1. Introduction

Nonreciprocal transmission of light plays an important role in all-optical information processing. Similar to electrical nonreciprocity, or the diode effect, all-optical diode (AOD) allows light to transmit only in one direction but blocks it in the opposite direction. The breaking of time-reversal symmetry of light can be achieved with various mechanisms, such as magneto-optic effect [1,2], optical nonlinearity [3–6], metamaterials [7,8], liquid crystals [9,10], optomechanical effects [11,12], and indirect interband photonic transitions [13,14]. As a key element in optical communications and computing, on-chip AOD should have broad operation bandwidth, high transmission contrast rate (which is defined as the ratio of the transmission intensities for forward and backward incident signal under the same conditions), and is compatible with semiconductor CMOS processes. One of the best ways for achieving high transmission contrast rate is based on Fano resonances, where the transmission can approach zero due to the destructive interference between the cascaded resonators [3,15,16]. However, the operation bandwidth for Fano diode is quite narrow (usually less than 0.005nm [3,6,15,16]), and several specially designed cavities have to be employed [3,6,15,16]. If a broad-band, high-contrast AOD can be realized via a single cavity, the experimental complexity will be greatly reduced, and it is more feasible in practice.

On the other hand, optical computing, optical interconnection systems and integrated photonic circuits sometimes might request the AOD reversible, that is, flipping the “conducted” state from one direction to the opposite one. Dynamical tuning of the nonreciprocity can make us manipulate the flow of light more robustly, and thus realize more advanced control of the light propagating systems, such as optical switching, router, and rectifier, etc [9,10]. So far, the proposed mechanisms to realize reversible optical diode are still very few, and most of them can flip the conducted direction only when the signal’s wavelength or power is changed [9,10,17]. To the best of our knowledge, the reversible AOD operating for the same signal (namely, the signal’s wavelength or power does not need to be altered) in a CMOS-compatible fashion has not been reported yet. Obviously, this case is rather important in information processing and quantum computing, since we usually have to robustly manipulate the same signal photon. In this paper, we propose a new mechanism to realize broad-bandwidth, high-contrast nonreciprocal light transmission, and report the first theoretical demonstration of flipping the conducted direction of AOD for the same signal light, via a single nanocavity and an auxiliary pump pulse. This new approach is based on precisely molding the dynamic interaction between the signal light, pump pulse, and resonant mode inside a nonlinear nanocavity connected with two asymmetric waveguides, which provides an extra freedom for the controlling of nonreciprocal light transmissions, as will be shown below.

2. Analytical model and numerical experiments

Without loss of generality, we start by considering a two dimensional (2D) photonic crystal (PC) structure, as sketched in Fig. 1. It consists a square lattice of silicon rods with a linear refractive index of n₀ = 3.48 and a radius of r = 0.2a, where a = 541nm is the lattice constant. The waveguide (WG) are created by reducing the radius of each rod in a line to r₀, which can be precisely tuned to change the transmission characteristics of the WG as desired. The nonlinear nanocavity locates asymmetrically at the WG, and divides the WG into a short one (WG1, with a length of 2a) and a relatively longer one (WG2, with a length of 8a), which is crucial for nonreciprocal light transmission, as will be shown below. The nonlinear nanocavity is made by polymer rod, which possesses instantaneous response time and much larger Kerr nonlinearity coefficient than silicon. In our calculations, the linear refractive index and Kerr nonlinearity coefficient of the polymer material are set to be n₀ = 1.59 and n₂ = 1.3×10⁻³ μm²/W [18], respectively. The radius of the cavity is selected to be 0.21a so that the cavity supports a single mode centered at ω₀ = 0.3491(2πc/a), where c is the speed of light in vacuum. We suppose the coupling rates of the cavity mode amplitude into the left WG (WG1) and right WG (WG2) are γ₁ and γ₂, respectively, and the intrinsic decay rate of the cavity is γ₀, which commonly originates from the linear absorption or radiation in PC cavity. Thus, the half of the linewidth of the cavity mode in linear case is γ = γ₁ + γ₂ + γ₀.

 

Fig. 1 Sketch of a PC AOD system composed of a nonlinear nanocavity coupled with two asymmetric WGs. The nonlinear nanocavity (the red circle) is made by polymer rod, which has a linear refractive index n₀ = 1.59 and a Kerr nonlinearity coefficient n₂ = 1.3×10⁻³ μm²/W, and all the other rods are made of silicon. p_s and p_p represent the incident powers of the signal light and the pump pulse, respectively, and t_d is the delay time of the pump pulse relative to the signal light.

