#DevOptical Part 28: The Achromatic Doublet

Up to now, we have always selected one default glass from the Schott catalog, the N-BK7 glass. On some exceptions, I used two glasses like when we made [»] doublets, but I did not tell how the glasses were chosen. This is precisely what I would like to address in this post.

Glass selection is one of the most important tasks of an optical designer. It may seem extremely opaque at first sight, but unless you are trying to achieve very high-performance optimizations, the basic rules underlying glass selection for achromatizing doublets are actually pretty straightforward.

Glass selection affects both first and third order optics (and higher orders as well but it’s less frequent to draw glass selection based on higher order aberrations theory – at least to my knowledge). For instance, some glasses will require less bending to achieve the same converging or diverging power, leading to less aberrations. Some glasses will spread blue and red light stronger than others. Finally, some glasses will be very resistant to climatic conditions, be easier to manufacture or will simply attenuate some wavelengths less than others.

Ultimately, from an optical design perspective, keeping the manufacturing and environmental resistance parts aside, glasses are entirely characterized by their refractive index, n. When electromagnetic radiation (e.g. light) passes through a transparent medium, the electric part of the wave makes the electron cloud of the material oscillate around the nucleus of the atoms composing that medium. This oscillation, like any oscillator in physics, has a fundamental frequency, w₀, and will react with some damping to any other frequency w according to the square of their difference, (w-w₀)². As such, an electromagnetic wave will be absorbed and re-emitted in random directions: a process known as scattering. In a transparent medium, the different atoms scattering contributions builds an output wave having the same direction as the input wave but lagging in phase. This phase lag is characterized by a complex expression, renamed the refractive index of the material, which is therefore proportional to this (w-w₀)² term governing the oscillation process. We can exchange the frequency by the wavelength through some mathematical computations, leading to a factor proportional to (λ-λ₀)². More information about the scattering process can be obtained in your classical textbooks such as Optics by Eugene Hect. The more advanced mathematical computations can be obtained in books such as Principles of Optics, by Max Born and Emil Wolf.

Very early optical researchers noted the dependence of the refractive index of a material with inverse squared wavelength and proposed an empirical relation known as the Cauchy formula

A typical glass will therefore have a λ^-2 dependence, as shown in Figure 1. Note the linear dependance in the bottom plot. The agreement of experimental data for N-BK7 with the model between 0.4 µm and 0.75 µm (visible spectrum) is ∆n_rms=0.0015 (0.1%).

Historically, refractive indices were measured by the deviation of the light through a prism for different wavelengths occurring in nature, like the peaks of a helium lamp (named the d line at 587.562 nm). Other important lines include the f line at 486.134 nm and the c line at 656.281 nm.

Instead of using the constant A and B of the Cauchy formula, scientists used the refractive index measured at the d line and the ratio of this index with the difference of index between the f and c line. This later term is called the Abbe number and is noted V_d to recall the reader that it is computed around the d line:

where

It is possible to relate the A and B constants of the Cauchy formula to the experimentally measured n_d and V_d terms:

Since the dependance of the refractive index in the Cauchy formula is governed by the B term, it is also governed by the ratio between n_d-1 and V_d. For similar n_d, a glass with a larger V_d will spread the different wavelengths less than a glass with a small V_d. In optics, we talk about the dispersion properties of the glass, translated by the V_d term (a glass with high V_d is therefore a glass with low dispersion and vice-versa).

One of the important consequences of the refractive index dependance to wavelength is that our first order system will behave differently at different wavelengths. Indeed, the power of a thinlens is proportional to its total curvature and its refractive index as explained [»] here:

The power of a single thinlens will therefore be different at two different wavelengths and will focus light at two different distances. In practice, this is undesirable when imaging with white light because only one color will be in focus and the others will produce a blurry spot.

This effect can be mitigated by using two thinlenses of different materials in a [»] thin-doublet (i.e. with no air gap). We now have two lenses of power P₁ and P₂, made of the refractive indices n₁ and n₂, and would like that, (1), their combined power is equal to some value P, and also that, (2), their combined power does not change with wavelength (i.e. the derivative of the power relative to the wavelength is zero):

and so

knowing that, from the Cauchy empirical formula,

The condition can be met only when

which gives, after replacing the B terms by the V_d expressions,

where I omitted the reference to the d line in the Abbe number notation for V₁ and V₂.

