1. Introduction

Guiding light in the empty core of a hollow core (HC) fiber is one promising branch in fiber optics, as these waveguide can provide low optical loss in regimes beyond solid core silica fibers and can allow applications in previously inaccessible areas such as biophotonics or ultrashort pulse management. HC photonic crystal fibers (HC-PCFs) rely on a periodic arrangement of micrometer-size air holes in the cladding with the result of photonic band gap formation, which can be used to confine the light in the HC. These fibers show losses as low as 1.2 dB/km in the near infrared (NIR) and about 1 dB/m in the visible (VIS) part of the spectrum [1, 2]. Since it is challenging to shift the photonic band gaps towards shorter wavelength, the ultraviolet (UV, < 400 nm) domain has only been seriously accessed with Kagome fibers [3]. In this kind of fiber the cladding exhibits a low density of cladding states, which allows waveguiding in the HC over a much larger spectral bandwidth than the HC-PCF design but with about 100 times larger optical attenuation. All mentioned fibers require sophisticated internal structures with hundreds of micro- or even nanometer size features, making them difficult to reproduce and, as a consequence, has discarded them from being widely used.

Another promising fiber design relies on “anti-matching” the core mode to all other modes of the system – the so-called antiresonant guidance mechanism. This mechanism can be illustrated by considering the coupling between two single waveguides: If the effective mode indices of two propagating modes do not match, the waveguides cannot effectively exchange their energy. Coupled-mode-theory implies that the energy transfer reduces towards larger index differences, which is employed in anti-resonant waveguide by anti-matching the core mode with all other modes as much as possible. Fibers of this kind of design are significantly simpler due to a strongly reduced number of involved capillaries in the fabrication. One important example is the negative curvature fiber consisting of eight capillaries within one jacket, guiding the light in a leaky mode with losses as low as 24.4 dB/km at 2.4 µm [4]. Almost all anti-resonant fibers investigated so far operate in the NIR [4,5], and there is an intensive debate if this guidance concept can be transferred to shorter wavelengths, where highly promising applications can be anticipated. Almost all anti-resonant fibers, however, reveal massively increasing loss in the UV above the level of several dB/m. The related discussion in the respective community is mostly focused on fabrication induced surface-roughness losses as being the origin of this increase.

One recent antiresonant design includes a thin silica wall surrounding a square-shaped HC, showing propagation loss as low as 2 dB/m in a UV transmission band centered at around 350 nm [6] (an image of this fiber is shown in Fig. 1(a). The mode in the hollow center is not resonant with the silica strand modes as well as with the leaky modes of the outer waveguides, and thus the guided mode is described as double-antiresonant modified tunneling leaky mode [6]. The fiber is made from four stacked capillaries inside a jacket tube, which is a significantly simpler design than for instance the HC-PCF or even the negative curvature fiber [6]. Corresponding transfer matrix method (TMM [7],) based simulations confirm the mentioned guidance mechanism and suggest loss of about 0.2 dB/m close to 350 nm wavelength, being ten times lower than observed in the experiment. The reason for the discrepancy remains, however, unclear.

 

Fig. 1 (a) Scanning electron microscope image of the fiber cross section (2a: core extension, t: strand thickness, b: radius of microstructured part, w: extension of outer waveguide). Dark area is air and bright area is silica. (b) Extended ring model (ERM) used for calculations shown in (c) including a high index ring and an infinite high index cladding (white is air, cyan is silica). Dashed line indicates extension to infinity. (c) Measured attenuation of the fundamental mode (blue line) and corresponding calculated leakage loss using the ERM (depicted in (b)) for a single wall thickness t = 560 nm (purple dashed line) and of a thickness variation of Δt = 70 nm (solid green line) is included. The grey vertical dotted lines refer to the order of strand resonance. Bands are labeled by the respective short-wavelength strand resonance (e.g. the band located with a minimum loss at 230 nm is named band 6).

