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Abstract
This paper presents the design and comprehensive analysis of an ultra-fast fractional-order temporal differentiator (DIFF) based on a plasmonic inner-wall Bragg grating microring resonator (BG-MRR). Due to the ring radius of 1.1 µm and the strong confinement of the surface plasmon polaritons in the plasmonic waveguide with very small cross-section, the overall footprint of the DIFF circuit is significantly small (approximately 4 × 2.5 µm2). By changing the coupling regimes of the microring resonator, a broad range for the differentiation order α, i.e., 0.7–1.7 and a wide 3 dB bandwidth of 3.1 THz [24.8 nm] for α = 0.7 and 3.9 THz [31.2 nm] for α = 1.7 have been realized. Comparing the outputs of the BG-MRR-based DIFF with the corresponding mathematical DIFF indicates that deviations for α > 1 are significantly larger than those of α < 1. Therefore, a fractional-order temporal DIFF circuit based on plasmonic cascaded BG-MRR has been proposed for α > 1.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In photonics, a temporal differentiator (DIFF) is one of the basic blocks for ultra-fast all-optical signal processing that provides α-th order temporal differentiation of the complex envelope of an arbitrary input optical pulse [1–3]. In general, the differentiation order α can be categorized into integer and fractional-orders [1,4].
One of the essential applications of the fractional-order DIFF is description of compact dynamic systems performance in micro and nano scales. For this reason, nowadays, the fractional-order DIFF has attracted a lot of attention in engineering applications such as signal processing [1,2]. More precisely, an optical fractional-order temporal DIFF can be used in various applications such as optical computing, differential equation solving, chaotic systems, image processing, ultra-fast coding, frequency measurement, pulse generation and shaping, and temporal pulse characterization [1,5–10].
To date, various devices have been proposed to implement the optical temporal DIFFs, which are mostly based on the fiber optics or silicon integrated waveguides. While directional couplers [11–13], photonic crystal cavities [14], semiconductor optical amplifiers [15], and twin-core fibers [16] have been employed to realize the optical temporal DIFFs, most applied devices can be categorized into three main groups: (1) grating structures [17–24]; (2) ring resonators [1–5,25–29]; and (3) Mach-Zehnder and multimode interferometers [28,30–33]. Grating structures can be constructed in various configurations, such as long-period fiber gratings [19], tilted fiber Bragg gratings [18,22] and phase-shifted Bragg gratings [20,23]. In recent years, it has been proven that on-chip microring resonator (MRR) is a well-qualified candidate to realize the optical temporal DIFF because of its compactness, maturity in fabrication, and opto-electronic integration. It has been proposed in various configurations, such as all-pass MRR (APMRR) [1,2], and MRR combined with Mach-Zehnder interferometer [29,30].
Although the optical fibers have been widely used in design of the optical DIFFs due to advantages such as simplicity, low loss, and low cost [17], advantages such as compactness, high stability, and optoelectronic integration can be realized with planar waveguides [31]. On the other hand, nowadays, increasing operation speed and compactness of the optical integrated circuits based on planar waveguides are particularly important. However, the optical diffraction limit prevents further compactness of conventional photonic devices [34–36]. The surface plasmon polaritons (SPPs) are electromagnetic excitations propagating at the interface between a dielectric and a conductor, which can localize light in the nano scale domain by overcoming its diffraction limit [37,38]. In recent years, with development of modern fabrication technologies, miniaturizing optical integrated circuits has significantly received attention. Therefore, the plasmonic waveguides can be considered as an ideal candidate for preparation of the optical integrated circuits in the nano scale [35,36]. In addition, signal processing devices based on plasmonic waveguides have a small footprint and high operation bandwidths that increases the processing speed of the optical pulses [39].
So far, several structures for plasmonic waveguides have been proposed. In [35], two fundamental planar plasmonic waveguides, metal-dielectric-metal waveguide and metal-dielectric-air waveguide have been investigated. Practically, to realize a plasmonic waveguide, a trade-off between optical mode confinement, propagation loss, structural dimensions, and fabrication technology should be considered. Recently, a nano scale three-dimensional plasmonic waveguide (TDPW) has been proposed, just by depositing an Ag stripe on a metal-dielectric-air waveguide. The core of TDPW is the region from bottom to top covered with the Ag stripe, which can provide high optical performance (mode confinement and propagation loss) at the telecommunication wavelengths. In addition, due to simple structure of the TDPW, its fabrication is easily possible [35,36]. Therefore, the TDPW can be considered as a suitable infrastructure for realizing optical integrated circuits in the nano scale.
Recently, a temporal DIFF based on a silicon hybrid plasmonic MRR with the ring radius of 1.2 µm has been proposed [26]. The differentiation orders and their corresponding 3 dB bandwidth achieved for this DIFF are respectively 0.6–1 and 1.74–2.5 THz (14 ‒ 20 nm). Also, a Gaussian waveform with temporal FWHM of 400 fs has been considered as an input pulse. However, a comparative analysis based on optical and mathematical results has not been provided in various coupling regimes of the MRR. On the other hand, the time-reversal characteristic has not been investigated in the temporal DIFF outputs. In fact, the output of the optical fractional-order DIFF should be compared with that of the time-reversed mathematical DIFF in order to evaluate performance of the designed plasmonic DIFF. Also, the time-delay characteristic of the optical DIFF outputs has not been studied. In addition, comparing phase response of the optical DIFF with that of the mathematical DIFF as an important criterion for determination of the differentiation order has not been presented. Moreover, the performance of the proposed DIFF for α > 1 has not been investigated. Eventually, deviations of the optical DIFF outputs from those of the mathematical DIFF have not been presented in the time domain.
