Near-threshold high spin amplification in a 1300 nm GaInNAs spin laser

Using the experimental set up described in section 1, the pump power and the fibre position were adjusted to achieve lasing, the latter using the mechanical and piezo adjustments on the stage. The fibre position was carefully adjusted to provide the best compromise of stability via anchoring on the wafer surface, a low threshold, and good coupling of 1300 nm laser light back into the fibre. Firstly, the LI curve was generated to accurately determine the laser threshold. The VCSEL power for the LI curve was integrated from the lasing spectra on the OSA, with a bandwidth of 2 nm. As this is much less than the gain bandwidth, the amount of spontaneous emission included was small, thus the lasing threshold was clear from the rapid increase in power, being at 740 ± 1 mA. The spectra and LI curve are shown in figures 2(a), (b) where the pump currents I used in the polarisation measurements of 760, 800, 850 and 900 mA are shown by the red vertical lines. The pump power range is 169 mW (760 mA) to 201 mW (900 mA). These correspond to pumping rates of 1.03, 1.08, 1.15 and 1.22 × P_th using equation (1). This is based on the pump current/power relation, which was determined as linear in this range

$$\begin{eqnarray}&&P/{P}_{{\rm{t}}{\rm{h}}}=\displaystyle \frac{I-{I}_{{\rm{t}}{\rm{h}},{\rm{p}}{\rm{u}}{\rm{m}}{\rm{p}}}}{{I}_{{\rm{t}}{\rm{h}},{\rm{V}}{\rm{C}}{\rm{S}}{\rm{E}}{\rm{L}}}-{I}_{{\rm{t}}{\rm{h}},{\rm{p}}{\rm{u}}{\rm{m}}{\rm{p}}}\,}.\end{eqnarray} \tag{ 1 }$$

Figures 2(c), (d) shows the measured variation of the VCSEL polarisation with that of the pump at the four pumping rates mentioned above; each response was measured twice. The polarisation responses show that for pumping rates close to threshold (red and black curves), the degree of circular polarisation, i.e. the Stokes S3 parameter, switches sharply from near LCP (S3 = −1) to near RCP (S3 = +1) and back for small changes in pump polarisation around pump S3 of zero. As the pumping rate is increased (blue, green curves), the characteristic softens to a more gradual change. Although the generatrend of softening RCP, LCP transition with increased pumping applies, it is seen that the transition is sharpest at 1.08 × P_th (red curve). Here the VCSEL S3 approaches a maximum value at pump S3 of around ±0.15, then falls back slightly. In fact the results at 1.03. and 1.15 × P_th also demonstrate this behaviour, although the effect is subtle in those cases by comparison. Further from pump linearity, i.e. S3 > 0.25, the VCSEL S3 values show a consistently reducing trend with increasing pump rate, which is the expected result.

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The absolute VCSEL S3 valuesear pump S3 = 0 in figures 2(c), (d) is seen to be considerably greater than the absolute pump S3 values. This means there is amplification of the spin i.e. $S{3}_{{\rm{o}}{\rm{u}}{\rm{t}}}/S{3}_{{\rm{i}}{\rm{n}}}\equiv S{3}_{{\rm{V}}{\rm{C}}{\rm{S}}{\rm{E}}{\rm{L}}}/S{3}_{{\rm{p}}{\rm{u}}{\rm{m}}{\rm{p}}}\gt 1.$ The cause of this amplification and sharp switching is the proximity to threshold of the RCP, LCP modes at low pump rates: the LCP or RCP lasing output increases very rapidly as threshold is achieved, whilst the other mode is suppressed. However higher pumping rates soften this change. This rapid lasing transition is achieved despite the spin-up, spin-down electron populations being very similar, even for circular pumping. This is due to the rapid balancing of up, down populations from the spin relaxation time being much shorter (∼1/100) than the electron recombination lifetime. The spin amplification for the polarisation responses in figures 2(c), (d) were calculated, and these are shown in figure 3. The maximum amplification values (which occur closest to pump linearity) increase from approximately 2.8 at the highest pump rate to over 20 at t lowest pump rate (closest to threshold). However away from this trend, the particularly sharp LCP, RCP transition at 1.08 × P_th. discussed above produces very strong amplification, with calculated values there of over 1000 (figure 3(b)). This spin amplification is potentially very useful in applications. It should be noted that to determine accurately the peak amplification very close to pump S3 = 0 requires the pump S3 to be measured very accurately, as any systematic error in the pump S3 will strongly distort the calculated amplification. Here, any systematic errors identified in the measured pump S3 were eliminated prior calculation of the spin amplification, by a correction in the pump S3 of typically 0.02.

