1. Introduction

Due to their wavelength-scale dimensions and high Q factors, microring resonators are quite attractive devices and can be widely used as basic building blocks for future high density optical circuits, such as optical filters[1–3], switches, modulators[4, 5], memories[6], delay lines[7], sensors[8], pulse restorers[9] and polarization splitters[10]. Recently, there has been considerable interests in developing novel structures based on microring resonators, such as nested microring resonators[11], cross-connect microring resonators[12] and modified microring resonators or so-called microgear resonators[13]. Cascaded microring resonators have also received considerable attentions due to their abilities of tailoring spectra, compared to a single microring resonator by varying coupling coefficients and ring size.[1, 2, 7, 9, 12, 14–20] The system of two resonators is a simple scheme to realize cascaded resonators. For conventional dual microring resonators, resonators are formally series-coupled via a inter-resonator 2×2 coupler[12, 15–17] and parallel-coupled via feedback waveguides.[1, 7, 14, 18]

 

Fig. 1. (a).All-pass and (b)add-drop microring resonators (c) 2×2 coupler (d) 3×3 coupler. The dashed boxes are coupling regions correspond to couplers. Arrows represent optical transmission directions

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Figures 1(a) and 1(b) show two basic types of a single microring resonator, the all-pass and add-drop types, respectively. The latter distinguishes from the former by an added waveguide for dropping signals. They can be generally constructed by 2×2 couplers and ring resonators. The all-pass microring resonator has just one bus waveguide and hence one coupler, while the add-drop type has two bus waveguides and hence two couplers. The 2×2 coupler illustrated in Fig. 1(c) corresponds to coupling regions between the ring resonator and the straight waveguide, which serves the same function as a partial reflector. The coupling coefficient is critical for performances of the micoring resonator, such as the full width at half maximum(FWHM), the finesse and the Q factor[21].

In this paper, we propose new types of dual microring resonators or called 3×3 coupler based dual microring resonators. The proposed dual microring resonators are coupled via 3×3 couplers illustrated as shown in Fig. 1(d), and therefore different from other reported dual microring resonators[1, 7, 12, 14–18]. Due to the special transmission characteristics of 3×3 couplers[22], 3×3 coupler based dual microring resonators show interesting transmission characteristics. In section 2, we develop an analytical model for 3×3 coupler based dual microring resonators following the transfer matrix method. In section 3, transmission characteristics are studied and effects of coupling coefficients on them are investigated. Finally, effects of loss on transmission characteristics are discussed in section 4.

2. Theory

When assembling two microring resonators coupled via a 3×3 coupler, the add-drop type can be combined with the all-pass or the same type. The 3×3 coupler possesses different properties compared to the 2×2 coupler. Hence, transmission characteristics with the input at the 3×3 and 2×2 couplers are different. Considering the input at the center or upper port, 3×3 coupler based dual microring resonators can be grouped into four types as shown in Figs. 2(a)–2(d), namely, Types I-IV, respectively. Types I and II are combinations of the all-pass and the add-drop types with the input at the center and upper ports, respectively. Types III and IV are combinations of two add-drop types with the input at the center and upper ports, respectively. There is just one drop in types I and II, while two drops in types III and IV. Figure 2(e) presents a transfer matrix model for these types. The dashed line exists for type III and IV corresponding to the lower bus waveguide. Since components of these types are couplers (the 2×2 and 3×3 couplers) and resonators, transfer matrixes of these components will be given firstly. Then a detailed model for these proposed structures is developed.

 

Fig. 2. Schematic diagram and model of 3×3 coupler based dual microring resonators (a) combination of the all-pass and the add-drop types with the input at the center port (type I). (b) combination of the all-pass and the add-drop types with the input at the upper port (type II). (c) combination of two add-drop types with the input at the center port (type III). (d) combination of two add-drop types with the input at the upper port (type IV). (e) Transfer matrix model for these types. The dashed line exists for type III and IV corresponding to the lower bus waveguide. The dashed boxes correspond to the 2×2 and 3×3 couplers. Arrows represent optical transmission directions.

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2.1. Couplers and resonators

As shown in Fig. 2(e), complex amplitudes of the 2×2 coupler in the microring resonator n are denoted as c_n, d_n, e_n and f_n , with n = 1 and 2. d_n and f_n are complex amplitudes at the input port of the 2×2 coupler in the resonator n, while c_n and e_n at the output port. Their interactions can be described by the matrix relation [21]

where t_n = cos η_n and k_n = sinη_n are transmission and coupling coefficients of the 2·2 coupler, respectively. The parameter η_n characterizes the coupling strength between two adjacent waveguides of the 2×2 coupler.

