Low-threshold bistability in nonlinear microring tower resonator

Mehdi Shafiei; Mohammad Khanzadeh

doi:10.1364/OE.18.025509

1. Introduction

Recently, optical bistability has been attracted increasing attention due to several applications in fast all-optical devices; including switches, logic gates, transistors, flip-flops, and optical memories [1–5]. Microring arrays, among a large number of devices that provide bistable behavior, have more interesting features because these structures can simultaneously take the unique advantages of the microring resonators and photonic crystal structures [6, 7]. In this paper, we focus on vertically stacked multi-ring resonator [8], illustrated in Fig. 1(a) , and demonstrate wavelength and power bistability in these 3D arrays. In this paper, this configuration will be called microring tower (MRT) resonator.

Fig. 1 a) Sketch of MRT with N rings. b) Schematic of an infinite periodic VMR structure with a defect.

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In our previous study [9], it was proved that in the linear regime a defect site in a periodic MRT creates a very high quality factor resonant mode. Also, this structure has extreme compactness in comparison with in-plane microring arrays. Furthermore, it offers lower group velocity and higher flexibility in the transmission spectrum, while it doesn’t suffer from short coupling length problem of other devices such as CROW and SCISSOR with circular resonator units [9]. This paper is the first effort to study the microring tower in nonlinear regime. We begin by introducing a simple theoretical model to find defect modes of MRT in the linear regime. To increase the quality factor of the device, then we study the influence of different geometry cavity parameters on the Q factor. In passing, we derive the nonlinear coupled wave equations for light propagation along the structure. Finally, the resulting equations are solved numerically. Calculations show that optical bistability can be formed in the structure with a very low input power of $165 μ W .$

2. Linear response

2.1. Simple model for fast calculation of cavity- modes

Inspiring from [10], we first suggest a theoretical model for defect modes in an infinite array of stacked and coupled microrings. This model is applicable to linear regime of a MRT which we abbreviate it as LMRT. Thus, let us consider an infinite array of vertically coupled microrings with the same radii, R. Making use of coupled mode theory, the fields can be described by following equations for the amplitudes $A_{n} (s)$ in the n^th ring [9]:

i d A_{n} / d s + κ_{n - 1, n} A_{n - 1} e^{i (β_{n - 1} - β_{n}) s} + κ_{n, n + 1} A_{n + 1} e^{i (β_{n + 1} - β_{n}) s} = 0, n = 0, \pm 1, \pm 2, \cdot \cdot \cdot

where

β_{n}

is the propagation constant of the mode in the n^th ring,

κ_{n - 1, n}

is the coupling coefficient between adjoining rings, and s is the coordinate along each ring. Furthermore, the amplitude

A_{n} (s)

at n^th ring satisfies the condition of closure of rings as follows:

A_{n} (0) = A_{n} (L) \exp (i β_{n} L),

where

L = 2 π R .

Now, consider a uniform structure with the same coupling coefficients and propagation constants,

(κ_{n - 1, n} = κ, β_{n} = β),

apart from a single defect located at

n = 0

(as shown in Fig. 1(b)), which its propagation constant differs from the rest by

δ β_{def} .

Moreover, we assume that the coupling coefficient between the defect site and its nearest neighbors is

κ^{'} .

Making use of variable changes

a_{0} (s) = A_{0} (s) \exp (i δ β_{d e f} s)

and

a_{n} (s) = A_{n} (s)

for

n \neq 0,

Eq. (1) for the middle three sites can be rewritten as follows:

{\begin{cases} i d a_{1} / d s + κ^{'} a_{0} + κ a_{2} = 0 \\ i d a_{0} / d s + δ β_{d e f} a_{0} + κ^{'} (a_{- 1} + a_{1}) = 0 \\ i d a_{- 1} / d s + κ a_{0} + κ^{'} a_{- 2} = 0 \end{cases} .

