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In a [»] previous post, I have shown how to derive the shape and stop position of a single lens system with no coma and no astigmatism. Here, I will come back on these results under a different perspective which will ultimately enable us to derive more generic principles of aberration cancelation at the interface level. We will finally derive important equations to design single lens elements with no spherical aberration and no coma, which will later serve as building blocks in optical systems.

As a reminder, the two anastigmat, acomatic, lens systems we previously derived are repeated in Figure 1. They both show no coma (*S _{2}=0*) and no astigmatism (

*S*) when some lens shape is respected

_{3}=0__and__the stop is placed at a specific position relative to the lens.

These systems were derived using the [»] stop-shift theory and the expression of [»] aberration of thinlenses through some lengthy mathematical procedures. We will use a different strategy here that is more straightforward.

Let’s come back to the expressions of the [»] Seidel aberrations which I have repeated here below

where *h* is the ray intercept height, *c* the surface curvature, *u* the incoming ray angle, *u’* the exit ray angle, *n* the initial refractive index, *n’* the refractive index after the interface and

where *(h,u)* refers to the marginal ray and *( h,u)* to the chief ray,

*i*and

*being then the ray incidence angle on the interface for the marginal and chief ray respectively.*

__i__We can derive a few interesting cases which are listed in Figure 2.

The case *i=0* occurs when the object is located at the center of curvature of the surface (the marginal ray strikes the surface at a normal angle). **When an object is located at the center of curvature of an interface, there is no spherical nor coma contribution of that interface.**

The case * i=0* occurs when the pupil is located at the center of curvature of the surface (the chief ray strikes the surface at a normal angle).

**When the pupil is located at the center of curvature of an interface, there is no coma, no astigmatism and no distortion contribution of that interface.**

The case *h=0* occurs when the surface is located on a plane conjugated with the object plane (the marginal ray strikes the surface at zero height). **When a surface is located at a conjugate plane of the object plane, there is no spherical, no coma and no astigmatism contribution of that interface.**

Finally, **when u’/n’=u/n, there is no spherical, no coma and no astigmatism contribution of that interface as well**. We will come back to this case later on.

Revising our first system in Figure 1 (the planoconvex case), we can see that the first interface has u’/n’*=u/n* (because *u=u’=0*) which yields no spherical aberration, no coma and no astigmatism. Our math also shows that the second interface has *i=0* which produces some spherical aberration but no coma and no astigmatism. This is not obvious at first sight because our math showed that the stop is located at a distance *d=(n-1)/(nP)* from the second surface (see [»] here the derivations). This actually makes sense when noting that the radius of curvature is *R=(n-1)/P*. The extra *n* factor comes from the fact that the pupil image position is actually shifted by the refraction of the first planar interface. Overall, our lens produces some spherical aberration but no coma nor astigmatism which is in line with our former study. A similar analysis can be performed for the second system of Figure 1 except this time we first meet the condition * i=0* and then the condition

*u’/n’=u/n*.

By combining the various cases of Figure 2 for both interfaces to make a singlet lens, we can therefore produce an element that has reduced aberrations. Limiting ourselves to the three first aberrations, we can produce lenses that have either (1) no coma, (2) no spherical and no coma, (3) no coma and no astigmatism, (4) no spherical, no coma and no astigmatism. The various combinations are listed in Figure 3. Note that some combinations are technically infeasible or meaningless (those on the diagonal).

Before we investigate each of the combinations, let’s come back to the case *u’/n’=u/n* which is known as the **Young-Weierstrass point**.

When a ray of direction *u* intercepts a surface of curvature *c* at a height *y*, its refracted direction, *u’*, can be derived as

where *n* is the refractive index of the medium before the interface and *n’* is the refractive index of the medium after the interface.

The condition *u’/n’=u/n* then becomes

and therefore

with *z* the paraxial focus position

Given some paraxial focus position of an initial system, it is then possible to compute the radius of curvature, *c*, of a surface that will bend the rays by a factor *n’/n* but without introducing any spherical aberration, coma or astigmatism.

We can now come back to our table of Figure 3. Note that we have already presented the case of acomatic, anastigmat, lenses so we know how to design such lens system (although we did it using a different mathematical framework, the results will be the same and so the interest is somewhat limited here). The case of acomatic lens is not more interesting either because we already know from the [»] thinlens theory and the [»] stop-shift equations how to compute the shape of a lens to produce zero coma.

