PIONEERING EXPERIMENT
FOCUSING COHERENT Light
Figure 1.( a) When a plane wave is incident on a scattering medium, a disordered speckle pattern is transmitted.( b) The wavefront is adaptively shaped so that a clear focus is formed. Reprinted with permission from [ 1 ] © Optical Society of America.
On paper the idea was simple enough. First, measure the transmission matrix of the sample that relates the complex amplitude of the light in all incident modes to the transmitted amplitudes. Next, perform a singular value decomposition of this matrix to find singular vectors(‘ channels’) that experience a high transmission.
Experiment
The experiment was based on a spatial light modulator( SLM) to control the incident light, a device of which each pixel controls the phase of the light reflected from it. Due to the finite size of the optics, however, the SLM could only control a tiny fraction of the incident modes. At the time, there was no theory available to extract properties of the open channels with such incomplete control. As a first step toward obtaining an increased total transmission, we therefore started simple: use the SLM to maximize the intensity of the scattered light at a single point behind a scattering medium, as depicted in Fig. 1. In this case, only a single row of the transmission matrix needs to be considered. Measuring the matrix elements of this row can be done by modulating the phase of each pixel( or group of pixels) in sequence, while monitoring the intensity in the target point behind the sample; in essence performing self-referencing phase stepping holography.
After the measurement is completed, the SLM adjusts the phases so that the light from all pixels interferes constructively at the target. The first sample in the experiment was a layer of titanium dioxide, one of the strongest scattering materials for visible light, making a 10-micrometer layer as opaque as a thick sheet of printing paper. When illuminated by a laser beam, without any shaping, the sample transmits only disordered speckle, as shown in Fig. 2a, with no indication of any focus. When we turned on the SLM, the most unlikely thing happened: the experiment worked at the first try. And, in accordance with our theory( but still to our surprise), shaping the wavefront led to a focus that was easily discernible with the naked eye, as seen in Fig. 2b. Thanks to the superposition principle, one can show that this algorithm finds the global optimal wavefront for focusing on the target
in a fixed number of steps. This is true regardless of the number of pixels, and regardless of the scattering strength of the sample. This feature really separated our approach from those used in the adaptive optics( AO) community at the time, because AO traditionally uses Zernike polynomials that are orthogonal in the optical path length of the light. While this works extremely well for low-frequency aberrations, for stronger scattering the relation between the amplitude of the Zernike modes and the intensity in the target becomes highly non-linear and non-convex, making efficient optimization hard or even impossible [ 2 ]. Under the assumption that the complex transmission matrix elements are uncorrelated, the intensity in the focus is predicted to grow linearly with the number of controlled inputs, according to, η = π – 4
( N – 1) + 1. Here, η is known as the intensity enhancement, defined as the ensemble-averaged intensity in the focus normalized to the
Figure 2. Intensity image( log scale) of the transmitted light( a) before and( b) after shaping the incident wavefront. Adapted with permission from [ 1 ] © Optical Society of America.
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