The searchlight effect in hyperbolic materials

Graeme W. Milton; Ross C. McPhedran; Ari Sihvola

doi:10.1364/OE.21.014926

1. Introduction

Interest in hyperbolic materials, in which the dielectric tensor is real but with its eigenvalues taking different signs, has surged following the discovery of superlensing. The story of super-lenses itself had its genesis in three pivotal papers.

The first by Veselago [1] suggested that a slab of dielectric constant ε = −1 and magnetic permeability μ = −1 could act as a lens. This is indicated by ray tracing, taking into account the negative refractive index of the slab.

The second by Nicorovici, McPhedran, and Milton [2] (see also [3]) showed that in two-dimensional quasistatics one could have apparent point singularities appear in the field surrounding a coated cylinder with coating having dielectric constant ε = −1 + iδ, in the limit δ → 0. Specifically, with r_s and r_c denoting the shell and core radii, and with a dipole source at x = z₀ outside the coated cylinder located in the annulus a²/r_s > z₀ > r_s where $a = r_{s}^{2} / r_{c}$ it was proved in that paper that the complex potential V(z) outside the coated cylinder, where z = x + iy converged as δ → 0 to the potential

{\tilde{V}}_{e} (z) = \frac{1}{z - z_{0}} - (\frac{1 - ε_{c}}{1 + ε_{c}}) \frac{a^{2}}{z_{0}^{2} (z - a^{2} / z_{0})},

for all z > a²/z₀, where ε_c is the dielectric constant of the core. [The physical potential is Re{[V(z) +V(z̄)]e^−iωt/2} where ω is the frequency and t is the time.] Thus an apparent point singularity appears at the point z = a²/z₀ which lies outside the coated cylinder. In the limit δ → 0, the shell acts to magnify the core by a factor of r_s/r_c so it has the same response as a cylinder of radius a and dielectric constant ε_c (becoming invisible if ε_c = 1) but now the ”image dipole” lies in the matrix. Within the radius a²/z₀ > |z| > r_s it was numerically found that the potential develops enormous oscillations. This blowing up of the field within a localized region dependent on the position of the source, now called localized anomalous resonance, may be physically regarded as a localized surface plasmon and is responsible for a type of invisibility cloaking [4, 5], that has been the subject of considerable study [6–15].

The third pivotal paper by Pendry [16], which served as a catalyst for the field, made the bold claim that the Veselago lens would be a superlens, capable of focussing much finer than the wavelength of the radiation. The appearance of apparent point singularities caused by localized anomalous resonance was later found to justify this claim for fixed amplitude point sources [3, 17–27] though a single polarizable dipole is cloaked rather than perfectly imaged when it is sufficiently close to the superlens [4, 14, 28], and a small dielectric inclusion is at least partially cloaked [6, 9, 29]. Despite all the work on this topic (the paper of Pendry has over 6,700 citations) it remains an open question as to whether large dielectric inclusions (which interact with the surface plasmons) are perfectly imaged when they are close to a superlens: work by Bruno and Lintner [6], would indicate they are not perfectly imaged (in the limit in which the loss in the lens goes zero) while work of [29] suggests that they may be perfectly imaged, though it is not clear if sufficiently small loss has been taken in this latter investigation.

It was suggested by Pendry and Ramakrishna [30] that a stack of layers of equal thicknesses alternating between ε = +1 and ε = −1, would have an effective dielectric constant of infinity perpendicular to the layers and zero parallel to the layers, thus channelling the field like a set of infinitely conducting wires in an insulating matrix with ε = 0. If the two constituent materials have different thicknesses and/or the dielectric constants with unequal magnitudes but opposite signs then the effective dielectric tensor can be a hyperbolic material (one needs to add a small loss to the material with negative dielectric constant to justify this, both physically and mathematically) and the dispersion relation in such materials allows for real wavevectors with arbitrarily large wavenumbers, thus allowing for propagation of waves with arbitrarily small wavelength [31]. The breakthrough came with the independent recognition of Jacob, Alekseyev and Narimanov [32] and Salandrino and Engheta [33] that a multicoated cylinder or sphere with many thin coatings with dielectric constants of alternating signs would correspond to a hyperbolic material with radial symmetry and be capable of magnifying an image from the subwavelength scale to a scale where conventional imaging would work: the hyperlens was born. This was subsequently verified experimentally [34, 35].

In this paper, our interest is in the two-dimensional quasistatic dielectric equation

\nabla \cdot ε \nabla V = 0, ε = [\begin{matrix} ε_{x} & 0 \\ 0 & ε_{y} \end{matrix}] .

In the hyperbolic medium we consider ε_x is real and ε_y = ε_x/c² with c = −iμ + η where η is a small positive parameter (μ and η are real constants). In this medium, as η goes to zero, (1.2) formally approaches the wave equation,

\frac{\partial^{2} V}{\partial y^{2}} = μ^{2} \frac{\partial^{2} V}{\partial x^{2}},

and thus in this limit one should expect wavelike solutions in the hyperbolic medium (think of y as the time and μ as the wave velocity). In this paper we study how field singularities arise in the limit η → 0 when there is a circular hole in a hyperbolic material, and we study how a pair of polarizable point dipoles interact in a hyperbolic medium when a uniform field is applied at infinity.

It is known that both temporal and spatial dispersion play important parts in the design of hyperbolic metamaterials in the near visible spectral regions [36]. However, the incorporation of temporal and spatial dispersion into our analysis would lead to significant complications, and might obscure the interesting physical effects we discuss on an in-principle basis. We stress also that our derivations do not assume properties of materials appropriate to any specific spectral region, and are thus quite general.

2. The field around a circular hole in an anisotropic lossless medium

Here we review the known solution for the field surrounding a circular hole in a two-dimensional anisotropic medium, when the dielectric constants of the medium are real and positive. This problem has been treated before by Yang and Chou [37] in the context of the equivalent problem of antiplane elasticity, but their solution is more general than we need and they gave the potential only as an integral. In the next section we will use analytic continuation to obtain the solution when the dielectric constants of the medium are complex. For this it is important to express the solution in cartesian coordinates rather than in stretched elliptical coordinates (which would be dependent on the dielectric constants of the medium and become unphysical when the dielectric constants of the medium become complex).

Consider the transformation

z + \frac{r^{2}}{z} = 2 w,

which maps the circle |z| = r in the z plane onto the slit −r ≤ w ≤ r on the real w axis, and maps a larger circle |z| = r₀, with r₀ > r to an ellipse E, which in the w = u + iv plane intersects the axes at

u = \pm (r_{0} + r^{2} / r_{0}) / 2, v = \pm (r_{0} - r^{2} / r_{0}) / 2 .

