# Near-perfect absorption in epsilon-near-zero structures with hyperbolic
dispersion

Klaus Halterman1,∗ and J. Merle Elson2

###### Abstract

We investigate the interaction of polarized electromagnetic waves with
hyperbolic metamaterial structures, whereby the in-plane permittivity
component $\epsilon_{x}$ is opposite in sign to the normal component
$\epsilon_{z}$. We find that when the thickness of the metamaterial is smaller
than the wavelength of the incident wave, hyperbolic metamaterials can absorb
significantly higher amounts of electromagnetic energy compared to their
conventional counterparts. We also demonstrate that for wavelengths leading to
$\Re(\epsilon_{z})\approx 0$, near-perfect absorption arises and persists over
a range of frequencies and subwavelength structure thicknesses.

1Michelson Lab, Physics Division, Naval Air Warfare Center, China Lake,
California 93555, USA
2PO Box 965, Tijeras, New Mexico 87059, USA<EMAIL_ADDRESS>
(160.3918) Metamaterials; (160.1190) Anisotropic optical materials.

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##  1 Introduction

With recent advances in nanoscale fabrication of metal-dielectric multilayers
and arrays of rods, hybrid structures can now be created that absorb a
substantial portion of incident electromagnetic (EM) radiation. In
conventional approaches, strong absorption was achieved by utilizing materials
that had either high loss or large thickness. Nowadays, with the advent of
metamaterials, absorbing structures can be created that harness plasmonic
excitations or implement high impedance components [1] that have extreme
values [2, 3, 4, 5] of the permittivity $\bm{\epsilon}$ or permeability
$\bm{\mu}$. In close connection with these developments, there has also been a
substantial amount of research lately involving anisotropic metamaterials,
where now $\bm{\epsilon}$ and $\bm{\mu}$ are tensors that have in general
differing components along the three coordinate axes. An important type of
anisotropic metamaterial is one whose corresponding orthogonal tensor
components are of opposite sign, sometimes referred to as indefinite media
[6]. When such structures are described by a diagonal tensor, the
corresponding dispersion relation permits wavevectors that lie within a
hyperbolic isofrequency surface, and hence such a material is also called a
hyperbolic metamaterial (HMM). The inclusion of HMM elements in many designs
can be beneficial due to their inherent nonresonant character, thus limiting
loss effects [7].

The earliest HMM construct involving bilayers of anisotropic media was
discussed in the context of bandpass spatial filters with tunable cutoffs [6].
For wavelengths $\lambda$ in the visible spectrum, an effective HMM was
modeled using arrays of metallic nanowires [8] spaced apart distances much
smaller than $\lambda$, thus avoiding the usual problems associated with
resonances. Periodic arrays of carbon nanotubes [9] have been shown to exhibit
HMM characteristics in the THz spectral range. Other possibilities involve
metal-dielectric layers: The inclusion of active media in metal-dielectric
multilayers can result in improved HMM-based imaging devices [10]. For certain
layer configurations, nonlocal effects [11], which depending on geometry [12],
can limit the number of accessible photonic states [13]. The absorption in
thin films has been shown experimentally to be enhanced when in contact with a
multilayered HMM substrate [14]. Rather than using metallic components,
tunable graphene can switch between a hyperbolic and conventional material via
a gate voltage [15]. The HMM dispersion can be tuned in
gyromagnetic/dielectric [16] and semiconductor/dielectric structures [17].
Slabs of semiconductors can also exhibit tunability by photogenerating a
grating via variations in the carrier density caused by two incident beams,
revealing a hyperbolic character [18].

[width=0.45]Fig1at.pdf

Figure 1: (a) Schematic of the hyperbolic metamaterial configuration: The HMM
layer of thickness $\tau$ is bordered by a semi-infinite superstrate and
substrate. The permittivites ${\bm{\epsilon}}_{i}$ $(i$$=$$1,2,3)$ are in
general anisotropic. The incident field is polarized in the $x$$-$$z$ plane at
an angle $\theta$. (b) Dispersion contours for a type-2 HMM where
$\epsilon_{x}$$<$$0$, and $\epsilon_{z}$$>$$0$ and (c) for a type-1 HMM with
$\epsilon_{x}$$>$$0$, and $\epsilon_{z}$$<$$0$.

Increased absorption can also be achieved by incorporating a grating with the
HMM, so that by introducing surface corrugations, or grooves, light can
diffract and generate a broad spectrum of wave vectors into the HMM layer.
These wavevectors can couple via surface modes [19] due to the impedance
mismatch at the various openings. Grating lines were patterned above a layered
${\rm Au/TiO_{2}}$ HMM structure, creating a “hypergrating” capable of
exciting both surface and bulk plasmons [20]. By judiciously designing the
materials below the grating, it can be possible to absorb a considerable
fraction of the diffracted EM field. Indeed, a HMM comprised of arrays of
silver nanowires was experimentally shown to reduce the reflectance by
introducing surface corrugations [21]. Spherical nanoparticles deposited on
planar HMM structures also resulted in reduced reflectance due to the
increased density of photonic states [22].

In this paper we show that near-perfect absorption of EM radiation can arise
in a simple HMM structure adjacent to a metal. We investigate a range of
frequencies where the permittivity components perpendicular and parallel to
the interfaces are of opposite sign. We consider two possibilities: when the
HMM dispersion relation is of type-1 or type-2, which for our geometry
corresponds to $\epsilon_{x}$ $>$ $0$, $\epsilon_{z}$ $<$ $0$ or
$\epsilon_{x}$ $<$ $0$, $\epsilon_{z}$ $>$ $0$ respectively (see Fig. 1). We
show that for those $\lambda$ leading to the real part of the permittivity
component perpendicular to the interfaces ($\epsilon_{z}$) nearly vanishing,
an intricate balance between material loss and structure thickness ($\tau$)
yields a broad range of incident angles $\theta$ and $\tau$ in which nearly
the entire EM wave is absorbed. These findings are absent in conventional
anisotropic “elliptical” structures.

##  2 Methods

We assume that the incident EM wave propagates with wave vector in the $x-z$
plane with polarization ($E_{x},E_{z},B_{y}$) ($p$-polarized) or
($E_{y},B_{x},B_{z}$) ($s$-polarized). Once the wave enters the anisotropic
medium, its polarization state can then split into linear combinations of both
TE and TM polarizations [23]. Consider an unbounded diagonally anisotropic
medium described by homogeneous parameters
($\epsilon_{x},\epsilon_{y},\epsilon_{z}$) and ($\mu_{x},\mu_{y},\mu_{z}$),
where it is always possible to choose principal coordinate axes so that the
permittivity and permeability are diagonal. Assuming a harmonic time
dependence, $\exp(-i\omega t)$, for the EM fields, Maxwell’s equations give
the corresponding wave equations for the electric field components $E_{x}$ and
$E_{y}$:

$\displaystyle\frac{\partial^{2}E_{x}}{\partial
z^{2}}+\left[\left(\frac{\omega}{c}\right)^{2}\epsilon_{x}\mu_{y}-\left(\frac{\epsilon_{x}}{\epsilon_{z}}\right)k_{x}^{2}\right]E_{x}=0,$
(1) $\displaystyle\frac{\partial^{2}E_{y}}{\partial
z^{2}}+\left[\left(\frac{\omega}{c}\right)^{2}\epsilon_{y}\mu_{x}-\left(\frac{\mu_{x}}{\mu_{z}}\right)k_{x}^{2}\right]E_{y}=0.$
(2)

Equations (1) and (2) illustrate that the wave equations are different for
$E_{x}$ and $E_{y}$, resulting in two different wave vectors. In this work, we
focus exclusively on $p$-polarization from which the nature of the HMM
dispersion can be qualitatively understood. From Eq. (1),
${\hat{k}}_{z}^{2}=\epsilon_{x}\mu_{y}-\left({\epsilon_{x}}/{\epsilon_{z}}\right){\hat{k}}_{x}^{2}$
(the caret symbol signifies normalization by $\omega/c$). For this discussion
we assume real valued material parameters and positive $\mu_{y}$. Focusing on
$\epsilon_{x}$$>$$0$, we consider two scenarios (a) $\epsilon_{z}$$>$$0$ and
(b) $\epsilon_{z}$$<$$0$, yielding the respective dispersion relations
${{\hat{k}}_{z}^{2}}/({\epsilon_{x}\mu_{y}})+{{\hat{k}}_{x}^{2}}/({\epsilon_{z}\mu_{y}})=1$
and
${{\hat{k}}_{z}^{2}}/({\epsilon_{x}\mu_{y}})-{{\hat{k}}_{x}^{2}}/({|\epsilon_{z}|\mu_{y}})=1$.
Thus the isofrequency contours are (a) ellipses and (b) hyperbola (see e.g.,
Fig. 1(c) when $k_{y}=0$). Moreover, for the ellipsoidal case, as
${\hat{k}}_{x}$ increases there will be a frequency cutoff since
${\hat{k}}_{z}^{2}$ eventually becomes negative. On the other hand, for the
hyperbolic case, when ${\hat{k}}_{x}$ increases, there is no cutoff since
${\hat{k}}_{z}^{2}$ remains positive. If $\epsilon_{x}<0$ and
$\epsilon_{z}>0$, we then have the possibility of a connected hyperbola (see
Fig. 2(b)).

