Image size magnification and power and dilation factors for optical instruments in general

Traditional treatments of spectacle magnification for distant objects consider only stigmatic spectacle lenses and they compare the retinal image size in a refractively fully compensated eye with the image size in the uncompensated eye. Spectacle magnification is expressed as a product of two factors, the power and shape factors of the lens. The power factor depends on the position of the entrance pupil of the eye. For an eye with an astigmatic cornea, however, the position of the entrance pupil is not well defined. Thus, the traditional approach to spectacle magnification does not generalize properly to allow for astigmatism. Within the constraints of linear optics and subject to the restriction that the eye's iris remains the aperture stop, this paper provides a complete, unified and exact treatment for optical instruments in general. It compares retinal image size in a generalized sense (including image shape and orientation) for any instrument in front of an eye with that of the eye alone irrespective of whether the instrument compensates or not. The approach does not make use of the concept of the entrance pupil at all and it allows for astigmatism and for non-alignment of refracting elements in the instrument and in the eye. The concept of spectacle magnification generalizes to the concept of instrument size magnification. Instrument size magnification can be expressed as the product of two matrix factors one of which can be interpreted as a power factor (as back-vertex power) and the other factor for which the name dilation factor is more appropriate in general. The general treatment is then applied to a number of special cases including afocal instruments, spectacle lenses (including obliquely crossing thick bitoric lenses), contact lenses, stigmatic systems and stigmatic eyes. In the case of spectacle lenses, the dilation factor reduces to the usual shape factor.