The transformation (2.4) can be inverted to express z and 1/z in terms of w:

z = w + \sqrt{w^{2} - r^{2}}, \frac{1}{z} = \frac{w - \sqrt{w^{2} - r^{2}}}{r^{2}} .

It is to be emphasized that the square root needs to be taken so that the branch cut is a straight line between w = r and w = −r: for computational purposes one can set

\sqrt{w^{2} - r^{2}} = (w + r) \sqrt{(w - r) / (w + r)},

where the square root on the right hand side is defined with the branch cut along the negative real axis. Let us suppose the material outside E is isotropic with real positive dielectric constant ε_x, and now let us make some observations. First consider the potential γw which at the surface z = r₀e^iθ equals

γ (z + r^{2} / z) / 2 = γ (r_{0} e^{i θ} + r^{2} e^{- i θ} / r_{0}) / 2,

and has complex conjugate

\frac{\bar{γ}}{2} (\frac{r_{0}^{2}}{r_{0} e^{i θ}} + \frac{r^{2} r_{0} e^{i θ}}{r_{0}^{2}}) = \bar{γ} t,

where γ̄ is the complex conjugate of γ and

\begin{array}{l} t & = & \frac{1}{2} (\frac{r_{0}^{2}}{z} + \frac{r^{2} z}{r_{0}^{2}}) \\ = & \frac{w}{2} (\frac{r^{2}}{r_{0}^{2}} + \frac{r_{0}^{2}}{r^{2}}) + \frac{\sqrt{w^{2} - r^{2}}}{2} (\frac{r^{2}}{r_{0}^{2}} - \frac{r_{0}^{2}}{r^{2}}) . \end{array}

Thus the potential Re(γw − γ̄t) vanishes on the boundary ∂E, while the potential Re(γw + γ̄t) has no flux of displacement field across ∂E (because the conjugate potential Im(γw +γ̄t) vanishes on ∂E).

Let us take a parameter c which to begin with we assume is real with 1 > c > 0 and let us make the additional stretching transformation

x = u, y = v / c,

which transforms the ellipse which intersects the u and v axes at the points (2.5) to an ellipse which intersects the x and y axes at the points

x = \pm (r_{0} + r^{2} / r_{0}) / 2, y = \pm (r_{0} - r^{2} / r_{0}) / (2 c) .

This final transformed ellipse will be the unit circle x² + y² = 1 if we choose

r_{0} = 1 + c, r = \sqrt{1 - c^{2}},

so that

(r_{0} + r^{2} / r_{0}) = 2, (r_{0} - r^{2} / r_{0}) = 2 c .

Inside this unit circle we put an isotropic medium with unit dielectric constant ε₀ = 1.

After this stretching transformation the dielectric tensor in the exterior medium becomes

ε = [\begin{matrix} ε_{x} & 0 \\ 0 & ε_{y} \end{matrix}],

where

ε_{y} = ε_{x} / c^{2} .

The potential Re(γw−γ̄t), expressed as a function of x and y will still vanish on the unit circle, while the potential Re(γw + γ̄t) will still have no associated flux of displacement field across this boundary (assuming the dielectric tensor of the exterior medium is transformed to the anisotropic value (2.15)) This trick of making an affine coordinate transformation to convert an isotropic matrix to an anistropic one has been used to find the solution for an isotropic sphere in an anisotropic medium [38] and to find the effective dielectric tensor of assemblages of stretched confocal coated ellipsoid assemblages (Section 8.4 of [39]). More generally it can be used to obtain explicit solutions for the fields around three-dimensional ellipsoidal inclusions in a uniform applied field with an anisotropic core and anisotropic matrix, each with arbitrary orientation.

Now consider a potential V(x,y) given by

\begin{array}{l} V (x, y) & = & Re (β w + \bar{γ} t) for x^{2} + y^{2} \geq 1, \\ = & δ_{x} x + δ_{y} y for x^{2} + y^{2} < 1, \end{array}

where w = u + iv = x + icy and from (2.10) and (2.13) t is given by

t = \frac{(1 + c^{2}) (x + i c y)}{1 - c^{2}} + \frac{2 c \sqrt{{(x + i c y)}^{2} + c^{2} - 1}}{c^{2} - 1} .

At the boundary of the unit circle Re(γ̄t) = Re(γw) so continuity of the potential V(x,y) requires

Re [(β + γ) w] = δ_{x} x + δ_{y} y .

Also continuity of the normal component of the displacement field requires

n_{x} ε_{x} \frac{\partial}{\partial x} Re (β w + \bar{γ} t) + n_{y} ε_{y} \frac{\partial}{\partial y} Re (β w + \bar{γ} t) = n_{x} δ_{x} + n_{y} δ_{y},

where

n = (n_{x}, n_{y}) = (x / \sqrt{x^{2} + y^{2}}, y / \sqrt{x^{2} + y^{2}})

is the unit outward normal to the boundary of the unit disk. Since the potential Re(γw+γ̄t) has no associated flux of displacement field across this boundary we have

n_{x} ε_{x} \frac{\partial}{\partial x} Re (\bar{γ} t) + n_{y} ε_{y} \frac{\partial}{\partial y} Re (\bar{γ} t) = n_{x} ε_{x} \frac{\partial}{\partial x} Re (- γ w) + n_{y} ε_{y} \frac{\partial}{\partial y} Re (- γ w),

so the flux continuity condition (2.20) reduces to

n_{x} ε_{x} \frac{\partial}{\partial x} Re [(β - γ) w] + n_{y} ε_{y} \frac{\partial}{\partial y} Re [(β - γ) w] = n_{x} δ_{x} + n_{y} δ_{y} .

When β and γ are real, β = β′ and γ = γ′, corresponding to an applied field acting in the x-direction, then δ_y = 0 and (2.19) and (2.22) are satisfied when

β^{'} + γ^{'} = δ_{x}, ε_{x} (β^{'} - γ^{'}) = δ_{x} .

These have the solution

δ_{x} = \frac{2 γ^{'} ε_{x}}{ε_{x} - 1}, β^{'} = \frac{γ^{'} (ε_{x} + 1)}{ε_{x} - 1},

and from (2.17) and (2.13) the potential outside the inclusion,

V (x, y) = β^{'} x + γ^{'} Re (t),

equates to

\begin{array}{l} V (x, y) & = & \frac{γ^{'} (ε_{x} + 1) x}{ε_{x} - 1} + \frac{γ^{'} (1 + c^{2}) x}{1 - c^{2}} \\ + \frac{γ^{'} c [\sqrt{{(x + i c y)}^{2} + c^{2} - 1} + \sqrt{{(x - i c y)}^{2} + c^{2} - 1}]}{c^{2} - 1}, \end{array}

while the field inside is

V (x, y) = δ_{x} x = \frac{2 γ^{'} ε_{x} x}{ε_{x} - 1} .

