Generation of cylindrical vector beam from GaAs/InGaAs/GaAs core-multishell nanowire cavity

We investigated the beam profiles and polarization states in the low-temperature photoluminescence from vertical GaAs/InGaAs/GaAs core-multishell nanowire (NW) under continuous-wave and pulsed excitations. In the beam profile under pulsed excitation, a doughnut-shaped intensity distribution was confirmed. The beam was shown to exhibit an axisymmetric distribution in the polarization. These observations indicate that cylindrical vector beams were generated from the NW. The observed polarization did not correspond to low-order vector beams but suggested the generation of higher-order beams.


Introduction
Recently, light waves such as optical vortices [1][2][3] (or Laguerre-Gaussian beams) having spiral wavefronts or polarization vortices 3,4) having axisymmetric polarization distributions have attracted much attention. They have singularities in phase, intensity, or polarizations, and behave differently from Hermite-Gaussian beams, which are eigenmodes in ordinary lasers. One of the strong interests in optical vortices is in their optical angular momentum, and various applications are proposed utilizing their unique properties. 3) The polarization vortices are also referred to as cylindrical vector beams because they have axisymmetric polarization distribution and are representing forms of generalized vector beams 5) defined as optical beams with inhomogeneous distribution of the polarization. The most fundamental vector beams (we omit "cylindrical" hereafter) are the radially or azimuthally polarized beams. In addition to the singularity at its beam center, one of their unique features is that the focusing characteristics differ significantly from those of ordinary light waves. That is, when a radially polarized beam is focused, an electric field is generated along the beam axis near the center of the beam. 6,7) Because of these characteristics, vector beams are expected to be used in a wide variety of fields, such as optical communication, 8) optical trapping and manipulation, [9][10][11][12] imaging, [13][14][15] laser processing, [16][17][18] and particle acceleration, 19,20) as well as in the search for physical properties based on the interaction between light and matters. 21) Various methods are reported for the generation of vector beams, 5) including conversion and synthesis of ordinary laser beams using external elements [22][23][24][25][26][27][28][29] and direct generation from photonic crystal lasers. 30,31) Some of the approaches utilize mode conversion in an axisymmetric structure such as a cylindrical waveguide or an optical fiber. [32][33][34][35] This is because radially and azimuthally polarized states are directly correlated with the TE and TM modes of the optical fiber. [35][36][37] However, due to the small index contrast between the core and clad in the conventional optical fiber, TE and TM modes are degenerated. Therefore, some sort of mechanisms should be introduced to select a particular mode. In addition, mode conversion is generally less efficient than the method using direct generation via laser oscillation. Therefore, compact and efficient vector beam sources, which do not rely on a large laser system or external elements, are expected for the applications. However, such reports are limited except for a few reports on the generation of radially 38) and azimuthally 39) polarized beams using nanolasers based on plasmonic nanocavities.
Semiconductor nanowires (NWs) are one-dimensional thin wire structures with diameters ranging from several nanometers to sub-micrometers, and lengths ranging from several hundred nanometers to several micrometers. 40) This anisotropic and high aspect ratio of the structure allows various kinds of device applications with the capability of high-density integration. Thanks to the material degree of freedom, NWs are realized using many varieties of semiconductors, and it is relatively easy to fabricate heterostructures having three-dimensional forms, such as a core-shell structure. 41,42) Owing to their small diameters, it is possible to suppress misfit dislocations originating from lattice-mismatch between the constituent semiconductors, and this enables us to produce high-quality crystals going beyond the limit in the thin-film layer structures. [43][44][45][46] In addition, as materials for photonic applications, they exhibit unique optical properties that differ from the materials with macroscopic or nanoscale dimensions in all three dimensions. For instance, it can be considered a natural waveguide similar to optical fiber. This means NWs allow both confinement of the light in the lateral direction and light propagation along the elongation axis. Furthermore, the light confinement can be tuned by their lateral size and shapes to achieve high light extraction efficiencies [47][48][49][50] or emission patterns. 51,52) Therefore, NWs are expected to be used in various photonic devices to achieve higher performance and to realize a new type of devices, such as NW-based light-emitting diodes, [51][52][53][54][55][56][57][58] lasers, 41,42,[59][60][61][62][63][64][65][66] and so on.
