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Abstract
Miniaturization of a conventional spectrometer is challenging because of the tradeoffs of size, cost, signal-to-noise ratio, and spectral resolution, etc. Here, a new type of miniaturized infrared spectrometer based on the integration of tunable graphene plasmonic filters and infrared detectors is proposed. The transmittance spectrum of a graphene plasmonic filter can be tuned by varying the Fermi energy of the graphene, allowing light incident on the graphene plasmonic filter to be dynamically modulated in a way that encodes its spectral information in the receiving infrared detector. The incident spectrum can then be reconstructed by using decoding algorithms such as ridge regression and neural networks. The factors that influence spectrometer performance are investigated in detail. It is found that the graphene carrier mobility and the signal-to-noise ratio are two key parameters in determining the resolution and precision of the spectrum reconstruction. The mechanism behind our observations can be well understood in the framework of the Wiener deconvolution theory. Moreover, a hybrid decoding (or recovery) algorithm that combines ridge regression and a neural network is proposed that demonstrates a better spectral recovery performance than either the ridge regression or a deep neural network alone, being able to achieve a sub-hundred nanometer spectral resolution across the 8∼14 µm wavelength range. The size of the proposed spectrometer is comparable to a microchip and has the potential to be integrated within portable devices for infrared spectral imaging applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
One essential property of light is its spectrum, which yields information about the differences between energy levels corresponding to electronic transitions, molecular vibrations and rotations, etc. Such information can not only reveal the constituent elements of a sample and how they combine together to form it, but also yield the geometric properties of material micro-structures [1–3]. Therefore, with the help of spectroscopy, we can identify the element components and the state of a sample. Furthermore, if spatial information (in addition to the spectral information) is obtained, the material composition in each region of a scene could be distinguished, which will improve the accuracy of target identification and image restoration [4,5].
Long wave infrared at wavelengths between 8 and 14 µm is a valuable spectral range to study because it is an atmospheric window and many characteristic absorption lines of gases and organic matters are distributed within this range [6,7]. Nowadays, spectra in this waveband can be measured with high resolution and precision, mainly with two types of infrared spectrometers, i.e., the grating spectrometer and the Fourier transform infrared spectrometer (FTIR). The grating spectrometer relies on the dispersion characteristics of diffractive optics (usually a grating) to separate different frequency components of the light [8,9] while the FTIR typically utilizes a Michelson interferometer to get the interferogram of the sampling light which is then Fourier-transformed to recover the spectrum [10]. For both techniques, dependence of resolution on the optical element size, the optical path, or the precision mechanical scanning device hinders the miniaturization. Despite this, in applications such as portable biological monitoring and spectral imaging, it is highly desirable to further miniaturize the spectrometer down to sub-millimeter sizes.
Recently, a new type of miniaturized spectrometer based on a micro-sized filter array and a recovery algorithm has been developed [11–19]. Each unit of the filter array has a different transmission spectral response, which controls the weightings of the different frequency components irradiated on the detector beneath, resulting in an indirect spectral sampling of the incident light. After reading out the photocurrents of the detectors underneath the filters, the incident spectrum can be reconstructed based on the known optical response characteristics of the filters and a recovery algorithm. To improve the performance of this type of spectrometer, it is vital to find a set of uncorrelated filter arrays. The most widely used filter arrays are metal surface plasmon arrays [11–13], photonic crystal arrays [14–16], colloidal quantum dot arrays [17,18], and alloy nanowires [19], based on which a variety of on-chip spectrometers have been demonstrated.
Nevertheless, since such spectrometers require dozens or even hundreds of filter combinations to achieve high-precision spectral recovery, and each filter needs at least one detector pixel. Therefore, a compromise has to be made between the spectral resolution and the spatial resolution. To tackle this problem, it seems promising to use tunable filters to replace the original spatially distributed filter arrays, and this approaching has attracted many research interests [20–22]. In the mid and far-infrared frequency range, micro and nano devices that can achieve a broad tunability mainly include temperature-controlled phase change material devices [23], narrow bandgap heterojunction structures [24], and electrically tunable graphene plasmonic devices [25–30]. Among them, graphene plasmonic devices can realize frequency and amplitude modulation over a wide spectral range and are also readily to be integrated with room temperature detectors. Hence, it would be of interest to investigate whether graphene plasmonic devices could be used in miniaturized infrared spectroscopies and which factors are crucial in determining the spectral recovery performance.
In this paper, we report a miniaturized infrared spectrometer based on a tunable graphene plasmonic filter and a recovery algorithm combining ridge regression and a fully connected neural network. The infrared transmission response of the filter is first analyzed with the finite element method to get its response matrix for spectral recovery, where the influences of graphene nanoribbon width, array duty cycle, graphene Fermi energy, and carrier mobility are considered. The ridge regression algorithm (RGA) is then used to recover the spectral information of random incident spectra within the 8∼14 µm range and a criterion of choosing the proper RGA coefficient at different SNR is established for the best recovery performance. To optimize the spectral recovery performance, the relationship between the spectral resolution and the filter resonance bandwidth as well as the detection signal-to-noise ratio is theoretically analyzed. Finally, we demonstrate that integrating a ridge regression layer with a fully connected neural network could further improve the recovered spectral resolution.
