Condition of Optical Systems Independent of Frequency for Wide Field-of-View Radio Telescopes

In this paper, we present a condition of optical systems to form images independent of frequency. The condition independent of frequency in case of a single Gaussian beam on an optical axis has been known, but that in case of a beam propagated off the axis or tilted beam, which appears in a wide field-of-view (FOV) telescope, has not been known. We first show relations between an arbitrary electric field on an object plane and an induced one on an image plane after passing through a lens with calculating Fresnel diffraction integrals. If the lens formula is met, there is a one-to-one correspondence between points on the object and image plane. This result enables to use geometrical optical methods. The condition and relations derived here are confirmed by simulations. We also applied them to a wide-FOV telescope.