Modulated photonic-crystal surface-emitting lasers (M-PCSELs) are new semiconductor lasers that can emit a laser beam in arbitrary directions. Therefore, they can be regarded as a next-generation light source for light detection and ranging. In this manuscript, we numerically investigate the characteristics of M-PCSELs, including one- and two-dimensional optical coupling constants and radiation constants, by using three-dimensional coupled wave theory. We also carry out lattice point designs and show that the two-dimensional optical coupling can be enhanced by more than four times over our previous M-PCSEL devices by employing elliptical lattice points. Moreover, we find that two-dimensional optical coupling can be maintained even when large lattice point position modulations are introduced for large radiation constants. These results indicate that more stable two-dimensional oscillation and a higher slope efficiency are expected in M-PCSELs with elliptical lattice points.

Currently, light detection and ranging (LiDAR)^{1–3} is attracting considerable attention as it is required for various applications including self-driving cars,^{4–6} autonomous mobile robots,^{7,8} and security sensing. The key technique of LiDAR is two-dimensional (2D) scanning of laser light. Generally, this functionality is implemented by mechanical systems such as rotating mirrors and microelectromechanical systems (MEMS). However, the inclusion of the mechanical system necessitates a large device size. In addition, the scanning speed is typically slow and the reliability is relatively poor. If a laser that could scan the emission direction electrically and in 2D without the use of a mechanical system were realized, such drawbacks could be resolved.

Recently, the authors proposed and demonstrated modulated photonic-crystal surface-emitting lasers (M-PCSELs), which can emit laser light in arbitrarily angled directions.^{9} A schematic image of the M-PCSEL device structure is shown in Fig. 1(a). This structure is based on the PCSEL concept, in which the band-edge mode of a 2D photonic crystal is used as the laser cavity of a semiconductor laser.^{10–15} An example of the M-PCSEL photonic crystal structure is shown in the lower half of Fig. 1(a). At a glance, it appears that the photonic crystals do not have any periodicity, but in actuality, the air hole positions are modulated by shifting them by a distance (*d*) and at an angle (ψ) from the original lattice position (**r**); this modulation allows the laser beam to be emitted in arbitrary directions. We have also integrated various modulated photonic crystals into one M-PCSEL chip with individual electrodes, as shown in Fig. 1(b).^{9} The far-field patterns obtained while sequentially changing the driving electrodes are shown on the left- and right-hand sides of Fig. 1(b). These patterns indicate that on-chip beam scanning has been realized electrically. These M-PCSELs are expected to be used as future light sources for LiDAR applications.

However, a pressing issue in our previous M-PCSEL devices is the occasional appearance of an X-shaped beam among the single-lobed ones, as observable in Fig. 1(b). This X-shaped beam indicates that lasing is unstable: unintentional 1D resonance modes are lasing in addition to the intended 2D one, leading to the emission of a beam with a large divergence angle and a low quality. To suppress such 1D resonance, enhancement of the 2D optical coupling by adjustment of the photonic crystal structure is worthwhile. In this paper, we numerically investigate these constants by using 3D coupled wave theory and carry out an investigation of lattice point designs for enhancing 2D optical coupling in M-PCSELs. In addition, we also investigate the effect of the lattice position modulation on radiation constants and optical coupling constants in M-PCSELs.

