Absolute instruments (AIs) in optics mean devices that bring stigmatically an infinite number of light rays from a source to its image, which can perform perfect imaging in the perspective of geometrical optics [1,2]. Two well-known examples of AIs are a plane mirror and Maxwell’s fish-eye lens [with gradient refractive index profile; see Fig. 1(a)]. Actually, there are a lot of AIs, such as the Eaton lens, Luneburg lens, and invisible lens . The invisible lens has a spherically symmetric index profile that forces light rays to make loops around its center and then propagate in their original directions, which makes it invisible. Recently, one author proposed a general method to design AIs with the help of the Hamilton–Jacobi equation , which has flourished in the family of AIs. No matter how perfectly stigmatic the geometrical-optics image might be, in the wave-optics regime, the resolution is always limited by diffraction. Owing to this limitation, “perfect imaging” in the perspective of geometrical optics and wave optics is quite different . As far as we know, the only AI in both perspectives is Pendry’s slab . However, the frequency spectrum of other AIs has been investigated by numerical method [6,7] and the Wentzel–Kramers–Brillouin (WKB) approximation [8,9]. It is found that their spectrum is (at least approximately) degenerate and equidistant, which contributes to periodical evolution of waves in AIs [6,8].