1. Introduction

Spectroscopic instruments such as spectrometers and wavemeters typically employ a dispersive element (e.g. a prism or grating), an interferometer (e.g. the Fabry-Perot multiple beam interferometer), or a complex scattering medium (e.g. a ‘speckle’ plate [1]). Many optical devices are not confined to a single category; for example, the Fizeau or ‘wedge’ interferometer is a multiple beam interferometer but also imparts spectral-spatial dispersion to an input signal [2]. Furthermore, recently studied ‘random’ spectrometers [3–8] are essentially predicated on the interference of multiple beams at an image plane, where the ‘beams’ represent the arrival of light over a diversity of paths (or modes) through a complex but stationary optical medium.

We previously reported [9–11] some unique characteristics of tapered, omnidirectional-clad, hollow Bragg waveguides as enabling elements for spectrometry and wavelength interrogation, especially for applications involving guided or confined light. Light launched into the wide end of such a waveguide typically excites a large number of modes, which can be sub-divided into families of discrete vertical mode orders [9]. As light in a given mode propagates down the taper, it remains confined by the omnidirectional mirrors until nearing a wavelength-dependent cutoff point. Near and at cutoff, a significant portion of the light is radiated in a nearly out-of-plane direction through the dielectric mirrors [10]. The radiation of light from multiple modes, each mode subject to significant back-reflection as it approaches cutoff, and each mode achieving cutoff at a unique position for a given wavelength, produces a complex ‘radiation streak’, which can be efficiently projected onto a nearby image sensor using simple optics. In previous work, we have relied solely on the position of the ‘terminal’ cutoff point (associated with the fundamental lateral mode) as a mechanism for spectral-to-spatial mapping [10–11]. Here, we show that the intricate, spectrally sensitive details contained in a typical radiation streak, combined with calibration-based extraction algorithms, can greatly enhance the resolving power.

2. Conceptual description

Figure 1(a) illustrates the underlying principle, described in detail elsewhere [9–11]. Light confined to the air core of a hollow waveguide can be subject to omnidirectional reflection by Bragg mirrors of sufficiently high index contrast [12]. If the waveguide is tapered, as depicted using a ray-optics slab model in Fig. 1(a), light coupled into the wide end of the taper will remain highly confined (for sufficiently reflective claddings) until nearing a wavelength- and mode-dependent critical core dimension (i.e. cutoff). One can view this as the adiabatic transformation of a waveguide mode into a normal-incidence (relative to the cladding layers) Fabry-Perot mode [10]. This simplified model ignores some details, of course. For example, as a given mode approaches cutoff, there is both a divergence of radiation loss and significant reflection into the reverse-propagating mode [9–11]. Moreover, in practice we employ tapered channel waveguides (with the air-core tapered in both width and height, see Fig. 1(b)) fabricated using a buckling self-assembly process [13–14]. Typically, these waveguides support multiple (up to several dozen [9]) low-loss, air-guided modes. For TE polarized modes, these can be labeled TE_pq (p,q = 0,1,2…), where p and q denote the number of electric field anti-nodes in the vertical (out-of-plane) and horizontal (in-plane) directions, respectively [10].

 

Fig. 1. (a) Schematic illustration of the experimental setup. Light is coupled into the wide end of a tapered hollow waveguide, exciting multiple modes. The ray trajectories depict the adiabatic evolution of guided light into a vertical cavity resonance (within the omnidirectional band of the claddings), for one particular mode at two different wavelengths. Radiation loss diverges as the cut-off point is approached. (b) Overhead-view microscope image showing portions of three tapered waveguides (scale bar: 250 μm). (c) (Visualization 1) Typical radiation streak for a monochromatic (980 nm) input signal, captured using a 40x objective lens (NA = 0.65). The streak comprises light radiating near and at cutoff for a family of horizontal (in-plane) modes belonging to one vertical mode order (TE_1q for the case shown). The terminal cutoff spot is associated with near-vertical radiation from the fundamental horizontal (i.e. TE₁₀) mode.

