1. Introduction

The measurement of the refractive index (RI) of liquids is important in various fields of application, such as industrial process monitoring, food quality control, medical diagnostics and environmental contamination monitoring. Fiber-based RI sensors are widely used and have been implemented in various configurations based on plasmonic resonances [1–6], Bragg gratings [7–9], long period gratings [10, 11], photonic crystal fibers [12, 13], fibers with micro-channel [14], Fresnel reflection [15] or interferometry [16, 17]. Fiber-integrated Fabry-Perot resonators (FPRs) are particularly attractive since they provide high sensitivities, large dynamic ranges and linear response functions. Fiber-based FPRs are created by deeply cut air slots penetrating the fiber’s core thus allowing the formation of standing waves inside the hollow section. Such sensor reveal almost negligible temperature cross-sensitivity (< 5·10⁻⁶ RIU/°C) [18, 19]. Typical fabrication methods rely on sawing [20], chemical etching [21–23], laser micromachining (using femtosecond (fsec) [18, 19] and ultraviolet laser [24, 25]) and focused ion beam (FIB) milling [26–28]).

Femtosecond laser micromachining is a versatile tool for microstructuring but reveals practical limits in terms of achievable quality and smoothness of the machined surfaces. The rough surfaces lead to strong light scattering, resulting in poor mirror reflectivity and poor fringe contrasts, being severe in the case of a FPR. Diffraction limits the spatial accuracy and thus fsec-machining cannot be used to realize feature sizes with sub-wavelength dimensions. In contrast, FIB milling provides a precise and powerful processing tool due to a fine and controllable spot size (a few tens of nanometer). Open notch resonators are preferable than tunnel geometries [25, 28] as they allow straightforward access for fast analyte exchange in direct real-time refractive index measurement [18, 20, 22, 26]. A key problem of such FPR systems is the low refractive index contrast between the liquid (typically water) and silica (of which the fibers are made from), resulting in a poor reflectivity and thus very small fringe contrast.

Here we show a fiber-integrated slot-type microresonator with strongly improved mirror reflectivity for precise refractive index sensing. It is straightforward to implement and allows very simple and fast access to the actual sensing area. This resonator is formed by a vertical, micrometer-sized slot deeply milled into the core of a step-index fiber (SMF 28 type). The resonator has its sides coated with a high refractive index (HI) layer, which strongly increases the amplitude of the reflected light and leads to a significantly enhanced fringe reflectivity difference (ΔR_F = R_max-R_min). Our sensor allows measuring refractive indices within the range of silica glass, which is impossible using an uncoated resonator due to diminishing ΔR_F.

We precisely analyzed our sensor, which in fact acts as a sophisticated FPR, by two different mathematical approaches. We show that the fundamental sensing limit of FPRs in general is given by the detection limit factor, which ultimately limits the minimum detectable wavelength shift, i.e. index change. The FPR was been fabricated by precise FIB milling and operates in the telecommunication range. This wavelength range was chosen because of the availability of inexpensive components and negligible material dispersions. Residual light from the free fiber end is removed by a “mild taper” at the output. Taking into account an induced surface roughness and beam divergence within the slot the optical reflection spectra show excellent agreement with the simulations, revealing a sensitivity of about 1.15 µm/ RIU and a minimum detectable refractive index change of roughly 2.2·10⁻⁴ (see Sec. 4).

