*4.1. Polarization from Uniform Jets*

In uniform axisymmetric jets, the comoving spectral luminosity, *L* 0 *ν* <sup>0</sup> and the bulk-Γ do not vary with polar angle *θ* measured from the jet axis, e.g., in a top-hat jet,

$$\frac{L\_{\nu'}'(\theta)}{L\_0'} = \frac{\Gamma(\theta)}{\Gamma\_0} = \begin{cases} 1, & \theta \le \theta\_j \\ 0, & \theta > \theta\_j. \end{cases} \tag{9}$$

It is further assumed that Γ, *θ<sup>j</sup>* , *θ*obs, and the spectrum (assumed here to be a power law) remain constant with the radius during emission of the prompt GRB (while *L* 0 *ν* <sup>0</sup> can vary with the radius). Since the emission arises in an ultrarelativistic jet (Γ 1), it is strongly beamed along the direction of motion primarily into a cone of angular size 1/Γ. Consequently, most of the observed radiation arrives from angles ˜*θ* . 1/Γ around the LOS. If the LOS intersects the jet surface and is more than a beaming cone away from the edge of the jet, i.e., if *θ*obs . *θ<sup>j</sup>* − Γ <sup>−</sup><sup>1</sup> or equivalently if *<sup>q</sup>* <sup>≡</sup> *<sup>θ</sup>*obs/*θ<sup>j</sup>* . <sup>1</sup> <sup>−</sup> *<sup>ξ</sup>* −1/2 *<sup>j</sup>* where *ξ<sup>j</sup>* ≡ (Γ*θj*) 2 , then the observer remains unaware of the jet's edge (however, see Section 4.3), and the emission can be approximated as if arising from a spherical flow. In this instance, after averaging over the GRB image on the plane of the sky, a finite net polarization will only be obtained if the direction of polarization is not axisymmetric around the LOS. Hence, it becomes necessary to break this symmetry in order to obtain any net polarization. This naturally happens if the LOS lies near the edge of the jet. Therefore, in such cases a special

alignment between the flow direction and the observer is needed. This and other effects that break the symmetry and yield finite net polarization are highlighted below.
