We present time-resolved emission experiments of semiconductor quantum dots in silicon 3D inverse-woodpile photonic band gap crystals. A systematic study is made of crystals with a range of pore radii to tune the band gap relative to the emission frequency. The decay rates averaged over all dipole orientations are inhibited by a factor of 10 in the photonic band gap and enhanced up to 2× outside the gap, in agreement with theory. We discuss the effects of spatial inhomogeneity, nonradiative decay, and transition dipole orientations on the observed inhibition in the band gap.

We study the local density of states (LDOS) in a finite photonic crystal, in particular in the frequency range of the band gap. We propose a new point of view on the band gap, which we consider to be the result of vacuum fluctuations in free space that tunnel in the forbidden range in the crystal. As a result, we arrive at a model for the LDOS that is in two major items modified compared to the well-known expression for infinite crystals. Firstly, we modify the Dirac delta functions to become Lorentzians with a width set by the crystal size. Secondly, building on characterization of the fields versus frequency and position we calculated the fields in the band gap.We start from the fields at the band edges, interpolated in space and position, and incorporating the exponential damping in the band gap. We compare our proposed model to exact calculations in one dimension using the transfer matrix method and find very good agreement. Notably, we find that in finite crystals, the LDOS depends on frequency, on position, and on crystal size, in stark contrast to the well-known results for infinite crystals. * e.yeganegidastgerdi@utwente.nl 1 arXiv:1309.5730v2 [physics.optics]

The famous vanishing of the density of states (DOS) in a band gap, be it photonic or electronic, pertains to crystals in the infinite-size limit. In contrast, all experiments and applications pertain to finite crystals, which raises the question: Upon increasing the size L of a crystal, how fast does the DOS approach the infinite-crystal limit? Answering this question, however, requires an understanding of how linewidth of the modes in crystal with finite support scales as a function of crystal length L. We develop such a theory in a finite support crystal using Bloch-mode broadening due to the crystal boundaries. Our results suggest that total DOS inside a bandgap has the same scale dependence irrespective of the number of dimensions which the crystal is defined in. This can pave the way to establishing design rules for the usage of vanishing density of states, notably to cavity QED, quantum information processing, and Anderson localization.The discovery brought about by crystallography that a crystal consists of an infinitely extended periodic array of basic units with perfect periodic symmetry [1] has led to the birth of modern condensed matter physics [2][3][4]. The quantum-mechanical description of the electronic degrees of freedom has led to the notion of density of states (DOS), and to the characterization of semiconductors as having a range of vanishing density of states, bandgap, with associated band edges [3,4]. Remarkably, the most important observable -the electric conductance -is only defined for systems that deviate from perfect crystalline symmetry in that they have a finite size instead of infinite translational symmetry [2]. The size-dependence of the conductance is one of the pillars of condensed matter physics, and the study of finite-size scaling in general plays a central role in condensed matter and statistical physics [3,[5][6][7]].An analogy can be drawn between electronic condensed-matter and photonic condensed-matter phenomena, as the underlying mechanism is in both cases wave interference [3,4]. Indeed photonic crystals reveal Bragg reflections for light, which are apparent as a beautiful iridescence [8]. When the light-matter interaction is sufficiently strong, photonic crystals can develop a bandgap analogous to electronic semiconductors and insulators [9][10][11]. The nanophotonic analogue of a semiconductor is widely considered to be a photonic crystal with a complete 3D band gap in the photonic DOS [9][10][11].Most theories of the density of states in condensed matter and nanophotonics consider infinite samples (L → ∞). Examples are the plane-wave expansion for wave states -both electronic [2] or photonic [11] -or the thermodynamic limit in liquid state theory [12], that all maximally exploit the underlying periodic or continuous symmetry. Theories for the density of states of waves, electrons and photons, that address a finite sample are rare [13]. To the best of our knowledge, there are no theories that address a sample with finite support, that is, a sample where a finite "crys...

Light propagates inside an ideal and infinite photonic crystal when the Bloch condition is satisfied and any allowed light wave is decomposed in a basis of propagating Bloch waves [1]. In real photonic crystals, light propagation is modified due to unavoidable fabrication-induced structural imperfections in size, positions, and permittivity of the building blocks, as well as the finite size of the crystal. The unavoidable deviations from perfect periodicity result in scattering [2,3]. Here, we seek to understand the nature of waves inside a real photonic crystal since inevitable disorder plays a significant role in wave propagation.We investigate experimentally three-dimensional (3D) opals and two-dimensional (2D) silicon photonic crystals. We measured reflected intensity from the crystals with a high numerical aperture. The statistics of the reflected intensity differs from that of uncorrelated scattered waves typical of statistically homogeneous media (disordered ZnO nanoparticles). Thus, the reflected intensity is correlated, which is caused by the presence of correlated Bloch waves. We quantify the optical correlations by measuring the position-dependent degree of correlation across each crystal surface. We find that the correlation coefficient depends on disorder in different regions within the opal photonic crystal (see Fig. 1(a)). In contrast, the CMOS-fabricated 2D crystal shows a very high correlation of about 0.9 over a large range as shown in Fig. 1(b). We propose a model that assumes that spherical waves are generated due to disorder and are superimposed with propagating Bloch waves. By comparing our model with experimental result in Fig. 1(b), we extract that the ratio of the intensity of the scattered spherical waves and Bloch waves is 0.36, which is in agreement with 0.32 from the sample parameters. Our results therefore reveal the actual form of wave propagation in real photonic crystals, which are crucial in several applications [4,5] and fundamental research [6].

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