Optics with respect to lens manufacturing This may not be the right place to ask, but I'm sure someone knows something about it. How does an eyeglass prescription translate into the exact dimensions and curvature of the lens?
 A: You'll need to research details but eyeglasses do five main things:


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*Change the optical power of the eye's lens; ("Sphere" correction)

*Correct optical aberration, mostly astigmatism; ("Cylinder" correction)

*Allow for differences in parameters 1. and 2. between the two eyes; (Differentiation of left (OS) and right (OD) eyes)

*Align the sight of both eyes; ("Prismatic" correction, i.e. tilting the optical axis of one eye relative to the other)

*Allow for several different versions of each lens with different optical powers for each eye (multifocal lenses: the "Add" part of a prescription)


Excluding (5), (1) through (4) define two ellipsiodal surfaces with a "wedge" angle between them.
To discuss these, I shall use a cylindrical polar description, i.e. the $z$ axis along the lens's optic axis and $r$ and $\phi$ the radial and azimuthal co-ordinates describing points in the transverse direction.
The basic equation is the Lens Maker's Formula (see the HyperPhysics page for definition and signs of symbols):
$$P = \frac{n_{lens}-n_0}{n_0}\left(\frac{1}{R_1}-\frac{1}{R_2}\right)\tag{1}$$
If the surface radiusses $R_1,\,R_2$ are in meters, then the power $P$ will be in inverse meters (diopters), which is the same units as used on a prescription.
A pure "Sphere" prescription defines two spherical surfaces with $R_1$ and $R_2$ chosen to give the right amount of optical power whilst (1) keeping spherical aberration acceptably small, (2) defining a lens that will fit into spatial envelope available for it and (3) heeding prevailing fashion considerations.
"Cylinder" means that the power $P$ depends on the azimuthal angle $\phi$. This means that the surfaces are no longer spherical but ellipsoids with sectional curvatures that have a $\cos(2\,(\phi-\phi_0))$ dependence when the sectioning plane contains the z-axis. Thus there is no unique focal point: the focus moves as a fan of rays confined to a lone azimuthal plane changes their $\phi$ value. The "Axis" part of the prescription tells you the orientation $\phi_0$ of the azimuthal angle of maximum/ minimum optical power. Thus, a lens with a "Sphere" prescription $s$ diopters, "Cylinder" prescription $c$ diopters and "Axis" $\phi_0$ has a power versus azimuthal angle dependence:
$$P = s + c\,\cos(2\,(\phi-\phi_0))\tag{2}$$
where $\phi_0=0$ corresponds to horizontal (for a standing person) and $\phi$ is measured in an anticlockwise sense from the standpoint of someone looking into the patient's face.
The "Prism" part of the prescription adds a tilt between the two spherical surfaces to correct the tendency of the two eyes to focus in different directions and make them see along a common optical axis. The tilt is the angular displacement imparted to a plane wave: you can work this out from first principles from Snell's law. The "Prism" is one hundred times the radian angle of tilt (strictly speaking, the tangent of the angle of tilt, but the values are so small that it doesn't matter, and the including of "tangent" in the definition makes it independent of angular units).
Point 5 means that several different copies of the same lens but with different values of $s$ (in Eq. (2)) in each of the two eyeglasses. The "Add" parts of a prescription tells the optometrist how much $s$ to add / subtract for the multifocal copies of the lens. In refractive optics, this is done by literally stacking different versions of the lens, which the wearer chooses by their gaze. In diffractive prosthetics that surgically replace the eye's natural lens, there are several holograms written over each other. The brain then swiftly learns to choose the in-focus image on the retina and ignore the others, so that the wearer simply subconsciously focusses by choosing which image to heed at will.
