## Abstract

On-chip spectrometers with tailored spectral range and compact footprint have been pursued widely in the last decade. Splitting different frequencies typically requires a propagation length that scales inversely with the frequency resolution, which leads to a trade-off between resolution and size. Scattering media in the diffusive regime provide a long light path and multipath interference in a compact area, resulting in strong dispersive properties that can be used for on-chip compressive spectrometry. However, the performance suffers from the low light transmission through the diffusive medium. It has been found that there exist “open channels” such that light with certain wavefronts can pass through the medium with high transmission. Here we show that a scattering structure can be designed so that open channels match target input/output channels in order to maximize transmission while keeping the dispersive properties typical of diffusive media. Specifically, we use inverse design to generate a scattering structure where the open channels match the output waveguides at desired wavelengths. For a proof of concept, we propose a ${{1}} \times {{10}}$ multiplexer covering a band of 500 nm in the mid-infrared spectrum, with a footprint of only ${9.4}\;\unicode{x00B5}{\rm m} \times {14.4}\;\unicode{x00B5}{\rm m}$. We also show that filters with nearly arbitrary spectral response can be designed, enabling a new degree of freedom in on-chip spectrometer design, and we investigate the ultimate resolution limits of these structures. The structures can also be designed based on a simple geometry consisting of circular holes with diameters from 200 to 700 nm etched in a dielectric slab, making them highly suited for fabrication. With the help of compressive sensing, the proposed method represents an important tool in the quest towards integrated lab-on-a-chip spectroscopy.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Motivated by the vast potential applications in farming, the food industry, and healthcare, the miniaturization and integration of spectrometers has become a hot research topic in the last decade. A straightforward solution for integration is to replace free-space diffraction gratings with on-chip dispersive components, e.g., planar waveguide curved gratings [1,2] and arrayed waveguide gratings [3–5]. However, the resolution of such integrated spectrometers is directly related to the optical path length, and therefore to the lateral dimension, typically resulting in millimeter (mm)- to centimeter (cm)-sized structures. One way to overcome this challenge is to fold the optical path in a resonant cavity. Spectral information can then be collected using an array of microcavities with different resonant frequencies [6–8] or a tunable microcavity [9]. The drawback of these devices is that each cavity (or cavity setting) only gets a small portion of the incoming light, resulting in a low efficiency. Another approach is inspired by Fourier transform infrared spectroscopy (FTIR), in which light is split and sent to an array of on-chip Mach–Zehnder interferometers (MZIs) with different free spectral ranges [10–12], to mimic the effect of optical path tuning in a FTIR. Alternatively, switch circuits are used to tune the optical path on a chip to avoid the large array of MZIs [13]. Also, in this case, the need for large path differences results in relatively large structures. Photonic crystals [14,15] and digital planar holography [16–18] have also been explored for strong dispersion on chip, with limited sensitivity or bandwidth. Despite substantial effort, building an integrated broadband spectrometer with a small footprint is still a challenging task.

Disordered scattering media in the diffusive regime are known to provide long light path in a compact area due to multiple scattering. It has been demonstrated that such property can be exploited for spectroscopy in free space [19] and on a chip [20,21] using disordered photonic crystals in a small footprint. However, the performance suffers from the low light transmission through the diffusive media. Recently, it has been found that there exist “open channels” in random scattering media or waveguides, i.e., light with certain wavefront and wavelength can pass through the structures with optimal transmission [22–28]. So far, work in this area has been focused on engineering the wavefront of the incoming light to match the open channels occurring in a random scattering medium. In this paper, we show that, using inverse design [29], it is possible to *engineer* the open channels to match the desired wavefront by optimizing the size, position, and/or shape of the scatterers. In this way it is possible to combine the strong dispersive properties of a scattering medium with the high transmission typical of designed dispersive structures. For a proof of concept, we propose a ${{1}} \times {{10}}$ multiplexer covering a band of 500 nm in the mid-infrared spectrum, with a footprint of only ${9.4}\;\unicode{x00B5}{\rm m} \times {14.4}\;\unicode{x00B5}{\rm m}$. We also show that by allowing a deformation of the holes, filters with nearly arbitrary spectral response can be designed, enabling a new degree of freedom in on-chip spectrometer design. The ultimate resolution limit in these structures is shown to be related to the spectral correlation function of the underlying scattering medium. We further discuss the experimental feasibility of the designed structures and propose an adapted inverse design method based on a simple geometry consisting of circular holes with variable diameters and positions.