Similarly, complex amplitudes of the 3×3 coupler in microring resonators at the input and the through ports are denoted as a_m and b_m , respectively, with m=0,1 and 2. a ₀, a ₁ and a ₂ (or b ₀, b ₁ and b ₂) correspond to complex amplitudes at the center, upper and lower waveguides of the 3×3 coupler, respectively. Assuming that the 3×3 coupler is in a planar structure and the coupling between the upper and lower waveguides is negligible, complex amplitude transfer characteristics can be expressed as [22]

where s ₁=(t ₀+1)/2 , s ₂ = i√2k ₀/2 , s ₃=(t ₀-1)/2 , s ₄=t ₀ and t ₀ =cos(√2η₀) and k ₀ =sin(√2η₀) ; χ₀ also characterizes the coupling strength between two adjacent waveguides of the 3×3 coupler. Generally, η_min Eqs. (1) and (2) can be derived using the well-known coupled-mode theory. In Ref [23], there is a detailed derivation of general mode coupling equations for the symmetric 3×3 coupler, which can be modified for the planar 3×3 coupler here. However, the specific form of η_m is not important for the purpose of our general analysis in this paper. Moreover, purely real or purely imaginary matrix elements in Eqs. (1) and (2) implies that the propagation constants of all the guides and resonators are matched. This is reasonable and some careful fabrications are required. The coupling strength coefficient η_m can be varied by changing distances between waveguides or interaction lengths of couplers. Because η_m decreases exponentially as the distance increases, changing interaction lengths is a more effective method by using racetrack-like microring resonator configurations. Hence, the coupling coefficient k_m can be tailored to any value in the range from 0 to 1.

The half-trip complex amplitude change factor in the resonator n is given by p_n = exp(iγ_nπR_n), where γ_n = β + iα_n is the complex propagation constant, β = 2πn_eff /λ is the real propagation constant, n_eff is the effective index and α_n=-ln(τ_n)/(2πR_n) is the amplitude attenuation coefficient of the resonator n τ_n and R_n are the round-trip transmission coefficient and the radius of the resonator n, respectively. Relations of complex amplitudes a_n, b_n, c_n and d_n in resonators are given by

To simplify the formalizations, we denote θ = 4π² n _eff R ₁/λ. and ∆θ(~[-π,π]) as the total and normalized round-trip phases of resonator 1, respectively, where θ = ∆θ+ 2mπ and m is the resonance order. The resonator size ratio is ρ_n= R_n/R ₁. Then the half-trip complex amplitude change factor can be rewritten as p_n = √τ_n exp(iρ_nθ/2). θ ₀ = 0 is denoted as the normalized round-trip phase resonance corresponding to the resonance in the range [-π, π] and 2π is the normalized free spectrum range(FSR) for a single microring resonator.

2.2. Normalized complex amplitude responses

Assuming that there are only input signals and no add signals, a ₀ is the input for types I and III, while f ₁ is the input for types II and IV. For types I and III, ξ₀, ξ₁ and ξ₂ are normalized complex amplitude responses at the though, the upper drop and the lower drop ports, respectively. For types II and IV, $\tilde{\xi }$ ₀, $\tilde{\xi }$ ₁ and $\tilde{\xi }$ ₂ are normalized complex amplitude responses at the though, the upper drop and the lower drop ports, respectively. By solving Eqs. (1)-(3) and noting that s ₁ -s ₃ = 1 , s ₁ + s ₃ = s ₄ , s ² ₁ -s ₃ ² = s ₄ , s ₁ s ₄ -s ₂ ² = s ₁ and s ₄ ² -2s ₂ ²=1 (these identities are useful for deriving and simplifying expressions), normalized complex amplitude responses for these types are found as

where S = 1-s ₁(t ₁ p ² ₁ + t ₂ p ² ₂) + s ₄ t ₁ t ₂ p ₁ ² p ₂ ² is the denominator for all the expressions. Equation (4a) corresponds to types I ( k ₂ = 0 , t ₂ = 1 , c ₂ = d ₂ and ξ₂ = e ₂ = f ₁ = f ₂ = 0 )and III ( f ₁ = f ₂= 0 ), while Eq. (4b) corresponds to types II ( k ₂ = 0 , t ₂ = 1 , c ₂ = d ₂ and$\tilde{\xi }$ ₂ = a ₀ = e ₂ = f ₂ = 0 ) and IV (a ₀ = f ₂ = 0).

