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First, problem background: I'm attempting to source lenses for use in eyeglasses to deal with extreme sensitivity to chromatic aberration, and likely need either achromatic doublets or single lenses in very low dispersion material (e.g. various fluorite crown glasses with Abbe numbers up to 90 or so).

I've found several optical manufacturers/labs that advertise single-item custom optics and seem to have the capabilities to produce what I want, but I don't have any background in lens design to know what kind of design parameters I'd need to provide to make a meaningful inquiry, much less an order.

My prescription is moderately high negative, -6.25 and -7 diopters, with cylinder components (yes, let's make it harder) of -1 and -2, respectively. Most of the non-ophthalmic optics sources I've found list their regular stock (non-custom) items in just focal length and diameter. Obviously pure spherical diopters can be converted to a focal length if needed, but I'm not sure how the cylinder components come into play and possibly complicate or preclude certain designs (for doublets?), and what other parameters might be relevant/important (for example, outer curvature which seems to be something of a free parameter but maybe there are reasons it should correspond in some way to the eyes).

What do I need to learn to go about translating my prescription into something I might be able to actually get made?

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  • $\begingroup$ Cylinder components imply a different focal length in different planes. For example, a +4 sphere with an additional +1 vertical axis cylinder means a 25-cm focal length in the vertical plane while a 20-cm focal length in the horizontal plane. In a doublet, obviously the inner curvature of the outer lens must match the outer curvature of the inner lens. This is probably achieved by default by these curvatures being spherical, which conceptually may introduce chromatic aberrations. The idea of the overall outer curvature probably is to have its center in the center of the eye rotation. $\endgroup$ – safesphere Dec 21 '17 at 4:53
  • $\begingroup$ There is also a relation among the prescription, lens thickness, refractive index, and reflections. Stronger prescriptions, like yours, make the glass thick (along the edge for negative). For this reason often manufacturing of specialty glasses is limited by a certain optical power, especially of the cylinder part. You can reduce the thickness by increasing the refractive index, but this in turn increases the amount of light reflected. So an anti-reflective coating is a must, but it is not 100% effective. $\endgroup$ – safesphere Dec 21 '17 at 5:10
  • $\begingroup$ @safesphere: Are you talking about single lenses or doublets? As a general rule (not always true; the big exception which I'm actually considering and have some potential sources for is sapphire) higher index has increasingly bad dispersion/low Abbe and wouldn't be a potential solution for single lenses anyway. $\endgroup$ – R.. GitHub STOP HELPING ICE Dec 21 '17 at 5:20
  • $\begingroup$ Sorry, which part of my comments are you asking about? I think most of what I mentioned would apply to both types, except for stronger reflection that applies mostly to the inner lens of the doublet, because the anti-reflective coating is not applied on the outer lens. Do you know what it would cost to make a pair of sapphire lenses? They would outlast any frame with no scratches, except for the anti-reflective coating that would eventually wear off. How would you shape them to the frame? Only diamond can sand them, I assume, no regular lab would agree or have the skills? $\endgroup$ – safesphere Dec 21 '17 at 5:37
  • $\begingroup$ @safesphere: I meant the second comment where you mentioned thickness and reducing it. I'm content with all options (except of course sapphire if I consider that a real option) being very thick - the single lenses because low-index, and the doublets because they're doublets. $\endgroup$ – R.. GitHub STOP HELPING ICE Dec 21 '17 at 5:47
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You will need to learn about the generalized Coddington Equations, which are simply Snell's law applied to an interface between two mediums and relate the Hessian matrix (called the "curvature matrix" in the jargon that people who use the Coddington equations speak) of a general quadric refracting surface and the direction cosines of the rays refracted by that surface. See, for example,

Charles E. Campbell, "Generalized Coddington equations found via an operator method", JOSA-A, 23, #7, p1691

You're going to have to apply them to several wavelengths at once to come up with the design of doublet lens with the correct powers in each direction and whose powers are set equal at two target wavelengths.

You will probably benefit from a commercial raytracer software such as Zemax; unfortunately these are all grotesquely overpriced given that they are merely glorified Snell law calculators, with an optimized attached.

You'll also need to be fluent in the language of the ophthalmologist's prescription, although it sounds as though you may be more advanced along this road than I; for what it's worth, my answer here is a primer.

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