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Abstract
The optical implementation of mathematical spatial operators is a critical step toward achieving practical high-speed, low-energy analog optical processors. In recent years, it has been shown that using fractional derivatives in many engineering and science applications leads to more accurate results. In the case of optical spatial mathematical operators, the derivatives of the first and second orders have been investigated. But no research has been performed on fractional derivatives. On the other hand, in previous studies, each structure is dedicated to a single integer order derivative. This paper proposes a tunable structure made of graphene arrays on silica to implement fractional derivative orders smaller than two, as well as first and second orders. The approach used for derivatives implementation is based on the Fourier transform with two graded index lenses positioned at the structure's sides and three stacked periodic graphene-based transmit arrays in middle. The distance between the graded index lenses and the nearest graphene array is different for the derivatives of order smaller than one and between one and two. In fact, to implement all derivatives, we need two devices with the same structure having a slight difference in parameters. Simulation results based on the finite element method closely match the desired values. Given the tunability of the transmission coefficient of the proposed structure in the approximate amplitude range of [0,1] and phase range of [-180, 180], on top of the acceptable implementation of the derivative operator, this structure allows obtaining other spatial multi-purpose operators, which are a prelude to achieving analog optical processors and even improving the optical studies performed in image processing.
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1. Introduction
With extensive advances in various branches of science and an increase in the amount of calculations as a result of optimizations using big data sets becoming an integral part of many scientific and industrial research projects, the need for faster processors is felt more than ever [1]. Meanwhile, existing electronic processors are limited in speed. The use of optical-electronic processors, on the other hand, will result in a decrease in speed and accuracy due to the need for optical-to-electronic and electronic-to-optical conversions [2]. Therefore, many efforts have been made to realize all-optical processors, which, in addition to outperforming their electronic counterparts in terms of speed, have high processing power and low power consumption [3]. To achieve high-speed and low-energy-consumption processors using modern methods, optical mathematical operators (for example, derivation, integral, convolution, Hilbert transformer, etc.) must first be created as building blocks of all-optical processors.
Initially, optical components such as lenses and filters were used to implement mathematical operators [4]. In such arrangements, it is necessary to spread light over a long distance, hence they have large dimensions, and as a result of light passing through free space and loss phenomena such as absorption and scattering, efficiency decreases [5]. Furthermore, these structures lack the ability to integrate, and it is quite challenging to align their elements [1]. Therefore, these operators perform inefficiently when it comes to implement the optical processors. Extensive research on the processing of optical signals and their implementation in integrated platforms was conducted in order to improve these flaws. Among all, micro- and nano-scale processors have always been attractive to researchers [6].
Optical operator research can be divided into two broad categories: temporal operators and spatial operators. Many advances have been made in the temporal domain, and some studies have not only focused on performing a single operator, but also on operators with tunable and reconfigurable capabilities [2,7]. Additionally, in this field, solving equations, including differential equations, has been thoroughly considered [8–10]. Despite being a significant improvement over their predecessors, these implementations have inherent limitations of their constituent microelectronic components, such as large dimensions, high energy consumption, and limited operation speed [11]. Following the studies on metamaterials and their applications in wave manipulation [12], Silva et al. presented the concept of “computational metamaterials” in 2014, which can be considered an important first step toward the realization of spatial optical analog calculators [13]. In addition to their small size, such operators have the ability to process in parallel, making them ideal for processing big data and images [14].
The first step in implementing a spatial mathematical operator is to extract its transfer function in the Fourier domain, which is then implemented using one of two approaches based on Fourier transform and Green's function [14]. Silva et al. implemented first and second derivative, integral, and convolution operators using both methods [13]. This work inspired researchers, and many studies in the field of optical implementation of mathematical operators in experiment and theory followed. Among the works done using the Fourier method are [15–23,3] and likewise for the Green's function method we can refer to [6,10,24–28]. The majority of these significant and influential studies have been on a few spatial operators, with a focus on first- and second-order derivatives, as well as integrals. Additionally, in most of the conducted studies, neither tunable nor reconfigurable systems were considered.
