above, the pulse profile depends on emission location and viewing angle. Here we use Crab parameters and the viewing angle $\zeta=82^{\circ}$ ( the dashed line in panel A of Figure 2) to construct the light curve. The intensity of the light curve at each phase (panel B of Figure 2) is proportional to the number of interceptions between the dashed line and the solid lines in panel A of Figure 2.

PHASE-RESOLVED SPECTRA OF THE CRAB PULSAR
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In this section, we describe how to calculate the phase-dependent spectra of pulsars. We will use the parameters of the Crab pulsar as example. Because the Crab pulsar outer gaps are thin, we use the electric field of the CHR model: $$E_{||}(r)={\Omega B(r)a^2(r)\over c s(r)}= {\Omega B(r)f^2(r)R^2_L\over c
s(r)},$$ where $a(r)$ is the thickness of the outer gap at position $r$, the radius of the curvature $s\sim (rR_L)^{1/2}$, and $f(r)\equiv a(r)/R_L$ is the local fractional size of the outer gap. Since the magnetic flux subtended in the outer gap should be constant in the assumed steady state, $$f(r)\sim f(R_L)\left({r\over R_L}\right)^{3/2},$$ where $f(R_L)$ is estimated by the pair production condition described in previous section(N.B. There are two possible ways to determine $f(R_L)$, Eq. 