Unidirectional lasing in nonlinear Taiji micro-ring resonators

We begin by studying the solutions to Eq. (II) in the simplest backscattering-free ring resonator case (i.e. $t=1$, $\beta_{\pm,\mp}=0$). Fig. 2 shows the steady-state intensities $|\tilde{a}^{0}_{\pm}|^{2}$ of both modes (panels (a) and (b)) together with the imaginary parts of the eigenvalues of the fluctuation dynamics matrix (II) (panels (c) and (d)) for growing values of the pump rate from $P_{0}=0$ to $P_{0}=4\gamma_{\rm T}$ in the $g=1$ (panels (a) and (c)) and $g=2$ (panels (b) and (d)) cases. In both situations, below the lasing threshold $P_{0}<\gamma_{\rm T}$ the intensity $|\tilde{a}^{(0)}_{\pm}|^{2}$ of both modes is zero and the four eigenvalues

feature the same negative imaginary part. Since gain saturates with $\tilde{a}^{(0)}_{\pm}$, in this regime the only possible stable solution is that no coherent light is present. Above the lasing threshold $P_{0}>\gamma_{\rm T}$ the eigenvalues change to

The $g=1$ case presents a particular behaviour as it is the only situation in which the system shows effective gain in both directions and features a threefold degenerate Goldstone mode with $\lambda_{1,2,3}=0$. This is due to the phase freedom of each direction and the CW $\leftrightarrow$ CCW symmetry which is present in this case only. Beyond the bifurcation point both intensities depart linearly with randomly-chosen values satisfying $(|\tilde{a}^{(0)}_{+}|^{2}+|\tilde{a}^{(0)}_{-}|^{2})/n_{\rm S}=P_{0}/\gamma_{\rm T}-1$.

For any other value $1<g\leq 2$ the resonator lases randomly in one mode only with equal probability as $|\tilde{a}^{(0)}_{\pm}|^{2}/n_{\rm S}=P_{0}/\gamma_{\rm T}-1$, while the other one remains unamplified. In contrast to the $g=1$ case, now the linearized analysis features a single Goldstone mode ($\lambda_{3}=0$, $\lambda_{1,2,4}\neq 0$). Due to the asymmetric gain amplification terms in Eq. (II) a larger intensity in a certain direction randomly determined by the initial conditions and the background random noise $\eta_{\pm}$ leads to a smaller amplification in the other one. As $t$ evolves this asymmetry drives the system into a unidirectional lasing state in the randomly favored mode, while the other is killed.

By employing a Fourier transform we were able to ascertain that for $P_{0}>\gamma_{\rm T}$ the steady-state field amplitudes $a^{(0)}_{\pm}$ oscillate in time at the nonlinear-shifted resonance frequency of the resonator, which we take as the reference frequency $\omega$. In the $g=1$ case this is given by

while for $g>1$ the intensity in one of the two directions vanishes and one has either

for unidirectional lasing in the CW or CCW direction, respectively.

The fact that no bidirectional lasing can be observed for $g>1$ can be put on more solid grounds as follows. From Eq. (II) one sees that if both field amplitudes $\tilde{a}^{(0)}_{\pm}$ are taken as real numbers a finite intensity in the two directions (i.e. $\tilde{a}^{(0)}_{\pm}\neq 0$) would simultaneously imply

which necessarily sets $|\tilde{a}^{(0)}_{+}|^{2}=|\tilde{a}^{(0)}_{-}|^{2}$. Diagonalizing the matrix in this case, one obtains the eigenvalues

Having $g>1$ implies $\Im{\lambda_{3}}>0$ and therefore this solution is unstable. We conclude that for $g>1$ the only possible stable solutions involve unidirectional lasing in one of the two counter-propagating modes, randomly chosen by the initial conditions.

We now move on to the study of the stability of an active TJR under the same ramp in the pump rate employed in the previous Subsection. For these simulations we set $\gamma_{\rm S}=0.2\gamma_{\rm A}$, $\beta_{\pm}=0$, and $\beta_{\mp}=-i2ck^{2}_{\rm S}e^{i\omega_{0}n_{\rm L}L_{\rm S}/c}/L_{0}n_{\rm L}$. We have neglected the nonlinear phase shift inside the S waveguide as the optical power inside it is very small compared to that circulating along the external waveguide.