To determine the absorbed EM energy, it is convenient to first determine the
Fresnel reflection coefficient, $r$. The corresponding reflectance $R$ is then
given by $R=|r|^{2}$. For a $p$-polarized plane wave incident at an angle
$\theta$ relative to the normal of a planar layer of thickness $\tau$, we
find,

$\displaystyle
r=\beta\left[\frac{({\hat{k}}_{z1}\epsilon_{x2}-{\hat{k}}_{z2}\epsilon_{x1})({\hat{k}}_{z2}\epsilon_{x3}+{\hat{k}}_{z3}\epsilon_{x2})e^{i\phi_{2}}+({\hat{k}}_{z1}\epsilon_{x2}+{\hat{k}}_{z2}\epsilon_{x1})({\hat{k}}_{z2}\epsilon_{x3}-{\hat{k}}_{z3}\epsilon_{x2})e^{-i\phi_{2}}}{({\hat{k}}_{z1}\epsilon_{x2}-{\hat{k}}_{z2}\epsilon_{x1})({\hat{k}}_{z2}\epsilon_{x3}-{\hat{k}}_{z3}\epsilon_{x2})e^{i\phi_{2}}+({\hat{k}}_{z1}\epsilon_{x2}+{\hat{k}}_{z2}\epsilon_{x1})({\hat{k}}_{z2}\epsilon_{x3}+{\hat{k}}_{z3}\epsilon_{x2})e^{-i\phi_{2}}}\right],$
(3)

where the semi-infinite substrate and superstrate are in general anisotropic
(see Fig. 1). The details can be found in Sec. Appendix: Poynting’s Theorem.
We define,

$\beta=\exp({-2i\phi_{3}}),$ (4)

where

$\phi_{j}\equiv(\omega/c){\hat{k}}_{zj}\tau,$ (5)

and

${\hat{k}}_{zj}^{2}\equiv\epsilon_{xj}\mu_{yj}-\left(\epsilon_{xj}/\epsilon_{zj}\right){\hat{k}}_{x}^{2}.$
(6)

The index $j$ labels the regions $1,2$ or $3$ (see Fig. 1). In all cases
below, the incident beam is in vacuum (region 3) so that
${\hat{k}}_{x}=\sin\theta$, which is conserved across the interface. The
frequency dispersion in the HMM takes the Drude-like form:
$\epsilon_{z2}=a+ib$, where $a=1-\alpha^{2}/[1+(\alpha f)^{2}]$, and
$b=\alpha^{3}f/[1+(\alpha f)^{2}]$. Here, $\alpha\equiv\lambda/\lambda_{z}$,
$f=0.02$, and the characteristic wavelength, $\lambda_{z}=1.6$$\mu$m. When
discussing the two types of HMM, the permittivity parallel to the interface is
described using $\epsilon_{x2}=\pm 4+0.1i$ for type-1 ($+$) and type-2 ($-$).
The wavelength range considered here, where the system exhibits HMM behavior
is consistent with experimental work involving HMM semiconductor hybrids [24].

When the surrounding media is air and the central layer is a diagonally
anisotropic HMM, setting the numerator of (3) to zero leads to a set of
conditions on the wavevector components that results in a complete absence of
reflection ($R=0$):

$\displaystyle{\hat{k}}_{x}^{2}=\
\epsilon_{z2}\left(\frac{\epsilon_{x2}-\mu_{y2}}{\epsilon_{z2}\epsilon_{x2}-1}\right);\quad{\hat{k}}_{z2}^{2}=\epsilon_{x2}^{2}\left(\frac{\epsilon_{z2}\mu_{y2}-1}{\epsilon_{z2}\epsilon_{x2}-1}\right)$
(7) $\displaystyle{\hat{k}}_{x}^{2}=\
\epsilon_{z2}\mu_{y2}-\left(\frac{\epsilon_{z2}}{\epsilon_{x2}}\right)\left(\frac{n\lambda}{2\tau}\right)^{2};\quad{\hat{k}}_{z2}^{2}=\left(\frac{n\lambda}{2\tau}\right)^{2},$
(8)

where $n$ is an integer. The $\hat{k}_{x}$ in Eq. (7) corresponds to the
classic Brewster angle condition for isotropic media:
${\hat{k}}_{z}^{2}\epsilon_{x2}^{2}={\hat{k}}_{z2}^{2}$, and the $\hat{k}_{x}$
in Eq. (8) corresponds to a standing wave condition in the $z$-direction. In
either case, when Eqs. (7) or (8) is satisfied, a minimum in $R$ arises. Under
the Brewster angle condition in Eq. (7), a simple rearrangement shows that
${\hat{k}}_{x}^{2}(\epsilon_{x2}-\epsilon_{z2}^{-1})$ $=$
$\epsilon_{x2}-\mu_{y2}$. This implies that if we choose
$\epsilon_{x2}=\mu_{y2}$ and $\epsilon_{x2}\epsilon_{z2}=1$, then we should
have $R=0$ for any value of $\hat{k}_{x}=\sin\theta$. This choice of
anisotropic material parameters is similar to the perfectly matched layer
(PML) approach to eliminating unwanted reflection from absorbing computational
domain boundaries, especially in time-domain [25, 26] and frequency-domain
algorithms [27, 28]. Note that such a PML medium is somewhat artificial since
$\epsilon_{x2}=\epsilon_{z2}^{-1}$ implies sources in region 2. Nonetheless
such a concept is successful for absorbing layers designed to simulate an
infinite computational domain.

[width=.34]1a.pdf [width=.34]1b.pdf

[width=.34]1c.pdf [width=.34]1d.pdf

Figure 2: Absorption as a function of incident angle $\theta$. The superstrate
is air, and the HMM layer is supported by a perfectly conducting substrate. In
(a) and (b) a range of HMM widths $\tau$ are studied (legend units are in
microns). In (a) $\Re(\epsilon_{2x})>0$ and $\Re(\epsilon_{2z})<0$ (type-1
HMM), and in (b) $\Re(\epsilon_{2x})<0$, and $\Re(\epsilon_{2z})>0$ (type-2
HMM). For both panels (a) and (b), $\lambda\approx\lambda_{z}$ so that
$\Re(\epsilon_{2z})\approx 0$. Panels (c) and (d) show the effects of varying
$\lambda$ for both the type-1 and type-2 cases respectively. For those cases
$\tau$ is fixed at $0.16$ $\mu$m. For normal incidence ($\theta=0^{\circ}$),
there is generally little absorption (high reflectance). Remarkably, for a
range of HMM widths and wavelengths there are strong absorption peaks spanning
a broad range of $\theta$.

We now illustrate the important case of a HMM backed by a perfectly conducting
metal, and the near-perfect absorption that can arise. As is appropriate for
HMM structures, we also consider the regime where all materials are
nonmagnetic (${\bm{\mu}}=1$). The reflection coefficient in Eq. (3) then
becomes,

$\displaystyle
r=e^{-2i\phi}\left[\frac{({\hat{k}}_{z2}+{\hat{k}}_{z}\epsilon_{x2})e^{i\phi_{2}}-({\hat{k}}_{z2}-{\hat{k}}_{z}\epsilon_{x2})e^{-i\phi_{2}}}{({\hat{k}}_{z2}-{\hat{k}}_{z}\epsilon_{x2})e^{i\phi_{2}}-({\hat{k}}_{z2}+{\hat{k}}_{z}\epsilon_{x2})e^{-i\phi_{2}}}\right],$
(9)

where $\phi\equiv(\omega/c){\hat{k}}_{z}\tau,$ and ${\hat{k}}_{z}=\cos\theta.$
It is readily verified that for lossless media, Eq. (9) yields perfect
reflection ($|r|^{2}$$=$$1$) as expected. In the absence of transmission, the
absorption, $A$, is simply written as $A=1-R$.

