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Abstract
We develop a self-consistent theoretical model for simulating the lasing characteristics of photonic-crystal surface-emitting lasers (PCSELs) under continuous-wave (CW) operation that takes into account thermal effects caused by current injection. Our model enables us to analyze the lasing characteristics of PCSELs under CW operation by solving self-consistently the changes in the in-plane optical gain and refractive index distribution, which is associated with heat generation and temperature rise, and the change in the oscillation modes. We reveal that the lasing band-edge selectivity and beam quality of the PCSELs are affected by the spatial distribution of the band-edge frequency of the photonic crystal formed by the refractive index distribution, which depends on the temperature distribution in the resonator. Furthermore, we show that single-mode lasing with narrow beam divergence can be realized even at high current injection under CW operation by introducing a photonic-crystal structure with an artificially formed lattice constant distribution, which compensates such band-edge frequency distribution.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photonic-crystal surface-emitting lasers (PCSELs) [1–9] are surface-emitting semiconductor lasers which use two dimensional (2D) in-plane optical feedback at the band-edge of a photonic-crystal for lasing. In principle, these lasers are capable of high power and high beam quality operation, that is, high brightness operation, owing to broad-area single-mode resonance, which is difficult to achieve with conventional semiconductor lasers. Recently, we have proposed a double-lattice photonic-crystal resonator [9], which is suitable for large-area resonance, and have succeeded in operating a device with a broad circular emission area of ≥500 µm diameter with a high beam quality and high power of around 10 W under pulsed operation. In addition, we have successfully demonstrated an even higher peak power of >20 W with a shorter pulse width of ∼35 ps by introducing two-dimensionally arranged gain and loss sections [10]. We have also demonstrated the application to light detection and ranging (LiDAR) as an example of an important application of such a high-brightness laser under pulsed operation [11]. In order to apply PCSELs in a wider range of fields such as laser processing, it is necessary to achieve high-brightness CW operation. PCSELs under CW operation generate heat due to continuous current injection, unlike those under pulsed operation. Accordingly, we have mounted the device in a water-cooled package with a high thermal conductivity sub-mount for proper thermal management, and have achieved a high power of 7 W to 8 W under CW operation with a device with an emitting area of 800-µm to 1-mm diameter [9,12]. However, it has been experimentally indicated that the temperature distribution in the resonator caused by heat generation forms a non-uniform refractive index distribution (i.e., non-uniform photonic crystal), which leads to the change of the oscillation mode and a degradation of beam quality. In order to realize the high brightness of PCSELs under CW operation, it is important to analyze such effects of temperature distribution on the lasing characteristics and clarify the device physics of the PCSEL under CW operation.
We have developed a three-dimensional coupled-wave theory (3D-CWT) [13,14] for the analysis of large-area PCSELs. This theory focuses on the coupling between Bloch waves in a photonic crystal, which enables analysis that takes into account the 3D structure of the devices by reducing computation resources and provides physical insights for the lasing characteristics. Recently, we have also developed time-dependent 3D-CWT for a more comprehensive analysis of the lasing characteristics of the PCSELs [15], wherein we account for not only the coupling of light, but also the interaction between photons and carriers, and the temporal and spatial changes of their distributions. However, the analysis assumes a uniform temperature distribution in the resonator and has not taken into account the change in optical gain and refractive index caused by in-plane temperature distribution, which is caused by heat generation during CW current injection.
In this paper, we analyze the lasing characteristics of PCSELs under CW operation taking into account the heat generation of the device linked to CW current injection by further developing the above-mentioned time-dependent 3D-CWT. Our model enables us to analyze the lasing characteristics of PCSELs under CW operation by solving self-consistently the changes in the in-plane optical gain and refractive index distribution, which is associated with heat generation and temperature rise, and the change in the oscillation modes. Section 2 describes a self-consistent theoretical model for PCSELs under CW operation. In Section 3, we analyze the lasing characteristics of PCSELs under CW operation by this method. Then, we clarify that mode hopping can occur and the beam divergence can increase following the appearance of inhomogeneous in-plane refractive index distribution under high CW current injection. This distribution causes a spatial photonic bandgap to form, which affects the optical gain of each mode differently. In Section 4, we show that high power and high beam quality of PCSELs under CW operation is achievable by introducing a temperature compensating structure by forming an intentional in-plane band-edge frequency distribution. Section 5 is a summary.
