Transmission zeros with topological symmetry in complex systems

Understanding vanishing transmission in Fano resonances in quantum systems and metamaterials and perfect and ultralow transmission in disordered media has advanced the knowledge and applications of wave interactions. Here, we use analytic theory and numerical simulations to understand and control the transmission and transmission time in complex systems by deforming a medium and adjusting the level of gain or loss. Unlike the zeros of the scattering matrix, the position and motion of the zeros of the determinant of the transmission matrix (TM) in the complex plane of frequency and field decay rate have robust topological properties. In systems without loss or gain, the transmission zeros appear either singly on the real axis or as conjugate pairs in the complex plane. As the structure is modified, two single zeros and a complex conjugate pair of zeros may interconvert when they meet at a square root singularity in the rate of change of the distance between the transmission zeros in the complex plane with sample deformation. The transmission time is the spectral derivative of the argument of the determinant of the TM. It is a sum over Lorentzian functions associated with the resonances of the medium, which is the density of states, and with the zeros of the TM. Transmission vanishes, and the transmission time diverges as zeros are brought near the real axis. Monitoring the transmission and transmission time when two zeros are close may open new possibilities for ultrasensitive detection.

Figure 1

Transmission zeros in a random quasi-one-dimensional (1D) sample. Simulations are carried out in a sample of width 20 and length 60 which supports five channels in the energy range of the simulation. (a) Intensity profile for the perfectly transmitting eigenchannel at $E=0.6845$ and the eigenchannel with vanishing transmission at $E=0.6873$ indicated, respectively, by the blue and red squares, in (b). The color bar is linear for the first pattern and logarithmic for the second pattern. (b) Spectra of the five transmission eigenvalues in the lossless system. (c) Transmission times for onsite loss $\gamma =0.002$. ${\tau }_p$ is calculated by integrating the imaginary part of the local Green's function ${\tau }_p=-\int \mathrm{Im}G(r,r,E)d\vec{r}$, which is the local density of states (DOS). ${\tau }_z={\tau }_{\mathrm{T}}-{\tau }_p$ is given by the black curve. The inset shows that ${\tau }_p$ and ${\tau }_{\mathrm{T}}$ coincide, with ${\tau }_z=0$ in the lossless system. (d) Lossy system with $\gamma =0.0053$ in which one transmission zero lies on the real axis at $E=0.666$ with ${\tau }_N=0$. (e) Displacement of three transmission zeros in the complex plane for the losses in (b)–(d). (f) Phase map of $argdet\left(t\right)$ in the complex energy plane. Red (blue) squares indicate transmission zeros (poles).

Figure 2

Interconversion of single and paired transmission zeros in a lossy billiard. (a) Profiles of energy flow and field amplitude at the T-zero indicated by the star in (b). The length of the arrows indicates the logarithm of the flux. The amplitude of the field is given by the color bar on the right. The permittivity in the system is $\varepsilon =1-5\times {10}^{-4}i$. The sample width and length are 10 and 20 cm. The leads have a width of 1 cm. The diameter of the disk is 4 cm. (b) Trajectories of two transmission zeros with $x$ coordinates of the center of the lowest disk at 1.996, 1.998, 2, 2.002, 2.0024, 2.0028, 2.003, 2.005, and 2.008 (cm). The two zeros meet at zero point (ZP) when $x=2.003\mathrm{cm}$. The star indicates the transmissionless mode in the lossy system. (c) Phase diagram of transmission between 18.21 and 18.22 GHz when the lowest disk is at 2.000 cm (left), at 2.003 cm (middle, ZP), and 2.0033 cm (right). Red (blue) squares indicate transmission zeros (poles).

Figure 3

Sensitivity of near degenerate transmission zeros in lossless billiard. (a) Left panel shows the trajectory of two T-zeros relative to the coordinate of the lowest disk. Right panel shows the variation of the frequency of the zeros with displacement of the disk. The green circles and the curve drawn through them give the real frequencies of the pair of T-zeros which meet at $x=2.003\mathrm{cm}$. The trajectories for the two real zeros after they are created at $x\sim 2.003\mathrm{cm}$ are shown in the red and blue curves. The red (blue) curve in the inset of the right panel shows the transmission spectrum when the $x$ coordinate of the lowest disk is 2.003 (2.004) cm. (b) The trajectories of two transmission zeros with radius $r$ of the lowest disk with center at $x=2\mathrm{cm}$. Two curves meet at $r=1.9872\mathrm{cm}$. The inset shows transmission spectra for a radius of the disk of 1.9872 cm (red) and 1.9850 cm (blue).