What should the two lenses of achromatic doublet have?

It is not the intention of this chapter to study lens aberrations. However, the design of an achromatic doublet lens lends itself to the sort of calculation we are doing in this chapter.

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A combination of two lenses in contact, a converging lens made of crown glass and a weaker diverging lens made of flint glass, can be designed so that the combination is a converging lens that is almost achromatic. Flint glass is a little denser than crown glass, and has a higher refractive index and a greater dispersive power.

The dispersive power \(\omega\) of glass is usually defined as

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

Here C, D and F refer to the wavelengths of the C, D and F Fraunhofer lines in the solar spectrum, which are respectively, H\(\alpha\) (656.3 nm), Na I (589.3 nm), H\(\beta\) (486.1 nm), and which may be loosely referred to as &#;red&#;, &#;yellow&#; and &#;blue&#;. A typical value for a crown glass would be about 0.016, and a typical value for a flint glass would be about 0.028.

An achromatic doublet is typically made of a positive crown glass lens whose power is positive but which decreases with increasing wavelength (i.e. toward the red), cemented to a weaker flint glass lens whose power is negative and also decreases (in magnitude) with increasing wavelength. The sum of the two powers is positive, and varies little with wavelength, going through a shallow minimum. Typically, in designing an achromatic doublet, there will be two requirements to be satisfied: 1. The power or focal length in yellow will be specified, and 2. You would like the power in red to be the same as the power in blue, and to vary little in between.

Consider the doublet illustrated in Figure II.15, constructed of a biconvex crown lens and a biconcave flint lens.

I have indicated the indices and the radii of curvature. The power (reciprocal of the focal length) of the first lens by itself is

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

and the power of the second lens is

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

I shall write these for short, in obvious notation, as

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

But we need equations like these for each of the three wavelengths, thus:

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

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

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

Now we want to satisfy two conditions. One is that the total power be specified:

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

The other is that the total power in the red is to equal the total power in the blue, and I now make use of equations \(\ref{eq:2.10.5a,b}\) and \(\ref{eq:2.10.7a,b}\):

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

On rearrangement, this becomes

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

Now, making use of equations \(\ref{eq:2.10.1}\) and \(\ref{eq:2.10.6a,b}\), we obtain the condition that the powers will be the same in red and blue:

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

For example, suppose that we want the focal length in yellow to be 16 cm ( \(P^{(D)}= 0.\) cm-1) and that the dispersive powers are 0.016 and 0.028. Equations \(\ref{eq:2.10.8}\) and \(\ref{eq:2.10.11}\) then tell us that we must have \( P_1^{(D)}= 0.\) cm-1 \) and \(P_2^{(D)}= -0.083\) cm-1. (\(f_1 =6.86\) cm and \(f_2 = -12.0\) cm).

If we want to make the first lens equibiconvex, so that \(a = b\), and if \(n_1 = 1.5\), Equation \(\ref{eq:2.10.2}\) tells us that \(a\) = 6.86 cm. If \(n_2 = 1.6\), Equation \(\ref{eq:2.10.3}\) then tells us that \(c = &#;144\) cm. That \(c\) is negative tells us that our assumption that the flint lens was concave to the right was wrong; it is convex to the right.

Exercise \(\PageIndex{1}\)

Suppose that, instead of making the crown lens equibiconvex, you elect to make the last surface flat &#; i.e. \(c\) = &#;. What, then, must \(a\) and \(b\) be?

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

Achromatic doublet lenses ("achromat") are designed to eliminate chromatic and spherical aberrations inherent in singlet lenses.  When used on-axis,  an achromatic lens focuses an parallel input beam to a perfect "point", limited only by the effects of diffraction. This performance can be achieved over a broadband of wavelength.   Achromatic lenses can be used to collimate and focus laser beams.  They can also be used for high-quality imaging on-axis.  However,  the off-axis performance is significantly worse than the on-axis performance.  If your application requires good optical performance off-axis as well as on-axis,  multi-element lenses such as our digital imaging lenses are recommended.

For some applications it is desirable to have a high-quality lens with a lower f/# (for example,  collimating a diode laser with large divergence angle).  This can be done by adding a matching aplanatic meniscus lens to a doublet lens.  An aplanatic meniscus lens will shorten the focal length of the doublet lens (thus, lowering the F-num) without introducing additional spherical aberrations.

The following is the listing of our standard achromatic lenses.  They are all designed to achieve a wave-front aberration <1/4 wavelength.  You can also use lens design wizards to design custom achromatic lenses. Achromatic elements have a Minimum Order Quantity (MOQ) ranging from 100 to 1,000 units.

General Specifications:

Parameter Value Default unit mm Material Cemented crown and flint glasses Focal length tolerance +/-2% Diameter tolerance +0, -0.15mm Center thickness tolerance +/-0.25mm Centration 3 arc min Surface quality 40-20 Coating Single layer anti-reflection coating centered at 550nm

PN: ACH006-025 Description: Achromatic doublet lens, D=6.3, EFL=25 Diameter: 6.3 EFL: 25 BFL: 23.45 Tc1: 2.3 Tc2: 0.9 Tc: 3.2 Te: 2.73 Radius1: 12.37 Radius2: 11.16 Radius3: 32.17 Material1: BK7 Material2: SF5 Meniscus Lens: MNP006-048 Volume Price: $29.50 (MOQ Range Applies) Request Volume Quote