2. Results and discussion
2.1 Structure and principle
Figure 1(a) shows the schematic of the proposed infrared spectrometer. It adopts a graphene nanoribbon array as an infrared filter and a layer of ion gel on the graphene as the top gate. The gate voltage controls the number of ions in the gel that migrate to the graphene layer, and hence could tune the infrared transmittance of the device by changing the carrier concentration (and also the Fermi energy) in graphene. More specifically, the relationship between the gate voltage and the Fermi energy could be described as [31]: Vg-VCNP = |Ef| / e + 4πEf2e / (h2vf2Cg), where Vg is the gate voltage, VCNP is the Dirac voltage, h is the Planck constant, νf is the Fermi velocity equaling to 106 m/s, e is the unit charge and Cg is the capacitance of the ion gel gate in a unit area. A dielectric layer with a thickness of h = 0.5 µm and refractive index n = 1.5 lies between the nanoribbon and the infrared detector, acting as a protective layer.
Fig. 1. (a) Schematic of the proposed miniaturized spectrometer composed of a tunable graphene nanoribbon filter and an infrared detector. (b) Schematic illustration of the spectral measurement work flow. As a light with a specific spectrum shape is incident onto the device, its spectrum will be modulated by the tunable graphene filter. This provides a time-variant sampling of the spectral information, and the detector also outputs a time-variant photocurrent correspondingly. Then, the spectrum is recovered from the measured photocurrent using the ridge regression algorithm or a deep neural network. Note that the frequency range of 22∼37 THz in the figure corresponds to the wavelength range of 8∼14 µm, namely, the long wave infrared region.
Figure 1(b) shows the workflow of such an infrared spectrometer. As a beam of light with the spectrum of s(λ) is incident on the graphene nanoribbon filter whose transmittance is ti(λ) at gate voltage Vi, the transmitted spectrum becomes s(λ)ti(λ). The transmitted light then reaches the detector and is converted to a photocurrent with a quantum efficiency of η(λ). By integrating over the wavelength λ, the output photocurrent of the detector at a filter gate voltage of Vi could be obtained as:
Normally, the data are processed and stored in discrete forms, and therefore Eq. (1) should be discretized as follows,
where tij, sj, and ηj are discrete forms of ti(λ), s(λ), and η(λ) with respect to the wavelength, and M is the total number of the discrete points. We apply a varying gate voltage Vi to the graphene filter and measure the photocurrent of the detector underneath, thereby acquiring multiple samples of the incident spectrum. Assuming the number of measurements is N (i.e., the total number of Vi is N), then Eq. (2) can be further written into a matrix form as:Here, I is the N-dimensional photocurrent column vector whose i-th row is Ii, S is the M dimensional spectrum column vector with the j-th row being Sj, and T is called the N × M dimensional response matrix with the nth row being [η1ti1, η2ti2, …ηitik…, ηMtiM]. In practical applications, the response matrix T can be obtained by measuring the photocurrent of an infrared detector irradiated by a frequency continuously tunable laser. When I and T are known, the incident spectrum S can be recovered with S = T−1I. However, normally, when the number of measurements (i.e., N) is not large enough, the T matrix cannot be guaranteed to be full column rank, meaning that a direct calculation of S can be hardly achieved through the inversion operation. Therefore, in most cases, only an estimated value of the spectrum S could be obtained by using recovery algorithms. In the following, we will use the RGA combined with a fully connected neural network to recover the spectrum.
2.2 Determination of the filter structure parameters
For the spectrometer to realize high precision spectral measurements in the long wave infrared range of 8∼14 µm (about 22∼37 THz), a broadband tunable transmission is essential. Hence, we first analyze the infrared transmission property of the tunable graphene filter to determine the structure parameters of it.
According to previous studies [32–34], in the long wave infrared range, graphene can be approximated as a layer of conductive plane whose conductivity is described by the Drude-like model,
Fig. 2. (a) Dispersion curves of the graphene plasmons under different Fermi energies. (b) Standing wave conditions of the graphene plasmon resonance (the upper panel) and the transmittances of the 100 nm wide nanoribbons simulated by the finite element method (the lower panel). The frequencies meeting the standing wave conditions are marked with red hollow circles, and they align precisely with the transmission valleys beneath. The insets are the z component electric fields at the transmission dips.
Equation (5) is the dispersion relation of propagating surface plasmons on graphene monolayer. Taking the resonance as the constructive interference of the propagating surface plasmons, the resonant frequencies of graphene nanoribbons can be determined by combing Eq. (5) with the standing wave condition of localized graphene plasmons [35,36],
Note that the numerical data in Fig. 2(b) and the subsequent figures are simulated using finite element method with COMSOL Multiphysics, where perfectly matched layers (PMLs) are applied at two ends of the computational space in the z-direction to avoid back-reflection at the interfaces. The continuity periodic boundary conditions are imposed in the x-direction to simulate a periodic array. The graphene ribbon is modelled as a line segment with a current boundary condition of J = σE. In addition, two ports are placed at the top and bottom interfaces for applying the light source (a TM-polarized light) and measuring the transmission coefficients, respectively.
