1. INTRODUCTION

Structuring light fields in all three dimensions is a topical research direction in optics [1], as it has enabled the discovery of new physical concepts [2] and significantly expanded the range of photonic applications [3], including life science (e.g., optical manipulation and trapping [4], microscopy [5]), nanoscale interferometric detection [6], optical communication (e.g., multiplexing transmission [7], mode division multiplexing [8]) and quantum technology (e.g., modulation of atomic spin [9], quantum light in complex media [10]). Furthermore, 3D light structuring is particularly essential to the field of nanoscience, allowing unique phenomena such as the Kerker effect or bounded states in the continuum to be explored and exploited [11,12]. Sophisticated concepts for complex beam shaping in all three dimensions are currently being explored, examples of which include metasurface [13,14] or digital holograms [15,16]. One particular challenge is the generation of constant intensity profiles, i.e., intensity distributions, which show no spatial variations. In this respect, concepts such as mode mixing [17], diffractive optical elements [18], or resonant effects [19] represent examples of possible pathways to generate spatially invariant radiation patterns over limited distances. The realization of such a scenario is particularly challenging on the microscale, which can be illustrated on the example of a focused beam: even though a lens can efficiently focus light, the spatial domain of almost constant intensity is limited to a few micrometers or even below in case of tight focusing. Therefore, the generation of a domain with constant light intensity along all three spatial directions (including the direction of propagation)—a light strand—is thus principally challenging.

One strategy to overcome divergence within defined spatial domains relies on using modes of optical waveguides. These photonic devices, implemented in numerous ways on-chip or in fiber form, confine the light inside a core domain. Due to the invariant cross-section, characteristic Eigenstates, so-called optical modes, form with identical intensity distributions at any axial location of the waveguide. As long as the waveguide’s geometry remains unchanged, the only source that reduces the intensity during propagation along the waveguide is attenuation. Nanostructuring of waveguides presents one pathway for tailoring mode fields at the nanoscale, examples of which include strong field enhancement in slot waveguides [20,21] or nanochannels in optical fibers [22,23].

 

Fig. 1. Concept of light-strand formation via a flat-field mode, created inside a fluidic nanostrand enhanced step-index fiber, used for nanoparticle tracking analysis. (a) Illustration of the flat-field mode, optofluidic nanobore fiber, and light scattering process. Note that the light blue area refers to the light strand. (b) Measured scattered intensity (normalized to the mean intensity) of a 50 nm gold nanosphere diffusing inside the nanochannel (green: index adjusted liquid yielding flat-fields; purple: water) as a function of the frame index (bottom axis) or time (top axis). (c) Histograms showing the corresponding distribution of intensities.

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One nanoscience-related research field that especially benefits from tailored light fields is nanoparticle tracking analysis (NTA), an example being the real-time detection and counting of nano-objects via optical forces [24]. Through image-based tracking and statistically analyzing diffusive properties, NTA allows characterizing individual nanoparticles (NPs) in terms of size or dynamics at the single-species level, with applications in fields such as bioanalytics [25] or nanoscale material science [26]. In this context, interfacing of NTA and optical fibers (fiber-assisted NTA, FaNTA) has attracted increasing attention [27–29]. In addition to advantages such as fast readout or longitudinal invariant NP illumination by optical modes, the simultaneous confinement of NP diffusion and light to selected spatial areas (e.g., micro- or nanofluidic channels) enables recording of extraordinarily long trajectories and thus unprecedented statistical significance. One example is a step-index fiber with an axial nanochannel, the nanobore fiber, that allows 50 nm gold nanospheres to be tracked over 100,000 frames [30]. A major disadvantage of all currently used fibers is the reduction of modal intensity by several orders of magnitude along the radial direction. This problem, which also occurs in free-space arrangements, degrades localization accuracy, limits trajectory length and, in extreme cases, terminates the trajectory at all, substantially reducing statistical significance and preventing accurate characterization of nanoscale specimens. Furthermore, due to the spatially changing mode field, the scattering intensity of a single NP cannot be used as a parameter for characterization because the particle experience different local intensity while performing Brownian motion. These examples emphasize that there is a great demand for modes with constant fields, ideally without any transverse and axial dependence.

