Electromagnetically induced transparency (EIT) like transmission based on 3 × 3 cascaded multimode interference resonators

micromachines Article Electromagnetically Induced Transparency (EIT) Like Transmission Based on × Cascaded Multimode Interference Resonators Trung-Thanh Le International School (VNU-IS), Vietnam National University (VNU), Hanoi 1000, Vietnam; thanh.le@vnu.edu.vn; Tel.: +84-985-848-193 Received: July 2018; Accepted: August 2018; Published: August 2018 Abstract: We propose a method for generating the electromagnetically induced transparency (EIT) like-transmission by using microring resonator based on cascaded × multimode interference (MMI) structures Based on the Fano resonance unit created from a × MMI coupler with a feedback waveguide, two schemes of two coupled Fano resonator unit (FRU) are investigated to generate the EIT like transmission The theoretical and numerical analysis based on the coupled mode theory and transfer matrix is used for the designs Our proposed structure has advantages of compactness and ease of fabrication We use silicon waveguide for the design of the whole device so it is compatible with the existing Complementary Metal-Oxide-Semiconductor (CMOS) circuitry foundry The fabrication tolerance and design parameters are also investigated in this study Keywords: optical microring resonator; electromagnetically induced transparency (EIT); multimode interference (MMI); transfer matrix method (TMM); finite difference time difference (FDTD); beam propagation method (BPM) Introduction The electromagnetically induced transparency (EIT) effect is a nonlinear effect found in the interaction process between light and material The EIT effect has been intensively investigated in recent years [1,2] The EIT has wide applications such as in quantum information [3], lasing without inversion [4], optical delay, slow light [5], nonlinearity enhancement [6] and precise spectroscopy [7], pushing frontiers in quantum mechanics and photonics and sensing technology [8] In order to create the EIT effect, there are some suggested approaches There is a significant benefit to determine the optical EIT like transmissions with high modulation depth, which is defined by the difference in intensities between the peak and the dip at resonant wavelengths The EIT was first observed in atomic media [2] Then, the EIT-like effects were found in optical coupled resonant systems [9–11], mechanics, electrical circuits [12], plasmonics, metamaterials [7,13] and hybrid configurations [14] In the coupled resonant systems, the basic underlying physical principle is the interference of fields instead of the probability of amplitudes, as in a three-level atomic system [15,16] Most of the proposed structures so far for the optical EIT generation use metal-insulator-metal (MIM) plasmonic waveguide resonators [17,18], array of fiber optic resonators [19], microspheres [20], metallic arrays of asymmetric dual stripes [21], heptamer-hole array [22], plasmonic nanoring pentamers [23] and microtoroid resonator coupled system [24] For these systems, the fiber beam splitters, directional couplers or MIM plasmonic waveguide must be used As a result, such structures bring large size, the complexity of the fabrication process to control exactly the coupling ratios of the directional couplers and sensitivity to fabrication tolerance The transparency window of the EIT is caused by reduced absorption, due to the quantum destructive interference between the transitions from the two dressed states, into a common energy Micromachines 2018, 9, 0; doi:10.3390/mi9080000 www.mdpi.com/journal/micromachines Micromachines 2018, 9, of 10 level Similarly, the EIT-like effect generated by optical resonators work by the means of coherent interference between the resonating modes which produce optical transparency inside the absorption window [25] Compared to the EIT in atomic systems, the analogue of electromagnetically induced transparency with optical resonators based on directional couplers has many remarkable advantages such as simpler structure, smaller device size and easier design However, due to the small size of these structures, it is challenging to detune optical resonator for controlling the resonant interaction between the two optical pathways and controlling the coupling ratio of the