High-resolution optical spectroscopy using multimode interference in a compact tapered fibre

Optical spectroscopy is a fundamental tool in numerous areas of science and technology. Much effort has focused on miniaturizing spectrometers, but thus far at the cost of spectral resolution and broad operating range. Here we describe a compact spectrometer that achieves both high spectral resolution and broad bandwidth. The device relies on imaging multimode interference from leaky modes along a multimode tapered optical fibre, resulting in spectrally distinguishable spatial patterns over a wide range of wavelengths from 500 to 1,600 nm. This tapered fibre multimode interference spectrometer achieves a spectral resolution down to 40 pm in the visible spectrum and 10 pm in the near-infrared spectrum (corresponding to resolving powers of 104–105). Multimode interference spectroscopy is suitable in a variety of device geometries, including planar waveguides in a broad range of transparent materials. While desirable for compact solutions, the miniaturization of spectrometers comes at the cost of spectral resolution and operating range. Here, Wanet al. propose a tapered fibre multimode interference spectrometer exhibiting high spectral resolution from the visible to the near infrared in a compact configuration.

where do is the spectral resolution at frequency o 0 . However, high-resolution grating spectrometers are inevitably bulky because their spectral resolution scales inversely with optical path length. Fourier transform spectrometers can be more compact in size, but their scanning interferometric configuration makes them slow for large bandwidth applications.
In recent years, there has been a strong effort to develop novel spectrometers that are compact and have high resolving powers and operational bandwidths 5,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] . In addition to miniature grating spectrometers 12 , resonant structures such as nanocavities [13][14][15][16][17][18] have also been employed to separate spectral components into unique detectors. This research has produced millimetre-scale spectrometers with high resolving powers of R410 4 , but with limited fractional bandwidth, ðo max þ o min Þ=2 % 10 À 2 . Another class of compact spectrometers relies on imaging the speckle patterns of a photonic bandgap fibre bundle 19 or a multimode optical fibre [20][21][22] . However, these spectrometers are centimetres to metres long, which can result in the instability of the interference pattern due to environmental fluctuations, hence limiting their long-term operation. Recently, a silicon-based spectrometer 23 that measures the multimode transmission profile from a disordered photonic crystal structure achieved a resolving power comparable to typical grating spectrometers, but it requires coupling to a subwavelength input waveguide and has a limited bandwidth from 1,500 to 1,525 nm. A high-resolution Fourier transform spectrometer 24 has also been demonstrated on-chip, but its narrow free spectral range limits its operation bandwidth to 0.75 at 1,550 nm.
Although some of these recently developed compact spectrometers can already match the spectral resolution of grating spectrometers, a compact device with both high resolving power and broad operating bandwidth remains elusive. Here we introduce a compact spectrometer based on multimode interference (MMI) that simultaneously achieves high resolving power of R410 4 (R410 5 ) in the visible (near-infrared) spectrum, and bandwidth BB1 in a millimetre-scale device.

Results
Using MMI for spectroscopy. Figure 1 compares the resolving powers and fractional bandwidth of several bench-top and compact spectrometers, including the device introduced in this work. Supplementary Table 1 provides more details about these spectrometers. The device is based on a tapered multimode fibre, as illustrated in Fig. 2a, produced by tapering a multimode optical fibre (Thorlabs AFS105/125y) using the flame-brush technique with taper lengths B5 mm. A non-adiabatic taper angle causes light at frequency o to couple to a large number of spatial modes with different propagation constants, resulting in a unique interference profile that can be imaged as a 'fingerprint', or basis state, of a given frequency o. Specifically, in the tapered optical fibre, the electric field varies in the propagation direction z as where E l,m are the spatial profiles of the Bessel modes l and m in the fibre, f l,m are the initial phase offsets, and b l,m (z) are the phase propagation constants, plotted in Fig. 2b. The tapering of the fibre leads to a non-uniform modal dispersion, which modifies the interference along the taper length, as reflected in the z-dependent propagation constant b l,m (z) and E l,m (z), while also changing the mode confinement, leading to radiative losses. Imaging of the far-field interference pattern I(x,y,z,o) ¼ |E(x,y,z,o,t)| 2 and subsequent basis mapping enables resolving power in excess of R ¼ 10 5 and broad bandwidth BB1. This tapered fibre MMI (TFMMI) imaging approach enables unique performance compared with other spectroscopy tools (Fig. 1).