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For simplicity, we temporally do not consider any nonlinear absorptions in the cavity, and only electrical Kerr nonlinearity is taken into account, which provides the highest possible speed owing to its inherently instantaneous response nature [19,20]. Here, the new approach for AOD is based on the controllable interaction between the signal, pump pulse, and the cavity mode, as will be demonstrated below.

To show this more clearly, we consider a continuous-wave (CW) signal light, superposed with a pump pulse, which acts as a control “trigger”. We suppose the stable field of the CW signal light has the form of $s_{in}=\sqrt{p_{\text{s}}}{\mathrm{e}}^{j\omega t}$, and the pump pulse has a Gaussian shape of $s_p=\sqrt{p_p}{\mathrm{e}}^{j\omega t}\cdot {\mathrm{e}}^{-{(t-t_d)}^2/t_0^2}$, where p_s is the signal power, p_p is the peak power of the pump pulse and is significantly higher than p_s; ω is the carrier frequency, which is commonly set lower than the resonant frequency ω₀; t₀ is the pulse during time, and t_d is the delay time of the pump pulse relative to the signal light. For better comparing the forward and backward signal transmissions under the same conditions, we can fix the pump pulse at a given place (e.g., the leftmost end of WG1), and just shift the CW signal light from one side to another and inverse its launch direction. According to temporal coupled-mode theory [21] the dynamic equation for the amplitude of the cavity mode can be written as

Although the “walls” at the two side of the cavity have the same thickness, γ₁ may not equal to γ₂. Actually, due to the partial reflections of the air-WG interfaces at the input and output ports of the WGs, WG1 and WG2 can act as effective F-P cavities with different lengths, and therefore different transmission spectra. These transmission spectra for WG1 and WG2 can be intentionally changed by finely tuning r₀. By using finite-difference time-domain (FDTD) method [24], we obtain the transmission spectra for isolated individual WG1 and WG2 (in absence of the central cavity) when r₀ = 0.2r, 0.3r and 0.5r, as shown in Fig. 2.

 

Fig. 2 Transmission spectra for isolated individual WG1 and WG2 when r₀ = 0.2r, 0.3r and 0.5r. The vertical dashed line denotes the location of the resonant frequency at ω₀, and the green curve at the bottom is the total transmission spectra for the AOD system in linear case.

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We see at the frequency region near ω = ω₀ (marked by a vertical dashed line), the relative magnitudes of the transmission rates for WG1 and WG2 are: less than 1when r₀ = 0.2r, equal to 1 when r₀ = 0.3r, and greater than 1 when r₀ = 0.5r. Obviously, a relatively higher transmission rate near ω = ω₀ indicates a “smoother” waveguide for these frequencies, so that the coupling rate between the cavity and WG is greater, vice versa. Using this method, we can freely set the ratio of γ₁ to γ₂ to any desired value just by finely tuning r₀. For example, when r₀ = 0.2r, 0.3r and 0.5r, we get γ₁/ γ₂ = 0.52, 1 and 2.07, respectively. This feature is very useful for AOD design.

In addition, we should point out that although WG1 and WG2 can act as effective F-P cavities, the confinements of the F-P WG modes are commonly weak since the partial reflections of the air-WG interfaces at the two ends of the WGs are not strong. As a result, the spectral widths of the F-P WG modes are far broader than that of the nanocavity mode. Thus, in the transmission spectra of this AOD system, the background of the F-P WG modes will be filtered out due to the filtering effect of the inline-coupled nanocavity, so that the total transmission spectra have standard Lorentzian lineshapes (see the green curve at the bottom of Fig. 2), just as if these effective F-P cavities never existed, except for their contributions on tuning γ₁ and γ₂. Consequently, for this inline-coupled cavity-WG system, the interactions between these effective F-P cavities and the central nanocavity can still be safely treated as a common WGs-cavity coupling problem, as described by Eq. (1).

