Enhanced modulation characteristics in broken symmetric coupled microring lasers

Chi Xu; William E. Hayenga; Hossein Hodaei; Demetrios N. Christodoulides; Mercedeh Khajavikhan; Patrick LiKamWa

doi:10.1364/OE.394830

1. Introduction

In optical communication systems, light is primarily modulated by two methods, externally and directly. In direct schemes, the bandwidth is generally limited to a value close to the relaxation oscillation frequency [1,2]. Over the years, several strategies have been proposed to boost the direct modulation bandwidth of semiconductor lasers. These include, for example, cavity designs that promote the Purcell factor [3–6], push-pull modulation of a composite-resonator laser [7], injection-locking techniques [8,9], and materials with higher differential gain [10], to name a few. Clearly, of interest will be to devise new techniques that can be readily applied to a variety of cavity geometries and gain systems in order to systematically enhance the modulation response of semiconductor lasers.

In the past decade, parity time (PT) symmetry has been proposed as a means to mold the flow of light via enhancing light-matter interactions [11,12]. Progress in this area has so far enabled new effects such as unidirectional invisibility [13–15], sensitivity enhancement at an exceptional point [16–18], and chiral mode conversion [19–21], to name a few. In this respect, recently, single mode PT-symmetric lasers have been introduced [22–24] that can potentially be used as chip-scale light sources in on-chip optical communication links. In this vein, one may question how fast such coupled dual cavity lasers can be directly modulated and what role if any does the symmetry breaking point, also known as an exceptional point (EP), play in enhancing the frequency response.

In this work, we investigate the effect of non-Hermitian degeneracies on the direct modulation bandwidth of microring semiconductor lasers. In doing so, in Section 2 we provide a rate equation model to study the dynamics of carriers and photons in a PT-symmetric microring laser system. In Section 3, we derive the response function of this configuration by applying a small signal analysis. In Section 4, we demonstrate an enhancement in modulation bandwidth when the laser approaches an exceptional point. In Section 5, we consider the laser working in non-PT-symmetric domain and its modulation performances. Finally, Section 6 concludes the paper.

2. Rate equation model for lasers operating in a PT-broken phase

Figure 1 depicts a schematic of a PT-symmetric laser, comprising of a pair of identical microring resonators. The two cavities are coupled, leading to an energy exchange in time. One may also incorporate a pair of bus waveguides on either side of the rings in order to facilitate light outcoupling or extraction. The cross sections of the rings are designed in such a way so as to support the fundamental transverse electric mode (TE₁₀), which exhibits the largest overlap with the gain region. As shown in Fig. 1, one of the resonators is subjected to gain, while the other one experiences loss (or a lower level of gain).

Fig. 1. A schematic of a PT-symmetric dual microring laser.

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Using temporal coupled mode theory, fields in the two resonators can be described by the following system of differential equations [25]:

(1)$$\begin{aligned}\mathop{\dot E_{1}} &= ({G - \gamma } )({1 - \textrm{j}\alpha} ){E_1} + \textrm{j}\kappa {E_2}, \\ \mathop{\dot E_{2}} &= - ({F + \gamma } )({1 - \textrm{j}\alpha } ){E_2} + \textrm{j}\kappa {E_1}, \end{aligned}$$

where ${E_i}\; ({i = 1,2} )$ are the modal fields in the cavities with gain and loss, respectively, $\mathrm{\gamma }$ denotes the linear loss in each resonator, which is mainly due to scattering, bending and output coupling; G and F represent carrier induced gain and loss in the respective resonators, and $\kappa $ is the strength of the temporal coupling coefficient, which is related to spatial coupling $\kappa ^{\prime}$ by $\kappa = \kappa ^{\prime}{v_g}/({2\pi R} )$ [26]. $\alpha $ stands for linewidth enhancement factor that is zero for an ideal PT-symmetric system. The frequency detuning resulting from $\alpha $ can be neglected compared with a strong coupling. In addition, lasers with zero $\alpha $ factor have been realized in a quantum dot (QD) platform [27]. In what follows, we discuss the laser operating in PT-symmetry domain by assuming $\alpha $ is too small. And in Section 5 we analyze the same arrangement with a nonzero linewidth enhancement factor.