Now, using the fact that

we get

which can only occur when

And since

we get (after some mathematical manipulations)

We can also state that the [»] power partition ratio of the achromatized thin-doublet must be equal to

These relationships show that it is possible to build a system that has the same power at multiple wavelengths provided we use two thinlenses whose power partition ratio is fixed by the values of the respective Abbe numbers of the glasses as given by the formula above.

Note that the power partition ratio is inversely proportional to the difference in Abbe number. To avoid selecting individual lens with too much power (leading to high curvature and therefore higher overall aberrations), it is therefore wise to always pick glasses that have very dissimilar Abbe number. The traditional glass chart in Figure 2 is of good help to select glasses based on their Abbe number differences. For historical reasons, glasses have been classified between flints and crowns based on their Abbe number. While this makes little sense today due to the numerous glasses available, the names are still frequently met among glass manufacturers.

The method presented here-above is particular to this post. You will usually find a different approach in your classical textbooks that I will briefly discuss now to view things along different perspective.

You can solve the same problem by noting that you want the total power at some wavelength λ_d to be equal to some constant P:

and that the total powers at two other wavelengths to be equal:

Here we say that λ_d is the design wavelength and λ_f, λ_c are the auxiliary wavelengths. These names will make sense as we progress into the post.

Simplifying the last equation, we get

substituting one more time the curvatures by their respective power computed at the design wavelength, λ_d,

we get

which gives the same results without having to postulate the Cauchy relation for the refractive index.

It is very important to note that there is a strong shortcoming with the relations built on the Cauchy formula, which is that the correction is true for all wavelengths (the derivative of power to the wavelength is always zero at any wavelength). In practice, this is not the case. Early scientists already knew that some glasses did not follow the Cauchy formula and were then called abnormal dispersion glasses. They also knew that the chromatic correction was not perfect at all wavelengths, even for glasses with normal dispersion (glasses following the Cauchy formula).

Although the power of the thin-doublet is equal at the auxiliary wavelengths λ_f and λ_c, it has a slightly different value at the design wavelength, λ_d. A typical power variation plot with wavelength is given in Figure 2. The difference in focus between the design wavelength and the auxiliary wavelengths is called the secondary spectrum. Note that the defocus varies by more than the reported secondary spectrum, even in the range defined by the auxiliary wavelengths [λ_f, λ_c].

We can compute the value of the secondary spectrum as the power difference between the auxiliary and design wavelengths

and knowing that

we have

and, from the definition of the Abbe number,

we have the relation

where we introduced the partial dispersion measured at the f line, ρ_f, as

The secondary spectrum is therefore proportional to the difference in partial dispersion between the two glasses and inversely proportional to the difference in Abbe number between the two glasses. To reduce the secondary spectrum and increase the achromaticity of our thin-doublet, we must then select two glasses with similar partial dispersions but very dissimilar Abbe numbers.

To assist the optical designer in this task, a new chart is introduced in Figure 4 to sort the glasses based on their partial dispersions and Abbe numbers. The goal is to select glasses that lies on a horizontal line (∆ρ_f≈0) but widely spaced apart (∆V≠0).

We see that only a few glasses meet these criteria, and they are usually all expensive (highlighted in red in Figure 4). All other glasses lay very close to what is called the normal line which is directly linked to the concept of normal/abnormal dispersion glasses that we met previously. You will frequently hear the term ED glasses for “extra-low dispersion glasses” in marketing of telescope or camera objectives to emphasize the presence of an abnormal dispersion glass – usually leading to superior performances of the overall assembly in terms of achromaticity.

One of the questions that may arise is “how bad does the secondary spectrum influence the design performances?”. Since most glasses follow the normal line, the secondary spectrum is moderately independent from the actual glass pair selected. For glasses selected on the normal line, most authors uses the value of ~2,200 as a ∆V/∆ρ_f  (slope of the normal line), leading to a secondary spectrum of about

For an achromat imaging a collimated beam, the defocus is therefore about the focal length divided by 2,200 (e.g. 90 µm for a 200 mm focal length lens). On the other hand, we know that the depth of focus of an optical system is about 4λf_#² with f_# the f-number of the system (this formula is strictly equivalent to λ/NA²). This means the effect of the defocus can be mitigated by a proper choice of aperture of the system:

As an illustration, for a 200 mm length achromat in the visible, the maximum f-number authorized by the secondary spectrum is f/6.4. If this value is not wide enough for your system (and taking some margin!), you should consider selecting glasses outside of the normal line. For a 100 mm achromat in the same conditions, the limiting value decreases to f/4.5. The formula therefore clearly favors slow lenses (lenses with low f-numbers).

