GRIN lens description on Hamilton's phase plane

Abstract

Gradient index (GRIN) lenses are used both in imaging and beam shaping optical systems. Usually, to design a GRIN lens system, we would find the equivalent lens effective focal length and the principal planes position. Then if we design an imaging system the Newton formula gives the image position or, if we design a beam shaping system the Kogelnik's ABCD law gives the Gaussian beam waist size and position. The Hamilton's phase plane approach gives the same result as these sets of formulas, but is more spectacular and clearer. Hamilton's phase plane approach gives a vivid and clear illustration of GRIN lens imaging and beam shaping properties and does not require substitution by an equivalent lens. Hamilton's phase plane approach allows easy derivation of transparent and simple direct formulas for a GRIN lens created image and beam transformation, including Gaussian beams and multimode fiber beams. These formulas and graphical illustrations facilitate solving, and give a clearer understanding of practical problems.