##  3 Results

Figure 2 shows the absorption as a function of incident angle $\theta$ for
both types of HMM: type-1, $\Re\\{\epsilon_{2x}\\}>0$,
$\Re\\{\epsilon_{2z}\\}<0$ (panels a and c), and type-2,
$\Re\\{\epsilon_{2x}\\}<0$, $\Re\\{\epsilon_{2z}\\}>0$ (panels b and d). Since
$\lambda\approx\lambda_{z}$, we have also the condition,
$\Re(\epsilon_{2z})\approx 0$. There cannot be any substrate transmission and
thus $R<1$ is due to intrinsic HMM losses. In terms of practical designs, it
is important to determine the range of sub-wavelength HMM layer thicknesses
that can admit perfect absorption. Thus Figs. 2(a) and (b) explore differing
$\tau$ ranging from 0.001 to 0.156 $\mu$m. Although the relative sign of
$\epsilon_{x2}$ and $\epsilon_{z2}$ usually plays a pivotal role, for
extremely thin HMM widths this is not the case. Indeed in the regime of small
$\phi_{2}$, Eq. (9) simplifies to,

$\displaystyle
r=\frac{{\hat{k}}_{z}+i2\pi\hat{\tau}(1-{\hat{k}}_{x}^{2}/\epsilon_{z2})}{-{\hat{k}}_{z}+i2\pi\hat{\tau}(1-{\hat{k}}_{x}^{2}/\epsilon_{z2})},$
(10)

which is independent of $\epsilon_{x2}$. Here we have introduced the
dimensionless thickness: $\hat{\tau}\equiv\omega\tau/c$. Setting the numerator
of $r$ to zero gives the angle, $\theta_{c}$, where the reflectance vanishes:

$\displaystyle\theta_{c}=\cos^{-1}\bigl{[}\bigl{(}i\epsilon_{z2}+\sqrt{(2\hat{\tau})^{2}(1-\epsilon_{z2})-\epsilon_{z2}^{2}}\,\,\bigr{)}/({2\hat{\tau}})\bigr{]}.$
(11)

In Fig. 2(a), for the incident wavelength of $1.601$$\mu$m, the approximate
absorption angles are found from taking the real part of Eq. (11), giving,
$\theta_{c}\approx$ $79^{\circ}$,$66^{\circ}$,$52^{\circ}$,$28^{\circ}$, and
$11^{\circ}$, in order of increasing $\tau$. Deviations in the angle predicted
from Eq. (11) arise for larger $\tau$ as higher order corrections are needed.
As the thickness $\tau$ decreases, near-perfect absorption shifts towards
grazing incidences, in agreement with Eq. (11) where as $\tau\rightarrow 0$,
$\theta_{c}\rightarrow\pi/2$. For the type-2 HMM, similar trends are seen in
Fig. 2(b), where $\lambda=1.59\mu$m and the near-perfect absorption angles
were found to agree well with Eq. (11). It is apparent that for a type-2 HMM,
the angular range of near-perfect absorption exhibits a greater sensitivity to
$\tau$ than the type-1 case shown. In both cases (a) and (b), near-perfect
absorption can be controlled over nearly the whole angular range by properly
choosing the effective material thicknesses. When calculating the regions of
high absorption, $\Im(\epsilon_{z2})$ plays a significant role when
$\Re(\epsilon_{z2})\approx 0$. This is consistent with anisotropic leaky-wave
structures [4] and coherent perfect absorbers [5]. Although more difficult to
achieve in practice, subwavelength isotropic slabs where the permittivity and
permeability simultaneously vanish, can exhibit perfect absorption for small
loss and a perfectly conducting metal backing [29]. Additional control of
absorption may also be possible with the introduction of gain media [10].

[width=8.0cm]f3a.pdf

[width=8.0cm]f3b.pdf

Figure 3: Density plots showing absorption as a function of incident
wavelength $\lambda$ and angle $\theta$. Bright regions correspond to high
absorption. The HMM thickness in both plots is $\tau=0.16\mu$m. The
characteristic wavelength, $\lambda_{z}=1.6\mu$m separates the HMM regions
according to (a) type-1: $\epsilon_{x2}>0$ and $\epsilon_{z2}<0$ for
$\lambda>1.6\mu$m, and (b) type 2: $\epsilon_{x2}<0$, and $\epsilon_{z2}>0$
for $\lambda<1.6\mu$m. Thus we find that when the metamaterial is effectively
hyperbolic, absorption can be strongly enhanced.

[width=7cm]fig3c_v3.pdf

[width=7cm]fig3d_v3.pdf

Figure 4: Absorption as a function of incident angle $\theta$ (a) and
wavelength $\lambda$ (b) extracted from the high absorption regions of the
density plots in Fig. 3(a) and (b).

Next we investigate how varying the wavelength of the source beam affects the
absorption features. In Figs. 2(c) and (d) the thickness $\tau$ is set to
$0.16\mu$m for both the type-1 and type-2 HMM cases respectively. For the
type-1 HMM (panel c), as $\lambda$ increases beyond $\lambda_{z}$, the
wavelength-dependent $\epsilon_{z2}$ shifts so that its real part becomes more
negative. The corresponding absorption peaks then migrate towards
$\theta=90^{\circ}$. The opposite trend occurs for the type-2 case, where
increasing $\lambda$ from $\lambda=1.5\mu$m causes $\Re(\epsilon_{z2})$ (which
is positive at this wavelength) to approach zero. Consequently, the observed
double-peaked absorption shifts towards normal incidence, consistent with the
trends above and Eq. (11), where as $\lambda\rightarrow\lambda_{z}$ (and hence
$\Re(\epsilon_{2z})\rightarrow 0$), the angle of near-perfect absorption tends
to zero. It is worth noting that if the HMM is replaced by an isotropic
metallic layer like silver, the condition where the permittivity is near zero
is consistent with the generation of bulk longitudinal collective oscillations
of the free electrons. This type of excitation can produce moderate (but less
than $100\%$) absorption when there is minimal intrinsic material loss.

For the case of vacuum superstrate and substrate, Eq. (3) reveals that when
$\sin\phi_{2}=0$, then $R=0$. If on the other hand, both substrate and
superstrate are perfectly conducting, then setting the denominator of (3)
equal to zero also yields $\sin\phi_{2}=0$, which coincides with the
dispersion relation for guided waves in an HMM layer. Equation (8) shows that
when $n=0$, ${\hat{k}}_{x}^{2}=\epsilon_{z2}\mu_{y2}$ and
${\hat{k}}_{z}^{2}=0$, corresponding to a TEM mode which is essentially a
plane wave confined to propagate in the $x$-direction. Thus if
$\phi_{2}={\hat{k}}_{z2}\hat{\tau}=n\pi$, this assertion is valid if
$\phi_{2}<<n\pi$. If however $\epsilon_{z2}/\epsilon_{x2}<0$, Eq. (8) reveals
that there is no guided mode cutoff for ${\hat{k}}_{x}^{2}$.

To present a global view of the parameter space in which our anisotropic
structure can absorb unusually large portions of incident energy, we present
in Figs. 3(a) and (b), 2-D density plots that map the absorption versus
$\lambda$ and $\theta$. The HMM thickness is fixed at $\tau$$=$ $0.16\mu$m, as
in Figs. 2(c) and (d). In Fig. 3(a) $\epsilon_{x2}=(4,0.1)$, so that the HMM
region where $\Re(\epsilon_{z2})<0$ corresponds to $\lambda$ $>$ $\lambda_{z}$
(recall that $\lambda_{z}$=$1.6\mu$m). Similarly for (b), $\epsilon_{x2}=(-4,\
0.1)$, and thus the HMM region there corresponds to $\lambda$ $<$ $1.6\mu$m.
Figs. 4(a) and (b) are slices from Figs. 3(a) and (b). In Fig. 4(a) near-
perfect absorption occurs at $\theta=65^{\circ}$ for both HMM types. For
$\lambda=1.7\mu$m, $\Re(\epsilon_{x2})=4$ and $\Re(\epsilon_{z2})=-0.128$
corresponding to a Type-1 HMM. For $\lambda=1.5\mu$m, $\Re(\epsilon_{x2})=-4$
and $\Re(\epsilon_{z2})=0.121$, corresponding to a Type-2 HMM. In Fig. 4(b),
the Type-1 absorption peak occurs at $\lambda=1.66\mu$m, where
$\Re(\epsilon_{z2})=-0.076$, and the Type-2 case peaks at $\lambda=1.55\mu$m,
where $\Re(\epsilon_{z2})=0.062$.

