Multi-Spectral Transmission Matrix

• Multi-Spectral Transmission Matrix is a frequency-resolved linear operator that maps input optical fields to output fields across multiple wavelengths in complex media.
• It enables precise spatiospectral and spatiotemporal control, facilitating applications such as programmable dispersive optics, hyperspectral imaging, and enhanced nonlinear imaging.
• Its experimental realization uses discrete spectral sampling, phase conjugation algorithms, and advanced calibration protocols to achieve deterministic light control.

A multi-spectral transmission matrix (MSTM) provides a deterministic, frequency-resolved linear mapping between the input and output optical fields of a complex system, enabling measurement-based spatiospectral and spatiotemporal control of broadband light through strongly scattering or dispersive media. The core concept generalizes the monochromatic transmission matrix formalism to encompass the full spectral bandwidth of ultrashort pulses, non-monochromatic fields, or complex objects, supporting not only spatial but also spectral and even polarization-domain manipulations. MSTM methodologies underpin a range of advanced applications in coherent control, nonlinear imaging, programmable dispersive optics, and hyperspectral imaging.

1. Formal Definition and Mathematical Structure

Let $E_{\mathrm{in}}(r_i,\lambda)$ denote the complex amplitude of the incident field at input coordinate $r_i$ and wavelength $\lambda$, and $E_{\mathrm{out}}(r_o,\lambda)$ that of the field emerging at output coordinate $r_o$. The MSTM $T(r_o, r_i; \lambda)$ is defined by the linear relation: $$E_{\mathrm{out}}(r_o, \lambda) = \int T(r_o, r_i; \lambda)\, E_{\mathrm{in}}(r_i, \lambda)\, dr_i$$ Discrete implementations assign indices $i$ (output pixels), $j$ (input modulator elements), and $k$ (wavelengths or frequencies), representing the MSTM as a rank-3 tensor $h_{i,j,k}$. The universal form is: $$E_{i}^{\mathrm{out}}(\omega_k) = \sum_{j=1}^{N} h_{i,j,k}\, E_{j}^{\mathrm{in}}(\omega_k)$$ where $N$ is the number of controlled spatial modes, and $N_\lambda$ is the number of sampled wavelengths spanning the source bandwidth $\Delta\lambda_L$ in steps of the medium's spectral correlation bandwidth $\Delta\lambda_c$. The MSTM encapsulates both the medium’s spatial diversity and its wavelength-dependent Green’s function, forming a complete measurement-based operator over the optical bandwidth (Andreoli et al., 2014, Mounaix et al., 2015, Mounaix et al., 2018).

2. Experimental Realization and Calibration Protocols

MSTM measurement protocols sample the optical system at discrete spectral intervals—each determined by the spectral correlation bandwidth of the medium ($\Delta\lambda_c \sim 1\,\mathrm{nm}$), which is governed by the photon confinement time $\tau_m$. Typical setups employ:

• Sources: Tunable CW or pulsed Ti:Sapphire lasers (e.g., 710–920 nm, 0.2 nm step, ~100 fs, $\Delta\lambda_L\sim10\,\mathrm{nm}$)
• Modulators: 2D phase-only SLM, partitioned into $N$ macro-pixels, each programmable with phase patterns (e.g., Hadamard basis)
• Sample: Multiple-scattering layer (e.g., ZnO or TiO$_2$), with thickness dictating $\Delta\lambda_c$
• Detection: CCD imaging (spatially resolved) and, optionally, polarization and spectral analysis via a spectrometer
• Spectral Sampling: $N_\lambda = \Delta\lambda_L/\Delta\lambda_c$ (e.g., $N_\lambda \sim 6 - 11$ for $\Delta\lambda_L \sim 10\,\mathrm{nm}$; $\Delta\lambda_c \sim 1\,\mathrm{nm}$)
• Calibration: At each wavelength, monochromatic TMs are measured by sequentially displaying known spatial phase patterns and acquiring output fields via interferometric (e.g., phase-stepping holography) or intensity-based schemes (Andreoli et al., 2014, Mounaix et al., 2015, Mounaix et al., 2018, Li et al., 2021).

In the context of incoherent or broadband imaging (e.g., through fiber bundles), the MSTM maps a "stacked" object vector containing intensities for all spectral and polarization channels to a concatenated measurement vector (Li et al., 2021).

3. Algorithms for Spatiospectral and Spatiotemporal Control

MSTM measurements grant deterministic control over the transmitted field via tailored wavefront shaping. Key protocols include:

• Single-wavelength focusing: At output pixel $i_t$ and wavelength $\omega_k$, optimal input is the phase conjugate:

$$E_j^{\mathrm{in}} = \exp\left[ -i\,\arg h_{i_t,j,k} \right]$$

• Multispectral (simultaneous) focusing: To focus all sampled spectral components at the same location:

$$\phi_j = \arg \left( \sum_{k=1}^{N_\lambda} h^*_{i_t,j,k} \right)$$

• Programmable dispersive optics: Routing each $\omega_k$ to an arbitrary output pixel $i_t(k)$ is achieved by coherent superposition of phase-conjugated inputs for each channel, transforming a random medium into a software-defined, reconfigurable grating (Andreoli et al., 2014).

