1. Introduction

In optical measurements the index of refraction of the medium the optical radiation is penetrating has to be considered. Especially in the case were the optical radiation passes different media this fact is of high concern, as also refraction at the contact surface between the two media has to be considered. Working at usual laboratory conditions, the medium the radiation is penetrating is ambient air and the formulas for the index of refraction of air derived by Edlén and modified by others can be used to calculate the refractive index with low uncertainty [1, 2, 3, 4, 5]. However, the situation is more complex when dealing with the measurement of blackbody radiation generated in a cavity radiator. Due to the different temperature inside and outside the blackbody cavity and due to purging of the cavity with argon or other inert gases, the index of refraction of the media inside and outside the blackbody cavity are, in general, different. In literature usually simply the index of refraction of ambient air is taken into consideration, accounting for the optical path between the opening of the cavity and the detecting instrument. For standard reference laboratory conditions the refractive index of ambient air is shown in Fig. 1 [2]. However, this accounts only for the travelling part of the optical radiation outside the cavity, neglecting all effects originating inside the cavity. In some radiometric and photometric experiments reported in literature the refractive index is even neither stated nor considered at all.

 

Fig. 1. Refractive index of air a 1013 hPa, 0.03 % CO2 content and 50 % relative humidity.

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For the generation of the blackbody radiation inside the cavity the index of refraction of the medium inside the cavity has to be taken into account [6] and this might differ significantly from the value of the ambient medium, especially if the blackbody cavity is at elevated temperatures or purged by an inert gas.

2. Consideration of the index of refraction for blackbody radiation

Planck’s formula for the spectral radiance L _s,ν of a blackbody emitted per unit area and frequency interval in a solid angle is given as a function of frequency ν as [6]

In Eq. (1), n is the index of refraction of the medium inside the cavity at the cavity temperature T, ν is the frequency of the optical radiation, h is Planck’s constant, k is the Boltzmann constant and c the velocity of light in vacuum. For optical radiation inside a medium with refractive index n the following equation for the wavelength λ, the velocity of light in the medium c_n and the frequency ν holds

resulting in

Rewriting Eq. (1) as a function of wavelength λ inside the medium requires a relation for L _s,λ and L _s,ν. This can be found by using Eq. (3) resulting in the following expression

Using Eq. (4) and Eq. (3), Eq. (1) can be written as follows

Equation (5) describes the spectral radiance at the exit aperture of a cavity radiator containing a medium with refractive index n inside as a function of the wavelength λ inside this medium. Equation (5) is identical to the results given in literature [6].

The radiation emitted by the opening of the blackbody cavity has to travel the distance between the exit aperture and the detector through the ambient medium, usually air at ambient conditions. This will cause a wavelength shift according to the index of refraction of the ambient medium, n _amb. To account for this, Eq. (5) is rewritten in terms of vacuum wavelength λ₀= n·λ resulting in

Accounting for the wavelength shift in the ambient medium requires replacement of λ₀ by n _amb·λ

Comparing Eq. (7) with Eq. (5) reveals that the effect of including the transmission of the radiation through the ambient medium with refractive index n _amb introduced the factor (n/n_amb)³, changing the effective spectral radiance L _s,λ by this factor.

Another effect which has to be accounted for is the reflection occurring at the contact area between the media inside of the blackbody cavity and the outside media, usually ambient air. For perpendicular incidence the corresponding reflectivity R is

In Eq. (8) it is assumed that the transition between the two media inside and outside the blackbody is abrupt. However, in reality, as it is a transition between two gaseous phases, the transition is not abrupt but smooth and diffuse, therefore, Eq. (8) is an upper limit for the expected reflectivity. As the two media inside and outside the blackbody cavity are usually both gases the reflectivity at the contact between these two media will be negligible small. Therefore, Eq. (8) as a worst case assumption will be used for the purpose of this letter.

Consideration of Eq. (8) leads to the final formula for the spectral radiance emitted by a blackbody radiator at temperature T with a medium of refractive index n into a medium of refractive index n_amb

with the factor $a(n_{\mathrm{amb}},n)=\left[1-{\left(\frac{n_{\mathit{amb}}-n}{n_{\mathit{amb}}+n}\right)}^2\right]\frac{n^3}{n_{\mathit{amb}}^3},$ which accounts for the effect of the two different media inside and outside the blackbody cavity. For the correct consideration the factor a(n_amb, n) has to be taken into account, when calculating the spectral radiance emitted by a blackbody radiator according Planck’s law.

3. Discussion

For an estimation of the error introduced by simply using Eq. (5) with the refractive index of the ambient medium instead of using Eq. (9) calculations have been performed. The most pronounced effect is expected, if the refractive index inside and outside the blackbody cavity differs significantly. This will be the case for blackbodies at elevated temperatures. Usually such blackbodies are operated in an argon gas atmosphere [7]. As the refractive index will approach unity for highest gas temperatures, for a worst case estimation it is assumed that the refractive index of the argon gas inside a blackbody cavity at temperatures of 3000 °C is unity. Using the modified Edlén equation for the refractive index of air as the outside medium given in Ref. 2 the relative difference (i.e. the ratio of the difference of a value a and a value b with respect to the value a) between Eq. (9), considering the two different media inside and outside the blackbody cavity correctly, and Eq. (5), considering only the refractive index of ambient air, has been calculated. The resulting relative difference as a function of wavelength is shown in Fig. 2.

 

Fig. 2. Relative difference (solid line) between the spectral radiance of a blackbody radiator with the correct consideration of the medium with refractive index of unity inside and of ambient air outside [according Eq. (9)] with respect to the spectral radiance only considering the refractive index of the ambient air [according Eq. (5)]. Also shown is the resulting error (dashed line) in determining the radiance temperature of a blackbody at 3000 °C (a) overview, b) detail at short wavelengths; the arrow in the graphics indicates that the right ordinate is valid for the dashed line while the left ordinate is valid for the solid line).

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Figure 2 shows that neglecting the difference in refractive index inside and outside the cavity will introduce a nearly constant relative difference of about 0.08 % over the whole wavelength range from 200 nm to 15 μm, which is theoretically increasing to 0.1 % at wavelengths down to 100 nm, well within the vacuum ultraviolet regime. This error neglecting the different refractive indices might be tolerable in some photometric experiments, however it has to be taken into account for accurate radiance temperature measurements. The introduced error in determining the temperature of a blackbody at 3000 °C using radiometric methods is also shown in Fig. 2. In this worst case estimation this error is in the order of several tenths of a Kelvin in the visible, increasing to several Kelvin in the infrared region. In literature it is stated that the obtained uncertainty in radiometric temperature measurements nowadays is as low as a few tenths of a Kelvin for temperatures around 3000 °C [8, 9, 10]. The correct consideration of the refractive index is, therefore, mandatory for radiometric temperature measurements. This is especially important for the determination of the thermodynamic phase transition temperatures of the novel metal-carbon and metalcarbide-carbon eutectics for an improved International Temperature Scale, where an uncertainty in temperature of about 0.1 K is required [11].