TIE-19. Temperature Coefficient of the Refractive Index. Technical Information Advanced Optics. Introduction. 1. Fundamentals.

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1 Version July TIE- Introduction The refractive index of optical glasses changes with temperature, the extend of which depending on the glass type and on the wavelength. It also changes with air pressure, therefore in the following we distinguish between the refractive index relative to vacuum or absolute refractive index and the refractive index at normal air pressure, called refractive index relative to normal air (relative refractive index, all values given in the Pocket Catalog are relative refractive index values). The standard measurement temperature for refractive index values given on test certificates is C []. By taking into account the temperature coefficients of the glasses it is possible to calculate the change for any wavelength from near UV to near IR in a temperature range from C up to + C. The constants of the formula are listed in the data sheets of the respective optical glass []. The calculation of the actual refractive index at a given temperature often seems not to be straight forward and the TIE- [] displays only the formula for the temperature coefficients for. Fundamentals.... Derivation and Characterization of the Formula.... Temperature Coefficients of Optical Glasses.... Athermal Glasses.... Literature.... the absolute refractive index. This will give a guideline how to calculate the refractive index at different temperatures. Additionally some results of temperature coefficients for optical glasses will be given and discussed. Some special glasses with negative temperature coefficients in an optical system can help to keep wave front deformations caused by temperature changes on a minimum level. Such glasses are called athermal glasses.. Fundamentals The refractive index of glasses relative to vacuum (absolute refractive index) can be described with sufficient approximation as a function of wavelength from the near IR through the visible to the near UV spectral region by a Sellmeier-type formula []. The product a N V f as well as λ can be determined for a chosen temperature (for example room temperature) by fitting the measured data for n abs (λ) at this temperature. This parameter is temperature independent. n abs(λ) = a N V f λ λ λ λ () with: n abs (λ) refractive index relative to vacuum λ wavelength of the electromagnetic wave in vacuum a constant N average number of oscillators in volume V f average oscillator strength λ average resonant wavelength of the electronic oscillator in the UV

2 TIE-. Derivation and Characterization of the Formula The derivative of the Sellmeier function () with temperature yields equation () after simplification by omitting terms with negligible contributions. The coefficients D, D, D, E, E and λ TK for a given glass type will be determined by fitting the experimentally obtained data []. dn abs (λ, T) = n (λ, T ) n (λ, T ) ( D + D ΔT + D ΔT + E + E ΔT λ λ TK ) () with: T Reference temperature ( C) T Temperature (in C) ΔT Temperature difference T-T λ Wavelength of the electromagnetic wave in a vacuum (in µm) D, D, D, E, E and λ TK : constants depending on glass type The refractive index values given in the Optical Glass Pocket Catalog apply for an air pressure of. Pa. They are called relative refractive index values n rel. These values can be used for n(λ, T ) in equation () with sufficient accuracy. When the function of temperature is closed integrable, the change in the absolute refractive index n abs (λ, T ), with the temperature difference ΔT can be derived from equation (): Δn abs (λ, T) = n (λ, T ) n (λ, T ) ( D ΔT + D ΔT + D ΔT + E ΔT + E ΔT λ λ TK ) and () The refractive index relative to air dn rel(λ, T) and Δn rel (λ, T) can be calculated from: n rel (λ, T) = and dn abs (λ, T) n rel (λ, T) or dn rel (λ, T) n abs (λ, T) n air (λ, T, p) = n air (λ, T, p) dn rel(λ, T) dn air (λ, T, p) = dn abs (λ, T) + dn n rel (λ, T) air (λ, T, p) n air (λ, T, p) In () and () n rel (λ, T ) can be substituted for n rel (λ, T) with sufficient accuracy. The value for n air (λ, T, p) and dn air(λ, T, p) can be calculated with good accuracy with [] n air (λ, T, P) = + n air (λ, C, P ) = + (. + () () () n air (λ, C, P ) ( +. C (T C) ) P () P µm λ ( µm λ ) + µm ) λ () ( µm λ ) n abs (λ, T) = n abs (λ, T ) + Δn abs (λ, T) () Consequently, two simple closed expressions for the calculation of the temperature coefficient of the absolute refractive index: dn abs (λ, T), as well as the change of the absolute refractive index with temperature: Δn abs (λ, T) are available.

