Joint control of coherent transmission, reflection, and absorption

Abstract

Controlling multiple wave properties simultaneously poses a key challenge in coherent control of wave transport. We present a theory for joint coherent control of transmission, reflection, and absorption in linear systems. We prove that the numerical range provides the mathematical structure governing achievable responses, and reveal non-abelian effects due to non-commutativity between transmission, reflection, and absorption matrices. We provide an algorithm to achieve arbitrary target responses. Our results establish a theoretical foundation for joint coherent control of waves.

Figures (7)

• Figure 1: (a) An $(l+m)$-port linear time-invariant system with $l$ ports on the left side and $m$ ports on the right side. A coherent wave $\bm{a}$ input into $n$ left-side ports produces transmitted and reflected waves $\bm{b_t} = t\bm{a}$ and $\bm{b_r} = r\bm{a}$. (b) $r$ and $t$ are block submatrices of the entire $S$-matrix. (c) A silicon waveguide ($n_i=3.48$) embedded in silica ($n_0=1.444$). (d) Modified waveguide section with random lossy silica scatterers ($n_s=1.444+0.100i$). (e) Band dispersion of TE modes in the uniform waveguide with width $w=0.39~\mu\text{m}$, supporting two guided modes at wavelength $\lambda_0=1.55~\mu\text{m}$. (f) Schematic illustration of the set $\Omega$ containing all attainable tuples $(\tau[\bm{a}],\rho[\bm{a}],\alpha[\bm{a}])$ under varying input states $\bm{a}$.
• Figure 2: Attainable responses for a two-mode ($n=2$) disordered waveguide. (a) Ternary plot showing the set $\Omega$ of achievable $(\tau,\rho,\alpha)$ tuples, which forms an elliptic disk (purple boundary). Cyan stars mark the foci. Purple dots show numerical results from 30,000 random input states. Red, green, and blue lines indicate bounds on $\tau$, $\rho$, and $\alpha$ from Eq. (\ref{['eq:def-hexagon']}). (b-d) Projections of $\Omega$ onto the $(\tau,\rho)$, $(\tau,\alpha)$, and $(\rho,\alpha)$ planes coincide with the numerical ranges $W(T+iR)$, $W(T+iA)$, and $W(R+iA)$. Each projection is an elliptic disk with foci determined by the eigenvalues of the corresponding matrix (marked with distinct symbols).
• Figure 3: Attainable responses for a three-mode ($n=3$) disordered waveguide. (a) Ternary plot showing $\Omega$, which forms an ovular shape bounded by a smooth curve (purple). Purple dots show numerical results from random input states. Lines indicate bounds as in Fig. \ref{['fig:example-2modes']}(a). Purple cross indicates the assigned goal in Eq. (\ref{['eq:goal_tuple']}). (b-d) Projections of $\Omega$ onto the $(\tau,\rho)$, $(\tau,\alpha)$, and $(\rho,\alpha)$ planes coincide with the numerical ranges of $T+iR$, $T+iA$, and $R+iA$. Each projection is ovular and contains the three eigenvalues of the corresponding matrix (marked with distinct symbols).
• Figure 4: Possible shapes of $\Omega$ for $n=3$ beyond the ovular shape shown in Fig. \ref{['fig:example-3modes']}(a). (a) A triangle. (b) The convex hull of an ellipse and an external point. (c) A two-dimensional shape with one flat boundary segment. (d) An elliptic disk. (e) A line segment. (f) A point. The $T$ and $R$ matrices used to generate each panel are listed in Appendix \ref{['SI-subsec:T_R_A_matrices-Fig4']}.
• Figure 5: Non-abelian bounds on the numerical range $W(T+iA)$ (gray). The convex hull $C(T+iA)$ (pink) of eigenvalues (black dots) provides an inner bound, while the set $B(T+iA)$ (brown) gives an outer bound. Examples are shown for: (a) The $n=2$ case in Fig. \ref{['fig:example-2modes']}(c). (b) The $n=3$ case in Fig. \ref{['fig:example-3modes']}(c).

Theorems & Definitions (2)

• Theorem 1
• proof