2. Device analysis

2.1 Models for the reflection properties of the resonator

The in-fiber resonator (Figs. 1(a) and 1(b)) we introduce here consists of a step index fiber with a milled resonator. Due to the great depth of the slot it acts as an FPR with no waveguide confinement within the milled section. The sides of the resonator have been improved by high-reflecting dielectric layers, which effectively increases mirror reflectivity and, thus, the resulting fringe reflectivity difference (FRD). The actual sensing area is the volume of the electromagnetic mode within the slot. Changes of the refractive index within the slot, i.e. the sensing area, result in a modification of the phase of the light circulating inside the cavity, leading to a shift of the actual resonance wavelengths. The phase of the circulating wave is given by:

 

Fig. 1 The reflectivity-enhanced in-fiber microresonator for precise refractive index sensing. (a) Schematic of the in-fiber microresonator with the deeply cut slot and the tapered section. (b) Scanning-electron-micrograph of the fabricated fiber resonator (base fiber: SMF-28). (c) One-dimensional transfer-matrix-model of the resonator (green: slot representing the actual sensing area, pink: high-refractive index layers, light blue: silica). The parameters d_C and d_L indicate the extensions of cavity and layers, n_F, n_L, and n_C the refractive indices of fiber, layer and slot. (d) Extended Fabry-Perot model with cavity extension Δd and additional phase shift Δϕ (color code is identical to that in (c)).

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The device itself is supposed to operate in the reflection rather than the transmission mode, which allows very flexible handling and thus makes it very attractive for many applications. Residual reflection from the free end of the fiber is reduced to a minimum by a shallow tapered section (Fig. 1(a)) out-coupling the transmitted light through the resonator.

Since the slot entirely penetrates the core of the fiber and low-NA fibers (NA = 0.14) with small beam divergence area are considered, our resonator actually resembles a Fabry-Pérot-type cavity and can thus be modeled by the transfer-matrix method (TMM) [29]. This straightforward-to-implement approach relies on approximating the individual longitudinal material sections by laterally infinitely extended layers with complex refractive indices. Since the light entering the layers is always perpendicular to the layers’ planes, it is sufficient to use the TMM in one-dimensional configuration (Fig. 1(c)). The low-NA fibers considered here have intrinsically very small core/cladding refractive index contrasts (in our case the contrast is about Δn = 0.0054 (0.36%) allowing the modal dispersion of the core mode to be neglected and use of the refractive index of silica for the outermost layers. The reflectivity of the reflectivity-enhanced resonator is then calculated by multiplying the respective transfer and propagation matrices, taking into account that no light is entering the device from the output side of the resonator, which is ensured by the tapered free fiber end. Losses inside the air cavity arising from the beam divergence of the circulating mode or particular substance absorption are taken into account by assuming a complex refractive index inside the slot. The TMM also allows the inclusion of losses originating from the layer interface only, as the FIB-milling is a potential source of surface roughness.

It is interesting to note that the spectral distribution of the reflectivity (green curve in Fig. 2) is, for our reflectivity-enhanced FPR, a beating between the slow-oscillating reflection function of the HI-layers (purple curve in Fig. 2) and the fast-oscillating reflection function of the cavity – a result of the very different widths of layers and slot (Fig. 2). Within a limited spectral interval the reflectivity reveals a purely sinusoidal behavior (inset in Fig. 2 and Fig. 9).

 

Fig. 2 Reflection of the reflectivity-enhanced FPR (resonator filled with water) showing the beating between the reflection functions of the HI-layer and of the central slot (green curve: reflection function of entire FPR, purple curve: reflection function of the isolated HI-layer d_L = 200 nm, d_C = 24.5 µm). The purple dots and dashed lines refer to the position of maximum reflection of the HI-layers calculated using Eq. (2) (numbers on top of the diagram indicate the corresponding mode order). The yellow section indicates the range at which the sensing experiments have been performed (Inset is a close-up of this yellow interval).

