  In 2005 Bourgain gave the first explicit construction of a two-source
extractor family with min-entropy rate less than $1/2$. His approach combined
Fourier analysis with innovative but inefficient tools from arithmetic
combinatorics and yielded an unspecified min-entropy rate which was greater
than $.499$. This remained essentially the state of the art until a 2015
breakthrough of Chattopadhyay and Zuckerman in which they gave an alternative
approach which produced extractors with arbitrarily small min-entropy rate. In
the current work, we revisit the Fourier analytic approach. We give an improved
analysis of one of Bourgain's extractors which shows that it in fact extracts
from sources with min-entropy rate near $\frac{21}{44} =.477\ldots$, moreover
we construct a variant of this extractor which we show extracts from sources
with min-entropy rate near $4/9 $ = $.444\ldots$. While this min-entropy rate
is inferior to Chattopadhyay and Zuckerman's construction, our extractors have
the advantage of exponential small error which is important in some
applications. The key ingredient in these arguments is recent progress
connected to the restriction theory of the finite field paraboloid by Rudnev
and Shkredov. This in turn relies on a Rudnev's point-plane incidence estimate,
which in turn relies on Koll\'ar's generalization of the Guth-Katz incidence
theorem.