From the application point of view, we believe that NWs grown vertically on a substrate are one of the candidates for compact light sources of vector beams for the following reasons: first, NWs can be a gain medium for light emitters. There are several demonstrations of the lasing oscillation in NWs, as just described. The light emission by current injection is also reported [51][52][53][54][55][56][57][58] as light-emitting diodes owing to the feasibility of the formation of pn-junctions and electrical contacts. Most importantly, NWs have similar waveguiding properties as optical fiber as mentioned above, where the NW and surrounding air have the role of a core and a clad, respectively. It is noted that, because of the large difference in refractive indices between NWs and air, the degeneracy of the TE and TM modes is lifted. Although the TE and TM modes are not the fundamental modes of the waveguide, radially or azimuthally polarized laser beams can be obtained without external optics if cavity mode, edge reflectivity, 67) Q-factors, and gain spectra are designed appropriately.
In this paper, we report on the generation of vector beams from NWs without relying on external optics. We investigated low-temperature photoluminescence (PL) from relatively thick vertical GaAs/InGaAs/GaAs core-multishell NWs possessing cavity mode under continuous-wave (CW) and pulsed photoexcitation and conducted an observation of the beam profiles and polarization states as well as their spectra. Hollow beam was obtained under the intense pulsed excitation condition, which indicated the existence of the singularity in the center of the beam. Analysis of the polarization of the hollow beam indicated that it had an axisymmetric distribution of polarization. Comparisons of the experimental results with the simulation indicate the generation of a high-order vector beam from the NWs.
The part of results in this paper is reported in the extended abstract of the 2022 International Conference on Solid State Devices and Materials (SSDM2022). 68) In the extended abstract of SSDM2022, the polarization distribution of the beam under pulsed excitation is shown as well as the excitation intensity dependence of the PL, and part of the simulation results is also presented. In addition to these results disclosed in SSDM2022, this paper includes the analysis of excitation intensity dependence of the PL, which confirmed the signature of lasing oscillation in NWs. Based on more detailed data on the distribution of polarization in the beam obtained by both CW and pulsed excitation, we identified the mode of the vector beam in the present work.

Experimental procedure
GaAs/InGaAs/GaAs core-multishell NWs standing vertically from the substrate were used in the present study. Schematic and SEM images of the sample are shown in Fig. 1(a). The NWs were grown by selective-area metal-organic vaporphase epitaxy (SA-MOVPE) using a two-step growth sequence. It is the same as that used in our previous studies. In Refs. 69, 70, we described the growth process and the existence of cavity modes in NWs based on the measurements of the temperature dependence of PL spectra. The Fabry-Perot-like cavity modes in the vertical geometry originate from the light propagating along the NW and its reflection at the NW top surface and the interface between NW and substrate. Note that the propagating mode is the waveguide mode of the NW. The gain medium, i.e. InGaAs, lays the sidewall of the GaAs core NW, which is formed along the direction of light propagation inside the waveguide, expecting that the gain area spreads along the direction of light propagation. Estimating from the observed SEM image of Fig. 1(a), average diameter d (face-to-face distance of hexagon) and the length L of the NWs in the present NW array measured 742 nm and 6.6 μm with variation Δd = 36.6 nm and ΔL = 0.139 μm, respectively. Thick NWs shown here were intentionally chosen to ensure low loss and high-Q cavities in NWs grown by SA-MOVPE. 69,70) For spectroscopy and observation of the beam profile, lowtemperature μ-PL measurement was carried out by a conventional setup with HeNe laser and mode-locked Ti: sapphire laser as CW and pulsed excitation sources, respectively [ Fig. 1(b)]. In this setup, the laser light for excitation was reflected by a beam splitter at an angle as close as to the normal incidence and focused onto a single NW using an ×50 microscope objective, which also collected the PL from the sample. The polarization state of the PL was analyzed by using either λ/2 or λ/4 waveplates (HWP and QWP, respectively) placed in front of the linear polarizer (LP). LP was set vertically to the laboratory frame, while the principal axis of the HWP (QWP) is changed from the vertical by angle Analyzed light was fed into a spectrometer equipped with cooled CCD to obtain PL spectra. The CCD camera was used to directly observe and analyze the beam profiles. In this case, parts of the PL spectra were filtered out using bandpass filters (BPFs) with a typical bandwidth of 5 nm. Figure 2 shows the PL spectra of three different NWs (NW (a), (b), and (c)) under CW (solid lines) and intense pulsed (dotted lines) excitations. Spectra and the peak positions differed from NWs to NWs. Although the size uniformity of the NW seemed to be relatively high from the SEM observation as described in the previous section, the PL spectra exhibited large NW to NW variations. This indicated the existence of microscopic variations of NWs, such as in the thickness and indium content of the InGaAs shell layer as well as the variation of the diameter d of NWs. With all these variations, however, multiple sharp lines with a typical linewidth of 0.5 nm were observed in each NWs under CW excitation. Similar sharp peaks were observed in our previous study. They were shown to be originated from the cavity modes of NWs, 70) although the nature of the modes was not identified. The peaks observed in the PL spectra in the current work also presumably originated from the cavity modes. As we will discuss later, their origin is considered mostly to be from the Fabry-Perot-like modes of fundamental HE modes propagating along the NW and partly from that of the higher order modes, such as TE, TM, and so on.

