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Abstract
Terahertz (THz) diffractive optical neural networks (DONNs) highlight a new route toward intelligent THz imaging, where the image capture and classification happen simultaneously. However, the state-of-the-art implementation mostly relies on passive components and thus the functionalities are limited. The reconfigurability can be achieved through spatial light modulators (SLMs), while it is not clear what device specifications are required and how challenging the associated device implementation is. Here, we show that a complex-valued modulation with a π/2 phase modulation in an active reflective graphene-plasmonics-based SLM can be employed for realizing the reconfigurability in THz DONNs. By coupling the plasmonic resonance in graphene nanoribbons with the reflected Fabry-Pérot (F-P) mode from a back reflector, we achieve a minor amplitude modulation of large reflection and a substantial π/2 phase modulation. Furthermore, the constructed reconfigurable reflective THz DONNs consisting of designed SLMs demonstrate >94.0% validation accuracy of the MNIST dataset. The results suggest that the relaxation of requirements on the specifications of SLMs should significantly simplify and enable varieties of SLM designs for versatile DONN functionalities.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Terahertz (THz) images, which are generated in a broad range from ∼0.1 THz to 30 THz, can provide complementary information to that obtained with microwaves, infrared, visible, and x-ray radiations [1]. The unique spectral properties of THz waves, such as excellent penetration ability in non-conducting materials, low non-ionizing photon energy, and a wide variety of spectral fingerprint features to identify specific molecule species [2,3] enable a variety of applications of THz imaging and sensing in security [4–6], agriculture [7–9], biomedical sensing [10,11], and pharmaceutics [12,13].
This process of forming THz images has mostly relied upon a pixel-by-pixel raster-scan serial data acquisition, typically with the mechanical movement of either the object under study [14] or of the illumination beam [15]. The process is generally slow and limits the image acquisition speed. Computational imaging, particularly compressive sensing through a few measurements by reconfiguring spatial light modulators (SLMs), has the advantages of impressively accelerating the image acquisition and facilitating the effective utilization of scarce and sophisticated THz components, such as single-photodetector imaging [16,17]. Moreover, the unprecedented growth of machine learning (ML) algorithms in recent years, such as convolutional neural networks, has revolutionized computer vision [18] and offered fresh perspectives for imaging technologies [19]. Similar to the post analysis employed to images obtained in other frequencies, a variety of ML algorithms have been applied to THz images as well [20].
The breakthroughs in the pioneering work by Lin et al. [21] judiciously combine ML algorithms with computational imaging in a unique all-optical hardware, diffractive optical neural networks (DONNs). This system has achieved THz image analysis such as classifications in a simultaneous process of image acquisition, when THz waves propagate through multiple carefully crafted diffractive optical components. The advantages of DONNs include few-shot measurements instead of pixel-by-pixel raster scans, and no need of post-image analysis or high-resolution high-end photodetector arrays. Furthermore, the redesign of DONN architectures enables simultaneous handling of multiple tasks [22]. However, the major limitation is the lack of reconfigurability in current systems. The conventional design wisdom assumes that a full-range 2π phase-only modulation is needed for SLMs, which is stringent for practical implementations [21,23].
Metasurfaces, two-dimensional planar surfaces with subwavelength functional optical structures, have become as a viable route to spatially regulate light amplitude, phase, and polarization in a versatile and compact manner and as a promising candidate for providing reconfigurability in optical ML hardware [24,25]. In this paper, based on the metasurface, we design a reflective SLM consisting of graphene nanoribbons resonantly oscillating in the THz region, which possesses a complex-valued modulation with a moderate phase tunable range. Furthermore, we show that the DONNs built from the arrays of designed SLM can achieve >94.0% validation accuracy of MNIST dataset. The obtained accuracy is comparable to the similar architectures built from passive components with a 2π phase range. This suggests the relaxation of the requirement of full-range 2π phase-only modulation in SLMs for DONNs, which will significantly simplify and enable varieties of SLM designs to diversify functionalities of DONNs.
2. Device and result
The structure of the graphene-plasmonics-based reflective SLM unit cell is illustrated in Fig. 1(a). The localized plasmonic oscillations are bounded by the boundaries of nanoribbons with width, W = 0.2 µm. A periodic array of these ribbons in a period, P = 0.3 µm, leads to a significant coupling of free-space THz radiation to the plasmonic waves in graphene. On top of these ribbons is a dielectric layer such as high-κ dielectrics Al2O3, and a top gate electrode is deposited for controlling the carrier densities or Fermi level, EF, of graphene ribbons. Underneath these ribbons is a high-resistance silicon (Si) substrate with thickness d = 6.0 µm. A back metallic reflector is coated to form a reflective device. The resonant graphene ribbons, the Si dielectric layer, and perfect metallic reflector together form an Fabry-Pérot (F-P) resonator. This sandwiched architecture can not only enhance the light-matter interaction for large phase response but also be capable of maintaining large amplitude response through engineering the interference of incident light and reflected light by carefully optimizing Si layer thickness.