When it comes to the practical applications of optical mathematical operators, studies are limited to the first and second order derivatives in edge detection [29–31]. This is while fractional derivatives are among the most useful operators in science and engineering topics, such as electromagnetics, fluid mechanics, biological, optics, and signals processing [32], and many studies have been conducted in recent years about its applications in the field of image processing and the elimination of some of the challenges faced in image processing using integer-order derivative methods, such as image enhancement, image compression, and image edge detections [33].
To the best of our knowledge, no research has been carried out on the optical implementation of spatial fractional derivatives, hence it seems necessary to provide a device for the fractional derivative implementation. Moreover, no tunable structure with the capacity to perform derivatives of different orders has been introduced for spatial optical operators.
In this work, for the first time, the optical implementation of spatial fractional derivatives is studied. First, a tunable structure is introduced to implement fractional derivatives of orders smaller than 2. Naturally, the proposed structure can also implement derivatives of integer degrees; in this case, the derivatives of the first and second orders are also implemented. The surface conductivity of graphene depends on the gate voltage and this allows to tune the derivative orders, which makes graphene an appropriate material for the proposed structure. Out of the methodologies that have been presented, the Fourier transform is used for the implementation. Then the, performance of the proposed structure is evaluated using the finite element approach after analyzing correlations and extracting the required parameters. Several simulations are performed, and eventually, it is confirmed that the simulation results match the desired values quite well.
2. Proposed structure and theoretical background
Before proceeding with the optical implementation of an operator, it is important to first extract the transfer function of that mathematical operator. To design a system with input f(x) and output g(x) that can perform fractional derivatives, we have:
where α is the order of the derivative and a positive rational number. Several mathematical techniques for calculating fractional derivatives, such as the Riemann-Liouville, Granold-Letnikov, and Caputa methods, are better suited for use in physical and engineering applications [34]. However, due to the nature of optical calculations, the Fourier transform method is very effective for optical implementations. Despite this method's shortcomings in analytical computations, it is quite useful for our purpose. To find the transfer function, it is necessary to rewrite Eq. (1) in Fourier domain, that is:The transfer function is therefore of the form ${({i}{{k}_{x}})^\alpha }$ or ${({i}{x})^\alpha }$.
2.1 Proposed structure
Figure 1 shows our proposed structure. As can be seen, this structure has 3 main parts and the TM wave enters it along the y axis. In this illustration f(x) is the input, and g(x) is the output. To implement the transfer function, the Fourier transform must first be applied to the input wave. To this end, a Graded-Index (GRIN) lens with a length of Lg and a width of WS has been chosen. In Ref. [13], the following equation is used to calculate the relative permittivity of the GRIN lens:
where εc = 2.01 and x is the longitudinal distance of each point to the origin. Similarly, in our proposed structure, due to the difference in the wavelength used compared to the Ref. [13], to perform the appropriate Fourier transform in the GRIN lens, it is necessary to select the relative permittivity value of this area as follows:
Fig. 1. (a) A schematic view of the proposed structure which is constructed of two GRIN lenses and an operator zone. f and g represent the input and output waves, respectively. Ws, Lg, and Loz are the proposed structure width, the GRIN lens length, and the operator zone length, respectively. The operator zone is an array of unit cells made of graphene nanoribbons placed on silica. (b) Each unit cell of the operator zone with a width of Λ=400 nm consists of three graphene nanoribbons with a width of W = 280 nm and thickness of 1 nm and four spacer layers of silica. lP1 and lP2 are the distance of the nanoribbons from the first and last GRIN respectively, while ld=λd/4 is the distance between the graphene nanoribbons and λd is the wavelength in silica.
The input wave first enters the GRIN lens, and the Fourier transform is applied to it, and then the resulting wave enters the operator zone with a length of Loz and a width of Ws. This zone, which is responsible for performing the transfer function, applies the necessary changes on the Fourier transform of the initial wave. Finally, the wave leaving the operator zone is transformed into a spatial wave through the inverse Fourier transform. This is done by the final GRIN layer which has the same relative permittivity as the first GRIN layer.
