---
title: "Thick Lens Designer Interactive Lab | Sandbox Physics"
description: "Where inside a thick lens does the light seem to bend, and how do you find that place?"
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last_modified: "2026-10-29"
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Optics 022 · Where a thick lens bends the light

# Thick Lens Designer

**Where inside a thick lens does the light seem to bend, and how do you find that place?** At the rear principal plane, which for a lens in air also holds the rear nodal point. It can lie inside the glass or outside it, and the focal length is measured from it, not from the glass. To find it, rock the lens on a turntable in a parallel beam: the focused spot stands still only when the turning axis passes through the rear nodal point.

Rock a thick lens in a parallel beam and slide its turning axis until the focused spot stands still: the axis is then on the rear nodal point.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Every ray is traced exactly through both spherical surfaces with Snell’s law in three dimensions, at one wavelength, 587.6 nm, with no dispersion, diffraction, reflection loss or stray light. The focal lengths, principal planes and nodal points are the paraxial ones of the two surfaces, so they describe rays close to the axis; the traced spot also shows the lens’s spherical aberration. The lens is in air, so its nodal points coincide with its principal points. The default lens matches a Thorlabs LA1951: N-BK7, 25.4 mm across, convex radius 13.1 mm, 11.7 mm thick, focal length 25.4 mm and back focal length 17.6 mm. Every lens is cut to that 25.4 mm diameter, or smaller where a surface becomes a hemisphere or the two surfaces meet. The collimator’s beam is perfectly parallel, the turntable turns about an exactly vertical axis, and the camera is focused once with the lens square to the beam and stays put while the lens rocks; where the focus is too close to the glass, the camera stands just clear of the turning lens. The thin-lens error compares the lensmaker’s equation without its thickness term, with the lens taken to sit at its middle, against the true focus. The scene is drawn to scale.

Physics tutorial

## Find a thick lens’s rear nodal point

**Background** A real lens has two curved glass surfaces a few millimetres apart, and light bends at each. Seen from outside, the pair still acts as if the light bent once, at a plane that can sit inside the glass or even outside it. Optical test benches find that plane with a nodal slide: they rock the lens in a parallel beam and move the turning axis until the focused spot stops moving.

**Why it matters** The focal length printed on a lens is measured from that plane, not from the glass. Knowing where it is tells you where to put a sensor behind a lens, and why the same lens flipped round focuses at a different distance from the glass.

Start with the essentials

- **Focus question**: Where inside a thick lens does the light seem to bend, and how do you find that place?
- **One-sentence intuition**: For a lens in air, a ray aimed at the front nodal point leaves from the rear nodal point $N'$ in its old direction. The focus of a parallel beam lies on that undeviated ray, so if the lens turns, the focus follows $N'$ sideways. Only when the turning axis passes through $N'$ does the spot stand still, and then the distance from the axis to the focus is the focal length.

### Core mathematical model

#### The lensmaker’s equation with the thickness term

$$
\frac{1}{f}=(n-1)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}+\frac{(n-1)\,d}{n\,R_{1}R_{2}}\right)
$$

Each surface refracts as in OpenStax University Physics Vol. 3, Eq. 2.11; following a ray through the first surface, across the glass and through the second gives this focal length. $R_{1}$ and $R_{2}$ are the radii of the front and rear surfaces, positive when the centre of curvature lies behind the surface, $d$ is the centre thickness and $n$ the index. Drop the last term and you have the thin-lens version. The Lab’s sliders give the curvatures $1/R$ in $\mathrm{m^{-1}}$, so that a flat surface is simply zero.

#### Where the rear principal plane sits

$$
x_{H'}=-\frac{f\,(n-1)\,d}{n\,R_{1}},\qquad f_{\mathrm{B}}=f+x_{H'}
$$

Measured from the rear surface, negative in front of it. The back focal length $f_{\mathrm{B}}$ is the distance from the glass to the focus. For a plano-convex lens with its curved side to the beam, $f=R_{1}/(n-1)$ and $x_{H'}=-d/n$: the LA1951’s plane is $11.7\,\mathrm{mm}/1.5168=7.71\,\mathrm{mm}$ inside the glass, and $f_{\mathrm{B}}=25.36-7.71=17.65\,\mathrm{mm}$ against the 17.6 mm Thorlabs lists. In air the rear nodal point $N'$ sits on this plane.

#### The spot follows the rear nodal point

$$
\Delta y\approx\left(x_{N'}-p\right)\sin\alpha
$$

Turning the lens by $\alpha$ about an axis at $p$ carries $N'$ sideways by this much, and the focus with it. The spot stands still when $p=x_{N'}$; that is the nodal-slide test described by P. J. Reardon and by S. Morel. The traced spot in the Lab includes the lens’s aberrations, so it departs slightly from this rule, by about 5 % at the default settings.

#### A glass ball

$$
f=\frac{nD}{4(n-1)}
$$

With $R_{1}=D/2$, $R_{2}=-D/2$ and $d=D$ the first formula gives this focal length, measured from the centre, where both principal planes and both nodal points sit. A 25.4 mm N-BK7 ball, $n=1.5168$, has $f=18.6\,\mathrm{mm}$, and its focus is only 5.9 mm beyond the glass.