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For a practical device there will be growth variations in the different mirror, passive and active material layers across the wafer, and this will affect device parameters such as birefringence, dichroism and photon lifetime. These parameters are defined in appendix A1 and are discussed further in section 3.2. Furthermore there may be varying strain from the anchoring of the fibre on the wafer, which may suppress or excite different lateral modes of the device, noting that there is no lateral mode confinement in the sample and these modes have been shown to have associated polarisation behaviour [24, 25]. Because our wafer is optically pumped with a fibre, imaging of the lateral modes was not possible (as opposed to electrical pumping, where a camera can be positioned to image the output relatively easily e.g. [26]). With these influences in mind, to ascertain that the sharp LCP-RCP transitions and corresponding amplification found above were repeatable, the experiment was repeated in different areas of the wafer, and figure 4 presents typical results. In figure 4(a) for pumping rates from 1.04 to 1.13 × P_th, the polarisation responses show the expected softening of the LCP-RCP transition with increasing pump rate. In figure 4(b) for the pump rate increasing from 1.13 to 1.32 × P_th, the softening is seen to reverse however, with the transition becoming sharpest at the highest pump rate. Comparing the resultant spin amplification rates in figure 4(c), the lowest and highest pumping rates of 1.04 and 1.32 × P_th therefore show the highest amplification, of approximately 8 (black and grey lines there, note the different colour keys in these figures). Here there is no anomalous increase then decrease in $S{3}_{{\rm{V}}{\rm{C}}S{\rm{E}}{\rm{L}}}$ with increasing $S{3}_{{\rm{p}}{\rm{u}}{\rm{m}}{\rm{p}}}$ that was seen at 1.08 × P_th see figures 2(c), (d). In other measurements, we have consistently observed that effect, and have also observed a more consistent softening of the LCP-RCP transition with increasing pump rate. These results demonstrate that good amplification and a sharp LCP-RCP transition can still be achieved consistently across a large sample (5 × 5 mm), despite the different influences on results. The maximum spin amplification of 8 in figure 4, and above 20 in figure 3 compare to a spin amplification corresponding to around 3.8 reported in [5]. It is noted that very high amplification vues are in evidence in [9], achieved with pulsed optical pumping in an 880 nm device.

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A comprehensive model of the simple experimental measurements which have been taken using a small polarised photopumped spot (typically 6–10 μm in diameter) to excite a planar wafer is a complex task because there are a number of competing and interdependent effects. For example, to determine the electromagnetic field behaviour in such a weakly guided system, account must be taken of the spatial distribution of the pump and the resultant distribution of carriers. This feeds into the field solutions both via the gain and refractive index [27] using the SFM (to include polarisation), in a similar way to the approach in the extended SFM that has been used for conventional electrically-pumped VCSELs [24], but now including the polarisation of the pump. It would also be necessary to include thermal lensing effects in such a model to account for the spatial temperature distribution [25]. To do all this accurately in a single self-consistent model requires knowledge of many key parameters, many of which are not well established. Thus in order to gain some insight into the experimental behaviour we make the simplifying assumption that the responses can be modelled using the SFM alone. While clearly limited in applicability this does provide an accessible approach to assessing the main features of the experimental results.

To compare the modelled results with those from experiment, the VCSEL S3 response to pump state was generated using the SFM (see appendix A1 for definition of equations and terms). It has been shown [28] that the polarisation response of the SFM is dependent on the ratio of a number of key parameters, most notably the ratios of birefringence and field decay rates, γ_p/κ, and electron spin relaxation and recombination rates, γ_s/γ. Therefore variation of these two ratios whilst maintaining reasonable individual parameter values in the SFM, and comparing modelled and experiment results is a useful method to initially determine parameters. For a ratio γ_s/γ of 64, a ratio γ_p/κ of 0.00125 gave reasonable agreement experiment (figures 2, 3 results) to model, with further optimisation producing a γ_p/κ value of 0.0014. The γ_s/γ ratio of 64 is in good agreement with values of these two parameters reported in the literature see appendix A2. Calculation of the field decay rate κ for a device is again relatively straightforward and accurate, and a value of 68.5 ns⁻¹ is derived in appendix A2. The remaining parameter of the linear birefringence γ_p (and the linear dichroism, γ_a) is more difficult to predict and must generally be measured for a particular VCSEL. This has been done in e.g. [29, 30], but depends on the lasing linewidth being relatively small (here they are around 0.4 nm or 40 GHz see figure 2(a)) and use of a high resolution spectrometer, which was not available here. Therefore the birefringence was simply derived from the calculated κ at the optimised ratio γ_p/κ of 0.000 14, being 0.1 ns⁻¹ for κ of 68.5 ns⁻¹. This birefringence is low for a VCSEL, but is thought to be reasonable for the residual value found in the planar wafer. A small, positive dichroism value γ_a of +0.1 ns⁻¹ was initially estimated as part of the optimisation process. This fits reasonably with reported values [29, 30]. A somewhat higher dichroism of 0.5 ns⁻¹ was also investigated, and this is detailed below.