Fig. 3 shows the steady-state intensities $|\tilde{a}^{(0)}_{\pm}|^{2}$ (in panels (a) and (b) for $g=1$ and $g=2$, respectively) and the imaginary part of the eigenvalues (in panels (c) and (d) for $g=1$ and $g=2$, respectively) for the family of steady-state solutions that are most commonly reached. Below the lasing threshold $P_{0}<\gamma_{\rm T}$ the two directions feature a zero intensity for both interaction types. Above the lasing threshold $P_{0}>\gamma_{\rm T}$ the intensity of the mode in which the S-waveguide coupling is directed (in our case the CCW) departs linearly from the bifurcation as $|\tilde{a}^{(0)}_{-}|^{2}/n_{\rm S}=P_{0}/\gamma_{\rm T}-1$, while the other mode (the CW one) remains empty. The off-diagonal S-waveguide coupling in Eq. (4) does not change the eigenvalues of the linearized theory w.r.t. the ring resonator case and therefore they are given again by Eqs. (5-6) below threshold and by Eqs. (III.1-10) above threshold. As opposed to the ring resonator case, not even for $g=1$ one can have a finite intensity in both modes simultaneously: the presence of the S waveguide breaks in fact the CW $\leftrightarrow$ CCW symmetry and favors unidirectional lasing in the CCW direction. As in the previous Subsection using a Fourier transform we found that above $P_{0}>\gamma_{\rm T}$ the steady-state field amplitudes $a^{(0)}_{\pm}$ oscillate at the nonlinear-shifted resonance frequency of the resonator, which is taken as the reference frequency $\omega$. For our TJR this is given by

for any value of $g$.

While the unidirectional CCW-only solutions depicted in Fig. 3 are the steady-state that is most frequently reached by the numerics, it is important to verify whether other solutions are possible in specific parameter regimes. To this purpose, we considered Eq. (II) assuming a finite intensity in both modes, i.e. $\tilde{a}^{(0)}_{\pm}\neq 0$. Without loss of generality $\tilde{a}^{(0)}_{+}$ is taken to be a real number. From Eq. (II) this would require

In the $g=1$ case the first two conditions impose the denominator in Eq. (20) to vanish, which implies that $\tilde{a}^{(0)}_{+}=0$.

To carry on this analysis in the $g>1$ case we explored the phase space of the problem by numerically solving Eq. (II) using different values of $P_{0}$ and $\gamma_{\rm S}$ in order to explore its possible steady-state solutions. We found that stable bidirectional lasing is possible for values of $P_{0}$ and $\gamma_{\rm S}$ within the dark blue region of Fig. 4a. This kind of solutions involve a strong lasing in the CW direction and a smaller intensity populating the CCW due to the presence of a weak S coupling. For larger values of $\gamma_{\rm S}$, $|\tilde{a}^{(0)}_{-}|^{2}$ increases up to a threshold in which the $\tilde{a}^{(0)}_{+}\neq 0$ solution disappears and only the $\tilde{a}^{(0)}_{-}\neq 0$ one remains. While this figure is plotted in the linear ($n_{\rm NL}=0$) regime, we have verified that the same conclusion holds in the presence of nonlinearities, $n_{\rm NL}\neq 0$. However, the stronger is the nonlinearity $n_{\rm NL}$, the smaller is $|\tilde{a}^{(0)}_{-}|^{2}$ in the bidirectional solution.

The light blue parameter range of Fig. 2 features a single solution with emission in the CCW mode only. The lower limit is given by the pump threshold $P_{0}=\gamma_{\rm T}$. Of course, this CCW-mode lasing solution also extends in the dark-blue region of Fig. 2 where it coexist with the other, bidirectional solution. Which of the two solutions is actually chosen by the system will depend on the initial conditions.

Panels (b) and (c) show the lasing intensity and the imaginary part of the eigenvalues (calculated by diagonalizing the matrix $A$ in Eq. (4)) for the ramp depicted as the dashed line in panel (a), respectively. The initial conditions correspond to unidirectional lasing in the CW direction. As $\gamma_{\rm S}$ is increased the initially unfavored CCW mode gets rapidly populated and the emission intensity in the CW mode decreases. As this happens the imaginary part of one of the eigenvalues grows and eventually crosses zero towards positive values. At this point the solution becomes unstable and the system experiences a transition towards a unidirectional lasing regime in the CCW direction. Note that as $\gamma_{\rm S}$ grows, the total loss rate $\gamma_{\rm T}$ is also enhanced so that the CCW intensity decreases until the system ends up in the trivial solution.

