Prove that $f=0$ if $\varphi(y)=\int_\mathbb{R} |f(x+iy)|^2 d x$ is a bounded function.

Question

Suppose that $f$ is an entire function of exponential type and \begin{equation} \varphi(y)=\int_{-\infty}^\infty |f(x+iy)|^2 d x \end{equation} is a bounded function on $\mathbb{R}$. Prove that $f=0$. The first theorem came to my mind is the Paley-Wiener Theorem which implies the existence of an $F \in L^2(0,\infty)$ such that \begin{equation} f(z)=\int_0^\infty F(t) e^{itz} dt \end{equation} in the upper half plane. However, I can't go further. Please comment! Thanks!