Phase mapping of ultrashort pulses in bimodal photonic structures: A window on local group velocity dispersion

The amplitude and phase evolution of ultrashort pulses in a bimodal waveguide structure has been studied with a time-resolved photon scanning tunneling microscope (PSTM). When waveguide modes overlap in time intriguing phase patterns are observed. Phase singularities, arising from interference between different modes, are normally expected at equidistant intervals determined by the difference in effective index for the two modes. However, in the pulsed experiments the distance between individual singularities is found to change not only within one measurement frame, but even depends strongly on the reference time. To understand this observation it is necessary to take into account that the actual pulses generating the interference signal change shape upon propagation through a dispersive medium. This implies that the spatial distribution of phase singularities contains direct information on local dispersion characteristics. At the same time also the mode profiles, wave vectors, pulse lengths, and group velocities of all excited modes in the waveguide are directly measured. The combination of these parameters with an analytical model for the time-resolved PSTM measurements shows that the unique spatial phase information indeed gives a direct measure for the group velocity dispersion of individual modes. As a result interesting and useful effects, such as pulse compression, pulse spreading, and pulse reshaping become accessible in a local measurement.

Figure 1

(Color) Schematic representation of a pulse tracking experiment. The evanescent field of the pulse traveling inside a waveguide is picked up by a fiber probe with subwavelength dimensions. The photon tunneling signal picked up by the probe is interferometrically mixed with part of the same pulse that has propagated through the reference branch. The length of the reference branch, and thus the time that it takes the reference pulse to travel through this branch, is controlled by an optical delay line. Each subsequent measurement shown in this paper is obtained by raster scanning the optical probe across the photonic structure while the height above the structure $(<10\mathrm{nm})$ is kept constant by a feedback mechanism. The simultaneous measurement of optics and topography makes it possible to directly relate optical information to the structural properties.

Figure 2

(Color) A time resolved heterodyne interference PSTM measurement on a bimodal ${\mathrm{Si}}_3{\mathrm{N}}_4$ channel waveguide for a fixed position of the optical delay (image size: $253.8\times 10.2\mu {\mathrm{m}}^2$). Linear polarized light has been coupled in the waveguide such that only the $\mathrm{T}{\mathrm{E}}_{00}$ and $\mathrm{T}{\mathrm{E}}_{01}$ modes are excited. (a) The topography of the waveguide. The measured height and width of the waveguide are $39\pm 3\mathrm{nm}$, and $3.1\pm 0.1\mu \mathrm{m}$. The slab thickness is determined to be $170\pm 5\mathrm{nm}$. (b) The optical amplitude of the pulse while propagating through the waveguide as derived from the measured LIA signal as a function of the lateral position in the plane of the sample. It is apparent that the amplitude is confined to the waveguide. A clear beating pattern due to multiple propagating modes is observed at the right side of the image. (c) The corresponding $\cos \Phi$ of the optical interference. The red dotted line depicts the center of the waveguide. As a result of the beating between different modes phase singularities occur at the position of zero amplitude. These singularities consist of pairs with opposite topological charge (indicated by arrows). It is striking to observe that the distance between the singularities is not constant, in sharp contrast to measurements performed with continuous wavelength laser sources. (d) The optical amplitude of the $\mathrm{T}{\mathrm{E}}_{00}$ as retrieved from the LIA data through Fourier filtering. (e) Optical amplitude of the $\mathrm{T}{\mathrm{E}}_{01}$ mode. We observe that the $\mathrm{T}{\mathrm{E}}_{01}$ is clearly in front of the $\mathrm{T}{\mathrm{E}}_{00}$ mode, which demonstrates that the $\mathrm{T}{\mathrm{E}}_{01}$ mode travels faster in the waveguide structure.

Figure 3

(Color) (a) A Fourier transform of the $\mathcal{A}\cos \Phi$ signal along the waveguide. The positions of the maxima immediately yield the phase velocity for the respective modes and hence the effective indices of $1.52\pm 0.02$ and $1.49\pm 0.03$ for the $\mathrm{T}{\mathrm{E}}_{00}$ and $\mathrm{T}{\mathrm{E}}_{01}$ modes, respectively. Note that the $\mathrm{T}{\mathrm{E}}_{01}$ is very weak compared to the $\mathrm{T}{\mathrm{E}}_{00}$ mode. (b) The mode profile for each individual mode measured by taking a cross section through the waveguide at the maximum amplitude of the individual modes in Figs. 2d, 2e, respectively. (c) The simultaneously acquired topography along such a cross section. (d) A line trace along the waveguide which shows the pulse shape for the respective modes. The FWHM of the pulse profile is determined by the GVD in both the signal and reference branch. Fitting a Gaussian line shape (black and red line) for the respective modes therefore allows a retrieval of the difference in GVD between the two modes.

