Direct Observation of Bloch Harmonics and Negative Phase Velocity in Photonic Crystal Waveguides

The eigenfield distribution and the band structure of a photonic crystal waveguide have been measured with a phase-sensitive near-field scanning optical microscope. Bloch modes, which consist of more than one spatial frequency, are visualized in the waveguide. In the band structure, multiple Brillouin zones due to zone folding are observed, in which positive and negative dispersion is seen. The negative slopes are shown to correspond to a negative phase velocity but a positive group velocity. The lateral mode profile for modes separated by one reciprocal lattice vector is found to be different.

Figure 1

(a) Schematic representation of a phase-sensitive NSOM experiment on a W3 PhCW. The evanescent field of a pulse traveling inside the photonic structure is picked up by a coated fiber probe with a subwavelength aperture as shown in (c). At each position the signal picked up by the probe is interferometrically mixed with part of the same pulse that has propagated through the reference branch giving access to the local field amplitude and phase. (b) A scanning electron micrograph of the sample under study.

Figure 2

A phase-sensitive NSOM measurement on a W3 PhCW for a fixed position of the optical delay. Short pulses were launched at a wavelength of $1245\pm 0.5\mathrm{nm}$ ($\omega =0.369$; $a=460\mathrm{nm}$). (a) Topography of the structure. (b) Raw data from the lock-in amplifier corresponding to the amplitude times the cosine of the optical field. In the waveguide, guided light with a large spread in spatial frequencies is visible. (c) The optical field amplitude. It is apparent that pulses with different modal distributions are excited in the W3 waveguide. The dominant periodicity observed in (c) corresponds to 460 nm, as expected for Bloch modes.

Figure 3

(a) Experimentally mapped dispersion diagram of the modes excited in the W3 waveguide. Displayed is the Fourier transform of the data for multiple excitation wavelengths on a logarithmic scale. Clearly modes below and above the light cone (dotted line) are visible. (b) Displays the lateral mode distribution for $\omega =0.369$ by displaying $F(k_x)$ for each $y$ in Fig. 2b on a linear scale. Here we can directly see the modal distribution for each of the lines in the dispersion diagram at $\omega =0.369$. It is clear that for Bloch harmonics that belong to the same Bloch mode, e.g., indicated by arrows, the modal distribution is not the same.

Figure 4

(a) Spatial Fourier spectrum $F(k_x)$ for $\omega =0.369$ summed over all $y$. Arrows indicate peaks that belong to the same Bloch mode ($\mathrm{FFT}=\mathrm{\text{fast}}$ Fourier transform). (b) Lateral mode profiles for the Bloch harmonics indicated by the arrows in Fig. 3b and in panel (a). The modal distribution is different for each individual Bloch harmonic. (c) A Fourier filter allows retrieval of the phase evolution of individual modes. Top panel in (c) shows the sine and cosine channel of the LIA for $k_x=0.393$, while the bottom panel corresponds to $k_x=0.607$. For the Bloch harmonic at $k_x=0.393$ the sine is in front, while it is lagging at $k_x=0.607$. As a result, the wave vectors for these two harmonics from the same Bloch mode have an opposite sign.