a) Overlap integral given by $\eta=\int_{-\infty}^{\infty} \Psi_{m^{'}} ^{output} \Psi_{m}^{input} dx$ The paper by tong et al, explains that they adjusted the overlap until the output is maximized. I think that for that occur then say the input was a gaussian $$E=E(0)e^{ax^2}$$ where $$E(0)$$ would be a central maximum, then overlap adjusted until the output was $$E(0)$$, making the overlap integral just integrating a gaussian.
b) Using the mode solver for this part. I wanted to see how modes would look after $$LP_{01}$$, not single moded. So I went back to the paper and looked at the equation for the diameters that allow for single mode operation, $$D< \frac{2.4 \lambda}{\pi \sqrt{n_0^2-n_1^2}}$$, where $$n_1$$ is 1 for air, and $$n_0$$ is the index of refraction for the medium in use. From the tong paper, the second page of the paper, or pg.817 of the nature journal it was published in, index of refraction for Silica is said to be $$n_0=1.46$$. This gives $$NA=1.063$$, using $$\lambda=633 nm$$, the max diameter is then given to be 454.6 nm for single mode operation. So I will put diameters larger then this for non single modes. Using 600 nm for the diameter. The image generated looks just like two very sharp gaussians