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Abstract
Multimode interference patterns are strongly dependent on spectral components and can be used as fingerprints to reconstruct a spectrum with random amplitudes. Motivated by this concept, we designed and realized a high-performance compact spectrometer based on a tapered spiral-shaped waveguide with a detector array integrated directly on top. The device relies on imaging the multimode interference from leaky modes, resulting in a resolution of 20 pm in the visible range and a bandwidth from 545 to 725 nm with a 250 µm radius structure. Spectra of multiple narrow lines and synthesized broadband are well reconstructed. The ability to achieve such high resolution and broad bandwidth in a compact footprint is expected to have a significant role in low-cost and multifunctional integrated systems.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the past few decades, optical spectrometers have been widely used in fields of biochemistry, optics and material science [1–4]. Traditional spectrometers which based on the configuration of spectral-to-spatial mapping disperse different spectral components of a probe signal onto spatially calibrated detectors. Then, the corresponding distribution of amplitudes as a function of wavelength could be achieved as the target spectrum. The commercial spectrometers possess moderate spectral resolution and bandwidth in both visible and infrared region, but still suffer from their bulky size and high cost. In the pursuit of compact and high-performance spectrometers, various kinds of spectrometers were realized based on photonic crystal cavities [5,6], arrayed waveguide grating [7,8] and ring-grating [9]. According to their demonstrations, the resolution is strongly deteriorated by fabrication imperfection and the bandwidth increases proportionally with devices’ footprint.
In addition to these carefully designed systems, devices based on disordered scattering structures [10] and multimode waveguides [11–13] have also been revolutionized for spectroscopy applications. Unlike traditional spectrometers [14,15], these spectrometers record a series of unique and wavelength-dependent speckle patterns and form a calibration matrix to reconstruct the probe signal. Redding et al. [10] used compact disordered patterns on a photonic chip to achieve a resolution of 0.75 nm, but it has a limited bandwidth from 1500 to 1525 nm. To further improve the resolution, we put forward a tapered fiber multimode interference (TFMMI) spectrometer to achieve a spectral resolution of 40 pm and operation bandwidth from 400 to 2400 nm [13]. As the TFMMI spectrometer has a footprint of 1.5 mm × π × 40 µm2, it still needs complicated packaging and coupling with on-chip photonic components.
Although the figures of merit of the recently developed devices are competitive to the commercial grating spectrometers, but a compact spectrometer with high-resolution, broadband and low-cost remains elusive. Here, we introduce a tapered spiral-shaped waveguide (TSSW) spectrometer that coiled into a spiral shape could largely decrease its footprint but maintains high-performance. Using the leaky multimode interference (MMI) patterns as spatial channels to provide uniquely wavelength-dependent fingerprints, we can perfectly reconstruct any probe signals. Experimentally, this TSSW spectrometer has a spectral resolution of 20 pm in visible region and an operation bandwidth from 545 to 725 nm at least with a radius of 250 µm. For better comparison, the figures of merit of a spectrometer can also be quantified by the resolving power R = ω0/δω (where δω is the resolution at frequency ω0, corresponding to the resolution) and the fractional bandwidth B = 2(ωmax -ωmin)/(ωmax -ωmin) (where ωmax and ωmin are the maximum and minimal values of the operational frequency range, corresponding to bandwidth). Figure 1 compares the R and B of several typical compact spectrometers with our TSSW spectrometer (R = 31750, B = 0.28) where the top-right device possesses the best performance. Although the TFMMI spectrometer is better than our device, it has a much larger footprint as mentioned above.
Fig. 1. Comparison of various compact spectrometers. Two important metrics of R and B are used to compare the spectrometers, and the arrow line from bottom-left to top-right indicates the performance of the spectrometers becomes better.
