Achromatic doublet design and optimization

Basic Concepts

The achromatic doublet is a widely used optical structure that reduces chromatic aberrations. At its core, it consists of a two-lens system: one lens is concave, typically made from flint glass, while the other is convex, usually crafted from crown glass.

For further details, please visit our website.

But why choose these specific shapes and materials? We will explore the reasoning behind this design.

Chromatic Aberration

Understanding chromatic aberration is essential to comprehend its effects. Chromatic aberration arises due to varying refractive indices of materials at different wavelengths. For instance, BK7 exhibits a refractive index of 1. for blue light (480 nm) and 1. for red light (700 nm). This disparity means that a biconvex lens made with BK7 will focus red and blue light at different points, creating chromatic aberration, as illustrated in Figure 1.

FIGURE 1. Chromatic Aberration on a Biconvex Lens (50MM)

To mitigate chromatic aberration, it is crucial to match the focal lengths of the lens elements across the used wavelengths. The achromatic doublet uses a negative lens (concave) combined with a positive lens (convex) to counteract their respective dispersions, effectively canceling each other out.

Flint Glass vs Crown Glass

The negative lens component is generally constructed from "Flint glass." This type of glass has a high refractive index (above 1.55) and high chromatic aberration (indicated by an Abbe number below 50). Examples include F, LF, SF, KF, BaF, and BaLF. Lead-free flint glass is often indicated by an "N" prefix, such as N-SF8.

Crown glass, on the other hand, forms the positive lens. It has low dispersion (Abbe number above 50) and a refractive index typically below 1.5. However, some crown glasses, like BK7, have indices that overlap with certain flint glasses. Crown glass names frequently include a "K" suffix (from the German "kron," meaning crown). Examples include K, SK, BK, LaK, LaSK, and more.

Design of Doublets

To begin designing a doublet, one can approximate using a linear equations system:

Equation 1:
f1 + f2 = feq

where f1 and f2 represent the focal lengths of the first and second elements, respectively, while feq is the equivalent focal length of the system. The second equation required is:

Equation 2:
(V1 * f1) + (V2 * f2) = 0

where V1 and V2 denote the Abbe numbers for the first and second elements. Solving for f1 and f2 using Equations 1 and 2 leads to:

By correcting the lens shown in Figure 1 with an equivalent achromatic doublet, we can set feq to 50 mm and select appropriate materials for our lenses. For flint glass, we choose N-SF8 (Abbe number 31.31), and for crown glass, we select N-BK7 (Abbe number 64.17). Substituting these values into Equations 3 and 4 yields f1 = 25.61mm and f2 = -50.48mm. Figure 2 illustrates the chromatic aberration using these parameters (note: the lens is not yet optimized).

Figure 2. Achromatic Doublet (50MM) with Minimal Chromatic Aberration

By cleverly combining selected materials (flint glass and crown glass) and configuring the doublet shape, optical engineers can create lens solutions free from chromatic aberrations.

Achromatic Doublet Optimization

Now we will discuss the steps required to optimize an achromatic doublet.

It's important to note that various optimization methods exist; there is no single "formula" that guarantees the best results. Optimization is a process that demands revisions and feedback. It requires a level of artistry and technical expertise from the optical engineer. The method presented here has produced acceptable results in past experiences.

Our Variables

We begin with defined design goals, such as focal length, wavelength range, aperture diameter, and field of view (FOV). Based on these parameters, we can manipulate several variables:

1. Flint glass
2. Crown glass
3. Four radii of curvature (two for each lens)
4. Separation between lenses

Using these variables, we will optimize for three types of aberrations: chromatic, coma, and spherical aberrations.

Correcting Chromatic Aberration

The initial aberration to address is chromatic aberration, which is one of the primary reasons for using an achromatic doublet. As discussed earlier, an achromat consists of flint glass and crown glass. Selecting the appropriate glass combination will help minimize chromatic aberration, though finding the right glass can be challenging. How do we determine the correct glass types?

Understanding the Abbe Number (or V-number) is crucial, as it measures glass dispersion; a higher Abbe number indicates lower dispersion. Figure 1 displays the Abbe diagram, allowing us to plot V-number against refractive index for various glass types.

Theoretically, any combination of two glass types can be used. However, certain guidelines can ease the selection process:

1. The positive lens (crown glass) should have a higher Abbe number than the negative lens (flint glass).

2. A substantial difference in Abbe numbers leads to lower curvature surfaces, aiding in coma and spherical aberration optimization.

3. The negative lens (flint glass) should preferably have a higher refractive index than the positive lens (crown glass).

