6.2 Some Characteristic Features of MFE
6.2.5 Field Solutions
Both figures 6.4and 6.5are based on the same Green’s function equations:
∇2ϕ(φ, t)− 1
c2 ∂t
2ϕ(φ, t) =−δ(φ)δ(t) (6.11)
After applying a Fourier transformation on the equation and choosing a proper unit (c= 1) we have:
(∂ν2+ν2)ϕ(φ, ν) =−δ(φ) (6.12) The general solution has the following form:
G(±)(φ, τ) =
Z +∞ −∞
G(±)(φ, ν)e−ıνtdν (6.13)
Chapter 6. The Spectrum of The Maxwell Fisheye Medium 99
To adopt the analogy with time reversal imaging, we can carefully choose the solution. The boundary conditions on source and drain suggest choosing the retarded solution for the source and the advanced solution for the drain [64]. Choosing the retarded solution provides the casualty condition discussed in [60]:
G(+)(φ, τ) =
Z t0
−∞
G(+)(φ, ν)e−ıνtdν= 0 (6.15)
For the drain we choose the advanced solution as we demand the extraction of the field at the drain.
G(−)(φ, τ) =
Z +∞ t0
G(−)(φ, ν)e−ıνtdν = 0 (6.16)
Boundary conditions on the source point, φs = 0, and on the drain point, φd=π, lead
to:
Gs(φ, t) =A(+)cos (ν(π− |φ|))e−ıνt (6.17)
and
Gd(φ, t) =A(−)cos (ν(π− |φ−π|))e−ıνt (6.18)
A drain (represented by the advanced solution) is a time reversal source that swallows the field. A perfect point drain swallows all the energy from the system, while an imperfect drain mainly absorbs but partially re-emits (reflects) the energy back into the lens. It is reasonable to assume the reflection is negligible except for resonance frequencies. The total solution for the fields in a single source-single drain MFE, aligned in antipodal position, would be the summation over both solutions:
ψtotal(φ, t) = Z G(+)s (φ, t;φ0, t0)Js(φ0, t0)dφ0 dt0+ Z G(d−)(φ, t;φ00, t00)Jd(φ00, t00)dφ00dt00 (6.19) By substituting G(+)s and G( −)
d from equations 6.17 and 6.18, and the corresponding
currents from Equation 6.9, we can calculate the total field in the MFE lens as well as the currents in the cables. Methodologically, assuming a large value forzsand therefore
a negligible reaction of the source port to the drain field in off-resonance frequencies, we can calculate the field by recursive substitution. The final results simulated by Mathematica (Figure 6.4a) shows that the presence of source and drain together will change the boundary conditions. At resonance frequency, despite the exceeding high values of the of individual elements of he field (source field, drain field, and secondary passive stimulation of the source), the total field remains finite and a has a smooth flow from source to the drain (Figure 6.4a).
Chapter 6. The Spectrum of The Maxwell Fisheye Medium 100
(a) At resonance frequency: Despite
the exceedance by individual elements of the field, the total field remains fi- nite and a has a smooth flow from
source toward drain.
(b)Off resonance frequency: Secondary ex- citation of the source (orange graph) has a negligible effect, as the field is not exceeding
in off resonance frequencies.
Figure 6.4: A cross section of the MFE, including the source and an imperfect passive
drain port. Red, purple and orange show the source field, drain field, and the passive stimulation of the source port respectively. Blue is the overall propagating field in the
system.
6.3
1-Dimensional MFE; Matrix Model
In this section we focus on the injection and extraction of the field through the source and drain ports. We use the physics of the 1-dimensional MFE as described in section 6.1 but illustrate each port as a vertex and apply a matrix approach to calculate the transmission and reflection of the fields. We will see that assuming the interaction between source and drain can describe Mi˜nano’s observations of the sub-wavelength focusing of the field. Let us consider the simplest case of perfect imaging, the 1-D sphere illustrate as a circle inFigure6.5. Imagine that light is confined to a circle, say a fibre loop or ring resonator. Here light can go in only two directions, right or left. An “image” is formed when the two rays meeting have the same phase, which happens when both are antipodal. Light is coupled in and out of the circle by two 1-D channels that represent the source and the detector. These 1-D channels are idealisations of the cables used for injecting and extracting radiation in the simulation [1] and experiment [2]. Clearly, this 1-dimensional system represents a rather primitive model, but it does reproduce the findings of the experiment [2]: the model is simple, but not too simple.
We show how a point detector is able to sense minute displacements of a point source. Note that this is not imaging in the traditional sense of taking the image of a source
Chapter 6. The Spectrum of The Maxwell Fisheye Medium 101
Figure 6.5: 1-dimensional model of a perfect-imaging device. The device is repre-
sented by a circle where light can propagate clockwise and counter-clockwise. Light is coupled in and out through one-dimensional channels that represent the source with in-coming amplitudea0= 1 and the detector witha1 = 0. A wave with amplitudea00
is reflected back to the source and a wave with amplitudea01 is detected. We calculate the transmission|a01|2as a function of the wavenumber and the misalignmentδbetween
the actual detector position and the image of the source.
distribution all at once, but rather corresponds to scanning, as follows. Suppose that the detector is moved across the imaging region of the device. The detector would only produce a signal when it is close to the imaging point of the point source. The resolution is the distance from the actual imaging point where the detector begins to fire. We will show that, for light at the resonance frequency of the instrument, the resolution of the scan is infinitely fine. Well-known and widely-used examples of scanning methods that beat the diffraction limit are near-field scanning optical microscopy [96] and fluorescence microscopy [14]. In contrast to the former, here the detector is placed far away from the source; and in contrast to the latter, only linear optics are used. The idea of Mi˜nanoet al. [1] of turning an absolute optical instrument into a super resolving scanning device may thus find a place in the arsenal of methods for breaking the diffraction limit. Here we explain how it works.
Chapter 6. The Spectrum of The Maxwell Fisheye Medium 102