To accurately predict the amplified 2D beam profiles based on the given input profiles, we conducted a comprehensive study on modal gain. In order to achieve this, we developed a 2D gain model that was formulated as shown in Figure 5. According to this model, the amplified profile So(r) is the product of the input seed profile Si(r), the radial gain profile G0(r), and the radial coefficient C(r).
To determine the radial coefficient C(r) as a function of the radius, we performed curve fits using measured data for the seed profile, radial gain profile, and amplified profile. The fitting process allowed us to calculate the radial coefficient C(r) accurately. Figure 6 illustrates an example of a measured seed beam profile S(r) and the corresponding gain profile G0(r) at the 50 mm amplifier equivalent plane.
To simplify the analysis, we assumed that the beam profiles exhibit cylindrical symmetry. Therefore, we fitted the azimuthally averaged beam profile. For the input and output beam profiles, we employed a super-Gaussian fit, while the radial gain profiles were modeled using a fourth-degree polynomial function. The polynomial function decayed inward from the perimeter, following the principles of Beer's Law absorption of the pump light.
By employing this methodology, we were able to accurately estimate the input profile by reverse calculation, considering a top-hat output beam profile. Remarkably, the results exhibited a strong agreement with the current shaped seed profile, validating the efficacy of our approach.
Figure 5. Gain model: the amplified profile, So(r), is a product of input profile, Si(r), radial gain profile, G0(r), and radial coefficient, C(r).
Figure 6 Fits for the seed beam profile and small signal grain profile for the 50 mm rod amplifier. (a) Measured 2D seed beam profile S(r) at 50 mm near-field equivalent plane. (b) Measured 2D gain profile G0(r) (c) Lineout and fit to measured seed beam profile. (d) Lineout and fit to measured radial gain profile.