2. Theory
When a monochromatic light with frequency ω incidents into a non-adiabatic multimode waveguide, a number of spatial modes are excited with different propagation constants. The electric field E of each mode in this waveguide could be written as the summation of all the modes:
To verify that the MMI patterns of the TSSW provide a possibility to the usage of a spectrometer, we performed numerical simulations of the waveguide by the finite-difference frequency-domain (FDFD) method. In the simulation, the waveguide width is w = 10 µm, the height is h = 1.5 µm and the effective refractive index of the core (s1813)is n1 = 1.71atλ = 636 nm. The s1813 waveguide is fabricated on a CYTOP layer (n2 = 1.30) with a thickness of t = 3 µm, and it supports ∼300 spatial modes in the region of interests (ROI). The plots of propagation constants β at three wavelengths of λ = 535, 635 and 735 nm are shown in Fig. 2(a) where each set of β are linearly fitted. According to the dispersion plot, β of the first 50 non-degenerated spatial modes fall between 1.60×107 and 1.68×107 m−1. Noting that β is a function of wavelength and mode order, the phase difference (= Δβ·z) between the highest and the fundamental modes accumulate as light propagates along the waveguide, causing the MMI pattern strongly dependent on wavelength. To characterize its transmission bandwidth that could be used to estimate the TSSW spectrometer’s operation bandwidth, the β curve of the fundamental modes across the wavelength range from 545 to 725 nm is calculated in Fig. 2(b) which indicates the TSSW spectrometer having a relatively broad bandwidth than most of its counterparts [2,9,14], as shown in Fig. 1.
Fig. 2. (a) The plots of propagation constant β at wavelengths of 535, 635 and 735 nm, which shows a linear increment for each set of β, (b) the β curve of fundamental modes from 545 to 725 nm shows a broad transmission bandwidth of the s1813 waveguide, inset: the intensity distribution |I(R, L, ω)| of the fundamental mode at λ = 636 nm.
3. Fabrication and measurement
The structure of TSSW consists of a s1813 waveguide deposited on a CYTOP membrane which provides a perfect light confinement with air, and the fabrication process is low-cost and time-saving compared to most photonic devices. In Fig. 3, the fabrication procedures are illustrated in six steps. Firstly, the silicon wafer is sequentially cleaned in acetone, Isopropyl Alcohol and Piranha solutions. Secondly, the CYTOP material is spin-coated on the silicon wafer with a speed of 3000 rpm for 1 minute and dried in air to form a 3 µm thick CYTOP membrane. Thirdly, the s1813 photoresist (McGill Nanotools Company) is spin-coated on the CYTOP membrane with a speed of 3500 rpm for 1.5 minutes and baked on a 90 degree hot plate to form a 1.5 µm thick s1813 membrane. Fourthly, a diamond-knife is used to vertically cut the input section of the TSSW to achieve a smooth facet for light coupling. Fifthly, the UV light is used to expose the target pattern on s1813 membrane with 10 mW power and 30% dosage. Finally, the whole chip is soaked in CD26 solution for 45 seconds to develop the exposed part. The fabrication of the TSSW takes only ∼ 15 minutes and we can obtain 16 duplicates at a time.
The scanning electron microscope (SEM) image of the fabricated TSSW is shown in Fig. 4(a) and body defects and sidewall roughness resulting from fabrication imperfection can be seen from the magnified image. The diagrammatic sketch (shown as an optical image) in Fig. 4(b) indicates that integrated with a detector array (marked region) on top, our TSSW spectrometer could be realized. In the experiment, we use a commercial camera instead. One may also noticed that there is a steeply tapered region after a short narrow input section, which is particularly designed for two reasons: (1) the narrow input section (w = 2.5 µm) has a better mode matching condition and a higher coupling efficiency could be obtained, (2) the steep region provides the TSSW with a more gradually tapered angle from w = 10 to 4 µm, causing the uniform leakage of MMI pattern. We coupled light into the TSSW using a lensed fiber (SM630, OZ Optics) whose operation bandwidth covers the bandwidth of the device. The usage of a polarization maintaining fiber (PMF) and a polarization controller (PC) guarantees stable polarization states for calibration. We characterized the device in a ×17 microscope set-up and a series of MMI images were acquired by a charge-coupled device (CCD) camera (Photometrics CoolSNAP K4, integration time: 50 ms) while scanning the tunable line width laser. The beatings of MMI can be estimated by Δz = 2π/(Δβ) ≈ 180 1 m. For the magnification used here, this period is imaged onto ∼413 pixels on our camera with a pixel size of 7.4 µm and spatially resolved. This device is designed to image the scattered light above, which can capture the transmitting light throughout the whole waveguide. As the MMI pattern largely relies on the coupling light from the fiber to TSSW, the excited modes have unpredicted and variable amplitudes. In Fig. 4(d), a period of the intensity distribution is extracted from the MMI pattern, Eq. (1) and nonlinear optimization algorithm are conducted with the first 50 modes. An agreement of intensity distribution is achieved between the measured data and theoretical model, verifying the feasibility of our model. The deviation mainly results from the approximate model and fabrication imperfections of the TSSW.