On the other hand when β and γ are imaginary, β = iβ″ and γ = iγ″, corresponding to an applied field acting in the y-direction, then δ_x = 0 and (2.19) and (2.22) are satisfied when

- c (β^{″} + γ^{″}) = δ_{y}, - ε_{y} c (β^{″} - γ^{″}) = δ_{y} .

These have the solution

δ_{y} = \frac{2 c γ^{″} ε_{y}}{1 - ε_{y}}, β^{″} = \frac{γ^{″} (ε_{y} + 1)}{ε_{y} - 1},

and from (2.17) and (2.13) the potential outside the inclusion,

V (x, y) = - β^{″} c y + γ^{″} Im (t),

equates to

\begin{array}{l} V (x, y) & = & \frac{γ^{″} (ε_{y} + 1) c y}{1 - ε_{y}} + \frac{γ^{″} (1 + c^{2}) c y}{1 - c^{2}} \\ + \frac{γ^{″} c [\sqrt{{(x + i c y)}^{2} + c^{2} - 1} - \sqrt{{(x - i c y)}^{2} + c^{2} - 1}]}{i (c^{2} - 1)}, \end{array}

while the field inside is

V (x, y) = δ_{y} y = \frac{2 c γ^{″} ε_{y} y}{1 - ε_{y}} .

3. The field singularities around a circular hole in a hyperbolic medium

The potential V(x,y) given by (2.26) and (2.27), or by (2.31) and (2.32), solves the dielectric equations when c and ε_x take any real positive value. By analytic continuation these formulae also solve the dielectric equations when c and ε_x are complex, and in this case V (x, y) is given by substituting the complex values of c, ε_x and ε_y = ε_x/c² in these formula. The branch cuts in the square roots need to be taken so (for fixed non-zero η) there are no singularities in the field outside the cylinder. Guided by (2.7) we choose

\sqrt{{(x + i c y)}^{2} + c^{2} - 1} = (x + i c y + \sqrt{1 - c^{2}}) \sqrt{\frac{x + i c y - \sqrt{1 - c^{2}}}{x + i c y + \sqrt{1 - c^{2}}},}

\sqrt{{(x - i c y)}^{2} + c^{2} - 1} = (x - i c y + \sqrt{1 - c^{2}}) \sqrt{\frac{x - i c y - \sqrt{1 - c^{2}}}{x - i c y + \sqrt{1 - c^{2}}},}

where the square roots on the right hand side of these expressions have their branch cuts along the negative real axis. The expression beneath the square root in (3.33) will be real when

\frac{x + i c y - \sqrt{1 - c^{2}}}{x + i c y + \sqrt{1 - c^{2}}} = \frac{x + i \bar{c} y - \bar{\sqrt{1 - c^{2}}}}{x - i \bar{c} y + \bar{\sqrt{1 - c^{2}}}},

(where the bar denotes complex conjugation) which is satisfied when (x,y) lies on the line

y Re [\bar{c} \sqrt{1 - c^{2}}] = x Im [\sqrt{1 - c^{2}}] .

Along this line the ratio in (3.35) will be negative along the interval between the points where

x + i c y = \pm \sqrt{1 - c^{2}}

, i.e. between the points

(x, y) = \pm (\frac{Re [\bar{c} \sqrt{1 - c^{2}}]}{Re [c]}, \frac{Im [\sqrt{1 - c^{2}}]}{Re [c]}),

which will lie in the unit disk x² + y² < 1 if and only if

{(Re [c])}^{2} - {(Re [\bar{c} \sqrt{1 - c^{2}}])}^{2} - {(Im [\sqrt{1 - c^{2}}])}^{2} > 0 .

Numerically we have checked that the left hand side is always non-negative and zero only when c is purely imaginary. This shows that there are no branch cuts in the potential outside the circular hole when η > 0.

The parameters γ′, γ″, β′, β″, δ_x, δ_y, r and r₀ are also generally complex. Let us take ε_x to be real and positive and c = −iμ + η where η is a small positive parameter. Then

ε_{y} = ε_{x} / {(- i μ + η)}^{2} \approx ε_{x} / μ^{2} + 2 i ε_{x} η / μ^{3}

is close to being real and negative. Since

\begin{array}{l} \sqrt{{(x + i c y)}^{2} + c^{2} - 1} = \sqrt{(x + μ y + i η y + \sqrt{1 - c^{2}}) (x + μ y + i η y - \sqrt{1 - c^{2}}),} \\ \sqrt{{(x - i c y)}^{2} + c^{2} - 1} = \sqrt{(x - μ y - i η y + \sqrt{1 - c^{2}}) (x - μ y - i η y - \sqrt{1 - c^{2}})}, \end{array}

we see that the potential V(x,y) given by (2.26) or (2.31) develops singularities as η → 0 along the four characteristic lines

\begin{array}{l} x + μ y + \sqrt{1 + μ^{2}} = 0, x + μ y - \sqrt{1 + μ^{2}} = 0, \\ x - μ y + \sqrt{1 + μ^{2}} = 0, x - μ y - \sqrt{1 + μ^{2}} = 0, \end{array}

which are tangent to the unit disk touching it at the four points

(x, y) = (\pm 1 / \sqrt{1 + μ^{2}}, \pm μ / \sqrt{1 + μ^{2}}) .

Figure 1 shows a plot of the absolute value of the potential V(x,y) showing how kinks develop along the characteristic lines

Fig. 1 Plot of the absolute value of the potential V(x,y) around the disk given by (2.26) with an applied field directed along the x-axis and with parameters γ′ = 1, ε_x = 3 and c = 0.01 − i. The potential has kinks at the four characteristic lines given by (3.41) which are also drawn. Near these lines the electric field is huge.