PL spectra under CW and pulsed excitations
The spectra looked completely different under the intense pulsed excitation condition. The peak intensity under the pulsed excitation condition was stronger by about 6 than the CW intensity at a similar excitation level, and the spectra were dominated by a few sharp peaks, which were absent under CW excitations. It is also noted that these new sharp lines under the pulsed excitation condition were at the shorter wavelength regions than those under CW excitations in each NWs. To clarify the origin of the difference in the spectra, we investigated the excitation intensity dependence of the PL in detail. In the following, we focused our investigation on NW (c). Figure 3(a) shows the PL spectra of the NW (c) and their excitation intensity dependence under the pulsed excitation condition. Several sharp peaks were observed in the PL spectra at the lowest excitation intensity studied here. They were the same as those observed at CW excitation. (Fig. 2). With the increase of the excitation intensity, their peak intensity increased linearly. However, as described above, new peaks appeared in the shorter wavelength region as the excitation light intensity was increased. Figure 3(b) shows the excitation light intensity integrated peak intensity curves ( i.e. L-L curves) and full width at half maximum (FWHM) for the peak near the wavelength of 895 nm [indicated by blue arrows in Fig. 3(a)] and the peak observed at 875 nm [indicated by red arrows in Fig. 3(a)]. Note that the latter peak with a linewidth of about 0.34 nm started to appear at an excitation intensity of 250 μW superimposed on a broad   background [see inset of Fig. 3(b)], while the former was observed even at the lowest excitation intensity investigated here. Furthermore, the intensity of the 875 nm peak developed nonlinearly, while that of the peak at 890 nm was nearly linear. To gain further insight into the origin of the nonlinear increase of the peak at 875 nm, we fitted the L-L dependence with the rate equation of the microcavity laser described in Refs. 63, 71 We were able to obtain reasonable fitting of the L-L curve with spontaneous emission coefficient 0.010. b = (see supplemental information for the detail of rate equation and fitting). The nonlinear behavior of the peaks under the pulsed excitation condition was already reported in Ref. 70, and we believed that it was a signature of the stimulated emission or lasing. However, no further evidence was given in the previous report. The L-L curve obtained in the previous study was also fitted with the similar fitting parameters used in Fig. 3(b), and the well-fitted results were confirmed (see the supplemental information). Although it seemed to be in the regime of transition from spontaneous emission and lasing oscillation, the results strongly support the lasing of NW (c) at 875 nm.
On the other hand, if one considered the results of L-L characteristics and excitation intensity dependence of the FWHM, it seemed the peak at 895 nm did not contribute to the lasing. Thus, we conclude that the 875 nm peak was originating from lasing oscillation but the 895 nm peak was not. This difference in the nature of the peaks was due to their different origin of the radial mode in the NW cavity, which was confirmed by their difference in the beam shape discussed later.

Beam profiles and polarization states
Having known the basic nature of the PL peaks, we investigated the beam profiles and polarization nature of the emissions from core-multishell NWs. Figures 4 and 5 summarize the CCD images of the PL emission from NW (c) obtained under various experimental conditions with the CW and pulsed excitation, respectively. Figure 4 shows the beam observed with vertical LP and HWP, and without BPF under CW excitation. Rotation angle 2 / q l of the HWP was 0°a nd 45°, respectively for Figs. 4(a) and 4(b). The intensity was the strongest at the center of the beam and the beam shape was anisotropic. The center spot was ellipsoidally elongated along the vertical axis with sidelobes in the horizontal direction when 0 . In the case of high-intensity pulsed excitation, a donutshaped beam profile was observed; the donut-shaped beam was confirmed without spectral filtering [ Fig. 5(a)]. When the 875 nm peak was extracted, this central singularity became clearer, as shown in Fig. 5(b). The observation of the emission image through LP and rotating HWP revealed that the beam had an axisymmetric distribution in polarization, as shown in Figs. 5(c), 5(d), and 5(e) for 2 / q l = 0°, 22.5°, and 45°, respectively. Note that the direction of the rotation of the image is clockwise (see detail for supplemental information); that is, the direction of the image rotation is opposite to that of HWP and the case of the image of CW excitation. On the other hand, the beam profile of the 875 nm peak, which is shown in Fig. 5(f), exhibited the strongest intensity at its center with anisotropic shape. This is similar to the results shown in Fig. 4. When This clearly indicates that waveguide modes were different between 895 and 875 nm peaks.