The field propagation in a uniform infinite array is described by

e x p (i μ s + i ξ z_{n})

where ξ is effective propagation constant along the vertical direction,

μ = 2 κ \cos (ξ d)

,

z_{n} = n d

, and d is the period of the structure [9]. Therefore the following solution may be proposed:

a_{n} = {\begin{cases} e^{i μ s} (e^{i ξ z_{n}} + r e^{- i ξ z_{n}}) ​, n < 0 \\ q e^{i μ s}, n = 0 \\ t e^{i μ s} e^{i ξ z_{n}} ​, n > 0 \end{cases}

where t and r, to be determined, are transmission and reflection coefficients associated with a forward and reflected wave, respectively. Substituting Eq. (4) into Eqs. (3) and applying some straightforward manipulations, one arrives at the following equations for the coefficients r and q:

r = t - 1, q = κ t / κ^{'},

where

t = \frac{2 i {κ^{'}}^{2} \sin ξ d}{2 ({κ^{'}}^{2} - κ^{2}) \cos ξ d + 2 i {κ^{'}}^{2} \sin ξ d + κ δ β_{d e f}} .

In steady state, we can also expect that the defect modes assure the conditions of ring closure and therefore satisfy the following dispersion relation [9]:

B = 2 π l - 2 K \cos (ξ d),

where

K = κ L,

B = β L

and l is an integer number. Note that for defect modes placed in the bandgap, the propagation constantξ is imaginary. Inserting the expression for

\sin (ξ d)

from Eq. (7) into Eq. (6) one obtains

t = \frac{- {K^{'}}^{2} {[{(B - 2 π l)}^{2} - 4 K^{2}]}^{1 / 2}}{({K^{'}}^{2} - K^{2}) (2 π l - B) - {K^{'}}^{2} {[{(B - 2 π l)}^{2} - 4 K^{2}]}^{1 / 2} + K^{2} δ B_{d e f}},

where,

K^{'} = κ^{'} L

. Defect modes are obtained by Eq. (8) when the denominator is set to zero. Accordingly, the defect modes are given by the following simple equation:

B_{d e f} = 2 π l - \frac{δ B_{d e f} ({K^{'}}^{2} - K^{2}) \pm {K^{'}}^{2} \sqrt{δ B_{d e f}^{2} - 4 (K^{2} - 2 {K^{'}}^{2})}}{K^{2} - 2 {K^{'}}^{2}} .

In the case of

κ^{'} = κ

this equation reduces to

B_{d e f} = 2 π l \pm \sqrt{δ B_{d e f}^{2} + 4 K^{2}} .

The two solutions conform to the two possible defect modes in each bandgap; the positive (negative) sign represent the modes in the left (right) of the band with

δ β_{d e f} < 0

(

δ β_{d e f} > 0

). For the defect modes the coefficients t and r are infinitely large, so Eq. (4) shows that the field is localized at the defect position and decays exponentially away with the rate of

\exp (| i ξ z_{n} |)

. These situations are in agreement with results in ref [9]. In order to determine the appropriateness of the model we compare it with numerical calculations. The results are shown in Fig. 2 . Except for small defects, there is an excellent agreement between numerical calculations and the model. Note that, in numerical calculations the number of the rings is inevitably finite. However, for sufficiently high number of the rings its behaviors are very similar to the infinite one [9].

Fig. 2 Defect modes $B_{d e f f}$ versus $δ B_{d e f}$ for two different defect modes: (right) positive and (left) negative defects. The Curves denote the model results assuming $l = 68;$ and K = 0.8. Filled circles denote the numerical results where the number of rings is 9 and the coupling parameter between the input waveguide and its adjacent ring, $K_{0}$ , is 0.8.

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While the numerical calculations are time consuming and need to sweep whole spectrum, this model delivers defect mode only with a simple algebraic equation. Note that, the linear solution is the first step in nonlinear calculations which will be performed in the next section.

