= Focal ratio
{title2=$F=f/D$}
The <focal ratio>, or f-number, of a converging <optical instrument> focused at infinity is the <dimensionless quantity>
$$
N=F=\frac{f}{D},
$$
where $f>0$ is effective <focal length> and $D$ is the on-axis <entrance pupil> <diameter>. The notation f/4 means $N=4$, not $f=4$ in a particular unit. A $100\,\mathrm{mm}$ <focal length> with $D=25\,\mathrm{mm}$ gives f/4. Closing the <aperture stop> until $D=12.5\,\mathrm{mm}$ gives f/8 while retaining the same <focal length>. The mechanical opening need not equal $D$. https://www.edmundoptics.com/knowledge-center/application-notes/imaging/lens-iris-aperture-setting/[Gregory Hollows and Nicholas James, Edmund Optics, System Throughput, f-number, and Numerical Aperture]{ref}.
For a <thin lens> in air with a circular <aperture stop> at the <thin lens>, the converging bundle from a distant on-axis point has half-<angle> $\alpha$ satisfying
$$
\tan\alpha=\frac{D}{2f}=\frac{1}{2N},\qquad \alpha\simeq\frac{1}{2N}\quad\hbox{in radians}.
$$
The last step uses the <small-angle approximation>. For a compound <optical lens>, the cone is set by the image-side <optical pupil> and its distance from the receiving plane; the simple triangle is not an exact high-<angle> rule. At finite object distance a <thin lens> instead gives $v=f(1+|m|)$, by its imaging equation and <transverse magnification>, so its working cone corresponds to $v/D=N(1+|m|)$. Nominal <focal ratio> therefore does not alone specify close-range image <irradiance>. https://www.edmundoptics.com/knowledge-center/application-notes/imaging/lens-iris-aperture-setting/[Hollows and James, working f-number and image-space cone]{ref}.
Two useful scalings follow under stated conditions. For a fixed uniform incident beam and unchanged transmission, collected <radiant flux> scales as $D^2$. For an extended source with fixed <radiance>, the central image <irradiance> scales approximately as $N^{-2}$ in the <paraxial approximation>: the <solid angle> of the cone scales as $(D/f)^2$. For an ideal uniformly illuminated unobscured circular <entrance pupil>, the first dark ring of the <point spread function> has angular <radius> and <focal plane> <radius>
$$
\theta_1\simeq1.22\frac{\lambda}{D},\qquad r_1\simeq f\theta_1=1.22\lambda N.
$$
Here $\lambda$ is <wavelength>. These are first-zero radii, not diameters or a universal rule for resolving arbitrary objects. A proof outline evaluates the circular <aperture>'s <Fraunhofer diffraction> using its <Fourier transform>, then uses the first zero of the resulting <Bessel function>. <Oscillations and Waves by Richard Fitzpatrick/Two-Dimensional Fourier Optics>[Richard Fitzpatrick, University of Texas at Austin, Two-Dimensional Fourier Optics, circular-aperture first minimum]{ref}.
At $\lambda=0.55\,\mathrm{\mu m}$, the example's f/4 and f/8 first-ring radii are $2.684$ and $5.368\,\mathrm{\mu m}$. Closing the <aperture stop> halves $D$, quarters collected <radiant flux> and doubles this <diffraction> radius. It can also reduce <optical aberrations> by excluding outer <light rays>; actual sharpness reflects both effects. This explains why a smaller <aperture> need not produce a sharper image. https://www.edmundoptics.com/knowledge-center/application-notes/imaging/limitations-on-resolution-and-contrast-the-airy-disk/[Gregory Hollows and Nicholas James, Edmund Optics, The Airy Disk and Diffraction Limit]{ref}.
\Image[/aperture-focal-ratio-field-view-cone-scaling.png]
{title=<Aperture> and <focal ratio>: a common <optical focus>, reduced collection and increased <diffraction> radius}
{description=Original analytical illustration. Left: equal-scale <thin lens> geometry with $f=100\,\mathrm{mm}$ and $D=25$ or $12.5\,\mathrm{mm}$. Right: collected <radiant flux> relative to f/4 is $(4/N)^2$, and first-ring <radius> relative to f/4 is $N/4$. The <wavelength> is $0.55\,\mathrm{\mu m}$; the <entrance pupil> is circular, uniformly illuminated and unobscured. These are calculated quantities, not measurements.}
A proposed <optical bench> check puts a fixed <collimator> before a converging <thin lens> and a <photodetector> large enough to collect the entire image behind it. Compare two centred <aperture stops> of known <diameters> while fixing source output, detector gain and integration time. The integrated signal tests the $D^2$ collection law; a spatially resolved <optical camera> at the <focal plane> can separately measure the <point spread function>. Repeat a dark reading and the first <aperture stop>, and check that neither the incoming beam nor detector clips either bundle. A resolved pinhole source, imperfect collimation, <optical aberrations> and detector pixel size can dominate the apparent spot width; a single peak reading does not measure total collected <radiant flux>.
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