2.3. Transmission spectra and phase responses

For types I and III T, D ₁ and D ₂ are denoted as transmission spectra at the through, the upper drop and the lower drop ports, respectively; and ϕ₀, ϕ₁ and ϕ₂ are corresponding phase responses, respectively. For types II and IV, T̃ , D̃ ₁ and D̃ ₂ are denoted as transmission spectra at the through, the upper drop and the lower drop ports, respectively; $\tilde{\varphi }$ ₀, $\tilde{\varphi }$ ₁ and $\tilde{\varphi }$ ₂ are corresponding phase responses, respectively. Using Eq. (5), transmission spectra and phase responses can be written as

where Arg(X) = atan[Im(X)/Re(X)] and X is an arbitrary complex number. Eqs. (5a) and (5b) are equations for types I (D ₂ = 0 and ϕ ₂ = 0 ) and III, while Eqs. (5c) and (5d) for types II (D̃ ₂ = 0 and $\tilde{\varphi }$ ₂ = 0 ) and IV. Due to ξ₁ = $\tilde{\xi }$ ₁ in Eqs. (4a) and (4b), we can obtain

When there is no loss, the sum of transmission spectra at the through and drop ports equals to unity due to the energy conservation, meaning that

Equation (4) affords a universal description of these types with arbitrary resonator and waveguide parameters. Transmission characteristics can be calculated for structures with arbitrary ring sizes, coupling coefficients and loss using Eqs. (4) and (5). However, we study the symmetrical dual microring resonators in this paper. Related assumptions are given as: (i) Resonators have identical sizes (i.e., ρ₁ = ρ₂ = 1 , hence R ₁=R ₂=R and p ₁ = p ₂ = p = √τ exp(iθ/2) ) and round-trip transmission coefficients (i.e., τ₁=τ₂=τ) (ii)The two 2×2 couplers are identical (i.e., η₁=η₂=η, hence k ₁=k ₂ =η and t ₁=t ₂=t) for types III and IV. Using above conditions, simplified expressions of normalized complex amplitude responses are rewritten as follows.

Type I:

Type II:

Type III:

Type IV:

The above Eqs. (8)–(11) are in forms of four parameters: the 2×2 coupler coupling coefficient k(or $t=\sqrt{1-k^2}$ ), the 3×3 coupler coupling coefficient k ₀ (or $t_0=\sqrt{1-k_0^2}$ ), the normalized round-trip phase ∆θ and the round-trip transmission coefficient τ of microring resonators.

3. Transmission characteristics

In this section, we focus on the lossless cases of these types (i.e., τ = 1). General transmission spectra and phase responses are analyzed and calculated firstly. Then effects of coupling coefficients on transmission spectra at the drop ports are discussed.

3.1. General transmission characteristics

Noting that η₀ as well as η characterizes the coupling strength of two adjacent waveguides, k ₀ = sin(√2η₀) and k = sin(η), thus k ₀ is slightly greater than k when η ₀ ≈ η. To give a description of general transmission characteristics, coupling coefficients are chosen as k ₀=0.5 and k = 0.4 , corresponding to η₀ = asin(0.5)/√2 ≈ 0.37 and η = asin(0.4) ≈ 0.41, respectively. Transmission spectra and phase responses are calculated as shown in Figs. 3(a)–3(d) and Figs. 3(e)–3(h), respectively. Transmission spectra at the through and the drop ports are complementary and their sum equals to unity, validating the transfer matrix model. Hence, studies on transmission spectra at the drop ports are sufficient for mastering properties of these types.

As shown in Fig. 3(a), type I holds interesting transmission spectra. Taking T as example, the spectrum has a narrow peak within a notch. Generally, there are two minima in T and two maxima in D ₁ . This is similar to the phenomenon, namely, coupled-resonator-induced transparency (CRIT).[1, 7, 14, 15, 17, 18] In series-coupled dual microring resonators with small loss[15, 17], the mode splitting and the classical destructive interference give rise to CRIT. Series-coupled dual microring resonators[15, 17] can be reduced as a single microring resonator with an equivalent embedded 1×1 coupler (or phasor). Thus, the resonance condition is changed compared to that of a single microring resonator and hence the so-called CRIT appears. In parallel-coupled dual microring resonators with a slight nonzero detuning of the two ring resonances[1, 7, 14, 18], CRIT is caused by the interference between direct and indirect pathways for two cavities’ decays. The transparency resonance coincides with the resonant wavelengths of these dual microring resonators. By introduction of 3×3 couplers, CRIT is also obtained here. The 3×3 coupler and the lower microring resonator in proposed types can also be reduced as an equivalent embedded 2×2 coupler, whose transfer matrix is more complicated than a normal 2×2 coupler. From the equivalent approach, CRIT here is similar to that in series-coupled dual microring resonators, with effective transfer matrixes different. The resonance condition is changed due to the phase changes in the equivalent embedded couplers and hence the CRIT is resulted. The inset in Fig. 3(a) shows a magnified transmission spectrum of D ₁ . When D ₁ reaches the maximum transmission D ₁ ^max , dD ₁ / d(∆θ) = 0 with solutions as the splitting phases θ ₊ and θ _- . Thus, we find that

