Optics 022 · Imaging, instruments, and visual systems

Thick Lens Designer

A two-surface N-BK7 design bay couples visibly changing curvature and center thickness to exact meridional Snell tracing, local normals, Gaussian principal planes, effective focal length, and a live finite-thickness error map.

Effective focal length

f_{\mathrm e}

$0.50$

Principal-plane locations

H_1,\,H_2

$50\%$

Finite-thickness focal correction

f_{\mathrm e}-f_{d\to0}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Thick Lens Designer

BackgroundA thick lens is two separated refracting surfaces, not a thin-lens icon with a width slider. For spherical vertices V1 and V2, the first-order system power is $\Phi=\Phi_1+\Phi_2-\frac{d}{n}\Phi_1\Phi_2$ , where each surface power uses its signed curvature radius. The exact colored paths independently intersect both rendered spheres and apply vector Snell refraction twice.

Why it mattersWhat does the thin-lens approximation miss when curvature and thickness are both adjustable?

Start with the essentials

Focus question: What does the thin-lens approximation miss when curvature and thickness are both adjustable?
One-sentence intuition: Object and image distances for a thick lens belong to its principal planes, whose vertex-relative locations are $H_1=V_1+\frac{\Phi_2}{\Phi}\frac{d}{n},\qquad H_2=V_2-\frac{\Phi_1}{\Phi}\frac{d}{n}$ . Changing curvature radius or center thickness therefore moves both the glass geometry and the cardinal planes.

Core mathematical model

Two-surface power

\Phi_1=\frac{n-1}{R_1},\quad \Phi_2=\frac{1-n}{R_2},\quad f_{\mathrm e}=\frac{1}{\Phi}

The thickness coupling vanishes only as d tends to zero. The displayed d-to-zero comparison is recomputed from the same surface powers.

Principal planes

P=\frac{\Phi_2}{\Phi}\frac{d}{n},\qquad P''=-\frac{\Phi_1}{\Phi}\frac{d}{n}

P is measured from the front vertex and P double-prime from the rear vertex. They are not automatically located at the mechanical center.

Surface-by-surface Snell invariant

n_i(\hat{\mathbf k}_i\!\cdot\!\hat{\mathbf t})=n_t(\hat{\mathbf k}_t\!\cdot\!\hat{\mathbf t})

The exact ray tracer conserves tangential optical momentum at each local spherical normal; it does not bend the ray at the lens center.

Common difficulties

Confusing aperture radius with curvature radius

Typical misconceptionChanging R only needs to update a focal-length number; the visible lens can keep the same surface.

Better mental modelR is the radius of the generating sphere. This apparatus rebuilds both spherical meshes, their clear edge, every intersection, local normal, refraction, system power, and principal plane whenever R changes.

Run the experiment

01
Scene 1: Two spherical refractions
Use the two-refraction scene. Sweep surface curvature radius while orbiting the lens and verify that the rendered sag, edge aperture, hit points, normals, and outgoing rays all change together.
What to observe: A smaller curvature radius creates visibly deeper surfaces and larger local normal changes. Marginal rays cease to share the paraxial crossing, exposing spherical aberration instead of hiding it.
02
Scene 2: Principal-plane tracking
Enter principal-plane tracking. Increase center thickness and compare the physical vertices with H1 and H2 before reading the Gaussian image position.
What to observe: For an asymmetric lens, H1 and H2 shift by different amounts. Measuring both conjugates from the mechanical midpoint produces a systematic error even when the Gaussian approximation itself is valid.
03
Scene 3: Thin-lens error map
Open the thin-lens error map. Move from a shallow lens toward the thick, strongly curved preset and compare effective focal length with the d-to-zero limit.
What to observe: The focal-length correction grows continuously with thickness and surface power. It is a model term with units, not a decorative displacement of the lens mesh.