The optical diode is essential in optical computation and communication,^[1–4] as well as in the quantum information processing area. Until now, kinds of non-reciprocal schemes have been fabricated, such as the magneto-optic^[5–7] and acousto-optic effects systems,^[8] the chiral structures,^[9,10] nonlinear parity asymmetric systems,^[11–14] etc. However, most of them were either active which need strong external electromagnetic field, or very complicated which need an accurate micro/nano-scale processing technique.

Recently, non-reciprocal light transport has been realized experimentally in a parity-time asymmetric system.^[15,16] Parity-time symmetry can be broken by taking into account the gain saturation in an active resonator.^[16] Although the coupled active-passive whispering-gallery microcavities can provide a non-reciprocal isolation ratio from −8 dB to +8 dB, high-Q resonator is needed and it is not passive. In this work, we will show a high contrast optical diode up to 22 dB in a pure passive system comprised of an asymmetric ring cavity containing a saturable absorptive atomic ensemble, which indicates that active medium, parity-time asymmetry and high-Q resonators are not always necessary.

In our previous work,^[17,18] we have shown that the time-reversal asymmetry, spatial asymmetry and optical nonlinearity are three significant factors for optical non-reciprocity, which is consistent with the result of Ref. [19]. In the present work, we will show a giant non-reciprocity in a classical atomic gas-cavity coupling system, which easily fits these three factors. Specifically, optical nonlinearity naturally happens due to saturable absorption of gas, and spatial symmetry can be broken by furnishing different reflectivity walls, meanwhile time-reversal asymmetry is naturally satisfied by taking into account non-radiative dissipations of the cavity and the resonant absorption of atoms.

Although optical absorptive bistability was realized first in 1969,^[20] and since then on, a great deal of attention has been paid to observing and understanding the phenomenon of optical bistability^[21–24] in symmetric or one-side cavity, there was little work on controllable bistability within asymmetric cavity.^[18,25] In our previous work,^[18] we have presented a new proposal to obtain giant optical non-reciprocity up to 13 dB by using single-atom controllable bistability in an asymmetric cavity. We have shown that single-atom bistable states were controlled by the asymmetric walls of the cavity and it is feasible to realize the smallest all-optical diode. However, the manipulation of single atom in nano-cavity is still tough in a quantum optical system. It will be much convenient if we can realize this kind of optical diode in a classical optical bistable system.

In this work, we pay attention to the asymmetric ring cavity containing a two-level atomic ensemble which can be performed much more easily under the present technology. More interestingly, we also find an apparent direction-dependent bistability within the asymmetric ring cavity, which can be used to realize a much higher contrast optical diode up to 22 dB. Specifically, under a certain input power, it shall operate in the linear regime (I_in < I_l) for the forward input case while it shall work in the saturated regime (I_in > I_u) for the backward input case. Here I_in, I_l, and I_u are the input intensity, the lower, and upper threshold intensities of bistability regime, respectively.

The model considered is shown in Fig. 1. This model is similar to the prototype of a Fabry-Perot resonator filled with a saturable absorber presented in Refs. [20] and [26], but it has been modified with asymmetric ring cavity now. To simplify the analysis, its four mirrors M₁, M₂, M₃, and M₄ are lossless. M₃ and M₄ are ideal mirrors with 100% reflectivity, while M₁ and M₂ are different from each other to present the asymmetry of the ring cavity, which are characterized by different reflected and transmitted coefficients, i.e., r₁, r₂, t₁, and t₂, respectively. Meanwhile, their reflectivities and transmittivities are defined by T_j = |t_j|² and R_j = |r_j|² (j = 1, 2), respectively. On the upper branch of the ring cavity, it is filled with a saturable two-level atomic ensemble with length l, which is marked by a gray shadow.

Here, we distinguish the forward input case from the backward input case. The forward input case is shown in Fig. 1, in which the input field E_in is incident from M₁ to M₂, while the backward input case refers to opposite one incident from M₂ to M₁. Taking the forward input case for example, after a similar derivation to that in Ref. [20], we can obtain the relation between input power I_in ∝ |E_in|² and output power I_t ∝ |E_t|²,

where

refers to the multi-reflection in cavity. Unlike the usual linear optics, the absorption coefficient α(x) is dependent on the field intensity in the saturable two-level atomic ensemble I_f ∝ |E_f|² and it is also position-dependent in general, which can be expressed as

Here, α₀ is the linear absorption coefficient and I_s is the critical power of saturable atomic ensemble. The field intensity in the saturable medium I_f is also related to nonlinear absorption coefficient, writing

By using the iterative method we can solve Eqs. (1)–(3) numerically and obtain the steady relationship between I_t and I_in. The method adapted here is a little different from the mean field approximation adopted in Refs. [20] and [26] concerning the optical bistability in symmetry cavity, in which the nonlinear absorption coefficient α was spatially averaged and independent of its position. It should be pointed out that for the backward input case, equations (1) and (2) are not changed, but T₂ should be replaced by T₁ in Eq. (3).