Equation (1) supports two solutions in the form of ${({{E_1},\; {E_2}} )^T} = {({{E_{01}},{E_{02}}} )^T}\textrm{exp}({ - \textrm{j}\xi t} )$, where ${E_{01}}$ and ${E_{02}}$ are complex constants and $\xi $ is the eigenvalue that can be expressed as $\xi = \textrm{j}({G - F - 2\gamma } )/2 \pm \sqrt {{{({G + F} )}^2}/4 - {\kappa ^2}} $. Two regimes of operation can be identified, depending on whether $\delta {\mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle> }\vphantom{_x}}$}} }2\kappa $, where $\delta = G + F$ represents the gain-loss contrast. When $\delta < 2\kappa $, the laser operates in the unbroken PT-symmetry regime where the modal solutions of the two rings exhibit identical amplitudes and a real frequency splitting of $2\kappa \sqrt {1 - {\delta ^2}/({4{\kappa^2}} )} $. On the other hand, when $\delta > 2\kappa $, the frequency splitting occurs along the imaginary axis, and the modal amplitudes become unbalanced in terms of both amplitude and phase (displaying a $\mathrm{\pi }/2$ phase shift). This latter situation is a characteristic of a broken PT-symmetry phase. The point at which $\delta = 2\kappa $ marks the boundary between these two regimes, and is better known as an exceptional point.

Our goal here is to investigate the frequency response of PT-symmetric lasers operating in the broken phase regime. To simplify our analysis and without any loss of generality, here we consider only the interaction between the two cavities when each is emitting at a single longitudinal mode-as expected from properly designed PT-symmetric lasers operating in the broken phase. We will also assume that light circulates in each ring in a unidirectional fashion. This can be accomplished by incorporating appropriately designed S-bends in the resonators [28]. Based on these assumptions, we can derive the laser rate equations in the broken PT-symmetric phase. Writing the complex field ${E_i}$ by amplitude ${A_i}$ and phase ${\phi _i}$ i.e. ${E_i}(t )= {A_i}(t ){e^{ - j{\phi _i}}}$, the coupled mode equations are separately given by $d{A_i}/dt = ({{G_i} - \gamma } ){A_i} + {({ - 1} )^i}\kappa sin\phi {A_{3 - i}}$, $d{\phi _i}/dt ={-} \kappa cos\phi {A_{3 - i}}/{A_i}$, where $\phi $ is the phase difference. Dynamics of photon densities can be obtained by calculating $d{|{{A_i}} |^2}/dt = A_i^\ast d{A_i}/dt + {A_i}dA_i^\ast{/}dt$. Then laser rate equations in the broken PT-symmetric phase can be written by:

(2)$$\begin{aligned} \mathop{\dot N_{1}} &= {\eta _{\textrm{I}}}{I_1}/({qV} )- {v_g}g{S_1} - {N_1}/{\tau _{\textrm{N}}}, \\ \mathop{\dot S_{1}} &= \Gamma {v_{\textrm{g}}}g{S_1} - {S_1}/{\tau _{\textrm{p}}} - 2\kappa \textrm{sin}\phi \sqrt {{S_1}{S_2}} , \\ {\dot{\phi}} &= \kappa \textrm{cos}\phi ({1/\rho - \rho } ), \\ \mathop{\dot S_{2}} &={-} \Gamma {v_{\textrm{g}}}f{S_2} - {S_2}/{\tau _{\textrm{p}}} + 2\kappa \textrm{sin}\phi \sqrt {{S_1}{S_2}} , \\ \mathop{\dot N_{2}} &= {\eta _{\textrm{I}}}{I_2}/({qV} )+ {v_{\textrm{g}}}f{S_2} - {N_2}/{\tau _{\textrm{N}}}, \end{aligned}$$

