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 Dnair/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 opto-mechanical 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 thermo-optic 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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 Updating... Scott Sparrold, Oct 9, 2011, 7:22 AM Scott Sparrold, Oct 9, 2011, 8:43 AM Scott Sparrold, Oct 9, 2011, 9:07 AM Scott Sparrold, Oct 9, 2011, 8:55 AM Scott Sparrold, Oct 9, 2011, 8:53 AM
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