Fourier transform of surface measure on half unit sphere

Question

Let $\mathbb{S}^{1}$ denote the unit sphere in $\mathbb{R}^{2},$ i.e. $\mathbb{S}^{1}=\{x\in\mathbb{R}^{2}: \vert x\vert=1\},$ and $\sigma_{1}$ the surface measure on $\mathbb{S}^{1}.$ It is a well-known fact that the Fourier transform $\widehat{\sigma_{1}}$ of $\sigma_{1}$ is given by $$\widehat{\sigma_{1}}(x)=\displaystyle\int\limits_{\mathbb{S}^{1}} e^{ixt} \, d\sigma_{1}(t)=2\pi J_{0}(\vert x\vert) \quad (x\in\mathbb{R}^{2})$$ where $J_0$ is the Bessel function (of the first kind) of order $0.$

My question is: What happens if we restrict $\sigma_{1}$ to $\mathbb{S}^{1}_{+}:=\{\xi\in\mathbb{S}^{1}: \xi_{1}>0\}?$ More precisely: What can be said about the Fourier transform of $\sigma_{1}|_{\mathbb{S}^{1}_{+}}$, i.e. on the integral $$\displaystyle\int\limits_{\mathbb{S}^{1}_{+}} e^{ixt} \, d\sigma_{1}(t).$$ Does anyone have an idea or a reference?