We conclude that the presence of the S-shaped element breaking $\mathcal{P}$-symmetry in Eq. (II) rules out the possibility of pure unidirectional lasing solutions in the unfavored direction. For $g=1$ not even bidirectional emission is possible and beyond the lasing threshold the system compulsory lases in a single direction with the preferential chirality introduced by the S waveguide. For $g>1$ bidirectional lasing is possible for a small enough S-coupling, but only for particular values of the parameters and always giving $|\tilde{a}^{(0)}_{+}|^{2}\gg|\tilde{a}^{(0)}_{-}|^{2}$.

From a more abstract perspective, we can conclude this section by noting that, for any value of $g$, in contrast to what happens in a passive device or for a pump rate $P_{0}<\gamma_{\rm T}$, the solution for $P_{0}>\gamma_{\rm T}$ is not invariant under $\mathcal{T}$-reversal even though the underlying equations of motion are fully $\mathcal{T}$-reversal symmetric. We can therefore state that the presence of the $\mathcal{P}$-breaking S-shaped element induces a dynamical breaking of $\mathcal{T}$-symmetry above the lasing threshold.

Prior to the numerics we discuss the perturbative analytical solution of Eq. (II) valid for both the ring resonator and the TJR as microscopic backscattering is added. We assume that this is so small that the backscattering-free solution of Eq. (II) involving unidirectional lasing does not substantially change. In order to account for the small intensity in the suppressed mode we considered the backscattering couplings $\tilde{\beta}_{\pm,\mp}$ as the small perturbations to the backscattering-free ring resonator and TJR solutions explored in Sec. III. The new coupling parameters for the ring resonator are $\beta_{\pm,\mp}=\tilde{\beta}_{\pm,\mp}$, while those for the TJR are $\beta_{\pm}=\tilde{\beta}_{\pm}$ and $\beta_{\mp}=-i2ck^{2}_{\rm S}e^{i\omega_{0}n_{\rm L}L_{\rm S}/c}/L_{0}n_{\rm L}+\tilde{\beta}_{\mp}$. Without loss of generality, and in order to use the same indices in both cases, we considered an unperturbed solution in which the ring resonator lases unidirectionally in the CCW mode. We can then write the perturbed solutions as $\tilde{a}^{(0)}_{\pm}=\tilde{a}^{(0)}_{\pm,\rm un}+\delta\tilde{a}^{(0)}_{\pm}$, where $\tilde{a}^{(0)}_{\pm,\rm un}$ are the unperturbed solutions with $\tilde{\beta}_{\pm,\mp}=0$. As shown in Sec. III these are given by $\tilde{a}^{(0)}_{+,\rm un}=0$ and $|\tilde{a}^{(0)}_{-,\rm un}|^{2}/n_{\rm S}=P_{0}/\gamma_{\rm T}-1$. On the other hand $\delta\tilde{a}^{(0)}_{\pm}$ are the first-order perturbative corrections. Introducing the perturbed solution into the steady-state given by Eq. (II) and staying at linear order $\mathcal{O}(\tilde{\beta}_{\pm,\mp})$, $\mathcal{O}(\delta\tilde{a}^{(0)}_{\pm})$ in the perturbation we arrive at

The intensity in the perturbatively populated CW mode is given by $|\delta\tilde{a}^{(0)}_{+}|^{2}$. This means that as long as backscattering couples light from the CCW into the CW mode (i.e. for $\tilde{\beta}_{\pm}\neq 0$) the CW mode will host a finite emission. Note that the intensity in the CW mode increases whenever the system features a smaller value of $g$, that is a more nonlocal gain saturation. The perturbation theory breaks down for a pure thermo-optically driven gain saturation as Eq. (23) diverges for $g=1$. This is a consequence of the threefold-degenerate Goldstone mode that appears in this case.

Our simulations confirm that Eq. (23) correctly accounts for the intensity of the perturbatively populated CW mode as long as $|\tilde{\beta}_{\pm}|\ll\gamma_{\rm T}$. The intensity in the preferred mode is still given with a large precision by the unperturbed solution $|\tilde{a}^{(0)}_{-,\rm un}|^{2}$ and the eigenvalues of the matrix $A$ in Eq. (4) do not significantly change. Therefore we conclude that in the presence of perturbative backscattering both the ring resonator and the TJR lase unidirectionally to a great extent, although some light is also present in the unfavored mode at the same frequency, with an intensity proportional to the backscattering coupling in its direction.

