1. Introduction

Spectrometers are a widely used, indispensable tool for the measurement of light transmission, reflection, scattering and emission across a broad range of disciplines, including material science, chemical and biological sciences, and astronomy. However, current state-of-the-art high-resolution spectrometry relies on bulky and expensive bench-top systems, making them inaccessible to many application settings. Recent developments in mass-scalable, silicon PICs have enabled the realization of compact and lower cost ‘lab-on-a-chip’ devices, operating over a wide optical wavelength range. By leveraging the small footprint of these PICs, various high-performance spectrometers integrated on the silicon platform [1–4] have now been demonstrated. The traditional approach has been to disperse the incident light using gratings into an array of channels or by using filters to measure the spectral power with an array of photodetectors. Since the resolution scales with the linear dimension of these types of grating-based spectrometers, the device footprint tends to be relatively large. Recent examples are an echelle grating-based spectrometer with a footprint of 3 mm $\times$ 3 mm and an arrayed waveguide grating (AWG)-type spectrometer with a footprint varying from $\sim$1 cm$^2$ to $\sim$1 mm$^2$ depending on refractive index contrast of the PIC platform [3,5–7]. Several devices with improved resolution and/or smaller footprint, based on ring resonator arrays and photonic crystal defect cavities have also been demonstrated, but these proven to be overly sensitive to even small fabrication errors [4].

In addition to these more standard device architectures, the incorporation of disordered structures in the spectrometer design have also been explored [8,9]. Redding et al. presented such a device in which a semi-circular structure with a disordered array of air-filled holes (also fabricated in SOI) acts as a diffuse component, surrounded by a full-bandgap photonic-crystal boundary to channel the light onto the detector. This device was able to achieve 0.75 nm resolution at a wavelength of 1500 nm with a 25 nm bandwidth [8]. Extending the operational wavelength range, a similar PIC-based device was demonstrated by Hartmann et al., this time in the silicon nitride (Si$_3$N$_4$) platform, with successful reconstruction of several signal probes [9]. Although these approaches can be useful in various spectroscopy applications, they have tended to require a complexity of components (such as photonic-crystal waveguide channels) and high-resolution fabrication based on electron beam lithography, (EBL). The use of EBL can increase the fabrication cost and complexity, potentially limiting mass-scale fabrication. The cost comparison between the two methods may vary and can depend on the specific application [10].

In a channel-waveguide-based photonics framework, speckle patterns due to multimode interference can be used to reduce the size and complexity, and enhance the bandwidth of operation. We previously reported the design of a disorder-based spectrometer by incorporating a randomized holes array within an MMI device [11]. These devices can be more tolerant to fabrication error, and can be implemented, along with integrated photodetector, in a more compact manner at potentially lower cost, thanks to the possibility of mass manufacturing via mature complementary metal-oxide-semiconductor (CMOS) compatible UV-lithography.

Our design is based on analysis of the spectral character of speckle patterns formed as light transits the MMI with integrated scattering hole array, which provides a unique fingerprint of the input (probe) signal. These wavelength-dependent speckle patterns are measured and stored in a transmission matrix describing the spectral-to-spatial mapping of the spectrometer. By engineering the input-to-output relation, a base of orthogonal elements can be generated, represented by the number of independent outputs of the MMI. These orthogonal base elements can be used to represent any signal in a unique way and can facilitate discrimination between different signals [12].

For the spectral-to-spatial mapping of the input-to-output signals, the speckle signal intensity $I(r)$ can be expressed as:

Following on from our original simulation work, here we experimentally demonstrate the operation of the compact spectrometer device built around this design. The spectrometer is based on a simple $1\times 16$ MMI splitter with an integrated scattering hole array element, fabricated in the SOI and operating in the near-infrared regime $\sim$ 1310 nm, key to several applications; methane sensing [14], optical coherence tomography (OCT) based imaging [15] and optical gyroscopes [16]. We first calibrated the transmission matrix $T$ over the full wavelength range of our swept source (1281 nm to 1348 nm) and used this to create a generic reconstruction algorithm based on a truncated inversion technique [13], capable of accurately reconstructing the spectra from any signal source with an operating wavelength in this band.