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The overall reflection coefficient (envelope function resulting from the HI-layer) is thus maximized if the following condition is fulfilled:

The configuration of our in-fiber FPR (Fig. 1a) intuitively suggests using the well-known reflectivity function of a FPR with one single mirror reflectivity [30], which is given in the case of complex refractive index within the cavity by:

2.2 Detection limit of the fiber integrated microresonator

Our proposed fiber sensor operates by precisely measuring the spectral shifts of the reflection minima in the case the refractive index inside the slot is changed. From a device point of view one of the most important parameters is the minimum detectable wavelength shift Δλ_min (or the minimum resolvable refractive index Δn_min) which has to be as small as possible. Both quantities crucially depend on the minimum measurable change in reflection ΔR_min which is predefined by the actual measurement system (precise definition of this quantity is given in the inset of Fig. 3(b). Since, in a real experiment, ΔR_min is typically much smaller than ΔR_F, Eq. (3) can be expanded with respect to ϕ_R in the vicinity of the resonances (defined by Eq. (4)), leading to

 

Fig. 3 (a) Example calculation showing the parabolic approximation (using Eq. (5)) of the FP-reflection function in the vicinity of the reflection resonances (Red: reflection function of a FPR calculated using Eq. (3). Blue: parabolic approximations in the vicinity of the resonances (example parameters: R = 0.05, n_C = 1 + i0.001, d_C = 24.45 µm)). (b) Normalized detection limit factor as a function of single mirror reflectivity. The inset shows the general definition of the smallest measureable wavelength shift.

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Rearranging Eq. (5) allows then expressing Δλ_min and Δn_min in the following compact and analytical form:

It is interesting to note that in case of high mirror reflectivity, the reflection minima are well separated and each resonance can be approximated by a Lorentzian-type oscillator function [30], revealing that the quality factor of the cavity Q is correlated to the finesse by Q = mF. The detection limit factor is therefore fundamentally limited by the photon lifetime inside the cavity (which is proportional to Q). The appearance of F inside Eq. (6) allows calculating D via F = Δλ_FSR / Δλ_FWHM, both straightforwardly experimentally accessible observables (Δλ_FSR: free-spectral range, Δλ_FWHM: full-width-half-maximum reflection dip width), which we use later to reveal the potential performance of our sensor. As shown in Fig. 4 D_N decreases towards larger values of FRD, thus clearly revealing that a larger ΔR_F (i.e. larger reflectivity as shown in the upper right-handed inset of Fig. 4) is favorable in sensor applications. The influence of the cavity loss can be neglected over a large range of values (lower left-handed inset of Fig. 4), only being relevant when ϕ_Ι approaches unity.

 

Fig. 4 Detection limit factor as function of fringe reflectivity difference for a fixed value of resonator loss (ϕ_I = 0.01). Upper right-handed inset: single mirror reflectivity versus ΔR_F. Lower left-handed inset: Dependence of D_N on ϕ_I for two different single mirror reflectivities R.

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2.3 Signal-to-noise ratio analysis

Equation (6) allows performing a noise analysis by investigating the dependency of Δn_min and Δλ_min on the signal-to-noise ratio (SNR). For this purpose we have assumed that ΔR_min corresponds to the standard deviation of the total noise of the output signal σ_tot divided by the amplitude of the reference signal I_out. The SNR is then given in a logarithmical representation by: SNR = 10·lg(I_out/ σ_tot) = 10·log (1/ΔR_min)). In increasing SNR therefore induces a strong decrease of ΔR_min (inset of Fig. 5) and, as a consequence, a smaller value of minimum measurable refractive index (Fig. 5). Despite of small output noise, a high SNR is generally obtained for high light intensities (associated with high photon flux P) and small losses within the fiber circuit η, leading to I_out~(1-η)⋅P. High levels of light intensities can, however, thermally induced undesired changes in the refractive index due to the light absorption in the analyte, especially in the case when broadband light sources are used.

 

Fig. 5 Minimum resolvable refractive index change as function of SNR. Inset: ΔR_min as a function of SNR.

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The total output noise in the FPR sensor system σ_tot arises from four different sources: (i) intensity fluctuations of the light source (σ_{light source} ~measured intensity), (ii) photon (shot) noise (σ_shot ~measured intensity^1/2), (iii) thermal (dark current) noise, (σ_thermal independent of light intensity, temperature dependent) and noise of the electronic (σ_readout temperature dependent). All noise sources are uncorrelated so that the variance of the total noise current is the sum of the variances of the four contributing noise currents: σ_tot² = Σ(σ_i)².