## 2. METHODS

As a starting point of the study, a 2D simulation model of ${9.4}\;\unicode{x00B5}{\rm m} \times {14.4}\;\unicode{x00B5}{\rm m}$ planar silicon slab embedded in air is built. Light is guided to the slab through an input waveguide and coupled out from 10 output waveguides. A constant refractive index of 3.43 is used for silicon in the model. Inspired by [20], an array of aperiodic (golden spiral lattice) circular holes with radius of 240 nm is distributed over the slab, producing multiple scattering. The method described below does not depend on specific distribution of holes, as long as they are in the diffusive regime. Light passing through the slab forms a wavelength-dependent speckle pattern, which can be used to retrieve the spectral information of incoming light. In this proof-of-concept design, we target the mid-infrared wavelength range around 3250 nm, where many important organic gas molecules feature absorption lines associated to the C-H bond. In this range, the ratio of hole-diameter to wavelength (${{2}}\pi nr/\lambda\; \approx \;{1.5}$) corresponds to the Mie scattering regime. This indicates that scattering direction is sensitive to any change in the shape or size of the scatterers, leaving room for geometry optimization. The transport mean free path in the diffusing medium is estimated to be ${l_{t}} = {1.7}\;\unicode{x00B5}{\rm m}$ with $T = 4{l_t}/3L$ [30] (${{L}}$ is transmission and ${{L}}$ is length of the waveguide) by averaging the calculated transmission over different realizations. The length of the simulated structure (9.4 µm) is between ${l_{t}}$ and the localization length, implying that it operates in the diffusive regime. We will consider a transverse-magnetic (TM) polarization for all simulations in the following. The method also works for the transverse-electric (TE) polarization, and an example of that is given in Supplement 1.

In this paper we use the device described above as an initial solution, and we optimize it to meet multiple design targets in a broadband wavelength range by allowing the free deformation of all the holes and boundaries of the slab. To ensure the optimized device meets all the design targets at different wavelengths, the performance is first evaluated at all the targeting wavelengths, and the worst of them is chosen in each iteration as the figure of merit (FoM). The adjoint method [31–35] is used to calculate the gradient of FoM, and the geometry of the device is updated once in each iteration to maximize the FoM. The electromagnetic field is calculated by the finite-difference time domain (FDTD) method with a commercial software (Lumerical FDTD Solutions), so the performance in a concerned spectral range is retrieved by a Fourier transformation of the signal in the time domain by a single simulation. For 2D models, each simulation takes less than 1 min on a desktop computer with an Intel Core i7-6700 processor, and the total time for optimization depends on the design target and is usually a few hours. Though we mostly use 2D models in this paper for proof of the concept, 3D optimization can be done with the same principle as shown below, but takes a few days. The advantage of using FDTD rather than frequency-domain methods here is that transmission of all the target wavelengths can be obtained in a single simulation run, and hence the computation load in each iteration is reduced. A rectangular mesh grid with a ${{30}}\;{\rm{nm}} \times {{30}}\;{\rm{nm}}$ unit mesh cell is used. A grayscale refractive index at the boundary between two materials (silicon and air) is applied for antialiasing. The level-set method [36] is used to represent the topology of the device, and the steepest descent method is applied to update it. A smoothing function is also included to avoid tiny features.