2. Devices and methods

2.1 Device design and fabrication

We designed the spectrometer around the commercial standard SOI platform, with a $220 \pm 20$ nm silicon layer atop a 2 $\mu$m SiO$_2$ buried oxide (BOX) on a silicon substrate [17]. The devices were fabricated via UV lithography at Southampton University under the CORNERSTONE program. A post-fabrication protective layer of SiO$_2$ was deposited above the device layer. Figure 1(b) shows the optical microscope images of our disordered MMI spectrometer, consisting of a single channel on the input side into which light is launched and 16 output channels from where the diffuse light is collected. Vertical grating couplers were included in the device design, based on an optimized coupling angle of between 9$^{\circ }$ and 11$^{\circ }$(off normal) for efficient coupling of light into the fundamental TE guided mode, at a design centre wavelength of 1310 nm with a 100 nm bandwidth. Such grating couplers are now a standard feature in PICs owing to the straightforward (alignment tolerant) fiber-to-chip delivery and collection of light for rapid, wafer scale testing.

 

Fig. 1. (a) Numerical simulation of TE polarized light at $\lambda$ = 1310 nm through the MMI structure calculated using the FDTD method (color bar is in normalized log scale). (b) Optical microscope images of the fabricated spectrometer along with its dimensions. The dispersive element is an array of randomly positioned SiO$_2$ holes. The probe signal is coupled to the random structure via a tapered waveguide at the right of the MMI. The light then diffuses through the random array via multiple scattering and eventually reaches the 16 tapered waveguides at the end of the rectangular MMI. These tapered waveguides will couple the signals to the detectors (not integrated). The distribution of intensities over the detectors is used to identify the input spectrum.

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The two dimensional (2D) scattering array in our device, consists of 150 SiO$_2$ filled holes, position-randomized using the Bridson algorithm to achieve a uniform blue noise distribution [18]. The holes have a design radius of 0.2 ${\mu }$m and are located in the silicon slab waveguide of the main MMI body. The overall MMI device geometry, position and number of holes and total length of the hole array was optimized through 2D-finite difference time domain (FDTD) simulations. The simulations were used to analyze the performance of the device in terms of power transmission, scattering loss, and speckle correlation. The optimized values, based on the simulation results, were embedded in the final design to enable us to achieve a similar operational bandwidth ($\sim$ 67 nm) to that of our swept laser.

To verify that the holes were fabricated according to our designs, an Axion M2M Zeiss microscope with a Zeiss EC epiplan-neofluar 100$\times$, 0.9NA objective lens, giving an overall magnification of $\times$1000, was used to capture an image of the hole array in the MMI. The resolving power of this microscope configuration is stated as 34 nm. Images from the optical microscope were converted to 8-Bit in ImageJ software, and the threshold for ‘binarizing’ the image was set by manual calibration of the measurement of several of the holes. All 150 holes were detected and their pixel areas measured, using ImageJ. Assuming approximate circularity of the holes, their radius was determined to be 184 nm with a measurement uncertainty (set by the microscope) of $\pm$ 30 nm, as shown in Fig. 2(a). 1$^{st}$ and 2$^{nd}$ nearest neighbour (NN) distances were calculated using the MATLAB function; regionprops (data, ‘centroid’), which returned ‘centre of mass’ of each hole, given by 8-connected components in the binary image. This yielded a 1$^{st}$ NN mean separation of 1490 nm (standard deviation 190 nm) and 2$^{nd}$ NN mean separation of 1750 nm (standard deviation 260 nm) as shown in Fig. 2(b).

