For lenses, what is the practical difference between plano- and bi-?

Question

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.

Answer 1

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.

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Examples:

• https://www.thorlabs.com/navigation.cfm?guide_id=2272

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.

• https://www.lasercomponents.com/de/?embedded=1&file=fileadmin/user_upload/home/Datasheets/lc-optics/substrates/bestform_lenses.pdf&no_cache=1

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.