First-order optics: cardinal points

The previous article laid the foundations of geometrical optics: the nature of light, the electromagnetic spectrum and Snell's law. With that in place, it's time for the next step: how to predict where an image forms and how big it is, without tracing every ray surface by surface. In optical system design, before tackling aberrations or manufacturing, it's essential to establish the system's basic architecture. First-order optics —also called Gaussian optics— describes the behaviour of "perfect" optical systems in the paraxial region: an infinitesimally thin strip around the optical axis where every angle is small enough that its sine and tangent can be set equal to the angle itself, in radians.

sin θ ≈ tan θ ≈ θ

It sounds like an excessive simplification, but a well-corrected system follows these laws almost exactly —which makes these calculations the foundation of any serious optical design.

The black-box concept and the cardinal points

A complex optical system, made up of several lenses and mirrors, can be treated as a black box whose image-forming properties are fully defined by its cardinal points. Knowing only the entry and exit points, these let you calculate the path of light through the whole system without worrying about the refractions happening in between.

Fig. 1 — The system is treated as a black box: the cardinal points —F₁, P₁, P₂, F₂— are enough to follow the ray from entry to exit, without knowing the internal surfaces.

There are three fundamental pairs of cardinal points:

Focal points (F₁, F₂). Rays entering the system parallel to the optical axis —coming from an object at infinity— cross at the second focal point F₂. Conversely, rays passing through the first focal point F₁ emerge parallel to the axis.
Principal points (P₁, P₂). These are where the principal planes intersect the optical axis. Those planes have unit magnification: a ray appears to enter the system at some height y above the first principal plane, and appears to leave the second principal plane at that same height y.
Nodal points (N₁, N₂). A ray aimed at the first nodal point emerges from the system as if it came from the second nodal point, at the same angle it went in with. Key fact for the engineer: when the system sits in air —the most common case— the nodal points coincide exactly with the principal points.

Focal length definitions

These points define the distances that actually matter for mechanical and optical design:

Effective focal length (EFL). The distance from the principal point to the corresponding focal point (f or f′). It's the value that defines the system's power.
Back focal length (BFL). The distance from the vertex of the lens's last physical surface to the second focal point. Vital for making sure there's room for a sensor or prisms behind the lens.
Front focal length (FFL). The distance from the first physical surface to the first focal point.

Fig. 2 — EFL, BFL and FFL: EFL is measured from the principal plane, while BFL and FFL are measured from the lens's physical vertex —the dimension that actually limits available mechanical space.

Fundamental equations for locating images

To avoid tracing rays surface by surface, the engineer relies on two mathematical forms of the same law.

Newton's equation

Ideal when the distances are known relative to the foci. If x is the distance from the object to the first focus and x′ the distance from the image to the second focus:

x · x′ = −f · f′

In air, where f = f′, the equation simplifies to x′ = −f² / x.

The Gauss equation

The form most used in practice: it measures the distances s (object) and s′ (image) from the principal planes.

1/s′ = 1/f + 1/s

Following the usual sign convention, an object placed to the left of the lens gives a negative value of s.

Fig. 3 — Two rays are enough to locate the image: the one parallel to the axis that passes through F₂, and the one through the centre that isn't deviated. The image is inverted and real.

Lateral and longitudinal magnification

Lateral magnification (m) is the ratio between the image size (h′) and the object size (h):

m = h′/h = s′/s

A negative value of m indicates an inverted (real) image; a positive value indicates an upright (virtual) image.

Longitudinal magnification (m_L), meanwhile, describes how much an object's thickness is magnified along the optical axis, and relates to the lateral one through the approximation:

m_L ≈ m²

In other words, if a system has a lateral magnification of 2, an object with depth will appear 4 times deeper in image space —a common trap for anyone designing depth relays or 3D systems.

Fig. 4 — Lateral magnification: the image comes out inverted and twice as tall (m = −2). Longitudinal magnification grows with the square of m: the same scene, four times deeper in image space.

Sign convention: the engineering standard

For these equations to work without ambiguity, the engineer follows a strict convention:

Light travels from left to right.
Distances measured to the right of a reference point are positive; to the left, negative.
Heights above the optical axis are positive.
The focal length of a converging lens is positive, and that of a diverging one is negative.

Half a dozen points and two equations are enough to lay out a telescope without tracing a single ray through a single lens.

From black box to real aberrations

Mastering first-order optics lets the engineer carry out the preliminary design —the layout— of complex instruments such as telescopes and microscopes, treating each lens group as a simple component defined by its cardinal points. Once this foundation is in place, the natural next step is to analyse Seidel aberrations: the real deviations any built system shows against this ideal paraxial-ray model.

END