Download Full Size | PDF

To better understand the behavior of the fields near these singularities let us consider the potential V(x,y) given by (2.26) in a region near the characteristic line $x + μ y + \sqrt{1 + μ^{2}} = 0$ but away from the disk and away from the three other characteristic lines. In this region V(x,y) takes the form

V (x, y) = H (x, y) + G (x, y) \sqrt{x + μ y + i η y + r},

where

\begin{array}{l} r & = & \sqrt{1 - c^{2}} \approx \sqrt{1 + μ^{2}} + \frac{i μ η}{\sqrt{1 + μ^{2}}}, \\ H (x, y) & = & \frac{γ^{'} (ε_{x} + 1) x}{ε_{x} - 1} + \frac{γ^{'} (1 + c^{2}) x}{1 - c^{2}} + \frac{γ^{'} c \sqrt{{(x - i c y)}^{2} - r^{2}}}{c^{2} - 1}, \\ G (x, y) & = & \frac{γ^{'} c \sqrt{x + i c y - r}}{c^{2} - 1} . \end{array}

Near the characteristic line the electric field blows up as η → 0 and

\frac{\partial V}{\partial y} \approx \frac{μ G_{0}}{\sqrt{x + μ y + i η y + r}} \approx \frac{μ G_{0}}{\sqrt{h + i η s}},

where

G_{0} = \frac{γ^{'} i μ \sqrt{- 2 \sqrt{1 + μ^{2}}}}{1 + μ^{2}} = - γ^{'} μ \sqrt{2} {(1 + μ^{2})}^{- 3 / 4}

is the limit as η → 0 of G(x,y) on the line

x + μ y + \sqrt{1 + μ^{2}} = 0

, and where

h = x + μ y + \sqrt{1 + μ^{2}}, s = y + \frac{μ}{\sqrt{1 + μ^{2}}} .

(Thus h measures the distance from the characteristic line

x + μ y + \sqrt{1 + μ^{2}} = 0

.) So the local time averaged power dissipated in this region per unit area is proportional to

\begin{array}{l} Im (ε_{y}) {| \frac{\partial V}{\partial y} |}^{2} & \approx & \frac{{| G_{0} |}^{2} μ^{2} Im (ε_{y})}{| h + i η s |} \\ \approx & \frac{2 η ε_{x} {| G_{0} |}^{2}}{μ \sqrt{h^{2} + η^{2} s^{2}}}, \end{array}

in which (3.39) has been used to estimate Im(ε_y). Let us change variables from (x, y) to (h, s), so that dxdy = dhds. Observe that the right side of (3.47) is an even function of h and that the integral

\int_{0}^{h_{0}} \frac{d h}{\sqrt{h^{2} + η^{2} s^{2}}} = ln (h_{0} + \sqrt{h^{2} + η^{2} s^{2}}) - ln (η | s |),

when η is very small, and h₀ is not too large (so the approximation of being near the characteristic line is still valid) has a dominant contribution of −ln(η|s|). So when η is very small we have

\int Im (ε_{y}) {| \frac{\partial V}{\partial y} |}^{2} d h \approx - 4 ε_{x} {| G_{0} |}^{2} η ln (η | s |) .

Hence along the characteristic line the power absorption (integrated across the line), per unit length of the characteristic line, goes as η ln(η|s|) which goes to zero as η → 0. However it extends a long way out along these characteristic lines so the total contribution does not tend to zero. To see this first note that the approximation (3.45) will clearly break down at large values of s, specifically when ηs is of the order of one, since then the right hand side of (3.45) becomes comparable to ∂H/∂y. Thus a ball park estimate of the total absorption coming from this characteristic line is

2 \int_{0}^{1 / η} - 4 ε_{x} {| G_{0} |}^{2} η ln (η | s |) d s = 8 ε_{x} {| G_{0} |}^{2} .

The interesting point is that this total absorption remains finite and non-zero as η → 0. This explains the discovery of Sihvola [40] that a hole in a hyperbolic medium may have loss even though the medium is essentially lossless.

4. The dipole approximation for the far field around a circular hole in a hyperbolic medium

Consider the potential V(x,y) given by equation (2.26) corresponding to an applied field in the x-direction. When x + icy and x − icy are both large we can use the approximations

\begin{array}{l} \sqrt{{(x + i c y)}^{2} + c^{2} - 1} \approx x + i c y + \frac{c^{2} - 1}{2 (x + i c y)}, \\ \sqrt{{(x - i c y)}^{2} + c^{2} - 1} \approx x - i c y + \frac{c^{2} - 1}{2 (x - i c y)}, \end{array}

to obtain

\begin{array}{l} V (x, y) & \approx & \frac{γ^{'} (ε_{x} + 1) x}{ε_{x} - 1} + \frac{γ^{'} (1 + c^{2}) x}{1 - c^{2}} - \frac{2 γ^{'} c x}{1 - c^{2}} + \frac{γ^{'} c}{2 (x + i c y)} + \frac{γ^{'} c}{2 (x - i c y)} \\ \approx & γ^{'} x (\frac{ε_{x} + 1}{ε_{x} - 1} + \frac{1 - c}{1 + c}) + \frac{γ^{'} c x}{x^{2} + c^{2} y^{2}} \\ \approx & γ^{'} (\frac{ε_{x} + 1}{ε_{x} - 1} + \frac{1 - c}{1 + c}) (x - \frac{x α_{x}}{x^{2} + c^{2} y^{2}}), \end{array}

where α_x is the polarizability

α_{x} = \frac{- c}{(\frac{ε_{x} + 1}{ε_{x} - 1} + \frac{1 - c}{1 + c})} = \frac{c (1 + c) (1 - ε_{x})}{2 (ε_{x} + c)},

which has been normalized to make the last bracketed expression in (4.53) as simple as possible.

Similarly, when the applied field is in the y direction and x + icy and x − icy are both large the potential given by (2.31) has the far field behavior

\begin{array}{l} V (x, y) & \approx & \frac{γ^{″} (ε_{y} + 1) c y}{1 - ε_{y}} + \frac{γ^{″} (1 + c^{2}) c y}{1 - c^{2}} - \frac{2 γ^{″} c^{2} y}{1 - c^{2}} + \frac{γ^{″} c}{2 i (x + i c y)} - \frac{γ^{″} c}{2 i (x - i c y)} \\ \approx & γ^{″} c y (\frac{ε_{y} + 1}{1 - ε_{y}} + \frac{1 - c}{1 + c}) - \frac{γ^{″} c^{2} y}{x^{2} + c^{2} y^{2}} \\ \approx & γ^{″} c y (\frac{ε_{y} + 1}{1 - ε_{y}} + \frac{1 - c}{1 + c}) (y - \frac{y α_{y}}{x^{2} + c^{2} y^{2}}), \end{array}

where α_y is the normalized polarizability

α_{y} = \frac{c}{(\frac{ε_{y} + 1}{1 - ε_{y}} + \frac{1 - c}{1 + c})} = \frac{c (1 + c) (1 - ε_{y})}{2 (1 + c ε_{y})} .