Apart from guaranteeing a broadband tunable range, another benefit of choosing a relatively high Fermi energy for the graphene filter is that a larger transmission modulation of the incident spectrum could be obtained. Figure 3(a) shows the transmission spectra of the graphene filter as the Fermi energy increases from 0.2 to 0.8 eV. It can be seen that the modulation amplitude of the spectra, defined as the difference between the transmission dip depths at the resonant frequency and that of a non-resonant frequency, increases from about 15% at 0.2 eV to about 60% at 0.8 eV. Meanwhile, the half width at half minimum (HWHM) of the transmission dip is decreased from 4.6 THz at 0.2 eV to 2.4 THz at 0.8 eV. These trends could be more clearly seen in Fig. 3(c), where the modulation amplitude and the HWHM are extracted from Fig. 3(a). Such results could be well explained by the temporal coupled mode theory, which shows that the transmittance T of the graphene plasmonic filter could be given by [37,38],
Here, ω is the frequency of the incident light, and ωa is the resonant frequency of the graphene plasmonic filter. γ0 and γ1 are the intrinsic loss rate and the out-coupling rate of the graphene plasmonic mode, respectively, and they have semi-analytical forms as γ0 = 1/2τ, γ1 ∝ Ef due to the negligible magnetic energy of the graphene plasmonic mode at infrared frequency range [39,40]. Equation (7) suggests that the transmittance is a Lorentz-like curve with the depth of the transmittance dip at the resonant frequency (i.e., ω = ωa) being γ02 / ( γ0 + 2γ1)2, and the HWHM being 2( γ0 + 2γ1). As the Fermi energy increases, γ1 becomes larger since γ1 ∝ Ef, but γ0 would decrease because γ0 = e·vf2 / 2μEf (note that such relation is deduced from γ0 = 1/2τ and τ = μEf / (e·vf2)). In our case, the decrease rate of γ0 is faster than the increase rate of γ1, resulting in an increase in γ02 / ( γ0 + 2γ1)2 (i.e., the depth of the transmission dip) and a decrease in 2( γ0 + 2γ1) (i.e., the HWHM), which agrees with the simulation results in Fig. 3(a) and (c). Therefore, a relatively high Fermi energy of graphene gives a large transmittance modulation, which contributes to a larger signal-to-noise (SNR) ratio in the receiving infrared detector. As demonstrated later, it is important for the proposed spectrometer to work under a large SNR ratio to achieve a high spectral recovery performance.
Fig. 3. (a) Transmittances of the graphene plasmonic filter versus the Fermi energy of graphene. (b) Transmittances versus the duty cycle of the graphene ribbon arrays. (c) The extracted HWHMs of the transmittance and depth of transmission dips from (a). (d) The extracted HWHMs and depth of transmission dips from (b). In the calculation, the graphene ribbon width is fixed at 100 nm, and the mobility and Fermi energy of graphene are set to 1500 cm2/(Vs) and 0.5 eV, respectively.
To further enhance the transmitted amplitude, so as to increase the SNR ratio of the spectrometer, we can optimize the duty cycle of graphene nanoribbons. Figure 3(b) displays the transmission spectra when the duty cycle increases from 0.35 to 0.98, and Fig. 3(d) shows the depth of the transmission dip and the HWHM extracted from Fig. 3(b). It is evident that the smaller the duty cycle, the larger the transmission dip depth, indicating that a narrower gap between adjacent nanoribbons is favored if a larger transmission modulation is desirable. However, when the gap becomes too narrow (e.g., less than 50 nm), it is difficult to fabricate and the HWHM would broaden due to the enhanced coupling between adjacent nanoribbons. The resonance frequency would red-shift significantly as well. With all these in consideration, the gap between adjacent graphene nanoribbons is set to be 50 nm.
2.3 Crucial factors affecting the spectrometer performance
Based on the determined structural parameters of the graphene plasmonic filter, the crucial factors that influence the spectral recovery performance of the spectrometer are analyzed. To this end, the complete spectral measurement process is simulated in two steps,
- 1) generating the measured photocurrent data as a function of varying the Fermi energy of the graphene plasmonic filter for a given fixed input spectrum;
- 2) recovering the input spectrum from the measured photocurrent data using optimization algorithms.
Firstly, to generate the detector measured currents for an input spectrum, the Fermi energy of the graphene plasmonic filter is varied to tune its transmission characteristics, resulting in a time series of the measured photocurrent. Such a process is detailed in section 2.2 and can be described by Eq. (2). However, in practical situations there is always a noise current that may originate from the detector noise or the system noise or a combination of them. Hence, Eq. (2) should be modified to take the noise current into account,
and Eq. (8) can be further cast into a matrix form:Here, Inoise is the noise term, and we assume the noise is a Gaussian white noise whose amplitude follows a normal distribution, i.e., Inoise ∼ N(0, TS/SNR1/2). To obtain the T matrix in Eq. (9), the Fermi energy of the graphene plasmonic filter is gradually increased from 0.2 to 0.8 eV with a step of 0.01 eV (i.e., the number of measurements is 61), and the corresponding transmittances of the filter in the frequency range of 22∼37 THz are calculated with a sampling interval of 0.025 THz (i.e., the transmittance is a row vector with 601 elements). Then the T matrix is constructed by stacking the transmittances of the filter at different Fermi energy into a $61 \times 601$ matrix.
