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Abstract
The lowest threshold lasing mode in a rounded D-shape microcavity is theoretically analyzed and experimentally demonstrated. To identify the lowest threshold lasing mode, we investigate threshold conditions of different periodic orbits by considering the linear gain condition due to the effective pumping region and total loss consisting of internal and scattering losses in ray dynamics. We compare the ray dynamical result with resonance mode analysis, including gain and loss. We find that the resonance modes localized on the pentagonal marginally unstable periodic orbit have the lowest threshold in our fabrication configuration. Our findings are verified by obtaining the path lengths and far-field patterns of lasing modes.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Microcavity lasers have attracted much attention owing to the high-quality factor of their whispering gallery modes, which are supported by total internal reflection. A circular microcavity has been spotlighted because of its extremely high-quality factor [1] and the potential for highly sensitive label-free detection [2–5] and gyroscopy [6,7]. Currently, it has been reported that deformed microcavities from the circular one have various outstanding characteristics such as directionality [8–16] and chirality [13,17] with minimized degradation of high-quality factor, which are crucial for wide-range applications.
When a deformed semiconductor microcavity laser is excited by current injection, it is hard to excite a whole cavity region because the area of the contact region for pumping is smaller than the cavity surface due to fabricating limitation. Because of this reason, periodic orbits in a deformed microcavity are partially overlapped with the contact region. Accordingly, the modes localized on them are excited only along the overlapped parts of the periodic orbits near the threshold (i.e., we rule out the current diffusion at higher current injection). This “partial excitation” makes different threshold conditions for lasing of resonance modes localized on the periodic orbits. In other words, the lasing modes are conditionally determined by pumping circumstances: the lasing threshold of a certain resonance mode is strongly dependent on its loss and the effective pumping rate [18]. The smaller contact region compared to the cavity for injection current is very natural and general in fabricating two-dimensional semiconductor lasers. However, a lasing threshold condition connecting the loss and partial excitation of periodic orbits in a microcavity laser has not been well analyzed so far. Hence, in the present paper, we aim to study their relations explicitly.
To study the lowest threshold lasing mode, we take a semiconductor microcavity defined by a rounded D-shape (RDS) boundary morphology. In this shape of the microcavity, there exist unstable periodic orbits (UPOs) and marginally unstable periodic orbits (MUPOs). Here, MUPOs exhibit segmented-continuous curve structures originating from the invariant curves of non-isolated periodic orbits in phase space of a circular cavity. While, the modes localized on MUPOs slowly decay to leaky regions owing to the stickiness of classical rays in the vicinity of them [19,20], the ones on UPOs decay quickly. This implies that, modes localized on MUPOs are advantageous for lasing over those localized on UPOs, known as scarred modes [21]. Through numerical and experimental analyses, we investigate lasing threshold conditions of resonance modes localized on UPOs and MUPOs. In numerical investigations, we make a prediction for the lowest threshold lasing modes, which is deduced in terms of the effective pumping region (i.e., partial excitation) in both ray dynamics and resonance mode analysis. To obtain the threshold condition in ray dynamics, we calculate the gain due to the effective pumping region and the total loss consisting of internal and scattering losses. Also, in the resonance mode analysis, the threshold pumping strength is obtained by using the spatial wave function and complex wavenumber of resonance modes. This numerical prediction is clearly confirmed in the experiments.