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The nearly out-of-plane radiation from these multiple counter-propagating and interfering modes produces a complex ‘radiation streak’, such as shown in Fig. 1(c). The intricate details of the radiation pattern are extremely sensitive to changes in the input coupling conditions or source wavelength, due to changes in the mode coupling and cutoff conditions. A visual illustration of this sensitivity is provided in supplementary video files, and also in figures below. The portion of the radiation streak captured by the image sensor depends on the numerical aperture (NA) of the collection optics, owing to the evolution of the radiation angle along the taper axis depicted in Fig. 1(a). In fact, by using a sufficiently low-NA collection optic, it is possible to image only the terminal cutoff spot of the radiation pattern. In previous work, we have shown that this can be employed as a straightforward strategy for reasonably high-resolution spectrometry [10] and wavelength interrogation [11]. However, it clearly discards a significant amount of information about the input signal contained in the remainder of the radiation streak.

Information about the source wavelength is encoded in multiple aspects of the radiation streak, including the position of the terminal cutoff point, the spatial period of the standing waves, and more generally in the overall multimodal interference pattern. Accordingly, these waveguides are potentially powerful spectral discrimination components, combining features of a dispersive medium, a Fabry-Perot, and a ‘speckle’ interferometer. Furthermore, these multiple ‘axes’ of information content make the radiation patterns well-suited for principle component analysis (PCA) [15] of the spectral-spatial relationship, as described below.

The operating principle of the proposed wavemeter is depicted in Fig. 2. As shown in Fig. 2(a), a large change in wavelength results in a correspondingly large spatial shift of the radiation streak due to the shift in cutoff position z. This spatial dispersion provides a straightforward means for extracting a coarse wavelength estimate with ∼ 1 nm resolution [10]. The operating range is limited by the omnidirectional bandwidth of the cladding mirrors and the sensitivity range of the image sensor, and can be several hundred nanometers [9–11]. This range is sub-divided into some number W of coarse wavelength bins centered at wavelengths λ_Ci, each having width Δλ_C ∼ 1 nm. Within each coarse bin, a set of calibration images is captured using a tunable laser stepped in fine increments Δλ_F on the picometer scale. For an unknown monochromatic input signal, the coarse wavelength is first estimated in order to establish the appropriate bin, and then a more precise wavelength estimate is obtained using the radiation streak image (i.e. a set of pixels P_i assigned to coarse wavelength bin λ_Ci, as indicated by the trapezoidal regions in Fig. 2(a)) and the corresponding set of calibration data for that bin.

 

Fig. 2. (a) (Visualization 2) Radiation streaks captured using a 20x objective lens (NA = 0.40) at 3 different wavelengths (λ = 940, 945, and 950 nm), illustrating the spatial dispersion imparted by the taper. (b) Block diagram illustrating the two-step procedure used to extract the wavelength of a nominally monochromatic input signal. The spectral dependence of the terminal cutoff position is used to extract a coarse wavelength estimate. This estimate is then fed into a PCA-based estimation of the precise wavelength using the ‘speckle’ pattern of the overall radiation streak. (c) Further illustration of the procedure: the operating range B is determined by the waveguides and the image sensor. This range is sub-divided into W ‘coarse’ bins, and a pre-determined calibration set within each of these bins enables the algorithmic determination of a ‘fine’ wavelength estimate.

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3. Experiment

The tapered waveguides used here (see Fig. 1(b)) were fabricated using the same materials (sputtered a-Si and SiO₂) and procedures (buckling self-assembly) as described in previous work [13–14]. However, here the thicknesses of the a-Si and SiO₂ layers were nominally set to ∼160 nm and ∼60 nm, respectively, to act as quarter-wave layers centered at ∼950 nm wavelength. Both the upper and lower mirrors of the buckled waveguides have 5 periods, with a high index (a-Si) layer next to the air core [13]. We experimentally verified (not shown) that straight channel waveguides on the wafer support a low-loss propagation band in the ∼ 800–1200 nm wavelength range for air-guided, TE-polarized modes. As described elsewhere [10], these waveguides are highly polarizing due to very high leakage loss for TM-polarized modes. Thus, all of the results described below pertain to families of TE-polarized modes. Moreover, the cladding mirrors provide nearly omnidirectional reflection for TE-polarized light over the entire 800–1200 nm range, which thus represents the ultimate limit on the operating range for these particular tapers. For the system described below, however, the sensitivity range of the silicon CMOS image sensor is limited to wavelengths below ∼ 1100 nm. Moreover, the results reported here are restricted to the 915–985 nm wavelength range of the available tunable laser (Toptica CTL-950). This laser has a minimum step size of 5 pm, which placed a limit on our ability to discern the ultimate resolution of our wavemeter prototype. For the experiments below, ∼ 0.5 mW laser power was coupled into the tapered waveguide via a cleaved single mode fiber, which was epoxied to the chip in order to improve the stability of the coupling conditions. A commercial wavemeter (HighFinesse WS-6) with sub-pm-scale resolution was used to monitor the laser output and also for comparison with the prototype wavemeter results.