Characterization and spectroscopy scheme. We coupled light into the TFMMI using a butt-coupled single-mode fibre with infibre polarizer (OZ Optics). This coupling was long-term stable and robust against coupling misalignment, mechanical vibrations and polarization drifts, as will be described in the section below. We then characterized the device in a Â17 microscope set-up (Fig. 2a). For calibration, we acquired a series of images of 1.5 mm of the tapered region with a waist ranging from 30 to 10 mm in diameter while scanning a narrow linewidth (o10 MHz) laser from 634.800 to 639.400 nm in steps of 2 pm and from 1,500.000 to 1,580.000 nm in steps of 1 pm. Figure 2d shows examples of these calibration images at wavelengths of (638, 595, 532, 450) nm in the visible spectrum and (1,550, 1,530) nm in near-infrared. The images associated with each wavelength formed the column vectors in our calibration matrix that converts between spatial patterns and input frequencies.   Various metrics can be used to compare compact spectrometers, two of which-the fractional bandwidth, B and resolving power, R-are compared in this figure. For the purpose of comparison, we also include five commercial bench-top spectrometers (green): optical spectrum analysers (OSA), a stimulated Brillouin scattering spectrometer, an Echelle spectrometer and a single-grating spectrometer (note that multi-grating spectrometers can have wider operating bandwidth). Shown in blue are some of the recently developed compact spectrometers. The TFMMI spectrometer (red) that is introduced in this work achieves a resolving power of R410 4 (R410 5 ) in the visible (near-infrared) spectrum, and bandwidth BB1. AWG, arrayed waveguide grating, FT, Fourier transform spectrometer. Figure 3a,b shows the resulting calibration matrices for visible and near-infrared wavelengths, respectively. Equation (1) predicts that different colours produce distinguishable camera images in the limit of a very long tapered fibre. The finite pixel size of the camera and fibre length, and the number of imaged modes could result in basis images that may not be fully orthogonal. To verify that the images (column vectors) are distinguishable, we evaluated their cross-correlation by computing every permutation of the inner product between two different wavelengths vector u i and u j : for an ideal TFMMI of long length, u i Á u i ¼ 1 and u i Á u j ¼ 0 for all iaj after normalization. In practice, there is background noise and crosstalk between these states due to the finite length of the fibre, but as seen from Fig. 3c,d, the inner products u i Á u i monotonically decrease to an isolation level of B10 À 1 as the wavelength separation u j À u i increases. In principle, the TFMMI spectrometer supports a broad wavelength range from 400 nm up to 2,400 nm due to the broad transparency window of silica fibre 28 . However, we did not have the instruments to test this full range. Instead, we used a monochromator (Acton SP2500) and a supercontinuum source (NKT SuperK Extreme) to produce a 1-nm wide probe signal to extend our previous analysis to 500 nmolo800 nm and 1,100 nmolo1,600 nm. Figure 3e indicates a nearly orthonormal basis set at these wavelengths. To resolve an arbitrary input, the signal's spectrum s is related by W ¼ Ls to the detected MMI patterns W, where L is the calibration matrix. Matrix left-inversion gives a least-square solution s ¼ L À 1 W, which applies in situations even when W = 2 spanfu i g i where u i are the calibrated frequencies, that is, the spectral reconstruction works even for a continuous input spectrum. In addition, this solution can be treated as an initial solution s 0 to the more general minimization problem kLs À Wk 2 where || Á || 2 denotes the 2-norm [19][20][21][22][23]29 . By eliminating the non-physical negative solutions and then solving this minimization problem using a nonlinear optimization algorithm, we can reconstruct the spectra of a measured signal more accurately (see Supplementary Methods for further discussion).
Experimental results. On the basis of the spectroscopy scheme discussed in the previous section, we used our TFMMI spectrometer to reconstruct the spectra of multiple probe beams. Figure 4a shows the successful reconstruction of four laser lines (red dashed line) using our constructed spectrometer (coloured solid lines).
We can estimate the spectral resolution of the device from a correlation function [20][21][22][23] m) is the intensity of pixel m for image vector of wavelength l. The average correlation over all pixels was normalized as shown in Fig. 4b. The half-width at half-maximum of dlE0.06 nm approximates the spectral resolution, that is, wavelength vectors that are approximately separated by 0.06 nm at lE637 nm are sufficiently distinguishable.
We directly determined the spectral resolution of the TFMMI spectrometer by measuring the intensity profiles of two nearby input wavelengths at various power levels. Images were acquired sequentially and summed to produce the addition of two frequency states. Over tens-of-milliseconds integration times of these experiments, there would be no interference between these laser fields given that they are separated by more than 1 pm (0.7 GHz) at 637 nm. Therefore, adding sequential measurements at different frequencies is justified (though twice the readout noise is introduced compared with a single exposure). The detuning between the pair of wavelengths was gradually increased to find the wavelength separation at which the reconstruction algorithm (e) Imaging the fibre using a smartphone camera shows a potential for portable and field applications. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8762 ARTICLE could resolve them. This is the case at Dl ¼ 40 pm in the visible regime (Fig. 4c) and Dl ¼ 10 pm in the NIR regime (Fig. 4d), providing the upper bounds of the TFMMI resolution in these spectral ranges. Using the same device and calibration matrix, we reconstructed multiple narrow laser lines as shown in Fig. 4a and multiple broader synthesized Gaussian spectra (Fig. 4e).