2.1. Influence of cavity parameters on the Q factor

Now let us discuss one important cavity property that enhances processes in nonlinear optics, namely very high quality factor. To study the dependence of the Q factor to the cavity geometry, we first study the influence of $δ β_{d e f}$ on the Q factor. The results are shown in Fig. 3(a) where we consider a uniform MRT with 9 rings and coupling parameters $K_{0} = K = 0.6,$ and see that for larger defects, the Q factor is higher. This can readily explain by considering the position of the defect modes in the bandgap. By increasing the defect size, the position of the defect modes goes far from the band edges. As the defect mode approaches the band edges, the linewidth broadens so that the defect mode has a smaller Q. Next, we study the influence of the coupling parameters. Calculation show that if the coupling parameter between adjacent microrings is equal to the coupling parameter between input/output waveguides and adjacent microring, i. e. $K_{0} = K,$ the Q factor has a minimum value (see Fig. 3(b)) whilst increasing the coupling parameter difference, $| K_{0} - K |,$ noticeably increases the quality factor.

Fig. 3 dependence of the Q factor a) on the $δ β_{d e f},$ and b) on the K₀ and K.

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Also, the Q factor strongly depends on the number of the microrings. For example, for the choice of $K = K_{0} = 0.8,$ and $δ B_{d e f e c t} = 1.6,$ while for $N = 9,$ $Q = 4.25 \times 10^{4},$ for $N = 11,$ calculations show that $Q = 3 \times 10^{5} .$ we attribute this improvement mainly to the fact that the structure with more microring give better transmission characteristics and simulate ideal infinite structures more exactly.

3. Nonlinear response

3.1. Coupled wave equations

Following our previous work [9], we now take nonlinear effects into account. As before, we keep using the local coordinate system (x, y, s), where s is the coordinate along a ring waveguide and (x, y) are the coordinates in the surface perpendicular to the s coordinate, so the propagation of light in nonlinear MRT can be explained by coupled mode theory too. The structure consists of N microrings with an input and an output waveguide as shown in Fig. 1(a). let us seprate the transverse and axial dependencies of the electric fields in the form $A_{n} (s) E_{n} (x, y) \exp (i {\tilde{β}}_{n} s - i ω t)$ , where $E_{n} (x, y)$ is the normalized eigen mode in the ring n [10]. Here, loss is introduced in the analysis by adding an imaginary part to the propagation constant in the form $\tilde{β} = β + i α$ , where αis the loss (or gain) per unit length in the ring. Therefore, using conventional nonlinear coupled mode theory in the presence of the optical Kerr effect, the following nonlinear coupled equations are obtained [11, 12]:

\begin{array}{l} i \frac{d A_{m}}{d s} + κ_{m, m - 1} A_{m - 1} e^{i (β_{m - 1} - β_{m}) s} + γ {| A_{m} |}^{2} A_{m} = 0 \\ i {\frac{d A_{n}}{d s}}_{n} + κ_{n, n - 1} A_{n - 1} e^{i (β_{n - 1} - β_{n}) s} + κ_{n + 1, n} A_{n + 1} e^{i (β_{n + 1} - β_{n}) s} + γ {| A_{n} |}^{2} A_{n} = 0 \\ i {\frac{d A_{- m}}{d s}}_{- m} + κ_{- m + 1, - m} A_{- m + 1} e^{i (β_{- m + 1} - β_{- m}) s} + γ {| A_{- m} |}^{2} A_{- m} = 0 \end{array}

where

n = 0, \pm 1, \dots, \pm (m - 1),

γ = n_{2} ω_{0} / c A_{e f f},

ω_{0}

is the optical angular frequency,

n_{2}

is the nonlinear refractive-index coefficient, and

A_{eff}

is the common effective area of waveguide modes. Now, the boundary condition of closure of each ring has the form

A_{n} (0) = A_{n} (L) \exp (i β_{n} L - α L), n = 0, \pm 1, \dots, \pm (m - 1),

and in the first and last ring we have the following equations of the coupling between waveguides and adjacent rings:

\begin{array}{l} A_{\pm (m + 1)}^{(o u t)} = (1 - ρ)^{1 / 2} [\cos (K_{0}) A_{\pm (m + 1)}^{(i n)} + i \sin (K_{0}) A_{\pm m} (L)] \\ A_{\pm m} (0) = (1 - ρ)^{1 / 2} [i \sin (K_{0}) A_{\pm (m + 1)}^{(i n)} + \cos (K_{0}) A_{\pm m} (L)] \end{array}

Where ρ denote the intensity-insertion-loss coefficient [11]. In the following, for simplicity we assume that the ingoing amplitude

A_{m + 1}^{(i n)} = 0

and also,

ρ = 0.

