Effective focal length (EFL)

The effective focal length (EFL) of a lens or a multi-element system is the distance from its rear principal plane to the rear focal point, where a collimated beam parallel to the optical axis comes to focus. It is the reciprocal of the system's optical power, $f = 1/\Phi$, and is the $f$ that enters the imaging equation, the magnification, the f-number and the Gaussian-beam focusing formulas. For a thin lens it equals the focal length; for thick lenses and assemblies the principal planes can lie inside the glass, between elements or outside the system, so the EFL is generally not a distance to any physical surface. Examples run from 10 mm for a 20× microscope objective (with a 200 mm tube lens) to hundreds of millimeters in telephoto camera lenses.

EFL, BFL and FFL

The back focal length (BFL) is measured from the last physical surface to the rear focal point, and the front focal length (FFL) from the first surface to the front focal point. They locate the foci for mounting; the EFL sets the scale of the image. An object at infinity at field angle $\theta$ is imaged at height

$$h' = f \tan\theta$$

in a distortion-free lens, which is a working definition of the EFL. In air the EFL is the same whichever way round the lens is used; the BFL and FFL are not. In an immersion objective, where the two sides have different indices, the front and rear focal lengths, measured from the principal planes, differ in proportion to them.

Two-lens combination

Two thin lenses of focal lengths $f_1$ and $f_2$ separated by $d$ have

$$\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2},$$

with the back focal length

$$\text{BFL} = f\left(1 - \frac{d}{f_1}\right).$$

For $f_1$ = 100 mm, $f_2$ = 50 mm and $d$ = 30 mm, $1/f$ = 0.0100 + 0.0200 − 0.0060 = 0.0240 mm⁻¹, so $f$ = 41.7 mm, against 33.3 mm for the same lenses in contact. The BFL is 41.7 × 0.70 = 29.2 mm, so the rear principal plane lies 12.5 mm in front of the second lens, in the air gap. The same result follows from the system's ray-transfer matrix, whose element $C = -1/f$ (ABCD matrix).

Telephoto and retrofocus designs

Placing a negative lens behind a positive one moves the rear principal plane forward, ahead of the first element. With $f_1$ = 100 mm, $f_2$ = −50 mm and $d$ = 60 mm, the EFL is 500 mm but the BFL is only 200 mm, so the whole system from front lens to focus is 260 mm long, 0.52 of its focal length. A lens whose physical length is shorter than its EFL is a telephoto design in the optical sense. The reverse arrangement, a negative group in front, gives a retrofocus lens with a BFL longer than its EFL.

Camera lenses and the 35 mm equivalent

Camera lenses are marked with their EFL. The "35 mm equivalent focal length" quoted for compact cameras and phones is a different quantity: the EFL multiplied by the crop factor, the ratio of the 43.3 mm diagonal of a 36 × 24 mm frame to the sensor's diagonal. A 25 mm lens on a Four Thirds sensor (17.3 × 13.0 mm, crop factor 2.0) gives the field of view of a 50 mm lens on a full frame, but its EFL, and the spot it focuses a beam of given diameter to, remain those of a 25 mm lens. Only the actual EFL belongs in optical calculations.

Measurement

On the bench the EFL is found without locating the principal planes. On a nodal slide, the lens is rotated about a vertical axis while imaging a collimated source; when the rotation axis passes through the rear nodal point (the rear principal point, for a lens in air) the image does not move, and the distance from there to the focus is the EFL. The magnification method uses the rate, $1/f$, at which magnification grows with screen distance. The field-angle method uses $h' = f\tan\theta$: a target at infinity subtending 2° produces an image 1.75 mm high with a 50 mm lens. The nodal-slide, magnification and Bessel methods are described in How to Measure the Focal Length of a Lens.

Pitfalls

• Using the EFL as a mounting distance, which places a detector or fiber off focus by the gap between principal plane and last surface.
• Comparing a catalog EFL, usually specified at 587.6 nm, with a measurement at another wavelength: a plano-convex N-BK7 lens of 50.0 mm EFL in the visible has 51.6 mm at 1550 nm. Achromatic doublets reduce this shift.

Common questions

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

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

Why is it called "effective" focal length?

It is the focal length of the single equivalent thin lens, placed at the rear principal plane, that focuses and magnifies as the whole system does.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).