By using four-order Runge-Kutta method [25] to solve Eq. (1), we can calculate the dynamic evolution of the nonlinear transmission coefficient T (defined as p_out / p_s), under the pump of a control pulse. It is found that the dynamic evolution process is very sensitive to the power of the pump pulse: even a slight change in power might lead to a quite distinct result. As an example, Fig. 3(a) depicts the calculated dynamic transmissions for a given incident CW signal light, under the excitation of a control pulse with quite slightly different pump powers just around one of the critical points. One can see that when p_p = 222.884W/μm, the transmission coefficient eventually gets to a steady “high” state of T = 0.83; while when we increase p_p slightly to 222.887 W/μm, the steady value of T sharply drops to 0.008, i.e., a “low” state. The evolution processes are determined by the dynamic interactions between three light components inside the nonlinear cavity: the signal, pump pulse and the excited cavity mode. The cavity mode has a red-shifted resonant frequency of ω₀ - 2γγ_out |A|² / p₀, and will decay at a rate of γ. By comparing the two curves in Fig. 3(a), it is found that they begin to separate from each other at the moment just behind the first “transmission valley” marked by a black star (see the inset): after which, if the survived cavity energy is not high enough to ensure the transmission rate to rise over T₀ (see the gray dashed line, which is corresponding to a steady “high” state when t→∞), then the evolution of the transmission will eventually get to a “low” state; otherwise, it will finally reach a “high” state.

 

Fig. 3 (a) Time evolutions of the nonlinear transmissions under different pump powers of the control pulse. High state: p_p = 222.884W/μm; low state: p_p = 222.887W/μm. Inset is the enlarged part of the region inside the dashed box. The other physical parameters used for calculations are: ω₀ = 0.3491, γ₁ = γ₂ = 8.67×10⁻⁵, γ₀ = 6.69×10⁻⁶ (all in units of 2πc/a), p₀ = 0.0026W/μm, p_s = 0.032W/μm, t₀ = 20(a/c), and t_d = 1000(a/c). These system parameters correspond to the case of r₀ = 0.3r, and can be obtained via numerical experiments. The detailed method of calculating p₀ can be found in [26,27]. (b) Steady transmission as a function of the pump pulse power.

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Fig. 3(b) further shows the steady value of T as a function of the pump pulse power. We can see clearly there are several “critical points” (the edges of the rectangular areas), at which the “low” state and “high” state can be flipped sharply. Recalling that it is commonly thought the bistability can always be trigged from low state to a high state only if the power of the pump pulse is high enough [22]. However, our theoretical results shown in Fig. 3 indicate that the critical condition is actually much more complicated. These rich dynamical features may find applications in optical switching and nonreciprocal transmissions.

To study the steady state of this AOD system, we can let t→∞ and substitute it into Eq. (1). In this case, the term of “${\text{e}}^{-{(t-t_d)}^2/t_0^2}$” can actually be neglected, so that we can derive an analytical solution from Eq. (1):

Equation (2) clearly indicates the bistable states for nonlinear transmission will occur, as shown by the solid curves in Fig. 4 (where the gray dashed-dotted lines denote unstable states, which mean that even a tiny perturbation will cause the solution to decay to either the upper or lower branches). The bistable states mean there are two possible solutions for a given incident signal power and wavelength. However, which state (or solution) the system will select is not determined by Eq. (2), but by the external pump pulse. Therefore, we should turn to Eq. (1) if we want to predict the only final state.

 

Fig. 4 (a) Forward and backward transmission spectra for (a) r₀ = 0.2r (p₀₁ > p₀₂) and (b) r₀ = 0.3r (p₀₁ = p₀₂). The pink region marked by “I” denotes backward-conduction AOD, while the pale blue region marked by “II” implies reversible AOD. The solid and dash curves represent analytical results from Eq. (2), while the scatted dots are from FDTD simulation. In the theoretical calculations, ω₀ = 0.3491(2πc/a) and p_s = 0.032W/μm are taken. In addition, for (a): γ₁ = 9.31×10⁻⁵, γ₂ = 1.79×10⁻⁴, γ₀ = 1.78×10⁻⁵ (all in units of 2πc/a), p₀₁ = 0.01W/μm and p₀₂ = 0.0052W/μm; for (b): γ₁ = γ₂ = 8.67×10⁻⁵, γ₀ = 6.69×10⁻⁶ (all in units of 2πc/a), and p₀₁ = p₀₂ = 0.0026W/μm. In the FDTD simulations, the grid sizes in the horizontal and vertical directions are all chosen to be a/30, and a perfectly matched layer (PML) of 1μm is employed as absorbing boundary. The empty circles and triangles represent a forward conducted AOD, while the solid ones denote a backward AOD. The reversible AOD shown in (b) has a maximum contrast rate C_max = 112 (over 20dB), with a working bandwidth over 7.5nm.

