# Static Structures in Leaky Mode Waveguides +
## 1 Introduction
Leaky mode devices comprise a transparent slab of a piezoelectric substrate, such as lithium niobate, with a waveguide indiffused on the surface which sits adjacent to an interdigital transducer. When an RF signal excites the transducer, a surface acoustic wave is generated that travels across the surface of the waveguide. Light trapped in the waveguide, traveling contra-linearly or collinearly with the SAW pattern, may be mode-coupled from a guided mode to a leaky mode which can be steered and shaped by the SAW pattern to form a holographic image. The use of such a device in a near-eye application is shown in Figure 2. A chirped SAW pattern is generated by the transducer which is designed to mode couple light so that it appears to have originated from a point in space. The leaky mode light travels at a shallow angle through the substrate. This angle is usually very shallow, typically between 1\({}^{\circ}\) and 12\({}^{\circ}\) from parallel to the substrate surface. However, this angle can be modified by passing the leaky mode light from one substrate to another with another index (or the same index, but canted at an angle) thereby increasing the angle by an angular bias if desired (will assume an angular bias of 10\({}^{\circ}\) for the analysis in this paper). The lateral translation of the light ends at the bottom surface of the substrate where the light is out-coupled by a high-spatial frequency out-coupling grating usually patterned by interference lithography and etched into the substrate. This grating has only one diffracted order into the air (all others are evanescent) and toward the viewer's eye. The light  
Figure 1: Low-profile near-eye display concept with a goal of large field of view. rays that reach the viewer's eye can be back cast to a distant image point. As the surface acoustic wave moves across the device's SAW aperture, the image point moves across the viewer's field of view. The illumination light in the waveguide pulses in coordination with this movement to draw or emit points as the image point 'cursor' travels. When it has completed its scan, a new SAW chirp pattern is generated with a different depth and the field of view that is scanned again for as many depth planes as desired (we recently demonstrated a device with arbitrary focus from 4 ft to 10 ft [4]). Several leaky mode waveguide/transducer 'channels' can be placed adjacent to one another to create a vertically multiplexed array. Each channel becomes responsible for the one line of the display. In this way, images can be created at multiple depth planes to form a 3D image. In this paper, we propose a simple modification to the leaky mode device by adding a laser-induced, static grating in the substrate of the leaky mode device. Femtosecond lasers can induce phase changes and 'catastrophic' lattice changes in the bulk of lithium niobate. The local nature of the change is aided by two photon up-conversion in the substrate. The result is that relatively high contrast gratings can be formed in the substrate with critical dimensions down below a micron and write depth of several tens of microns (see Figure 3). When coupled with high-accuracy, large-travel stages, whole wafers can be written with sub-surface gratings [10; 11; 12]. These gratings can be uniform, chirped, Raman-Nath, or Bragg grating. In our application, we explore a uniform thin grating underneath our leaky mode channels. It would be possible to have different periods underneath each channel, but for this analysis we will assume that the induced grating has only one period everywhere. Figure 3: A laser-induced grating in x-cut lithium niobate and the resulting diffraction pattern with several high-order modes. Figure 2: Reduction of view angel for bottom-exit leaky mode devices: (**a**) light diffracting from a surface acoustic wave (SAW) pattern is out-coupled with a grating; (**b**) The K-vector analysis shows a significant reduction in \(\theta_{out}\) vs. \(\theta_{in}\). The former of these angles dictates the display view zone and field of view.