Comment on: Reply to comment on `Perfect imaging without negative refraction'

Whether or not perfect imaging is obtained in the mirrored version of Maxwell's fisheye lens is debated in the comment/reply sequence [Blaikie-2010njp, Leonhardt-2010njp] discussing Leonhardt's original paper [Leonhardt-2009njp]. Here we show that causal solutions can be obtained without the need for an"active localized drain", contrary to the claims in [Leonhardt-2010njp].

Whether or not perfect imaging is obtained in the mirrored version of Maxwell's fisheye lens is debated in the comment/reply sequence [1, 2] discussing Leonhardt's original paper [3]. Here we show that causal solutions can be obtained without the need for an "active localized drain", contrary to the claims in [2]. R.J. Blaikie (RJB) notes in [1] that that Ulf Leonhardt's (UL) setup in [3] incorporates an "active localized drain" at the image point. It is this drain, modeled as a phase-delayed mirror image of the source, which provides the sub-wavelength detail of the source's image. RJB showed that a steady-state numerical simulation of the fields without the drain did not show the perfect superresolution image of the source. In response, UL noted [2] that steady-state solutions neglect causality and that inclusion of a sink solves the problem of energy buildup.
We now address both of UL's concerns about RJB's simulations using a time-domain numerical solution with a source only active for a finite time. In figs. 1 & 2 the results from such a simulation, calculated using the open source MEEP [4] implementation of FDTD [5], are shown. In FDTD, the real valued electric and magnetic fields are defined over space, and an algorithm is applied which propagates these fields, step by step, forwards in time. It is thus explicitly causal, and entirely independent of any decomposition into plane waves or modes.
x y source FIG. 1: Snapshot of the electric field Ez from a simulation of the cylindrical mirrored fisheye, as the light first reaches the image point. Parameters are taken from RJB but with zero losses: n(r) = 2/(1 + (r/r0) 2 ) for r0 = 10µm, f = 100THz (i.e. λ0 = 3µm).
The point-like source used is independent of all other properties of the simulation [12]. We follow UL and RJB and use a frequency independent refractive index profile, so there is no temporal dispersion. Our simulation therefore correctly tests for the geometric response and achievable spatial resolution, irrespective of the fact that transients are used instead of steady-states. Fig. 2 shows  fig. 1 showing the field Ez along the x-axis. The intensity FWHM of the source and image(s) are given, and can be compared to either λ0 or the local wavelength (λ0/n ≃ 2.3µm) at the image point. Further bounces gradually degrade the image quality, and more so for briefer source durations.
that the image is not as sharp as the source, and matches the steady-state results of RJB [1].
We do not include an active drain in our simulations because we believe it unlikely to form a part of any actual device; e.g. if this mirrored fisheye replaced the elliptical cavity used in lamp-pumped lasers. Active drains often appear in the literature when systems with sources are designed using folded-space or mirror-imaged transformations: e.g. the transformation optics slab-lens [6]. However some authors insist that such active drains are unphysical and/or mathematically ambiguous [7-10]. RJB was able to replace the drain in his steady-state simulation with a carefully-phased source [1], but this will fail in general for both time domain simulations and physical devices [11]. Whatever the method, we consider achieving super-resolution by such means to be of little utility, since it requires a customised element to enhance and 'image' the field at each precisely tuned pixel.
In summary, despite the claims in [2], causality does not require the presence of an active drain, irrespective of whether or not an active drain might be otherwise useful.