Control of Light Diffusion in a Disordered Photonic Waveguide
CControl of Light Diffusion in a Disordered Photonic Waveguide
Raktim Sarma, Timofey Golubev, Alexey Yamilov, ∗ and Hui Cao † Department of Applied Physics, Yale University, New Haven, CT, 06520, USA Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409,USA (Dated: May 13, 2019)We control the diffusion of light in a disordered photonic waveguide by modulating the waveguidegeometry. In a single waveguide of varying cross-section, the diffusion coefficient changes spatially intwo dimensions due to localization effects. The intensity distribution inside the waveguide agrees tothe prediction of the self-consistent theory of localization. Our work shows that wave diffusion canbe efficiently manipulated without modifying the structural disorder.
The concept of diffusion is widely used to study thepropagation of light through multiple scattering mediasuch as clouds, colloidal solutions, paint, and biologicaltissue . The diffusion, however, is an approximationas it neglects wave interference effects . Most of thescattered waves go through independent paths and haveuncorrelated phases, so their interference is averaged out.However, a wave may return to a position it has previouslyvisited after multiple scattering, and there always existsthe time-reversed path which yields identical phase delay.Constructive interference between the waves travelingin the time-reversed paths increases the energy densityat the original position, thus suppressing diffusion andeventually leading to localization . This effect has beenaccounted for by a renormalized diffusion coefficient D inthe self-consistent theory of localization . The amountof renormalization depends on the return probability,which is determined by the size of a random mediumas well as the position inside . We recently reporteda direct observation of the position-dependent diffusioncoefficient in disordered waveguides . By changing thewaveguide length and width, we tuned the diffusion coef-ficient by varying the strength of wave interference. How-ever, the width of each waveguide was kept constant, andwe switched between the waveguides to control diffusion.In this Letter, we fabricate disordered waveguides witha variable cross-section and thus achieve control of lighttransport in the same system.These new structures makeit necessary to account for spatial variation of diffusioncoefficient D in two dimensions (2D) due to the modula-tion of the waveguide width. Experimentally we fabricatea random array of air holes in a waveguide geometry on asilicon wafer, and probe light propagation inside the 2Dstructure from the third dimension. The measured spatialdistribution of light intensity inside the disordered waveg-uide agrees well to the prediction of the self-consistenttheory of localization . Instead of changing the degreeof disorder, we demonstrate that the wave diffusion canbe manipulated by changing geometry (cross-section) ofthe random waveguide nano-structures.The disordered waveguides were fabricated with asilicon-on-insulator (SOI) wafer where the thickness ofsilicon layer and the buried oxide were 220 nm and 3 µ mrespectively. The patterns were written by electron beamlithography and etched in an inductively-coupled-plasma FIG. 1: Top-view scanning electron microscope (SEM) imageof a quasi-2D disordered photonic waveguide. Light is injectedfrom the left end of the waveguide and incident onto therandom array of air holes. The waveguide wall is made of atriangle lattice of air holes which forms a 2D photonic bandgapto confine light inside the waveguide. The width of the randomwaveguide is changed gradually from 40 µ m to 5 µ m througha tapered region. (ICP) reactive-ion-etcher (RIE). Figure 1 is the scanningelectron microscope (SEM) image of a fabricated sample.The waveguide contained a 2D random array of air holes.The hole diameters were 120 nm, and the average (center-to-center) distance of neighboring holes was about 385nm. The total length L of the random waveguide was 120 µ m, and the waveguide width was changed from W =40 µ m to W = 5 µ m via a tapered region. The lengthsof wider ( W ) and narrower ( W ) sections were L = 52 µ m and L = 58 µ m respectively. The tapered sectionwas 10 µ m long, with a tapering angle of 60 degrees. Thewaveguide walls were made of triangle lattice of air holes(lattice constant 440 nm, hole radius 154 nm) that hadcomplete 2D photonic bandgap.In the optical experiment, we used a lensed fiber tocouple monochromatic light (wavelength ∼ a r X i v : . [ phy s i c s . op ti c s ] M a y waveguide plane). Light was