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The scope of this paper is to evaluate two potential sources of additional losses on the example of the UV-guiding fiber just mentioned. These loss contributions have not seriously been investigated within the context of anti-resonant fibers. The first source is an accumulated variation of the silica wall thickness, which lead to an overall reduced bandwidth of the transmission bands and, as a consequence, to increased modal attenuation. The second contribution is surface roughness induced loss resulting from fabrication-induced frozen-in surface capillary waves. It is known that such kind of roughness represents the physical loss limit of state-of-the-art HC-PCFs in the NIR [1,8]. Since this contribution is known to increase with frequency, surface scattering is potentially relevant for antiresonant fibers especially in the UV. Here, we theoretically analyze the impact of these two effects on the UV loss of the double-antiresonant HC-fiber, whereas the arguments presented also hold qualitatively for any kind of anti-resonant fiber.

2. State of experimental and theoretical results

The fiber investigated here has a square-shaped empty core (Fig. 1(a), edge length 2a = 16.6 µm) surrounded by thin silica strands with an average thickness of t_c = 560 nm (details of the fiber fabrication can be found in [6] and in the appendix A1). This square is centrally hold in place by a microstructured cladding with an outer radius of b = 37 µm where the solid unstructured cladding begins (outer fiber diameter: 125 µm). Loss maxima are observed at those wavelengths where the core mode is phase-matched to strand resonances given by

This fiber exhibits several transmission bands with minimum attenuations of about 2 dB/m at 500 nm and 350 nm. Here, each band is identified by the appropriate strand resonance at the short wavelength side of the band. For instance, the band with a minimum loss at 230 nm is named band 6. Attenuation steadily increases towards shorter wavelengths reaching 10 dB/m around 280 nm and 50 dB/m around 230 nm. Corresponding TMM simulations consider a concentric ring model including not only the hollow core but an outer air ring, as well, with a silica cladding extended to infinity (Fig. 1(b), 2a = 16.6 µm, t = 560 nm, b = 37 µm.

The outer air ring is crucial to correctly model the behavior of the tunneling leaky modes as demonstrated in [6]. This simulations show a decreasing loss towards shorter wavelength starting around 10 dB/m in the NIR range and reducing to 0.1 dB/m for UV wavelengths (Fig. 1(c), purple dashed line). While there is a reasonable agreement for NIR and VIS wavelengths, the discrepancy towards shorter wavelengths increases significantly. Moreover the experiments show a pronounced decrease of spectral bandwidth of the transmission bands. This reduction gets more severe towards shorter wavelength and is not yet observed in the TMM simulations.

3. Influence of wall thickness variation

We found in the experiment that within one fiber cross section the wall thickness is not constant, but rather varies between 520 nm and 590 nm (t _± = t_c ± Δt/2, Δt = 70 nm). As suggested by the above mentioned strand resonance conditions (Eq. (1) it can be anticipated that a varying wall thickness variation impose the core mode to sample over all thicknesses within Δt. A simple estimation of the resulting bandwidth of each transmission band for a given value of Δt can be performed by analyzing the shift of the strand resonances adjoining one transmission band (e.g. the resonance m = 5 and m = 6 of band 6, Fig. 1(c), see appendix A2 for details).

This allows deriving an equation for the bandwidth ratio K = Δλ/Δλ₀ (Δλ and Δλ₀: reduced and initial band bandwidth; Eq. (3), which is a measure how strong the individual transmission band is affected by Δt. Bands of higher number show a significantly stronger bandwidth reduction, especially for large values of thickness variation (Fig. 2), which is a result of the increasing slope in the expression for K. A critical thickness variation Δt_crit at which the respective band fully disappears can be defined (Eq. (4). The actual value of this critical variation is given by Δt_crit = 2t_c/(2m-1) and decreases towards higher band number (inset of Fig. 2) revealing that transmission bands at shorter wavelength are stronger influenced by such thickness variations, which is especially problematic in the UV domain.

 

Fig. 2 Bandwidth ratio as function of band number (purple: Δt = 2 nm, green: Δt = 40 nm, blue: Δt = 100 nm, t_c = 560nm). The parameter Δt_crit indicates the thickness variation at which the transmission band fully disappears. Inset: critical thickness variation (normalized to t_c) as function of band number.

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To estimate the influence of the thickness variation on the modal attenuation we use the following approach using the TMM simulations: We calculate the spectral distributions of all five transmission bands for the two extreme values of the thickness interval observed in the cross section image (t₁ = 520 nm and t₂ = 590 nm) and averaged the resulting spectra at fixed wavelength, leading to a modified loss parameter (γ_mod = [γ(t₁) + γ(t₂)]/2, green curves in Fig. 1(c)). This procedure is a rough estimate, as it fundamentally neglects the distributions of the strand thicknesses and the overlap of the core mode with those. However, we believe that this simple procedure is a reasonable estimation for a qualitative understanding, as the bandwidth of γ_mod fairly matches to the experimental data when using the two extreme thickness values.