where ε ₀ = (1- t ₀)(1-t) /(4√t ₀ t). Equation (12a) is accurate when ε ₀<2. When ε ₀≥2, θ ₊ and θ _- are degenerated and equal to π and -π, respectively. Therefore, D ₁ ^max is modified as D ₁ ^max =D ₁(θ ₊=π) = 2k ² ₀ k ² /(1+(1 + t ₀)(1+t)/2 + t ₀ t)² . Figure 3(e) shows the phase responses of type I. Across the resonance θ ₀, the phase response ϕ₀ shifts 2π, which is the same as that of a single microring resonator at the through port. However, ϕ ₁ shifts 3π, which possesses additional 2π compared to that of a single microring resonator at the drop port [20]. This is caused by the lower resonator. Transmission spectra of Type II is shown in (b) and they are the same as those of type I, evident from Eqs. (6) and (7a). This suggests that transmission spectra of type I and II are the same no matter the input is at the center or the upper port when there is no loss. Phase response $\tilde{\varphi }$ ₁ of type II is also the same as ϕ₁ of type I, which is also evident from Eq. (6). However, phase responses of $\tilde{\varphi }$ ₀ is not the same as ϕ₀ [see Eqs. (8a) and (9a)]. The slope coefficient of $\tilde{\varphi }$ ₀ is larger than that of ϕ₀ near the resonance θ₀, hence results a larger normalized group delay of resonance light. This is under the condition of k ₀ > k . When k ₀ < k, the slope coefficient of $\tilde{\varphi }$ ₀ is smaller than that of ϕ₀ near the resonance θ ₀. The the slope coefficient of $\tilde{\varphi }$ ₀ is the same as that of ϕ₀ near the resonance θ ₀ when k ₀ = k.

From Eqs. (10b) and (10c), we can obtain D ₁=D ₂, ϕ ₁=ϕ ₂. Therefore, transmission spectra and phase responses of two drops of type III are the same as shown in Figs. 3(c) and 3(g). From Eqs. (10), we can conclude that the normalized complex amplitude responses are the same as those of a single microring resonator with two 2×2 couplers’s coupling coefficients as k and k ₀, while there is just a modified factor 1/ √2 for two drops. Thus, type III just act as two add-drop microring resonators with a corporate through and identical drop transmissions. Type IV holds the same transmission characteristics as Type III at the upper drop port, i.e., D ₁ = D ₁ and ϕ₁ = $\tilde{\varphi }$ ₀, as shown in Figs. 3(d) and 3(h). The slope coefficient of $\tilde{\varphi }$ ₀ is also larger than that of ϕ₀ of type III near the resonance θ ₀. Compared to $\tilde{\varphi }$ ₁, $\tilde{\varphi }$ ₂ shifts additional π across the resonance θ ₀. The inset in Fig. 3(d) shows that transmission spectra of two drop ports possess different resonance widths or so-called full width half maximums (FWHMs). This can be explained that the upper drop D̃ ₁ is a first-order filter and the lower drop D ₂ is a second-order filter [see Eqs. (11b) and (11c)]. FWHMs for the upper and lower drop ports W̃ ₁and W̃ ₂ satisfy D̃ _n(W̃ _n/2) = [D̃ _n(θ ₀) + D̃_n(π)) /2 . They are found as

where A = 2ε ₂, B = -ε ₁(1+ε ₂) , C = 2ε _l ²(1+ε ₂ )² /[ε ₁ ² +(1 + ε ₂)²)-2ε ₂ , ε ₁(1+t ₀)t and ε ₂ =t ₀ t ₂ . when ∆θ = θ ₀, D ₁ and D ₂ reach the maxima D̃ ₁ ^max and D̃ ₂ ^max , respectively. Expressions for these maxima are given as

 

Fig. 3. Transmission spectra of (a) type I (b) type II (c) type III (d) type IV; Phase responses of (e) type I (f) type II (g) type III (h)type IV. Parameters are chosen as τ = 1 ,k ₀ = 0.5 and k = 0.4

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3.2. Effects of coupling coefficients for types I and II

In the previous section, we have demonstrated that D ₁ = D̃ ₁ and T = T̃ for types I and II, and θ ₊ and D ₁ ^max are also the same for them. Therefore, discussions on θ ₊, D ₁ ^max and D ₁ of type I are sufficient for these two types.

