Bloch modes and disorder phenomena in coupled resonator chains

Coherent photon coupling and disorder effects in chains of coupled resonators are studied both theoretically and experimentally, aiming at applications in the visible spectral range. Coupling effects have been explored utilizing the Bloch-mode formation in linear structures of quantum dot doped microsphere resonators. Coherent coupling in chains of 14 resonators is evidenced experimentally by spectrally and spatially resolved mode spectroscopy. Realizing a coupling constant of $\kappa =1.1\times {10}^{-3}$ ensures spectral distinction of the collective resonances. Experimentally, a slowing factor of $S=31$ is derived for the group velocity of light. A coupled oscillator model for arbitrarily disordered chains of coupled resonator delay lines is presented. The transition between order and disorder effects is investigated with a formalism combining Bloch-mode formation with mutual anticrossing relations.

Figure 1

Trade-offs between the wavelength of the guided light, the cavity radius, and the coupling constant in coupled resonators are evaluated for a fixed slowing factor $S=31$ after Eq. (2). Well-separated coupled-cavity modes can be unambiguously explored utilizing sufficiently large coupling constants; the corresponding Bloch regime is indicated in the plot with increased shading.

Figure 2

(Color) Mode maps in a chain of 14 linearly aligned microsphere resonators. The geometry of the structure is given in the spectrally integrated photoluminescence maps in diagrams (a) In diagram (b) intensity mode maps resonant to a class of TE polarized resonances are displayed, revealing a spatially varying intensity profile. Diagram (c) shows a perturbed mode due to a detuned resonator size. In diagrams (d) and (e), delocalized modes due to weak interresonator coupling are shown. (The arrows indicate spatial detection regions for the spectra shown in Fig. 3.)

Figure 3

Spectrum at two characteristic points in a chain of 14 linearly aligned microsphere resonators. The spatial detection regions are indicated with arrows in Fig. 2. A uniform splitting of the ${\mathrm{TE}}_{19}^1$ can be observed at both spatially separated areas. For comparison, a Mie scattering calculation for the mode structure of a single, isolated sphere resonator has been added.

Figure 4

Scheme of a finite chain of coupled microresonators. The disorder is introduced into the coupled resonator model via coupling photonic molecules consisting of differently sized resonators.

Figure 5

Light localization and eigenfrequencies in a chain of five resonators for different detuning values of the fifth resonator’s frequency. The frequency detuning is $-\frac{10}{3}\kappa \Omega$ in case of diagram (a), $-\kappa \Omega$ (b), $0\kappa \Omega$ (c), and $\frac{1}{2}\kappa \Omega$ (d). The gray scale legend below the diagrams relates the gray scale to relative intensity values.

Figure 6

Calculation of the coupled resonator eigenfrequencies dependent on the frequency detuning of the fifth resonator. Dashed lines indicate Bloch-mode frequencies for a four-resonator Bloch mode induced by a coupling constant of $\kappa =0.003$ (horizontal lines) and the single-resonator eigenfrequency of the detuned resonator (linearly increasing). Apparently, multiple anticrossing relations between the four-resonator Bloch modes and the tuned resonance can be observed.

Figure 7

Calculated light localization in a chain of 13 linearly detuned resonators for different frequency detunings. In diagram (a), a detuning of $\frac{10}{9}\kappa \Omega$ between adjacent resonators has been applied, $\frac{4}{9}\kappa \Omega$ in (b), $0\kappa \Omega$ in (c), and $-\frac{1}{9}\kappa \Omega$ in (d). The gray scale legend below the diagrams relates the gray scale to relative intensity values.

Figure 8

Spatiospectrally resolved intensity profile of mode splittings in a linear three-resonator system (see scheme in the middle). The left diagram shows the measured signature; the right image displays a calculation of mode splittings after Eq. (10).

Figure 9

(Color) Mode map resonant to a ${\mathrm{TE}}_{30}^2$ mode at $\lambda =572.9\mathrm{nm}$ of a bent six-resonator structure. The second resonator remains almost completely field-free, although it is coherently coupled to the neighboring microresonators, as indicated by the measured spectral splittings and the calculated peak intensities (see Fig. 10).

Figure 10

Calculation of the mode structure of a six-resonator system with nonuniform coupling. The coupling between the fifth and the sixth resonator is reduced with a weighing factor of 60% compared to the overall coupling. For a subset of modes, the second resonator displays an almost vanishing light field.