Spherical mirrors

We call spherical mirror any spherical cap that is polished and has high reflecting power.

It is easy to see that the sphere of which the above cap is part has two faces, one internal and one external. When the reflective surface considered is the inner one, the mirror is called concave. In cases where the reflective face is external, the mirror is called convex.

Light reflection in spherical mirrors

As for flat mirrors, the two laws of reflection are also obeyed in spherical mirrors, that is, the angles of incidence and reflection are equal, and the rays incident, reflected, and the line normal to the incident point.

Geometric aspects of spherical mirrors

For the study of spherical mirrors it is useful to know the elements that compose them, outlined in the figure below:

  • Ç it's the center of the sphere;
  • V it's the vertex of the hubcap;
  • The axis that goes through the center and the vertex of the cap is called main axis.
  • The other lines that cross the center of the sphere are called secondary axes.
  • The angle , which measures the angular distance between the two minor axes crossing the two outermost points of the cap, is the opening from the mirror.
  • The radius of the sphere R that originates the hubcap is called bend radii from the mirror.

An optical system that can conjugate to an object point, a single point as an image is called stigmatic. Spherical mirrors are usually neither stigmatic nor applanetic or orthoscopic like flat mirrors.

However, spherical mirrors are only stigmatic for rays that strike close to their vertex V and with a slight inclination to the main axis. A mirror with these properties is known as a Gauss mirror.

A mirror that does not satisfy Gauss conditions (near-vertex incidence and small inclination in relation to the main axis) is called astigmatic. An astigmatic mirror conjugates to a point an image that looks like a stain.

Spherical mirror spotlights

For the mirrors concave From Gauss, it can be seen that all light rays that strike along a direction parallel to the secondary axis pass through (or converge on) the same point F - the main focus of the mirror.

In the case of convex mirrors, the continuation of the reflected ray passes through the focus. Everything happens as if the reflected rays originate from the focus.