We call (4.53) and (4.54) the dipole approximation for the far field. Note that in a hyperbolic medium it does not suffice for x² + y² to be large to ensure that both x + icy and x − icy are large. One must also be sufficiently distant from the lines x = ±μy since along these lines either x + icy or x − icy is close to zero when η is small. Thus for the dipole approximation for the far field to be valid one must be sufficiently far from the four characteristic lines (3.41): this makes sense as the electric field diverges to infinity along these lines, whereas the dipole field only diverges on the two lines x = ±μy as η → 0.

The expressions (4.54) and (4.56) for the polarizabilities could have been obtained more easily from the far field expressions for the potential outside an elliptical hole in an isotropic medium. When c is real and positive and before the stretching, the elliptical hole has axis lengths of 2 and 2c and in the (u,v) plane. With an applied field in the u-direction the far field in the isotropic medium with dielectric constant ε_x has potential

u - \frac{π c u (1 - ε_{x})}{2 π (u^{2} + v^{2}) [ε_{x} + (1 - ε_{x}) \frac{c}{1 + c}]},

where πc is the area of the ellipse and c/(1+ c) is the depolarization factor of the ellipse in the u-direction. By making the transformation x = u, y = v/c this potential gets mapped to

x - \frac{x α_{x}}{x^{2} + c^{2} y^{2}},

where the polarizability α_x is given by (4.54). It is similarly easy to derive (4.56) using the depolarization factor 1/(1 + c) of the ellipse in the v direction. When c = 1 and ε_y = ε_x (4.54) and (4.56) reduce to α_x = α_y = (1 − ε_x)/(1 + ε_x) which (within a proportionality factor) is the polarizability of a hole in an isotropic medium having dielectric constant ε_x.

5. The response of two interacting polarizable dipoles in a hyperbolic medium

Here we study the mathematics of the interaction of two ideal polarizable dipoles in a hyperbolic medium. We leave open the question as to whether these ideal polarizable dipoles have any physical significance. Nevertheless the searchlight effect discussed here should motivate future studies to see, say, whether a distant very small circular disk, if appropriately positioned, can substantially influence the response of a large circular disk.

By definition a polarizable dipole with rectangular symmetry located at the origin responds to a local field acting on it in the x direction so that the potential V(x,y) close to the origin has the expansion

V (x, y) \approx x + a_{1} - \frac{α_{x} / 2}{x + i c y} - \frac{α_{x} / 2}{x - i c y} = x + a_{1} - \frac{x α_{x}}{x^{2} + c^{2} y^{2}},

and responds to an local field acting on it in the y direction so that the potential V(x,y) close to the origin has the expansion

V (x, y) \approx y + a_{2} + \frac{α_{y} / (2 i c)}{x + i c y} - \frac{α_{y} / (2 i c)}{x - i c y} = y + a_{2} - \frac{y α_{y}}{x^{2} + c^{2} y^{2}} .

Here we call α_x and α_y the polarizabilities of the polarizable dipole, a₁ and a₂ are constants, and c = iμ − η with η being small. By taking linear combinations, the response to an arbitrarily oriented local field is such that the potential close to the origin has the expansion

V (x, y) \approx γ_{x} x + γ_{y} y + a - \frac{x γ_{x} α_{x} + y γ_{y} α_{y}}{x^{2} + c^{2} y^{2}} .

With c = −iμ + η and η > 0 small, the potential on the right hand side of (5.59) now has a local time averaged power dissipation near the characteristic line x +μy = 0 of

\begin{array}{l} Im (ε_{y}) {| \frac{\partial V}{\partial y} |}^{2} & \approx & \frac{μ^{2} {| α_{x} |}^{2} Im (ε_{y})}{4 {| g + i η y |}^{4}} \\ \approx & \frac{η ε_{x} {| α_{x} |}^{2}}{2 μ {(g^{2} + η^{2} y^{2})}^{2}}, \end{array}

where g = x + μy. So when η is very small the dissipation integrated with respect to g, in the range −g₀ ≥ g ≥ g₀, is approximately

{\int_{- g_{0}}^{g_{0}} Im (ε_{y}) | \frac{\partial V}{\partial y} |}^{2} d g \approx \frac{ε_{x} {| α_{x} |}^{2}}{2 μ y^{3} η^{2}} \int_{- \infty}^{\infty} \frac{d ν}{{(ν^{2} + 1)}^{2}},

where ν = g/(ηy). Thus this power dissipation integrated across the characteristic line, per unit length of the characteristic line, blows up as η → 0.

Now consider a uniform applied field in the x-direction acting on two polarizable dipoles, each with rectangular symmetry, one located at the origin and the other at the point (x₀, y₀). The total field V(x,y) is the sum of the uniform field plus the two dipolar fields:

V (x, y) = x + \frac{x β_{1 x} + y β_{1 y}}{x^{2} + c^{2} y^{2}} + \frac{(x - x_{0}) β_{2 x} + (y - y_{0}) β_{2 y}}{{(x - x_{0})}^{2} + c^{2} {(y - y_{0})}^{2}} .

Expanding this around the origin x = y = 0 gives

\begin{array}{l} V (x, y) & \approx & \frac{x β_{1 x} + y β_{1 y}}{x^{2} + c^{2} y^{2}} - \frac{(x_{0} β_{2 x} + y_{0} β_{2 y})}{x_{0}^{2} + c^{2} y_{0}^{2}} \\ + x + \frac{x β_{2 x} + y β_{2 y}}{x_{0}^{2} + c^{2} y_{0}^{2}} - \frac{(2 x x_{0} + 2 y y_{0}) (x_{0} β_{2 x} + y_{0} β_{2 y})}{{(x_{0}^{2} + c^{2} y_{0}^{2})}^{2}}, \end{array}

which allows us to identify the local field acting on the dipole at the origin. If (α_1x, α_1y) are the polarizability coefficients of the dipole at the origin then from (5.61) we have

\begin{array}{l} β_{1 x} = [- 1 - \frac{β_{2 x}}{x_{0}^{2} + c^{2} y_{0}^{2}} + \frac{2 x_{0} (x_{0} β_{2 x} + y_{0} β_{2 y})}{{(x_{0}^{2} + c^{2} y_{0}^{2})}^{2}}] α_{1 x}, \\ β_{1 y} = [- \frac{β_{2 y}}{x_{0}^{2} + c^{2} y_{0}^{2}} + \frac{2 c^{2} y_{0} (x_{0} β_{2 x} + y_{0} β_{2 y})}{{(x_{0}^{2} + c^{2} y_{0}^{2})}^{2}}] α_{1 y} . \end{array}

