Relation between the QR decomposition of matrix A and the QR of the augmented version of A

Question

suppose I have a matrix $A_{1}\in C^{m*n}, m\geqslant n$ so the QR decomposition of $A_{1}$ is $A_{1}=Q_{1}*R_{1}$. Now, define the augmented matrix $A_{2}=\begin{bmatrix} A_{1}\\ I_{n*n} \end{bmatrix}$ where $I_{n*n}$ is $n*n$ identity matrix, so the QR decomposition of $A_{2}$ is $A_{2}=Q_{2}*R_{2}$.
Is there any relation between $Q_{1}$ and $Q_{2}$, $R_{1}$ and $R_{2}$, and especially between the diagonal elements of $R_{1}$ and $R_{2}$ knowing that $det(I_{n*n}+A_{1}^{H}*A_{1})=det(A_{2}^{H}*A_{2})$.