Achromatic doublet

An achromatic doublet, or achromat, is a compound lens made of two elements of different glasses: a converging element of low-dispersion crown glass and a weaker diverging element of high-dispersion flint glass. Because the flint element disperses light more strongly per unit of optical power, it cancels the crown element's color error while taking away only part of its power. The result brings two chosen wavelengths, conventionally the blue F line (486.1 nm) and the red C line (656.3 nm), to the same focus, and the bending of the two elements is also used to reduce spherical aberration and coma. Catalog doublets run from about 10 mm to over 1 m in focal length at diameters of a few millimetres to 50 mm or more, and they are the default imaging and focusing lens in most optical labs.

Design condition

For two thin elements in contact with powers $\phi_1 = 1/f_1$ and $\phi_2 = 1/f_2$, the combined power and the achromatic condition are

$$\phi = \phi_1 + \phi_2 ,$$ $$\frac{\phi_1}{V_1} + \frac{\phi_2}{V_2} = 0 ,$$

where $V_1$ and $V_2$ are the Abbe numbers of the two glasses. Solving,

$$f_1 = f\,\frac{V_1 - V_2}{V_1}, \qquad f_2 = -f\,\frac{V_1 - V_2}{V_2} .$$

Worked example. For $f = 100$ mm with N-BK7 ($n_d = 1.5168$, $V_d = 64.17$) and F2 flint ($n_d = 1.6200$, $V_d = 36.37$), the crown element needs $f_1 = 43.3$ mm and the flint element $f_2 = -76.4$ mm. Making the crown equiconvex requires radii of $\pm 44.8$ mm; cementing a flint element to its rear face and solving for the flint's power leaves an outer flint radius of about $-811$ mm, nearly flat. This thin-lens solution is only the starting point: a real design adjusts the bending and thicknesses to correct spherical aberration and coma as well. The larger the difference $V_1 - V_2$, the weaker both elements can be, which is why glass pairs with widely separated Abbe numbers are preferred.

What the doublet corrects, and what remains

An N-BK7 singlet of 100 mm focal length has an F-to-C focal shift of about $f/V = 1.56$ mm, the axial chromatic aberration that blurs white-light images and forces refocusing when switching lasers. A doublet removes this to first order, leaving the secondary spectrum: the wavelengths between and beyond F and C still focus at slightly different points, because the two glasses' dispersion curves do not have the same shape. With ordinary crown and flint pairs the secondary spectrum is about $f/2000$, some 50 µm for the 100 mm lens, a reduction of about 30 times. Apochromats, which bring three wavelengths together using anomalous-dispersion glasses, reduce it further, as summarized in aberrations.

The same two-element structure gives the designer enough freedom to control spherical aberration at one conjugate. A well-made doublet focusing a collimated beam on axis at a modest f-number has far lower wavefront error than a plano-convex singlet of the same focal length and aperture, and at slow enough apertures it is diffraction-limited.

Measurement

The chromatic correction is checked by measuring back focal length at several wavelengths with a collimated beam and a knife edge or shear plate. Plotted against wavelength, an achromat's focal shift is a parabola-like curve with equal values at F and C and a turning point in the green, while a singlet's is monotonic.

Where doublets are used

Doublets form the lenses of Keplerian beam expanders, relay and tube lenses, telescope objectives, and the focusing lenses of spectrometers and imaging systems. They are also the usual choice when a system is aligned with a visible laser and used at another wavelength. Near-infrared versions use different glass pairs.

Pitfalls

Orientation matters: for a collimated input, the side with the stronger curvature, usually the crown element, faces the collimated beam; reversed, spherical aberration rises markedly. The design is optimized for infinite conjugates unless stated otherwise, so for 1:1 imaging two identical doublets back to back perform better than one used at finite conjugates. Cemented doublets have an adhesive layer that absorbs in the ultraviolet and limits the tolerable laser power, so high-power systems use air-spaced doublets. The quoted focal length is the effective focal length at a design wavelength; the back focal length, measured from the last vertex, is shorter and sets the physical spacing. For a single wavelength at high NA, an aspheric lens usually does better.

Common questions

Why are achromatic doublets made of two different glasses?

Two thin elements of the same glass in contact have the same Abbe number, so any combination with net positive power still has a net color error. Using a crown and a flint with different dispersion lets the negative element cancel the color while cancelling only part of the power.

What is the difference between an achromat and an apochromat?

An achromat brings two wavelengths to a common focus and leaves a secondary spectrum of roughly $f/2000$. An apochromat brings three wavelengths together, usually with fluorite or extra-low-dispersion glass, and its residual color is much smaller.

Do achromatic doublets help with a single-wavelength laser?

Yes. Even at one wavelength, a doublet's extra surfaces correct spherical aberration, so it focuses a collimated beam to a smaller spot than a plano-convex singlet of the same focal length and aperture.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 6; W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 5.