## 3. RESULTS AND DISCUSSION

Figure 1(a) shows the dielectric slab with aperiodic distributed holes as described in the Methods section. Light passing through the diffusive medium experiences multiple scatterings and generates a wavelength-dependent speckle pattern. By sweeping the wavelength in the range of interest, a matrix of transmittance values for the different wavelengths and output ports is generated. This matrix is the key to reconstruct the spectrum of incoming light. For scattering media in the diffusive regime, two columns in the matrix become uncorrelated when their wavelength difference is larger than the “spectral correlation width” $\delta \lambda$, which scales with ${l_t}/{L^2}$, where ${{L}}$ is dimension of the device [20]. Such a matrix can be used to reconstruct the spectrum of input light, with spectral resolution depending on $\delta \lambda$, using numerical optimization algorithms, which work particularly well when the spectrum is sparse on a certain basis. A main challenge of this method is the low signal-to-noise ratio due to the low transmission, which degrades the spectrum’s reconstruction quality. In order to solve this challenge, we assign a target wavelength in the band of concern (2975 to 3475 nm) to each output and optimize their transmission by inverse design, with the device shown in Fig. 1(a) as an initial solution. For comparison, Fig. 1(b) shows the optimized structure and its speckle pattern after 900 iterations. It can be seen that the device becomes a multiplexer, with the transmission at the target wavelength greatly improved in all the outputs. Compared to the original matrix, the average transmittance over the entire wavelength range is 2.7 times higher, while the condition number of the matrix is 16 times smaller. That means that the quality of spectral reconstruction with an optimized device will be significantly improved due to a higher signal-to-noise ratio and a better conditioned matrix. Figure 1(c) shows that the transmittance increases quickly from around 0.02 to 0.5 in about 400 iterations and then slowly converges to 0.6 in the next 500 iterations. The difference between the average and minimum transmittance in all the outputs is below 3%, indicating that even though in each iteration only the wavelength with the lowest transmittance is optimized, the target wavelengths maintain about the same transmittance during and after the optimization. In actual structures the transmission is likely to be lower due to the out-of-plane scattering as shown in the result of 3D modelling in Fig. 1(c); however, the FoM still experiences significant improvement during the optimization of over a factor of 10 during optimization. In view of computation time, we employ 2D modelling in the rest of this paper, but all the conclusions can be translated to 3D, with lower expected transmission (depending also on the chosen geometry). Additionally, the out-of-plane scattering loss also broadens the spectral width of transmission channels and thereby limits the highest spectral resolution. We note that the out-of-plane scattering depends on the spatial wavevector distribution of the structure [20]. More details on the effect of out-of-plane loss are provided in Supplement 1. The field distribution after the optimization at one of the target wavelengths is shown in Fig. 1(d). Unlike conventional photonic crystal waveguides, light here is not confined in a given channel but rather spreads over a large part of whole device, depending on the wavelength. The complex interference of all possible light paths leads to high transmission in the target output waveguide. The area of footprint of the optimized device is several orders of magnitude smaller than that of traditional waveguide multiplexers.

We now focus on the understanding of the effect of optimization on the properties of the diffusive medium. To this aim, we simplified the model to a “diffusive waveguide,” i.e., a quasi-1D dielectric waveguide in the $x$- direction with a perfect electric conductor boundary condition and filled with holes described in the Methods section [Fig. 2(a)]. Normally light has low transmission through such a waveguide due to scattering. However, there are some exceptions in which high transmittance can occur when the wavefront of incoming light meets certain conditions, the so-called “open channels” [22–28]. In order to find these open channels, the transmission matrix $t$ needs to be constructed first. This is done by using the guided modes of the empty waveguide as a basis [22]. A guided mode $E_{\rm {in}}^{(j)}(y)$ is launched at the input plane (${{x}} = {{0}}$), then the transmitted field $E_t^{(j)}(y)$ at the output end (${{x}} = {{L}}$) is calculated with the FDTD method and decomposed in the basis of the empty waveguide modes using $E_t^{(j)}(y) = \mathop \sum \nolimits_{i = 1}^N {t_{\textit{ij}}}E_{{\rm in}}^{(j)}(y)$. The coefficient ${t_{\textit{ij}}}$ relates the field incident in waveguide mode ${{j}}$ to the field transmitted to waveguide mode $i$. By repeating the procedure for $j = {{1}},\;{{2}},\;{\ldots}\;N$, the full transmission matrix $t$ is obtained, where $N$ is the number of modes supported by the empty waveguide. Transmission channels are defined from a singular value decomposition of the transmission matrix $t = U{{\Sigma}}{V^\dagger}$, where ${{\Sigma}}$ a real diagonal matrix containing the transmission coefficients of the eigenchannels. $U$ and $V$ are unitary matrices that map the eigenchannels onto the guided modes at the input and output sides. The eigenchannels with transmission close to 1 are referred to as the open channels of the system.

