<blockquote>
<p>I don't know if this is a trivially obvious question for astronomers...</p>
</blockquote>
<p>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 <a href="https://en.wikipedia.org/wiki/Higgs_boson" rel="nofollow noreferrer">Higgs bosons</a> and <a href="https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)" rel="nofollow noreferrer">metric tensors</a>. So no, your question is not trivial for any group.</p>
<p>In fact, <em>it's a great question!</em> Because it highlights that lens design is all about surfaces and <em>ratios of refractive indices</em>, not lenses.</p>
<p>The problem is that the surfaces of a lens based on <a href="https://en.wikipedia.org/wiki/Snell%27s_law" rel="nofollow noreferrer">Snell's law</a></p>
<p><span class="math-container">$$\sin\theta_2 = \frac{n_1}{n_2} \ \sin\theta_1$$</span></p>
<p>would not be spherical, but spherical surfaces are what we get when grinding (unless one goes out of one's way).</p>
<p>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.</p>
<p>Spherical surfaces contribute to spherical aberration, and Wikipedia's <a href="https://en.wikipedia.org/wiki/Spherical_aberration#Correction" rel="nofollow noreferrer">Spherical aberration; correction</a> 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.</p>
<p>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 <a href="https://en.wikipedia.org/wiki/Chromatic_aberration" rel="nofollow noreferrer">chromatic aberration</a>. 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.</p>
<p>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.</p>
<p>The shape and design of a single bi-convex lens calculated for point-to-parallel conjugation is called a <em>best form lens</em>.</p>
<hr />
<p>Examples:</p>
<ul>
<li><a href="https://www.thorlabs.com/navigation.cfm?guide_id=2272" rel="nofollow noreferrer">https://www.thorlabs.com/navigation.cfm?guide_id=2272</a></li>
</ul>
<blockquote>
<p>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.</p>
</blockquote>
<ul>
<li><a href="https://www.lasercomponents.com/de/?embedded=1&amp;file=fileadmin/user_upload/home/Datasheets/lc-optics/substrates/bestform_lenses.pdf&amp;no_cache=1" rel="nofollow noreferrer">https://www.lasercomponents.com/de/?embedded=1&amp;file=fileadmin/user_upload/home/Datasheets/lc-optics/substrates/bestform_lenses.pdf&amp;no_cache=1</a></li>
</ul>
<blockquote>
<p>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.</p>
<p>Best form lenses generally have better imaging qualities than
conventional positive lenses.</p>
</blockquote>

CC BY-SA 4.0
2023-01-12T10:47:24.860Z
The mismatched parentheses in the third paragraph [are making me uncomfortable](https://xkcd.com/859/).
IMSoP
CC BY-SA 4.0
2023-01-12T14:29:55.903Z
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.
nflemming2004
CC BY-SA 4.0
2023-01-12T21:51:04.203Z
@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.
uhoh
CC BY-SA 4.0
2023-01-12T22:00:10.860Z
@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!
uhoh
CC BY-SA 4.0
2023-01-12T22:23:56.893Z
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).
DrZ214
CC BY-SA 4.0
2023-01-12T22:36:23.993Z
@DrZ214 Side note first: an optimized parallel to point aspherical *mirror* will be a parabola, but for a lens, not even a parabola (nor any other conic) is the exact correct shape. Okay as far as plano-convex vs bi-convex, try to think of it differently. All lenses have two surfaces with radii of curvature $R_1, R_2$ and from time to time one of the two radii just happens to be infinite, meaning a flat surface. It doesn't become a different kind of lens, those distinctions, as well as that of a *meniscus lens* (with one positive and one negative surface) are all arbitrary.
uhoh
CC BY-SA 4.0
2023-01-12T22:40:47.350Z
We have those different classifications to make it easer for people to talk about the shapes, but in terms of optics, it's only the surfaces that matter. Each optical surface doesn't have any information about the other surfaces. But if you really want to treat a lens as a single unit, then you can use  $$1/R_{eff} = 1/R_1 + 1/R_2$$ where $R_{eff}$ is the radius of curvature for a one-side plano lens of equivalent focal length. So if you have a lens with convex surfaces of radii +20 and +30 cm, then it has the same focal length as a plano-convex lens with a radius of 12 cm.
uhoh
CC BY-SA 4.0
2023-01-12T22:46:10.793Z
@DrZ214 note that I've treated positive radius as convex. In formal lens design, light is supposed to come from one direction and for a bi-convex for example, the radius of curvature of the first side is positive (because the center of curvature is downstream) and the second radius of curvature is defined as negative because the center is upstream. You will see both ways of defining the sign of radius of curvature as you read further elsewhere. To convert from radius of curvatures to focal length, see [thin lens;focal length](https://en.wikipedia.org/wiki/Thin_lens#Focal_length):
uhoh
CC BY-SA 4.0
2023-01-12T22:46:52.283Z
@DrZ214 $$\frac{1}{f} \approx (n-1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$ assuming the index of the glass is $n$, the index of the air on either side is 1, and the signs of $R_1, R_2$ are defined using the upstream-negative, downstream-positive method. So if your glass has an index $n=1.5$, and the biconvex lens has radii of +20 and +30 cm, then the focal length is about 24 cm.
uhoh
CC BY-SA 4.0
2023-01-11T22:29:52.843Z
2023-01-12T22:12:35.477Z
2023-01-12T22:12:35.477Z
uhoh
<p>I don't know if this is a trivially obvious question for astronomers. If so please forgive my ignorance.</p>
<p>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.</p>
<p>What is the practical difference? Why would you choose a bi- instead of a plano- for any given purpose?</p>
<p>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.</p>