directional couplers [26] In the literature, only × directional coupler was used for microring resonator based on the EIT effects [25] However, such structure is very sensitive to the fabrication It has a large size and requires a complicated fabrication process It was shown that the integration of multimode interference (MMI) and resonators can provide new physical characteristics By using the MMIs, we can overcome the disadvantages of devices based on directional couplers such as compactness, ease of fabrication and large fabrication tolerance [27] One of such structures is a × MMI based microring resonator We have proposed for the first time microring resonator structures based on × and × MMI couplers for Fano resonance generation [28–30] In this study, we further develop new structures based on only cascaded × multimode interference coupler based microring resonators to produce the EIT resonance like transmissions The proposed device is analyzed and optimized using the transfer matrix method, the beam propagation method (BPM) and finite difference time difference (FDTD) [31] A description of the theory behind the use of multimode structures to achieve the FRU and EIT effect is presented in Sections and A brief summary of the results of this research is given in Section Single Fano Resonance Unit (FRU) Fano resonance can be created by many approaches such as integrated waveguide-coupled microcavities [32,33], prism-coupled square micro-pillar resonators, multimode tapered fiber coupled micro-spheres and Mach Zehnder interferometer (MZI) coupled micro-cavities [34], plasmonic waveguide structure [35,36] We have proposed integrated photonic circuits for realizing Fano resonance based on × MMI and × MMI microring resonator [29,37] Figure 1a shows a scheme for Fano resonance unit (FRU) based on only one × MMI coupler with a feedback waveguide Figure 1b,c shows the FDTD simulation for the FRU with input signal presented at input ports and 2, respectively In the time domain, the Fano resonance system created by × MMI coupler based microring resonator in Figure can be expressed by the coupled mode equations [38] dan = [j(ω0 + δωn ) − ] an + d f n + dgn+1 dt τ (1) gn = exp(jφ) f n + dan (2) √ where n = 1, and d = j exp(jφ/2)/ τ; φ is the phase of the resonator depending on the feedback waveguide, δωn is the nonlinear phase shift, an is the amplitude in the resonator mode; f n , gn are the complex amplitudes at input and output ports; ω0 and τ are resonant frequency and lifetime of the resonator In the frequency domain, the × MMI coupler can be described by a transfer matrix M = [mij ]3×3 which describes the relationships between the input and output complex amplitudes (fields) of the coupler [39] The length of the MMI coupler is to be LMMI = Lπ , Lπ is the beat length of the MMI coupler The relationship between the output complex amplitudes bj (j = 1, 2, 3) and the input complex amplitudes (i = 1, 2, 3) of the coupler can be expressed by [39]   −e−j2π/3 b1  −j2π/3    b2  = √  e −1 b3  e−j2π/3 −1 e−j2π/3 −1 e−j2π/3 −e−j2π/3     a1 a1      a2  = M a2  a3 a3 (3) Micromachines 2018, 9,   b  −e − j2π/3 e − j2π/3 −1    − j2π/3 − j2π/3  e −1 e  b2  = 3   − j2π/3 −1 e −e − j2π/3  b3    a   a        a2  = M a2       a    a  (3) of 10 (a) (b) (c) Figure × 3×MMI (multimode interference) based resonator (b) Figure 1 (a) (a)Fano Fanoresonance resonanceunit unitcreated createdbyby3 3 MMI (multimode interference) based resonator FDTD simulation for for input and (c) (c) finite difference time difference (FDTD) simulation forforinput (b) FDTD simulation input and finite difference time difference (FDTD) simulation input2.2 The complex amplitudes at output ports and of the first microring resonator of Figure are The complex amplitudes at output ports and of the first microring resonator of Figure are given by given by mm m32 aexp(jq) (jq) aexp(jq) m m13aexp(jq) 13 mm 31 aexp b1 = (bm1 =(m )a11+(m +(m (4) + 13 31 )a + +13 32 )a )a2 (4) 11 +111 12 12 −1-m m33 aexp(jq) 33 −aexp(jq) m33 aexp(jq ) 1-m 33aexp(jq) m m32 aexp(jq) m23 m31 aexp(jq) )a1 +(m 22 m + m23 aexp(jq) )a (5) b2 = (m 21 + m m aexp(jq) − 23 m3331aexp(jq) )a +(m + 231 −32m33 aexp)a (jq) b2 =(m 21 + (5) 22 1-m 33aexp(jq) 1-m 33aexp(jq) where α = exp(−α0 L) is the transmission loss along the ring waveguide, L is the length of the