Near-field mapping of optical eigenstates in coupled disk microresonators

We report on the real-space experimental observation of whispering-gallery modes in coupled disk microresonators excited by a tapered optical fiber. Using a high-resolution scattering scanning near-field optical microscope (SNOM) technique, the resonator modes can be perturbed, leading to altered transmission and reflection through the tapered fiber. The correlation of the perturbed signals with the position of the perturbing SNOM tip results in intensity maps of the optical resonator modes under investigation. A combination of a coupled-mode theory of the excited microresonator including the SNOM tip and a two-dimensional modal expansion technique can be used to understand different regimes of the perturbation, leading to qualitatively different transmission and reflection maps for the same optical resonator mode. Detailed measurements of the intensity distributions of eigenmodes of two and three coupled disks in different arrangements are presented and compared to theoretical calculations.

Figure 1

Different samples under investigation: (a) single disk, (b) two coupled disks, and (c) three coupled disks in a triangular configuration (linear arrangement not shown). (d) A sketch of the setup showing the tunable laser source (TLS, with modulation for lock-in), a fiber polarization controller (FPC), and a circulator (C) for extraction of the reflection signal (R). The sample, the tapered fiber (TF), and the SNOM tip (Tip) are placed in a nitrogen-purged box. The reflection and transmission (T) signals are fed into a lock-in amplifier (L-In) with the amplitude outputs connected to the SNOM controller (SNOM-contr) where the signals are correlated to the position of the tip. A PC is used to control all devices and to read out the measured transmission and reflection mode maps.

Figure 2

Sketch of used quantities and coordinate relations. The red arrow marks the direction of the Gaussian distribution of the incident field through the tapered fiber.

Figure 3

Calculation of the spectrum and intensity distribution of three coupled disks of equal size (radius $7.5\mu$m, thickness $1\mu$m, refractive index 1.445, all gaps 400 nm). The splitting of the fundamental mode ($m=35$) is observable for (a) the resonant approximation and (b) the full model taking into account coupling to higher-order modes. Here the expansion coefficient $b_{m=35}$ for the field in each disk (disk 1, blue; disk 2, red; disk 3, green) is plotted. The intensity distribution of the mode marked by an arrow in (b) is shown in (e) using the full model. A comparison to a rigorous three-dimensional FDTD simulation (spatial resolution 30 nm) shows good agreement for (c) the spectrum, as well as for (f) the intensity distribution [same mode marked by the arrow (c)]. Only two field monitors in disks 1 and 2 were used for the spectrum calculation, so the green line for disk 3 is missing in (c). A comparison of the magnetic field (which needs to be continuous across boundaries) along the dashed line in (e) for the resonant approximation and the full modal expansion in (d) underlines the need for the full model in order to calculate the fields correctly.

Figure 4

Typical measurement signals for scanning across the gap of two coupled disks: (a) overlaid reflection and topography images, (b) only transmission signal, and (c) only reflection signal measured by scattering SNOM. A comparison with an aperture SNOM measurement in collection mode shows the relation between (d) the real mode structure and the measured (e) transmission and (f) reflection signals.

Figure 5

Successive mode mappings for a stepped wavelength scan through a resonance of a single disk using the perturbation of a scattering SNOM tip. For each of the 11 wavelengths marked in the transmission spectrum, (a) a scan window of $1.5\times 1.5\mu {\mathrm{m}}^2$ [which is highlighted in (f), frame 11] was mapped and arranged horizontally to a picture series (numbered 1 to 11) for (c) transmission and (e) reflection signals. For the same area of the disk (and arranged in the same way), the intensity distributions of the mode were calculated in (b), and from this the theoretical (d) transmission and (f) reflection signal maps were obtained. The dotted lines in (b)–(f) indicate the phase shift that the mode distribution shows when scanning through the resonance. A good agreement is obtained for the shift of the signal maximum in transmission [(e), (f), frames 6, 7] and reflection [(c), (d), frames 8, 9] compared to the expected intensity maximum of the mode [(b), frame 5]. Also, complex patterned maps are obtained from the calculations in agreement with the measured signals [ring structure in (c), (d), frame 9].

Figure 6

(a) The measured transmission for two coupled disks shows splitting in two eigenmodes: antisymmetric (resonance 1) and symmetric (resonance 2) mode. (b) In the top view of the measured sample with the tapered fiber for excitation of disk 1, the scan region of $10\times 10\mu {\mathrm{m}}^2$ across the gap is highlighted. The measured reflection mode maps and calculated mode intensity distributions are compared for the (c) antisymmetric and (b) symmetric mode. In each measurement square in (c) and (d), the numbers highlight the particular disk, whereas in the corresponding simulation, the green lines mark the disk's rim.

Figure 7

(a) The measured transmission for three coupled disks in a linear arrangement shows splitting in three eigenmodes. (b) In the top view of the measured sample, the scan windows are highlighted. On the left of the picture the shadow of the SNOM head is seen while scanning the gap between disks 2 and 3. The measured reflection mode maps and simulated mode intensity distributions are compared for both gaps and all resonance dips in (a), starting from the (c) antisymmetric mode or resonance 1, over the (d) middle resonance 2, to the (e) symmetric mode or resonance 3. In each measurement square in (c)–(e), the numbers highlight the particular disk, whereas in the corresponding simulation, the green lines mark the disk's rim.

Figure 8

(a) Measured transmission spectrum for three coupled disks in a triangular arrangement shows splitting in six eigenmodes. The unnumbered dips belong to the other polarization state, which was not suppressed completely. (b) A top view of the measured sample with tapered fiber for excitation of disk 1. The white squares highlight the SNOM scan window ($5\times 5\mu {\mathrm{m}}^2$) with the resulting intensity distribution shown in (c)–(h) for each of the gaps (1-2, 2-3, 1-3) and the numbered resonances from (a). In each measurement square in (c)–(h), the numbers highlight the particular disk, whereas in the corresponding simulation, the green lines mark the disk's rim.