4(a)–4(c) can be quite large, however they suffer from the same property of inducing low index modulation in the fiber. This is in contrast to about ∼ 3 × 102 Bragg periods (∼ 100 μm penetration depth) required for > 99.99% reflectivity using the 700 nm thick resist film coating the fiber core. Consequently, resulting large penetration depths that occur will of course require longer mirrors to reach a given reflectivity. Additionally, if a pair of such mirrors is used to form a cavity within the fiber, the cavity mode volume would be larger for mirrors with larger penetration depth, which would in turn decrease the atom-field coupling strength in cavity quantum electrodynamics (QED) experiments [35].
4. Crystal holes filling method
Another technique that could be used to modulate the effective index of the fiber in order to create a Bragg grating is to selectively fill the outer PC holes with a photo-sensitive polymer, as shown in Figs. 1(c) and 1(d). The disadvantage of this approach to fabricating Bragg gratings, however, is that the relatively smaller size of the PC holes (∼ 1 μm) will likely prevent material from being removed by flushing the holes with a developer after exposure. As a result, only the difference in index between the exposed and unexposed polymer can be relied on, rather than the absence of material (in contrast to the method described in the previous section). This results in a reduction of the effective index contrast for the propagating mode by about an order of magnitude compared to the previous method based on coating the walls of the hollow core with photoresist. Filling the PC holes will thus require more Bragg periods in order to produce a given reflectivity (even in the absence of fiber loss) and a larger penetration depth will occur. Similar to coating the inner wall of the hollow core, the presence of material in the PC region will disrupt the finely-tuned bandgap effect and will increase propagation losses in the fiber.
For this technique, we investigate the use of a UV-curable epoxy (glue), in which a standing wave of UV light can be used to cure the glue in a periodic pattern, resulting in a modulated index of the material. The three indices we simulated are 1.62, which will imitate UV-glues such as Norland optical adhesives NOA162 epoxy, as well as 1.45, and 1.30 in order to again span the range of potential indices. Curing the epoxy is assumed to increase its refractive index by ∼ 2 × 10−2.
Simulations are done as each hole is filled individually in a circularly consecutive pattern. Other filling sequences investigated were found to produce larger attenuations, however there may still exist a more optimal manner in which to consecutively fill the holes but this is not further explored in this paper. We present two implementations of this technique, one in which the first layer holes in the PC region (closest to the fiber core) are filled, and the other for when the second layer of holes are filled. Figures 5(a)/5(c) and 5(b)/5(d) show the loss and effective index, respectively, associated with the number of filled holes in the first/second layers of the PC region with uncured UV epoxy for the hexagonal hole model.
Fig. 5 Simulation results for attenuation and effective refractive index in the fiber with the first or second layer of PC holes filled with epoxy: (a) and (b) show the simulation results for the filled first layer of PC holes using the hexagonal hole model. (c) and (d) show results for the filled second layer of PC holes. All simulations are performed at a wavelength of 851 nm. The ’hole number’ signifies the number of filled holes in a clockwise direction. Three different indices for the uncured polymer material were used: 1.62 (representative of Norland NOA162 optical adhesive), 1.45 and 1.30. All plot lines are not extrapolations of the data, but instead provide an easier visual distinction between data sets.
Figures 6(a) and 6(b) show the maximum reflectivities calculated for the hexagonal hole model by filling the first and second layers of the PC region, respectively. These calculated reflectivities are obtained by combining the numerically calculated attenuation and effective index of the fiber with injected epoxy (cured and uncured) using the two methods introduced in the previous section. It can be seen that these two methods for determining the reflectivity are again relatively comparable for the hexagonal model simulation. The filled PC holes give a maximum reflectivity of ∼ 99.8% for the simulated model (using 1.30 uncured epoxy index and one hole filled in the first layer of the PC region). Reflectivities exceeding 99% appear to be achievable by filling one or two holes in the first ring with epoxy of either one of the three refractive indices we considered, while filling the holes in the second ring seems to result in reflectivities barely exceeding ∼ 90%.