 

Fig. 4. Effects of coupling coefficients on θ ₊ (a) Contour plot of θ ₊ /k as functions of k ₀ and k . The dashed lines (i)–(iii) correspond to k ₀ = 0.2, 0.5 and 0.8, respectively. The solid line (iv) corresponds to k ₀=k . (b) θ ₊ as a function of k when k ₀ = 0.2, 0.5, 0.8 and k , respectively. The inset shows θ ₊ when k ₀=k and k near 0.995 .

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Fig. 5. Effects of coupling coefficients on D ₁ ^max . (a) Contour plot of D ₁ ^max as functions of k ₀ and k . The dashed lines (i)–(iii) correspond to k ₀ = 0.2, 0.5 and 0.8, respectively. The solid line (iv) corresponds to k ₀=k . (b) D ₁ ^max as a function of k when k ₀ = 0.2, 0.5, 0.8 and k , respectively.

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Figure 4 illustrates effects of coupling coefficients on θ ₊ . As seen from Eq. (12a), k ₀ and k have the same effects on θ ₊. Therefore, the contour plot of θ ₊ is symmetrical along the solid line (iv) corresponding to k ₀ = k as shown in Fig. 4(a). θ ₊ increases along directions parallel to lines (i)–(iv) as k increases. Figure 4(b) shows θ ₊ as a function of k when k ₀ = 0.2, 0.5, 0.8 and k, respectively. For the cases of k ₀ = 0.2, 0.5 and 0.8, θ ₊ is greater with greater k ₀ and increases as k increases. For the case k ₀ =k, θ ₊ is very small and θ ₊ =(1-t)/√2t using the Taylor expansion when the coupling coefficient is small. However, θ ₊ increases rapidly when the coupling coefficient k is greater. When k = 0.5 , θ ₊ ≈ 0.033π. When $k=2\sqrt{5\sqrt{6}-12}$ (0.995), ε ₀ = 2 and hence θ ₊ = π meaning that two peaks are merged into one peak. The inset in Fig. 4(b) shows that θ ₊= π when k is greater than 0.995.

Figure 5 shows effects of coupling coefficients on D ₁ ^max . The contour plot of D ₁ ^max as functions of k ₀ and k is shown in Fig. 5(a). k ₀ and k also have the same effects on D ₁ ^max, which can be seen from Eq. (12b). Hence, D ₁ ^max is also symmetrical along the solid line (iv) corresponding to k ₀ =k. Figure 5(b) shows D ₁ ^max as a function of k when k ₀ = 0.2,0.5,0.8 and k , respectively. For the cases of k ₀ = 0.2, 0.5 and 0.8, D ₁ ^max reaches maxima when k = 0.2, 0.5 and 0.8, respectively. This implies that the maximum of D ₁ ^max can be obtained when k = k ₀. For the case of k ₀ = k, D ₁ ^max - 1/(2 - k ⁴ /16) and hence is about 0.5 when the coupling coefficient k is small (<0.5). When k>0.8, D ₁ ^max increases rapidly as k increases. D ₁ ^max reaches the maximum 8/9 (0.8889) when k approaches 1.

 

Fig. 6. (a)-(c) Contour plots of D ₁ as functions of k when k ₀ equals to 0.2, 0.5 and 0.8, respectively. The dashed lines (i)–(iii) correspond to k = 0.2, 0.5 and 0.8, respectively, when k ₀ =0.5 . (d) D ₁ under k= 0.2, 0.5 and 0.8, respectively, when k ₀ =0.5 .(e) D ₁ under k = k ₀ equaling to 0.2, 0.5, 0.8, 0.9, 0.95 and 0.995, respectively.

