This is a method to show how to select two optical materials to use in a doublet configuration. They will be achromatic (two wavelengths with a common focus) and passive athermal. Really the thermal defocus has to be less than the diffraction limited depth of focus or ±2*λ*(F/#)^{2} Here is a quick summary of how an doublet is used to correct color, creating an achromat. As applied to the thermal equations, on can calculate the lens system therm optic coefficient and sets it equal to zero: Knowing materials color V number ("V") and the thermal V number ("V_{T}"), the governing equations can be combined to solve for passive athermalization and an achromat: The trick is locating two materials which have the same ratio of color to thermal V numbers. The easiest way to locate these pairs is with a nomograph! Plot thermal V number (V_{T}) versus color V number. Find a set of materials whose slope goes throught the origin. Let us suppose we have four refractive materials: apple, peach, orange and artichoke. Their color and thermal V numbers are plotted below Orange and Apple have matched thermal / color V numbers ratios. Therefore they can be designed to produce a doublet that is achromatic and passively athermal. References

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