Here, we present a waveguide-based concept for creating light strands inside optofluidic fibers containing liquid-filled nanochannels located at the center of the core [Fig. 1(a)]. Key to the concept is the integrated nanochannel, which is externally accessible and allows for studying light–matter interaction in a nanofluidic environment. As demonstrated in simulations and experiments, the modal field inside the liquid shows no spatial intensity dependence along all three spatial directions. All relevant properties and a generic flat-field condition are revealed. Clear evidence of the light strand is demonstrated in FaNTA experiments through analyzing the Brownian motion of a single gold nanosphere inside the liquid-filled nanochannel [Figs. 1(b) and 1(c)].

2. CONCEPT

Generally, transverse flat fields, i.e., electric fields with no radial dependence ($dE(r)/dr = 0$) inside the center region of a waveguide do not appear under typical circumstances. In the following, a characteristic equation—the flat-field condition—for the hypothetical systems of a multilayer symmetric slab waveguide (MSW) in TE-polarization [inset of Fig. 2(a), details in the Supplement 1 Section 6] is derived. In the following, the subscript $i$ refers to the respective layers that correlated with the different sections of the fiber considered in the experiments ($i = 1$: central layer (bore), $i = 2$: high-RI layer (core), $i = 3$: most outer layer (cladding)).

 

Fig. 2. Flat-field modes in a symmetric slab waveguide (TE polarization). (a) Spectral dependence of the thickness of the high-index layer ${d_2}$ (${d_1} = 0.4\;{\unicode{x00B5}{\rm m}}$), calculated using the flat-field condition (Eq. (1)). The different curves refer to different mode orders $m$ (indicated in the legend). The inset on the top illustrates the waveguide considered (${n_3} = 1.44 \lt {n_1} = 1.445 \lt {n_2} = 1.45$). Right: transverse intensity distributions at ${\lambda _0} = 633\;{\rm nm} $ (indicated by the vertical dashed red line in [a]). The semitransparent blue area in the center of each plot refers to the domain of medium 1. (b) Fundamental flat-field mode ($m = 0$, ${d_2} = 0.657\;{\unicode{x00B5}{\rm m}}$). (c) Higher-order layer mode ($m = 3$, ${d_2} = 8.549\;{\unicode{x00B5}{\rm m}}$).

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The electromagnetic fields of the supported modes in the central region of the MSW have either the form ${\sim}{e^{(\pm i{k_1}x)}}$ or ${\sim}{e^{(\pm {k_1}x)}}$ (${k_1}$: wave vector in layer 1, $x$: transverse coordinate). In case a flat field should be obtained, the argument of these trigonometric functions ($\arg = {k_1}x$) should not be spatially dependent, implying a vanishing wave vector (${k_1} = 0$). Using the definition of the wave vector in the center medium ${k_1} = 2\pi /{\lambda _0}\sqrt {n_1^2 - n_{{\rm eff}}^2}$ (${\lambda _0}$: vacuum wavelength, ${n_1}$: refractive index (RI) of medium 1, ${n_{{\rm eff}}}$: effective mode index), the flat-field condition reads as

 

Fig. 3. Model properties of the nanobore fiber geometry in case the flat-field condition [Eq. (1)] is applied. (a) Sketch of the cross-section of the nanobore fiber (dark blue: bore filled with liquid, dark green: doped silica core, light gray: silica cladding). (b) Scanning-electron-microscopic image of the implemented fiber structure. (c) Spatial distribution of the Poynting vector at the operation wavelength (${\lambda _0} = 632\;{\rm nm} $). The magenta circle indicates the boundary of the nanobore. (d) Intensity distribution along the two lines indicated in (c) (dashed orange: $y = 0$, light green: $x = 0$). The light blue area indicates the domain of the bore. For comparison, the dotted grey line refers to the mode in case the nanochannel is filled with water.

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This condition implies that flat fields are obtained in case the effective index of the mode is equal to the RI of medium 1. This condition has interesting consequences from the waveguiding perspective. First, the flat-field condition cannot be fulfilled for the guided modes of a single-interface step-index waveguide, demonstrating the relevance of a multi-interface system. Moreover, Eq. (1) implies that the phase velocities of the light inside medium 1 is equal to that of an associated plane wave in the bulk medium ($\nu _1^{{\rm bulk}} = {\nu _{{\rm mode}}}$). Mathematically, Eq. (1) imposes that the underlying differential equation to be solved for medium 1 changes from the wave equation to a regular second-order differential equation, with solutions of the form ${\sim}{c_1} + {c_2} \cdot x$ (${c_1}$ and ${c_2}$: constants). Due to the symmetry of the assumed waveguide geometry, ${c_2} = 0$ and the field in medium 1 is constant.