2. Rounded D-shape microcavity laser
An RDS microcavity comprises three circular arcs and one linear section, as shown in Fig. 1(a). The main circular arc with a radius of $R=30$ $\mu$m is tangentially connected to two smaller circular arcs with a radius of $r=0.5R$ at an angle $\phi$ from the y-axis. The linear section is tangentially connected to the two smaller arcs. We set the angle $\phi =80^{\circ }$, where the cavity has various MUPOs and UPOs. Our system is fabricated based on seven-layered multi-quantum-well (MQW) surrounded by InGaAsP separate-confinement-heterostructure (SCH). This core structure is interposed between the n-doped InP substrate at the bottom layer with a buffer zone and by a p-doped InP cladding layer at the top layer. Finally, a p-doped InGaAs ohmic layer is grown at the uppermost surface. The whole structure is illustrated in Fig. 1(b). The active layer consists seven pairs of a 0.8% compressively strained InGaAsP well ($\lambda _{g}=1.68$ $\mu$m, $72$ $ {\AA }$ thick) and a 0.60% tensile strained InGaAsP barrier ($\lambda _{g}=1.31$ $\mu$m, $105$ $ {\AA }$ thick), and undoped InGaAsP separate confinement layers (see Ref. [22,23] for details). The laser was etched 4 $\mu$m deep into an RDS cross-section by inductively coupled plasma etching. The SiO$_2$ passivation layer is deposited by using the PECVD to define the current contact region where the electrode and ohmic layer adjoin. The boundary of this contact region was defined along the sidewall of the microcavity with a 3 $\mu$m inward gap. A p-contact Ti/Au metal electrode was deposited through the lift-off process [24] to set the current contact layer. The scanning electron microscope (SEM) image of the full structure of our laser is shown in Fig. 1(c); Fig. 1(d) shows the 3 $\mu$m gap between the sidewall and electrode. Although the electrode was slightly misaligned, this misalignment does not affect the experimental results because the electrode is large enough to cover the entire contact region.
Fig. 1. An RDS semiconductor microcavity laser. (a) is a schematic of our RDS microcavity with $\phi =80^{\circ }$ and $r=0.5R$. A linear section between the smaller arcs is highlighted by red color. (b) shows a stack layer diagram of an InP-based semiconductor laser along the vertical cross-section. (c) is an SEM image of a fabricated RDS microcavity laser. (d) is a zoomed image of the red square region in (c) showing the $3$ $\mu$m inward gap between the contact region and the sidewall. (e) shows a vertical cross-section of RDS embedding an SCH-MQW active layer marked by a red band. Green bars in (d) and (e) represent the length of $\lambda$.
3. UPOs and MUPOs in the RDS cavity
The phase space $\left (q,p\right )\equiv \left (S/S_{\textrm {max}}, p_{0}\sin \chi \right )$ of our cavity is constructed by the canonical conjugate variables $S$ and $p$: $S$ is the boundary length from the point $\theta =0^{\circ }$, $S_{\textrm {max}}$ the total boundary length, and $p$ the tangential momentum of the ray for the incident angle $\chi$ at $S$ with the initial momentum of $p_0\equiv 1$. Because there is no recognizable stable periodic orbits, the overall structure of phase space follows a fully chaotic property. To reveal MUPOs and UPOs, we numerically obtain the chaotic repeller distribution (CRD) [25] and survival probability distribution (SPD) [19,20,26], as shown in Fig. 2 through ray dynamical simulations. For calculations of CRD, we follow the schemes used in Ref. [25]. MUPOs and UPOs above the critical line are observed and highlighted by red-straight curves and green squares, respectively. MUPOs and UPOs in Fig. 2(a) are labeled by the number of rotations ($r$) and reflections ($t$) as follows: $p_{r,t}=p_{1,3}$, $p_{1,4}$, $p_{1,5}$, $p_{1,6}$ MUPOs and $p_{1,4}$, $p_{1,5}$ UPOs.
Fig. 2. Ray dynamical simulation of the RDS cavity. (a) is CRD, which shows chaotic saddles possessing marginally unstable periodic orbits (red) and unstable periodic orbits (green). (b) is normalized SPD in a logarithmic scale. Three dotted circles around $S/S_{\textrm {max}}=0.4$, $0.8$, and $1.0$ indicate emission windows.
The emission properties of the RDS cavity are investigated by means of the survival probability distributions for TE polarized rays applying the Fresnel equations [19,20,26]. SPD in a logarithmic scale in Fig. 2(b) elucidates the emission windows below the critical line through the unstable manifolds of chaotic saddles [19,20,27–29]. The figure shows three emission windows marked by dotted circles around $S/S_{\textrm {max}}=0.4$, 0.8, and 1.0 for the counter-clockwise (CCW) traveling rays $(p>0)$. Due to the mirror-reflection symmetry of RDS, three emission windows for clockwise (CW) traveling rays are located at $S/S_{\textrm {max}}=0.0$, 0.2, and 0.6.