<telescope><optics><telescope-lens>
For lenses, what is the practical difference between plano- and bi-?
DrZ214
# For lenses, what is the practical difference between plano- and bi-?

(Asked by: username_0 on 2023-01-11)

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.

**Tags:** telescope, optics, telescope-lens


---

## 1 answers

### Answer by username_1 on 2023-01-11. Score: 10

> 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](https://en.wikipedia.org/wiki/Higgs_boson) and [metric tensors](https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)). 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](https://en.wikipedia.org/wiki/Snell%27s_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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/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:

* <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.

> **Comments:**
> **username_2**: The mismatched parentheses in the third paragraph [are making me uncomfortable](https://xkcd.com/859/).  
> **username_3**: 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.  
> **username_1**: @username_2 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.  
> **username_1**: @username_3 *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!  
> **username_0**: 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).  
> **username_1**: @username_0 Side note first: an optimized parallel to point aspherical *mirror* will be a parabola, but for a lens, not even a parabola (nor any other conic) is the exact correct shape. Okay as far as plano-convex vs bi-convex, try to think of it differently. All lenses have two surfaces with radii of curvature $R_1, R_2$ and from time to time one of the two radii just happens to be infinite, meaning a flat surface. It doesn't become a different kind of lens, those distinctions, as well as that of a *meniscus lens* (with one positive and one negative surface) are all arbitrary.  
> **username_1**: We have those different classifications to make it easer for people to talk about the shapes, but in terms of optics, it's only the surfaces that matter. Each optical surface doesn't have any information about the other surfaces. But if you really want to treat a lens as a single unit, then you can use  $$1/R_{eff} = 1/R_1 + 1/R_2$$ where $R_{eff}$ is the radius of curvature for a one-side plano lens of equivalent focal length. So if you have a lens with convex surfaces of radii +20 and +30 cm, then it has the same focal length as a plano-convex lens with a radius of 12 cm.  
> **username_1**: @username_0 note that I've treated positive radius as convex. In formal lens design, light is supposed to come from one direction and for a bi-convex for example, the radius of curvature of the first side is positive (because the center of curvature is downstream) and the second radius of curvature is defined as negative because the center is upstream. You will see both ways of defining the sign of radius of curvature as you read further elsewhere. To convert from radius of curvatures to focal length, see [thin lens;focal length](https://en.wikipedia.org/wiki/Thin_lens#Focal_length):  
> **username_1**: @username_0 $$\frac{1}{f} \approx (n-1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$ assuming the index of the glass is $n$, the index of the air on either side is 1, and the signs of $R_1, R_2$ are defined using the upstream-negative, downstream-positive method. So if your glass has an index $n=1.5$, and the biconvex lens has radii of +20 and +30 cm, then the focal length is about 24 cm.  

---

