The evanescent fields of subwavelength waveguides can strongly interact with the surrounding medium. Such light-matter interfaces have enabled a number of advancements in sensing and quantum information science. For example, ultracold atoms have been trapped and manipulated in evanescent fields of single-mode optical nanofibers for single-photon switches and ultralow power nonlinear optics , , , , . The strength of the atom-photon interaction depends critically on the waveguide dimensions, and optimization of the light-matter interaction requires exquisite control of the evanescent field by tailoring the precise geometry of the waveguide , . Indeed, recent proposals have expanded the trapping possibilities by suggesting the use of superpositions of higher order modes for complex potential landscapes , , but such proposals are especially challenging to realize in optical nanofibers because they require a mechanically robust implementation to eliminate the spatial intensity fluctuations.
The success of the nanofiber platform for fundamental studies has led to proposals to use integrated waveguides, which offer scalability, structural integrity, and control of complex waveguide geometries , , . An added benefit provided by the structural integrity is that not only are spatial intensity fluctuations substantially reduced but also, the exact intensity profiles of guided modes can be directly imaged. In previous works, computational methods have been used to explore the specific near-field structure of propagating modes above the waveguide , . These works demonstrate the importance of both the in-plane field structure and the out-of-plane decay length of propagating modes for applications such as cold atom trapping ,  and molecular spectroscopy . In other works, near-field scanning optical microscopy (NSOM) has been used to experimentally measure the propagating fields but not in highly evanescent nanophotonic waveguides relevant to atom trapping , , , . These works have demonstrated that NSOM has the potential to be an alignment-free, non-destructive imaging technique for rapid characterization of waveguide devices .
In this work, we demonstrate imaging of stable intensity profiles in silicon-nitride (SiN) nanophotonic waveguides using NSOM. We fabricated thin core waveguides over a range of widths that support higher order modes. By launching superpositions of these modes, we observed beating by NSOM that is in excellent agreement with calculations. In particular, not only do the beating frequencies agree with calculated value but also, stable spatial maps of the intensity profiles are recorded at a resolution and contrast that individual beats can be analyzed and theoretical beat lengths can be accurately measured. Furthermore, weaker mechanisms, such as standing modes generated by back reflections and modal interference, are easily observable. Our three-dimensional (3-D) imaging results include detection of the evanescent decay of the fields, as well as a direct observation of standing wave patterns often used in these one-dimensional systems for cold atom trapping. Three-dimensional NSOM images combine nanoscale resolution with an ability to quantitatively measure the evanescent electric fields in devices that include perturbations due to fabrication defects and coupling. Such high accuracy in all dimensions demonstrates the potential for optimizing and inspecting complex and extended waveguide geometries. The understanding of the 3-D character of evanescent fields in subwavelength waveguides has applications in disciplines ranging from chemical and biological sensing  and optical computing ,  to ultra-high precision metrology via interacting near fields , .
2 Materials and methods
2.1 Simulation methods
We modeled the waveguide properties using finite-element analysis (Comsol Multiphysics software: Comsol Multiphysics – Electromagnetic Waves, Comsol Inc, Burlington, MA, USA). These modal simulations provide a range of relevant parameters for different wavelengths and waveguide dimensions, such as phase velocity (equal to c/neff, where neff is the effective refractive index for a particular propagating waveguide mode), evanescent field extent, and polarization content. From neff, the beat length formed between modes (i) and (j), yb(ij), is given by
Modes are labeled according to their dominant electric field component, such that quasi-transverse electric (TE) modes have a stronger in-plane (x) electric field component than the out-of-plane component (z). Similarly, quasi-transverse magnetic modes (TM) have a stronger out-of-plane electric field component (z) than in-plane transverse component (x). A TEmn (TMmn) mode has m nodes in the x direction and n nodes in the z direction for the x (z) electric field component.
