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Three-dimensional spatially resolved optical energy density enhanced by wavefront shaping

Peilong Hong, Oluwafemi S. Ojambati, Ad Lagendijk, Allard P. Mosk, and Willem L. Vos

Open Access Open Access

Abstract

While a three-dimensional (3D) scattering medium is from the outset opaque, such a medium sustains intriguing transport channels with near-unity transmission that are pursued for fundamental reasons and for applications in solid-state lighting, random lasers, solar cells, and biomedical optics. Here, we study the 3D spatially resolved distribution of the energy density of light in a 3D scattering medium upon the excitation of highly transmitting channels. The coupling into these channels is excited by spatially shaping the incident optical wavefronts to a focus on the back surface. To probe the local energy density, we excite isolated fluorescent nanospheres distributed inside the medium. From the spatial fluorescent intensity pattern we obtain the position of each nanosphere, while the total fluorescent intensity gauges the energy density. Our 3D spatially resolved measurements reveal that the differential fluorescent enhancement changes with depth, up to 26× at the back surface of the medium, and the enhancement reveals a strong peak versus transverse position. We successfully interpret our results with a newly developed 3D model without adjustable parameters that considers the time-reversed diffusion starting from a point source at the back surface.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Supplementary Material (1)

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Figures (7)
Fig. 1.
Fig. 1. Schematic of our method to probe the 3D ( x , y , z ) spatially resolved local energy density that is enhanced by wavefront shaping. Incident green light is wavefront shaped to an optimized focus at the back surface of a scattering medium (ensemble of ZnO nanospheres) to preferentially excite open transmission channels. The scattering medium is sparsely doped with fluorescent nanospheres that probe the local energy density of the green incident light at different positions ( x , y , z ) by emitting orange light in proportion to the local energy density of the green excitation light.

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Fig. 2.
Fig. 2. Measured differential fluorescence enhancement ∂ η f / ∂ F (blue circles with error bars) versus depth z and scaled depth z / ℓ for two samples with thicknesses of (a)  L = 8   μm and (b)  L = 16   μm . The vertical error bars are comparable to the symbol size. The fluorescent nanospheres are centered on the optical axis at ( x 0 , y 0 ) . The red curve is the energy density enhancement predicted by our 3D model. The green dashed–dotted curve indicates a hypothetical absence of control of the energy density.

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Fig. 3.
Fig. 3. Differential fluorescence enhancement ∂ η f / ∂ F versus transverse displacement Δ x relative to the optical axis for scattering samples with thicknesses of (a)  L = 8   μm [nanosphere at ( y , z ) = ( 0 , 5.9 ± 0.1 )   μm ] and (b)  L = 16   μm [nanosphere at ( y , z ) = ( 0 , 14.7 ± 0.2 )   μm ]. Red circles are the measured data with error bars. The light blue surface map and the blue solid line are the differential energy densities along the transverse x position at y = 0 , as predicted by our 3D model.

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Fig. 4.
Fig. 4. Determining the depth z of a fluorescent sphere in the L = 8   μm sample. (a) Fluorescence image measured in real space by averaging m 1 = 41 data sets, each with a different random incident wavefront. (b) Fourier transform of the data in panel (a). Intensities in (a) and (b) are normalized to their maxima. (c) Solution of the diffusion equation with a single fluorescent nanosphere at depth z = 3.3 ± 0.3   μm . (d) Cross sections through the centers of (b) (blue circles) and (c) (red line), respectively, showing a good match.

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Fig. 5.
Fig. 5. Measured fidelity F versus the phase perturbation factor δ ϕ m on the optimized phase pattern. The red curve is a guide to the eye.

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Fig. 6.
Fig. 6. Measured fluorescence enhancement η f versus fidelity F (blue squares) at a depth z = 3.3 ± 0.3   μm in the L = 8   μm thick sample. Red dashed line: Eq. (2) with unity intercept and differential fluorescence enhancement ∂ η f / ∂ F = 6.9 ± 0.7 as the only adjustable parameter, with the error at the 95% confidence interval (red dotted lines). Magenta circles are rebinned data (101 raw data per bin), which give a similar result: ∂ η f / ∂ F = 6.7 ± 0.6 (green dashed–dotted line).

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Fig. 7.
Fig. 7. (a) Transverse ( x , y ) cross sections of the calculated energy densities of optimized light at various depths z inside a scattering medium with thickness L = 8   μm . The scale bar is 10 μm. (b) Energy densities of optimized light W o integrated over the transverse ( x , y ) -plane (red solid curve), unoptimized light W uo (green short-dashed curve), and the energy density of the fundamental diffusion mode W m = 1 as a function of depth z . (c) The energy densities W o ( x = 0 , y = 0 , z ) (red solid curve) and W uo ( x = 0 , y = 0 , z ) (green dashed curve) as a function of z . I 0 and D are the incident intensity and the diffusion constant, respectively.

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Equations (4)

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W e ( x , y , z ) = F · W o ( x , y , z ) + ( 1 − F ) · W uo ( x , y , z ) .
η f ( x , y , z ) = 1 + F · ∂ η f ∂ F ,
ϕ p ( i , j ) = ϕ o ( i , j ) + δ ϕ ( i , j ) ,
W o ( x , y , z ) = W of ( x , y , z ) + W bg ( x , y , z ) ,
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