Refractive index compensating interferometry

Interferometric length measurements require the consideration of the refractive index of the air. The refractive index of the air varies with the temperature, pressure, humidity and composition of the air. The refractive index can be determined by means of a refractometer or is calculated from measured values of the air parameters temperature, pressure, humidity and CO₂ content using the Edlen formula. It is difficult to measure the refractive index in the interferometer path as, for example, the temperature sensors cannot be applied in the interferometer path. Due to temperature gradients and the time constants of the sensors, deviations may occur and affect the measurement uncertainty.

For approximately 40 years, a compensation method has been known which is based on a measurement with two different wavelengths. When the refractive index is disregarded, slightly different lengths L₁ and L₂ result due to the dispersion. As a result, the mechanical length is obtained as L = L₁ - A (L₂ - L₁). The correction term A (L₂ - L₁) describes the influence of the refractive index of the air and lies in the order of magnitude of 2.7^.10⁴L. In the case of dry air, the ratio A = (n₁ - 1)/(n₁-n₂) of the two refractive indices n_1,2 now only depends on the wavelengths and no longer on the environment parameters. For wavelength pairs in the range of the visible light and the near infrared, parameter A is larger than 10 (e.g. A~65 for the wavelengths 1064 nm and 532 nm). By this, measurement uncertainties in the length difference (L₂ - L₁) in accordance with the formula for the length are degraded by at least one order of magnitude.

Another disadvantage is that in the case of moist air, parameter A depends on the pressure, temperature and air humidity. In this case, the compensation method furnishes systematic deviations whose magnitudes depend on the variation of the air parameters. To avoid the systematic deviations, a modified procedure was applied. The refractive index depends approximately on the wavelength and on the three parameters pressure, temperature and air humidity (the influence of the CO₂ content is minute). If two of these parameters are measured, the third parameter - and thus the refractive index - can be calculated from the two lengths L₁ and L₂. Due to its homogeneity, the air pressure in the beam path is easy to measure. The sensors for air humidity and temperature work, however, relatively slow which complicates the measurement in the case of air turbulences.

To investigate the procedure, an interferometer, composed of a linear stage with a translation of 25 cm and a dead length of approx. 2.85 m, was installed in a closed measuring chamber. The dead length increases the differences in the compensation methods, as these depend on the length. A frequency-doubled Nd:YVO₄ laser with the wavelengths 1064 nm and 532 nm served as light source for the interferometer. Air pressure, temperature and humidity were measured separately.


Evaluation was performed in three different ways:

1. Classical calculation of the refractive index using the Edlen formula
2. Classical two-colour method without using the measured air parameters
3. Improved method, making use of the measured pressure and humidity

The figure shows the time dependence of a measurement at increasing air humidity. The red curve depicts the difference between the standard two colour method and the result from classical calculation while the black curve is the difference between the improved two colour method and the classical calculation. Whereas the standard two colour method deviates at increasing air humidity from the result based on the Edlen formula by up to 2.5 µm, the new procedure furnishes deviations below 1 µm. The relatively strong scattering of the data of the two-colour interference method is to be attributed to the factor A~65 available in the experiment, which increases the uncertainties of the two length measurements.

Fig. 1:

Contact: Dr. Karl Meiners-Hagen, Dr. Ahmed Abou-Zeid

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© Physikalisch-Technische Bundesanstalt Page created: 26-Sep-2006, last update: 01-Nov-2007 11:33 AM, K. Eggert