Fig. 6 The calculated maximum reflectivities for a Bragg grating caused by periodic exposure of UV-curable epoxy selectively injected into the photonic crystal region. The reflectivities are shown for the hexagonal hole model when the (a) first and (b) second layers in the PC region are filled with modulated material. See the caption of Fig. 4 for descriptions of the penetration depth method (PDM) and the method of single expression (MSE) for the maximum reflectivity calculations. The three different material indices used for the uncured epoxy are 1.62, 1.45, and 1.30, while the cured epoxy indices are 1.64, 1.47, and 1.32. All plot lines are not extrapolations of the data, but instead provide an easier visual distinction between data sets.
Maximum reflectivities are observed to overall be lower when the Bragg layers are formed by filling the second layer, rather than the first layer, of PC holes. However, the maximum reflectivity for a mirror created by filling the PC holes is, in general, predicted to be substantially lower than when a resist film is added to the hollow fiber core. This can be explained by the low effective index modulation and high attenuation of the Bragg layers caused by filling the PC holes. Disruption of the PC region reduces the fiber ability to trap light in the transverse direction, leading to larger losses. The differentiation between Bragg layers due only to the small difference in cured and uncured epoxy contributes to the extremely low index modulation of the mirror. This low effective index modulation also implies that a larger penetration, and thus mirror length, would be required to produce the given reflectivities caused by filled PC holes, as compared to a mirror created by modulation of a resist layer in the hollow core.
5. Discussion
The results of our numerical simulations of the hexagonal model are summarized and juxtaposed in Fig. 7, which shows the reflectivity of a Bragg mirror with lossy layers estimated using the penetration depth to determine the total loss (PDM, see equation 1). Here, |Δn| = |n2 − n1| with n1,2 being the values of the (complex) effective refractive index of each layer. The highest reflectivities predicted for the simulated fiber models is marked in the plots for both the Bragg mirror approaches based on modulated resist coating the wall of the hollow core, as well as for the approach based on filing the crystal holes. For comparison, the result arising from modulating the refractive index of only the silica material of the photonic crystal is identified as well. Although introducing even small amounts of the photosensitive polymer into the fiber structure results in significant increase of the propagation loss, the increased contrast in effective refractive index between the grating layers offers seems to offer distinct advantages for the mirror performance.
Fig. 7 The estimated maximum reflectivity of a lossy Bragg mirror for a range of effective refractive index contrast between layer pairs and average loss per unit length of the structure. The effective index contrast and loss ranges for the hexagonal hole model is plotted and the corresponding resist layer thickness that produced the highest reflectivities in the core coating method for the different refractive indices of the photoresist (1.61, 1.45, and 1.30) are marked in the plot (black circles), together with the points (purple diamonds) corresponding to the largest reflectivity predicted for the hole filling method (with the number of filled holes specified).
For the circular hole model, the highest reflectivity occurs by coating the hollow core with a resist thickness of 700 nm (at a resist index of 1.61), giving a reflectivity of > 96%. The filled PC holes instead give a maximum reflectivity of ∼ 44.4% using the 1.30 uncured epoxy index and one hole filled in the first layer of the PC region. The number of Bragg periods required for the largest reflectivities occurring from coating the hollow core periodically is ∼ 2 × 102 periods (corresponding to ∼ 90μm penetration depth). Much like the hexagonal hole model results, the Bragg mirrors formed by modulation of the silica material itself using the circular hole model produce large reflectivities of > 97%, however they again require a much larger number of Bragg periods of ∼ 2 × 104 (∼ 0.9 cm penetration depth) since the effective index contrast between layers is greatly reduced.
Our reflectivity calculations can be further validated by bidirectional eigenmode expansion simulations solving Maxwell’s equations in the frequency domain for a Bragg grating using the Lumerical software package. Figure 8(a) shows the simulated spectrum of the hexagonal hole model forming a Bragg grating with 100,000 periods for a resist coating thickness of 700 nm (and resist index of 1.61) of the hollow core walls, and produces similar reflectivity values to those found using the penetration depth method (PDM, Fig. 4(a)) that relies on plane-wave approximation of the light propagation in the structure. Additionally, this simulation predicts the range of frequencies reflected by this grating.
