expression for the transition probability is given as $${\cal F}(\omega) = \int_{-\infty}^{\infty} du \, \chi(u)  \int_{-\infty}^{\infty} ds \, \chi(u -s) e^{- i \omega s} \, W_{\Theta_0, \epsilon}(u,u-s)
\label{transitionprobability}$$ where $\chi(\tau)$ is the smooth switching function which vanishes for $\tau < \tau_0$ and $\tau > \tau_f$ while it is unity for $\tau_0 < \tau < \tau_f$ and is smooth in its transition. For the stationary Rindler trajectory, we have $W_{\Theta_0, \epsilon}(u,u-s) = W_{\Theta_0, \epsilon}(s)$ as shown in Eqs.(\[TXRindler\]) and (\[wfinalcompact\]). Hence in the above equation Eq.(\[transitionprobability\]) for transition probability, we can interchange the sequence of integration and perform the $u$ integral first to get $${\cal F}(\omega) =  \int_{-\infty}^{\infty} ds \, e^{- i \omega s} \, W_{\Theta_0, \epsilon}(s) \, Q(s)
\label{transitionprobability2}$$ where $Q(s) = \int_{-\infty}^{\infty} du \, \chi(u) \chi(u -s)$ is also a smooth analytic function in complex $s$ plane. One can then perform the contour integral in Eq.(\[transitionprobability2\]), by choosing an appropriate contour and evaluating the residues of the expression at the poles of the Wightman function $W_{\Theta_0, \epsilon}(s)$ to obtain a finite result. However, from the expression in Eq.(\[denominator\]), one can see that the combination of $\epsilon$ and $|\cos{\Theta_0}|$ always appears as product, hence evaluating the contour integral in Eq.(\[transitionprobability2\]) would preserve the product structure which would imply that taking the point-like limit $\epsilon \rightarrow 0^+$ would make the $\Theta_0$ dependence to go away. In-fact, one can also verify that even in the non-stationary case, the product feature of the $\epsilon$ and $|\cos{\Theta_0}|$ would still hold and the direction dependence would vanish again in the the point-like limit $\epsilon \rightarrow 0^+$.

A spatially extended detector in the Rindler frame {#rindlerframesection}
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Our aim 