Derivation of frequencies and radiation constants of modes A and C
The frequency and radiation constant (δA, αA) of mode A, which has the lowest radiation constant among all four modes, and those (δC, αC) of its counterpart mode C can be obtained by solving the following equation based on three-dimensional coupled-wave theory16,18:
$$begin{array}{l}left(delta +{rm{i}}frac{alpha }{2}right),left(begin{array}{c}{R}_{x}\ {S}_{x}\ {R}_{y}\ {S}_{y}end{array}right),=,left[left(begin{array}{cccc}{kappa }_{11} & {kappa }_{1{rm{D}}} & {kappa }_{2{rm{D}}+} & {kappa }_{2{rm{D}}-}\ {kappa }_{1{rm{D}}}^{ast } & {kappa }_{11} & {kappa }_{2{rm{D}}-}^{ast } & {kappa }_{2{rm{D}}+}\ {kappa }_{2{rm{D}}+} & {kappa }_{2{rm{D}}-} & {kappa }_{11} & {kappa }_{1{rm{D}}}\ {kappa }_{2{rm{D}}-}^{ast } & {kappa }_{2{rm{D}}+} & {kappa }_{1{rm{D}}}^{ast } & {kappa }_{11}end{array}right)right.\ ,,,+,left(begin{array}{cccc}{rm{i}}mu & {rm{i}}mu {{rm{e}}}^{{rm{i}}{theta }_{{rm{p}}{rm{c}}}} & 0 & 0\ {rm{i}}mu {{rm{e}}}^{-{rm{i}}{theta }_{{rm{p}}{rm{c}}}} & {rm{i}}mu & 0 & 0\ 0 & 0 & {rm{i}}mu & {rm{i}}mu {{rm{e}}}^{{rm{i}}{theta }_{{rm{p}}{rm{c}}}}\ 0 & 0 & {rm{i}}mu {{rm{e}}}^{-{rm{i}}{theta }_{{rm{p}}{rm{c}}}} & {rm{i}}mu end{array}right)\ ,,,left.+,left(begin{array}{cccc}{k}_{x} & 0 & 0 & 0\ 0 & -{k}_{x} & 0 & 0\ 0 & 0 & {k}_{y} & 0\ 0 & 0 & 0 & -{k}_{y}end{array}right)right],left(begin{array}{c}{R}_{x}\ {S}_{x}\ {R}_{y}\ {S}_{y}end{array}right).end{array}$$
(2)
The first and second terms on the right-hand side of equation (2) represent the Hermitian and non-Hermitian coupling processes described in Fig. 1c,d, respectively. The third term on the right-hand side of equation (2) represents the deviation of the wave number from the Γ point in an arbitrary direction represented by wave numbers ({k}_{x}) and ({k}_{y}), which induces a change in the eigenfrequency of each mode.
Here, we consider the eigenfrequencies of the modes in the Γ–M direction (({k}_{x}={k}_{y}=k/sqrt{2})), which is parallel to the axis of symmetry of the double-lattice photonic crystal (y = x). Based on the symmetry along y = x, the coupled-wave matrices on the right-hand side of equation (2) can be block diagonalized using the basis-transformation matrix P:
$$P=frac{1}{sqrt{2}}left(begin{array}{cccc}1 & 0 & 1 & 0\ 0 & 1 & 0 & 1\ 1 & 0 & -1 & 0\ 0 & 1 & 0 & -1end{array}right),$$
(3)
as