Fig. 8 (a) The simulated spectrum of a fiber Bragg grating using a 700nm thick resist (1.61 index) coating the hollow core walls for 100,000 periods. The maximum reflectivity is ∼ 99.9969%. (b) Estimated reflectivity bandwidth of the Bragg mirror (dashed blue, Eq. (9)) and corresponding penetration depth (solid orange, Eq. (2)) plotted against refractive index contrast between the layer pairs forming the mirror.
Note that for a low-contrast Bragg mirror such as discussed here, the mirror bandwidth, Δω, can also be quickly estimated using the Fourier coefficients of the spatially varying dielectric constant
where ω0 denotes the angular frequency of light resonant with the Bragg condition. Figure 8(b) then shows the bandwidth and corresponding penetration depth for a range of index contrast values that we expect to arise for a Bragg grating formed by the methods proposed here (using the hexagonal hole model). We can see from Figs. 7 and 8(b) that as the index modulation is increased, both the maximum reflectivity and the width of the bandgap increase, while the penetration depth decreases for an ideal Bragg mirror at a given average loss. At the same time, when loss in the fiber becomes more significant, the reflectivity can be reduced drastically, however the bandgap and penetration depth remain relatively constant. Lastly, while our calculations predict that a relatively high reflectivity can be achieved by modulating just the fiber material (e.g. by high intensity laser pulses), the benefit of using Bragg gratings formed by adding photosensitive material to the fiber structure is that the penetration depth, and thus the length of mirror required to achieve a given reflectivity, would be significantly smaller in this case. In addition to size-related advantages, a shorter mirror would allow tighter longitudinal confinement of light in a cavity formed by a pair of such mirrors, which is preferable for a number of applications, especially those related to cavity QED.6. Conclusion and outlook
To summarize, two methods for fabricating reflective Bragg gratings in HCPCFs have been proposed and numerically simulated. The numerical simulations and subsequent reflectivity calculations performed for our fiber models predict a HCPCF-integrated Bragg mirror that has a maximum reflectivity of > 99.99% and that does not obstruct the hollow core. Given that our fiber models have propagation loss exceeding that of the manufacturer specifications, our reflectivity calculations should represent a reasonable estimate of the achievable performance.
This Bragg reflection is possible despite the small overlap between the propagating mode confined to the hollow core and the grating itself. A relatively similar scenario occurs in other well-known photonic devices, such as distributed Bragg reflectors (DBR) which are used in certain diode lasers. In this case, the reflectors are formed by the periodic presence of material on the surface of a waveguide containing the diode to create Bragg mirrors which form the laser cavity, even though the overlap between the guided mode and the grating structure is again minimal [36].
Periodic coating of the inner wall of the hollow core of the fiber with photoresist appears to be the most promising way to produce a high reflectivity Bragg mirror due to the relatively low loss and high index contrast between Bragg layers resulting from this approach, although finding techniques to deposit a photoresist layer of controllable thickness onto the core wall might present a challenge. On the other hand, creating Bragg layers by selectively filling the PC holes of the fiber with a periodically exposed UV-curable polymer results in a mirror with smaller contrast between its Bragg layers but the resist injection methods required for this have already been demonstrated [20–22].
While our calculations focused on a specific HCPCF model, we anticipate similar results for other photonic-bandgap HCPCFs. This work can be further extended by modelling these Bragg gratings for HCPCFs based on inhibited coupling between core and cladding modes, also known as ’kagome’ fibers [26]. Given the distinctly different guiding mechanism in these fibers, it would be interesting to see the effects of polymer injection into the fibers’ microstructure on propagation losses and effective refractive index.
We expect these mirrors to be utilized for novel devices, such as fiber-integrated lasers with atomic or molecular vapors serving as the gain medium, sensors, and frequency standards, as well as for fundamental studies of light-matter interactions.