In our previous paper [1], an optical fractional-order temporal DIFF based on an ultra-compact silicon MRR has been proposed with a small radius of 4 µm and 3 dB bandwidth of 139.643 GHz (1.1 nm). Also, its performance and accuracy have been evaluated considering various optical input pulses.
In this paper, for the first time to the best of our knowledge, design and comprehensive analysis of a fractional-order temporal DIFF based on plasmonic inner-wall Bragg grating MRR (BG-MRR) have been presented for sub-picosecond signal processing. The proposed plasmonic temporal DIFF has a significant bandwidth that realizes the first and fractional-orders derivatives of the complex envelope of the input pulses. Furthermore, thanks to the valuable features of the TDPW, the proposed DIFF circuit has a small footprint (approximately 4 × 2.5 µm2) compared to previous work [26]. In addition, to validate the designed plasmonic temporal DIFF and investigate its performance in the frequency and time domains, its simulation results obtained by the 3D-FDTD numerical method have been compared with the results calculated from the BG-MRR transfer function and the results computed from the formulas of the mathematical DIFF. The 3 dB bandwidth of the proposed plasmonic BG-MRR-based temporal DIFF is 3.1 THz [24.8 nm] for α = 0.7 and 3.9 THz [31.2 nm] for α = 1.7, which to the best of our knowledge, the obtained bandwidths are the highest values among the plasmonic MRR-based temporal DIFFs proposed to date, while the radius of the utilized MRR is only 1.1 µm. Due to the wide bandwidth of the proposed DIFF, ultra-fast optical pulses can be used for signal processing. Therefore, here, an ultra-short Gaussian pulse with the temporal FWHM of 300 fs has been launched as the input pulse to the DIFF. Also, for the various differentiation orders, key factors such as time-reversal, time-delay, and deviation of the plasmonic DIFF outputs have been computed and investigated with respect to the mathematical DIFF.
Comparing the outputs of the proposed and their corresponding mathematical DIFFs shows that with increasing α, deviation of the optical DIFF output from that of the mathematical DIFF increases. In addition, for α > 1, for example α = 1.7, the plasmonic DIFF cannot produce the three-peak output waveform same as the mathematical counterpart, which leads to increase significantly its output deviation. Therefore, using the mathematical properties of the fractional derivatives, a fractional-order temporal DIFF based on plasmonic cascaded BG-MRR (CBG-MRR) has been designed taking into account the propagation length of SPPs (Lspp). In order to investigate performance of the CBG-MRR-based DIFF, its results have been compared with those of the mathematical counterpart in the frequency and time domains.
The rest of this paper is organized as follows. In Section 2, the design and analysis of the proposed temporal DIFF are studied. Sections 3 and 4 present the analysis of the temporal DIFF in the frequency and time domains, respectively. Also, in Section 5, design of a temporal DIFF with differentiation order of α > 1 is proposed. Finally, the highlights of the proposed plasmonic DIFFs are summarized in Section 6.
2. Design and analysis of the plasmonic temporal DIFF
The α-order temporal derivative of f(t) is represented by Dtαf(t), where α is a positive real number. It has been proven that [1,40]
where ω and F(ω) represent the angular frequency and the Fourier transform of f(t), respectively. According to Eq. (1), the transfer function of the mathematical DIFF as a filter can be considered as follows [1,2]It has been shown that the MRR can be performed as a temporal DIFF with α = 1, α > 1, and α < 1 [1]. Therefore, in Sections 3 and 4, the transfer function of the mathematical DIFF, Eq. (2), will be used to evaluate the numerical simulation results of the temporal DIFF based on plasmonic BG-MRR.
The bent waveguide based on the TDPW with 90°-bending has excellent optical transmission properties, which can be used to design the MRR [35,36]. The scheme of the temporal DIFF based on plasmonic APMRR and its transverse cross-section along the dashed line are shown in Figs. 1(a), (b). The width (w) and height (hAg) of the Ag stripe and the thickness of the SiO2 layer (hSiO2) are three important geometrical parameters of the TDPW. In order to realize a trade-off between the optical mode confinement and the propagation loss, the values of hAg, w and hSiO2 have been considered as 200 nm, 80 nm and 70 nm, respectively [35,36]. The air gap between the ring and the bus waveguide is shown as g, and the length of the bus waveguide is 4 µm. Also, the refractive index of Ag has been obtained from [41].
Fig. 1. (a) Top view; (b) Transverse cross-section of the temporal DIFF based on the plasmonic APMRR; and (c) Electric field profile for the fundamental quasi-TM guided mode of the silver strip TDPW at λ = 1548 nm based on 3D-FDTD simulations. Effective mode area and physical area are 0.0190597 µm2 and 1.4 × 0.675 µm2, respectively.