Radiation streak images were captured using a silicon CMOS camera (Thorlabs DCC1545M), with pixel size ∼ 5.2 μm and sensor area ∼ 6.7 × 5.5 mm. As discussed further below, the spatial dispersion of the cutoff position is ∼ 2 μm/nm for these tapers [11]. Given the target operating range of ∼ 100 nm, we chose to magnify the radiation streak by ∼ 10–20x, in order to make use of a large portion of the sensor area and also to increase the pixel resolution of the images. As depicted in Fig. 1(a), this magnification was achieved simply by using a standard objective lens at long working distance. For most of the results presented below, we used a 40x objective lens (NA = 0.65) operating at ∼ 20x magnification. Very similar results were obtained using a 20x objective lens (NA = 0.40) operating at ∼ 10x magnification. As noted above, the NA of the lens determines the extent of the radiation streak captured by the image sensor. Thus, the radiation streaks imaged using the 40x lens are longer and have a more detailed interference pattern (compare Figs. 1(c) and 2(a), for example), typically resulting in somewhat improved resolution.

As shown in Fig. 1(a), a polarizer was also placed in front of the sensor to suppress any residual TM-polarized light, which would only degrade the SNR of the detection. Nevertheless, due to the highly polarizing properties of the waveguides, we observed negligible differences in images captured with and without the polarizer in place. Finally, note that the camera outputs 8-bit digitized images and has a relatively low quantum efficiency (< 10%) in the ∼950 nm wavelength range used here. In spite of this, it was typically necessary to place a neutral density filter (optical density ∼ 0.6) in front of the image sensor in order to prevent pixel saturation, corresponding to an effective laser input power in the ∼ 100 μW range. Further discussion of efficiency and throughput considerations is provided in Section 4.

4. Algorithm description and results

4.1 Analysis of radiation streaks as speckle patterns

Since the terminal cutoff point tracks along the taper with changing wavelength, the radiation streaks are essentially moving speckle patterns. Figure 3(a) shows a typical dispersion plot, with spatial dispersion ∼ 2 μm/nm over most of the range. Regions of anomalously high dispersion (near ∼ 925 and ∼ 965 nm for the case shown) are associated with ‘defects’, which are essentially short sections of the waveguide exhibiting low taper slope [11]. We attribute these defects to the experimental nature of our fabrication process, and expect they could be eliminated through process refinement.

 

Fig. 3. (a) Plot showing the terminal cutoff position versus wavelength for a typical tapered waveguide, as extracted manually (symbols) and using an automated image processing algorithm (solid line). Inset: typical plot of pixel intensity versus distance for a radiation streak image. The large peak (circled) and subsequent sharp drop-off associated with the terminal cutoff point are characteristic features that occur at a wavelength-dependent position. (b) Inner product covariance plot for images captured at 1 nm intervals across the entire tunable range of the laser.

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Automated extraction of the coarse wavelength estimate was carried out as follows. First, a pre-calibration step was performed in which the pixel value associated with the terminal cutoff point was determined at 1 nm intervals across the full range of the tunable laser. This data was used as a look-up table in assigning a coarse wavelength estimate to a monochromatic input signal of unknown wavelength. An automated image analysis routine was created, using characteristic features in the pixel intensity profile near the terminal cutoff point (see the inset of Fig. 3(a)). Specifically, the algorithm sequentially analyzes pixel values starting at the right hand side of an image (i.e. the narrow end of the taper, which is always dark) and searches for the sharp falling edge and peak of the terminal cutoff spot. Based on this, the algorithm places the data set into the appropriate coarse wavelength bin. For example, Fig. 3(a) is a typical plot of the extracted pixel position versus input wavelength, where the laser was stepped in 0.1 nm increments. As evident from the plot, erroneous binning occurred in a few cases and was also traced to defects on the tapered waveguide.