### Common difficulties

#### A focal length is not measured from the glass

**Typical misconception** The focal length is the distance from the lens to the focus.

**Better mental model** From which part of the lens? The LA1951’s focus is 17.6 mm behind its flat face, but its focal length is 25.4 mm, measured from the rear principal plane 7.7 mm inside the glass. The distance from the glass is the back focal length, a different number.

#### The nodal point is not the middle of the lens

**Typical misconception** A lens turns about its centre without moving the image.

**Better mental model** The LA1951’s middle is 5.85 mm in front of its flat face, but its rear nodal point is 7.71 mm in front of it, and turning about the middle leaves the spot swaying. Only a symmetric lens, such as the glass ball, has its nodal points at the middle.

#### Flipping a lens keeps its focal length

**Typical misconception** Turning a plano-convex lens round changes its focal length.

**Better mental model** The focal length stays 25.4 mm; what moves is where the principal planes sit in the glass, so the focus moves from 17.6 mm behind the glass to 25.4 mm. The aberration changes too: with the curved side to the parallel beam both surfaces share the bending, and the spot’s radius is 0.054 mm instead of 0.24 mm.

#### Nodal points and principal points are the same in air

**Typical misconception** The nodal points are a separate pair of points with their own positions.

**Better mental model** When the same medium lies on both sides, as for a lens in air, the nodal points coincide with the principal points. They part only when the two sides differ, as in the eye, where the back is filled with fluid.

### Run the experiment

1. 01

   #### Watch the spot sway

   Keep the default preset, Axis behind the lens: the spot sways.

   **What to observe:** The turning axis is 10 mm behind the flat face, 17.7 mm from the rear nodal point. As the lens rocks ±8°, the spot on the camera swings ±2.6 mm, close to the 2.5 mm the rule gives.

2. 02

   #### Drag the axis

   Drag the turning axis on the turntable towards the lens and into the glass.

   **What to observe:** The swing shrinks, stops, then reverses once the axis passes the purple bead that marks the rear nodal point. The traced curve in the data row crosses zero there.

3. 03

   #### Stop the spot

   Load Axis on the nodal point: the spot stands still.

   **What to observe:** With the axis 7.7 mm inside the glass the spot moves by less than 0.002 mm. From the axis to the camera is 7.7 mm of glass plus 17.6 mm of air, 25.4 mm: the focal length.

4. 04

   #### Flip the lens

   Load Flat side first: the nodal point moves.

   **What to observe:** The focal length is still 25.4 mm, but the focus is now 25.4 mm behind the curved face and the rear nodal point sits on that face. With the axis left 7.7 mm inside the glass, the spot sways ±1.0 mm, and its radius grows from 0.054 to 0.24 mm.

5. 05

   #### Try a glass ball

   Load Glass ball: nodal point at the centre.

   **What to observe:** Turning about the centre leaves the spot still. The focal length is 18.6 mm and the focus 5.9 mm beyond the glass; the thin-lens formula, with 12.3 mm, puts the focus 6.4 mm too close.

6. 06

   #### Change the glass

   Load Dense glass: a shorter focus, then change the glass refractive index.

   **What to observe:** N-SF11, with a refractive index of 1.785, shortens the focal length to 16.7 mm and the back focal length to 10.1 mm. The rear nodal point moves to 6.6 mm inside the glass.

7. 07

   #### Make the lens thin

   Load Thin lens: the simple formula works, then thicken it with the centre thickness.

   **What to observe:** The 3.6 mm LA1509 has a focal length of 99.7 mm, and the thin-lens formula misses its focus by 0.57 mm. Thickening the glass leaves the focal length alone, because the flat face adds no power, but pulls the focus and the principal planes towards the beam, as the process plots show.

Physical conclusion

The default LA1951 lens, 11.7 mm thick, has a focal length of 25.4 mm, but its focus is only 17.6 mm behind its flat face: the rear nodal point is 7.7 mm inside the glass. With the turning axis 10 mm behind the lens the spot sways ±2.6 mm over ±8°; with the axis at 7.7 mm inside the glass it moves by less than 0.002 mm. Flipping the lens keeps the 25.4 mm focal length, moves the focus to 25.4 mm behind the curved face and widens the spot from 0.054 to 0.24 mm in radius. A glass ball has its nodal points at its centre and a focal length of 18.6 mm, where the thin-lens formula gives 12.3 mm; for the 3.6 mm thick LA1509 the same formula misses the focus by only 0.57 mm in 100 mm.

## Related experiments and articles

Continue exploring

### When lens thickness matters

Keep the surface curvature fixed and change thickness. Does the focus stay put?

1. [Lab **Thick Lens Ray Tracing** Follow refraction at the front and rear surfaces.](https://sandboxphysics.com/labs/thick-lens-refraction/)
2. [You are here **Thick Lens Designer** Rock the lens to find its rear nodal point, and watch thickness move the focus.](https://sandboxphysics.com/labs/thick-lens-designer/)

Probes and measurement · Reading path

### Diffraction, phase and surface structure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Zernike: a reference wave makes transparent structure visible**](https://sandboxphysics.com/stories/probe-evidence/zernike-phase-reference/)
2. [Lab **Phase Microscopy Lab** Compare phase and amplitude imaging](https://sandboxphysics.com/labs/phase-microscopy-lab/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)