2. Self-consistent theoretical model of PCSELs under CW operation
The steady-state lasing characteristics of PCSELs under CW operation are determined by the interaction between the distributions of the oscillation mode and the temperature. When lasing does not occur, almost all the input power transforms into heat. In this case, the heat density distribution in the photonic-crystal resonator is determined by the current density distribution, and the temperature distribution is in turn determined by the heat density distribution. Next, when lasing occurs, a portion of the input power is converted into optical power and emitted as laser light over an area equivalent to the spatial profile of the oscillation mode. Consequently, the heat density, and therefore temperature, over this area decreases. Then, due to the temperature dependence of the effective refractive index of the photonic crystal, the in-plane refractive index distribution is changed and the spatial profile of the oscillation mode is also changed. Through the above interaction, the steady-state lasing characteristics are determined. As described above, it is necessary to solve the distribution of the oscillation mode and the temperature self-consistently when analyzing the lasing characteristics of PCSELs under CW operation. Next, we describe the details of a self-consistent theoretical model.
Figure 1 shows the calculation flow of the self-consistent analysis of PCSELs under CW operation. First, we set the current density distribution to $J({\boldsymbol r})$ and the initial temperature distribution in the photonic-crystal layer to $T({\boldsymbol r})$. Next, using time-dependent 3D-CWT, we calculate the carrier density $N({\boldsymbol r})$ and the photon density $U({\boldsymbol r})$ in the active layer taking into account the temperature dependence of the optical gain and the refractive index and given the initial temperature distribution $T({\boldsymbol r})$. When we express the complex amplitude vectors of basic light waves propagating in the + x, -x, +y and -y directions (shown in Fig. 2), respectively, as ${[\begin{array}{{cccc}} {{R_x}}&{{S_x}}&{{R_y}}&{{S_y}} \end{array}]^T}$, the rate equation of the vectors and the rate equation of carriers are given as follows [15]:
Fig. 2. Bloch wave states represented by wave vectors (arrows) in reciprocal space including four basic waves (red arrows), high-order waves, and radiative waves.
Here, $c$ is the speed of light in vacuum, ${n_\textrm{g}}$ is the group index of the guided mode, ${\Gamma _{\textrm{active}}}$ is the optical confinement factor in the active layer, $g(N,T)$ is the temperature dependent optical gain, ${\alpha _{\textrm{in}}}$ is the internal loss, $\delta (T)$ is the deviation from the Bragg condition due to the temperature-induced refractive index change, ${\bf C}$ is a 4×4 matrix which represents the cross coupling of the waves shown in Fig. 2, including basic, high order, and radiative waves, $\gamma $ is the velocity of the refractive index change, which has only a minor effect, and ${\left[ {\begin{array}{{cccc}} {{f_1}}&{{f_2}}&{{f_3}}&{{f_4}} \end{array}} \right]^T}$ is the random noise expressing the spontaneous emission. ${n_{\textrm{eff}}}$ is the effective refractive index of the guided mode, ${d_{\textrm{active}}}$ is the thickness of the active layer, ${\eta _i}$ is the current injection efficiency, ${\tau _\textrm{c}}$ is the carrier lifetime, and $D$ is the diffusion coefficient of the carriers. Here, the temperature-induced non-uniformity of the refractive index distribution (and band-edge frequency distribution) is automatically considered by the position-dependent $\delta (T)$ in Eq. (1).