 

Fig. 2. (a) Histogram of the distribution of the holes radius: A fitted Gaussian function reveals a mean radius of 184 nm with a standard deviation of 7 nm. (b) Histogram of the 1$^{st}$ and 2$^{nd}$ nearest neighbour (NN) distance: 1$^{st}$ NN mean 1490 nm (standard deviation 190 nm) 2$^{nd}$ NN mean 1750 nm (standard deviation 260 nm).

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2.2 Experimental Setup

The experimental setup for the charaterization of our spectrometer device is shown in Fig. 3. To initially calibrate the device, we used a TE polarized, 100 nm bandwidth swept laser source (Santec HSL-20) with a center wavelength of 1310 nm and optical power of 20 mW, and 100 kHz sweeping rate. The laser can act as a broadband light source, when a slow (>1 ms response time, compared to 1$\mu$s repetition time of the swept source laser) detector is used or when the power is averaged over > 1 ms time [19]. We used single mode fibers for coupling the light into and out of the PIC, and achieved precise fiber-to-chip alignment by mounting the cleaved ends of the fibers to $xyz$ translation stages (Thorlabs NanoMax313D). An InGaAs detector (Thorlabs S154C) was used in conjunction with a power meter (Thorlabs PM100D) to calibrate the optical alignment, by maximizing the power collected at each detector before taking spectral measurements. The spectral data was acquired using a single port of a balanced photodetector (Thorlabs PDB430C) and a high speed data acquisition (DAQ) card (AlazarTech ATS 9350) at 500 MS/s (Mega Samples per second) for each output channel.

 

Fig. 3. Schematic of the experimental setup.

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2.3 Characterization methodology

For the spectral characterization of the spectrometer, we first used the aforementioned swept laser source and recorded the wavelength-dependent intensity distribution for each of the 16 MMI output channels. Figure 4(b) shows the speckle pattern recorded from the device where it is evident that a shift in the input wavelength has a notable effect on the transmitted intensities. The spectral resolution of the spectrometer depends on the uncorrelated speckle pattern generated on the detectors. The output correlation study was done using the spectral correlation function of the intensity on the detector plane as:

 

Fig. 4. (a) The spectral correlation function of light intensities averaged over all detection channels of the spectrometer. The half-width at half-maximum is 3.2 nm, meaning a wavelength shift of 3.2 nm reduces the degree of spectral correlation to half. (b) The measured intensity distribution on the detection channels as a function of the input wavelength.

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This disordered speckle pattern is capable of converting each output signal into uncorrelated but deterministic signals at N different channels. Mathematically, this is similar to decomposing the incoming signal into N orthogonal spectra. The number of orthogonal base elements is given by the number of independent output channels responsible for the bandwidth of operation and the spectral resolution. For N detectors, and the operational bandwidth B in which the output channels remain uncorrelated, the spectral resolution $\delta \lambda$ is limited to $\delta \lambda$ =B/(2N) [8]. For our MMI based spectrometer with 16 independent detectors, we chose the operational bandwidth of 67 nm (from 1281 nm to 1348 nm). To test the limit of the spectrometer resolution, we specify the spectral channel spacing at 0.1 nm which is less than the estimated spectral resolution of 3.2 nm. However, actual resolution also depends on the reconstruction algorithm and the experimental noise of the measurements [8].

After discretizing the spectral channels with the spectral channel spacing of the input $S$, we stored the calibration in the transmission matrix, $T = IS^{-1}$ where, $I$ is a vector representing the wavelength-dependent intensities measured in 16 spatial channels of the output [13]. Each column in the $T$-matrix represents the intensities measured at the 16 spatial channels for a given spectral channel.