Using the SFM parameters described above, the VCSEL S3 values were generated from the SFM for the four pumping rates from figures 2, 3, γ_a of +0.1 ns⁻¹ and the varying pump ellipticity. The SFM pumping rates η are derived from the scaled experimental pumping rates, as described in appendix A2. The SFM results are compared with the experimental S3 values and spin amplification rates (as per figures 2(c), (d) and 3) in figure 5. The SFM S3 values (in red) in figures 5(a), (c), (e), (g) follow the experimental results reasonably well, particularly near pump S3 = 0 where the spin amplifying properties are defined. The corresponding spin amplification rates in figures 5(b), (d), (f), (h) are also reasonably close in experiment and model. For the chosen dichroism in the SFM of 0.1 ns⁻¹, the amplification is somewhat higher at lower pumping rate in the experiment, and lower at higher pumping rates.

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A further comparison experiment to model is shown in figure 6 describing the change in VCSEL ellipticity for a finite pump ellipticity of +0.31. This value of ellipticity was chosen here to reveal the spin amplification properties of the VCSEL. The experimental results were collected for the same wafer area and fibre alignment as per figures 2, 3, 5 results. The VCSEL S3 values in figure 6 are seen to match those values well (at the respective pump rates and pump ellipticity) and this again reveals the spin amplification, as the VCSEL ellipticity remains above 90% from 1.02 to 1.12 × P_th, for the pump ellipticity of only 0.31. The experimental pumping rates have been scaled up to the equivalent SFM pumping rates η here. The SFM results are for γ_a of 0.1 ns⁻¹, and additionally a higher γ_a of 0.5 ns⁻¹, which was mentioned above. The lower value of dichroism is seen to produce a better fit at the lower pumping rates, and the higher dichroism is seen to produce a better fit at the higher pumping rates. These modelled results indicate reasonable boundaries of the suitable dichroism value to model these results in the SFM. As the dichroism could not be measured and verified in this case, the divergence between experiment and model might lie in changes such as increases to birefringence (which will also soften the LCP-RCP transition), dichroism, or indeed excitation of different lateral modes of the wafer. A final comment on the SFM parameter values is that the tuning of 0.18 nm over the pumping range seen in figure 2(a) indicates a change in temperature of the overall VCSEL of around 1.6° using a cavity shift of 0.11 nm per degree [31]. Therefore it is not reasonable here to ascribe the divergence in modelled to experiment results to a reduction in spin lifetime from the temperature-dependent D'Yakanov–Perel effect [32], which might otherwise be posited as a cause.

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In order to explore further the spin amplification and its dependence on parameters, an approximate solution of the SFM equations has been developed. The approach follows that in [33] and the only requirements on the range of parameter values are $k\gg \gamma$ and ${\gamma }_{s}\gg \gamma$ , both which are satisfied by the values used in fitting the experimental results here. Details are provided in [34], giving the derivation of a useful algebraic relation between the normalised pumping rate η, the ellipticity of the pump P ($S{3}_{{\rm{p}}{\rm{u}}{\rm{m}}{\rm{p}}}$) and the ellipticity of the output ε ($S{3}_{{\rm{V}}{\rm{C}}{\rm{S}}{\rm{E}}{\rm{L}}}$). This relation gives excellent agreement with results from numerical solution of the full SFM for plots of P versus ε over the full range from −1 to +1, similar to those in figures 4 and 5. We give here the limiting expression that is valid for the spin amplification at small ellipticities:

$$\begin{eqnarray}&&\varepsilon \approx \displaystyle \frac{P\eta }{\eta -1+\tfrac{{\gamma }_{s}({\gamma }_{a}^{2}+{\gamma }_{p}^{2})}{\gamma \kappa ({\gamma }_{a}+\alpha {\gamma }_{p})}}.\end{eqnarray} \tag{ 2 }$$

The simple form of equation (2) makes it easy to see the dependence of maximum spin amplification (ε/P) on the SFM parameters. Figure 7 gives an example of its application in demonstrating the dependence on total pump rate and dichroism.

