Non-Hermitian systems are open systems that can exchange energy, matter, or information with their surroundings through the engineering of gain or loss materials. While quantum mechanics traditionally requires Hamiltonians to be Hermitian to produce real eigenvalues, it's been discovered that a system with a spatially varying complex potential energy can still have real eigenvalues if it obeys a certain symmetry known as Parity-Time (PT) symmetry.

One concept that has gained particular attention in non-Hermitian photonics is the phenomenon of "exceptional points." These are points in parameter space at which the eigenvalues and the eigenvectors of a non-Hermitian system coincide. Around these exceptional points, systems can have unique and often counterintuitive properties. For instance, light in these systems can exhibit uni-directional propagation, enhanced sensitivity to external parameters, or unusual scattering behavior. Non-Hermitian photonics has potential applications in many areas, including the design of new types of lasers, sensors, and photonic devices with novel properties.

This notebook demonstrates the unidirectional surface plasmon polariton (SPP) propagation from non-Hermitian metagratings. The design is adapted from Yihao Xu et al. ,Subwavelength control of light transport at the exceptional point by non-Hermitian metagratings. Sci. Adv. 9, eadf3510(2023).

For more interesting contents in the realm of nanophotonics, please check out the case studies on Anderson localization, hyperbolic polaritons, and plasmonic nanoantennas. For more simulation examples, please visit our examples page. If you are new to the finite-difference time-domain (FDTD) method, we highly recommend going through our FDTD101 tutorials. FDTD simulations can diverge due to various reasons. If you run into any simulation divergence issues, please follow the steps outlined in our troubleshooting guide to resolve it.

Simulation Setup¶

Define the basic model parameters including the simulation wavelength/frequency and the metagrating geometric parameters.

The metagrating utilizes four materials: a silicon dioxide substrate, a gold thin film, germanium nanostrips, and chromium nanostrips. These materials can be defined in different ways. Specifically, we use the gold medium directly from the material library. The germanium is defined using the from_nk method by specifying the real and imaginary parts of the refractive index at the simulation frequency. Silicon dioxide is considered as a lossless medium. Chromium is modeled as a dispersive medium by fitting the raw data from experimental measurements.

In this simulation, we are only concerned about one frequency, so in principle the dispersive fitting is not necessary. However, chromium has $Re(\varepsilon)<1$ at 1150 nm. In Tidy3D, we require the nondispersive medium to have $Re(\varepsilon)\ge 1$. Therefore, it can not be defined using the from_nk method directly and dispersive fitting is the only option.

The raw data file Cr_Sytchkova.csv can be found on the refractive index database or on our github repo.

After the fitting, we see a warning that the fitting RMS error didn't get lower than the threshold. We can inspect the fitting result of chromium to see if the fitting is sufficiently good. Here we do see the fitting is quite reasonable.

Next, we define the metagrating structures. The N unit cells can be defined by using a simple for loop.

For excitation, we define a GaussianBeam incident on the metagrating. The waist_radius is set to five grating periods to ensure the entire grating array is covered by the beam.

To visualize the SPP propagation, a FieldMonitor is defined. Two FluxMonitors are defined to measure the power transmissions of the SPP to the left and right.

Define the Tidy3D Simulation. The simulation is 2D so periodic boundary condition is applied to the direction with zero size (the $y$ direction in this case).

Visualize the simulation setup.

Since the grating structures are small compared to the simulation domain, we can zoom in to see them more clearly.

The simulation settings look correct. We are ready to submit the simulation to the server.

Result Analysis¶

Plot $|H_y|^2$ to visualize the SPP propagation. We can see the SPP is only propagating to the right, i.e. unidirectional propagation.

To quantify the directionality, we define a contrast parameter $C_{exc} = \frac{I_r-I_l}{I_r+I_l}$, where $I_r$ and $I_l$ are the transmitted power to the right and to the left, respectively. $C_{exc}=0$ corresponds to an isotropic propagation. For the ideal unidirectional propagation, $C_{exc}=1$ or $C_{exc}=-1$. In this case, we observe $C_{exc}>0.98$.

Sweeping Nanostrip Spacing¶

The SPP propagating can be effectively tuned by changing the center-to-center distance between the nanostrips within the grating unit cell. We would like to perform a parameter sweep of the center-to-center distance $s$ to investigate how the unidirectional SPP propagation changes. To do so, we first define a function make_sim that returns a simulation given a $s$ value.

The range of interest for $s$ is from 350 nm to 770 nm. A batch of 5 simulations is created.

Run the simulations in the batch in parallel.

Plot $|H_y|^2$ for each simulation. We can see that the SPP propagates to the right for smaller $s$ and to the left for larger $s$.

$C_{exc}$ is also computed for each simulation.

Plot $C_{exc}$ as a function of $s$. $C_{exc}$ can be tuned from >0.98 to <-0.98 by changing $s$. Compared to the original paper, the result here is slightly different, which we attribute to slightly different material properties used.