Further insight into this anomalous absorption can be gained from studying the
balance of energy [30]. For our structure and material parameters, it suffices
to compute,

$\displaystyle\frac{4\pi}{c}\int_{V}dv\ {\bm{E}}\cdot{\bm{J}}^{*}=-\int_{V}dv\
\nabla\cdot\left({\bm{E}}\times{\bm{H}}^{*}\right)-\frac{i\omega}{c}\int_{V}dv\left[\epsilon^{*}_{x2}|E_{x2}|^{2}+\epsilon^{*}_{z2}|E_{z2}|^{2}-|H_{y2}|^{2}\right].$
(12)

Since we have incorporated the conductive part of the HMM into the dielectric
response, the $\bm{J}$ term is absent. In all of the near-perfect absorption
examples investigated here, evaluation of Eq. (12) confirmed that the net
energy flow into the HMM volume, $V$, is converted into heat.

[width=7.4cm]Fig4a_avg.pdf

[width=7.4cm]Fig4b_avg.pdf

Figure 5:  The power ${\bm{P}}$ in the HMM, normalized by the incident power
in the $z$-direction, and plotted as a function of $\theta$. In (a)
$\lambda=1.7$ $\mu$m (type-1 HMM) and in (b) $\lambda=1.5$ $\mu$m (type-2
HMM). The material parameters are the same as in Fig. 4(c). Panel (a) reveals
that energy flow parallel to the interface ($P_{x2}$) in the type-1 HMM is
negative, which is opposite that of the vacuum region containing the incident
beam.

To explore further the behavior of the energy flow, we present in Fig. 5 the
average power ${\bm{P}}$ in the HMM as a function of $\theta$ for the two
cases in Fig. 4(a). Thus, panel (a) is for $\lambda=1.7$$\mu$m (type-1 HMM),
and panel (b) corresponds to $\lambda=1.5$$\mu$m (type-2 HMM). The average
power along the $x$ and $z$ directions, $P_{x2}$ and $P_{z2}$, is found from
averaging the corresponding components of the Poynting vector over the HMM
region (see Eqs. (29)). It is evident that the direction of energy flow
depends on the sign of $\epsilon_{z2}$ (or equivalently whether $\lambda$ is
above or below $\lambda_{z}$). The component of $\bm{P}$ normal to the
interfaces ($P_{z2}$) must always have the same sign on both sides of the
interface [31]. Its direction parallel to the interface ($P_{x2}$) however can
be negative if the HMM is of type-1, as seen in Fig. 5(a), and is clearly
opposite in direction to $k_{x}$, which is always positive. This manifestation
of “negative refraction” was discussed in the context of uniaxially
anisotropic media [32], certain nanowire structures [8], and observed in ZnO-
based multilayers [24]. Comparing the peaks in panels (a) and (b) with Fig.
4(a), we see the correlation with the angles where near-perfect absorption
occurs and those where $|P_{z2}|$ is maximal.

##  4 Conclusion

In conclusion, we have investigated the absorption properties of both type-1
and type-2 hyperbolic metamaterials. We found that HMMs can absorb
significantly higher amounts of electromagnetic energy compared to their
conventional counterparts, where $\Re(\epsilon_{x2})$ and $\Re(\epsilon_{z2})$
are both of the same sign. Our results show that the incident beam can couple
to the HMM structure without recourse for a second compensating layer. We also
revealed that the condition $\Re(\epsilon_{z})\approx 0$ leads to near-perfect
absorption over a range of frequencies, angles of incidence, and subwavelength
structure thicknesses, making the proposed structures experimentally
achievable. Alternate methods exist to achieve perfect absorption, including
periodic layers of silver and conventional dielectrics that depending on the
direction of incident wave propagation and loss, can exhibit anisotropic
behavior that cancels the reflected and transmitted waves simultaneously [33].
Our HMM with metallic backing is a different configuration in which no energy
can be transmitted, and the inherently finite width of the structure means
that there are no Bloch wave excitations. Arrays of metal-dielectric films can
serve as an effective HMM waveguide taper, resulting in light localization and
enhanced absorption [34], however, the modes responsible for “slow-light” are
very sensitive to the presence of loss [35].

When the incident wavelength results in the dielectric response of the
metamaterial possessing a nearly vanishing component of the permittivity,
contributions from nonlinear effects and/or spatial dispersions can become
important. Nonlinear effects can in this case generate interesting phenomena
such as two-peaked or flat solitons [36], as well as additional venues for
second- and third-harmonic generation [37], and guided waves whose Poynting
vector undergoes localized reversal [38]. Since the nonlinear part of the
dielectric response can now be of the same order as the (small) linear part,
the transmissivity can exhibit directional hysteresis behavior [39]. Spatial
dispersion can moreover lead to the appearance of additional EM waves, as was
reported for nanorods [40]. For metal-dielectric structures, nonlocality
arising from the excitation of surface plasmons can also lead to significant
corrections [41] to conventional effective medium theories [42].

## Appendix: Poynting’s Theorem

In this section we present the details on how the EM fields are
straightforwardly calculated for determining the reflectance and energy flow
in HMM structures. We have considered in this paper diagonally anisotropic HMM
layers ($\epsilon_{x},\epsilon_{z},\mu_{y}$). We also assume that EM wave
propagation and polarization is in the $x$-$z$ plane. The wave equation for
$E_{x}$ is thus,

$\displaystyle\frac{\partial^{2}E_{x}}{\partial
z^{2}}+\left[\left(\frac{\omega}{c}\right)^{2}\epsilon_{x}\mu_{y}-\left(\frac{\epsilon_{x}}{\epsilon_{z}}\right)k_{x}^{2}\right]E_{x}=0.$
(13)

Taking into account that $\nabla\cdot{\bm{D}}=\nabla\cdot{\bm{B}}=0$, this
yields the electric field solutions in their respective media as,

$\displaystyle{\bm{E}}_{1}$ $\displaystyle=$
$\displaystyle\left[A\left\\{{\hat{x}}+{\hat{z}}\left(\frac{{\hat{k}}_{x}\epsilon_{x1}}{{\hat{k}}_{z1}\epsilon_{z1}}\right)\right\\}e^{-ik_{z1}z}\right]e^{ik_{x}x},$
(14) $\displaystyle{\bm{E}}_{2}$ $\displaystyle=$
$\displaystyle\left[G\left\\{{\hat{x}}-{\hat{z}}\left(\frac{{\hat{k}}_{x}\epsilon_{x2}}{{\hat{k}}_{z2}\epsilon_{z2}}\right)\right\\}e^{ik_{z2}z}+F\left\\{{\hat{x}}+{\hat{z}}\left(\frac{{\hat{k}}_{x}\epsilon_{x2}}{{\hat{k}}_{z2}\epsilon_{z2}}\right)\right\\}e^{-ik_{z2}z}\right]e^{ik_{x}x},$
(15) $\displaystyle{\bm{E}}_{3}$ $\displaystyle=$
$\displaystyle\left[C\left\\{{\hat{x}}-{\hat{z}}\left(\frac{{\hat{k}}_{x}\epsilon_{x3}}{{\hat{k}}_{z3}\epsilon_{z3}}\right)\right\\}e^{ik_{z3}z}+I\left\\{{\hat{x}}+{\hat{z}}\left(\frac{{\hat{k}}_{x}\epsilon_{x3}}{{\hat{k}}_{z3}\epsilon_{z3}}\right)\right\\}e^{-ik_{z3}z}\right]e^{ik_{x}x}.$
(16)