The chip-scale rib waveguides used in this work comprised a 175-nm-thick (Si3N4) layer deposited by low-pressure chemical vapor deposition (LPCVD) over a 5-μm-thick thermal silicon oxide layer, all on a 325-μm-thick silicon wafer. LPCVD Si3N4 allows for low-loss light propagation at visible and near-infrared wavelengths. After patterning a thin-film resist on the Si3N4 surface using an electron beam in fixed-beam moving stage mode, the straight 0.65–2.36-μm-wide rib waveguides (cross-section shown in Figure 1A) were formed by a 100-nm-deep reactive ion etch using SF6/C4F8 in an ICP/RIE system. The thickness of the waveguide core, 175 nm, was etched only at 100 nm as a balance between the need to maximize lateral confinement of the propagating modes, using the etched portion, and minimize sidewall scattering losses from roughness, by allowing the mode to extend vertically into the 75-nm unetched layer. The chip was then laser-scribed and cleaved along a silicon crystal plane to produce smooth waveguide ends (facets) without the need for additional polishing or substrate thinning. The 2.4-mm-long cleaved sample was then cleaned using oxygen plasma ashing and a piranha rinse. Fabry-Perot fringe analysis  shows optical losses of approximately 1–3 dB/cm at wavelengths between 765 and1640 nm for the waveguide modes excited in this study. We performed loss measurements at λ=765–790 nm, λ=965–995 nm, λ=1370–1490 nm, and λ=1440–1640 nm using various tunable laser sources. The loss was obtained from different samples with lengths ranging from Lwaveguide=2.4 mm to 9.6 mm. In general, we observed higher losses at shorter wavelengths. We attribute this to the increased effect of waveguide sidewall roughness at smaller λ; the insertion loss was also lower at longer wavelengths due to better mode-matching between the lensed fibers (used to couple light to/from the devices) and the waveguide at longer wavelengths. We used lensed fibers optimized for 1550 nm for λ=1370–1640 nm (standard lensed fiber), 980 nm fibers for the range of λ=965–995 nm (custom fiber), and 780 nm fibers for λ=765–790 nm (custom fiber). Concerning polarization, we find that TM in general results in lower loss compared to TE, although this difference is within the error of our measured loss. We attribute the polarization-dependent loss to the modal overlap with the sidewalls, which is stronger for the TE mode than for the TM mode.
NSOM measurements were performed on a WITec alpha300 NSOM, which operates in a fixed-tip, scanning sample configuration. The NSOM probe was an aluminum-coated cantilever AFM probe with a 90-nm aperture etched. All scans were performed in contact mode. Scans were set to a 100-nm pixel size laterally and 4-nm vertical resolution. Due to the mechanical limit of the piezo-electric scanning stage, the maximum lateral extent of all scans was 100 μm. The tip, operating in a collection mode, directly collected light into a 100-μm core optical fiber coupled to an Excelitas photon counting module (Figure 1B).
For all optical measurements, a tunable continuous-wave (CW) diode laser, New Focus Velocity model 6300-LN, operating at a wavelength of 780 nm at ~14 mW, was coupled to a custom polarization-maintaining focusing optical fiber with 2-μm spot size (Oz Optics). The lensed fiber had a working distance of 10 μm and was aligned to the waveguides using a Newport 5-axis stage, with ~30-nm minimum incremental movement. Incident light polarization was set by rotating the lensed end of the fiber to 0° (horizontal), 45°, and 90° (vertical) relative to the wafer surface, with 45° generating the greatest selection of beats as it simultaneously excites both TE and TM modes. The coupling system and the waveguides were rigidly mounted on the scanning stage so that the illumination fiber and waveguide sample would remain immobile with respect to one another while being raster scanned relative to the NSOM tip. A second polarizer was set between the NSOM tip and detector at the preferred polarization of passage through the tip in order to reduce noise. NSOM images with and without the second polarizer (analyzer) were compared to ensure there was no significant loss of signal. Once aligned, the system was stable to within ~1% of the in-coupled power for more than 5 h. The image collection time was ~30 min, enabling multiple measurements to be performed at approximately identical waveguide coupling conditions.