As shown in Fig. 1(c), with applying the 3D-FDTD numerical technique for the modal analysis of the proposed TDPW, a good optical confinement for the fundamental quasi-TM guided mode has been realized in the silver strip TDPW at λ = 1548 nm. Its effective refractive index (i.e., β/k0, where β and k0 are the propagation constant and the wave number in free space, respectively [42]) is 1.842512 + 0.004477i and the propagation length of the SPPs is equal to 27.515 µm that can be calculated by [35]
The resonance condition of the MRR can be expressed by [42]
where m is the azimuthal mode number [42]. In order to realize the resonance wavelength of the deigned APMRR-based DIFF approximately equal to the telecommunication wavelength of 1550 nm (193.414 THz), the average radius of the APMRR has been designed equal to 1.2 µm.The intensity transmission and phase response curves of the designed APMRR-based DIFF are shown in Fig. 2 based on 3D-FDTD simulations. It is apparent that the phase jump at the resonance frequency of 193.798 THz (1546.93 nm) is equal to π. Therefore, it can be concluded that the designed APMRR has been located at the critical-coupling regime.
Fig. 2. 3D-FDTD simulation results for (a) intensity transmission; (b) phase response of the temporal DIFF based on the plasmonic APMRR. The inset shows |E|2 profile in the critical-coupling regime. (IL: Insertion loss)
In order to design an ultra-fast optical temporal DIFF based on plasmonic APMRR, 3 dB bandwidth of the APMRR should be enhanced. This goal can be achieved using grating structure, which effectively realizes desired 3 dB bandwidth and operation frequency [43,44].
According to Fig. 3(a), by applying inner-wall gratings with the period of Λ in the ring structure, the scheme of the temporal DIFF based on the plasmonic BG-MRR is proposed. The height and width of the gratings teeth are h and d = Λ/2, respectively. The resonance frequency of 193.664 THz (1548 nm) can be realized for the designed DIFF considering Λ = 60 nm and h = 40 nm. In order to investigate effect of the fabrication tolerances on the performance of the proposed DIFF, its geometrical parameters have been considered as d = 30 ± 5 nm, h = 40 ± 5 nm, and w = 80 ± 5 nm. The results obtained in Figs. 3(b), (c) show that the grating period (Λ = 2d) and the width of the waveguide (w) have small impacts on the resonance frequency. In addition, Fig. 3(b) shows that the resonance frequency changes considerably by variations of the height of grating teeth (h). In general, among the studied geometrical parameters, h has the most severe effect on the DIFF response.
Fig. 3. (a) Top view of the temporal DIFF based on the plasmonic BG-MRR; 3D-FDTD simulation results for (b) variations of the resonance frequency with respect to the gratings period considering various values for h; and (c) intensity transmission considering various values for w.
Based on the geometrical parameters shown in Fig. 3(a), it can be concluded that the distance traveled by the plasmonic coupled mode between the input and output of the DIFF structure (≈ 11 µm) is less than the value of LSPP (27.515 µm). Therefore, the plasmonic mode can well travel in the DIFF structure without tolerating considerable loss.
The intensity transmission and phase response of the BG-MRR-based DIFF in the critical-coupling regime are shown in Fig. 4, which indicate realization of a large 3 dB bandwidth equal to 27.9 nm. With comparing the obtained results for the APMRR-based and BG-MRR-based DIFFs shown in Figs. 2(a) and 4(a), it can be concluded that the BG-MRR-based DIFF has more 3 dB bandwidth and signal processing speed than the APMRR-based DIFF. Therefore, our goal has been achieved without increasing R (even with reducing it by 100 nm).
Fig. 4. Simulation results for the BG-MRR-based DIFF in its critical-coupling regime based on the 3D-FDTD (solid line) and analytical (dotted line) methods, (a) intensity transmission; (b) phase response.
In order to validate the above-mentioned results obtained from the 3D-FDTD method, simulations based on an analytical method using transfer function of the BG-MRR in its critical-coupling regime have been performed. The transfer function can be expressed by [45–47]
The simulation results obtained by the analytical method based on transfer function of the BG-MRR (HBG-MRR) have been shown in Fig. 4. Fortunately, the simulation results based on the 3D-FDTD and the analytical methods are in good agreement that validates our designed DIFF structure.
From the fabrication point of view, the bottom 150 nm Ag layer is deposited on top of a silicon wafer using metal evaporation tool. The SiO2 spacer with 70 nm thickness is deposited with sputtering method; then an approximately 300 nm thickness layer of poly-methyl methacrylate (PMMA) resist is spin-coated on top of the fabricated nanostructure. Next, electron beam lithography with sub-10 nm resolution is applied to structure pattern on PMMA resist. The patterned PMMA layer is used as a positive photoresist to design Ag strip nanostructures on the SiO2. Then 200 nm thickness Ag layer is provided using electron beam evaporation. Finally, a lift-off process is used to remove the PMMA from the structure [48–50].
3. Analysis of the temporal DIFF in the frequency domain
In this section, the intensity transmission and phase response of the temporal BG-MRR-based DIFF (simulated by the 3D-FDTD method) and the magnitude and phase responses of the corresponding mathematical DIFF (Eq. (2)) have been studied in detail. The BG-MRR has three coupling regimes, namely under-coupling with τ > art, critical-coupling with τ = art, and over-coupling with τ < art [42]. Figure 5 shows the intensity transmission and phase response of the BG-MRR-based DIFF. By altering the air gap, the coupling regime of the BG-MRR changes, which leads to the variation of 3 dB bandwidth, notch depth and phase jump at the resonance frequency. Also, same as Fig. 4, it is well apparent that for g = 93 nm, the critical-coupling regime has been realized with the deepest notch in the intensity transmission and the phase jump equal to π at the resonance frequency.