2.2 Transfer functions implementation process in the operator zone
The operator zone shown in Fig. 1(a) is made by putting together several unit cells. The number of cells can be obtained from the ratio of the total width of the structure (Ws) to the width of each unit cell (Λ). This area is responsible for creating the transformation function, which leads to the desired output by manipulating the amplitude and phase of the input wave. Since this area comprises several unit cells, each cell implements a point of the amplitude and phase diagram of the transfer function.
As previously stated, the transfer function of the derivative operator with order α was obtained as (ix)α. Since the transfer function of each operator must have a maximum value of 1 according to the law of conservation of energy, it is necessary to normalize the obtained transfer function in the applied range. Therefore, for the fractional derivative with order α, the transfer function is as follows:
where -Ws/2 ≤ x ≤ Ws/2. For any value of x, this function is a complex number that, like other its amplitude is in the range of [-1, 1] and its phase is in the range of [-180, 180]. Therefore, the structure of the middle layer must have full control over the amplitude and phase. In this regard, the two methods of MTA [36] and MRA [37] are commonly used, providing the aforementioned full control but lacking tunability. To give a tunable structure, graphene is a suitable alternative for this part [38]. Due to its distinctive electrical, mechanical, and optical characteristics, particularly its low resistance and tunable surface conductivity, this substance is essential [39]. In addition to tunability, it will result in loss reduction. The chemical potential of graphene, and accordingly its optical properties, can be altered by applying a gate voltage [40,41]. Graphene can be modeled as an optical circuit element with RLC admittance [42]. To design a unit cell (Fig. 1(b)) there are several conditions to be met: The unit cell must have full control over the amplitude and phase, fulfill the impedance matching condition, and finally have the maximum transfer. As a result, here if we use only one layer of graphene, according to 1 + r = t, the maximum transfer is established if r = 0. But that also implies that we’ll have only one degree of freedom, and hence there would be no control over the phase. If two layers are used, despite the amplitude improvement, it still provides no control over the phase, so the optimal option is to employ three layers [36]. By choosing graphene, in addition to having complete control over amplitude and phase, a structure with tunability and lower loss can be constructed, which is impossible with previous conventional structures. To implement the concept of MTA proposed in Ref. [36], we use three substrated periodic graphene nanoribbons as shown in Fig. 1(a).To have the minimum transfer amplitude ripple and the required range of admittance, the best length for silica separator layers is λd/4 [43], where λd is the wavelength of light in silica. Also, in the simulations, the initial and final layers of graphene must be separated from GRIN. These distances are named lP1 and lP2, and their selected value will be assessed in the following.
2.3 Transmission line model of the operator zone’s unit cell
Figure 2 displays the transmission line model of a unit cell in the operator zone.
Fig. 2. Transmission line model of the proposed structure building block for the operating zone consisting of graphene and quarter-wavelength sections of silica. Tp1, Tp2, TG1, TG2, TG3, and Td respectively represent ABCD matrices for the distance between the input GRIN and first graphene, distance between the output GRIN and last graphene, the first graphene, the middle graphene, the last graphene, and quarter-wavelength distance. Y1, Y2, and Y3 are the equivalent admittance of graphene of the first, middle, and last layers, respectively.
Graphene can be regarded as a circuit component that simultaneously possesses the properties of an inductor, a resistance, and a capacitor [42], with the values of the resistance, inductance, and capacitance as follows:
Scenario 1: When one is nλ and the other is (2n + 1)λ/2. In this case, the final matrix will be as follows:
Scenario 2: When both are equal in value. As a result:
The transfer coefficient of structure S21 is then determined using the following equation [44]:
where η0 is the free space impedance. Considering that scenarios 1 or 2 have different ABCD matrix elements, the amplitude and phase of the transfer coefficient must be calculated for each one. In our structure, the desired transfer function is built by putting together several unit cells, each of which has an S21 with a pair of amplitude and phase values. Then, the amplitude and phase diagram of the transfer function are accordingly formed by connecting the corresponding values of each unit cell.Our goal is to obtain the values μc,1 and μc,2 for each cell. For this purpose, we break the amplitude and phase diagram of the transfer function into as many data points as the number of unit cells that we want to have. Then, we assign each pair of amplitude and phase values to S21 of each corresponding unit cell. Finally, based on Eq. (13), we obtain the μc,1 and μc,2 of each graphene nanoribbon should have to achieve the target S21 for each specific unit cell.