Interestingly, when the aperture diameter is fixed (f_#=f/D), the system is now favored for longer focal lengths since

Historically, this favored telescopes with very long focal lengths (several meters) such as to be completely tolerant to secondary spectrum. For instance, a 100 mm aperture telescope with a standard achromat would require a focal length of at least 3 meters in the visible range! Things can be greatly improved by using glasses outside the normal line. Such achromats are usually referred to as super-achromats although the name is not standardized so you must be careful when it pops up at optics suppliers. An example of a f=50 mm super-achromat vs. standard achromat is given in Figure 5. The standard achromat uses the glasses along the normal line (N-BK7/N-SF2) while the super-achromat uses glasses outside of the normal line (N-PAK51/N-KZSF2). The secondary spectrum is decreased from 25 µm (standard achromat) to only 3 µm (super achromat).

Here we also restricted ourselves to thin-doublets. Just like we matched the powers at two different wavelengths, it is possible to match the power at three different wavelengths through triplets (systems of three lenses). In such cases, the secondary spectrum tends to be much thinner than for ordinary achromats. When the power is matched at three different wavelengths, we talk about apochromatic systems. Some vendors also refer to super-apochromats for highly optimized lenses but, again, you should be very careful due to the lack of standardization for that name. We could continue like this forever, but it usually makes little sense in practice because a 3-wavelengths corrected system is usually enough to maintain the secondary spectrum well within the acceptable defocus range of 4λf_#². For the sake of completeness, you should also know that there are different means of approaching achromaticity such as through diffractive optics (especially useful in the infrared region!). Again, we’re covering relatively conventional systems here, but you might meet more complex beasts occasionally in your career.

Let’s now come back to why our Cauchy approach predicted perfect achromatism while experience shows it’s not the case.

The reason behind this is that, while the Cauchy formula would be valid for media containing only one oscillator, but glasses actually contain multiple oscillators. At the very least, the movement of the nucleus introduces a second resonance which we need to take care of. Typically, one of the oscillators (electronic) will fall in the UV range while the second oscillator (nuclei) will fall in the IR range. This is also the reason why most glasses are opaque in the UV and IR range. A more correct formula, which includes multiple oscillators, is the Sellmeier formulas. Almost all glass manufacturers and optical design softwares use Sellmeier formulas with three oscillators as they allow a fit of the refractive index better than the reproducibility of the glass blend during manufacturing (melt repeatability is on the order of ∆n≈10^-5).

With three oscillators, the expression for the refractive index now becomes

where the C_i are the squared resonant wavelengths (i.e. λ_0,i=√C_i).

As for the Cauchy formula, the different coefficients are obtained by fitting this model to actual refractive indices measured at specific wavelength lines (typically 10 wavelengths across the VIS-NIR range). Glass manufacturers like Schott can extend the analysis to more wavelengths if you pay them for the job. This is a very delicate process because every pieces of the measurement process must be controlled with extremely high precision, so extra measurements like these are usually expensive.

Most of the properties of glasses in the visible are linked to the oscillator in the UV range, the IR oscillators having less impact. This explains why a lot of glasses closely follow the Cauchy formula in the visible. This is, however, not true anymore in the infrared range and properties like V_d lose their meaning as they are computed through indices measured in the visible range which is influenced by a different oscillator. For instance, it is impossible to perform a design in the SWIR range (λ=0.9-1.7 µm) using the provided values of V_d according to Schott themselves. This has important implications because most of the design tools (charts but also optical design CAD softwares) were built based on the historical notions of n_d and V_d, whereas the trend in optics these days seems to go towards more IR (or UV) rather than the visible.

It is therefore important to replace the definitions of the Abbe number based on the actual wavelength range covered by the instrument. You should always replace the design and auxiliary wavelengths in the formula with those needed in your design if you wish to compute the proper partition ratios for your achromatized thin-doublet.