To estimate the scattering loss of each periodic orbit, it is useful to implement the survival probability time distribution (SPTD) [30] of ray dynamical simulation. Because the scattering loss is usually expressed in decibels per centimeter (dB/cm), our SPTD at time $t=l/c$, where $c$ is the speed of light, is obtained as an averaged survival intensity normalized by an initial intensity of $I_0$ in decibel unit as follows :
Fig. 3. Semi-logarithmic plots of survival probability time distributions (discrete symbols) and their linear fittings (straight curves) for each periodic orbit. (a) $p_{1,3}$MUPO, (b) $p_{1,4}$MUPO, (c) $p_{1,4}$UPO, (d) $p_{1,5}$MUPO, (e) $p_{1,5}$UPO, and (f) $p_{1,6}$MUPO. From the slopes of the fitting curves, the decaying exponent $\alpha _{s}$ with respect to the effective propagation time of the propagation length with a constant velocity of rays is obtained.
Resonance modes in our RDS cavity are obtained by using the boundary element method [32] in the region $n_{e}kR\sim 396$, which matches the experimental one. Here, $n_e$, $k$, and $R$ denote the effective refractive index of $3.3$, the vacuum wavenumber, and the radius of the main circle in Fig. 1(a), respectively. Wavenumbers of resonance mode groups localized on MUPOs and UPOs are shown in Fig. 4(a). The regular interval $\textrm {Re}({\Delta kR})$ of each mode group identified in Fig. 4 corresponds to a periodic orbit in the sense of the relation $L_{p}^{\textrm {res}}/R=2\pi /n_{e}\Delta kR$, where $L_p^{\textrm {res}}$ is the periodic orbit length. The average intervals $\langle \Delta kR \rangle$ of the resonance mode groups and corresponding path lengths are shown in Table 1 for $R=30$ $\mu$m. This result agrees well with the classical path lengths $L_p^{\textrm {ray}}$ obtained by ray dynamical calculations, i.e., geometries of periodic orbits. By exemplifying representative highest-Q modes, which are marked by dotted-red squares in Fig. 4(a), it is confirmed that spatial wave patterns and their Husimi distributions are well-matched with the periodic orbits inferred, as are shown in Fig. 4(b) [32,33].
Fig. 4. Passive resonance modes for the RDS cavity. (a) is the eigenvalue distribution of $p_{1,3}$ (blue triangle), $p_{1,4}$ (cyan square), $p_{1,5}$ (red pentagram), and $p_{1,6}$ MUPO (black circle) and $p_{1,4}$ (green diamond) and $p_{1,5}$ UPO (purple hexagram) in the region $n_{e}kR\sim 396$. The eigenvalues of representative modes marked with dotted-red boxes are $120.2018-i3.8006\times 10^{-5}$ for $p_{1,3}$ MUPO, $119.0782-i3.4248\times 10^{-5}$ for $p_{1,4}$ MUPO, $119.7007-i3.2675\times 10^{-5}$ for $p_{1,5}$ MUPO, $119.7075-i3.1429\times 10^{-5}$ for $p_{1,6}$ MUPO, $119.8308-i3.4238\times 10^{-5}$ for $p_{1,4}$ UPO, and $119.1448-i3.1848\times 10^{-5}$ for $p_{1,5}$ UPO. (b) is the spatial wave patterns and their Husimi distributions of representative modes. We overlap the Husimi distribution on the periodic orbits in CRD to make clear the mode localization.