Secondly, to recover the input spectrum from the measured photocurrents, we adopt the RGA, which is widely used to solve underdetermined linear equations by minimizing the loss function:
In Eq. (10), α is an adjustable coefficient called ridge regression parameter, which is used to limit the amplitude of the recovered spectrum so that an accurate result could be obtained. For example, in the case of too few measurement numbers or too large measurement noise, a large α can suppress oscillation in the recovered spectrum. The RGA has an analytical solution in the form of:
In the RGA algorithm, the standard of choosing a proper ridge regression parameter α should be established. Figure 4(a) displays the recovered spectra with α increased from 1 × 10−3 to 1 × 101 for the same input spectra of 70 dB SNR. It can be seen that when α is too small (e.g., α = 1 × 10−3 in the figure), the recovered spectrum suffers from high-frequency oscillation noises, which affects the recovery accuracy. On the contrary, when α is too large (e.g., α = 1 × 101), the recovered spectrum tends to be flattened, and the fine sharp structures of the original spectrum cannot be reconstructed, leading to a low spectral resolution. Therefore, to achieve a reasonable spectral recovery, it is necessary to select the value of α through a trade-off between accuracy and resolution. We choose the value of α so that the power of the noise spectrum Snoise = MRGA(α)In is a small portion κ (e.g., κ = 2.5 × 10−3) of the power of the original spectrum:
In this situation, the oscillation of the recovered spectrum can be suppressed because its relative height is about κ1/2 of the original spectrum, and simultaneously an acceptable accuracy is obtained since the value of α may be kept small. According to this criterion, we can determine the optimal α value in the RGA according to the SNR of the spectrometer, so as to obtain a higher spectral resolution or accuracy than that at a fixed α value.
Fig. 4. (a) The recovered results of the same original spectrum as the value of the ridge regression parameter α varies. (b) Schematic diagram of the Rayleigh criterion. (c) The ridge regression parameter α and the spectral resolution plotted as a function of the SNR. (d) The dependence of the spectral resolution and HWHM of the transmittance curve on the carrier mobility of graphene.
The key parameter to evaluate the performance of a spectrometer is the spectral resolution. Here the Rayleigh criterion is employed to determine the spectral resolution of the recovered spectra, where the resolution is defined as the distance between two adjacent Gaussian peaks in the input spectra whose recovered spectra are just about to merge into one peak. For example, as shown in Fig. 4(b), the red, green, and blue solid lines are the input spectra containing two Gaussian peaks with a frequency interval of 1.4, 1, and 1.2 THz, respectively, and the dashed lines are their recovered spectra. As the frequency interval decreases from 1.4 to 1 THz, the recovered spectrum changes from two discrete peaks to a combined wave packet, signifying a transition from the resolvable state to the unresolvable state. In this process, there is an intermediate critical state (shown by the blue lines) where the two peaks in the recovered spectrum can just be distinguished, and the difference between the central frequencies of their input spectra (i.e., 1.2 THz) is defined as the spectral resolution according to the Rayleigh criterion.
On the basis of the two-step simulation of the spectral measurement process and the Rayleigh criterion, we now investigate the essential parameters that affect the performance of the tunable graphene filter-based spectrometer.
The impact of the SNR of the measurement process is firstly investigated because it is obvious that a spectrum can hardly be recovered if a large noise is introduced in the measurement process. Here, we select a spectrum consisting of two adjacent Gaussian peaks whose maximums are symmetrically distributed on the two sides of 30 THz as the input test spectra, and then scan α from 1 × 10−5 to 1 × 102 to determine the optimum α value at a specific SNR according to Eq. (12). Then the frequency spacing between the Gaussian peaks is gradually increased from 0 to 5 THz. When the peaks can be just resolved according to the Rayleigh criterion, the corresponding frequency spacing is taken as the spectral resolution of the spectrometer under that specific SNR condition. Figure 4(c) shows the simulation results, where the SNR is in the range of 40 dB to 80 dB and the mobility of graphene is fixed at 1500 cm2 /(V·s). Clearly, as the SNR increases (i.e., the noise decreases), α decreases gradually from 10 to 2 × 10−3, this is because a smaller α can satisfy the condition in Eq. (12) in a lower noise situation. Meanwhile, as indicated by the red curve in the figure, the spectral resolution at a center frequency of 30 THz improves from 3 to 1 THz, proving that a larger SNR indeed leads to a better spectral resolution.
Another important parameter is the HWHM of the filter transmittance curve. A smaller HWHM could give a finer modulation of the incident spectrum, thus giving a better spectral resolution. According to Eq. (7), the HWHM is given by 2(γ0 + 2γ1), which is in turn related to the graphene mobility through γ0 = e·vf2 / 2μEf. This suggests that a larger mobility μ would give a smaller intrinsic loss rate γ0, and therefore a smaller HWHM. As shown in Fig. 4(d), when the mobility of graphene is increased from 300 to 3000 cm2 /(V·s) under a fixed SNR of 70 dB, the spectral resolution improves from 3.7 to 0.65 THz. Also, the improvement is more evident in the low mobility region (i.e., 300 to 1000 cm2 /(V·s)), and decreases towards the high mobility region (i.e., 1000 to 3000 cm2 /(V·s)). Such a variation trend matches that for the HWHM of the graphene plasmonic filter transmittance curve, which is plotted as the red line in the figure.