Fig. 4. (a) The scanning electron microscope (SEM) image of the fabricated TSSW, inset: the ×10 magnified image, (b) the diagrammatic sketch of the TSSW, the red-marked region indicates the integrated position of the detector array, (c) the setup for TSSW measurement, SMF: single mode fiber, MMF: multimode fiber, and PMF: polarization maintaining fiber, (d) the intensity distribution is extracted from MMI pattern for the purpose of illustration, showing qualitative agreement between experiment and model.
4. Spectrometer operation
In order to apply this TSSW as a spectrometer, we first calibrated its calibration matrix by acquiring the MMI patterns of input wavelength which form the column vectors or basis states. We scanned a narrow linewidth (< 10 MHz) laser from 634.80 to 639.40 nm in steps of 10 pm and the resulting calibration matrix is shown in the left figure of Fig. 5(a), and two regions are selected from it (marked in white) for clear observation. Although Eq. (1) predicts that different wavelengths produce distinguishable images in the multimode waveguide, the finite pixel size of the camera, the limited 5.3 mm long TSSW and the number of imaged modes could result in the images (basis states) not fully orthogonal. To verify that the basis states are distinguishable, we evaluated the cross-correlation by computing every permutation of the inner product between two different wavelength states ui and uj. For the ideal case, ui·uj = 1 (i = j) and ui·uj = 0 (i ≠ j) after normalization. In practice, there are experimental noise and crosstalk between these states due to the finite length of TSSW, but as seen from Fig. 5(b), the inner products ui·uj monotonically decrease to an isolation level of ∼ 10−2 as the wavelength separation |λi - λj| increases.
Fig. 5. The basis states of wavelength corresponding to MMI patterns describe a spectral-to-spatial mapping. (a) Left: the calibration matrix shows the intensity distributions of a wavelength range from 634.80 to 639.40 nm in steps of 10 pm of the TSSW spectrometer, right: two magnified regions for clear observation, (b) the monotonical decrease of inner products of the basis states in (a) indicating decreasing overlap of the spatial distribution, (c) the inner products between wavelengths from 545.0 to 725.0 nm decay, showing that no two distinguishable wavelengths have the same associated fingerprint.
In principle, the TSSW spectrometer could support a relatively broad wavelength range at least from 500 to 800 nm due to the broad transparency window of s1813 waveguide as illustrated in Fig. 1(b). However, we did not have the proper instruments to test its maximum bandwidth. We used a monochromator (Acton SP2500) and a super-continuum source (NKT SuperK Extreme) to produce a 1 nm wide probe signal to extend our previous analysis from 545.0 to 725.0 nm as shown in Fig. 5(c), indicating a nearly orthonormal basis set at these wavelengths. The optimization of signal-to-noise ratio (SNR) of the inner products helps to obtain the uncorrelated spatial positions for further demonstration. To reconstruct an arbitrary probe signal, its spectrum s is related by ψ = Λ·s to the detected MMI pattern ψ, where Λ is the calibration matrix. Matrix left-inversion (using Moore-Penrose pseudo-inversion method) gives a least-square solution s =Λ−1·ψ, which can be applied in situations even when ψ isn’t fully belong to the span{ui} where ui corresponds to the calibrated wavelengths, that is, the spectral reconstruction works even for a continuous input spectrum. To overcome the perturbations that exist in singular values of the pseudo-inverse matrix, a truncated singular value decomposition (SVD) of Λ was used to obtain the initial solution s0 which is applied to the general minimization problem ||Λs - ψ||2, where || · ||2 denotes the 2-norm [16,17]. By eliminating the non-physical negative solutions and solving the minimization problem with nonlinear optimization algorithm, we can reconstruct the spectra of a measured signal more accurately.