Table 1. Path lengths of periodic orbits for our RDS microcavity
4. Linear gain condition at the threshold
Now, we discuss the robustness of the lasing of resonance modes supported by MUPOs and UPOs in terms of the lowest threshold conditions. The general lasing condition for a semiconductor laser is expressed by the following inequality:
where $G$ is the gain, $\alpha _{i}$ the internal loss, $\alpha _{s}$ the scattering loss, $L_p$ the orbit path length of a mode, and R the survival ratio per cycle. This inequality simply implies that when the gain of certain modes in the active medium is greater than the total loss of the modes, the mode can lase above the threshold, i.e, $G=\alpha _{i}+\alpha _{s}$. From now on, we will use the single-mode approximation and disregard modal interactions to simplify our problems in analyzing a lasing mode at the lowest threshold condition of the RDS microcavity laser [18,34]. Note that, strictly speaking, the full version of the Maxwell–Bloch equations should be solved numerically taking nonlinear modal interactions [34,35] into account to investigate the general lasing modes.Because an inward gap exists between the contact region and the sidewall of our microcavity laser, the effective pumping region is limited by the contact region, and only partial trajectories of periodic orbits are overlapped with this effective pumping region. To reflect this partial pumping condition in our derivations, we define an effective excited length $L_e$, then the gain in the threshold condition can be modified as follows:
As are shown in Fig. 4(b), wave functions $\psi (x,y)$ of resonance modes are broadly distributed, though they are localized mainly along the classical periodic orbits. Hence, based on the schemes used in Ref. [18], we can derive the threshold condition of lasing modes by further assuming that the real part of a wavenumber (frequency) of a mode is almost the same as the center of the gain spectrum (or the gain spectrum is broad enough to cover the resonance modes that we obtained) [18,34]. Subsequently, the threshold pumping strength $W_{\textrm {th}}$ is expressed by:
Obviously, the main factor to determine the threshold pumping strength $W_{\textrm {th}}$ is the threshold parameter $p_w$, which depends on the fraction of wave functions overlapped with the effective pumping region. This effective pumping region is controlled by adjusting the profiles of the function $\Theta (x,y)$ in Eq. (6) for theoretical prediction. Figure 5 shows the parameters $p_c$ in Eq. (5) and $p_w$ in Eq. (6) as a function of the “effective inward gap” $\equiv l_g$: distance between the effective pumping region and the sidewall. In obtaining $p_c$ and $p_w$ in Fig. 5, we exemplify the case of $\alpha _{i}=6.8$ cm$^{-1}$, which is reported in Ref. [38] for the eight-quantum-well structures. The mode, which has the lowest threshold parameter, can be the first-appearing lasing mode for each $l_g$. In the range 0.7 $\mu$m < $l_{g}$ < 1.8 $\mu$m, both parameters of $p_c$ and $p_w$ for $p_{1,5}$ MUPO exhibit the lowest values compared with the ones of other periodic orbits.
Fig. 5. Threshold parameters depending on the length of the inward gap between the sidewall and the effective pumping region at the gain layer. (a) and (b) are threshold parameters of $p_c$ and $p_w$, respectively, which are defined by the classical excited length and the ratio of the sum of wave function inside the cavity and the sum of the overlapped wave function. Both results show a range of 0.7 $\mu$m < $l_{g}$ < 1.8 $\mu$m for the lowest threshold parameter of $p_{1,5}$ MUPO. The insets are the magnified graphs, which show the lowest threshold parameters of $p_{1,5}$ MUPO.
Now, we compute the true effective pumping region and, in turn, the effective inward gap $l_g$ taking into account the realistic parameters, which are used in our experiments. The effective pumping region inside the cavity is well defined, considering Ohm’s law (${\boldsymbol {J}}=q\rho \mu {\boldsymbol {E}}$) and the continuity equation for drift current. In the procedures, the effects of small amount of direct diffusion current is neglected. Further assuming that the electric field inside a semiconductor drift region decreases linearly as the depth increases from the top electrode to the bottom ground, and that the effective pumping region has a discrete profile instead of a Gaussian profile, the effective pumping region $A_{\textrm {eff}}$ is expressed by [39–41]
5. Experimental results and emission characteristics
We experimentally investigate the first lasing mode to verify our theoretical prediction. Our RDS laser is uniformly pumped through the electrode with continuous-wave injection current at the temperature of $20^{\circ }$C. In order to take the maximized laser output, we align a single-mode lensed fiber at $\theta =150^{\circ }$, which is the predetermined direction fixed by preliminary experiments. The laser output is collected $10$ $\mu$m away from the sidewall for the measurement of output power and emission spectra. Then, the collected laser output is split with a 50:50 optical coupler and equally distributed to an optical power meter and an optical spectrum analyzer. A far-field pattern (FFP) is measured $470$ $\mu$m away from the sidewall.