where ${I_i}$ is the injected current, ${\eta _{\textrm{I}}}$ is the current injection efficiency, q is the electron charge, V is the volume of the active region, ${N_i}$ and ${S_i}$ are the carrier density and photon density in each ring, respectively. $\rho $ stands for the ratio of modal field amplitudes. ${v_{\textrm{g}}}$ is the group velocity, ${\tau _{\textrm{N}}}$ is the carrier lifetime, and $\Gamma $ is the confinement factor. g and f are the optical gain and loss per unit length, which are related to the temporal modal gain and loss coefficients G and F via $2G = \Gamma {v_{\textrm{g}}}g$ and $2F = \Gamma {v_{\textrm{g}}}f$, respectively. Finally, ${\tau _{\textrm{p}}}$ represents the photon lifetime which is given by $1/({2\gamma } )$. Since the two cavities are identical in shape and size, $\Gamma $, ${\tau _{\textrm{p}}}$ and ${v_{\textrm{g}}}$ are assumed to be the same for both rings. This classical rate equation model is used to describe the dynamics of the coupled ring laser system based on III-V material. A rigorous gain model can be determined from Fermi’s Golden Rule and momentum matrix elements of various gain materials are calibrated [29,30].

3. Small signal modulation

The rate equations are now used to study the modulation frequency response function of this composite system. In this regard, we take the derivative of Eq. (2) with respect to the variables ${I_i}$, ${N_i}$, ${S_i}$, $\phi $. By assuming an infinite small sinusoidal modulation upon the laser working at steady state, the frequency response can be obtained by solving equations:

(3)$$({\textrm{j}\omega \textbf{I} - \textbf{J}} )\mathrm{\Delta }X = {\eta _{\textrm{I}}}{({qV} )^{ - 1}}\mathrm{\Delta }Y$$

where $X = ({{N_1},\; {S_1},\; \phi ,\; {S_2},\; {N_2}} )$, and $Y = ({{I_1},\; 0,\; 0,\; 0,\; 0} )$ can be expressed as $X = {X_0} + \mathrm{\Delta }X{e^{\textrm{j}\omega t}}$, $Y = {Y_0} + \mathrm{\Delta }Y{e^{\textrm{j}\omega t}}$ under the modulation. $\textbf{J}$ is the Jacobian of Eq. (2), where each nonzero term is expressed as follows:

(4)$$\textbf{J} = \left[ {\begin{array}{ccccc} {{m_{NN}}}&{{m_{N1}}}&{{m_{N\phi }}}&{{m_{N2}}}&{{m_{Nn}}}\\ {{m_{1N}}}&{{m_{11}}}&{{m_{1\phi }}}&{{m_{12}}}&{{m_{1n}}}\\ {{m_{\phi N}}}&{{m_{\phi 1}}}&{{m_{\phi \phi }}}&{{m_{\phi 2}}}&{{m_{\phi n}}}\\ {{m_{2N}}}&{{m_{21}}}&{{m_{2\phi }}}&{{m_{22}}}&{{m_{2n}}}\\ {{m_{nN}}}&{{m_{n1}}}&{{m_{n\phi }}}&{{m_{n2}}}&{{m_{nn}}} \end{array}} \right]$$

{m_{NN}} ={-} 1/{\tau _{\Delta N}} - {v_{\textrm{g}}}a{S_1},

{m_{N1}} ={-} {v_{\textrm{g}}}g,

{m_{1N}} = {m_{2n}}/{\rho ^2} = \mathrm{\Gamma }{v_{\textrm{g}}}a{S_1},

{m_{11}} ={-} \kappa \rho \textrm{sin}\phi - 1/{\tau _{\textrm{p}}} + \mathrm{\Gamma }{v_{\textrm{g}}}g

{m_{1\phi }} ={-} {m_{2\phi }} ={-} 2\kappa \sqrt {{S_1}{S_2}} \textrm{cos}\phi ,

{m_{12}} ={-} {m_{21}}/{\rho ^2} ={-} \kappa \textrm{sin}\phi /\rho ,

{m_{\phi 1}} ={-} {\rho ^2}{m_{\phi 2}} = ({1/{S_1} + 1/{S_2}} )\kappa \rho \textrm{cos}\phi /2,