Fig. 1. Graphene-plasmonics-based reflective SLM unit cell. The period of ribbon arrays is P, the width of each ribbon is W, and the distance between array and back reflector is d.
The optical response is then obtained through finite-difference-time-domain (FDTD) simulations. Here, in order to cascade multiple layers of diffractive SLMs, the calculation is done at oblique incident angle of 5°. Figure 2 (a) and (b) show the amplitude reflection spectra and phase spectra, respectively, when the graphene carrier mobility, µ = 5000 cm2/(Vs) and EF = 0.5 eV. On each reflection spectrum, there are two notches (points A and C in Fig. 2(a) and (b)), at which occur a large phase shift (∼π) and significant amplitude tuning with respect to the Fermi level. In contrast, in the middle of these two points, point B, the reflection amplitude tuning is small while the phase modulation is strong. Moreover, the spectra and the resonance frequencies of these two notches as a function of Fermi level are displayed in Fig. 2(c). One can find that the resonance frequencies increase when the Fermi level changes from 0.47 eV to 0.55 eV. Here, the clear anti-crossing behavior suggests a strong coupling of two photonic modes originating from the F-P cavity architecture, with one from graphene plasmonic resonance in the array of nanoribbons and their reflected counterparts.
Fig. 2. Amplitude and phase spectral response of graphene plasmonic resonators. (a) Amplitude reflection spectra of graphene nanoribbon structures at EF = 0.47 eV (green lines), 0.50 eV (blue lines), and 0.55 eV (red lines). Dashed lines are from FDTD simulations and solid lines are from the analytical model (details in analysis section). (b) Corresponding phase spectra. (c) All FDTD-simulated reflection spectra as a function of Fermi level. Blue and red dots indicate the resonance frequency positions. (d) Amplitude reflection and phase response as a function of Fermi level at specific frequency 10.8 THz.
Furthermore, at point B with a specific frequency of 10.8 THz, the reflection and phase response as a function of Fermi level is shown in Fig. 2(d). These results possess large and nearly unchanged reflection and significant phase tuning range. As the Fermi level is tuned between 0.47 eV and 0.55 eV, the reflection amplitude always maintains a high value of > 0.54 at the mobility of 5000 cm2/(Vs), while the phase tunable range is ∼ π/2. Note that, since nearly unchanged reflection (> 0.54) and significant phase tuning range ∼ π/2 is obtained at point B, such frequency (10.8 THz) can thus be used for image processing in DONNs.
3. Analysis
To clearly illustrate the physical process of the design structure, we further analyze the spectral response obtained through FDTD simulations by using coupled mode theory and the transfer-matrix method [26,27]. As depicted in Fig. 3(a), the total reflection rtotal consists of graphene reflection coefficient (rc1 from channel 1 or CH1) and the one propagating through the F-P cavity and being reflected (rc2 from channel 2 or CH2), so that rtotal = rc1 + rc2. To extract these reflection coefficients, the device can then be modeled as a multilayer dielectric structure as shown in Fig. 3(b). In contrast to the Si layer, the graphene-nanoribbon-array layer has both resonant and non-resonant behavior. As a result, it is modeled as a three-layer structure. For the non-resonant behavior, the graphene-nanoribbon-array layer can be treated as homogeneous layers with an effective refractive index, nmat. The resonant behavior can be considered as a virtual layer in the middle of two homogeneous layers.
Fig. 3. Analysis of the reflective graphene plasmonic resonator (SLM unit cell). (a) Illustration of interference of reflected light directly from graphene plasmonic resonance and from the F-P cavity formed with the back reflector. (b) The multilayer dielectric structure of SLM unit cell. (c) Three trajectories of two reflection channels and total signal at EF = 0.5 eV.
The propagation property of the dielectric layers without resonance behavior can be described by transfer matrices, which can be divided into interface transfer matrices and intra-layer transfer matrices based on transfer matrix method. The l-th dielectric interface can be represented by a transfer matrix:
The resonant behavior of the graphene-nanoribbon-array layer can be described by the transfer matrix, Tr, according to coupled mode theory. Since the graphene plasmonic resonance mode can couple to both the upward- and downward-traveling waves, the resonator can be considered as a single-mode optical resonator coupled with two ports as shown in Fig. 3(b). The equations for the resonance mode can be written as [28,29]
At the steady state (da/dt = 0), a can be given as
Therefore, we can obtain the transfer matrix Tr, which describes the relationship between downward and upward propagating waves through
Since propagation characteristics of each layer can be represented by (2 × 2) transfer matrices, the behavior of the device can be calculated based on this model. Thus, we can then obtain the parameters of the resonant cavity by fitting the calculated data and the simulated data. As shown in Fig. 2(a) and (b), the calculated spectra (solid lines) show excellent agreement with FDTD simulations (dash lines) with the Fermi level as 0.47 eV, 0.50 eV, and 0.55 eV. Specifically, the extracted intrinsic quality factor Qi increases and coupling quality factor Qc decreases as the Fermi level increases. At EF = 0.5 eV, Qi is ∼ 16.8 and Qc is ∼ 10.7.