The wavelength of minimum loss of each band is close to those when assuming a constant thickness (t_c = 560 nm, purple dashed curve in Fig. 1(c)) and to those from the experiment. The behavior of the magnitude of the loss is two-fold: in the NIR, hardly any influence of the thickness variation is visible (bands 2 and 3). This situation changes at shorter wavelength (bands 4 to 6): Here the minimum attenuation value at the transmission band center does not decreases but rather increases towards shorter wavelength, which is in strong contrast to the results from the simulations with constant t (they show about one order of magnitude lower loss). This behavior qualitatively agrees with the experimental observations. However, this procedure does not explain the experimentally measured attenuation values, which are about one to two orders of magnitude higher (blue curve in Fig. 1(c)).

Both presented estimations show that the thickness variation is a key loss mechanism which can significantly increase the modal attenuation and decrease bandwidth especially the UV, which needs to be minimized as much as possible.

4. Influence of surface roughness

Another potential source of loss is surface roughness at the silica/air interfaces. Techniques to calculate this contribution are relying for instance on a Green’s function method [10] or on coupled mode theory-based models [11]. In the framework of this work the contribution of roughness induced loss is analyzed by a straightforward-to-implement model introduced in [8,12]. It relies on calculating the overall far field distribution of the scattered light generated by induced dipoles originating from protrusions and indentations of the silica surface. These irregularities result from thermally excited surface capillary waves frozen-in during fiber drawing, which impose intrinsic roughness on the glass surfaces. The statistical distribution of the irregularities is included into the model by a particular power spectral density function [8], allowing calculating the total scattered power for an arbitrary input field. This method is thus applicable to any cross section and any kind of optical mode and does not rely on any particular symmetry conditions or on the type of mode considered. The current loss limit of state-of-the-art HC-PCF was successfully explained using this approach [8]. A direct comparison of results calculated by our implemented code and those provided in [12] is shown in the appendix A3 (Fig. 4).

We applied this method to our extended concentric ring model (Fig. 1(b), TMM simulations) and compare the results to the experimental findings (Fig. 1(c)): First, we calculate the electric field of the fundamental HE₁₁ mode at the two silica-air interfaces surrounding the core. This field is then used as input field for the roughness-induced loss calculations using the above-mentioned approach. All material properties such as polarizability, surface tension, transition temperature, and power spectral density are taken from [8].

The distribution of scattered light shows rotational symmetry (Fig. 3(a)), and most of the light is scattered in the forward and backward directions close to the fiber axis, with only little amount of light being scattered perpendicularly. The visible oscillations are interferences caused by the overlap of coherent waves originating from different positions along the air/silica boundaries.

 

Fig. 3 (a) Example of the distribution of the scattered intensity at a fixed radial distance from the center of the fiber (Only the inner two surfaces are considered. Figure shows the part of the sphere at which significant scattering occurs. The fiber is aligned along the z-axis. Wavelength: 830 nm). (b) Spectral distribution of the roughness-induced scattering loss contribution calculated using the ERM model and the approach from [8] (green: simulations assuming a constant electric field amplitude (wavelength: 830 nm), purple: simulations taking into account the full wavelength dependence of the field). The red circles refer to the minimum loss values and wavelengths of the respective transmission band from constant thickness simulation (purple dashed curve in Fig. 1(b)), the brown square to the calculations including Δt = 70 nm (green solid curve in Fig. 1(b)). For comparison the experimental results are shown in blue.

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As first step we neglect the wavelength dependence of the modal field distribution i.e. we assume constant field amplitudes at the silica surfaces using the structural parameters of our fiber (2a = 16.6 µm, t = 560 nm, b = 37 µm, green curve in Fig. 3(b)). The electric field was calculated at a wavelength of 830 nm close to the scattering loss minimum of the corresponding transmission band. Starting at a loss value of about 0.1 dB/km we observed a steadily increasing loss with decreasing wavelength by more than one order of magnitude up to a level of 4 dB/km for wavelengths close to 200 nm. This reflects the common expectation that shorter wavelengths are more susceptible to roughness induced loss.

