  We propose a new approach for high-contrast imaging at the diffraction limit
using segmented telescopes in a modest observation bandwidth. This concept,
named "spectroscopic fourth-order coronagraphy", is based on a fourth-order
coronagraph with a focal-plane mask that modulates the complex amplitude of the
Airy disk along one direction. While coronagraphs applying the complex
amplitude mask can achieve the theoretical limit performance for any arbitrary
pupils, the focal plane mask severely limits the bandwidth. Here, focusing on
the fact that the focal-plane mask modulates the complex amplitude along one
direction, we noticed that the mask can be optimized for each spectral element
generated by a spectrograph. We combine the fourth-order coronagraph with two
spectrographs to produce a stellar spectrum on the focal plane and reconstruct
a white pupil on the Lyot stop. Based on the wavefront analysis of an optical
design applying an Offner-type imaging spectrograph, we found that the
achievable contrast of this concept is 10^{-10} at 1.2 - 1.5 times the
diffraction limit over the wavelength range of 650 - 750 nm for the entrance
pupil of the LUVOIR telescope. Thus, this coronagraph concept could bring new
habitable planet candidates not only around G- and K-type stars beyond 20 - 30
pc but also around very nearby M-type stars. This approach potentially promotes
the characterization of the atmospheres of nearby terrestrial planets with
future on- and off-axis segmented large telescopes.
