  We present a new method for determining the frequencies and wavefunctions of
black hole quasinormal modes (QNMs) and Regge poles. The key idea is a novel
ansatz for the wavefunction, which relates the high-$l$ wavefunctions to null
geodesics which start at infinity and end in perpetual orbit on the photon
sphere. Our ansatz leads naturally to the expansion of QNMs in inverse powers
of angular momentum $L = l + 1/2$ (in 4D), and to the expansion of Regge poles
in inverse powers of frequency. The expansions can be taken to high orders. We
begin by applying the method to the Schwarzschild spacetime, and validate our
results against existing numerical and WKB methods. Next, we generalise the
method to treat static spherically-symmetric spacetimes of arbitrary spatial
dimension. We confirm that, at lowest order, the real and imaginary components
of the QNM frequency are related to the orbital frequency and the Lyapunov
exponent for geodesics at the unstable orbit. We apply the method to five
spacetimes of current interest, and conclude with a discussion of the
advantages and limitations of the new approach, and its practical applications.