Fig. 5. (a) Intensity transmission; (b) phase response of the BG-MRR-based DIFF for various values of the air gap based on 3D-FDTD simulations.
The phase response and intensity transmission of the BG-MRR-based DIFF and the phase and magnitude responses of the mathematical DIFF are shown respectively in Figs. 6 and 7.
Fig. 6. Phase response curves of (a), (b) the BG-MRR-based DIFF in the under-coupling regime, and the mathematical DIFF with α = 0.7 and 0.9; (c) the BG-MRR-based DIFF in the critical-coupling regime, and the mathematical DIFF with α = 1; (d)-(f) the BG-MRR-based DIFF in the over-coupling regime, and the mathematical DIFF with α = 1.2, 1.4, and 1.7.
Fig. 7. Intensity transmission curves of (a), (b) the BG-MRR-based DIFF in the under-coupling regime, and the mathematical DIFF with α = 0.7 and 0.9; (c) the BG-MRR-based DIFF in the critical-coupling regime, and the mathematical DIFF with α = 1; (d)-(f) the BG-MRR-based DIFF in the over-coupling regime, and the mathematical DIFF with α = 1.2, 1.4, and 1.7. (Δ: Difference between notch depths)
It should be emphasized that necessary conditions for comparing spectral performances of the optical and the mathematical DIFFs are: 1) Resonance frequency at the intensity transmission and magnitude response curves be the same; and 2) Phase jump in the overlap region of the phase responses (in the neighborhood of the resonance frequency) be the same. Once these conditions are met, agreement between the intensity transmission and phase response of the optical DIFF with the magnitude and phase responses of the corresponding mathematical DIFF should be investigated only in the neighborhood of the resonance frequency. Also, the phase response is only criterion to determine the order of differentiation α. In fact, α is equal to the phase jump in the overlap region of the optical and mathematical DIFFs phase responses divided by π.
It is apparent from Figs. 6 and 7 that decreasing the air gap g from 101 nm to 84 nm causes change in the BG-MRR coupling regime from under to over and consequently increase in α from 0.7 to 1.7 and 3 dB bandwidth from 3.1 THz to 3.9 THz (significantly more than that of the previous work [26]). On the other hand, from theoretical point of view, decreasing the air gap increases the coupling coefficient, which leads to decrease the transmission coefficient [42]. Given that art is constant, the operating regime of the BG-MRR tends towards the over coupling, and the 3 dB bandwidth of the BG-MRR increases. So, simulation results shown in Figs. 6 and 7 are valid. However, it should be noted that with increasing α and 3 dB bandwidth, the agreement between the intensity transmission and magnitude response curves decreases. This phenomenon leads to increase in deviation of the output results of the plasmonic DIFF from the mathematical counterpart in the time domain that will be investigated in Section 4.
The insets depicted in Fig. 7 represent respectively the first-order differentiation (Fig. 7(c)) and the fractional-order differentiations (Figs. 7(a), (b), (d)-(f)) for a constant input pulse.
4. Analysis of the temporal DIFF in the time domain
To realize proper operation for the designed plasmonic DIFF, bandwidth of the input pulse should be smaller than the operation bandwidth of the DIFF [1,2,9]. So, according to Fig. 8, an ultra-fast Gaussian pulse with temporal FWHM of 300 fs (spectral bandwidth of 1.47 THz) has been considered as the input pulse applied to the optical and mathematical DIFFs.
Figure 9 shows the temporal outputs of the plasmonic BG-MRR-based DIFF (simulated by the 3D-FDTD technique) and the mathematical DIFF (inverse Fourier transform of Eq. (2)) for various differentiation orders. These outputs have two important characteristics: time-reversal and time delay.
Fig. 9. (a) and (b) Outputs of the BG-MRR-based DIFF in the under-coupling regime and mathematical DIFF with α = 0.7 and 0.9; (c) Output of the BG-MRR-based DIFF in the critical-coupling regime and mathematical DIFF with α = 1; (d)-(f) Outputs of the BG-MRR-based DIFF in the over-coupling regime and mathematical DIFF with α = 1.2, 1.4, and 1.7. (ATR: after time reversing; BTR: before time reversing)
The time-reversal characteristic obtained for the outputs of the mathematical DIFF with fractional differentiation orders of α < 1 and α > 1 is due to the inverse behavior of the phase response of the mathematical DIFF for fractional differentiation orders [5].
According to this characteristic, the time variable t replaces with ‒t. Therefore, the temporal outputs of the plasmonic fractional-order DIFF (the blue dashed line) obtained in the under and over-coupling regimes of the BG-MRR should be modeled as the time-reversed mathematical fractional-order DIFF (the red solid line), while the temporal output of the plasmonic first-order DIFF achieved in the critical-coupling regime of the BG-MRR can be modeled as the mathematical first-order DIFF without reversing its output pulse in time.
