Reflection of waves from slowly decaying complex permittivity profiles

Wave propagation through rapidly but continuously varying media is surprisingly subtle, and in a pair of recent papers [Horsley et al., J. Opt. 18, 044016 (2016); Longhi, Eur. Phys. Lett. 112, 64001 (2015)] it was found that planar media with a spatially varying permittivity $\epsilon \left(x\right)$ obeying the spatial Kramers-Kronig relations do not reflect waves incident from one side, however rapid the changes in $\epsilon \left(x\right)$. Within this large class of media there are some examples where the dissipation or gain is not asymptotically negligible and it has been pointed out [Longhi, Eur. Phys. Lett. 112, 64001 (2015)] that it is impossible to define meaningful reflection and transmission coefficients in such cases. Here we show—using an exactly soluble example—that despite the lack of any meaningful reflection and transmission coefficients, these profiles are still reflectionless from one side in the sense that the profile generates no counterpropagating wave for incidence from one side. This finding is demonstrated through examining the propagation of pulses through the profile, where from one side we find that no second reflected pulse is generated, while from the other there is. We conclude with a discussion of the effect of truncating these infinitely extended profiles, illustrating how the reflectionless behavior may be retained over a wide range of incident angles.

Figure 1

Behavior of the integrand of $w\left(z\right)$ [see (7)] as a function of the dimensionless variable $t$. The brightness is proportional to the magnitude of the function, and the color indicates phase (advancing from zero to $\pi$ in the succession, red, yellow, green, and light blue). The integrand has a branch cut between the points $t=0$ and $t=-1$, shown as a red dashed line, and points of stationary phase $t=t_{\pm }$ (9) are indicated by red dots. The dashed arrows in (a) and (b) show the motion of the end points of the contours $C_1$ and $C_2$ as the position $x$ is increased or decreased, respectively. Comparing (a) and (b) we see that as $x$ is increased (units $\lambda =2\pi /k_0$) from negative to positive, the end points of the two contours move counterclockwise from the lower to upper half of the $t$ plane. This leaves the contour $C_1$ (waves incident from the left) still passing through $t_{+}$ only, while $C_2$ (waves incident from the right) becomes tangled around the branch cut, leading to contributions from both $t_{-}$ and $t_{+}$ (inset showing how this tangled contour can be deformed into the succession of contours $C_1, C_2^{\prime }$, and $C_1^{\prime }$, with the numbers at the top indicating the Riemann sheet). When $x_0<0$, the sense of rotation of the contours is reversed.

Figure 2

(a) Shows the two exact solutions $E_{z1}$ and $E_{z2}$ of the Helmholtz Eq. (4) in the permittivity profile (2), numerically computed from (5) and the contour integral representation (7) with the two contours shown in Fig. 1 (the two plots are offset vertically for the purposes of visualization, and the units on the $x$ axis are $\lambda =2\pi /k_0$). These two forms for the electric field correspond to waves incident from the left and right of the profile, respectively. The choices of parameters defining the profile are $k_{0}A=1.5-0.2i$ and $k_0{x}_0=0.15$, and the waves are at normal incidence $k_y=0$. Blue solid and green dashed lines are the real and imaginary parts of $E_z\left(x\right)$ and the thinner red line is the absolute value. This particular profile is an example where the wave amplitude is not asymptotically constant, and the two smaller middle panels show the behavior of the waves over a larger scale where a slow change in the overall wave amplitude is evident. (b) Shows the normalized Poynting vector $\langle S_x\rangle \left(x\right)/\left|\langle S_{x,\max }\rangle \right|$ calculated from $\langle S_x\rangle =-(1/\left(2{\mu }_0\right))\mathrm{Re}\left[E_z^{☆}{B}_y\right]$, normalized such that the maximum value in the plot is $\pm 1$. The upper blue line is calculated for the wave incident from the left ($E_{z1}$), whereas the lower thinner green line is for the wave incident from the right ($E_{z2}$). The lack of any rapid oscillations in the former case indicates a lack of reflection [c.f. Eq. (23)], whereas the rapid oscillations in the latter indicates reflection [c.f. Eq. (25)].

Figure 3

A Gaussian pulse is launched with central frequency ${\mathrm{\Omega }}_0=c{K}_0=2\pi c/\lambda$ generated by a current localized at positions $x=-10\lambda$ (a)–(c), and $x=10\lambda$ (d)–(f) with a bandwidth of $\mathrm{\Delta }\mathrm{\Omega }=0.3{\mathrm{\Omega }}_0$. The electric field (thicker blue line) and Poynting vector (thinner red line) are given in units of $E_0={\mu }_{0}c{\mathcal{J}}_0{\mathrm{\Omega }}_0$ and $S_0=E_0^2/\left({\mu }_{0}c\right)$, respectively. The vertical dashed lines indicate the region over which the permittivity is plotted in the inset of (d) (real part, blue solid and imaginary part green with circles). It is imagined that over this bandwidth the permittivity is given by (2) for the parameters $K_{0}A=0.7+0.2i$ and $K_0{x}_0=0.1$ (a profile that contains regions of gain). It is useful to list these parameters for a realistic case. For example, for a central frequency in the microwave regime ${\mathrm{\Omega }}_0={10}^{10}\mathrm{rad}{\mathrm{s}}^{-1}$ we have $\lambda =0.18\mathrm{m}, \mathrm{\Delta }\mathrm{\Omega }=3\times {10}^9\mathrm{rad}{\mathrm{s}}^{-1}, A/x_0=7+2i$ and $x_0=0.003\mathrm{m}$, which illustrates that these large index variations on the scale of millimeters will not reflect centimeter scale waves. Notice that the reflectionless behavior is clearly identified: (a)–(c) For a source on the left there is no reflected pulse, while (d)–(f) for a source on the right there is.

Figure 4

As in Fig. 3, but for a narrower pulse bandwidth of $\mathrm{\Delta }\mathrm{\Omega }=0.1{\mathrm{\Omega }}_0$, demonstrating that the nonreflecting behavior from the left is not dependent on the pulse width.

Figure 5

As in Fig. 3, but for sources located at $x=-30\lambda$ (a) and $x=+30\lambda$ (b). The slow decay to zero of the imaginary part of $\epsilon \left(x\right)$ causes the amplitude of the pulse to slowly grow or diminish as it propagates towards the central region of the profile (numbers within the plots indicate different times), but the qualitative character of the nonreflecting behavior from the left is not affected.

Figure 6

A pulse is launched at $x=-20\lambda$ (a) and propagates through the permittivity profile (31) (real part blue solid, imaginary part green with circles), for the values $K_{0}A=1.2-0.5i, K_0{x}_0=0.1, L=1.6\lambda$, and $\mathrm{\Delta }x=0.39\lambda$, truncated to a value of 1 at the vertical dashed lines. The inset of (a) shows the permittivity profile within the dashed lines, along with the Gaussian envelope function (black solid line). (b) Shows the pulse shape after propagation through the truncated profile, with only very slight reflection evident (inset plot is a 300 times magnification of the reflected pulse, plotted with the same vertical range as the main panel).