Cardinal point (optics)

In geometrical optics, cardinal points are a set of points that define the imaging properties of an optical system, such as a lens or a system of lenses. These points, together with the focal length(s), uniquely determine the location and size of the image formed by the system for any given object. The most commonly used cardinal points are the two focal points, the two principal points, and the two nodal points.

• Focal Points (F and F'): The front focal point (F) is the object point for which the image is formed at infinity. The rear focal point (F') is the image point for an object at infinity. In other words, parallel rays entering the system from the left converge at F', and parallel rays entering from the right converge at F.

• Principal Points (P and P'): The principal points are the points where the principal planes intersect the optical axis. The principal planes are imaginary surfaces perpendicular to the optical axis. A ray aimed at the front principal point (P) will emerge from the rear principal point (P') at the same height from the axis and at the same angle relative to the axis. In thin lens approximation, the principal points coincide at the center of the lens.

• Nodal Points (N and N'): The nodal points are a pair of points on the optical axis such that a ray directed at the front nodal point (N) will emerge from the rear nodal point (N') at the same angle to the axis, effectively shifting the ray parallel to itself. If the refractive index on both sides of the optical system is the same (e.g., air), then the nodal points coincide with the principal points.

The distances between the principal points and the focal points determine the focal lengths of the optical system. Specifically, the distance from the rear principal point (P') to the rear focal point (F') is the rear focal length, and the distance from the front principal point (P) to the front focal point (F) is the front focal length. These focal lengths are crucial for calculating image magnification and location. The knowledge of the cardinal points is essential for understanding and designing optical systems, enabling the prediction of image formation characteristics.