We expected a very low correlation between images captured across this entire range, due to the ‘moving’ nature of the speckle pattern. To verify this, we stored the pixel values from a rectangular ‘box’ sufficiently large to encapsulate the entire radiation streak over the entire wavelength range. The pixel values in each image were normalized and arranged into a vector, and the inner product [7] between vectors for every pair of wavelengths was computed. An inner product of 1 indicates completely correlated (i.e. identical) vectors and an inner product of 0 indicates completely un-correlated (i.e. orthogonal) vectors. The result of such a calculation is shown in Fig. 3(b), indicating the images have little correlation over the entire range as expected.

We previously [11] applied a sub-pixel centroid detection algorithm to the terminal cutoff spot, and showed it was possible to track wavelength changes with resolution on the order of 20 pm. A primary goal of the present work was to achieve improved resolution by extracting more information from the images. As discussed above, the radiation streaks, when captured using sufficiently high-NA collection optics, are essentially speckle patterns. For example, the images in Fig. 4(a) exhibit clear pattern variations for wavelength changes of 100 pm, even though movement of the terminal cutoff spot is not discernable at this scale. An oft-used figure of merit [3–8] for quantifying the resolution of speckle spectrometers is the spectral correlation width δλ, which is the half-width-at-half-maximum (HWHM) of the following spectral correlation function [3]:

 

Fig. 4. (a) Series of images captured at 100 pm increments, showing significant variation in the speckle pattern. (b) Spectral correlation function for a set of images captured in 5 pm increments over various ranges as indicated. For the 1 nm range, the HWHM resolution is δλ ∼ 15 pm. (c) Result of SVD matrix inversion on a synthesized image produced by a spectrum comprising lines at 960.30 and 960.31 nm. (d) As in part (c), but for spectral lines spaced by 5 pm.

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Here, I(x,λ) is the pixel intensity of pixel x at wavelength λ, and Δλ is an increment of wavelength. The spectral correlation is averaged over all pixels, normalized with respect to C(Δλ = 0) and plotted versus Δλ. Note that it is important to exclude dark regions of the image sensor from this calculation, since the uncorrelated pixel noise results in a misleadingly low estimate of δλ. As shown in Fig. 2(a), we used trapezoidal regions encompassing the brightest portion of the radiation streak near cutoff. Figure 4(b) shows typical results of such a calculation applied to our speckle images, for images captured at 5 pm intervals over various ranges. Interestingly, δλ was observed to scale approximately linearly with the range of integration (i.e. bandwidth), from δλ ∼ 5 pm for a 0.25 nm range to δλ ∼ 15 pm for a 1 nm range. This is consistent with the statistical rule of thumb that the number of simultaneously discernible spectral channels is equal to the number of orthogonal spatial channels (i.e. ‘speckles’) in the images [3]. Mathematically, this can be expressed as R = S·δλ, where R is the range of contiguous bandwidth contributing to a given speckle image and S is the number of spatial channels or modes. This furthermore implies S ∼ 50 for our speckle images (captured with the 40x objective lens), which we confirmed from a spatial correlation integral (not shown) similar to Eq. (1) but with x and λ inter-changed [6]. It is notable that to achieve δλ < 50 pm typically requires a device with significantly larger footprint, for example a multimode fiber 5 m in length [3]. However, here the speckle images are produced by a collection of highly dispersive modes approaching cutoff, which effectively increases the range of propagation delays contributing to the interference pattern [6–7].