3. Fabrication and measurement setup

The FPR was implemented by FIB-milling (dual-beam system Lyra XMU (TESCAN)) a silica step index fiber (SMF-28, Fig. 6). Before milling the fiber was polished at an angle of ~3 ° with respect to the fiber axis. The fiber was presputtered with platinum and electrically grounded to avoid electrostatic charge accumulation on the fiber surface during the FIB processing. A two-step process was used: (i) high-current milling (10 nA) for cavity excavation (ii) low-current processing (1 nA) to improve the parallelism and flatness of the cavity walls. The final cavity had a width and height of ~24.85 µm and ~18 µm, respectively.

 

Fig. 6 Scanning-electron micrograph of the focused-ion-beam milled and refined fiber-based micro-cavity (side view). The dashed yellow line represents the section of the guiding core.

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The chosen cavity length represents a good trade-off between FIB machining time, achievable refractive index resolution Δn_min and resonator losses. A greater cavity is advantageous due to smaller fringes, hence higher resolvable refractive index (see Sec. 2). However, a larger cavity results in higher resonator losses due to diffraction and significantly longer FIB machining time.

The walls of the cavity were coated with a thick hafnium oxide layer (HfO₂, d_L = 190 nm) using atomic layer deposition (ALD), which is ideal for the deposition of layers on such complex geometries (details of the deposition can be found in Appendix A1). We chose HfO₂ since it shows a good compromise between high deposition rate and refractive index (about 2). The dielectric function of the HfO₂ layer was experimentally determined by means of a combination of ellipsometric and transmission measurement (SENTECH variable-angle spectroscopic ellipsometer SE850). These measurements show that the hafnium oxide layer has no significant absorption for wavelengths greater than ~400 nm. The material dispersion of HfO₂ can be neglected in the relevant wavelength range, giving a constant refractive index of 1.983. We chose an HfO₂ thickness of 190 nm, giving a maximum of the slow-oscillating reflection function according to Eq. (1) (mode order zero, Eq. (2)).

We measured the reflection spectra of the in-fiber FPR as function of the analyte, i.e. refractive index inside the cavity in the spectral interval 1450-1750 nm, using a monolithic fiber setup (Fig. 7). Light from a supercontinuum source (SC) (KOHERAS Versa) was coupled to the FPR-sensor (FPS) via a 3 dB coupler (C). Reflections from the free coupler end were suppressed by means of an index-matching gel (MG). The reflected intensity was measured by an optical spectrum analyzer (Ando, AQ6317B), which was set to a spectral resolution of 1 nm. Commercially available refractive index liquids (Cargille Laboratories, Inc.) were applied as analytes (the material dispersions of the liquids were included into calculations). The absolute reflectivity of the FPR R_FP was obtained by normalizing the reflected spectra to that of a cleaved fiber end face in air. A reflectance of 3.36% (λ = 1550 nm) for an ideal cleaved fiber end face was assumed.

 

Fig. 7 Scheme of the setup to measure the optical response of the sensor (SC: supercontinuum light source, OSA: optical spectrum analyzer, C: fiber coupler, IMF: index matching fluid, FPR: Fabry-Perot-Resonator).

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4. Analysis of experimental data

We measure a series of well separated and clear reflection dips with a FRD (defined in Fig. 8) of about ΔR_F = 0.21 in the case the slot is being filled with water (green curve in Fig. 8). A FPR without HI-layer results in a FRD about 0.008 (see Fig. 9(a) compared to a FPR with a HI-layer), showing that by using our concept we are able to boost ΔR_F by a factor of 26.

