Estimate of the Nonlinear Coupling Between Circular Eigenstates in a Spin-VCSEL

Spintronic devices such as spin-LEDs and spin-lasers have raised great interest as they allow encoding the electronic spin state in the output optical polarization offering new applications in telecommunications and quantum technologies [1]. Despite important advances in controlling the polarization switching [2], there are still limiting factors, such as the required level of spin injected electrons needed to induce these switches. In a laser, this crucial aspect is leveraged by the coupling strength between the two circularly polarized eigenmodes, which is conveniently quantified by the Lamb's coupling constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C$</tex> [3]. When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C &lt; 1$</tex>, simultaneous oscillation of the two polarizations eigenstates is possible. Conversely, when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C &gt; 1$</tex>, the laser is bi-stable and a hysteresis cycle appears. The closer to 1 is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C$</tex>, the lower electronic spin unbalance is required to flip the polarization. In this work, we estimate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C$</tex> for circularly polarized eigenstates, for the first time to the best of our knowledge [5]. The experimental setup is shown in Fig. 1(a). Our gain structure, a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1 / 2$</tex> -VCSEL (Vertical Cavity Surface Emitting Laser) emitting around 1550 nm, is pumped by a fibred single-mode 980 nm diode. A bi-convex lens with a focal length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f_3=20$</tex> mm is inserted to guarantee the optical stability of our plan-plan 11.3 cm-length cavity. A 10 mm-thick AR-coated <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{Y}\mathrm{V}\mathrm{O}_4$</tex> birefringent crystal, cut at 45° from its optical axis, is inserted inside the cavity to obtain a spatial separation, on the output mirror side, of 1 mm between the two orthogonally polarized eigenmodes. To predict the laser architecture ending with superimposed circular polarizations in the active medium, we rely on the use of generalized [4×4] Jones formalism associated with the field resonant condition [4]. It turns out that a quarter wave-plate and a 45°-rotation Faraday rotator have to be inserted between the gain chip and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{Y}\mathrm{V}\mathrm{O}_4$</tex>. Razor blades are mounted on two piezo-actuators in order to apply diffraction losses independently on each mode.