Focal length

The focal length of a lens is the distance from the lens to the point where it brings a collimated beam to focus. The focal length formula for a thin lens in air links it to the object distance $s$ and image distance $s'$:

with both distances positive for a real object and a real image, and the transverse magnification is $m = -s'/s$. The lensmaker's equation gives the focal length from the glass and the surfaces, $1/f = (n-1)(1/R_1 - 1/R_2)$ with the usual sign convention for the radii. A plano-convex lens reduces this to $f = R/(n-1)$, and a symmetric biconvex lens to $f = R/[2(n-1)]$: a plano-convex N-BK7 lens ($n$ = 1.5168 at 587.6 nm) with a 25.84 mm radius has $f$ = 50.0 mm.

The same definition holds for a curved mirror, whose focal length is half its radius of curvature. The focal length sets the element's power, $1/f$, usually quoted in dioptres (inverse metres), and together with the aperture diameter it fixes the f-number and so the smallest spot the element can make.

Dependence on wavelength

Because the refractive index falls with wavelength, a singlet's focal length grows toward the infrared. N-BK7 has an index of 1.5168 at 587.6 nm, 1.5066 at 1064 nm and 1.5007 at 1550 nm, so a plano-convex lens with a 25.84 mm radius has a focal length of 50.0 mm in the visible, 51.0 mm at 1064 nm and 51.6 mm at 1550 nm. Catalogue focal lengths are normally given at 587.6 nm, and a lens used at a telecom wavelength needs this correction; achromatic doublets reduce the variation over the band they are designed for.

Effective and back focal length

For a thick lens or a multi-element assembly, the effective focal length is measured from a principal plane, which may lie inside the glass or outside it altogether, and is the quantity that enters the imaging and Gaussian beam formulas. The back focal length is the distance from the last physical surface to the focus, the number needed to position a fiber or detector. Datasheets list both; for a plano-convex lens with its flat side toward the focus they differ by the centre thickness divided by the index.

Measurement

The quickest bench method is to send a well collimated beam through the lens and find the position of the smallest spot, which gives the back focal length directly. The Bessel method avoids locating principal planes: with an object and a screen a fixed distance $D$ apart ($D > 4f$), there are two lens positions, a distance $d$ apart, that form a sharp image, and $f = (D^2 - d^2)/(4D)$. A 50 mm lens with the screen 250 mm from the object gives two positions 111.8 mm apart. Precision measurements use a nodal slide or an autocollimator, which locate the principal planes as well as the focus.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 5; SCHOTT N-BK7 datasheet (Sellmeier coefficients); W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008), Ch. 2.