local pair production processes. The pair production per unit length inside the gap is a decreasing function of $r$. According to CHR model, $E_{||}\propto r^{-1/2}$ for the thin outer gap (e.g. the Crab pulsar), which gives $E_{\gamma}(r)\propto
r^{-1/8}$ after using the large $r$ limit $s(r)=(rR_L)^{1/2}$. Since $E_{\gamma}$ is only weakly dependent on $r$, they assume $\sigma_{\gamma\gamma}\approx const$. The local pair production per unit length is $N_{e^{\pm}}(r)=(1-e^{-\tau_{\gamma\gamma}})N_{\gamma}(r)
\approx \tau_{\gamma\gamma}N_{\gamma}(r)$, where $\tau_{\gamma\gamma}=n_X(r)\sigma_{\gamma\gamma}l(r)$ is the local optical depth, $n_X=R^2T^4_s\sigma/r^2kT_sc$ is the X-ray number density at $r$, $l(r)\approx (2s(r)f(r)R_L)^{1/2}$ is the local optical path, $f(r)=h(r)/R_L$ is the local vertical extension of the gap (since $B(r)h^2(r)$ is a constant, which gives $f(r)\propto r^{3/2}$ and $f_0\sim
f(R_L/2)$ ), and $N_{\gamma}=eE_{||}(r)/E_{\gamma}(r)$ is the number of curvature photons emitted at $r$ per $e^+/e^-$ per unit length. Then $N_{e^{\pm}}(r)\propto r^{-11/8}$. Since most pairs are produced near the null surface where $r=r_{in}$, so the pair production take place mainly in the range $r_{in}\leq r\leq r_{lim}$ where $r_{lim}$ is estimated as $r_{lim}N_{e^{\pm}}(r_{lim})/r_{in}N_{e^{\pm}}(r_{in})\sim
(r_{lim}/r_{in})^{-3/8}\sim 1/2$, which gives $r_{lim}\sim 6r_{in}$. This limits pair production both along the field lines and in transverse directions, and gives $\Delta\Phi\sim 160^{\circ}$ by using the parameters of the Crab pulsar.

Within the pair production regions, outgoing and incoming 