Description
A temporal boundary occurs when a material’s property changes abruptly everywhere at the same time.¹ Applications include frequency conversion and nonreciprocal devices.²˒³ When a wave encounters a temporal boundary at which the refractive index jumps from $n_1$ to $n_2$, the incident wave splits into a forward and a backward wave at unchanged $k$, but at a new frequency $\omega_2=(n_1/n_2)\omega_1$.⁴ At a temporal boundary the spatial pulse width $w$ is unchanged but its temporal duration $wn/c$ changes by $n_2/n_1$.
Depending on the applied boundary conditions there are two sets of Fresnel coefficients (FCs).⁴˒⁵ One set, $t_{DB}=\frac{n_1(n_1+n_2)}{2n_2^2}$, $r_{DB}=\frac{n_1(n_1-n_2)}{2n_2^2}$ uses continuity of $\mathbf{D}$ and $\mathbf{B}$ and is rigorously derived assuming no surface charges. The set $t_{EH}=\frac{n_1+n_2}{2n_2}$ and $r_{EH}=\frac{n_1-n_2}{2n_2}$ follow from continuity of $\mathbf{E}$ and $\mathbf{H}$.
Which FCs are correct? Though we can directly determine the ratio of the electric field amplitudes for a long initial pulse, this is not very accurate and requires separation of the incoming and reflected fields. In FDTD calculations the results agree best with the DB FCs. Instead, we take the spatial Fourier transforms of the electric field before and after the time boundary, which allows unambiguous separation of the propagation directions. The results are consistent with the DB FCs. When instead we take a temporal Fourier transform the results match $t_{EH}, r_{EH}$ due to the change in temporal pulse length. Since the spatial parameters are unchanged across the boundary, we need to take the spatial Fourier transform.
References
[1] Morgenthaler, F. R. (1958). IEEE Transactions Microwave Theory and Techniques 6, 167.
[2] Sharabi, Y., et al. (2021). Physical Review Letters 126, 163902.
[3] Lustig, E., et al. (2018). Optica 5, 1390.
[4] Xiao, Y., et al. (2014). Optics Letters 39, 574.
[5] Mendonça, J. T., et al. (2002). Physica Scripta 65, 160.
| I am the presenting author | Yes |
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