Purity of Vector Vortex Beams through a Birefringent Amplifier

Creating high-quality vector vortex (VV) beams is possible with a myriad of techniques at low power, and while a few studies have produced such beams at high power, none have considered the impact of amplification on the vector purity. Here we employ tools to study the amplification of VV beams and, in particular, the purity of such modes. We outline a versatile toolbox for such investigations and demonstrate its use in the general case of VV beams through a birefringent gain medium. Intriguingly, we show that it is possible to enhance the purity of such beams during amplification, paving the way for high-brightness VV beams, a requirement for their use in high-power applications such as optical communication and laser-enabled manufacturing.

Figure 1

Illustration of (a) a HOPS representing VV modes in the left and right polarization basis and (b) a modified HOPS representing the VV modes in the horizontal and vertical basis. The output beam profile is displayed at different positions on the sphere with arrows indicating polarization distribution.

Figure 2

(a) The vector quality factor $V_{\mathrm{out}}$, as a function of crystal angle $\theta$, for three cases: $\alpha =0.5$ and ${\eta }_R=2$ (blue), $\alpha =0.8$ and ${\eta }_R=0.1$ (orange), and $\alpha =0.4$ and ${\eta }_R=0.1$ (green). The 3D plots of the parameter space show the richness of the theory, with the input state fixed in (b) and (c) at $\alpha =0.5$ and $\alpha =0.7$, respectively. In (d) and (e), the amplification factor is fixed at ${\eta }_R=0.2$ and ${\eta }_R=1$, respectively.

Figure 3

The fundamental Gaussian mode of the source laser is modified in a generation step with a polarizer (Pol) and quarter wave plate (QWP) and passed through a $q$ plate ($q_{1/2}$) to produce vector beams of variable purity, from 0 to 1. The resultant beam is passed through the amplification stage with half wave plates (HWP3 and HWP4) dove prisms (DP) on either side of the crystal gain medium in order to rotate the beam relative to the crystal axes (effectively rotating the crystal). Finally, the amplified beam is passed through a detection system to determine the VQF by a state tomography: After passing through a polarizing beam splitter (PBS), each component of the field is measured in the OAM basis by holograms encoded on a spatial light modulator (SLM). The outputs are measured in the Fourier plane using a charge-coupled device (CCD) camera.

Figure 4

Illustration of the actual intensity measurements $I_{ij}$, corresponding to (a) a perfect input VV beam with $V=1$ and (b) an amplified VV beam with $V=0.7$.

Figure 5

Evolution of the horizontal and vertical amplification factors ${\eta }_h$ (blue curve) and ${\eta }_v$ (red curve) as a function of the orientation of the $\mathrm{Nd}:\mathrm{YLF}$ crystal, $\theta$. The points show the experimental data with the theoretical curves overlaid. The initial amplification factors at $\theta =0$ are taken as our two parameters ${\eta }_{h0}$ and ${\eta }_{v0}$ for all calculations.

Figure 6

Purity of the output beam over different orientations of the $\mathrm{Nd}:\mathrm{YLF}$ amplifier for an input purity of $V=1$. The theory is shown as the solid curve together with the experimental data points. The theory predicts $V=0.99$ and independent of angle, which is corroborated by the experiment. The inset, an enlarged data sequence, shows a slight modulation due to a systematic experimental uncertainty in the form of rotation-dependent misalignment, an error on the order of 1%.

Figure 7

(a) The vector purity of two VV beams, $U_1$ (blue) and $U_2$ (red), as the crystal axis is rotated (for identical $V_{\mathrm{in}}=0.5$). The purity oscillates about the initial value of 0.5 (black solid line) but in some cases is improved due to equalization of the modal weightings. (b) The same oscillation in purity is plotted on the HOPS as trajectories for the two beams, show in red and blue.

Figure 8

The change in the purity of the amplified beam ($V_{\mathrm{out}}$) with respect to the purity of the injected seed beam ($V_{\mathrm{in}}$). By adjusting the QWP, the state of the input field continuously changes. The purity of the amplified beam $V_{\mathrm{out}}$ is calculated for QWP angle $\theta \in [0,45]$ (blue dots) and $\theta \in [0,-45]$ (red dots). The experimental data are plotted against the theoretical predictions (solid lines), showing excellent agreement.

Figure 9

The output power for the CVV beams as a function of the pump power for a given injected power of 164 mW. Point $A$ represents the threshold point for the amplifier.