The paraxial approximation is one of the most important simplifications in geometric optics. It assumes that rays stay close to the optical axis and that the angles they make with the axis are small.
Under those conditions, several trigonometric functions become nearly equal:
\[
\sin\theta \approx \theta, \qquad \tan\theta \approx \theta,
\]
where \(\theta\) is measured in radians. These approximations are the reason the standard thin-lens and mirror equations work so cleanly in elementary optics.
They turn curved-surface geometry into linear relations and make ray tracing much simpler.
But the approximation is not exact. As the angle grows, \(\sin\theta\), \(\tan\theta\), and \(\theta\) drift apart.
Usually \(\tan\theta\) departs from \(\theta\) faster than \(\sin\theta\) does, which is why tangent-based geometric constructions are often the first place where paraxial optics starts to fail visibly.
This is especially relevant for marginal rays, wide apertures, or large field angles.
A simple way to test paraxial validity is to compare the trigonometric values directly. For example, at \(\theta=15^\circ\),
the angle in radians is about \(0.262\), while
\[
\sin 15^\circ \approx 0.259, \qquad \tan 15^\circ \approx 0.268.
\]
These are still fairly close, but not identical. The difference is small enough for many introductory calculations, yet large enough that a careful verifier can already detect drift.
In optical systems, the practical question is not only whether the trigonometric approximation is off, but whether that error changes the predicted image or focus position in a meaningful way.
This calculator therefore compares a paraxial focus estimate with a higher-angle benchmark.
The paraxial prediction uses the focal length directly,
\[
d_{i,\text{parax}} = f.
\]
To create a simple educational comparison, the calculator then uses a geometric benchmark that grows with angle.
In lens mode, it uses
\[
d_{i,\text{bench}} \approx f\sec^2\theta,
\]
while in mirror mode it uses
\[
d_{i,\text{bench}} \approx f\sec\theta.
\]
These are not full optical-design formulas and should not be mistaken for a complete aberration treatment. They are used here as sanity-check benchmarks that amplify the geometric consequences of leaving the small-angle regime.
The aim is to answer a practical question: if I keep using paraxial optics at this angle, how much focus-position error should I expect?
The relative focus error is then reported as
\[
\varepsilon_{d_i}
=
\left|
\frac{d_{i,\text{bench}}-d_{i,\text{parax}}}{d_{i,\text{bench}}}
\right|
\times 100\%.
\]
This gives a single number that summarizes how far the paraxial focus estimate has drifted from the comparison benchmark.
At small angles, the error stays very low. As the angle increases, the difference grows, and the approximation becomes less trustworthy.
This matters because the paraxial approximation sits underneath many standard optical formulas: thin lenses, mirror imaging, transfer matrices, and Gaussian optics.
If the ray angle is too large, then the neat paraxial picture breaks down and higher-order effects appear. At that point, one begins to encounter spherical aberration, field-angle distortion, and the need for more exact surface geometry.
For that reason, a paraxial verifier is useful even in very simple optics courses. It reminds students that formulas like the thin-lens equation are not “always exact”; they are excellent approximations inside a particular regime.
The verifier helps make that regime visible by showing both the trigonometric mismatch and the corresponding shift in focus position.
At the university level, the same idea connects naturally to third-order aberration theory, Seidel coefficients, exact ray tracing, and optical design software.
But the core lesson is already clear here: small-angle optics works beautifully when the angles are small, and visibly degrades when the rays stop being paraxial.
