In this chapter, we give a complete spectral analysis of the Schrödinger operatorSchrödinger operator with the PT-symmetric periodic optical potential $$4\cos ^{2}x+4iV\sin 2x$$ . We investigate in detail the shape of the spectrum for all $$V\in (0,\infty ).$$ We prove that the second critical point $$V_{2}$$ , after which the real parts of the first and second band disappear, is a number between 0.8884370025 and 0.8884370117. Besides, we give a scheme by which one can find arbitrary precise value of the second critical point as well as the k-th critical points $$V_{k}$$ after which the real parts of the $$(2k-3)$$ -th and $$\left( 2k-2\right) $$ -th bands disappear, where $$k=3,4,\ldots $$ We study the critical points $$V_{k}$$ and describe the changes of the spectrum of L(q) when V moves from the left to the right of the critical points. Moreover, we investigate spectral singularitiesSpectral singularities and essential spectral singularitiesEssential spectral singularity (ESS). We prove that if $$V>1/2,$$ then the operator L(V) has infinitely many spectral singularitiesSpectral singularities. Moreover, it has no ESS and has ESSEssential spectral singularity respectively if and only if $$V\ne V_{k}$$ and $$V=V_{k}$$ for $$k\ge 2,$$ where $$V_{k}\rightarrow \infty $$ as $$k\rightarrow \infty $$ and $$V_{2}