In a similar fashion, by rewriting (5.64) as

V (x, y) = x_{0} + (x - x_{0}) + \frac{[x_{0} + (x - x_{0})] β_{1 x} + [y_{0} + (y - y_{0})] β_{1 y}}{{[x_{0} + (x - x_{0})]}^{2} + c^{2} {[y_{0} + (y - y_{0})]}^{2}} + \frac{(x - x_{0}) β_{2 x} + (y - y_{0}) β_{2 y}}{{(x - x_{0})}^{2} + c^{2} {(y - y_{0})}^{2}},

and expanding this around the point (x₀, y₀) we obtain

\begin{array}{l} V (x, y) & \approx & \frac{(x - x_{0}) β_{2 x} + (y - y_{0}) β_{2 y}}{{(x - x_{0})}^{2} + c^{2} {(y - y_{0})}^{2}} + x_{0} + \frac{(x_{0} β_{1 x} + y_{0} β_{1 y})}{x_{0}^{2} + c^{2} y_{0}^{2}} \\ + (x - x_{0}) + \frac{(x - x_{0}) β_{1 x} + (y - y_{0}) β_{1 y}}{x_{0}^{2} + c^{2} y_{0}^{2}} \\ - \frac{[2 (x - x_{0}) x_{0} + 2 (y - y_{0}) y_{0}] (x_{0} β_{1 x} + y_{0} β_{1 y})}{{(x_{0}^{2} + c^{2} y_{0}^{2})}^{2}} . \end{array}

So if (α_2x, α_2y) are the polarizability coefficients of the dipole at the point (x₀, y₀) then we have

\begin{array}{l} β_{2 x} = [- 1 - \frac{β_{1 x}}{x_{0}^{2} + c^{2} y_{0}^{2}} + \frac{2 x_{0} (x_{0} β_{1 x} + y_{0} β_{1 y})}{{(x_{0}^{2} + c^{2} y_{0}^{2})}^{2}}] α_{2 x}, \\ β_{2 y} = [- \frac{β_{1 y}}{x_{0}^{2} + c^{2} y_{0}^{2}} + \frac{2 c^{2} y_{0} (x_{0} β_{1 x} + y_{0} β_{1 y})}{{(x_{0}^{2} + c^{2} y_{0}^{2})}^{2}}] α_{2 y} . \end{array}

Introducing

e = x_{0}^{2} + c^{2} y_{0}^{2} \approx x_{0}^{2} - μ^{2} y_{0}^{2} - 2 i μ η y_{0}^{2},

in terms of which

c^{2} = (e - x_{0}^{2}) / y_{0}^{2}

, the equations (5.66) and (5.69) take the equivalent form

\begin{array}{l} \frac{β_{1 x}}{α_{1 x}} = - 1 - \frac{β_{2 x}}{e} + \frac{2 x_{0} (x_{0} β_{2 x} + y_{0} β_{2 y})}{e^{2}}, \\ \frac{β_{1 y}}{α_{1 y}} = - \frac{β_{2 y}}{e} + \frac{2 (e - x_{0}^{2}) (x_{0} β_{2 x} + y_{0} β_{2 y})}{y_{0} e^{2}}, \\ \frac{β_{2 x}}{α_{2 x}} = - 1 - \frac{β_{1 x}}{e} + \frac{2 x_{0} (x_{0} β_{1 x} + y_{0} β_{1 y})}{e^{2}}, \\ \frac{β_{2 y}}{α_{2 y}} = - \frac{β_{1 y}}{e} + \frac{2 (e - x_{0}^{2}) (x_{0} β_{1 x} + y_{0} β_{1 y})}{y_{0} e^{2}} . \end{array}

These four equations have the solution

β \equiv [\begin{matrix} β_{1 x} \\ β_{1 y} \\ β_{2 x} \\ β_{2 y} \end{matrix}] = A^{- 1} [\begin{matrix} - 1 \\ 0 \\ - 1 \\ 0 \end{matrix}],

where A is the matrix

A = [\begin{matrix} \frac{1}{α_{1 x}} & 0 & \frac{1}{e} - \frac{2 {x_{0}}^{2}}{e^{2}} & - \frac{2 x_{0} y_{0}}{e^{2}} \\ 0 & \frac{1}{α_{1 y}} & - \frac{2 x_{0} (e - {x_{0}}^{2})}{e^{2} y_{0}} & - \frac{1}{e} + \frac{2 {x_{0}}^{2}}{e^{2}} \\ \frac{1}{e} - \frac{2 {x_{0}}^{2}}{e^{2}} & - \frac{2 x_{0} y_{0}}{e^{2}} & \frac{1}{α_{2 x}} & 0 \\ - \frac{2 x_{0} (e - {x_{0}}^{2})}{e^{2} y_{0}} & - \frac{1}{e} + \frac{2 {x_{0}}^{2}}{e^{2}} & 0 & \frac{1}{α_{2 y}} \end{matrix}],

which has determinant

det (A) = \frac{[4 {x_{0}}^{2} (e - {x_{0}}^{2}) (α_{1 x} - α_{1 y}) (α_{2 x} - α_{2 y}) + (e^{2} - α_{1 x} α_{2 x}) (e^{2} - α_{1 y} α_{2 y})]}{e^{4} α_{1 x} α_{1 y} α_{2 x} α_{2 y}},

that vanishes when

x_{0}^{2}

solves the quadratic

4 {x_{0}}^{2} (e - {x_{0}}^{2}) (α_{1 x} - α_{1 y}) (α_{2 x} - α_{2 y}) + (e^{2} - α_{1 x} α_{2 x}) (e^{2} - α_{1 y} α_{2 y}) = 0 .

We are interested in what happens to this solution when x₀ and y₀ are such that e is very small. Using Maple to compute the matrix inverse and taking the limit e → 0 we find that

β \to [\begin{matrix} \frac{2 {x_{0}}^{2} α_{1 x} α_{1 y} [2 {x_{0}}^{2} (α_{2 y} - α_{2 x}) + α_{2 y} α_{2 x}]}{α_{1 y} α_{2 y} α_{1 x} α_{2 x} - 4 {x_{0}}^{4} (α_{1 x} - α_{1 y}) (α_{2 x} - α_{2 y})} \\ - \frac{2 {x_{0}}^{3} α_{1 x} α_{1 y} [2 {x_{0}}^{2} (α_{2 y} - α_{2 x}) + α_{2 y} α_{2 x}]}{y_{0} [α_{1 y} α_{2 y} α_{1 x} α_{2 x} - 4 {x_{0}}^{4} (α_{1 x} - α_{1 y}) (α_{2 x} - α_{2 y})]} \\ \frac{2 {x_{0}}^{2} α_{2 x} α_{2 y} [2 {x_{0}}^{2} (α_{1 y} - α_{1 x}) + α_{1 x} α_{1 y}]}{α_{1 y} α_{2 y} α_{1 x} α_{2 x} - 4 {x_{0}}^{4} (α_{1 x} - α_{1 y}) (α_{2 x} - α_{2 y})} \\ - \frac{2 {x_{0}}^{3} α_{2 y} α_{2 x} [2 {x_{0}}^{2} (α_{1 y} - α_{1 x}) + α_{1 x} α_{1 y}]}{y_{0} [α_{1 y} α_{2 y} α_{1 x} α_{2 x} - 4 {x_{0}}^{4} (α_{1 x} - α_{1 y}) (α_{2 x} - α_{2 y})]} \end{matrix}] .