{m_{\phi \phi }} = \kappa ({\rho - 1/\rho } )\textrm{sin}\phi ,

{m_{22}} ={-} \mathrm{\Gamma }{v_{\textrm{g}}}f - 1/{\tau _{\textrm{p}}} + \kappa \textrm{sin}\phi /\rho ,

{m_{n2}} = {v_{\textrm{g}}}f,

{m_{nn}} ={-} 1/{\tau _{\Delta N}} - {v_{\textrm{g}}}a{S_2},

where ${\tau _{\Delta N}}$ is the differential carrier lifetime, a is the differential gain. At steady state, $\phi = \mathrm{\pi }/2$ indicating a stable frequency detuning and a power flow from gain cavity to the loss one. In fact, there is another phase locked state at $\phi ={-} \pi /2$. However, it is difficult to experimentally achieve due to the instability and requiring simultaneous control of pumping. We are interested in lasers with a stable and single longitude mode output. In this case, g and f can be expressed by the modal ratio $\rho $ via

(5)$$\Gamma {v_{\textrm{g}}}g = 2G = 1/{\tau _{\textrm{p}}} + 2\kappa \rho ,$$

(6)$$\Gamma {v_{\textrm{g}}}f = 2F = 2\kappa /\rho - 1/{\tau _{\textrm{p}}},$$

The gain-loss contrast is given by $\delta = \kappa ({\rho + 1/\rho } )$, that is actually controlled by unequal pumping rates. Equations (5) and (6) implies that the threshold gain ${g_{\textrm{th}}}$ of a laser in the PT-broken symmetry regime is a function of $\kappa $ through $\Gamma {v_{\textrm{g}}}{g_{\textrm{th}}} = 1/{\tau _{\textrm{p}}} + 4{\kappa ^2}/({\mathrm{\Gamma }{v_{\textrm{g}}}f + 1/{\tau_{\textrm{p}}}} )$. As expected, for a single ring arrangement, i.e., $\kappa = 0$, $\Gamma {v_{\textrm{g}}}{g_{\textrm{th}}} = 1/{\tau _{\textrm{p}}}$. Clearly, by increasing the coupling strength $\kappa $, more photons enter into the lossy ring, and hence a higher value of gain is required in order to keep the system above threshold.

The modulation frequency response $|{{H_1}} |= |{\mathrm{\Delta }{S_1}/\mathrm{\Delta }{I_1}} |$ is obtained by solving Eq. (3), whose features also depend on $\delta $ and $\kappa $. In the following section, we study the ramifications of the above analysis using numerical results. In our study we focus on the characteristics of the frequency response of cavity 1 i.e. ${H_1}({\textrm{j}\omega } )$ and in particular the effect the exceptional point has on the modulation bandwidth of the laser.

4. Direct modulation bandwidth enhancement at EP

The effect of PT-symmetry on the frequency response (${H_1}({\textrm{j}\omega } ))$ of these coupled microring lasers will be further elucidated via numerical simulations. The QD laser parameters used in our simulations are given in Table 1, where the radius and width of cavities are 50 µm and 2 µm, respectively [31]. And ${I_{th}}$ is the threshold current of the single ring laser. Figure 2(a) depicts the frequency response of a PT-symmetric laser with a coupling strength of ${\kappa _0} = $600 GHz at a current injection level of 2 mA for various values of $\rho $. Compared to a single ring configuration $({\textrm{where}\; \kappa = 0} )$, the modulation bandwidth of the coupled ring laser broadens as $\rho $ increases. This enhancement is attributed to the presence of an additional photon decay channel through ring 2, i.e. coupling-related term $2\kappa \rho .$ Fig. 2(c) displays the modulation bandwidth (${f_{3\textrm{dB}}}$) in such systems as a function of $\delta /({2\kappa } ),$ where the system’s exceptional point (EP) occurs at $\delta /({2\kappa } )= 1$. In this case, the ${f_{3\textrm{dB}}}$ follows the square root of the injection current above threshold (${f_{3\textrm{dB}}} \propto \sqrt {{I_1} - {I_{\textrm{th}}}} $).