Furthermore, Fig. 3(c) shows the trajectories of three reflection coefficients (blue line, rtotal, pink line, rc1, orange line, rc2) with increasing frequency at EF = 0.5 eV, which are obtained from the analytical model. It is clear that the shape of blue trajectory is similar to the orange one, and the pink trajectory only plays a major role in the real part of the blue trajectory. From the blue trajectory, one can find that as the frequency increases, the magnitude of the total reflection coefficient decreases to the minimum (point A), then increases to the maximum (point B) and finally decreases to the other minimum (point C). This trend is consistent with the two notches of the reflection spectrum at EF = 0.5 eV shown in Fig. 2(a).
4. Application
A reconfigurable THz DONNs system can then be built based on these abovementioned active SLMs, as shown in Fig. 4. When an input image formed under coherent illumination, such as in remote-sensing scenarios, is incident upon the SLMs of each layer, the wave front is reflected and diffracted to the next SLMs. The diffraction provides interconnections between pixels in adjacent layers, and the input and output waves of each SLM layer are controlled by the reconfigurable complex-valued reflection coefficients of graphene plasmonic modulators. This cascaded structure mimics the interconnection and weights of conventional neural networks, and the proof-of-concept experiment [21] based on passive components has shown the capability of classifying input images on detectors.
In our simulation, the design of DONNs consists of 5 layers of reconfigurable graphene-based SLMs with an array size of 200 × 200. The pixel size is 60 µm × 60 µm, which means there are 200 periods of graphene ribbons in each pixel (SLM unit cell). The oblique incident angle is set to 5° as abovementioned. Note that, with such a small incident angle, each SLM unit cell can control incident light independently with minimal pixel crosstalk. Furthermore, the conventional scalar diffraction theory under normal incidence can still be accurate enough [30]. In the DONNs, the distance between layers is set to be 8 cm to make sure there is enough room between adjacent layers to allow light propagation at the oblique angle 5°. For the device in each pixel, we use the amplitude and phase response of the device shown in Fig. 2(d) under conditions of μ = 5000 cm2/(Vs) and the frequency of 10.8 THz. And the device response is represented using the technique of categorical reparameterization with Gumbel-Softmax [31], which avoids the procedures of curve fitting and quantization. And, the analog-to-digital conversion is assumed to be 8 bits at the interface between driving circuits and the graphene gate. Meanwhile, in the driving circuit, the speed of the SLM can be estimated by using the formula 1/(2πRC), where C is the effective capacitance and R is the equivalent resistance for each SLM unit cell [32,33]. Furthermore, C is ε0εrAc/dg ≈ 0.26 pF, where ε0 = 8.85 × 10−12 F/m and εr = 9.34 are the vacuum permittivity and the relative permittivity of Al2O3, respectively. Ac = 32 µm2 is the effective area composed of metal area at the end of graphene ribbons, and dg = 10 nm is the thickness of the Al2O3 layer. Moreover, R can be estimated as ∼ 218 Ω with a 125 Ω/sq sheet resistance [34]. Thus, the speed of the SLMs in reconfiguration process is estimated to be∼2.8 GHz.
In the image analysis process, we explored handwritten digits in MNIST dataset. Figure 5(a) displays both the binarized input images of digits modified from MNIST dataset and the captured image obtained from the detector array. The input images are binarized for facile experimental implementation in the THz range [21]. For example, they can be produced through manually cutting metal foils. Ten sub-regions on the detector plane are defined to represent the labels for ten digits, and a softmax function is applied on the average intensity in each sub-region for classification. As the THz wave propagates through multiple SLMs, a clear focused beam spot on the detector plane suggests a correct classification of digit 7. Moreover, Fig. 5(b) displays the softmax probability corresponding to the detector reading for digit 7 in Fig. 5(a).
Fig. 5. The application of graphene-plasmonics-based THz DONNs on the classification of MNIST handwritten digits. (a) The classification patterns after propagating through 5 layers of graphene-based SLMs as well as the input pattern. The examples include digits 0 and 7 and the red squares in the detector plane indicate the area of interest for classification. (b) The corresponding label prediction probability after taking the softmax function of the average detector intensity over the area of interest. (c) The validation accuracy as a function of epochs. (d) The confusion matrix of 10000 input images of handwritten digits.
