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Up to now I have shown how to make systems with [»] zero coma and zero astigmatism, [»] zero coma and zero spherical and [»] zero coma, astigmatism and spherical. But all of them still suffered from [»] Petzval field-curvature. This is normal because we have shown [»] here that single-lens systems cannot be corrected for Petzval and that [»] moving the STOP does not affect the Petzval term either.

When the system has remaining Petzval curvature, the off-axis rays will focus at a different longitudinal position that is proportional to the square of the image height. The point will therefore focus on a sphere whose radius is proportional to the value of the Petzval term. At moderate field angles this is not an issue but it will quickly limit the system performances at wider angles.

We will see here how to cancel the Petzval term, leaving only the last Seidel aberration, distortion. Note that distortion is rarely an issue because it can be post-corrected in imaging systems as it only affects the metric of the system and still produces sharp images.

The Petzval introduced by a [»] thinlens is directly proportional to its power, *P*, and inversely proportional to the refractive index of its glass material, *n*,

with *L* the Lagrange invariant of the system.

Cancelling Petzval is only useful if there is no astigmatism in the system since the [»] total field curvature of the wavefront is given by the sum of astigmatism and Petzval. Correcting astigmatism is a bit trickier since we have effect of both the thinlens and stopshift equations

with *S _{1}* the spherical aberration,

*S*the coma and

_{2}*Q*the eccentricity factor.

For a lens with no spherical nor coma or with the STOP placed at the lens (*Q=0*), this simplifies to

While we cannot correct Petzval with a single thinlens, we can do it with a system of two or more lenses.

A **dialyte** is an optical system made of one positive and one negative power elements widely spaced apart (if the distance is small, the term **split doublet** is more frequently used). The elements powers are chosen such that the ratio between their power and refractive index are equal in magnitude but with opposite sign such as to balance each other. They are then spaced apart to control the overall power of the system since

Here, I propose to use the same glass for the two lenses, that is *n _{1}=n_{2}* such that we have

which balance both the system Petzval and astigmatism at the same time.

Since we cannot cancel *Q* for both lens at the same time, this also implies that each individual element must be corrected for spherical and coma. If there was remaining coma and spherical, the stop-shift theory tells us that we will introduce extra astigmatism. This is not an issue because we have shown how to design aspheric lenses and singlet lenses such that they are aplanatic (no coma and no spherical aberration).

While we can produce an aplanatic lens using an asphere for any power (see [»] here), there are some restrictions for singlet lenses as we have shown [»] here. We could have used two asphere, but that would be more expensive to build. Nonetheless, we will study that case as well in a future post.

Recall that the power of a singlet aplanatic lens is bound to the distance at which the lens is placed to the paraxial focus position of the system

and so

or, put differently,

For an object at infinity, we therefore have the condition

because the paraxial focus position of the first lens is given by *1/P _{1}*.

We find *P _{1}* through the total power of the system

and so

An example system using an asphere and a singlet element is given in Figure 1. I called it **the perfect dialyte** due to the beauty of the equations symmetry and the fact that it can produce perfectly focused images provided the two lenses are aplanatic. I’m certainly not the first to come across these equations, but in none of my optical design textbooks did I ever met a description of such system. Tell me if you have literature references of an equivalent system!

The system is diffraction-limited at f/4, 8° total FOV (4° half-FOV) as can be seen in Figure 2. I haven’t presented this kind of figure yet but it shows in blue the [»] RMS spot size as a function of field angle. The black-line represents the radius of the Airy disk such that the system is diffraction-limited if the blue line is well below. Note that the system can be slightly extended, either up to 10° FOV or by opening the aperture a touch but higher-order aberrations quickly degrade the system performances.

A traditional spot diagram is shown in Figure 3 for three different field angles. We already see some higher-order aberrations in off-axis rays although the system is still diffraction-limited. Note that the worst result is not obtained at the maximum field angle but at some lower value, close to 3°, which is typical of zonal aberrations (more on that later). This is where Figure 2 brings-in important information that you may have skipped in your traditional spot diagram.

Note that the system of Figure 1 is only one member of my perfect dialyte family. Here, the distance *L* between the two lenses was fixed because I wanted to use a singlet as second aplanatic lens. If we had used a different type of aplanatic lens, for instance a second aspheric lens, we could have chosen any value for *L*.

One option is to select *L* directly, such that you fix the total track of your layout. It is the most straightforward as you have

and therefore

But we could also have fixed the [»] back-focal length to some fixed value, to accommodate for a camera mount for instance. In this case we would have had

Remember that the only requirements of the perfect dialyte system is that the lenses are made of the same glass, that *P _{2}=-P_{1}*, and that the two lenses must be aplanatic.

Up to now we have seen only two ways to produce aplanatic lens but I’m about to expand your portfolio in the next post with the introduction of the thin-doublet!

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

I would also like to give a big thanks to **Young**, **Naif**,** Samuel**,** James**,** ****Sebastian**,** Lilith**,** ****Alex**,** Hitesh**,** ****Stephen**,** ****Jesse**, **Jon**, **Sivaraman**,** Cory**,** Karel**,** ****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 18: Thinlenses Aberrations

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[»] #DevOptical Part 23: Plane Windows