First, we describe the operating principle of M-PCSELs and the physical mechanism for obtaining the laser beam in angled directions. Figure 2(a) shows the band structure of a square-lattice photonic crystal with a lattice constant of *a*; within this band structure, there exist several singularity points (band edges). The point circled in red is called the Γ-point band edge, at which four fundamental light waves, propagating in the ±*x* and ±*y* directions, couple between each other. These four fundamental waves, which form the band-edge mode, can also be coupled to radiation modes that propagate in a direction perpendicular to the surface of the photonic crystal. This Γ point is used in PCSELs for vertical emission. Conversely, in M-PCSELs, for which only beams emitted in oblique directions are desired, such vertical emission should be suppressed. For this purpose, we employed the M-point band edge, circled in blue in Fig. 2(a). Figure 2(b) shows a coupling schematic at the M-point band edge, where there exist four fundamental light waves with wavenumbers [±π/*a*, ±π/*a*] in reciprocal lattice space (*a*: lattice constant). These light waves couple to each other through the reciprocal lattice vectors **G** = [±2π*m*_{1}/*a*, ±2π*m*_{2}/*a*] (*m*_{1}, *m*_{2}: integers) of the photonic crystal. Because the M-point lies outside the air light cone, no coupling to radiation modes is possible. However, when position modulation is introduced, a diffraction vector **k** appears in the reciprocal lattice space of the photonic crystal [red arrow in Fig. 2(b)]. The fundamental light waves are diffracted by **k** in reciprocal lattice space. By designing **k** so as to diffract the fundamental light waves into the air light cone, the light waves can couple to the radiation modes and be emitted to free space in a direction arbitrarily determined by **k**.

Next, we describe concrete design rules for the introduction of **k**. In the M-PCSEL, the air hole positions are shifted by a distance (*d*) at an angle (ψ) from their original lattice positions (**r**), as shown in Fig. 1(a). For the introduction of **k**, the angle ψ is set to satisfy

In order to emit a beam into free space in a direction at angles of *θ*_{x} and *θ*_{y} from the surface normal of the photonic crystal in the *x*- and *y*-directions, respectively, **k** can be derived as

where the in-plane wavenumber conservation rule is considered and *n*_{eff} is the effective refractive index of the device. This requires that

Note that when the position modulation is introduced, diffraction vectors **k** + **H** (**H** = [±2π*n*_{1}/*a*, ±2π*n*_{2}/*a*], *n*_{1},*n*_{2}: integers) also appear in the reciprocal lattice space (orange arrows in Fig. 2) and therefore diffraction to a single designated point occurs for all four fundamental light waves. A diffraction vector −**k** also appears by the modulation, and therefore, twin beams are obtained as shown in Fig. 1(b). We note that the technique of shifting holes of a photonic crystal for modifying the coupling of light to free space has been used elsewhere in nanocavity designs. In most of these designs, the holes are shifted to adjust the spatial envelope function of the electric field of the nanocavity mode to suppress its leaky components.^{16,17} Other nanocavity designs adjust these leaky components by employing photonic supercell structures, wherein multiple lattice points or lattice constants are used, but the periodicity of the original photonic crystals are maintained.^{18,19} Such supercell structures can also introduce **k** vectors, but in this case, **k** must satisfy **k** = [±2π/(*ma*), ±2π/(*na*)], where integers *m* and *n* are the number of lattice periods in the supercell in the *x* and *y* directions, respectively. Conversely, the hole position modulation in M-PCSELs can introduce arbitrary **k** because the modulation is not restricted by the periodicity of the original photonic crystal; thus, the hole position modulation used here and these nanocavity designs are conceptually different.

To investigate the coupling effect of the resonant mode, we developed a 3D coupled mode theory for the M-point resonant mode and modulated photonic crystals. Note that we previously developed a 3D coupled mode theory for the Γ-point resonant mode, which is applicable to PCSELs for vertical emission.^{20–22} In this work, we extended this theory to the M-point resonant mode and modulated photonic crystals. The obtained coupled mode equation for the infinite system is as follows:

where

Here, *R*_{1} to *R*_{4} are the electric field amplitudes of the four fundamental waves shown in Fig. 2(b), and κ_{1D} and κ_{2D} are 1D and 2D coupling constants, respectively. κ_{0} is the self-coupling term. These coupling constants are derived by considering both direct couplings between fundamental waves and indirect couplings via higher order M-points. The concrete expression of matrix *C* is described in the Appendix. δ is the deviation from the Bragg condition, and α is the threshold gain. Then, we numerically analyzed the characteristics of our previously developed M-PCSEL device [Fig. 1(b)] using the newly developed coupled mode theory. The PCSEL layer structure and photonic crystal structure [equilateral triangle lattice points with an air filling factor (FF) of 10%] assumed in the analysis were identical to those of the M-PCSEL device shown in Fig. 1(b). To analyze the characteristics of the lattice structure itself, we refrained from introducing position modulation. κ_{1D} was estimated to be ∼1200 cm^{−1}; however, κ_{2D} was estimated to be as small as 60 cm^{−1}. When 2D coupling is weak, the threshold gain of 1D resonance modes is smaller than that of the 2D resonance modes at the M-point band edge, leading to 1D oscillation and an X-shaped beam pattern; therefore, enhancement of κ_{2D} is important in order to ensure stable 2D oscillation at the M-point band edge.

With the aim of increasing κ_{2D}, we investigated various photonic-crystal lattice patterns. Here, we report the results of our numerical analysis on elliptical lattice points. Figure 3(a) is a schematic image of the photonic-crystal structure, which consists of elliptical lattice points for which the directions of the major and minor axes are parallel to the Γ-X direction of the photonic crystal. The lattice-point ellipticity was defined as the ratio of the minor axis radius to the major axis. Figures 3(b) and 3(c) show κ_{2D} and κ_{1D} calculated for various ellipticities and the FF values of the lattice points without position modulation. The results for equilateral triangle lattice points are also shown in these figures. Figure 3(b) indicates that the maximum κ_{2D} for perfectly circular lattice points (ellipticity: 1) is as small as 140 cm^{−1}, even for large air holes (FF: 30%–40%). Similarly, the maximum κ_{2D} for the equilateral triangle lattice points is as small as 120 cm^{−1} for large air holes. In contrast, large κ_{2D} is obtained by employing elliptical lattice points. For example, κ_{2D} > 220 cm^{−1} can be obtained for an elliptical lattice point with an ellipticity of 0.5 and a relatively small FF of 10%–15%. Figure 3(c) indicates that the ellipticity dependence of κ_{1D} is small compared with κ_{2D}.

Here, we would like to explain the reason why κ_{2D} is enhanced by employing elliptical lattice points. The origin of the 2D optical coupling is indirect coupling via various higher-order M points of the photonic crystal. We note that direct 2D coupling by **G** vectors does not occur for transverse electric (TE) mode because the electric fields of light waves propagating in perpendicular directions are orthogonal to each other.^{20–22} We investigated the higher-order M points which make the largest contributions to 2D optical coupling and found that the contributions of path I (by $G0,\xb11=[0,\xb12\pi /a]$ and $G0,\u22132=[0,\u22132\pi \u22c52/a]$) and path II (by $G\xb11,0=[\xb12\pi /a,0]$ and $G\u22131,\u22131[\u22132\pi /a,\u22132\pi /a]$), shown in Fig. 3(d), are particularly large, for the coupling of fundamental wavevectors [π/*a*, π/*a*] and [π/*a*, −π/*a*]. For circular lattice points, the coupling strengths of these two paths are similar in value but opposite in sign; therefore, their contributions are canceled, leading to weak 2D coupling. Conversely, for the elliptical lattice points used here, the balance between these two paths is lost, which alleviates their cancellation and thereby enhances the 2D optical coupling.