Inserting the flat-field condition [Eq. (1)] into the dispersion equation of the symmetric three-layer slab waveguide (see Supplement 1 Section 6 for details) yields an equation that relates the thickness of the high-RI layer ${d_2}$ to all relevant parameters:

 

Fig. 4. Statistical analysis of a single gold nanosphere performing a Brownian motion inside water-filled nanobore fiber ($d = 50\;{\rm nm} $). The plots show the probability distribution of the scattered intensity in the central (green, CB) and side (purple, SB) bins ([a] experiments, [b] simulations). The bars in the charts refer to histogrammic representations, while the dashed lines have been obtained using nonparametric kernel density estimations ($N_{{\rm CB}}^{{\rm exp}} = 6715$, $N_{{\rm SB}}^{{\rm exp}} = 579$, $N_{{\rm CB}}^{{\rm sim}} = 4771$, $N_{{\rm SB}}^{{\rm sim}} = 2263$). The dotted horizontal lines indicate the intensities of maximum probability $I_{{\rm max}}^{{\rm bin}}$. The inset in (b) shows the mode profile along the $x$-direction (orange line, ${n_1} = 1.3317$) with the bins indicated by the light-colored bars (CB (width: 80 nm): green, SB (width: 80 nm; purple). More details can be found in the Supplement 1 Section 2 (Fig. S2).

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To demonstrate flat-fields experimentally and their benefits in the context of FaNTA, Eq. (1) was applied to the case of a suitable nanostructured optical fiber, which is the nanobore fiber (NBF, Figs. 1(a) and 3(a), all the following parameters refer to ${\lambda _0} = 632\;{\rm nm} $). This type of silica-based step-index fiber (cladding: fused silica (${n_3} = 1.457$); core: ${{\rm GeO}_2}$-doped fused silica (doping level 5.3 mol. %, core diameter $b = 3\;{\unicode{x00B5}{\rm m}}$, ${n_2} = {n_3} + \Delta n$, $\Delta n = 8 \times {10^{- 3}}$, designed and fabricated in collaboration with Heraeus Quarzglas GmbH & Co. KG) contains a nanochannel of RI ${n_1}$ in the center of the cross-section (diameter $a = 400\;{\rm nm} $) and was successfully used in previous FaNTA-related experiments [27,30]. Precise tuning of the RI of the channel can be achieved through infiltration of a composite liquid with a predefined RI [31]. To establish flat fields, the RI in the nanochannel was numerically determined (using the finite-element modeling and via a self-written code, details can be found in [30]) to ${n_1} = {n_{{\rm eff}}} = 1.46088$, with the corresponding intensity distribution confirming the flat fields [Figs. 3(c) and 3(d)].

Fiber implementation is based on high-temperature fiber drawing of a doped silica cane containing a central hole (SEM in Fig. 3(b), details in [27]). The parameters of the resulting fiber correspond to those mentioned above. Note that the optofluidic NBF is single-mode at ${\lambda _0}$ and thus is insensitive to bending (dispersion properties can be found in the Supplement 1 Section 5).

3. RESULTS

The formation of a light strand in the NBF is verified here by applying the FaNTA concept. The overall idea is based on the temporal analysis of the intensity of the scattered signal of a single nanoscale scatterer diffusion inside the nanofluidic channel. Thus, in the case of NP diffusion within the light strand, the same intensity is present at all locations inside the channel and the width of the intensity distributions of the scattered light should be substantially narrower than for non-flat fields. To demonstrate this, the available FaNTA-related optical setup (details in the Methods section) was used to measure the scattered intensity and in-plane position of a gold nanosphere (mean physical diameter: 50 nm, nanoComposix, details in the Methods section) diffusing inside an NBF. It is important to note that different to other NTA implementations, FaNTA generally allows observing the NP for a very long time (see [30]), making it possible to accurately sample the modal fields. An estimation based on comparing diffusion length and channel diameter (Supplement 1 Section 3 for estimation of diffusion length) shows that in the present experiments, the NP transversely crosses the channel 24.4 times during observation. Different refractive index scenarios, i.e., field patterns inside the liquid channels, have been realized by using different liquids or via an added Peltier element, which change the local temperature via the thermo-optic response of the liquid.