On the other hand, the output pulses of the BG-MRR-based DIFF have time delays compared to the associated mathematical DIFF. This characteristic is due to photon lifetime in the coupling regimes of the MRR [1].
In addition, output pulses of the plasmonic DIFF have deviations from those of the mathematical counterparts. This deviation can be evaluated quantitatively using the formula defined as bellow [2]
According to the results illustrated in Figs. 9(a)-(c), deviations equal to 1.46%, 3.6%, and 6.1% have been computed respectively for α = 0.7, 0.9, and 1. Therefore, the output pulses of the BG-MRR-based DIFF for α ≤ 1 have a very good agreement with those of the mathematical DIFF considering negligible deviations. In addition, according to the results shown in Figs. 9(d)-(f), relatively large deviations equal to 12%, 18%, and 21.3% have been obtained respectively for α = 1.2, 1.4, and 1.7. Therefore, in contrast to the case α ≤ 1, the output pulses of the BG-MRR-based DIFFs for α > 1 have considerable deviations. These significant values for the deviation grow proportionally with increasing α. In addition, for the fractional differentiation orders close to 2, the plasmonic temporal DIFF cannot produce the three-peak output pulse as the mathematical DIFF.
The deviation is due to three important factors: (1) temporal broadening of the output optical pulse due to inherent dispersion of the plasmonic waveguide; (2) partial disagreement between the phase response curves of the optical and mathematical DIFFs in the neighborhood of the resonance frequency; and (3) partial disagreement between the intensity transmission of the optical DIFF and magnitude response of the mathematical DIFF in the neighborhood of the resonance frequency, especially the difference between their notch depths; which the last two factors were discussed in Section 3. There are different methods for dispersion compensation, such as using the combination of a chirp filter and a phase modulator, which can be utilized to reduce the temporal broadening of the optical output pulses [51].
Therefore, it can be concluded that the designed plasmonic temporal DIFF based on the BG-MRR is very suitable for differentiation orders of α ≤ 1. In order to overcome the mentioned challenge for realization of appropriate plasmonic temporal DIFF for α > 1 (especially for α close to 2), another design should be proposed. This idea has been presented in Section 5.
5. Design a temporal DIFF with differentiation order of α > 1
It was shown that the designed temporal DIFF based on the plasmonic BG-MRR could not perform adequately for differentiation orders close to 2. Here, another plasmonic fractional-order temporal DIFF has been designed for α > 1 based on the mathematical principles of fractional derivative.
The fractional derivative formulas are categorized into three common types: Grunwald-Letnikov, Riemann-Liouville, and Caputo. It has been proven that the Caputo derivative has a good accuracy in the fractional calculus. One of the properties of the Caputo derivative is [40]
According to this formula, the fractional-order derivative of f(t) with the order of α + m is equal to the fractional-order derivative of mth derivative of f(t) with the order of α. Utilizing this property, a new configuration for fractional-order temporal DIFF based on plasmonic CBG-MRR has been proposed as shown in Fig. 10. This circuit consists of two cascaded plasmonic BG-MRR-based temporal DIFFs with air gaps g1 = 93 nm and g2 = 101 nm, respectively. The materials and geometrical parameters for this structure, except g1 and g2, are same as those of the BG-MRR-based-DIFF. The designed distance between the two rings, L, is equal to 1.72 µm, which prevents the coupling of the optical pulses between two rings. So, total footprint of this structure is approximately equal to 8 × 2.5 µm2. In addition, based on the geometrical parameters, it can be concluded that the distance traveled by the plasmonic coupled mode between the input and output of the DIFF structure (≈ 22 µm) is less than the value of LSPP. Therefore, the plasmonic mode can well travel in the DIFF structure without tolerating considerable loss.
The phase response and intensity transmission of the CBG-MRR-based DIFF have been shown in Fig. 11. Also, Fig. 12 shows the temporal outputs of the plasmonic CBG-MRR-based DIFF and the mathematical DIFF for α = 1.7.
Fig. 11. Simulation results for the CBG-MRR-based DIFF and the mathematical DIFF, (a) phase response; (b) intensity transmission.
Comparing Figs. 9(f) and 12 indicates that temporal broadening of the differentiated optical pulse for the CBG-MRR circuit is larger than that of the BG-MRR circuit. This is because of the nearly doubling the CBG-MRR length compared to the BG-MRR length, and consequently doubling the impact of waveguide dispersion on temporal broadening of the optical output pulse. It is interesting to note that despite this undesirable enhancement in the temporal broadening, the deviation has been reduced from 21.3% to 9.8%. The reason for this behavior should be sought in factors 2 and 3 expressed in Section 4.
About the 2nd factor, it can be seen that the overlap of the phase response curve of the optical DIFF with that of the mathematical DIFF in the neighborhood of the resonance frequency has been increased for the CBG-MRR-based optical DIFF (Fig. 11(a); α = 1.7) compared to the BG-MRR-based optical DIFF (Fig. 6(f); α = 1.7). So, this phenomenon has played a constructive role in reducing the deviation. The reason for this effect is superposition of the performances of the cascaded BG-MRRs with g1 = 93 nm (Fig. 6(c); α = 1 with perfect overlap) and g2 = 101 nm (Fig. 6(a); α = 0.7 with partial overlap) and consequently, the major role of the perfect overlap realized in the critical coupling regime (Fig. 6(c)).