Figure 6(a) shows the total output power and major peak intensity versus injection current. From the total output power (red) in Fig. 6(a), we can identify the lasing threshold near 44 mA. To see the threshold of the first lasing mode, the peak intensity of the first main lasing mode is measured as a function of the injection current (blue), and it is revealed that the threshold of this mode is almost the same as the one of the total output power as 44 mA. Because the sum of the internal and scattering losses is inversely proportional to the quality factor ($=Q$) of modes [37,42], we estimate it to obtain the total loss at the lasing threshold. By measuring a center wavelength, $\lambda _{c}\sim 1551$ nm, of the major peak and its full width at half maximum, $w=0.077$ nm, [see Fig. 6(b)], it is obtained that $Q=\lambda _{c}/w\approx 2.0\times 10^{4}$. Now, the internal loss $\alpha _{i}$ can be extracted by subtracting the scattering losses of $p_{1,5}$MUPO, $\alpha _{s}=1.253{\textrm { cm}}^{-1}$ [see Fig. 3(d)], from the total loss deduced from the experimental quality factor [37,42], as follows
Fig. 6. Experimental results of the RDS semiconductor microcavity laser. (a) shows the total output power (red) and major peak intensity (blue) as functions of injection current. A vertical dotted line shows a threshold of lasing at $44$ mA. (b) shows the line width $w$ of the major peak $\lambda =\lambda _c$ (black) fitted by Gaussian curve (red) at the threshold current. The spectra in (c), (d), and (e) are measured at $44$ mA, $47$ mA, and $51$ mA, respectively. Two mode groups in the spectra are marked by dashed-red lines (Group 1) and dashed-green lines (Group 2). (f) is the far-field pattern from the ray dynamical simulation, (g) is the average far-field pattern from the resonance modes localized on $p_{1,5}$ MUPO, and (h) is the far-field pattern from the experiments. All of the far-field patterns indicate three emission directions around $\theta =10^{\circ }$, $70^{\circ }$, and $150^{\circ }$.
This value of the internal loss is consistent with the previous results of semiconductor lasers [38,43,44] as well as the one used in Fig. 5.
Figure 6(c) shows the spectrum at the threshold current, where the prominent lasing peak is not noticeable. Above the threshold, a main lasing mode group (Group 1) is observed at 47 mA, as shown in Fig. 6(d), which lases at $\left \{1547.40, 1551.14, 1555.93 \textrm { nm}\right \}$. Also, we can find another mode group (Group 2) at $\left \{1547.90, 1551.57, 1555.47 \textrm { nm}\right \}$. The path length of each mode group can be calculated using the relation between the path length and the free spectral range as follows:
As a final remark, FFPs of the RDS microcavity laser obtained experimentally and numerically are compared. Figures 6(f)-(h) show FFPs of the ray dynamical simulation, the resonance modes localized on $p_{1,5}$ MUPO, and the experiments, respectively. All FFPs clearly reveal three emission directions toward $\sim 150^{\circ }$, $\sim 70^{\circ }$, and $\sim 10^{\circ }$ with the strongest emission at $\theta =150^{\circ }$. These emission directions originate from the emission windows of $S/S_{\textrm {max}}=0.6$ (CW), $1.0$ (CCW), and $0.2$ (CW), respectively. Although the emission directions of the FFPs from the ray and wave simulation are slightly different, the nice overall agreement among the three FFPs indicates that the first lasing mode in our RDS microcavity laser is the mode localized on $p_{1,5}$ MUPO.
6. Conclusion
We have studied a rounded D-shape semiconductor microcavity laser excited by the continuous-wave injection current. Based on the fundamental linear gain condition with a single-mode lasing approximation, we predict that the modes localized on the pentagonal-shaped marginally unstable periodic orbit has the lowest threshold for lasing in our rounded D-shape microcavity laser, which has a 3 $\mu$m inward gap between the contact region and the sidewall. This prediction of the lowest threshold condition for lasing has been transparently verified in experiments. The path length deduced from ray dynamics, resonance modes, and experiments agreed well with the length of the pentagonal-shaped unstable and marginally unstable periodic orbits. Nice agreement of emission directions toward $\theta \sim 150^{\circ }$, $\sim 70^{\circ }$, and $\sim 10^{\circ }$ is obtained for all results of ray dynamics, resonance modes, and experiments. The results of the present work clearly reveal the fact that the lasing thresholds of the resonance modes are decisively affected by the effective pumping region and cavity morphology. In the same context, the lasing of scars reported in our previous study [46] can be explained without contradictions.
Funding
Ministry of Health and Welfare (HG18C0069).
Disclosures
The authors declare no conflicts of interest.
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