Due to the wavelength-dependent characteristics of this TSSW device, a shift of wavelength δλ is sufficient to produce uncorrelated patterns. To quantify this spectral sensitivity, we define the spectral correlation function of the intensity distribution, C(Δλ, x) = <I(λ, x)I(λ + Δλ, x)>/[ < I(λ, x)><I(λ + Δλ, x)>] - 1, where I(λ, x) denotes the intensity at a position x for input wavelength λ, and <···> represents the average over λ. Then, we averaged C(Δλ, x) over each position x across the patterns to obtain the spectral correlation. The half-width at half-maximum of 30 pm indicates two patterns are sufficiently distinguishable at δλ = 30 pm, which also provides an estimate for the spectral resolution. As the resolution depends on the image’s SNR and reconstruction algorithm, δλ should be larger than the actual resolution. The spectral resolution is the capability of resolving two closely spaced wavelengths, and as the MMI patterns between two different wavelengths do not interfere with each other, the spectral resolution can be calculated by adding them together and conducting the nonlinear reconstruction algorithm. In order to synthesize such a test probe signal, we separately recorded the intensity patterns at two closely spaced wavelengths. The detuning between the pair of wavelengths was gradually increased to find the wavelength separation δλ at which one could resolve them. In Fig. 6(a), the TSSW spectrometer hardly resolves two wavelengths when δλ = 10 pm while clearly resolves them when δλ = 20pm. A further increase of resolution may be realized by increasing the length of the TSSW. Considering the bandwidth we used in the calibration matrix, there are 230 independent spectral channels that work in spectral reconstruction. One could choose 20 pm (230 channels) or an even larger (less) wavelength separation for reconstruction which relies on the practical needs. However, for a given footprint, this will decrease the width which in turn affects its broad bandwidth. Meanwhile, we also investigated the linearity of the spectral measurement by setting the probe signals at various power levels of 0.1, 0.5 and 1 W. As a result, the integration of photon counts increases proportionally and the reconstructed spectra overlap after normalization. On the basis of the spectroscopy scheme, we used our spectrometer to successfully reconstruct multiple narrow lines (“sparse” spectrum) as shown in Fig. 6(b) and multiple broader synthesized Gaussian spectra (“dense” spectrum) in Fig. 6(c). One can conclude that the device can preciously resolve the spectrum with few spectral components, the spectral contrast is high and the optimization algorithm quickly finds the correct combination of spectral channels. For broadband inputs, due to the summation of many uncorrelated MMI patterns, the contrast of interference fringes diminishes, resulting in a lower SNR and worse spectral reconstruction [12]. This reconstruction operation indicates the capability of the device to detect any arbitrary spectrum within its operation bandwidth, as shown in Fig. 7 which shows the same detecting result between the commercial and TSSW spectrometers. In principle, the more basis states we use in the calibration matrix, the more precise we obtain the reconstructed spectrum. Meanwhile, more memory and time are consumed. Here, we used the coarse calibration matrix from 635 to 655 nm in steps of 2 nm to reconstruct the probe spectrum less than 0.1 s. Limited by the instruments in the lab, we only reconstructed the spectra within the calibration range (634.80 to 639.40 nm) and didn’t show the entire reconstructed spectra.