Here we demonstrate how the values of the Hermitian and non-Hermitian coefficients describing backscattering (21-22) control the lasing chirality in a ring resonator. This fact can be employed to construct unidirectional lasers by properly engineering the resonator’s microscopic backscattering, for example by means of one or more nanotips coupled with the evanescent field of the resonator, similarly to the experiment of [22]. Note that in a ring resonator $\gamma_{\rm S}=0$ and therefore $\gamma_{\rm T}=\gamma_{\rm A}$.

As was shown in Sec. IV.1 the presence of perturbative backscattering does not appreciably change the fluctuation dynamics eigenvalues and the system lases unidirectionally to a great extent. Nevertheless, for values $|\tilde{\beta}_{\pm,\mp}|\gtrsim 10^{-2}|\gamma_{\rm A}|$ the perturbation theory (23) breaks down, a significant intensity populates the unfavored lasing direction and the intensity in the preferred direction falls below its usual value $P_{0}/\gamma_{\rm A}-1$. Fig. 5 shows the intensities (panel (a)) and imaginary part of the eigenvalues of matrix $A$ (panel (b)) for a ring resonator featuring $g=2$, $h=0$ and $|n|=0.1\gamma_{\rm A}$. Above the lasing threshold located at $P_{0}\simeq 0.9\gamma_{\rm A}$ both counterpropagating modes are amplified with the same intensity and oscillate simultaneously in a coherent way. As $P_{0}$ and $|\tilde{a}^{(0)}_{\pm}|^{2}$ grow the effect of mode competition in the gain saturation gets reinforced and intensity fluctuations become more susceptible of breaking the symmetry of the laser emission. This is evident from panel (b), where one of the imaginary parts quickly grows as $P_{0}$ increases and at some point turns positive. Here, the system undergoes a transition towards a regime with a larger lasing intensity in a preferred mode. The larger $g$ (that is, the larger the local character of the gain saturation), the smaller the value of $P_{0}$ necessary to drive the system out of the bidirectional emission state as gain saturation asymmetry becomes more important. Nevertheless, the unfavored mode maintains a small intensity due to the finite backscattering in both directions.

The smaller lasing threshold obtained in Fig. 5 can be understood by diagonalizing the matrix $A$ of Eq. (4). In the general coupling case $\beta_{\pm,\mp}\neq 0$ below the lasing threshold (i.e. for $|\tilde{a}^{(0)}_{\pm}|^{2}=0$) the eigenvalues are

The terms in the square roots can modify their imaginary parts, therefore shifting the threshold position w.r.t. the backscattering-free case, where these terms were zero. The new lasing threshold takes place at a power $P_{0}=\gamma_{\rm A}-Im\{(\beta_{\pm}\beta_{\mp})^{1/2}\}$. If $n=0$ one has that $\beta_{\mp}=\beta^{*}_{\pm}$ and the square roots are purely real. In this case the lasing threshold remains at its usual position $P_{0}=\gamma_{\rm A}$. The same is true for a unidirectional coupling (featuring either $\beta_{\pm}=0$ or $\beta_{\mp}=0$). The maximum shift for fixed $|\beta_{\pm,\mp}|$ occurs for $h=0$ and is given by $P_{0}=\gamma_{\rm A}-|\beta_{\pm}|$.

In spite of the finite intensity emitted in the unfavored direction, in the small backscattering regime the majority of the ring resonator’s emission still takes place in a preferential direction determined by the particular choice of backscattering coefficients. Fig. 6 shows the probability of a ring resonator featuring $g>1$ to lase preferentially in a certain direction as a function of the Hermitian and non-Hermitian parameters. Here, the probability is calculated by averaging over many independent realizations of the initial noise used to seed the laser operation.

