Temporal light control in complex media through the singular-value decomposition of the time-gated transmission matrix

The complex temporal behavior of an ultrashort pulse of light propagating through a multiple scattering medium can be characterized experimentally through a time-gated transmission matrix. Using a spatial light modulator, we demonstrate here that injecting singular vectors of this matrix allows us to optimally control the energy deposition at any controllable delay time. Our approach provides insights into the fundamental aspects of multispectral light scattering and could find applications in imaging or coherent control.

Figure 1

Scheme of the experimental setup. An ultrashort pulse of light, of central wavelength ${\lambda }_0=808\mathrm{n}\mathrm{m}$, delivered from a Ti:sapphire pulsed laser (MaiTai HP, Spectra Physics, $\simeq 100\mathrm{f}\mathrm{s}$ pulse length) is divided upon two paths by a polarizing beam splitter (PBS). On one path the pulse wavefront is modulated by a reflective phase-only SLM (HSP512L-1064, Meadowlarks) and passes through a $\sim 10\mu \mathrm{m}$ scattering layer of ${\mathrm{TiO}}_2$ particles (transmittance of $\sim 0.3$, suspended on a glass slide and static on the timescale of experiments) where it gets elongated. On the second path, the pulse is sent on a controlled delay line. Both pulses get recombined on a beam splitter (BS) and are imaged on a CCD camera (Manta, G-046, Allied Vision). Two half-wave plates ($\lambda /2$) adjust the polarization of the power of both arms and a polarizer (P) before the camera selects the desired polarization.

Figure 2

Temporal control of scattered light with the time-gated TM. (a) Temporal profile of the spatially averaged output pulse amplitude $\left|E\right|$ for the first singular vector $v_1$ (peaked blue) and the last singular vector $v_{225}$ (dented orange) of a time-gated TM measured at ${\tau }_0=1.1$ ps. The pulse shape obtained for a plane-wave input is shown in gray. In the inset, the amplitude enhancement ${\eta }_{\mathrm{E}}$ (measured at ${\tau }_0$) relative to the plane-wave input is shown for the whole range of singular vectors (blue dots). The expected field enhancement for phase and amplitude control $\tilde{s}=s/\sqrt{\langle s^2\rangle }$ corresponds to the normalized singular values (see SM [26] Sec. VI.) and is indicated by the red solid line. (b) Enhancements obtained with the first ($v_1$, blue dots) and last ($v_{225}$, orange squares) singular vectors for different delay times ${\tau }_0$. The plane-wave output pulse is shown as a visual aid (arbitrary units, gray line). Here, the individual time-gated TMs have been measured with ten phase steps per mode (instead of four—see SM [26] Sec. I) to reduce measurement noise in the pulse tail. (c) Simultaneous enhancement at three different delay times by projecting the sum of the corresponding TM's first singular vectors $v_1^{({\tau }_1,{\tau }_2,{\tau }_3)}$ (three peaks, green) compared to the plane-wave input (no peaks, gray) and to a single time control $v_1^{\left({\tau }_2\right)}$ (single peak, blue). All TMs used here were measured for $N_{\mathrm{SLM}}\simeq 640$ (see SM [26] Sec. III A) and $N_{\mathrm{CCD}}=225$. Data displayed in (a) are averaged over four disorder realizations whereas (b) and (c) correspond to a single realization.

Figure 3

Simulation results in a waveguide geometry with a similar scattering strength and using 50 input and output modes. Solid lines indicate the output pulse shapes for amplitude and phase control of the singular vectors $v_1$ (peaked blue) and $v_{50}$ (dented orange) corresponding to the largest and smallest singular value of the TM at the value of ${\tau }_0$ (indicated by the vertical dashed black line). The corresponding results for phase-only control are depicted by dashed lines. The gray line represents the time-of-flight reference when injecting a set of random inputs. The simulation results are averaged over ten realizations of the disorder.

Figure 4

(a) Amplitude enhancement ${\eta }_{\mathrm{E}}$ over the degree of control scaled as $1/\sqrt{\gamma }$ for the first singular vector $v_1$ (blue dots) and the global-focusing vector (red squares). Here, $N_{\mathrm{SLM}}\simeq 120$ was fixed and $N_{\mathrm{CCD}}$ was varied from $N_{\mathrm{CCD}}=25$ ($1/\sqrt{\gamma }=2.25$) to $N_{\mathrm{CCD}}=225$ ($1/\sqrt{\gamma }=0.75$). The insets show typical examples of the output field amplitude patterns for the first singular vector (top left) and the global-focusing vector (bottom right). Output fields are only affected in the region where the TM was measured, a central square, and the edges are displayed for visual reference. (No average over disorder realizations was performed.) (b) Temporal evolution of the speckle grain size for the first singular vector $v_1$ (peaked blue) and an intermediate singular vector $v_{121}$ (dented yellow) in the same configuration used in Figs. 2 and 2. At each $\tau$ the speckle grain size is normalized to the one obtained for the plane-wave input pulse. For all data presented in this figure the TM was measured at ${\tau }_0=1.1\mathrm{p}\mathrm{s}$. (An average over four disorder realizations was performed.)