Fig. 6. The performance of the TSSW spectrometer. (a) We verified the spectral resolution by resolving two closely spaced wavelengths using our constructed spectrometer: sharply peaked input wavelengths of 636.12 and 636.14 nm are clearly resolved, while input wavelengths of 636.02 and 636.03 nm are not fully resolved, indicating the resolution to 20 pm at 636 nm. (b) Multiple narrow laser lines and (c) multiple broadband synthesized Gaussian spectra are well reconstructed using the basis of our spectroscopy scheme.
Fig. 7. The reconstructed broadband spectra from an unknown source are detected by the commercial and TSSW spectrometers.
5. Discussion and conclusion
To investigate the operation of the TSSW spectrometer, we use a lensed fiber mounted on a three-axis stage to couple light into the waveguide, which inevitably suffers from the coupling misalignment after a period time of measurement. This can be improved by gluing or fixing the lensed fiber on the three-axis stage. And in practical applications, the TSSW spectrometer can be either integrated with on-chip components or the coupling and imaging conditions are automatically optimized. Ignoring the misalignment, we acquired three calibration matrices at a time period of 10 min, 16 h and 48 h, and obtained an average correlation value of 0.998, 0.983 and 0.959, which indicates the long-term stability of the device. The degradation in correlation was due to the drifts of the three-axis stage and microscope set-up, and when repositioned, the correlation increased back to > 0.980. The device also showed robust against mechanical vibration and polarization shift over the tested time of several months. Another possible constraint is that this device requires the input light to be coupled through a single mode fiber, which also needs consideration in future. In addition, the temperature sensitivity depends on the resolution: the higher its resolution, the more sensitive it is to the temperature shift. The change in temperature can alter the refractive index of s1813. From the thermo-optic coefficient of dn/dT = ∼ −3.4×10−4 and Eq. (1), a small temperature variation ΔT causes a shift of wavelength in the calibration matrix and therefore deteriorates the accuracy but not the precision of the reconstructed spectrum. In particular, this TSSW device would be insusceptible to temperature fluctuations if ΔT < Δn/(dn/dT) ∼ 0.37 K where Δn causes a 50% decorrelation of the spectral correlation. Alternatively, this deterioration can be suppressed by calibration correction or active thermal stabilization.
Another important quality of a spectrometer is photon detection efficiency which can be defined as the detected photon on camera per incident photon into the TSSW. The TSSW used in our experiments has a signal transmission (refers to the power or intensity ratio of the output port and input port of the device) of 70 ∼ 80%, which also encounters losses of fiber coupling and fiber connection of 2.46 dB, and linear attenuation in waveguide of 2 dB. And by assuming an ideal quantum efficiency of the camera η1 = 0.9 and the collecting efficiency of the set-up η2 = 0.04 (NA = 0.4), the device has a detection efficiency (the ratio of imaging integral intensity and input integral intensity)of ∼ 0.49%. Various methods can be used to further enhance its photon detection, for example, optimizing the structural parameters, depositing a reflective layer beneath the TSSW and introducing nanoscatters.
In conclusion, a TSSW spectrometer consisting of a carefully designed TSSW and a detector array integrated on top, can provide both high resolving power (R = 31750) and broad bandwidth (B = 0.28) with a 250 µm radius. Benefited from the strongly wavelength-dependent MMI patterns and the nonlinear reconstruction algorithm, spectra with both of the sparse and dense details are perfectly reconstructed with a single-shot measurement. Once the TSSW spectrometer is integrated with multiple functional devices, they are expected to have a significant role on high-performance, low-cost and portable systems.
Funding
National Natural Science Foundation of China (61702347); Excellent Young Scientist Foundation of Hebei Province (F2018210100).
Disclosures
The authors declared that they have no conflicts of interest to this work. We declare that all sources of funding for the research reported, all relevant commercial relationships and non-financial, professional and personal relationships could reasonably be perceived as a conflict of interest in connection with the work submitted.
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