Also, about the 3rd factor, it can be seen from comparing Figs. 7(f) and 11(b) that the difference between notch depths (Δ), has been decreased from 102 dB for the BG-MRR with g = 84 nm (α = 1.7) to 86 dB for the CBG-MRR with g1 = 93 nm and g2 = 101 nm (α = 1.7). So, this phenomenon has also played a constructive role in reducing the deviation. The reason for this effect is superposition of the performances of the cascaded BG-MRRs with g1 = 93 nm (Fig. 7(c); α = 1 with Δ = 41 dB) and g2 = 101 nm (Fig. 7(a); α = 0.7 with Δ = 29 dB).
Therefore, it can be concluded that the technique applied in order to overcome the challenges arising from the above-mentioned 2nd and 3rd factors, i.e., utilizing CBG-MRR instead of BG-MRR for DIFFs with α > 1 has led to enhance the agreement between the intensity transmission of the optical DIFF and the magnitude response of the mathematical DIFF and also the agreement between the phase response curves of the optical and mathematical DIFFs in the neighborhood of the resonance frequency. It is interesting to note that these constructive effects have led to a significant reduction in the deviation for the CBG-MRR-based DIFF compared to the deviation for the BG-MRR-based DIFF. This effect is so strong that not only overcomes the deviations arising from 2nd and 3rd factors, but also it overcomes the deviation enhancement arising from the waveguide dispersion.
Also, the three-peak output waveform has been produced by the CBG-MRR-based temporal DIFF same as the mathematical counterpart. Therefore, the quality of the differentiated pulse has been improved remarkably. As a result, the proposed CBG-MRR-based temporal DIFF is a successful idea to overcome the above-mentioned challenge for realization of appropriate temporal DIFF for α > 1 (especially for α close to 2).
6. Conclusion
In this paper, design, simulation, and comprehensive accurate analysis of an ultra-fast fractional-order temporal DIFF based on the plasmonic BG-MRR were presented for sub-picosecond signal processing. Thanks to the small transverse cross-section of the utilized plasmonic waveguide and the small ring radius, the designed ultra-compact DIFF has the footprint of approximately 4 × 2.5 µm2. It was demonstrated that the BG-MRR-based DIFF has more 3 dB bandwidth than the APMRR-based DIFF. In fact, a broad range for the differentiation order α, i.e., 0.7–1.7 and a wide 3 dB bandwidth of 3.1 THz [24.8 nm] for α = 0.7 and 3.9 THz [31.2 nm] for α = 1.7 were realized. Furthermore, with analyzing the time-reversal and time-delay characteristics and computing deviations of the plasmonic DIFFs output pulses from those of the mathematical counterparts in the various differentiation orders (0 < α < 2), the temporal outputs of the BG-MRR-based DIFF were compared with those of the mathematical DIFF. Moreover, by comparing the outputs of the BG-MRR-based DIFF with the corresponding mathematical DIFF, it was illustrated that the deviations for α > 1 are significantly larger than those of α < 1. Therefore, a fractional-order temporal DIFF circuit based on the CBG-MRR was designed for α > 1 with overall footprint of approximately 8 × 2.5 µm2. According to the simulation results for the differentiation order α = 1.7, the deviation in the CBG-MRR-based DIFF compared to the BG-MRR-based DIFF was reduced dramatically from 21.3% to 9.8%, realizing a good agreement between the output pulses of the CBG-MRR-based DIFF and the mathematical DIFF. Overall, it can be concluded that the comprehensive and accurate analysis presented here in the frequency and time domains plays an important role in obtaining a deep insight into the design of the plasmonic MRR-based fractional-order temporal DIFF.
Acknowledgments
We thank members of the Electromagnetics and Photonics Research Group (EPRG) at Sahand University of Technology for useful discussions.
