Abstract
Realization of onechip, ultralargearea, coherent semiconductor lasers has been one of the ultimate goals of laser physics and photonics for decades. Surfaceemitting lasers with twodimensional photonic crystal resonators, referred to as photoniccrystal surfaceemitting lasers (PCSELs), are expected to show promise for this purpose. However, neither the general conditions nor the concrete photonic crystal structures to realize 100Wto1kWclass singlemode operation in PCSELs have yet to be clarified. Here, we analytically derive the general conditions for ultralargearea (3~10 mm) singlemode operation in PCSELs. By considering not only the Hermitian but also the nonHermitian optical couplings inside PCSELs, we mathematically derive the complex eigenfrequencies of the four photonic bands around the Γ point as well as the radiation constant difference between the fundamental and higherorder modes in a finitesize device. We then reveal concrete photonic crystal structures which allow the control of both Hermitian and nonHermitian coupling coefficients to achieve 100Wto1kWclass singlemode lasing.
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Introduction
Realization of onechip singlemode highpower semiconductor lasers, which surpass the performance of all other lasers such as solidstate lasers, fiber lasers, and gas lasers, has been one of the ultimate goals of laser physics and photonics for decades. The demand for such semiconductor lasers has been rapidly increasing for a wide variety of applications including nextgeneration laser processing^{1,2} and remote sensing^{3,4}. Conventional semiconductor lasers such as edgeemitting lasers and verticalcavity surfaceemitting lasers involve fundamental difficulties for singlemode highpower operation because an increase of the device size inevitably results in the onset of multiple transversemode lasing^{5,6,7,8}. On the other hand, photoniccrystal surfaceemitting lasers (PCSELs)^{9,10,11,12,13,14,15,16}, which utilize a twodimensional standingwave resonance at a singularity point (Γ point, etc.) of the photonic band for lasing oscillation, show promise for overcoming this difficulty; the mutual coupling coefficients among propagating waves and radiative waves inside PCSELs can be manipulated by the unit cell design^{11,14}, which can greatly enhance the threshold margin between the fundamental mode and the other higherorder modes. Towards the realization of ultralargearea singlemode PCSELs, a doublelattice photonic crystal, in which two latticepoint groups are shifted in the x and y directions by one quarter of the lattice constant a (which is almost equal to the wavelength in the material), was recently proposed^{14}. In the doublelattice photonic crystal, the optical feedback (coupling coefficient) is weakened by the destructive interference for 180° diffraction of the waves reflected by each lattice point, with which the optical losses of the higherorder modes are increased more than those of the fundamental mode. Based on this concept, 10Wto20Wclass singlemode lasing was experimentally demonstrated with PCSELs with a diameter of as large as 400–500 µm^{15,16}. In addition, by introducing destructive interference of not only 180° diffraction but also 90° diffraction, the possibility of singlemode lasing in a larger device area with a diameter up to 2.5 mm was suggested^{14}. However, the conditions to realize singlemode lasing in an even larger area (3–10 mm), which are essential to realize 100Wto1kW singlemode lasing, have not yet been derived, nor have the concrete photonic crystal structures to satisfy the conditions been clarified. This is because the nonHermitian process inside photonic crystals, which accompanies energy loss, has not been so far utilized effectively.
In this paper, we first mathematically derive the complex eigenfrequencies of the four photonic bands around the Γ point, which form twodimensional broadarea cavity modes, by considering not only the Hermitian but also the nonHermitian optical couplings inside PCSELs using the threedimensional coupledwave theory^{17,18}. Next, we provide a formula for the threshold margin of the fundamental mode over the higherorder modes in the finitesize device, and reveal the general conditions for the Hermitian and nonHermitian optical coupling coefficients to realize broadarea singlemode lasing. We show that the key to realize such broadarea singlemode lasing is in the utilization of a carefully designed doublelattice photonic crystal structure with an appropriate backside reflection, by which the flexible control of both the Hermitian and nonHermitian coupling coefficients becomes possible. Finally, we investigate the lasing stability of the designed PCSELs through a comprehensive analysis by considering carrierphoton interactions, and reveal more detailed requirements for the Hermitian and nonHermitian coupling coefficients, with which stable 100Wto1kWclass singlemode lasing can be expected in an ultralarge lasing diameter (3–10 mm).
Results
Hermitian and nonHermitian optical couplings inside PCSELs
Figure 1 shows a schematic of mutual couplings of waves inside a squarelattice PCSEL, where a doublelattice photonic crystal is employed. The Hermitian couplings, which express the optical couplings without accompanying energy loss (or vertical radiation loss), are shown in Fig. 1a, while the nonHermitian couplings, which express the optical couplings with accompanying energy loss (or vertical radiation loss), are shown in Fig. 1b. Here, a backside reflector is placed beneath the photonic crystal to reflect the downward radiation to the upward direction and to control especially the magnitude of nonHermitian coupling coefficient. In the analysis below, we focus on the transverseelectriclike (TElike) modes of the photonic crystal since the active layer of the typical PCSELs consists of multiple quantum wells with TE gain. Note that the analysis below can be also applied to other squarelattice photonic crystals that have the same reflection symmetry as the doublelattice photonic crystal (i.e. reflection symmetry along y = x).