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The SFM in terms of the complex linearly polarised fields ${\tilde{E}}_{x},\,{\tilde{E}}_{y}$ is written as in equations (A1.1)–(A1.4) [21], with N as the normalised total spin up and down electron density $({N}_{+}+{N}_{-}),$ m the difference in normalised electron densities $({N}_{+}-{N}_{-}),$ ${\eta }_{+},{\eta }_{-}$ the spin up, down pumping rates, γ the electron recombination lifetime, γ_s the electron spin lifetime, κ the field decay rate, α the linewidth enhancement factor, γ_a the linear dichroism and γ_p the linear birefringence

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\tilde{E}}_{x}}{{\rm{d}}t}=\kappa (1+{\rm{i}}\alpha )[(N-1){\tilde{E}}_{x}+{\rm{i}}m{\tilde{E}}_{y}]-({\gamma }_{a}+{\rm{i}}{\gamma }_{p}){\tilde{E}}_{x},\end{eqnarray} \tag{ A1.1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\tilde{E}}_{y}}{{\rm{d}}t}=\kappa (1+{\rm{i}}\alpha )[(N-1){\tilde{E}}_{x}-{\rm{i}}m{\tilde{E}}_{y}]+({\gamma }_{a}+{\rm{i}}{\gamma }_{p}){\tilde{E}}_{y},\end{eqnarray} \tag{ A1.2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}N}{{\rm{d}}t}=-\gamma [N(1+| {\tilde{E}}_{x}{| }^{2}+| {\tilde{E}}_{y}{| }^{2})-({\eta }_{+}+{\eta }_{-})\\ &&\,\,\,+\,{\rm{i}}m({\tilde{E}}_{y}{\tilde{E}}_{x}^{* }-{\tilde{E}}_{x}{\tilde{E}}_{y}^{* })],\end{eqnarray} \tag{ A1.3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}m}{{\rm{d}}t}=-{\gamma }_{s}m-\gamma [m(| {\tilde{E}}_{x}{| }^{2}+| {\tilde{E}}_{y}{| }^{2})-({\eta }_{+}-{\eta }_{-})\\ &&\,\,\,\,+\,{\rm{i}}N({\tilde{E}}_{y}{\tilde{E}}_{x}^{* }-{\tilde{E}}_{x}{\tilde{E}}_{y}^{* })].\end{eqnarray} \tag{ A1.4 }$$

The threshold modal gain for a laser is derived as per equation (A1.5), with ${g}_{{\rm{t}}{\rm{h}}}$ the threshold gain, ${\alpha }_{{\rm{l}}{\rm{o}}{\rm{s}}{\rm{s}}}$ the material losses, ${\alpha }_{{\rm{m}}{\rm{i}}{\rm{r}}{\rm{r}}{\rm{o}}{\rm{r}}}$ the losses from the mirror transmission out of the cavity, ${L}_{{\rm{c}}{\rm{a}}{\rm{v}}}$ the effective cavity length and ${R}_{B},\,{R}_{t}$ the bottom and top mirror reflectivities

$$\begin{eqnarray}&&{g}_{{\rm{t}}{\rm{h}}}={\alpha }_{{\rm{l}}{\rm{o}}{\rm{s}}{\rm{s}}}+{\alpha }_{{\rm{m}}{\rm{i}}{\rm{r}}{\rm{r}}{\rm{o}}{\rm{r}}}={\alpha }_{{\rm{l}}{\rm{o}}{\rm{s}}{\rm{s}}}-\displaystyle \frac{1}{2{L}_{{\rm{c}}{\rm{a}}{\rm{v}}}}\,\mathrm{ln}({R}_{B}{R}_{t}).\end{eqnarray} \tag{ A1.5 }$$

The material losses ${\alpha }_{{\rm{l}}{\rm{o}}{\rm{s}}{\rm{s}}}$ for GaInNAs have been reported as 700 m⁻¹ [35, 36], and the mirror reflectivities ${R}_{B},\,{R}_{t}$ for this device are 99.2% and 99.8% [7]. The effective cavity length ${L}_{{\rm{c}}{\rm{a}}{\rm{v}}}$ is the 3-λ cavity length [7] of 3.9 μm. With the mirror penetration depths calculated as 0.61 μm [37], the cavity length ${L}_{{\rm{c}}{\rm{a}}{\rm{v}}}$ is then 5.12, giving the threshold gain ${g}_{{\rm{t}}{\rm{h}}}$ as 1644 m⁻¹.

