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A dioptre, also spelled diopter, is the unit of optical power equal to one reciprocal metre. In $\Phi =1/f$, convert focal length to metres before taking the reciprocal: $f=25\mathrm{cm}=0.25\mathrm{m}$ gives $\Phi =4\mathrm{D}$, not $0.04\mathrm{D}$. A dioptre measures focusing strength, distinct from the transverse magnification of an image.

The optical power is reciprocal signed focal length, $\Phi =1/f$. With $f$ expressed in metres, optical power is expressed in dioptres: $f=1\mathrm{m}$ gives $\Phi =+1\mathrm{D}$, while $f=-0.5\mathrm{m}$ gives $\Phi =-2\mathrm{D}$. The sign distinguishes converging and diverging optical lens elements; the magnitude measures how strongly they change the direction of a light ray at a given height. ^[ref].

For two ideal thin lens elements in contact in air, optical powers add: ${\Phi }_{\mathrm{total}}={\Phi }_1+{\Phi }_2$. Thus $+2\mathrm{D}$ and $-1\mathrm{D}$ combine to give $+1\mathrm{D}$, or $f_{\mathrm{total}}=1\mathrm{m}$. Separation changes the rule: for two ideal thin lens elements separated by $d$ in air, their equivalent optical power is ${\Phi }_1+{\Phi }_2-d{\Phi }_1{\Phi }_2$. The contact rule is the $d=0$ case.^[ref]

For regular astigmatism, the two principal meridians of the eye are perpendicular and have different optical powers, commonly because the cornea has unequal curvatures. The crystalline lens can also contribute. A point object consequently does not form one sharp point image: the idealised paraxial optics gives two focal lines at different distances, perpendicular to one another, with an intervening region of blur. An ordinary spherical lens changes the focus in both meridians together and cannot by itself remove their difference. Irregular astigmatism, for example with a scarred cornea, lacks this simple pair of uniform principal powers.^[ref]

Several complementary observations reveal it:

The prescription must specify both the difference in optical power and its orientation. If the required correcting powers in the two principal directions are $R_1$ and $R_2$, choose the cylinder axis along direction 1. In the ideal thin lens model:Here $S$ is the spherical correcting power, $C$ the cylindrical correcting power and $\alpha$ the cylinder axis, all powers measured in dioptres. A cylindrical lens contributes no extra power along its axis and its full power perpendicular to that axis. For example, required corrections of $-1$ and $-3$ dioptres are supplied by a sphere of $-1$ dioptre and a cylinder of $-2$ dioptres, with its axis in the direction requiring the smaller negative correction. These are correcting powers, not the uncorrected eye's intrinsic powers; confusing the two reverses the intended correction.

Spectacles with this sphere-and-cylinder combination are the central historical remedy. Present options also include toric contact lenses, appropriate rigid contact lenses for some irregular cases, and selected refractive surgery. The suitability of these depends on the cause and condition of the cornea; a spectacle cylinder cannot generally cancel all irregularities of an irregular cornea.^[ref]

Radiant flux is the power carried by electromagnetic radiation: radiation energy transferred per unit time, measured in watts, $\mathrm{W}=\mathrm{J}{\mathrm{s}}^{-1}$. Specify what is counted: all radiation emitted by a source, radiation within one beam, or radiation intercepted by a photodetector. Writing $Q(t)$ for the cumulative energy crossing the specified boundary givesThe first expression is an instantaneous derivative; the second is an average over the stated time interval. If $0.020\mathrm{J}$ arrives during $0.010\mathrm{s}$, the mean radiant flux is $2\mathrm{W}$. The photodetector's absorbed power equals this incident radiant flux only if it absorbs all of the incident radiation.

Keeping radiant flux fixed while widening a beam reduces its irradiance, the incident power per receiving area. Radiance additionally resolves the propagation direction and has units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$; an optical camera can collect different radiant fluxes from sources with the same radiance but different apparent solid angles. Luminous flux, measured in lumens, weights the optical spectrum by the human eye's standardized response instead of counting energy equally at every wavelength. Equal radiant fluxes need not give equal luminous fluxes.^[ref]

Radiant power also differs from the optical power of an optical lens, which is reciprocal focal length in dioptres. Likewise, astronomical radiative flux is power per area. Stating the units prevents these shared words from hiding different physical quantities.