We first study the situation in which both of them are finite and of equal strength, i.e. $|h|=|n|\neq 0$. In panel (a) these are kept at a fixed modulus $|h|=|n|\simeq 5\times 10^{-4}\gamma_{\rm A}$ while the phase angle between them $\Delta\phi=\phi_{n}-\phi_{h}$ is varied. For $\Delta\phi=0,\pi$ (which implies $|\beta_{\pm}|=|\beta_{\mp}|$) the system has equal probabilities of lasing in each mode, while for $\Delta\phi=\pm\pi/2$ the emission preferentially takes place in one particular direction with 100$\%$ probability. It is easy to realize that $\Delta\phi=+\pi/2$ implies $\beta_{\mp}=0$ while $\Delta\phi=-\pi/2$ corresponds to having $\beta_{\pm}=0$. These results are in perfect agreement with the behavior of passive microdisk resonators reported in [31].

However, this is only valid when $|h|=|n|\neq 0$. If one of the two parameters is kept fixed and the other is reduced to zero, the lasing probability in each direction tends to $\mathcal{P}=0.5$ regardless of the phase difference $\Delta\phi$, as shown in panels (b) and (c). This is easily understood by exploring Eqs. (21-22): either $n=0$ or $h=0$ imply in fact an equal coupling strength in the two directions, i.e. $|\beta_{\pm}|=|\beta_{\mp}|$.

In the case of a TJR coupling unidirectionally the CW into the CCW direction one has that $|\beta_{\mp}|\gg|\beta_{\pm}|$ and therefore

which falls into the $|n|=|h|$, $\Delta\phi=-\pi/2$ case. As already shown in this Section, this implies that even though the resonator hosts a small intensity in the CW mode due to the finite coupling $\beta_{\pm}\neq 0$, the majority of the system emission takes place in the CCW mode. The presence of additional backscattering coupling light into the CCW direction does not have any visible effect as it does not modify the coupling $\beta_{\mp}$ significatively ²²2For a TJR with an oppositely oriented S-element, one still has $|n|=|h|$ but $\Delta\phi=\pi/2$. As expected, this implies a preferred emission in the CW direction..

In order to shine more light into the role of the S waveguide we performed a coupling ramp from $\gamma_{\rm S}=0$ (which corresponds to a ring resonator) to $\sqrt{\gamma_{\rm S}/\gamma_{\rm A}}=1.5$ in a resonator featuring $g=2$ and a fixed pump rate $P_{0}\simeq 36\gamma_{\rm A}$. The initial situation is described by the backscattering parameters employed in Fig. 5, namely $|n|=0.1\gamma_{A}$ and $h=0$, for which the two directions have equal probabilities of hosting the preferential lasing emission. Our results are displayed in Fig. 7. As $\gamma_{\rm S}$ grows $\beta_{\mp}$ increases: This leads to an intensity transfer from the CW to the CCW mode. For $\sqrt{\gamma_{\rm S}/\gamma_{\rm A}}\gtrsim 0.25$ the S-waveguide coupling strength grows beyond the backscattering couplings and the imaginary part of one of the eigenvalues of matrix (4) crosses zero from below. The system then experiences a transition towards a state with preferential lasing in the CCW direction. As $\gamma_{\rm S}$ continues growing the relative importance of backscattering w.r.t. the S coupling decreases. For $\sqrt{\gamma_{\rm S}/\gamma_{\rm A}}\simeq 1$ the ratio between backscattering in the CW direction $\beta_{\pm}$ and the total loss rate $\gamma_{\rm T}$ already gives $|\beta_{\pm}|\simeq 5\times 10^{-2}\gamma_{\rm T}$, which means that the system is approaching the perturbative backscattering regime described in Sec. IV.1. As this happens the spectrum of eigenvalues of the matrix $A$ in Eq. (4) approaches the backscattering-free eigenvalues (III.1-10) and the intensity ratio between the two directions already gives a sizable value $|\tilde{a}^{(0)}_{-}|^{2}\simeq 250|\tilde{a}^{(0)}_{+}|^{2}$ for $\sqrt{\gamma_{\rm S}/\gamma_{\rm A}}=1.5$. The eventual decrease of the intensity also in the CCW direction that is observed at higher $\gamma_{\rm S}$ is due to the growth of the total loss rate $\gamma_{\rm T}$ that is naturally associated to the increasing $\gamma_{\rm S}$.

These results confirm the possibility to use the S waveguide in order to guarantee unidirectional lasing. The necessary condition is to implement a sufficiently large coupling allowing to treat backscattering as a microscopic perturbation to unidirectional lasing, which according to our model is legitimate at least up to $|\tilde{\beta}_{\pm,\mp}|\lesssim 10^{-2}\gamma_{\rm T}$.