On the other hand, very little can be done concerning the tolerance analysis of your system because software like ZEMAX OpticStudio uses a Cauchy-like formula with a tolerance on V_d as given by the glass manufacturer. These don’t translate into the SWIR/MIR/LIR ranges, and you should never trust the values returned by your software because the behavior of the system is then mainly affected by the IR oscillators while the values of V_d are mostly affected by the UV oscillator. For most of the design job in the visible, this is not an issue, and you can trust the software outputs without any problem.

Now that we have a better formula to compute the refractive indices, it would be interesting to have a look at what happens at the vicinity of the center wavelength of our thin-doublet. I will however keep things general and not assume any particular mathematical expression for the following derivations. On the other hand, if you wish to compute any quantitative value based on these formulas, you will need to assume a specific model such as the Sellmeier equations.

Let’s perform a Taylor series expansion of the power around the center wavelength, λ₀,

We will focus our attention on the first two derivatives here, ∂P/∂λ and ∂²P/∂λ².

Knowing that

we get, for the first derivative,

which we can re-express in terms of individual power P₁ and P₂ as

where

The first derivative plays a similar role to the overall chromatic dispersion trend we met previously and ν can be seen as the instantaneous version of the Abbe number, V_d. The only difference between the two definitions is that the Abbe number is arbitrarily normalized over the λ_f-λ_c wavelength range.

Indeed

Similarly, we get for the second derivative,

And, knowing that the following conditions should hold for achromaticity,

we get

and

where the instantaneous partial dispersion plays a similar role to the partial dispersion we met previously:

As mentioned previously, you will need a mathematical expression to compute quantitative values for these instantaneous terms. For the Sellmeier equations, the derivatives are

You may ask what is the purpose of having instantaneous definitions of the dispersion and partial dispersion properties in the optical design process. The role they play is actually very similar to the definition of the instantaneous speed in physics. Speed, by definition, is the ratio of a travelled distance by the time required for the travel. As the time shortens, and ultimately becomes zero, we get the definition of the instantaneous speed. It is worth noting that this notion troubled scientists at the time it was introduced because speed is inherently linked to the concept of motion (e.g. if you would freeze a frame of a video of a moving car, what meaning could you give to the word “speed” since the car is not moving at all on the isolated frame?). It is therefore important to grasp the concept of these instantaneous properties as indicators of how the system will behave at the direct vicinity of the design wavelength – reason for which I introduced them through a Taylor expansion of the power. On the other hand, the traditional dispersion and partial dispersion parameters give information on a very broad range of wavelengths. Using traditional dispersion and partial dispersion will give you a very poor estimate of how the system will behave at the vicinity of the design wavelength.

In consequence, when designing a system for narrowband sources such as a laser or LEDs, it is interesting to substitute the definition of the Abbe number and partial dispersion by their instantaneous counter-part since we know we will always be working at the direct vicinity of the design wavelength. All the design rules, charts, and formula can then be used just like with the regular version of these numbers.

One of the advantages of these new formulas is that they guarantee that the wavelength dependency of the power near the design wavelength is minimal thanks to the zero first-order derivative and small second-order derivative. The resulting shape is therefore close to a parabola with its minimum located at λ₀ and with very little curvature, i.e pretty insensitive to small changes in the working wavelength that might be due to current, temperature or batch-to-batch variations.

An example of a doublet lens optimized for desensitization at the laser wavelength is given in Figure 5. The bottom system is clearly optimized for minimum wavelength sensitivity near its design wavelength, λ₀, at contrario to the top system which has more sensitivity (non-zero slope) at λ₀ because it was built using the standard definition of the Abbe number. This can be seen in the RMS plot diagrams where the spread is quite different in the standard Abbe version compared to the instantaneous Abbe version which keeps the same spreading. Note the bending of the different lenses between the top and bottom solutions!

Until now, we focused the design process exclusively on thin-doublets but we know that these are theoretical objects only because real doublets will always have some thickness between their elements.

To illustrate this, I will consider the telescope system of Figure 6 made of a split doublet made of elements P₁ and P₂ with an aperture in front of it. The distance between the elements is noted L and the image is formed at some distance B from the second lens.

The [»] ABCD matrix of this system is

where the total power is now

Using P directly as a shorthand we have

Let’s have a look at what happens to a ray of height y and direction u traversing this system:

Since the topic of this post is about the chromatic behavior in optical systems, let’s study how the interception coordinate, y’, changes with wavelength:

Several interesting conclusions can be drawn from this mathematical expression.