Apart from the boundary conditions, Eq. (10) in the case of uniform structures reduces to the well known discrete nonlinear Schrödinger equation (DNSE). At low powers, the nonlinear term of Eq. (10) can be ignored and the results of previous section can be used. Note that, Eq. (10) can also explain the propagation of light in nonlinear waveguide arrays [13,14]. This is not surprising because the coupling mechanism in two structures is very similar. In fact, one can directly arrive at Eq. (10) by this comparison.

3. 2. Nonlinear transmission properties

We next perform the same transmission analysis as Ref [9]. to investigate the nonlinear transmission properties of a nonlinear MRT (NMRT) with a defect at the middle of the structure. Here, we have a typical example of a boundary-value problem. Such problems usually present difficulties of greater magnitude than of the initial-value problems. However, we propose an iterative method to solve Eqs. (10)-(12) based on the following idea: Using finite difference (FD) approximation, Eq. (10) converts to a system of nonlinear algebraic equations that can be rewritten in matrix form for the vector $x = (A_{n})$ of mode amplitudes:

A \cdot x = b

where the complex matrix A collect the constant coefficient as well as terms proportional to

| A_{n} |^{2},

and b is a vector which depends on the boundary conditions. Because the matrix elements of A are not constant and depend on the solution as well, this is not an ordinary linear system and therefore cannot be solved directly. It is, however, possible to find a solution iteratively. Substituting an initial guess for

A_{n}

, the elements of matrix A are known, therefore, Eq. (13) transforms to an ordinary system that can be solved for x. An appropriate initial guess

x_{0},

can be selected (for example, the solution of linear limit) and then the iteration process starts and continues to a prescribed accuracy. As an important advantage, this method is compatible with sparse methods for solving the linear systems and hence it can be very fast. Note that, for sufficiently high input amplitudes, this method may not converge. This behavior, we believe, depend on the nature of DNSE and the boundary conditions in Eq. (11).

Being equipped with a proper method, the properties of NMRT are investigated. In this section, we ignore the loss in the Eq. (10)-(12) to identify the most general physical effects. Also, we use material parameters of AlGaAs at $λ = 1.55 μ m$ with $n_{2} = 1.5 \times 10^{- 13} c m^{2} W^{- 1}$ [15]. For conventional microring size the mode effective area is about $1 μ m^{2}$ . And hence, $γ \sim 60 m^{- 1} W^{- 1}$ . At first, we consider a uniform NMRT consisting of 9 rings with coupling parameters $K_{0} = 0.2,$ and $K = 2.$ Fig. 4(a) shows the transmission around a resonant mode with $Q \sim 1.6 \times 10^{4}$ for deferent input powers. As can be seen, increasing the input power results in asymmetric resonance transmission pattern. From the steep declines in the transmission, it can be seen that optical bistability is obtained for powers of approximately $90 m W$ and above for this configuration. In Fig. 4(b) we plot the input power versus output power while fixed input wavelength for different detuning parameters $δ = B_{i n} - B_{0},$ where $B_{0}$ and $B_{i n}$ are normalized propagation constant of linear resonance mode and input field respectively. Note that, the device in this mode shows counterclockwise hysteresis cycles. As the detuning parameter increase, the bistability threshold and the width of the hysteresis cycle reduce.

Fig. 4 a) Detailed spectrum of a resonant mode at different input powers for a lossless uniform MRT with N = 9, $K_{0} = 0.2,$ and $K = 2.$ b) Output power $(P_{o u t})$ versus input power $(P_{i n})$ for various detuning parameters.

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Another example is shown in Fig. 5 , where we plot input/output power of a mode close to $B = 425.389$ for different detuning parameters. As can be seen, by increasing the detuning parameter sufficiently, optical field experiences two clockwise hysteresis cycles in transmitted power. Also, the switching power is lowered to about $40 m W$ which is lower than previous one in Fig. 4. This is because of higher quality factor ( $\sim 3 \times 10^{4}$ ) of this mode.