Figure 4

(Color) Pulse tracking experiment for different reference times. Linear polarized light has been coupled in the waveguide to excite the $\mathrm{T}{\mathrm{E}}_{00}$ and $\mathrm{T}{\mathrm{E}}_{01}$ modes. (a) The simultaneously measured topography (image size: $253.8\times 10.2\mu {\mathrm{m}}^2$). [(b)–(j)] The optical field amplitude as measured by the instrument for nine different positions of the optical delay line. From (b)–(j) the optical path length of the reference branch is increased in steps of $60\pm 0.6\mu \mathrm{m}$. This results in steps of the reference time of $200\pm 2\mathrm{fs}$. For each frame a similar pattern as in Fig. 2b is found, but further along the waveguide. Note that the occurrence of a beating between the modes prevents direct pinpointing of the position of the individual pulses in these images.

Figure 5

(Color) The optical amplitude for the $\mathrm{T}{\mathrm{E}}_{00}$ mode as derived by applying a Fourier filter on the raw data of Figs. 4d, 4e, 4f, 4g, 4h, 4i, 4j. This result shows that the position of the $\mathrm{T}{\mathrm{E}}_{00}$ pulse at a reference time can be pinpointed in space. The pulses can be seen to propagate through the structure as a function of time giving a direct local measurement of the group velocity for this mode. By determining the “center of mass” for each frame a local value of the group velocity $v_g$ for the pulse of $1.46\pm 0.04\times {10}^8\mathrm{m}∕\mathrm{s}$ is found.

Figure 6

(Color) (a) Schematic drawing of the amplitude of both a $\mathrm{T}{\mathrm{E}}_{00}$ and a $\mathrm{T}{\mathrm{E}}_{01}$ mode in an isoline representation. Positions where the amplitude for both modes is equal are indicated by the red lines. Singularities can only occur at locations where the amplitude is equal. Therefore it is clear that singularities move to the center of the waveguide. (b) For a Fourier limited pulse the two interfering modes consist of two discrete periods. Green is used for the $\mathrm{T}{\mathrm{E}}_{00}$ mode, while blue represents the $\mathrm{T}{\mathrm{E}}_{01}$ mode which has a longer wavelength. Depicted are lines where the phase for the modes is 0 (solid lines) and $\pi$ (dotted lines). Note that at the center of the waveguide, depicted by the black line, a phase jump of $\pi$ occurs for the $\mathrm{T}{\mathrm{E}}_{01}$ mode. At regular intervals the dotted and solid lines for the respective modes coincide, as indicated by the depicted arrows, representing lines of opposite phase. Depicting the red line for locations of equal absolute amplitude shows that a phase singularity will occur at the locations indicated by the crosses. Note that the phase singularities occur at equidistant intervals when the phase difference $\Delta \Phi$ between the modes is equal to $\pi$.

Figure 7

(Color) Measured distance between subsequent phase singularities as a function of reference time. The horizontal axis depicts the number of a singularity, which corresponds to an effective phase difference between the two modes of $\pi$ as illustrated in Fig. 6b. The first singularity visible in Fig. 4b (see arrow) is used as the reference point and numbered as 1. Subsequent measurements are lettered corresponding to the frames used in Fig. 4. It is striking to observe that the distance between individual singularities varies within one measurement frame in sharp contrast with the expectation put forward in Fig. 6. Furthermore, this separation between singularities also shows a dependence on the reference time.

Figure 8

Mach-Zehnder model for the time resolved photon scanning tunneling microscope. The fiber length difference $\Delta z_{\mathit{\text{fiber}}}$ between the two arms is compensated by the air path length difference $\Delta z_{\mathit{\text{air}}}$. Heterodyne detection is applied, using acousto-optic modulation of the light in the reference branch. The resulting interference signal is measured with a LIA and allows one to extract the amplitude and phase of the local optical field: AO, acousto-optic modulator; and LIA, lock-in amplifier.

Figure 9

(a) Calculated optical phase [Eq. (4)] for the two modes present in our measurement as a function of position on the waveguide. Experimental parameters used in this calculation are given in the text. The $\mathrm{T}{\mathrm{E}}_{00}$ (solid line) and $\mathrm{T}{\mathrm{E}}_{01}$ (dashed line) modes show an almost linear behavior. Deviation from linearity is however obvious when plotting the phase difference $\Delta \Phi$ between those two modes (dotted line). Note that between subsequent locations where singularities occur (indicated by crosses) the change for $\Delta \Phi$ has to be exactly equal to $\pi$. The shape of the $\Delta \Phi$ curve depends only on the ${\beta }^{″}s$ for the individual modes. Therefore adjusting these parameters in the calculation so that the phase difference at the measured locations of the singularities is equal to $\pi$ allows a direct determination of the GVD for the individual modes.