The above simulation results can be understood in a unified framework based on the Wiener deconvolution theory, which deals with the signal recovery problem in a noisy measurement process. According to the theory, the recovered spectrum in the Fourier domain can be expressed as follows,
where k is the Fourier frequency, and SF(k), TF(k), and IF(k) are the Fourier counterparts of the original spectrum S(λ), the filter transmission function T(λ), and the measured photocurrent I(V), respectively. In Eq. (13), the relative strength between TF(k) and 1/SNR(k) in the denominator of the right-hand expression determines whether the spectrum component at the Fourier frequency k can be properly recovered. Specifically, only in the k range where TF(k) >> 1/SNR(k) (for example, TF(k) is more than 10 times 1/SNR(k)), the influence of noise could be safely neglected, and hence the SF(k) can be properly recovered.In our case, TF(k) is a bandlimited function as shown in Fig. 5(a), and according to the characteristic of the Fourier transform, the bandwidth of TF(k) (denoted as BW) is inversely related to the HWHM of T(λ), i.e., BW × HWHM ∼ 1. Therefore, in the figure, the TF(k) curve with a broader BW (i.e., the red curve) corresponds to a T(λ) of a narrower HWHM. Also plotted in the figure is the 1/SNR(k) function. Since the noise is assumed to be a Gaussian white noise, the function is a constant in the whole k space. The colored areas in the figure correspond to the unrecoverable k regions because the noise at these Fourier frequencies surpasses the signal, or quantitatively, the condition TF(k) >> 1/SNR(k) at these k regions cannot be fulfilled. The figure clearly shows that TF(k) with a larger BW has a smaller unrecoverable k region. Therefore, its Fourier counterpart T(λ) could have higher frequency k components, which correspond to a higher resolution of the spectrum. In other words, the filter transmittance curve T(λ) with a narrower HWHM gives a better spectral resolution, which is in accordance with the aforementioned simulation results.
Fig. 5. Recoverable and unrecoverable regions in the k space for different HWHMs of the transmittance curve (a), and different SNRs of the measurement process (b). The criterion for judging the transition from the recoverable to the unrecoverable state is TF(k) = C/SNR(k) where the value of C is much larger than 1, for example, C = 10. In other words, if TF(k) > C/SNR(k), the signal is considered to be much larger than the noise, and the corresponding k component of the spectrum is recoverable, and vice versa.
Similarly, for a fixed graphene filter transmittance function T(λ), the unrecoverable k region in the Fourier space also varies with the measurement noise (or 1/SNR). As shown in Fig. 5(b), the green and brown areas represent the unrecoverable k regions for a large and a small SNR, respectively. Clearly, a larger noise (i.e., a larger 1/SNR) would result in a narrower recoverable k region, and therefore a lower spectral resolution, which is also in accordance with the previous simulation results.
2.4 Neural network-based recovery algorithm
To further improve the spectral recovery performance, a neural network is combined with the RGA to form a hybrid network for restoring the spectral information. The structure of the hybrid network is shown in Fig. 6(a). It is composed of 6 connected neural layers with the number of neurons in each layer being 61, 601, 2000, 5000, 10000, and 601, respectively. The first layer is the input layer where the detector measured photocurrent is fed. The second layer is the RGA layer, and it has 61${\times}$601 lines connected to the first layer which are called weight parameters. If we initialize these weight parameters as the element of the RGA matrix MRGA(α), then the layer acts as the solver of the RGA algorithm as expressed in Eq. (11). The third to the fifth layers are ordinary fully connected layers with Relu activation functions, and the last layer is the output layer. In such a hybrid neural network, the RGA layer could firstly generate a preliminary spectrum that is already similar to the target spectrum, and then the subsequent fully connected layers can further fine-tune the spectrum to improve its resolution and accuracy. Because the hybrid neural network can be trained to learn the characteristics of the input and recovered spectra, it can break the spectral resolution limit set by the HWHM of the filter and the SNR of the measurement process in the conventional RGA.
Fig. 6. (a) The structure of the hybrid neural network which consisted of an input layer, an RGA layer, 3 fully connected layers, and an output layer. (b) Comparison between the original three random spectra and the recovered three spectra by the RGA algorithm (the first row), the hybrid neural network without RGA layer uninitialized (the second row), the hybrid neural network with the RGA parameters initialized but untrainable (the third row), and the hybrid neural network with the RGA parameters initialized and trainable (the fourth row). (c) The training MSE loss function versus training epochs for the three hybrid neural networks.
The training process of the hybrid neural network is as follows: At first, 100000 input spectra are randomly generated, which are composed of 1∼5 Lorentz pulses with the HWHMs, the amplitudes, and central frequencies in the range of 0.375∼1.5 THz, 0.25∼1, and 22∼37 THz, respectively. Then, the measured photocurrent is calculated according to Eq. (9) by feeding into these generated spectra. In the calculations, the response matrix T in Eq. (9) is generated by assuming that the mobility of graphene is 1500 cm2 /(V·s) and the Fermi energy varies in the range of 0.2∼0.8 eV, the noise Inoise in Eq. (9) is set to a value to guarantee that the SNR is 60 dB. To evaluate the spectral recovery performance, a loss function is defined as the mean square error (MSE) between the estimated spectra and the actual input spectra. A smaller MSE means a higher similarity and thus a better spectral recovery performance. Furthermore, the network parameters are updated with the inverse gradient descent algorithm during the training.