Thus when α_1x ≠ α_1y and α_2x ≠ α_2y we see that in the limit e → 0 all components of β blow up to infinity when x₀ is such that

x_{0}^{4} = \frac{α_{1 x} α_{1 y} α_{2 x} α_{2 y}}{4 (α_{1 x} - α_{1 y}) (α_{2 x} - α_{2 y})},

which is in agreement with (5.75) when one sets e = 0. Of course this equation generally will not have a solution for real x₀ if any of the polarizabilities are complex.

Additionally let us suppose that the polarizabilities α_2x and α_2y are very small, and α_2x ≠ α_2y. Specifically let suppose that α_2y = kα_2x, where k is a fixed constant not equal to 1. Then if the limit α_2x → 0 is taken after the limit e → 0, (5.76) implies

β \to [\begin{matrix} \frac{α_{1 x} α_{1 y}}{α_{1 x} - α_{1 y}} \\ - \frac{α_{1 x} α_{1 y} x_{0}}{(α_{1 x} - α_{1 y}) y_{0}} \\ 0 \\ 0 \end{matrix}] .

By contrast if α_2y = kα_2x, where k is a fixed constant, and we take the limit α_2x → 0 directly in (5.72), keeping e fixed and non-zero, then we obtain

β \to [\begin{matrix} - α_{1 x} \\ 0 \\ 0 \\ 0 \end{matrix}] .

Thus even a polarizable dipole at (x₀, y₀) with very small polarizability can have a very large effect on the net dipole moments of the system if it is positioned close to one of the characteristic lines where e is small. Even though its polarizability is small the field exerted by this dipole on the dipole at the origin is still significant. We call this the searchlight effect since μ and thus the angle of the characteristic lines will depend on frequency, so by varying the frequency and observing the net dipole moments of the system one may hope to detect something about the relative location of the polarizable dipoles, even though one of the polarizable dipoles, by itself, is difficult to detect. The effect is illustrated in Figs. 2 and 3.

Fig. 2 Plot of the absolute value of the dipole amplitude β_1x near the line where e ≈ 0 with polarizabilities α_1x = 2, and α_1y = 1, α_2x = 0.2, and α_2y = 0.1 and parameter c = 0.01 − i. We use rotated coordinates $ξ = (x_{0} + y_{0}) / \sqrt{2}$ and $τ = (x_{0} - y_{0}) / \sqrt{2}$ . Note the long-range interaction.

Download Full Size | PDF

Fig. 3 Same as Fig. 2 but with polarizabilities α_1x = 2, and α_1y = 1, α_2x = 0.1, and α_2y = 0.2 and parameter c = 0.01 − i.

Download Full Size | PDF

The case k = 1 when α_2y = α_2x is rather special, but very interesting. In this case, with e → 0 (5.76) simplifies to

β \to [\begin{matrix} 2 {x_{0}}^{2} \\ - \frac{2 {x_{0}}^{3}}{y_{0}} \\ \frac{2 {x_{0}}^{2} (α_{1 x} α_{1 y} - 2 {x_{0}}^{2} α_{1 x} + 2 {x_{0}}^{2} α_{1 y})}{α_{1 x} α_{1 y}} \\ - \frac{2 {x_{0}}^{3} (α_{1 x} α_{1 y} - 2 {x_{0}}^{2} α_{1 x} + 2 {x_{0}}^{2} α_{1 y})}{y_{0} α_{1 y} α_{1 x}} \end{matrix}],

and this result does not depend on the magnitude of α_2x. Remarkably, note that the magnitudes of the components of β increase as x₀ increases: recalling that when e = 0, y₀ = ±x₀/(ic) we see that β_1x and β_1y increase as

x_{0}^{2}

, while β_2x and β_2y increase as

x_{0}^{4}

when α_1x ≠ α_1y and as

x_{0}^{2}

when α_1x = α_1y. Thus the interaction increases the further the polarizable dipoles are apart!

To shed more light on this one can, using Maple, directly compute the right hand side of (5.72) when α_2y = α_2x to obtain

β = [\begin{matrix} \frac{(- e^{2} + e α_{2 x} - 2 {x_{0}}^{2} α_{2 x}) α_{1 x}}{e^{2} - α_{1 x} α_{2 x}} \\ - \frac{2 α_{1 y} x_{0} (e - {x_{0}}^{2}) α_{2 x}}{(e^{2} - α_{2 x} α_{1 y}) y_{0}} \\ \frac{α_{2 x} f}{(e^{2} - α_{2 x} α_{1 y}) (e^{2} - α_{1 x} α_{2 x})} \\ - \frac{2 x_{0} (e - {x_{0}}^{2}) α_{2 x} g}{y_{0} (e^{2} - α_{2 x} α_{1 y}) (e^{2} - α_{1 x} α_{2 x})} \end{matrix}],

where

\begin{array}{l} f & = & e^{3} α_{1 x} - 2 {x_{0}}^{2} α_{1 x} e^{2} - e α_{1 x} α_{2 x} α_{1 y} + 2 {x_{0}}^{2} α_{1 x} α_{2 x} α_{1 y} - e^{4} \\ + 4 α_{1 x} α_{2 x} e {x_{0}}^{2} - 4 α_{1 x} α_{2 x} {x_{0}}^{4} + α_{2 x} α_{1 y} e^{4} - 4 {x_{0}}^{2} α_{1 y} α_{2 x} e + 4 {x_{0}}^{4} α_{1 y} α_{2 x}, \\ g & = & α_{1 x} e^{2} - α_{1 y} α_{1 x} α_{2 x} - α_{1 x} α_{2 x} e + α_{1 y} α_{2 x} e + 2 α_{1 x} α_{2 x} {x_{0}}^{2} - 2 {x_{0}}^{2} α_{1 y} α_{2 x} . \end{array}

If the limit e → 0 is taken in this expression we recover (5.80). On the other hand it is evident from (5.81) that there are resonances when e² equals α_1xα_2x or α_1yα_2x, and that the resulting expression for β depends crucially on the ratio of the magnitude of e² to these two quantities. Also if η is non-zero the interaction decreases for sufficiently large separations of the polarizable dipoles. On the characteristic lines x₀ = ± μy₀ we have $e \approx - 2 i μ η y_{0}^{2}$ . So for large x₀ = ±μy₀ (5.81) implies that, for example,

β_{1 x} \approx \frac{x_{0}^{2} α_{2 x} α_{1 x}}{2 μ^{2} η^{2} y_{0}^{4}} = \frac{α_{2 x} α_{1 x}}{2 η^{2} y_{0}^{2}},

which goes to zero as

1 / y_{0}^{2}

as y₀ → ∞. Figure 4 shows how, as the separation increases, the interaction along the characteristic line first increases, then decreases.