 

Fig. 8 Spectral response of the fiber-integrated FPR in reflection mode when exposed to water (single mode operation range 1.45 µm to 1.75 µm). The green curve refers to the experimentally measured data, the purple one to the results of the transfer-matrix simulations. The green numbers indicate the respective mode order m, the arrows point the minimum at which the detection limit factor has been exemplarily calculated. The grey arrow refers to the definition of the fringe reflectivity difference ΔR_F.

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Fig. 9 Measured spectral distribution of the reflectivity (normalized) of our FPR in the case of a water-filled slot. (a) comparison two resonators (w: with HfO₂ layer (blue curve), w/o: without HfO₂ layer (red curve), OSA resolution 1 nm) (b) Close-ups of the interference peaks and valleys (sensor probe includes hafnium oxide layer (OSA resolution 0.1 nm).

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We found that fringe valleys show fewer fluctuations and very clear dips, whereas the reflectivity peaks include strong ripples which cause problems in the later analysis effectively reducing D_N (Fig. 9(b)). Moreover we found in simulations that for increasing ϕ_I, the reflectivity maxima significantly broaden leading to a non-sinusoidal shape, whereas the minima remain sharp. These two observations therefore represent the fundamental reasons why we have chosen the reflectivity minima for our sensing experiments.

The resonator then reveals a normalized detection limit factor of D_N = 0.221 (using Eq. (6)). For example if the resonance with m = 42 (hence λ_R = ~1535 nm, F = 2.23) and ΔR_min = 0.1% (SNR = 30 dB) is considered (corresponding to the minimum measureable reflectivity change of our setup), our sensor reveals a minimum detectable wavelength shift of Δλ_min = 0.26 nm i.e. a minimum resolvable refractive index change of Δn_min = 2.2 × 10⁻⁴ taking into account the corresponding refractive index of water at the respective resonance [23]. If curve-fitting algorithms (e.g. based on polynomial or other specific functions) are used for determining the resonance wavelength, further improvement of the minimum resolvable refractive index change can be achieved.

The ion-milling process intrinsically induces surface roughness onto the silica/HfO₂ and HfO₂/water interfaces, reducing fringe contrast by a significant amount compared to perfectly smooth layer interfaces. Such a roughness can be mathematically included into the TMM by reducing the single interface reflection and transmission factors r and t. As shown in Appendix A2, surface roughness mainly affects r rather than t, allowing the assumption that only r is reduced by a particular factor a, whereas t is left unchanged. A detailed TMM analysis reveals that a has a dominant influence on the amplitude of the reflection maxima and not on the minima. Therefore we vary a to fit the TMM maxima to the experimental values with the optimal results achieved for a = 0.75 with no wavelength dependence. No particular roughness distribution model was employed for our study, as we are not able to characterize the roughness within the resonator by means of any non-destructive technique. A rough estimation using a white noise approach (see Appendix A2 for details) suggests an rms amplitude of about 48 nm, which appears to be realistic in the scope of FIB milling. An optimal match to the resonances (reflection minima) is achieved if one takes into account an imaginary part of the dielectric function inside the slot of about Im(ε_C) = 0.002. The origin of this contribution, which corresponds to an intensity reduction of approximately 7% per pass, can presumably be attributed to the beam divergence in the slot, where the light is not confined by any waveguide boundaries.

With these two empiric parameters, the TMM calculations very accurately reproduce the experimental findings even on a logarithmic scale (Fig. 8, dielectric functions are taken from [31, 32]), which shows that our device in fact represents a sophisticated, fiber-integrated version of a reflectivity-enhanced FPR with precisely manufactured mirrors.

The spectral positions of the experimentally determined resonance wavelengths decrease almost purely linearly with increasing mode order (Fig. 10), again resembling the typical behavior of a FPR. These experimental findings have been compared to the reflection resonances obtained from the TMM and from the resonances calculated using Eq. (4). For the FPR approach we assume two conditions: (i) extended FP model configuration with Δϕ ≠ 0 and Δd ≠ 0 and (ii) the regular FP model condition with Δϕ = 0 and Δd = 0. In addition to the expected good match with the TMM results, the experimental resonances agree to a very high degree with those from the extended FP model (optimal parameters Δϕ = 0.2085 × 2π and Δd = 125.628 nm) within a precision less than 0.3 nm (inset of Fig. 10, Δλ_R is the absolute difference between experimental and calculated resonance wavelength). The HI-layers therefore act as FP-mirrors with extended electromagnetic field penetration depth and additional phase shift.