A. Segmented Data Analysis

To analyze the flat fields using FaNTA, the analysis of the statistical data of the scattering intensity is performed in two different spatial regions (with respect to one transverse direction [here x-direction]). To ensure the highest possible sensitivity of the scattering intensity on the mode profile, the intensity data within the side bins (SB) and the central bin (CB) were analyzed separately (the locations of the bins are illustrated by the colored bars in the inset of Fig. 4(b) and are shown in Fig. S2) using histograms and nonparametric kernel density estimations. The main parameters resulting from the latter representation, and used in the following discussion, are (i) the intensity at which the kernel estimator is maximal, i.e., the histogram reveals the highest probability, $I_{{\rm max}}^{{\rm bin}}$; and (ii) the relative full-width-half-maximum (FWHM) intensity bandwidth of the individual distribution $\Delta I_{{\rm FWHM}}^{{\rm bin}}$ defined at the intensities left and right to $I_{{\rm max}}^{{\rm bin}}$ at a height $p = {p_{{\rm max}}}/2$ (bin: SB or CB). Note that all intensity values presented in the following have been normalized with respect to the intensity with highest probability in CB: ${I_n} = I/I_{{\rm max}}^{{\rm bin}}$. The relative distance between the intensities of maximum probability of CB and SB is defined by $\delta {I_{{\rm max}}} = (I_{{\rm max}}^{{\rm CB}} - I_{{\rm max}}^{{\rm SB}})/I_{{\rm max}}^{{\rm CB}}$. Two situations have to be considered. In the case the fields inside the NBF are evanescent (${n_1} \lt {n_{{\rm eff}}}$), higher intensities are present in the SBs than in the CB, leading to different intensities at which the counts are maximum ($I_{{\rm max}}^{{\rm SB}} \gt I_{{\rm max}}^{{\rm CB}}$) (details in the Supplement 1 Section 2 and the Method section). A complementary behavior is present for case ${n_1} \gt {n_{{\rm eff}}}$, corresponding to Gaussian-shaped fields in the nanochannel ($I_{{\rm max}}^{{\rm SB}} \lt I_{{\rm max}}^{{\rm CB}}$). This data analysis procedure was verified by simulations before the actual experiments and is applied to different RI scenarios in the following.

B. Case I: Water Core

The first scenario considers water containing gold nanospheres (nanoComposix, ultra-uniform, $d = 50\;{\rm nm} $, concentration $c = 4.2 \times {10^{10}}/{\rm ml}$) inside the nanochannel, leading to evanescent fields in the central medium due to the RI distribution (${n_1} \lt {n_{{\rm eff}}}$). The results of the NTA experiments (histograms in Fig. 4(a), values given in Table 1) show asymmetric distributions around the respective maximum probability at $I_{{\rm max}}^{{\rm bin}}$. Comparing the two histograms reveals that the intensity of maximum probability for the SB is localized at much larger values compared to the CB ($I_{{\rm max}}^{{\rm SB}} \gt I_{{\rm max}}^{{\rm CB}}$, $\delta {I_{{\rm max}}} = - 0.332$). This behavior, which is also observed in simulations [Fig. 4(b)], clearly indicates the presence of an evanescent field within the water-filled nanochannel (inset of Fig. 4(b)). Another feature is the relatively large intensity bandwidth of both distributions ($\Delta I_{{\rm FWHM}}^{{\rm CB}} = 0.275$, $\Delta I_{{\rm FWHM}}^{{\rm SB}} = 0.251$), resulting from the strong spatial dependence of the evanescent field within the bins. Note that the analysis crucially depends on the high number of frames in the bins considered (${N_{{\rm CB}}} = 4771$, ${N_{{\rm SB}}} = 579$), which results from the confinement of the NPs to the nanochannel and represents a key advantage of FaNTA.

Table 1. Comparison of the Key Benchmark Figures of the RI-Adjusted and Water-Based Experiments^a

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C. Case II: Temperature Scan

To investigate the flat-field condition in the experiments, a second FaNTA scenario with adjusted RI was established. In detail, the RI of the liquid in the nanochannel was adjusted through local thermo-optical tuning (using the Peltier element), allowing to realize RIs just above and below that associated with the flat-field condition. In detail, a mixture of water and dimethyl sulfoxide (DMSO, wt% 9/91, ${n_1} = 1.4646$ at ${\lambda _0} = 632\;{\rm nm} $, more details in the Supplement 1 Section 4) including gold nanospheres (nanoComposix, $d = 50\;{\rm nm} $, concentration $c = 4.2 \times {10^{10}}/{\rm ml}$) was infiltrated into the NBF and the scattering intensity in CB and SB were analyzed separately.