Subject to the bandwidth and resolution restrictions mentioned, speckle patterns produced by stationary random media can be used for spectral analysis of arbitrary signals [3–7], and such devices have been termed as ‘speckle’ or ‘random’ spectrometers. Typically, a tunable laser is used to create a calibration data set. An unknown spectrum, which can in principle contain an arbitrary mix of frequencies within the specified bandwidth, can then be extracted from its measured speckle pattern, by using matrix inversion combined with nonlinear optimization algorithms [4,5]. To corroborate the estimate of resolution obtained from the spectral correlation function, we synthesized a spectrum corresponding to a pair of closely spaced spectral lines, by linear addition of the individual calibration images at two discrete wavelengths [6,7]. We then employed a truncated singular value decomposition (SVD) [3,7–8] matrix inversion to reconstruct the spectrum from the composite image. Figure 4(c) shows a typical result for spectral lines spaced by 10 pm. The lines are clearly resolved according to a Rayleigh criterion, corroborating the estimate from above. Figure 4(d) shows that spectral lines separated by 5 pm could not be resolved, as expected since 5 pm is the minimum step size of the laser used to create the calibration images. The ultimate resolution, which is impacted by signal noise and post-processing errors in addition to speckle de-correlation, is left for future study. Nevertheless, a resolution < 10 pm compares favorably to other reported integrated optics devices [4,6].

It is well known [3–7] that a nonlinear optimization algorithm can be added in order to reduce the reconstruction error, and can even enable the reconstruction of continuous spectra. However, since speckle contrast must exceed noise in practice, it has also been shown [16] that these ‘random’ spectrometers are typically best suited to the spectral analysis of narrowband and/or sparse signals; i.e. for use as wavemeters [1,8] or wavelength interrogators [11] rather than as spectrometers. Accordingly, we focus mainly on the wavemeter application in the following.

Given this, it is important to point out that the spectral correlation width (δλ), while a meaningful parameter for comparing speckle devices, is a somewhat arbitrary measure of the resolution. For wavemeter applications, differences in wavelength much smaller than δλ can be extracted, provided the degree of de-correlation between the speckle patterns is sufficiently large compared to the sources of noise present. For example, Klaus Metzger et al. [8] recently used principal component analysis (PCA) to track wavelength changes on the order of femtometers, in spite of the fact that their speckle patterns showed no significant de-correlation (in the sense described above) on this scale. We employed a similar strategy here, as discussed below.

4.2 PCA-based extraction of the wavelength

PCA is a powerful data analysis technique [1,15], often applied to data of a high dimensionality (such as an image with thousands of pixels), in order to reveal features that exhibit a large variation across a parameter of interest, such as wavelength. These features, or principal components (PC’s), can be studied directly to obtain insight into the nature of a system, or used in an inferential manner to extract specific information of interest. We expected that PCA might be well-suited to our devices, since it is clear that numerous features embedded within the radiation streak will change predictably with wavelength. For example, the cutoff position for each interfering mode in the radiation streak varies approximately linearly with the input wavelength. While these simple relationships are somewhat difficult to see in the complexity of the overall pattern, especially for the higher-order modes, this is precisely the type of task for which PCA is designed [15].

We applied PCA to sets of calibration images, captured at 5 pm intervals (the minimum increment possible with the laser used) over the 1 nm range of each coarse wavelength bin (see Fig. 2(c)). Within each bin, the input to the PCA algorithm was a fixed set of pixels from a cropped, trapezoidal region of the image aligned with the radiation streak location for a given coarse wavelength (see Fig. 2(a)). These cropped images (when captured with the 40x objective at ∼ 20x magnification) corresponded to an area of ∼ 10 μm × ∼ 100 μm on the tapered waveguide, and encompassed ∼ 10000 pixels on the image sensor. From these images, a virtual 1D array of pixels, X_j = [x₁,x₂, …x_N], is created by stacking columns of pixels within the trapezoid on top of one another. The order in which these pixels are arranged is arbitrary and has no effect on the final result, but must be consistent for a particular calibration set. Each 1D array is then arranged as rows in a M × N matrix T, where M is the total number of wavelengths captured for calibration. Each pixel in T is then zero-meaned across all wavelengths, in order to reduce noise and remove any features in the pixels that do not change with respect to wavelength. Finally, the symmetric N × N covariance matrix C_V = T·T^T/(N−1) [3,7] is calculated.

