  In this paper, we study the structure of the spectra for the Sierpi\'{n}ski
type spectral measure $\mu_{A,\mathcal{D}}$ on $\mathbb{R}^2$. We give a
sufficient and necessary condition for the family of exponential functions
  $\{e^{-2\pi i\langle\lambda, x\rangle}: \lambda\in\Lambda\}$ to be a maximal
orthogonal set in $L^2(\mu_{A,\mathcal{D}})$. Based on this result, we obtain a
class of regular spectra of $\mu_{A,\mathcal{D}}$. Moreover, we discuss the
Beurling dimensions of the spectra and obtain the optimal upper bound of
Beurling dimensions of all spectra, which is in stark contrast with the case of
self-similar spectral measure. An intermediate property about the Beurling
dimension of the spectra is obtained.