Fig. 5 nonlinear response of the structure of Fig. 4 for another mode near $B = 425.389$ for three different detuning parameters.

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Since, it is desirable an all-optical device to operate at a very low power, let us consider how can lower bistability threshold in our device. In order to achieve very low bistability threshold, we use high Q-factor defect modes. An example is shown in Fig. 6 , where we consider an NMRT consisting of 9 rings with coupling parameters $K = K_{0} = 0.8,$ in which the dimensionless propagation constant of the center ring at $n = 5$ differs from those of others rings by $δ B_{d e f e c t} = 1.6.$ A section of the through transmission spectrum in linear or a low power case is shown in Fig. 6(a). A sharp resonant mode with quality factor of $Q = 4.25 \times 10^{4}$ can be seen inside the gap near $B = 424.998$ . Figure 6(b) shows the details of transmission around the defect mode for different input powers, $P_{i n},$ in nonlinear case with $γ = 60$ . One can observe, as $P_{i n}$ increases, the defect mode shifts to the left or according to $B = β L \propto 1 / λ$ shifts to longer wavelengths. Also, for defect modes located at the left side of the gap; i.e. $δ B_{d e f e c t} < 0$ , calculations show that the red-shift occurs again. These red-shifts can readily be explained by considering the Kerr effect and localization of the field in the defect site. Also, as shown in Fig. 6(b) the bistability threshold is about 10 mW. furthermore, the power transmission in Fig. (c) - (d) show that again, we have multiple hysteresis cycle for sufficiently high $δ .$ This behavior is useful in some types of optical switches and memories.

Fig. 6 a) A section of transmission spectrum of a lossless MRT ( $N = 9$ , $K = K_{0} = 0.8,$ and $δ B_{d e f e c t} = 1.6$ ) for low input powers, a sharp dip shows a defect mode near $B = 424.998$ . b) Nonlinear transmission for different input powers. c) - d) Input/Output characteristic for two different $δ .$

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Now the results in section 2 can be used to reduce the bistable threshold by increasing the Q factor. As mentioned above, the Q factor is very sensitive to the number of the rings, defect size, $K,$ and $K_{0} .$ with proper choice of this parameter, one can increase the Q factor considerably and thus, the threshold of optical bistability will be lower. For example, with the choice of $N = 11$ , $K = 0.6,$ $K_{0} = 0.4,$ and $δ B_{d e f e c t} = 1.3,$ a $Q \approx 8.5 \times 10^{5}$ is obtained. Figure 7(a) shows that the optical bistability in this configuration is formed at a very low input power of $16 μ W .$ This represents very low switching threshold, which has often been introduce in optimized photonic crystals [16-18]. This value is very small, of course, in comparison with similar structures such as CROWs and microcoil resonators [19, 20]. For example, a recent work on microcoil resonators showed that the input power to see the bistable effects is about $16 W$ [20], which is considerably larger than the similar value in MRT. We believe that very low bistability threshold in our structure is due to the implementation of the microring resonators as well as the high Q defect modes. By modifying the structure, it is possible even more to reduce the bistability threshold. For example, to increase the Q factor and decrease the bistability threshold further, we change the dimensionless propagation constant of the first and last ring in the previous configuration by the amount of $Δ B = + 0.85.$ This improve mode-matching between the rings and input/output waveguides. The results are shown in the Fig. 7(b), where the bistability threshold reduced to the value of $P_{i n} = 9 μ W .$ Of course, we expect that optical bistability could be attained at even lower input power by more optimization of the structure.

Fig. 7 a) Nonlinear transmission response of a) a lossless MRT with parameters: $N = 11$ , $K = 0.6,$ $K_{0} = 0.4,$ and $δ B_{d e f e c t} = 1.3,$ at different input powers. b) A modified design, where the dimensionless propagation constant of first and last rings are changed by the amount of $Δ B = + 0.85.$

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4. Effects of the loss upon device performance

In the previous section we focused on the behavior of ideal devices where we assume lossless propagation in the structure. It is clear that the microcavity performance is limited by various loss mechanisms in the structure. The loss is comprised of surface scattering, material absorption and waveguide-bending radiation. An important advantage of the microring tower resonator is that its structure enables the modes vertically and continuously coupled together at any point of ring trajectory. This lead to the overall loss becomes less sensitive to intrinsic loss in the individual resonators. Also, the implementation of a simpler shaped microresonator such as microdisk resonators instead of microring resonators in addition to simplifying the fabrication techniques can reduce the scattering losses. Furthermore, microdisk resonators support the extremely high-Q whispering gallery mode that can reduce the bistability threshold.

