Focal plane

The focal plane of a lens or mirror is the plane perpendicular to the optical axis that passes through a focal point. A collimated beam traveling along the axis focuses on axis in the back focal plane; a collimated beam arriving at an angle $\theta$ focuses in the same plane, displaced from the axis by $f\tan\theta$, where $f$ is the focal length. For a 100 mm lens and a 1° tilt the spot moves 1.75 mm. A lens therefore converts the directions of incoming light into positions in its back focal plane, which is the basis of spectrometers, far-field measurements, scanning systems and cameras focused at infinity.

Front and back focal planes

A lens has two focal planes, one on each side. The back (rear) focal plane lies one effective focal length beyond the rear principal plane, and collimated light entering from the front focuses there. The front focal plane lies one focal length before the front principal plane: a point source placed in it leaves the lens as a collimated beam, which is how fiber outputs and laser diodes are collimated. In a thick lens or a multi-element system the principal planes can lie well inside or outside the glass, so the position of the back focal plane relative to the last surface is quoted separately as the back focal length.

Angle-to-position mapping

A ray arriving at angle $\theta$ crosses the back focal plane at

$$x = f\tan\theta \approx f\theta,$$

independent of where it strikes the lens. With $f$ = 100 mm, $\theta$ = 1° gives $x$ = 1.7455 mm, and the small-angle value $f\theta$ = 1.7453 mm differs by 0.01%; at 5° the two are 8.749 mm and 8.727 mm, a 0.25% difference. Scan lenses designed for galvanometer scanners are corrected to follow $x = f\theta$ so that the spot moves linearly with mirror angle. The same mapping measures beam divergence: a beam's width in the back focal plane, divided by $f$, gives its far-field angle wherever the lens is placed.

Fourier-transform property

A field placed in the front focal plane appears in the back focal plane as its two-dimensional Fourier transform, each spatial frequency $\nu$ mapped to a position $x = \lambda f \nu$. With $f$ = 200 mm and 632.8 nm light, a 100 line/mm grating sends its first orders to points 12.7 mm from the axis. For an input placed elsewhere the intensity pattern is the same and only a quadratic phase changes. This property underlies spatial filtering, 4f image processing and the rest of Fourier optics.

Focal plane and image plane

The image plane is where the lens forms a sharp image of a particular object plane. It coincides with the back focal plane only for an object at infinity. For a thin 50 mm lens, an object 250 mm away is imaged 62.5 mm behind the lens, 12.5 mm beyond the focal plane; this is why camera lenses move away from the sensor to focus close. Chromatic dispersion also moves the focal plane: a plano-convex N-BK7 lens with $f$ = 50.0 mm at 587.6 nm has $f$ = 51.6 mm at 1550 nm, so a lens aligned in the visible must be refocused by 1.6 mm for an infrared beam.

Curved focal surface

The surface of best focus of a real lens is curved, an aberration called field curvature. For a single thin N-BK7 lens of 50 mm focal length the Petzval radius is 76 mm, and 5 mm off axis the Petzval surface departs from the flat focal plane by 0.16 mm, far more than the ±18 µm depth of focus at f/4 and 550 nm. Flat-field objectives and scan lenses are corrected to flatten it.

Focal plane arrays and microscopy

In infrared imaging and astronomy, "focal plane" also names the detector placed there. A focal plane array (FPA) is a two-dimensional array of detector pixels, such as the InGaAs arrays of a SWIR camera, with formats of 320 × 256 to 1280 × 1024 pixels, or the HgCdTe arrays used in the mid- and long-wave infrared. Each pixel views one direction: with a 15 µm pixel pitch behind a 50 mm lens, each pixel subtends 0.3 mrad.

In a microscope objective, the back focal plane contains the pupil, of radius $f\,\text{NA}$: 4 mm for a 20×/0.40 objective of 10 mm focal length. Imaging this plane shows the angular distribution of light leaving the specimen, which is used in conoscopy, Fourier-plane imaging and optical-trap detection.

Common pitfalls

A distance measured from the lens surface is the back focal length, which differs from the effective focal length. The smallest spot of a focused beam lies at the focal plane only for a collimated or nearly collimated input; a strongly diverging input focuses beyond it.

Common questions

Is the focal plane the same as the image plane?

Only for an object at infinity. For a finite object the image plane lies farther from the lens than the back focal plane, by an amount given by the thin-lens equation.

What is a focal plane array?

A two-dimensional detector array placed in the focal plane of an imaging lens, the term used mainly for infrared cameras. Each pixel corresponds to one viewing angle, with an angular size equal to the pixel pitch divided by the focal length.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W. H. Freeman, 2017); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).