Next, we calculate the heat density distribution $P({\boldsymbol r})$ using the following equation:
Next, we calculate the steady-state temperature distribution ${T_{\textrm{final}}}({\boldsymbol r})$ using the following equation:
We calculate the carrier density $N({\boldsymbol r})$ and photon density $U({\boldsymbol r})$ again under the updated temperature distribution $T({\boldsymbol r})$. Although the time constant ${\tau _\textrm{T}}$ in the real device is determined by the thermal capacity of the device and is relatively long ($\mathrm{\mu}\textrm{s}\sim \textrm{ms}$), here we assumed ${\tau _\textrm{T}} = 5\textrm{ ns}$, which is sufficiently longer than the relaxation oscillation period ($< 1\textrm{ ns}$), to speed up the calculation without affecting the lasing characteristics. We repeat calculations above until we obtain a self-consistent solution and get the steady-state lasing characteristics.
3. Numerical results of CW lasing characteristics of PCSELs
Using the theoretical model described in Section 2, we analyzed the lasing characteristics of a PCSEL with a large resonator of 1-mm diameter under CW operation. The vertical structure of the PCSEL assumed in the analysis is shown in Table 1. Figure 3 shows the assumed photonic-crystal structure, the photonic band structure, and the radiation constant of each band in an infinite system. Here, the double-lattice photonic crystal is designed to operate at band-edge A, which has the lowest frequency and radiation constant among the four band edges (${\alpha _\textrm{A}}$, ${\alpha _\textrm{D}}$, ${\alpha _\textrm{C}}$ and ${\alpha _\textrm{B}}$) in the absence of heating, which is possible during pulsed operation [9].
Fig. 3. (a) Structure of the photonic crystal. (b) The photonic-band diagram. (c) The radiation constant of each band in an infinite system.
Table 1. Structural parameters of the PCSEL
The photonic-crystal structure used in our calculations is based on a fabricated structure, whose air-hole filling factors and ellipticities vary over the height of the air hole; the values of filling factor and ellipticity shown in Fig. 3(a) correspond to the air-hole height at which the filling factors are largest. Here, lattice separation $d$ is the distance between two air holes in the x (or y) direction. The difference of the radiation constant between band-edges A and B is $47.5\textrm{ c}{\textrm{m}^{ - 1}}$ when the refractive index distribution is uniform inside the photonic crystal. It should be noted that the effective difference of the threshold gain between band-edges A and B is modified by the temperature-induced non-uniformity of the refractive index inside the PCSEL under CW operation, which determines the ease of occurrence of mode-hopping as discussed in this section and in Appendix C. We set the current density distribution $J({\boldsymbol r})$ to be uniform over a circular area of 1-mm diameter, taking into account a current spread of $25\;\mathrm{\mu}\textrm{m}$ at the edge of the current injection area. In addition, we set the current injection efficiency ${\eta _i}$ to be 1. Based on the I-V characteristics of the previously fabricated PCSELs, we set the threshold voltage ${V_{\textrm{th}}} = 1.5\;\textrm{V}$ and the differential resistance $R = 0.06\;\Omega $. The temperature rise of the photonic-crystal resonator is taken into account by appropriately setting the temperature distribution of a point heating source $f({\boldsymbol r})$ in Eq. (6). This distribution $f({\boldsymbol r})$ should be changed when analyzing the CW lasing characteristics under different chip-to-package mounting conditions, including the sub-mount material and the performance of the cooling package. Here, we express the temperature distribution of a point heating source as $f({\boldsymbol r}) = {1 / {(4\pi {\kappa _{\textrm{eff}}}|{\boldsymbol r}|)}}$ using the effective thermal conductivity ${\kappa _{\textrm{eff}}}$ [16]. Based on a thermal analysis considering the chip-to-package mounting conditions and the assumption that all the heating occurs in the active layer, we set ${\kappa _{\textrm{eff}}} = 284.2\;\textrm{W}{\textrm{m}^{\textrm{ - 1}}}{\textrm{K}^{\textrm{ - 1}}}$. (See Appendix A for details.) We set the initial temperature to $20^\circ \textrm{C}$. The details on the temperature-dependent gain and refractive index as well as the other parameters used for the 3D-CWT analysis are shown in Appendix B.