According to Eq. (8), the chemical potential of graphene should match the structural parameters such as the period length (Λ), and fill factor (W/Λ). On the other hand, these parameters are related to the resolution of the wavefront (the smaller the period length, the higher the resolution of the wavefront). Therefore, in the design process of the structure, in addition to the possible fabrication constraints, other points should also be considered. To effectively manipulate the amplitude and phase of the transmitted wave, the admittance mentioned in Eq. (9) should follow smooth changes around resonance frequency in a suitable range [38]. As stated in [42], a higher fill factor leads to a lower quality factor for each R-L-C branch. This issue emerges in the form of high bandwidth resonance and smooth changes in admittance. Selecting an operating frequency of 20 THz and a fill factor of 0.7, the period length should then be chosen such that the desired resonance occurs for a reasonable value of chemical potential, and moreover, full control over transmission amplitude and phase is achievable. In our design, these requirements are met by choosing Λ= 400 nm for µc = 0.85 eV. In other words, to increase the resolution of the wavefront, we need to reduce the period length. However, the extent of this reduction is limited by: having a reasonable amount of chemical potential, maintaining full control over the amplitude and phase of the transfer function, and compliance with manufacturing constraints.
For the optical simulation of the structure, we use the optical characteristics of each nanoribbon. The bridge between the chemical potential and permittivity of graphene is its surface conductivity. Graphene can be modeled using a surface conductivity of σs. According to Kubo's formula, σs can be written as follows [45]:
For full-wave simulations, graphene is used as an ultrathin layer. The thickness value (Δ) can be chosen arbitrarily, but it should be substantially lower than the wavelength.
The relative permittivity of graphene is calculated as follows [40]:
It indicates that chemical potential and permittivity coefficient are connected. For each derivative order, we extract the associated transfer function, and then the chemical potential that creates that transfer function. Therefore, what changes by altering the derivative’s order is the chemical potential of each graphene nanoribbon. Consequently, to change the derivative’s order, it is enough to adjust the chemical potential of each graphene nanoribbon, which is feasible due to the chemical potential's dependence on the gate voltage. The relation between gate voltage and chemical potential is as follows [46]:
where e is the electron charge, h is the substrate thickness, εr is the relative permittivity of the substrate, and vf is the Fermi velocity.3. Results and discussion
After extracting the equations related to the transfer function of the operator and the transfer coefficient of the operator zone, we can achieve the desired operator by stacking graphene nanoribbons together. In the following, we will discuss the implementation of fractional and integer derivatives. In the implementations for the input function of f(x)=ax exp(-x2/b), we have set b=λ02/0.9, a = 2.1/λ0 [13], Δ=1 nm, f = 20 THz, T = 300 K, λ0 = c/f, τ=1 ps, Ws = 5λ0+ 2λ0 ⁄nd, Lg = 12λ0-5 µm, and chemical potential in the range of [0.1 eV, 1.2 eV].
To implement the derivative function of a specific order, we first obtain the amplitude and phase values of the transfer function for that order. The amplitude and phase values for transfer function of derivatives of orders α = 0.3, 0.5, 0.7, 1, 1.5, 1.7, and 2 which, are plotted in Fig. 3. As can be seen, the amplitude values are symmetrical with respect to the x = 0 axis for all derivative orders, however, for phase values, this holds only for the 2nd order derivative.
Fig. 3. (a) Amplitude and, (b) phase of desired transfer functions for various derivative orders as functions of the structure’s width (Ws).
Figure 4 shows the phase and amplitude of S21 for both scenarios as a function of various internal and external chemical potential values. The next step is to obtain the chemical potential for each nanoribbon. For this, we split the amplitude and phase diagrams of the transfer function into the total count of unit cells, and according to the amplitude and phase of each unit cell in Fig. 3, we obtain the chemical potential from Fig. 4.