feedback waveguide and α0 L) (dB/cm) the loss coefficient in the the optical waveguide; θ = of β0 Lthe is where is the is transmission loss along thecore ringofwaveguide, L is the length α = exp(−α the phase accumulated over the racetrack waveguide, where β0 = 2πneff /λ and neff is the effective feedback waveguide and α (dB/cm) is the loss coefficient in the core of the optical waveguide; refractive index, λ is the wavelength is the phase over the racetrack waveguide, where is θ = βIn β0SiO = 22πn study, weaccumulated use silicon waveguide for the design, where SiO2 (n = 1.46) usednas L this eff / λisand eff the upper cladding material The parameters used in the designs are as follows: the waveguide has a the effective refractive index, λ is the wavelength standard silicon thickness of h = 220 nm and access waveguide widths are W = 500 nm for single co waveguide for the design, where SiO ( n a = 1.46) is used as the In this study, we use silicon SiO mode operation It is assumed that the designs are for the transverse electric2 (TE) polarization at a upper material The in thewe designs as dimensional follows: the beam waveguide has a centralcladding optical wavelength λ = parameters 1550 nm In used this study, use theare three propagation h = 220 nm W = 500 nm standard silicon thickness of and access waveguide widths are co a method (3D-BPM) and Finite Difference Time Domain (FDTD) to design the whole structure [40] for singleFirstly, mode we operation It isthe assumed that are for the transverse electric polarization optimize position of the the designs access waveguide ports of the × 3(TE) MMI coupler to at a centralthe optical wavelength this study, we use the three(3) dimensional beam λ =31550 nm In determine proper matrix of the × MMI coupler expressed by Equation The normalized propagation method (3D-BPM) and Finite Difference Time Domain (FDTD) to design the whole output powers at output ports of the × MMI varying with the location of input port are shown in structure Figure 2a.[40] Figure 2b shows the normalized output powers at output ports for different locations of optimize theand position waveguide ports of the ×BPM MMI coupler to inputFirstly, port we Here, the width lengthofofthe theaccess MMI coupler are optimized by the simulations to determine the proper matrix of the × MMI coupler expressed by Equation (3) The normalized be WMMI = µm and LMMI = 99.8 µm As a result, the optimal positions of the input ports and output at µm, output ports of the × MMI varying with the location of input port are shown are p1,3powers = ∓ 2.05 respectively in Figure 2a Figure 2b shows the normalized output powers at output ports for different locations of input port Here, the width and length of the MMI coupler are optimized by the BPM simulations Micromachines 2018, 9, x Micromachines 2018, 9, x of 10 of 10 to be WMMI = μm and LMMI = 99.8 μm As a result, the optimal positions of the input ports and = μm and L = 99.8 μm As a result, the optimal positions of the input ports and to be W of 10 MMI2018, 9, MMI Micromachines 2.05 μm , respectively are p1,3 =  are p1,3 =  2.05 μm , respectively (a) (a) (b) (b) Figure2.2.Normalized Normalizedoutput outputpowers powers fordifferent different positionsofof(a)(a)input input port1 1and and (b)input input port2.2 Figure Figure Normalized output powers for for different positions positions of (a) input port port and (b) (b) input port port The phase sensitivity of the output signals to the length variation of the × MMI coupler based phase sensitivity sensitivity of of the the output outputsignals signalsto tothe thelength lengthvariation variationof ofthe the33× × MMI coupler based The phase microring resonators is particularly important to device performance We use the BPM to investigate resonators is particularly important to device performance We We use the BPM to investigate microring resonators the effect of the MMI length on the phase sensitivity Figure 3a shows the phases at output ports of effect of ofthe theMMI MMIlength lengthon onthe thephase phasesensitivity sensitivity Figure shows phases at output ports of the effect Figure 3a 3a shows thethe phases at output ports of the the × MMI coupler at different MMI lengths We see that a change of ± 10 nm in MMI length causes MMI coupler at different MMI lengths We see thata achange