In our system, covariance is a statistical measure of the relative change of one pixel with respect to another as the wavelength is changed. Thus, C_V contains a complete set of information across the entirety of pixel space about how the images vary with wavelength. The goal of PCA is to determine an appropriate transformation of pixel space into a new basis that brings out the behavior of interest in the image – i.e. the changes with respect to wavelength. One way that this can be achieved is through solving for the eigenvectors and eigenvalues of C_V. This new complete basis of eigenvectors forms PC space, denoted P = {P₁,P₂, …., P_N : P_i ∈ ℝ^N }. All vectors in this space are normalized and orthogonal to one another. The PC’s are ordered according to the magnitude of their corresponding eigenvalues. Those with the largest eigenvalues correspond to the greatest variations with respect to wavelength (i.e. relevant features), and those with the smallest correspond to minimal or uncorrelated variations (i.e. static features or noise).

Since PC space forms a complete basis for the set of images, a linear combination of these PC’s can exactly express every image used in the calibration set. For a particular image at wavelength λ_j:

We collect these coefficients for each calibration wavelength λ_j. One way to interpret these coefficients is as points within higher dimensional PC space [1]. Figure 5(a) visualizes the coefficients for the first three PC’s plotted as points in a 3D space, for a typical set of calibration data captured at 5 pm intervals, and with each sequential wavelength in the set connected by a blue line. While this 3D space only represents a fraction of the higher dimensional data, it can be used to qualitatively judge the resolution of the data to be analyzed. For our device, a spread-out and snaking path is indicative of well de-correlated data, while a jumble of points with large overlap may indicate little to no changes outside of noise within the wavelengths of interest.

 

Fig. 5. (a) Plot showing the trajectory of the first 3 principle components extracted from a set of images with 5 pm spacing in the 930–931 nm range. (b) Schematic illustration of wavelength assignment for an unknown input, based on analysis of the nearest calibration point in multi-dimensional PCA space. (c) Plot of extracted wavelength versus input wavelength. (d) Histogram showing the distributions of errors for the extracted wavelength set from part (c). (e),(f) As for parts (c),(d), except in the 950–951 nm range.

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With the system calibrated, a new image to be analyzed is captured, cropped, and placed into a 1D array X_sample, as for the calibration images. It is then projected onto each of the PC’s and plotted in PC space, as per Eq. (2). The Euclidean distance from this point to all of the stored calibration points is calculated, and the closest point is used to assign the wavelength, as depicted schematically in Fig. 5(b). More sophisticated techniques, such as measuring Mahalanobis distance [1], might improve the extraction accuracy but were not explored here. In practice, only a small number of the largest PC’s - typically on the order of ∼ 10 - were found necessary to achieve accurate results. The choice of the optimal number of PC’s is dependent on the system being analyzed, but is generally a tradeoff between the size of the calibration data set, the algorithm accuracy, and the system noise. It is desirable to use as few PC’s as minimally needed to obtain accurate results, in order to minimize the size of the calibration data set.

We tested PCA using fine-step calibration data, as described above, in multiple coarse wavelength bins spanning the range of our tunable laser. Two different sets, typical of the behavior in all cases tested, are presented in Figs. 5 (c-f). The devices were calibrated, as described above, within the selected bin ranges. Within a few minutes, a second set of images, separate from those used in the calibration step, were captured over the same wavelength range. These test images were used as input data for the PCA algorithm, and the statistics of the errors between the actual wavelength and the extracted wavelength were recorded. Resolution was estimated by calculating the standard deviation of the error across the entire set of test images. This produced a resolution of ∼ 0.6 pm and ∼ 0.8 pm for the first and second sets of test data shown, covering the 930–931 nm and 950–951 nm coarse wavelength bins, respectively. Very similar results were obtained using the same procedure in coarse wavelength bins spanning the entire range of the laser. Notably, the maximum recorded error was typically 5 pm, which is the minimum step size of the laser used.

4.3 Stability and throughput

A well-known challenge [3–8] for speckle-based interferometric devices is their sensitivity to mechanical and thermal changes in the environment, which can impact the stability and long-term repeatability of measurements. For our prototype system, several factors might contribute to limited stability, including changes in polarization for the input light, changes in temperature of the tapered waveguide chip [17], and drifting alignment of the optical path between the taper chip and the image sensor. Similar to Ref. [7], we found that mechanical drift was the dominant issue here, since the taper chip, objective lens, and camera were each mounted on a separate 3-axis positioner attached to a vertically oriented rail. The mechanical instability resulted in small variations in the observed images, which mainly affects the fine PCA component of our algorithm.

