= Field of view
{wiki}
The <field of view> is the region of object directions sampled by an <optical instrument> in one pointing. State whether it is horizontal, vertical or diagonal, and whether it is a full <angle> or a half-<angle>. An object-plane width in millimetres also describes a <field of view>, but requires a specified object distance. Neither that width nor the angular <field of view> is the <aperture> <diameter> or the <diffraction-limited resolution>.
For a centred flat rectangular <photodetector> of width $w$ in the <focal plane>, with positive effective <focal length> $f$, infinity focus and ideal rectilinear imaging without <optical distortion>, the full horizontal <field of view> is
$$
\Theta_x=2\arctan\left(\frac{w}{2f}\right).
$$
The model maps an object direction $\theta$ to image height of magnitude $|y|=f|\tan\theta|$; set $|y|=w/2$ and use the <inverse tangent>. This retains exact <right triangle> geometry within the stipulated imaging model; a paraxial <focal length> alone does not guarantee that a real wide-field <optical lens> obeys it. With detector height $h$, replace $w$ by $h$ for the vertical <field of view>, and by $\sqrt{w^2+h^2}$ for the diagonal <field of view>. The <small-angle approximation> gives $\Theta_x\simeq w/f$ in radians. https://www.edmundoptics.com/knowledge-center/application-notes/imaging/understanding-focal-length-and-field-of-view/[Gregory Hollows and Nicholas James, Edmund Optics, Understanding Focal Length and Field of View]{ref}.
For an original worked example, a $36\,\mathrm{mm}$ detector width gives $\Theta_x\approx39.60^\circ$ at $f=50\,\mathrm{mm}$, and $20.41^\circ$ at $f=100\,\mathrm{mm}$. Keeping $D=25\,\mathrm{mm}$ makes these f/2 and f/4 respectively: the <field of view> changes because $f$ changes. By contrast, closing $D$ from $25$ to $12.5\,\mathrm{mm}$ at $f=100\,\mathrm{mm}$ changes f/4 to f/8 while retaining $20.41^\circ$ in this model. The clean separation is useful: <entrance pupil> <area> controls collection, <focal ratio> controls the paraxial cone, and $w/f$ controls angular image coverage. Edge clipping, <optical distortion>, <field curvature> and <optical aberrations> can restrict the practically usable <field of view>.
\Image[/aperture-focal-ratio-field-view-sensor-geometry.png]
{title=<Field of view>: a fixed detector width subtends a smaller <angle> at longer <focal length>}
{description=Original ideal rectilinear model. Left: central <light rays> cross the <thin lens> centre and reach the two detector edges, with equal coordinate scales in both panels; the opening is $25\,\mathrm{mm}$ in each. Right: calculated full horizontal <field of view> for $w=36\,\mathrm{mm}$, with the $50$ and $100\,\mathrm{mm}$ examples marked. The detector lies in the <focal plane>, the object is at infinity, and <optical distortion> is absent. No plotted quantity is measured data.}
A proposed angular readout on an <optical bench> illuminates a small asymmetric target at the <optical focus> of a <collimator>, providing a bundle with a known direction. Rotate the <optical camera> about the centre of its <entrance pupil>, reading a graduated rotation stage, and record the target image position on its fixed <photodetector>. Centre the target to establish zero; turn left and right until its centre reaches each detector edge. The difference between stage readings measures the full horizontal <field of view>. Repeat with a smaller <aperture stop>: unchanged edge directions but reduced signal support the distinction between collection and angular coverage. Keep the same active detector width, infinity focus, illumination, integration time and gain; approach each reading from the same direction to check mechanical backlash. A finite target blurs the edge criterion, detector thresholds can imitate lost coverage, and a wide bundle must illuminate the whole <entrance pupil> at every tested <angle>. These checks distinguish a geometrical boundary from insufficient signal. The analytical diagram is the prediction for this protocol, not a record of its execution.
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