The results (histograms in Fig. 5) show symmetric probability distributions around $I_{{\rm max}}^{{\rm bin}}$, with significant differences to the water-related measurements (Fig. 4, a direct comparison of the different benchmark parameters can be found in Table 1): The intensity bandwidths $\Delta I_{{\rm FWHM}}^{{\rm bin}}$ are substantially narrower than in the pure water case (RI-adjusted mixture: $\Delta I_{{\rm FWHM}}^{{\rm bin}} \approx 0.093$; water: $\Delta I_{{\rm FWHM}}^{{\rm bin}} \approx 0.26$), indicating a significantly smaller spatial dependence of the field within the bins considered. Note that the histograms of the water-related experiments (Fig. 4) cover a much larger intensity range ($0.65 \lt {I_n} \lt 1.7$) than their RI-adjusted counterparts (Fig. 5, $0.85 \lt {I_n} \lt 1.15$). In general, the probability distributions are much more symmetric and resemble Gaussian distributions for all temperatures, suggesting very little spatial variation in intensity. Note that due to intrinsic errors of the tracking process (e.g., detector noise) a Gaussian distribution is to be expected even in the case of a completely constant intensity. Another decisive difference between both measurement series are the intensities of maximum probability, which are much closer in the RI-adjusted experiments compared to those of the water-based measurements. This behavior is quantified by the parameter $\delta {I_{{\rm max}}}$, being more than a factor of 10 smaller for the RI-adjusted experiments (RI-adjusted mixture: $| {\delta {I_{{\rm max}}}} | \le 0.02$, water: $| {\delta {I_{{\rm max}}}} | \le 0.332$, Table 1).

 

Fig. 5. Statistical analysis of the RI-adjusted FaNTA experiments used to study the flat-field condition. Refractive index tunability has been achieved by locally heating a mixture of DMSO and water (9 wt. % water, 91 wt. % DMSO). Similar to Fig. 4, the plots show the probability distribution of the scattered intensity in the central (cyan, CB) and side (magenta, SB) bins. Each row refers to the one temperature applied to the NBF (from top to bottom: 20°C, 23°C, 25°C, 30°C, 35°C.). Each histogram contains (colored bars) the corresponding continuous curves from a nonparametric kernel density estimation. The respective number of frames used in each plot is represented by ${N_{{\rm CB}}}$ and ${N_{{\rm SB}}}$. The dotted horizontal lines indicate the intensities of maximum probability $I_{{\rm max}}^{{\rm bin}}$. The bottom plot shows the relative intensity distance of maximum probability of CB and SB $\delta {I_{{\rm max}}} = (I_{{\rm max}}^{{\rm CB}} - I_{{\rm max}}^{{\rm SB}})/I_{{\rm max}}^{{\rm CB}}$ as a function of applied temperature $T$ (dots: experiments, dashed lines: guide-to-the-eye). The top $x$-axis indicates the corresponding refractive index of the liquid at the operation wavelength (${\lambda _0} = 632\;{\rm nm} $). The colored backgrounds refer to the character of the mode inside the nanochannel (light green: Gaussian-type, light blue: evanescent). The vertical blue dashed lines and the blue dot indicate the temperature at which the flat-field condition occurs.

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Comparing the RI-adjusted results for temperatures below and above the flat field condition reveals an opposite shifting behavior of the intensities of maximum probability in both bins $I_{{\rm max}}^{{\rm bin}}$, indicating a fundamental change of the character of the mode field inside the nanochannel. This effect can be quantified by considering the dependency of the difference of the relative intensities in two bins on temperature ($\delta {I_{{\rm max}}} = \delta {I_{{\rm max}}}(T)$, Fig. 5 bottom). A sign change of $\delta {I_{{\rm max}}}$ is observed, showing that through modulating the temperature, the system transfers from a Gaussian-type distribution (${n_1} \gt {n_{{\rm eff}}}$) inside the nanochannel to an evanescent-type behavior (${n_1} \lt {n_{{\rm eff}}}$). This result can be explained by the negative thermo-optic coefficient of the liquid (more details in the Supplement 1 Section 4), leading to a RI reduction in case the temperature is increased. The temperature at which the mode in the channel changes its character, i.e., a constant intensity, is $T = 28.5^\circ {\rm C}$ (vertical dashed blue line and blue dot in Fig. 5 bottom), corresponding to the flat field condition and the formation of a light strand (${n_1} \approx 1.461$).