We characterized the instability by setting the laser at a fixed wavelength and power, and periodically acquiring images across the span of a few hours. The inner product correlation of these images with respect to the image at ${\Delta }t = 0 $ was calculated. Typical results over two different wavelength ranges are shown in Fig. 6, revealing that the system was reasonably stable on a time-scale of only several minutes. We observed significant defocusing and lateral shifting of the images with time, which could be attributed to drifts in the multi-axis positioners [7]. There was also some evidence of cyclical variations on time-scales similar to those of the building HVAC systems. We anticipate that the small size of the system is well-suited to the development of a compact, integrated, and possibly hermetically sealed package, with the optics rigidly mounted to a compact camera, and with active temperature control of the taper chip. These strategies, combined with the use of a stable reference frequency source [8], offer significant scope for improving the long-term stability, but are left for future work.

 

Fig. 6. Normalized correlations of calibration images at various time intervals with respect to t = 0, for (a) 930–931 nm at 15 minute intervals up to 1 hour and (b) 960–961 nm at 1 hour intervals for 3 hours.

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Finally, we note that the optical throughput for these tapers can be approximated as η_T ∼ η_in·η_prop·η_rad [10], where η_in is a factor accounting for coupling efficiency between the input fiber and the particular family of modes contributing to the radiation streak of interest, η_prop is the fraction of the radiation streak captured within the NA of the collection optic, and η_rad is the fraction of light radiated (versus absorbed) by the waveguide claddings. Here, η_in is estimated to be on the order of 30%, since the input light is divided between ∼3 vertical mode orders (i.e. 3 radiation streaks [9]). Furthermore, η_prop is estimated to be ∼ 30–40%, mainly limited by the fact that half the light is radiated through the bottom mirror in the nominally symmetric tapered waveguides. Finally, we estimate η_rad ∼ 5–10%, due to the relatively high absorption loss by the a-Si layers in the ∼950 nm wavelength range. This could be greatly improved through the use of a-Si:H layers, or by operating at longer wavelengths [11]. Nevertheless, the overall throughput η_T ∼ 1% already compares favorably to the reported efficiency for related devices from the literature [4,7].

5. Discussion and conclusions

Our proposed system leverages some of the same principles as recently reported speckle-based devices [3–8], but with added functionality provided by the dispersion of the mode cutoff positions. It also bears some conceptual similarities to the so-called ‘SWIFTS’ devices [18,19], which extract spectral information from standing-wave patterns produced by counter-propagating waves in a high-index single-mode waveguide. However, those devices require patterning of nano-scale scattering elements along a relatively long waveguide, in order to enable sampling of a sub-micron-scale standing wave by an adjacent image sensor. Conversely, the standing wave patterns in our tapers arise from ‘slow light’ modes near cutoff [12], producing spatial features on the ∼10 μm scale, easily resolved by standard optics. Furthermore, the main part of the radiation streak is typically much less than 1 mm in length, and the radiated light is preferentially emitted in an out-of-plane direction, favorable for efficient detection by an adjacent image sensor.

There are numerous avenues for improving the performance (resolution, stability, repeatability, etc.) and reducing the size of the system, including through optimization of the tapered waveguides to increase throughput and dispersion. Notably, the ‘optical part’ at the heart of the system is a section of a tapered waveguide with a footprint on the order of ∼50 μm × ∼ 200 μm. The overall size of the prototype is thus determined mainly by the optics (objective lens) used to magnify the radiation streak onto the image sensor. With improvements in the waveguide and image sensor chips (e.g. smaller pixels), it might eventually be feasible to couple the tapered waveguide chip rigidly and directly to the image sensor chip [20]. It is also worth noting that image sensors are available with significantly better specifications compared to the low-cost camera used here. By using a camera with lower-noise and higher efficiency, further improvements in sensitivity and resolution are anticipated. We hope to explore some of these options in future work.

In conclusion, we have described a proof-of-principle wavemeter system based on the dispersive (i.e. moving), multimode-interference-based ‘speckle’ pattern produced by radiation from a tapered, hollow waveguide clad by omnidirectional Bragg mirrors. We demonstrated the capability for < 10 pm resolution over a 70 nm operating range, both limited by the instrumentation available for system calibration.