Motivated by the example of surreal numbers, we introduce a particular class of real closed valued fields, the omega fields, for which the valuation is encoded by a so-called omega map. In particular, we investigate them in the key situation of bounded power series fields (over the reals, or more generally any model of $\mathbb{R}_{\textrm{an},\exp}$). We show that every such omega field of bounded series admits many exponential functions, some of them making it into a model of $\mathbb{R}_{\textrm{an},\exp}$, and others that are not even o-minimal. On the other hand, there are exponential bounded series fields which are not omega-fields. This is obtained using a key formula linking the omega map and the logarithm, and by the construction of general examples.