In the following, we briefly investigate the effects of loss on quality factor and bistability threshold. To avoid unnecessary complications, we assume that coupling between waveguides and adjacent rings are lossless. The filled circle in Fig. 8(a) shows the calculated relationship between the quality factor and the loss for the microcavity of the Fig. 7(a). These circles can be fitted very well with an inverse decay curve. As similar to the cases of a ring resonator and microcoil resonators [6, 21–23], this result approximately suggests a relation of the form $Q \propto 1 / α$ between the quality factor and the loss (especially for $α ≫ 1$ ). The constant of the proportionality depend on the geometrical parameters (e. g. the number of the rings) and the chosen eigen mode.

Fig. 8 a) Filled circles show the calculated quality factors with different losses for the microcavity of the Fig. 7(a). The solid line is the curve $8.51 / (0.13 α + 1)$ . b) The relationship between bistability threshold and loss. The solid line is the curve $c \times α^{2}$ with appropriate constant c.

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Figure 8(b) depicts the variation of the bistability threshold versus the loss. The calculated values can be fitted well with a binomial curve with a second order term. Therefore, the relation between the bistability threshold and loss can be approximated in the form of $p_{t r} \propto α^{2}$ , which agrees well with the result $p_{t r} \propto Q^{- 2}$ in photonic crystal microcavities [24]. Assuming that the loss α in the example of the Fig. 7(a) is $0.01 m m^{- 1}$ (which is considerably smaller than those of used in the Refs [22, 23].), we find that the bistability threshold is $P_{t r} = 165 μ W .$

5. Conclusion

In this paper we present an alternative bistable structure based on the vertically stacked microring arrays. This compact configuration is of interest to nonlinear processes and slow light applications, which has often been accomplished in CROWs. The implementation of the microring resonators along with the defect modes with very high quality factors significantly reduces the bistable threshold in these configurations. We observed optical bistability at very low input power of $165 μ W .$ The similarity between the nonlinear coupled wave equation in this structure and the DNSE is an interesting feature that can lead to many useful phenomena.

Owing to the small spatial period of MRT in comparison with similar planar structures such as CROWs, this structure exhibit lower group velocity and subsequently can improve the efficiency of the nonlinear processes much better. The constant coupling coefficient along entire length of each resonator makes it easy to analyze. Furthermore, this removes the challenging short coupling length problems that usually appear in similar structures such as CROWs and SISCORs.

We also introduce a theoretical model in linear regime for finding defect modes in the transmission spectrum. This model is in very good agreement with numerical calculations. The algebraic formula that is presented makes it possible to find defect modes exactly, without any need to the numerical calculations. We also study the dependence of the quality factor on the cavity geometry. We find in our calculations that if the cavity parameters carefully be chosen the quality factor can noticeably increase. Specially, the quality factor strongly depends on the number of the microrings and the magnitude of the defect.

Finally, it should be noted that many features of this structure are not known at this stage. Therefore the results obtained in this work are at the beginning and we expect that its many features warrant further investigations.

Acknowledgment

The authors gratefully acknowledge Mohammad Agha-bolorizadeh and Raza Farrahi-Moghaddam for fruitful discussions and their help in the course of this work. This research was supported by the Vali-e-Asr University of Rafsanjan under grant No. P. 4561.

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Low-threshold bistability in nonlinear microring tower resonator

Abstract

1. Introduction

2. Linear response

2.1. Simple model for fast calculation of cavity- modes

2.1. Influence of cavity parameters on the Q factor

3. Nonlinear response

3.1. Coupled wave equations

3. 2. Nonlinear transmission properties

4. Effects of the loss upon device performance

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (8)

Equations (13)

Optics Express