Deterministic Terahertz Wave Control in Scattering Media

Abstract

Scattering-assisted synthesis of broadband optical pulses is recognized to have a cross-disciplinary fundamental and application importance. Achieving full-waveform synthesis generally requires means for assessing the instantaneous electric field, i.e., the absolute electromagnetic phase. These are generally not accessible to established methodologies for scattering-assisted pulse envelope and phase shaping. The lack of field sensitivity also results in complex indirect approaches to evaluate the scattering space-time properties. The terahertz frequency domain potentially offers some distinctive new possibilities, thanks to the availability of methods to perform absolute measurements of the scattered electric field, as opposed to optical intensity-based diagnostics. An interesting conceptual question is whether this additional degree of freedom can lead to different types of methodologies toward wave shaping and direct field-waveform control. In this work, we theoretically investigate a deterministic scheme to achieve broadband, spatiotemporal waveform control of terahertz fields mediated by a scattering medium. Direct field access via time-domain spectroscopy enables a process in which the field and scattering matrix of the medium are assessed with minimal experimental efforts. Then, illumination conditions for an arbitrary targeted output field waveform are deterministically retrieved through numerical inversion. In addition, complete field knowledge enables reconstructing field distributions with complex phase profiles, as in the case of phase-only masks and optical vortices, a significantly challenging task for traditional implementations at optical frequencies based on intensity measurements aided with interferometric techniques.

Figure 1

Schematic of experimental-driven methodology (a) Conceptual overview of methodology, including the nonlinear conversion of optical patterns to THz structural waves and the retrieval of transmission properties of the scattering medium defined in terms of a coherent transfer matrix. The full knowledge of the coherent transfer matrix retrieved using an orthogonal set of patterns can be used to achieve scattering-assisted focusing at the output of the scattering medium. (b) Input THz pulse electric-field profile. (c) Scattered THz pulse collected at a generic mth output pixel. (d) Intensity spectral density of the input THz field. (e) Intensity spectral density of the scattered THz pulse as collected at a generic mth pixel. In our simulations, we considered a 1 nJ THz pulse of duration 250 fs at the input with 40 dB SNR per pixel. The 6.4 × 6.4 mm² sample illumination area is spatially sampled at 200 μm resolution, corresponding to a number of pixels of 32 × 32.

Figure 2

Transfer Matrix reconstruction. Mean squared error (MSE) of the coherent transfer matrix elements as reconstructed through a Walsh–Hadamard decomposition. The input and output planes are divided into 16 × 16 pixels, corresponding to a scattering matrix composed of 256 × 256 entries.

Figure 3

Spatiotemporal focusing of THz field: (a) optimized intensity spectral density distribution showing the focus spot in THz band. (b) Comparison between intensity spectral density profiles of the perturbed THz spectrum (blue) and optimized spectrum (green) at the mth pixel. (c) THz pulse profile of scattered field (blue) and optimized field (green) from the mth pixel of the output plane. (d) Intensity spectral density distribution of the output THz field showing two simultaneous focus spots at mth pixel and m^′th pixel. (e) Intensity spectral density distribution of optimized THz field for two simultaneous focus spots at mth and m′th pixel with two different spectra centered around 0.7 and 1.3 THz. SNR per pixel: 40 dB. The 6.4 × 6.4 mm² sample illumination area is spatially sampled at 200 μm resolution, corresponding to a number of pixels of 32 × 32.

Figure 4

Deterministic coherent control without previous knowledge of the source. (a) Input field-temporal profile corresponding to a chirped pulse. The temporal profile includes the 40 dB noise applied at the detection. (b) Intensity spectral density (blue line) and spectral phase (orange line) for the field profile in panel (a). (c, d) Same as panels (a) and (b) but for a nonoptimized incident pattern. (e, f) Same as panels (a) and (b) for the optimized profile targeted by our optimization routine. (g, h) Same as panels a and b for the optimized output field. The 6.4 × 6.4 mm² sample illumination area is spatially sampled at 200 μm resolution, corresponding to a number of pixels of 32 × 32.

Figure 5

Time-resolved THz phase-sensitive imaging through the scattering medium. (a) Schematic of imaging methodology. Inset: time-resolved reconstruction of the image corresponds to the measurement at t = 0 ps in panel (e). (b) Temporal evolution of output speckles corresponding to two different pixels (red and cyan dots shown in panel (a)) before deconvolution. (c) Temporal evolution of reconstructed THz pulse (after deconvolution) for two different pixels (red and cyan dots shown in panel (a)). (d) Structural Similarity (SSIM) index in the time-resolved reconstruction of an image object. (e) Reconstructed phase images at t = −0.52, 0, and 0.12 ps. The 6.4 × 6.4 mm² sample illumination area is spatially sampled at 200 μm resolution, corresponding to a number of pixels of 32 × 32 (see Video S1). Logo used with permission from the University of Sussex.

Figure 6

Complex propagation of THz vortex beam through the scattering medium. (a, b) Retrieved spatial field and phase distribution of L₀¹ vortex beam at t = 0 ps. (c, d) Temporal profile of the retrieved THz pulse corresponding to two different pixels for the L₀¹ vortex beam in panels (a) and (b). (e–h) Same as panels (a–d) for an L₁¹ vortex beam. The 6.4 × 6.4 mm² sample illumination area is spatially sampled at 200 μm resolution, corresponding to a number of pixels of 32 × 32 (see Video S2).