Figure 4(a) shows the calculated optical output characteristics under CW operation together with a reference result obtained by neglecting the rise in temperature, i.e., by considering pulsed operation. The output power under CW operation is smaller than that under pulsed operation and a kink can be seen at injection current of 7 A to 9 A. Figure 4(b) and (c) show the calculated spectra at various injection currents under pulsed operation and CW operation, respectively. The chip-to-package mounting conditions considered in these calculations are described in Appendix A. Under pulsed operation, single-mode oscillation at band-edge A can be seen over a wide range of injection current. On the other hand, under CW operation, single-mode oscillation at band-edge A can be seen when the injection current is below ∼7 A. It should be noted that the red shift of the oscillation wavelength as the injection current increases is due to the increase of refractive index associated with the rise of temperature. However, other modes with multiple peaks start to oscillate at 9 A. This mode hopping corresponds to the kink in the calculated optical output characteristics. A comparison of the lasing spectrum and the photonic-band diagram shown in Fig. 3(b) reveals that the group of peaks at the shorter wavelengths originates in band-edge B, which, in the absence of heating, has a larger radiation constant than that of the mode at band-edge A. Figure 4(d) and (e) show the calculated photon density distribution and the far-field pattern (FFP) under pulsed operation and CW operation, respectively. When single-mode oscillation at band-edge A is obtained (at injection currents of 5 A and 11 A under pulsed operation and at 5 A under CW operation), photons are spread uniformly over the entire oscillation area, and the beam divergence is around $0.2^\circ$. On the other hand, under CW operation at a high injection current of 11 A, when the modes at band-edge B oscillate, photons are localized in various microscopic locations in the oscillation area and the beam divergence increases to over $0.5^\circ$.
Fig. 4. Calculated lasing characteristics. (a) Output power versus injection current. (b) Spectra at various injection currents under pulsed operation. (c) Spectra at various injection currents under CW operation. (d) Photon density distribution and FFP under pulsed operation. (e) Photon density distribution and FFP under CW operation.
Next, we discuss the mechanism of the lasing characteristics of the above analysis. The mechanism of the oscillation of the short-wavelength modes, corresponding to the modes at band-edge B, under CW high current injection is explained based on the change in the in-plane band-edge frequency distribution caused by the in-plane temperature distribution, as follows. As shown in Fig. 5(a), under the assumption that the current distribution is uniform, the temperature at the periphery of the oscillation area becomes lower than that of the center because the heat of the resonator area dissipates laterally as well as vertically into the sub-mount and cooling package. It should be noted that this non-uniform temperature distribution is dependent on the chip-to-package mounting conditions of the device and the current distribution. Consequently, the in-plane band-edge frequency distribution has a lower frequency at the center of the oscillation area, where the effective refractive index of the photonic crystal is higher due to the temperature dependence of the refractive index, as shown in Fig. 5(b). For the mode at band-edge A, the center of the oscillation area acts as a photonic bandgap (PBG) and photons are pushed out to the surrounding area, which leads to an increase in the cavity loss (in-plane loss). This corresponds to the higher threshold current and lower slope efficiency under CW operation than those under pulsed operation at injection currents of around 5 A to 7 A in Fig. 4(a). On the other hand, the mode at band-edge B becomes strongly confined at the center of the current injection area by the surrounding PBG, and thus such an increase in the cavity loss does not occur. As a result, the effective difference of the cavity loss shrinks compared to the initial value without the temperature distribution shown in Fig. 3(c). In addition, the carriers in the central area are not consumed by the mode at band-edge A due to the influence of the central PBG, which provides a larger optical gain for the mode at band-edge B. Such changes in the cavity losses and modal gain of the band edges becomes more remarkable as the temperature increases, and thus the mode at band-edge B starts to oscillate at high injection current (around 9 A in the case of the above calculations).
