Direction-dependent Optical Modes in Nanoscale Silicon Waveguides

On-chip photonic networks have the potential to transmit and route information more efficiently than electronic circuits. Recently, a number of silicon-based optical devices including modulators, buffers, and wavelength converts have been reported. However, a number of technical challenges need to be overcome before these devices can be combined into network-level architectures. In particular, due to the high refractive index contrast between the core and cladding of semiconductor waveguides, nanoscale defects along the waveguide often scatter light into the backward-propagating mode. These reflections could result in unwanted feedback to optical sources or crosstalk in bidirectional interconnects such as those employed in fiber-optic networks. It is often assumed that these reflected waves spatially overlap the forward-propagating waves making it difficult to implement optical circulators or isolators which separate or attenuate light based on its propagation direction. Here, we individually identify and map the near-field mode profiles of forward-propagating and reflected light in a single-mode silicon waveguide using Transmission-based near-field scanning optical microscopy (TraNSOM). We show that unlike fiber-optic waveguides, the high-index-contrast and nanoscale dimensions of semiconductor waveguides create counter propagating waves with distinct spatial near-field profiles. These near-field differences are a previously-unobserved consequence of nanoscale light confinement and could provide a basis for novel elements to filter forward-propagating from reflected light.

experiment. Optical fibers are used to couple light into and out of the waveguide (shown in green). The waveguide is fabricated in silicon on insulator. Details of the experimental setup and device fabrication can be found in the Methods section. While scanning the waveguide with an Atomic Force Microscope (AFM) probe we constantly monitor the power transmitted through the device. This technique, Transmission-based Near-field Scanning Optical Microscopy (TraNSOM), was recently developed for imaging near field profiles in high index-contrastwaveguides 16,17 and subsequently applied to optical resonant cavities [19][20][21] . While phasesensitive 22,23 and time-resolved 24 near-field microscopy techniques can determine the direction of optical propagation based on the sign of the propagation constant, here we use TraNSOM to search for spatial differences in intensity profiles between forward-propagating and reflected modes. When the probe interacts with the evanescent field of the guided wave it scatters some of the light out of this mode. Since the probe is in the near field of the waveguide, much of this scattered light couples back into the guided mode and propagates in a direction opposite to the incident light 9 . This is measured as a probe-induced reflection and can be used to determine the propagation direction of the incident light. For instance, when the probe interacts with the forward-propagating mode, light is reflected away from the output and the power transmitted decreases. Conversely, when the probe interacts with the backward-propagating reflected light, probe-induced reflection redirects some of this light toward the output. Thus probe interaction with the reflected light results in an increase in the optical power detected at the output. By looking for transmission changes of opposite sign, we aim to distinguish forward-propagating from reflected light. If the mode profiles of the forward-propagating and reflected waves are identical (as is expected for low-index waveguides like fiber optics) we should observe that scattering by the probe only decreases the transmitted power. This is because at every point across the waveguide, the probe would interact simultaneously with both forward-propagating and reflected light. Due to propagation losses, the amplitude of the forward propagating mode is larger and would therefore dominate the measured signal. If, however, the forward-propagating and reflected waves are spatially separated (i.e. at specific points across the waveguide the probe interacts with one wave and not the other), we should be able to observe both a decrease and increase in transmission as the probe interacts individually with either the forward-propagating or reflected waves respectively. We verify that backward propagating light is responsible for the measured increase in transmission by eliminating its contribution to the measured signal and repeating the TraNSOM measurement. This is achieved by using an optical source with a short coherence length. Figure   2b shows the TraNSOM image using an optical source with a 1.4 mm coherence length. Because this coherence length is shorter than the path from the probe to the end of the waveguide and back, by the time the reflected light is scattered by the probe it has no well-defined phase relationship with the forward-propagating light. Therefore the scattered light from the backwardpropagating reflected wave is equally likely to constructively or destructively interfere with the forward-propagating mode and thus has no net effect on the transmitted power. Therefore, by using a short-coherence-length source we can selectively map the distribution of only the forward-propagating light. As expected, Fig. 2b shows no probe-induced increases in transmission. This verifies that the measured increase in transmission is the result of interaction with the backward-propagating reflected light in the guided mode. Figure 2c shows a theoretical TraNSOM signal calculated with contributions from both forward-propagating and reflected light. Figure 2d shows the same calculation considering only forward-propagating light. This corresponds to the measured data in Fig. 2a  We see in Fig. 3b that although the TE mode is polarized primarily in the horizontal (x) direction there are large minor field components in the y-direction at the waveguide corners. Due to the large minor field components these modes are frequently referred to as quasi-TE or quasi-TM. If both TE and TM modes are excited the total shape of the mode profile is the coherent sum of these two modes. This causes the total mode profile to "lean" to the left or right depending on the phase difference between the TM and TE modes (see Fig. 3c and d) 26 . When the forward propagating wave encounters a perturbation to the waveguide width, such as the narrow tapers used to couple light on and off chip 27 (see Figs. 1 and S1), the reflected TE mode acquires a π phase shift, while the TM mode does not 28 . This causes the backward-propagating wave to "lean" in the opposite direction as the forward-propagating light. This property of large minor field components at the waveguide corners is unique to high-index nanoscale waveguides. In fiber optic waveguides, the index contrast is too small to produce these field components, and thus the shape of the mode profile does not depend on the phase between the TE and TM modes.