3 TIE- dn air (λ, T, P) =. n air (λ, T, P) +. C T () index compared to the value at C can be calculated with equation (). The corresponding value for air can be calculated by () and (). with: P P λ T. Pa (normal pressure in Pascal) Air pressure Wavelength of the electromagnetic wave in a vacuum (in µm) Temperature (in C) With the help of glass specific parameters D, D, D, E, E and λ TK the value of the temperature coefficient of the absolute refractive index (compared to vacuum) as a function of temperature and wavelength can be calculated with equation () for each glass type. The variation of the absolute refractive The constants of formula () and () are given in the glass data sheets []. The constants are valid for a temperature range from C to + C and a wavelength range from. µm to. µm. The temperature coefficients in the data sheets are guideline values. Upon request, measurements can be performed on individual melts for the temperature and wavelength range as given above with a precision better than ± /K. The constants of the dispersion formula can be calculated from this measured data and will be listed on the test certificate for the individual glass melt on request.. Temperature Coefficients of Optical Glasses The extent of change of the refractive index at a given wavelength varies from glass to glass. Table shows the list of our current product range sorted by the relative temperature coef- ficients between + C and + C at the e-line (~. nm). The values reach from +. /K for P-SF down to. /K for N-PK. [+/+ deg. C] nm e-line g-line P-SF..... SF..... SF..... SFHTultra..... SF..... SFHT..... SFA..... N-LASF..... SF..... P-LASF..... N-LASFB..... P-LASF..... P-LASF..... N-LAF..... N-LAF..... Tab. : Product range sorted by relative temperature coefficients between + C and + C at the e-line. [+/+ deg. C] nm e-line g-line LAFN..... SF..... SF..... N-ZK..... N-LASFA..... N-LASF..... SF..... N-LASF..... N-LASFHT..... N-KZFS.....

4 TIE- [+/+ deg. C] nm e-line g-line LASF..... N-LAF..... N-LASF..... N-KZFS..... N-LASF..... N-SSK..... N-LAF..... N-LAK..... N-BALF..... N-LASF..... N-LASFHT..... N-LAK..... N-LASFA..... N-BAF..... N-KZFS..... F..... SF..... N-LAKB..... N-SK..... N-SKHT..... F..... FHT..... N-BASF..... N-BASF..... N-LAF..... K..... N-LAK..... N-KZFS..... N-SF..... N-BAK..... N-BAKHT..... N-LAK..... P-SKA..... Tab. : Product range sorted by relative temperature coefficients between + C and + C at the e-line. [+/+ deg. C] nm e-line g-line N-KZFS..... N-KZFSHT..... N-LAK..... N-SK..... P-SK..... P-SK..... N-F..... N-BK..... N-SF..... N-SF..... N-BAF..... N-BAF..... N-SSK..... N-SK..... N-LAK..... N-SK..... SCHOTT N-BK..... N-BKHT..... N-PSK..... N-BAF..... LLF..... N-BALF..... N-SSK..... N-SK..... N-SF..... N-BAK..... N-SF..... N-SF..... N-SK..... N-SF..... N-SFHT.....

5 TIE- [+/+ deg. C] nm e-line g-line N-SFHTultra..... N-K..... LFHTi..... LF..... N-KF..... N-SF..... P-LAK..... N-BAK..... K..... P-SF..... N-SF..... N-SF..... N-SFHT..... N-SFHTultra..... [+/+ deg. C] nm e-line g-line N-SF..... N-SF..... N-LAF..... N-LAK..... N-LAK..... N-LAK..... N-FK..... N-PSKA..... N-FKA..... N-FK..... N-PKA..... N-PK..... Tab. : List of our current product range sorted by the relative temperature coefficients between + C and + C at the e-line (~. nm). Figures to show the change of the temperature coefficients (absolute on the left side and relative on the right side) with temperature for different wavelengths and different glasses. It can be seen that the shape of the curve of the relative temperature coefficients differs from the absolute temperature coefficients. The reason is the influence of the temperature coefficients of the air. In general the influence of the temperature on the refractive index at shorter wavelengths is much higher compared to longer wavelengths. This dispersion of the temperature coefficient is more pronounced for high dispersion glasses.

6 TIE- dn abs (λ,t) dn rel (λ,t) dn abs / [ /K] dn rel / [ /K] glass type: SF n d :. v d :. glass type: SF n d:. v d:.. nm. nm. nm. nm. nm. nm. nm. nm. nm. nm Fig. : Temperature coefficient of the absolute (left figure) and relative (right figure) refractive index of SF for different wavelengths. dn abs (λ,t) dn rel (λ,t) dn abs / [ /K] dn rel / [ /K] glass type: F n d :. v d :. glass type: F n d :. v d :.. nm. nm. nm. nm. nm. nm. nm. nm. nm. nm Fig. : Temperature coefficient of the absolute (left figure) and relative (right figure) refractive index of F for different wavelengths.

