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The geometric optics approximation provides an interpretation for eikonalcorrespondence that, in black-hole-containing spacetimes, connectshigh-frequency black hole quasinormal modes with closed photon orbits aroundsaid black hole. This correspondence has been identified explicitly forSchwarzschild, Reissner-Nordstr\"om, Kerr, and Kerr-Newman black holes, theviolation of which can be a potential hint toward physics beyond GeneralRelativity. Notably, the aforementioned black hole spacetimes have sufficientsymmetries such that both the geodesic equations and the master wave equationsare separable. The identification of the correspondence seems to largely relyon these symmetries. One naturally asks how the eikonal correspondence wouldappear if the spacetime were less symmetric. For a pioneering work in thisdirection, we consider in this paper a deformed Schwarzschild spacetimeretaining only axisymmetry and stationarity. We show that up to the first orderof spacetime deformations the eikonal correspondence manifests through thedefinition of the \textit{averaged} radius of trapped photon orbits along theirone period. This averaged radius overlaps the potential peak in the master waveequation, which can be defined up to the first order of spacetime deformations,allowing the explicit identification of the eikonal correspondence.

Eikonal quasinormal modes and photon orbits of deformed Schwarzschild black holes