Diophantine Optics

LESIA, Observatoire de Paris, CNRS, UPMC, Université Paris-Diderot, Observatoire, 92195 Meudon Cedex, France

e-mail: daniel.rouan@obspm.fr

Abstract

What I call Diophantine optics is the exploitation in optics of some remarkable algebraic relations between powers of integers. The name comes from Diophantus of Alexandria, a greek mathematician, known as the father of algebra. He studied polynomial equations with integer coefficients and integer solutions, called diophantine equations. Since constructive or destructive interferences are playing with optical path differences which are multiple integer (odd or even) of λ/2 and that the complex amplitude is a highly non-linear function of the optical path difference (or equivalently of the phase), one can understand that any Taylor development of this amplitude implies powers of integers. This is the link with Diophantine equations. We show how, especially in the field of interferometry, remarkable relations between powers of integers can help to solve several problems, such as achromatization of a phase shifter or deep nulling efficiency. It appears that all the research that was conducted in this frame of thinking, relates to the field of detection of exoplanets, a very active domain of astrophysics today.