To predict how a lens will change power (inverse focal length) with temperature, the therm optic coefficient,β, is used. This applies to a refractive or diffractive lens.
The prime ' is used to denote that power has been thermally cycled. The DΦ/DT is a derivative. The equation simplified by changing the derivative to a linear shift (Δ) or DΦ/DT ≅ ΔΦ/ΔT.
If the therm optic coefficient, β, is inverted it becomes the thermal abbe number V _{t.} This concept is introduced because making a passively athermal lens has identical equations for color correcting. Said another way: if you understand the equations behind constructing an achromatic doublet, then making a passively athermal lens is identical!
Refractive therm optics coefficients (reference 1 to 4)
The full equation for the therm optic coefficient is
For infrared materials, the index of air is 1.0 and Dn_{air}/dT=0
Schott literature in the 1970s used another form of the therm optic coefficient. This is used for inhomogeneous temperature effects
Diffractive therm optic coefficients
If you are using a diffractive optical element (DOE) or a holographic optical element (HOE), then the therm optic coefficient is different.
(need to include n*dnair/dt.....) (reference 2)
So if you have plate window with a pure diffractive on it, the therm optic coefficient, β, is simple a function of the coefficient of thermal expansion, α, of the diffractive substrate.
If you are doing a detailed model in a ray trace program and you want to see how a specific optical prescription parameter is thermally cycled, see here.
References:
 Introduction to optomechanical design, D. Vukobratovich, SPIE press 1993
 "Introduction to Optomechanical Design", Course Notes Opti 690 University of Arizona, D. Vukobratovich, Spring 1995
 Private Communications with Dan Vukobratovich.
 Handbook of thermooptic coefficients of optical materials with applications By Gorachand Ghosh
 "The influence of temperature on diffractive lens performance", G. P. Behrmann, J. P. Bowen, Applied Optics, Vol. 32, Issue 14

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