Fig. 5. In-plane band-edge distribution under CW operation. (a) Calculated temperature distribution in the photonic-crystal layer. (b) Global change in band-edge frequency distribution due to change in the temperature distribution. (c) Local change in band-edge frequency distribution due to change in carrier density distribution.
In addition, the mechanism of the localized oscillation with large beam divergence of the mode at band-edge B is explained as follows. The refractive index of the area where photons are distributed increases as carriers are consumed by stimulated emission. Since the mode at band-edge B originates from a downward-convex band in k-space, the surroundings, where carriers are not consumed, work as a PBG for the mode at band-edge B. Therefore, the localization of photons is strengthened as shown in Fig. 5(c), and thus the beam divergence increases due to many small oscillation areas as shown in Fig. 4(e). Meanwhile, for the mode at band-edge A, the carrier-induced refractive index change described above works to spread photons, and thus the oscillation of band-edge A is stable [15].
It should be noted that the ease of occurrence of the oscillation at band-edge B also depends on the initial difference of the radiation constant between the modes at band-edges A and B. When this difference is small, the cavity loss difference between the modes at band-edge A and B becomes almost zero even with a small PBG, which induces the oscillation at band-edge B at low CW injection current. On the other hand, when the initial difference is large, the cavity loss of the mode at band-edge A remains smaller than that of band-edge B even with a large PBG, and the oscillation at band-edge B is suppressed. (See Appendix C for further details). However, even when the oscillation at band-edge B is suppressed, the PBG pushes out the mode at band-edge A to the edge of the oscillation area, resulting in a lower slope efficiency and degradation of beam quality.
The above results obtained by our theoretical model have recently been verified experimentally. For example, unlike the behavior of a device with a large radiation constant difference between band-edges A and B, where oscillation was obtained exclusively at band-edge A [9], mode hopping from band-edge A to B has been experimentally observed in a recently fabricated PCSEL with a small difference of radiation constant. The results of these experiments will be reported elsewhere in the future. In the next section, we use our theoretical model to investigate methods for improving the lasing characteristics of PCSELs under CW operation.
4. Introduction of the temperature compensation structure
In the analysis and discussion of the previous section, we have clarified that the in-plane temperature distribution in the photonic-crystal layer of PCSELs causes the band-edge frequency distribution to appear as shown in Fig. 6(a), and, consequently, can cause mode hopping and degradation of the beam quality. In order to avoid mode competition at high current injections, we next investigate the use of a temperature compensation structure in the photonic crystal, consisting of a pre-installed upward-convex band-edge frequency distribution shown in the left panel of Fig. 6(b).
Fig. 6. Schematic of temperature compensation. (a) Band-edge frequency distribution of a conventional photonic crystal without a temperature-compensation structure before (left) and after (right) a rise in temperature. (b) Band-edge frequency distribution of a photonic crystal with a temperature compensation structure before (left) and after (right) a rise in temperature.
With this structure, it is expected that the band-edge frequency distribution will be flat at a high injection current as shown in the right panel of Fig. 6(b). This structure is introduced by forming an artificial in-plane distribution of the lattice constants or air-hole sizes of the photonic crystal. Here, we investigate a structure with an artificial in-plane distribution of the lattice constant. The refractive index at the edge of the oscillation area, where the temperature is relatively low at CW current injection, is smaller than that at the center of the oscillation area; in response to this difference, we increase the lattice constant at the edges, as shown in Fig. 7(a). The distribution of the lattice constant change, $\Delta a({\boldsymbol r} )$, is determined as follows:
Fig. 7. A design of temperature compensation. (a) In-plane distribution of lattice constant change $\Delta a$. (b) Amount of the compensation temperature and the variation of lattice constant as a function of position.