Next, we investigated κ_{2D} and κ_{1D} when lattice position modulation was introduced for light emission in angled directions. We assumed emission angles of θ_{x} = 36° and θ_{y} = 0° from the surface normal of the photonic crystal in the *x*- and *y*-directions, respectively, and confirmed that κ_{2D} and κ_{1D} were almost unchanged. The lattice point shape was assumed to be elliptical with an FF of 10% and an ellipticity of 0.55. Blue curves in Figs. 4(a) and 4(b) show the calculated κ_{1D} and κ_{2D} as functions of the lattice position shift *d* for the modulation. These figures indicate that κ_{1D} decreases by 40% when *d* = 0.15*a*, whereas κ_{2D} decreases by only 20%; this result means that optical couplings should be maintained even when a large *d* is introduced. The reason why these couplings are relatively well maintained for large *d* is as follows. The schematic image of the 1D optical coupling for the case of κ_{1D} is shown in Fig. 4(c). This figure indicates the case of coupling between wavevectors [π/*a*, π/*a*] and [−π/*a*, −π/*a*]. The origin of the 1D optical coupling is direct coupling by $G\xb11,\xb11=[\xb12\pi /a,\xb12\pi /a]$ (orange arrow) and indirect coupling via higher-order modes (green arrows) by **G** vectors (in this figure case, by $G\xb12,0$ and $G\u22131,\xb11$) induced by the unmodulated photonic crystal structure. When position modulation is introduced, the Fourier components of the photonic-crystal structure contributing to the **G** vectors (related to these couplings) decrease in magnitude. However, the position modulation generates additional diffraction vectors **k** and **k** + **H**, and indirect coupling by two of these vectors, shown by red arrows in Fig. 4(c), occurs. We resolved κ_{1D} into the direct coupling, the indirect coupling via higher-order modes of the original photonic crystal, and the indirect coupling by diffraction vectors **k** and **k** + **H**. The results are shown in Fig. 4(a) by orange, green, and red lines, respectively, which indicates that the 1D direct coupling and indirect coupling via higher-order modes by **G** vectors of the original photonic crystal structure decrease drastically, whereas the enhancement of indirect coupling by **k** and **k** + **H** is small. Therefore, κ_{1D} decreases by 40% when *d* increases from zero to 0.15*a*. A schematic image of the 2D optical coupling for the case of κ_{2D} is provided in Fig. 4(d), which shows that indirect coupling via higher-order modes of the original photonic crystal (green arrows) and by diffraction vectors **k** and **k** + **H** exists (red arrows). We note that direct 2D coupling by **G** vectors does not occur for transverse electric (TE) mode, as mentioned before. The green and red lines in Fig. 4(b) show the κ_{2D} value resolved into the indirect coupling via higher-order modes of the original photonic crystal and into the indirect coupling by diffraction vectors **k** and **k** + **H**. This figure indicates that the increase in indirect coupling by the **k** and **k** + **H** introduced by the position modulation [red arrows in Fig. 4(d)] compensates the decrease in indirect coupling via higher-order modes of the original photonic crystal structure [green arrows in Fig. 4(d)], which maintains the κ_{2D} value.

We also calculated the radiation constant (α_{v}) for the portion of light emitted to air as a function of *d* at the M-point band edge. The results, shown in Fig. 5(a), indicate that α_{v} is almost proportional to the square of *d*. Therefore, larger *d* is expected to yield a M-PCSEL with higher output efficiency. We note that α_{v} obtained by the coupled mode theory we have originally developed for PCSELs with surface-normal emission includes all radiation components, which was sufficient to obtain the correct value of α_{v}. However, in M-PCSEL, obtaining α_{v} by the same method includes contributions which are confined within the cladding and are not emitted from the surface of the device. We have distinguished these components and show in Fig. 5(a) only the contribution of light emitted from the surface of the device. In addition, the α_{v} values of bands C and D are smaller than those of bands A and B. This characteristic is different from the result for the Γ-point band edge, which is used for PCSELs for vertical emission.^{15} Furthermore, operation in band C or band D is expected for M-PCSELs. An α_{v} of 2–4 cm^{−1} is predicted for band C or band D when *d* is set to 0.20*a*. To examine the differences between the Γ-point and M-point band edges, we considered the electric fields of bands A and D at these two points, as shown in Fig. 5(b). In this figure, the electric-field nodes are located at the centers of the lattice points for band A, whereas the electric-field antinodes are located at the centers of the lattice points for band D, for both the Γ-point and M-point band edges. The electric field distribution of band B (C) has the same characteristics as band A (D) regarding the relative positions between the nodes and lattice points. In the Γ-point case, the electric fields of bands C and D around the lattice points are strong and symmetric. Thus, the diffraction in the out-of-plane direction by the photonic-crystal lattice points is enhanced and the α_{v} value of these bands is large. Conversely, the α_{v} values of bands A and B are small because the electric fields of these modes around the lattice points are weak and antisymmetric, and the diffraction in the out-of-plane direction by the lattice points is smaller. On the other hand, in the M-point case, when lattice position modulation is not introduced, the directions of the electric fields around the neighboring lattice points are opposite. Therefore, the diffraction from the lattice points is canceled and the α_{v} values of all modes are zero. When lattice position modulation is introduced, the electric field nodes are located at the centers of the lattice points for bands A and B. Thus, the space derivatives of the electric fields exceed those for bands C and D, where the antinodes of the electric fields are located at the centers of the lattice points [see Fig. 5(c)]. Therefore, the out-of-plane diffraction due to position modulation for modes A and B exceeds that for modes C and D, and the α_{v} values for bands C and D are smaller than those for modes A and B in M-PCSELs.