This situation, however, changes significantly when the change of the mode field pattern of the leaky mode with wavelength is taken into account (purple curve in Fig. 3(b)). The resulting loss distribution actually resembles the evolution of the transmission bands to a very high degree, but at significantly lower loss level. The smallest scattering loss is located near the center of the individual band where the light is most strongly confined, i.e. the distance between fiber axis and radiation caustic is as large as possible [6]. For our fiber the wavelengths of smallest leakage and smallest scattering loss differ by 110 nm in the NIR (min. leakage loss at 720 nm, min. scattering loss at 830 nm) and by 2 nm in the UV (min. leakage loss at 231 nm, min. scattering loss at 233). A similar behavior was stated in [12], where one structural parameter was varied at a constant wavelength. Towards the edges of each band, the mode is less confined to the core and the electric field amplitude at the surfaces increases, with the result of an increasing scattering loss when approaching the strand resonances. The minimum scattering loss values of each band (purple triangles in Fig. 3(b)) do not show a pronounced dispersion as for the above mentioned fixed-field calculation (green curve in Fig. 3(b)). From the NIR down to the UV only a minor increase from 0.1 dB/km to 0.2 dB/km is found.

The overall magnitude of the roughness-induced scattering loss is at least three orders of magnitude below the expected leakage loss even in the UV (red dots and brown squares in Fig. 3(b)) and, as a consequence, does not play a significant role for the observed experimental values. A contribution from the most outer silica boundary (at b = 37µm) was found to be at least two orders of magnitude smaller than the one from the inner two surfaces and can thus be fully neglected.

5. Discussion

The above discussion reveals the importance of the different loss mechanisms on the overall attenuation of the leaky mode. For the thickness variation we indeed found that this loss contribution has a strong influence on the overall attenuation, as it explains the reduced bandwidth of the transmission bands and reverses the trend of decreasing loss towards shorter wavelengths which was observed in those simulations assuming a constant strand thickness (Fig. 1(b), Δt = 0). By analyzing the shift of the strand resonances (i.e. loss maxima), we were able to verify that both the bandwidth reduction as well as the loss enhancement is more severe in the UV than in the NIR (Fig. 2). This highlights the importance of structural imperfections, which ought to be avoided as much as possible by a careful and precise fiber fabrication.

Simulations related to surface scattering induced by the roughness of surface capillary waves reveal that this contribution increases towards shorter wavelength as expected, but is practically irrelevant for the loss in the UV. The increase towards shorter wavelength is practically nullified by the accompanied redistribution of the electric field of the leaky mode, showing that the contribution of surface roughness induced scattering in case of good confinement is of the order of 0.1 dB/km for all considered wavelengths and is therefore not the reason for the short wavelength loss increase, which is observed in many leaky waveguide systems. The typical assumption of the roughness being inevitably the origin of strongly increasing loss of anti-resonant optical fibers at the short wavelength side of the spectrum is, as a consequence, in fact wrong.

Since the scattering loss waves is practically irrelevant, the physical limit of our design is given by leakage loss only, which can be as small as 0.1 dB/m in the UV (if Δt = 0). Our fiber does not reach such values predicted even when including the variation of Δt = 70 nm in the model. One reason may be a loss mechanism utilizing the modes inside the struts which connect the inner core surround with the outer silica cladding in the real fiber which cannot be included in the TMM calculations. Another reason may be longitudinal inhomogenities not considered by the power spectral density of the investigated surface capillary waves. Extensive experimental investigations have shown the occasional emergence of structural imperfections of micrometer-size along the fiber axis, which constitute scattering centers and thus induce loss.

6. Conclusion and summary

Here we have discussed two possible contributions to the modal attenuation in anti-resonant HC fibers, namely (i) the variation of wall thickness of the silica strands surrounding the HC and (ii) scattering loss induced by surface roughness. By analyzing the respective magnitudes, we revealed that wall thickness variations are critical especially in the UV, whereas roughness-induced scattering loss has only a negligible contribution also at short wavelengths for this kind of fiber. The second mechanism is intrinsic to the fiber drawing process and thus cannot be avoided, whereas the contribution of the first mechanism can be minimized by careful fiber fabrication. The analysis reveals the potential of such leaky mode fiber waveguides also in the UV, and next steps will involve a more precise fiber fabrication for reaching loss figures as low as 1 dB/m by removing longitudinal and azimuthal imperfections (by improved fabrication procedure and assembling the structure in a clean environment).