We analyzed the lasing characteristics of a 1-mm-diameter PCSEL with the temperature compensation structure shown in Fig. 7. Here, we assume the same double-lattice photonic-crystal structure as that shown in Fig. 3(a). To analyze PCSELs with variable lattice constants, we employ a position-dependent coupled-wave matrix ${\bf C^{\prime}}$ as described in Appendix D. Figure 8(a) shows the calculated optical output characteristics under CW operation for the PCSEL with temperature compensation, together with those of a reference PCSEL without temperature compensation. It can be seen that the optical output power of the PCSEL with temperature compensation increases without kinking as the injection current increases. In addition, an output power of over 10 W is obtained at injection current of 15 A and higher. Figure 8(b) shows that single-mode oscillation only at band-edge A is maintained over a wide range of high injection current. Figure 8(c) shows the calculated photon density distribution. It can be seen that photons are not localized in microscopic regions even at a high injection current of 17 A. When the in-plane temperature difference is smaller than that targeted by temperature compensation, the photons of the mode at band-edge A are confined at the center owing to the surrounding PBG, which decreases the cavity loss (threshold current) and increases the slope efficiency. The calculated results show that photons spread over the entire oscillation area as the injection current increases. At an injection current of 17 A, the photon density at the center of the oscillation area is somewhat reduced. This is because the temperature distribution exceeds the designed temperature compensation and the central area starts to work as PBG for the mode at band-edge A. Figure 8(d) shows the calculated far-field pattern. A narrow beam divergence of $0.25^\circ$ (evaluated using the beam’s 1/e2 width) is obtained even at a high injection current of 17 A. As described above, high-brightness beam under CW operation with an output power of 10 W and a beam divergence of $0.25^\circ$ is expected by introducing the temperature compensation structure in the photonic crystal.
Fig. 8. Calculated lasing characteristics of the PCSEL with temperature compensation under CW operation. (a) Output power versus injection current with (blue) and without (red) temperature compensation. (b) Spectra at various injection currents. (c) Photon density distribution. (d) FFP.
5. Conclusion
We have developed a self-consistent theoretical model for simulating the lasing characteristics of PCSELs under CW operation that takes into account the interaction of the distribution of oscillation mode and temperature. By analyzing a PCSEL with a resonator of 1-mm diameter under CW operation, we have revealed that the refractive index distribution in the resonator due to the temperature distribution reduces the optical gain for band-edge A and increases the optical gain for band-edge B. These changes to the optical gain can cause mode hopping from band-edge A to band-edge B to occur and can degrade the beam quality as a consequence of unstable oscillation on band-edge B, which has a downward-convex band structure in k-space. We have also shown that single-mode oscillation and a narrow beam divergence of $0.25^\circ $ can be obtained at a high current injection of 17 A and a high output power of 10 W by introducing a temperature compensation structure in the photonic crystal. By introducing temperature compensation structures in PCSELs with resonators ten times broader than those analyzed in this paper, high CW output powers of around 100 W can be expected. Thus, our results will contribute to the realization of CW 100 W-class high-brightness PCSELs for use as single-chip semiconductor light sources in ultra-compact laser processing machines.
Appendix A: Thermal analysis for setting the temperature distribution of a point heat source $f({\boldsymbol r})$
For setting the model of temperature rise within the photonic-crystal layer of PCSELs, we performed a thermal analysis of the device using the commercial finite-element solver, COMSOL. Figure 9(a) shows a schematic of the model for the thermal analysis. This model consists of a water-cooling package with cooling fins, a sub-mount made of a highly thermally conductive material with a thermal expansion coefficient equal to that of the PCSEL, and a PCSEL on top of the sub-mount. A uniform circular heat source of 1-mm diameter is placed in the active layer of the PCSEL, and the temperature distribution in the photonic-crystal layer is analyzed. Figure 9(b) shows the calculated in-plane temperature profile in the photonic crystal layer. The temperature distribution of the point heat source $f({\boldsymbol r})$ is estimated from the results of this analysis by convoluting $f({\boldsymbol r})$ and a uniform heat density distribution, as shown in Fig. 9(c), and is used to calculate the temperature distribution using Eq. (6). The temperature rise at the center or temperature non-uniformity of this thermal analysis are affected by the chip-to-package mounting conditions, including the sub-mount material and the performance of the cooling package.
