is

$$\begin{aligned}
\label{ParticleNumber}
N(f) &=& \int d \omega \int d \omega^{\prime}
\langle 0_M |(A^*_{\omega} \hat{c}^{\prime \dagger}_{\omega} +B^*_{\omega}\hat{d}^{\prime \dagger}_{\omega})
(A_{\omega^{\prime}} \hat{c}^{\prime}_{\omega^{\prime}} +B_{\omega'}\hat{d}^{\prime}_{\omega^{\prime}}) |0_M \rangle \nonumber \\
\nonumber \\
&=& 2 \int d \omega (|A_{\omega}|^2+|B_{\omega}|^2)(1-\cos \theta_{\omega})\frac{e^{2\pi \omega/a}}{(e^{2\pi \omega/a}-1)^2}, \nonumber \\
\nonumber \\
&=& \sqrt{\frac{8}{\pi}}\frac{\sigma}{k_0} \int_0^{\infty} d \Omega 
\bigg\{\exp\bigg[-\frac{2\sigma^2(\Omega-k_0 V_0)^2}{k_0^2}\bigg]+\exp\bigg[-\frac{2\sigma^2(\Omega+k_0 V_0)^2}{k_0^2}\bigg] \bigg\}
(1-\cos \theta_{\Omega})\frac{e^{2\pi \Omega}}{(e^{2\pi \Omega}-1)^2}\end{aligned}$$

where $\Omega = \omega/a$ is the dimensionless Rindler frequency.

Two special cases are of particular interest. Consider first that the mirror is completely transparent for all modes, that is $\cos^2 \, \theta_{\omega} = 1$. From Eq. (\[ParticleNumber\]), the particle number vanishes, $N(f)=0$. This is not surprising because a completely transparent mirror does nothing to the Minkowski vacuum. The second case is that the mirror is perfect for all modes, that is, $\cos^2 \, \theta_{\omega} = 0$. When $\Omega \rightarrow 0$, $(e^{2\pi \Omega}-1)^{-2} \sim \Omega^{-2} $ and all other factors in the integrand of Eq. (\[ParticleNumber\]) are finite. Therefore, the particle number $N(f)$ is divergent.

This infrared divergence occurs because we naively assume that the mirror accelerates for an infinitely long time, which seems physically unreasonable. In the framework of the self-interaction model, the mirror is switched on and off so that one obtains finite particle flux [@Obadia01]. 