It can be seen from Fig. 2(b) that incident light with a Gaussian profile launched at the middle of the input plane does not pass through the diffusive waveguide well (transmittance = 0.16), because its wavefront does not match that of any of the high-transmission channels. In contrast, incident light matching a high-transmission channel can easily pass as shown in Fig. 2(c) (transmittance = 0.95). Then we optimize the geometry of the scatterers by setting the transmission through the middle of output plane as a target [Fig. 2(b)], resulting in a high transmittance = 0.69 for a Gaussian beam input [Fig. 2(f)]. When calculating the open channels of the structure after optimization, we observe that the second maximal transmission channel is tailored to match to the Gaussian input beam and the target output, which can be seen in the similarity between Figs. 2(f) and 2(g). This clearly shows that the geometry optimization increases the transmission not by changing the diffusive nature of the device but by tailoring the transmission matrix so that the open channels match the desired input/output modes. The algorithm automatically determines which channel or combination of channels are matched with the target wavefront. Figures 2(d) and 2(h) show that the transmission eigenvalues follow a similar bimodal distribution regardless of the geometry change. Therefore, it is clear that the enhanced transmittance in Fig. 2(f) is not obtained by changing the average transmission properties. The possibility of designing open channels in diffusive media demonstrated here is a powerful tool to optimize the performance of random or pseudorandom photonic structures for a wide range of functionalities beyond spectrometry, e.g., broadband light harvesting, filters with features in a broad wavelength and wide angular range.

On-chip filtering with an arbitrary transmission spectrum is of interest for sensing [37] and signal processing [38]. Previously it has been achieved with arrayed-waveguide gratings and modulators on each arm [38]. However, such a device has a large footprint. A more compact approach was proposed based on complex Bragg gratings on planar waveguides [37], but it is limited to discrete narrow spectral notches. Here we provide a method to generate planar filters with an arbitrary transmission spectrum using a scattering medium with optimized geometry. Similar to the case in Fig. 1, we start with a ${9.4}\;{\rm{\unicode{x00B5}{\rm m}}} \times {9.4}\;{\rm{\unicode{x00B5}{\rm m}}}$ dielectric slab with aperiodic distributed uniform holes, but we only keep one input and one output. To demonstrate the versatility of the method, we set two types of target spectra: low-pass and sinusoidal spectral shape in a wavelength range from 3000 to 3300 nm. Optimizing the performance at each wavelength simultaneously becomes impractical due to the broad spectrum. Alternatively, choosing a few fixed wavelength points to optimize would make the final result depend on how the points are chosen. To address this problem, we define the FoM as the largest difference between the target spectrum and the device spectrum in the current iteration, i.e., $\max\{{| {{T_{{\rm tar}}}(\lambda) - {T_{{\rm dev}}}(\lambda)} |} \},\quad {\lambda _1} \le \lambda \le {\lambda _2},$ where ${\lambda _1}$ and ${\lambda _2}$ are the lower and upper bounds of the wavelength. In this way, the target wavelength is not a preset constant but is chosen dynamically according to the result in the current iteration. Figures 3(a)–3(d) show the target spectra and the spectra of optimized filters, which closely match the target ones, with a maximum transmittance difference around 0.1. The number of sampling frequencies is chosen to be 60, since more points are not improving the performance. While Figs. 3(a)–3(d) show that the spectral responses of the designed filters fit the targets well, it is reasonable to assume that there is a limit on the narrowest spectral feature that can be realized by a device with finite area, limited by the optical path length it supports. To verify this idea, we set a series of target sinusoidal spectra with different free spectral range (FSR) and try to fit them with inverse designed filters from the same initial geometry. In this case, the smallest $\max\{{| {{T_{{\rm tar}}}(\lambda) - {T_{{\rm dev}}}(\lambda)} |} \}$ that one can get without capturing any spectral feature is $\max\{{| {{T_{{\rm tar}}}(\lambda)} |} \}/2$, by a “filter” with constant transmittance of $\max\{{| {{T_{{\rm tar}}}(\lambda)} |} \}/2$ in the wavelength range of interest. In our case, we target the peak transmittance to be 0.6, and therefore we set a FoM of 0.3 as a minimum requirement for the filter. Figure 3(f) shows the evolution of FoM as a function of the number of iterations for different targets. All of them have improved FoMs compared to their initial condition, because the transmittance is increased over the entire wavelength range. However, it can be seen that only those with period above 50 nm reach a FoM below 0.3, suggesting that the optimization approaches a resolution limit intrinsic to this structure. Such a limit is imposed by the optical path length supported by the device and scales with ${l_t}/{L^2}$ [20]. As the geometry optimization only modifies the scattering matrix but not the statistic transmission properties or $L$, the resolution limit is already determined before optimization. As evidence of this point, we calculate the spectral correlation function of the transmission on the filter before and after optimization using $C({{{\Delta}}\lambda}) = \langle T(\lambda)T({\lambda + {{\Delta}}\lambda})\rangle /[\langle {T(\lambda)\rangle \langle T({\lambda + {{\Delta}}\lambda})}\rangle ] - 1$, where $T(\lambda)$ is the transmittance in the output waveguide at wavelength $\lambda$ and $\langle \ldots \rangle$ represents the average over $\lambda$. To be representative, the correlation function is averaged over different detection channels by varying the position of the output waveguide. As shown in Fig. 3(f), C is normalized to 1 at ${{\Delta}}\lambda = 0$, and its half-width at half-maximum $\delta \lambda$ is about 25 nm for the unoptimized structure, meaning that a wavelength shift of 25 nm is sufficient to reduce the degree of correlation of transmittance to 0.5. The calculated correlation function matches well the theoretical correlation function C1 (green line; see Supplement 1). After optimization the value remains the same or becomes larger, depending on the target spectra. However, it is known that in similar diffusive media the spectral width of channels with high transmittance can be broader than those with lower transmittance due to a shorter time delay [39]. Therefore, $\delta \lambda$ calculated above indicates a lower limit to the spectral resolution of the device, e.g., two independent wavelength channels need to be separated from each other by at least ${{2}}\delta \lambda$, which is 50 nm, which corresponds to the resolution limit obtained from the geometry optimization. To conclude, the proposed method can be used to design filters with arbitrary transmission spectra with a resolution limit of ${{2}}\delta \lambda$, which is determined by the dimension and mean free path of the diffusive area.

