Deterministic Turing Machine with infine tape in both directions
You can design a Turing Machine that goes one step left, then two right, then 3 left and so on...
You don't need too much states for that: just put two markers on the tape: $b$ for the current left boundary and $c$ for the current right boundary of what you explored, and push them back from one cell each time you reach them.
You have one state $q_b$ going left until $b$ is reached, and one state $q_c$ going to the right until $c$ is reached. Plus special states to shift them by one position.

You will eventually find the $a$ letter, at which point you just go to $q_f$.