In Fig. 4f-h, we demonstrate that the device can reconstruct continuous spectra in a single exposure. In Fig. 4f, we measured a 2-nm bandwidth (full-width at half-maximum) input signal generated by filtering a supercontinuum light source (NKT SuperK Extreme). The red dashed line shows a measurement on a commercial grating spectrometer 30 , which matches closely the reconstructed spectrum from the TFMMI device. In Fig. 4g,h, we extend the input bandwidth to 10 and 15 nm, respectively. For this measurement, we calibrated the fibre spectrometer using an acousto-optic tunable filter in steps of 2 nm (filter bandwidth of 2 nm). We find good agreement with the measured spectra at 10-nm bandwidth (Fig. 4h) and qualitative agreement at 15-nm bandwidth (Fig. 4g).
Long-term stability and repeatability. It is also important to evaluate the long-term stability of the TFMMI spectrometer, as interference-based devices can be sensitive to changing environmental conditions. To minimize the susceptibility of this spectrometer to mechanical vibrations, we securely mounted the tapered fibre in an acrylic fixture with a cover slip window. The calibration matrix obtained at a time interval of Dt ¼ (5 min, 12 h, 48 h) has an average correlation huðl; tÞ Á uðl; t þ DtÞi l of (0.9977, 0.9835, 0.9588), indicating the long-term stability of the calibration. The decrease in correlation with time was primarily due to drifts in the three-axis stages; when the device was repositioned in the focal plane of the microscope, the correlation increased back to 40.98. Furthermore, the devices did not show noticeable degradation in performance over the maximum tested time of 4 months. The multimode section of the device is also sensitive to the spatial profile and polarization of the input mode. The spectrometer operates on the principle that the relative amplitudes of the excited MMF modes remain unchanged throughout the experiment. The single-mode input fibre with in-fibre polarizer provides high stability and repeatability-after 10 cycles of disconnecting and reconnecting the input channel with a monochromatic laser, the correlation between images, u i ðlÞ Á u j ðlÞ, was 0.9955 ± 0.0034, indicating that the constructed device was insensitive to fibre alignment and input polarization. In addition, changes in temperature would alter the refractive index of the silica fibre. From the thermo-optic effect and equation (1), one can see that small temperature variation DT causes an offset o-o þ Do in the calibration and therefore affects the accuracy but not the precision of the reconstructed spectrum. In particular, the device would be insusceptible to temperature fluctuations if DTo(dn/dT R) À 1 B5°C where n is the refractive index and dn/dT is the thermo-optic coefficient. Recent research 21 on the transverse speckle patterns in multimode fibres additionally indicates that the B15-cm-long fibre used in our experiment would remain correlated up to DTB10°C. A shift of Do may be applied 22 to correct the calibration for temperature variations 4DT, and active thermal stabilization could also be implemented in applications where high spectral accuracy is needed in the presence of large ambient temperature changes.

Discussion
An important factor in a spectrometer is the photon detection efficiency. In our device, o1% of incident photons are detected because the leakage out of the tapered fibre is weak as 70-80% of input light remains guided in the fibre, and we only imaged 20-30% of the tapered region (Supplementary Methods). Various methods can be used to improve the collection efficiency. For example, optimizing the tapering parameters of the fibre or depositing nanoscatterers 31 could enhance the detection. A spiralled fibre geometry would increase the fraction of the fibre taper that is imaged on the camera. Imaging a longer section of fibre would also likely increase the spectral resolution.
We have shown that the TFMMI device operates across a large spectral range from 500 to 1,600 nm and can precisely resolve sparse spectra, that is, spectra with few frequency components. For very broadband inputs, for example, spanning many tens of terahertz, the contrast of the interference fringes diminishes, resulting in lower signal-to-noise and worse frequency reconstruction. Improving the spatial resolution of the microscope imaging would improve the interference visibility. In addition, dispersion engineering in the tapered fibre could be used to produce distinguishable wavelength patterns for targeted spectral signatures across a broad spectrum.
The fibre configuration suggests that the TFMMI spectrometer would be most suitable in fibre-coupled applications in which signals could be efficiently channelled to a single-mode input fibre. Although the device was not optimized for any specific figures of merit, the spectrometer's narrow spectral resolution of 40 pm (10 pm)-corresponding to resolving powers of 15,950 (155,000) at 635 nm (1,550 nm)-and broad operational range from 500 to 1,600 nm show unique advantages of the MMF-based spectroscopy approach. The millimetre-scale size of our device is already comparable to those of recent compact spectrometers (Supplementary Table 1) and could be reduced further by directly integrating the detector arrays with the tapered optical fibre. Furthermore, the TFMMI spectrometer concept can be realized in a variety of on-chip configurations and materials, including tapered waveguides for the ultraviolet and mid-IR spectral ranges. Optimized geometries, such as on-chip tapered waveguide spirals, will likely allow for improved photon collection efficiency. Such compact spectrometers based on MMI in tapered fibres or in onchip waveguides promise picometer-class spectroscopy in lab-ona-chip and portable applications.