changeofof±±10 10nm nmin inMMI MMI length length causes 3the × 33×MMI coupler at −4 different MMI lengths We see that a change of 4.7 × 10−4 π (rad) in output phases For the existing CMOS circuitry with a fabrication a change change of of 4.7 4.7 × × 10 −4 ππ (rad) fabrication (rad) in in output phases For For the existing CMOS circuitry with a fabrication error of ± nm [41], this is feasible and has a very large fabrication tolerance Similarly, we consider ± nm [41], this is feasible and has a very large fabrication tolerance Similarly, we consider error of ± the effect of the positions of input waveguides on the phase sensitivity as shown in Figure 3b For a the positions positions of input input waveguides waveguides on the phase sensitivity as shown in Figure 3b 3b For a the effect of the fabrication tolerance in the MMI length of ±50 nm, the fluctuation in phases is nearly unchanged the MMI MMI length length of of ± ±50 fabrication tolerance in the 50 nm, nm, the the fluctuation fluctuation in in phases phases is is nearly nearly unchanged unchanged (a) (a) (b) (b) Figure The sensitivity of phases in output ports to (a) MMI (multimode interference) length and (b) Figure 3 The sensitivity of toto (a)(a) MMI (multimode interference) length andand (b) Figure sensitivity ofphases phasesininoutput outputports ports MMI (multimode interference) length position The of input position of input 2 (b) position of input Coupled Fano Resonances and Generation of the EIT Effect Coupled Fano Resonances and Generation of the EIT Effect Coupled Fano Resonances and Generation of the EIT Effect The schemes for coupled Fano resonances to generate the EIT effect is modeled in Figure 4, The schemes for coupled Fano resonances to generate the EIT effect is modeled in Figure 4, Thesingle schemes for coupled Fano resonances to generate theFigure EIT effect in Figure 4, where where Fano resonance and Fano resonance of 4a is modeled exactly the same and Fano where single Fano resonance and Fano resonance of Figure 4a is exactly the same and Fano single Fano 1resonance and Fano2resonance of 4awith is exactly the same resonance resonance and Fano 1resonance of Figure 4b is Figure different an exchange of and inputFano ports We show resonance and Fano resonance of Figure 4b is different with an exchange of input ports We show that and by Fano resonance of Figure 4b is different withinanFigure exchange of EIT input ports.can Webe show that by cascading two Fano resonances as shown 4, the effects created The that by cascading two Fano resonances as shown in Figure 4, the EIT effects can be created The cascading in Figure EIT effects can created The exchange exchangetwo unitFano can resonances be realized as byshown using only one 24,×the MMI coupler asbe shown in reference [42] unit exchange unit can be realized by using only one × MMI coupler as shown in reference [42] can be realized by using only one × MMI coupler as shown in reference [42] Micromachines 2018, 9, Micromachines 2018, 9, x Micromachines 2018, 9, x of 10 of 10 of 10 Figure of Coupled Coupled Fanoresonances resonances (a)bar bar connectand and (b)cross cross connect Figure4 4.Schemes Schemes of Figure Schemes of CoupledFano Fano resonances(a) (a) barconnect connect and(b) (b) crossconnect connect In our case, the Fano resonance and are the FRU created by × MMI coupler based In our ourcase, case,the the Fano resonance 2and are thecreated FRU created ×coupler MMIbased coupler based In Fano resonance and are 2the FRU by × 3by MMI microring microring resonator as shown in Figure The first schemes of Figure 4a can be made as shown in microring resonator as shown in Figure The first schemes of Figure 4a can be made as shown in resonator as shown in Figure The first schemes of Figure 4a can be made as shown in Figure 5a and Figure 5a and the second scheme of Figure 4b can be made as shown in Figure 5b Figure 5a and the second scheme ofbe Figure be made as shown the second scheme of Figure 4b can made4b ascan shown in Figure 5b in Figure 5b (a) (a) (b) (b) Figure Coupled Fano resonances based on microring resonators based on × cascaded MMI Figure 5 Coupled Coupled Fano Fano resonances resonances based based on microring microring resonators resonators based based on on 33×× 33 cascaded cascaded MMI MMI Figure couplers (a) cross without cross-connect