To evaluate the performance of the hybrid neural network, we compare it with the RGA algorithm and two other modified hybrid neural networks, which have the same structure as the original hybrid neural network, but with the weights of the RGA layer uninitialized and untrainable, respectively. After 50 epochs of training with the training dataset, they are used to recover three randomly generated spectra, and the results are presented in Fig. 6(b). The first row displays the spectra recovered by the RGA algorithm, and the three spectra cannot be well recovered with it. This is because the peak spacings observed in the three spectra are beyond the resolution of the RGA algorithm, which is limited by the relatively low mobility of graphene (1500 cm2 /(V·s)). The second row gives the recovered spectra of the modified neural network with an uninitialized RGA layer. The spectral recovery performance is poor, and the network converges very slowly during the training process, as indicated in Fig. 6(c). In fact, such a network can be taken as a pure fully connected neural network since the RGA is uninitialized and trainable. This means using the neural network alone (without the RGA layer) cannot recover the spectra satisfactorily. The third and fourth rows in Fig. 6(b) are the results recovered from the hybrid neural networks with and without the RGA layer weights fixed, respectively. It is evident that both of them look much more similar than the previous two groups of results and some peaks with frequency spacings under 1 THz can be distinguished, meaning that a resolution better than 1THz is obtained. Moreover, as shown in Fig. 6(c), both networks converge much faster than the network with the RGA layer uninitialized (i.e., the non-ridge network in the figure). When comparing the two, the recovered spectra in the fourth row are slightly better than those of the third row, indicating that keeping the RGA layer parameters unfixed could bring a small extra benefit. In a practical measurement, a trainable RGA layer is more tolerant to calibration errors. This is because the calibration process is actually the process of determining the parameters of the RGA layer, and calibration errors of these parameters are inevitable. Hence, a trainable RGA layer could update these parameters and reduce the errors during the training process. Therefore, these results demonstrate that the combination of the RGA algorithm with a neural network can give a better spectral recovery performance than the RGA or the neural network alone, and it could also break the spectral resolution limit set by the HWHM of the graphene plasmonic filter and the SNR of the measurement process.
At last, it is essential to discuss the potential of the tunable graphene plasmonic filter based spectrometer. It is found that if we increase the mobility of graphene to 10000 cm2 /(V·s) and the SNR to 70 dB, the resolution of the spectrometer could reach 0.25 THz, corresponding to about 80 nm in the 8-14 µm range, which is acceptable in many applications. For comparison, the resolutions of many mainstream infrared spectrometers used in aerospace are about 50 nm [41]. At the same time, a graphene mobility of 10000 cm2 /(V·s) is not difficult to realize in experiments as demonstrated previously [42,43]. The SNR of 70 dB (i.e., the amplitude of the signal-to-noise ratio is about 3162) in our device structure is also attainable, because our graphene plasmonic filter is a band-stop filter, which means that a large proportion of the incident energy is passed to the detector rather than wasted as in the grating-based spectrometers. The resolution could be further improved by engineering the shape of the graphene filter transmittance curve T(λ) (e.g., by employing coupling phenomena between graphene ribbons and other micro-nano structures [44–46]) so that its Fourier counterpart TF(k) declines more slowly in the k space. In this case, the spectrometer could be more robust to the impact of noise according to the Wiener deconvolution theory.
3. Conclusion
In summary, a miniature spectrometer based on a tunable graphene plasmonic filter is proposed. The infrared transmission properties of the graphene plasmonic filter is firstly analyzed. It is found that the Fermi energy of graphene, the graphene ribbon width and the duty cycle affect the resonance frequency, the depth of the transmission dip, and the HWHM of the transmittance curve for the filter. These parameters are then optimized so that the filter transmittance curve could be beneficial to the miniature spectrometer. The RGA is applied to realize the spectral recovery, and a criterion to determine the optimal RGA coefficient is proposed for the best recovery performance at different SNR. Furthermore, it is demonstrated that the HWHM of the filter transmittance curve and the SNR of the measurement process are two key factors determining the spectral resolution, the mechanism of which can also be well explained in the framework of the Wiener deconvolution theory. Finally, a hybrid neural network that integrates the RGA and a fully connected network is proposed for spectral recovery. It significantly improves the resolution and accuracy of the spectrum recovery process. Our study offers a new solution for infrared miniature spectrometers and could contribute to the development of algorithm-based miniature spectroscopy and spectral imaging.
Funding
National Natural Science Foundation of China (62005281, 62101586).
Acknowledgment
We appreciate Peter Spencer for proofreading the manuscript for us.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. S. Weiss, “Fluorescence spectroscopy of single biomolecules,” Science 283(5408), 1676–1683 (1999). [CrossRef]
2. D. J. Harding, P. Gruene, M. Haertelt, G. Meijer, A. Fielicke, S. M. Hamilton, W. S. Hopkins, S. R. MacKenzie, S. P. Neville, and T. R. Walsh, “Probing the structures of gas-phase rhodium cluster cations by far-infrared spectroscopy,” J. Chem. Phys. 133(21), 214304 (2010). [CrossRef]
3. M. Haertelt, V. J. F. Lapoutre, J. M. Bakker, B. Redlich, D. J. Harding, A. Fielicke, and G. Meijer, “Structure determination of anionic metal clusters via infrared resonance enhanced multiple photon electron detachment spectroscopy,” J. Phys. Chem. Lett. 2(14), 1720–1724 (2011). [CrossRef]
4. N. M. Nasrabadi, “Hyperspectral target detection: An overview of current and future challenges,” IEEE Signal Process. Mag. 31(1), 34–44 (2014). [CrossRef]
5. Z. Feng, J. Zhang, and J. Feng, “Spectral-Spatial Joint Target Detection of Hyperspectral Image Based on Transfer Learning,” IGARSS 2020 - 2020 IEEE International Geoscience and Remote Sensing Symposium 1770–1773 (2020).