The observation of the beam profile was also carried out with LP and rotating QWP for both types the excitation. It was confirmed that the change due to the change in the rotation angle 4 / q l of the QWP was very small in contrast to the change introduced by the rotation of HWP (the results are summarized in the supplemental information). Therefore, the beam and its spatial distribution contain virtually no component of the circular polarization irrespective of the excitation.
The observation of a hollow beam and the existence of an axisymmetric distribution in polarization clearly demonstrate the generation of vector beams from NWs under pulsed excitation. Although it is less convincing due to the absence of singularity, the vector beam was also obtained under the CW excitation condition. Since LP is set vertically, the intensity images in Figs    polarization distribution of TM modes. The absence of singularity in the beam center is probably due to the overwrap of multiple propagating modes in the NW waveguide, such as HE modes. On the other hand, the polarization distribution of the beam obtained under the pulsed excitation condition is different from that expected for the TM and TE modes. That is, the vertically (or horizontally) polarized portion of the beam does not appear at the vertical or horizontal position from the center of the beam, and appeared at the tilted position by about 45°, as shown in Figs. 5(d) and 5(f). For this reason, we believe that the origin of the beam under pulsed excitation originates from higher order modes, namely, HE 21 mode. HE 21 mode also can explain the clockwise rotation of the image shown in Figs. 5(d)-5(f), as we describe in the next section when HWP is rotated in the counterclockwise direction.

Simulation of the beam intensity profile
To understand the polarization state of the emitted vector beam from NWs in more detail, we investigated the polarization state of the raw, and measured beams by using simulations. The simulations were performed as follows: First, assuming that the NW is a cylindrical waveguide with the NW as a core and air as a clad, the eigenmode and electromagnetic (EM) field in the NW waveguide were calculated by solving the mode equation of step-index fiber. 37) The EM field was rearranged into the Cartesian coordinate system to visualize the spatial distribution of the direction of the electric field. The direction of the electric field is indicated by the red arrow in the following figures. Next, the spatial distribution of the Stokes parameters, S = (S 0 , S 1 , S 2 , S 3 ), [72][73][74][75] was calculated from the EM field. Then, Mueller calculus [72][73][74][75] was adopted to obtain the Stokes parameters of the beam passed through the polarization optics (HWP with rotation angle 2 / q l and vertical LP) by the experimental system. Finally, we calculated S 0 , which represents the total light intensity, and its spatial distribution was visualized as a beam profile.
Before showing the results of the simulation, we summarize the properties of waveguide modes in the present NWs and their relationship with vector beams. In the stepindex fiber with core radius R, the cutoff wavelength c l of TE and TM modes is given by n R 2 2 2.405, is the index contrast, n 1 and n 2 are the refractive index of core and air clad, respectively. For the NWs with diameter d = 2 R, n 3.6 1 = and n 1, 2 = d c / l is calculated to be 5.2. Thus, the NWs investigated in the present study support high-order propagation modes. Indeed, the calculation shows that the mode with azimuthal mode number up to m 6 = exists for wavelength 900 nm, lw here m is the azimuthal mode number. The EM fields in the NW have a dependence on the cylindrical coordinate system (r, θ) as in Ref. 37 Here, E r and E q represent the radial and azimuthal components of the electric field, respectively, and θ 0 is the phase factor. The phase factor arises from the cylindrical symmetry. The mode with m = 0 corresponds to TE 0n or TM 0n mode, and that with m≠1 is a hybrid mode, namely, HE mn or EH nn mode. Here, n is the radial mode number. The lowest order mode is HE 11 with m = 1, and modes with m = 0 or m ≧ 2 are the higher order modes. When m = 0, the phase factor θ 0 is fixed for the TM ( 0 q = 0°) and TE ( 0 q = 90°) modes, and E r and E q is zero at r = 0. These are the vector beams with radial or azimuthal polarization, respectively. When m ≠ 0, however, θ 0 is arbitrary, and the polarization states depend on θ 0 . If θ 0 is not fixed, the light is unpolarized. When m = 1 and θ 0 are a constant, the light is linearly polarized. The modes with m ≧ 2 and the fixed θ 0 are axisymmetric and correspond to higher-order vector beams.