andon (b) bar with cross-connect made from × MMI coupler couplers (a) and (b)(b) barbar withwith cross-connect mademade from from × MMI couplers (a) cross crosswithout withoutcross-connect cross-connect and cross-connect × coupler MMI [42] [42] coupler [42] By using analytical analysis, the transmissions at the output ports of Figure 5a, for the input By using analytical analysis, the transmissions at the output ports of Figure 5a, for the input signal at inputanalysis, port (the transmissions ) are expressed Bypresented using analytical at theby output ports of Figure 5a, for the input signal signal presented at input port ( ) are expressed by presented at input port (a2 = 0) are expressed by 2 m13m31αe jθ m13m31αe jθ m13m32αe jθ m 23m31αe jθ jθ jθ jθ jθ T1 = (m11 + m13m31αe )(m11 + m13m31αe ) + (m12 + m13m32αe )(m 21 + m 23m31αe ) jθ jθ jθ jθ T1 = (m11 + 1− m αe )(m11 + 1− m αe ) + (m12 + 1− m αe )(m 21 + 1− m αe ) jθ jθ jθ jθ 1− m33 1− m33 1− m33 1− m33 33αe 33αe 33αe 33αe (6) (6) Micromachines 2018, 2018, 9, 9, x0 Micromachines of 10 10 66 of 2 jθ m m m αemjθ αe jθ m13 mm mm13 mαe32jθαejθ mαe m31 αejθ 23jθ 31 αe m13 αe jθ) + (m + m 23 m 23+ m31 23 jθ31)(m11)(m 31 32 + )( m ) (6) T1 = (m11T+= (m13 + 31 12 21 )(m jθ21 + 21m33 αe 111+− m33 αejθjθ ) + (m 22 + − m33 1− −jθm)33 αejθ (7) 1− m αe jθ 1− m αe 1− m αe jθαe 1− m 1αe 33 αejθ 33 αejθ 33 33 αejθ jθ m23 m31 at these output m13 m31 of Figure 5b, for m m23 m31 The transmissions ports the input atαe input port 23 m 32 signal presented T2 = (m21 + )( m + ) + ( m + )( m + ) (7) 22 11 21 jθ jθ jθ jθ − m33 αe − m33 αe − m33 αe 1( ) are − m33 αe The transmissions at these output ports of Figure 5b, for the input signal presented at input port m 23m32αe jθ m13m32αe jθ m13m32αe jθ m13m31αe jθ (a2 = 0) are ' (8) T = (m11 + )(m 22 + ) + (m12 + )(m12 + ) T1 = (m11 + 1− m33αe jθ αejθ 1− m33αe jθ αejθ 1− m33αe jθ 1− m33αe jθ αejθ m13 m31 m m m m m m αejθ )(m22 + 23 32 jθ ) + (m12 + 13 32 jθ )(m12 + 13 32 jθ ) jθ − m33 αe − m33 αe − m33 αe − m33 αe 2 (8) m m αe jθ m m αe jθ m m αe jθ m m αe jθ (9) T' = (m 21 + 23 31 )(m 22 + 23 32 ) + (m 22 + 23 32 )(m12 + 13 32 ) jθ jθ jθ αejθ jθ jθ m23 m1− m231− mm3233 αe mm mαe m αejθjθ 1− 1− mm αe m33 αe 2333 32 αe 32 31 αe 13 T2 = (m21 + )(m22 + ) + (m22 + )(m12 + 33 ) (9) jθ jθ jθ jθ − m33 αe − m33 αe − m33 αe − m33 αe For our design, the silicon waveguide is used The effective refractive index calculated by the For our design, the silicon waveguide is used The effective refractive index calculated by the FDM (Finite Difference Method) is to be neff = 2.416299 for the TE polarization It assumed that the FDM (Finite Difference Method) is to be neff = 2.416299 for the TE polarization It assumed that the loss = 0.98 loss coefficient coefficient of of the the silicon silicon waveguide waveguide is is αα = 0.98 [43], [43], the the length length of of the the feedback feedback waveguide waveguide is is L = 700 μ m [25] For the first scheme of Figure 5a, the EIT effects shown in Figure can be LRR = 700 µm [25] For the first scheme of Figure 5a, the EIT effects shown in Figure 6a can be6a generated generated at output and while the input signal is port at the Figure 6b shows at output ports andports while the 2input signal is at the input input Figureport 6b shows the EIT effects the are EIT areatalso created at1output andinput while the is input signal is port alsoeffects created output ports and ports while1the signal presented at presented input portat2.input We see that We see that the modulation depth of 80% for these EIT like transmissions have been achieved As a the modulation depth of 80% for these EIT like transmissions have been achieved As a result, our result, our structure can generate both the W-shape and M-shape transmissions Such shapes can be structure can generate both the W-shape and M-shape transmissions Such shapes can be useful for useful optical switching, fast light and slow light and sensing applications opticalfor switching, fast and slow and sensing applications (a) (b) Figure of the the coupled coupled Fano Fanoresonances resonancesfor forFigure Figure5a5awith withinput inputsignal signal presented Figure 6 Transmissions Transmissions of is is presented at at (a) input port and (b) input port (a) input port and (b) input port For the second scheme of Figure 5b, the