After calibrating the transmission matrix, we can reconstruct any arbitrary input (probe) spectrum by measuring the wavelength-dependent intensity distribution matrix $I$ on each of the detectors and multiplying by the inverse of the previously calibrated transmission matrix as $S = T^{-1}I$. The arbitrary probe signal, however, must lie in the experimental range of the input signals used for calibrating the device since the bandwidth and the resolution of the device depends on it. For obtaining $T^{-1}$, we initially used the singular value decomposition technique to decompose the transmission matrix as $T = UDV^{T}$ (where, $U$ is a $M \times M$ unitary matrix, D is a $M \times N$ diagonal matrix and V is a $N \times N$ unitary matrix) and take the reciprocal of each diagonal element of $D$ and then transpose it to obtain a diagonal matrix $D^{\prime }$. The inverse of $T$ is then given as $T^{-1} = VD^{\prime }U^{T}$ [13]. Although in practice, we found this type of matrix inversion process to be overly error prone due to a rather high level of experimental noise. Thus, to improve the accuracy of the spectral reconstruction algorithm, we used the truncated inversion matrix method [13]. In this way, we only take the reciprocal of the elements of $D$ above a threshold value and set the remaining elements to zero. The truncated inverse matrix of $T$ then becomes $T^{-1}_{trunc} = VD^{\prime }_{trunc}U^{T}$, which is then used to reconstruct the input spectrum as $S=T^{-1}_{trunc}I$.

To verify the spectrometer performance, we used one device to generate the $T$-matrix, and another (replica) device (on a separate chip) to perform measurements at the output for source reconstruction. To compare the reconstructed spectra with that of true source, we measured the spectrum using a 4 mm reference waveguide terminated with identical grating couplers.

In addition to the reconstruction of the experimental broadband spectrum, we also numerically generated single and multiple Lorentzian probe signals of varying spectral bandwidths and varying peak separations, and calculated the speckle patterns of these using the $T$-matrix in order to test reconstruction capabilities and calculate the spectrometer resolution limit.

3. Results

Figure 5 shows the reconstruction of the continuous broadband spectrum in comparison with the measured reference waveguide spectrum in normalized units (necessary to account for the different losses associated with the reference waveguide and the 16-port MMI). The swept source laser has an average output power of 13.32 dBm and the average power measured at 16 output ports is −33.45 dBm, thus, giving an overall loss for the device of around −46.76 dB. The comparison between the reference and reconstructed spectra shows a close agreement over the full 67 nm bandwidth with a normalized Root Mean Square Error (RMSE) of approximately $\sim$ 0.08. The fabrication tolerance provided by the foundry is quoted as 0.2 $\mu$m. We observed that the average RMSE in the reconstruction of a broadband spectra for three different chips was 0.1, 0.08 and 0.13. The coefficient of variation of the RMSE calculated for broadband source reconstruction measured for three identical chips was $\sim$ 0.2 RMSE units. A direct correlation between the fabrication tolerance and the RMSE was not established in our study.

 

Fig. 5. Reconstructed spectrum for a continuous, broadband probe signal.

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We also tested the algorithm with numerically generated Lorentzian signals which simulate the spectral shape of our swept laser source, using the $T$-matrix generated from the physical device. We used Lorentzian probes of various full-width at half-maxima (FWHM) to determine the bandwidth limit of the device and varied the peak separation of the Lorentzian doublets to determine the resolution limit using Rayleigh-like criteria (the maxima of one peak coinciding with the minima of a nearby peak).

Figure 6(a) shows the spectrum at the bandwidth limit with a FWHM $\approx$ 1 nm, for a single Lorentzian peak centered at a wavelength 1312 nm, with an RMSE of 0.069. Figure 6(b) shows the reconstruction of a resolution limited Lorentzian doublet signal with FWHM of $\sim$ 1 nm (for each of the underlying Lorentzian signals). In this case, with a peak separation, $\Delta \lambda$ = 3 nm, the RMSE is 0.152. We also reconstructed the spectrum for multiple narrow spectral lines with varying amplitudes shown in Fig. 6(c).