According to Bloch’s theorem, the electric fields distribution E(r) inside the photonic crystal with a lattice constant of a is expressed by the superposition of many propagating plane waves as follows;
Here, E_{m,n} is an electric field vector of each Fourier component (m, n are integers), β_{0} = 2π/a is the magnitude of the reciprocal lattice vector, and k = (k_{x}, k_{y}) is a wavevector representing a deviation from the Γ point. Resonance at the Γ point, which is used for typical PCSELs, is composed of four fundamental waves expressed with (m, n) = (±1, 0), (0, ±1), where the complex electricfield amplitudes of these waves are expressed as R_{x}, S_{x}, R_{y}, and S_{y} as illustrated in Fig. 1a. These four waves are directly coupled with each other, and are also indirectly coupled via higherorder waves (m^{2} + n^{2} > 1) and radiative waves (m = n = 0). Considering the reflection symmetry along y = x in the doublelattice photonic crystal as shown in Fig. 1, the mutual couplings between these fundamental waves can be expressed with the following matrix equations in the framework of the threedimensional coupledwave theory (3DCWT)^{17,18};
Expressions for each term on the righthand side of Eqs. (3) and (4) are provided in Supplementary Section 1. The real and imaginary part of the eigenfrequency (δ and α) on the left side of Eq. (2) denote the wavenumber (frequency) and loss (radiation constant) of each resonant mode, respectively.
\({{{{{{\bf{C}}}}}}}_{{{{{{\rm{Hermitian}}}}}}}\)in Eq. (3) is a Hermitian matrix, where the condition \({{{{{{\bf{C}}}}}}}_{{{{{{\rm{Hermitian}}}}}}}={{{{{{\bf{C}}}}}}}_{{{{{{\rm{Hermitian}}}}}}}^{{{\dagger}} }\) is satisfied. \({{{{{{\bf{C}}}}}}}_{{{{{{\rm{Hermitian}}}}}}}\) expresses the couplings among the fundamental waves (Fig. 1a), which consists of a 180° (or 1D) coupling coefficient (\({\kappa }_{{{{{{\rm{1D}}}}}}}\)), 90° (or 2D) coupling coefficients (\({\kappa }_{{{{{{\rm{2D}}}}}}+}\), \({\kappa }_{{{{{{\rm{2D}}}}}}}\)), and selfcoupling coefficient (\({\kappa }_{11}\)), where all the couplings do not accompany vertical emission loss. Note that \({\kappa }_{{{{{{\rm{2D}}}}}}+}\) and \({\kappa }_{{{{{{\rm{2D}}}}}}}\) differ in value, because the doublelattice photonic crystal does not have C_{4} symmetry unlike a general singlelattice photonic crystal with circular lattice points. We should also note that \({\kappa }_{{{{{{\rm{1D}}}}}}}\) and \({\kappa }_{{{{{{\rm{2D}}}}}}}\) are complex numbers because the doublelattice structure does not have C_{2} symmetry, while \({\kappa }_{{{{{{\rm{2D}}}}}}+}\) is a real number because of the reflection symmetry about the line of y = x. \({\kappa }_{11}\) is a real number, which expresses selfcoupling for fundamental four waves without accompanying vertical emission loss.
\({{{{{{\bf{C}}}}}}}_{{{{{{\rm{non}}}}}}{{{{{\rm{Hermitian}}}}}}}\) in Eq. (4) shows nonHermitian couplings of the fundamental waves through radiative waves, where the condition \({{{{{{\bf{C}}}}}}}_{{{{{{\rm{non}}}}}}{{{{{\rm{Hermitian}}}}}}}={{{{{{\bf{C}}}}}}}_{{{{{{\rm{non}}}}}}{{{{{\rm{Hermitian}}}}}}}^{{{\dagger}} }\) is satisfied. Note that in previous references on 3DCWT^{17,18}, mutual couplings via radiative waves were expressed with another coupledwave matrix \({{{{{{\bf{C}}}}}}}_{{{{{{\rm{rad}}}}}}}\). The difference between \({{{{{{\bf{C}}}}}}}_{{{{{{\rm{rad}}}}}}}\) and \({{{{{{\bf{C}}}}}}}_{{{{{{\rm{non}}}}}}{{{{{\rm{Hermitian}}}}}}}\) is that the former contains both nonHermitian and Hermitian couplings, while the latter retains only nonHermitian couplings. Such reconstruction of the coupledwave matrices facilitates the derivation of analytical formulae of the radiation constants and threshold margin in a finitesized PCSEL, as shown later. In \({{{{{{\bf{C}}}}}}}_{{{{{{\rm{non}}}}}}{{{{{\rm{Hermitian}}}}}}}\), \(i\mu\) is a purely imaginary number, which expresses selfcoupling of four fundamental waves through radiative waves with accompanying vertical emission loss (Fig. 1b). The magnitude of \(\mu\) can be continuously changed by changing the phase difference between the upwardradiated wave and the downwardradiated wave that is reflected at the bottom reflector. \(i\mu {e}^{{\pm}\! i{\theta }_{{{{{{\rm{pc}}}}}}}}\) expresses ±180° coupling through radiative waves with accompanying vertical emission loss. θ_{pc} represents the phase change associated with ±180° coupling [see Supplementary Eq. (S14) and Supplementary Fig. S1 in Supplementary Section 1 for details].
