How to Measure the Focal Length of a Lens: Bench Methods and Accuracy

Scope

This article gives bench procedures for measuring the focal length of a positive lens or lens assembly with ordinary lab equipment: a rail, a light source, a screen or camera, and a ruler or calibrated translation stage. It covers the quick methods that give the back focal length, autocollimation, the Bessel (displacement) method, and the magnification method, with the accuracy each can reach. Negative lenses are measured by pairing them with a stronger positive lens of known focal length, as noted at the end.

Effective and back focal length

The effective focal length (EFL) $f$ is measured from a lens's rear principal plane to its focal point, and it is the $f$ in the thin-lens relation

$$\frac{1}{s} + \frac{1}{s'} \;=\; \frac{1}{f},$$

where $s$ and $s'$ are the object and image distances from the front and rear principal planes. The back focal length (BFL) is measured from the lens's last surface to the focal point. For a thin lens the two coincide; for a thick lens, an achromat or a multi-element objective they differ, sometimes by several millimetres, because the principal planes lie inside the glass or even outside the lens. Datasheets usually give both. A method that measures distances from the lens's surface or mount returns the BFL, or a distance offset from either; a method that uses only differences in position returns the EFL.

Methods

Distant object or collimated laser

Form an image of an object many focal lengths away (a window, a distant light) on a screen, or focus a collimated laser beam, and measure the distance from the lens's rear surface to the sharpest image or smallest spot. With the object at a distance $s$, the image lies beyond the focal point by about $f^2/s$, so an object 10 m from a 100 mm lens puts the image 1 mm beyond focus, a 1% error. A collimated laser removes that error, and the waist can be found precisely with a beam profiler or knife edge; the measured value is still the BFL.

Autocollimation

1. Place a point source (a pinhole illuminated from behind, or the end of a single-mode fiber) on the lens's axis.
2. Place a plane mirror behind the lens, facing it, roughly normal to the axis.
3. Move the lens along the axis until the light returned by the mirror comes to a sharp focus back on the source plane, observed on a card beside the pinhole or through a beamsplitter.
4. The source is then at the front focal point. The distance from the source to the lens's front surface is the front focal length.

Tilting the mirror slightly moves the returned spot sideways without affecting focus, which makes it easier to see beside the source. On a nodal slide, which rotates the lens about a vertical axis, the rotation point that leaves the image stationary locates the principal plane, and autocollimation then gives the EFL directly.

Bessel displacement method

The Bessel method uses only the separation of two lens positions, so it gives the EFL without locating the principal planes.

1. Fix an illuminated object (a slide, a grid or a crosshair) and a screen a distance $D$ apart on the rail, with $D$ greater than $4f$.
2. Move the lens between them to find the position that forms a sharp, magnified image on the screen. Record the lens position.
3. Move the lens toward the screen to find the second position that forms a sharp, reduced image. Record its position.
4. With $d$ the distance between the two lens positions,

$$f \;=\; \frac{D^2 - d^2}{4D}.$$

For $D$ = 1000 mm and $d$ = 600 mm, $f$ = 160.0 mm. The two positions are symmetric about the midpoint, so any offset between the carriage reading and the lens's principal planes cancels in $d$; for a thick lens, $D$ strictly needs a correction equal to the separation of the principal planes, which is small compared with $D$ when $D$ is several times $f$. An error of 1 mm in $d$ gives an error of $d/2D$ mm in $f$, 0.3 mm in the example. Repeating each focus setting several times, approaching from both sides, and averaging reduces the dominant error, which is judging best focus.

Magnification method

With an object of known size $h$ and its image of measured size $h'$, the magnification $m = h'/h = s'/s$. Measuring the image size at two screen positions a distance $\Delta$ apart, with the object and lens fixed, gives

$$f \;=\; \frac{\Delta}{m_2 - m_1},$$

since the magnification grows linearly with image distance at a rate $1/f$. A camera sensor with known pixel pitch makes a convenient screen, and the method, like the Bessel method, uses only a difference in position.

Common errors

Focus judgment. Every method depends on finding best focus, and the uncertainty in that position is roughly the depth of focus. Slow lenses (large f-number) have long depths of focus; use a sharp, high-contrast target, a magnifier or camera at the image, and average several settings approached from both directions.

Wavelength. A singlet's focal length changes with wavelength through the glass's dispersion, by about 1.5% between blue (486 nm) and red (656 nm) light for N-BK7 crown glass (see chromatic aberration). Measure with the wavelength the lens will be used at, or with a narrowband filter on a white source.

Aberrations. Spherical aberration spreads the focus along the axis, so paraxial and marginal rays focus at different points. Stopping the lens down to a small aperture gives the paraxial focal length, which is the value datasheets quote.

Lens orientation. The BFL of an asymmetric lens depends on which way it faces. Record the orientation with the result.

Negative lenses. A negative lens forms no real image. Combine it in contact with a positive lens of known focal length $f_1$ that is stronger than it, measure the pair's focal length $f_\text{pair}$, and compute $1/f_2 = 1/f_\text{pair} - 1/f_1$.

References: E. Hecht, Optics (5th ed., Pearson, 2017), chapters on geometrical optics; D. Malacara (ed.), Optical Shop Testing (3rd ed., Wiley, 2007), chapter on the measurement of optical parameters; F. L. Pedrotti, L. M. Pedrotti and L. S. Pedrotti, Introduction to Optics (3rd ed., Cambridge University Press, 2017).