Funding
Industry Canada; Canada’s Natural Sciences and Research Council (NSERC) under the Discovery Grants Program.
Acknowledgments
J. F. would like to thank Ontario Graduate Scholarship (OGS). V. B. was in part supported by IQC’s USEQIP program and O. A. would like to thank King Saud University for their generous support during the course of this research.
References and links
1. D. E. Chang, V. Vuletić, and M. D. Lukin, “Quantum nonlinear optics — photon by photon,” Nat. Photonics 8, 685–694 (2014). [CrossRef]
2. S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett. 94, 093902 (2005). [CrossRef] [PubMed]
3. T. Takekoshi and R. J. Knize, “Optical guiding of atoms through a hollow-core photonic band-gap fiber,” Phys. Rev. Lett. 98, 210404 (2007). [CrossRef] [PubMed]
4. A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Generation of large alkali vapor densities inside bare hollow-core photonic band-gap fibers,” Opt. Express 16, 18976 (2008). [CrossRef]
5. C. Christensen, S. Will, M. Saba, G.-B. Jo, Y.-I. Shin, W. Ketterle, and D. Pritchard, “Trapping of ultracold atoms in a hollow-core photonic crystal fiber,” Phys. Rev. A: At. Mol. Opt. Phys. 78, 4 (2008). [CrossRef]
6. M. Bajcsy, S. Hofferberth, T. Peyronel, V. Balic, Q. Liang, A. Zibrov, V. Vuletic, and M. Lukin, “Laser-cooled atoms inside a hollow-core photonic-crystal fiber,” Phys. Rev. A: At. Mol. Opt. Phys. 83, 063830 (2011). [CrossRef]
7. F. Blatt, T. Halfmann, and T. Peters, “One-dimensional ultracold medium of extreme optical depth,” Opt. Lett. 39, 446–449 (2014). [CrossRef] [PubMed]
8. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, a. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009). [CrossRef] [PubMed]
9. V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2012). [CrossRef]
10. M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, a. Abdolvand, P. S. J. Russell, and I. a. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014). [CrossRef]
11. P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, J. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005). [CrossRef] [PubMed]
12. K. Nagayama, M. Kakui, M. Matsui, T. Saitoh, and Y. Chigusa, “Ultra-low-loss (0.1484 db/km) pure silica core fibre and extension of transmission distance,” Electron. Lett. 38, 1168–1169 (2002). [CrossRef]
13. D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated arrow waveguides with hollow cores,” Opt. Express 12, 2710–2715 (2004). [CrossRef] [PubMed]
14. R. Folman, P. Kruger, J. Schmiedmayer, J. Denschlag, and C. Henkel, “Microscopic atom optics: From wires to an atom chip,” Advances in Atomic, Molecular and Optical Physics 48, 263–356 (2008). [CrossRef]
15. H. Tanji-Suzuki, I. D. Leroux, M. Schleier-Smith, M. Cetina, A. Grier, J. Simon, and V. Vuletić, “Interaction between Atomic Ensembles and Optical Resonators. Classical Description,” Adv. At. Mol. Opt. Phy. 60, 201–237 (2011). [CrossRef]
16. A. Othonos, “Fiber Bragg gratings,” Rev. Sci. Instrum. 68, 4309–4341 (1997). [CrossRef]
17. H. R. Sørensen, J. Canning, J. Lægsgaard, K. Hansen, and P. Varming, “Liquid filling of photonic crystal fibres for grating writing,” Opt. Commun. 270, 207–210 (2007). [CrossRef]
18. G. Zito and S. Pissadakis, “Holographic polymer-dispersed liquid crystal bragg grating integrated inside a solid core photonic crystal fiber,” Opt. Lett. 38, 3253–3256 (2013). [CrossRef] [PubMed]