\({{{{{{\bf{C}}}}}}}_{{{{{{\rm{non}}}}}}{{{{{\rm{gamma}}}}}}}\) in Eq. (5) denotes the deviation of the wavevector from the Γ point, which induces the change in frequency.
Frequencies and radiation constants at the Γ point
In this section, we consider a PCSEL with infinite size, for which the electric field distribution is periodic in the plane of a photonic crystal (the effect of a finite size is considered in the next section). For the infinitesize PCSEL, the loss originates from the vertical loss (nonHermitian process) expressed by a radiation constant. Because there are four possible bandedge modes at the Γ point (A, B, C, D), we derive the radiation constants for these modes, together with their resonant frequencies by using Eq. (2).
Before doing so, we first consider only the Hermitian process expressed by the first term of the right hand side of Eq. (2) (\({{{{{{\bf{C}}}}}}}_{{{{{{\rm{Hermitian}}}}}}}\)), while ignoring the nonHermitian process. In this case, the optical couplings for the four fundamental waves can be expressed as:
This equation can be divided into the following equations by considering the reflection symmetry of the photonic crystal along y = x:
The physical meanings of the above equations are shown in Fig. 2. Figure 2a shows the electric field vectors of the four fundamental waves in a general case, which are coupled with each other according to Eq. (6), where e_{x} and e_{y} are unit vectors in the x and y directions, respectively. Following the division of Eq. (6) into Eq.(7) and Eq. (8), Fig. 2a can be divided into Fig. 2b and Fig. 2c; Fig. 2b corresponds to Eq. (7), where the electricfield pairs that have antisymmetric vectors about the axis of y = x with the amplitudes of R_{x} + R_{y} and S_{x} + S_{y} are coupled with each other, while Fig. 2c corresponds to Eq. (8), where the electricfield pairs that have symmetric vectors about the axis of y = x with the amplitudes of R_{x} − R_{y} and S_{x} − S_{y} are coupled with each other. The coupling coefficients between the antisymmetric electricfield pairs (R_{x} + R_{y} and S_{x} + S_{y}) in Fig. 2b are represented by \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) and \({\kappa }_{{{{{{\rm{1D}}}}}}}^{\ast }+{\kappa }_{{{{{{\rm{2D}}}}}}}^{\ast }\) as shown in the nondiagonal terms in Eq. (7), and thus, \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) represents the Hermitian coupling coefficient for the antisymmetric modes (which we define as modes A and C). Similarly, the coupling coefficients between the symmetric electricfield pairs (R_{x} − R_{y} and S_{x} − S_{y}) in Fig. 2c are represented by \({\kappa }_{{{{{{\rm{1D}}}}}}}{\kappa }_{{{{{{\rm{2D}}}}}}}\) and \({\kappa }_{{{{{{\rm{1D}}}}}}}^{\ast }{\kappa }_{{{{{{\rm{2D}}}}}}}^{\ast }\), and thus \({\kappa }_{{{{{{\rm{1D}}}}}}}{\kappa }_{{{{{{\rm{2D}}}}}}}\) represents the Hermitian coupling coefficient for the symmetric modes (which we define as modes B and D).
Now, we introduce the nonHermitian process in addition to the Hermitian process to obtain the radiation constants for the infinite photonic crystal structure. If we consider the nonHermitian coupling coefficients (\(i\mu\) and \(i\mu {e}^{\!{\pm}\! i{\theta }_{{{{{{\rm{pc}}}}}}}}\)) in Eq. (4) together with the aforementioned Hermitian coupling coefficients in Eq. (3), the analytical formulae of the complex eigenfrequencies, which express the resonant frequencies δ_{i} and radiation constants α_{i}, of the four modes at the Γ point (i = A, B, C, D) can be derived as follows (the detailed derivation of these formulae are provided in Supplementary Section 2).