Appropriate simulations were performed to verify the experimental data (details in Supplement 1 Section 8), which shows a high degree of similarity with the measured data. In particular, the dependence $\delta {I_{{\rm max}}} = \delta {I_{{\rm max}}}(T)$ qualitatively corresponds to the shape of the plot in Fig. 5. Compared with the experiments, the flat-field condition found in simulations is established at a slightly higher temperature (${T_{{\rm sim}}}{= 30.45^ \circ}{\rm C}$, ${T_{{\rm exp}}}{= 28.5^ \circ}{\rm C}$), owing to differences between the geometry assumed in the simulations and the NBF used. Overall, the simulated intensity distributions are narrower than those of the experiments, which results from additional errors (e.g., localization inaccuracies) inevitably occurring in the experiments.

4. DISCUSSION

Through transitioning from evanescent to Gaussian-type fields, the RI-adjusted FaNTA measurements show a flat intensity distribution within the nanochannel, successfully demonstrating formation of a light strand as a new type of optical mode. Simulations (see Supplement 1 Section 7) show that for smaller nanochannels, the flat-field condition becomes increasingly less critical as the modal field decays over a smaller distance. Thus, further studies considering different RI distributions need to clarify how the flat fields behave in geometries with larger or smaller channels. Note that it is important to consider the small RI contrast between core and cladding in the case of the nanobore fiber. This configuration implies that the fiber mode has no relevant azimuthal dependence (weakly guiding approximation) and the flat field condition can be satisfied along any radial direction. Note that for large index contrast and small cores, the situation is different because the fundamental mode is not solely Gaussian [32].

As demonstrated here, one field of application benefiting from flat-fields is NTA. By generating a light strand, NPs can be illuminated and tracked with constant intensity along all three spatial directions in the field-of-view, reducing variations of intensity-related inaccuracies among all frames and preventing loosing the NPs while tracking. Having such fields enables for the first time effective exploitation of the scattered intensity of individual NPs in a confined environment, opening up new possibilities for NTA. For example, the scattered intensity can be used as an indicator to characterize the rotational diffusion of moving nonspherical NPs or for size and refractive index estimation. Note that the light strand enables pure intensity-dependent detection in a waveguide environment, which is an order-of-magnitude faster than conventional image-based tracking. Furthermore, extremely small NPs can be consistently detected, since insufficient intensities, typically occurring at the edge of the nanochannel, are prevented. Flat-field waveguides additionally open a new platform for nanorheology, e.g., to study complex flow phenomena or inhomogeneous media. Studies of individual NPs or fundamental investigations of light–matter interaction at the nanoscale can also be addressed. Here, one example is measuring photon pressure by constant illumination using the light strand, which could be useful in the context of elucidating the Abraham–Minkowski controversy.

Due to its generality, the flat-field condition can also be applied to other on-chip waveguides that employ total-internal reflection as guidance mechanism, one example being the nanoslot waveguide [20,21], where flat fields can be realized along a chosen direction. The strong modal dispersion of leaky waveguides may suggest that the flat field concept may be also applicable to leaky hollow-core systems such as on-chip waveguides (e.g., optofluidic light cages [33], microgap waveguides [34]) or ARROW [35] or fibers (e.g., hollow-core [29] or single-element anti-resonant-element fibers [36]). However, since the guiding mechanism of the nanobore fiber used here and that of leaky waveguides are fundamentally different, more studies need to be carried out to uncover the specificity of the flat-field concept in the context of leaky waveguides, being out of the scope of this work.

5. SUMMARY AND OUTLOOK

Light fields with tailored intensity profiles are important for a wide range of photonic applications. This work, to the best of our knowledge, presents a concept for the realization of light fields with constant intensity along all three spatial directions (including the direction of propagation), a so-called light strand, within a complex optofluidic nanostructured optical fiber. One unique feature of this concept is the integrated nanochannel located in the center of the fiber core, which is externally accessible and enables to study the light–matter interaction in a nanofluidic environment. This unexplored type of optical mode was realized through integrating liquid-filled nanochannels into the cores of step-index fibers, leading to fields that do not show transverse or longitudinal intensity variations inside the nanochannel. All relevant properties and a universally valid condition for obtaining flat fields that can be directly applied to other types of waveguides were revealed. Moreover, unambiguous experimental evidence of the light strand was demonstrated in FaNTA experiments through analyzing the Brownian motion of a single gold nanosphere inside the liquid-filled nanochannel, showing the advantageous properties of flat fields for NTA.