6. S. W. Sharpe, T. J. Johnson, R. L. Sams, P. M. Chu, G. C. Rhoderick, and P. A. Johnson, “Gas-Phase Databases for Quantitative Infrared Spectroscopy,” Appl. Spectrosc. 58(12), 1452–1461 (2004). [CrossRef]
7. J. Nong, L. Tang, G. Lan, P. Luo, Z. Li, D. Huang, J. Yi, H. Shi, and W. Wei, “Enhanced Graphene Plasmonic Mode Energy for Highly Sensitive Molecular Fingerprint Retrieval,” Laser Photonics Rev. 15(1), 2000300 (2021). [CrossRef]
8. D. R. Lobb, “Theory of Concentric Designs for Grating Spectrometers,” Appl. Opt. 33(13), 2648–2658 (1994). [CrossRef]
9. T. Pügner, J. Knobbe, and H. Grüger, “Near-Infrared Grating Spectrometer for Mobile Phone Applications,” Appl. Spectrosc. 70(5), 734–745 (2016). [CrossRef]
10. M.-L. Junttila, J. Kauppinen, and E. Ikonen, “Performance Limits of Stationary Fourier Spectrometers,” J. Opt. Soc. Am. A 8(9), 1457–1462 (1991). [CrossRef]
11. E. Laux, C. Genet, T. Skauli, and T. W. Ebbesen, “Plasmonic photon sorters for spectral and polarimetric imaging,” Nat. Photonics 2(3), 161–164 (2008). [CrossRef]
12. E. Li, X. Chong, F. Ren, and A. X. Wang, “Broadband on-chip near-infrared spectroscopy based on a plasmonic grating filter array,” Opt. Lett. 41(9), 1913–1916 (2016). [CrossRef]
13. C. Brown, A. Goncharov, Z. S. Ballard, M. Fordham, A. Clemens, Y. Qiu, Y. Rivenson, and A. Ozcan, “Neural Network-Based On-Chip Spectroscopy Using a Scalable Plasmonic Encoder,” ACS Nano 15(4), 6305–6315 (2021). [CrossRef]
14. N. K. Pervez, W. Cheng, Z. Jia, M. P. Cox, H. M. Edrees, and I. Kymissis, “Photonic crystal spectrometer,” Opt. Express 18(8), 8277–8285 (2010). [CrossRef]
15. B. Redding, S. F. Liew, R. Sarma, and H. Cao, “Compact spectrometer based on a disordered photonic chip,” Nat. Photonics 7(9), 746–751 (2013). [CrossRef]
16. Z. Wang, S. Yi, A. Chen, M. Zhou, T. S. Luk, A. James, J. Nogan, W. Ross, G. Joe, A. Shahsafi, K. X. Wang, M. A. Kats, and Z. Yu, “Single-shot on-chip spectral sensors based on photonic crystal slabs,” Nat. Commun. 10(1), 1–6 (2019). [CrossRef]
17. J. Bao and M. G. Bawendi, “A colloidal quantum dot spectrometer,” Nature 523(7558), 67–70 (2015). [CrossRef]
18. H. Li, L. Bian, K. Gu, H. Fu, G. Yang, H. Zhong, and J. Zhang, “A Near-Infrared Miniature Quantum Dot Spectrometer,” Adv. Opt. Mater. 9(15), 2100376 (2021). [CrossRef]
19. Z. Yang, T. Albrow-Owen, H. Cui, J. Alexander-Webber, F. Gu, X. Wang, T.-C. Wu, M. Zhuge, C. Williams, P. Wang, A. v Zayats, W. Cai, L. Dai, S. Hofmann, M. Overend, L. Tong, Q. Yang, Z. Sun, and T. Hasan, “Single-nanowire spectrometers,” Science 365(6457), 1017–1020 (2019). [CrossRef]
20. J. Zhang, Z. Cheng, J. Dong, and X. Zhang, “Cascaded nanobeam spectrometer with high resolution and scalability,” Optica 9(5), 517–521 (2022). [CrossRef]
21. A. C. Liapis, B. Gao, M. R. Siddiqui, Z. Shi, and R. W. Boyd, “On-chip spectroscopy with thermally tuned high-Q photonic crystal cavities,” Appl. Phys. Lett. 108(2), 021105 (2016). [CrossRef]
22. S. Yuan, D. Naveh, K. Watanabe, T. Taniguchi, and F. Xia, “A wavelength-scale black phosphorus spectrometer,” Nat. Photonics 15(8), 601–607 (2021). [CrossRef]
23. M. A. Kats, R. Blanchard, P. Genevet, Z. Yang, M. Mumtaz Qazilbash, D. N. Basov, S. Ramanathan, and F. Capasso, “Thermal Tuning of Mid-Infrared Plasmonic Antenna Arrays Using a Phase Change Material,” Opt. Lett. 38(3), 368–370 (2013). [CrossRef]
24. Z. Zhang, P. Lin, Q. Liao, Z. Kang, H. Si, and Y. Zhang, “Graphene-Based Mixed-Dimensional van der Waals Heterostructures for Advanced Optoelectronics,” Adv. Mater. 31(37), 1806411 (2019). [CrossRef]
25. Y. Yao, M. A. Kats, P. Genevet, N. Yu, Y. Song, J. Kong, and F. Capasso, “Broad electrical tuning of graphene-loaded plasmonic antennas,” Nano Lett. 13(3), 1257–1264 (2013). [CrossRef]
26. Z. Madadi, “Graphene-Based Tunable Plasmonic Perfect Absorber in FIR-Band Range,” Plasmonics 16(6), 1909–1914 (2021). [CrossRef]
27. Q. Guo, C. Li, B. Deng, S. Yuan, F. Guinea, and F. Xia, “Infrared Nanophotonics Based on Graphene Plasmonics,” ACS Photonics 4(12), 2989–2999 (2017). [CrossRef]