Table 1. Parameters of InGaAs QD microring laser

View Table

Fig. 2. (a) Frequency responses of a PT laser for various modal ratios. ${I_1} = 2\; \textrm{mA}$ and ${\kappa _0} = 600 $GHz. (b) Trajectories of poles of ${H_1}$. The blue markers indicate the resonance frequency of the constituent ring laser. The arrows show the direction of $\delta /({2\kappa } )\to 1 $ (the system approaching an EP). (c) Dependences of the modulation bandwidth versus $\delta /({2\kappa } )$, where the current injected above threshold $({{I_1} - {I_{\textrm{th}}}} )$ is 0.2 mA, 0.8 mA, and 1.8 mA, respectively. Red curves indicate the numerical solutions compared with estimations from Eq. (7) (dashed blue).

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To further understand the role of $\rho $ in broadening the modulation bandwidth of the PT-symmetric laser, we reconsider the expression of ${H_1}({\textrm{j}\omega } )$, whose characteristics depends on its zeros and poles. A close examination of $\textrm{j}\omega \textbf{I} - \textbf{J}$ reveals that, only a pair of complex-conjugate poles as a function of modal field coupling, play a pivotal role in radio frequency. To further visualize this new scheme of controlling the modulation characteristics, the pair of poles p of the impulse response function ${H_1}({\textrm{j}\omega } )$ are depicted in Fig. 2(b). The arrows show the trajectory of the poles when the system moves from a single cavity arrangement to the EP ($\rho $ varies from 0 to 1). By increasing $\rho $, poles are pulled away from the resonance frequency of the constituent laser (blue marks in Fig. 2(b)), implying a bandwidth broadening mainly caused by gain-loss contrast. In this regard, the modulation bandwidth ${f_{3\textrm{dB}}} = 1.55|p |$ can be estimated based on the distance of one of the poles p from the origin. As shown in Fig. 2(b), $|p |$ increases when the system approaches the EP, thus resulting in a broadening of the bandwidth. By considering the single ring relaxation resonance frequency as ${\omega _{\textrm{R}}} = \sqrt {\Gamma {v_g}a{\eta _{\textrm{I}}}({{I_1} - {I_{\textrm{th}}}} )/({\textrm{q}V} )} $, one can then arrive at the following expression of the 3 dB cut-off frequency of a PT-symmetric laser:

(7)$${f_{3\textrm{dB} - \textrm{PT}}} = \frac{{{f_{3\textrm{dB} - \textrm{single}}}}}{{\sqrt {1 - {\rho ^2}} }},$$

here ${f_{3\textrm{dB} - \textrm{single}}} \approx 1.55{\omega _{\textrm{R}}}/({2\mathrm{\pi }} )$ is the 3 dB bandwidth of the constituent ring laser. Equation (7) suggests a bandwidth enhancement factor of $1/\sqrt {1 - {\rho ^2}} $ for a PT-symmetric system operating in the broken regime. The enhancement is low when the gain-loss contrast is very high and increases as the system approaches the exceptional point. Beyond this point, our analysis is no longer valid since the two rings switch their functions. The dashed blue curves in Fig. 2(c) depict ${f_{3\textrm{dB}}}$vs. $\delta /({2\kappa } )$, demonstrating a full agreement with the numerical solutions (red solid curves) for all values before the EP.

In general, the transient response of a PT-symmetric laser in the broken phase is determined by the eigenvalues ${\mathrm{\Lambda }_m}\; ({m = 1,\; 2,\; 3,\; 4,\; 5} )$ of Jacobian matrix $\textbf{J}$, resulting in ${\textrm{e}^{{\mathrm{\Lambda }_m}t}}$ terms. The system is stable if the real components of all eigenvalues are negative; otherwise, it oscillates. An arrangement comprising only of a single ring is expected to be continuously stable since the real parts of the two poles are negative $({ - \zeta /2} )$, where $\zeta $ is the damping factor. However, in a PT-symmetric laser configuration, the two conjugate poles tend to move to the right-half plane when the system is approaching the exceptional point, as shown in Fig. 2(b). In this case, the real part of p becomes positive after $\rho $ reach a critical value ${\rho _{\textrm{so}}}$ shown in Fig. 3(a). Consequently, any small perturbation of the current will result in an oscillation that strengthens until the small signal approximation no longer applies. The effect of the injection current $({{I_1}} )$ on ${\rho _{\textrm{so}}}$ is depicted in Fig. 3(b). The area under each curve represents the stable region where a small signal modulation analysis holds. As shown in this figure, one can increase the stability of this system by strengthening the coupling between the two resonators.