 

Fig. 10 Distribution of the measurement reflection resonances as a function of mode order of the fiber-integrated FPR. Inset: derivations of the different models from the experimental data. (Purple triangles: transfer matrix method, red triangles: extended FP model, blue squares: regular FP model).

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One key performance parameter for optical sensors is the change of the resonance wavelength with respect to changes of the environmental refractive index, which is referred to as sensitivity and is generally given by S = dλ_R / dn_C. We measured S by filling the slot with various precisely characterized analytes (see Sec. 3) and measuring λ_R for the mode orders m = 41…44. In our case, the resonance wavelengths depend linearly on the refractive index for one fixed mode order (inset of Fig. 11), with the sensitivity accordingly defined by the respective slopes (S = S(m)). This linear dependence is fundamentally a result of the negligible dispersions of the materials involved particularly at near-infrared wavelengths, making our sensors very attractive for this spectral regime in particular. For all mode orders considered, the experimental sensitivities lie between 1.19 µm/ RIU and 1.10 µm/ RIU and decrease towards higher m. The numerically determined sensitivities obtained with the TMM almost perfectly agree with the experimental value (Fig. 10), again revealing that the matrix approach is a powerful tool to analyze our device. The sensitivities obtained from the extended and regular FP-model (using Eq. (4)) are given by S_FP = 2(d_C + 2Δd) / (m + Δϕ / π), with a slight overestimate by about 1%, which is negligible for any serious application. It is interesting to note that even though the absolute resonance positions are much better represented by the extended FP model (inset of Fig. 11), the sensitivities of the extended and regular models are almost identical – a result of the small contribution of extended cavity length and additional phase shift. Thus, assuming S_FP = 2d_C / m provides an efficient way to estimate the expected sensitivities for such kind of resonators.

 

Fig. 11 Comparison of the sensitivities calculated using the different models together with the experimentally determined values for S_FP (green circles: experimental data, purple triangles: transfer matrix method, red triangles: extended FP model, blue squares: regular FP model). The inset shows linear fits to the experimentally measured resonances for different mode orders (numbers and colors refer to the different orders of modes).

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From the practical point of view a less expensive sensor system can be implemented using a swept laser source with a photodiode or a superluminescent diode in combination with a low-cost spectrometer. Reduction of production efforts and costs can be achieved by a combination of fsec laser machining (coarse slot fabrication) and FIB machining (polishing mirror surfaces). Once the slot is being fabricated, ALD allows simultaneous deposition of HI-layers on a large number of samples within one deposition run.

5. Conclusion

This report introduces an in-fiber refractive index sensor based on an integrated microresonator with strongly enhanced fringe reflectivity difference. The resonator design is straightforward to implement and relies on a deeply milled air-slot inside a step-index fiber. The side walls are refined by high-reflective layers, which lead to an improvement of the fringe reflectivity difference by a factor of 26. The reflection spectra of that cavity show a beating between the reflection function of the layers and the reflection function of the air-cavity, which allowed us to define a particular design rule for device optimization. Two mathematical models were applied to our sensor and we found excellent agreement with theory. We generally analyzed the performance of Fabry-Perot-based sensors in terms of minimum detectable wavelength shift and found that it is determined by the detection limit function, a fundamental quantity of any Fabry-Perot resonator, which depends on the characteristics of the cavity itself and the detection capability of the measurement system. The analysis reveals that this function decreases by several orders of magnitude for increasing mirror reflectivity, which was the key motivation for refining the side walls of our sensor. Our device shows a sensitivity of about 1.15 µm/ RIU and a minimum detectable index change of Δn_min = 2.2 × 10⁻⁴, which are more than sufficient for any real-world application. We anticipate application of our device in various areas of fiber optical sensing such as biophotonics, environmental science or medicine.