Two methods of NSOM were used during these experiments. The first was standard contact-mode in-plane scanning microscopy with the tip engaged with the surface of the waveguide, which generated conventional two-dimensional (2-D) plan view images. The second technique used the force-distance technique  to produce a 3-D data set of near-field optical intensity. At each lateral pixel location, the tip begins ~1 μm above the surface and acquires three data sets as the stage is raised to meet the tip: (i) optical signal intensity collected through the tip, (ii) tip deflection as measured by a quadrant photo-diode, and (iii) vertical (Z-axis) position of the stage. The optical signal data are calibrated to the Z-axis position, which is defined as zero at the point of contact between the sample and the tip, identified by the initial upward deflection of the tip upon surface contact (see Supplementary Information).
3 Simulation results
The longitudinal component of the Poynting vector (propagating power density, ) for the TE00 and TM00 modes at 780 nm in a 1-μm-wide waveguide is shown in Figure 2A. Both modes have significant evanescent field present in the region above the Si3N4 core. Figure 2B shows the dependence of the effective indices for the supported modes on the waveguide rib width as extracted from simulation. It is important to note that even though modes are labeled as TE or TM, certain rib widths exist for which the polarization character of the modes is strongly mixed. For example, at a width near 0.6 μm, the TE10 and TM00 modes are strongly mixed, and at a width near 1.85 μm, the TE30 and TM00 modes are mixed (circled regions in Figure 2B). Mode-mixing manifests as an anti-crossing in the dispersion, whereas modes prevented from mixing due to symmetry (such as TM00 and TE20) show a crossing in their dispersion.
Multiple populated modes within a single waveguide beat against one another due to their different modal effective indices (i.e. propagating wavelengths). Beat lengths yb were calculated from Eq. (1) for a number of low-order modes and are presented in Figure 3 as solid curves. The number of mode pairs becomes large for a modest number of supported modes (e.g. 15 mode pairs for a 2-μm-wide waveguide), resulting in even more mode pairs than those shown for the widest waveguides considered. In general, beat lengths for mode pairs with the same polarization (e.g. TE-TE or TM-TM) tend to increase in wider waveguides due to the convergence of their effective indices towards the slab value. On the other hand, beat lengths for mode pairs with opposite polarizations (e.g. TE10-TM00) tend to increase as the waveguide is narrowed due to the degeneracy of their effective indices in a square waveguide. Secondly, as higher order modes are cutoff in the narrower waveguides, there are fewer beats to be observed, which simplifies beat image analysis. Lastly, we note that while most beats are dispersive, a few (most notably TE00-TM00) exhibit little dependence on waveguide width, which is a consequence of these fundamental modes being generally well-confined and low loss for the geometries considered here.
4 Experimental results
4.1 2-D NSOM scans and beating analysis
While the computations above allow one to predict the periodicity of beating between supported waveguide modes, experimental verification using conventional optical microscopy is difficult, as near fields do not radiate . Far-field detection must rely on elastic scattering from surface imperfections or Rayleigh scattering from the waveguide material, both of which are weak. Therefore, planar NSOM scans, taken at the surface of the waveguide, were acquired to verify the lateral evanescent field confinement and beat periodicity. Figures 4A,C and 5A show the evanescent field intensity measured at the surface of waveguides with widths ranging from 0.65 to 2.36 μm (measured by scanning electron microscopy). In Figure 4, NSOM images obtained with horizontally polarized waveguide excitation are shown alongside corresponding Comsol simulations of electric field-magnitude squared for 0.90- and 2.36-μm waveguide widths. Features with a number of different periodicities are evident in the signal along the longitudinal direction, with the smallest feature ~0.2 μm in size (indicated by an arrow on 2.36-μm waveguide in Figure 4C). These smallest features are due to Fabry-Perot interference fringes formed by the waveguide end facets. Such fringes would have a spacing of approximately λ/(2neff) ≈0.23 μm, a sub-diffraction limited feature resolvable due to our ~90-nm NSOM tip aperture. The images in Figure 4 all show periodic lateral oscillation of the measured and calculated field strength, which is a manifestation of interference between the two lowest order TE modes (TE00 and TE10). Indeed, the period of the oscillation matches the beat length for the TE00-TE10 beat for both waveguides, as calculated in Figure 3.