Similarly, the components of the magnetic field are written,

$\displaystyle{\bm{H}}_{1}$ $\displaystyle=$ $\displaystyle\
-{\hat{y}}\left(\frac{\epsilon_{x1}}{{\hat{k}}_{z1}}\right)Ae^{-ik_{z1}z}e^{ik_{x}x},$
(17) $\displaystyle{\bm{H}}_{2}$ $\displaystyle=$ $\displaystyle\
{\hat{y}}\left(\frac{\epsilon_{x2}}{{\hat{k}}_{z2}}\right)\Big{[}Ge^{ik_{z2}z}-Fe^{-ik_{z2}z}\Big{]}e^{ik_{x}x},$
(18) $\displaystyle{\bm{H}}_{3}$ $\displaystyle=$ $\displaystyle\
{\hat{y}}\left(\frac{\epsilon_{x3}}{{\hat{k}}_{z3}}\right)\Big{[}Ce^{ik_{z3}z}-Ie^{-ik_{z3}z}\Big{]}e^{ik_{x}x}.$
(19)

The $I$ terms represent the incident field. The quantities ${\hat{k}}_{zj}$
and $\phi_{j}$ are defined in Eqs. (5) and (6). Utilizing matching boundary
conditions for the tangential components of the electric and magnetic fields
permits calculation of the coefficients,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{4e^{-i\phi_{3}}{\hat{k}}_{z1}{\hat{k}}_{z2}\epsilon_{x2}\epsilon_{x3}}{{\cal
G}^{-}{\cal F}^{-}e^{i\phi_{2}}+{\cal G}^{+}{\cal F}^{+}e^{-i\phi_{2}}};\quad
C=\beta\left[\frac{{\cal G}^{-}{\cal F}^{+}e^{i\phi_{2}}+{\cal G}^{+}{\cal
F}^{-}e^{-i\phi_{2}}}{{\cal G}^{-}{\cal F}^{-}e^{i\phi_{2}}+{\cal G}^{+}{\cal
F}^{+}e^{-i\phi_{2}}}\right],$ (20) $\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{2e^{-i\phi_{3}}{\hat{k}}_{z2}\epsilon_{x3}{\cal
G}^{+}}{{\cal G}^{-}{\cal F}^{-}e^{i\phi_{2}}+{\cal G}^{+}{\cal
F}^{+}e^{-i\phi_{2}}};\quad
G=\frac{-2e^{-i\phi_{3}}{\hat{k}}_{z2}\epsilon_{x3}{\cal G}^{-}}{{\cal
G}^{-}{\cal F}^{-}e^{i\phi_{2}}+{\cal G}^{+}{\cal F}^{+}e^{-i\phi_{2}}},$ (21)

were $\beta$ is given in Eq. (4). We also define,

$\displaystyle{\cal F}^{\pm}$ $\displaystyle=$
$\displaystyle{\hat{k}}_{z3}\epsilon_{x2}\pm{\hat{k}}_{z2}\epsilon_{x3},$ (22)
$\displaystyle{\cal G}^{\pm}$ $\displaystyle=$
$\displaystyle{\hat{k}}_{z2}\epsilon_{x1}\pm{\hat{k}}_{z1}\epsilon_{x2},$ (23)

where $\epsilon_{xj}$ describe the media for regions $j=1,2,3$, and $k_{zj}$
is defined in Eq. (6). The caret symbol signifies that wavenumber components
$k_{x}$ and $k_{zj}$ have been normalized to $\omega/c$. In general,
${\hat{k}}_{x}$ can be any value, but for the case of an incident plane wave
in vacuum, ${\hat{k}}_{x}=\sin\theta$.

For time-harmonic fields, consider now the integral,

$\displaystyle\frac{1}{2}\frac{4\pi}{c}\int_{V}dv\
{\bm{E}}\cdot{\bm{J}}^{*}=-\frac{1}{2}\int_{V}dv\
\nabla\cdot\left({\bm{E}}\times{\bm{H}}^{*}\right)-\frac{i\omega}{2c}\int_{V}dv\left[{\bm{E}}\cdot{\bm{D}}^{*}-{\bm{H}}^{*}\cdot{\bm{B}}\right],$
(24)

where we have used,

$\displaystyle\nabla\times{\bm{H}}=\frac{4\pi}{c}{\bm{J}}-\frac{i\omega}{c}{\bm{D}}\quad{\rm;}\quad\nabla\times{\bm{E}}=\frac{i\omega}{c}{\bm{B}}.$
(25)

The media are diagonally anisotropic with
${\bm{D}}={{\bm{\epsilon}}}\cdot{\bm{E}}$ and
${\bm{B}}={\bm{\mu}}\cdot{\bm{H}}$. Inserting Eqs. (15) and (18) into Eq. (24)
yields the following energy conservation relationships,

$\displaystyle\frac{i\omega}{2c}\int_{0}^{\tau}\ dz\ {E_{2x}}{D^{*}_{2x}}$
$\displaystyle=$ $\displaystyle\
\frac{i\epsilon^{*}_{2x}}{2}\Bigl{[}|G|^{2}\Bigl{(}\frac{{\rm
e}^{-2\tau\Im({k_{2z}})}-1}{{-2\Im{({\hat{k}}_{2z})}}}\Bigr{)}+|F|^{2}\Bigl{(}\frac{{\rm
e}^{2\tau\Im({k_{2z}})}-1}{{2\Im{({\hat{k}}_{2z})}}}\Bigr{)}$ (26)
$\displaystyle+$ $\displaystyle 2\Re\Bigl{\\{}GF^{*}\Bigl{(}\frac{{\rm
e}^{2i\tau\Re({k_{2z}})}-1}{{2i\Re{({\hat{k}}_{2z})}}}\Bigr{)}\Bigr{\\}}\Bigr{]},$
$\displaystyle\frac{i\omega}{2c}\int_{0}^{\tau}\ dz\ {E_{2z}}{D^{*}_{2z}}$
$\displaystyle=$ $\displaystyle\
\frac{i\epsilon^{*}_{2z}}{2}\bigg{|}\frac{{\hat{k}}_{x}\epsilon_{2x}}{{\hat{k}}_{2z}\epsilon_{2z}}\bigg{|}^{2}\Bigl{[}|G|^{2}\Bigl{(}\frac{{\rm
e}^{-2\tau\Im{(k_{2z})}}-1}{{-2\Im{({\hat{k}}_{2z})}}}\Bigr{)}$ (27)
$\displaystyle+$ $\displaystyle|F|^{2}\Bigl{(}\frac{{\rm
e}^{2\tau\Im{(k_{2z})}}-1}{{2\Im{({\hat{k}}_{2z})}}}\Bigr{)}-2\Re\Bigl{\\{}GF^{*}\Bigl{(}\frac{{\rm
e}^{2i\tau\Re({k_{2z}})}-1}{{2i\Re{({\hat{k}}_{2z})}}}\Bigr{)}\Bigr{\\}}\Bigr{]},$
$\displaystyle\frac{i\omega}{2c}\int_{0}^{\tau}\ dz\ {H^{*}_{2y}}{B_{2y}}$
$\displaystyle=\
\frac{i\mu_{2y}}{2}\bigg{|}\frac{\epsilon_{2x}}{{\hat{k}}_{2z}}\bigg{|}^{2}\Bigl{[}|G|^{2}\Bigl{(}\frac{{\rm
e}^{-2\tau\Im{(k_{2z})}}-1}{{-2\Im{({\hat{k}}_{2z})}}}\Bigr{)}$ (28)
$\displaystyle+|F|^{2}\Bigl{(}\frac{{\rm
e}^{2\tau\Im{(k_{2z})}}-1}{{2\Im{({\hat{k}}_{2z})}}}\Bigr{)}-2\Re\Bigl{\\{}GF^{*}\Bigl{(}\frac{{\rm
e}^{2i\tau\Re{(k_{2z})}}-1}{{2i\Re{({\hat{k}}_{2z})}}}\Bigr{)}\Bigr{\\}}\Bigr{]},$

where the $x$ and $y$ integrations over $V$ are omitted.