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. F. Bahrami-Chenaghlou, A. Habibzadeh-Sharif, and A. Ahmadpour, “Full-wave analysis and design of optical fractional-order temporal differentiators based on ultra-compact microring resonator,” J. Modern Opt. 67(10), 880–889 (2020). [CrossRef]
2. A. Ahmadpour, A. Habibzadeh-Sharif, and F. Bahrami-Chenaghlou, “Electrically Tuned Fractional-Order Temporal Differentiator in Silicon Photonics,” Photo. Nan. – Funda. App. 47, 100969 (2021). [CrossRef]
3. G. Zhou, L. Zhang, F. Li, X. Hu, T. Wang, Q. Li, M. Qiu, and Y. Su, “All-optical temporal differentiation of ultra-high-speed picosecond pulses based on compact silicon microring resonator,” Electron.Lett. 47(14), 814–816 (2011). [CrossRef]
4. A. Zheng, J. Dong, L. Zhou, X. Xiao, Q. Yang, X. Zhang, and J. Chen, “Fractional-order photonic differentiator using an on-chip microring resonator,” Opt. Lett. 39(21), 6355–6358 (2014). [CrossRef]
5. B. Jin, J. Yuan, K. Wang, X. Sang, B. Yan, Q. Wu, F. Li, X. Zhou, G. Zhou, C. Yu, C. Lu, H. Yaw Tam, and P. K. A. Wai, “A comprehensive theoretical model for on-chip microring-based photonic fractional differentiators,” Sci. Rep. 5(1), 14216–14227 (2015). [CrossRef]
6. J. Hou, J. Dong, and X. Zhang, “Reconfigurable symmetric pulses generation using on-chip cascaded optical differentiators,” Opt. Express 24(18), 20529–20541 (2016). [CrossRef]
7. S. Liao, Y. Ding, J. Dong, T. Yang, X. Chen, D. Gao, and X. Zhang, “Arbitrary waveform generator and differentiator employing an integrated optical pulse shaper,” Opt. Express 23(9), 12161–12173 (2015). [CrossRef]
8. J. Hou, J. Dong, and X. Zhang, “Optical solver for a system of ordinary differential equations based on external feedback assisted microring resonator,” Opt. Lett. 42(12), 2310–2313 (2017). [CrossRef]
9. C. Cuadrado-Laborde and M. V. Andres, “Design of ultra-broadband all-optical fractional differentiator with a long-period fiber grating,” Opt. Quantum Electron. 42(9-10), 571–576 (2011). [CrossRef]
10. C. Cuadrado-Laborde, L. Poveda-Wong, A. Carrascosa, J. L. Cruz, M. V. Andres, and A. Diez, “Analog photonic fractional signal processing,” Prog. Opt. 63, 93–178 (2018). [CrossRef]
11. T. L. Huang, A. L. Zheng, J. J. Dong, D. S. Gao, and X. L. Zhang, “Terahertz-bandwidth photonic temporal differentiator based on a silicon-on-isolator directional coupler,” Opt. Lett. 40(23), 5614–5617 (2015). [CrossRef]
12. S. Sun, M. Li, J. Tang, N. H. Zhu, T. J. Ahn, and J. Azana, “Femtosecond pulse shaping using wavelength selective directional couplers: proposal and simulation,” Opt. Express 24(8), 7943–7950 (2016). [CrossRef]
13. T. T. Yan, W. H. Ren, and Y. C. Jiang, “Asymmetric Wavelength-Selective Directional Couplers as Fractional Order Optical Differentiators,” IEEE Access 7, 56533–56538 (2019). [CrossRef]
14. N. L. Kazanskiy, P. G. Serafimovich, and S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013). [CrossRef]
15. S. Sun, Y. Deng, N. Zhu, and M. Li, “Tunable fractional-order photonic differentiator using a distributed feedback semiconductor optical amplifier,” Opt. Eng. 55(3), 031105 (2015). [CrossRef]
16. H. You, T. Ning, L. Pei, W. Jian, J. Li, and X. Wen, “An all-fiber optical temporal differentiator for wavelength-division-multiplexed system based on twin-core fiber,” Opt. Quantum Electron. 46(11), 1481–1490 (2014). [CrossRef]
17. X. Liu and X. Shu, “Design of arbitrary-order photonic temporal differentiators based on phase-modulated fiber Bragg gratings in transmission,” J. Lightwave Technol. 35(14), 2926–2932 (2017). [CrossRef]
18. H. Shahoei, J. Albert, and J. Yao, “Tunable fractional order temporal differentiator by optically pumping a tilted fiber Bragg grating,” IEEE Photonics Technol. Lett. 24(9), 730–732 (2012). [CrossRef]
19. L. Poveda, A. Carrascosa, C. Cuadrado, and J. L. Cruz, “Long-period grating assisted fractional differentiation of highly chirped light pulses,” Opt. Commun. 363, 37–41 (2016). [CrossRef]
20. W. Zhang, W. Li, and J. Yao, “Optical differentiator based on an integrated sidewall phase-shifted Bragg grating,” IEEE Photonics Technol. Lett. 26(23), 2383–2386 (2014). [CrossRef]
21. X. Liu and X. Shu, “Design of an all-optical fractional-order differentiator with terahertz bandwidth based on a fiber Bragg grating in transmission,” Appl. Opt. 56(24), 6714–6719 (2017). [CrossRef]
22. M. Li, L. Y. Shao, J. Albert, and J. Yao, “Continuously tunable photonic fractional temporal differentiator based on a tilted fiber Bragg grating,” IEEE Photonics Technol. Lett. 23(4), 251–253 (2011). [CrossRef]
23. X. Liu, X. Shu, and H. Cao, “Proposal of a phase-shift fiber Bragg grating as an optical Differentiator and an optical integrator simultaneously,” IEEE Photon. J. 10(3), 1–7 (2018). [CrossRef]
24. A. Karimi, A. Zarifkar, and M. Miri, “Design of ultracompact tunable fractional-order temporal differentiators based on hybrid plasmonic phase-shifted Bragg gratings,” Appl. Opt. 57(25), 7402–7409 (2018). [CrossRef]