The corresponding field decay rate κ is half the intensity decay rate, as per (A1.6), with c the vacuum speed of light and n_g the group index. Using a group index n_g value of 3.6 from [38] then gives κ as 68.5 ns⁻¹

$$\begin{eqnarray}&&\kappa =\displaystyle \frac{c{g}_{{\rm{t}}{\rm{h}}}}{2{n}_{g}}.\end{eqnarray} \tag{ A1.6 }$$

The Stokes S3 parameter of the pump is defined in terms of the pump rates ${\eta }_{+}+{\eta }_{-}$ as per (A1.7). From this, the pump rates ${\eta }_{+}+{\eta }_{-}$ can easily be derived in terms of the total pump rate $\eta ={\eta }_{+}+{\eta }_{-}$ and the pump S3 polarisation

$$\begin{eqnarray}&&S3=\displaystyle \frac{{\eta }_{+}-{\eta }_{-}}{{\eta }_{+}+{\eta }_{-}}.\end{eqnarray} \tag{ A1.7 }$$

The total pump rate η is scaled from the experimental pump rate above threshold $P/{P}_{{\rm{t}}{\rm{h}}}$ as the ratio of transparency to threshold gain as per (A1.8), giving a 43% over-unity scaling of the pump rate

$$\begin{eqnarray}&&\eta =1+[P/{P}_{{\rm{t}}{\rm{h}}}-1]\left(1+\displaystyle \frac{{\alpha }_{{\rm{l}}{\rm{o}}{\rm{s}}{\rm{s}}}}{{g}_{{\rm{t}}{\rm{h}}}}\right).\end{eqnarray} \tag{ A1.8 }$$

The electron recombination lifetime γ was taken as 1 ns⁻¹ [7]. The electron spin relaxation rate γ_j has been measured as 33 ns⁻¹ in this material, when rapidly thermally-annealed [39]. As the mirror growing process is expected to anneal the active region [40], we have taken this as a reasonable value for our VCSEL. Using the relation between the measured spin relaxation rate γ_j, the recombination rate γ and the SFM spin relaxation rate ${\gamma }_{s}=2{\gamma }_{j}+\gamma$ gives a value for γ_s of 67 ns⁻¹, close to the value of 64 ns⁻¹ that was selected in section 3.2.

Finally, the (Henry) α parameter in GaInNAs quantum well lasers has been calculated in [31], and has also been measured experimentally in [41]. These show close agreement, with a value close to 2.0 at 1300 nm. It should be noted that both show a slightly reduced value of around 1.8 at 1290 nm, and significantly larger values at 1310 nm. As the value is sensitive to the exact material composition [31], 2.0 was chosen as a suitable value for this work.

The Stokes parameters S1–S3 can be generated from the fields ${\tilde{E}}_{x},\,{\tilde{E}}_{y}$ using equations (A1.5)–(A1.8) from [42] chapter 6, although there are a number of equivalent forms for this. In (A1.11), (A1.12) ${\tilde{E}}_{x}^{* }$ denotes the complex conjugate of the complex x—polarised field

$$\begin{eqnarray}&&S0={| {\tilde{E}}_{x}| }^{2}+{| {\tilde{E}}_{y}| }^{2},\end{eqnarray} \tag{ A1.9 }$$

$$\begin{eqnarray}&&S1=\displaystyle \frac{{| {\tilde{E}}_{x}| }^{2}-{| {\tilde{E}}_{y}| }^{2}}{S0},\end{eqnarray} \tag{ A1.10 }$$

$$\begin{eqnarray}&&S2=\displaystyle \frac{2{\rm{R}}{\rm{e}}({\tilde{E}}_{x}^{* }{\tilde{E}}_{y})}{S0},\end{eqnarray} \tag{ A1.11 }$$

$$\begin{eqnarray}&&S3=\displaystyle \frac{2{\rm{I}}{\rm{m}}({\tilde{E}}_{x}^{* }{\tilde{E}}_{y})}{S0}.\end{eqnarray} \tag{ A1.12 }$$