In order to check the availability of our method, we calculate the input-output relation for the case of symmetric cavity first, i.e., R₁ = R₂. By fixing the linear absorption coefficient as α₀l = 3.75 which was presented in Ref. [20], the curves of I_t versus I_in are plotted in Fig. 2(a) with different values of R₁ = R₂. The figure shows that when R₁ = R₂ > 0.8, the optical bistability happens. For any bistable state, bistable regime is characterized by two threshold intensities I_l and I_u. On the other hand, by fixing the mirrors of R₁ = R₂ = 0.9, we plot the curves of I_t versus I_in in Fig. 2(b) with different values of α₀l. It shows that the bistable states happen when α₀l > 1.25. These results are in accordance with the results in previous studies and show that the bistable states appear only in a definite range of input intensities with using appropriate parameters.^[21]

Now we discuss the direction-dependent optical bistability induced by spatial symmetry breaking in the asymmetric ring cavity. The reflectivities of mirrors M₁ and M₂ are set to be R₁ = 0.8 and R₂ = 0.9 to present asymmetric ring cavity, respectively. By fixing α₀l = 3.75, the relations between I_t and I_in for forward and backward input cases are plotted as the solid and dashed curves in Fig. 3(a), respectively. Comparing the curves in Fig. 3(a) with the blue and red curves in Fig. 2(a), we find that bistable phenomena in the asymmetric cavity are very different from those in the symmetric cavity. In a spatial asymmetric system, optical bistability is direction-dependent now.

For the forward input case, input field is incident on M₁ with R₁ = 0.8 at first, and the solid curve in Fig. 3(a) is similar to the red dashed curve in Fig. 2(a), but the bistable regime of I_in in asymmetric cavity is much wider than that in symmetric cavity with R₁ = R₂ = 0.8. For the backward input case, the input field is incident on M₂ with R₂ = 0.9 at first, and the dashed curve in Fig. 3(a) is more similar to the blue dotted curve in Fig. 2(a), but the bistable regime for backward input case is narrower than that of symmetric cavity with R₁ = R₂ = 0.9. Therefore, in the asymmetric cavity, optical bistability is highly direction-dependent. The result of direction-dependent absorptive bistability in classical system is in line with our previous prediction of controllable single atom bistability in quantum optical system.^[18]

The non-reciprocal transport in the present asymmetric system originates from the varying optical nonlinearities induced by absorption saturation in two input cases, which is similar to that in the active system.^[16] More importantly, our scheme works in the bistable regime while the active system does not, which is the reason why our scheme can generate much higher transmission contrast than previous ones. In this work, the optical bistable regime is separated significantly between the forward and backward input cases. Therefore, our scheme will provide excellent optical non-reciprocity. Furthermore, our scheme is passive and without high-Q resonators.

Here, we show how to construct an efficient optical diode under the optimal window of operation power. As shown in Fig. 3(a), when the input power is located in the optimal window, i.e., I_in ∈ [5.24,9.48], forward input field can transport the cavity easily but the backward input field is blocked completely. The values of normalized transmittivity T = I_t/I_in for the forward input case T_f and backward input case T_b are plotted in Fig. 3(b) in the optimal window. It is shown that the optical diode contrast C = 10 × log₁₀(T_f/T_b) can easily reach 22 dB in the present classical optical bistable system.

To explain such a direction-dependent optical bistability and non-reciprocity analytically, we explore the bistability condition in detail. Since the value of αl is near zero undergoing large input power, the exponential term e^−αl can be approximated by 1 − αl. Taking the forward input case for example, with the definition of X = I_in/(I_sT₂) and Y = I_t/(I_sT₂), the equation connecting the input power I_in and the transmitted power I_t can be simplified into

where p = (1 − r₁r₂)/(t₁t₂) and k₀ = α₀l·r₁r₂/(2t₁t₂). If Y has three solutions for a certain X, then bistability happens. The thresholds of bistability for Eq. (4) are the solutions of equation B² − 4AC = 0, where A = b² − 3ac, B = bc − 9ad, C = c² − 3bd, and a = p², b = 2p(p+k₀) − X, c = (p+k₀)² − 2X, d = −X. Notice that equation (4) is independent of the input direction because r₁ and r₂ as well as t₁ and t₂ always appear in couples, therefore the threshold of X for bistability is direction-independent.

