Field of view (FOV)

The field of view (FOV) is the part of the scene that an imaging system records: the range of directions it accepts, given as a full angle, or, when the object distance is fixed, the width of the object plane it covers. It is set by the size of the field stop, usually the sensor or an aperture in an intermediate image, and by the focal length that maps angles to positions on it. Camera lenses for a 12.8 mm wide sensor span from about 77° horizontally at 8 mm focal length to about 15° at 50 mm; a microscope objective at 40× shows a field about half a millimeter across.

Angular field of view

For an object far away compared with the focal length, a ray at angle $\theta$ to the axis lands at height $f\tan\theta$ in the focal plane. A sensor of width $d$ therefore accepts a full angle

$$\text{FOV} = 2\arctan\!\left(\frac{d}{2f}\right).$$

For a sensor 12.8 mm wide and 9.6 mm high (16.0 mm diagonal):

Doubling the focal length nearly halves the angle; the relation is exactly inverse only in the small-angle limit. At an object distance of 10 m, the 25 mm lens covers a strip 5.12 m wide and the 50 mm lens 2.56 m. Lenses with strong distortion, such as fisheye designs, do not follow $f\tan\theta$, and their FOV is read from the manufacturer's mapping or measured directly.

Each pixel also has its own instantaneous field of view, the pixel pitch divided by the focal length. A 10 µm pixel behind a 25 mm lens subtends 0.4 mrad, the figure usually quoted for thermal and SWIR cameras and for scanning lidar receivers, where the FOV of each detector element and the angular step of the scan together set the point density.

Field of view in microscopes

In a microscope the object distance is fixed, and the FOV is given as a diameter in the specimen plane. The eyepiece's field stop has a diameter called the field number (FN), in millimeters, and the object-side field is

$$\text{FOV} = \frac{\text{FN}}{M_\text{obj}} ,$$

where $M_\text{obj}$ is the microscope objective magnification, multiplied by any intermediate magnification. With FN 22, a 10× objective shows 2.2 mm and a 40× objective 0.55 mm. A camera sees the sensor width divided by the magnification at the camera port: a 12.8 mm sensor at 40× covers 0.32 mm, smaller than the eyepiece field, so the camera image is a crop of what the eye sees unless a reducing adapter is used.

Field of view and resolution

A wider field at fixed sensor size means fewer pixels per degree, and a wider field at fixed aperture eventually means more aberration at the edge. The useful measure is the number of resolvable elements across the field. A 40×/0.65 objective at 550 nm has an Airy disk radius of $0.61\lambda/\mathrm{NA}$ = 0.52 µm, so a 0.55 mm field holds about 1070 resolvable spots across its diameter, and sampling them at the Nyquist rate needs about 2100 pixels. Designs that combine large fields with high numerical aperture are expensive because the lens must keep diffraction-limited performance over the whole field.

The same constraint appears as conservation of étendue: the étendue an imaging system accepts is roughly the area of its entrance pupil times the solid angle of its field. For a given detector size and acceptance angle, enlarging the field requires a smaller entrance pupil or a larger detector, which is why wide-field, large-aperture systems such as survey telescopes and flash lidar receivers are physically large.

Telecentric lenses

In an object-space telecentric lens the chief rays are parallel to the axis in object space, so the FOV is a fixed linear width instead of an angle, and the image size does not change as the object moves along the axis. Machine-vision gauging relies on this; the front element must be at least as large as the field.

Pitfalls

• Edge quality. The nominal FOV includes the corners, where vignetting and field curvature reduce brightness and sharpness. Natural vignetting alone, roughly $\cos^4\theta$, leaves 88% of the axial brightness at the 14.4° horizontal edge of the 25 mm lens above, and 82% at its 17.7° corner.
• Region of interest. On CMOS sensors the region of interest can be cropped to raise frame rate, which narrows the FOV without any change to the optics.
• Focus distance. At close focus the lens extends, the image distance exceeds $f$, and the angular FOV shrinks below the value calculated at infinity.

Common questions

How is field of view calculated from focal length?

With $\text{FOV} = 2\arctan(d/2f)$, using the sensor width, height or diagonal for $d$. For a 12.8 mm sensor and a 25 mm lens, the horizontal FOV is 28.7°.

What is field number on an eyepiece?

The diameter in millimeters of the eyepiece's field stop, typically 18–26.5 mm. Divided by the objective magnification, it gives the diameter of the specimen area visible through the eyepiece.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).