Transmission-type photonic doping for high-efficiency epsilon-near-zero supercoupling

Supercoupling effect is an exotic and counterintuitive physical phenomenon of epsilon-near-zero (ENZ) media, in which the light can be “squeezed” and tunneled through flexible channels substantially narrower than its wavelength. Theoretically, ENZ channels with infinitely small widths perform ideal supercoupling with full energy transmission and zero-phase advance. As a feasible solution to demonstrate ENZ supercoupling through a finite-width channel, photonic doping can assist the light in squeezing, but the resonant dopant introduces inevitable losses. Here, we propose an approach of transmission-type photonic doping to achieve proximate ideal ENZ supercoupling. In contrast to the conventional resonance-type photonic doping, our proposed transmission-type doping replaces high-quality-factor two-dimensional resonant doping modes with low-quality-factor one-dimensional modes, such that obviously high transmission efficiency and zero-phase advance in ENZ supercoupling is achieved and observed in experiments. Benefiting from the high-efficiency ENZ supercoupling, waveguides with near-total energy transmission can be engineered with arbitrary dimensions and shapes, serving as flexible power conduits in the paradigm of waveguide integrated circuits for future millimeter-wave and terahertz integrated circuit innovations.


(
) ( ) Where c is the speed of light in vacuum.From Eq. (S1), we get Where λ is the wavelength of the PMC frequency in a vacuum, and λ d is the wavelength in the dopant.In other words, to accommodate a dopant with a height of h d , we need an ENZ channel with a thickness h at least h d .For a better understanding of this issue, researchers have demonstrated that the magnetic field distribution in 2D resonance-type doping follows a cylindrical or quasi-cylindrical mode, as shown in previous studies 1,2 .To accommodate this quasicylindrical mode in the resonance-type doping, the dopant needs to have a length of at least half a wavelength in both the horizontal and vertical directions.Furthermore, based on the image principle 3 , the minimum height should be greater than λ d / 4. With the given parameters (ε d = 37 and ω p = 2π×3×10 9 rad/s) in the main text, we have calculated that the minimum width of the ENZ channel is 4.11 mm, approaching the maximum thickness that can be achieved with the current SIW manufacturing technique.Resonance-type doping cannot meet the requirements of the ultranarrow thickness and profiles of substrates, which limits its practical application in the field of microwaveintegrated devices.
Moreover, in the simulation of lossy ENZ media, the transmission efficiency of resonance-type doping demonstrates a strong dependence on the remaining space between the top of the dopant and the PEC boundary.This is mainly because of the two singular points in the rectangular dopant of resonance-type doping 2 .Here we analyze this dependence through simulation, whose geometry is shown in the Fig. S1a.We set the parameters as h p = 30 mm, h = 6 mm, 2l = 80 mm, and ε d = 37 + 2×10 -5 i.We adjust the size of dopant flexibly to enable the EMNZ frequency ω p = 2π×3×10 9 rad/s.To emulate the plasmonic implementation for the general ENZ media, we set the relative permittivity of ENZ host satisfies the Durde-model: , where ω = 2πf is the angular frequency in radians per second 4 .Simulated results are shown in Fig. S1b.As can be observe, the ranking of the transmission amplitude from high to low for several doping scenarios is as follows: transmission-type doping, resonance-type doping, and the undoped general case.In other words, our transmission-type doping is currently the most efficient way to achieve ENZ supercoupling, as demonstrated by the high-efficiency ENZ supercoupling presented in the main text.For resonance-type doping with different remaining width w, the transmission amplitudes exhibit different behaviors.A larger remaining width has better transmission efficiency.This is because a too narrow remaining width may cause unnecessary reflection.As discussed earlier, the remaining width is space-constrained in resonance-type doping of SIW integration.Therefore, the resonance-type doping of ENZ supercoupling is unable to meet the requirement for long-distance transmission due to this space limitation.
The discussion in this section provides supplementary results for our transmission-type photonic doping approach in achieving high-efficiency ENZ supercoupling.