$$begin{array}{c}{P}^{-1}CP,=,left(begin{array}{cccc}{kappa }_{11}+{kappa }_{2{rm{D}}+} & {kappa }_{1{rm{D}}}+{kappa }_{2{rm{D}}-} & 0 & 0\ {kappa }_{1{rm{D}}}^{ast }+{kappa }_{2{rm{D}}-}^{ast } & {kappa }_{11}+{kappa }_{2{rm{D}}+} & 0 & 0\ 0 & 0 & {kappa }_{11}-{kappa }_{2{rm{D}}+} & {kappa }_{1{rm{D}}}-{kappa }_{2{rm{D}}-}\ 0 & 0 & {kappa }_{1{rm{D}}}^{ast }-{kappa }_{2{rm{D}}-}^{ast } & {kappa }_{11}-{kappa }_{2{rm{D}}+}end{array}right)\ ,,+,left(begin{array}{cccc}{rm{i}}mu & {rm{i}}mu {{rm{e}}}^{{rm{i}}{theta }_{{rm{p}}{rm{c}}}} & 0 & 0\ {rm{i}}mu {{rm{e}}}^{-{rm{i}}{theta }_{{rm{p}}{rm{c}}}} & {rm{i}}mu & 0 & 0\ 0 & 0 & {rm{i}}mu & {rm{i}}mu {{rm{e}}}^{{rm{i}}{theta }_{{rm{p}}{rm{c}}}}\ 0 & 0 & {rm{i}}mu {{rm{e}}}^{-{rm{i}}{theta }_{{rm{p}}{rm{c}}}} & {rm{i}}mu end{array}right)\ ,,+,frac{1}{sqrt{2}}left(begin{array}{cccc}k & 0 & 0 & 0\ 0 & -k & 0 & 0\ 0 & 0 & k & 0\ 0 & 0 & 0 & -kend{array}right).end{array}$$
(4)
Then, the coupled-wave equation (2) can be divided into the following two forms:
$$begin{array}{l}left(delta +{rm{i}}frac{alpha }{2}right),left(begin{array}{c}{R}_{x}+{R}_{y}\ {S}_{x}+{S}_{y}end{array}right)=left[left(begin{array}{cc}{kappa }_{11}+{kappa }_{2{rm{D}}+} & {kappa }_{1{rm{D}}}+{kappa }_{2{rm{D}}-}\ {kappa }_{1{rm{D}}}^{ast }+{kappa }_{2{rm{D}}-}^{ast } & {kappa }_{11}+{kappa }_{2{rm{D}}+}end{array}right)right.+left(begin{array}{cc}{rm{i}}mu & {rm{i}}mu {{rm{e}}}^{{rm{i}}{theta }_{{rm{p}}{rm{c}}}}\ {rm{i}}mu {{rm{e}}}^{-{rm{i}}{theta }_{{rm{p}}{rm{c}}}} & {rm{i}}mu end{array}right)\ ,,,,left.+,frac{1}{sqrt{2}}left(begin{array}{cc}k & 0\ 0 & -kend{array}right)right],left(begin{array}{c}{R}_{x}+{R}_{y}\ {S}_{x}+{S}_{y}end{array}right)end{array}$$
(5)
and
$$begin{array}{l}left(delta +{rm{i}}frac{alpha }{2}right),left(begin{array}{c}{R}_{x}-{R}_{y}\ {S}_{x}-{S}_{y}end{array}right)=left[left(begin{array}{cc}{kappa }_{11}-{kappa }_{2{rm{D}}+} & {kappa }_{1{rm{D}}}-{kappa }_{2{rm{D}}-}\ {kappa }_{1{rm{D}}}^{ast }-{kappa }_{2{rm{D}}-}^{ast } & {kappa }_{11}-{kappa }_{2{rm{D}}+}end{array}right)+left(begin{array}{cc}{rm{i}}mu & {rm{i}}mu {{rm{e}}}^{{rm{i}}{theta }_{{rm{p}}{rm{c}}}}\ {rm{i}}mu {{rm{e}}}^{-{rm{i}}{theta }_{{rm{p}}{rm{c}}}} & {rm{i}}mu end{array}right)right.\ ,,,,left.+,frac{1}{sqrt{2}}left(begin{array}{cc}k & 0\ 0 & -kend{array}right)right],left(begin{array}{c}{R}_{x}-{R}_{y}\ {S}_{x}-{S}_{y}end{array}right),end{array}$$
(6)
where equation (5) gives the coupling between a pair of electric-field vectors Rx + Ry and Sx + Sy, which leads to the formation of modes A and C (Supplementary Fig. 1).
