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**[»] Optics**.

Up to now we have been working exclusively with spherical surfaces which are the historical surfaces in optical design. But modern manufacturing techniques allow to produce *any* shape you want. This is usually referred to as **freeform optics**. Here, we will focus on a subclass of freeform optics which are the surfaces that have radial symmetry and whose surface sag (depth), *z*, is a function of the radial coordinate *r*. We call these surfaces **aspheric surfaces**.

Whereas there has been various attempts to produce non-spherical surfaces with more or less success, it is only in 1956 that the first aspheric lens was commercially produced in a reliable way (see [∞] Wikipedia’s page). By altering the shape of a spherical lens using a radial power series it was possible to correct the spherical aberration of a lens and therefore improves its imaging quality.

There are two ways to specify aspheric surfaces in both the design and manufacturing specification process: through a **conic section** or through a **power series**. Other method exists but are more relevant for generalized freeform optics that doesn’t have radial symmetry (Zernike or Chebyshev surfaces).

On one hand, we already covered conic sections in our post on [»] raytracing:

where *z* is the sag of the lens, *r*, the radial coordinate, *c*, the curvature and, *k*, the conic constant.

On the other hand, the power series looks more straightforward and is defined as

where *z _{sphere}* is the sag of a spherical surface (just use the conic surface with

*k=0*) and the

*a*are the aspheric coefficients.

_{i}There are a few remarks to make about the power series formula.

First, **it is recommended to use **** a_{2}=0** because that terms will compete with the curvature of the lens and may drive your optimization program crazy.

Second, some authors like to use power series on the conic surface itself with *k≠0*. **If you choose to do that, you should use a_{4}=0** as well because the term will compete with the conic constant.

Third, there is a direct link between the *a _{i}* coefficients and the order of the wavefront expansion that you can correct. For instance,

*a*allows to correct the third-order wavefront spherical aberration term,

_{4}*a*to correct the fifth-order wavefront term etc.

_{6}**There is then no need to toggle all the aspheric coefficients terms during optimization.**Instead, start with

*a*, then toggle

_{4}*a*if it’s not enough and so on.

_{6}As a conclusion, for third-order aberrations correction, you can use either the conic constant or a power series but not both. If you feel the need to correct higher aberration, the power series is best suited. However, there is no analytical solution to the power series surface *at contrario* to the conic section so its raytracing will be slower.

Just like with spherical surfaces, there exists [»] Seidel formulas for aspheric surface.

The aspheric contribution for an aspheric surface with the STOP placed at the lens is:

with *h* the intercept height of the marginal ray and *∆**n* the change in refractive index at the interface.

When the STOP is not at the lens, you can apply the [»] stop-shift equations.

The formulas are given for an *a _{4}* term and therefore the power-series version of the aspheric surface. You can get the equivalent

*a*term for a conic constant,

_{4}*k*, and curvature,

*c*, using the formula

A consequence of this latter formula is that conic surfaces can only produce aspheric behavior if they have some curvature. We will come back on this case in a future post where we will show an aspheric surface that has no power.

To illustrate how to use aspheric surfaces, we will see how to create an aspheric lenses that is aplanatic (no coma and no spherical aberration).

In our post on [»] thinlenses, we have seen that the bending of the lens affect both the spherical aberration and the coma. While there is always a bending that will produce zero coma, there is not always one that will produce zero spherical aberration.

We have seen that by [»] moving the STOP we can cancel the coma for a given known amount of spherical aberration that we can minimize for a given thinlens. But there are two objections to this method: first we have to fix the STOP to a given position which is not always possible (for instance, how do you do when you want to apply the same trick to multiple lenses in a system?), second the spherical aberration is never perfectly zero because if it were we could not correct coma by moving the stop (see [»] here why). There are a [»] few cases where a single lens can be aplanatic but they are very restrictive so I’m not counting them in.

With aspheric surface, we can correct both coma and spherical aberration.

First, select the lens bending that will cancel coma

Then, select the *a _{4}* term that will cancel the remaining spherical aberration

with

as we have shown in our previous post on [»] thinlenses.

The *h ^{4}* terms cancel out and the asphericity only depends on power, magnification ratio and refractive index since the partition factor has been fixed previously to cancel coma.

An example lens is given in Figure 1. It features a 10D lens optimized for no spherical aberration and no coma at 587 nm over a field of 2° for an aperture of 10 mm.

The MTF is given in Figure 2. As it can be expected by the remaining astigmatism and petzval, the performances are only good on-axis. At 2° a large drop can already be observed due to the strong field curvature of that lens. The performances are however diffraction-limited up to 1°. The off-axis aspect will be fixed in a later post, no need to rush for now.

The important aspect is that we have updated our portfolio of lenses designs that are aplanatic. And we know from the [»] STOP shift theory that aplanatic lenses only suffer from constant astigmatism and Petzval – a great step forward to a 3^{rd} order fully-corrected optical design.

So, are aspheric lenses the ultimate solution for lens design? Well… it depends.

Aspheric lenses are indeed a great tool but they come with a price. Because the manufacturing technology is different (CNC machining *vs* grinding) you will typically pay much more for aspheric lenses when ordering volume. In my experience, it is common to have a 6-fold price difference between the same lens in aspheric *vs* spheric form for a batch of 10 lenses. For only 1 lens the price shall be about the same due to the tooling cost of spheric lenses but it is uncommon to order only one lens, even if you intend to build only one instrument.

There is actually one advantage of aspheres: weight. Aspheric lenses are typically used in projects where weight and/or space are major constrains. Instruments for space application or photographic lenses often rely on aspheric surface to avoid putting multiple elements when they are not strictly necessary. There it is common to bring the aspheric lens into play.

Still, it is a very interesting tool in our arsenal and in the next post I will show one more inventive usage of aspheric surfaces by correcting the residual spherical aberration in our acomatic anastigmat lens!

Want to discuss this further? Check out our new [∞] community board!

I would also like to give a big thanks to **Naif**,** Young**,** Samuel**,** James**,** Andrew**,** Lilith**,** Sebastian**,** Hitesh**,** Jesse**, **Alex**,** Stephen**,**Sivaraman**,** Jon**,** Cory**,** Karel**,** Aviv**, **Themulticaster**,** Tayyab**,** Marcel**, **Kirk **and **Dennis** who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments!

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[»] #DevOptical Part 21: Aplanatic Lenses