scattered by the air holesinside the waveguide and undergoes diffusion. The waveg-uide walls provided in-plane confinement of the scatteredlight. However, some of the light was scattered out ofthe waveguide plane. This leakage allowed us to observelight propagation inside the disordered waveguide fromthe vertical direction. The spatial distribution of lightintensity across the waveguide was projected by a 50 × objective lens [numerical aperture (NA) = 0.42] onto anInGaAs camera (Xeva 1.7-320). Figure 2(b) shows a typ-ical near-field image, from which we extracted the 2Dintensity distribution inside the waveguide I ( y, z ). Theensemble averaging was done over three random config-urations of air holes and 25 input wavelengths equallyspaced between 1500 nm and 1510 nm. The wavelengthspacing was chosen to produce different intensity distri-butions. Further averaging was done by slightly movingthe input beam position along the transverse y direction.Nevertheless, the front surface of the random structureswas always uniformly illuminated by the incident light. FIG. 2: (Color online) (a) A schematic of the experimentalsetup. A lensed fiber couples the light to the structure andanother 50 × objective lens (NA = 0.42) collects the lightscattered by the air holes out of the waveguide plane andprojects onto a camera. (b) Near-field optical image of theintensity of light scattered out-of-plane from the disorderedwaveguide. The wavelength of the probe light is 1505 nm. The relevant parameters for light propagation in thedisordered waveguide are the transport mean free path (cid:96) and the diffusive dissipation length ξ a . The transportmean free path (cid:96) depends on the size and density ofthe air holes. The dissipation mostly comes from out-of-plane scattering as the silicon absorption at the probewavelength is negligible. As shown in our previous work ,this vertical loss of light can be treated a dissipation (asabsorption) and described by the characteristic length ξ a = √ D τ a , where τ a is the ballistic dissipation timeand D is the diffusion coefficient without localizationcorrections.There are three main advantages of using the planarwaveguide geometry. First, it allows a precise fabricationof the designed structure so that the parameters such as the transport mean free path can be accurately controlled.Second, we can easily monitor the in-plane diffusion bycollecting the out-of-the-plane scattered light. Third,the localization length ξ can be tuned by changing thewaveguide width W , because ξ = ( π/ N (cid:96) , where N =2 W/ ( λ/n e ) is the number of propagating modes in thewaveguide, which is proportional to W . By varying thewidth of a single waveguide, we adjust the strength oflocalization effect along the waveguide. The localizationlength in the wider section of the waveguide ( W = 40 µ m) is 8 times longer than that in the narrower section( W = 5 µ m). Hence, the suppression of diffusion bywave interference is enhanced approximately 8 times inthe narrower section of the waveguide.For a quantitative description of light transport in therandom waveguide of variable width, we used the self-consistent theory of localization to calculate the diffusioncoefficient D ( y, z ) inside the waveguide. The renormal-ization of D depends on the return probability, which isposition dependent . The maximum renormalizationhappens inside the random media at a location where thereturn probability is the highest, and the renormalizationis lowest near the open boundaries of the random media.As it will be shown below, the return probability takesthe maximum value in the narrow portion of the structureand not at the geometrical center as in waveguides witha uniform cross-section. The renormalization of the diffu-sion coefficient also depends on the amount of dissipation,which suppresses feedback from long propagation pathsand sets an effective system size beyond which the wavewill not return .Numerically we computed D ( y, z ) using the commercialpackage Comsol Multiphysics after setting the values ofthe transport mean free path (cid:96) and the diffusive dissipa-tion length ξ a . First the return probability was calculatedat every point in the waveguide . This was done bymoving a point source throughout the structure and cal-culating the light intensity at the source for each sourceposition. This intensity was taken as the return proba-bility which was then used to renormalize D ( y, z ). Themodified D ( y, z ) was then used to recalculate the returnprobability. Several iterations of this procedure were per-formed until the changes in D ( y, z ) between iterationsbecame