 

Fig. 6. (a) Reconstructed Lorentzian peak probe of FWHM at 1 nm. (b) Reconstructed Lorentzian peak probes separated by $\Delta \lambda$ = 3 nm.(c)Reconstructed spectrum for multiple narrow spectral lines with varying amplitude. (d) Reconstructed spectra (dashed lines) with FWHM at 0.25 nm (orange), 2.5 nm (red), and 10 nm (brown).

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We note that while the bandwidth limit for the single Lorentzian signal was found to be $\sim$ 1 nm, the RMSE increases significantly with the inverse bandwidth. This is illustrated, for three different Lorentzian signals in Fig. 6(d), with FWHM at 0.25 nm, 2.5 nm and 10 nm, all centered on 1312 nm.

4. Discussion and conclusions

In this work we have shown that a disordered $1\times 16$ MMI based photonic element can be employed as a compact, chip-based spectrometer. Combining MMI with enhanced optical path length associated with an array of spatially randomized SiO$_2$-filled scattering holes in the transmission path of the device, enables nanometer scale resolution in an extremely small device footprint. Table 1 shows the comparison between the current device and two half-circle design approaches [8, 9]. With this device, we were able to demonstrate spectral reconstruction of a broadband source with RMSE of $\sim$ 0.079 by calibrated $T$-matrix and using a truncated pseudo-inversion algorithm. It is observed that there are spikes at the edges of the spectrum in Fig. 5. This is due to the low signal-to-noise ratio present at those wavelengths due to weaker optical power at the edges. The performance of the device was consistent when the temperature in the laboratory changed from 20°C - 25°C.

Table 1. MMI-based approach compare to the half-circle approach

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The calibration of the spectrometer is the most critical step where a transmission matrix needs to be measured experimentally, as we have done here. It would have been ideal to show the comparison of the experimental probes with the reconstructed spectrum, which we reserve for future work. Nevertheless, we have demonstrated the reconstruction capability of our system using an experimental, broadband signal and have shown how it can be used to reconstruct various numerically generated narrowband spectra. The system exhibits a bandwidth of 67 nm, a lower bandwidth limit of $\sim$ 1 nm and peak-to-peak resolution of 3 nm. Our generic reconstruction algorithm proves to be robust over the experimental range of the input signals we tested, even in the presence of experimental noise. However, there are various factors that we deem to have limited the performance of our device; out-of-plane scattering loss, mainly due to the relatively large holes, radiation losses in the waveguide tapers, corner sections and junctions, grating coupler insertion losses, estimated to be $\sim$22 dB, and low contrast speckle pattern, due to the relatively small hole density in our prototype design. All these aspects can be improved by design optimisation, based on the initial results presented here. We examined devices with an increased scattering centre density of 324 holes, which should theoretically improve resolution (because this scales with the optical path length). However, because of the relatively large hole sizes in our device, $\sim$200 nm radius (limited by fabrication), the large scattering and radiation loss (from larger hole arrays) further reduces the in-plane transmitted optical signal intensity at the output channels below the detection limit of our experiment.

Ideally, if the pseudo-random perturbation acts in the wavelength range where the out-of-plane scattering and radiation losses are minimal, i.e. for the condition where the material wavelength ($\lambda$/n) < hole diameter, then in-plane scattering can be maximised by increasing the hole density of the array. Also, the precise placement of holes should ideally be co-located where the different planar modes interfere, which is wavelength dependent. Therefore, the dynamic ranges and resolution for different wavelengths within the measurement range will depend on the wavelength.

The demonstration of this integrated spectrometer, albeit using external coupling and detection here, demonstrates the potential for packaging the whole system into a compact device, by further integration of the detector, on-chip. Simultaneous measurement on each detector channel can also be implemented for practical usage, thereby reducing the contribution of measurement noise and increasing the sensitivity of the device. We anticipate that this type of device can add new functionality to the silicon photonics ‘tool-kit’ for a range of nanophotonic, PIC-based ‘on-chip’ spectroscopy applications.