First, for rays parallel to the optical axis (u=0), we find that the ray intercept at the image plane varies with wavelength by a quantity

The condition to get an achromatic system is therefore more complicated than simply cancelling the derivative of the power versus wavelength (dP/dλ). The true achromatic condition for this system is

from which we can solve the power partition ratio using the expression of the refractive indices like we did previously.

Note that when L=0, the thin-doublet case, the expression decays into dP/dλ=0 which is exactly what we solved in the first part of this post.

The second teaching that we get from our development is for off-axis rays (u≠0). If we assume the system is achromatic for on-axis rays (see here-above) we’re therefore left with

which cannot be zero unless dP₂/dλ itself is zero which is not the case for a regular lens (or we wouldn’t be in the trouble of achromatizing anything).

In conclusion, a thick doublet system that is chromatically corrected for on-axis rays cannot be corrected at the same time for off-axis rays. We could build a thick doublet system that is achromatic (dy’/dλ=0) for a given (y,u) couple but we cannot build one that would satisfy the achromatic condition for all (y,u) couples. This is the limitation of doublet systems. The solution to this problem is to add more lenses, at the very least one more.

To achromatize thick systems, instead of solving equations through ABCD matrices and ray interception computation, optical designers appended the [»] Seidel equations with two more terms

where it is very common to replace the derivative of the refractive indices by their Abbe number (classical or instantaneous):

The first term of the extended Seidel equations, C₁, is named longitudinal chromatic aberration while the second term, C₂, is named lateral chromatic aberration. C₁ plays a role similar to what we observed for on-axis rays in our achromatic doublet system of Figure 6 and C₂ plays the role of our observation for off-axis rays.

The [»] stop-shift equations are also appended to

When lateral chromatic aberrations are present, the colors tend to smear out as if they all had slightly different magnification ratio. The effect of longitudinal chromatic aberration is to blur the image depending on the wavelength by having a slight defocus on each color. Finally, when longitudinal chromatic aberration is present, the amount of lateral chromatic aberration can be cancelled by a proper position of the stop.

As for third-order aberrations, there is no generic formula to solve all the optical systems at once but there are well-known solutions to specific problems. And just like we can use thinlenses as an approximate starting point for real systems, we can use thin-doublets as a starting point to achromatization.

Let’s illustrate this with the example of Figure 7 showing an achromatic telescope system with a beamsplitter cube and its on-axis [»] rms performance plot. On top, you have the equivalent paraxial system. At the middle, we replace the paraxial lens by an achromatic doublet optimized for spherical aberration (S₁), coma (S₂), and longitudinal chromatic aberrations (C₁), ignoring the presence of the cube. Then, at the bottom, we optimize the whole system for best rms performance. The adjustments here are minor because of the limited aperture size but in faster systems the modifications would have been more important. This emphasizes that approximations can yield could results in systems that aren’t too fast.

The optimization process used to generate thick elements from thin ones or to optimize the whole system globally goes beyond the scope of this post and will be covered later. The only important point to consider is that we can first optimize elements locally (ignoring the other elements of the system) to generate an approximate solution before we optimize the whole system at once. For systems that aren’t too fast, the approximate solution can fall very close to the fully optimized one.

Also, in this post we only discussed achromatization of doublets but did not address in detail the more complex case of achromatizing a complete system neither did we discuss in detail how glass selection affects third order aberrations as we believe these topics are better suited by dedicated post on each aberration types. Also, we did not introduce the concept of aberrations variations with wavelength. Indeed, a system can be fully corrected for third-order aberrations at the design wavelength but have non-negligible aberrations at other wavelengths. A typical example of this is spherochromatism which is wavelength-dependent spherical aberration. Although there is a lot of topics left to cover, we already covered a lot of ground in this post, and you should now feel confident in designing your own achromatic doublets. Remember that isolated achromats represent a large part of the sales figures of optical suppliers and are used on a daily basis in optical labs all around the world!

I would like to give a big thanks to Young, Sebastian, Alex, Stephen, Lilith, James, Jesse, Jon, Cory, Karel, Sivaraman, Samy, David, Michael, Kausban, Shaun, Themulticaster, Tayyab, Kirk, Marcel, Onur, Dennis, Benjamin, Sunanda, Zach, M and Natan 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!