## 2 Materials and Methods
### Electromagnetic Parameters Measurement and Microwave Absorption Calculation  
The principle of transmission line calculates the input impedance and reflection loss of single-layer microwave absorbing material with a metal backplate. As shown in the figure below, when an incident wave hits on the surface of absorber, it is divided into two parts, one part reflects from the surface into air (R\({}_{1}\)), and the other part infiltrates into the absorber (Figure 1a). The absorber will absorb the entered microwaves and convert them into heat. The electromagnetic wave entering the material is reflected again at the interface between the absorber and the metal plate (R\({}_{2}\)). If R\({}_{1}\) and R\({}_{2}\) meet the conditions of interference cancellation, then extinction effect can be obtained, leading to a dissipation of them at the air-absorber interface and resulting in enhanced microwave absorption performance. At this time, the reflected microwave energy will reach the lowest value and the reflection loss reach the peak.  
The reflection loss of the material under different thickness can be calculated from the relative permeability and dielectric constant in a given frequency range. The relative complex permittivity (\(\epsilon_{r}=\epsilon^{\prime}-je^{\prime\prime}\)) and relative complex permeability (\(\mu_{r}=\mu^{\prime}-j\mu^{\prime\prime}\)) of OCGAs were measured on a vector network analyzer (Agilent N5224B PNA, Agilent, Santa Clara, CA, USA) at 2-18 GHz (Figure 1b). The test samples were fabricated using melting paraffin to impregnate the OCGAs and cut into standard coaxial rings with an outer diameter of 7.0 mm; an inner diameter of 3.04 mm and a thickness of 4.0 mm. To investigate the microwave absorption properties of aerogels; the reflection loss of the incident electromagnetic wave was calculated according to the following equations [19; 20]:  
Figure 1: (**a**) The schematic diagram of microwave absorption material. (**b**) Schematic diagram of vector network analyzer and coaxial sample.  
\[RL(\text{dB})=20 lg|\frac{Z_{in}-1}{Z_{in}+1}| \tag{1}\]  
\[Z_{in}=\sqrt{\frac{\mu_{r}}{\varepsilon_{r}}}tanh(\frac{j2\pi fd}{c}\sqrt{\mu_{r} \varepsilon_{r}}) \tag{2}\]  
where \(RL\) is the value of reflection loss, \(Z_{in}\) is the normalized input impedance, \(\mu_{r}\) and \(\varepsilon_{r}\) are the measured relative complex permeability and permittivity, respectively, \(c\) is the light speed in vacuum, \(f\) is the frequency of microwaves and \(d\) is the thickness of the absorber.  
The enhancement of the complex permittivities can be interpreted by [PERSON] relaxation theory as expressed in the following equations [24]:  
\[\epsilon^{\prime}(\omega)=\varepsilon_{\infty}+\frac{\varepsilon_{s}- \varepsilon_{\infty}}{1+\omega^{2}\tau^{2}} \tag{3}\]  
\[\varepsilon^{\prime\prime}(\omega)=\frac{\varepsilon_{s}-\varepsilon_{ \infty}}{1+\omega^{2}\tau^{2}}+\frac{\sigma}{\omega\varepsilon_{0}}= \varepsilon_{p}^{\prime\prime}+\varepsilon_{c}^{\prime\prime} \tag{4}\]  
where \(\omega\) is the angular frequency, \(\tau\) is the polarization relaxation time, \(\varepsilon_{s}\) is the static permittivity, \(\varepsilon_{\infty}\) is the relative permittivity in the high-frequency limit, \(\sigma\) is the conductivity, \(\varepsilon_{0}\) is the permittivity in vacuum (\(\varepsilon_{0}\) = 8.854 \(\times\) 10\({}^{-12}\) F/m), \(\varepsilon_{p}\)\({}^{\prime\prime}\) is the imaginary part corresponding to the relaxation polarization, and \(\varepsilon_{c}\)\({}^{\prime\prime}\) is the imaginary part corresponding to the conductivity loss.  
The quarter-wavelength matching model is used in this study, in which the relationship of absorbers thickness (\(t_{m}\)) with peak frequency (\(f_{m}\)) are described by the following equations [25]:  
\[t_{m}=\frac{nc}{4f_{m}\sqrt{|\varepsilon_{r}||_{Hr}}},\ \ (n=1,\ 3,\ 5) \tag{5}\]  
where \(\mid\varepsilon_{r}\mid\) and \(\mid\mu_{r}\mid\) are the modulus of measured \(\varepsilon_{r}\) and \(\mu_{r}\) at \(f_{m}\).