  In this paper, we consider the spectral theory of linear
differential-algebraic equations (DAEs) for periodic DAEs in canonical form,
i.e., \begin{equation*}
  J \frac{df}{dt}+Hf=\lambda Wf, \end{equation*} where $J$ is a constant
skew-Hermitian $n\times n$ matrix that is not invertible, both $H=H(t)$ and
$W=W(t)$ are $d$-periodic Hermitian $n\times n$-matrices with Lebesgue
measurable functions as entries, and $W(t)$ is positive semidefinite and
invertible for a.e. $t\in \mathbb{R}$ (i.e., Lebesgue almost everywhere). Under
some additional hypotheses on $H$ and $W$, called the local index-1 hypotheses,
we study the maximal and the minimal operators $L$ and $L_0'$, respectively,
associated with the differential-algebraic operator
$\mathcal{L}=W^{-1}(J\frac{d}{dt}+H)$, both treated as an unbounded operators
in a Hilbert space $L^2(\mathbb{R};W)$ of weighted square-integrable
vector-valued functions. We prove the following: (i) the minimal operator
$L_0'$ is a densely defined and closable operator; (ii) the maximal operator
$L$ is the closure of $L_0'$; (iii) $L$ is a self-adjoint operator on
$L^2(\mathbb{R};W)$ with no eigenvalues of finite multiplicity, but may have
eigenvalues of infinite multiplicity. As an important application, we show that
for 1D photonic crystals with passive lossless media, Maxwell's equations for
the electromagnetic fields become, under separation of variables, periodic DAEs
in canonical form satisfying our hypotheses so that our spectral theory applies
to them (a primary motivation for this paper).