EIT effects shown in Figure 7a can be generated at output For the second scheme of Figure 5b, the EIT effects shown in Figure 7a can be generated at output port and port while input signal is at input port Figure 7b shows the EIT effects are also created port and port while input signal is at input port Figure 7b shows the EIT effects are also created at output ports and while the input signal is presented at input port at output ports and while the input signal is presented at input port Micromachines 2018, 9, x of 10 Micromachines 2018, 9, Micromachines 2018, 9, x of 10 of 10 (a) (b) Figure Transmissions of the coupled Fano resonances for Figure 5b with input signal is presented at (a) input port and (b) (a)input port (b) Transmissions of the coupled Fano resonances for Figure 5b with input signal is presentedof In Figure order our proposed analytical wefor use the FDTD accurate Figureto verify Transmissions of the coupled Fano theory, resonances Figure 5b withfor input signal ispredictions presented at at (a) input port and (b) input port the device’s working principle Figure (a) input port and (b) input port 2.8 shows the FDTD simulations for the device of Figure 5a,b for input signal at port 1, respectively Figure shows the FDTD of Figure 5a,b for input signal at order to verify our proposed analytical theory, of we userefractive the FDTD for accurate predictions port In In our FDTD simulations, we take into account of silicon material of In order to verify our proposed analytical theory, wethe use the FDTDindex for accurate predictions of the device’s working principle.equation Figure 8[44,45]: shows the FDTD simulations for the device of Figure 5a,b calculated by using the Sellmeier the device’s working principle Figure shows the FDTD simulations for the device of Figure 5a,b for for input signal at port 1, respectively Figure shows the FDTD of Figure 5a,b for input signal at input signal at port 1, respectively Figure shows the FDTD of Figure 5a,b for input signal at port port In our FDTD simulations, we2 take intoAaccount refractive index of silicon material Bλ12 of the (λ) = ε + of2the + refractive n account In our FDTD simulations, we take into index of silicon material calculated (10) by 2 calculated by using the Sellmeier equation [44,45]: λ λ − λ using the Sellmeier equation [44,45]: Bλ A-3 2 1λ2 = 1.1071 mm xε +10A where ε = 11.6858, A = 0.939816mm , B = 8.10461 andBλ (λ) = + n (10) (10) n2 (λ) = ε + λ22 +λ22− λ112 In our FDTD simulations, a Gaussian light pulseλof 15 λfs − pulse width is launched from the input λ1 to investigate the transmission characteristics of the device The grid sizes Δx = Δy = nm and -3 λ1 = 1.1071 mm where ε = 11.6858, A = 0.939816mm , 2B = 8.10461 x 10 and − ε =are 11.6858, = 0.939816 mm , B = 8.10461 × 10 and λ1 = 1.1071 mm chosenA[46] Δzwhere = 10 nm In our FDTD simulations, a Gaussian light pulse of 15 fs pulse width is launched from the input to investigate the transmission characteristics of the device The grid sizes Δx = Δy = nm and Δz = 10 nm are chosen [46] (a) (a) (b) Figure FDTD simulations for for input signal at input portport for the the EITEIT scheme of Figure 5a,b5a,b at at Figure FDTD simulations input signal at input for scheme of Figure wavelength λ =λ 1550 nm nm (b) wavelength = 1550 Figure FDTD simulations for input signal at input port for the EIT scheme of Figure 5a,b at wavelength λ = 1550 nm Micromachines 2018, 9, of 10 Micromachines 2018, 9, x of 10 Micromachines 2018, 9, x of 10 (a) (b) Figure FDTD simulations for input input port for port the EIT scheme of Figure 5a,b =1550 5a,b nm Figure FDTD simulations forsignal inputatsignal at input for the EIT scheme of λ Figure λ = 1550 nm For the purpose of comparing the theoretical and FDTD analysis, we investigate a comparison of the EIT like transmission effect between the theory and FDTD simulations It is shown that the (a) (b)is launched from the input In our FDTD simulations, a Gaussian light pulse of 15 fs pulse width FDTD simulation has a good agreement with our theoretical analysis as presented in Figure 10 to investigate transmission characteristics the device gridofsizes ∆x5a,b = ∆y = 5nm nm and Figure FDTD the simulations for input signal at input of port for the EITThe scheme Figure λ =1550 ∆z = 10 nm are chosen [46] For the wewe investigate a comparison