Fig. 9. Schematic of thermal analysis. (a) Thermal analysis model. (b) Temperature profile in the plane of the photonic crystal layer with heating powers of 10 W, 20 W, 30 W, and 40 W. (c) Temperature profile calculated by convolution of a uniform heat density distribution and $f({\boldsymbol r})$.
Appendix B: Parameters used for 3D-CWT simulations
The temperature-dependent optical gain of the active layer $g({N,T} )$ at the lasing wavelength was calculated by using k-p perturbation theory [17] and then fitted by using the following linear fractional function of the carrier density [18]:
Here, ${N_{\textrm{tr}}}(T)$ is the transparency carrier density, ${g_{\textrm{max}}}(T)$ is the maximum gain, and $[{ - {g_0}(T)} ]$ is the absorption coefficient when there are no carriers. In our simulations, we assume that the lasing wavelength of the PCSEL coincides with the gain peak at $20^\circ \textrm{C}$. As the temperature rises, the transparency carrier density increases due to the change in carrier energy distribution and material bandgap. The temperature dependence of these three parameters were determined by the fitting with linear functions as follows:
The deviation from the Bragg condition due to the temperature-induced refractive index change $\delta (T)$ is given by
where ${{dn} / {dT}}$ is the temperature coefficient of the refractive index, and $\Delta T$ is the amount of the temperature change at each section.The other parameters used for the transient 3D-CWT in Section 3 and 4 are shown in Table 2.
Table 2. Parameters used for 3D-CWT simulations
Appendix C: The influence of radiation constants on the ease of occurrence of mode-hopping
As noted in Section 3, the ease of occurrence of oscillation at band-edge B depends on the difference of the radiation constant between band-edges A and B. Table 3 shows the radiation constant of each mode in three different photonic-crystal structures. The rightmost column shows the difference of radiation constants of the modes at band-edges B and A. Table 4 shows the structural parameters of these photonic crystals, including the thickness of the phase adjustment layer. Structure I is the same as that shown in Section 3. Structure II exhibits a larger difference of the radiation constant between band-edges A and B than that of Structure I, and thus more stable oscillation is expected using Structure II. Conversely, Structure III exhibits a smaller difference of the radiation constant between band-edges A and B, and thus less stable oscillation is expected using Structure III. Structures II and III are created by modifying Structure I, as described in Table 4.
Table 3. Radiation constant of each mode
Table 4. Structural parameters of three different photonic-crystal structures
Figure 10 shows the calculated lasing characteristics for Structures I, II and III. From these results, it can be seen that a larger difference in the radiation constant between band-edges A and B provides more stable oscillation in the mode of band-edge A at high injection current.
Appendix D: Coupled-wave matrix ${\bf C}$ of PCSELs with variable lattice constants
To analyze PCSELs with variable lattice constants for temperature compensation, we set a 4×4 matrix, ${\bf C}$, which represents the cross coupling of the waves as follows.
A lattice constant change at each microscopic region $\Delta a(x,y)$ is equivalent to a displacement in the x and y directions of each unit cell $(\Delta x(x,y),\Delta y(x,y))$;
When the unit cells are translated by $(\Delta x,\;\Delta y)$, the coupled-wave matrix ${\bf C}$ changes to ${\bf C^{\prime}}$ as follows:
By employing the above position-dependent matrix ${\bf C^{\prime}}$ in each microscopic region, we can analyze the oscillation characteristics of a PCSEL with an artificially formed lattice constant distribution.
Funding
Council for Science, Technology and Innovation Cross Ministerial Strategic Innovation Promotion Program Photonics and Quantum Technology for Society 5.0.
Disclosures
The authors declare no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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