Finally, the time-averaged Poynting vector in $V$ is
${\bm{S}}={\bm{E}}\times{\bm{H}}^{*}/2$, giving the result,

$\displaystyle S_{2x}(z)$ $\displaystyle=$ $\displaystyle\
\Bigl{(}\frac{{\hat{k}}_{x}}{\epsilon_{z2}}\Bigr{)}\bigg{|}\frac{\epsilon_{x2}}{{\hat{k}}_{z2}}\bigg{|}^{2}\bigg{[}|G|^{2}{\rm
e}^{-2z\Im(k_{z2})}+|F|^{2}{\rm e}^{2z\Im(k_{z2})}-2\Re\Bigl{(}GF^{*}{\rm
e}^{2iz\Re(k_{z2})}\Bigr{)}\bigg{]},$ (29) $\displaystyle S_{2z}(z)$
$\displaystyle=$ $\displaystyle\
\Bigl{(}\frac{\epsilon_{x2}}{{\hat{k}}_{z2}}\Bigr{)}^{*}\bigg{[}|G|^{2}{\rm
e}^{-2z\Im(k_{z2})}-|F|^{2}{\rm e}^{2z\Im(k_{z2})}-2i\Im\Bigl{(}GF^{*}{\rm
e}^{2iz\Re(k_{z2})}\Bigr{)}\bigg{]}.$ (30)


# Modification of Heisenberg uncertainty relations in
non-commutative Snyder space-time geometry

Marco Valerio Battisti<EMAIL_ADDRESS>ICRA - International Center for
Relativistic Astrophysics  Dipartimento di Fisica (G9), Università di Roma
“Sapienza” P.le A. Moro 5, 00185 Rome, Italy  Stjepan Meljanac
<EMAIL_ADDRESS>Rudjer Bovskovic Institute, Bijenivcka c.54, HR-10002 Zagreb,
Croatia

###### Abstract

We show that the Euclidean Snyder non-commutative space implies infinitely
many different physical predictions. The distinct frameworks are specified by
generalized uncertainty relations underlying deformed Heisenberg algebras.
Considering the one-dimensional case in the minisuperspace arena, the bouncing
Universe dynamics of loop quantum cosmology can be recovered.

###### pacs:

04.60.Bc; 02.40.Gh; 11.10.Nx

##  I Introduction

Non-commutative geometries are widely considered as plausible candidates for
describing physics at the Planck scale noncom  and have natural connections
with string theory SW . Moreover some of these models can be related to the
intuitions of doubly special relativity (DSR) AMS  where another invariant
scale (apart from the speed of light) is introduced ab initio in the theory.
Interest in DSR is also increased because such a framework can be regarded as
a semi-classical limit of quantum gravity (see RovSmo  and references
therein).

In this paper the Snyder proposal Sny  of a non-commutative space-time is
analyzed from a physical point of view. This model can be understood by means
of the projective geometry approach to the de Sitter space of momenta with two
universal constants and is relevant since it can be related to some of DSR
models Kov . Furthermore, it has some motivations from loop quantum gravity LO
and two-time physics tt .

The starting point of our analysis is the requirement that the only deformed
commutator in the Euclidean Snyder framework is one between the coordinates.
This way, the translation group is not deformed and the rotational symmetry is
preserved. We then show that, infinitely many commutators between the non-
commutative coordinates and momenta are possible, such that in all the cases
the algebra closes. This way, infinitely many different physical predictions
of the Snyder space are allowed. These are summarized in the deformed
symplectic geometry and in the generalized uncertainty principle at classical
and quantum level, respectively. The physical interesting framework of a
deformed quantum cosmology is also analyzed. Here we deal with a one-
dimensional system and our picture is almost uniquely fixed. We show that this
framework naturally leads to the non-singular (bouncing) Friedmann dynamics
obtained in recent issues of loop quantum cosmology (LQC) bloop .

The paper is organized as follows. In Section II the algebraic structure of
the Euclidean Snyder space is analyzed. Section III is devoted to discuss the
physical implications of this framework. Concluding remarks follow. Over the
paper we adopt units such that $\hbar=c=1$.

##  II Realizations of Snyder space

The algebraic structure of the non-commutative Snyder space is analyzed in
this Section. All possible realizations of this space, the general form of the
uncertainty principle and the required hermiticity conditions are showed. The
known algebras are then recovered as particular cases of our construction.

Realizations. Let us start by considering a $n$-dimensional non-commutative
(deformed) Euclidean space such that the commutator between the coordinates
has the non-trivial structure ($\\{i,j,...\\}\in\\{1,...,n\\}$)

$[\tilde{x}_{i},\tilde{x}_{j}]=\kappa M_{ij}\,,$ (1)

where with $\tilde{x}_{i}$ we refer to the non-commutative coordinates and
$\kappa\in\mathbb{R}$ is the deformation parameter with dimension of a squared
length. We then demand that the rotation generators
$M_{ij}=-M_{ji}=i(x_{i}p_{j}-x_{j}p_{i})$ satisfy the ordinary $SO(n)$ algebra

$[M_{ij},M_{kl}]=\delta_{jk}M_{il}-\delta_{ik}M_{jl}-\delta_{jl}M_{ik}+\delta_{il}M_{jk}$
(2)

and that the translation group is not deformed, i.e. $[p_{i},p_{j}]=0$. In
order to preserve the rotational symmetry the commutators between $M_{ij}$ and
the coordinates $\tilde{x}_{i}$, as well as between $M_{ij}$ and $p_{k}$, have
to be undeformed. Therefore, we assume that the relations

$\displaystyle[M_{ij},\tilde{x}_{k}]$ $\displaystyle=$
$\displaystyle\tilde{x}_{i}\delta_{jk}-\tilde{x}_{j}\delta_{ik},$ (3)
$\displaystyle[M_{ij},p_{k}]$ $\displaystyle=$ $\displaystyle
p_{i}\delta_{jk}-p_{j}\delta_{ik}$

hold. This way we deal with the (Euclidean) Snyder space Sny . The above
relations however do not uniquely fix the commutators between $\tilde{x}_{i}$
and $p_{j}$. In particular, there are infinitely many of such commutators
which are all compatible (in the sense that the algebra closes in virtue of
the Jacobi identities) with the above natural requirements.

This feature can be understood by analyzing the realizations Mel ; Luk ; Gosh
of such a non-commutative space. The concept of realization was developed in a
series of papers Mel  (for a similar approach in the $\kappa$-deformed space-
time see Luk  and a related analysis in the context of DSR can be found in
Gosh ). A realization of the Snyder algebra (1) is defined as a rescaling of
the non-commutative coordinates $\tilde{x}_{i}$ in terms of the ordinary phase
space variables ($x_{i},p_{j}$). The most general $SO(n)$ covariant
realization for $\tilde{x}_{i}$ is given by

$\tilde{x}_{i}=x_{i}\varphi_{1}(\mu,\nu)+\kappa(x_{j}p_{j})p_{i}\varphi_{2}(\mu,\nu),$
(4)

where the convention $a_{i}b_{i}=\sum_{i}a_{i}b_{i}$ is adopted and
$\varphi_{1}$ and $\varphi_{2}$ are two arbitrary finite functions depending
on the dimensionless quantities $\mu=\kappa p^{2}$ and $\nu=\kappa m^{2}$. In
particular, the second quantity accounts for a mass-like term $m^{2}$ which
can be positive, negative or zero. In order to recover the ordinary Heisenberg
algebra, suitable boundary conditions on these functions have to be imposed.
We have to demand that, in the $\kappa\rightarrow 0$ ($\mu,\nu\rightarrow 0$)
limit, $\varphi_{1}(0,0)=1$.

The realization above is, of course, not completely arbitrary since it depends
on the adopted algebraic structure. In particular, the two functions
$\varphi_{1}$ and $\varphi_{2}$ are constrained by the relations (1) and (3).
Inserting the formula (4) into the non-commutative coordinate commutator (1),
the first restriction we obtain reads

$2\left(\varphi_{1}^{\prime}\varphi_{1}+\mu\varphi_{1}^{\prime}\varphi_{2}\right)-\varphi_{1}\varphi_{2}+1=0,$
(5)

where $\varphi_{1}^{\prime}=\partial\varphi_{1}/\partial\mu$. The other
condition on $\varphi_{1}$ and $\varphi_{2}$ arises after considering the
realization (4) into the commutator $[M_{ij},\tilde{x}_{k}]$ in (3). Such
second constraint can be written as

$\left(x_{l}[M_{ij},p_{l}]p_{k}+[M_{ij},x_{l}]p_{l}p_{k}\right)\varphi_{2}=0$
(6)

and is immediate to verify that the argument in the brackets identically
vanishes. Therefore, only one condition on $\varphi_{1}$ and $\varphi_{2}$
appears. As a matter of fact, given any function $\varphi_{1}(\mu,\nu)$
satisfying the boundary condition $\varphi_{1}(0,0)=1$, the function
$\varphi_{2}(\mu,\nu)$ is uniquely determined by the equation (5) and reads
$\varphi_{2}=(1+2\varphi_{1}^{\prime}\varphi_{1})/(\varphi_{1}-2\mu\varphi_{1}^{\prime})$.
In other words, there are infinitely many ways to express, via $\varphi_{1}$,
the non-commutative coordinates (1) in terms of the ordinary ones without
deforming either the rotation and the translation groups.

