Prospects of coherent control in turbid media: Bounds on focusing broadband laser pulses

We study the prospects of controlling transmission of broadband and bichromatic laser pulses through turbid samples. The ability to focus transmitted broadband light is limited via both the scattering properties of the medium and the technical characteristics of the experimental setup. There are two time scales given by pulse stretching in the near- and far-field regions which define the maximum bandwidth of a pulse amenable to focusing. In the geometric-optics regime of wave propagation in the medium, a single setup can be optimal for focusing light at frequencies $\omega$ and $n\omega$ simultaneously, providing the basis for the $1+n$ coherent quantum control. Beyond the regime of geometric optics, we discuss a simple solution for the shaping, which provides the figure of merit for one’s ability to simultaneously focus several transmission modes.

Figure 1

Implied experimental setup. In panel (b), the SLM and imaging lens are replaced by the image of the SLM on the input surface of the sample. In addition, the direction of light propagation is reversed.

Figure 2

(a) Distribution of impurities in one of the four planes modelled in the calculation. The area shown is $770\times 770$ $\mu$m${}^2$. Panels (b) and (c) show the near-field intensity and phase distributions for a ${\lambda }_0=800$ nm beam after passing through four diffusing planes separated by regions of empty space. In panel (c), intensity is shown by brightness, and phase, between 0 and 2$\pi$, is shown by color. (d) A 25 fs input pulse exits the system in the far field stretched to 100 fs. (e) Far-field speckle pattern for 800 nm light.

Figure 3

(a) Phase mask applied by ISLM to correct the wave front shown in Fig. 2c. (b) Near-field intensity and phase distribution at 800 nm after the phase correction. (c) Corrected far-field intensity distribution.

Figure 4

(a) Laser intensity in the zeroth far-field mode as a function of frequency with the ISLM mask tuned for ${\omega }_0$. Panels (b) and (c) show the near-field wave fronts at $\omega =1.5{\omega }_0$ and $\omega =2{\omega }_0$.

Figure 5

Focusing as a function of frequency in the limit of low spatial dispersion.