The frequencies and radiation constants of modes A and C can be then derived from equation (5) as follows:
$${delta }_{{rm{A}},{rm{C}}}+{rm{i}}frac{{alpha }_{{rm{A}},{rm{C}}}}{2}={kappa }_{11}+{kappa }_{2{rm{D}}+}+{rm{i}}mu mp sqrt{[({kappa }_{1{rm{D}}}+{kappa }_{2{rm{D}}-})+{rm{i}}mu {{rm{e}}}^{{rm{i}}{theta }_{{rm{p}}{rm{c}}}}][{({kappa }_{1{rm{D}}}+{kappa }_{2{rm{D}}-})}^{ast }+{rm{i}}mu {{rm{e}}}^{-{rm{i}}{theta }_{{rm{p}}{rm{c}}}}]+{left(frac{{k}}{sqrt{2}}right)}^{2}}={kappa }_{11}+{kappa }_{2{rm{D}}+}+{rm{i}}mu mp sqrt{[({kappa }_{1{rm{D}}}+{kappa }_{2{rm{D}}-}){{rm{e}}}^{-{rm{i}}{theta }_{{rm{p}}{rm{c}}}}+{rm{i}}mu ][{{({kappa }_{1{rm{D}}}+{kappa }_{2{rm{D}}-}){{rm{e}}}^{-{rm{i}}{theta }_{{rm{p}}{rm{c}}}}}}^{ast }+{rm{i}}mu ]+{left(frac{{k}}{sqrt{2}}right)}^{2}}={kappa }_{11}+{kappa }_{2{rm{D}}+}+{rm{i}}mu mp sqrt{[{R}+{rm{i}}{I}+{rm{i}}mu ][{R}-{rm{i}}{I}+{rm{i}}mu ]+{left(frac{{k}}{sqrt{2}}right)}^{2}},$$
(7)
where δA + iαA/2 of mode A and δC + iαC/2 of mode C correspond to the negative and positive square-root terms, respectively, and R and I are defined as (Requiv {rm{Re}}left[left({kappa }_{1{rm{D}}}+{kappa }_{2{rm{D}}-}right){{rm{e}}}^{{rm{-i}}{theta }_{{rm{pc}}}}right]) and (Iequiv {rm{Im}}left[left({kappa }_{1{rm{D}}}+{kappa }_{2{rm{D}}-}right){{rm{e}}}^{{rm{-i}}{theta }_{{rm{pc}}}}right]), respectively.
Note on the threshold gain margin Δα
v
It is difficult to specify a general value of Δαv sufficient for single-mode oscillation in PCSELs. However, we have found that increasing Δαv by simultaneously reducing R and μ contributes to the preservation of single-mode oscillation even in the presence of a non-uniform in-plane refractive index distribution borne by various physical phenomena. These findings will be reported separately.
Estimation of the coupling coefficients R, I and μ of the fabricated devices
To estimate the coupling coefficients R, I and μ of the fabricated device shown in Fig. 2a, we derived the photonic band structure around the Γ point by measuring the subthreshold spontaneous emission spectra at various radiation angles (corresponding to in-plane wave numbers), whose peak emission wavelengths and line widths corresponded to the band frequencies and radiation constants, respectively. The frequencies and radiation constants of modes A and C are plotted in Fig. 2b,c. R, I and μ were then estimated by fitting the analytical frequencies and radiation constants given by equation (1) to their measured values.
On the other hand, it was difficult to estimate the coupling coefficients of the fabricated device shown in Fig. 4a by directly measuring the photonic band structure around the Γ point due to the pre-installed lattice-constant distribution. Thus, the coupling coefficients were instead estimated by evaluating the shape of the embedded air holes.
Design of the pre-installed lattice-constant distribution
The distribution of the lattice-constant variation Δa(x,y), which compensates for a temperature distribution ΔTcomp(x,y), is determined as follows:
$$frac{Delta a(x,y)}{a},=,frac{Delta {T}_{{rm{comp}}}(x,y)}{{n}_{{rm{eff}}}}frac{{rm{d}}n}{{rm{d}}T}.$$
(8)
Here, a is the original lattice constant, neff is the effective refractive index of the photonic crystal at room temperature, and dn/dT is the rate of change of refractive index with respect to temperature. Based on equation (8) and the self-consistent analysis of photon–carrier–thermal interactions (Supplementary Text Section 5), we introduced the lattice-constant distribution shown in Fig. 3e,f.
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