= Solution
<Photonic structures> control <light> through the spatial arrangement of <dielectric> material. <Waveguides> confine transverse <modes>; periodic <photonic crystals> combine <interference> and scattering to produce a <photonic band structure> $\omega_n(\mathbf k)$. A <photonic band gap> excludes propagating <modes> in specified directions and polarizations. \b[A one-dimensional stop band is not automatically a complete three-dimensional gap.]
A simple example is a <distributed Bragg reflector> with alternating high- and low-index layers. At normal incidence, the quarter-wave choice
$$
\boxed{n_Hd_H=n_Ld_L=\lambda_0/4}
$$
lets successive interface reflections reinforce near the design <wavelength>. A departure from periodicity can create a localized defect <mode> inside the stop band. An isolated defect forms a cavity; a line defect can guide <light> through otherwise forbidden bulk propagation. Finite structures leak and real materials absorb, so ideal band arguments do not ensure infinite confinement.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2014/paper-1-oe2014-photonic-stack.png]
{title=Original transfer-matrix calculation for lossless quarter-wave dielectric stacks, showing a stop band and a transmission resonance from a central defect}
Changing the photonic <density> of states can suppress or enhance <spontaneous emission> and control how emitters couple to selected <modes>. This is distinct from electronic band-gap engineering: it is the electromagnetic <modes> that are being structured. A complete electromagnetic gap overlapping an emitter's transition is the ideal mechanism proposed for suppressing otherwise allowed <spontaneous emission>.<Inhibited Spontaneous Emission in Solid-State Physics and Electronics by Eli Yablonovitch>[Physical Review Letters 58, 2059 (1987), primary abstract: three-dimensional periodic dielectric and overlapping electromagnetic gap]{ref}
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