The images in Figure 5A were collected with incident polarization set to 45° relative to the sample surface for all fabricated widths. All waveguides show significant confinement in the X-axis (transverse to the waveguide whose size is indicated by dotted lines). The weaker signal collected outside of the waveguides is unguided light scattering into the cladding. While the narrowest waveguide (0.65 μm) supports only a few modes, the wide waveguides (>0.65 μm) populate many modes, which results in increased scattering into the cladding. All waveguides show periodic longitudinal intensity variations at a period of approximately 5 μm, which, as detailed below, is due to beating between the TE00 and TM00 modes. Accurate extraction of the multiple overlaid beat periodicities requires a Fourier analysis of the NSOM intensity data. We applied a fast Fourier transform (FFT) to each single pixel column of data along the waveguide and then averaged the resulting FFTs across the waveguide . These results are shown in Figure 5B. In general, most waveguides show strong features near ~5-μm periodicity with other prominent features showing up and increasing in multitude as waveguide width increases. In order to identify the origins of these periodic near-field features, we have marked the periodicities predicted from simulation (see Figure 3) as colored dots and connecting lines (color of each dot corresponds to curve colors in Figure 3). There is strong correlation between the calculated beat periodicities and peaks in the FFTs. This correlation is also seen clearly in Figure 3 where the measured FFT peak positions, extracted from Figure 5B, have been added as points to the simulated dispersion curves. As the waveguide width increases, the FFT spectrum becomes particularly crowded near a periodicity of ~5 μm. For the widest waveguide inspected, there are nine beat lengths predicted over the plotted range with nearly half of them clustered near a periodicity of 5 μm, making specific correlation between predicted and observed beat identification ambiguous for the wider waveguides. Conversely, modal cut-off in the narrowest waveguide results in a simplified response and shows only two beating patterns for the 0.65-μm waveguide. The beating structure is so simple, in fact, that a beating pattern with a periodicity of ~6 μm can be identified through direct inspection of the planar NSOM scan of this waveguide (Figure 5A).
Excitation polarization determines what modes are launched. The data discussed above were taken with the source polarized at 45° to the substrate, as this would excite both TE and TM modes and enable measurements of TE-TE, TM-TM, and TE-TM beat lengths. NSOM scans were also recorded with the optical polarization of the excitation light aligned either horizontally or vertically to the substrate, which should induce a preference for launching TE and TM modes, respectively. Figure 6 shows an example of the resulting FFT spectra for the 0.65 waveguide. Under excitation at 45° (top blue curve), the 0.65-μm waveguide produces a strong TE00-TE10 beat and a weaker TE00-TM00 beat (positions identified with vertical dotted green lines). The weaker TE00-TM00 beating is clearly diminished when the waveguide is excited with horizontal polarization as the TM00 mode is not excited effectively in this geometry. The TE00-TE10 beating is observed regardless of incident polarization although, strictly speaking, there should be very little TE modal population content under vertical excitation (bottom red curve, which shows small contribution from TE). We believe that its weak presence is due to the strong mixing between the TM00 and TE10 modes in the 0.65-μm-wide waveguide, which enables some population of the TE10 mode with vertical polarization.