It is worth noting that: (i) The realizations (4) have sense if there exist
the inverse transformation $x_{i}=\tilde{x}_{j}(\varphi^{-1})_{ji}$ and the
necessary and sufficient condition is $\det|\delta_{ij}\varphi_{1}+\kappa
p_{i}p_{j}\varphi_{2}|>0$. If we deal with a $n\geq 2$ dimensional space, such
a condition reads $\varphi^{n-1}_{1}(\varphi_{1}+\mu\varphi_{2})>0$, i.e.
$\varphi_{1}>0$ and $\varphi_{1}+\mu\varphi_{2}>0$. (ii) Our analysis can be
straightforward generalized to a Snyder Minkowskian space-time. In this case,
all the relations above hold as soon as the following replacements are taken
into account ($\\{\alpha,\beta\\}\in\\{0,...,n\\}$): the $SO(n)$ generators
are substituted by the Lorentz generators $L_{\alpha\beta}$ and
$(\tilde{x}_{\alpha},p_{\alpha})$ now transform as four-vectors under Lorentz
algebra which indices are raised and lowered by the Minkowski metric
$\eta_{\alpha\beta}$, i.e $p^{2}=\eta^{\alpha\beta}p_{\alpha}p_{\beta}$ is
Lorentz invariant.

Uncertainty relations. In order to complete the analysis of the deformed
algebra we need to analyze the $(\tilde{x}-p)$ commutation relation. This way,
the general form of the uncertainty principle, and thus the physical
consequences of the model, can be discussed. The commutator between
$\tilde{x}_{i}$ and $p_{j}$ arises from the realization (4) and reads

$[\tilde{x}_{i},p_{j}]=i\left(\delta_{ij}\varphi_{1}+\kappa
p_{i}p_{j}\varphi_{2}\right).$ (7)

Of course, the ordinary one is recovered in the $\kappa\rightarrow 0$ limit.
From the commutator above we can immediately obtain the generalized
uncertainty principle underlying the Snyder non-commutative space, i.e.

$\Delta\tilde{x}_{i}\Delta
p_{j}\geq\frac{1}{2}|\delta_{ij}\langle\varphi_{1}\rangle+\kappa\langle
p_{i}p_{j}\varphi_{2}\rangle|.$ (8)

Three remarks are in order. (i) The algebra we obtain can be regarded as a
deformed Heisenberg algebra. More precisely, the deformation of the only
commutator between the spatial coordinates as in (1) leads to infinitely many
realizations of the algebra, and thus generalized uncertainty relations (8),
all consistent with the assumptions underlying the model. (ii) Unless
$\varphi_{2}=0$ no compatible observables exist. These are coupled with each
other and an exactly simultaneous measurable couple $(\tilde{x}_{i},p_{j})$ is
not longer allowed. A measure of the $i$-component of the (non-commutative)
position will always affect a measure of the $j(\neq i)$-component of the
momentum by an uncertainty $\Delta p_{j}\gtrsim|\kappa\langle
p_{i}p_{j}\varphi_{2}\rangle|/\Delta\tilde{x}_{i}$. (iii) For any fixed
$\varphi_{1}$ the non-commutative framework is unique, but we can realize the
commutator (7) in terms of any commutative coordinates $x_{i}^{\prime}$ and
corresponding canonical momenta $p_{i}^{\prime}$ satisfying
$[x_{i}^{\prime},p_{j}^{\prime}]=i\delta_{ij}$. Of course all these
descriptions lead to the same physical consequences.

Hermiticity conditions. The non-commutative coordinates $\tilde{x}_{i}$ have
to be hermitian operators in any given realization. All the commutators given
above are invariant under the formal anti-linear involution “${\dagger}$”

$\tilde{x}_{i}^{\dagger}=\tilde{x}_{i},\quad p_{i}^{\dagger}=p_{i},\quad
M_{ij}^{\dagger}=-M_{ij}\,,$ (9)

where the order of elements is inverted under the involution. However, the
realization (4) in general is not hermitian. The hermiticity condition can be
immediately implemented as soon as the expression

$\tilde{x}_{i}=\frac{1}{2}\left(x_{i}\varphi_{1}+\kappa(x_{j}p_{j})p_{i}\varphi_{2}+\varphi_{1}^{\dagger}x_{i}^{\dagger}+\kappa\varphi_{2}^{\dagger}p_{i}^{\dagger}(x_{j}p_{j})^{\dagger}\right)$
(10)

is taken into account. However, the physical result do not depend on the
choice of the representation as long as exist a smooth limit
$\tilde{x}_{i}\rightarrow x_{i}$ as $\kappa\rightarrow 0$. Therefore, we can
restrict our attention to non-hermitian realization only.

Recovering the know realizations. The non-commutative Snyder space has been
analyzed in literature from different points of view Kov ; LO ; tt  (see also
GB ), but only two particular realizations of its algebra are known: the
Snyder Sny  and the Maggiore Mag  ones. The original realization of Snyder, in
particular, expressed through the commutator between $\tilde{x}$ and $p$,
reads

$[\tilde{x}_{i},p_{j}]=i\left(\delta_{ij}+\kappa p_{i}p_{j}\right).$ (11)

It is not difficult to see that this is a particular case of our realization
(4) as soon as $\varphi_{1}=1$. From this condition, the function
$\varphi_{2}$ is fixed by (5) as $\varphi_{2}=1$ and the above commutation
relation is recovered. The condition on the inverse mapping implies that
$p^{2}>-1/\kappa$. On the other hand, the Maggiore algebra

$[\tilde{x}_{i},p_{j}]=i\delta_{ij}\sqrt{1-\kappa(p^{2}+m^{2})},$ (12)

can be regarded as the particular case of (4) when the condition
$\varphi_{2}=0$ is taken into account. But this requirement alone is not
enough. In fact, from the constraint (5), the function $\varphi_{1}$ is not
uniquely fixed but reads $\varphi_{1}=\sqrt{1-\mu+f(\nu)}$, where $f(\nu)$ is
a generic function of $\nu$ (the inverse mapping condition entails
$p^{2}<(1+f)/\kappa$). Only in the specific case $f(\nu)=-\nu$ the commutator
(12) is recovered. Finally, we note that, the deformed algebra proposed by
Kempf et al. in Kem  can be regarded as a particular case of (12) as $|\mu|\ll
1$ and $m=0$, i.e. $[\tilde{x}_{i},p_{j}]=i\delta_{ij}(1+\beta p^{2})$ where
$\beta=-\kappa/2$ with $\kappa<0$. In the one-dimensional framework (see
below), this algebra is the same of the Snyder one (11).

##  III Physical implications

As understood, the physical consequences of a non-commutative space geometry
are deeply and completely new scenarios are opened at both classical and
quantum levels. Two physically relevant frameworks are analyzed in this
Section: a generic mechanical system and the so-called quantum cosmological
arena.

