General Optics Principles — Abridged Guide

Quick-reference equations, tables, and rules of thumb for foundational optics. For full derivations, worked examples, and diagrams, see the Comprehensive Guide.

1.Introduction

Geometric optics applies when optical features are much larger than the wavelength of light (Fresnel number N_F ≫ 1). Below this threshold, diffraction and interference dominate and wave optics is required.

For visible light (400–700 nm) and standard optical components (≥ 10 mm aperture), geometric optics is almost always sufficient for system layout and component selection.

General optics establishes the foundational principles — reflection, refraction, Fermat's principle, paraxial approximations, stops and pupils, f-number, and ray matrix methods — thatfiber optics — how total internal reflection and NA govern beam acceptance →every other optics topic builds upon.

2.Fermat's Principle & OPL

Optical Path Length

\text{OPL} = \sum_i n_i \, d_i

n_i = refractive index of medium i, d_i = physical path through medium i

Fermat's principle: light follows the path of stationary optical path length. All laws of geometric optics — rectilinear propagation, reflection, and refraction — derive from this single principle.

For imaging systems, a perfect image requires all rays from an object point to its image point to have equal OPLs. Aberrations are the failure of this condition.

3.Reflection

Law of Reflection

\theta_r = \theta_i

Equal angles: the angle of reflection equals the angle of incidence, measured from the surface normal. The incident ray, reflected ray, and normal all lie in the same plane.

A surface is specular when its RMS roughness σ < λ/(8 cos θ). This is why IR optics tolerate rougher surfaces than UV optics.

4.Refraction & Snell's Law

Snell's Law

n_1 \sin\theta_1 = n_2 \sin\theta_2

Critical Angle (TIR)

\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right)

Light bends toward the normal when entering a denser medium. Total internal reflection occurs when the incidence angle exceeds the critical angle (only at higher-to-lower index transitions).

For N-BK7 glass, the critical angle is 41.3° — right-angle prisms exploit this because 45° > 41.3°, guaranteeing TIR without coatings.

Material	n_d (587.6 nm)	V_d	Type
N-BK7	1.517	64.2	Crown
Fused silica	1.458	67.8	Crown
N-SF11	1.785	25.8	Dense flint
CaF₂	1.434	95.1	Fluoride

5.Paraxial Approximation & Sign Conventions

Paraxial Snell's Law

n_1 \theta_1 = n_2 \theta_2

Under the paraxial approximation (sin θ ≈ θ), Snell's law becomes linear, enabling matrix methods and closed-form imaging equations. Accurate to within 1% for angles up to ~10°.

Small-Angle Explorer →

The Cartesian sign convention (light left-to-right, distances positive rightward, heights positive upward) is the safest choice for multi-element calculations. Never mix conventions within a single analysis.

6.Image Formation

Thin Lens Equation

\frac{1}{s'} - \frac{1}{s} = \frac{1}{f}

s = object distance (neg. for real objects), s' = image distance, f = focal length

Lateral Magnification

m = -\frac{s'}{s}

Real images (s' > 0) are formed where rays converge; virtual images (s' < 0) form where rays appear to diverge from. Inverted images have m < 0.

At 2f, magnification = −1 (unit magnification, inverted). Closer than f produces a virtual, magnified, upright image — the magnifying glass configuration.

7.Stops, Pupils & FOV

Angular Field of View

\text{FOV} = 2 \arctan\!\left(\frac{h}{f}\right)

h = half-height of detector/field stop, f = effective focal length

The aperture stop limits the cone of light from each object point (controls brightness and resolution). The field stop limits the extent of the scene imaged (controls field of view). Entrance and exit pupils are images of the stop in object and image space.

When coupling two optical systems, match the exit pupil of the first to the entrance pupil of the second to avoid vignetting.

8.F-Number, NA & Étendue

F-Number

f/\# = \frac{f}{D_{\text{EP}}}

Numerical Aperture

\text{NA} = n \sin\theta_{\max} \approx \frac{1}{2 \cdot f/\#}

Étendue

G = \pi \, n^2 \, A \, \sin^2\theta

Conservation law: étendue is conserved in lossless optical systems. Source étendue must match or be smaller than the system étendue for efficient light coupling.

Each f-stop step (f/1.4 → f/2 → f/2.8 → f/4...) halves the image illuminance. At finite conjugates, use the working f-number: f/\#_W = f/\# × (1 − m).

F/# & NA Calculator →lens types and focal length selection — from plano-convex to achromatic doublets →

9.Matrix Methods

Ray Transfer Matrix

\begin{pmatrix} y_{\text{out}} \\ u_{\text{out}} \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} y_{\text{in}} \\ u_{\text{in}} \end{pmatrix}

Cascade element matrices by multiplying in reverse encounter order: M_sys = M_N · ... · M₂ · M₁. The effective focal length of the system is f = −1/C.

Element	Matrix
Free space (d)	[[1, d], [0, 1]]
Thin lens (f)	[[1, 0], [−1/f, 1]]
Curved mirror (R)	[[1, 0], [−2/R, 1]]

For an afocal system (telescope), C = 0 in the system matrix, and the angular magnification is the D element.

Ray Matrix Calculator →

10.Limits of Geometric Optics

Airy Disk Radius

r_{\text{Airy}} = 1.22 \, \lambda \cdot f/\#

Even a perfect lens cannot beat the diffraction limit. A system is diffraction-limited when its wavefront error is below λ/14 (Maréchal criterion).

At f/4 with 550 nm light, the Airy disk radius is ~2.7 μm. If pixel pitch is larger than this, detector resolution — not optics — limits performance.

Continue Learning

Comprehensive Guide →F/# & NA Calculator →Ray Matrix Calculator →Small-Angle Explorer →

The Comprehensive Guide includes 7 worked examples, 6 SVG diagrams, and 8 references.