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Sandbox Physics

Geometrical optics · two-surface refraction

Thick Lens Ray Tracing

Each ray intersects the front sphere, refracts into glass by Snell's law, then leaves through the rear sphere. Separate bundles from the front and rear object faces form distinct image planes.

Interactive modelThick Lens Ray Tracing
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 neglects propagation between two curved surfaces. With greater thickness or stronger curvature, refractions occur at distinct locations and principal planes shift away from the lens center.

Why it mattersCamera groups, microscope objectives, eyes, and ball lenses cannot be understood as one abrupt bend. Objects with axial depth also send different depth planes to different image planes.

Start with the essentials

Focus question
How do lens thickness and object depth separately change image position and longitudinal image depth?
One-sentence intuition
Process each ray as front-surface intersection, refraction, glass propagation, rear-surface intersection, then refraction again. Every object depth needs its own ray bundle.

Core mathematical model

Local Snell law

n1sin⁡θ1=n2sin⁡θ2n_1\sin\theta_1=n_2\sin\theta_2

Angles are measured from the local surface normal at the hit point, not from the optical axis.

Thick-lens optical power

Φ=(n−1)(1R1−1R2+(n−1)tnR1R2)\Phi=(n-1)\left(\frac1{R_1}-\frac1{R_2}+\frac{(n-1)t}{nR_1R_2}\right)

The last term is the thickness correction, and effective focal length is f=1/Φf=1/\Phi.

Common difficulties

Real rays do not bend once at the center

Typical misconceptionOnce effective focal length is known, every physical ray can turn abruptly at the lens center.

Better mental modelEffective focal length summarizes a paraxial input-output relation; actual paths change direction at both surfaces.

Object depth is not object height

Typical misconceptionDrawing a taller arrow accounts for object thickness.

Better mental modelDepth here lies along the optical axis. Near and far faces have different object distances and therefore different image distances.

Run the experiment

  1. 01

    Recover the thin-lens limit

    Minimize lens thickness and set object depth to zero.

    What to observe: The two object and image planes merge, approaching the familiar thin-lens case.
  2. 02

    Increase only lens thickness

    Hold curvature and index fixed while increasing t.

    What to observe: The path inside glass grows and the principal planes and effective focal length shift.
  3. 03

    Expand object depth

    Increase object depth and compare amber with teal bundles.

    What to observe: Two object planes form two image planes; longitudinal image depth generally differs from object depth.