This leaves the case that produce lenses with zero spherical aberration and zero coma. These lenses are called **aplanatic**.

From Figure 3, we can list the following cases of aplanatic lens conditions:

**1/ A planoconvex lens located on the object plane.** When we place a planoconvex lens on the object plane (or one of its conjugates), we have *h _{1}=0* which produces the aplanatic condition. By using a radius of curvature equal to the thickness of the lens (typically a hemispheric ball lens),

*c*, we get

_{2}=-1/T*i*which also produces the aplanatic condition. Assuming the lens is placed in air, its power is then

_{2}=0*P=(n-1)/T*which is always a positive value.

**2/ A planoconvex lens located on the object plane (Young-Weierstrass variant).** Using a radius of curvature of *c _{2}=-(n+1)/(nT)* this time we can produce the same effect but with a power of

*P=(n²-1)/(nT)*which is still always a positive value. For common refractive indices, it tends to gives slightly more powerful lenses by using stronger curvatures. You can also obtain similar powers to case #1 by using smaller thicknesses.

**3/ A planoconvex lens located on the image conjugate plane.** This is similar to the case #1 except this time the planoside is placed on the image plane (or one of its conjugates).

**4/ A planoconvex lens located on the image conjugate plane (Young-Weierstrass variant).** This is similar to the case #2 with the planoside placed on the image plane (or one of its conjugates).

**5/ A meniscus lens placed at the paraxial focus position (divergent variant).** For a given system with paraxial focus z, placing a lens with a front curvature of *c _{1}=1/z* (case

*i=0*) and a rear curvature of

*c*(Young-Weierstrass case) yields a lens of power

_{2}=(n+1)/(nz)*P=-(n-1)/(nz)*. Furthermore, the partition factor of that lens can be computed as being

*–(2n+1)*.

**6/ A meniscus lens placed at the paraxial focus position (convergent variant).** Similar to case #5, by using this time the Young-Weierstrass case on the first surface, such that *c _{1}=(n+1)/z*, and the case

*i=0*on the second surface, such that

*c*, we produce a lens of power

_{2}=n/z*P=(n-1)/z*. The partition factor can be computed as being

*2n+1*.

It is important to note that case #5 and #6 only applies for systems where the light is already converging (or diverging). For infinite system, we would have z=∞ yielding lenses of no overall power. You can see these cases as lenses that increase or decrease the converging power of an already existing system. We will see applications of these lenses in a future post.

Also, cases #1 to #4 are somewhat dangerous to use in practice as working on conjugates of the object and image planes (or close to them) put very high strain on the surface quality of the planoside as any defects on these surface will directly be imaged on your sensor.

Finally, and worth to be mentioned, cases #2 and #4 have the extra advantage of showing no astigmatism in addition to being already aplanatic. These two lenses seems to be often met in high-resolution microscopy setups.

Before I end this post I would like to come back on cases #5 and #6.

An example of convergent aplanatic lens is given in Figure 4. I placed a paraxial lens in Zemax to produce perfect focusing, so don’t pay attention to the leftmost element on the figure.

In the convergent aplanatic case we have the radii of curvatures

with *T* the lens thickness and *n* its refractive index.

We can compute the equivalent focal length, *f*, of the lens as

which simplifies to the following expression for thinlenses (*T=0*)

To create a lens of given power *P=1/f*, the two here-above formula allow to compute at which distance *z* from the paraxial focus position of the system you need to place the lens. You can then compute the corresponding radii of curvatures.

The divergent case is treated similarly. An example is given in Figure 5.

The radii of curvatures are this time

and the equivalent focal length

which simplifies to the following expression for thinlenses

From the simplified equations of thinlenses, it is relatively evident that the convergent case always has *P>0* while the divergent case has *P<0* for system with real images (*z>0*).

In a future post I will show practical usage of a divergent aplanatic lens but we first have to cover some other important concepts first. In a few posts all the tiny bits will start to assemble…

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I would also like to give a big thanks to **Naif**,** Young**,** Eric**,** Samuel**,** James**,** Andrew**,** Lilith**,** Hitesh**,** Sebastian**,** Jesse**,** Sivaraman**,** Jon**,** Cory**,** Karel**,** Alex**,** Stephen**,** Themulticaster**,** Tayyab**,** Marcel** and **Kirk** 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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