Geometry for Optical Alignment — Abridged Guide

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

1.Overview

Geometric optics models light as rays obeying two laws: reflection (θᵢ = θᵣ) and refraction (Snell's law). Every alignment calculation reduces to geometry and trigonometry.

Key principle: a beam has 4 degrees of freedom (x, y, θₓ, θᵧ). Two kinematic mirrors provide exactly 4 adjustments (pitch + yaw each) — necessary and sufficient for complete alignment.

2.Coordinate Systems

Standard optics convention: z = beam propagation, x = lateral (table surface), y = vertical (beam height). Right-handed system.

Polar ↔ Cartesian

x = r\cos\varphi, \; y = r\sin\varphi \qquad r = \sqrt{x^2+y^2}

Spherical ↔ Cartesian

x = r\sin\theta\cos\varphi, \; y = r\sin\theta\sin\varphi, \; z = r\cos\theta

Sign conventions matter. This site uses the Cartesian convention: distances along +z are positive, heights above axis (+y) are positive. Never mix conventions in a single calculation.

3.Trigonometric Essentials

Core Definitions

\sin\theta = \frac{O}{H}, \quad \cos\theta = \frac{A}{H}, \quad \tan\theta = \frac{O}{A}

Degree ↔ Radian

\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \qquad 1\text{ rad} = 57.296°

All paraxial optics formulas require radians — never degrees.

Key Identities

\sin^2\theta + \cos^2\theta = 1 \qquad \sin(2\theta) = 2\sin\theta\cos\theta

The double-angle identity explains why a mirror tilted by α deflects the beam by 2α — the most important fact in mirror alignment.

4.Angular Measurement

Unit	Symbol	In Radians	Typical Use
Degree	°	0.01745 rad	Component specs
Arcminute	′	2.909 × 10⁻⁴ rad	Angular resolution
Arcsecond	″	4.848 × 10⁻⁶ rad	Precision alignment
Milliradian	mrad	10⁻³ rad	Beam divergence
Microradian	µrad	10⁻⁶ rad	Pointing stability

Memorize: 1 rad ≈ 57.3°, 1° ≈ 17.45 mrad, 1 mrad ≈ 3.44′, 1″ ≈ 4.85 µrad.

Angular Error → Positional Error

\Delta x = L \cdot \delta\theta

δθ in radians. 1 µrad = 1 µm displacement per 1 m distance.

Pitch = rotation about x (vertical steering). Yaw = rotation about y (horizontal steering). Roll = rotation about z (matters for polarizers, waveplates, cylindrical lenses).

5.Small-Angle Approximation

Paraxial Approximation

\sin\theta \approx \theta \qquad \tan\theta \approx \theta \qquad \cos\theta \approx 1

θ must be in radians. Valid for θ ≲ 10° (sin error < 0.5%).

Angle	sin θ error	tan θ error	cos θ ≈ 1 error
5°	0.13%	0.25%	0.38%
10°	0.51%	1.02%	1.52%
15°	1.15%	2.35%	3.41%
20°	2.06%	4.27%	6.03%

The paraxial approximation underpins: the thin lens equation, Gaussian beam divergence, and the ABCD matrix formalism. It fails for high-NA objectives, 45° mirrors, and large-apex prisms.

6.Geometry of Reflection

Law of Reflection

\theta_i = \theta_r

Angles measured from surface normal, not surface.

Angular doubling: Mirror tilts by α → beam deflects by 2α. This is the #1 alignment fact.

Retroreflection: Corner cube (3 orthogonal mirrors) returns beam antiparallel regardless of orientation. Two mirrors at exactly 90° do the same in 2D.

7.Geometry of Refraction

Snell's Law

n_1 \sin\theta_1 = n_2 \sin\theta_2

Parallel Plate Displacement

d = t \cdot \frac{\sin(\theta_1 - \theta_2)}{\cos\theta_2}

t = thickness, θ₂ from Snell's law. Exit beam is parallel but offset by d.

Critical Angle (TIR)

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

N-BK7: 41.3° | Fused silica: 43.3° | Diamond: 24.4°

Prism Deviation

\delta = \theta_1 + \theta_4 - A \qquad n = \frac{\sin\!\left(\frac{A+\delta_{\min}}{2}\right)}{\sin\!\left(\frac{A}{2}\right)}

8.Solid Angles & Étendue

Solid Angle of a Cone

\Omega = 2\pi(1-\cos\theta) \;\text{[sr]}

Small angle: Ω ≈ πθ² ≈ π·NA² ≈ π/[4(f/#)²]

Étendue (Throughput)

G = n^2 A \Omega = \pi n^2 A \sin^2\theta = \frac{\pi A}{4(f/\#)^2}

Conserved in lossless systems. Cannot decrease — limits light concentration.

If source étendue exceeds receiver étendue, excess light is lost regardless of optics in between. Match source and receiver étendue for optimal coupling.

9.Beam Path Geometry

Define optical axis with two irises, as far apart as practical. Walk the beam with two mirrors: M1 controls position at Iris 1, M2 controls angle at Iris 2. Iterate until converged.

Mirror Separation for Height Change

L = \frac{|y_2 - y_1|}{\sin(2\alpha)}

At 45° incidence: L = height difference. Z-fold is simplest configuration.

Cumulative Alignment Error

\Delta x_{\text{total}} = \sum_{i} L_i \cdot \delta\theta_i

Errors early in the path dominate — concentrate alignment effort on first elements.

Decentration sensitivity: Beam off-center by Δx on a lens of focal length f acquires angular error δθ = Δx/f. Short-FL lenses are much more sensitive.

10.Practical Quick-Reference

Formula	Expression	Use
Angle conversion	θ_rad = θ_deg × π/180	Required for all paraxial formulas
Positional error	Δx = L · δθ	Angular error → position at distance
Angular doubling	Deflection = 2 × mirror tilt	Key alignment fact
Snell's law	n₁ sin θ₁ = n₂ sin θ₂	Refraction at interfaces
Solid angle (cone)	Ω = 2π(1−cos θ)	Light collection
Small-angle	sin θ ≈ θ ≈ tan θ	< 10° in radians

Common pitfalls: Forgetting the 2× in angular doubling; mixing sign conventions; placing irises too close together; not centering beam on lenses.

Continue Learning

Comprehensive Geometry Guide →Optical Geometry Calculator →Small-Angle Approximation Explorer →Units & Conversions (Abridged) →

The Comprehensive Guide includes 7 worked examples, 7 SVG diagrams, detailed derivations, and 8 cited references covering all formulas on this page.