25. J. Dong, A. Zheng, D. Gao, S. Liao, L. Lei, D. Huang, and X. Zhang, “High-order photonic differentiator employing on-chip cascaded microring resonators,” Opt. Lett. 38(5), 628–630 (2013). [CrossRef]
26. A. Karimi, A. Zarifkar, and M. Miri, “Subpicosecond flat-top pulse shaping using a hybrid plasmonic microring-based temporal differentiator,” J. Opt. Soc. Am. B 36(7), 1738–1747 (2019). [CrossRef]
27. T. Yang, S. Liao, L. Liu, and J. Dong, “Large-range tunable fractional-order differentiator based on cascaded microring resonators,” Front. Opto. electron. 9(3), 399–405 (2016). [CrossRef]
28. H. Shahoei, D. X. Xu, J. H. Schmid, and J. Yao, “Photonic fractional order differentiator using an SOI microring resonator with an MMI coupler,” IEEE Photonics Technol. Lett. 25(15), 1408–1411 (2013). [CrossRef]
29. M. Liu, Y. Zhao, X. Wang, X. Zhang, S. Gao, J. Dong, and X. Cai, “Widely tunable fractional order photonic differentiator using a Mach–Zehnder interferometer coupled microring resonator,” Opt. Express 25(26), 33305–33314 (2017). [CrossRef]
30. W. Zhang, W. Liu, W. Li, H. Shahoei, and J. Yao, “Independently tunable multichannel fractional-order temporal differentiator based on a silicon-photonic symmetric Mach–Zehnder interferometer incorporating cascaded microring resonators,” J. Lightwave Technol. 33(2), 361–367 (2015). [CrossRef]
31. W. Liu, W. Zhang, and J. Yao, “Silicon-based integrated tunable fractional order photonic temporal differentiators,” J. Lightwave Technol. 35(12), 2487–2493 (2017). [CrossRef]
32. A. Zheng, T. Yang, X. Xiao, and Q. Yang, “Tunable fractional order differentiator using an electrically tuned silicon-on-isolator Mach–Zehnder interferometer,” Opt. Express 22(15), 18232–18237 (2014). [CrossRef]
33. J. Dong, A. Zheng, D. Gao, and L. Lei, “Compact, flexible and versatile photonic differentiator using silicon Mach–Zehnder interferometers,” Opt. Express 21(6), 7014–7024 (2013). [CrossRef]
34. P. Kumar, D. K. Singh, and R. Ranjan, “Optical performance of hybrid metal-insulator-metal plasmonic waveguide for low-loss and efficient photonic integration,” Microw. Opt. Technol. Lett. 62(4), 1489–1497 (2020). [CrossRef]
35. W. Wang, H. Ye, Q. Wang, and W. Lin, “Propagation Properties of Nanoscale Three-Dimensional Plasmonic Waveguide Based on Hybrid of Two Fundamental Planar Optical Metal Waveguides,” Plasmonics 13(5), 1615–1621 (2018). [CrossRef]
36. W. Wang and W. Lin, “Optical properties of plasmonic components based on a nanoscale three-dimensional plasmonic waveguide,” J. Opt. Soc. Am. B 36(8), 2045–2051 (2019). [CrossRef]
37. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). [CrossRef]
38. S. A Maier, Plasmonics: Fundamentals and Applications (Springer Science and Business Media, 2007).
39. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef]
40. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Elsevier, 1998).
41. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
42. V. Van, Optical Microring Resonators: Theory, Techniques, and Applications (Taylor & Francis Group, CRC Press, 2017).
43. Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quant. Electron. 41(9), 689–697 (2009). [CrossRef]
44. Y. M. Kang, A. Arbabi, and L. L. Goddard, “Engineering the spectral reflectance of microring resonators with integrated reflective elements,” Opt. Express 18(16), 16813–16825 (2010). [CrossRef]
45. C. E. Campanella, F. D. Leonardis, and V. M. N. Passaro, “Performance of Bragg grating ring resonator as high sensitivity refractive index sensor,” In 2014 Fotonica AEIT Italian Conference on Photonics Technologies (IEEE, 2014), pp. 1–4.
46. C. E. Campanella, A. Giorgini, S. Avino, P. Malara, R. Zullo, G. Gagliardi, and P. D. Natale, “Localized strain sensing with fiber Bragg-grating ring cavities,” Opt. Express 21(24), 29435–29441 (2013). [CrossRef]
47. H. Gu, H. Gong, C. Wang, X. Sun, X. Wang, Y. Yi, C. Chen, F. Wang, and D. Zhang, “Compact Inner-Wall Grating Slot Microring Resonator for Label-Free Sensing,” Sensors 19(22), 5038 (2019). [CrossRef]
48. F. Cheng, J. Gao, T. S. Luk, and X. Yang, “Structural color printing based on plasmonic metasurfaces of perfect light absorption,” Sci. Rep. 5(1), 11045 (2015). [CrossRef]
49. F. E. Camino, V. R. Manfrinato, A. Stein, L. Zhang, M. Lu, E. A. Stach, and C. T. Black, “Single-Digit Nanometer Electron-Beam Lithography with an AberrationCorrected Scanning Transmission Electron Microscope,” J. Visualized Exp. 139(139), e58272 (2018). [CrossRef]
50. X. Sun, D. Dai, L. Thylén, and L. Wosinski, “Double-Slot Hybrid Plasmonic Ring Resonator Used for Optical Sensors and Modulators,” Photonics 2(4), 1116–1130 (2015). [CrossRef]
51. D. Benedikovic, M. Berciano, C. Alonso-Ramos, X. L. Roux, E. Cassan, D. Marris-Morini, and L. Vivien, “Dispersion control of silicon nanophotonic waveguides using sub-wavelength grating metamaterials in near- and mid-IR wavelengths,” Opt. Express 25(16), 19468–19478 (2017). [CrossRef]