For modes A and C:
and, for modes B and D:
In Eqs. (9) and (10), the Hermitian coupling coefficients \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) and \({\kappa }_{{{{{{\rm{1D}}}}}}}{\kappa }_{{{{{{\rm{2D}}}}}}}\) in Eqs. (7) and (8) are rewritten as \(({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}){e}^{i{\theta }_{{{{{{\rm{pc}}}}}}}}\) and \(({\kappa }_{{{{{{\rm{1D}}}}}}}{\kappa }_{{{{{{\rm{2D}}}}}}}){e}^{i{\theta }_{{{{{{\rm{pc}}}}}}}}\) respectively. These modifications allow us to consider the relative phase of \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) and \({\kappa }_{{{{{{\rm{1D}}}}}}}{\kappa }_{{{{{{\rm{2D}}}}}}}\) with respect to the phase of nonHermitian ±180°coupling θ_{pc}, wherein the coefficients are made invariant with respect to global translation of the air holes inside the unit cell (see Supplementary Fig. S1c in the Supplementary Section 1 for details). Hereafter, we call \(({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}){e}^{i{\theta }_{{{{{{\rm{pc}}}}}}}}\) the “phaseinvariant effective Hermitian coupling coefficient” for modes A and C, and we call \(({\kappa }_{{{{{{\rm{1D}}}}}}}{\kappa }_{{{{{{\rm{2D}}}}}}}){e}^{i{\theta }_{{{{{{\rm{pc}}}}}}}}\) the phaseinvariant effective Hermitian coupling coefficient for modes B and D.
Next, we consider the specific case of \(({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}){e}^{i{\theta }_{{{{{{\rm{pc}}}}}}}}={\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}} \sim 0\), which corresponds to the case in which destructive interference of not only 180° diffraction but also 90° diffraction is achieved, as we discussed in our previous paper^{14}. In this case, the radiation constants of the four modes can be derived from Eqs. (9) and (10) as follows:
In Eq. (11), it is seen that the radiation constant \({\alpha }_{{{{{{\rm{A}}}}}}}\) of mode A is much smaller than those of the other modes (B, C, D). Specifically, the difference in the radiation constant between mode A and the other three modes can be shown to be sufficiently large (>20 cm^{−1}) (see Supplementary Fig. S2 in Supplementary Section 3 for detail), which indicates that the lasing oscillation occurs stably in mode A. Therefore, we hereafter focus on mode A along with mode C as its counterpart, since these two modes have the same symmetry as described above.
Next, we show that the value of \({\alpha }_{{{{{{\rm{A}}}}}}}\) is controllable by changing the value of the phaseinvariant effective Hermitian coupling coefficient \(({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}){e}^{i{\theta }_{{{{{{\rm{pc}}}}}}}}\) around zero, in addition to the value of the nonHermitian coupling coefficient \(i\mu\). Toward this purpose, we define here the real and imaginary parts of \(({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}){e}^{i{\theta }_{{{{{{\rm{pc}}}}}}}}\) as R and I, respectively, and we assume that I  is much smaller than R + iμ  . The physical meaning of this assumption and the role of I in radiation process are explained in Supplementary Section 4. Under this assumption, Eq. (9) for modes A and C can be then transformed as
Equation (12) gives the following analytical expressions for the frequency difference between modes A and C at the Γ point (\({\delta }_{{{{{{\rm{AC}}}}}}}\)) and the radiation constants (\({\alpha }_{{{{{{\rm{A}}}}}}}\) and \({\alpha }_{{{{{{\rm{C}}}}}}}\)):
It is seen in these equations that the frequency gap \({\delta }_{{{{{{\rm{AC}}}}}}}\) between modes A and C is mostly determined by the real part of the phaseinvariant effective Hermitian coupling coefficient, R, while \({\alpha }_{{{{{{\rm{A}}}}}}}\) is determined by both real and imaginary parts of the phaseinvariant effective Hermitian coupling coefficients, R and I, as well as the absolute value of the nonHermitian coupling coefficient μ. Thus, \({\alpha }_{{{{{{\rm{A}}}}}}}\) can be controlled by appropriately choosing the values of R, I, and μ.
Figures 3a, b show examples of the radiation constant \({\alpha }_{{{{{{\rm{A}}}}}}}\) and the mode gap frequency \({\delta }_{{{{{{\rm{AC}}}}}}}\), respectively, which are calculated using Eq. (9), as functions of \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) in the complex plane, where μ is fixed to 90 cm^{−1} (μ is also changed in the next section). Note that the axes of the phaseinvariant effective Hermitian coupling coefficients R and I are also drawn in the figures; these axes are rotated with respect to those for \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) by θ_{pc}, which is fixed to 0.92π as a typical value for the doublelattice structure whose midpoint is taken as the origin (see Supplementary Section 1). It is seen in Fig. 3a that the radiation constant \({\alpha }_{{{{{{\rm{A}}}}}}}\) becomes 0 cm^{−1} on the R axis (namely, when I = 0 cm^{−1}), and this value can be set to a desirable value (from 0 cm^{−1} to 30 cm^{−1}) by adjusting I  . From this result, a photonic crystal with a doublelattice structure, or more generally a structure without C_{2} symmetry, enables the flexible control of the radiation constant \({\alpha }_{{{{{{\rm{A}}}}}}}\). It is also seen in Fig. 3b that the frequency gap \({\delta }_{{{{{{\rm{AC}}}}}}}\) becomes 0 cm^{−1} on the I axis (namely, R = 0 cm^{−1}) in the range of \(I\le \mu\). The frequency degeneracy of the two bands A and C at the Γ point is due to the complete cancellation of the inplane optical feedback.