However, the threshold intensities I_l and I_u are direction-dependent due to the definition X = I_in/(I_sT₂). For the case of forward input, I_l (I_u) ∝ T₂. Meanwhile for the case of backward input, I_l (I_u) ∝ T₁. In Fig. 3, T₁ = 0.2 and T₂ = 0.1, the threshold I_l and I_u for the forward input case (solid curve) are 4.74 and 5.12, respectively; meanwhile I_l and I_u for the backward input case (dashed curve) are 9.48 and 10.24, respectively, which are double compared with the threshold intensities in the forward input case. It presents a remarkable direction-dependent bistability in the asymmetric cavity.

The direction-dependent optical bistability can be used not only as optical diode or isolator, but also as optical switching, memory, transistor,^[27] etc. Here we introduce a new application as optical logic AND gate. Despite the superiority of such a kind of logic gate, our proposal provides a new access to it at least. The block diagram of such a logic AND gate is shown in Fig. 4, where two optical signals I_a and I_b are incident on the left wall M₁ of saturated bistable optical device through an add/drop filter and output signal I_t transports from the right wall M₂. We present a random phase mask on the line of I_b to ensure the two input fields are fully decoherent. Therefore, the input power is I_in = I_a + I_b. After transmitting through the saturable absorptive bistable device, the output power I_t depends on the optical bistable state.

Here, optical logic states 0 and 1 are characterized by the intensity of the field, which can be defined as

where I₁ and δ_e (≪ I₁) are the threshold intensity of logic 1 and its tolerance, respectively. In order to obtain logic AND gate, I_in = I₁ should be located in the linear regime while I_in = 2I₁ in the saturated regime with I_t(2I₁) = I₁ to ensure the output logic state is true.

As is well known, the output power I_t in the saturated regime can be simplified into a linear function, i.e., I_t = a + b·(I_in − I_l), where a and b are the fitting parameters. In principle, a and b as well as I_l can be adjusted by the reflectivities of walls. Logic AND gate should fulfill the following conditions at least, i.e. δ_e < a/2 and I₁ ∈ (I_u/2 + a/2, I_l − a/2).

As an example, we set α₀l = 3.75, R₁ = 0.8, R₂ = 0.9 and plot curve of I_t versus I_in in Fig. 5 to illustrate the operating power of optical AND gate. After numerical simulation and linear fitting we obtain I_l = 4.74, I_u = 5.24, a = 1.07, and b = 1.017. Therefore, with the coded power I₁ = 3.69 and δ_e < 0.51, we can realize the optical logic AND gate. It can be easily verified that only for logic input state (1,1) the output logic state is true when we calculate the output I_t at different input states as shown in Table 1. It should be pointed out that although we take an asymmetric system for example here to illustrate logic AND gate, it is different from such a kind of optical diode. To construct such a kind of logic AND gate, the spatial asymmetry is not always necessary.

Finally, we discuss the experimental proposal to such an optical diode and a logic AND gate. Since there are many bistable experiments done in symmetric classical macro cavity, we can retain their main equipment presented in Ref. [20] but only replace the symmetric cavity with the asymmetric ring cavity. However, the bistable operating power was rather high in the past work and it should be lowered to an acceptable level. As is well known, the saturable critical power I_s ∝ 1/μ²τ,^[20,28,29] where μ and τ are atomic dipole moment and the atomic lifetime of the saturable medium in the resonator, respectively. Thus larger μ and τ are beneficial to lower I_s.^[22] Recently, there were a remarkable series of experiments in bistable multiple-quantum-well devices that lowered the critical power to an acceptable level.^[24,27] Since silicon-based optical micro devices are desired, we also present a possible experimental proposal based on micro/nano technology. Recently, on-chip whispering-gallery microcavities with gain and loss have been fabricated.^[16] Therefore, it is not difficult to construct such a four-mirror ring cavity with loss in one of the rectangle sides after careful designing. Four Bragg reflectors^[30] should be placed in the corners of ring cavity and the spatial asymmetry can be obtained by furnishing different Bragg reflectors. Therefore, it is feasible to realize such a kind of optical diode and logic AND gate.