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From Eq. (2), it is readily found that for a given frequency detuning, when 2|A|²γ_out/ p₀ = δ, on-resonance transmission occurs and T will reach its peak value T_max = η. In this case, p₀ = 2|A_c|²γ_out/δ, where |A_c|² is the critical cavity energy for “high” state. Considering that |A_c|² should be the same no matter for forward or backward transmission, so that for a given δ, p₀ is proportional to γ_out, which indicates different launch direction of signal will lead to a different p₀. If we denote p₀ for forward and backward transmissions as p₀₁ and p₀₂, respectively, we get

Equation (3) indicates that if γ₁ ≠ γ₂, then p₀₁ ≠ p₀₂. This will make the bistable regions for two opposite transmission directions apart from each other, as shown in Fig. 4(a). Obviously, if we can manage to let forward and backward transmissions lie at different branches (upper or lower), then AOD effect can be realized. For instances, in region I of Fig. 4(a), the forward transmission is always in low branch, so we just only need to trigger the backward transmission in high state, and this can be readily accomplished; while in region II, considering the two upper branches need different pump energy, and the dynamic evolution process is very sensitive to the power of the pump pulse (as has been verified above), so that under the same pump condition, we can always select an appropriate power of the pump pulse to make the forward transmission in high state while restrain the backward transmission (we call this as forward conducted AOD), or vice versa (backward conducted AOD). This has been testified by performing FDTD simulations on the AOD system sketched in Fig. 1(a), as shown by the scatted circles and triangles in region II of Fig. 4(a).

This approach is based on γ₁ ≠ γ₂, which results in p₀₁ ≠ p₀₂ and make the light transmission nonreciprocal. However, the transmission rate is relatively low due to γ₁ ≠ γ₂, and the separation between the two bistable regions make the working area for reversible AOD effect (namely, region II) narrow, as can be seen from Fig. (4a). To break this limit, a natural idea is to make γ₁ = γ₂, since γ₁ = γ₂ can ensure η maximal [23], which indicates a higher T [known by Eq. (2)]. In this case, according to Eq. (3), p₀₁ will equal to p₀₂, so that the two bistable regions will overlap completely, which ensure a biggest working bandwidth for reversible nonreciprocal transmissions, as shown in Fig. 4(b). But here comes another question: the complete overlap of the two bistable curves indicates they share the same critical pump power condition, so that it is very difficult for us to make forward and backward transmissions lie at different branches via selecting different pulse powers.

Here we find another mechanism to overcome this difficulty: taking advantage of the different lengths of WG1 and WG2. By comparing the forward and backward transmission cases, one of the most important changes is that the effective delay times between the signal light and the pump pulse when they arrive at the cavity respectively are actually different: because the length of WG2 is much longer than that of WG1. As a result, the time-dependent interaction between the signal and the pump pulse inside the nonlinear cavity should also be different for forward/backward-transmission cases, leading to a constructive/destructive interference and making the cavity energy above or below the critical switching value, which will significantly affect the evolution of the transmission state. For the structure sketched in Fig. 1(a), our FDTD simulations reveal that the signal traveling time from WG2 to cavity is about 16(a/c) longer than that along WG1.

To investigate the influence of the change in the effective delay time, we can alternatively change t_d and observe the dependence of the steady transmission rate on t_d for forward and backward transmissions. Figure 5(a) is the simulation results corresponding to the rightmost edge (λ = 1570.5nm) of the bistable region shown in Fig. 4(b). One can see that to achieve high state under a given pump pulse power, the required t_d values for forward (red circles) and backward transmissions (blue triangles) are not continuous, but discrete, and as predicted, almost all of the desired delay times for backward transmission are always 16(a/c) longer than that for forward case (as partially marked by the curved arrows). This can be seen clearly by simply rightward shifting the data of the red circles 16(a/c) away from their original positions, and it is found that the data of the red circles and blue triangles indeed overlap each other, as shown by the inset in Fig. 5(a). Thus, for a given pump power, we can always select an appropriate t_d to make the AOD forward or backward conducted.