Appendix

A1: fiber fabrication

The square core HC-fiber has been realized using a modified version of the stack-and-draw approach: First, a large air filling fraction silica tube has been drawn into a capillary. Four resulting capillaries with precisely equal dimensions were assembled in square arrangement and fitted into a suitable silica jacket tube. This entire preform was then drawn into a cane with an outer diameter of several millimeters. This cane was overcladded by another silica jacket and drawn to the final fiber. The different drawing steps were performed without applying any external pressure (details of the fabrication can be found in [6]).

A2: bandwidth reduction

The influence of thickness variation on the different transmission bands can be estimated by analyzing the shifts of the strand resonances (i.e. loss maxima). When the two extreme strand thicknesses are given by t_± = t_c ± Δt/2, Eq. (1) can be used to calculate the resulting transmission band bandwidth by

(2)$$\Delta {\lambda }_m(\Delta t)=2\sqrt{n^2-1}\left(\frac{t_c+\Delta t/2}{m-1}-\frac{t_c-\Delta t/2}{m}\right)=\frac{2\sqrt{n^2-1}}{m(m-1)}\left(t_c+\frac{\Delta t}{2}(1-2m)\right)$$

if the material dispersion of the refractive index across one transmission band is neglected (which is reasonable if the electronic resonances of the material are far away). The bandwidth ratio is then given by the ratio between the “disturbed” and “undisturbed” bandwidths, leading to

(3)$$K=\frac{\Delta {\lambda }_m(\Delta t)}{\Delta {\lambda }_m(0)}=1+\frac{\Delta t}{2t_c}(1-2m).$$

The critical thickness at which K diminishes is accordingly defined by

(4)$$K=0\Rightarrow \Delta t_{crit}=2t_c{(2m-1)}^{-1}.$$

A3: scattering loss model implementation

Here we use the method presented in [8] to calculate the contribution of the roughness related scattering loss. This approach relies on assuming that the roughness of the different silica interfaces induces subwavelength dipoles, being distributed according to a characteristic power spectral density function. The roughness itself (the dipoles) originates from thermally excited surface capillary waves frozen-in during fiber drawing [1,8]. The scattered radiation of all dipoles interferes in the far field forming a characteristic scattering pattern (Fig. 3(a) and Fig. 4).

 

Fig. 4 Comparison of the roughness-induced scattering loss contribution between the results from [12] and using our implementation with exactly the same parameters (radius: 10 µm, wall thickness: 372 nm, wavelength: 1550 nm). Top row: scattered power at a fixed radial distance from the center of the fiber ((a) simulations taken from [12], (b) data using our implementation, distance from fiber axis: 1 m). The corresponding angle distributions of the scattered power are compared in (c) and (d). The inset in (c) is an illustration of the simulated silica ring.

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Figure 4 shows a comparison between the results shown in [12] and our simulations when calculating the scattering loss contribution for a mode propagating inside a hollow silica ring using the same parameters. Most light is scattered along the forward direction of the fiber axis (direction defined by the propagating mode, Figs. 4(a) and 4(b)) with significantly less power being scattered perpendicularly. The angular distributions of the scattered power show a very similar scattering behavior for both cases with 28 local maxima (Figs. 4(c) and 4(d)). The results from [12] (Figs. 4(a) and 4(c)) suggest an azimuthal dependence of the scattered light even though this calculation method assumes full rotational symmetry as it does not include any polarization dependence but rather an averaging [8]. Our results, however, indicate full rotational symmetry (Figs. 4(b) and 4(d)) which appears to be more reasonable within the limits of the theory used. This discrepancy is of minor importance, because the regions of discrepancy are located within the domains of small intensity, i.e. in a direction almost perpendicular to the fiber axis. The authors of [12] state a loss value of 0.13 dB/km for the given parameters whereas our calculation suggests 0.05 dB/km.

Acknowledgments

Funding from the federal state of Thuringia (Forschergruppe Fasersensorik; FKZ: 2012 FGR 0013) and ESF is highly acknowledged.

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