7 TIE- dn abs (λ,t) dn rel (λ,t) dn abs / [ /K] dn rel / [ /K] glass type: SCHOTT N-BK n d :. v d :. glass type: SCHOTT N-BK n d :. v d :.. nm. nm. nm. nm. nm. nm. nm. nm. nm. nm Fig. : Temperature coefficient of the absolute (left figure) and relative (right figure) refractive index of SCHOTT N-BK for different wavelengths. dn abs (λ,t) dn rel (λ,t) dn abs / [ /K] dn rel / [ /K] glass type: N-LAF n d :. v d :. glass type: N-LAF n d :. v d :.. nm. nm. nm. nm. nm. nm. nm. nm. nm. nm Fig. : Temperature coefficient of the absolute (left figure) and relative (right figure) refractive index of N-LAF for different wavelengths.

8 TIE- Figure summarizes the above displayed examples in one diagram to better pronounce the different temperature coefficients of different glasses. Often it is easier to display the absolute values as a function of temperature. In figure the change of the relative refractive index with temperature is displayed for the above discussed glass types. It can be seen that a temperature increase of C changes the refractive index of SF at nm by.. Additionally the diagrams also contain the temperature change curves of N-PK. This glass has a negative temperature coefficient over the complete temperature range. dn abs / [ /K] dn abs (λ,t) SCHOTT N-BK SF N-PK N-LAF F dn rel / [ /K] dn rel (λ,t) SCHOTT N-BK SF N-PK N-LAF F Fig. : Temperature coefficient of the absolute (left figure) and relative (right figure) refractive index of different glass types in comparison at nm. n rel [ ] SCHOTT N-BK SF N-PK N-LAF F Fig. : Change of the relative refractive index (catalog value at C) with temperature of some different glass types at nm.

9 TIE-. Athermal Glasses In conditions, where the optical system is subjected to varying inhomogeneous temperature environments, wave front distortions can be expected due to changes in optical path length and refractive index with temperature. To compensate such effects, it is important that the effect of wave front distortions generated due to the inhomogeneous thermal expansion of the glass can be compensated by the refractive index change with temperature. The fact that the thermal expansion coefficient of optical glasses is always positive implies the need of optical glasses with a negative temperature coefficient to compensate thermal effect on the wave front. This can be expressed in the following equation: G = α (n rel (λ, T) ) + dn rel(λ, T) α is the thermal expansion coefficient of the glass, n rel (λ, T) the refractive index at the proposed wavelength. The value thermo optical constant G should be as low as possible. Glasses with a value of G near zero are called athermal glasses. Table shows that not all glasses exhibiting a negative temperature coefficient are excellent athermal glasses. () Optical glasses with negative temperature coefficients are N-LAK, N-PSKA, N-FK, N-FKA, N-PKA, N-FK and N-PK. But only N-PK, N-FK, N-PKA and N-FKA show a real athermal behavior. Glass n e Relative temperature coefficients of refractive index [+/+ deg. C], e-line [ K] alpha / [ K] G [ K] N-PK.... N-FK.... N-PKA.... N-FKA.... N-FK.... N-PSKA.... N-LAK.... SCHOTT N-BK.... N-LAF.... F.... SF.... Tab. : Optical glasses and their thermo optical constant G.

10 TIE- Figure shows the temperature coefficient of N-PK. It can be seen that the values are negative over the complete temperature region from C to + C. dn abs (λ,t) dn rel (λ,t) dn abs / [ /K] dn rel / [ /K] glass type: N-PK n d :. v d :. glass type: N-PK n d :. v d :. Fig. : Temperature coefficient of the absolute (left figure) and relative (right figure) refractive index of N-PK for different wavelengths.. Literature. nm. nm. nm. nm. nm [] SCHOTT Optical Glass Pocket Catalog [] Data Sheets Optical Glasses from SCHOTT [] SCHOTT TIE- Refractive index and dispersion [] The properties of optical glass, Bach, Neuroth (Eds.), Springer. nm. nm. nm. nm. nm Version July SCHOTT reserves the right to make specification changes in this product flyer without notice. SCHOTT AG Hattenbergstrasse Mainz Germany Phone + ()/- Fax + ()/- info.optics@schott.com