Back focal length (BFL)

The back focal length (BFL) is the distance from the last physical surface of a lens, or of a multi-element assembly, to the point where it focuses a collimated beam. It differs from the effective focal length (EFL), which is measured from the rear principal plane, an imaginary plane that can sit inside the glass or entirely outside it. For a thin lens the two coincide. For a 50 mm plano-convex singlet they differ by a few millimeters, and for a camera lens or microscope objective they can differ by tens of millimeters; in many infinity-corrected objectives the back focal plane lies inside the barrel, a negative BFL. The BFL is the number needed to mount a fiber, detector or camera sensor at the focus; the EFL is the number that enters the imaging, magnification and Gaussian beam formulas.

Thick-lens formulas

For a single lens of index $n$, surface radii $R_1$ and $R_2$ (positive when the center of curvature lies to the right of the surface) and center thickness $d$, the lensmaker's equation gives the EFL:

$$\frac{1}{f} = (n-1)\left[\frac{1}{R_1} - \frac{1}{R_2}\right]$$ $$\qquad + \frac{(n-1)^2\, d}{n R_1 R_2} .$$

The back and front focal lengths, measured from the rear and front vertices, are

$$\text{BFL} = f\left[1 - \frac{(n-1)\,d}{n R_1}\right],$$ $$\text{FFL} = f\left[1 + \frac{(n-1)\,d}{n R_2}\right].$$

Worked example: plano-convex lens

Take N-BK7 ($n$ = 1.5168 at 587.6 nm), $R$ = 25.84 mm and $d$ = 3.6 mm. With the curved side facing the collimated beam ($R_1$ = 25.84 mm, $R_2 = \infty$), the thickness term vanishes and $f$ = 50.0 mm. The rear principal plane then lies inside the glass, $d/n$ = 2.37 mm before the flat face, so

$$\text{BFL} = 50.0 - 2.37 = 47.6\ \text{mm},$$

while the front focal length is 50.0 mm, because the front principal plane touches the curved vertex. Turned around, with the flat side toward the beam, the EFL is still 50.0 mm but the BFL becomes 50.0 mm and the FFL 47.6 mm. The EFL of any lens is the same from either side; its BFL is not, which is why datasheets state the orientation.

The BFL also changes with wavelength through the index. At 1550 nm N-BK7 has $n$ = 1.5007, so the same lens has $f$ = 51.6 mm and, curved side first, a BFL of 49.2 mm: a fiber placed at the visible focus would sit about 1.6 mm short of the infrared one.

Multi-element systems

In an achromatic doublet, an objective or a camera lens, the principal planes can be placed almost anywhere by design. A retrofocus (inverted telephoto) design has a BFL longer than its EFL, which gives a short-focus lens room for a mirror or beam splitter behind it; a telephoto design places its rear principal plane in front of the first lens, so the whole assembly, from front vertex to focus, is shorter than its focal length. Camera mounts specify a flange focal distance, the distance from the mounting flange to the sensor, which plays the role of BFL for interchangeable lenses; the C-mount value is 17.526 mm. For any system, ray-transfer (ABCD) matrices give the EFL from the C element and the BFL from the distance at which a collimated input ray crosses the axis.

Back focal plane

The plane perpendicular to the axis through the rear focal point is the back focal plane. In Fourier optics a lens forms the angular spectrum of the input field there: each direction leaving the object is focused to its own point. In a microscope objective obeying the sine condition, light leaving the specimen at angle $\theta$ crosses the back focal plane at radius $f n\sin\theta$, so the plane contains the objective's pupil, of radius $f\,\text{NA}$: 4 mm for a 20×/0.40 objective of 10 mm focal length. Imaging this plane, by conoscopy with a Bertrand lens or by placing a relay lens in front of a camera, shows the angular distribution of the light, which is how diffraction orders, surface plasmon angles and emission patterns are measured.

Pitfalls

• Using the EFL as a mounting distance for a thick lens, which puts the detector or fiber a few millimeters off focus.
• Ignoring the BFL shift between the catalog design wavelength and the working wavelength.
• When the beam arriving at the lens has a Rayleigh range not much longer than the focal length, as for a small input beam, the focused waist sits slightly closer to the lens than the geometric focus; for a well collimated beam of a few millimeters the difference is negligible.
• Some datasheets quote a mechanical BFL, from the edge of a mount.

Common questions

What is the difference between back focal length and effective focal length?

The EFL is measured from the rear principal plane and sets magnification and focal spot size. The BFL is measured from the last lens surface and gives the physical position of the focus. For thin lenses they are nearly equal.

How is back focal length measured?

Send a well collimated beam through the lens and measure from its last surface to the smallest spot, with a camera or knife edge; How to Measure the Focal Length of a Lens gives the bench procedure.

What is the front focal length?

The distance from the first lens surface to the front focal point, where a point source produces a collimated output. For an asymmetric lens it differs from the BFL.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 5 and 6; W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W. H. Freeman, 2017).