On the other hand, unlike doublelattice photonic crystals, when a photonic crystal has a more general symmetry, such as C_{2} or C_{4} symmetry, \({\kappa }_{{{{{{\rm{1D}}}}}}}\), \({\kappa }_{{{{{{\rm{2D}}}}}}}\) are purely real numbers and θ_{pc} = π, so I = 0. In this case, Eqs. (9) and (10) can be transformed into the following simpler expressions:
The radiation constants of modes A and B are exactly zero, which prohibits laser emission in the vertical direction. Even when a small structural perturbation is added to the photonic crystal to enable vertical emission, the radiation constant difference between modes A and B remains small, which results in multiplemode lasing.
Threshold margin for singlemode lasing in a finitesized PCSEL
Next, we consider a finitesized PCSEL and derive the conditions for increasing the threshold margin between the fundamental mode and higherorder modes originating from the same bandedge mode (mode A). As we explained in the previous section, the difference of radiation constant between mode A and the other three modes can be shown to be large enough (>20 cm^{−1}) to ensure that lasing occurs on bandedge A, and that the threshold margin between the fundamental mode and the higherorder modes within bandedge A is the “global threshold margin” of the largearea device (see Supplementary Section 5 for details). Figure 4a shows typical electric field distributions of the fundamental mode and the first higherorder mode inside a finitesized device with a diameter of L, wherein the inplane wavenumbers of the envelope functions of the two modes are approximately π/L and 2π/L, respectively^{18,19} (the electric field distributions of other higherorder modes are shown in Supplementary Section 5). Therefore, by increasing the sensitivity of change of the radiation constant of mode A with respect to the inplane wavenumbers (or dα_{A}/dΔk), we can increase the threshold margin between the fundamental and higherorder modes. In the following analysis, we consider the band structure and radiation constants in the vicinity of the Γ point and derive the general conditions to maximize the threshold margin for singlemode lasing.
We first consider that the inplane wavevector of the Bloch waves are slightly shifted along the y = x axis of reflection symmetry (i.e., in the ΓM direction, for which \({k}_{x}={k}_{y}=\varDelta k/\sqrt{2}\)). By solving Eq. (2), we obtain the analytical formula of the complex eigenfrequencies as follows (see Supplementary Section 2);
where not only mode A but also mode C are considered as the counterpart. Figures 4b and 4c show the calculated frequencies (\({\delta }_{{{{{{\rm{A}}}}}}}\) and \({\delta }_{{{{{{\rm{C}}}}}}}\)) and radiation constants (\({\alpha }_{{{{{{\rm{A}}}}}}}\) and \({\alpha }_{{{{{{\rm{C}}}}}}}\)) as functions of Δk, where R is taken as a parameter and μ and I are fixed to the constant values (μ = 90 cm^{−1}, I = 30 cm^{−1} in Fig. 4b and μ = 40 cm^{−1}, I = 20 cm^{−1} in Fig. 4c). As seen in the left panels of Figs. 4b and 4c, the frequency gap \({\delta }_{{{{{{\rm{AC}}}}}}}\) between modes A and C decreases when R decreases [see also Eq. (13)], while the change in the radiation constant \({\alpha }_{{{{{{\rm{A}}}}}}}\) (and \({\alpha }_{{{{{{\rm{C}}}}}}}\)) becomes steeper when R decreases (see the right panels). Moreover, the change in \({\alpha }_{{{{{{\rm{A}}}}}}}\) and \({\delta }_{{{{{{\rm{AC}}}}}}}\) becomes more sensitive to a change of Δk in the vicinity of the Γ point especially for the smaller nonHermitian coupling coefficient μ (Fig. 4c). Although the most sensitive change can be obtained at R = 0 cm^{−1}, where both the frequencies and the radiation constants of modes A and C degenerate at \(\varDelta k=\pm \sqrt{2({\mu }^{2}{I}^{2})}\), forming an exceptional point^{20,21}, this point should be avoided for stable singlemode oscillation as shown later.