Light strands, realized through employing nanostructured waveguides, represents a conceptually new type of mode that has not been realized in the context of free-beam optics and waveguides, to the best of our knowledge, overall resolving the issue of spatially dependent fields. The proposed solution concretely addresses the problem of decaying modal amplitudes commonly encountered in many photonic applications. The flat-field condition is not limited to the systems discussed and can be extended to other waveguide systems, such as nanoslot waveguides, on-chip hollow-core waveguides, and anti-resonant fibers. The light strand concept will be particularly useful for life science (e.g., characterization of nano-objects), nanoscale material science (e.g., analysis of chemical reactions or rheological aspects on the nanoscale), and fundamental studies of light–matter interaction (e.g., nanoscale refractive index measurements).

6. METHODS AND MATERIALS

A. Preparation of Nanoparticle Solution

Gold nanospheres (mean diameter 50 nm, nanoComposix) were used for confirming the flat fields. To adjust the RI of the liquid, the NPs were dispersed in a solution consisting of water and DMSO, allowing to precisely tune the RI through the mixing ratio. The material dispersion of the mixture $n({\lambda _0},c)$ as a function of concentration ratio $c$ and operating wavelength ${\lambda _0}$ were calculated using the equations described in the Supplement 1 Section 4. For instance, at room temperature ($T{= 25^ \circ}{\rm C}$), the RI of the solution ranges between $1.33 \lt n \lt 1.47$ depending on $c$, allowing to tune the RI for fulfilling the flat-field condition. Note that DMSO has been chosen as a mixing component because it is chemically stable, mixes well with water, has well-known physical properties [37], and has been successfully employed in previous FaNTA experiment [28]. For filling, the NBF was inserted into the prepared solution and the liquid flowed into the nanochannel via capillary force.

B. Optical Setup

The optical experiments rely on an improved version of the setup shown in one of our previous works [30] (details can be found in the Supplement 1 Section 1). Laser light (${\lambda _0} = 632\;{\rm nm} $) was coupled into the liquid-filled NBF, with the fiber kept straight to prevent mode distortions and influence of polarization coupling. A Peltier element was placed below the fiber within the tracking area to fine-tune the RI of the liquid through adjusting the temperature. Note that the temperature coefficient of water/DMSO solutions is around $4 \times {10^{- 4}}/{\rm K}$, exceeding the value of silica by two orders of magnitude. This makes it possible to exclude the influence of the glass materials in the temperature-related experiments. More details on RI can be found in the Supplement 1 Section 4 (Fig. S3).

In the NP tracking experiments, $N = 20000$ frames are recorded to sample the intensity along the transverse direction of the channel (frame rate $\nu = 3592\;{\rm Hz} $, exposure time $\tau = 0.08\;{\rm ms} $). Here one unique feature of FaNTA plays an essential role: Due to the nanochannel, the diffusion is corralled, and extremely long NP-tracks can be recorded. As shown in the Supplement 1 Section 3, this, combined with the small channel diameter, leads the NP to traverse through the cross-section of the nanochannel multiple times, which is essential for accurately sampling the field.

C. Data Analysis

The image-based tracking of the nanospheres in the acquired movies relies on the Python package Trackpy, used for processing the images to obtain the scattered intensity and the position in transverse direction ($x$-direction) in each frame [30]. After image-processing, the intensity values are analyzed in selected sections of the nanochannel using histograms and nonparametric kernel density estimations (Fig. 4, more details in the Supplement 1 Section 2). The decisive factor here is the evaluation of the intensity dynamics in two selected areas (with respect to the transverse [$x$-] direction), which in the present study are the SB and the CB: For the case of ${n_1} \gt {n_{{\rm eff}}}$, the mode in the nanochannel has a Gaussian shape, with the consequence that the intensity values in the CB are slightly higher than those at the edges. The complementary situation is obtained for ${n_1} \lt {n_{{\rm eff}}}$, as the mode field in the nanochannel is evanescent. Thus the shape of the mode can be analyzed by comparing the relative peaks of the CB and SB histograms, while a flat-field situation is obtained when the peaks are located at the same peak intensity. More details can be found in the Supplement 1 Section 2.