Extensive properties

Extensive property changes, such as volume, alter the shape of optical and mechanical components. Systems are geometrically optimized for optical performance and are sensitive to components changing shape and orientation. While volume is a three dimensional parameter, thermal changes can be modeled in a single dimension with linear expansion, assuming an adequately small temperature range. For examples, glass manufacturer Schott provides the coefficient of linear thermal expansion for a temperature range of -30 C to 70 C. The change in length of a material is a function of the change in temperature with respect to the standard measurement temperature, ${\textstyle T_{0}}$. This temperature is typically room temperature or 22 degrees Celsius.

${\displaystyle \Delta T=T-T_{0}\,}$

${\displaystyle L_{\left(T\right)}=L_{0}+\Delta L=\left(1+\alpha \Delta T\right)L_{0}\,}$

Where ${\textstyle L_{\left(T\right)}}$ is the length of a material at temperature ${\textstyle T}$, ${\textstyle L_{0}}$ is the length of the material at temperature ${\textstyle T_{0}}$, ${\textstyle \Delta T}$ is the change in temperature, and ${\textstyle \alpha }$ is the coefficient of thermal expansion. These equations describe how diameter, thickness, radius of curvature, and element spacing change as a function of temperature.

Intensive properties

Dispersion of N-BK7 over the visible spectrum using coefficients from the Schott Optical Glass Data Sheets.^[4][5]

The dominant intensive property change, in terms of optical performance, is the index of refraction. The refractive index of glass is a function of wavelength and temperature.^[4] There are multiple formulas that can be used to define the wavelength dependence, or dispersion, of a glass. Following the notation from Schott, the empirical Sellmeier equation is shown below.^[5]

${\displaystyle n_{\left(\lambda ,\ T_{0}\right)}={\sqrt {1+{\frac {B_{1}\lambda ^{2}}{\lambda ^{2}-C_{1}}}+{\frac {B_{2}\lambda ^{2}}{\lambda ^{2}-C_{2}}}+{\frac {B_{3}\lambda ^{2}}{\lambda ^{2}-C_{3}}}}}\,}$

Temperature coefficients of the absolute refractive index of N-BK7 for the d, F, C spectrum.^[6]

Where ${\textstyle \lambda }$ is wavelength and ${\textstyle B_{1}}$, ${\textstyle B_{2}}$, ${\textstyle B_{3}}$, ${\textstyle C_{1}}$, ${\textstyle C_{2}}$, and ${\textstyle C_{3}}$ are the Sellmeier coefficients. These coefficients can be found in glass catalogs provided from manufacturers and are usually valid from the near-ultraviolet to the near-infrared. For wavelengths beyond this range, it is necessary to know the material's transmittance with respect to wavelength. From the dispersion formula, the temperature dependence of refractive index can be written:^[6][7]

${\displaystyle {\frac {dn_{abs\left(\lambda ,\ T\right)}}{dT}}={\frac {n_{\left(\lambda ,\ T_{0}\right)}^{2}-1}{2n_{\left(\lambda ,\ T_{0}\right)}}}\left(D_{0}+2D_{1}\Delta T+3D_{2}\Delta T^{2}+{\frac {E_{0}+2E_{1}\Delta T}{\lambda ^{2}-\lambda _{TK}^{2}}}\right)\,}$

and

${\displaystyle n_{\left(\lambda ,\ T\right)}=\left(1+\Delta T{\frac {dn_{abs\left(\lambda ,\ T\right)}}{dT}}\right)n_{\left(\lambda ,\ T_{0}\right)}\,}$

Where ${\textstyle D_{0}}$, ${\textstyle D_{1}}$, ${\textstyle D_{2}}$, ${\textstyle E_{0}}$, ${\textstyle E_{1}}$, and ${\textstyle \lambda _{TK}}$ are glass-dependent constants for an optic in vacuum. The power of an optic as a function of temperature can be written from the equations for extensive and intensive property changes, in addition to the lensmaker's equation.

${\displaystyle \Phi _{lens\left(\lambda ,\ T\right)}=\left(n_{\left(\lambda ,\ T\right)}-1\right)\left({\frac {1}{R_{1\left(T\right)}}}-{\frac {1}{R_{2\left(T\right)}}}+{\frac {\left(n_{\left(\lambda ,\ T\right)}-1\right)L_{\left(T\right)}}{{n_{\left(T\right)}R}_{1\left(T\right)}R_{2\left(T\right)}}}\right)\,}$

${\displaystyle \Phi _{mirror\ \left(T\right)}={\frac {-2}{R_{(T)}}}\,}$

Where ${\textstyle \Phi }$ is optical power, ${\textstyle R}$ is the radius of curvature, ${\textstyle L}$ is the thickness of the lens. These equations assume spherical surfaces of curvature. If a system is not in vacuum, the index of refraction for air will vary with temperature and pressure according to the Ciddor equation, a modified version of the Edlén equation.^[8][9]

Comparison of optical performance over temperature with different mechanical housing materials for a positive N-BK7 singlet designed at 22 C.^[10] "No thermal expansion" still includes effects from dn/dT.^[6]

Passive athermalization

Passive athermalization works by choosing materials for a system that will compensate the overall change in system performance. The simplest way to do this is to choose materials for the optics and mechanics which have low CTE and ${\textstyle {\frac {dn}{dT}}}$ values. This technique is not always possible as glass types are primarily chosen based on their refractive index and dispersion characteristics at operating temperature. Alternatively, mechanical materials can be chosen which have CTE values complementary to the change in focus introduced by the optics. A material with the preferred CTE is not always available, so two materials can be used in conjunction to effectively get the desired CTE value.^[1][10] Negative thermal expansion materials have recently increased the range of potential CTEs available, expanding passive athermalization options. ^[11]

Active athermalization

When optical designs do not permit the selection of materials based on their thermal characteristics, passive athermalization may not be a viable technique. For example, the use of germanium in mid to long wave infrared systems is common because of its exceptional optical properties (high index of refraction and low dispersion). Unfortunately, germanium is also known for its large ${\textstyle {\frac {dn}{dT}}}$ value, which makes it difficult to passively athermalize.^[12]

Because the primary aberration induced by temperature change is defocus, an optical element, group, or focal plane can be mechanically moved to refocus a system and account for thermal changes.^[13] Actively athermalized systems are designed with a feedback loop including a motor, for the focusing mechanism, and temperature sensor, to indicate the magnitude of the focus adjustment.^[10]