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Effects of coupling coefficients on the transmission spetrum D ₁ are illustrated in Fig. 6. Contour plots of D ₁ as functions of k are calculated as shown in Figs. 6(a)–6(c) and corresponding values of k ₀ are chosen as 0.2, 0.5 and 0.8, respectively. As k ₀ increases, maximum regions of D ₁ move upwards, corresponding to that D ₁ ^max reaches the maximum when k = k ₀ . Simultaneously, maximum regions of D ₁ become wider and larger. This corresponds to that θ ₊ and D ₁ ^max become greater, respectively. Taking k ₀ =0.5 shown in Fig. 6(b) as example, three dashed lines (i)-(iii) correspond to k = 0.2, 0.5 and 0.8, respectively. Corresponding transmission spectra are calculated as shown in Fig. 6(d). As k increases, θ ₊ also increases which can also be seen from Fig. 4. Moreover, D ₁ ^max reaches the maximum when k = k ₀. Figure 6(e) shows D ₁ under k=k ₀ equaling to 0.2, 0.5, 0.8, 0.9, 0.95 and 0.995, respectively. When k=k ₀ > 0.8 , D ₁ still preserves a obvious peak at θ ₊ . After k = k ₀ > 0.8 , D ₁ has a wide notch near the resonance and becomes very flat away from the resonance. D ₁(π) increases rapidly because the two peaks move towards a degenerated peak when k= 0.995 .

3.3. Effects of coupling coefficients for types III and IV

Similarly, discussions on W̃ ₁, W̃ ₂, D̃ ₁ ^max , D̃ ₂ ^max , D̃ ₁ and D̃ ₂ of type IV are sufficient for both types III and IV. W̃ ₁, D̃ ₁ ^max and D ₁ of type IV can be representative for the two drop ports of type III.

 

Fig. 7. (a). Contour plot of W̃ ₁ /π as functions of k ₀ and k . (b) W̃ ₁ as a function of k when k ₀ = 0.2, 0.5, 0.8 and k , respectively. (c) Contour plot of W̃ ₂/π as functions of k ₀ and k . (d) W̃ ₂ as a function of k when k ₀ = 0.2, 0.5, 0.8 and k , respectively. The inset shows W̃ ₂ when 0.7 < k < 0.9 . The dashed lines (i)-(iii) correspond to k ₀ = 0.2, 0.5 and 0.8, respectively. The solid line (iv) corresponds to k ₀=k .

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Fig. 8. (a) Contour plot of D̃ ₁ ^max as functions of k ₀ and k . (b) D̃ ₁ ^max as a function of k when k ₀ = 0.2, 0.5, 0.8 and k , respectively. (c) Contour plot of D̃ ₂ ^max as functions of k ₀ and k . (d) D̃ ₂ ^max as a function of k ₀ when k ₀ = 0.2, 0.5, 0.8 and k , respectively. The dashed lines (i)-(iii) correspond to k ₀ = 0.2, 0.5 and 0.8, respectively. The solid line (iv) corresponds to k ₀=k .

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Figure 7 shows effects of coupling coefficients on W̃ ₁ and W̃ ₂. The effects of k ₀ and k on W̃ ₁ are the same as shown in Fig. 7(a) [see Eq. (13a)]. Figure 7(b) shows W̃ ₁ as a function of k when k ₀ = 0.2, 0.5, 0.8 and k, respectively. For the cases of k ₀ = 0.2, 0.5 and 0.8, W̃ ₁ increases as k increases. When k approaches to 0, W̃ ₁ - 2(1-t ₀)/√t ₀ . For the case of k ₀=k, W̃ ₁ -2(1-t ²)/t when k is small (<0.5). Figure 7(c) shows effects of k ₀ and k on W̃ ₂, suggesting that k affects W̃ ₂ mainly while k ₀ has very less effects on W̃ ₂. Figure 7(d) shows W̃ ₂ as a function of k when k ₀ = 0.2, 0.5, 0.8 and k, respectively. W̃ ₂ increases as k increases and has small variations (<0.1π) when k ₀ varies. The inset in Fig. 7(d) shows small variations under different k ₀, when k ranges from 0.7 to 0.9. Thus, W̃ ₂ is generally dependent on k while nearly independent on k ₀.