Fig. 4. Amplitude and phase of S21 as a function of internal and external graphene chemical potential for both scenarios: (a) amplitude and (b) phase of S21 for the first scenario. (c) amplitude and (d) phase of S21 for the second scenario.
As can be seen in Fig. 4, the difference between these two scenarios lies in the phase values (see plots (b) and (d)), which is very useful for our designs. To get a better understanding, let's look back at Fig. 3. As shown, for α ≤ 1 and 1 ≤α ≤ 2, the absolute values of the phase are in the ranges (0, 90] and (90, 180], respectively. On the other hand, for all orders, the amplitude varies between 0 and 1. In the first scenario, it is not possible to fully control the amplitude for phases in (90, 180], so this scenario won’t be useful to implement derivatives of orders greater than 1. The same holds for scenario 2 and orders below one. Therefore, the fact that there is a phase difference between scenarios 1 and 2 actually offers us a way to fully control the amplitude and phase. In other words, to implement all derivatives, we need two devices with the same structure having a slight difference in parameters: for α ≤ 1 the first scenario, and for 1 ≤ α ≤ 2 the second scenario must be applied to the design of the operator zone. The difference between these two scenarios lies in the distance between the graded index lenses and the nearest graphene array (lP1 and lP2). After finding the values of the chemical potential, the values of the surface conductivity and permittivity of each nanoribbon are calculated according to Eqs. (14) and (15).
3.1 Implementation of fractional derivatives
To implement fractional derivatives, we first consider the derivative of order 0.3. Its transfer function is in the form of T(x) = (ix / (Ws/2)0.3, and in this case, the amplitude and phase of the transfer function are shown in Fig. 3. To obtain the values of chemical potential, the first scenario should be used and according to the values of μc,1 and μc,2, the final structure can be implemented.
Figure 5(a) shows the distribution of the magnetic field along the z-axis for the order of 0.3. Length of the operator zone has been chosen 4λ0/nd with regard to impedance matching and minimizing the reflection of graphene in the GRIN lens. The results of the simulation (red dash) and the expected numerical solution (black solid) along with the input wave profile (blue round dot) are also shown in Fig. 5(b). As can be observed, the system's result closely matches the desired outcome. To check the ability of the proposed structure in performing fractional derivatives of other orders, the implementations were done for each order, and the results can be seen in Fig. 6. The solid black lines represent the desired results, and the dashed red curves represent the simulation results. Here again, the two lines match quite well. For derivatives of orders higher than one, the second scenario must be applied in the design of the operator zone. For this purpose, the length of the operator zone was set to 4.5λ0/nd.
Fig. 5. (a) Distribution of the magnetic field Hz for the derivative of order 0.3 and (b) the diagram of the input magnetic field (blue round dot) along with the simulated output (red dash) and the expected output (black solid). For α ≤ 1 the first scenario, and for 1 ≤ α ≤ 2 the second scenario has been applied.
Fig. 6. Derivative curve of orders (a) 0.5, (b) 0.7, (c) 1.5 and (d) 1.7. The black solid curves correspond to the desired results, and the red dotted curves correspond to the simulation results.
3.2 Implementation of the 1st and 2nd order derivatives
To check the effectiveness of the proposed structure in implementing integer derivatives, we will investigate the integer derivatives of orders 1 and 2. The results are shown in Fig. 7. Figure 7(a) displays the distribution of the magnetic field Hz for the second derivative. Plots (b) and (c) compare the desired and simulation results for the second and first-order derivatives, respectively. Here, the solid black lines represent the analytical solution, and the dotted red lines represent the result of the simulations. As can be seen, the answers match again.
Fig. 7. (a) Distribution of the magnetic field, Hz, for the second-order derivative. Curves of (b) second-order derivative and (c) first-order derivative. The black and red curves represent the analytical and simulation results, respectively.