6. Appendix

Appendix A1 – deposition and characteristics of HfO₂-layer

The deposition of the HfO₂ layer was carried out by means of the plasma-enhanced atomic layer deposition (PEALD) technique with TEMAH (tetrakisethylmethylaminohafnium, Hf[N(C₂H₅)₂]₄) as a hafnium precursor and oxygen as a reactant gas. For this we used the OpAl ALD tool with plasma option (Oxford Plasma Technology). One ALD cycle consists of the following sequence: 2 s pulse of TEMAH precursor, followed by 7 s Ar-purge, 7.5 s oxygen plasma and finally 5 s Ar-purge. The growth rate of the hafnium oxide film at a temperature of 225 °C was 0.14 nm per cycle, thus 1340 cycles were required to realize a film thickness of 190 nm, giving a total deposition time of around 8 hours. The roughness of the deposited HfO₂ films was measured by atomic-force-microscopy (Park Scientific Instruments, probe tip radius 10 nm) on an equivalent planar quartz glass test substrate. The measurement was carried out in the contact mode. We obtained an rms roughness of 5.5 nm, with the roughness of the uncoated substrate being below the measurement limit.

Appendix A2 – influence of interface roughness

Interface roughness leads to incoherent scattering of light at the respective boundary and consequently to energy dissipation. Since the actual loss of electromagnetic energy is happening only at the location of the interface and not inside the layers, it is incorrect to describe this particular loss channel by assuming a complex phase inside the layer. The mathematically correct description relies on a modification of the single interface reflection and transmission factors r and t given by:

(7)$$r=a\frac{n_{\text{R}}-n_{\text{L}}}{n_{\text{R}}+n_{\text{L}}}$$

(8)$$t=b\frac{2n_{\text{R}}}{n_{\text{R}}+n_{\text{L}}}$$

with the refractive indices left and right of the interface n_L and n_R (propagation along the normal direction from the left-handed side is assumed) and the amplitude reduction factors a and b [33]. Depending on the characteristics of the surface roughness various mathematical models have been developed for describing a and b [34, 35]. In the analysis discussed here, we intended to reveal the general influence of surface roughness on the amplitude reduction factors rather than using a particular model. Therefore we assume the simplest case of an uncorrelated roughness (“white noise” approach) with a Gaussian distribution, which allowed us to describe the roughness by one single quantity, viz. the standard deviation σ, which leads to [33].

(9)$$a=\exp \left[-2{\left(n_{\text{R}}{k}_0\sigma \right)}^2\right]$$

(10)$$b=\exp \left\{-\frac{1}{2}{\left[\left(n_{\text{R}}-n_{\text{L}}\right)k_0\sigma \right]}^2\right\}$$

It is interesting to note that Eq. (9) represents the limit of Eq. 18 of reference [34] in the case the correlation distance vanishes. Since a depends on the actual value of the refractive index in the right-handed medium, it decreases much faster than b depending on the difference between the two indices (Fig. 12).

 

Fig. 12 Dependence of reflection and transmission amplitude reduction factor on rms roughness amplitude (λ₀ = 1.55 µm) for the silica/HfO₂ interface. The black dashed line corresponds to the optimum value of a = 0.75 and b = 0.995 obtained from fitting the experimental data to the TMM. The inset shows the spectral dependence of a and b for σ = 48 nm.

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Our TMM analysis shows that using a = 0.75 and b ≈ 1 gives the best agreement with the experiment, suggesting σ = 48 nm according to Fig. 12. This value of roughness has a realistic magnitude, since FIB-milling induced inaccuracies are of the order of several tens of nanometers.

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