Simulations using a 3D Finite Difference Time Domain (FDTD) method verify that forward and backward propagating waves "lean" in opposite directions. Since rectangular high-indexcontrast waveguides are typically highly birefringent, the TE and TM modes propagate with different phase velocities due to their different effective indices 29 . The evolving phase difference between the TE and TM modes causes the near-field intensity distribution to oscillate between left and right "leaning" profiles. This can be seen in 3D-FDTD simulations where both the TE and TM modes were launched from left to right (Fig. 3e). Here we plot an x-z cross section through the waveguide 200 nm above the silicon oxide substrate. The period of oscillation (L) for this experiment is determined by the birefringence (Δn eff =0.46 from finite element mode solver) and the wavelength (λ=1.532 µm): L= λ/ Δn eff = 3.3 µm. This period matches the beat period measured by TraNSOM in Fig. 2a and b. The beating observed here is similar to the polarization mode beating observed in birefringent fibers 18 ; however, in low-indexcontrast fibers, due to the negligible minor field components, the shape of the mode profile does not change as it propagates. The phase of the oscillation is determined by the initial polarization state (phase relationship between the TE and TM components). The backward-propagating reflected wave is simulated by adding a π phase shift to the TE mode and launching both TE and TM modes from the right of the waveguide (Fig. 3f). As expected, the backward-propagating and forward-propagating light oscillate out of phase with one another, i.e. at each point in the waveguide the modes "lean" in opposite directions. For example, the forward-propagating mode leans toward the point labeled A' (Fig. 3e) while the backward propagating mode leans away (Fig. 3f). Point B' shows the opposite behavior. These points correspond to points A and B in The two terms to the right of the equality represent the scattering of forward-propagating and reflected light respectively (note the sign difference). Also, note the phase (sign) change added to the TE mode in the second term, which is the result of reflection. Here the TM and TE subscripts denote the TE or TM mode respectively and E y is the y-component of the electric field. The cross-sectional profile of the probe is written as A (see Methods), x and z are Cartesian coordinates, k is the propagation constant, and Z 0 is the free space impedance. The scattering efficiency (Q) is the measured to be ~25 (see Supplementary Information (SI) and Fig. S2). Due to the large index contrast between the probe and air-cladding, the efficiency with which scattered light couples to the counter-propagating mode (η) is expected to be near be unity 9 . For simplicity, we will take this factor to be 1. The amplitudes of the forward and backward propagating modes are written as a and b respectively. We can write b in terms of a according to: where α is the waveguide loss per unit length, l is the distance between the probe and the source of reflection (the waveguide output in this case) and R is the reflectivity. We take the waveguide loss to be -6.64 dB/cm and -15.36 dB/cm for the TM and TE modes respectively. These values are taken from similar waveguides fabricated in silicon (see Methods) and measured using the cut-back method 30 . The coupling efficiency of each polarization and their respective reflection at the waveguide interfaces is difficult to measure directly; however, the reflection is known to be high, particularly for the TM mode, which despite having lower propagation loss, transmits -11 dB less power through the device as compared to the TE mode.
Based on our analytical model, we identify specific values of reflectivity and ratios of TE to TM excitation which allow forward-propagating and reflected light to be distinguished. Figure   4 shows the maximum change in transmission (max[ΔT/T 0 ]) as a function of reflectivity at the chip interface (R) and the relative amplitude of TE mode (|a TE | 2 ). This is calculated using equation (1) and (2) (1) is always larger than that of the second term. In other words, at each probe position more light is scattered from the forward-propagating light than from the reflected light. We refer to this as the "normal" scattering regime, since as expected, introduction of a scattering point results in a decrease in power transmitted through the waveguide. Conversely, where max[ΔT/T 0 ] > 0, forward-propagating and reflected light can be distinguished. In this regime, a scattering point can redirect the backward-propagating reflected light such that the power transmitted through the waveguide increases. Since this is an unexpected consequence of near-field scattering, we refer to this as the "anomalous" scattering regime. The waveguide used for these experiments was designed with 0.75 such that the anomalous regime is accessible for ratios of TE to TM excitation > 0.5 which is easily achieved using a polarization controller.
We verify our measured results by reproducing the data in Fig. 2a  This agreement between our model and the measured data confirms that unlike micron-scale low-index fiber optics, nanoscale waveguides can posses forward and backward traveling waves with unique near-field profiles.
The distinct near-field profiles for counter-propagating waves reveal fundamental differences between optical propagation in nanoscale waveguides compared to free-space and fiber optics.
While this phenomenon is dependent on input polarization and reflectivity of the TE and TM modes, it is solely a consequence of strong optical confinement and is likely to occur in the nanoscale high-index waveguides that are used widely in industrial and academic research labs.
In addition to potentially affecting device performance, this phenomenon could be utilized as a basis for selectively attenuating reflected waves. Active components or specific waveguide geometries could be developed to create uni-directional waveguides which could limit the intensity of reflected light. This would provide a path toward developing robust nanophotonic devices and architectures unhindered by optical reflections.