Intrinsic symmetry of refracting profiles derived from a parabola

The intrinsic symmetry of kinoform refracting profiles, which determines the possible types of the arrangement of parabolic segments constituting a planar lens on a plane, is considered. It is shown that the intrinsic symmetry of a compound kinoform profile is determined by a symmetry group G consisting of two subgroups G = G _t ? G _m . The elements of subgroup G _m = {E, m ₁, m ₂, m ₃} are symmetry planes such that m ₁ and m ₃ correspond to the reflection of the kinoform profile into itself and m ₂ is a black-white antisymmetry plane. A rule is stated for the assembling of segments in a kinoform lens following from the condition of the conservation of the focal length. For the first time, a subgroup of permutations is introduced into consideration such that its action is reduced to a permutation among single profiles within the common compound set. It is shown that the existence of subgroup G _t determines the properties of lenses that are observed experimentally in the spectral dependences of the lens gain in the focal spot.