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.

Focal length, from the rear principal plane $\text{—}$

Focus behind the rear glass surface $\text{—}$

Thin-lens formula’s focus error (positive: too far) $\text{—}$

Spot sway on the camera $\text{—}$

Rear nodal point from the rear glass surface $\text{—}$

Rear nodal point from the turning axis $\text{—}$

Focal length by the thin-lens formula $\text{—}$

Spot radius (rms), lens square to the beam $\text{—}$

Rays traced for this record $\text{—}$

Optics 022

Where can the lens turn without moving the focus?

Front surface curvature

c_{1}

76.3\,\mathrm{m^{-1}}\ (R=13.1\,\mathrm{mm})

Rear surface curvature

c_{2}

0\,\mathrm{m^{-1}}\ (R=\infty)

Centre thickness

d

11.7\,\mathrm{mm}

Glass refractive index

n

1.5168\ (\text{N-BK7})

Turning axis from the rear glass surface

p

10\,\mathrm{mm}

How far the lens rocks each way

\alpha_{\max}

8^{\circ}

Beam diameter

D

8\,\mathrm{mm}

In air, a ray aimed at the front nodal point leaves the rear nodal point in its old direction. Turn the lens about the rear one, and the focus stays where it was.

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

BackgroundA 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 mattersThe 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 misconceptionThe focal length is the distance from the lens to the focus.

Better mental modelFrom 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 misconceptionA lens turns about its centre without moving the image.

Better mental modelThe 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 misconceptionTurning a plano-convex lens round changes its focal length.

Better mental modelThe 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 misconceptionThe nodal points are a separate pair of points with their own positions.

Better mental modelWhen 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

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.
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.
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.
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.
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.
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.
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.

Thick Lens Designer

Find a thick lens’s rear nodal point

Core mathematical model

The lensmaker’s equation with the thickness term

Where the rear principal plane sits

The spot follows the rear nodal point

A glass ball

Common difficulties

A focal length is not measured from the glass

The nodal point is not the middle of the lens

Flipping a lens keeps its focal length

Nodal points and principal points are the same in air

Run the experiment

Watch the spot sway

Drag the axis

Stop the spot

Flip the lens

Try a glass ball

Change the glass

Make the lens thin

Related experiments and articles

When lens thickness matters

Diffraction, phase and surface structure