What Is the Difference Between Paraxial and Marginal Rays?

Paraxial rays and marginal rays are both used to describe light traveling through lenses, but the terms refer to different things. Paraxial describes a ray’s position and angle relative to the optical axis. Marginal describes where the ray passes through the lens or aperture.

This distinction matters because the standard thin-lens and mirror formulas are paraxial approximations. A ray near the edge of a lens is often less paraxial, but it is not automatically nonparaxial.

What is a paraxial ray?

The optical axis is the central reference line through an optical system. A paraxial ray travels close to that axis and makes a small angle with it. In this regime, calculations can use small-angle approximations such as

$$\sin\theta \approx \tan\theta \approx \theta,$$

where $\theta$ is measured in radians. These approximations make it possible to derive the familiar Gaussian imaging rules, including the thin-lens equation

$$\frac{1}{f}=\frac{1}{s_o}+\frac{1}{s_i}.$$

Here $f$ is the focal length, and $s_o$ and $s_i$ are the object and image distances under the usual thin-lens convention. The equation predicts the image location well when the rays involved are paraxial and the lens is being treated in the ideal thin-lens approximation.

“Close to the axis” is not a universal fixed distance. Whether a ray is sufficiently paraxial depends on the optical system and on the accuracy needed.

What is a marginal ray?

A marginal ray is a ray from an object point that passes through the outer edge of the system’s aperture stop (or, in a simple lens diagram, near the edge of the lens aperture). It represents an edge-of-aperture ray in the bundle from that object point.

The term is about the ray’s location across the aperture, not by itself about its angle or whether a small-angle approximation is accurate. A marginal ray through a small aperture can be close to the axis and have a small angle, making it paraxial too. With a wide aperture, the marginal ray is more likely to be nonparaxial.

Do not confuse a marginal ray with a chief ray. The chief ray from an object point passes through the center of the aperture stop; the marginal ray passes through its edge.

Worked example: a small versus a wide aperture

Consider an ideal thin lens with focal length $f=100\,\text{mm}$. In the paraxial model, a parallel incoming ray at height $y$ from the axis is bent by a small angle whose magnitude is approximately

$$|\theta| \approx \frac{|y|}{f}.$$

If the aperture edge is at $y=5\,\text{mm}$, the edge ray has an approximate angle

$$|\theta| \approx \frac{5}{100}=0.05\text{ rad}\approx 2.9^\circ.$$

It is a marginal ray because it passes through the aperture edge, and it is reasonably paraxial because the ratio of its height to the focal length is small and its angle is small.

If the aperture edge is instead at $y=50\,\text{mm}$, the paraxial estimate gives $|\theta|\approx0.5\text{ rad}$, or about $28.6^\circ$. That is a much larger angle, so the small-angle approximation is less reliable. The ray is still marginal: it still passes through the aperture edge. This calculation illustrates why the labels are not opposites; it does not predict the exact path through a particular thick or spherical lens.

Why do the rays sometimes focus at different points?

A real lens with spherical surfaces may not bring all rays from one object point to exactly the same image point. Rays farther from the axis can focus at a different axial position from paraxial rays. This is spherical aberration. It can make the image blurred, especially when a lens is used with a wide aperture.

Stopping down the lens blocks some outer rays. That reduces the contribution of marginal rays and often improves sharpness, though it also reduces the amount of light entering the system. Other aberrations may also affect image quality, so stopping down does not correct every possible defect.

In an idealized paraxial treatment, rays are assumed to meet at the predicted Gaussian image point. In a real-ray trace, marginal rays are followed through the actual optical surfaces to see how far their paths depart from that approximation.

Common mistakes

• Treating “marginal” as the opposite of “paraxial.” They describe different properties: aperture position versus closeness to the axis and smallness of angle.
• Assuming every edge ray has a large angle. The aperture may be small enough that even its edge ray is paraxial.
• Applying the thin-lens equation to every ray without qualification. It is a paraxial model; a wide-aperture real lens can show aberrations.
• Calling the central ray marginal. The ray through the center of the aperture stop is the chief ray, not the marginal ray.

When interpreting a ray diagram, first check where the ray passes through the aperture, then check its height and angle relative to the optical axis. Those two checks tell you whether it is marginal, paraxial, both, or neither.