of the purpose purpose of ofcomparing comparingthe thetheoretical theoreticaland andFDTD FDTDanalysis, analysis, investigate a comparison the EIT like transmission effect between the theory and FDTD simulations It is shown that the FDTD of the EIT like transmission effect between the theory and FDTD simulations It is shown that the simulation has a good with our theoretical analysisanalysis as presented in Figure FDTD simulation has aagreement good agreement with our theoretical as presented in 10 Figure 10 Figure 10 Comparison of theoretical and FDTD simulations Conclusions We have presented a new method for the generation of the EIT effect based on coupled × MMI based microring Figure resonators Both of the theoretical M-shape and W-shape like transmissions are created 10 Comparison and FDTD FDTD simulations Figure 10 Comparison of of theoretical and simulations The device based on silicon waveguide, that is compatible with the existing CMOS circuitry, has been optimally designed Our FDTD simulations show a good agreement with the theoretical analysis Conclusions Conclusions based on the transfer matrix method The EIT effect can be determined based on these structures with We have have presented new method the generation of EIT the effect EIT effect on coupled 3×3 We a anew method forfor thefabrication generation of the basedbased on coupled × MMI advantages ofpresented ease of fabrication and large tolerance MMI based microring resonators the M-shape and W-shape like transmissions are created based microring resonators Both Both of theofM-shape and W-shape like transmissions are created The The device based on silicon waveguide, that is compatible with the existing CMOS circuitry, has been been Funding: This research is funded by Ministry of Natural Resources and Environment of Vietnam under the device based on silicon waveguide, that is compatible with the existing CMOS circuitry, has optimally designed Our FDTD simulations show a good agreement with the theoretical analysis project BĐKH.30/16-20 optimally designed Our FDTD simulations show a good agreement with the theoretical analysis based on on the transfer matrix method The EIT can based theInterest: transferThe matrix method EIT effect effect can be be determined determined based based on on these these structures structures with with Conflicts of author declaresThe no conflict of interest advantages of ease of fabrication and large fabrication tolerance advantages of ease of fabrication and large fabrication tolerance References Funding: This This research researchisisfunded funded Ministry of Natural Resources and Environment of Vietnam under the Funding: byby Ministry of Natural Resources and Environment of Vietnam under the project Zhou, X.; Zhang, L.; Pang, W.; Zhang, H.; Yang, Q.; Zhang, D Phase characteristics of an project BĐKH.30/16-20 BĐKH.30/16-20 electromagnetically induced transparency analogue in coupled resonant systems New J Phys 2013, 15, Conflicts Conflicts of of Interest: Interest: The The author author declares declares no no conflict conflict of of interest interest 103033 Fleischhauer, M.; Imamoglu, A.; Marangos, J.P Electromagnetically induced transparency: Optics in References coherent media Rev Mod Phys 2005, 77, 633–673 Zhou, X.; Zhang, L.; Pang, W.; Zhang, H.; Yang, Q.; Zhang, D Phase characteristics of an electromagnetically induced transparency analogue in coupled resonant systems New J Phys 2013, 15, 103033 Micromachines 2018, 9, of 10 References 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Zhou, X.; Zhang, L.; Pang, W.; Zhang, H.; Yang, Q.; Zhang, D Phase characteristics of an electromagnetically induced transparency analogue in coupled resonant systems New J Phys 2013, 15, 103033 [CrossRef] Fleischhauer, M.; Imamoglu, A.; Marangos, J.P Electromagnetically induced transparency: Optics in coherent media Rev 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(m11 + 1− m 33? ?e jθ αejθ 1− m 33? ?e jθ αejθ 1− m 33? ?e jθ 1− m 33? ?e jθ αejθ m 13 m31 m m m m m m αejθ )(m22 + 23 32 jθ ) + (m12 + 13 32 jθ )(m12 + 13 32 jθ ) jθ − m 33 αe − m 33 αe − m 33 αe − m 33 αe 2 (8)... m 33 1− m 33 1− m 33 1− m 33 33? ?e 33 αe 33 αe 33 αe (6) (6) Micromachines 2018, 2018, 9, 9, x0 Micromachines of 10 10 66 of 2 jθ m m m αemjθ αe jθ m 13 mm mm 13 mαe32jθαejθ mαe m31 αejθ 23jθ 31 αe m13