Fully general spatiospectral waveforms (e.g., for temporal focusing or pulse shaping) are created by setting target amplitude and phase spectra at the output and performing matrix-phase-conjugation: $$\phi_{\mathrm{SLM}}(n) = \arg \left\{ \sum_{\ell=1}^{N_\omega} w_\ell T_{m_0,n}^*(\omega_\ell) e^{i\phi(\omega_\ell)} \right\}$$ allowing, for example, the imposition of flat, linear, or quadratic spectral phases to achieve Fourier-limited foci, specific delays, or controlled chirp (Mounaix et al., 2018, Mounaix et al., 2015).

4. Performance Metrics, Limitations, and Optimization

Key performance characteristics of MSTM-based control include:

• Spectral correlation bandwidth ($\Delta\lambda_c$): Typically 1–1.6 nm, setting the spectral sampling interval and resolution.
• Number of controllable spectral/spatial channels: E.g., $N_\lambda=6-11$, $N=256$–1024 spatial modes
• Temporal focus duration: Compression from average output speckle width (~2 ps) to Fourier-limited pulse (e.g., 120–200 fs)
• Nonlinear enhancement: Two-photon excitation SBR boosted from ∼8 in spatial-only focusing to >130 via full spatiotemporal focusing
• Focusing efficiency: SBR increases with the number of combined spectral channels (e.g., SBR ≈ 30 for six-wavelength focus)
• Acquisition time: Typically minutes for moderate matrix sizes (e.g., ~15 min for $N\times N_\lambda=256\times6$), limited by SLM rate, system stability, and interferometric drift

Principal limitations:

• Phase stability is required over the measurement duration
• Absolute spectral phase across the bandwidth may remain undetermined, preventing deterministic pulse compression in some implementations
• Scalability to higher $N, N_\lambda$ is limited by acquisition speed and hardware parallelization (Andreoli et al., 2014, Mounaix et al., 2015, Mounaix et al., 2018)

5. Physical Interpretation: Spectral Correlations, Eigenchannels, and Modal Basis

The MSTM description naturally extends to spectral analysis of transmission eigenchannels. At each frequency, the transmission matrix $T(\omega)$ admits a singular value decomposition: $$T(\omega) = U(\omega)\, \Lambda(\omega)\, V^\dagger(\omega)$$ yielding eigenchannels whose spectral correlation bandwidth is a function of their transmittance: high-transmission channels are narrowband (correlation width $\sim \delta\omega_{\mathrm{field}}$), while low-transmission channels are spectrally broader, as they are built from many widely detuned, interfering modal contributions.

The time-domain (impulse) response of an eigenchannel is the Fourier transform of its singular value spectrum. High-transmission channels possess longer group delays (wider temporal width), whereas low-transmission channels exhibit prompt, temporally narrow responses (Shi et al., 2015).

Underlying these properties is the modal expansion of $T(\omega)$ in terms of quasi-normal modes: $$T(\omega) = \sum_{m=1}^M T^m\, \frac{\Gamma_m/2}{\Gamma_m/2 + i(\omega - \omega_m)}$$ where each mode contributes with weight, linewidth, and distinct in/out eigenchannels.

6. Applications and Extensions

• Deterministic focusing and spatiospectral shaping: Precise control of spectral components at arbitrary spatial locations, including programmable dispersive elements with arbitrary wavelength-position mapping (Andreoli et al., 2014).
• Spatiotemporal focusing: Achieving Fourier-limited pulse recombination and enhanced nonlinear signals through strongly scattering media; demonstrated enhancements in two-photon fluorescence SBR and spatial selectivity deep within opaque environments (Mounaix et al., 2018, Mounaix et al., 2015).
• Hyperspectral imaging and polarization control: Recovery of broadband, polarized spectral information from single-shot, speckle-encoded images, enabling high-resolution, multiplexed imaging with minimal hardware (Li et al., 2021).
• Fundamental light-matter studies: Opening perspectives for coherent control, quantum pump–probe experiments, and mesoscopic transport studies in disordered systems (Mounaix et al., 2015, Shi et al., 2015).

7. Outlook and Future Directions

Ongoing research seeks to improve MSTM scalability via higher-speed spatial modulators, parallelized detection (e.g., off-axis holography, compressed sensing), and further exploitation of degrees of freedom (amplitude, polarization). Adaptive inversion algorithms, digitally emulated calibration, and incorporation of machine learning priors are proposed to accelerate and robustify reconstruction from complex scattering systems. Expanding the spectral coverage and miniaturization of calibration procedures will further broaden the impact in in vivo imaging, compact spectroscopy, and ultrafast light manipulation (Andreoli et al., 2014, Mounaix et al., 2018, Li et al., 2021).