2.10: Designing an Achromatic Doublet

In this section, our focus will not be on lens aberrations directly, but rather on the calculations pertinent to designing an achromatic doublet lens.

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An achromatic doublet generally comprises two lenses positioned in contact: a converging lens crafted from crown glass and a diverging lens made from flint glass. This arrangement can create a combined converging lens that minimizes chromatic aberration. Notably, flint glass, being denser than crown glass, possesses a higher refractive index and enhanced dispersive capabilities.

The dispersive power \(\omega\) of glass is mathematically expressed as:

\[\omega= \frac{n^{(F)}-n^{(C)}}{n^{(D)}-1}.\label{eq:2.10.1} \]

The terms C, D, and F correspond to the Fraunhofer lines associated with specific wavelengths in the solar spectrum (H\(\alpha\) at 656.3 nm, Na I at 589.3 nm, and H\(\beta\) at 486.1 nm), often referred to as 'red', 'yellow', and 'blue'. For crown glass, the typical dispersive power value is around 0.016, whereas flint glass is generally 0.028.

An achromatic doublet typically involves a positive crown glass lens that exhibits a power decreasing with increased wavelength (towards red) cemented to a weaker flint glass lens. This flint glass lens has a negative power that also decreases in magnitude as wavelength increases. The overall power of the assembly remains positive and changes minimally across wavelengths, exhibiting a shallow minimum. In the design process of an achromatic doublet, two main conditions must be fulfilled: Firstly, the power or focal length in yellow must be predetermined; secondly, the power in red should match that in blue, with minimal variation in between.

Let's examine the doublet shown in Figure II.15, constructed using a biconvex crown lens and a biconcave flint lens.

The indices of refraction and the radii of curvature have been indicated. The power for the first lens alone can be defined by:

\[ P_1 = (n_1-1) \left(\frac{1}{a}+\frac{1}{b}\right),\label{eq:2.10.2} \]

For the second lens:

\[ P_2 = -(n_2-1)\left( \frac{1}{b}+\frac{1}{c}\right). \label{eq:2.10.3} \]

Shortening notation, we have:

\[ P_1 = k_1(n_1-1), \qquad P_2 = -k_2(n_2-1). \label{eq:2.10.4a,b} \]

However, we require similar equations for each of the three wavelengths:

\[ P_1^{(C)} = k_1(n_1^{(C)}-1), \qquad P_2^{(C)} = -k_2(n_2^{(C)}-1), \label{eq:2.10.5a,b} \]

\[ P_1^{(D)} = k_1(n_1^{(D)}-1), \qquad P_2^{(D)} = -k_2(n_2^{(D)}-1),\label{eq:2.10.6a,b} \]

\[P_1^{(F)} = k_1(n_1^{(F)}-1), \qquad P_2^{(F)} = -k_2(n_2^{(F)}-1). \label{eq:2.10.7a,b} \]

To satisfy our requirements, we need to fulfill the following conditions:

\[P_1^{(D)} +P_2^{(D)} = P^{(D)} . \label{eq:2.10.8} \]

Additionally, we ensure that the total power in red equals that in blue, applying equations \(\ref{eq:2.10.5a,b}\) and \(\ref{eq:2.10.7a,b}\):

\[ k_1(n^{(C)}_1- 1) - k_2(n^{(C)}_2-1)= k_1(n^{(F)}_1- 1) - k_2(n^{(F)}_2-1).\label{eq:2.10.9} \]

Rearranging gives us:

\[ k_1(n^{(F)}_1- n^{(C)}_1) = k_2(n^{(F)}_2-n^{(C)}_2). \label{eq:2.10.10} \]

Utilizing equations \(\ref{eq:2.10.1}\) and \(\ref{eq:2.10.6a,b}\), we derive the condition that ensures equal power in red and blue:

For further insights, explore our content at Achromatic Lenses solution.

\[ \omega_1P_1 + \omega_2 P_2 = 0. \label{eq:2.10.11} \]

To illustrate, if we aim for a focal length of yellow light to be 16 cm (where \(P^{(D)}= 0.\) cm-1) and our dispersive powers are noted as 0.016 for crown glass and 0.028 for flint glass, equations \(\ref{eq:2.10.8}\) and \(\ref{eq:2.10.11}\) indicate that \( P_1^{(D)}= 0.\) cm-1 \) and \(P_2^{(D)}= -0.083\) cm-1. (\(f_1 =6.86\) cm and \(f_2 = -12.0\) cm).

If we assume the first lens is equibiconvex, resulting in \(a = b\), and substituting \(n_1 = 1.5\), we derive \(a\) to be 6.86 cm via Equation \(\ref{eq:2.10.2}\). For \(n_2 = 1.6\), applying Equation \(\ref{eq:2.10.3}\) yields \(c = '144\) cm, whereby the negative value denotes that we initially assumed the flint lens to be concave when it should indeed be convex to the right.

Exercise \(\PageIndex{1}\)

If instead of fashioning the crown lens equibiconvex, you choose to make the last surface flat, meaning \(c\) becomes 0, what must \(a\) and \(b\) then represent?

Answers: \(a\) = 6.55 cm, \(b\) = 7.20 cm.