Figure 8 illustrates effects of coupling coefficients on D̃ ₁ ^max and D̃ ₂ ^max . As shown in Fig. 8(a), the effects of k ₀ and k on D̃ ₁ ^max are also the same [see Eq. (14a)]. Figure 8(b) shows D̃ ₁ ^max as a function of k when k ₀ = 0.2, 0.5, 0.8 and k , respectively. For the cases of k ₀ = 0.2, 0.5 and 0.8, D̃ ₁ ^max reaches the maximum 0.5 when k equals to k ₀ and decreases when k keeps away from k ₀ . For the case of k ₀ = k, D̃ ₁ ^max keeps as 0.5 as k increases. However, effects of coupling coefficients k ₀ and k on D̃ ₂ ^max are different as shown in Fig. 8(c). When k is small, D̃ ₂ ^max is much larger and approaches to 1 and decreases as k increases. When k ₀ is small, D̃ ₂ ^max is very small and increases as k ₀ increases. Figure 8(d) shows D̃ ₂ ^max as a function of k when k ₀ = 0.2, 0.5, 0.8 and k, respectively. For cases of k ₀ = 0.2, 0.5 and 0.8, D̃ ₂ ^max becomes larger as k ₀ increases. For the case of k ₀ = k, D̃ ₂ ^max keeps the same value 0.25. From Eqs. (14), we can obtain D̃ ₂ ^max/D̃ ₁ ^max = (1 + t)(1-t ₀)/ ( 2(1-t)(1 + t ₀) ) . When k ₀=k, D̃ ₂ ^max / D̃ ₁ ^max =1/2, suggesting that the intensity of the resonance light at the upper drop port is two times of that at the lower drop port.

 

Fig. 9. (a)-(c) Contour plots of D̃ ₁ as functions of k when k ₀ equals to 0.2, 0.5 and 0.8, respectively. The dashed lines (i)-(iii) correspond to k = 0.2, 0.5 and 0.8, respectively. (d) D̃ ₁ under k equaling to 0.2, 0.5 and 0.8, respectively, when k ₀=0.5 . (e) D̃ ₁ under k = k ₀ equaling to 0.2, 0.5 and 0.8, respectively.

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Fig. 10. (a)-(c) Contour plots of D̃ ₂ as functions of k when k ₀ equals to 0.2, 0.5 and 0.8, respectively. The dashed lines (i)–(iii) correspond to k = 0.2, 0.5 and 0.8, respectively. (d) D̃ ₂ under k equaling to 0.2, 0.5 and 0.8, respectively, when k ₀=0.5 . (e) D̃ ₂ under k = k ₀ equaling to 0.2, 0.5 and 0.8, respectively.

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Effects of coupling coefficients on D̃ ₁ are illustrated as shown in Fig. 9. Figure 9(a)–(c) show contour plots of D̃ ₁ as functions of k when k ₀ equals to 0.2, 0.5 and 0.8, respectively. When k ₀ increases, the maximum region of D̃ ₁ moves upwards and to k =k ₀ . Taking k ₀ =0.5 shown in Fig. 9(b) as example, W̃ ₁ increases as k increases and D̃ ₁ ^max reaches the maximum when k = k ₀ as shown in Fig. 9(d). Figure 9(e) shows D̃ ₁ under the same k and k ₀ . W̃ ₁ increases as k increases while D̃ ₁ ^max keeps the same value 0.5. Figure 10 shows effects of coupling coefficients on D̃ ₂. Figures 10(a)–10(c) show contour plots of D̃ ₂ as functions of k when k ₀ equals to 0.2, 0.5 and 0.8, respectively. Taking k ₀ = 0.5 shown in Fig. 10(b) as example, W̃ ₂ increases and D̃ ₂ ^max decreases when k increases as shown in Fig.10(d). Figure 10(e) shows D̃ ₂ under the same k and k ₀ . W̃ ₂ increases as k increases while D̃ ₂ ^max keeps the same value 0.25. As shown in Figs. 9(e) and 10(e), W̃ ₂ is much smaller compared to W̃ ₁.

4. The effects of loss

In practical devices, possible loss caused by absorption, scattering or radiation is unavoidable. Thus, the effects of loss are discussed in this section.

Figure 11 shows effects of loss for types I and II. When there is loss, Eq. (11a) can be modified as: θ ₊= arccos(w) if w < 1 and θ ₊ = 0 if w ≤ 1. w = (-Y - √y ² -4XZ)/(2X) , where X = 4uτ , Y = -4u-4uτ ² , Z = v(1+u)-(v ² + (1-u)²)τ+v(1+u)τ ² , u = t ₀ tτ ² and v = (1 + t ₀)(1 + t)τ/2 . And D ₁ ^max = D ₁(∆θ = θ ₊). Figure 11(a) shows D ₁ when τ is adopted as 1.0(lossless), 0.9 and 0.8, respectively. Coupling coefficients k and k ₀ are adopted to be 0.5. Figure 11(b) gives D ₁ ^max and θ ₊ as a function of τ. As τ decreases, D _{1 max} decreases; θ ₊ increases to the maximum 0.049 π when τ decreases to 0.898, and then decreases to a minimum 0 when τ decreases to 0.77. When θ ₊=θ ₀=0( i.e., τ ≤ 0.77), the two peaks are merged at θ ₀ and only one peak exists. This implies that loss will seriously affect the transmission spectrum.

