## 2 Theory
### Measurement Model  
In order to obtain an accurate initial \(n_{0}\), the indirect measurement method of the air refractive index should be used first. Therefore, some high-sensitivity temperature, pressure and humidity sensors, and CO\({}_{2}\) water vapor sensors were arranged in the measuring optical path in order to measure the air temperature, pressure and humidity, and then the initial refractive index \(n_{0}\) was calculated and obtained by the refractive index correction formula.  
Because the conventional Edlen formula for calculating the air refractive index does not include the modified item of the CO\({}_{2}\) error [14;15], the improved Edlen formula is as follows:  
\[\left(n-1\right)_{s}=10^{-8}\cdot\left[A+B/\left(130-\sigma^{2}\right)+C/\left( 38.9-\sigma^{2}\right)\right] \tag{5}\]  
\[\left(n-1\right)_{x}=\left[1+0.540\cdot\left(x-0.0003\right)\right]\cdot\left( n-1\right)_{s} \tag{6}\]  
\[n_{TP}=1+P\cdot\left(n-1\right)_{x}/D\cdot\left[1+10^{-8}\cdot\left(E-F\cdot T \right)\cdot P\right]/\left(1+G\cdot T\right) \tag{7}\]  
\[Y=-J\cdot\left[1-\left(T/273.16\right)^{-1.5}\right]+K\cdot\left[1-\left(T/27 3.16\right)^{-1.25}\right] \tag{8}\]  
\[n_{TP}-n_{TP}=-\left(f/100\cdot 611.657\cdot e^{Y}\right)\cdot\left(292.75/T \right)\cdot\left[3.7345-0.0401\cdot\sigma^{2}\right]\cdot 10^{-10} \tag{9}\]  
Here, the constants can be defined as follows [11]:  
\(A=8342.54\); \(B=2,406,147\); \(C=15,998\); \(D=96,095.43\); \(E=0.601\); \(F=0.00972\); \(G=0.003661\); \(J=13.928169\); \(K=34.7078238\); \(\sigma=1/\lambda\); \(T=t+273.15\).  
Here \(\left(n-1\right)_{s}\) is the refractive index of air in the standard state, \(\sigma\) is the wave number in the vacuum, expressed in the unit of um\({}^{-1}\), \(\left(n-1\right)_{x}\) is the refractive index of air with the molar content of carbon dioxide, where the unit is \(\left(\times 10^{-6}\right)\), \(n_{TP}\) is the refractive index of air under the conditions of temperature, air pressure and carbon dioxide, and \(n_{TP}\) is the refractive index of air under the conditions of temperature, humidity, air pressure and carbon dioxide.  
Figure 2: The layout of light path based on the etalon. (**a**) Description of the layout of the reference light path based on the etalon. (**b**) Description of the layout of the measurement light path based on the etalon.  
After obtaining the initial value of the air refractive index, the etalon can be utilized to calculate the real-time wavelength compensation. The compensation factor \(C\) is:  
\[C=\frac{10^{6}}{n_{Tpf}+10^{6}} \tag{10}\]  
Suppose that the number \(P_{n}\) of preset wavelengths at the moment of measurement is:  
\[P_{n}=\frac{LRf}{C\lambda} \tag{11}\]  
where \(L\) is the effective length between the two mirrors of the etalon of the air refractive index, \(R\) is the electrical resolution of the interferometer, and f is the fold factor of the laser interference in the light path.  
The value of the compensation \(C_{n}\) is therefore:  
\[C_{n}=\frac{LRf}{(P_{A}+P_{n})\lambda} \tag{12}\]  
where \(P_{A}\) is cumulative count.  
The dead-path count \(C_{Dn}\) is:  
\[C_{Dn}=\frac{DRf}{C\lambda} \tag{13}\]  
where \(D\) is the dead-path length in the interferometer optical path.  
Then the initial actual displacement \(\mathrm{P_{s0}}\) after compensation is:  
\[\mathrm{P_{s0}}=\frac{C_{n}(\mathrm{P_{A}}+C_{Dn})\lambda}{\mathrm{Rf}} \tag{14}\]  
Finally, the current value \(n_{1}\) of the reflective index of air is:  
\[n_{1}=\left(1+\frac{\Delta L}{P_{s0}}\right) \tag{15}\]