Mechanical system. Let us start by considering a mechanical system, i.e. a
model with a finite number of degrees of freedom described by a Hamiltonian
$H=H(\tilde{x},p)$. At classical level the deformations induced on the system
appear as soon as the (classical) limit
$-i[\cdot,\cdot]\rightarrow\\{\cdot,\cdot\\}$ is taken into account. In doing
this the deformation parameter $\kappa$ is regarded as an independent constant
with respect to $\hbar$. Modifications of a non-commutative framework on the
classical dynamics are then summarized in the deformed Poisson brackets

$\\{F,G\\}=\left(\frac{\partial F}{\partial\tilde{x}_{i}}\frac{\partial
G}{\partial p_{j}}-\frac{\partial F}{\partial p_{i}}\frac{\partial
G}{\partial\tilde{x}_{j}}\right)\\{\tilde{x}_{i},p_{j}\\}+\frac{\partial
F}{\partial\tilde{x}_{i}}\frac{\partial
G}{\partial\tilde{x}_{j}}\\{\tilde{x}_{i},\tilde{x}_{j}\\}$ (13)

between any phase space functions. This symplectic structure is not fixed but
depends on the two functions $\varphi_{1}$ and $\varphi_{2}$ constrained by
(5) and $\varphi_{1}(0,0)=1$. From the relation above, the equations of motion
of a mechanical system are deformed as

$\displaystyle\dot{\tilde{x}}_{i}$ $\displaystyle=$
$\displaystyle\\{\tilde{x}_{i},H\\}=\frac{\partial H}{\partial
p_{j}}\left(\delta_{ij}\varphi_{1}+\kappa
p_{i}p_{j}\varphi_{2}\right)+\frac{\kappa}{i}\frac{\partial
H}{\partial\tilde{x}_{j}}M_{ij},$ $\displaystyle\dot{p}_{i}$ $\displaystyle=$
$\displaystyle\\{p_{i},H\\}=-\frac{\partial
H}{\partial\tilde{x}_{j}}\left(\delta_{ij}\varphi_{1}+\kappa
p_{i}p_{j}\varphi_{2}\right).$ (14)

When the deformation parameter vanishes ($\kappa\rightarrow 0$) the ordinary
Hamilton equations are recovered. At quantum level our picture implies either
modifications of the Ehrenfest theorem through (III), either deformations of
the canonical quantization prescription via the commutator (7). As we said,
this commutator is not fixed at all by the assumptions described above and for
any choice of the realization (4) of the non-commutative coordinates, the
corresponding Hilbert spaces are thus unitarily inequivalent. Each quantum
framework (Hilbert space) corresponds to a specific choice of the realization
(4). We also stress that given an eigenvalue problem
$\hat{H}(\tilde{x},p)\psi(x)=E\psi(x)$, the wave function $\psi$ and the
spectrum $E$ depend on $\varphi_{1}$.

This is not surprising since the deformation of the canonical commutation
relations can be viewed, from the realization (4), as an algebra homomorphism
which is a non-canonical transformation. In particular, it can not be
implemented at quantum level as an unitary transformation. From this point of
view, the set of predictions of any deformed Heisenberg algebra can not be
matched by the set of predictions arising from another one, i.e. neither by
the set of prediction of the ordinary framework (the $\kappa\rightarrow 0$
limit). New features are then introduced, for any fixed $\varphi_{1}$, at both
classical and quantum level. This way, a Snyder structure (1) in which the
translation and rotation groups are undeformed, leads to infinitely many
different physical predictions through (4).

A notable problem to be considered is the harmonic oscillator with non-
commutative quadratic potential, i.e. $H=p^{2}/2m+m\omega^{2}\tilde{x}^{2}/2$.
In the one-dimensional case the symmetry group is trivial ($SO(1)=\text{Id}$)
and the most general realization is given by $\tilde{x}=x\sqrt{1-\mu+f(\nu)}$.
Considering the representation of this algebra (we take $f=0$) in the momentum
space, the deformed stationary Schrödinger equation for this model is given by
the so-called Mathieu equation and the energy spectrum appears to be modified
as
$E_{n}=\omega(2n+1)/2-\omega\kappa(2n^{2}+2n+1)/8d^{2}+\mathcal{O}(\kappa^{2}/d^{4})$
where $d=1/\sqrt{m\omega}$ is the characteristic length scale (for more
details see Bat ). We note that, if $\kappa>0$ this is the spectrum obtained
in polymer (loop) quantum mechanics pol , while if $\kappa<0$ this result
resembles the one recovered in DJM03 .

Quantum cosmology. An interesting quantum mechanical arena to test such a
framework is the so-called minisuperspace reduction of the dynamics, i.e.
quantum cosmology. As well-know Wald , by imposing symmetries on the spatial
Cauchy surfaces which fill the space-time manifold, a considerable
simplification on the gravitational theory occurs. In particular, by requiring
the spatial homogeneity the phase space of general relativity reduces to six
dimensions. The system is described by three degrees of freedom, i.e. the
three scalar factors of the Bianchi models. Furthermore, by imposing also the
spatial isotropy, we deal with one-dimensional mechanical systems. These are
the Friedmann-Robertson-Walker (FRW) models which describe the observed
Universe and on which the standard model of cosmology is based.

In order to discuss the implications of the Snyder algebra on the FRW
Universes we consider the one-dimensional case of the scheme analyzed above.
If we assume the minisuperspace as Snyder-deformed, then the isotropic scale
factor $a$ (namely the radius of the Universe) and its conjugate momentum $p$
satisfy the commutation relation $[a,p]=i\sqrt{1-\mu+f(\nu)}$. It is worth
stressing that, when $\kappa>0$ (taking $f=0$) a natural cut-off on the
momentum arises, i.e. $|p|<\sqrt{1/\kappa}$, while as $\kappa<0$ the
uncertainty relation (8) predicts a minimal observable length
$\Delta{\tilde{x}}_{\text{min}}=\sqrt{-\kappa}$. Moreover, at the first order
in $\kappa$, the string theory result String
$\Delta{\tilde{x}}\gtrsim(1/\Delta p+l_{s}^{2}\Delta p)$, in which the string
length $l_{s}$ can be identify with $\sqrt{-\kappa/2}$, is recovered.

Following Bat  is possible to show that the effective Friedmann equation of
Snyder-deformed flat FRW cosmological model becomes

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G}{3}\rho\left(1-\frac{\rho}{\rho_{c}}+f(\nu)\right),$ (15)

where $G$ is the gravitational constant, $\rho=\rho(a)$ denotes a generic
matter energy density and $\rho_{c}=(2\pi G/3\kappa)\rho_{P}$ is the critical
energy density ($\rho_{P}$ being the Planck one). When the limit
$\kappa\rightarrow 0$ is taken into account, the critical energy density
diverges (the function $f(\nu)$ disappears) leading to the ordinary dynamics.
It is worth noting that, if $f(\nu)=0$ and $\kappa>0$, the equation (15)
resembles exactly the effective bouncing Friedmann equation of LQC bloop .
Such a dynamics is singularity-free since, when $\rho$ reaches the critical
energy density, $\dot{a}$ vanishes and the Universe experiences a (big)-bounce
instead of the classical big-bang. On the other hand, if $f(\nu)=0$ and
$\kappa<0$, the effective braneworlds dynamics is recovered Roy .

Summarizing, the non-commutative Snyder minisuperspace framework can clarify
similarities and differences between different quantum gravity theories. Other
comparisons between deformed and loop-polymer quantum cosmology, in view of
discussing the fate of the cosmological singularity at quantum level, were
performed considering the flat FRW model filled with a massless scalar field
BM07a  and the Taub cosmological model BM07b . Such investigations deserve
interest either in clarifying the role of loop quantization techniques in
cosmology, either in establishing a phenomenological contact with some
frameworks relevant in a flat space-time limit of quantum gravity.

##  IV Concluding remarks

In this paper we have shown how there are infinitely many realizations of the
Snyder algebra, equations (1-3), implying different commutation relations
between the non-commutative coordinates $\tilde{x}$ and momenta $p$, i.e. we
deal with deformed Heisenberg algebras. These depend on an arbitrary function
$\varphi_{1}(\mu,\nu)$ such that $\varphi_{1}(0,0)=1$ ensuring the correctness
of the picture. Therefore, different non-commutative spaces, described by
distinct commutations relations (7), imply different (unitarily inequivalent)
physical consequences. On the other hand, in the one-dimensional case the
commutator between $\tilde{x}$ and $p$ is fixed (up to a function of the mass-
like term) and, when implemented in the minisuperspace dynamics, the loop as
well as the braneworlds cosmological evolutions are recovered.

Acknowledgments. We thank Daniel Meljanac for comments. M.V.B. thanks S.M. for
the warm hospitality in Zagreb during which this paper was carried out. This
work was supported in part by Ministry of Science and Technology of the
Republic of Croatia under contract No. 098-0000000-2865.

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