Local Snell law
Angles are measured from the local surface normal at the hit point, not from the optical axis.
Geometrical optics · two-surface refraction
Thick Lens Ray Tracing
What does the thin-lens model miss when thickness matters? The second refraction, and the glass between the two surfaces. A real ray bends at the front surface, travels through the glass and bends again at the back. Points at different depths of the object also image at different distances, so each depth needs its own bundle of rays.
Each ray hits the front sphere, refracts into the glass by Snell's law and leaves through the rear sphere. The front and rear faces of the object each send their own bundle of rays, which form images on two different planes.
Effective focal length104 mm
Front-face image+156 mm
Rear-face image+169 mm
Image depth13 mm
Ray trace6 rays transmitted
Physics tutorial
Thick lenses: return from one refraction plane to real interfaces
BackgroundThe thin-lens model ignores the distance light travels between the two curved surfaces. When the lens is thicker or more strongly curved, the two refractions happen at clearly different places, and the principal planes move away from the lens center.
Why it mattersCamera lens groups, microscope objectives, eyes and ball lenses cannot be explained as one sudden bend. An object with depth along the axis also images each depth onto a different image plane.
Start with the essentials
- Focus question
- How do a thicker lens and a deeper object each change where the image forms and how deep it is along the axis?
- One-sentence intuition
- Each ray hits the front surface and refracts, travels through the glass, then hits the rear surface and refracts again; every depth in the object needs its own bundle of rays.
Core mathematical model
Thick-lens optical power
The last term is the correction for thickness. The effective focal length is .
Common difficulties
Real rays do not bend once at the center
Typical misconceptionOnce I know the effective focal length, I can draw every real ray turning sharply at the lens center.
Better mental modelThe effective focal length only summarizes how paraxial rays enter and leave the lens. Real rays still change direction at each of the two surfaces.
Object depth is not object height
Typical misconceptionDrawing a taller arrow takes care of the object thickness.
Better mental modelDepth here runs along the optical axis. The near and far faces sit at different object distances, so they form images at different distances.
Run the experiment
- 01
Recover the thin-lens limit
Set Lens thickness to its minimum and Object depth to zero.
What to observe: The two object planes merge, the two image planes merge, and the result approaches the familiar thin lens. - 02
Increase only lens thickness
Keep both radii and the Glass index fixed, and increase Lens thickness.
What to observe: Light travels farther inside the glass, the principal planes move and the effective focal length changes. - 03
Give the object depth
Increase Object depth and compare the amber and teal bundles.
What to observe: The two object planes form two image planes. The image depth along the axis is generally not the same as the object depth.
Related experiments and articles
Continue exploring
When lens thickness matters
Keep the surface curvature fixed and change thickness. Does the focus stay put?