The paraxially traced marginal ray (thick line) has before the first surface of the group of L lenses the angle $u_1=\alpha$with the optical axis and after the last surface the angle $u_{2L+1}=\beta$. The refractive index before surface k is $n_k$, after the surface it is $n_{k+1}$.The surface numbering for the marginal ray angles $u_k$is the same. Inside each lens (i.e. for even k values) the refractive index is n, as shown here for the first lens (m=1) with surfaces 1 and 2. Outside the lenses we have $n_1=n_3=...=n_{2L+1}=1$. In this figure, the L lenses of interest form the entire optical system, but the same notation is used when these lenses are part of a larger system. The planes of the object, image, entrance pupil and exit pupil are denoted by O, I, EP and XP respectively. In the thin-lens approximation, all axial distances between surfaces 1 and 2L will be set equal to zero in the aberration formulas.