19. J. Zhang, C. Con, and B. Cui, “Electron beam lithography on irregular surfaces using an evaporated resist,” ACS Nano 8, 3483–3489 (2014). [CrossRef] [PubMed]
20. Y. Wang, C. R. Liao, and D. N. Wang, “Femtosecond laser-assisted selective infiltration of microstructured optical fibers,” Opt. Express 18, 18056–18060 (2010). [CrossRef] [PubMed]
21. Y. Huang, Y. Xu, and A. Yariv, “Fabrication of functional microstructured optical fibers through a selective-filling technique,” Appl. Phys. Lett. 85, 5182–5184 (2004). [CrossRef]
22. L. Xiao, W. Jin, M. Demokan, H. Ho, Y. Hoo, and C. Zhao, “Fabrication of selective injection microstructured optical fibers with a conventional fusion splicer,” Opt. Express 13, 9014–9022 (2005). [CrossRef] [PubMed]
23. C. M. B. Cordeiro, E. M. dos Santos, C. H. Brito Cruz, C. J. de Matos, and D. S. Ferreiira, “Lateral access to the holes of photonic crystal fibers – selective filling and sensing applications,” Opt. Express 14, 8403–8412 (2006). [CrossRef] [PubMed]
24. B. T. Kuhlmey, B. J. Eggleton, and D. K. C. Wu, “Fluid-filled solid-core photonic bandgap fibers,” J. Lightwave Technol. 27, 1617–1630 (2009). [CrossRef]
25. S. Shabahang, G. Tao, J. J. Kaufman, Y. Qiao, L. Wei, T. Bouchenot, A. P. Gordon, Y. Fink, Y. Bai, R. S. Hoy, and A. F. Abouraddy, “Controlled fragmentation of multimaterial fibres and films via polymer cold-drawing,” Nature 534, 1–24 (2016). [CrossRef]
26. F. Benabid and P. J. Roberts, “Linear and nonlinear optical properties of hollow core photonic crystal fiber,” J. Mod. Opt. 58, 87–124 (2011). [CrossRef]
27. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]
28. J. Laegsgaard and P. J. Roberts, “Dispersive pulse compression in hollow-core photonic bandgap fibers,” Opt. Express 16, 9628–9644 (2008). [CrossRef] [PubMed]
29. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, “Photonic crystals: Molding the flow of light,” Princeton University Press (2008).
30. H. V. Baghdasaryan, T. M. Knyazyan, T. H. Baghdasaryan, B. Witzigmann, and F. Roemer, “Absorption loss influence on optical characteristics of multilayer distributed bragg reflector: wavelength-scale analysis by the method of single expression,” Opto-Electron. Rev. 18, 438–445 (2010). [CrossRef]
31. K. Hirao and K. Miura, “Writing waveguides and gratings in silica and related materials by a femtosecond laser,” J. Non-Cryst. Solids 239, 91–95 (1998). [CrossRef]
32. K. Sugioka and Y. Cheng, “Femtosecond laser processing for optofluidic fabrication,” Lab Chip 12, 3576–3589 (2012). [CrossRef] [PubMed]
33. L. B. Fu, G. D. Marshall, J. A. Bolger, P. Steinvurzel, E. C. Mägi, M. J. Withford, and B. J. Eggleton, “Femtosecond laser writing bragg gratings in pure silica photonic crystal fibres,” Electron. Lett. 41, 638–640 (2005). [CrossRef]
34. C. Martelli, J. Canning, N. Groothoff, and K. Lyytikainen, “Strain and temperature characterization of photonic crystal fiber bragg gratings,” Opt. Lett. 30, 1785–1787 (2005). [CrossRef] [PubMed]
35. C. J. Hood, H. J. Kimble, and J. Ye, “Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity,” Phys. Rev. A: At. Mol. Opt. Phys. 64, 033804 (2001). [CrossRef]
36. G. Hasnain, K. Tai, L. Yang, Y. H. Wang, R. J. Fischer, J. D. Wynn, B. Weir, N. K. Dutta, and A. Y. Cho, “Performance of gain-guided surface emitting lasers with semiconductor distributed bragg reflectors,” IEEE J. Quantum Electron. 27, 1377–1385 (1991). [CrossRef]