Effects of loss for types III and IV are illustrated as shown in Fig. 12. Modified expressions of maximal transmissions are D̃ ₁ ^max=k ₀ ² k ² τ/ ( 2(1-t ₀ tτ)²) and D̃ ₂ ^max = (1-t ₀)² k ⁴ τ ²/[4(1-tτ)²(1-t ₀ tτ)²) . And modified expressions of W̃ ₁ and W̃ ₂ are Eqs. (13) with t substituted by tτ . Figure 12(a) shows D̃ ₁ when τ is adopted as 1.0(lossless), 0.9 and 0.8, respectively. Figure 12(b) shows D̃ ₁ ^max and W̃ ₁ as a function of τ . When τ decreases, D̃ ₂ ^max decreases while W̃ ₁ increases. Figure 12(c) shows D̃ ₂ when τ is adopted as 1.0(lossless), 0.9 and 0.8, respectively. Figure 12(d) shows D̃ ₂ ^max and W̃ ₂ as a function of τ . When τ decreases, D̃ ₂ ^max also decreases while W̃ ₂ increases. Compared with D̃ ₁ ^max, D̃ ₂ ^max decreases more rapidly due to signals undergoing more round-trip distances. However, W̃ ₂ keeps smaller than W̃ ₁ for the same τ.

 

Fig. 11. Effects of loss for types I and II (a) D ₁ when τ is adopted as 1.0(lossless), 0.9 and 0.8, respectively. (b) D ₁ ^max and θ ₊ as a function of τ. k=k ₀= 0.5

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Fig. 12. Effects of loss for types III and IV (a) D̃ ₁ when τ is adopted as 1.0(lossless), 0.9 and 0.8, respectively. (b) D̃ _max and W̃ ₁ as a function of τ (c) D̃ ₂ when τ is adopted as 1.0(lossless), 0.9 and 0.8, respectively. (d) D̃ ₂ ^max and W̃ ₂ are as a function of τ . k = k ₀ =0.5

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5. Conclusions and discussions

In conclusion, we have proposed four types of 3×3 coupler based dual microring resonators. A transfer matrix model has been developed for studying them. After that, general transmission characteristics are discussed with analytical expressions given. Effects of coupling coefficients and loss on transmission spectra are studied detailed. Coupling coefficients of the 3×3 and 2×2 couplers and loss of resonators determine their transmission spectra.

The types I and II show a coupled-resonator-induced-transparency-like transmission spectrum at the through port. At the drop port, coupling coefficients of the 3×3 and 2×2 couplers have the same effects on the maximum transmission and the splitting phase. Maximum transmissions can be obtained when coupling coefficients of the 3×3 and 2×2 couplers are the same. If the coupling coefficient of the 3×3 coupler keeps invariable, the splitting phase shows a dependence relation with the coupling coefficient of the 2×2 coupler. When the coupling coefficient of the 2×2 coupler is changed by environment conditions in the range from 50% to 100%, the splitting phase changes in a large range. Thus, these two types are expected to have potential applications in sensors.

The type III shows symmetrical transmission spectra at two drop ports, which also serves as two add-drop micoring resonators with a corporate through and identical drop transmissions. The type IV shows the simultaneous realizations of transmission spectra with different intensities and FWHMs at two drop ports, corresponding to a first-order and a second-order filters. For the first-order filter, the maximum transmission is 50% when coupling coefficients of the 3×3 and 2×2 couplers are the same; and coupling coefficients of the 3×3 and 2×2 couplers have the same effects on its FWHM. For the second-order filter, the maximum transmission can be as high as 100% when the coupling coefficient of the 2×2 couplers is very small; the coupling coefficient of the 2×2 couplers mainly affects its FWHM, while the coupling coefficient of the 3×3 couplers has very small effect on it. Types III and IV may be found applications in optical filters.

In practical microring resonator based devices, loss and detuning due to physical limitations and fabrication errors may be important issues restricting practical using. However, employing heating or electro-optical effects can change the effective index to realize tuning and adopting materials with strong confinements and low absorptions can also overcome the loss limitation. Recently, coupling-induced resonance frequency shifts (CIFS)[24] are demonstrated to be an important fundamental source of resonance frequency mismatch. This may also appear in the proposed structures, if the ring-bus waveguide spacing is very narrow (~100nm). To overcome this effect in the proposed 3×3 coupler based dual microring resonators, similar designs of CIFS-free coupled-resonator systems can also follow that discussed in Ref.[24].