Higher-order vector beams produced by photonic-crystal lasers

: We have successfully generated vector beams with higher-order polarization states using photonic-crystal lasers. We have analyzed and designed lattice structures that provide cavity modes with different symmetries. Fabricated devices based on these lattice structures produced doughnut-shaped vector beams, with symmetries corresponding to the cavity modes. Our study enables the systematic analysis of vector beams, which we expect will lead to applications such as high-resolution microscopy, laser processing, and optical trapping.


Introduction
Vector beams, which possess an inhomogeneous distribution of polarization directions, have attracted much attention because they can provide novel functional properties at their focus. For example, azimuthally polarized ring-shaped vector beams enable the optical trapping of metal particles [1][2][3], and radially polarized beams are expected to produce sharp focal spots [4][5][6]. These characteristics are expected to be useful in various fields of biological science and material science [7,8]. Several methods have been proposed for the generation of such vector beams [9][10][11][12][13][14][15], among which we believe that photonic crystal (PC) surface-emitting lasers ( Fig. 1) [16][17][18][19] are highly promising because they can directly produce vector beams from a small laser diode chip. We have already demonstrated the direct emission of radially and azimuthally polarized vector beams from a device based on a square-lattice PC, utilizing transverse magnetic (TM) and transverse electric (TE) modes, respectively [15]. The next challenge is the generation of vector beams with other polarization states, which may attract both academic and industrial interest. Here, we investigate the potential of PC lasers to produce novel types of vector beams by altering the design of the PC. It is expected that the characteristics of vector beams are essentially determined by the symmetry of the cavity mode that the PC structure provides.
In Section 2, we analyze some basic properties of PC cavities such as the symmetries of the fundamental waves and the electromagnetic fields generated by three different PC structures, and we demonstrate their potential to produce vector beams. In Section 3, we discuss the characteristics of the output beams obtained from fabricated devices based on these structures including beam patterns and polarization. Finally, concluding remarks are given in Section 4.

Photonic crystal design
We discuss three kinds of PC structures (shown in Figs. 2(a)-2(c)) that we have designed to produce vector beams: (a) the square lattice structure, which we have previously analyzed [15]; (b) the triangular lattice and (c) the square lattice with a longer lattice period. Here, we set the filling factor to be 12% for structures (a) and (b), and 3% for (c). The PC consists of a periodic distribution of air holes (ε = 1.0) within a GaAs host material (ε = 12.89).

Reciprocal lattice space and fundamental waves
In order to understand the resonant mode of each lattice structure in Fig. 2, we first discuss the fundamental waves of the cavity modes in reciprocal lattice space, which are shown in Figs. 3(a)-3(c). Superscripts have been assigned to each Γ-point in order of increasing distance from the origin of reciprocal space. The yellow arrows represent the fundamental waves of the cavity modes of interest in this paper. For each of the three lattice types, the fundamental waves at identical Γ-points are equivalent; they can thus couple with each other by the reciprocal lattice vectors and produce cavity modes [19]. That is, the square lattice in Fig. 3(a) has 4 equivalent fundamental waves at the Γ (2) point, whereas the triangular lattice in Fig. 3(b) has 6 equivalent waves at the Γ (2) point and the square lattice in Fig. 3(c) has 8 equivalent waves at the Γ (5) point. In order to obtain the same lasing wavelength (λ d ) for each structure, the lattice constant (a) is designed in such a way that the magnitude of the fundamental waves (2π/λ d ) is identical. For example, in Fig. 3(a), the magnitude of the fundamental waves is equal to the shortest distance between the reciprocal lattice points (b); because b = 2π/a for the square lattice structure, the lattice constant is determined as a = λ d . In the same way, the magnitudes of the fundamental waves in Figs respectively. Because each set of fundamental waves possesses different symmetry, the respective optical characteristics such as the photonic band structure, electromagnetic field distribution, and beam polarization should be symmetry dependent. We calculate these characteristics in Section 2.2 using the plane wave expansion method. (4) G (5) G (5) G (4) G (5) G (5) G (4) G (5) G (5) G (4) G (5) G (5) fundamental waves Fig. 3. Reciprocal space representations of (a) square lattice with 4 fundamental waves at lowest-order Γ (2) points, (b) triangular lattice with 6 fundamental waves at lowest-order Γ (2) points, and (c) square lattice with 8 fundamental waves at higher-order Γ (5) points.   Fig. 4(a), the electromagnetic field possesses 4-fold rotational symmetry, whereas in Figs. 4(b) and 4(c), 6 and 8 magnetic field antinodes (red and blue) are distributed around the lattice hole. The output beam is emitted in the direction normal to the PC surface by the PC diffraction effect [8,18]. Therefore, the beam pattern can be determined by taking the Fourier transformation of the electromagnetic field distribution, reflecting the symmetry of the system. If lasing oscillation arises from these cavity modes, one may expect vector beams with the corresponding symmetry of polarization to be produced. We note that the TM mode shows similar characteristics to the TE mode with respect to the number of band edges and cavity symmetry (see Appendix).

Experimental results
We have fabricated GaAs-based lasers with the PC structures designed in Section 2, using previously reported device parameters [18]. The directions and symmetries of the beam polarization correspond well to the calculated electric field around a lattice hole, shown in Fig. 4 and Fig. 7. Thus, we have successfully generated higher-order vector beams with small divergence angles directly from a laser diode chip.

Conclusion
We have analyzed cavity modes at the Γ-points of square and triangular lattice photonic crystals, not only for the fundamental order but also for higher-order band edges. The symmetry of the fundamental waves is important in establishing the basic laser characteristics because it determines the number of band edges and the symmetry of the electromagnetic field. The beam patterns obtained from the three devices that we have designed and fabricated were doughnut-shaped with small divergence angles of approximately 1°. The polarization direction is rotated between one and three times along the circumference of the doughnut depending on the symmetry of the photonic crystal structure. Therefore, we have succeeded in the generation of various novel vector beams, including higher-order beams. The design of such vector beams based on the cavity symmetry of photonic-crystal lasers will facilitate further variation and systematization of vector beams, which will in turn lead to versatile applications such as high-resolution microscopy, advanced laser processing and optical trapping.

Appendix: Band structure and electromagnetic field for TM mode
In Fig. 7 we show the photonic band diagrams and electromagnetic field distributions for the TM mode, in analogous fashion to those for the TE mode in Fig. 4. The electromagnetic field distributions are shown for the lowest frequency modes indicated by red arrows. The color scale represents the electric field and the black arrows indicate the magnetic field. These field distributions reflect the symmetry of the systems in a similar way to the TE mode, although the electric and magnetic fields are inverse in the two cases.