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I don't know if this is a trivially obvious question for astronomers. If so please forgive my ignorance.

I've looked around at a lot of diagrams of lenses in telescopes. The plano-convex and bi-convex lenses both seem to converge light, but the plano-concave and bi-concave lenses both seem to diverge the light.

What is the practical difference? Why would you choose a bi- instead of a plano- for any given purpose?

The bi- is obviously twice as much work to manufacture so I would like to know what are its benefits. Note I'm assuming a thin lens in air for telescopes.

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I don't know if this is a trivially obvious question for astronomers...

Astronomers (amateur, pro, or otherwise) are a diverse bunch; some are intimately involved in telescope and instrument design, others spend most of their time thinking about Higgs bosons and metric tensors. So no, your question is not trivial for any group.

In fact, it's a great question! Because it highlights that lens design is all about surfaces and ratios of refractive indices, not lenses.

The problem is that the surfaces of a lens based on Snell's law

$$\sin\theta_2 = \frac{n_1}{n_2} \ \sin\theta_1$$

would not be spherical, but spherical surfaces are what we get when grinding (unless one goes out of one's way).

Spherical surfaces are almost never the optimum shape for a lens system design, but it is so much easier to make them than aspherical surfaces that some lens designs will even include extra spherical lenses to avoid (or at least minimize) aspherical surfaces.

Spherical surfaces contribute to spherical aberration, and Wikipedia's Spherical aberration; correction explains that for a given geometry (for example point-to parallel, like a telescope objective) you can choose the radii of the two surfaces to minimize the spherical aberration while keeping the focal length constant.

That's why if you take the objective lens of a refractor apart, you'll see that both the positive and negative lenses have curved surfaces on both sides for correcting chromatic aberration. There are three radii of curvature (if the middle surfaces are matched (and sometimes glued together) or four radii if they are air-gapped are calculated to minimize aberration.

Of course there are constraints on these; the designer is likely to have decided the final focal length ahead of time, and the relationship between the focal lengths of two elements in the achromat is determined by their relative indices of refraction.

The shape and design of a single bi-convex lens calculated for point-to-parallel conjugation is called a best form lens.


Examples:

Thorlabs' N-BK7 Best Form lenses are designed to minimize spherical aberration while still using spherical surfaces to form the lens. They provide the best possible performance from a spherical lens for collimating and focusing beams. We offer best form lenses uncoated, or with an AR coating for 350 - 700 nm, 650 - 1050 nm, or 1050 - 1700 nm.

Best form lenses are positive lenses with minimized spherical aberrations. They are used if the highest demands are made of the spot image. The spherical aberration is clearly defined by the diameter of the incident beam and its wavelength. If these values are known, then the radii of curvature of the lens can be designed to create as low an aberration as possible.

Best form lenses generally have better imaging qualities than conventional positive lenses.

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    $\begingroup$ The mismatched parentheses in the third paragraph are making me uncomfortable. $\endgroup$
    – IMSoP
    Jan 12, 2023 at 10:47
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    $\begingroup$ Worth noting, in regards to best form lenses, that the two surfaces are not the same radius, and it matters which face is towards the parallel/collimated light (i.e. towards the "infinite" distance). When I was in school, the professor actually said that if you were stuck between choosing a plano-convex and a double-convex lens (where the radius is the same on both sides) for a telescope objective, its better to choose the plano-convex and have the radiused side towards infinity, so the amount of refraction is divided more evenly between the two surfaces. $\endgroup$ Jan 12, 2023 at 14:29
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    $\begingroup$ @IMSoP Yikes! It must have been a difficult day, exacerbated by my straying from SE duties for almost 24 hours! :-) I've refactored the offending paragraph; thanks for the heads up. $\endgroup$
    – uhoh
    Jan 12, 2023 at 21:51
  • $\begingroup$ @nflemming2004 Interesting! Yes for a parallel to point configuration, "curvy-side-out" is definitely better; it has lower $\sin(\theta)$ meaning less deviation from the small angle approximation. But I didn't realize that a plano-convex in that configuration is better (wrt spherical aberration) than a symmetric bi-convex of the same focal length; thanks for that! $\endgroup$
    – uhoh
    Jan 12, 2023 at 22:00
  • $\begingroup$ I appreciate the details but I'm still missing something. First off, I understand the difference between spherical and aspherical. Ideally you want the parabola which focuses to an exact point, whereas spherical does not. But I'm still not sure on the difference between plano-convex and bi-convex. They both seem to do the same thing. Does the bi-convex focus more quickly (smaller focal-length) for a given diameter than the plano-convex? That doesn't seem possible because you could just make a sharper curve for the plano-convex (I think). $\endgroup$
    – DrZ214
    Jan 12, 2023 at 22:23

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