Spin-VCSELs with Local Optical Anisotropies: Toward Terahertz Polarization Modulation

We present a semiclassical model for spin-injected vertical-cavity surface-emitting lasers (spin-VCSELs) with local optical anisotropies. Particular focus is put on highly anisotropic spin lasers with broad application potential. A generalized matrix formalism for extraction of the laser modes is introduced, which enables us to calculate the spatial distribution of vectorial modes in arbitrary spin-VCSELs. The time dependence of such laser modes is further studied by means of the generalized coupled-mode theory, which is the natural anisotropic generalization of the conventional mode-decomposition approach. We use the circularly polarized optical modes as the basis for coupled-mode theory, which leads to extension of the well-known spin-flip model. In contrast to the conventional spin-flip model, the only input parameters are the geometric and local optical properties of the multilayer structure and properties of the gain media. The advantages of the theory are demonstrated in the design and optimization of spin-VCSEL structures with a high-contrast grating. We show that the proposed structures can be used for (i) polarization modulation in the terahertz range with tremendous applications for future ultrafast optical communication and (ii) as prospective compact terahertz sources.

Figure 1

(a) Approximate band structure inside a semiconductor QW with spin degree of freedom. It is assumed that radiative recombinations occur only between conduction-band (CB) electrons and heavy-hole (HH) states near the $\mathrm{\Gamma }$ point. (b) Spin-VCSEL structure considered here, in which only the propagation of electromagnetic waves parallel to the $z$ axis is considered. LH, light hole.

Figure 2

Birefringence $\delta {\epsilon }_{a,r}$ induced by phase-amplitude coupling as a function of the gain dichroism $\mathcal{D}$ and the Henry factor $\alpha$.

Figure 3

A muliple-QW spin-VCSEL device consisting of $N+3$ layers. Each layer is described by its own permittivity tensor ${\hat{\epsilon }}^{(n)}$ and thickness $d^{(n)}$.

Figure 4

The spatial distribution of the basis functions ${\mathit{\phi }}_k(z)=⟨z|k⟩$ within the laser cavity.

Figure 5

A simple single-QW structure used for numerical validation, with the electric field distribution calculated using the modified matrix formalism. The inset shows the position of the active layer and two possible positions [red ($\zeta =0.1$), blue ($\zeta =0.15$)] of the anisotropic passive layer within the $\lambda$-cavity field. The local reference frame is described by $\zeta$.

Figure 6

Frequency splitting calculated as a function of the linear birefringence (LB) parameter $\delta {\epsilon }_r$ (a), of the linear dichroism (LD) parameter $\delta {\epsilon }_i$, due to a nonzero Henry factor $\alpha$ (b), and of both anisotropy parameters (c). The results obtained with our scattering matrix (S-matrix) formalism are used as a reference to evaluate the precision of the method, based on coupled-mode theory (CMT).

Figure 7

(a) Calculated $N_{[110]}/N_{[1\overline{1}0]}$ ratio as a function of the linear dichroism (LD) parameter $\delta {\epsilon }_i$. (b) The calculated $N_{[110]}/N_{[1\overline{1}0]}$ ratio as a function of the linear gain dichroism (LGD) $\mathcal{D}=\delta {\epsilon }_{a,i}/(2\overline{\chi })$. (c) The ratio of threshold pumping rates $N_{0\uparrow }/N_{0\downarrow }$ in the structure with circular gain dichroism, due to electron spin imbalance, but without additional linear anisotropies. S-matrix stands for scattering matrix and CMT for coupled-mode theory.

Figure 8

Components of the reduced Stokes vector $\mathcal{S}=[{\mathcal{S}}_1,{\mathcal{S}}_2,{\mathcal{S}}_3{]}^T$ of a single laser eigenmode calculated with the rigorous matrix formalism (left) and with the extended spin-flip model based on the coupled-mode theory (right).

Figure 9

(a) A $\lambda =1300\mathrm{nm}$ VCSEL with four ($\mathrm{In}$,$\mathrm{Ga}$)$\mathrm{As}$ QWs and an intracavity ($\mathrm{Al}$,$\mathrm{Ga}$)$\mathrm{As}$/air grating and (b) a $\lambda =940\mathrm{nm}$ half-VCSEL with three ($\mathrm{In}$,$\mathrm{Ga}$)$\mathrm{As}$ QWs, a $\mathrm{Ga}\mathrm{As}$/air surface grating, and a microcavity of length $L$.

Figure 10

Compact source of monochromatic terahertz radiation based on a highly anisotropic spin laser.

Figure 11

(a) Frequency splitting between orthogonally polarized laser modes as a function of intracavity ($\mathrm{Al}$,$\mathrm{Ga}$)$\mathrm{As}$ grating thickness and fill factor. (b) Frequency splitting of modes in a vertical-external-cavity surface-emitting laser (VeCSEL) with a $\mathrm{Ga}\mathrm{As}$ surface grating and a microcavity as a function of grating thickness and microcavity length $L$.

Figure 12

Simulated polarization modulation of a spin-VCSEL with a grating for two different positions of the grating within the cavity.

Figure 13

Calculated cavity-decay-rate contribution coming from internal absorptions ${\kappa }_{\mathrm{abs}}$ as a function of the imaginary part of the permittivity ${\epsilon }_{m,i}$. Results onbatined with the robust method based on the matrix formalism are compared with results obtained with the simple analytic expression.