# Vergence (optics)

In geometrical optics , vergence is the reciprocal of the radius of curvature of a wave front . It is given in diopters (1 / m).

For a ( divergent ) bundle of rays emanating from a point , the wave fronts are spherical and the vergence is the negative reciprocal of the distance r from the starting point:

${\ displaystyle V = - {\ frac {1} {r}} \ ,.}$
The converging lens collimates the divergent beam.

If this bundle hits a converging lens and the starting point is at the focal point of the lens, the vergence is immediately in front of the lens

${\ displaystyle V = - {\ frac {1} {f}} \,}$   ( is the focal length of the lens ).${\ displaystyle f}$

The lens collimates the light , the rays become parallel, the wave fronts flat , the vergence zero.

The change in vergence, from to , is the power D of the lens. It is positive for converging lenses, negative for diverging lenses and is also given in diopters. ${\ displaystyle V = - {\ frac {1} {f}} \,}$${\ displaystyle V = 0 \,}$

In the opposite direction, a plane wave becomes a convergent bundle.

Conversely, if a plane wave ( , parallel rays) hits a converging lens, the bundle receives the positive refractive power D of the lens as vergence: immediately behind the lens applies ${\ displaystyle V = 0 \,}$

${\ displaystyle V = D = + {\ frac {1} {f '}} \ ,.}$

Positive vergence means convergence: the rays of the bundle run towards a focus. That is at a distance behind the lens. ${\ displaystyle f '\,}$

Curvature of the wave fronts when passing through a focus, geometric-optical (red) or wave-optical (blue).

As the focus approaches, the vergence increases faster and faster. The concept of vergence only makes sense in dimensions far above the wavelength . At a focus, vergence has a mathematical pole , that is, it diverges and changes its sign : Towards the focal point it would be positive infinite, then negative infinite. With the Gaussian beam of wave optics, on the other hand, the curvature of the wave fronts initially increases with vergence, but becomes smaller again close to the focus. As it passes through the focus, the curvature changes steadily from positive to negative, i.e. H. The wave fronts are flat directly at the intersection of the rays of geometric optics . Sufficiently far behind the focus, the vergence again coincides with the curvature of the wave fronts (both negative).