2.1 Simulation of the phase aberration using a quasi-Hilbert transform

Figure 1 shows the schematic diagram of a Mach–Zehnder interferometer setup in the experiment. A He–Ne laser is employed as the probe laser whose beam size is enlarged by an inversed telescopic system. The vapor cell (4 cm length, 1.5 cm diameter), in which 500 Torr methane was beforehand filled at the room temperature, is placed in one arm of the Mach–Zehnder interferometer. During the experiment, a heater was used to warm the cell and a thermocouple was employed to detect the temperature on the outside surface of a cell. Employing a CMOS camera with a 4f-optical imaging system, we record intensity distributions of the sample beam, reference beam, and interferential pattern at different heating conditions. Thus, the interferential irradiance intensity at the camera can be written as^{[Reference Wong and Chan16]}

(1) $$\begin{eqnarray}\displaystyle I_{\mathit{CCD}}(x,y) & = & \displaystyle I_{S}(x,y)+I_{R}(x,y)\nonumber\\ \displaystyle & & \displaystyle +\,2\sqrt{I_{S}(x,y)\cdot I_{R}(x,y)}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\cos [\unicode[STIX]{x1D719}(x,y)+(q_{x}x+q_{y}y)],\end{eqnarray}$$

where $I_{S}(x,y)$ and $I_{R}(x,y)$ are the sample and reference beam intensities, $\unicode[STIX]{x1D719}(x,y)$ is the spatial phase associated with the cell, $q_{x}$ and $q_{y}$ are the spatial frequencies of the interference fringes, respectively. Note that $\unicode[STIX]{x1D719}(x,y)$ is a parameter that we most concern about in this subsection.

Solving Equation (1), we can get a cosine term as expressed by

(2) $$\begin{eqnarray}\displaystyle & & \displaystyle \cos [\unicode[STIX]{x1D719}(x,y)+(q_{x}x+q_{y}y)]\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{I_{\mathit{CCD}}(x,y)-[I_{S}(x,y)+I_{R}(x,y)]}{2\sqrt{I_{S}(x,y)\cdot I_{R}(x,y)}}.\end{eqnarray}$$

Next, the cosine term is changed to frequency domain by means of a procedure of fast Fourier transform (FFT):

(3) $$\begin{eqnarray}F(u,v)=\text{FFT}(\cos [\unicode[STIX]{x1D719}(x,y)+(q_{x}x+q_{y}y)]).\end{eqnarray}$$

Then, the low frequency background noise and one of the conjugate patterns in $F(u,v)$ are removed by use of a bandpass filter. After that, we can have a new function through an inverse fast Fourier transform (IFFT)^{[Reference Ikeda, Popescu, Dasari and Feld13]}:

(4) $$\begin{eqnarray}\displaystyle z(x,y) & = & \displaystyle {\textstyle \frac{1}{2}}\cos [(q_{x}x+q_{y}y)+\unicode[STIX]{x1D719}(x,y)]\nonumber\\ \displaystyle & & \displaystyle +\,\text{i}\cdot \mathit{HT}\left\{{\textstyle \frac{1}{2}}\cos [(q_{x}x+q_{y}y)+\unicode[STIX]{x1D719}(x,y)]\right\},\qquad\end{eqnarray}$$

where the imaginary part of the right side of the above equation stands for the Hilbert transform of the real part, which is presented by

(5) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathit{HT}\left\{{\textstyle \frac{1}{2}}\cos [(q_{x}x+q_{y}y)+\unicode[STIX]{x1D719}(x,y)]\right\}\nonumber\\ \displaystyle & & \displaystyle \quad ={\textstyle \frac{1}{2}}\sin [(q_{x}x+q_{y}y)+\unicode[STIX]{x1D719}(x,y)].\end{eqnarray}$$

By combining Equations (4) and (5), we can calculate the value of a wrapped phase in the range of $-\unicode[STIX]{x1D70B}$ and $\unicode[STIX]{x1D70B}$ as expressed by

(6) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(x,y)=\arctan \left(\frac{\text{Im}[z(x,y)]}{\text{Re}[z(x,y)]}\right).\end{eqnarray}$$

Finally, the phase distortion associated with the vapor cell can be retrieved by an unwrapping theme and an incline correction procedure by using the calculation as follows:

(7) $$\begin{eqnarray}\unicode[STIX]{x1D719}(x,y)=\unicode[STIX]{x1D6F7}(x,y)-(q_{x}x+q_{y}y).\end{eqnarray}$$

2.2 Evaluation of the temperature distribution inside a vapor cell

Next, we focus on evaluating the temperature distribution in the cross-section of a cell while the temperature distribution is assumed to be uniform along the longitudinal axis. Thus, the optical path length $L(x,y)$ in waves for the probe laser can be expressed by

(8) $$\begin{eqnarray}L(x,y)=L_{\text{cell}}kn(x,y),\end{eqnarray}$$

where $L_{\text{cell}}$ is the length of a vapor cell, $k$ stands for the wavenumber of the probe laser, and $n(x,y)$ is the refractive index distribution inside a cell, respectively.

Generally, the temperature distribution will become nonuniform when the cell is heated. Nevertheless, the temperature at the cell edge can be directly measured through a commercial thermocouple. Thus, when a probe laser passes through the vapor cell, the phase difference of a wavefront is expressed by

(9) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}N(x,y)=L(x,y)-L_{\text{edge}}=L_{\text{cell}}kn(x,y)-L_{\text{cell}}kn_{\text{edge}},\end{eqnarray}$$

where $L_{\text{edge}}$ and $n_{\text{edge}}$ are the length in waves and the refractive index at the edge of a vapor cell. $\unicode[STIX]{x1D6E5}N(x,y)$ can be calculated by the theory introduced in last subsection with the following formula:

(10) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}N(x,y)=\unicode[STIX]{x1D719}(x,y)-\unicode[STIX]{x1D719}_{\text{edge}},\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{\text{edge}}$ is the spatial phase at the edge of a vapor cell. According to the ideal gas law and the Gladstone–Dale relation, the relationship between the temperature of the arbitrary point at the cross-section and that at the cell edge yields^{[Reference Shaffer, Lilly, Zhdanov and Knize12]}

(11) $$\begin{eqnarray}T(x,y)=T_{\text{edge}}\frac{n_{\text{edge}}-1}{n(x,y)-1}.\end{eqnarray}$$

Uniting Equations (9) and (11), we can obtain the temperature distribution of a heated cell expressed by

(12) $$\begin{eqnarray}T(x,y)=T_{\text{edge}}\frac{n_{\text{edge}}-1}{n_{\text{edge}}-1+\frac{\unicode[STIX]{x1D6E5}N(x,y)}{L_{\text{cell}}k}}.\end{eqnarray}$$