After selecting glass types, we can compute the focal length of each optical element in our doublet using the following equations:

Hongsheng Product Page

Equation 3:
f1 = (k1 * n1) / V1

Equation 4:
f2 = (k2 * n2) / V2

With an appropriate glass combination, we can significantly minimize chromatic aberration.

Correcting Coma Aberration

After calculating the focal lengths, we must now define the radii of curvature. To simplify this explanation, we will assume symmetric lenses with equal radii of curvature on the front and back surfaces of each lens. An initial approximation can be achieved using the following equations:

Equation 5

Equation 6

To correct coma, we will adjust the radii of curvature on the first and fourth surfaces (i.e., the first surface of the first lens and the second surface of the second lens). To maintain consistent optical power, we want (1/R1) + (1/R4) to remain constant. An increase in one radius will decrease the other. This process will be repeated until we achieve an acceptable reduction in coma.

Correcting Spherical Aberration

The final aberration we will address is spherical aberration. Once the first two aberrations are corrected, handling spherical aberration becomes relatively straightforward. We can utilize two tuning variables for adjustment: if we encounter a significant amount of spherical aberration, we can modify one of the internal radii of curvature of our lenses (i.e., surfaces two and three). If spherical aberration is minor, adjusting the spacing between lenses can help.

As we correct one aberration, we must continually monitor the entire optical system. Often, optimizing lenses involves compromise; it may not be feasible to eliminate all aberrations simultaneously.

The presented method serves as a foundation for simple optimization. More advanced techniques can reduce chromatic aberration for specific wavelengths and determine coma coefficients for each optical element, minimizing their combined effects. This approach can provide a starting point for further aberration reduction.

Different configurations of achromatic doublets can also be explored, such as Littrow, Fraunhofer, and Clark doublets.

Example of Achromatic Doublet Optimization

In this section, we will demonstrate the achromatic doublet optimization process. We seek a lens with a 50mm focal length, 20-degree FOV, and 30 mm diameter.

We begin with a single BK7 biconvex lens, as shown in Figure 2 alongside its spot diagram.

Examination of the spot diagram reveals significant coma (especially at full field) along with some spherical aberration. For further details on spot diagrams, refer to this entry.

We calculate specific Seidel aberrations and analyze the Ray Analysis graphs (discussions on those will take place later). We measure a chromatic focal shift of 1.5 mm along the axis, a maximum spherical aberration of 4mm at 20 degrees, and a Seidel coefficient for coma (third-degree) of 0.157.

Correction of Chromatic Aberration

In this example, we will optimize an achromatic doublet without a specific application. We will utilize two common glasses: BK7 (Abbe number = 64.166 and n_green = 1.) and SF2 (Abbe number = 33.819 and n_green = 1.). To calculate the radius of curvature for our lenses, we revert to Equations 3 and 4.

From those equations, we derive R1 = 24.77 mm and R2 = -59.33 mm. At this stage, both lenses have identical radii of curvature for the front and back surfaces, resulting in a biconvex and biconcave lens configuration, as illustrated in Figure 3.

Significant coma and spherical aberration still remain at this point, while chromatic focus shift has decreased from 1.5 mm to less than 0.1 mm. No considerable changes were made regarding spherical or coma aberrations, prompting the decision to rectify coma next.

Coma Correction

Now, we will adjust the radii of curvature of the first and fourth surfaces while striving to keep the focal length as close as possible to 50mm. Multiple optimization approaches exist, with tools like OSLO and Zemax offering diverse algorithms for various optimization scenarios. However, for this example, we will simply make incremental adjustments.

Figure 3 depicts alterations made to the first and fourth surfaces. We adjusted the first surface from 24.77mm to 23.8mm and the fourth surface from 59.33mm to 60.2mm. Following these modifications, chromatic aberration remained below 0.1mm, yet the Seidel Coma coefficient dropped to 0 (from the original 0.157). Additionally, spherical aberration lessened to nearly 2mm at 20 degrees, warranting further refinements.

Spherical Aberration

Next, we aim to achieve two objectives: reduce spherical aberrations while keeping the focal point close to 50 mm.

We will modify the spacing between the first and second lenses, along with the radius of curvature of the first surface of the second lens. This iteration will continue until we reach the final lens combination.

Results indicate our focal length is now nearer to the desired 50 mm. The coma coefficient hovers around 0.02 (just slightly reduced), with spherical aberration kept below 3 mm at 20 degrees.

We will conclude our discussion on optimization here. As previously mentioned, multiple approaches exist for tackling this problem, and numerous tools are available for optical system analysis. Future blogs will delve into ray intercept graphs and possibly revisit optimization examples or rudimentary analyses.