Fig. 3. (a) The dependence of Re($p$) versus $\rho $. ${\rho _{\textrm{so}}}$ is the criterion of stability. The small signal modulation is stable if $\rho < {\rho _{\textrm{so}}}$. (b) Dependences of ${\rho _{\textrm{so}}}$ versus ${I_1} - {I_{\textrm{th}}}$ for various coupling coefficients $\kappa $.

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5. Non-PT-symmetric system

In this section, we discuss the dynamics of the coupled ring laser with a finite linewidth enhancement factor $\alpha $, factoring its Hamiltonian out of PT-symmetry configurations. Reconsidering the coupled mode equations with a nonzero $\alpha $ factor, one can express the rate equations of this asymmetric pumped laser system as follows:

(8)$$\begin{aligned} \mathop{\dot N_{1}} &= {\eta _{\textrm{I}}}{I_1}/({qV} )- {v_{\textrm{g}}}g{S_1} - {N_1}/{\tau _N},\\ \mathop{\dot S_{1}}\ &= \Gamma {v_{\textrm{g}}}g{S_1} - {S_1}/{\tau _{\textrm{p}}} - 2\kappa \textrm{sin}\phi \sqrt {{S_1}{S_2}} ,\\ {\dot{\phi}} &= \alpha \Gamma {v_{\textrm{g}}}({g + f} )/2 + \kappa \textrm{cos}\phi ({1/\rho - \rho } ),\\ \mathop{\dot S_{2}} &={-} \Gamma {v_{\textrm{g}}}f{S_2} - {S_2}/{\tau_{\textrm{p}}} + 2\kappa \textrm{sin}\phi \sqrt {{S_1}{S_2}} ,\\ \mathop{\dot N_{2}} &= {\eta _{\textrm{I}}}{I_2}/({qV} )+ {v_{\textrm{g}}}f{S_2} - {N_2}/{\tau _N}, \end{aligned}$$

where the $\alpha $ factor affects the phase dynamic. At steady state, the phase difference can be determined from

(9)$$\phi = \pi + \textrm{arctan}[{{\alpha^{ - 1}}({{\rho^2} - 1} )/({{\rho^2} + 1} )} ].$$

Compared with the PT-symmetry arrangement, the modal phase shifts from $\mathrm{\pi }/2$, diminishing the coupling-related photon decay rate as well as the bandwidth enhancement illustrated in Section 4.

The frequency response of modulating the gain cavity can be also obtained by solving Eq. (3), with extra nonzero terms of Jacobian matrix J i.e. ${m_{\phi N}} ={-} {m_{\phi n}} = \alpha \Gamma {v_{\textrm{g}}}a/2$, and the phase-amplitude coupling term $({{m_{\phi 1}},\; {m_{1\phi }},\; {m_{\phi 2}},\; {m_{2\phi }}} )$ compared with the PT-symmetric case. These could result in more zeros and poles shaping $|{{H_1}({\textrm{j}\omega } )} |$. Therefore, we plot the trajectories of characteristic poles when $\alpha = 4$ [31], and two poles locate in RF domain, as shown in Fig. 4(a). One locates at the oscillation frequency of the constituent laser ${\omega _{\textrm{osc}}} = \sqrt {\omega _{\textrm{R}}^2 - {\zeta ^2}/4} $ (black mark in Fig. 4(a)) keeping as a invariant of $\kappa $ and $\rho $. Solid curves of Fig. 4(a) show the trajectories of the other pole p in various of $\rho $ with different $\kappa $. It is expected to generate another resonance frequency peak at ${\omega _{\textrm{p}}} = $ Im(p) in frequency response when damping 2|Re(p)| is weak. This hints at an effective way to broaden the modulation bandwidth of a single ring laser beyond its normal limit that is a result of strong damping at higher current injection levels.