4.2 3-D NSOM imaging of evanescent decay
The trap position above the waveguide depends critically on the strength of the evanescent fields. Quantitative measurement of the characteristic decay distance cannot be made with far-field techniques. We performed 3-D NSOM imaging of the evanescent fields above from the 0.65-μm-wide waveguide. Sections of this 3-D data set, taken along the longitudinal and transverse directions, are shown in Figure 7A and B (note that the vertical axes are stretched relative to the horizontal in order to make the features observable). There are two important features to point out in the longitudinal cross-section (Figure 7A). First, the beating structure supported on this narrow waveguide, evident through simple inspection, exhibits a period of ~6.5 μm, which matches quite well with the dominant features of the FFTs of 2-D scan for this same waveguide (0.65 μm waveguide in Figure 5). The second important observation is that the beating periodicity clearly persists well above the waveguide surface. This persistence is a natural consequence of all the evanescent fields decaying with approximately the same characteristic decay distance. The higher frequency features are due to Fabry-Perot fringes, as discussed previously. A section taken along the transverse direction, Figure 7B, shows fields primarily localized to the waveguide width with very little divergence of the fields above the guide thus maintaining confinement of the trapping potential in the transverse direction to the region directly above the guide. In addition to verifying the integrity of the beating structure above the waveguide, this 3-D data set allows for extraction of the characteristic decay rate of the evanescent fields. Field decay associated with regions of maximum and minimum field strength all show well-behaved exponential behavior. Two example decays are plotted along with exponential fits in Figure 7C. Fits for multiple positions including at the beat maximum and minimum yield 1/e decay lengths of 62±5 nm and 59±3 nm, respectively, and a contrast, defined as the ratio of the peak to valley field intensity, of 4±0.57. These decay lengths are similar to the computational results, which produced a decay length of 46 nm for the TE00 mode and 53 nm for the TM00 mode. The small discrepancy (~17%) may be due to the vertical resolution with which we define the absolute z-position of the collecting tip, which is determined by the data density in this direction (the pixel size of the acquired data in the z-direction is 4 nm).
The characteristic decay constants of tens of nanometers place these modes in an interesting regime for establishing trapping potentials. Plasmonic systems exhibit considerably different evanescent decays, with local surface plasmons  and surface plasmon polaritons , ,  exhibiting characteristic decay lengths for similar environments and excitation wavelength of ~5 and ~200 nm, respectively. Another common platform, symmetric waveguides, exhibits characteristic decay lengths of ~100 nm . We reiterate here that the field intensity in the high and low field regions decays with approximately the same decay constant, implying that any beating structure generated at the waveguide surface should extend above the waveguide. In addition to providing direct near-field measurements of lateral and vertical near-field structure, these results reveal that the trapping potential above the waveguide is well-modeled by finite-element methods and should create a robust platform for trapping.
In this work, we show the viability of 3-D NSOM as a technique capable of analyzing the near-field structure of propagating modes in thin core waveguides. The high resolution and low impact on the samples make this method ideal for identifying optical dipole near-field traps for integrated applications such as lab-on-chip or quantum information systems. It was found that the periodicity and selection of modal beats, generated by interference between low-loss co-propagating modes, can be tailored via waveguide design. We have measured modal beat periods as a function of waveguide width using a 3-D NSOM technique augmented by FFT analysis. These measurements agree with our finite-element calculations over a number of modes and waveguide widths. We extend conventional 2-D near-field imaging into a third dimension through a rastered force-distance technique and verify that the periodic beating occurring along the waveguide persists above the waveguide. The measured evanescent field decay length, a property only measurable using near-field techniques, also agrees with our calculations.
The combination of low propagation losses, lateral confinement, and significant penetration of the fields above the waveguide demonstrates the potential of thin core waveguides for an on-chip system to trap and manipulate nanoscale materials. The single-step lithographic fabrication of the waveguides enables scalability and robust production as an integrated trapping platform. Such a platform will allow the transmission of quantum information by moving either cold atoms or photons from one physical location to another, assisting in the development of future large-scale integrated quantum systems on a chip.
J. Ziegler gratefully acknowledges Research Associateships administered by the National Research Council. This work is supported by the Office of Naval Research through the Institute for Nanoscience at the U. S. Naval Research laboratory and by the Army Research Laboratory.
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Published Online: 2017-04-19
Citation Information: Nanophotonics, Volume 6, Issue 5, Pages 1141–1149, ISSN (Online) 2192-8614, DOI: https://doi.org/10.1515/nanoph-2016-0187.
©2017, Jed I. Ziegler et al., published by De Gruyter.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0