Optofluidic waveguide as a transformation optics device for lightwave bending and manipulation

Abstract

Transformation optics represents a new paradigm for designing light-manipulating devices, such as cloaks and field concentrators, through the engineering of electromagnetic space using materials with spatially variable parameters. Here we analyse liquid flowing in an optofluidic waveguide as a new type of controllable transformation optics medium. We show that a laminar liquid flow in an optofluidic channel exhibits spatially variable dielectric properties that support novel wave-focussing and interference phenomena, which are distinctively different from the discrete diffraction observed in solid waveguide arrays. Our work provides new insight into the unique optical properties of optofluidic waveguides and their potential applications.

Figure 1. Design of light focussing and interference in an optofluidic waveguide.

(a) Schematic illustration of the optofluidic waveguide with three laminar flows in the microchannel (the surrounding PDMS structures are not shown). The white arrow represents the flow direction. (b) GRIN profile of the microchannel in the transverse direction used to achieve light focussing. (c) Bidirectional GRIN profile used to achieve light interference. The intensity scale shows the refractive index distribution. (d) Simulated light propagation in the waveguide for the index profile shown in (b), where n(0)=1.432, n_c=1.332. Light is bent and periodically focussed in the optofluidic waveguide. The horizontal dashed lines represent the boundaries between the waveguide core and claddings. The vertical solid line represents the wavefront. (e) Simulated light propagation in the waveguide with the index profile shown in (c), where r=1/3 and Q_clad=0.5 μl min⁻¹. Light interference is clearly shown in the optofluidic waveguide. For (d) and (e), scale bar equals 100 μm and the vertical direction is magnified by two.

Figure 2. Control of the diffusion-induced bidirectional GRIN profile.

(a) Calculated refractive index profile for r=1/3 as a result of diffusion at a low flow rate of Q_clad=0.5 μl min⁻¹. The dashed black line indicates the x=0 line. (b) Cross-sectional refractive index profiles of (a) at x=0 (solid black), L/2 (solid blue) and L (solid red), where the dashed blue and red lines represent the fitted power law curves. (c) Calculated refractive index n_core at the core central line (upper lines) and n_clad at the core/cladding interface (lower lines) with Q_clad=0.5 μl min⁻¹, with r=1/2 (dashed red), r=1/3 (solid green) and r=1/6 (dash-dotted blue). The dotted black lines represent the high-flow-rate-induced step-index profile in (b).

Figure 3. Focussing of light in the optofluidic waveguides.

(a) Observed light propagation under different flow conditions. At a high flow rate of Q_clad=50 μl min⁻¹, the light is confined in the core due to the step-index distribution. At Q_clad=0.5 μl min⁻¹, the diffusion-induced gradient of the refractive index causes the light to repeatedly merge. With increasing r, the length of the first section becomes larger, for example, P_1/6=270 μm, P_1/3=300 μm and P_1/2=340 μm. The focussing period also increases, for example, the first period for P_1/3 is 300 μm, the second is extended to 490 μm and the third is 590 μm. (Scale bar equals 300 μm). (b) The first section lengths as a function of the flow rate of the core fluid at r=1/6 (square), r=1/3 (circle) and r=1/2 (triangle), showing an approximately linearly decreasing relationship over a Q_clad range of 0.1–1.0 μl min⁻¹.

Figure 4. Interference patterns in the optofluidic waveguide.

(a) Zoomed-in views of the first section from the experimental observation (top) and from the simulation using the finite-difference time-domain method (bottom) for Q_clad=0.5 μl min⁻¹, r=1/3. In (a), the vertical direction is magnified by two and the dashed lines indicate the region of core liquid without diffusion. A rough matching can be observed between the measured ray trajectories and the simulated field distribution patterns. (Scale bar equals 100 μm). (b) Intensity profiles along the central line in the first section (solid red), second section (dash-dotted blue) and third section (dotted black), respectively, showing that the light is not evenly distributed in the off-focal regions, which suggests the presence of interference patterns in the optofluidic waveguides. (c) Observed (top) and simulated (bottom) interference patterns in the optofluidic waveguide. (Scale bar equals 15 μm). (d) Comparison of the intensity profiles along the monitoring lines 1 and 2 in (c) for the measured (solid black) and simulated (dotted red) patterns. The positions and relative intensities of the bright spots show a rough agreement. The discrepancy is mainly attributed to the imperfection of the flow control (a.u. stands for arbitrary units).