The simulation for TM and TE modes reproduces the beam intensity profile for radially and azimuthally polarized beams, respectively; the intensity was zero at the center, and the strongest intensity was arranged vertically (horizontally) for TM (TE) modes when  2 / q l The degree of freedom in phase factor appears only for m ≠ 0, and neither TM nor TE modes can explain the tilt. We also carried out simulations for the mode with m ≧ 3, and found that the image exhibited multiple numbers [given by 2(m−1)] of bright spots, which reflected the vertical linearly polarized part of the beam. Therefore, we conclude that the azimuthal mode number m of the vector beam observed in our experiment is 2, or HE 21 mode of the vector beam was observed under the pulsed excitation condition.

Discussions
As explained above, hollow beams obtained from the NW under the pulsed excitation condition are due to high-order vector beams with m = 2. This higher order beam was not observed under the CW excitation condition, but the signature of generation of the fundamental mode of the vector beams was obtained. In turn, fundamental vector beams were not confirmed under the pulsed excitation condition. The reasons are explained as follows. As described in Sect. 3.1, cavity peaks were observed for the CW excitation. These are believed to be originated from low-order modes of the NW waveguide, namely, HE 1n , TE 0n , and TM 0n modes. On the other hand, nonlinearity in L-L characteristics was observed at the pulsed excitation, and this is due to the lasing. However, the correlation between cavity modes observed in CW excitation and lasing mode at the pulsed excitation is absent. Thus, the lasing originates from higher order modes with m ≧ 2. The cavity Q factor is sensitive to the azimuthal mode number m and larger m generally yields a larger Q factor. 69) This is presumably because of the cavity loss, which is determined by the reflectivity at the NW edges at the top and the substrate interfaces in vertical NWs, and larger m introduces a larger mismatch between propagating modes in NW and air/substrates. Thus, lasing with higher azimuthal mode took place. Thus, the vector beam obtained under lasing conditions corresponded to the higher order modes, while we observed the mixture of HE, TE, and TM modes under the CW excitation condition.
The constant phase factor θ 0 suggests the coherent nature of the beam. In other words, it is accompanied by the lasing. Yet, it is not clear why it is fixed at a particular angle, but it is speculated that the main reason for the non-uniform distribution is the intensity of the doughnut-shaped beam. This could be the result of the non-uniform excitation of NW in the experiment, or due to non-uniformity of the InGaAs shell layer, or due to the non-circular (hexagonal) shape of the NW, which resulted in the deviation from the situation of complete axisymmetric symmetry.
In the present experiment, the lasing and vector beam was obtained for mode m = 2. Towards the compact light source of the vector beam, several improvements should be pointed out. The lasing is possible when the gain overcomes the loss in the cavity. As discussed above, the loss, particularly, the one due to the edge reflectivity, 67) for modes with m = 0 or 1 is not considered to be good enough in the present NWs. To reduce the reflectivity loss and to achieve a lower lasing threshold, the approaches to form NWs on high-reflectivity mirrors would be effective. 75) Alternatively, it is important to increase the gain. This is possible by stacking the InGaAs layer on the sidewall to introduce more inner shells as the gain medium. It is also possible to increase the confinement factor, which is determined by the overwrap of the electric field and InGaAs layer. The electric field distribution depends on mode number m. Thus, an appropriate design of the coreshell structure would be able to achieve control of the radial mode for lasing and vector beam. Finally, the lasing in NWs by current injection is still a challenging task at present, 76,77) and requires further investigation.

Conclusion
We investigated the low-temperature PL from GaAs/InGaAs/ GaAs core-multishell NWs and its polarization state under the CW and pulsed excitation conditions. The signature of lasing was confirmed under intense pulsed excitation. The observation of the beam profile showed the generation of cylindrical vector beams with a singularity at the center under the lasing conditions. The detailed observation of polarization states and the comparison with the simulation showed that the origin of the vector beam was the mode with radial mode number m = 2, namely, HE 21 mode. Our results are promising for a compact light source of the vector beams with an appropriate design of the NW cavity structure.