Therefore, the proposed structure works well to implement not only fractional derivatives but also integer derivatives. An analytical approach was also used to find the desired results for the derivatives of the first and second orders. Of note, fractional orders, a numerical approach using the Grundwald- Letnikov method was applied. As shown in Figs. (6) and (7), there are slight discrepancies between the expected values and the simulation results in some derivative orders. The main reason lies in the fact that it is not always possible to find the ideal chemical potential. As mentioned earlier, the ideal amplitude and phase values for each point of the transfer function are extracted from Fig. 3. Then according to the suitable scenario and based on Fig. 4, the internal and external chemical potentials for nanoribbons are selected such that the ideal amplitude and phase extracted from Fig. 3 are achievable. But for some nanoribbons, it was not possible to choose the exact chemical potential and we inevitably chose the closest possible values.
Finally, the tunability of the presented structure is considered. According to Eq. (16), the gate voltage is directly proportional to the substrate thickness, h. Then, from experimental point of view, it is required to choose a very small gap between the graphene arrays and the gate electrode to achieve practical voltages between 100 and 200 V [47]. To this end, it is better to place each graphene-based transmit array on top of a thick silica substrate separated by a thin SiO2 dielectric spacer. As shown in Fig. 8(a), the dc bias voltage is applied between graphene nanoribbons and the gate electrode. The chemical potential of graphene can be locally tuned by adjusting the thickness of the spacer layer, h. Various derivative simulations were performed for thicknesses of h = 10, 25, 50, and 75 nm. Thicknesses of 25 and 50 nm (as opposed to thicknesses of 10 and 75 nm) led to results similar to the cases without spacers. In other words, the presence of a spacer layer with a thickness of 25 or 50 nm has a negligible effect on the previous simulation results. Therefore, the optimal thickness of h = 25 nm is chosen, which leads to a smaller gate voltage. It is worth noting that in the simulations we can use a graphene layer with a chemical potential close to zero as the gate electrode. So, its transfer matrix is in the form of Identity Matrix and there is no change in the ABCD matrix and as a result in S21. Since h << λd /4 the transfer matrix for distances between initial graphene nanoribbons remains unchanged. In other words, the transmission line model remains valid. Based on Eq. (16), due to the graphene/silica structure vf = 1.2x106 m/s [48], we can obtain the relation between chemical potential (in eV) and gate voltage as ${V_G} = 102.26\,\mu _c^2$. Figures 8(b) and 8(c) show the required values of gate voltage for external and internal nanoribbons along the structure’s width for various derivative orders of 0.5, 1.7, and 2. For the second-order derivative, due to the symmetry of the amplitude and phase of the transfer function, the values of the chemical potential are also symmetrical. By tuning the gate voltage applied on the graphene nanoribbons, the desired values for the chemical potential and consequently the desired derivative order can be achieved. Due to the large number of nanoribbons in this figure, it may be thought that overlap occurs in specific widths. Therefore, a part of Fig. 8(b) is enlarged to avoid this ambiguity.
Fig. 8. (a) 2D schematic view of the applying gate voltage to the proposed structure. The required values of gate voltage for (b) external nanoribbons and (c) internal nanoribbons along the structure’s width for various derivative orders of 0.5, 1.7, and 2.
4. Conclusion
In this work, for the first time in the field of spatial operators, a structure with the capacity to be tuned for the derivatives of different fractional and integer orders was proposed. In this way, we studied and analyzed the derivatives of orders 0 <α ≤ 2 because this range covers the entire breadth of phase transitions and for any other orders, the phase stays within the same bounds [-180,180]. For example, the 3rd order derivative’s phase is symmetric to that of the 1st order, and the same holds for orders 2.5 and 1.5. As for the amplitude, all derivative orders have a domain in the interval [0, 1]. As a result, it can be claimed that our proposed structure can perform all derivative orders. Like any research, this work can be improved. For example, we face challenges in creating impedance matching in this approach. Nevertheless, this research is a step towards the realization of analog optical processors, and it can be employed to improve optical image processing, especially in the case of edge detection. Also, due to the complete coverage of amplitude and phase by the two scenarios presented in this work, the proposed structure can likely implement other operators.
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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