2. Let the function be
$$
f(x)=\sqrt{\ln x+x^{2}-a}(a \in \mathbf{R}) .
$$

If there exists $b \in[1, \mathrm{e}]$, such that $f(f(b))=b$, then the range of the real number $a$ is . $\qquad$
2. $a \in [0,1]$.

Given that there exists $b \in [1, \mathrm{e}]$ such that $f(f(b))=b$, we get $f(b)=b$.
Thus, there exists $x \in [1, \mathrm{e}]$ such that
$$
\sqrt{\ln x + x^2 - a} = x,
$$

which implies $a = \ln x$ for $x \in [1, \mathbf{e}]$.
Therefore, $a \in [0,1]$.