Fig. 4. (a) Trajectories of two characteristic poles in direction of increasing $\rho $ (indicated by arrows), when coupling coefficient is 15 GHz, 45 GHz, and 100 GHz, respectively. Black marker indicates one pole that always locates at the oscillation frequency of the single ring laser. (b) Normalized frequency responses of ring 1 with different coupling coefficients.

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Figure 4(b) shows the normalized frequency responses of ring 1 for various coupling $\kappa $. The dashed curve exhibits the response of a single ring laser as a comparison. The resonance peak flattens out indicating the laser is overdamped and closed to its modulation limit. By coupling the laser to an identical one, modulation frequency responses exhibit one more resonance peak whose frequency ${\omega _{\textrm{p}}}$ can be controlled by $\kappa $. The modal field ratios are optimized at $\rho = 0.23$ (red curve), and $\rho = 0.41$ (blue curve) considering stability. For a weak coupling, i.e. 15 GHz and 45 GHz, that ${\omega _{\textrm{p}}}$ is slightly greater than ${\omega _{\textrm{osc}}}$, a roll-off beyond ${\omega _{\textrm{osc}}}$ is almost compensated by the next resonance peak. Therefore, the frequency response exhibits a broadened bandwidth with a peak in frequency response well beyond the limit of the single ring laser.

6. Conclusion

In conclusion, we have introduced a new paradigm for systematically increasing the direct modulation bandwidth of semiconductor lasers. By operating a coupled laser system in PT-broken phase, we can increase the 3-dB cut-off frequency beyond what is expected from each constituent single ring cavity. This enhancement in terms of modulation bandwidth can be attributed to tuning the relaxation resonance frequency by gain-loss contrast. The resulting boost in modulation bandwidth tends to reach its highest value when the system approaches the exceptional point, which can be reached while still operating far above threshold. In a non-PT-symmetric phase, the modulation characteristics are extremely fashioned by phase-amplitude coupling which generates an additional resonance frequency and bandwidth broadening that can break the modulation limit of each constituent ring laser.

Funding

Office of Naval Research (N0001416-1-2640, N00014-18-1-2347); National Science Foundation (DMR-1420620, ECCS 1454531, ECCS 1757025); Defense Advanced Research Projects Agency (D18AP00058, HR00111820038, HR00111820042); Army Research Office (W911NF-16-1-0013, W911NF-17-1-0481); Air Force Office of Scientific Research (FA9550-14-1-0037); United States-Israel Binational Science Foundation (2016381).

Disclosures

The authors declare no conflicts of interest.

References

1. G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, and J. Bowers, “III-V/silicon photonics for on-chip and intra-chip optical interconnects,” Laser Photonics Rev. 4(6), 751–779 (2010). [CrossRef]

2. G. P. Agrawal, Fiber-optic communication systems (John Wiley & Sons, 2012), Vol. 222.

3. H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2(7), 484–488 (2006). [CrossRef]

4. E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express 17(10), 7790–7799 (2009). [CrossRef]

5. C.-Y. A. Ni and S. L. Chuang, “Theory of high-speed nanolasers and nanoLEDs,” Opt. Express 20(15), 16450–16470 (2012). [CrossRef]

6. W. E. Hayenga, H. Garcia-Gracia, H. Hodaei, Y. Fainman, and M. Khajavikhan, “Metallic coaxial nanolasers,” Adv. Phys.: X 1(2), 262–275 (2016). [CrossRef]

7. C. Chen, K. L. Johnson, M. Hibbs-Brenner, and K. D. Choquette, “Push-pull modulation of a composite-resonator vertical-cavity laser,” IEEE J. Quantum Electron. 46(4), 438–446 (2010). [CrossRef]

