Lecture 5

Achromatic lenses

An achromatic lens or achromat is a lens that is designed to limit the effects of chromatic and spherical aberration. Achromatic lenses are corrected to bring two wavelengths (typically red and blue) into focus on the same plane.

The most common type of achromat is the achromatic doublet, which is composed of two individual lenses made from glasses with different amounts of dispersion. Typically, one element is a negative (concave) element made out of flint glass such as F2, which has relatively high dispersion, and the other is a positive (convex) element made of crown glass such as BK7, which has lower dispersion. The lens elements are mounted next to each other, often cemented together, as shown in figure (1-7), and shaped so that the chromatic aberration of one is counterbalanced by that of the other.

Figure (1-7) Example of an achromatic lens

0

The powers of the two lenses for the yellow center of the visible spectrum, conveniently represented by the Fraunhofer wavelength,

nm, are

( ) * + ( ) ……….. (1-1)

( ) * + ( ) ……….. (1-2)

where the radii of curvature are designated in Figure (1-8).

The focal length f of a thin-lens doublet satisfies the relation:

………….(1-3)

Here, and are the focal lengths of the two lenses in the doublet. Consequently, the power of the doublet is:

…………. (1-4)

1

r

11 r

21

r 22

r 12

Figure (1-8) Achromatic doublet, consisting of (1) crown glass equiconvex lens cemented to (2) a negative flint glass lens. The four radii of curvature Incorporating Eqs. (1-1are) and indicated (1-2) into Eq. (1-4) gives:

( ) ( ) ….. (1-5)

Chromatic aberration is absent at the wavelength if the power is

independent of wavelength, or ( ) , Applied to Eq. (1-5), this

condition is:

…………. (1-7)

The variation of with in the neighborhood of may be approximated using the red and blue Fraunhofer wavelengths,

nm and nm respectively:

…………… (1-8)

2

The dispersion constant for the glasses may be introduced by expressing the terms of Eq. (1-7) as:

( ) ( ) …………… (1-9) ( )

( ) ( ) …………… (1-10) ( )

where we have used Eqs. (1-1) and 1-2) as well as a dispersive constant , defined as the reciprocal of the dispersive power and given by:

……….. (1-11)

Substituting Eqs. (1-9) and (1-10) into Eq. (1-7), the condition for the absence of chromatic aberration may be written as:

…………… (1-12)

Combining Eqs. (1-4) and (1-12), the powers of the individual elements may be expressed in terms of the desired power of the combination

, ………. (1-13)

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The curvature factors expressed in Eqs. (1-1) and (1-2) may then be calculated using:

, …………. (1-14)

Finally, from the values of and the four radii of curvature of the lens faces may be determined. For simplicity of construction, the crown glass lens (1) may be chosen to be equiconvex. In addition, the curvature of the two lenses must match at their interface. The radii of curvature thus satisfy:

…… (1-15)

Example

Design an achromatic doublet of 5-cm focal length using 638/555 crown and 805/255 flint glass. Determine (a) radii of curvature; (b) focal lengths for D, C, and F Fraunhofer lines; (c) powers and dispersive powers of the individual elements. (d) Is Eq. (1-12) satisfied?

Glass type 638/555 crown 1.63461 1.63810 1.64611 805/255 flint 1.79608 1.80518 1.82771

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