Supplementary Note 2. Conventional analysis of transmission-type photonic doping
In this section, we attempt to analyze transmission-type doping using classical photonic doping theory to illustrate the limitations of conventional theory in addressing this issue 1 .We notice that the magnetic field distribution in the width direction in Fig. 2c of the main text is uniform, thus only the field in the length direction satisfies the passive scalar Helmholtz equation (SHE) where k 2 = ω 2 με.As the dielectric block is placed in the center, the magnetic field has an even symmetric distribution on both sides of the center.Therefore, there is only one SHE eigenvalue of the cosine function for the H z (x) component.
According to the theory of conventional photonic doping, we obtain the normalized magnetic field ψ z (x) = cos(kx)/cos(kd) in the dopant which satisfies the boundary conditions of the ENZ host.Here With the same variable definitions as those in the main text, we generate Fig. S2a to display the calculated results of effective permeability and transmission amplitude, along with the corresponding simulated result.We notice that the predicted PMC mode in the conventional theory is prohibited in the simulated result.We provide an explanation from the perspective of boundary condition analysis.Here we replot the dopant area of transmission-type doping in Fig. S2b and label the four vertices of the rectangular dopant with A, B, C, and D. At the PMC scenario, the tangent components of the magnetic field on boundary A-B-C-D are exactly zero.However, A-B and C-D are also in contact with the PEC boundaries, causing the tangent components of the electric field on A-B and C-D to also vanish.
According to the Uniqueness Theorem 3 , only the mode with zero electric fields or zero magnetic fields can be supported within the dopant.Because the mode is meaningless, the PEC effect cannot appear.The discussion in this section demonstrates that the conventional resonance-type doping theory cannot be used to describe the new approach of transmission-type doping.
From the transmission coefficient, we can get the amplitude and angle as which the length 2d of the dopant satisfies Eq. ( 6) in the main text.The phase Arg(T(ω)) = 0 at this moment, heralding that we have got the EMNZ state.
For the general case, we accurately determine the difference between our approximation and the actual situations through numerical calculations.Our findings reveal that our approximations are highly accurate.Here we prove this through two calculated results which has shown in Fig. S3.In Fig. S3a, we calculate the transmission amplitude at different dopant lengths to determine the dopant lengths corresponding to the occurrence of the EMNZ states.By normalizing the length d using the Eq. ( 6) in the main text, we obtain a series of multiplier factors m which corresponding to the EMNZ states.The specific values are marked at above the figure.These m values are very close to the integer values as our theoretical prediction, indicating that the approximation is feasible under the simulation conditions.In order to analyze the shift of the EMNZ frequency due to this approximation, we calculate the normalized EMNZ frequency changing with the length of ENZ channel 2l.As we can see, with the gradual increasing in the length channel length 2l, the EMNZ frequency first undergoes a blue shift and then a red shift.The maximum frequency shift occurs at l = 0.85λ, with a shift of 0.08ω p .This error is completely acceptable in engineering terms.In other words, integer multiples of half-wavelength modes are formed within the dopant at EMNZ frequencies.Since the dopant serves as a crucial path for EM waves in ENZ supercoupling, it introduces an additional phase shift of 180° or 360°.
In the main text, we have calibrated this phase to ensure consistency with concepts presented in previous work 1,2 .In a short summary, in this section, we show the basis for the approximate estimate of Eq. ( 6) in the main text and illustrate the accuracy of our approximate estimate through precise calculations.
of (π/2d) 2 > 0 for finite d, we can calculate the range of h as

Supplementary Figure 2 |
Conventional analysis of transmission-type photonic doping.(a) Calculated results for both effective relative permeability and transmission amplitude of transmission-type doping using traditional photonic doping theory.A simulated transmission amplitude is demonstrated for comparison.(b) Schematic for boundary conditions analysis for transmission-type doping.
permittivity of the dielectric dopant.According to the general photonic doping formula equation,