The radiation constant \({\alpha }_{{{{{{\rm{A}}}}}}}\) of mode A in the vicinity of the Γ point and under the condition of \((\sqrt{{I}^{2}+{(\varDelta k/\sqrt{2})}^{2}} < \sqrt{{R}^{2}+{\mu }^{2}})\), Eq. (16) can be approximated as follows;
The first term of the right side of Eq. (17) denotes the radiation constant equal to Eq. (14) that corresponds to that of the infinite size PCSEL, while the second term denotes the increase of the radiation constant owing to the deviation from the Γ point due to the finitesize effect. Using Eq. (17), the radiation constant difference between the fundamental mode and the 1st higherorder mode (shown in Fig. 4a) can be expressed with the following simple formula:
Although Eq. (18) is derived for the higherorder mode in the ΓM direction, it can be approximately applied to the higherorder mode in its orthogonal direction (ΓM′) because the band structures in the ΓM direction and ΓM’ direction are almost equal in the vicinity of the Γ point (see Supplementary Section 2). It should be also noted that the threshold margin between the fundamental mode and the higherorder modes depends not only on \(\varDelta {\alpha }_{{{{{{\rm{v}}}}}}}\) in Eq. (18) but also on the difference of their inplane losses (\(\varDelta {\alpha }_{//}\)). However, the contribution of the former is dominant in the case of largearea PCSELs (L ≥ 3 mm) because the portion of the electric field penetrating outside the active region becomes small (see Supplementary Section 6). We should note that the threshold margin might be better expressed by the exponential of the product of \(\varDelta {\alpha }_{{{{{{\rm{v}}}}}}}\) in Eq. (18) and the device diameter L:
This is because \({e}^{\varDelta {\alpha }_{{{{{{\rm{v}}}}}}}\cdot L}\) directly express the ratio between the light amplification rate of the fundamental mode and the 1^{st} higherorder mode when the light propagates from one edge to the other edge of the device.
The calculated threshold margins [\(\varDelta {\alpha }_{{{{{{\rm{v}}}}}}}\) in Eq. (18) and \({e}^{\varDelta {\alpha }_{{{{{{\rm{v}}}}}}}\cdot L}\) in Eq. (19)] are shown in Fig. 4d for PCSELs with L = 3 mm for various R and μ. It is seen that \(\varDelta {\alpha }_{{{{{{\rm{v}}}}}}}\) (and \({e}^{\varDelta {\alpha }_{{{{{{\rm{v}}}}}}}\cdot L}\)) increase as R decreases, which can be ascribed to the destructive interference of not only 180° but also 90° diffractions and weakened inplane optical feedback inside the photonic crystal. In addition, \(\varDelta {\alpha }_{{{{{{\rm{v}}}}}}}\) (and \({e}^{\varDelta {\alpha }_{{{{{{\rm{v}}}}}}}\cdot L}\)) are maximized when the absolute value of nonHermitian coupling coefficient (μ) balances with the real part of the phaseinvariant effective Hermitian coupling coefficient (R) for a constant R. It should be noted that the value of μ should have a certain value to ensure a sufficient radiation constant difference among the four bandedge modes (A~D) in Eq. (11). Therefore, for the realization of singlemode lasing operation in an ultralargearea PCSEL while keeping the large threshold margin, it is important to manipulate not only the phaseinvariant effective Hermitian optical couplings (R and I) but also the nonHermitian optical couplings (iμ) by controlling both the latticepoint design and the complex reflectivity of the backside reflector (shown in Fig. 1).
Although Eqs. (18) and (19) provide a general guideline for increasing the threshold margin between the fundamental mode and the 1st higherorder mode inside PCSELs, its derivation does not consider the carrierphoton interactions and the spatial nonuniformity of the carrier distribution inside the device, which is inevitably caused by the spatial hole burning effect at high current injection levels. Therefore, in the next section, we design a concrete photonic crystal structure appropriate for ultralargearea (L = 3–10 mm) PCSELs with various Hermitian and nonHermitian optical coefficients, and we discuss the lasing stability of the designed PCSELs through a comprehensive analysis of lasing characteristics considering the carrierphoton interactions.
Concrete design of singlemode ultralargearea PCSELs
Figures 5a, b show the cross section and the top view of the designed PCSEL for singlemode ultralargearea operation. In Fig. 5a, a GaAs photonic crystal layer is placed near the active layer (InGaAs/AlGaAs quantum wells) and is sandwiched by pclad and nclad layers. The light emitted downward from the photonic crystal layer is reflected at the ptype distributed Bragg reflector (DBR) below the pclad layer and then interferes with the upward emission. Figure 5b shows a doublelattice structure composed of an elliptic and circular hole, where the tuning parameters are a lattice separation of d and holesize balance 2x.