In summary, we numerically investigated the characteristics of M-PCSELs by using three-dimensional coupled wave theory and carried out lattice point designs for enhancing 2D optical coupling. We found that a κ_{2D} value of ∼220 cm^{−1} can be obtained for elliptical lattice points with an ellipticity of ∼0.5 and a relatively small FF of 10%–15%. This κ_{2D} value is about four times larger than those of our previously developed M-PCSEL devices. We investigated the influence of the lattice position modulation (*d*) and found that a radiation constant α_{v} of ∼2 to 4 cm^{−1} can be obtained when *d* is set to 0.20*a*. We also found that the κ_{2D} value remains high even when the large lattice point position modulation (*d* = 0.15*a*) is introduced. Our numerical investigation also predicted that the operation band is C or D; these characteristics differ from the Γ-point case. The obtained α_{v} values are smaller than that of PCSELs for vertical emission; we are currently investigating a new modulation method to increase these values. We expect that these results can be applied to the development of high-power beam scanners without mechanical systems and to the development of next-generation LiDAR systems.

This work was partly supported by the Core Research for Evolutional Science and Technology (CREST) from the Japan Science and Technology Agency (JST), Grant No. JPMJCR17N3, and by the 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).

### APPENDIX: CONCRETE EXPRESSION OF COUPLING MATRIX

The matrix *C* in the coupled mode equation [Eq. (5)] can be expressed as *C* = *C*_{d} + *C*_{i}, where *C*_{d} and *C*_{i} indicate direct coupling between fundamental waves and indirect coupling via higher order M-points, respectively.

The direct coupling term *C*_{d} can be expressed as

where

and $k0$ is the free-space wavenumber, $\beta 0=2\pi /a$, and $\Theta z$ is the electric field profile of fundamental waves in the z direction. The subscript PC of the integral specifies integration over the photonic crystal region. $\xi k,l$ indicates the Fourier coefficients of the dielectric constant distribution $\epsilon r$ of the photonic crystal and can be expressed as

We note that direct 2D coupling terms are zero because the electric fields of light waves propagating in perpendicular directions are orthogonal to each other.

The indirect coupling term *C*_{i} for the M-PCSEL without lattice point modulations can be expressed as

where

and *m*, *n*, *p*, *q*, *r*, and *s* are half integers. $\u03c2$ is expressed as

where

Here, *n*_{0} is the square root of the average dielectric constant of the material, and $Gm,nz,z\u2032$ is the Green function,

with $\beta z,m,n=m2+n2\beta 02\u2212k02n02j$.

*C*_{i} for the M-PCSEL with lattice point modulations can be derived in a similar way, by considering the indirect coupling via singularity points generated by **k**, in addition to the higher order M-points.