 

Fig. 5 (a) Required t_d values to achieve high state for forward (red circles) and backward transmissions (blue triangles), with λ = 1570.5nm, p_s = 0.032W/μm and p_p = 222.53W/μm. Inset shows that the data of the red circles and blue triangles will overlap each other if the data of the red circles are rightward shifted 16(a/c), as denoted by the curved arrows. (b) and (c) are the dynamic evolutions of the transmission rates for t_d = 1250 (a/c) and t_d = 1266 (a/c), respectively. These two t_d values correspond to the points inside the little dashed box in (a). The transmission contrast rates for forward conducted AOD in (b) and backward conducted AOD in (c) are 110 and 112, respectively. (d) Temporal evolutions of the intracavity energy for different t_d. The horizontal gray dashed line denotes the critical energy value for switching.

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As an example, Figs. 5(b) and 5(c) further present the FDTD-calculated dynamic evolution of transmissions for t_d = 1250 (a/c) and t_d = 1266 (a/c) (which mimics the change in the effective delay time for forward and backward transmissions). The corresponding temporal evolutions of the intracavity energy for different t_d are also plotted in Fig. 5(d), where the horizontal gray dashed line denotes the critical energy value for switching. From these curves, we can know that the constructive/destructive interference between the signal and pump pulse inside the nonlinear cavity is very sensitive to the pulse delay time relative to the signal light. Accordingly, for an AOD system consists of a nanocavity and two WGs with different lengths, t_d can really provide another freedom to manipulate nonreciprocal light transmissions, including the case of γ₁ = γ₂ as testified above, which means the whole bistable region shown in Fig. 4(b) can be utilized to realize reversible AOD.

By employing this approach, we can achieve broad-bandwidth, high-contrast nonreciprocal light transmissions. For a direct visualization, Fig. 6 depicts the steady field patterns of E_z (along the rods direction) for the forward/backward conducted AOD at an arbitrary signal wavelength of λ = 1563nm, which is about 7.5nm away from the rightmost edge of the bistable region shown in Fig. 4(b). Figure 6 intuitively demonstrates that high-contrast, and high-unidirectional-transmission-rate nonreciprocal AOD can be achieved via selecting appropriate pump pulse power and t_d. More importantly, Figs. 5 and 6 provide the direct demonstration that the nonreciprocal light transmission can be controllably reversed for the signal light under same wavelength and same signal power, with a broad operation bandwidth over 7.5nm (to the best of our knowledge, the reversible AOD operating for a same signal in a CMOS-compatible fashion has not been reported yet). This point has potential applications in optical computing, optical interconnection systems and integrated photonic circuits.

 

Fig. 6 Steady field patterns of E_z (along the rods direction) for reversible AOD with same wavelength (λ = 1563.2nm), same signal power (p_s = 0.032W/μm), and same pump pulse power (p_p = 77W/μm), but different pulse delay times. (a) and (b) Forward-conduction AOD: t_d = 994(a/c). (c) and (d) Backward-conduction AOD: t_d = 1010(a/c).

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The theory and approach presented in this paper is quite general and can also be realized in other resonator (e.g., microring, whispering-gallery mode cavity) systems, and is still effective for picosecond pulsed signal light, since our AOD system has a broad working bandwidth. Except for tuning r₀, changing the relative lengths of the input/output WGs is another alternative to tailor γ₁ and γ₂. In this paper, polymer materials are used to create nanocavity due to their instantaneous response feature and much larger Kerr nonlinearity coefficient than silicon. However, the use of polymer is not essential, and can be replaced with silicon at the expense of relatively higher operation power, while the physical mechanism is the same. In our simulations, for illustration purpose, the signal traveling time from WG2 to cavity is only 16(a/c) longer than that along WG1. However, this value can be arbitrary larger by introducing “slow light” technology into the WG design [28,29]. This would be helpful for actual experimental operations.

3. Conclusion

We present an extremely general yet simple method for achieving broad-band, high-contrast-ratio, reversible and high-unidirectional-transmission-rate AOD, based on a single nonlinear nanocavity with two asymmetric WGs. We demonstrate that the transiting of the signal light between the bistable states can be precisely molded by finely tuning the pump pulse power and the pulse delay time relative to the signal light, which provides extra freedom for the controlling of nonreciprocal light transmissions. Using this mechanism, we report the first theoretical demonstration that the nonreciprocal light transmission can be controllably reversed for the signal light at same wavelength, with a broad operation bandwidth (over 7.5 nm), which is promising in the fields of advanced photonic circuits, all-optical information processing, and optical communications.