The magnitude of the nonHermitian coupling coefficient μ can be continuously changed by adjusting the phase difference between the upward and downward emission with the thickness of the pclad layer (Fig. 5a). On the other hand, the real and imaginary parts of the Hermitian coupling coefficient \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) can be manipulated by tuning a lattice separation of d and holesize balance 2x in the doublelattice structure (Fig. 5b). The most dominant Fourier component that determines \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) is \({\xi }_{2,0}\) (see Supplementary Section 7), which induces the direct coupling between R_{x} and S_{x} (R_{y} and S_{y}). \({\xi }_{2,0}\) of a doublelattice photonic crystal with d = 0.25a + Δd can be approximated as follows (see also Supplementary Section 7);
where \({n}_{{{{{{\rm{GaAs}}}}}}}^{2}{n}_{{{{{{\rm{air}}}}}}}^{2}\) is the permittivity difference between GaAs and air, and FF_{total} and ΔFF denote the sum and the difference of the filling factors of the two holes, respectively. As is apparent from Eq. (20), the doublelattice photonic crystal enables the independent control of the real and imaginary parts of \({\xi }_{2,0}\), and thus \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\), by changing Δd and ΔFF separately. More concretely, by changing d and 2x in Fig. 5b, which induces the change in Δd and ΔFF, \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) (or R and I) can be controlled in the entire complex plane.
We numerically calculate the magnitude of nonHermitian coupling coefficient μ and the Hermitian coupling coefficient \({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}}\) (or R and I) by changing the thickness of the pclad layer t_{pclad} and the structural parameters of the doublelattice photonic crystal (d, 2x), respectively. The results are shown in Fig. 5c, d. The details of the simulation model and the parameters are given in Supplementary Section 8. As shown in these numerical simulations, the doublelattice photonic crystal resonators with backside reflectors allow the arbitrary control of the Hermitian and nonHermitian coupling coefficients. It is worth emphasizing that the arbitrary control of I, which is enabled by the doublelattice structure, leads to the ondemand control of the radiation constant as shown in Fig. 3a.
Lasing stability analysis considering carrierphoton interactions
Finally, we analyze the lasing characteristics of the designed ultralargearea PCSELs with the timedependent 3DCWT^{22}, which considers not only the mutual coupling of light but also the carrierphoton interactions and the spatial nonuniformity of the gain and refractive index distributions. The details of the simulation method are explained in Supplementary Section 8. We first consider 3mmdiameter doublelattice PCSELs with a fixed pclad thickness (which gives μ = 87 cm^{−1}) and fixed hole sizes (which gives \({{\mbox{Im}}}({\kappa }_{{{{{{\rm{1D}}}}}}}+{\kappa }_{{{{{{\rm{2D}}}}}}})\) ~ −25 cm^{−1}), and calculate the output power and lasing spectra by varying a lattice separation d or the real part of the phaseinvariant effective Hermitian coupling coefficient R. The above parameters are chosen so that a moderate radiation constant (~20 cm^{−1}) is obtained for mode A while much higher radiation constants (>40 cm^{−1}) are obtained for the other bandedge modes.
Figure 6a, b show the calculated currentlightoutput (IL) characteristics and lasing spectra of the four devices with different R, where the farfield beam pattern at each current is shown in the inset. In Fig. 6a, the threshold current and slope efficiency are almost the same for the designed devices, except for the one with a nearzero frequency gap between modes A and C (R = −5 cm^{−1}), which exhibits unstable lasing as explained later. The lasing spectra and the farfield beam patterns shown in Fig. 6b are completely different in the four devices. When R = 86 cm^{−1}, which almost equals μ, singlemode lasing with a nearly diffractionlimited divergence angle (θ_{1/e}^{2} ~ 0.03°) is obtained at an injection current of 140 A, demonstrating the possibility of 100Wclass singlemode lasing in a PCSEL with a diameter as large as 3 mm. When R = 178 cm^{−1}, which is much larger than μ, the lasing spectra broaden and the beam divergence angles increase, clearly showing the evidence of multimode lasing. This result agrees with the theoretical result shown in Fig. 4d, where the threshold margin between the fundamental mode and the higherorder mode decreases as R increases.
Interestingly enough, the broadening of the lasing spectra with the increase in the beam divergence also arises when R = 25 cm^{−1}, which is smaller than one third of μ, and the spectra broaden even further for the device with a nearzero bandgap (R = − 5 cm^{−1}). Such unstable lasing is caused by the carrierinduced refractive index change and the resultant frequency change inside the device. According to Eq. (16), the carrierinduced frequency change of mode A \((\varDelta {\delta }_{{{{{{\rm{A}}}}}}}/\varDelta N)\) and the carrierinduced radiation constant change \(\varDelta {\alpha }_{{{{{{\rm{A}}}}}}}/\varDelta N\) are related as follows (see Supplementary Section 9);
From this equation, one can understand that when R becomes much smaller than μ, the change in the radiation constant \(\varDelta {\alpha }_{{{{{{\rm{A}}}}}}}/\varDelta N\) becomes more drastic than the change in the frequency \(\varDelta {\delta }_{{{{{{\rm{A}}}}}}}/\varDelta N\). For example, let us consider the case where the photon density locally decreases from the steadystate value. In a normal case, the photon density returns to the steadystate value immediately. This is because the local decrease of photon density induces the local increase of carrier density, which leads to the increase of local optical gain that increases the photon density again. However, for a device with a smaller R, the carrierinduced local frequency change causes the drastic increase in the radiation loss through Eq. (21), thereby leading to the further reduction of the local photon density and to the unstable oscillation. Therefore, for realizing stable singlemode lasing in the entire device, it is important not only to increase the threshold margin according to Eq. (18) [or Eq. (19)] but also to balance the real part of the phaseinvariant effective Hermitian coupling coefficient and the magnitude of nonHermitian coupling coefficient (R~μ). This fact indicates that the ultimate case of R = 0 cm^{−1}, which forms the exceptional point discussed before, is not appropriate for stable lasing oscillation.