8. T. Simpson, J. Liu, and A. Gavrielides, “Bandwidth enhancement and broadband noise reduction in injection-locked semiconductor lasers,” IEEE Photonics Technol. Lett. 7(7), 709–711 (1995). [CrossRef]

9. J. Wang, M. Haldar, L. Li, and F. Mendis, “Enhancement of modulation bandwidth of laser diodes by injection locking,” IEEE Photonics Technol. Lett. 8(1), 34–36 (1996). [CrossRef]

10. J. D. Ralston, S. Weisser, I. Esquivias, E. C. Larkins, J. Rosenzweig, P. J. Tasker, and J. Fleissner, “Control of differential gain, nonlinear gain and damping factor for high-speed application of GaAs-based MQW lasers,” IEEE J. Quantum Electron. 29(6), 1648–1659 (1993). [CrossRef]

11. K. G. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in P T symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef]

12. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018). [CrossRef]

13. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by P T-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011). [CrossRef]

14. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef]

15. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef]

16. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112(20), 203901 (2014). [CrossRef]

17. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. J. N. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017). [CrossRef]

18. W. Chen, ŞK Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017). [CrossRef]

19. J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016). [CrossRef]

20. H. Xu, D. Mason, L. Jiang, and J. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016). [CrossRef]

21. A. U. Hassan, B. Zhen, M. Soljačić, M. Khajavikhan, and D. N. Christodoulides, “Dynamically encircling exceptional points: exact evolution and polarization state conversion,” Phys. Rev. Lett. 118(9), 093002 (2017). [CrossRef]

22. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346(6212), 975–978 (2014). [CrossRef]

23. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014). [CrossRef]

24. Y. Kominis, K. D. Choquette, A. Bountis, and V. Kovanis, “Exceptional points in two dissimilar coupled diode lasers,” Appl. Phys. Lett. 113(8), 081103 (2018). [CrossRef]

25. A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015). [CrossRef]

26. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]

27. Z. Xu, D. Birkedal, M. Juhl, and J. M. Hvam, “Submonolayer InGaAs∕ GaAs quantum-dot lasers with high modal gain and zero-linewidth enhancement factor,” Appl. Phys. Lett. 85(15), 3259–3261 (2004). [CrossRef]

28. J. Ren, Y. G. Liu, M. Parto, W. E. Hayenga, M. P. Hokmabadi, D. N. Christodoulides, and M. Khajavikhan, “Unidirectional light emission in PT-symmetric microring lasers,” Opt. Express 26(21), 27153–27160 (2018). [CrossRef]

29. C. Hermann and C. Weisbuch, “k→· p→ perturbation theory in III-V compounds and alloys: a reexamination,” Phys. Rev. B 15(2), 823–833 (1977). [CrossRef]

30. R. Nicholas, J. Portal, C. Houlbert, P. Perrier, and T. Pearsall, “An experimental determination of the effective masses for Ga x In1− x As y P1− y alloys grown on InP,” Appl. Phys. Lett. 34(8), 492–494 (1979). [CrossRef]

31. L. A. Coldren, S. W. Corzine, and M. L. Mashanovitch, Diode lasers and photonic integrated circuits (John Wiley & Sons, 2012), Vol. 218.

Symbol	Unit	Value
$V$	cm³	$1.57 \times 10^{- 12}$
$Γ$	1	0.008
$η_{I}$	1	0.2
$τ_{p}$	ps	11
$I_{th}$	mA	1.8
$a$	cm²	$1.6 \times 10^{- 15}$
$τ_{Δ N}$	ns	1.57
$λ$	nm	965
$v_{g}$	cm/s	$7.1 \times 10^{9}$

Enhanced modulation characteristics in broken symmetric coupled microring lasers

Abstract

1. Introduction

2. Rate equation model for lasers operating in a PT-broken phase

3. Small signal modulation

4. Direct modulation bandwidth enhancement at EP

5. Non-PT-symmetric system

6. Conclusion

Funding

Disclosures

References

Cited By

Figures (4)

Tables (1)

Equations (20)

Optics Express