Fig. 4 Plot of the absolute value of the dipole amplitude β_1x near the line where e ≈ 0 with polarizabilities α_1x = 2, and α_1y = 1, α_2x = α_2y = 0.1 and parameter c = 0.01 − i. We use rotated coordinates $ξ = (x_{0} + y_{0}) / \sqrt{2}$ and $τ = (x_{0} - y_{0}) / \sqrt{2}$ . Note that the interaction along the line τ = 0 first becomes stronger as ξ increases, then weakens.

Download Full Size | PDF

Acknowledgments

Graeme Milton is thankful to Ben Eggleton, CUDOS and the University of Sydney for the provision of office space during his visit there and to the National Science Foundation for support through grant DMS-1211359. R.C. McPhedran acknowledges the support of the Australian Research Council through its Discovery Grant Scheme. The Centre for Ultrahighband-width Devices for Optical Systems (CUDOS) is an ARC Centre of Excellence (Project No. CE110001018).

References and links

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of εand μ,”Uspekhi Fizicheskikh Nauk 92, 517–526 (1967). English translation in Sov. Phys. Uspekhi10:509–514 (1968) [CrossRef] .

2. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994) [CrossRef] .

3. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. Roy. Soc. A 461, 3999–4034 (2005) [CrossRef] .

4. G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” P. R. Soc. A 462, 3027–3059 (2006) [CrossRef] .

5. N.-A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express 15, 6314–6323 (2007) [CrossRef] [PubMed] .

6. O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys. 102, 124502 (2007) [CrossRef] .

7. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys. 10, 115021 (2008) [CrossRef] .

8. N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. 10, 115020 (2008) [CrossRef] .

9. G. Bouchitté and B. Schweizer, “Cloaking of small objects by anomalous localized resonance,” Quantum J. Mech. Appl. Math. 63, 437–463 (2010) [CrossRef] .

10. N.-A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states and cloaking in finite clusters of coated cylinders,” Wave. Random Complex 21, 248–277 (2011) [CrossRef] .

11. H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance,” Arch. Ration. Mech. Anal. 208, 667–692 (2013) [CrossRef] .

12. R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, “A variational perspective on cloaking by anomalous localized resonance,” (2012). ArXiv:1210.4823 [math.AP].

13. H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance ii,” (2013). ArXiv:1212.5066 [math.AP].

14. M. Xiao, X. Huang, J. W. Dong, and C. T. Chan, “On the time evolution of the cloaking effect of a metamaterial slab,” Opt. Lett. 37, 4594–4596 (2012) [CrossRef] [PubMed] .

15. H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Anomalous localized resonance using a folded geometry in three dimensions,” (2013). ArXiv:1301.5712 [math-ph].

16. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000) [CrossRef] [PubMed] .

17. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001) [CrossRef] .

18. F. D. M. Haldane, “Electromagnetic surface modes at interfaces with negative refractive index make a ’not-quite-perfect’ lens,” (2002). ArXiv:cond-mat/0206420 v3 (2002).

19. N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88, 207403 (2002) [CrossRef] .

20. A. L. Pokrovsky and A. L. Efros, “Diffraction in left-handed materials and theory of veselago lens,” (2002). ArXiv:cond-mat/0202078 v2 (2002).

21. S. A. Cummer, “Simulated causal subwavelength focusing by a negative refractive index slab,” Appl. Phys. Lett. 82, 1503–1505 (2003) [CrossRef] .

22. A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B 338, 333–337 (2003). See also arXiv:cond-mat/0202078 v2 (2002) [CrossRef] .

23. X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B 68, 113103 (2003) [CrossRef] .

24. G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67, 035109 (2003) [CrossRef] .

25. R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. 84, 1290–1292 (2004) [CrossRef] .

26. S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005) [CrossRef] [PubMed] .

27. V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30, 75–77 (2005) [CrossRef] [PubMed] .

28. G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Opaque perfect lenses,” Physica B 394, 171–175 (2007) [CrossRef] .

29. J. W. Dong, H. H. Zheng, Y. Lai, H. Z. Wang, and C. T. Chan, “Metamaterial slab as a lens, a cloak, or an intermediate,” Phys. Rev. B 83, 115124 (2011) [CrossRef] .

30. J. B. Pendry and S. A. Ramakrishna, “Refining the perfect lens,” Physica B 338, 329–332 (2003) [CrossRef] .

31. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003) [CrossRef] [PubMed] .

32. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006) [CrossRef] [PubMed] .

33. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74, 075103 (2006) [CrossRef] .

34. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007) [CrossRef] [PubMed] .

35. J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature 1, 143 (2010).

36. S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express 20, 15101–15105 (2012) [CrossRef] .

37. H. C. Yang and Y. T. Chou, “Antiplane strain problems of an elliptic inclusion in an anisotropic medium,” J. Appl. Mech. 44, 437–441 (1977) [CrossRef] .

38. A. H. Sihvola, “On the dielectric problem of isotropic sphere in anisotropic medium,” Electromagnetics 17, 69–74 (1997) [CrossRef] .

39. G. W. Milton, The Theory of Composites (Cambridge University Press, 2002) [CrossRef] .

40. A. Sihvola, “Metamaterials and depolarization factors,” Prog. Electromagn. Res. 51, 65–82 (2005) [CrossRef] .

The searchlight effect in hyperbolic materials

Abstract

1. Introduction

2. The field around a circular hole in an anisotropic lossless medium

3. The field singularities around a circular hole in a hyperbolic medium

4. The dipole approximation for the far field around a circular hole in a hyperbolic medium

5. The response of two interacting polarizable dipoles in a hyperbolic medium

Acknowledgments

References and links

Cited By

Figures (4)

Equations (83)

Optics Express