Based on the above discussion of the lasing stability, we finally investigate the feasibility of higherpower singlemode lasing in an even largersize PCSEL (L = 10 mm). Here, we consider two different designs: (1) R = 86 cm^{−1}, μ = 87 cm^{−1}, I = 54 cm^{−1} and (2) R = 45 cm^{−1}, μ = 44 cm^{−1}, I = 32 cm^{−1}. The former structure is the same as the one which enables singlemode lasing in a 3mmdiameter PCSEL in Fig. 6b, and the latter structure has a larger threshold margin owing to the smaller R and μ. The calculated lasing spectra and the farfield beam patterns for the two structures are shown in Fig. 6c. While multimode lasing occurs in the former structure, singlemode lasing with a divergence angle θ_{1/e}^{2} < 0.01° is obtained in the latter structure. The calculated IL characteristic for the latter device is shown in Fig. 6d, demonstrating the feasibility of kWclass singlemode lasing in a centimetersize PCSEL. The potential challenges and solutions for experimentally realizing the 100Wto1kW PCSELs are discussed in Supplementary Section 10.
Discussion
We have analytically provided general formulae for the complex eigenfrequencies of the four photonic bands around the Γ point and derived the general conditions for ultralargearea singlemode operation in PCSELs. We have proven that the threshold margin between the fundamental mode and the higherorder modes can be increased through the reasonable reduction of both the real part of phaseinvariant effective Hermitian coupling coefficient and the magnitude of nonHermitian coupling coefficients (R and μ), while the balance between the two coefficients should be maintained to ensure stable lasing. In this context, it is shown that in the case of R = 0 cm^{−1}, where an exceptional point appears, the device performance becomes unstable owing to the carrierinduced refractive index change. Through the detailed numerical simulations, we have demonstrated that PCSELs with doublelattice photonic crystals and backside reflectors allow the arbitrary control of both Hermitian and nonHermitian optical coupling coefficients, enabling 100Wto1kWclass singlemode lasing with an ultralarge lasing diameter (≥3∼10 mm). Our results provide universal guidelines towards the realization of onechip kWclass nextgeneration semiconductor lasers, which are expected to replace conventional bulky highpower lasers, such as gas lasers, solidstate lasers and fiber lasers. Such ultracompact highpower semiconductor lasers will bring innovation to a wide variety of industries using lasers, such as material processing^{1,2}, mobility^{4}, medicine^{23}, and even aerospace^{24}. Our theoretical analysis, which considers not only Hermitian optical couplings but also nonHermitian ones, also enables the detailed analysis of photonic bands around frequency gaps and exceptional points, which are attracting increasing attention in nonHermitian photonics^{25,26}. We believe that the theory established in this work will contribute to the development of a wide variety of research fields from fundamental laser physics and nonHermitian wave physics in general to industrial applications.
Data availability
The data that supports the plots within this paper and other findings of this study are available within this article and its Supplementary Information files, and are also available from the corresponding author upon reasonable request.
Code availability
The mathematical formulae of 3DCWT simulations are available within this article and its Supplementary Information files, and their associated codes are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was carried out under the project of Council for Science, Technology and Innovation (CSTI), Cross ministerial Strategic Innovation Promotion Program (SIP), “Photonics and Quantum Technology for Society 5.0” (Funding agency: QST) (S.N.). This work was partially supported by a GrantinAid for Scientific Research [22H04915 (S.N.), 20H02655 (T.I.)] from the Japan Society for the Promotion of Science (JSPS).
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S.N. supervised the entire project. T.I. established the theory with M.Y., J.G., and K.Y.; T.I. performed the numerical simulations with M.Y., J.G., and K.I.; S.N. and T.I. discussed the results and wrote the paper with J.G., M.Y., K.I., and M.D.Z.
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Inoue, T., Yoshida, M., Gelleta, J. et al. General recipe to realize photoniccrystal surfaceemitting lasers with 100Wto1kW singlemode operation. Nat Commun 13, 3262 (2022). https://doi.org/10.1038/s41467022309107
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DOI: https://doi.org/10.1038/s41467022309107
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