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Transfer ABCD Matrix for Time-Varying Media and Time Crystals

Carlos Molero, Pablo H. Zapata-Cano, Antonio Alex-Amor

Abstract

This paper introduces a formal definition of the transfer ABCD parameters in time-varying electromagnetic systems. The formal definition comes after the rearrangement of the fields $D$ and $B$ at the inputs and outputs of the temporal system based on the time-varying boundary conditions. Then, we derive the ABCD parameters of a temporal transmission line, i.e., a temporal slab, and compute the associated scattering parameters (reflection and transmission coefficients). The results presented here open up an alternative way, based on network theory, to analyze multilayer temporal configurations. Moreover, we show that the ABCD parameters can be used to compute the dispersion diagram ($ω$ vs $k$) of time crystals.

Transfer ABCD Matrix for Time-Varying Media and Time Crystals

Abstract

This paper introduces a formal definition of the transfer ABCD parameters in time-varying electromagnetic systems. The formal definition comes after the rearrangement of the fields and at the inputs and outputs of the temporal system based on the time-varying boundary conditions. Then, we derive the ABCD parameters of a temporal transmission line, i.e., a temporal slab, and compute the associated scattering parameters (reflection and transmission coefficients). The results presented here open up an alternative way, based on network theory, to analyze multilayer temporal configurations. Moreover, we show that the ABCD parameters can be used to compute the dispersion diagram ( vs ) of time crystals.

Paper Structure

This paper contains 10 sections, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Temporal Transfer ABCD Matrix
  3. Temporal Transmission Line (Temporal Slab)
  4. Associated Scattering Parameters
  5. Considerations about Conservation of Momentum Density
  6. Time Crystal: Infinite Temporal Transmission Lines
  7. Results
  8. Temporal multilayer slab
  9. Time Crystal
  10. Conclusion

Figures (3)

  • Figure 1: Use of transfer ABCD parameters in multilayer spatial and temporal interfaces. In both scenarios, ABCD matrices can be used to connect the EM fields at the input and output of the system. By exploiting the continuity of tangential {$E_t$,$H_t$} (total {$D$, $B$}) fields in the spatial (temporal) problems, the fields can be rearranged such that the ABCD matrix of each layer can be multiplied to obtain the global response of the time-invariant (time-varying) system. The multilayer composite is formed by $N+2$ layers: the input layer [$(i)$, $n=0$], $N$ intermediate layers, and the output layer [$(o)$, $n=N+1$].
  • Figure 2: Scattering parameters of an equally travel-distance multi-layered temporal structure with identical input and output media. $N=4$, $Z_1=Z_0/9$, $Z_2=Z_0/3$, and $Z^{(i)} = Z^{(o)} = Z_0$. Frequencies are normalized to the central frequency $f_0$.
  • Figure 3: Dispersion diagram, computed with the ABCD formalism (white dotted lines) and the exact numerical solution (colored plot), for a time crystal given by the temporal modulation $\varepsilon_r(t) = \varepsilon_{r0} \left[1 + m \cos(\Omega t) \right]$. (a) Discretization of the continuous temporal modulation (blue line) into $N$ steps (black lines). Dispersion diagram in the cases: (b) $N = 5$, (c) $N = 30$. Parameters: $\varepsilon_{r0} = 1$, $T = 0.1$ ns, $m = 0.7$, and $\mu_r = 1$.