One of the main challenges that hinders the practical application of inverse-designed photonic devices is fabrication feasibility. While a smoothing function can be used to avoid any features smaller than the lithographic resolution, the irregular shapes of holes or islands of material are still challenging in view of proximity effect correction and would require extensive optimization and possibly degrade performance. One option to avoid irregular shapes is to use the pixelated structure [36,40,41]; however, this limits the degrees of freedom and also excludes gradient-based optimization. Here we propose an intermediate solution by limiting the element geometry to circles with various sizes and positions. To implement this constraint, we start with a uniform circular hole array, and in each iteration translate the gradient into the amount of radius and position changes needed by each hole. The maximum/minimum allowed radius can be easily set in the algorithm and are here chosen as 100 nm and 350 nm. In the initial array, a square lattice with a lattice constant of 720 nm is used for simplicity, while other types of lattice are also viable. Figure 4(a) shows the final geometry of the device after 1000 iterations, with the same optimization target as in Fig. 1. A close look at the radii distribution shows that the average radius is almost the same as the original one, with less than 1% difference. While the radii of holes changed a lot from each other, the lattice was only slightly deformed. This indicates that the scattering properties of the individual scatterer are more critical to the performance than their positions. It can be seen in Fig. 4(d) that, although the constraint is very strong, high performance can still be obtained with a small compromise of the transmittance with respect to the optimization of Fig. 1. Even though the target function is not set to minimize the transmission at wavelengths other than the target wavelengths, the crosstalk is below 1:6 at all wavelengths. Though there are minor peaks in the background of the spectra, it has been previously shown that the spectral reconstruction can be as good as that with filters with nearly perfect line shape through the use of compressive sensing [42]. The influence of local and universal lithography distortion to the performance is investigated and the result is given in Supplement 1, showing that the design has a feasible tolerance to fabrication errors.

## 4. CONCLUSION

In summary, we propose a method to design the transmission properties of scattering media in the diffusive regime in order to obtain a spectral response adequate for spectrometry. High transmission through the scattering system is achieved by tailoring the open channels to match the designed wavefront at target wavelengths. The footprint of device is only ${9.4}\;\unicode{x00B5}{\rm m} \times {14.4}\;\unicode{x00B5}{\rm m}$, which is orders of magnitude smaller than conventional planar multiplexers. Furthermore, a method to design filters with an arbitrary transmission spectrum using a scattering medium is presented, and the ultimate resolution limits are explored. A modified adjoint method is developed to generate simplified geometries composed of circular scattering elements with varied sizes and positions. The method provides a third option, alongside pixelate geometry and freeform geometry, combining the benefits of gradient-based optimization and a simple geometry amenable to nanofabrication. The proposed method provides an avenue to design and fabricate compact and yet high-performance on-chip microspectrometers.

## Funding

Nederlandse Organisatie voor Wetenschappelijk Onderzoek (022.005.001).

## Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

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