Figure 5(c) shows the validation accuracy as a function of training epochs and Fig. 5(d) displays the confusion matrix after computing 10000 input images, from which the overall validation accuracy is 94.0%. In contrast to the similar architecture reported in Ref. 21 that consists of 5 diffractive layers with 200 × 200 pixels on each layer and 2π phase tuning range for each pixel, the complex-valued modulation with a π/2 phase tuning range in graphene-plasmonics-based diffractive layers can achieve nearly the same prediction accuracy (94.0% in our case with respect to 93.4% in Ref. 21). This suggests a relaxation of requirements on SLM specifications and offers new perspectives of designing undemanding and scalable SLMs for THz DONNs. Moreover, when the frequency of incident wave varies around ±0.55 THz (±5.0%), the validation accuracy is still close to 94.0%, which indicates that the DONNs are robust over substantial spectral variations.
5. Discussion
The advancement of material synthesis processes continues improving the quality of large area graphene with high carrier mobility [35], which can also have an influence on graphene-based optoelectronic devices. In addition to the devices with graphene carrier mobility 5000 cm2/(Vs), we evaluated the performance of the SLM and DONNs under different carrier motilities of graphene. Figure 6(a) and 6(b) display the amplitude reflection and phase response at 10.8 THz as a function of Fermi level with two other mobilities of 7500 cm2/(Vs) and 10000 cm2/(Vs). As shown in Fig. 6(a), the amplitude reflection increases with respect to increasing mobility because of reduced carrier loss inside graphene. In contrast, the phase response is nearly identical. The corresponding validation accuracies are 96.1% and 95.1% for the mobility of 7500 cm2/(Vs) and the mobility of 10000 cm2/(Vs), respectively. This suggests that mobility have little effect on the performance of DONNs, since the phase response is nearly identical. However, in practice, the higher carrier mobility leads to higher light reflection and lower optical loss, and thus the total power efficiency of DONNs is higher.
Fig. 6. The effect of graphene mobility on device response. (a) The amplitude reflection and (b) phase shift as a function of graphene Fermi level at specific frequency 10.8 THz, and (c) The validation accuracy as a function of epochs for carrier mobilities of 5000 (red line), 7500 (green line), and 10000 (blue line) cm2/(Vs).
We further explored the relationship between the number of SLM layers in DONNs and phase modulation range of individual device in the SLMs. We artificially changed the phase modulation range of SLMs from 0.5π to 0.1π and 2π, while maintaining the reflection amplitude response. As shown in Fig. 7(a), more diffractive layers generally lead to better validation accuracy [21]. For the 0.5π phase tuning range, the accuracy has a moderate dependence on the number of SLM layers, while for the 2π range, the accuracy is always high. In the DONNs consisting of 5 and 6 layers of SLMs, the classification accuracies for phase ranges of 0.5π to 2π are quite close. However, it is challenging to implement SLMs with a full 2π phase modulation. The reasonably good performance of DONNs implemented using SLMs with 0.5π phase modulation range further suggests the relaxation of requirements on SLM specifications. In the case of 0.1π range, the accuracy shows a substantial dependence on the number of diffractive layers, and it is much lower than that obtained for 0.5π and 2π ranges. The accuracy substantially drops to 69.6% for 6 SLM layers and 48.0% for 3 SLM layers. Furthermore, the validation accuracy of DONNs system depends on the array size of SLMs. As shown in Fig. 7(b), the accuracy drops to 80.2% when using 100 × 100 size SLMs. To have a high validation accuracy, large-size SLMs should be employed in DONNs.
Fig. 7. The effect of phase shift, layer and size of SLMs on DONNs. (a) The validation accuracy of DONNs as a function of the number of SLM layers with different phase modulation ranges of individual SLM cells. (b) The validation accuracy as a function of epochs with 100 × 100 (blue line) and 200 × 200 (red line) size SLMs.
6. Conclusion
We have investigated an active, reflective graphene-plasmonics-based SLM, which can be employed to implement reconfigurable THz DONNs. By coupling the plasmonic resonance in graphene nanoribbons with reflected F-P modes from the back reflector, we achieved a small modulation of reflection and a significant π/2 phase modulation. This structure was simulated by using FDTD and analyzed through coupled mode theory and transfer matrix method. Furthermore, with such devices we have proposed a reconfigurable reflective THz DONNs consisting of SLMs with 200 × 200 pixels on each layer. The overall validation accuracy of MNIST dataset is >94.0%. Our results and further discussions suggest the relaxation of full-range 2π phase-only modulation requirements for DONNs, which will significantly simplify and enable varieties of SLM designs for versatile DONN functionalities.
Funding
National Key R&D Program of China (2018YFE0201000); National Natural Science Foundation of China (61875120); “Shuguang Program” supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission.
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.
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