28. Y. Fan, N. H. Shen, F. Zhang, Q. Zhao, H. Wu, Q. Fu, Z. Wei, H. Li, and C. M. Soukoulis, “Graphene Plasmonics: A Platform for 2D Optics,” Adv. Opt. Mater. 7(3), 1800537 (2019). [CrossRef]
29. V. W. Brar, M. C. Sherrott, M. S. Jang, S. Kim, L. Kim, M. Choi, L. A. Sweatlock, and H. A. Atwater, “Electronic modulation of infrared radiation in graphene plasmonic resonators,” Nat. Commun. 6(1), 7032 (2015). [CrossRef]
30. Z. Fang, Y. Wang, A. Schlather, Z. Liu, M. Ajayan, F. J. Garcia de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active Tunable Absorption Enhancement with Graphene Nanodisk Arrays,” Nano Lett. 14(1), 299–304 (2014). [CrossRef]
31. A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S. K. Saha, U. V. Waghmare, K. S. Novoselov, H. R. Krishnamurthy, A. K. Geim, A. C. Ferrari, and A. K. Sood, “Monitoring dopants by Raman scattering in an electrochemically top-gated graphene transistor,” Nat. Nanotechnol. 3(4), 210–215 (2008). [CrossRef]
32. J. Nong, W. Wei, G. Lan, P. Luo, C. Guo, J. Yi, and L. Tang, “Resolved Infrared Spectroscopy of Aqueous Molecules Employing Tunable Graphene Plasmons in an Otto Prism,” Anal. Chem. 92(23), 15370–15378 (2020). [CrossRef]
33. W. Yao, L. Tang, J. Nong, J. Wang, J. Yang, Y. Jiang, H. Shi, and X. Wei, “Electrically tunable graphene metamaterial with strong broadband absorption,” Nanotechnology 32(7), 075703 (2021). [CrossRef]
34. J. Nong, L. Tang, G. Lan, P. Luo, C. Guo, J. Yi, and W. Wei, “Wideband tunable perfect absorption of graphene plasmons via attenuated total reflection in Otto prism configuration,” Nanophotonics 9(3), 645–655 (2020). [CrossRef]
35. A. Y. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons,” Phys. Rev. B 85(8), 081405 (2012). [CrossRef]
36. H. Zhuang, F. Kong, K. Li, and S. Sheng, “Plasmonic bandpass filter based on graphene nanoribbon,” Appl. Opt. 54(10), 2558–2564 (2015). [CrossRef]
37. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal Coupled-Mode Theory for the Fano Resonance in Optical Resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]
38. L. Tang, J. Du, C. Du, P. Zhu, and H. Shi, “Scaling phenomenon of graphene surface plasmon modes in grating-spacer-graphene hybrid systems,” Opt. Express 22(17), 20214–20222 (2014). [CrossRef]
39. L. Tang, W. Wei, X. Wei, J. Nong, C. Du, and H. Shi, “Mechanism of propagating graphene plasmons excitation for tunable infrared photonic devices,” Opt. Express 26(3), 3709–3722 (2018). [CrossRef]
40. L. Tang, J. Nong, W. Wei, S. Zhang, Y. Zhu, Z. Shang, J. Yi, and W. Wang, “Mode energy of graphene plasmons and its role in determining the local field magnitudes,” Opt. Express 26(5), 6214–6221 (2018). [CrossRef]
41. C. Liu, R. Xu, F. Xie, J. Jin, L. Yuan, G. Lv, Y. Wang, C. Li, and J. Wang, “New Airborne Thermal-Infrared Hyperspectral Imager System: Initial Validation,” IEEE J. Sel. Top. Appl. Earth Observat. Rem. Sens. 13, 4149–4165 (2020). [CrossRef]
42. A. Tyagi, V. Mišeikis, L. Martini, S. Forti, N. Mishra, Z. M. Gebeyehu, M. A. Giambra, J. Zribi, M. Frégnaux, D. Aureau, M. Romagnoli, F. Beltram, and C. Coletti, “Ultra-clean high-mobility graphene on technologically relevant substrates,” Nanoscale 14(6), 2167–2176 (2022). [CrossRef]
43. R. Nashed, C. Pan, K. Brenner, and A. Naeemi, “Ultra-High Mobility in Dielectrically Pinned CVD Graphene,” IEEE J. Electron Devices Soc. 4(6), 466–472 (2016). [CrossRef]
44. H. Li, Y. Zhang, H. Xiao, M. Qin, S. Xia, and L. Wang, “Investigation of acoustic plasmons in vertically stacked metal/dielectric/graphene heterostructures for multiband coherent perfect absorption,” Opt. Express 28(25), 37577–37589 (2020). [CrossRef]
45. H. J. Li, L. L. Wang, B. Sun, Z. R. Huang, and X. Zhai, “Tunable mid-infrared plasmonic band-pass filter based on a single graphene sheet with cavities,” J. Appl. Phys. 116(22), 224505 (2014). [CrossRef]
46. H. Li, C. Ji, Y. Ren, J. Hu, M. Qin, and L. Wang, “Investigation of multiband plasmonic metamaterial perfect absorbers based on graphene ribbons by the phase-coupled method,” Carbon 141, 481–487 (2019). [CrossRef]
