Purpose: The purpose of this study is to assess the impact of decentration of astigmatic intra-ocular lenses on the residual refraction after cataract surgery, using a computing scheme with 5 x 5 system matrices.
Methods: Based on the definition of an optical system in the paraxial Gaussian space containing astigmatic surfaces without restrictions to coaxiality, we derived a method (using 5 x 5 refraction and translation matrices) for calculating the residual refraction and the compensating prism in the spectacle plane after decentred implantation of thin and thick astigmatic intra-ocular lenses. The 'optical system eye' may contain astigmatic refractive surfaces with their axes at random.
Results: The capabilities of this computing scheme are demonstrated with two examples. In example 1 we calculate the residual refraction of a decentred 'thin astigmatic lens' for compensation of corneal astigmatism to achieve a spherical target refraction. In example 2 we compute the residual refraction after implantation of a 'thick astigmatic lens', where the spherical and cylindrical power as well as the implantation axis of the lens do not fully match the pre-operative recommendations and the lens is decentred relative to the optical axis. For both examples, we derive the residual prismatic effect in the spectacle plane and the lateral displacement of a ray exiting the spectacle correction when starting coaxially at the retina.
Conclusions: We have presented an en bloc matrix-based strategy for the calculation of the residual spherocylindrical refraction at the spectacle plane after implantation of a decentred thin or thick astigmatic intra-ocular lens without restrictions to coaxiality. The resulting system matrix is written as a product of 5 x 5 refraction and translation matrices.