small enough to be negligible. Once we obtainedthe final value of D ( y, z ), it was used to calculate theintensity I ( y, z ) inside the waveguide. The above calcu-lation was repeated for various combinations of (cid:96) and ξ a until the calculated I ( y, z ) matched the measured in-tensity distribution. The parameters that gave the bestagreement were (cid:96) = 2.9 µ m and ξ a = 35 µ m. Figure 3(a)plots the calculated return probability, which is greatlyenhanced by the stronger transverse confinement (along y direction) in the narrower section of the waveguide. Con-sequently, the renormalized diffusion coefficient D ( y, z ),shown in Fig. 3(b), reaches the minimum value closeto the middle of the narrower section. Note that in thetapered region, D changes not only along z , but alsoalong y . The smaller D near the boundary is attributedto the enhancement of return probability due to reflectionfrom the photonic crystal wall. Figure 3(c) shows thespatial distribution of in-plane diffusive light intensity I ( y, z ) inside the waveguide. FIG. 3: (Color online) (a) Calculated return probability inthe disordered waveguide shown in Fig. 1. (cid:96) = 2 . µ m, and ξ a = 35 µ m. (b) 2D renormalized position dependent diffusioncoefficient D ( y, z ) /D for the same structure as in (a). (c)Intensity distribution I ( y, z ) /I inside the random structureobtained from D ( y, z ) /D in (b). From the measured I ( y, z ), we computed the cross-section integrated intensity I t ( z ) = (cid:82) W ( z ) / − W ( z ) / I ( y, z ) dy and the cross-section averaged intensity I v ( z ) = I t ( z ) /W ( z ). The former quantity is proportional to the z -component of the energy flux through the cross-section ofthe waveguide, while the latter quantity, I v ( z ), is relatedto the energy density. As shown in Fig. 4(a), I t ( z ) decaysmore slowly with z in the wider section of the waveg-uide than in the narrower one. This is attributed to twofactors, (i) reflection from the boundary of the taperedregion, (ii) stronger localization effect in the narrowersection of waveguide. The narrowing of the waveguidewidth leads to a sharp drop of I t (energy flux), as partof the diffusive light is reflected back. The dashed curvein Fig. 4(a) is the calculated I t ( z ), which agrees well tothe experimental data. Figure 4(b) plots the measured I v ( z ) together with the calculated one. Again we see agood agreement except at z ∼ µ m. The near-fieldoptical image [Fig. 2(b)] reveals that near the photoniccrystal wall of the tapered section, the abrupt backwardscattering leads to formation of standing wave, thus theintensity is enhanced compared to the diffusive predic-tion. The spatial extent of this effect is determined by thetransport mean free path (cid:96) beyond which the directionof the reflected wave is randomized. Inherent inabilityof a diffusive description to describe transport on scalesshorter than (cid:96) explains the deviation of the experimen-tally measured intensity from the theoretical predictionas exhibited in Fig. 4(b) by a small bump at z ∼ µ m.In conclusion, we demonstrated an effective way of ma-nipulating light diffusion in a disordered photonic waveg- FIG. 4: (Color online) (a) Comparison of the measured cross-section integrated intensity I t ( z ) (solid blue line) to numericalsimulations (dashed red line). (b) Measured cross-sectionaveraged intensity I v ( z ) (solid blue line) in comparison withthe calculated one (dashed red line). The vertical dotted linesin (a,b) marks the starting point and the end point of thetapered region. uide. Instead of changing the degree of structural disorder,we varied the waveguide geometry (its cross-section). Bymodulating the width in a single waveguide, we manip-ulated the interference of scattered light and made thediffusion coefficient vary spatially in two dimensions. Wemeasured the intensity distribution inside the quasi-2Drandom structures by probing from the third dimensionand the experimental results agreed well to the predictionsof the self-consistent theory of localization. Although, theexperiments in this work were done with light, the out-lined approach to control diffusion is also applicable toother types of waves, such as acoustic wave, microwaveand the de Broglie wave of electrons.We acknowledge Douglas Stone and Arthur Goetschyfor useful discussions and suggestions. We also thankMichael Rooks for suggestions regarding fabrication ofthe sample. This work was supported by the NationalScience Foundation under grants nos. DMR-1205307,DMR-1205223 and ECCS-1128542. 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